WP18 phase 4 : audit final, réconciliation théorie-simulation, manuscrit v1.0
- research/results_audit.md final : étiquettes identité / théorie / simulation GE / PE / conjecture, décompte exact des runs et des résidus, classification des trois désaccords théorie-simulation - Prop. 9 restreinte (Assumption 7 : positions nominales) + Corollary 2 (créanciers nominaux nets : signe de dp = signe de Xi) ; Prop. 7 sous demande unitaire ; Prop. 2 avec contrainte LTV serrée + décision une fois pour toutes ; énoncés dans sections/equilibrium.tex, preuves dans appendix_proofs.tex (standalone : modèle + énoncés + preuves) - paper/main.tex v1.0 : 65 p. (45 p. de texte principal), résumé 217 mots, 7 mots-clés, JEL R21 R31 D31 D91 E31 G51 ; 13+1 figures, 5+4 tables ; chaque chiffre = macro de numbers.tex (2 571 macros) - Makefile : paper-inputs (copies figures/tables), simulate sauté si résultats présents ; figures sans titres dans l'image ; tables larges en resizebox ; README et progress.md Phase 4 Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com>
89 changed files +6,898 −854
modified
Makefile
+19 −4
@@ -3,9 +3,10 @@ PY ?= .venv/bin/python | ||
| 3 | 3 | PIP ?= .venv/bin/pip |
| 4 | 4 | WORKERS ?= 6 |
| 5 | 5 | |
| 6 | −.PHONY: all setup test test-fast simulate simulate-quick robustness figures tables paper clean clean-results | |
| 6 | +.PHONY: all setup test test-fast simulate simulate-force simulate-quick robustness figures tables paper paper-inputs clean clean-results | |
| 7 | 7 | |
| 8 | −all: test simulate robustness figures tables paper | |
| 8 | +# simulate is skipped when results/data/diagnostics.json exists (60-min run); `make clean-results` first to redo it | |
| 9 | +all: test simulate figures tables paper | |
| 9 | 10 | |
| 10 | 11 | setup: |
| 11 | 12 | python3 -m venv .venv |
@@ -18,8 +19,14 @@ test: | ||
| 18 | 19 | test-fast: |
| 19 | 20 | $(PY) -m pytest tests/ -q -m "not slow" |
| 20 | 21 | |
| 21 | −# Full run: baseline + all scenario groups + transitions + welfare (about 1.5 h on 18 cores) | |
| 22 | +# Full run: baseline + all scenario groups + transitions + welfare (about 60 min on 18 cores). | |
| 23 | +# Skipped if results/data/diagnostics.json is present; `make clean-results` (or `make simulate-force`) reruns it. | |
| 22 | 24 | simulate: |
| 25 | + @if [ -f results/data/diagnostics.json ]; then \ | |
| 26 | + echo "simulate: results/data/diagnostics.json present -> skipping the 60-min run (make clean-results to redo)"; \ | |
| 27 | + else $(MAKE) simulate-force; fi | |
| 28 | + | |
| 29 | +simulate-force: | |
| 23 | 30 | mkdir -p results/data results/logs results/figures results/tables |
| 24 | 31 | $(PY) -m src.simulation --workers $(WORKERS) 2>&1 | tee results/logs/simulate.log |
| 25 | 32 | $(PY) -m src.robustness |
@@ -39,13 +46,21 @@ figures: | ||
| 39 | 46 | tables: |
| 40 | 47 | $(PY) -m src.tables |
| 41 | 48 | |
| 42 | −paper: | |
| 49 | +# Copy the generated exhibits into paper/ (HARMONISATION: paper/figures, paper/tables are what main.tex includes) | |
| 50 | +paper-inputs: | |
| 51 | + mkdir -p paper/figures paper/tables | |
| 52 | + cp results/figures/fig*.pdf paper/figures/ | |
| 53 | + cp results/tables/*.tex paper/tables/ | |
| 54 | + | |
| 55 | +paper: paper-inputs | |
| 43 | 56 | cd paper && latexmk -pdf main.tex |
| 44 | 57 | |
| 45 | 58 | clean: |
| 46 | 59 | find . -name "__pycache__" -type d -prune -exec rm -rf {} + |
| 47 | 60 | rm -rf .pytest_cache |
| 48 | 61 | cd paper && latexmk -c 2>/dev/null || true |
| 62 | + cd paper && latexmk -c appendix_proofs_standalone.tex 2>/dev/null || true | |
| 63 | + rm -f paper/appendix_proofs_standalone.pdf | |
| 49 | 64 | |
| 50 | 65 | clean-results: |
| 51 | 66 | rm -rf results/data/* results/figures/* results/tables/* results/logs/* |
modified
README.md
+106 −30
@@ -3,12 +3,14 @@ | ||
| 3 | 3 | |
| 4 | 4 | **UQO Working Paper No. 18** — *A Dynamic General Equilibrium Theory* |
| 5 | 5 | |
| 6 | +[-b31b1b?logo=latex&logoColor=white)](paper/main.pdf) | |
| 6 | 7 | [](requirements.txt) |
| 8 | +[](results/data/calibration_targets.csv) | |
| 7 | 9 | [](Makefile) |
| 8 | 10 | [](https://uqo.ca) |
| 9 | 11 | [](mailto:simon-pierre.boucher@uqo.ca) |
| 10 | 12 | |
| 11 | −> **TL;DR** — A theoretical paper (no empirical estimation) on the conditions under which inherited wealth, mortgage leverage, nominal debt, housing supply and down-payment constraints generate persistent owner–renter wealth gaps among households with the same labour-income process, and on what an unexpected inflation changes. Part A: a three-stage lifecycle benchmark with nine propositions (closed-form ownership-entry threshold, leverage with transaction costs, intergenerational tenure persistence, existence/uniqueness of the stationary equilibrium, price neutrality without segmentation, surprise-inflation accounting and general-equilibrium impact). Part B: a 12-period OLG model with nominal annuity mortgages (FRM / 5-year term / ARM), gifts and estates, segmented rental and construction sectors, solved numerically (stationary general equilibrium and perfect-foresight transitions) with an **illustrative** calibration. | |
| 13 | +> **TL;DR** — A theoretical paper (no empirical estimation) on the conditions under which inherited wealth, mortgage leverage, nominal debt, housing supply and down-payment constraints generate persistent owner–renter wealth gaps among households with the same labour-income process, and on what an unanticipated inflation changes. A three-stage benchmark yields nine propositions (closed-form ownership-entry threshold, leverage multiplier with LTV-independent break-even, closed-form tenure persistence, existence/uniqueness, surprise-inflation accounting and impact). A 12-period OLG model with nominal annuity mortgages (FRM / 5-year term / ARM), gifts and estates, and segmented rental and construction sectors is solved in stationary general equilibrium and along perfect-foresight transitions. Transfers generate a 0.95 income-year wealth gap at mid-life and a rank–rank slope of 0.34 that vanishes without transfers; prices absorb 64–66 % of the partial-equilibrium ownership response to the down-payment constraint; no multiple equilibria; and, because households are net nominal creditors, a 10 % surprise **lowers** the real house price by 0.45 % where the benchmark predicted a rise. | |
| 12 | 14 | |
| 13 | 15 | --- |
| 14 | 16 | |
@@ -16,9 +18,32 @@ | ||
| 16 | 18 | |
| 17 | 19 | | Item | Value | |
| 18 | 20 | |---|---| |
| 19 | −| Phase | 3 completed (numerical implementation, 10 experiments, figures, tables); Phase 4 (audit + manuscript) pending | | |
| 20 | −| Code & results | reproducible from `make all`; raw results in `results/data/`, logs in `results/logs/` | | |
| 21 | −| Calibration | illustrative (verified Canadian comparison points in `results/data/calibration_targets.csv`; no verified supply elasticity) | | |
| 21 | +| Version | 1.0 (October 2026) | | |
| 22 | +| SSRN | not yet submitted | | |
| 23 | +| Journal | — (target: a field journal in housing or macro-finance; see `research/progress.md`, Phase 4 verdict) | | |
| 24 | +| Code & data | fully reproducible from `make all`; no microdata (illustrative calibration with verified Canadian comparison points) | | |
| 25 | + | |
| 26 | +## 🎯 What this paper does | |
| 27 | + | |
| 28 | +The paper asks under what conditions inherited wealth, mortgage leverage, nominal debt, housing supply and down-payment constraints generate persistent owner–renter wealth inequality among households with identical income processes, and what an unanticipated inflation changes. It answers in two steps. | |
| 29 | + | |
| 30 | +A **benchmark** (three stages of life, two-period generations, unit housing demand, nominal mortgage, inter-vivos transfer at the child's purchase) has closed forms. The ownership threshold is the down-payment bound plus the stage-1 consumption of the marginal buyer, $\hat b = (dp - \Lambda\rho)/(1-\Lambda) - y_1$; the parent–child tenure persistence is the second eigenvalue of the dynasty chain, $\lambda = G(\hat y_R) - G(\hat y_R - e_3 p)$, positive only if the house is net wealth at the end of life; a stationary equilibrium exists and is unique when the entry-threshold channel dominates the ladder channel; the revaluation of a price-level surprise is zero-sum with the lender and its general-equilibrium sign is that of a density-weighted net nominal position of the parents whose children are at the tenure margin. | |
| 31 | + | |
| 32 | +A **quantitative OLG economy** (12 five-year periods, four house sizes and continuous rented services, Rouwenhorst income risk, 25-year constant-payment nominal mortgages under three contracts with LTV and nominal payment-to-income constraints, gift at the child's entry and estate at the parent's death, segmented durable stocks, lender balance sheet) is solved numerically with an illustrative calibration and every result at fixed prices is labelled as such. | |
| 33 | + | |
| 34 | +## 🏆 Headline results | |
| 35 | + | |
| 36 | +| Object | Value (baseline / experiment) | | |
| 37 | +|---|---| | |
| 38 | +| Owner–renter net-worth gap at 40–44, equal income process | 0.95 years of income | | |
| 39 | +| P(child owns at 35–39 \| parent owner / renter) | 0.54 / 0.35; rank–rank wealth slope 0.34 | | |
| 40 | +| Same, without bequests | 0.38 / 0.38; rank–rank −0.01 (persistence vanishes) | | |
| 41 | +| Share of the PE ownership effect of θ absorbed by prices | 64 % (θ = 0.05), 66 % (θ = 0.40) | | |
| 42 | +| Multiple stationary equilibria (3 price scans incl. ε = 0.05) | none (monotone excess demand, one root) | | |
| 43 | +| Real house price on impact, 10 % surprise, FRM / term / ARM (level) | −0.45 / −0.47 / −0.79 % (benchmark predicted a rise) | | |
| 44 | +| CEV of mortgaged owners, persistent surprise, FRM / term / ARM | +1.14 / +0.46 / −1.21 % (ordering confirmed) | | |
| 45 | +| Same with a real-rate response, FRM / term | +1.43 / +0.20 % (the Canadian term loses its advantage) | | |
| 46 | +| Share of the PE ownership response to the surprise absorbed by prices | 77–92 % | | |
| 22 | 47 | |
| 23 | 48 | ## 🗂 Repository layout |
| 24 | 49 | |
@@ -26,58 +51,109 @@ | ||
| 26 | 51 | ├── CLAUDE.md # author's specification of the project |
| 27 | 52 | ├── configs/ # baseline.yaml (parameters, units, sources), scenarios.yaml (experiments) |
| 28 | 53 | ├── src/ |
| 29 | −│ ├── toy_model.py # Part A benchmark (closed forms, 44 tests) | |
| 30 | −│ ├── model.py # Part B primitives: conversions, grids, income process, mortgages, supply | |
| 54 | +│ ├── toy_model.py # benchmark (closed forms, 44 tests) | |
| 55 | +│ ├── model.py # quantitative primitives: conversions, grids, income process, mortgages, supply | |
| 31 | 56 | │ ├── household_solver.py # backward VFI with discrete tenure/size/loan/gift choices (numba) |
| 32 | 57 | │ ├── distributions.py # Monte-Carlo dynasty panel (gift + estate links), moments |
| 33 | −│ ├── equilibrium.py # stationary GE (nested Illinois + Broyden), MIT transitions (shooting) | |
| 58 | +│ ├── equilibrium.py # stationary GE (nested Illinois + lottery fixed point), MIT transitions (shooting) | |
| 34 | 59 | │ ├── calibration.py # verified comparison points and model moments |
| 35 | 60 | │ ├── simulation.py # experiments 1-10 (python -m src.simulation) |
| 36 | −│ ├── robustness.py / figures.py / tables.py | |
| 37 | −├── tests/ # pytest: identities, debt dynamics, feasibility, clearing, residuals | |
| 38 | −├── research/ # literature, model specification, propositions, failed hypotheses, audits, progress | |
| 39 | −├── results/{data,figures,tables,logs}/ | |
| 40 | −└── paper/ # main.tex, references.bib, appendix_proofs.tex (Phase 4) | |
| 61 | +│ ├── robustness.py / figures.py / tables.py # tables.py also writes numbers.tex (one macro per number) | |
| 62 | +├── tests/ # pytest: identities, debt dynamics, feasibility, clearing, residuals (69 tests) | |
| 63 | +├── research/ # literature (72 verified refs), model specification, propositions, failed | |
| 64 | +│ # hypotheses, results audit (integrity gate), style audit, progress | |
| 65 | +├── results/{data,figures,tables,logs}/ # raw CSV/JSON, 14 figures (PDF+PNG), 9 tables + numbers.tex | |
| 66 | +└── paper/ # main.tex → main.pdf (65 p.: 45 p. main text + references + appendices), | |
| 67 | + # sections/, appendix_proofs.tex, appendix_computation.tex, appendix/, | |
| 68 | + # figures/ and tables/ (copies made by `make paper`), references.bib | |
| 41 | 69 | ``` |
| 42 | 70 | |
| 43 | −## 🚀 Quick start (replication) | |
| 71 | +## 🚀 Quick start | |
| 44 | 72 | |
| 45 | 73 | ```bash |
| 46 | 74 | make setup # python3 -m venv .venv && pip install -r requirements.txt (pinned) |
| 47 | −make test # pytest (fast tests + two slow general-equilibrium tests) | |
| 48 | −make simulate # baseline, 48 stationary scenarios, 3 multiplicity scans, 16 transitions, welfare (60 min on 18 cores) | |
| 75 | +make test # pytest: 69 tests (≈ 20 s; two slow general-equilibrium tests included) | |
| 76 | +make simulate # 45 stationary GE + 5 PE + 3 scans + 16 transitions + welfare (≈ 60 min on 18 cores); | |
| 77 | + # skipped when results/data/diagnostics.json exists (make clean-results to redo) | |
| 49 | 78 | make figures tables # results/figures/fig01-14.{pdf,png}, results/tables/tab01-09.tex + numbers.tex |
| 50 | −make simulate-quick # 10-minute smoke test of the whole pipeline (small panel, subset of scenarios) | |
| 79 | +make paper # copies figures/tables into paper/ and runs latexmk → paper/main.pdf | |
| 80 | +make all # test → simulate (skipped if done) → figures → tables → paper | |
| 81 | +cd paper && latexmk -pdf appendix_proofs_standalone.tex # theory only (model + propositions + proofs) | |
| 51 | 82 | ``` |
| 52 | 83 | |
| 53 | −Every number in a table or figure maps to a file under `results/data/` (file names are given in the table notes and in `research/results_audit.md`). Results at fixed prices are always labelled `PE (prices fixed)`; general-equilibrium results carry their market-clearing residuals. | |
| 84 | +Runtime on an Apple-silicon machine: tests 20 s; full simulation 60 min (18 cores); figures + tables + paper under 2 min. | |
| 54 | 85 | |
| 55 | −## 🧮 Model and method (Part B) | |
| 86 | +## 🗃 Data | |
| 56 | 87 | |
| 57 | −* 12 five-year periods (25–84), retirement at 65, one child per household born when the parent is 55–59; inter-vivos gift at the child's entry and estate at the parent's death (child aged 55–59), both chosen with warm-glow utility. | |
| 58 | −* Owners choose among four house sizes with nominal constant-payment mortgages (25-year amortisation), LTV and payment-to-income constraints at a qualifying rate, refinancing; renters choose services continuously; selling, buying and maintenance costs. | |
| 59 | −* Segmented owner-occupied and rental stocks with durable stock-flow supply (elasticity illustrative), competitive landlords, small-open-economy real rate, Fisher nominal rate; lender balance sheet; three contract types (FRM, 5-year term, ARM) that differ only after an unanticipated inflation. | |
| 60 | −* Solution: discrete-choice VFI by age (numba), Monte-Carlo panel of 96,000 dynasties with common random numbers and a small unanticipated transitory income component (breaks the atoms of the cash distribution), nested regula-falsi market clearing with a damped fixed point on the anticipated-estate lottery, sequence-space shooting (tâtonnement pre-phase, parallel finite-difference Jacobian, Broyden) for MIT transitions. Diagnostics (Bellman residuals, clearing residuals, grid / sample / seed sensitivity) are in `results/data/diagnostics.csv` and Table 4. All 48 stationary equilibria clear to 5e-4; 12 of 16 transitions reach 1.5e-3 and the others are labelled with their residual. | |
| 88 | +There are no microdata. The calibration is **illustrative** (no moment is fitted). Verified Canadian comparison points (Survey of Financial Security 2023 owner–renter net worth and mortgage share; Hou 2010 ownership profile; CIBC gift statistics via Mirdamadi–Khalid 2023) are listed in `results/data/calibration_targets.csv` and reported next to the model's moments (Table 1 of the paper) without being targeted. No verified Canadian estimate of the housing-supply elasticity was found; the parameter is swept over 0.05–2. All simulation outputs are committed under `results/data/` (358 CSV/JSON files, 5.6 MB; `.npz`/`.pkl` and run logs are regenerable and excluded). | |
| 61 | 89 | |
| 62 | −## 📊 Figures and tables inventory | |
| 90 | +## 🔬 Methodology | |
| 63 | 91 | |
| 64 | −Figures (`results/figures/`): 01 model timing · 02 policy functions · 03 ownership thresholds · 04 lifecycle profiles · 05 equal-income paths · 06 inheritance-quintile trajectories · 07 inequality and mobility · 08 inflation transition · 09 inflation redistribution · 10 supply × down-payment heat map · 11 welfare · 12 solver diagnostics · 13 multiplicity search · 14 down-payment sweep. | |
| 92 | +1. **Benchmark theory** — `src/toy_model.py`: closed forms, general optimiser used to *verify* the hand-to-mouth assumption, dynasty chain, stationary equilibrium, surprise accounting and impact; proofs in `paper/appendix_proofs.tex`. | |
| 93 | +2. **Household problem** — `src/household_solver.py`: backward VFI over 12 ages; rent/buy/refinance/sell branches on a cash-on-hand grid, *stay* on the full state grid; exhaustive search + golden section on liquid assets; loans on 11 fractions of the binding constraint; gifts on 12 fractions of cash; 5-point estate lottery. | |
| 94 | +3. **Distribution** — `src/distributions.py`: 96,000-dynasty Monte-Carlo panel with common random numbers and a small unanticipated transitory income component (removes atoms of the cash distribution that made excess demand discontinuous). | |
| 95 | +4. **Stationary equilibrium** — `src/equilibrium.py`: nested Illinois regula falsi (rental inside owner market) + damped fixed point on the lottery and the transfer-tax rebate; tolerance 5e-4; supply scales normalised so that $p^* = \tilde p_r^* = 1$. | |
| 96 | +5. **Transitions** — sequence-space shooting over 14 periods with durable stocks, parallel finite-difference Jacobian, Broyden with refresh, damped tâtonnement pre-phase for large shocks; tolerance 1.5e-3 per period; every run labelled with its residual (12/16 within tolerance). | |
| 97 | +6. **Welfare** — consumption-equivalent variation with $U = V - \Phi$ (flow utility net of warm glow), entrants by transfer quintile and households alive at a shock by age and tenure. | |
| 98 | +7. **Traceability** — `src/tables.py` writes `numbers.tex` (≈ 2,500 macros); every number in the manuscript's prose is a macro read from `results/data/`. | |
| 65 | 99 | |
| 66 | −Tables (`results/tables/`): 01 notation · 02 parameters · 03 baseline moments · 04 convergence · 05 scenarios · 06 inflation · 07 mobility and inequality · 08 welfare · 09 robustness · `numbers.tex` (macros). | |
| 100 | +## 🖼 Figures & tables inventory | |
| 101 | + | |
| 102 | +| Figure (`results/figures/`) | Content | | |
| 103 | +|---|---| | |
| 104 | +| `fig01_model_timing` | timing of the OLG model (parent, child, gift, estate, mortgage term) | | |
| 105 | +| `fig02_policy_functions` | buy-minus-rent value, liquid saving, house size on the cash grid | | |
| 106 | +| `fig03_ownership_thresholds` | threshold by age and productivity; threshold vs θ in GE and PE | | |
| 107 | +| `fig04_lifecycle_profiles` | tenure, mortgages, net worth, balance sheets by age | | |
| 108 | +| `fig05_equal_income_paths` | equal-income wealth paths by gift quantile; two households straddling the threshold | | |
| 109 | +| `fig06_inheritance_quantile_trajectories` | ownership and median net worth by transfer quintile and age | | |
| 110 | +| `fig07_inequality_mobility` | Gini by age; tenure and wealth persistence across scenarios | | |
| 111 | +| `fig08_inflation_transition` | price, rent, ownership and real debt after a 10 % surprise, three contracts | | |
| 112 | +| `fig09_inflation_redistribution` | revaluation, consumption and CEV by age and tenure | | |
| 113 | +| `fig10_supply_elasticity_heatmap` | ε × θ: price, ownership, mobility, rank–rank | | |
| 114 | +| `fig11_welfare` | CEV of entrants by transfer quintile; CEV by age after the surprise | | |
| 115 | +| `fig12_solver_diagnostics` | market clearing, Bellman residuals, grid/sample sensitivity (Appendix B) | | |
| 116 | +| `fig13_multiplicity_search` | excess demand and ownership along the price (uniqueness scan) | | |
| 117 | +| `fig14_down_payment_sweep` | ownership, prices and persistence vs θ in GE and PE | | |
| 118 | + | |
| 119 | +Tables (`results/tables/*.tex`, copied to `paper/tables/`): 01 notation (Appendix C) · 02 parameters (Appendix B) · 03 baseline moments · 04 convergence (Appendix B) · 05 scenarios · 06 inflation · 07 mobility and inequality · 08 welfare · 09 robustness (Appendix B) · `numbers.tex` (macros). | |
| 67 | 120 | |
| 68 | 121 | ## ⚠️ Limitations |
| 69 | 122 | |
| 70 | −Illustrative calibration; no aggregate risk, default, survival risk or endogenous real rate; the estate is anticipated as an unconditional lottery; ARM = stylised within-period repricing at the 5-year frequency; Monte-Carlo noise bounds the market-clearing precision (equilibrium prices determined to about ±0.05 %). See `research/results_audit.md`. | |
| 123 | +- Illustrative calibration; no fitted moment; supply elasticity without a verified Canadian source. | |
| 124 | +- Ownership falls after 65 (warm glow over a *liquid* estate) whereas Canadian owners keep their houses; 72 % of entrants receive a gift vs 28 % in the Canadian source. | |
| 125 | +- No aggregate risk, default, survival risk or endogenous real rate; the estate is anticipated as an unconditional lottery and held fixed along transitions; the ARM is a stylised within-period repricing; the 5-year period understates peak LTV (nobody is ever underwater). | |
| 126 | +- Monte-Carlo noise bounds market clearing (≈ 5e-4 stationary, ≈ 1e-3 transitions); 4 of 16 transitions are reported with residuals above tolerance. | |
| 127 | +- Theory: single crossing proved only on the hand-to-mouth region; Prop. 2 holds for a once-and-for-all tenure decision with a binding LTV constraint; Prop. 7 under unit demand; Prop. 9 under pure debtor positions (Corollary 2 covers net creditors). The three simulation results that contradicted the broader statements are reported as results (`research/results_audit.md`). | |
| 71 | 128 | |
| 72 | −## 📖 Citation | |
| 129 | +## 📝 Citation | |
| 73 | 130 | |
| 74 | 131 | ```bibtex |
| 75 | −@techreport{boucher2026wp18, | |
| 132 | +@techreport{boucher2026housing, | |
| 76 | 133 | author = {Boucher, Simon-Pierre}, |
| 77 | 134 | title = {Housing, Leverage, and Intergenerational Wealth Inequality: A Dynamic General Equilibrium Theory}, |
| 78 | 135 | institution = {Universit\'e du Qu\'ebec en Outaouais, D\'epartement des sciences administratives}, |
| 79 | − type = {UQO Working Paper}, | |
| 136 | + type = {Working Paper}, | |
| 80 | 137 | number = {18}, |
| 81 | − year = {2026} | |
| 138 | + year = {2026}, | |
| 139 | + month = {October} | |
| 82 | 140 | } |
| 83 | 141 | ``` |
| 142 | + | |
| 143 | +## 👤 Author & contact | |
| 144 | + | |
| 145 | +**Simon-Pierre Boucher** — Département des sciences administratives, | |
| 146 | +Université du Québec en Outaouais (UQO), Gatineau, QC. | |
| 147 | +📧 [simon-pierre.boucher@uqo.ca](mailto:simon-pierre.boucher@uqo.ca) | |
| 148 | + | |
| 149 | +## 🗒 Notes | |
| 150 | + | |
| 151 | +Deviations from `Recherche-UQO/HARMONISATION.md` (all required by the project specification in `CLAUDE.md`): | |
| 152 | + | |
| 153 | +- Theoretical paper: no "Data" section; a "Numerical algorithm, calibration and verification" section replaces it. | |
| 154 | +- amsthm environments extended beyond WP8 (Assumption, Definition, Proposition, Lemma, Corollary, Remark, `proof`); statements in `paper/sections/equilibrium.tex`, proofs in `paper/appendix_proofs.tex` (compiles stand-alone with the model section through `appendix_proofs_standalone.tex`, PDF not committed). | |
| 155 | +- Appendices `appendix_proofs.tex` and `appendix_computation.tex` at the root of `paper/`, plus `appendix/appendix.tex` (notation); appendix tables and figures numbered A/B/C. | |
| 156 | +- `src/*.py` + `Makefile` + `configs/*.yaml` instead of numbered `scripts/`; `make simulate` is skipped when results exist. | |
| 157 | +- Wide tables (scenarios, inflation, mobility, robustness) are scaled with `\resizebox` and carry their note in a `minipage` (threeparttable cannot wrap a resized tabular). | |
| 158 | +- Figures are generated without in-image titles (LaTeX captions only); `WP18_FIG_TITLES=1 make figures` restores them for previews. | |
| 159 | +- Numbering history: WP 18 assigned by `HARMONISATION.md` §1; spbgit repository `wp18_uqo.git`. | |
modified
paper/appendix/appendix.tex
+5 −88
@@ -1,93 +1,10 @@ | ||
| 1 | 1 | % ============================================================================ |
| 2 | −% Appendix | |
| 2 | +% WP18 — Appendix C: supplementary table (notation) | |
| 3 | 3 | % ============================================================================ |
| 4 | −\newpage | |
| 5 | −\begin{appendices} | |
| 6 | − | |
| 7 | −\renewcommand{\thetable}{A\arabic{table}} | |
| 8 | −\renewcommand{\thefigure}{A\arabic{figure}} | |
| 9 | −\setcounter{table}{0} | |
| 10 | −\setcounter{figure}{0} | |
| 11 | − | |
| 12 | −\section{Additional Tables and Figures} | |
| 4 | +\section{Notation} | |
| 13 | 5 | \label{app:tables} |
| 6 | +\setcounter{table}{0}\setcounter{figure}{0} | |
| 14 | 7 | |
| 15 | −% --- Appendix Table: Sub-sample Analysis --- | |
| 16 | −\begin{table}[htbp] | |
| 17 | − \centering | |
| 18 | − \begin{threeparttable} | |
| 19 | − \caption{Sub-sample Analysis} | |
| 20 | − \label{tab:app_subsample} | |
| 21 | − \begin{tabular}{@{}lccc@{}} | |
| 22 | − \toprule | |
| 23 | − & (1) & (2) & (3) \\ | |
| 24 | − & Full Sample & Pre-2008 & Post-2008 \\ | |
| 25 | − \midrule | |
| 26 | − $X_{i,t}$ & 0.172$^{***}$ & 0.195$^{***}$ & 0.148$^{***}$ \\ | |
| 27 | − & (0.033) & (0.048) & (0.041) \\[6pt] | |
| 28 | − \midrule | |
| 29 | − Controls & Yes & Yes & Yes \\ | |
| 30 | − Firm FE & Yes & Yes & Yes \\ | |
| 31 | − Time FE & Yes & Yes & Yes \\[3pt] | |
| 32 | − Observations & 125{,}400 & 58{,}200 & 67{,}200 \\ | |
| 33 | − Adjusted $R^2$ & 0.190 & 0.175 & 0.203 \\ | |
| 34 | − \bottomrule | |
| 35 | − \end{tabular} | |
| 36 | − \begin{tablenotes}[flushleft] | |
| 37 | − \footnotesize | |
| 38 | − \item \textit{Notes:} This table reports sub-sample results. Column~(1) reproduces the baseline estimate from Table~\ref{tab:regression}, Column~(4). Columns~(2) and~(3) split the sample before and after January~2008. All specifications include the full set of controls and firm and time fixed effects. Standard errors clustered at the firm level are in parentheses. $^{***}$, $^{**}$, and $^{*}$ denote significance at the 1\%, 5\%, and 10\% levels. | |
| 39 | − \end{tablenotes} | |
| 40 | − \end{threeparttable} | |
| 41 | −\end{table} | |
| 42 | − | |
| 43 | −% --- Appendix Table: Robustness Checks --- | |
| 44 | −\begin{table}[htbp] | |
| 45 | − \centering | |
| 46 | − \begin{threeparttable} | |
| 47 | − \caption{Robustness Checks} | |
| 48 | − \label{tab:app_robustness} | |
| 49 | − \begin{tabular}{@{}lccc@{}} | |
| 50 | − \toprule | |
| 51 | − & (1) & (2) & (3) \\ | |
| 52 | − & Alt.\ Dep.\ Var. & Alt.\ Controls & Winsorized 5\% \\ | |
| 53 | − \midrule | |
| 54 | − $X_{i,t}$ & 0.165$^{***}$ & 0.178$^{***}$ & 0.169$^{***}$ \\ | |
| 55 | − & (0.036) & (0.034) & (0.032) \\[6pt] | |
| 56 | − \midrule | |
| 57 | − Controls & Yes & Yes & Yes \\ | |
| 58 | − Firm FE & Yes & Yes & Yes \\ | |
| 59 | − Time FE & Yes & Yes & Yes \\[3pt] | |
| 60 | − Observations & 125{,}400 & 125{,}400 & 119{,}800 \\ | |
| 61 | − Adjusted $R^2$ & 0.185 & 0.192 & 0.195 \\ | |
| 62 | − \bottomrule | |
| 63 | − \end{tabular} | |
| 64 | − \begin{tablenotes}[flushleft] | |
| 65 | − \footnotesize | |
| 66 | − \item \textit{Notes:} This table reports robustness checks. Column~(1) uses an alternative dependent variable. Column~(2) includes additional control variables. Column~(3) winsorizes all continuous variables at the 5th and 95th percentiles. All specifications include firm and time fixed effects. Standard errors clustered at the firm level are in parentheses. $^{***}$, $^{**}$, and $^{*}$ denote significance at the 1\%, 5\%, and 10\% levels. | |
| 67 | − \end{tablenotes} | |
| 68 | − \end{threeparttable} | |
| 69 | −\end{table} | |
| 70 | − | |
| 71 | −\section{Variable Definitions} | |
| 72 | −\label{app:variables} | |
| 73 | − | |
| 74 | −\begin{table}[htbp] | |
| 75 | − \centering | |
| 76 | − \caption{Variable Definitions} | |
| 77 | − \label{tab:app_variables} | |
| 78 | − \begin{tabular}{@{}p{3.5cm}p{10cm}@{}} | |
| 79 | − \toprule | |
| 80 | − Variable & Definition \\ | |
| 81 | − \midrule | |
| 82 | − $r_{i,t}$ & Monthly log return of firm $i$ in month $t$, computed as $\ln(P_{i,t}/P_{i,t-1})$. \\[4pt] | |
| 83 | − $\sigma_{i,t}$ & Monthly realized volatility of firm $i$, computed as the standard deviation of daily returns within month $t$, annualized by multiplying by $\sqrt{252}$. \\[4pt] | |
| 84 | − $X_{i,t}$ & Main explanatory variable. [Provide specific definition.] \\[4pt] | |
| 85 | − Size & Natural logarithm of market capitalization (in millions of USD) at the end of month $t-1$. \\[4pt] | |
| 86 | − Book-to-Market & Ratio of book equity to market equity, using the most recent fiscal year-end book value and the market value at the end of June. Updated annually following \citet{fama1993common}. \\[4pt] | |
| 87 | − Leverage & Total debt divided by total assets, measured at the most recent fiscal year-end. \\[4pt] | |
| 88 | − ROA & Return on assets, defined as net income divided by total assets. \\ | |
| 89 | − \bottomrule | |
| 90 | − \end{tabular} | |
| 91 | −\end{table} | |
| 8 | +Table~\ref{tab:notation} collects the notation of the quantitative model of Section~\ref{subsec:quant}; the benchmark of Section~\ref{subsec:benchmark} uses the same symbols with the stage-specific definitions of \eqref{eq:deltas}. | |
| 92 | 9 | |
| 93 | −\end{appendices} | |
| 10 | +\input{tables/tab01_notation} | |
added
paper/appendix_computation.tex
+34 −0
@@ -0,0 +1,34 @@ | ||
| 1 | +% ============================================================================ | |
| 2 | +% WP18 — Appendix B: numerical algorithm, grids, tolerances and diagnostics | |
| 3 | +% ============================================================================ | |
| 4 | +\section{Numerical algorithm and diagnostics} | |
| 5 | +\label{app:computation} | |
| 6 | +\setcounter{table}{0}\setcounter{figure}{0} | |
| 7 | + | |
| 8 | +\subsection{Algorithm} | |
| 9 | + | |
| 10 | +The stationary equilibrium is computed as follows. (1) Given prices $(p,\tilde p_r)$, the rent from \eqref{eq:rent}, the nominal rate, the estate lottery and the rebate, solve the household problem backward from $J$ to $1$: for each age, compute the values of the cash-only branches (rent, buy, refinance, sell-and-rent, move) on the cash-on-hand grid, with the loan chosen among eleven fractions of the binding origination constraint and the house size by comparison of the stored size-specific values, and the value of staying on the full state grid; next-period liquid assets are chosen by exhaustive search on the asset grid followed by a golden-section refinement; the gift at $j_g$ is chosen among twelve fractions of cash; the estate at $J$ is the terminal wealth. (2) Simulate the dynasty panel with common random numbers: \NDynasties{} slots, each holding a parent and a child whose phases are equidistributed, for ten generations of burn-in and six recorded periods, with the gift and the estate transmitted within the slot. (3) Compute excess demands in the two markets, the conditional means of the simulated estates by quintile (the lottery) and the transfer-tax revenue (the rebate). (4) Clear the rental market in $\tilde p_r$ by the Illinois regula falsi inside a regula falsi on $p$ for the owner-occupied market, then update the lottery and the rebate with damping 0.7 and repeat until the relative excess demands are below $5\times10^{-4}$ and the lottery changes by less than $10^{-3}$. The supply scales $\bar H$, $\bar H_r$ are set once, at the baseline, so that $p^* = \tilde p^*_r = 1$ (\SupplyScaleOwner{} and \SupplyScaleRental{} units). | |
| 11 | + | |
| 12 | +A transition after an unanticipated shock at $t = 0$ is computed by shooting in the sequence space: guess the log price paths $(\log p_t,\log\tilde p_{r,t})_{t<T}$ with $T = 14$, solve the household problem backward from the stationary terminal values with the time-varying prices, rates and contracts, simulate forward from the stationary cross-section with the realised shock, update the stocks by \eqref{eq:supply}, evaluate the excess demands at every date, and update the paths by Broyden steps from a finite-difference Jacobian (step 0.01 in log price, $2(T-1)$ columns computed in parallel) with periodic refresh; for large shocks a damped t\^atonnement pre-phase precedes the first Jacobian. The tolerance is $1.5\times10^{-3}$ on the maximum relative excess demand along the path; runs that stall at the Monte-Carlo noise floor above this value are labelled with their residual in every table. Partial-equilibrium counterparts repeat the forward simulation with prices fixed at the stationary values. The static-expectations variant solves temporary equilibria period by period with stationary continuation values. | |
| 13 | + | |
| 14 | +\subsection{Parameters, grids, tolerances and run times} | |
| 15 | + | |
| 16 | +\input{tables/tab02_parameters} | |
| 17 | + | |
| 18 | + | |
| 19 | +Grids: liquid assets, 50 points with double-exponential spacing on $[0,15]$; cash on hand, 120 points on $[0,22]$; mortgage balance per unit of house, 20 points on $[0,(1-\theta)p_{\max}]$ with $p_{\max} = 1.6$; four house sizes; five remaining-payment counts; five productivity states; eleven loan fractions; twelve gift fractions; a five-point estate lottery; a consumption floor of 0.01. Units are the average five-year labour income. The baseline stationary equilibrium takes \BaselineSeconds{} seconds, other stationary equilibria 45--300 seconds, a transition 2.5--8 minutes with the Jacobian on eight processes, and the full pipeline \SimulateMinutesOne{} minutes on 18 cores: \NScenariosSS{} stationary general equilibria (\NScenariosSSConverged{} within tolerance; the 24{,}000-dynasty variant stopped at the noise floor), five partial-equilibrium re-solutions, three multiplicity scans, \NTransitions{} transition experiments (\NTransitionsConverged{} within tolerance) and the welfare computations. | |
| 20 | + | |
| 21 | +\subsection{Diagnostics} | |
| 22 | + | |
| 23 | +\input{tables/tab04_convergence} | |
| 24 | + | |
| 25 | +\begin{figure}[htbp] | |
| 26 | +\centering | |
| 27 | +\includegraphics[width=\textwidth]{fig12_solver_diagnostics.pdf} | |
| 28 | +\caption{Solver diagnostics: absolute excess demand after each lottery round in the baseline (a), mean Bellman residuals by scenario (b), and percentage deviations of the key statistics from the baseline under the numerical variants (c). Source: \texttt{results/data/baseline/summary.json}, \texttt{diagnostics.json}, \texttt{grid\_sensitivity.csv}.} | |
| 29 | +\label{fig:diagnostics} | |
| 30 | +\end{figure} | |
| 31 | + | |
| 32 | +Table~\ref{tab:convergence} and Figure~\ref{fig:diagnostics} report the market-clearing residuals, the lottery convergence, the Bellman residuals at off-grid states, the forced-sale and clipped-policy shares and the numerical variants. Table~\ref{tab:robustness} lists the economic and numerical robustness runs with their equilibrium statistics and run times. Two events during the computation are recorded because they changed the design. A first full run revealed a genuine discontinuity in the rental excess demand, a jump of 1.5\% of rental demand between two adjacent prices in the near-fixed-supply, 5\%-down-payment scenario, caused by atoms of the cash distribution: entrants without a gift and liquidity-constrained renters at the same productivity state hold exactly the same cash and switch tenure together. The remedy was the small unanticipated transitory income component of Section~\ref{sec:computation}, after which the excess-demand functions were continuous and the roots bracketed; the budget identities were re-tested with the realised income. And the cold-start shooting of the real-rate shock under a 5\% down payment diverged (initial residual 0.17); a warm start from the 20\% path and an 80-step damped t\^atonnement followed by a fresh Jacobian brought the residual to $\RateFiveMaxRes$, where Broyden steps were again rejected, so that run is reported as not converged. | |
| 33 | + | |
| 34 | +\input{tables/tab09_robustness} | |
modified
paper/appendix_proofs.tex
+65 −226
@@ -1,68 +1,16 @@ | ||
| 1 | 1 | % ============================================================================ |
| 2 | −% WP18 — Appendix: assumptions, propositions and proofs (Part A benchmark model) | |
| 3 | −% Input from main.tex after \begin{appendices}. Theorem environments are defined | |
| 4 | −% in the preamble of main.tex (see research/wp_style_audit.md, §6) and reproduced | |
| 5 | −% in appendix_proofs_standalone.tex for stand-alone compilation. | |
| 2 | +% WP18 — Appendix A: lemmas and proofs of the propositions of Section 6. | |
| 3 | +% The statements (assumptions, propositions, corollaries) are in | |
| 4 | +% sections/equilibrium.tex; this file contains the preliminary lemmas, the | |
| 5 | +% proofs and the proof-gap remarks. Compiles stand-alone together with | |
| 6 | +% sections/model.tex and sections/equilibrium.tex through | |
| 7 | +% appendix_proofs_standalone.tex. Theorem environments: preamble of main.tex. | |
| 6 | 8 | % ============================================================================ |
| 7 | −\section{Assumptions, propositions and proofs} | |
| 9 | +\section{Proofs} | |
| 8 | 10 | \label{app:proofs} |
| 11 | +\setcounter{table}{0}\setcounter{figure}{0} | |
| 9 | 12 | |
| 10 | −This appendix states and proves the analytical results of the benchmark model of Section~\ref{sec:model} (three stages of life, two-period generations; full specification in \texttt{research/model\_specification.md}, Part~A). Throughout, $R = 1+r$, $d \equiv \theta + \kappa_b + m$ is the stage-1 cash outlay per unit of house price, and, in the stationary state with inflation $\bar\pi$, | |
| 11 | −\begin{equation} | |
| 12 | −\delta_2 \equiv (1-\theta)\Big[R - \frac{1-a}{1+\bar\pi}\Big], \qquad | |
| 13 | −\delta_3 \equiv \frac{(1-a)(1-\theta)R}{1+\bar\pi}, \qquad | |
| 14 | −e_3 \equiv (1-\kappa_s) - \delta_3, | |
| 15 | −\label{eq:app_deltas} | |
| 16 | −\end{equation} | |
| 17 | −so that an owner who bought one unit at price $p$ pays $D_2 = \delta_2 p$ at stage~2 and $D_3 = \delta_3 p$ at stage~3, and sells at stage~3 for $(1-\kappa_s)p$ in a stationary state. The identity | |
| 18 | −\begin{equation} | |
| 19 | −\delta_2 + \frac{\delta_3}{R} = (1-\theta)R | |
| 20 | −\label{eq:app_fisher} | |
| 21 | −\end{equation} | |
| 22 | −follows from $1+i = R(1+\bar\pi)$ and is the present-value (Fisher) neutrality of the nominal contract. Consumption of a hand-to-mouth household is written $c^O_1(b) = b + y_1 - dp$, $c^R_1(b) = b + y_1 - \rho$, $c^O_2 = y_2 - \delta_2 p - mp$, $c^R_2 = y_2 - \rho$, $W^O_3(y_3) = y_3 + e_3 p$, $W^R_3(y_3) = y_3$. | |
| 23 | − | |
| 24 | −\subsection{Assumptions} | |
| 25 | − | |
| 26 | −\begin{assumption}[Preferences, demography, housing need] | |
| 27 | −\label{ass:pref} | |
| 28 | −A household born at $t$ lives three stages and has utility $\ln c_1 + \chi\mathbf 1\{\text{owner}\} + \beta\,\mathbb E\ln c_2 + \beta^2\,\mathbb E[\ln c_3 + \nu \ln b']$ with $\beta\in(0,1)$, $\nu>0$, $\chi\ge 0$, where $b'$ is the inter-vivos transfer to its child, who enters the model at the parent's stage~3. Each household needs one unit of housing services at stages 1 and 2, obtained by owning one unit of the owner-occupied stock or renting one unit of the rental stock. Stage-3 households do not demand housing in the two markets. Incomes $y_1, y_2 > 0$ are common and deterministic; $y_3$ is drawn, independently across households and generations, from a distribution $G$ with a continuous density $g>0$ on $[\underline y_3, \bar y_3]$, $0 < \underline y_3 < \bar y_3 < \infty$. | |
| 29 | −\end{assumption} | |
| 30 | − | |
| 31 | −\begin{assumption}[Prices, contracts, lender] | |
| 32 | −\label{ass:prices} | |
| 33 | −The real rate $r\ge 0$ is exogenous. The one-period nominal rate satisfies $1+i_t = R(1+\pi^e_{t+1})$. A mortgage of real principal $B \le (1-\theta)p_t$ is a nominal contract paying $(i_1 + a)B^{nom}$ at stage~2 and $(1+i_2)(1-a)B^{nom}$ at stage~3, with $a\in(0,1]$, $i_1$ fixed at origination and $i_2 = i_1$ (FRM) or $i_2$ reset at stage~2 to the then-current nominal rate (ARM). Deposits pay $i_t$ and cannot be negative. The lender is competitive and risk neutral, holds the mortgages and the deposits, and its realised real profit accrues to rentiers outside the dynasties. The rental stock is segmented from the owner-occupied stock; per cohort, owner-occupied supply is $S(p) = \bar H p^{\varepsilon}$, $\varepsilon\in[0,\infty)$, and rental supply is $S_r(\rho) = \bar H_r \rho^{\varepsilon_r}$, $\varepsilon_r\in(0,\infty]$. | |
| 34 | −\end{assumption} | |
| 35 | − | |
| 36 | −\begin{assumption}[Feasibility and no default] | |
| 37 | −\label{ass:feas} | |
| 38 | −For all prices considered, $c^O_2 > 0$, $c^R_2 > 0$ and $\underline y_3 + \min\{e_3 p, 0\} > 0$. (Every stage-2 payment can be met out of income and every owner can repay the stage-3 balance out of income plus the sale proceeds; there is no default.) | |
| 39 | −\end{assumption} | |
| 40 | − | |
| 41 | −\begin{assumption}[Liquidity-constrained at the margin: hand-to-mouth, HTM] | |
| 42 | −\label{ass:htm} | |
| 43 | −On the wealth interval $\mathcal B = (dp - y_1, \bar b]$ relevant for the tenure decision, the following strict Euler inequalities hold for both tenures at the corner allocation $n_1 = n_2 = 0$ (and $B = (1-\theta)p$ for owners): | |
| 44 | −\begin{equation} | |
| 45 | −\frac{1}{c^o_1(b)} > \beta R\,\frac{1}{c^o_2}, \qquad | |
| 46 | −\frac{1}{c^o_2} > \beta R (1+\nu)\,\mathbb E\Big[\frac{1}{W^o_3(y_3)}\Big], \qquad o\in\{O,R\},\ b\in\mathcal B . | |
| 47 | −\label{eq:app_euler} | |
| 48 | −\end{equation} | |
| 49 | −\end{assumption} | |
| 50 | − | |
| 51 | −\begin{assumption}[Interior tenure margin] | |
| 52 | −\label{ass:interior} | |
| 53 | −$\Delta > 0$ and $dp > \rho$, where | |
| 54 | −\begin{equation} | |
| 55 | −\Delta \equiv \chi + \beta\big[\ln c^O_2 - \ln c^R_2\big] + \beta^2(1+\nu)\,\mathbb E\big[\ln W^O_3(y_3) - \ln W^R_3(y_3)\big], \qquad \Lambda \equiv e^{-\Delta} \in (0,1). | |
| 56 | −\label{eq:app_Delta} | |
| 57 | −\end{equation} | |
| 58 | −\end{assumption} | |
| 59 | − | |
| 60 | −\begin{assumption}[Static expectations on impact] | |
| 61 | −\label{ass:static} | |
| 62 | −In the period of an unanticipated inflation surprise, stage-1 households evaluate their stage-2 rent and stage-3 resale price at the pre-shock stationary values $(\rho^*, p^*)$, while buying, borrowing and renting at the current prices $(p_{t+1}, \rho_{t+1})$. | |
| 63 | −\end{assumption} | |
| 64 | − | |
| 65 | −Assumption~\ref{ass:htm} is verified, not imposed, in the numerical benchmark (\texttt{tests/test\_toy\_model.py}): the Euler slacks are computed on the support of the stationary transfer distribution and the general optimiser returns the corner. Assumption~\ref{ass:static} is only used in Proposition~\ref{prop:impact}; the perfect-foresight transition is computed numerically in Phase~3. | |
| 13 | +Throughout, $R = 1+r$, $d$, $\delta_2$, $\delta_3$ and $e_3$ are as in \eqref{eq:deltas}, the identity \eqref{eq:fisher} holds, and the hand-to-mouth consumption levels $c^o_j$ and terminal resources $W^o_3$ are those of Section~\ref{subsec:benchmark}. An owner who bought one unit at price $p$ pays $D_2 = \delta_2p$ at stage~2 and $D_3 = \delta_3p$ at stage~3, and sells at stage~3 for $(1-\kappa_s)p$ in a stationary state. | |
| 66 | 14 | |
| 67 | 15 | \subsection{Preliminary lemmas} |
| 68 | 16 | |
@@ -84,237 +32,128 @@ V^R(b) &= \ln c^R_1(b) + \beta\ln c^R_2 + \beta^2(1+\nu)\,\mathbb E\ln W^R_3(y_3 | ||
| 84 | 32 | with $K = \beta^2[\nu\ln\nu - (1+\nu)\ln(1+\nu)]$, and $V^O(b) - V^R(b) = \ln c^O_1(b) - \ln c^R_1(b) + \Delta$. |
| 85 | 33 | \end{lemma} |
| 86 | 34 | \begin{proof} |
| 87 | −Using Lemma~\ref{lem:split}, the stage-3 continuation is $(1+\nu)\ln W_3 + K$. For a renter the problem is to maximise $\ln(b+y_1-\rho-n_1) + \beta\ln(y_2-\rho+n_1R-n_2) + \beta^2(1+\nu)\mathbb E\ln(y_3+n_2R)$ over $n_1,n_2\ge 0$ (the stage-2 choice is made before $y_3$ is known; $n_1$ is chosen at stage 1 and $n_2$ at stage 2, but since $c_2$ is deterministic given $n_1$ the two-stage problem equals this static problem). The objective is strictly concave in $(n_1,n_2)$ and the constraint set is convex, so the Karush--Kuhn--Tucker conditions are necessary and sufficient. At $n_1 = n_2 = 0$ they read $1/c^R_1 \ge \beta R/c^R_2$ and $1/c^R_2 \ge \beta R(1+\nu)\mathbb E[1/W^R_3]$, which hold strictly by~\eqref{eq:app_euler}. For an owner, the choice variables are $(B, n_1, n_2)$ with $0\le B\le (1-\theta)p$; lowering $B$ by $\epsilon$ lowers $c_1$ by $\epsilon$ and raises $c_2$ by $\epsilon\,\delta_2/(1-\theta)$ and $W_3$ by $\epsilon\,\delta_3/(1-\theta)$. The KKT condition for $B = (1-\theta)p$ is | |
| 35 | +Using Lemma~\ref{lem:split}, the stage-3 continuation is $(1+\nu)\ln W_3 + K$. For a renter the problem is to maximise $\ln(b+y_1-\rho-n_1) + \beta\ln(y_2-\rho+n_1R-n_2) + \beta^2(1+\nu)\mathbb E\ln(y_3+n_2R)$ over $n_1,n_2\ge 0$ ($n_1$ is chosen at stage~1 and $n_2$ at stage~2, but since $c_2$ is deterministic given $n_1$ the two-stage problem equals this static problem). The objective is strictly concave in $(n_1,n_2)$ and the constraint set is convex, so the Karush--Kuhn--Tucker conditions are necessary and sufficient. At $n_1 = n_2 = 0$ they read $1/c^R_1\ge\beta R/c^R_2$ and $1/c^R_2\ge\beta R(1+\nu)\mathbb E[1/W^R_3]$, which hold strictly by~\eqref{eq:euler}. For an owner, the choice variables are $(B,n_1,n_2)$ with $0\le B\le(1-\theta)p$; lowering $B$ by $\epsilon$ lowers $c_1$ by $\epsilon$ and raises $c_2$ by $\epsilon\,\delta_2/(1-\theta)$ and $W_3$ by $\epsilon\,\delta_3/(1-\theta)$. The KKT condition for $B = (1-\theta)p$ is | |
| 88 | 36 | \[ |
| 89 | −\frac{1}{c^O_1} \;\ge\; \beta\frac{\delta_2}{1-\theta}\frac{1}{c^O_2} + \beta^2(1+\nu)\frac{\delta_3}{1-\theta}\,\mathbb E\Big[\frac{1}{W^O_3}\Big]. | |
| 37 | +\frac{1}{c^O_1}\;\ge\;\beta\frac{\delta_2}{1-\theta}\frac{1}{c^O_2} + \beta^2(1+\nu)\frac{\delta_3}{1-\theta}\,\mathbb E\Big[\frac{1}{W^O_3}\Big]. | |
| 90 | 38 | \] |
| 91 | −By~\eqref{eq:app_euler}, $\beta/c^O_2 < 1/(Rc^O_1)$ and $\beta^2(1+\nu)\mathbb E[1/W^O_3] = \beta\cdot\beta(1+\nu)\mathbb E[1/W^O_3] < \beta/(Rc^O_2) < 1/(R^2 c^O_1)$, so the right-hand side is strictly below $\frac{1}{c^O_1}\cdot\frac{\delta_2/R + \delta_3/R^2}{1-\theta} = \frac{1}{c^O_1}$ by~\eqref{eq:app_fisher}. The conditions for $n_1 = n_2 = 0$ are as for the renter. Substituting the corner allocations gives \eqref{eq:app_VO}--\eqref{eq:app_VR}, and the difference follows from \eqref{eq:app_Delta}. | |
| 39 | +By~\eqref{eq:euler}, $\beta/c^O_2 < 1/(Rc^O_1)$ and $\beta^2(1+\nu)\mathbb E[1/W^O_3] < \beta/(Rc^O_2) < 1/(R^2c^O_1)$, so the right-hand side is strictly below $\frac{1}{c^O_1}\cdot\frac{\delta_2/R+\delta_3/R^2}{1-\theta} = \frac{1}{c^O_1}$ by~\eqref{eq:fisher}. The conditions for $n_1 = n_2 = 0$ are as for the renter. Substituting the corner allocations gives \eqref{eq:app_VO}--\eqref{eq:app_VR}, and the difference follows from \eqref{eq:Delta}. | |
| 92 | 40 | \end{proof} |
| 93 | 41 | |
| 94 | −\subsection{Ownership-entry threshold} | |
| 42 | +\subsection{Threshold, comparative statics and tilt} | |
| 95 | 43 | |
| 96 | −\begin{proposition}[Ownership threshold] | |
| 97 | −\label{prop:threshold} | |
| 98 | −Under Assumptions~\ref{ass:pref}--\ref{ass:interior}, there is a unique $\hat b\in\mathcal B$ such that a household with transfer $b\in\mathcal B$ owns if and only if $b\ge\hat b$, where | |
| 99 | −\begin{equation} | |
| 100 | −\hat b \;=\; \frac{d\,p - \Lambda\,\rho}{1-\Lambda} - y_1, \qquad | |
| 101 | −\hat b - (dp - y_1) = \frac{\Lambda\,(dp-\rho)}{1-\Lambda} > 0, \qquad c^O_1(\hat b) = \frac{\Lambda(dp-\rho)}{1-\Lambda}. | |
| 102 | −\label{eq:app_bhat} | |
| 103 | −\end{equation} | |
| 104 | −If instead $dp \le \rho$ (owning costs no more cash than renting), every household with $b > dp - y_1$ owns and the threshold is the feasibility bound $dp - y_1$. | |
| 105 | −\end{proposition} | |
| 106 | −\begin{proof} | |
| 107 | −Let $D(b) \equiv V^O(b) - V^R(b) = \ln c^O_1(b) - \ln c^R_1(b) + \Delta$ on $\mathcal B$ (Lemma~\ref{lem:htm}). $D$ is continuous, $D(b)\to-\infty$ as $b\downarrow dp - y_1$ (since $c^O_1\to 0$ while $c^R_1 \to dp - \rho > 0$), and $D'(b) = 1/c^O_1(b) - 1/c^R_1(b) > 0$ because $c^O_1(b) = c^R_1(b) - (dp-\rho) < c^R_1(b)$ by Assumption~\ref{ass:interior}. As $b\to\infty$, $\ln(c^O_1/c^R_1)\to 0$ so $D(b)\to\Delta>0$. Hence $D$ has exactly one zero $\hat b$, at which $c^O_1/c^R_1 = \Lambda$, i.e. $b + y_1 - dp = \Lambda(b+y_1-\rho)$, which gives~\eqref{eq:app_bhat}; the two identities follow by substitution. If $dp\le\rho$, $c^O_1\ge c^R_1$ so $D(b)\ge\Delta>0$ for all feasible $b$. | |
| 44 | +\begin{proof}[Proof of Proposition~\ref{prop:threshold}] | |
| 45 | +Let $D(b)\equiv V^O(b)-V^R(b) = \ln c^O_1(b) - \ln c^R_1(b) + \Delta$ on $\mathcal B$ (Lemma~\ref{lem:htm}). $D$ is continuous, $D(b)\to-\infty$ as $b\downarrow dp-y_1$ (since $c^O_1\to 0$ while $c^R_1\to dp-\rho>0$), and $D'(b) = 1/c^O_1(b) - 1/c^R_1(b) > 0$ because $c^O_1(b) = c^R_1(b) - (dp-\rho) < c^R_1(b)$ by Assumption~\ref{ass:interior}. As $b\to\infty$, $\ln(c^O_1/c^R_1)\to 0$ so $D(b)\to\Delta>0$. Hence $D$ has exactly one zero $\hat b$, at which $c^O_1/c^R_1 = \Lambda$, i.e.\ $b+y_1-dp = \Lambda(b+y_1-\rho)$, which gives~\eqref{eq:bhat}; the two identities follow by substitution. If $dp\le\rho$, $c^O_1\ge c^R_1$ so $D(b)\ge\Delta>0$ for all feasible $b$. | |
| 108 | 46 | \end{proof} |
| 109 | 47 | |
| 110 | −\begin{remark}[Proof gap: single crossing outside the HTM region] | |
| 48 | +\begin{remark}[Proof gap: single crossing outside the hand-to-mouth region] | |
| 111 | 49 | \label{rem:gap_single} |
| 112 | −Proposition~\ref{prop:threshold} characterises the tenure decision on $\mathcal B$, where Assumption~\ref{ass:htm} holds. For $b>\bar b$ the household saves and the values are not the closed forms \eqref{eq:app_VO}--\eqref{eq:app_VR}; the envelope derivative is still $1/c^o_1(b)$ for each tenure, so $V^O-V^R$ is increasing wherever $c^O_1(b) < c^R_1(b)$, but we have not proved that this holds at every crossing point. The numerical benchmark finds a single crossing on $[dp-y_1, 3]$ with the general optimiser; a general proof is left open. | |
| 50 | +Proposition~\ref{prop:threshold} characterises the tenure decision on $\mathcal B$, where Assumption~\ref{ass:htm} holds. For $b>\bar b$ the household saves and the values are not the closed forms \eqref{eq:app_VO}--\eqref{eq:app_VR}; the envelope derivative is still $1/c^o_1(b)$ for each tenure, so $V^O-V^R$ is increasing wherever $c^O_1(b)<c^R_1(b)$, but this has not been shown at every crossing point. The numerical benchmark finds a single crossing on $[dp-y_1,3]$ with the general optimiser, and the quantitative model finds exactly one sign change of the buy-minus-rent value along cash at every age and productivity state (Section~\ref{sec:baseline}); a general proof is left open. | |
| 113 | 51 | \end{remark} |
| 114 | 52 | |
| 115 | −\begin{proposition}[Comparative statics of the threshold] | |
| 116 | −\label{prop:cs} | |
| 117 | −Under Assumptions~\ref{ass:pref}--\ref{ass:interior}, with the rent $\rho$ held fixed and the resale price moving with $p$ (stationary comparative statics), | |
| 118 | −\[ | |
| 119 | −\frac{\partial\hat b}{\partial p}>0,\quad | |
| 120 | −\frac{\partial\hat b}{\partial\rho}<0,\quad | |
| 121 | −\frac{\partial\hat b}{\partial\theta}>0,\quad | |
| 122 | −\frac{\partial\hat b}{\partial\kappa_b}>0,\quad | |
| 123 | −\frac{\partial\hat b}{\partial\kappa_s}>0,\quad | |
| 124 | −\frac{\partial\hat b}{\partial m}>0,\quad | |
| 125 | −\frac{\partial\hat b}{\partial\chi}<0,\quad | |
| 126 | −\frac{\partial\hat b}{\partial y_1}=-1,\quad | |
| 127 | −\frac{\partial\hat b}{\partial\bar\pi}>0 . | |
| 128 | −\] | |
| 129 | −\end{proposition} | |
| 130 | −\begin{proof} | |
| 131 | −$\hat b$ is defined by $D(\hat b;\xi) = 0$ for a parameter $\xi$, with $\partial D/\partial b = 1/c^O_1 - 1/c^R_1 > 0$ at $\hat b$ (proof of Proposition~\ref{prop:threshold}). Because the Euler inequalities are strict, a small change in $\xi$ leaves the corner allocation optimal, so the derivative of each value function with respect to $\xi$ is the partial derivative of \eqref{eq:app_VO}--\eqref{eq:app_VR} at the fixed allocation. By the implicit function theorem $\operatorname{sign}(\partial\hat b/\partial\xi) = -\operatorname{sign}(\partial D/\partial\xi)$. We compute $\partial D/\partial\xi = \partial V^O/\partial\xi - \partial V^R/\partial\xi$ for each parameter, using the shorthand $E_3 \equiv \mathbb E[1/W^O_3(y_3)]$ and the bounds implied by~\eqref{eq:app_euler}: | |
| 53 | +\begin{proof}[Proof of Proposition~\ref{prop:cs}] | |
| 54 | +$\hat b$ is defined by $D(\hat b;\xi) = 0$ for a parameter $\xi$, with $\partial D/\partial b = 1/c^O_1 - 1/c^R_1 > 0$ at $\hat b$ (proof of Proposition~\ref{prop:threshold}). Because the Euler inequalities are strict, a small change in $\xi$ leaves the corner allocation optimal, so the derivative of each value function with respect to $\xi$ is the partial derivative of \eqref{eq:app_VO}--\eqref{eq:app_VR} at the fixed allocation. By the implicit function theorem $\operatorname{sign}(\partial\hat b/\partial\xi) = -\operatorname{sign}(\partial D/\partial\xi)$. The benchmark has no later purchase, so $V^R$ depends on none of $\theta,\kappa_b,\kappa_s,m,\bar\pi$; this is the restriction named in the statement. We compute $\partial D/\partial\xi$ for each parameter, using $E_3\equiv\mathbb E[1/W^O_3(y_3)]$ and the bounds implied by~\eqref{eq:euler}: | |
| 132 | 55 | \begin{equation} |
| 133 | −\frac{\beta}{c^O_2} < \frac{1}{R\,c^O_1}, \qquad | |
| 134 | −\beta^2(1+\nu)E_3 < \frac{\beta}{R\,c^O_2} < \frac{1}{R^2 c^O_1}. | |
| 56 | +\frac{\beta}{c^O_2} < \frac{1}{Rc^O_1}, \qquad | |
| 57 | +\beta^2(1+\nu)E_3 < \frac{\beta}{Rc^O_2} < \frac{1}{R^2c^O_1}. | |
| 135 | 58 | \label{eq:app_bounds} |
| 136 | 59 | \end{equation} |
| 137 | 60 | |
| 138 | −\emph{Rent.} $\partial V^R/\partial\rho = -1/c^R_1 - \beta/c^R_2 < 0$ and $V^O$ does not depend on $\rho$, so $\partial D/\partial\rho > 0$. | |
| 61 | +\emph{Rent.} $\partial V^R/\partial\rho = -1/c^R_1 - \beta/c^R_2 < 0$ and $V^O$ does not depend on $\rho$, so $\partial D/\partial\rho>0$. | |
| 139 | 62 | |
| 140 | 63 | \emph{Premium and stage-1 income.} $\partial D/\partial\chi = 1$. $\partial D/\partial y_1 = 1/c^O_1 - 1/c^R_1 = \partial D/\partial b$, hence $\partial\hat b/\partial y_1 = -1$. |
| 141 | 64 | |
| 142 | 65 | \emph{Costs.} $\partial V^O/\partial\kappa_b = -p/c^O_1 < 0$; $\partial V^O/\partial m = -p/c^O_1 - \beta p/c^O_2 < 0$; $\partial V^O/\partial\kappa_s = -\beta^2(1+\nu)\,p\,E_3 < 0$; $V^R$ is unaffected. |
| 143 | 66 | |
| 144 | −\emph{Price.} $\partial V^O/\partial p = -d/c^O_1 - \beta(\delta_2+m)/c^O_2 + \beta^2(1+\nu)e_3E_3$. If $e_3\le 0$ all three terms are non-positive and the first is strictly negative. If $e_3>0$, use the second Euler inequality in the form $\beta/c^O_2 > \beta^2 R(1+\nu)E_3$ on the middle term: | |
| 67 | +\emph{Price.} $\partial V^O/\partial p = -d/c^O_1 - \beta(\delta_2+m)/c^O_2 + \beta^2(1+\nu)e_3E_3$. If $e_3\le 0$ all three terms are non-positive and the first is strictly negative. If $e_3>0$, use the second Euler inequality in the form $\beta/c^O_2 > \beta^2R(1+\nu)E_3$ on the middle term: | |
| 145 | 68 | \[ |
| 146 | −\frac{\partial V^O}{\partial p} < -\frac{d}{c^O_1} + \beta^2(1+\nu)E_3\,\big[e_3 - R(\delta_2+m)\big], \qquad e_3 - R(\delta_2+m) = (1-\kappa_s) - (1-\theta)R^2 - Rm, | |
| 69 | +\frac{\partial V^O}{\partial p} < -\frac{d}{c^O_1} + \beta^2(1+\nu)E_3\big[e_3 - R(\delta_2+m)\big], \qquad e_3 - R(\delta_2+m) = (1-\kappa_s) - (1-\theta)R^2 - Rm, | |
| 147 | 70 | \] |
| 148 | −where the last equality uses $\delta_3 + R\delta_2 = (1-\theta)R^2$ from~\eqref{eq:app_fisher}. If the bracket is non-positive we are done. Otherwise apply~\eqref{eq:app_bounds}: | |
| 71 | +where the last equality uses $\delta_3 + R\delta_2 = (1-\theta)R^2$ from~\eqref{eq:fisher}. If the bracket is non-positive we are done. Otherwise apply~\eqref{eq:app_bounds}: | |
| 149 | 72 | \[ |
| 150 | −\frac{\partial V^O}{\partial p} < -\frac{1}{c^O_1}\Big[d - \frac{(1-\kappa_s) - (1-\theta)R^2 - Rm}{R^2}\Big] | |
| 151 | −= -\frac{1}{c^O_1}\Big[1 + \kappa_b + m + \frac{m}{R} - \frac{1-\kappa_s}{R^2}\Big] < 0, | |
| 73 | +\frac{\partial V^O}{\partial p} < -\frac{1}{c^O_1}\Big[d - \frac{(1-\kappa_s)-(1-\theta)R^2-Rm}{R^2}\Big] | |
| 74 | += -\frac{1}{c^O_1}\Big[1+\kappa_b+m+\frac{m}{R}-\frac{1-\kappa_s}{R^2}\Big] < 0, | |
| 152 | 75 | \] |
| 153 | −because $R\ge 1$ implies $(1-\kappa_s)/R^2 < 1$. Hence $\partial D/\partial p < 0$. | |
| 76 | +because $R\ge 1$ implies $(1-\kappa_s)/R^2<1$. Hence $\partial D/\partial p<0$. | |
| 154 | 77 | |
| 155 | −\emph{Down payment.} With $\tilde\delta_2 \equiv -\partial\delta_2/\partial\theta = R - (1-a)/(1+\bar\pi) > 0$ and $\tilde\delta_3 \equiv -\partial\delta_3/\partial\theta = (1-a)R/(1+\bar\pi)$, note $\tilde\delta_2 + \tilde\delta_3/R = R$. Then | |
| 78 | +\emph{Down payment.} With the loan-to-value constraint binding, $B = (1-\theta)p$ and the payments are $\delta_2p$, $\delta_3p$. Let $\tilde\delta_2\equiv-\partial\delta_2/\partial\theta = R-(1-a)/(1+\bar\pi)>0$ and $\tilde\delta_3\equiv-\partial\delta_3/\partial\theta = (1-a)R/(1+\bar\pi)$, so that $\tilde\delta_2+\tilde\delta_3/R = R$. Then | |
| 156 | 79 | \[ |
| 157 | −\frac{\partial V^O}{\partial\theta} = p\Big[-\frac{1}{c^O_1} + \frac{\beta\tilde\delta_2}{c^O_2} + \beta^2(1+\nu)\tilde\delta_3 E_3\Big] | |
| 158 | −< \frac{p}{c^O_1}\Big[-1 + \frac{\tilde\delta_2}{R} + \frac{\tilde\delta_3}{R^2}\Big] = 0 , | |
| 80 | +\frac{\partial V^O}{\partial\theta} = p\Big[-\frac{1}{c^O_1} + \frac{\beta\tilde\delta_2}{c^O_2} + \beta^2(1+\nu)\tilde\delta_3E_3\Big] | |
| 81 | +< \frac{p}{c^O_1}\Big[-1+\frac{\tilde\delta_2}{R}+\frac{\tilde\delta_3}{R^2}\Big] = 0, | |
| 159 | 82 | \] |
| 160 | 83 | by~\eqref{eq:app_bounds}. A higher down payment is present-value neutral and hurts only through liquidity. |
| 161 | 84 | |
| 162 | −\emph{Anticipated inflation.} $\partial\delta_2/\partial\bar\pi = (1-\theta)(1-a)/(1+\bar\pi)^2 \equiv x > 0$ and $\partial\delta_3/\partial\bar\pi = -Rx$. Hence | |
| 85 | +\emph{Anticipated inflation.} $\partial\delta_2/\partial\bar\pi = (1-\theta)(1-a)/(1+\bar\pi)^2\equiv x>0$ and $\partial\delta_3/\partial\bar\pi = -Rx$. Hence | |
| 163 | 86 | \[ |
| 164 | −\frac{\partial V^O}{\partial\bar\pi} = xp\Big[-\frac{\beta}{c^O_2} + \beta^2 R(1+\nu)E_3\Big] < 0 | |
| 87 | +\frac{\partial V^O}{\partial\bar\pi} = xp\Big[-\frac{\beta}{c^O_2} + \beta^2R(1+\nu)E_3\Big] < 0 | |
| 165 | 88 | \] |
| 166 | −by the second inequality in~\eqref{eq:app_euler}; $V^R$ does not depend on $\bar\pi$. | |
| 89 | +by the second inequality in~\eqref{eq:euler}; $V^R$ does not depend on $\bar\pi$. | |
| 167 | 90 | \end{proof} |
| 168 | 91 | |
| 169 | −\begin{proposition}[Anticipated inflation and the down payment: tilt versus neutrality] | |
| 170 | −\label{prop:tilt} | |
| 171 | −(i) Under Assumptions~\ref{ass:pref}--\ref{ass:interior}, an increase in anticipated inflation $\bar\pi$ raises the real stage-2 payment and lowers the real stage-3 payment of a nominal mortgage, leaves their present value unchanged, and raises the ownership threshold $\hat b$. (ii) For a household whose optimal plan has $n_1>0$ and $n_2>0$ (slack liquidity constraints), a small change in $\bar\pi$ or in $\theta$ leaves its value, consumption and transfer unchanged. | |
| 172 | −\end{proposition} | |
| 173 | −\begin{proof} | |
| 174 | −(i) The signs of $\partial\delta_2/\partial\bar\pi$, $\partial\delta_3/\partial\bar\pi$ and the present-value identity are in the proof of Proposition~\ref{prop:cs}; the threshold result is the last item of that proposition. (ii) Let the stage-2 real payment rise by $x$ and the stage-3 payment fall by $Rx$ (a change in $\bar\pi$). Replace $n_2$ by $n_2 - x$, feasible for $x < n_2$: $c_2$ is unchanged, and $W_3 = y_3 + (1-\kappa_s)p - D_3 + n_2R$ changes by $+Rx - Rx = 0$. The original allocation of $(c_1,c_2,c_3,b')$ is therefore still feasible; conversely any allocation feasible after the change is feasible before it with $n_2 + x$, so the feasible sets of consumption plans coincide and the optimum is unchanged. For $\theta$: an increase by $x$ raises the down payment by $px$ and lowers the stage-2 and stage-3 payments by $\tilde\delta_2 px$ and $\tilde\delta_3 px$. Replace $n_1$ by $n_1 - px$ and $n_2$ by $n_2 - \tilde\delta_3 px/R$ (feasible for small $x$): $c_1$ is unchanged; $c_2$ changes by $+\tilde\delta_2 px - Rpx + \tilde\delta_3 px/R = 0$ by $\tilde\delta_2+\tilde\delta_3/R = R$; $W_3$ changes by $+\tilde\delta_3 px - \tilde\delta_3 px = 0$. The same argument in reverse shows the feasible sets coincide. | |
| 92 | +\begin{proof}[Proof of Proposition~\ref{prop:tilt}] | |
| 93 | +(i) The signs of $\partial\delta_2/\partial\bar\pi$, $\partial\delta_3/\partial\bar\pi$ and the present-value identity are in the proof of Proposition~\ref{prop:cs}; the threshold result is the last item of that proposition. (ii) Let the stage-2 real payment rise by $x$ and the stage-3 payment fall by $Rx$ (a change in $\bar\pi$). Replace $n_2$ by $n_2-x$, feasible for $x<n_2$: $c_2$ is unchanged, and $W_3 = y_3+(1-\kappa_s)p-D_3+n_2R$ changes by $+Rx-Rx = 0$. The original allocation of $(c_1,c_2,c_3,b')$ is therefore still feasible; conversely any allocation feasible after the change is feasible before it with $n_2+x$, so the feasible sets of consumption plans coincide and the optimum is unchanged. For $\theta$: an increase by $x$ raises the down payment by $px$ and lowers the stage-2 and stage-3 payments by $\tilde\delta_2px$ and $\tilde\delta_3px$. Replace $n_1$ by $n_1-px$ and $n_2$ by $n_2-\tilde\delta_3px/R$ (feasible for small $x$): $c_1$ is unchanged; $c_2$ changes by $+\tilde\delta_2px-Rpx+\tilde\delta_3px/R = 0$ by $\tilde\delta_2+\tilde\delta_3/R = R$; $W_3$ changes by $+\tilde\delta_3px-\tilde\delta_3px = 0$. The same argument in reverse shows the feasible sets coincide. | |
| 175 | 94 | \end{proof} |
| 176 | 95 | |
| 177 | −\subsection{Leverage amplification with transaction costs} | |
| 96 | +\subsection{Leverage} | |
| 178 | 97 | |
| 179 | −\begin{proposition}[Leverage multiplier, break-even growth, underwater threshold] | |
| 180 | −\label{prop:leverage} | |
| 181 | −Let an owner buy at $p$, hold a mortgage $(1-\theta)p$ and resell at $(1+g)p$ at stage~3. Under Assumption~\ref{ass:prices}: | |
| 182 | −\begin{enumerate} | |
| 183 | −\item End-of-life wealth responds to the resale price with $\partial W^O_3/\partial(\text{resale price}) = 1-\kappa_s$ per unit of price, i.e. $(1-\kappa_s)/d$ per unit of cash invested at purchase; this multiplier is decreasing in $\theta$, $\kappa_b$, $\kappa_s$ and $m$. | |
| 184 | −\item Compared with a renter who follows the same consumption path and invests the cash-flow differences at $R$, the owner has higher stage-3 wealth if and only if $g\ge g^*$ with | |
| 185 | −\begin{equation} | |
| 186 | −1+g^* = \frac{R^2(1+\kappa_b+m) + mR - (\rho/p)(R+R^2)}{1-\kappa_s}, | |
| 187 | −\label{eq:app_gstar} | |
| 188 | −\end{equation} | |
| 189 | −which is increasing in $\kappa_b$, $\kappa_s$ and $m$, decreasing in $\rho/p$, and independent of $\theta$. | |
| 190 | −\item A price decline of $x$ (resale at $(1-x)p$) leaves the owner with no housing equity if and only if $x\ge x^* \equiv 1 - \delta_3/(1-\kappa_s)$; a renter's wealth is unaffected. | |
| 191 | −\end{enumerate} | |
| 192 | −\end{proposition} | |
| 193 | −\begin{proof} | |
| 194 | −(1) $W^O_3 = y_3 + (1-\kappa_s)(1+g)p - \delta_3 p$, so $\partial W^O_3/\partial[(1+g)p] = 1-\kappa_s$; dividing by the stage-1 cash outlay $dp$ gives the multiplier, whose monotonicity is immediate. (2) The owner's housing-related stage-3 wealth is $(1-\kappa_s)(1+g)p - \delta_3 p$. The renter-investor pays $\rho$ at stages 1 and 2 instead of $dp$ and $(\delta_2+m)p$, so saves $dp-\rho$ at stage 1 and $(\delta_2+m)p-\rho$ at stage 2, worth $(dp-\rho)R^2 + ((\delta_2+m)p-\rho)R$ at stage 3. The owner does at least as well iff | |
| 98 | +\begin{proof}[Proof of Proposition~\ref{prop:leverage}] | |
| 99 | +(1) $W^O_3 = y_3+(1-\kappa_s)(1+g)p-\delta_3p$, so $\partial W^O_3/\partial[(1+g)p] = 1-\kappa_s$; dividing by the stage-1 cash outlay $dp$ gives the multiplier, whose monotonicity is immediate. (2) The owner's housing-related stage-3 wealth is $(1-\kappa_s)(1+g)p-\delta_3p$. The renter-investor pays $\rho$ at stages 1 and 2 instead of $dp$ and $(\delta_2+m)p$, so saves $dp-\rho$ at stage~1 and $(\delta_2+m)p-\rho$ at stage~2, worth $(dp-\rho)R^2+((\delta_2+m)p-\rho)R$ at stage~3. The owner does at least as well iff | |
| 195 | 100 | \[ |
| 196 | −(1-\kappa_s)(1+g) \ge dR^2 + (\delta_2+m)R + \delta_3 - \frac{\rho}{p}(R^2+R) = R^2(1+\kappa_b+m) + mR - \frac{\rho}{p}(R^2+R), | |
| 101 | +(1-\kappa_s)(1+g)\ge dR^2+(\delta_2+m)R+\delta_3-\frac{\rho}{p}(R^2+R) = R^2(1+\kappa_b+m)+mR-\frac{\rho}{p}(R^2+R), | |
| 197 | 102 | \] |
| 198 | −using $dR^2 + \delta_2 R + \delta_3 = (\theta+\kappa_b+m)R^2 + (1-\theta)R^2$ from~\eqref{eq:app_fisher}; $\theta$ cancels. (3) Housing equity at resale is $(1-\kappa_s)(1-x)p - \delta_3 p \le 0$ iff $x\ge x^*$. | |
| 103 | +using $dR^2+\delta_2R+\delta_3 = (\theta+\kappa_b+m)R^2+(1-\theta)R^2$ from~\eqref{eq:fisher}; $\theta$ cancels. (3) Housing equity at resale is $(1-\kappa_s)(1-x)p-\delta_3p\le 0$ iff $x\ge x^*$. | |
| 199 | 104 | \end{proof} |
| 200 | 105 | |
| 201 | 106 | \subsection{Dynasties and persistence} |
| 202 | 107 | |
| 203 | −\begin{proposition}[Tenure transitions, persistence and the equal-income wealth gap] | |
| 204 | −\label{prop:persist} | |
| 205 | −Under Assumptions~\ref{ass:pref}--\ref{ass:interior}, with stationary prices and the stationary transfer support contained in $\mathcal B$, let $\hat y_R \equiv (1+\nu)\hat b/\nu$ and $\hat y_O \equiv \hat y_R - e_3 p$. Then: | |
| 206 | −\begin{enumerate} | |
| 207 | −\item The child of a renter owns iff $y_3\ge\hat y_R$ and the child of an owner owns iff $y_3\ge\hat y_O$; the tenure of a dynasty is a Markov chain with $q_{RO} = 1-G(\hat y_R)$ and $q_{OO} = 1 - G(\hat y_O)$. | |
| 208 | −\item The second eigenvalue of the chain is $\lambda = q_{OO} - q_{RO} = G(\hat y_R) - G(\hat y_R - e_3 p)$; it equals the parent--child correlation of the ownership indicator in the stationary distribution and the per-generation persistence of any difference in ownership rates between groups of dynasties. $\lambda>0$ iff $e_3>0$ and $g>0$ somewhere on $(\hat y_O, \hat y_R)$; for fixed $\hat b$, $\lambda$ is non-decreasing in $e_3 p$. | |
| 209 | −\item For $\lambda<1$ the chain has the unique stationary owner share $\mu^* = q_{RO}/(1-\lambda)$ and converges to it geometrically at rate $\lambda$ from any initial distribution. | |
| 210 | −\item Among dynasties with identical income processes, the expected transfer received by an owner's child exceeds that of a renter's child by $\nu e_3 p/(1+\nu)$, and the end-of-life wealth gap between an owner and a renter with the same $y_3$ is $e_3 p$. | |
| 211 | −\end{enumerate} | |
| 212 | −\end{proposition} | |
| 213 | −\begin{proof} | |
| 214 | −(1) By Lemma~\ref{lem:split} and Lemma~\ref{lem:htm}, $b' = \nu W_3/(1+\nu)$ with $W^R_3 = y_3$ and $W^O_3 = y_3 + e_3p$; the child owns iff $b'\ge\hat b$ (Proposition~\ref{prop:threshold}), which gives the two cutoffs. Independence of $y_3$ across generations makes the tenure sequence Markov. (2) The transition matrix $\big(\begin{smallmatrix}1-q_{RO} & q_{RO}\\ 1-q_{OO} & q_{OO}\end{smallmatrix}\big)$ has trace $1-q_{RO}+q_{OO}$ and determinant $q_{OO}-q_{RO}$, hence eigenvalues $1$ and $\lambda$. For a two-state chain in its stationary distribution, $\mathbb E[O'\mid O=1]-\mathbb E[O'\mid O=0] = q_{OO}-q_{RO} = \lambda$, and $\operatorname{Corr}(O,O') = \lambda$ (direct computation with $\Pr(O=1)=\mu^*$). If $\mu_t$ is the owner share of a group, $\mu_{t+1} - \mu^* = \lambda(\mu_t-\mu^*)$. The sign and monotonicity statements follow from $\lambda = G(\hat y_R) - G(\hat y_R - e_3 p)$ with $G$ non-decreasing and strictly increasing where $g>0$. (3) Standard for an irreducible aperiodic two-state chain ($0<q_{RO}$, $q_{OO}<1$ or $\lambda<1$): $\mu^*$ solves $\mu^* = \mu^* q_{OO} + (1-\mu^*)q_{RO}$. (4) Immediate from $W^O_3 - W^R_3 = e_3 p$ and Lemma~\ref{lem:split}. | |
| 108 | +\begin{proof}[Proof of Proposition~\ref{prop:persist}] | |
| 109 | +(1) By Lemmas~\ref{lem:split} and~\ref{lem:htm}, $b' = \nu W_3/(1+\nu)$ with $W^R_3 = y_3$ and $W^O_3 = y_3+e_3p$; the child owns iff $b'\ge\hat b$ (Proposition~\ref{prop:threshold}), which gives the two cutoffs. Independence of $y_3$ across generations makes the tenure sequence Markov. (2) The transition matrix $\big(\begin{smallmatrix}1-q_{RO} & q_{RO}\\ 1-q_{OO} & q_{OO}\end{smallmatrix}\big)$ has trace $1-q_{RO}+q_{OO}$ and determinant $q_{OO}-q_{RO}$, hence eigenvalues $1$ and $\lambda$. For a two-state chain in its stationary distribution, $\mathbb E[O'\mid O = 1]-\mathbb E[O'\mid O = 0] = q_{OO}-q_{RO} = \lambda$, and $\operatorname{Corr}(O,O') = \lambda$ (direct computation with $\Pr(O = 1) = \mu^*$). If $\mu_t$ is the owner share of a group, $\mu_{t+1}-\mu^* = \lambda(\mu_t-\mu^*)$. The sign and monotonicity statements follow from \eqref{eq:lambda} with $G$ non-decreasing and strictly increasing where $g>0$. (3) Standard for an irreducible aperiodic two-state chain: $\mu^*$ solves $\mu^* = \mu^*q_{OO}+(1-\mu^*)q_{RO}$. (4) Immediate from $W^O_3-W^R_3 = e_3p$ and Lemma~\ref{lem:split}. | |
| 215 | 110 | \end{proof} |
| 216 | 111 | |
| 217 | 112 | \begin{remark}[No income risk: absorbing classes and history dependence] |
| 218 | 113 | \label{rem:norisk} |
| 219 | −If $G$ is degenerate at $\bar y_3$, then $\lambda\in\{0,1\}$: when $\nu\bar y_3/(1+\nu) < \hat b \le \nu(\bar y_3+e_3p)/(1+\nu)$ the two tenure classes are absorbing, every initial owner share $\mu_0$ compatible with that inequality at $p = S^{-1}(\mu_0)$ is a stationary equilibrium, and the long-run ownership rate is determined by the initial distribution of transfers. This continuum of history-dependent steady states is a knife-edge of the no-risk case: any $G$ with positive density on $(\hat y_O,\hat y_R)$ gives $\lambda<1$ and a unique stationary distribution given prices (part~3). We therefore state persistence as a half-life, $\ln 2/(-\ln\lambda)$, not as multiplicity (see \texttt{research/failed\_hypotheses.md}). | |
| 114 | +If $G$ is degenerate at $\bar y_3$, then $\lambda\in\{0,1\}$: when $\nu\bar y_3/(1+\nu)<\hat b\le\nu(\bar y_3+e_3p)/(1+\nu)$ the two tenure classes are absorbing, every initial owner share $\mu_0$ compatible with that inequality at $p = S^{-1}(\mu_0)$ is a stationary equilibrium, and the long-run ownership rate is determined by the initial distribution of transfers. This continuum of history-dependent steady states is a knife-edge of the no-risk case: any $G$ with positive density on $(\hat y_O,\hat y_R)$ gives $\lambda<1$ and a unique stationary distribution given prices (part~3). Persistence is therefore stated as a half-life, $\ln 2/(-\ln\lambda)$, not as multiplicity. | |
| 220 | 115 | \end{remark} |
| 221 | 116 | |
| 222 | 117 | \subsection{Stationary equilibrium} |
| 223 | 118 | |
| 224 | −\begin{definition}[Stationary equilibrium of the benchmark] | |
| 225 | −\label{def:se} | |
| 226 | −A stationary equilibrium is $(p^*,\rho^*,\mu^*)$ with $\hat b^* = \hat b(p^*,\rho^*)$ such that $\mu^* = q_{RO}/(1-\lambda)$ evaluated at $(p^*,\rho^*)$, $S(p^*) = \mu^*$ and $S_r(\rho^*) = 1-\mu^*$ (per cohort), the lender earns zero expected real profit, and the distribution of transfers is the stationary distribution of the dynasty chain. | |
| 227 | −\end{definition} | |
| 228 | − | |
| 229 | −Write $f(\mu) \equiv \mu^*\big(S^{-1}(\mu), S_r^{-1}(1-\mu)\big)$ for the owner share generated by the dynasty chain when prices clear both markets at a cohort owner share $\mu$, with the convention $\mu^* = 0$ when $\Delta\le 0$ (nobody owns) and $\mu^*=1$ when $\hat b < 0$. | |
| 119 | +\begin{proof}[Proof of Proposition~\ref{prop:exist}] | |
| 120 | +\emph{Existence.} $S^{-1}$ and $S_r^{-1}$ are continuous on $[0,1]$; $\Delta$, $\Lambda$ and $\hat b$ are continuous in $(p,\rho)$ where Assumption~\ref{ass:feas} holds (compositions of continuous functions and of the expectation of a continuous function over a compact support), with $\hat b\to+\infty$ as $\Delta\downarrow 0$ when $dp>\rho$, so the convention $\mu^* = 0$ at $\Delta\le 0$ makes $\mu^*$ continuous; $G$ is continuous. Hence $f:[0,1]\to[0,1]$ is continuous. At $\mu = 0$, $p = 0$, so $dp = 0\le\rho$ and every feasible household owns ($\hat b = -y_1<0$): $f(0) = 1\ge 0$. At $\mu = 1$, $f(1)\le 1$. By the intermediate value theorem $f(\mu)-\mu$ has a zero on $[0,1]$, which is a stationary equilibrium with $p^* = S^{-1}(\mu)$, $\rho^* = S_r^{-1}(1-\mu)$. | |
| 230 | 121 | |
| 231 | −\begin{proposition}[Existence; uniqueness under a dominance condition] | |
| 232 | −\label{prop:exist} | |
| 233 | −Let Assumptions~\ref{ass:pref}--\ref{ass:feas} hold for all $(p,\rho)$ with $p\in[0,S^{-1}(1)]$ and $\rho\in[0,S_r^{-1}(1)]$, with $\varepsilon>0$ and $\varepsilon_r<\infty$. Then a stationary equilibrium exists. If, in addition, Assumptions~\ref{ass:htm}--\ref{ass:interior} hold along the clearing curve and | |
| 234 | −\begin{equation} | |
| 235 | −\text{(C)}\qquad \frac{1+\nu}{\nu}\,\frac{\partial\hat b}{\partial p}\;\ge\; e_3 \qquad\text{for all } \mu\in(0,1), | |
| 236 | −\label{eq:app_C} | |
| 237 | −\end{equation} | |
| 238 | −then $f$ is non-increasing, strictly decreasing wherever $g(\hat y_R)>0$, and the stationary equilibrium is unique. | |
| 239 | −\end{proposition} | |
| 240 | −\begin{proof} | |
| 241 | −\emph{Existence.} $S^{-1}$ and $S_r^{-1}$ are continuous on $[0,1]$; $\Delta$, $\Lambda$ and $\hat b$ are continuous in $(p,\rho)$ where Assumption~\ref{ass:feas} holds (compositions of continuous functions and of the expectation of a continuous function over a compact support), with $\hat b\to+\infty$ as $\Delta\downarrow 0$ when $dp>\rho$, so the convention $\mu^*=0$ at $\Delta\le 0$ makes $\mu^*$ continuous; $G$ is continuous. Hence $f:[0,1]\to[0,1]$ is continuous. At $\mu = 0$, $p = 0$, so $dp = 0\le\rho$ and every feasible household owns ($\hat b = -y_1 < 0$): $f(0) = 1\ge 0$. At $\mu=1$, $f(1)\le 1$. By the intermediate value theorem $f(\mu)-\mu$ has a zero on $[0,1]$, which is a stationary equilibrium with $p^* = S^{-1}(\mu)$, $\rho^* = S_r^{-1}(1-\mu)$. | |
| 242 | − | |
| 243 | −\emph{Uniqueness.} $\mu^* = q_{RO}/(1-q_{OO}+q_{RO})$ is increasing in $q_{RO}$ and in $q_{OO}$; $q_{RO} = 1-G(\hat y_R)$ is non-increasing in $\hat y_R = (1+\nu)\hat b/\nu$ and $q_{OO} = 1-G(\hat y_O)$ is non-increasing in $\hat y_O = \hat y_R - e_3p$. Along the clearing curve, $dp/d\mu>0$ and $d\rho/d\mu<0$, so by Proposition~\ref{prop:cs} | |
| 122 | +\emph{Uniqueness.} $\mu^* = q_{RO}/(1-q_{OO}+q_{RO})$ is increasing in $q_{RO}$ and in $q_{OO}$; $q_{RO} = 1-G(\hat y_R)$ is non-increasing in $\hat y_R = (1+\nu)\hat b/\nu$ and $q_{OO} = 1-G(\hat y_O)$ is non-increasing in $\hat y_O = \hat y_R-e_3p$. Along the clearing curve, $dp/d\mu>0$ and $d\rho/d\mu<0$, so by Proposition~\ref{prop:cs} | |
| 244 | 123 | \[ |
| 245 | −\frac{d\hat b}{d\mu} = \frac{\partial\hat b}{\partial p}\frac{dp}{d\mu} + \frac{\partial\hat b}{\partial\rho}\frac{d\rho}{d\mu} \;\ge\; \frac{\partial\hat b}{\partial p}\frac{dp}{d\mu} > 0, | |
| 124 | +\frac{d\hat b}{d\mu} = \frac{\partial\hat b}{\partial p}\frac{dp}{d\mu}+\frac{\partial\hat b}{\partial\rho}\frac{d\rho}{d\mu}\;\ge\;\frac{\partial\hat b}{\partial p}\frac{dp}{d\mu}>0, | |
| 246 | 125 | \qquad |
| 247 | −\frac{d\hat y_O}{d\mu} = \frac{1+\nu}{\nu}\frac{d\hat b}{d\mu} - e_3\frac{dp}{d\mu} \;\ge\; \Big[\frac{1+\nu}{\nu}\frac{\partial\hat b}{\partial p} - e_3\Big]\frac{dp}{d\mu} \ge 0 | |
| 126 | +\frac{d\hat y_O}{d\mu} = \frac{1+\nu}{\nu}\frac{d\hat b}{d\mu}-e_3\frac{dp}{d\mu}\;\ge\;\Big[\frac{1+\nu}{\nu}\frac{\partial\hat b}{\partial p}-e_3\Big]\frac{dp}{d\mu}\ge 0 | |
| 248 | 127 | \] |
| 249 | −under (C). Thus both cutoffs are non-decreasing in $\mu$, $f$ is non-increasing, and strictly decreasing where $g(\hat y_R)>0$ (then $q_{RO}$ strictly falls). Since $f$ is non-increasing and $\mu\mapsto\mu$ is strictly increasing, $f(\mu)-\mu$ is strictly decreasing on $[0,1]$ and has at most one zero; existence gives exactly one. | |
| 128 | +under (C). Thus both cutoffs are non-decreasing in $\mu$, $f$ is non-increasing, and strictly decreasing where $g(\hat y_R)>0$. Since $f$ is non-increasing and $\mu\mapsto\mu$ is strictly increasing, $f(\mu)-\mu$ is strictly decreasing on $[0,1]$ and has at most one zero; existence gives exactly one. | |
| 250 | 129 | \end{proof} |
| 251 | 130 | |
| 252 | −\begin{corollary}[Fixed owner-occupied stock] | |
| 253 | −\label{cor:fixed} | |
| 254 | −If $\varepsilon = 0$ (so $S(p)=\bar H$), every stationary equilibrium has $\mu^* = \bar H$, and under (C) the price $p^*$ is the unique solution of $\mu^*(p,S_r^{-1}(1-\bar H)) = \bar H$. A change in any parameter that raises $\mu^*$ at given prices (a lower $\theta$, lower transaction costs, a higher $\chi$ or $y_1$) raises $p^*$ and leaves the ownership rate unchanged; it changes the composition of owners and, through $e_3p^*$ and $\hat b^*$, the persistence $\lambda$. | |
| 255 | −\end{corollary} | |
| 256 | −\begin{proof} | |
| 131 | +\begin{proof}[Proof of Corollary~\ref{cor:fixed}] | |
| 257 | 132 | Market clearing forces $\mu^* = \bar H$. Under (C), $\mu^*(p,\rho)$ is strictly decreasing in $p$ at fixed $\rho$ (proof of Proposition~\ref{prop:exist} with $d\rho = 0$), so the clearing price is unique and, by the implicit function theorem, moves in the direction of the demand shift. |
| 258 | 133 | \end{proof} |
| 259 | 134 | |
| 260 | −\begin{proposition}[Price neutrality without segmentation] | |
| 261 | −\label{prop:noseg} | |
| 262 | −Suppose instead of Assumption~\ref{ass:prices} that owner-occupied and rental units are the same stock, converted costlessly by competitive risk-neutral landlords with maintenance $m$ and operating wedge $\phi\ge 0$. Then in any stationary equilibrium $p^*$ solves $S(p^*) = 1$ (per cohort) and $\rho^* = p^*(1+m+\phi-1/R)$: the price and the rent are independent of $\theta$, $\kappa_b$, $\kappa_s$, $\chi$, $\nu$, $G$, the distribution of transfers, and of inflation surprises; tenure shares, $\hat b^*$ and $\lambda$ adjust to these prices without feeding back on them. | |
| 263 | −\end{proposition} | |
| 264 | −\begin{proof} | |
| 265 | −With unit demand, every stage-1 and stage-2 household occupies exactly one unit whatever its tenure, so the total demand for units per cohort is $1$ and $S(p^*)=1$ pins $p^*$. Landlord arbitrage (buy at $p$, pay $(m+\phi)p$, collect $\rho$, sell at $p$ next period, discount at $R$) gives $\rho^* = p^*(1+m+\phi) - p^*/R$. Nothing else enters either equation. | |
| 135 | +\begin{proof}[Proof of Proposition~\ref{prop:noseg}] | |
| 136 | +With unit demand, every stage-1 and stage-2 household occupies exactly one unit whatever its tenure, so the total demand for units per cohort is $1$ and $S(p^*) = 1$ pins $p^*$. Landlord arbitrage (buy at $p$, pay $(m+\phi)p$, collect $\rho$, sell at $p$ next period, discount at $R$) gives $\rho^* = p^*(1+m+\phi)-p^*/R$. Nothing else enters either equation. The argument uses unit demand at the first step: if the number of units demanded by a household depended on its wealth or on prices, $S(p^*)$ would equal an aggregate demand that depends on the wealth distribution, and the neutrality would fail (Remark~\ref{rem:noseg_scope}). | |
| 266 | 137 | \end{proof} |
| 267 | 138 | |
| 268 | 139 | \subsection{Surprise inflation} |
| 269 | 140 | |
| 270 | −\begin{proposition}[Direct revaluation and its incidence] | |
| 271 | −\label{prop:accounting} | |
| 272 | −Let the economy be in the stationary equilibrium up to $t$ and let a surprise $\pi^u$ occur at $t+1$: $1+\pi_{t+1} = (1+\bar\pi)(1+\pi^u)$. Consider (E1) a level surprise, with $\pi_{t+2} = \pi^e_{t+2} = \bar\pi$, and (E2) a persistent surprise, with $\pi_{t+2} = \pi^e_{t+2|t+1} = \pi_{t+1}$ and real rate $r' = r+\phi_r\pi^u$ for $(t+1,t+2)$. Holding real prices fixed: | |
| 273 | −\begin{enumerate} | |
| 274 | −\item For the cohort aged~2 at $t+1$, the real stage-2 payment is $\delta_2 p/(1+\pi^u)$ for every contract. The real stage-3 payment is $\delta_3 p/(1+\pi^u)$ for every contract under (E1); under (E2) it is $\delta_3 p/(1+\pi^u)^2$ for an FRM and $\delta_3 p\,(R'/R)/(1+\pi^u)$ for an ARM, $R' = 1+r'$. Hence the FRM borrower gains at least as much as the ARM borrower, strictly under (E2), and the ARM stage-3 payment rises in real terms iff $\phi_r > R$. | |
| 275 | −\item For the cohort aged~3 at $t+1$, the real stage-3 payment is $\delta_3 p/(1+\pi^u)$ for every contract. | |
| 276 | −\item A household holding nominal deposits $n$ at $t$ loses $nR\pi^u/(1+\pi^u)$ in real terms; a household whose nominal wage is indexed with elasticity $\eta_w$ loses $y[1-(1+\pi^u)^{-(1-\eta_w)}]$ of real income in the surprise period; a sitting tenant whose rent is indexed with elasticity $\eta_\rho$ gains $\rho[1-(1+\pi^u)^{-(1-\eta_\rho)}]$ at the landlord's expense. | |
| 277 | −\item The sum over households of real gains on nominal positions (mortgages minus deposits) equals the lender's real loss: the revaluation is zero-sum. Let $\Gamma_3$ denote the real stage-3 gain of the cohort aged~2: $\Gamma_3 = \delta_3p\,\pi^u/(1+\pi^u)$ under (E1) for both contracts, $\Gamma_3 = \delta_3p\,[1-(1+\pi^u)^{-2}]$ for an FRM under (E2), and $\Gamma_3 = \delta_3p\,[1-(R'/R)/(1+\pi^u)]$ for an ARM under (E2). An owner of that cohort with deposits $n$ gains on net over its remaining life iff | |
| 278 | −\begin{equation} | |
| 279 | −\delta_2 p\,\frac{\pi^u}{1+\pi^u} + \Gamma_3 \;>\; y_2\Big[1-(1+\pi^u)^{-(1-\eta_w)}\Big] + nR\,\frac{\pi^u}{1+\pi^u} . | |
| 280 | −\label{eq:app_gaincond} | |
| 281 | −\end{equation} | |
| 282 | −With full wage indexation and no deposits the left-hand side is positive for an FRM and for an ARM with $\phi_r \le R$; with $\eta_w<1$ the owner loses whenever the real-income loss exceeds the debt erosion, which happens for any $\pi^u>0$ if $y_2(1-\eta_w)$ is large relative to $\delta_2p+\delta_3p$. | |
| 283 | −\end{enumerate} | |
| 284 | −\end{proposition} | |
| 285 | −\begin{proof} | |
| 286 | −(1)--(2) Substitute the inflation paths into $D_2 = (i_1+a)(1-\theta)p/(1+\pi_{t+1})$ and $D_3 = (1+i_2)(1-a)(1-\theta)p/((1+\pi_{t+1})(1+\pi_{t+2}))$ with $1+i_1 = R(1+\bar\pi)$ and, for the ARM under (E2), $1+i_2 = R'(1+\pi_{t+2})$: the $(1+\pi_{t+2})$ factors cancel and leave $\delta_3p\,(R'/R)/(1+\pi^u)$; the FRM keeps $1+i_2 = R(1+\bar\pi)$ and the extra $(1+\pi^u)$ in the denominator remains. The real stage-3 payment of the ARM exceeds $\delta_3 p$ iff $R' > R(1+\pi^u)$, i.e. $1+r+\phi_r\pi^u > (1+r)(1+\pi^u)$, i.e. $\phi_r > 1+r = R$. The cohort aged~3 has a nominal stage-3 payment fixed before the shock, deflated by the realised price level. (3) Real deposit return $R/(1+\pi^u)$ against $R$; the wage and rent factors are definitions. (4) The lender's nominal claims are the mortgages net of deposits, so its real revaluation is minus the households' aggregate real revaluation. Condition \eqref{eq:app_gaincond} is the household's net real gain written out, the stage-3 term being the contract-specific erosion of items 1--2; the sign statements follow because $\Gamma_3\ge 0$ when $\phi_r\le R$ and the right-hand side is $O(\pi^u)$ with coefficient $y_2(1-\eta_w)$ versus $\delta_2p+\delta_3p$ on the left. | |
| 141 | +\begin{proof}[Proof of Proposition~\ref{prop:accounting}] | |
| 142 | +(1)--(2) Substitute the inflation paths into $D_2 = (i_1+a)(1-\theta)p/(1+\pi_{t+1})$ and $D_3 = (1+i_2)(1-a)(1-\theta)p/((1+\pi_{t+1})(1+\pi_{t+2}))$ with $1+i_1 = R(1+\bar\pi)$ and, for the ARM under (E2), $1+i_2 = R'(1+\pi_{t+2})$: the $(1+\pi_{t+2})$ factors cancel and leave $\delta_3p\,(R'/R)/(1+\pi^u)$; the FRM keeps $1+i_2 = R(1+\bar\pi)$ and the extra $(1+\pi^u)$ in the denominator remains. The real stage-3 payment of the ARM exceeds $\delta_3p$ iff $R'>R(1+\pi^u)$, i.e.\ $1+r+\phi_r\pi^u>(1+r)(1+\pi^u)$, i.e.\ $\phi_r>1+r = R$. The cohort aged~3 has a nominal stage-3 payment fixed before the shock, deflated by the realised price level. (3) Real deposit return $R/(1+\pi^u)$ against $R$; the wage and rent factors are definitions. (4) The lender's nominal claims are the mortgages net of deposits, so its real revaluation is minus the households' aggregate real revaluation. Condition \eqref{eq:gaincond} is the household's net real gain written out, the stage-3 term being the contract-specific erosion of items 1--2; $\Gamma_3\ge 0$ when $\phi_r\le R$, and the right-hand side is $O(\pi^u)$ with coefficient $y_2(1-\eta_w)+nR$ against $\delta_2p+\delta_3p$ on the left. | |
| 287 | 143 | \end{proof} |
| 288 | 144 | |
| 289 | −\begin{proposition}[General-equilibrium impact of a level surprise] | |
| 290 | −\label{prop:impact} | |
| 291 | −Let the economy be in a stationary equilibrium satisfying Assumptions~\ref{ass:pref}--\ref{ass:interior} with $g(\hat y_O)>0$, and let a level surprise $\pi^u>0$ occur at $t+1$ (any contract). Under Assumption~\ref{ass:static}, with stage-2 owners not transacting, owner-market clearing $2S(p_{t+1}) = \mu^* + \mu_{t+1}$, rental clearing $2S_r(\rho_{t+1}) = 2-\mu^*-\mu_{t+1}$, and the static-expectations dominance condition | |
| 292 | −\begin{equation} | |
| 293 | −\text{(C$'$)}\qquad \frac{1+\nu}{\nu}\,\frac{\partial\hat b}{\partial p}\Big|_{p''=p^*} \;\ge\; 1-\kappa_s , | |
| 294 | −\label{eq:app_Cprime} | |
| 295 | −\end{equation} | |
| 296 | −the impact responses satisfy: | |
| 297 | −\begin{enumerate} | |
| 298 | −\item $dp_{t+1}/d\pi^u \ge 0$ and $d\rho_{t+1}/d\pi^u\le 0$, both strict if $\varepsilon<\infty$ (respectively $\varepsilon_r<\infty$); if $\varepsilon=0$ the price response is strictly positive and $\mu_{t+1} = \mu^*$. | |
| 299 | −\item $d\mu_{t+1}/d\pi^u\ge 0$, strictly if $\varepsilon>0$. | |
| 300 | −\item $d\hat b_{t+1}/d\pi^u\ge 0$; the renter-to-owner transition probability $q_{RO}$ falls, the owner-to-owner probability $q_{OO}$ rises, and the persistence $\lambda$ rises. | |
| 301 | −\item Stage-3 owner-sellers gain $\delta_3p^*\pi^u/(1+\pi^u) + (1-\kappa_s)\,dp_{t+1} \ge 0$; the lender loses the nominal erosion; the stage-1 cash of an inframarginal owner-dynasty child changes by $\frac{\nu}{1+\nu}\big[\delta_3p^*\frac{\pi^u}{1+\pi^u} + (1-\kappa_s)dp_{t+1}\big] - d\,dp_{t+1}$, which is negative whenever $dp_{t+1}$ is large relative to the erosion (inelastic supply) and $\nu/(1+\nu) < d/(1-\kappa_s)$; renter-dynasty children who rent in both situations gain iff $\rho_{t+1}<\rho^*$. | |
| 302 | −\item Without segmentation (Proposition~\ref{prop:noseg}) items 1--3 are zero and only Proposition~\ref{prop:accounting} applies. | |
| 303 | −\end{enumerate} | |
| 304 | −\end{proposition} | |
| 305 | −\begin{proof} | |
| 306 | −Stage-3 owner-parents at $t+1$ bought at $p^*$ and sell at $p_{t+1}$, with nominal stage-3 payment fixed before the shock, so their end-of-life wealth is $W_3 = y_3 + E$ with $E(p_{t+1},\pi^u) \equiv (1-\kappa_s)p_{t+1} - \delta_3p^*/(1+\pi^u)$; renter-parents have $W_3 = y_3$. Under Assumption~\ref{ass:static} the children's threshold is $\hat b(p_{t+1},\rho_{t+1})$ computed with $p''=p^*$, $\rho'=\rho^*$; its partial derivative in $p$ is $-\big[d/c^O_1 + \beta(\delta_2+m)/c^O_2\big]\big/\big[1/c^O_1 - 1/c^R_1\big] > 0$ (no resale term) and in $\rho$ it is negative as in Proposition~\ref{prop:cs}. With $\hat y_R = (1+\nu)\hat b/\nu$ and $\hat y_O = \hat y_R - E$, the stage-1 owner share is | |
| 145 | +\begin{proof}[Proof of Proposition~\ref{prop:impact} and Corollary~\ref{cor:creditor}] | |
| 146 | +We prove the corollary; the proposition is the case $n^O = n^R = 0$. Stage-3 owner-parents at $t+1$ bought at $p^*$ and sell at $p_{t+1}$, with nominal stage-3 payment fixed before the shock and predetermined deposits $n^O$, so their end-of-life wealth is $W_3 = y_3+E+Rn^O/(1+\pi^u)$ with $E(p_{t+1},\pi^u)\equiv(1-\kappa_s)p_{t+1}-\delta_3p^*/(1+\pi^u)$; renter-parents have $W_3 = y_3+Rn^R/(1+\pi^u)$. Under Assumption~\ref{ass:static} the children's threshold is $\hat b(p_{t+1},\rho_{t+1})$ computed with $p'' = p^*$, $\rho' = \rho^*$; its partial derivative in $p$ is $-\big[d/c^O_1+\beta(\delta_2+m)/c^O_2\big]\big/\big[1/c^O_1-1/c^R_1\big]>0$ (no resale term) and in $\rho$ it is negative as in Proposition~\ref{prop:cs}. Let $Y\equiv(1+\nu)\hat b/\nu$. By Lemma~\ref{lem:split} the child of a renter-parent owns iff $y_3\ge\hat y_R\equiv Y-Rn^R/(1+\pi^u)$ and the child of an owner-parent iff $y_3\ge\hat y_O\equiv Y-E-Rn^O/(1+\pi^u)$, so the stage-1 owner share is | |
| 147 | +\[ | |
| 148 | +\mu^d(p,\rho,\pi^u) = \mu^*\big[1-G(\hat y_O)\big]+(1-\mu^*)\big[1-G(\hat y_R)\big]. | |
| 149 | +\] | |
| 150 | +Let $Q\equiv(\mu^*+\mu_{t+1})/2$, $p = S^{-1}(Q)$, $\rho = S_r^{-1}(1-Q)$ and $\Psi(Q,\pi^u)\equiv\mu^d(S^{-1}(Q),S_r^{-1}(1-Q),\pi^u)-(2Q-\mu^*)$; an impact equilibrium is a zero of $\Psi$. Differentiating the cutoffs at fixed prices, $\partial\hat y_O/\partial\pi^u = [-\delta_3p^*+Rn^O]/(1+\pi^u)^2$ and $\partial\hat y_R/\partial\pi^u = Rn^R/(1+\pi^u)^2$, so | |
| 307 | 151 | \[ |
| 308 | −\mu^d(p,\rho,\pi^u) = \mu^*\big[1-G(\hat y_O)\big] + (1-\mu^*)\big[1-G(\hat y_R)\big]. | |
| 152 | +\frac{\partial\Psi}{\partial\pi^u} = \frac{\mu^*g(\hat y_O)\big(\delta_3p^*-Rn^O\big)-(1-\mu^*)g(\hat y_R)Rn^R}{(1+\pi^u)^2} = \frac{\Xi}{(1+\pi^u)^2}, | |
| 309 | 153 | \] |
| 310 | −Let $Q\equiv(\mu^*+\mu_{t+1})/2$, $p = S^{-1}(Q)$, $\rho = S_r^{-1}(1-Q)$ and $\Psi(Q,\pi^u)\equiv \mu^d(S^{-1}(Q),S_r^{-1}(1-Q),\pi^u) - (2Q-\mu^*)$; an impact equilibrium is a zero of $\Psi$. Then $\partial\Psi/\partial\pi^u = \mu^* g(\hat y_O)\,\delta_3p^*/(1+\pi^u)^2 > 0$, and | |
| 154 | +which is strictly positive under Assumption~\ref{ass:nominal} because $g(\hat y_O)>0$. The deposits do not enter the price derivatives: | |
| 311 | 155 | \[ |
| 312 | −\frac{\partial\mu^d}{\partial p} = -\Big[\mu^*g(\hat y_O) + (1-\mu^*)g(\hat y_R)\Big]\frac{1+\nu}{\nu}\frac{\partial\hat b}{\partial p} + \mu^* g(\hat y_O)(1-\kappa_s) \le 0 | |
| 156 | +\frac{\partial\mu^d}{\partial p} = -\Big[\mu^*g(\hat y_O)+(1-\mu^*)g(\hat y_R)\Big]\frac{1+\nu}{\nu}\frac{\partial\hat b}{\partial p}+\mu^*g(\hat y_O)(1-\kappa_s)\le 0 | |
| 313 | 157 | \] |
| 314 | −under (C$'$), while $\partial\mu^d/\partial\rho = -[\cdots]\frac{1+\nu}{\nu}\partial\hat b/\partial\rho \ge 0$ multiplies $d\rho/dQ<0$. Hence $\partial\Psi/\partial Q \le -2 < 0$ and, by the implicit function theorem, $dQ/d\pi^u = -(\partial\Psi/\partial\pi^u)/(\partial\Psi/\partial Q) > 0$. Items 1--2 follow from $dp/dQ = p/(\varepsilon Q) > 0$ for $\varepsilon\in(0,\infty)$, $d\rho/dQ<0$ for $\varepsilon_r<\infty$, and $\mu_{t+1} = 2Q-\mu^*$. If $\varepsilon = 0$ the owner stock is fixed, $\mu_{t+1}=\mu^*$, and $p_{t+1}$ solves $\mu^d(p,\rho^*,\pi^u)=\mu^*$ with $\partial\mu^d/\partial p<0$ (strict because $g(\hat y_O)>0$ and (C$'$) with the $(1-\mu^*)g(\hat y_R)$ term), so $dp/d\pi^u = -(\partial\mu^d/\partial\pi^u)/(\partial\mu^d/\partial p) > 0$. Item 3: $d\hat b = (\partial\hat b/\partial p)dp + (\partial\hat b/\partial\rho)d\rho \ge 0$ since both products are non-negative; hence $\hat y_R$ rises and $q_{RO}$ falls. Since $\mu_{t+1}-\mu^* = \mu^*(q_{OO}-q^*_{OO}) + (1-\mu^*)(q_{RO}-q^*_{RO}) \ge 0$ with the second term non-positive, $q_{OO}\ge q^*_{OO}$, and $\lambda = q_{OO}-q_{RO}$ rises. Item 4 is the change in $W_3$ of stage-3 owners (both terms non-negative), the lender's side of Proposition~\ref{prop:accounting}, Lemma~\ref{lem:split} applied to the parents' gain minus the higher outlay $d\,dp_{t+1}$ of the child, and the stage-1 budget of a renter. Item 5: with one stock, $p_{t+1}$ solves $S(p)=1$ and does not depend on $\pi^u$. | |
| 158 | +under (C$'$), while $\partial\mu^d/\partial\rho = -[\cdots]\frac{1+\nu}{\nu}\partial\hat b/\partial\rho\ge 0$ multiplies $d\rho/dQ<0$. Hence $\partial\Psi/\partial Q\le-2<0$ and, by the implicit function theorem, $dQ/d\pi^u = -(\partial\Psi/\partial\pi^u)/(\partial\Psi/\partial Q)$ has the sign of $\Xi$. Items 1--2 of the proposition and the price statements of the corollary follow from $dp/dQ = p/(\varepsilon Q)>0$ for $\varepsilon\in(0,\infty)$, $d\rho/dQ<0$ for $\varepsilon_r<\infty$, and $\mu_{t+1} = 2Q-\mu^*$. If $\varepsilon = 0$ the owner stock is fixed, $\mu_{t+1} = \mu^*$, and $p_{t+1}$ solves $\mu^d(p,\rho^*,\pi^u) = \mu^*$ with $\partial\mu^d/\partial p<0$ (strict because $g(\hat y_O)>0$ and (C$'$) with the $(1-\mu^*)g(\hat y_R)$ term), so $dp/d\pi^u = -(\partial\mu^d/\partial\pi^u)/(\partial\mu^d/\partial p)$ again has the sign of $\Xi$. Item~3 of the proposition: $d\hat b = (\partial\hat b/\partial p)dp+(\partial\hat b/\partial\rho)d\rho\ge 0$ since both products are non-negative when $\Xi>0$; hence $\hat y_R = Y$ rises and $q_{RO}$ falls. Since $\mu_{t+1}-\mu^* = \mu^*(q_{OO}-q^*_{OO})+(1-\mu^*)(q_{RO}-q^*_{RO})\ge 0$ with the second term non-positive, $q_{OO}\ge q^*_{OO}$, and $\lambda = q_{OO}-q_{RO}$ rises. When $\Xi<0$ both products are non-positive, so $\hat b$ falls; the total derivative of $\hat y_R$ is then $\frac{1+\nu}{\nu}d\hat b/d\pi^u+Rn^R/(1+\pi^u)^2$, the sum of a negative price term and a non-negative deposit term, which gives the unsigned statement of the corollary and its sign when $n^R = 0$. Item~4 is the change in $W_3$ of stage-3 owners (both terms non-negative), the lender's side of Proposition~\ref{prop:accounting}, Lemma~\ref{lem:split} applied to the parents' gain minus the higher outlay $d\,dp_{t+1}$ of the child, and the stage-1 budget of a renter. Item~5: with one stock and unit demand, $p_{t+1}$ solves $S(p) = 1$ and does not depend on $\pi^u$. | |
| 315 | 159 | \end{proof} |
| 316 | − | |
| 317 | −\begin{remark}[Scope of Proposition~\ref{prop:impact}] | |
| 318 | −\label{rem:impact_scope} | |
| 319 | −The proposition covers the first period after the shock under per-period supply and static expectations. Contracts do not matter on impact because the stage-3 parents' nominal payment was fixed before the shock under every contract; they matter for the cohort aged~2 (Proposition~\ref{prop:accounting}), whose children buy at $t+2$. With units held for two periods, the stage-2 cohort's holdings crowd the next cohort ($\mu_{t+1}+\mu_t \approx 2S(p)$), which generates an alternating adjustment in the myopic simulation of \texttt{src/toy\_model.py}; the perfect-foresight transition with a durable stock, the second-round effects, and the owner--renter welfare comparison for marginal households are left to the numerical model and are not claimed here. | |
| 320 | −\end{remark} | |
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@@ -1,7 +1,8 @@ | ||
| 1 | −% Stand-alone wrapper to compile appendix_proofs.tex on its own: | |
| 1 | +% Stand-alone wrapper to compile the theory of WP18 on its own: | |
| 2 | 2 | % cd paper && latexmk -pdf appendix_proofs_standalone.tex |
| 3 | −% The preamble mirrors main.tex (template) plus the WP18 theorem environments | |
| 4 | −% proposed in research/wp_style_audit.md, §6. | |
| 3 | +% It inputs the model section, the statements (sections/equilibrium.tex) and the | |
| 4 | +% proofs (appendix_proofs.tex). Cross-references to the numerical sections and | |
| 5 | +% figures of the full paper are undefined here (warnings only, no errors). | |
| 5 | 6 | \documentclass[12pt,letterpaper]{article} |
| 6 | 7 | \usepackage[utf8]{inputenc} |
| 7 | 8 | \usepackage[T1]{fontenc} |
@@ -13,8 +14,12 @@ | ||
| 13 | 14 | \onehalfspacing |
| 14 | 15 | \usepackage{amsmath,amssymb,amsthm} |
| 15 | 16 | \usepackage{booktabs} |
| 17 | +\usepackage{graphicx} | |
| 18 | +\graphicspath{{figures/}} | |
| 16 | 19 | \usepackage[dvipsnames]{xcolor} |
| 17 | 20 | \usepackage[colorlinks=true,linkcolor=NavyBlue,citecolor=NavyBlue,urlcolor=NavyBlue]{hyperref} |
| 21 | +\usepackage[authoryear,round,semicolon]{natbib} | |
| 22 | +\bibliographystyle{aer} | |
| 18 | 23 | \usepackage{enumitem} |
| 19 | 24 | \setlist{nosep,leftmargin=*} |
| 20 | 25 | \usepackage[toc,page]{appendix} |
@@ -31,15 +36,18 @@ | ||
| 31 | 36 | \theoremstyle{remark} |
| 32 | 37 | \newtheorem{remark}{Remark} |
| 33 | 38 | |
| 34 | −\title{WP18 --- Appendix: assumptions, propositions and proofs\\ \large (stand-alone compilation of \texttt{appendix\_proofs.tex})} | |
| 39 | +\input{tables/numbers} | |
| 40 | + | |
| 41 | +\title{WP18 --- Model, propositions and proofs\\ \large (stand-alone compilation of the theory sections)} | |
| 35 | 42 | \author{Simon-Pierre Boucher} |
| 36 | −\date{Phase 2 draft, October 2026} | |
| 43 | +\date{October 2026} | |
| 37 | 44 | |
| 38 | 45 | \begin{document} |
| 39 | 46 | \maketitle |
| 40 | −\section{Model (placeholder for the main text)} | |
| 41 | −\label{sec:model} | |
| 42 | −This section stands in for the model section of the main text; the benchmark model is specified in \texttt{research/model\_specification.md}, Part~A. | |
| 47 | +\setcounter{section}{4} | |
| 48 | +\input{sections/model} | |
| 49 | +\input{sections/equilibrium} | |
| 50 | +\bibliography{references} | |
| 43 | 51 | \begin{appendices} |
| 44 | 52 | \input{appendix_proofs} |
| 45 | 53 | \end{appendices} |
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@@ -1,5 +1,7 @@ | ||
| 1 | 1 | % ============================================================================ |
| 2 | −% UQO Working Paper Template — Economics / Finance | |
| 2 | +% UQO Working Paper No. 18 — Housing, Leverage, and Intergenerational Wealth | |
| 3 | +% Inequality: A Dynamic General Equilibrium Theory (Simon-Pierre Boucher) | |
| 4 | +% Preamble = UQO template (HARMONISATION.md) + amsthm environments (WP18) | |
| 3 | 5 | % ============================================================================ |
| 4 | 6 | \documentclass[12pt,letterpaper]{article} |
| 5 | 7 | |
@@ -26,6 +28,18 @@ | ||
| 26 | 28 | % --- Math --- |
| 27 | 29 | \usepackage{amsmath,amssymb,amsthm} |
| 28 | 30 | |
| 31 | +% --- Theorem Environments (WP18; research/wp_style_audit.md, §6) --- | |
| 32 | +\theoremstyle{plain} | |
| 33 | +\newtheorem{proposition}{Proposition} | |
| 34 | +\newtheorem{lemma}{Lemma} | |
| 35 | +\newtheorem{corollary}{Corollary} | |
| 36 | +\newtheorem{theorem}{Theorem} | |
| 37 | +\theoremstyle{definition} | |
| 38 | +\newtheorem{assumption}{Assumption} | |
| 39 | +\newtheorem{definition}{Definition} | |
| 40 | +\theoremstyle{remark} | |
| 41 | +\newtheorem{remark}{Remark} | |
| 42 | + | |
| 29 | 43 | % --- Tables --- |
| 30 | 44 | \usepackage{booktabs} |
| 31 | 45 | \usepackage{tabularx} |
@@ -40,6 +54,7 @@ | ||
| 40 | 54 | |
| 41 | 55 | % --- Figures --- |
| 42 | 56 | \usepackage{graphicx} |
| 57 | +\graphicspath{{figures/}} | |
| 43 | 58 | \usepackage{subcaption} |
| 44 | 59 | \usepackage{float} |
| 45 | 60 | |
@@ -87,21 +102,18 @@ | ||
| 87 | 102 | \setlist{nosep,leftmargin=*} |
| 88 | 103 | |
| 89 | 104 | % ============================================================================ |
| 90 | −% METADATA — Edit these fields for each working paper | |
| 105 | +% METADATA | |
| 91 | 106 | % ============================================================================ |
| 92 | −\newcommand{\WPnumber}{XX} | |
| 93 | −\newcommand{\WPtitle}{Your Working Paper Title Here} | |
| 94 | −\newcommand{\WPsubtitle}{} % Leave empty if no subtitle | |
| 95 | −\newcommand{\WPdate}{\today} | |
| 107 | +\newcommand{\WPnumber}{18} | |
| 108 | +\newcommand{\WPtitle}{Housing, Leverage, and Intergenerational Wealth Inequality} | |
| 109 | +\newcommand{\WPsubtitle}{A Dynamic General Equilibrium Theory} | |
| 110 | +\newcommand{\WPdate}{October 2026} | |
| 96 | 111 | \newcommand{\WPversion}{1.0} |
| 97 | 112 | \newcommand{\WPabstract}{% |
| 98 | − This paper examines \ldots{} | |
| 99 | − We find that \ldots{} | |
| 100 | − Our results suggest \ldots{} | |
| 101 | − These findings contribute to the literature on \ldots{} | |
| 113 | + This paper asks under what conditions inherited wealth, mortgage leverage, nominal debt, housing supply and down-payment constraints generate persistent owner--renter wealth gaps among households with the same labour-income process, and what an unanticipated inflation changes. A three-stage benchmark with two-period generations yields a closed-form ownership-entry threshold that adds a liquidity component to the down-payment bound, a leverage multiplier whose break-even price growth is independent of the loan-to-value ratio, a closed-form intergenerational tenure persistence that is positive only if the house is net wealth at the end of life, existence and a uniqueness condition for the stationary equilibrium, and the accounting of a price-level surprise under three mortgage contracts. A twelve-period overlapping-generations model with nominal annuity mortgages, gifts and estates, and segmented rental and construction sectors is solved in stationary general equilibrium and along perfect-foresight transitions under an illustrative calibration. Transfers generate a \GapAgeFour{} income-year wealth gap at mid-life and a rank--rank wealth slope of \RankRankNW{} that vanishes without transfers; price adjustment removes \AbsorbThetaFivePct{}--\AbsorbThetaFortyPct\% of the partial-equilibrium ownership response to the down-payment constraint; no multiple equilibria arise under near-fixed supply. Because households are net nominal creditors, a 10\% surprise moves the real house price by \FrmLevelPzero\%, a fall where the benchmark predicts a rise; the contract ordering, fixed-rate above renewable term above adjustable, is confirmed.% | |
| 102 | 114 | } |
| 103 | −\newcommand{\WPkeywords}{Keyword 1, Keyword 2, Keyword 3, Keyword 4} | |
| 104 | −\newcommand{\WPjel}{G12, G14, C58} | |
| 115 | +\newcommand{\WPkeywords}{housing tenure, mortgage leverage, intergenerational wealth, bequests, down-payment constraint, surprise inflation, overlapping generations} | |
| 116 | +\newcommand{\WPjel}{R21, R31, D31, D91, E31, G51} | |
| 105 | 117 | |
| 106 | 118 | % --- Author --- |
| 107 | 119 | \newcommand{\WPauthor}{Simon-Pierre Boucher} |
@@ -116,6 +128,11 @@ | ||
| 116 | 128 | Gatineau, Québec, Canada J9A 1L8% |
| 117 | 129 | } |
| 118 | 130 | |
| 131 | +\hypersetup{pdftitle={\WPtitle: \WPsubtitle}, pdfauthor={\WPauthor}} | |
| 132 | + | |
| 133 | +% --- Generated numbers (src/tables.py -> results/tables/numbers.tex -> paper/tables/) --- | |
| 134 | +\input{tables/numbers} | |
| 135 | + | |
| 119 | 136 | % ============================================================================ |
| 120 | 137 | % DOCUMENT |
| 121 | 138 | % ============================================================================ |
@@ -127,17 +144,26 @@ | ||
| 127 | 144 | % --- Main Body --- |
| 128 | 145 | \input{sections/introduction} |
| 129 | 146 | \input{sections/literature} |
| 130 | −\input{sections/data} | |
| 131 | −\input{sections/methodology} | |
| 132 | −\input{sections/results} | |
| 133 | −\input{sections/robustness} | |
| 134 | −\input{sections/conclusion} | |
| 147 | +\input{sections/model} | |
| 148 | +\input{sections/equilibrium} | |
| 149 | +\input{sections/computation} | |
| 150 | +\input{sections/baseline} | |
| 151 | +\input{sections/inflation} | |
| 152 | +\input{sections/supply_policy_welfare} | |
| 153 | +\input{sections/sensitivity_conclusion} | |
| 135 | 154 | |
| 136 | 155 | % --- References --- |
| 137 | 156 | \newpage |
| 138 | 157 | \bibliography{references} |
| 139 | 158 | |
| 140 | −% --- Appendix --- | |
| 159 | +% --- Appendices --- | |
| 160 | +\newpage | |
| 161 | +\begin{appendices} | |
| 162 | +\renewcommand{\thetable}{\Alph{section}\arabic{table}} | |
| 163 | +\renewcommand{\thefigure}{\Alph{section}\arabic{figure}} | |
| 164 | +\input{appendix_proofs} | |
| 165 | +\input{appendix_computation} | |
| 141 | 166 | \input{appendix/appendix} |
| 167 | +\end{appendices} | |
| 142 | 168 | |
| 143 | 169 | \end{document} |
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paper/sections/baseline.tex
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@@ -0,0 +1,86 @@ | ||
| 1 | +% ============================================================================ | |
| 2 | +\section{Baseline simulations: inheritance, leverage and inequality} | |
| 3 | +\label{sec:baseline} | |
| 4 | +% ============================================================================ | |
| 5 | + | |
| 6 | +This section reports the baseline stationary equilibrium and the experiments that isolate the inheritance mechanism: life-cycle profiles, the entry threshold, equal-income wealth paths with different initial transfers, the dependence of inequality and mobility on transfers, the down-payment sweep in general and partial equilibrium, and the accounting of negative price shocks under leverage. All magnitudes are statements about this parameterisation. | |
| 7 | + | |
| 8 | +\subsection{Life-cycle profiles and the aggregate nominal position} | |
| 9 | +\label{subsec:profiles} | |
| 10 | + | |
| 11 | +\begin{figure}[htbp] | |
| 12 | +\centering | |
| 13 | +\includegraphics[width=\textwidth]{fig04_lifecycle_profiles.pdf} | |
| 14 | +\caption{Baseline life-cycle profiles in the stationary general equilibrium: tenure and mortgage status (a), mean net worth of owners and renters (b) and balance sheets (c), by five-year age group. Units: average five-year labour income. Source: \texttt{results/data/baseline/profiles\_by\_age.csv}.} | |
| 15 | +\label{fig:profiles} | |
| 16 | +\end{figure} | |
| 17 | + | |
| 18 | +Figure~\ref{fig:profiles} shows the life-cycle profiles. Ownership rises from \OwnAgeOne{} at 25--29 to \OwnPeak{} at 55--64 and falls to \OwnAgeTwelve{} at 80--84; a share \ShareOwnersMortgage{} of owners carry a mortgage, with a mean loan-to-value ratio at origination of \MeanLTVNew{} and a median house value of \HouseValueIncome{} years of income. The net-worth gap between owners and renters, who have the same income process by construction, is \GapAgeTwo{} years of income at 30--34, \GapAgeFour{} at 40--44, \GapAgeSeven{} at 55--59 and \GapAgeTen{} at 70--74. The gap is housing equity carried forward: forced amortisation and the leverage multiplier of Proposition~\ref{prop:leverage} accumulate it for owners while renters, whose median net worth under 35 is zero, save little before purchase. The balance-sheet panel contains the fact that governs the inflation results of Section~\ref{sec:inflation}: aggregate real mortgage debt is \TotalDebt{} per unit of age mass against real deposits of \TotalDeposits{}, so households are net nominal creditors by a ratio of \DebtDepositRatio{} to one. Mortgage debt is concentrated among owners aged 30--50; deposits are held by the young saving for the down payment and, mostly, by owners aged 55--75 who have repaid their loans and are accumulating a liquid estate. The old-age sale of houses is the moment where the model departs from the Canadian profile, as noted in Section~\ref{sec:computation}, and it also means that the estate is liquid: \ShareEstatePositivePct\% of children receive a positive estate at 55--59, of mean size \MeanEstate{}, and \ShareGiftPositivePct\% receive a positive gift at entry, of mean size \MeanGift{} (\MeanGiftAnnual{} years of average income, median \MedianGift). | |
| 19 | + | |
| 20 | +\subsection{The ownership-entry threshold} | |
| 21 | +\label{subsec:threshold_num} | |
| 22 | + | |
| 23 | +\begin{figure}[htbp] | |
| 24 | +\centering | |
| 25 | +\includegraphics[width=\textwidth]{fig03_ownership_thresholds.pdf} | |
| 26 | +\caption{Ownership-entry thresholds (units: average five-year income). Panel (a): the smallest initial liquid wealth at which a renter buys, by age and productivity state, in the baseline. Panel (b): the age-1 threshold at the median productivity state as a function of the minimum down payment, in general equilibrium (prices re-solved) and at fixed prices. The grid sensitivity of this statistic is about $\pm0.07$. Source: \texttt{results/data/baseline/thresholds.csv}, \texttt{results/data/theta\_sweep/*.json}.} | |
| 27 | +\label{fig:thresholds} | |
| 28 | +\end{figure} | |
| 29 | + | |
| 30 | +The quantitative counterpart of $\hat b$ is the smallest liquid wealth at which the age-$j$ renter buys, $\hat a_j(z)$. Figure~\ref{fig:thresholds}(a) plots it. At age~1 it is \ThresholdAgeOneLowZ, \ThresholdAgeOneZtwo, \ThresholdAgeOneMidZ, \ThresholdAgeOneZfour{} and \ThresholdAgeOneHighZ{} years of income for the five productivity states from the lowest to the highest; it falls with age, and at ages~2 and~3 it is negative for the two highest states (\ThresholdAgeTwoHighZ{} and \ThresholdAgeThreeHighZ{} at the top), meaning that those households buy out of income alone. The dependence on productivity is the feasibility component of Proposition~\ref{prop:threshold} operating through the payment-to-income cap: at the lowest state the cap, not the loan-to-value constraint, binds, and the threshold is the cash needed to buy the smallest house with the largest loan the cap allows. The threshold is the least robust statistic of the paper, moving by \DevCoarseThr{} between the coarse and baseline grids and \DevFineThr{} between the baseline and fine grids (Table~\ref{tab:convergence}), which is why statements about it are confined to differences larger than that. | |
| 31 | + | |
| 32 | +Panel (b) is the check of Proposition~\ref{prop:cs} and the first of the three discrepancies. At fixed prices the age-1 threshold at the median state is \ThetaFivePEThr{} with a 5\% minimum down payment, \ThresholdAgeOneMidZ{} with 20\% and \ThetaFortyPEThr{} with 40\%: it rises with $\theta$ between 20\% and 40\%, as the proposition predicts, but it is not lower at 5\% than at 20\%. The difference is within the grid sensitivity, and its economic source is the one named in Remark~\ref{rem:cs_scope}: with a small down payment the option of renting now and buying later is more valuable, which raises the renter's continuation value and the cash at which buying now is preferred. For the highest productivity state the threshold falls monotonically with $\theta$ at fixed prices (\ThetaFivePEThrHighZ, \ThresholdAgeOneHighZ, \ThetaFortyPEThrHighZ); for the lowest it is unchanged (\ThetaFivePEThrLowZ, \ThresholdAgeOneLowZ, \ThetaFortyPEThrLowZ) because the loan-to-value constraint does not bind there. Proposition~\ref{prop:cs} is therefore supported where its binding-constraint and once-and-for-all premises hold and contradicted, within noise, where they do not. In general equilibrium the price and rent responses add a second layer, and the threshold is \ThetaFiveThr, \ThetaTenThr, \ThresholdAgeOneMidZ, \ThetaThirtyThr{} and \ThetaFortyThr{} for the five values of $\theta$. | |
| 33 | + | |
| 34 | +\subsection{Equal-income wealth paths} | |
| 35 | +\label{subsec:equalincome} | |
| 36 | + | |
| 37 | +\begin{figure}[htbp] | |
| 38 | +\centering | |
| 39 | +\includegraphics[width=\textwidth]{fig05_equal_income_paths.pdf} | |
| 40 | +\caption{Net worth along a deterministic median-income path under baseline policies, for entrants with initial transfers at the 10th to 90th percentiles of the gift distribution (a) and for two entrants whose transfers straddle the age-1 threshold, $a_1 = \EqIncAoneMinus$ and $\EqIncAonePlus$ (b). Units: average five-year income; the estate at 55--59 is set at the median lottery point. Source: \texttt{results/data/baseline/equal\_income\_paths.csv}.} | |
| 41 | +\label{fig:equal} | |
| 42 | +\end{figure} | |
| 43 | + | |
| 44 | +Figure~\ref{fig:equal} isolates the mechanism of Propositions~\ref{prop:leverage} and~\ref{prop:persist} by holding the income path fixed at the median and varying only the transfer at entry. Entrants with gifts at or below the median ($a_1\le\EqIncAoneMedian$) rent until 45--49 and then buy the smallest house with the maximum loan; the entrant just above the age-1 threshold ($a_1 = \EqIncAonePlus$) buys at 25--29 and the one just below ($a_1 = \EqIncAoneMinus$) at 40--44. Net worth of the two straddling households is \EqIncNWPlusAgeFour{} against \EqIncNWMinusAgeFour{} at 40--44 and \EqIncNWPlusAgeSeven{} against \EqIncNWMinusAgeSeven{} at 55--59: a \EqIncEndowDiffPct\% difference in endowment becomes a \EqIncGapPctAgeFour\% difference in wealth at 40--44 and a \EqIncGapPctAgeSeven\% difference at 55--59, before the paths converge after retirement when both households sell. The threshold converts small endowment differences into two-decade wealth gaps through earlier leverage and forced amortisation; the sign is that of Proposition~\ref{prop:leverage}'s multiplier and the magnitude is illustrative. | |
| 45 | + | |
| 46 | +\subsection{Transfers, inequality and mobility} | |
| 47 | +\label{subsec:transfers} | |
| 48 | + | |
| 49 | +\begin{figure}[htbp] | |
| 50 | +\centering | |
| 51 | +\includegraphics[width=\textwidth]{fig06_inheritance_quantile_trajectories.pdf} | |
| 52 | +\caption{Ownership rate (a) and median net worth (b) by age and by quintile of the inter-vivos transfer received at entry, baseline cross-section. The bottom two quintiles receive no transfer. Source: \texttt{results/data/baseline/by\_gift\_quintile.csv}.} | |
| 53 | +\label{fig:quintiles} | |
| 54 | +\end{figure} | |
| 55 | + | |
| 56 | +Figure~\ref{fig:quintiles} shows the same mechanism in the cross-section. Ownership at 25--29 is \OwnAgeOneQOne, \OwnAgeOneQTwo, \OwnAgeOneQThree, \OwnAgeOneQFour{} and \OwnAgeOneQFive{} from the bottom to the top quintile of the transfer received; the bottom two quintiles receive nothing. The gap between the second and the fifth quintile narrows from \GapQtwoQfiveAgeOnePP{} percentage points at 25--29 to \GapQtwoQfiveAgeSevenPP{} at 55--59 as the untransferred households buy later, but median net worth at 55--59 is still \MedNWAgeSevenQTwo{} in the second quintile against \MedNWAgeSevenQFive{} in the fifth. Measured at the age at which the gift is made, 55--59, the probability that the child owns is \qOOageSeven{} if the parent owns and \qROageSeven{} if the parent rents, so the persistence statistic of Proposition~\ref{prop:persist} is $\lambda_7 = \LambdaAgeSeven$, a half-life of \HalfLifeGen{} generations; it is weak at that age because \ShareParentOwnerPct\% of parents own by then. At 35--39 the same probabilities are \qOOageThree{} and \qROageThree{}, $\lambda_3 = \LambdaAgeThree$. The rank--rank slope of net worth between parents and children at 55--59 is \RankRankNW{} and the Gini of net worth is \GiniNW{} over all ages and \GiniAgeSeven{} at 55--59. | |
| 57 | + | |
| 58 | +\begin{figure}[htbp] | |
| 59 | +\centering | |
| 60 | +\includegraphics[width=\textwidth]{fig07_inequality_mobility.pdf} | |
| 61 | +\caption{Inequality and mobility across stationary equilibria (general equilibrium unless stated): Gini of net worth by age (a) and intergenerational tenure and wealth persistence (b) for the baseline, without bequests, with a 50\% rebated transfer tax, under near-fixed supply, and with minimum down payments of 5\% and 40\%. Source: \texttt{results/data/\{baseline,bequests,sensitivity\}/*.json}.} | |
| 62 | +\label{fig:inequality} | |
| 63 | +\end{figure} | |
| 64 | + | |
| 65 | +\input{tables/tab07_mobility_inequality} | |
| 66 | + | |
| 67 | +Table~\ref{tab:mobility} and Figure~\ref{fig:inequality} vary the transfers. Without bequests ($\nu = 0$) the price falls to \BeqOffP{}, the rent rises by \BeqOffRhoDevPct\%, ownership falls to \BeqOffOwn{} (\BeqOffOwnUthirtyfive{} under 35, against \OwnershipUnderThirtyFive), and the transition probabilities of the child become equal, $q_{RO} = \BeqOffqRO$ and $q_{OO} = \BeqOffqOO$, with a rank--rank slope of \BeqOffRR: intergenerational persistence disappears entirely, as Proposition~\ref{prop:persist} requires, and here it is the whole of it because incomes are independent across generations by design. The Gini rises to \BeqOffGini, since transfers lift the bottom of the young distribution more than the top, and the equal-income gap at 40--44 falls from \GapAgeFour{} to \BeqOffGapFour. At fixed prices the same change lowers ownership by \BeqOffPEDropPP{} percentage points; the general-equilibrium price fall offsets all but \BeqOffGEDropPP{} of them. Halving the bequest motive gives intermediate values (price \BeqHalfP, $q_{OO} = \BeqHalfqOO$, rank--rank \BeqHalfRR). The comparison with an intergenerational income correlation of 0.4 is the caveat of \citet{fagereng2021why} in model form: $q_{RO}$ falls to \RhoGqRO{} and $\lambda_3$ rises to \RhoGLamThree{} with the rank--rank slope at \RhoGRR, so most of the tenure persistence beyond the baseline comes from income transmission, not from housing, and the model's transfer channel should be read as one mechanism among several. | |
| 68 | + | |
| 69 | +\subsection{The down-payment constraint in general and partial equilibrium} | |
| 70 | +\label{subsec:thetasweep} | |
| 71 | + | |
| 72 | +\input{tables/tab05_scenarios} | |
| 73 | + | |
| 74 | +\begin{figure}[htbp] | |
| 75 | +\centering | |
| 76 | +\includegraphics[width=\textwidth]{fig14_down_payment_sweep.pdf} | |
| 77 | +\caption{Down-payment sweep: ownership rate (a), equilibrium prices (b) and intergenerational persistence (c) as functions of the minimum down payment $\theta$, in general equilibrium (prices, lottery and rebate re-solved) and with prices fixed at the baseline. Source: \texttt{results/data/theta\_sweep/*.json}.} | |
| 78 | +\label{fig:theta} | |
| 79 | +\end{figure} | |
| 80 | + | |
| 81 | +Table~\ref{tab:scenarios} and Figure~\ref{fig:theta} report the down-payment sweep. In general equilibrium a minimum down payment of 5, 10, 20, 30 and 40\% gives prices of \ThetaFiveP, \ThetaTenP, \PriceBaseline, \ThetaThirtyP{} and \ThetaFortyP{} and ownership rates of \ThetaFiveOwn, \ThetaTenOwn, \OwnBaseline, \ThetaThirtyOwn{} and \ThetaFortyOwn, with under-35 ownership from \ThetaFiveOwnUthirtyfive{} to \ThetaFortyOwnUthirtyfive. At fixed prices the 5\% constraint raises ownership to \ThetaFivePEOwn{} (\ThetaFivePEOwnUthirtyfive{} under 35) and the 40\% constraint lowers it to \ThetaFortyPEOwn{} (\ThetaFortyPEOwnUthirtyfive): the price and rent responses absorb \AbsorbThetaFivePct\% and \AbsorbThetaFortyPct\% of the partial-equilibrium ownership effect, the direction of Corollary~\ref{cor:fixed} with a unit supply elasticity rather than a fixed stock. Mobility moves with the constraint, $q_{RO}$ at 35--39 falling from \ThetaFiveqRO{} to \ThetaFortyqRO{} and $\lambda_7$ rising from \ThetaFiveLamSeven{} to \ThetaFortyLamSeven, while the rank--rank slope is flat at \ThetaFiveRR--\ThetaFortyRR: a tighter constraint lowers upward tenure mobility without changing the rank persistence of wealth, because the latter is dominated by the size of transfers rather than by who crosses the threshold. The payment-to-income cap has a smaller effect in the same direction (ownership \PsiThirtyOwn{} and \PsiFiftyOwn{} for $\psi = 0.30$ and $0.50$). | |
| 82 | + | |
| 83 | +\subsection{Leverage and negative price shocks: accounting} | |
| 84 | +\label{subsec:pricedrop} | |
| 85 | + | |
| 86 | +Proposition~\ref{prop:leverage} is an accounting statement and is checked as one, on the baseline cross-section with prices fixed after the drop. Existing owners carry a mean loan-to-value ratio of \MeanLTVOwners{} and no household is underwater after a 10, 20 or 30\% fall in the house price, because the first annuity payment, a fifth of the loan, is made within the purchase period and the balance carried across five-year periods never exceeds the underwater threshold $x^*$; the mean loss relative to net worth is \LossNWAllTen, \LossNWAllTwenty{} and \LossNWAllThirty\% for the three drops. New borrowers evaluated at origination, with a mean loan-to-value of \MeanLTVNew, are \UnderwaterNewTen, \UnderwaterNewTwenty{} and \UnderwaterNewThirty\% underwater, consistent with $x^* = 1-0.8/0.95\approx 0.16$ at the 80\% cap with a 5\% selling cost, and lose \LossNWNewTen, \LossNWNewTwenty{} and \LossNWNewThirty\% of net worth. The five-year period therefore understates peak exposure, a limitation noted in Section~\ref{sec:sensitivity}; the general-equilibrium counterpart of a price fall, a persistent rise in the real rate, is in Section~\ref{subsec:rateshock}. | |
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paper/sections/computation.tex
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| 1 | +% ============================================================================ | |
| 2 | +\section{Numerical algorithm, calibration and verification} | |
| 3 | +\label{sec:computation} | |
| 4 | +% ============================================================================ | |
| 5 | + | |
| 6 | +\subsection{Solution method} | |
| 7 | +\label{subsec:algorithm} | |
| 8 | + | |
| 9 | +The household problem \eqref{eq:bellman} is solved by backward induction over the twelve ages. The discrete branches other than \emph{stay} depend on the state only through cash on hand, so the values of renting, buying, refinancing and selling are computed on a cash-on-hand grid of $120$ points and the value of staying on the full state grid (five productivity states, four house sizes, $20$ mortgage balances, five remaining-payment counts, $50$ liquid-asset points). The continuous choice of next-period liquid assets is an exhaustive search on the grid refined by a golden-section step; loans are chosen among eleven fractions of the binding constraint, gifts among twelve fractions of cash, and the estate is anticipated as a five-point lottery whose points are the conditional means of the simulated distribution of estates by quintile. Endogenous-grid methods were not used because the combination of discrete tenure and size choices with the two origination constraints and a mortgage grid makes the discrete-choice upper envelope fragile, while the cost of the exhaustive search under just-in-time compilation is low (one evaluation of the household block and the distribution takes about one second). | |
| 10 | + | |
| 11 | +The distribution is a Monte-Carlo panel of \NDynasties{} dynasty slots, each holding a parent and a child whose life-cycle phases are equidistributed, simulated with common random numbers for \HouseholdPeriods{} household-periods after a burn-in of ten generations. A panel rather than a histogram is used because the dynastic link, a gift at the child's age~1 and an estate at age~7 both correlated with the parent's state, requires the joint parent--child distribution, which a histogram over individual states does not carry. Two approximations are deliberate: the estate is anticipated as the unconditional lottery, although the realised estate in the panel is exactly linked to the parent; and along a transition the lottery is held at its stationary value. A small unanticipated transitory income component, realised before decisions, is added because entrants without a gift and liquidity-constrained renters otherwise share the same cash at a given productivity state and switch tenure simultaneously, which produced a discontinuity in aggregate rental demand of about 1.5\% over a 0.05\% price interval in a first run; with the component, excess demand is continuous and the root is bracketed. | |
| 12 | + | |
| 13 | +Market clearing uses a nested regula-falsi (Illinois) search, the rental price inside the owner-occupied price, followed by a damped fixed point on the estate lottery and on the transfer-tax rebate. A Broyden update on the two prices was tried first and abandoned: the excess-demand functions are smooth but strongly non-linear (tenure shares move by 15--20 percentage points for a 5\% change in the relative price) and the Monte-Carlo noise of about $10^{-3}$ in differences of excess demand for a 0.5\% price step corrupted the secant updates. The supply scales $\bar H$ and $\bar H_r$ are set so that the baseline clears at $p^* = p^{r*} = 1$, the normalisation of \citet{kaplan2020housing}; every counterfactual keeps these scales. Transitions are solved by shooting in the sequence space on the log price paths over $T = 14$ periods (70 years) with the durable stocks \eqref{eq:supply}, a finite-difference Jacobian computed in parallel, Broyden updates with periodic refresh, and, for large shocks, a damped t\^atonnement pre-phase. The tolerances are $5\times10^{-4}$ on relative excess demand in the stationary equilibrium, which pins the equilibrium price to about $\pm0.05\%$ given the Monte-Carlo noise, and $1.5\times10^{-3}$ per period along a transition, where each cross-section has one sixth of the stationary sample. Every run records its residuals; runs that stall above tolerance are labelled with their residual wherever they appear. | |
| 14 | + | |
| 15 | +Welfare is measured by the consumption-equivalent variation $\lambda^{CE}$ such that scaling non-housing consumption by $1+\lambda^{CE}$ in every period of the baseline makes an age-1 household with state $(z_1,a_1)$ as well off as in the alternative, | |
| 16 | +\begin{equation} | |
| 17 | +V^{alt}_1(z_1,a_1) = (1+\lambda^{CE})^{(1-\alpha)(1-\gamma)}\,U^{base}_1(z_1,a_1) + \Phi^{base}_1(z_1,a_1), | |
| 18 | +\label{eq:cev} | |
| 19 | +\end{equation} | |
| 20 | +where $U$ is discounted flow utility and $\Phi$ the discounted warm glow, the latter obtained from cohort simulations under baseline policies and the former as $U = V-\Phi$, an identity at the baseline; the simulated $U$ is kept as a check and differs from $V-\Phi$ by at most \WelfareCheckMaxPct\% because simulated policies use nearest-grid loan and size choices, a discrepancy that would inflate the variation if used directly. For households alive at a shock the same measure is applied to their remaining lifetime. Comparisons across scenarios that change preference parameters are reported but are not welfare statements, since they compare value functions under different utilities. | |
| 21 | + | |
| 22 | +\subsection{Calibration} | |
| 23 | +\label{subsec:calibration} | |
| 24 | + | |
| 25 | +The calibration is illustrative. Table~\ref{tab:parameters} in Appendix~\ref{app:computation} lists every parameter, its unit and its status; no moment is fitted, and the verified Canadian comparison points of Table~\ref{tab:moments} are reported next to the model's moments without being targeted. The period is five years. The annual discount factor is $0.96$, the curvature $\gamma = 2$, the housing share $\alpha = 0.20$; the warm-glow weight $\nu = 2$ and luxury shift $\underline x = 1$ were chosen so that mean gifts are about two years of income. Productivity follows an annual AR(1) with persistence $0.95$ and innovation standard deviation $0.15$, converted to the five-year period, with a hump-shaped age profile and a 50\% replacement rate in retirement. The four house sizes are $\{1,1.5,2.25,3.4\}$ units and rented services are continuous above $0.3$ units. Maintenance is 1\% of value per year, buying costs 2.5\% of the price, selling 5\%, refinancing 1\%, depreciation of the stocks 1\% per year. Mortgages are amortised over $N = 5$ periods, the Canadian 25-year standard; the baseline minimum down payment is $\theta = 0.20$, the uninsured threshold; the payment-to-income cap is $\psi = 0.39$ at a qualifying rate two percentage points above the contract rate. The real rate and trend inflation are 2\% per year, the latter the Bank of Canada target, so the five-year nominal rate is $0.219$. The landlord wedge is 5\% of value per period, segmentation is full and both supply elasticities are set to one, a value with no verified Canadian source, which is why Section~\ref{sec:supply} sweeps it from $0.05$ to $2$. | |
| 26 | + | |
| 27 | + | |
| 28 | +\subsection{Baseline equilibrium and verification} | |
| 29 | +\label{subsec:verification} | |
| 30 | + | |
| 31 | +The baseline stationary equilibrium clears at $p^* = \Pstar$ and $\tilde p_r^* = 1$ by normalisation, with an annual rent--price ratio of \RentPriceAnnual{}; relative excess demands are $\ExcessOwner$ in the owner-occupied market and $\ExcessRental$ in the rental market, the lottery converges in \LotteryRounds{} rounds and the whole equilibrium takes \BaselineSeconds{} seconds. Bellman residuals, measured as the relative difference between the interpolated value function and a re-optimisation on a 400-point asset grid at 400 random off-grid states, average $\BellmanOwnerMean$ for owners (99th percentile \BellmanOwnerPnn\%, maximum \BellmanOwnerMax\%) and $\BellmanRenterMean$ for renters; no household hits the forced-sale rule and \ClipSharePct\% of household-periods use a clipped interpolated policy. Table~\ref{tab:convergence} in Appendix~\ref{app:computation} collects these diagnostics with the grid, tolerance, sample-size and seed variants, and Figure~\ref{fig:diagnostics} plots them. The budget identities of every branch, the law of motion of nominal and real debt, the origination constraints, the terminal condition, the dynastic link, the lottery, the common random numbers, the monotonicity of the value functions and the single crossing of the tenure decision are unit-tested, and the zero-sum revaluation identity holds to machine precision in every transition. | |
| 32 | + | |
| 33 | +\input{tables/tab03_baseline_moments} | |
| 34 | + | |
| 35 | +Table~\ref{tab:moments} compares the model's moments with the verified comparison points. The ownership rate is \OwnershipAll{} overall, \OwnershipUnderThirtyFive{} under 35, \OwnThirtyFiveSixtyFour{} at 35--64 and \OwnSixtyFivePlus{} at 65 and over; \ShareMortgage{} of households hold a mortgage against 0.39 in the Survey of Financial Security; the mean net worth of owners is \NWRatioYoung{} times that of renters under 35, against a median ratio of about 10 in the survey. Two moments are flagged as not supported by the comparison points. The ownership profile peaks at \OwnPeak{} at 55--64 and falls to \OwnAgeTwelve{} at 80--84, because households with a warm glow over a liquid estate sell late in life to consume and to bequeath cash, whereas \citet{hou2010homeownership} report a plateau near three quarters after 65; this is a known consequence of the preference specification and is discussed in Section~\ref{sec:sensitivity}. And \ShareFirstBuyersGiftPct\% of first-time buyers receive a positive gift, well above the 28\% quoted by \citet{mirdamadi2023parents}, because every parent in the model has a warm-glow motive and the gift is the only transfer available at entry. These discrepancies do not affect the mechanisms the paper studies but they bound what the magnitudes can claim. | |
| 36 | + | |
| 37 | +\begin{figure}[htbp] | |
| 38 | +\centering | |
| 39 | +\includegraphics[width=\textwidth]{fig02_policy_functions.pdf} | |
| 40 | +\caption{Policy functions on the cash-on-hand grid in the baseline stationary equilibrium. Panel (a) plots the value of buying minus the value of renting at age~1 (25--29) by productivity state, clipped at $-0.6$; the single sign change along cash is the ownership-entry threshold. Panel (b) plots liquid saving at age~1 and panel (c) the house size bought at age~3 (0 = rent). Source: \texttt{results/data/baseline/policy\_age\{1,3\}.csv}.} | |
| 41 | +\label{fig:policy} | |
| 42 | +\end{figure} | |
| 43 | + | |
| 44 | +Figure~\ref{fig:policy} shows the policy functions at the entry ages. The buy-minus-rent value crosses zero exactly once along cash at every age from 1 to 3 and every productivity state, which is the single-crossing property that Proposition~\ref{prop:threshold} proves only on the hand-to-mouth region. At the crossing the buyer sets next-period liquid assets to zero for every state, so Assumption~\ref{ass:htm} holds for the marginal buyer; the marginal renter, however, saves, because in the quantitative economy renters accumulate towards a later purchase, which the benchmark rules out. This asymmetry is the origin of the scope restriction in Remark~\ref{rem:cs_scope}. | |
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paper/sections/conclusion.tex
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| 1 | −% ============================================================================ | |
| 2 | −\section{Conclusion} | |
| 3 | −\label{sec:conclusion} | |
| 4 | −% ============================================================================ | |
| 5 | − | |
| 6 | −This paper investigates \ldots{} Using data from \ldots{} over the period \ldots{}, we find that \ldots{} | |
| 7 | − | |
| 8 | −Our results contribute to the literature in several ways. First, \ldots{} Second, \ldots{} Third, \ldots{} | |
| 9 | − | |
| 10 | −These findings have important implications for \ldots{} From a policy perspective, \ldots{} For practitioners, \ldots{} | |
| 11 | − | |
| 12 | −Several limitations of the present study suggest directions for future research. First, \ldots{} Second, \ldots{} We leave these questions for future work. | |
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paper/sections/data.tex
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@@ -1,32 +0,0 @@ | ||
| 1 | −% ============================================================================ | |
| 2 | −\section{Data} | |
| 3 | −\label{sec:data} | |
| 4 | −% ============================================================================ | |
| 5 | − | |
| 6 | −\subsection{Data Sources} | |
| 7 | −\label{subsec:data_sources} | |
| 8 | − | |
| 9 | −We obtain data from \ldots{} Our sample spans the period from January~1990 to December~2023, covering \ldots{} We exclude \ldots{} consistent with standard practice in the literature \citep{fama1993common}. | |
| 10 | − | |
| 11 | −\subsection{Variable Construction} | |
| 12 | −\label{subsec:variables} | |
| 13 | − | |
| 14 | −Our dependent variable is \ldots{} defined as: | |
| 15 | −% | |
| 16 | −\begin{equation} | |
| 17 | − r_{i,t} = \ln\!\left(\frac{P_{i,t}}{P_{i,t-1}}\right), | |
| 18 | − \label{eq:returns} | |
| 19 | −\end{equation} | |
| 20 | −% | |
| 21 | −where $P_{i,t}$ denotes the adjusted closing price of asset~$i$ at time~$t$. | |
| 22 | − | |
| 23 | −The main explanatory variables include \ldots{} | |
| 24 | − | |
| 25 | −\subsection{Summary Statistics} | |
| 26 | −\label{subsec:summary_stats} | |
| 27 | − | |
| 28 | −Table~\ref{tab:summary_stats} presents the descriptive statistics for our key variables. | |
| 29 | − | |
| 30 | −\input{tables/summary_stats} | |
| 31 | − | |
| 32 | −Several patterns emerge from the summary statistics. First, \ldots{} Second, \ldots{} The standard deviations indicate \ldots{} | |
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paper/sections/equilibrium.tex
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| 1 | +% ============================================================================ | |
| 2 | +\section{Equilibrium and propositions} | |
| 3 | +\label{sec:equilibrium} | |
| 4 | +% ============================================================================ | |
| 5 | + | |
| 6 | +This section states the analytical results of the benchmark economy. Proofs are in Appendix~\ref{app:proofs}. Each proposition is followed by the assumption that the quantitative economy relaxes and, where the simulations of Sections~\ref{sec:baseline}--\ref{sec:supply} contradict the benchmark, by a remark that identifies the responsible assumption. Three results are stated more narrowly than in a first draft of this paper precisely because the simulations contradicted their broader versions: the monotonicity of the entry threshold in the down payment (Proposition~\ref{prop:cs}), price neutrality without segmentation (Proposition~\ref{prop:noseg}) and the sign of the house-price response to a surprise (Proposition~\ref{prop:impact} and Corollary~\ref{cor:creditor}). Simulation findings are never called theorems. | |
| 7 | + | |
| 8 | +\subsection{Assumptions} | |
| 9 | +\label{subsec:assumptions} | |
| 10 | + | |
| 11 | +\begin{assumption}[Preferences, demography, housing need] | |
| 12 | +\label{ass:pref} | |
| 13 | +Households have the preferences \eqref{eq:utilityA}, live three stages, need one unit of housing services at stages~1 and~2 (unit demand), and have the income process of Section~\ref{subsec:benchmark}: $y_1,y_2>0$ common and deterministic, $y_3\sim G$ with continuous density $g>0$ on $[\underline y_3,\bar y_3]$, $0<\underline y_3<\bar y_3<\infty$, independent across households and generations. | |
| 14 | +\end{assumption} | |
| 15 | + | |
| 16 | +\begin{assumption}[Prices, contracts, lender] | |
| 17 | +\label{ass:prices} | |
| 18 | +The real rate $r\ge 0$ is exogenous and $1+i_t = R(1+\pi^e_{t+1})$. A mortgage of real principal $B\le(1-\theta)p_t$ is the nominal contract of Section~\ref{subsec:benchmark} with $a\in(0,1]$; deposits pay $i_t$ and cannot be negative; the lender is competitive and risk neutral and its realised real profit accrues outside the dynasties; the rental stock is segmented, with $S(p) = \bar Hp^{\varepsilon}$, $\varepsilon\in[0,\infty)$, and $S_r(\rho) = \bar H_r\rho^{\varepsilon_r}$, $\varepsilon_r\in(0,\infty]$, per cohort. | |
| 19 | +\end{assumption} | |
| 20 | + | |
| 21 | +\begin{assumption}[Feasibility and no default] | |
| 22 | +\label{ass:feas} | |
| 23 | +For all prices considered, $c^O_2>0$, $c^R_2>0$ and $\underline y_3 + \min\{e_3p,0\}>0$. | |
| 24 | +\end{assumption} | |
| 25 | + | |
| 26 | +\begin{assumption}[Liquidity-constrained at the margin: hand-to-mouth] | |
| 27 | +\label{ass:htm} | |
| 28 | +On the wealth interval $\mathcal B = (dp-y_1,\bar b]$ relevant for the tenure decision, the strict Euler inequalities | |
| 29 | +\begin{equation} | |
| 30 | +\frac{1}{c^o_1(b)} > \beta R\,\frac{1}{c^o_2}, \qquad | |
| 31 | +\frac{1}{c^o_2} > \beta R(1+\nu)\,\mathbb E\Big[\frac{1}{W^o_3(y_3)}\Big], \qquad o\in\{O,R\},\ b\in\mathcal B, | |
| 32 | +\label{eq:euler} | |
| 33 | +\end{equation} | |
| 34 | +hold for both tenures at the corner allocation $n_1 = n_2 = 0$ (and $B = (1-\theta)p$ for owners). | |
| 35 | +\end{assumption} | |
| 36 | + | |
| 37 | +\begin{assumption}[Interior tenure margin] | |
| 38 | +\label{ass:interior} | |
| 39 | +$\Delta>0$ and $dp>\rho$, where | |
| 40 | +\begin{equation} | |
| 41 | +\Delta \equiv \chi + \beta\big[\ln c^O_2 - \ln c^R_2\big] + \beta^2(1+\nu)\,\mathbb E\big[\ln W^O_3(y_3) - \ln W^R_3(y_3)\big], \qquad \Lambda\equiv e^{-\Delta}\in(0,1). | |
| 42 | +\label{eq:Delta} | |
| 43 | +\end{equation} | |
| 44 | +\end{assumption} | |
| 45 | + | |
| 46 | +\begin{assumption}[Static expectations on impact] | |
| 47 | +\label{ass:static} | |
| 48 | +In the period of an unanticipated surprise, stage-1 households evaluate their stage-2 rent and stage-3 resale price at the pre-shock stationary values $(\rho^*,p^*)$ while transacting at the current prices. | |
| 49 | +\end{assumption} | |
| 50 | + | |
| 51 | +\begin{assumption}[Nominal positions at the shock date] | |
| 52 | +\label{ass:nominal} | |
| 53 | +At the date of a surprise the only nominal claims held by households are mortgage balances: stage-3 parents and stage-1 entrants hold no deposits. | |
| 54 | +\end{assumption} | |
| 55 | + | |
| 56 | +Assumption~\ref{ass:htm} is verified, not imposed, in the numerical benchmark: the Euler slacks are computed on the support of the stationary transfer distribution and a general optimiser returns the corner (\texttt{tests/test\_toy\_model.py}). Assumption~\ref{ass:htm} implies Assumption~\ref{ass:nominal} on the hand-to-mouth region, but the two are stated separately because the sign of the inflation result depends on the second and not on the first (Corollary~\ref{cor:creditor}). $\Delta$ collects everything that happens after the purchase, the stage-2 cash-flow comparison and the stage-3 equity; the tenure decision compares the stage-1 cash sacrifice $dp-\rho$ with $\Delta$. | |
| 57 | + | |
| 58 | +\subsection{The ownership-entry threshold} | |
| 59 | +\label{subsec:threshold} | |
| 60 | + | |
| 61 | +\begin{proposition}[Ownership threshold] | |
| 62 | +\label{prop:threshold} | |
| 63 | +Under Assumptions~\ref{ass:pref}--\ref{ass:interior} there is a unique $\hat b\in\mathcal B$ such that a household with transfer $b\in\mathcal B$ owns if and only if $b\ge\hat b$, where | |
| 64 | +\begin{equation} | |
| 65 | +\hat b = \frac{d\,p - \Lambda\,\rho}{1-\Lambda} - y_1, \qquad | |
| 66 | +\hat b - (dp-y_1) = \frac{\Lambda(dp-\rho)}{1-\Lambda} > 0, \qquad | |
| 67 | +c^O_1(\hat b) = \frac{\Lambda(dp-\rho)}{1-\Lambda}. | |
| 68 | +\label{eq:bhat} | |
| 69 | +\end{equation} | |
| 70 | +If instead $dp\le\rho$, every household with $b>dp-y_1$ owns and the threshold is the feasibility bound $dp-y_1$. | |
| 71 | +\end{proposition} | |
| 72 | + | |
| 73 | +The threshold is the sum of a feasibility component, the down payment and costs net of stage-1 income, and a preference--liquidity component equal to the stage-1 consumption of the marginal buyer. The latter depends on the rent, the costs, the amortisation schedule, expected inflation and the bequest motive through $\Lambda$. The proposition is proved on the hand-to-mouth region; for households rich enough to save, the envelope derivative of each value is still $1/c^o_1$, so $V^O-V^R$ increases wherever the would-be owner consumes less than the would-be renter, but single crossing at every wealth level is not proved (Remark~\ref{rem:gap_single} in Appendix~\ref{app:proofs}). The quantitative economy finds exactly one sign change of the buy-minus-rent value along cash for every age and productivity state (Section~\ref{sec:baseline}). | |
| 74 | + | |
| 75 | +\begin{proposition}[Comparative statics of the threshold] | |
| 76 | +\label{prop:cs} | |
| 77 | +Under Assumptions~\ref{ass:pref}--\ref{ass:interior}, with the rent held fixed, the resale price moving with $p$ (stationary comparative statics), and the loan-to-value constraint binding, | |
| 78 | +\begin{gather*} | |
| 79 | +\frac{\partial\hat b}{\partial p}>0,\qquad | |
| 80 | +\frac{\partial\hat b}{\partial\rho}<0,\qquad | |
| 81 | +\frac{\partial\hat b}{\partial\theta}>0,\qquad | |
| 82 | +\frac{\partial\hat b}{\partial\kappa_b}>0,\qquad | |
| 83 | +\frac{\partial\hat b}{\partial\kappa_s}>0,\\ | |
| 84 | +\frac{\partial\hat b}{\partial m}>0,\qquad | |
| 85 | +\frac{\partial\hat b}{\partial\chi}<0,\qquad | |
| 86 | +\frac{\partial\hat b}{\partial y_1}=-1,\qquad | |
| 87 | +\frac{\partial\hat b}{\partial\bar\pi}>0 . | |
| 88 | +\end{gather*} | |
| 89 | +The signs of $\partial\hat b/\partial\theta$, $\partial\hat b/\partial\kappa_s$, $\partial\hat b/\partial m$ and $\partial\hat b/\partial\bar\pi$ use the fact that, in the benchmark, a renter never buys later, so that $V^R$ does not depend on these parameters. | |
| 90 | +\end{proposition} | |
| 91 | + | |
| 92 | +All signs are liquidity effects. A higher down payment or a more front-loaded real payment schedule is present-value neutral under \eqref{eq:fisher} but moves resources from the constrained young to the unconstrained old; $r\ge 0$ guarantees that the price effect on the outlay dominates the price effect on future equity. | |
| 93 | + | |
| 94 | +\begin{remark}[Scope of Proposition~\ref{prop:cs}: the option to buy later] | |
| 95 | +\label{rem:cs_scope} | |
| 96 | +The sign of $\partial\hat b/\partial\theta$ is proved under the benchmark's structure in which the tenure decision is made once: a renter at stage~1 rents for life, so the value of renting is independent of the mortgage parameters. In the quantitative economy a renter can buy at any later age and the renter's continuation value therefore rises when $\theta$ falls, through the option of buying later with a smaller down payment. The entry threshold at a given age is then the crossing of two functions that both move with $\theta$, and its monotonicity is not implied by the proposition. Section~\ref{sec:baseline} finds the threshold at the median productivity state to be non-monotone in $\theta$ at fixed prices, within the grid sensitivity of that statistic, and finds it unchanged at the lowest state because the payment-to-income cap, not the loan-to-value cap, binds there; it rises with $\theta$ from $0.20$ to $0.40$ as the proposition predicts. The proposition should therefore be read as a statement about the once-and-for-all tenure decision of a liquidity-constrained household whose loan-to-value constraint binds. | |
| 97 | +\end{remark} | |
| 98 | + | |
| 99 | +\begin{proposition}[Anticipated inflation: tilt versus neutrality] | |
| 100 | +\label{prop:tilt} | |
| 101 | +(i) Under Assumptions~\ref{ass:pref}--\ref{ass:interior}, an increase in $\bar\pi$ raises the real stage-2 payment and lowers the real stage-3 payment of a nominal mortgage, leaves their present value unchanged, and raises $\hat b$. (ii) For a household whose optimal plan has $n_1>0$ and $n_2>0$, a small change in $\bar\pi$ or in $\theta$ leaves its value, consumption and transfer unchanged. | |
| 102 | +\end{proposition} | |
| 103 | + | |
| 104 | +\subsection{Leverage with transaction costs} | |
| 105 | +\label{subsec:leverage} | |
| 106 | + | |
| 107 | +\begin{proposition}[Leverage multiplier, break-even growth, underwater threshold] | |
| 108 | +\label{prop:leverage} | |
| 109 | +Let an owner buy at $p$, hold a mortgage $(1-\theta)p$ and resell at $(1+g)p$ at stage~3. Under Assumption~\ref{ass:prices}: | |
| 110 | +\begin{enumerate} | |
| 111 | +\item End-of-life wealth responds to the resale price with $(1-\kappa_s)$ per unit of price, i.e.\ $(1-\kappa_s)/d$ per unit of cash invested at purchase; this multiplier is decreasing in $\theta$, $\kappa_b$, $\kappa_s$ and $m$. | |
| 112 | +\item Compared with a renter who follows the same consumption path and invests the cash-flow differences at $R$, the owner has higher stage-3 wealth if and only if $g\ge g^*$, with | |
| 113 | +\begin{equation} | |
| 114 | +1+g^* = \frac{R^2(1+\kappa_b+m) + mR - (\rho/p)(R+R^2)}{1-\kappa_s}, | |
| 115 | +\label{eq:gstar} | |
| 116 | +\end{equation} | |
| 117 | +increasing in $\kappa_b$, $\kappa_s$ and $m$, decreasing in $\rho/p$, and independent of $\theta$. | |
| 118 | +\item A price decline of $x$ leaves the owner with no housing equity if and only if $x\ge x^*\equiv 1-\delta_3/(1-\kappa_s)$; a renter's wealth is unaffected. | |
| 119 | +\end{enumerate} | |
| 120 | +\end{proposition} | |
| 121 | + | |
| 122 | +Leverage scales gains and losses but, with borrowing at the deposit rate, does not change the break-even: the brief's ``leverage amplification'' is a statement about the multiplier and the underwater threshold, not about the mean return. In the stationary state the owner--renter end-of-life wealth gap among equal-income hand-to-mouth households is exactly $e_3p$ and the transfer gap is $\nu e_3p/(1+\nu)$ (Proposition~\ref{prop:persist}). | |
| 123 | + | |
| 124 | +\subsection{Dynasties and persistence} | |
| 125 | +\label{subsec:persistence} | |
| 126 | + | |
| 127 | +\begin{proposition}[Tenure transitions, persistence and the equal-income wealth gap] | |
| 128 | +\label{prop:persist} | |
| 129 | +Under Assumptions~\ref{ass:pref}--\ref{ass:interior}, with stationary prices and the stationary transfer support contained in $\mathcal B$, let $\hat y_R\equiv(1+\nu)\hat b/\nu$ and $\hat y_O\equiv\hat y_R - e_3p$. Then: | |
| 130 | +\begin{enumerate} | |
| 131 | +\item The child of a renter owns iff $y_3\ge\hat y_R$ and the child of an owner owns iff $y_3\ge\hat y_O$; the tenure of a dynasty is a Markov chain with $q_{RO} = 1-G(\hat y_R)$ and $q_{OO} = 1-G(\hat y_O)$. | |
| 132 | +\item The second eigenvalue of the chain, | |
| 133 | +\begin{equation} | |
| 134 | +\lambda = q_{OO}-q_{RO} = G(\hat y_R) - G(\hat y_R - e_3p), | |
| 135 | +\label{eq:lambda} | |
| 136 | +\end{equation} | |
| 137 | +equals the parent--child correlation of the ownership indicator and the per-generation persistence of any difference in ownership rates between groups of dynasties; $\lambda>0$ iff $e_3>0$ and $g>0$ somewhere on $(\hat y_O,\hat y_R)$; for fixed $\hat b$, $\lambda$ is non-decreasing in $e_3p$. | |
| 138 | +\item For $\lambda<1$ the chain has the unique stationary owner share $\mu^* = q_{RO}/(1-\lambda)$ and converges to it geometrically at rate $\lambda$. | |
| 139 | +\item Among dynasties with identical income processes, the expected transfer received by an owner's child exceeds that of a renter's child by $\nu e_3p/(1+\nu)$, and the end-of-life wealth gap between an owner and a renter with the same $y_3$ is $e_3p$. | |
| 140 | +\end{enumerate} | |
| 141 | +\end{proposition} | |
| 142 | + | |
| 143 | +Persistence is the probability mass of income draws for which parental housing equity is pivotal for the child's tenure. It is zero if the house is not net wealth at the end of life, and it is a half-life, $\ln 2/(-\ln\lambda)$, not a trap: without income risk the two tenure classes would be absorbing and a continuum of history-dependent steady states would exist, but any $G$ with positive density between the cutoffs makes the chain ergodic (Remark~\ref{rem:norisk} in Appendix~\ref{app:proofs}). The paper therefore does not claim path dependence or multiplicity as a generic result. | |
| 144 | + | |
| 145 | +\subsection{Stationary equilibrium} | |
| 146 | +\label{subsec:se} | |
| 147 | + | |
| 148 | +\begin{definition}[Stationary equilibrium of the benchmark] | |
| 149 | +\label{def:se} | |
| 150 | +A stationary equilibrium is $(p^*,\rho^*,\mu^*)$ with $\hat b^* = \hat b(p^*,\rho^*)$ such that $\mu^* = q_{RO}/(1-\lambda)$ evaluated at $(p^*,\rho^*)$, $S(p^*) = \mu^*$ and $S_r(\rho^*) = 1-\mu^*$ per cohort, the lender earns zero expected real profit, and the distribution of transfers is the stationary distribution of the dynasty chain. | |
| 151 | +\end{definition} | |
| 152 | + | |
| 153 | +Write $f(\mu)\equiv\mu^*\big(S^{-1}(\mu),S_r^{-1}(1-\mu)\big)$ for the owner share generated by the dynasty chain when both markets clear at a cohort owner share $\mu$, with $\mu^* = 0$ when $\Delta\le 0$ and $\mu^* = 1$ when $\hat b<0$. | |
| 154 | + | |
| 155 | +\begin{proposition}[Existence; uniqueness under a dominance condition] | |
| 156 | +\label{prop:exist} | |
| 157 | +Let Assumptions~\ref{ass:pref}--\ref{ass:feas} hold for all $(p,\rho)$ with $p\in[0,S^{-1}(1)]$, $\rho\in[0,S_r^{-1}(1)]$, with $\varepsilon>0$ and $\varepsilon_r<\infty$. Then a stationary equilibrium exists. If, in addition, Assumptions~\ref{ass:htm}--\ref{ass:interior} hold along the clearing curve and | |
| 158 | +\begin{equation} | |
| 159 | +\text{(C)}\qquad \frac{1+\nu}{\nu}\,\frac{\partial\hat b}{\partial p}\;\ge\;e_3\qquad\text{for all }\mu\in(0,1), | |
| 160 | +\label{eq:C} | |
| 161 | +\end{equation} | |
| 162 | +then $f$ is non-increasing, strictly decreasing wherever $g(\hat y_R)>0$, and the stationary equilibrium is unique. | |
| 163 | +\end{proposition} | |
| 164 | + | |
| 165 | +Condition (C) says that the entry-threshold channel, a higher price raises $\hat b$ and excludes renter-dynasty children, dominates the ladder channel, a higher price raises the equity $e_3p$ that owner-parents bequeath and includes owner-dynasty children. If (C) fails, multiple stationary equilibria of the \citet{stein1995prices} type are possible; they are not established here, and Section~\ref{sec:supply} searches for them numerically and finds none. | |
| 166 | + | |
| 167 | +\begin{corollary}[Fixed owner-occupied stock] | |
| 168 | +\label{cor:fixed} | |
| 169 | +If $\varepsilon = 0$, every stationary equilibrium has $\mu^* = \bar H$, and under (C) the price $p^*$ is the unique solution of $\mu^*(p,S_r^{-1}(1-\bar H)) = \bar H$. A parameter change that raises $\mu^*$ at given prices (a lower $\theta$, lower transaction costs, a higher $\chi$ or $y_1$) raises $p^*$ and leaves the ownership rate unchanged; it changes the composition of owners and, through $e_3p^*$ and $\hat b^*$, the persistence $\lambda$. | |
| 170 | +\end{corollary} | |
| 171 | + | |
| 172 | +\begin{proposition}[Price neutrality without segmentation, under unit demand] | |
| 173 | +\label{prop:noseg} | |
| 174 | +Suppose, instead of the segmented stocks of Assumption~\ref{ass:prices}, that owner-occupied and rental units are the same stock, converted costlessly by competitive risk-neutral landlords with maintenance $m$ and operating wedge $\phi\ge 0$, and that Assumption~\ref{ass:pref} (unit demand) holds. Then in any stationary equilibrium $p^*$ solves $S(p^*) = 1$ per cohort and $\rho^* = p^*(1+m+\phi-1/R)$: the price and the rent are independent of $\theta$, $\kappa_b$, $\kappa_s$, $\chi$, $\nu$, $G$, the distribution of transfers and of inflation surprises; tenure shares, $\hat b^*$ and $\lambda$ adjust to these prices without feeding back on them. | |
| 175 | +\end{proposition} | |
| 176 | + | |
| 177 | +\begin{remark}[Scope of Proposition~\ref{prop:noseg}: the intensive margin] | |
| 178 | +\label{rem:noseg_scope} | |
| 179 | +The neutrality is a consequence of unit demand: every household occupies one unit whatever its tenure, so credit, transfers and inflation can only reshuffle who owns, not how many units are demanded. With an intensive margin, house sizes and rented services that respond to wealth and prices, total housing demand depends on the wealth distribution, and a shock that lowers household wealth lowers the single-stock price. Section~\ref{sec:inflation} finds that the price response to a surprise in the unsegmented quantitative economy ($\PImpactfrmlevelnoseg\%$) is almost as large as in the segmented one ($\PImpactfrmlevel\%$): in the quantitative model the price feedback is mostly a wealth effect on total housing demand, not the tenure-composition effect that the benchmark isolates. The \citet{kaplan2020housing} neutrality is thus recovered only in its unit-demand form. | |
| 180 | +\end{remark} | |
| 181 | + | |
| 182 | +\subsection{Surprise inflation} | |
| 183 | +\label{subsec:surprise} | |
| 184 | + | |
| 185 | +Two experiments are considered from a stationary equilibrium. (E1) A \emph{level} surprise at $t+1$: $1+\pi_{t+1} = (1+\bar\pi)(1+\pi^u)$ and expectations for $t+2$ onward unchanged. (E2) A \emph{persistent} surprise: $\pi_{t+1} = \pi_{t+2} = (1+\bar\pi)(1+\pi^u)-1$, learned at $t+1$, so that ARMs reset to the higher nominal rate while FRMs do not, with an optional real-rate response $r' = r+\phi_r\pi^u$ for $(t+1,t+2)$. | |
| 186 | + | |
| 187 | +\begin{proposition}[Direct revaluation and its incidence] | |
| 188 | +\label{prop:accounting} | |
| 189 | +Holding real prices fixed: | |
| 190 | +\begin{enumerate} | |
| 191 | +\item For the cohort aged~2 at $t+1$, the real stage-2 payment is $\delta_2p/(1+\pi^u)$ for every contract. The real stage-3 payment is $\delta_3p/(1+\pi^u)$ for every contract under (E1); under (E2) it is $\delta_3p/(1+\pi^u)^2$ for an FRM and $\delta_3p\,(R'/R)/(1+\pi^u)$ for an ARM, $R' = 1+r'$. The FRM borrower gains at least as much as the ARM borrower, strictly under (E2), and the ARM stage-3 payment rises in real terms iff $\phi_r>R$. | |
| 192 | +\item For the cohort aged~3 at $t+1$, the real stage-3 payment is $\delta_3p/(1+\pi^u)$ for every contract. | |
| 193 | +\item A household holding nominal deposits $n$ loses $nR\pi^u/(1+\pi^u)$ in real terms; a household whose nominal wage is indexed with elasticity $\eta_w$ loses $y[1-(1+\pi^u)^{-(1-\eta_w)}]$ of real income in the surprise period; a sitting tenant whose rent is indexed with elasticity $\eta_\rho$ gains $\rho[1-(1+\pi^u)^{-(1-\eta_\rho)}]$ at the landlord's expense. | |
| 194 | +\item The sum over households of real gains on nominal positions equals the lender's real loss: the revaluation is zero-sum. With $\Gamma_3$ the real stage-3 gain of the cohort aged~2 ($\Gamma_3 = \delta_3p\,\pi^u/(1+\pi^u)$ under (E1) for both contracts, $\delta_3p[1-(1+\pi^u)^{-2}]$ for an FRM and $\delta_3p[1-(R'/R)/(1+\pi^u)]$ for an ARM under (E2)), an owner of that cohort with deposits $n$ gains on net over its remaining life iff | |
| 195 | +\begin{equation} | |
| 196 | +\delta_2p\,\frac{\pi^u}{1+\pi^u} + \Gamma_3 \;>\; y_2\Big[1-(1+\pi^u)^{-(1-\eta_w)}\Big] + nR\,\frac{\pi^u}{1+\pi^u}. | |
| 197 | +\label{eq:gaincond} | |
| 198 | +\end{equation} | |
| 199 | +\end{enumerate} | |
| 200 | +\end{proposition} | |
| 201 | + | |
| 202 | +Four classes of owners lose from a surprise: ARM borrowers when the real rate rises by more than $R$ per unit of surprise, owners with imperfectly indexed wages when the real-income loss exceeds the debt erosion, owners holding nominal deposits larger than their mortgage, and, in general equilibrium, new buyers in inelastic markets (Proposition~\ref{prop:impact}, item~4). The claim that inflation enriches owners is false in general; condition \eqref{eq:gaincond} is what replaces it. | |
| 203 | + | |
| 204 | +\begin{proposition}[General-equilibrium impact of a level surprise under pure debtor positions] | |
| 205 | +\label{prop:impact} | |
| 206 | +Let the economy be in a stationary equilibrium satisfying Assumptions~\ref{ass:pref}--\ref{ass:interior} and \ref{ass:nominal} with $g(\hat y_O)>0$, and let a level surprise $\pi^u>0$ occur at $t+1$ (any contract). Under Assumption~\ref{ass:static}, with stage-2 owners not transacting, owner-market clearing $2S(p_{t+1}) = \mu^*+\mu_{t+1}$, rental clearing $2S_r(\rho_{t+1}) = 2-\mu^*-\mu_{t+1}$, and the static-expectations dominance condition | |
| 207 | +\begin{equation} | |
| 208 | +\text{(C$'$)}\qquad \frac{1+\nu}{\nu}\,\frac{\partial\hat b}{\partial p}\Big|_{p''=p^*}\;\ge\;1-\kappa_s, | |
| 209 | +\label{eq:Cprime} | |
| 210 | +\end{equation} | |
| 211 | +the impact responses satisfy: | |
| 212 | +\begin{enumerate} | |
| 213 | +\item $dp_{t+1}/d\pi^u\ge 0$ and $d\rho_{t+1}/d\pi^u\le 0$, both strict if $\varepsilon<\infty$ (respectively $\varepsilon_r<\infty$); if $\varepsilon = 0$ the price response is strictly positive and $\mu_{t+1} = \mu^*$. | |
| 214 | +\item $d\mu_{t+1}/d\pi^u\ge 0$, strictly if $\varepsilon>0$. | |
| 215 | +\item $d\hat b_{t+1}/d\pi^u\ge 0$; $q_{RO}$ falls, $q_{OO}$ rises, and $\lambda$ rises. | |
| 216 | +\item Stage-3 owner-sellers gain $\delta_3p^*\pi^u/(1+\pi^u) + (1-\kappa_s)dp_{t+1}\ge 0$; the lender loses the nominal erosion; the stage-1 cash of an inframarginal owner-dynasty child changes by $\frac{\nu}{1+\nu}\big[\delta_3p^*\frac{\pi^u}{1+\pi^u}+(1-\kappa_s)dp_{t+1}\big] - d\,dp_{t+1}$, which is negative when $dp_{t+1}$ is large relative to the erosion and $\nu/(1+\nu)<d/(1-\kappa_s)$; renter-dynasty children who rent in both situations gain iff $\rho_{t+1}<\rho^*$. | |
| 217 | +\item Without segmentation (Proposition~\ref{prop:noseg}) items 1--3 are zero and only Proposition~\ref{prop:accounting} applies. | |
| 218 | +\end{enumerate} | |
| 219 | +\end{proposition} | |
| 220 | + | |
| 221 | +The mechanism runs entirely through the parents whose debt is eroded: their larger transfers let more owner-dynasty children qualify, the price rises and the rent falls, the threshold rises, and renter-dynasty children at the margin are displaced. Assumption~\ref{ass:nominal} is what makes the parents' revaluation a gain. The next result drops it. | |
| 222 | + | |
| 223 | +\begin{corollary}[Net nominal creditors reverse the sign] | |
| 224 | +\label{cor:creditor} | |
| 225 | +Replace Assumption~\ref{ass:nominal} by predetermined nominal deposits $n^O\ge 0$ and $n^R\ge 0$ held at the shock date by owner-parents and renter-parents of the cohort aged~3 at $t+1$, chosen before the surprise, and keep the other assumptions of Proposition~\ref{prop:impact}. Define | |
| 226 | +\begin{equation} | |
| 227 | +\Xi \;\equiv\; \mu^*\,g(\hat y_O)\,\big(\delta_3p^* - Rn^O\big) \;-\; (1-\mu^*)\,g(\hat y_R)\,Rn^R . | |
| 228 | +\label{eq:Xi} | |
| 229 | +\end{equation} | |
| 230 | +Then $dp_{t+1}/d\pi^u$, $d\mu_{t+1}/d\pi^u$ and $-d\rho_{t+1}/d\pi^u$ have the sign of $\Xi$, with the same strictness conditions as in Proposition~\ref{prop:impact}; Proposition~\ref{prop:impact} is the case $n^O = n^R = 0$. If $\Xi<0$, the real house price and the stage-1 ownership rate fall and the rent rises on impact, and $\hat b$ falls; the change in $q_{RO}$ is the sum of a price term, positive when $\Xi<0$, and a direct deposit-loss term $-g(\hat y_R)Rn^R/(1+\pi^u)^2\le 0$, so it is unsigned unless $n^R = 0$, in which case $q_{RO}$ rises and $\lambda$ falls. | |
| 231 | +\end{corollary} | |
| 232 | + | |
| 233 | +\begin{remark}[Scope of Proposition~\ref{prop:impact}: the sign is a statement about net nominal positions] | |
| 234 | +\label{rem:impact_scope} | |
| 235 | +$\Xi$ is a density-weighted net nominal debtor position of the parents whose children are at the tenure margin. In the quantitative economy households are net nominal creditors in aggregate (real deposits \TotalDeposits{} against mortgage debt \TotalDebt{} per unit age mass), so the premise of Proposition~\ref{prop:impact} fails and Section~\ref{sec:inflation} finds, as the corollary indicates, that the real house price \emph{falls} on impact. Two further differences from the corollary are also traceable to benchmark reductions. The rent falls rather than rises, because with an intensive margin the wealth loss of nominal creditors lowers the total demand for housing services, and the two markets are no longer mirror images of a fixed unit demand. And the entry threshold rises rather than falls, because would-be buyers are themselves nominal creditors whose saving for the down payment is eroded, a channel absent from a benchmark in which entrants hold nothing before the transfer arrives. The proposition also covers only the first period under per-period supply and static expectations; the perfect-foresight transition with durable stocks, the second-round effects through the cohort aged~2, which are contract-dependent, and the welfare of marginal households are numerical objects. | |
| 236 | +\end{remark} | |
| 237 | + | |
| 238 | +\subsection{Equilibrium concepts for the quantitative economy} | |
| 239 | +\label{subsec:quant_eq} | |
| 240 | + | |
| 241 | +A \emph{stationary recursive competitive equilibrium} with constant inflation $\bar\pi$ and exogenous $r$ consists of value functions and policies solving \eqref{eq:bellman} at prices $(p^*,p^{r*},\rho^*)$ with $i^* = R(1+\bar\pi)-1$, stocks $(H^*,H^{r*})$ and a distribution $\Phi^*$ over states such that: the owner-occupied market clears, $\int h\,\mathbf 1\{o = O\}\,d\Phi^* = H^* = \iota(p^*)^{\varepsilon}/\delta_H$; the rental market clears, $\int s\,\mathbf 1\{o = R\}\,d\Phi^* = H^{r*}$ with \eqref{eq:rent} and $H^{r*} = \iota_r(p^{r*})^{\varepsilon_r}/\delta_H$ on the segmented margin; the distribution of gifts received at age~1 equals the distribution of gifts chosen at $j_g$, and likewise for estates at $j_e$, with the estate lottery used by households equal to the distribution of estates in $\Phi^*$; $\Phi^*$ is invariant under the policies, the income chain and the dynasty link; and the lender earns zero expected real profit. A \emph{transition equilibrium} after an unanticipated shock at $t = 0$ is a path of prices, stocks and distributions such that households optimise with perfect foresight of the post-shock path, markets clear every period, pre-shock FRM cohorts keep their origination rate, and the economy is within tolerance of the stationary equilibrium at a terminal date $T$. Partial-equilibrium counterparts, which re-solve the household block and the distribution at the stationary prices, are reported separately and are never labelled general equilibrium. | |
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paper/sections/inflation.tex
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| 1 | +% ============================================================================ | |
| 2 | +\section{Surprise inflation, mortgage contracts and a real-rate shock} | |
| 3 | +\label{sec:inflation} | |
| 4 | +% ============================================================================ | |
| 5 | + | |
| 6 | +\subsection{Design} | |
| 7 | +\label{subsec:infl_design} | |
| 8 | + | |
| 9 | +The economy starts in the baseline stationary equilibrium and at $t = 0$ the price level turns out 10\% above expectation over the shock period (a level surprise, about two percentage points per year for five years), or 10\% above in each of two consecutive periods with the second known at $t = 0$ (a persistent surprise); inflation then returns to target and expectations are correct thereafter. The transition is solved with perfect foresight of the post-shock price path over 70 years, durable stocks, and the three contracts: a fixed-rate mortgage whose nominal payment is fixed for the remaining term, the five-year term whose rate resets at the next period boundary, and a stylised adjustable-rate mortgage repriced within the period so that its balance is not eroded. Each transition is reported next to its partial-equilibrium counterpart, the same shock with prices fixed at the stationary values, and is labelled with its maximum market-clearing residual: the fixed-rate level surprise stalled at $\FrmLevelMaxRes$ after a Jacobian refresh and is labelled as not converged, while the term and adjustable level surprises and the three persistent surprises converged to $\TermLevelMaxRes$--$\FrmPersMaxRes$. For a level surprise the fixed-rate and five-year-term economies are identical at the five-year frequency, since both erode the balance once and both face the same post-shock rate, and every impact statistic of the two runs agrees to two decimals ($p_0 = \FrmLevelPzero$ against $\TermLevelPzero\%$); the pair therefore bounds the solver's price noise at about 0.02 percentage points. The price paths are back within 0.1\% of the stationary state after \FrmLevelPriceBackYears{} years and the residual at the terminal date is $\FrmLevelResT$. Zero-sum revaluation holds to machine precision. | |
| 10 | + | |
| 11 | +\input{tables/tab06_inflation} | |
| 12 | + | |
| 13 | +\subsection{Impact: the real house price falls} | |
| 14 | +\label{subsec:infl_impact} | |
| 15 | + | |
| 16 | +\begin{figure}[htbp] | |
| 17 | +\centering | |
| 18 | +\includegraphics[width=\textwidth]{fig08_inflation_transition.pdf} | |
| 19 | +\caption{Perfect-foresight transitions after a 10\% unanticipated price-level surprise (general equilibrium unless labelled): real house price (a), real rent (b), ownership rate (c) and real mortgage debt (d), in deviation from the stationary state, for the three contracts and the two persistence profiles; the dotted line in (c) is the fixed-rate level surprise with prices fixed. The fixed-rate level run is reported with its residual. Source: \texttt{results/data/inflation/*\_path.csv}.} | |
| 20 | +\label{fig:irf} | |
| 21 | +\end{figure} | |
| 22 | + | |
| 23 | +Table~\ref{tab:inflation} and Figure~\ref{fig:irf} give the central negative result of the paper. A level surprise lowers the real house price on impact by \FrmLevelPzeroAbs\% under the fixed-rate and term contracts and by \ArmLevelPzeroAbs\% under the adjustable contract, and lowers the real rent by \FrmLevelRhoZeroAbs{} and \ArmLevelRhoZeroAbs\%; a persistent surprise lowers the price by \FrmPersPzeroAbs\% (fixed rate), \TermPersPzeroAbs\% (term) and \ArmPersPzeroAbs\% (adjustable). Proposition~\ref{prop:impact} predicts a rise. The reason is the one stated in Corollary~\ref{cor:creditor} and visible in the aggregate balance sheet of Section~\ref{sec:baseline}: households are net nominal creditors. The surprise erodes real mortgage debt by \FrmLevelErosion{} per unit of age mass under the fixed-rate contract (and not at all under the adjustable one) but erodes real deposits by \FrmLevelDepLoss, so the lender gains \FrmLevelLender{} (\ArmLevelLender{} under the adjustable contract) and aggregate household wealth falls; the demand for housing, on both the extensive and the intensive margin, falls with it. The direction is the corollary's; the magnitude of the rent response is not, because the corollary's unit demand makes the two markets mirror images of a fixed number of units, whereas here the wealth loss lowers the demand for rented services as well, so both prices fall. | |
| 24 | + | |
| 25 | +The ownership rate barely moves in general equilibrium, by \FrmLevelDOwnGE{} percentage points on impact for the fixed-rate level surprise against \FrmLevelDOwnPE{} with prices fixed, \ArmLevelDOwnGE{} against \ArmLevelDOwnPE{} for the adjustable contract, and \FrmPersDOwnGE{} against \FrmPersDOwnPE{} for the persistent fixed-rate surprise: the price adjustment absorbs \ArmLevelAbsorbPct--\FrmLevelAbsorbPct\% of the partial-equilibrium ownership response, the composition result of Corollary~\ref{cor:fixed} in a durable-stock economy whose impact supply elasticity is $\varepsilon\delta_H\approx 0.05$. The price paths return with \FrmLevelPriceSignChanges{} sign change for the level surprise and \FrmPersPriceSignChanges{} for the persistent one, all after the impact and all smaller than 0.1\%; the cohort-crowding oscillation found in a myopic simulation of the benchmark does not appear under perfect foresight with durable stocks. | |
| 26 | + | |
| 27 | +The persistent surprise differs from the level surprise through the nominal rate expected for the second period, which jumps from \FrmLevelNominalRateZero{} to \FrmPersNominalRateZero{} per period. Real debt falls on impact by \FrmPersDebtZeroPctAbs\% under the fixed-rate contract, whose nominal payments are fixed through both inflation periods, by \TermPersDebtZeroPctAbs\% under the term contract and by \ArmPersDebtZeroPctAbs\% under the adjustable contract, where the fall is lower origination rather than erosion: the qualifying payment at the higher nominal rate tightens the payment-to-income cap, ownership falls by \FrmPersDOwnPEAbs{} to \ArmPersDOwnPEAbs{} percentage points with prices fixed and by \FrmPersDOwnGEAbs{} to \ArmPersDOwnGEAbs{} in general equilibrium. This is Proposition~\ref{prop:tilt}'s tilt operating along the transition through the nominal constraint, and it is why the persistent surprise lowers the price four times more than the level surprise. | |
| 28 | + | |
| 29 | +\subsection{Redistribution and the contract ordering} | |
| 30 | +\label{subsec:infl_redistribution} | |
| 31 | + | |
| 32 | +\begin{figure}[htbp] | |
| 33 | +\centering | |
| 34 | +\includegraphics[width=\textwidth]{fig09_inflation_redistribution.pdf} | |
| 35 | +\caption{Redistribution in the shock period of the fixed-rate level surprise: revaluation of nominal positions by age and tenure (a), consumption at impact relative to the no-shock path (b), and consumption-equivalent welfare of households alive at the shock by age and tenure (c), general equilibrium. Source: \texttt{results/data/inflation/frm\_level\_impact.csv}, \texttt{frm\_level\_welfare.csv}.} | |
| 36 | +\label{fig:redistribution} | |
| 37 | +\end{figure} | |
| 38 | + | |
| 39 | +Figure~\ref{fig:redistribution} shows who gains and who loses within the fixed-rate level surprise. Owners aged 30--44 gain on their nominal position, up to \FrmLevelOwnerYoungGainMax{} of a five-year income, and raise impact consumption by \FrmLevelOwnerYoungConsMin{} to \FrmLevelOwnerYoungConsMax\%; owners aged 60--74, deposit-rich and debt-free, lose \FrmLevelOwnerOldLossMin{} to \FrmLevelOwnerOldLossMax{} and cut consumption by \FrmLevelOwnerOldConsMinAbs{} to \FrmLevelOwnerOldConsMaxAbs\%; renters lose \FrmLevelNetRAbs{} on average and cut consumption by \FrmLevelConsRAbs\%. In lifetime terms, owners with a mortgage gain \FrmLevelCEVOM\% of consumption (\FrmLevelBetterOMPct\% of them are better off), owners without a mortgage lose \FrmLevelCEVONMAbs\%, renters lose \FrmLevelCEVRAbs\% and all households alive at the shock lose \FrmLevelCEVAllAbs\% on average, against \FrmLevelCEVAllPEAbs\% with prices fixed: the price response barely changes the incidence because prices move little. This is the age and tenure pattern of \citet{doepke2006inflation} and \citet{meh2011inflation}, with the model-specific addition that the losers include the young renters saving for a down payment. | |
| 40 | + | |
| 41 | +The contract ordering of Proposition~\ref{prop:accounting} is confirmed. For a level surprise the fixed-rate and term contracts give identical gains to mortgaged owners (\FrmLevelCEVOM\%) while the adjustable contract turns the gain into a loss (\ArmLevelCEVOM\%, \ArmLevelBetterOMPct\% better off), since no balance is eroded and the deposit loss remains. For a persistent surprise, mortgaged owners gain \FrmPersCEVOM\% under the fixed-rate contract (\FrmPersBetterOMPct\% better off), \TermPersCEVOM\% under the five-year term (\TermPersBetterOMPct\%) and lose \ArmPersCEVOMAbs\% under the adjustable contract, the ordering fixed rate above term above adjustable, with debt-free owners losing about \FrmPersCEVONMAbs\% and renters about \FrmPersCEVRAbs\% in every case. The Canadian renewable term is thus intermediate, as the institutional reading of Section~\ref{sec:literature} suggested, and the gap between it and the long fixed-rate contract opens only when the surprise outlasts the term. | |
| 42 | + | |
| 43 | +\subsection{Wage indexation, rent rigidity and a real-rate response} | |
| 44 | +\label{subsec:infl_variants} | |
| 45 | + | |
| 46 | +Three variants test the conditions of Proposition~\ref{prop:accounting}. With unindexed nominal wages ($\eta_w = 0$; the run stalled at $\UnindexedMaxRes$ and is labelled) the one-period real wage loss of \UnindexedWageAll{} per household dwarfs the nominal revaluation, the price falls by \UnindexedPzeroAbs\% and the rent by \UnindexedRhoZeroAbs\%, consumption falls by \UnindexedConsAllAbs\% on impact, and every group loses, mortgaged owners by \UnindexedCEVOMAbs\% with \UnindexedBetterOMPct\% better off: condition \eqref{eq:gaincond} fails for every indebted owner once the real-income loss exceeds the debt erosion. With rigid rents for sitting tenants ($\eta_\rho = 0$), incumbents receive a one-period transfer of \RigidRentsRentGainR{} from landlords, the market rent for new tenants rises by \RigidRentsRhoZero\% on impact because incumbents stay put, ownership falls by \RigidRentsDOwnPEAbs{} percentage points with prices fixed but only \RigidRentsDOwnGEAbs{} in general equilibrium, and the renters' welfare loss improves only from \FrmLevelCEVR{} to \RigidRentsCEVR\%: the hedge of \citet{pallotti2024who} is present but, at the five-year frequency, small. With a real-rate response to the persistent surprise ($\phi_r = 0.5$, a rise of five percentage points per period in the two surprise periods), the rate response dominates: the price falls by \FrmTaylorPzeroAbs\% (fixed rate) and \TermTaylorPzeroAbs\% (term), mortgaged owners gain \FrmTaylorCEVOM\% under the fixed-rate contract but only \TermTaylorCEVOM\% under the five-year term (\FrmTaylorBetterOMPct{} against \TermTaylorBetterOMPct\% better off), because the renewal at the higher real rate claws back most of the term borrowers' gain, exactly as Proposition~\ref{prop:accounting} states for $\phi_r>0$; debt-free owners and renters are cushioned by the higher deposit return, and the average household gains \FrmTaylorCEVAll\% under the fixed-rate contract and loses \TermTaylorCEVAllAbs\% under the term. The distinction between the Canadian term and the long fixed-rate contract therefore matters precisely when the surprise is persistent and monetary policy reacts. | |
| 47 | + | |
| 48 | +\subsection{Segmentation, supply elasticity and expectations} | |
| 49 | +\label{subsec:infl_structure} | |
| 50 | + | |
| 51 | +Three further variants check the structural assumptions of Proposition~\ref{prop:impact}. Without segmentation ($\sigma = 0$, one stock) the price falls by \NoSegPzeroAbs\% against \FrmLevelPzeroAbs\% with segmented stocks and the rent by \NoSegRhoZeroAbs\%, with ownership moving by \NoSegDOwnGE{} percentage points in general equilibrium against \NoSegDOwnPE{} with prices fixed and identical welfare: the neutrality of Proposition~\ref{prop:noseg} does not carry over, for the reason given in Remark~\ref{rem:noseg_scope}, and segmentation itself adds only about a seventh to the price response. With near-fixed long-run supply ($\varepsilon = \varepsilon_r = 0.05$; the run stalled at $\FixedSupplyMaxRes$ and is labelled) the price falls by \FixedSupplyPzeroAbs\% against \FrmLevelPzeroAbs\%, and ownership moves by \FixedSupplyDOwnGE{} percentage points against \FixedSupplyDOwnPE{} at fixed prices: the long-run elasticity barely matters on impact because the stock is durable and the shock-period supply is essentially fixed in both cases, so the price absorbs the demand shift and the composition result of Corollary~\ref{cor:fixed} is what the numbers show. Under static expectations (Assumption~\ref{ass:static}, with stationary continuation values and prices clearing period by period) the price falls by \StaticPzeroAbs\% against \FrmLevelPzeroAbs\% under perfect foresight, ownership moves by \StaticDOwnGE{} against \FrmLevelDOwnGE{} percentage points, welfare agrees to two decimals and the gift-age transition probability $q_{RO}$ falls from \FrmLevelqROsevenSS{} to \StaticqROsevenZero{} in both: the expectation scheme is immaterial for the size of the effects in this calibration, and the sign of the price response is negative under both. Mobility on impact is as Proposition~\ref{prop:impact} predicts in direction and negligible in size, $q_{RO}$ at the gift age falling to \FrmLevelqROsevenZero{} and $\lambda_7$ rising from \FrmLevelLamSevenSS{} to \FrmLevelLamSevenZero; the age-1 threshold at the median state rises from \FrmLevelThrSS{} to \FrmLevelThrZero{}, by \FrmLevelThrChangePct\%, through the erosion of the savings of would-be buyers rather than through a higher price, the third channel named in Remark~\ref{rem:impact_scope}. | |
| 52 | + | |
| 53 | +\subsection{A real-rate shock and leverage} | |
| 54 | +\label{subsec:rateshock} | |
| 55 | + | |
| 56 | +The general-equilibrium counterpart of a house-price fall is a persistent rise in the real rate: two percentage points per year for twenty years, unanticipated at $t = 0$, from the stationary equilibrium of each of three down-payment regimes, with fixed-rate contracts keeping their origination rate. The runs for $\theta = 0.20$ and $0.40$ converged ($\RateTwentyMaxRes$ and $\RateFortyMaxRes$); the run for $\theta = 0.05$ diverged from a cold start and, after a warm start and a damped t\^atonnement pre-phase, stalled at $\RateFiveMaxRes$, so its numbers are indicative only. The real house price falls on impact by \RateTwentyPzeroAbs\% at $\theta = 0.20$ and \RateFortyPzeroAbs\% at $\theta = 0.40$ (\RateFivePzeroAbs\% in the unconverged run), overshoots by \RateTwentyPriceMaxPos\% when the rate returns to 2\% twenty years later, and the rent falls by \RateTwentyRhoZeroAbs\%. Ownership on impact moves by \RateTwentyDOwnGE{} and \RateFortyDOwnGE{} percentage points in general equilibrium against \RateTwentyDOwnPE{} and \RateFortyDOwnPE{} with prices fixed: the ranking of the partial-equilibrium responses across $\theta$ (\RateFiveDOwnPE, \RateTwentyDOwnPE, \RateFortyDOwnPE{} for 5, 20 and 40\%) is the leverage-exposure ordering of Proposition~\ref{prop:leverage}, and general equilibrium removes it. Owners lose \RateTwentyNWOAbs{} of a five-year income of net worth on impact and cut consumption by \RateTwentyConsOAbs\%; renters' net worth rises slightly and their consumption falls by \RateTwentyConsRAbs\%. Welfare is nevertheless positive on average, \RateTwentyCEVAll\% for households alive at the shock, because net nominal creditors gain from the higher real return, which dominates the housing-wealth loss: mortgaged owners gain \RateTwentyCEVOM\% (\RateTwentyBetterOMPct\% better off, shielded by the fixed rate while their deposits earn more), debt-free owners \RateTwentyCEVONM\% and renters \RateTwentyCEVR\%. The price fall lowers the entry threshold (from \RateFortyThrSS{} to \RateFortyThrZero{} at the median state for $\theta = 0.40$) and raises upward mobility ($q_{RO}$ at 35--39 from \RateTwentyqROthreeSS{} to \RateTwentyqROthreeZero{} at $\theta = 0.20$). Leverage thus amplifies the housing-equity loss, as the accounting says, but with fixed-rate contracts, net creditor positions and no default the channel operates through consumption and entry rather than through foreclosure. | |
modified
paper/sections/introduction.tex
+9 −5
@@ -3,12 +3,16 @@ | ||
| 3 | 3 | \label{sec:introduction} |
| 4 | 4 | % ============================================================================ |
| 5 | 5 | |
| 6 | −The question of \ldots{} has attracted considerable attention in the finance and economics literature. Despite the growing body of evidence, the mechanisms underlying \ldots{} remain poorly understood. This paper contributes to this line of research by \ldots{} | |
| 6 | +Among Canadian families under 35, the median net worth of those who own their home is about ten times that of those who do not, and the gap is of the same order among families aged 55--64 \citep{statcan2024survey}. Differences in labour income between the two groups are far smaller than this. Housing tenure is, in other words, a sorting device for wealth: it converts a modest difference in initial resources, often a parental transfer at the time of first purchase, into a large and durable difference in balance sheets, through the down payment that gates entry, the leverage that scales the return on the house, and the bequest that carries the result into the next generation. This paper asks under what conditions inherited wealth, mortgage leverage, nominal debt, housing supply and down-payment constraints generate persistent owner--renter wealth inequality among households with the same labour-income process, and what an unanticipated inflation changes in that mechanism. | |
| 7 | 7 | |
| 8 | −The motivation for this study is threefold. First, \ldots{} Second, \ldots{} Third, \ldots{} | |
| 8 | +The question is theoretical and the answer is given in two steps. The first is a benchmark economy in which a household lives three stages, buys or rents once, repays a nominal mortgage, sells, and transfers part of the proceeds to a child who enters the model at that moment. Because the benchmark has closed forms, it delivers nine propositions: an ownership-entry threshold that equals the down-payment bound plus the stage-1 consumption of the marginal buyer (Proposition~\ref{prop:threshold}); the signs of its comparative statics, all of them liquidity effects of present-value-neutral parameters (Propositions~\ref{prop:cs}--\ref{prop:tilt}); a leverage multiplier net of transaction costs whose break-even price growth is independent of the loan-to-value ratio (Proposition~\ref{prop:leverage}); a closed-form intergenerational tenure persistence, the second eigenvalue of the dynasty chain, which is positive only if the house is net wealth at the end of life and which is a half-life rather than a trap (Proposition~\ref{prop:persist}); existence of a stationary equilibrium and a dominance condition for its uniqueness, with the corollary that a fixed stock converts every credit or transfer shift into price and composition (Proposition~\ref{prop:exist}, Corollary~\ref{cor:fixed}); price neutrality without segmentation under unit demand (Proposition~\ref{prop:noseg}); the full accounting of a price-level surprise under fixed-rate, renewable-term and adjustable-rate contracts, with the exact condition under which an indebted owner gains (Proposition~\ref{prop:accounting}); and the general-equilibrium impact of a surprise, whose sign is a statement about the net nominal position of the parents whose children are at the tenure margin (Proposition~\ref{prop:impact}, Corollary~\ref{cor:creditor}). | |
| 9 | 9 | |
| 10 | −Our main findings can be summarized as follows. We show that \ldots{} Furthermore, we document that \ldots{} Finally, our robustness checks confirm that \ldots{} | |
| 10 | +The second step is a quantitative overlapping-generations economy that relaxes the benchmark along the dimensions that matter for the question: twelve five-year periods with housing at every age and on an intensive margin, persistent income risk, nominal constant-payment mortgages amortised over 25 years under three contract types with a loan-to-value and a nominal payment-to-income constraint, an inter-vivos gift at the child's entry and an estate at the parent's death, segmented rental and construction sectors with durable stocks, and a lender with a balance sheet. It is solved in stationary general equilibrium and along perfect-foresight transitions after an unanticipated inflation, with the two housing markets clearing to a stated tolerance, and every result at fixed prices is labelled as such. The calibration is illustrative: no moment is fitted, verified Canadian comparison points are reported next to the model's moments, and no verified estimate of the Canadian housing-supply elasticity was found, so that parameter is swept rather than set. | |
| 11 | 11 | |
| 12 | −This paper relates to several strands of the literature. First, it contributes to the literature on \ldots{} \citep{fama1993common}. Second, it builds on the work of \citet{bollerslev1986generalized} by \ldots{} Third, our methodological approach draws on \citet{newey1987simple}. | |
| 12 | +Three findings are the paper's main results and all three are stated with their equilibrium status. First, transfers generate the equal-income gaps the question asks about and are the whole of their intergenerational persistence in this economy. In the baseline, owners hold \GapAgeFour{} years of average income more than renters at 40--44, children of owners are \qOOageThree{} rather than \qROageThree{} likely to own at 35--39, and the rank--rank slope of net worth between parents and children is \RankRankNW{}; switching the bequest motive off lowers the house price from 1 to \BeqOffP{} and sets the tenure persistence and the rank--rank slope to zero. The persistence is a half-life of \HalfLifeGen{} generations, not a trap: the search for multiple stationary equilibria, including under near-fixed supply, finds a monotone excess-demand function with a single root. Second, prices absorb most of what credit constraints would otherwise do to ownership. Lowering the minimum down payment from 20\% to 5\% raises ownership among households under 35 from \OwnershipUnderThirtyFive{} to \ThetaFivePEOwnUthirtyfive{} at fixed prices, but general equilibrium removes \AbsorbThetaFivePct\% of the aggregate ownership effect through the price and the rent; the long-run supply elasticity matters little for this because the demand shift is small and, in the transitions, because the impact elasticity of a durable stock is the product of the long-run elasticity and the depreciation rate. Third, and contrary to the benchmark's Proposition~\ref{prop:impact}, a 10\% unanticipated rise in the price level \emph{lowers} the real house price on impact, by \FrmLevelPzeroAbs\% under fixed-rate contracts and \FrmPersPzeroAbs\% when the surprise persists, because households in this economy are net nominal creditors: their deposits exceed their mortgages, so the surprise lowers aggregate household wealth and with it housing demand. The direct redistribution is as the accounting predicts, owners with a mortgage gain \FrmLevelCEVOM\% of lifetime consumption and renters lose \FrmLevelCEVRAbs\%, and the contract ordering, fixed-rate at least as good as the renewable term and both better than the adjustable rate, holds, with the renewable term penalised precisely when the surprise is persistent and the real rate responds. General equilibrium absorbs \FrmLevelAbsorbPct\% of the partial-equilibrium ownership response to the surprise, and the perfect-foresight transition differs from the static-expectations impact of the benchmark by a tenth of a percentage point. | |
| 13 | 13 | |
| 14 | −The remainder of the paper is organized as follows. Section~\ref{sec:literature} reviews the related literature. Section~\ref{sec:data} describes the data. Section~\ref{sec:methodology} presents the empirical methodology. Section~\ref{sec:results} reports the main results. Section~\ref{sec:robustness} provides robustness checks. Section~\ref{sec:conclusion} concludes. | |
| 14 | +The contributions are stated relative to the closest prior work and without inflation. The entry threshold with a liquidity component is an incremental extension of \citet{ortalomagne2006housing} and \citet{kaplan2020housing} to transfers, a rental sector and nominal debt; its general-equilibrium offset is a numerical statement, and the price feedback is shown to be governed by segmentation and the intensive margin, as \citet{greenwald2025do} would predict. The persistence result rationalises in closed form what \citet{charles2002transition} and \citet{daysal2023intergenerational} document empirically; it is new as theory and is deliberately not a multiplicity result. The inflation results are incremental on the direct revaluation of \citet{doepke2006inflation} and \citet{meh2011inflation} and on the contract comparison of \citet{garriga2017mortgages}, and new in two respects: the sign of the general-equilibrium price response is tied to a density-weighted net nominal position, and the Canadian five-year term is placed between the long fixed-rate and the adjustable contract and shown to lose its advantage when a persistent surprise is met by a real-rate response. The supply-elasticity and contract effects on mobility are comparative statics of a mobility measure, incremental on \citet{kiyotaki2011winners} and \citet{chambers2009accounting}, and small in this calibration. | |
| 15 | + | |
| 16 | +Three results in this paper are negative relative to the hypotheses with which it started, and they are reported as results. Surprise inflation lowers rather than raises the real house price when households are net nominal creditors; price neutrality without segmentation does not carry over to an economy with an intensive housing margin; and the entry threshold is not monotone in the down payment at every productivity state once households can buy later. In each case the responsible benchmark assumption is identified and the proposition is restated under it, rather than the theory being adjusted to fit the simulations or the simulations being discarded. | |
| 17 | + | |
| 18 | +The paper proceeds as follows. Section~\ref{sec:literature} reviews the literature by mechanism. Section~\ref{sec:model} sets out the benchmark and the quantitative economy. Section~\ref{sec:equilibrium} states the propositions and the remarks that scope them. Section~\ref{sec:computation} describes the algorithm, the calibration and the verification. Section~\ref{sec:baseline} reports the baseline, the equal-income paths and the down-payment sweep; Section~\ref{sec:inflation} the inflation and real-rate transitions; Section~\ref{sec:supply} supply, bequests, policy and welfare; Section~\ref{sec:sensitivity} sensitivity, limitations and conclusions. Appendix~\ref{app:proofs} contains the proofs, Appendix~\ref{app:computation} the numerical details and Appendix~\ref{app:tables} the supplementary exhibits. | |
modified
paper/sections/literature.tex
+38 −11
@@ -1,21 +1,48 @@ | ||
| 1 | 1 | % ============================================================================ |
| 2 | −\section{Literature Review} | |
| 2 | +\section{Related literature} | |
| 3 | 3 | \label{sec:literature} |
| 4 | 4 | % ============================================================================ |
| 5 | 5 | |
| 6 | −This section reviews the main theoretical and empirical contributions related to our research question. | |
| 6 | +The literature that bears on the question is large but fragmented along five mechanisms that are rarely combined in one general-equilibrium model: the collateral or down-payment constraint that gates entry into ownership; the leveraged return on housing and the resulting divergence of owner and renter balance sheets; the revaluation of nominal mortgage debt under surprise inflation; the supply side that converts demand shocks into prices rather than quantities; and the intergenerational transmission of wealth through transfers. This section reviews each mechanism, then the Canadian institutional setting, and closes with the gaps that the model is built to address. Only references whose metadata were verified are cited. | |
| 7 | 7 | |
| 8 | −\subsection{Theoretical Framework} | |
| 9 | −\label{subsec:theory} | |
| 8 | +\subsection{Collateral, leverage and the price feedback} | |
| 10 | 9 | |
| 11 | −The theoretical foundations of \ldots{} were established by \citet{cochrane2005asset}. In their framework, \ldots{} | |
| 10 | +The amplification mechanism descends from \citet{kiyotaki1997credit}: when durable assets serve as collateral, a shock that raises their price relaxes borrowing limits, which raises demand and the price again. \citet{iacoviello2005house} transplanted the mechanism into a monetary model with housing collateral and nominal debt, obtaining two results this paper inherits, that collateral constraints amplify demand shocks and that debt deflation transfers wealth from borrowers to lenders. Its limits are equally instructive: two representative households, no tenure choice, no wealth distribution and no inheritance, so that the redistribution it identifies is between two types rather than within a cross-section of households that differ in initial wealth. | |
| 12 | 11 | |
| 13 | −\subsection{Empirical Evidence} | |
| 14 | −\label{subsec:empirical_evidence} | |
| 12 | +The quantitative heterogeneous-household literature supplies the architecture. \citet{favilukis2017macroeconomic} build an overlapping-generations economy with aggregate and idiosyncratic risk, collateral constraints and heterogeneous bequest motives, and show that relaxing financing constraints produces a large price boom through a fall in the housing risk premium. \citet{kaplan2020housing} add a rental market, long-term mortgages and tenure choice and reach the conclusion that disciplines any claim about the collateral--price feedback: in their calibration, shifts in credit conditions move ownership, leverage and foreclosures but not prices, because the rental sector and long-term mortgages absorb the demand shift. \citet{greenwald2025do} resolve the apparent conflict with the credit-driven-boom literature by showing that everything hinges on the segmentation between the owner-occupied and rental stocks. \citet{sommer2013equilibrium} and \citet{sommer2018implications} provide the template for a rental equilibrium with landlords and an endogenous rent--price ratio; the latter is a standing warning that general-equilibrium price responses can reverse partial-equilibrium intuition. For the present paper this literature means that the strength of any price feedback is not a free parameter: it is governed by the segmentation parameter and the supply elasticity, both of which the model makes explicit, and Proposition~\ref{prop:noseg} restates the \citeauthor{kaplan2020housing} neutrality in the model's language while Remark~\ref{rem:noseg_scope} records the condition, unit demand, under which it holds. | |
| 15 | 13 | |
| 16 | −A growing body of empirical literature examines \ldots{} \citet{fama1993common} document that \ldots{} More recently, \citet{petersen2009estimating} show that \ldots{} | |
| 14 | +The empirical side of the collateral channel is well established. \citet{mian2011house} document home-equity borrowing of 25--30 cents per dollar of house-price gain, concentrated among constrained owners; \citet{gete2018mortgage} show that tighter mortgage supply pushes households into renting and raises rents; \citet{cloyne2020monetary} show that the consumption response to interest rates is driven by mortgagors, with outright owners not responding. None of these speak to intergenerational persistence, but together they justify two modelling choices: owners must be allowed to be liquidity-constrained despite high net worth, and the rental market must be an explicit margin rather than an outside option. | |
| 17 | 15 | |
| 18 | −\subsection{Contribution to the Literature} | |
| 19 | −\label{subsec:contribution} | |
| 16 | +The mortgage-contract literature is the second pillar. \citet{campbell2003household} formalise the household's choice between a nominal fixed-rate mortgage, which carries inflation risk, and an adjustable-rate mortgage, which carries income risk. \citet{chambers2009accounting} embed contract choice in an overlapping-generations economy with tenure choice and attribute a large share of the rise in US ownership to mortgage innovation: contract design changes who can own, which is the sense in which the present paper treats the contract. \citet{guren2021mortgage} and \citet{campbell2021structuring} study contract design for macroeconomic stabilisation in equilibrium life-cycle models; the latter prices mortgages off a risk-averse lender, a device that keeps lender wealth in the accounting of inflation redistribution, as the model here does with a simpler risk-neutral lender. | |
| 20 | 17 | |
| 21 | −Our paper departs from the existing literature in several important ways. First, \ldots{} Second, \ldots{} Third, \ldots{} | |
| 18 | +\subsection{Nominal debt and surprise inflation} | |
| 19 | + | |
| 20 | +The Fisher channel \citep{fisher1933debt} is the oldest element of the paper. Its modern quantification is \citet{doepke2006inflation}, who revalue the nominal positions of US households by age and wealth and find that a moderate inflation episode transfers wealth from rich, old bondholders and from foreigners to young middle-class households with fixed-rate mortgages. \citet{adam2016price} replicate the exercise for the euro area and \citet{meh2011inflation} for Canada, finding that the main winners are young middle-class households with mortgage debt while the old, the rich and holders of non-indexed pensions lose. \citet{pallotti2024who} bring the accounting up to the 2021--2023 episode and add a nuance that matters for a model with renters: rigid rents acted as a hedge for lower-consumption households, so the episode worked like an age-dependent tax. | |
| 21 | + | |
| 22 | +These are accounting exercises. They hold prices, rents, interest rates and behaviour fixed and therefore cannot answer whether the direct revaluation survives once rents, house prices and lender wealth respond. Three papers go further. \citet{meh2010aggregate} compute the general-equilibrium response to an unexpected price-level increase in an overlapping-generations economy with nominal positions calibrated for Canada and find that the young, the poor and the government gain and that the policy regime matters; the model has no housing market, so the owner--renter dimension is absent. \citet{auclert2019monetary} provides the sufficient-statistic language of unhedged interest-rate exposure and the Fisher channel but has no tenure structure or intergenerational dynamics. \citet{garriga2017mortgages} is the closest precedent for the inflation results: in a general-equilibrium model with long-term nominal mortgages they show that persistent inflation benefits fixed-rate borrowers over time but hurts adjustable-rate borrowers immediately. Their borrowers and lenders are two representative types; there is no renter priced out when nominal rates rise, no ownership-entry margin and no inheritance. The lesson this paper draws is that the sign of the general-equilibrium effect of a surprise cannot be assumed: \citeauthor{garriga2017mortgages} show that it is negative for adjustable-rate borrowers, \citeauthor{pallotti2024who} that rent rigidity can protect renters, and \citeauthor{kaplan2020housing} that the rental sector can absorb demand. Proposition~\ref{prop:impact} and Corollary~\ref{cor:creditor} therefore state conditions rather than a sign, and Section~\ref{sec:inflation} finds the sign that the brief expected to be positive to be negative. | |
| 23 | + | |
| 24 | +\subsection{Land scarcity, supply elasticity and the rent--price ratio} | |
| 25 | + | |
| 26 | +The case for treating land as a scarce factor is strong. \citet{knoll2017no} show that real house prices in fourteen countries were roughly flat until the mid-twentieth century and then tripled, with most of the rise attributable to land; \citet{davis2007price} show that the land share of US house value rose and drives price volatility; \citet{rognlie2015deciphering} shows that the post-war rise in the net capital share comes from housing; \citet{piketty2014capital} document the rise of wealth--income ratios in which housing is the main component. In the cross-section, \citet{saiz2010geographic} and \citet{glaeser2018economic} establish that geography and regulation determine supply elasticities, and \citet{gyourko2013superstar} show that inelastic supply with rising high-income demand produces persistent price divergence and sorting. | |
| 27 | + | |
| 28 | +Two theoretical papers connect supply scarcity to the owner--renter distribution. \citet{kiyotaki2011winners} build a life-cycle production economy in which land and capital produce real estate and find that when the land share is large, prices react strongly to productivity and world-interest-rate shocks, redistributing between net buyers and net sellers, while financing constraints have limited price effects. \citet{grossmann2021distributional} model land as a fixed factor in a growing economy and show that surging rents redistribute from renters to landowners. \citet{borri2021optimal} analyse optimal taxation when appreciation widens the owner--renter gap, and \citet{favilukis2023affordable} offer the most complete architecture, owners, renters, construction and endogenous wealth inequality, for evaluating affordability policies. What is missing is the interaction of supply elasticity with the other mechanisms: \citeauthor{kiyotaki2011winners} have no bequests and no nominal debt, \citeauthor{grossmann2021distributional} have representative owner and renter types, and neither models the entry threshold. The conjecture that lower supply elasticity raises the price response to any demand shift and therefore both the leveraged return of incumbents and the entry threshold of renters is plausible but, as Section~\ref{sec:supply} shows, quantitatively weak when the demand shift itself is small and when the stock is durable. | |
| 29 | + | |
| 30 | +\subsection{Tenure choice, the down-payment constraint and the property ladder} | |
| 31 | + | |
| 32 | +The microeconomics of tenure choice begins with \citet{henderson1983model}, whose rental externality rationalises an ownership premium, and with \citet{linneman1989impacts}, who show that it is the wealth constraint rather than the income constraint that blocks ownership. \citet{stein1995prices} shows that down-payment effects generate multiple equilibria and price--volume comovement in a static setting. The paper closest to the threshold result is \citet{ortalomagne2006housing}: in a life-cycle model with a property ladder and a down-payment constraint, the ability of young households to finance the down payment on a starter home drives prices, and capital gains on starter homes relax the constraints of existing owners. It has no bequests, no nominal debt, no inflation and no supply side, and the distribution of wealth across households of equal income is not an object of analysis. | |
| 33 | + | |
| 34 | +The quantitative life-cycle literature fills in the behavioural detail: \citet{gervais2002housing}, \citet{iacoviello2013housing}, \citet{halket2014saving}, \citet{diaz2010wealth}, \citet{kaas2021low}, \citet{floetotto2016government} and \citet{halket2015homeownership}. \citet{pazpardo2024homeownership} is especially relevant because it identifies a competing explanation for falling young ownership and wealth, riskier and more unequal earnings for later cohorts; the claim that inheritance generates gaps among households with similar labour income must therefore be isolated from the earnings channel, which is why the earnings process is held fixed across inheritance groups in the baseline and varied separately. \citet{nakajima2020home} close the life cycle: retired owners dissave slowly, so housing is the main asset bequeathed, which ties the tenure block to the bequest block. The quantitative model here departs from that finding in one respect that Section~\ref{sec:baseline} flags: with a warm glow over a liquid estate, old households sell their house before death. | |
| 35 | + | |
| 36 | +\subsection{Incomplete markets, bequests and intergenerational wealth} | |
| 37 | + | |
| 38 | +The incomplete-markets tradition \citep{samuelson1958exact,diamond1965national,aiyagari1994uninsured,huggett1996wealth} provides the equilibrium concept. \citet{denardi2004wealth} is the canonical bequest specification, voluntary bequests as a luxury good, which the model adopts and extends to housing. \citet{benhabib2011distribution} and \citet{gabaix2016dynamics} supply analytical benchmarks for persistence, Pareto tails from capital-income risk and the slow convergence of random-growth processes; they concern the tail of the wealth distribution rather than a tenure-based structure, and the half-life statements of Proposition~\ref{prop:persist} are the model's counterpart. \citet{kuhn2020income} provide the macro-historical motivation: over 1949--2016 the portfolio composition of US households, leveraged housing for the middle class and equity for the top, makes asset-price movements the main driver of wealth-inequality dynamics. | |
| 39 | + | |
| 40 | +The empirical literature on intergenerational wealth has converged on three findings the model must respect. Parental transfers at the time of purchase are a first-order determinant of entry: \citet{engelhardt1998intergenerational} find that about a fifth of US first-time buyers receive transfers, \citet{guiso2002private} that transfers advance first purchase in Italy, \citet{blickle2019borrowing} that a transfer raises the renter-to-owner transition probability through relaxed constraints rather than wealth effects, and \citet{charles2002transition} that parental wealth and transfers explain much of the Black--White ownership gap at equal income. Parental housing wealth specifically transmits to children: \citet{daysal2023intergenerational} estimate on Danish registers that about a quarter of a parental housing-wealth gain in early childhood passes through to the child's adult housing wealth, with earnings and education explaining only a fraction; \citet{benetton2022dynastic} study the US parent--child link through home equity. And transfers are not the whole story: \citet{fagereng2021why} show with adoptees that family environment raises child wealth causally and not mainly through gifts, \citet{nekoei2023how} show that inheritances reduce inequality on receipt but raise it within a decade because wealthy heirs earn higher returns, and \citet{boar2021dynastic} show that parents save to insure their children's income risk. The implication is that the inheritance channel formalised here is real and measurable but is not claimed to be the dominant source of intergenerational wealth persistence; the model isolates one mechanism, transfers interacting with the down-payment constraint and leveraged housing returns, while holding preferences, financial literacy and earnings transmission fixed. The \citeauthor{nekoei2023how} finding that returns on inherited wealth drive the reversal from equalising to disequalising is exactly what the ownership threshold does: it is a discontinuity in the return to the first dollar of transfer. | |
| 41 | + | |
| 42 | +\subsection{The Canadian institutional setting} | |
| 43 | + | |
| 44 | +Canada differs from the United States in ways that matter for every mechanism above. Mortgages are amortised over 25 years but renewed every five years or less at the prevailing rate, so the fixed-rate contract is fixed only for the term: \citet{kartashova2020how} exploit scheduled resets to show asymmetric spending and deleveraging responses; \citet{bounajm2024modelling} document how the Bank of Canada's projection model represents the term structure of mortgage debt; \citet{godbout2025how} estimate that a majority of outstanding mortgages renew in 2025--2026 at higher payments; and \citet{cmhc2026residential} show that traditional five-year fixed contracts were a small minority of new extensions in early 2026. The consequence for the inflation results is that the Canadian contract is intermediate between the US 30-year fixed-rate mortgage of \citeauthor{doepke2006inflation} and the adjustable contract of \citeauthor{garriga2017mortgages}: the inflation gain accrues only for the remaining term and is partly clawed back at renewal unless the surprise is transitory. \citet{allen2019search} document search frictions and rate dispersion in the insured segment, which are noted but not modelled. On the household side, \citet{meh2011inflation} provide nominal-exposure profiles by age and tenure, \citet{hou2010homeownership} the age profile of ownership, \citet{mirdamadi2023parents} the parental-ownership channel and the share of first-time buyers receiving family gifts, and \citet{statcan2024survey} the owner--renter wealth gaps quoted in the introduction. No Canadian analogue of the \citeauthor{saiz2010geographic} elasticity estimates was found among verified sources, so supply elasticity is treated as an illustrative parameter with a range. | |
| 45 | + | |
| 46 | +\subsection{Gaps} | |
| 47 | + | |
| 48 | +Reading the five strands together, four gaps are not filled by any verified paper. No general-equilibrium model combines bequest heterogeneity, a down-payment-gated ownership margin, an explicit rental sector and price feedback, and characterises the entry threshold analytically: \citeauthor{ortalomagne2006housing} have the threshold and the feedback but not bequests or a rental sector, \citeauthor{favilukis2017macroeconomic} have bequest heterogeneity and feedback but no tenure margin, \citeauthor{kaplan2020housing} have everything except inheritance heterogeneity and do not characterise the threshold. Persistence of the ownership gap across generations is documented empirically but not derived; the theoretical persistence results that exist concern the tail of the wealth distribution. Surprise inflation with fixed nominal mortgages has been studied in general equilibrium but never with renters, an entry margin and inheritance, and never with the Canadian renewable term placed between the long fixed-rate and the adjustable contract. And supply elasticity has been linked to owner--renter redistribution and contracts to ownership rates, but neither to intergenerational mobility in tenure or wealth. The model of Section~\ref{sec:model} is built to fill these gaps in a form that can be proved where possible and simulated where not; the verdicts on which results are new and which are incremental are given in the introduction and are not revisited in the conclusion. | |
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paper/sections/methodology.tex
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| 1 | −% ============================================================================ | |
| 2 | −\section{Methodology} | |
| 3 | −\label{sec:methodology} | |
| 4 | −% ============================================================================ | |
| 5 | − | |
| 6 | −\subsection{Empirical Model} | |
| 7 | −\label{subsec:model} | |
| 8 | − | |
| 9 | −To test our hypotheses, we estimate the following baseline specification: | |
| 10 | −% | |
| 11 | −\begin{equation} | |
| 12 | − y_{i,t} = \alpha + \beta_1 X_{i,t} + \beta_2 Z_{i,t} + \gamma_i + \delta_t + \varepsilon_{i,t}, | |
| 13 | − \label{eq:baseline} | |
| 14 | −\end{equation} | |
| 15 | −% | |
| 16 | −where $y_{i,t}$ is the dependent variable for firm~$i$ in period~$t$, $X_{i,t}$ is the main variable of interest, $Z_{i,t}$ is a vector of control variables, $\gamma_i$ represents firm fixed effects, $\delta_t$ captures time fixed effects, and $\varepsilon_{i,t}$ is the error term. | |
| 17 | − | |
| 18 | −Our coefficient of interest is $\beta_1$. Under the null hypothesis that \ldots{}, we expect $\beta_1 = 0$. A positive (negative) estimate of $\beta_1$ would indicate \ldots{} | |
| 19 | − | |
| 20 | −\subsection{Identification Strategy} | |
| 21 | −\label{subsec:identification} | |
| 22 | − | |
| 23 | −The main identification challenge arises from \ldots{} To address potential endogeneity concerns, we employ \ldots{} | |
| 24 | − | |
| 25 | −We cluster standard errors at the firm level to account for within-firm serial correlation, following \citet{petersen2009estimating}. | |
| 26 | − | |
| 27 | −\subsection{Extended Specifications} | |
| 28 | −\label{subsec:extended} | |
| 29 | − | |
| 30 | −We extend the baseline model in several directions. First, we allow for nonlinear effects: | |
| 31 | −% | |
| 32 | −\begin{equation} | |
| 33 | − y_{i,t} = \alpha + \beta_1 X_{i,t} + \beta_2 X_{i,t}^2 + \boldsymbol{\beta}_3' \mathbf{Z}_{i,t} + \gamma_i + \delta_t + \varepsilon_{i,t}. | |
| 34 | − \label{eq:nonlinear} | |
| 35 | −\end{equation} | |
| 36 | − | |
| 37 | −Second, we examine heterogeneous effects by interacting $X_{i,t}$ with \ldots{}: | |
| 38 | −% | |
| 39 | −\begin{equation} | |
| 40 | − y_{i,t} = \alpha + \beta_1 X_{i,t} + \beta_2 D_{i,t} + \beta_3 (X_{i,t} \times D_{i,t}) + \boldsymbol{\beta}_4' \mathbf{Z}_{i,t} + \gamma_i + \delta_t + \varepsilon_{i,t}, | |
| 41 | − \label{eq:interaction} | |
| 42 | −\end{equation} | |
| 43 | −% | |
| 44 | −where $D_{i,t}$ is an indicator variable equal to one if \ldots{} The coefficient $\beta_3$ captures the differential effect of $X$ for \ldots{} | |
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paper/sections/model.tex
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| 1 | +% ============================================================================ | |
| 2 | +\section{The model} | |
| 3 | +\label{sec:model} | |
| 4 | +% ============================================================================ | |
| 5 | + | |
| 6 | +The analysis uses two economies that share notation and mechanisms. The \emph{benchmark} (Section~\ref{subsec:benchmark}) compresses the life cycle into three stages and a generation into two periods; it delivers closed forms and the propositions of Section~\ref{sec:equilibrium}. The \emph{quantitative} economy (Section~\ref{subsec:quant}) has twelve five-year periods, housing on an intensive margin, persistent income risk, long-term nominal annuity mortgages, two bequest dates, durable housing stocks and an explicit lender; it is solved numerically in Sections~\ref{sec:computation}--\ref{sec:sensitivity}. Every reduction made in the benchmark is a candidate explanation for a discrepancy between the propositions and the simulations, and Section~\ref{sec:equilibrium} names those reductions explicitly. | |
| 7 | + | |
| 8 | +\subsection{Units and timing conventions} | |
| 9 | +\label{subsec:units} | |
| 10 | + | |
| 11 | +Time is discrete. All quantities labelled ``real'' are in units of the period consumption good. The consumer price index is $P_t$, gross inflation $1+\pi_t = P_t/P_{t-1}$, expected inflation over $(t-1,t)$ as of $t-1$ is $\pi^e_t$, and the surprise is defined multiplicatively, $1+\pi^u_t \equiv (1+\pi_t)/(1+\pi^e_t)$; in a stationary equilibrium $\pi_t = \pi^e_t = \bar\pi$. The real price of one unit of owner-occupied housing is $p_t$, so the house-price index $P_t p_t$ is distinct from the CPI. The real rent per unit of rental services is $\rho_t$. A mortgage originated at $t$ has nominal principal $B^{nom}_t$; its real value at a later date $t+k$ is $b^m_{t+k} \equiv B^{nom}_{t+k}/P_{t+k}$, so the nominal balance is what the contract fixes and the real balance is what matters for consumption. The real gross rate is $R \equiv 1+r$, exogenous (a small open economy), and the nominal rate on contracts written at $t$ satisfies the Fisher relation $1+i_t = R(1+\pi^e_{t+1})$ because the lender is competitive and risk neutral. A fixed-rate mortgage (FRM) keeps $i_t$ for the life of the loan; an adjustable-rate mortgage (ARM) resets every period; a term-$k$ contract resets every $k$ periods, which in the five-year calibration of the quantitative model represents the Canadian renewable term. Households belong to dynasties, infinite sequences of parent--child pairs; the labour-income process is identical across dynasties, so dynasties differ only through the transfers they receive. This ``equal-income'' design is what makes the owner--renter gaps of the paper attributable to transfers, leverage and constraints rather than to earnings. | |
| 12 | + | |
| 13 | +\subsection{The benchmark economy} | |
| 14 | +\label{subsec:benchmark} | |
| 15 | + | |
| 16 | +\paragraph{Demographics and timing.} | |
| 17 | +A household lives three stages, $j = 1,2,3$: a \emph{buyer} (young adult, who chooses tenure, buys or rents, and borrows), a \emph{payer} (prime age, who pays rent or mortgage interest and amortisation; under an ARM or term contract the rate is reset at the start of this stage) and a \emph{seller--donor} (who sells the house, repays the remaining balance, consumes and gives an inter-vivos transfer to the child). The child enters the model at stage~1 exactly when the parent enters stage~3, so a generation lasts two periods and the transfer arrives at the moment of the tenure decision, the ``gift at first purchase'' channel documented by \citet{engelhardt1998intergenerational} and \citet{blickle2019borrowing}. A unit mass of stage-1 households is born each period. Stage-3 households do not demand housing in the two markets (their retirement housing is outside the model and netted out of $y_3$); this is the one demographic simplification of the benchmark, and the quantitative model removes it. Within a period: the price level and the stage-3 income shock are realised; real prices $(p_t,\rho_t)$ are determined; stage-3 households sell, repay, consume and transfer; stage-1 households receive the transfer and choose tenure, housing and the mortgage while stage-2 households pay and consume; the owner-occupied and rental markets clear. | |
| 18 | + | |
| 19 | +\paragraph{Preferences and the housing need.} | |
| 20 | +A household born at $t$ with tenure $o \in \{O,R\}$ has lifetime utility | |
| 21 | +\begin{equation} | |
| 22 | +U^o = \ln c_1 + \chi\,\mathbf 1\{o = O\} + \beta\,\mathbb E[\ln c_2] + \beta^2\,\mathbb E[\ln c_3 + \nu\ln b'], | |
| 23 | +\qquad \beta\in(0,1),\ \nu>0,\ \chi\ge 0, | |
| 24 | +\label{eq:utilityA} | |
| 25 | +\end{equation} | |
| 26 | +where $b'$ is the transfer to the child. Each household needs exactly one unit of housing services at stages~1 and~2 (unit demand): an owner holds one unit of the owner-occupied stock, a renter rents one unit of the rental stock. Housing therefore enters utility only through the constant ownership premium $\chi$ \citep{henderson1983model}; $\chi = 0$ is allowed. The warm-glow term is the \citet{denardi2004wealth} specification with the luxury shift set to zero so that the transfer rule is linear. Incomes $y_1, y_2 > 0$ are deterministic and common; $y_3$ is drawn from a distribution $G$ with a continuous density $g > 0$ on $[\underline y_3, \bar y_3]$, independently across households and generations. | |
| 27 | + | |
| 28 | +\paragraph{Mortgage, deposits and the lender.} | |
| 29 | +A stage-1 owner buys one unit at $p_t$, pays the buying cost $\kappa_b p_t$ and maintenance $m p_t$, makes a down payment of at least $\theta p_t$ and borrows the nominal principal $B^{nom}_t \le P_t(1-\theta)p_t$. The contract pays interest $i_1 B^{nom}_t$ plus amortisation $a B^{nom}_t$ at stage~2, $a\in(0,1]$, and $(1+i_2)(1-a)B^{nom}_t$ at stage~3, with $i_1 = R(1+\pi^e_{t+1})-1$ fixed at origination for every contract and $i_2 = i_1$ (FRM) or $i_2 = R(1+\pi^e_{t+2|t+1})-1$ (ARM or term contract renewed at $t+1$). Real payments are $D_2 = (i_1+a)(1-\theta)p_t/(1+\pi_{t+1})$ and $D_3 = (1+i_2)(1-a)(1-\theta)p_t/[(1+\pi_{t+1})(1+\pi_{t+2})]$. In the stationary state write $D_2 = \delta_2 p$, $D_3 = \delta_3 p$ with | |
| 30 | +\begin{equation} | |
| 31 | +\delta_2 \equiv (1-\theta)\Big[R - \frac{1-a}{1+\bar\pi}\Big], \qquad | |
| 32 | +\delta_3 \equiv \frac{(1-a)(1-\theta)R}{1+\bar\pi}, \qquad | |
| 33 | +e_3 \equiv (1-\kappa_s) - \delta_3, \qquad d \equiv \theta + \kappa_b + m, | |
| 34 | +\label{eq:deltas} | |
| 35 | +\end{equation} | |
| 36 | +so that $d$ is the stage-1 cash outlay per unit of price and $e_3$ the net equity share at sale. The identity | |
| 37 | +\begin{equation} | |
| 38 | +\delta_2 + \frac{\delta_3}{R} = (1-\theta)R | |
| 39 | +\label{eq:fisher} | |
| 40 | +\end{equation} | |
| 41 | +is the present-value (Fisher) neutrality of the nominal contract: anticipated inflation front-loads real payments ($\partial\delta_2/\partial\bar\pi > 0$, $\partial\delta_3/\partial\bar\pi < 0$), the ``tilt'', without changing their present value. The house is net wealth at the end of life, $e_3 > 0$, if and only if $(1-\kappa_s)(1+\bar\pi) > (1-a)(1-\theta)R$, that is, if amortisation and the down payment are large enough; a two-period version in which the whole mortgage is a balloon fails this condition for generation-length periods, which is why the benchmark has three stages. The liquid asset is a one-period nominal deposit $n\ge 0$ paying $i_t$; its real gross return is $R/(1+\pi^u_{t+1})$, so a surprise erodes nominal savings exactly as it erodes nominal debt. The lender is a competitive risk-neutral intermediary that funds mortgages and takes deposits at the same nominal rate; its realised real profit in a surprise period accrues to rentiers outside the dynasties, and is the mirror image of the households' revaluation gains. | |
| 42 | + | |
| 43 | +\paragraph{Rental and owner-occupied supply.} | |
| 44 | +Landlords are competitive and risk neutral and the rental stock is segmented from the owner-occupied stock. Per cohort, owner-occupied supply is $S(p) = \bar H p^{\varepsilon}$, $\varepsilon\in[0,\infty)$, with $\varepsilon = 0$ a fixed stock, and rental supply is $S_r(\rho) = \bar H_r\rho^{\varepsilon_r}$, $\varepsilon_r\in(0,\infty]$. In the benchmark $S$ is a per-period supply curve (a reversible land-conversion technology); the quantitative model uses the durable stock-flow version whose impact elasticity is $\varepsilon\delta_H$. Nominal wages and rents are indexed to a surprise with elasticities $\eta_w,\eta_\rho\in[0,1]$ (full indexation in the baseline). The no-segmentation alternative, one stock converted costlessly by landlords, is the \citet{kaplan2020housing} and \citet{greenwald2025do} benchmark and is analysed in Proposition~\ref{prop:noseg}. | |
| 45 | + | |
| 46 | +\paragraph{Budgets.} | |
| 47 | +For an owner born at $t$, with $n_1$ and $n_2$ the deposits carried out of stages~1 and~2, | |
| 48 | +\begin{align} | |
| 49 | +c_1 + d\,p_t + n_1 &= b + y_1 + \big[(1-\theta)p_t - B_t\big], \label{eq:budgetO1}\\ | |
| 50 | +c_2 + D_2 + m\,p_t + n_2 &= y_2 + n_1 \tfrac{R}{1+\pi^u_{t+1}}, \label{eq:budgetO2}\\ | |
| 51 | +c_3 + b' &= W^O_3 \equiv y_3 + (1-\kappa_s)\,p_{t+2} - D_3 + n_2\tfrac{R}{1+\pi^u_{t+2}}, \label{eq:budgetO3} | |
| 52 | +\end{align} | |
| 53 | +where the bracket in \eqref{eq:budgetO1} is non-negative and zero when the loan-to-value constraint binds. For a renter, $c_1 + \rho_t + n_1 = b + y_1$, $c_2 + \rho_{t+1} + n_2 = y_2 + n_1R/(1+\pi^u_{t+1})$ and $c_3 + b' = W^R_3 \equiv y_3 + n_2R/(1+\pi^u_{t+2})$. Deposits cannot be negative, there is no refinancing, and no default: every stage-2 payment can be met out of income and every owner can repay the stage-3 balance out of income plus the sale proceeds (Assumption~\ref{ass:feas} below). Consumption of a hand-to-mouth household, which sets $n_1 = n_2 = 0$ and borrows the maximum, is written $c^O_1(b) = b + y_1 - dp$, $c^R_1(b) = b + y_1 - \rho$, $c^O_2 = y_2 - \delta_2 p - mp$, $c^R_2 = y_2 - \rho$, $W^O_3(y_3) = y_3 + e_3p$ and $W^R_3(y_3) = y_3$. | |
| 54 | + | |
| 55 | +\subsection{The quantitative economy} | |
| 56 | +\label{subsec:quant} | |
| 57 | + | |
| 58 | +The quantitative economy keeps the dynasty structure, the nominal contracts and the segmented markets of the benchmark and relaxes everything else. Figure~\ref{fig:timing} summarises its timing and Table~\ref{tab:notation} in Appendix~\ref{app:tables} collects the notation. | |
| 59 | + | |
| 60 | +\begin{figure}[htbp] | |
| 61 | +\centering | |
| 62 | +\includegraphics[width=\textwidth]{fig01_model_timing.pdf} | |
| 63 | +\caption{Timing of the overlapping-generations model. One period is five years; a household lives twelve periods (ages 25--84) and retires at 65; its child enters when the parent is 55--59, receives the inter-vivos gift $b^g$ at entry and the estate $b^e$ at the parent's death, when the child is itself 55--59. A mortgage originated at age 2 is amortised over $N = 5$ periods.} | |
| 64 | +\label{fig:timing} | |
| 65 | +\end{figure} | |
| 66 | + | |
| 67 | +\paragraph{Demographics and transfers.} | |
| 68 | +Ages are $j = 1,\dots,J = 12$ (periods of five years, 25--84), retirement at $j_R = 9$. Each household has one child who enters the model at the parent's age $j_g = 7$; the child receives an inter-vivos gift $b^g$ at its age~1, chosen by the parent at $j_g$, and the parent's estate $b^e$ at the parent's death, when the child is $j_e = 7$. Both are chosen with warm-glow utility; ``bequests off'' sets the warm-glow weight to zero for both. Population is constant with a unit mass per age, and there is no survival risk in the baseline. | |
| 69 | + | |
| 70 | +\paragraph{Preferences and income.} | |
| 71 | +Period utility over non-housing consumption and housing services is | |
| 72 | +\begin{equation} | |
| 73 | +u(c,s) = \frac{\big(c^{1-\alpha}s^{\alpha}\big)^{1-\gamma}}{1-\gamma}, \qquad | |
| 74 | +\phi(x) = \nu\,\frac{(x+\underline x)^{1-\gamma}}{1-\gamma}, | |
| 75 | +\label{eq:uB} | |
| 76 | +\end{equation} | |
| 77 | +with $s = (1+\chi)h$ for an owner of size $h$ and $s$ the rented size for a renter, and $\phi$ the warm glow over a transfer $x\in\{b^g,b^e\}$ with luxury shift $\underline x > 0$ \citep{denardi2004wealth}; the benchmark's linear rule is the limit $\underline x\to 0$, $\gamma\to 1$. Real labour income at $j < j_R$ is $y_j = \bar y_j z$, with $\bar y_j$ a hump-shaped age profile and $z$ a five-state Rouwenhorst chain for the five-year aggregate of an annual AR(1); the pension is a flat replacement of the last working-age income. The child's initial state $z_1$ is drawn from the stationary distribution independently of the parent in the baseline (equal-income design); an intergenerational earnings correlation $\rho_g$ is a sensitivity switch. A small unanticipated transitory component of income, realised before decisions, spreads the atoms of the cash-on-hand distribution that would otherwise make aggregate demand discontinuous in prices (Section~\ref{sec:computation}). | |
| 78 | + | |
| 79 | +\paragraph{Housing, tenure and transactions.} | |
| 80 | +Owner-occupied houses come in four sizes $h\in\mathcal H$, the smallest being the indivisibility that creates an entry threshold; rented services $s\ge\underline s$ are continuous with $\underline s < \underline h$. Maintenance costs $mph$ per period; buying and selling cost $\kappa_b ph$ and $\kappa_s ph$; moving between owned sizes is a sale plus a purchase; renters move costlessly. Each period an owner chooses among \emph{stay} (possibly refinancing at cost $\kappa_f ph$), \emph{move} and \emph{sell-and-rent}; a renter chooses between \emph{rent} and \emph{buy}. | |
| 81 | + | |
| 82 | +\paragraph{Nominal annuity mortgages and origination constraints.} | |
| 83 | +A mortgage with real principal $B$, nominal contract rate $i_0$ and $N = 5$ remaining payments (25 years) has the constant nominal payment $\text{pay}^{nom} = B^{nom} i_0/[1-(1+i_0)^{-N}]$, and the real balance evolves as | |
| 84 | +\begin{equation} | |
| 85 | +b^{m\prime} = \frac{b^m(1+i_c) - \text{pay}_t}{1+\pi_{t+1}}, \qquad \text{pay}_t = \text{pay}^{nom}_t/P_t, | |
| 86 | +\label{eq:debtlaw} | |
| 87 | +\end{equation} | |
| 88 | +where $i_c = i_0$ for an FRM, $i_c$ is the current Fisher rate for an ARM (with the nominal payment recomputed on the remaining balance and term), and $i_c$ resets at every period boundary for the five-year term contract. The real payment of an FRM falls with realised inflation; that of an ARM is re-priced. At origination and refinancing the loan must satisfy the loan-to-value constraint $B\le(1-\theta)p_th$ and a nominal payment-to-income constraint at a qualifying rate $i_q = i_t+\Delta_q$ (the stress test): $\text{pay}^{nom}(i_q)/(P_ty) \le \psi$. Because the qualifying payment is nominal and income is current, expected inflation raises $i_t$ and tightens this constraint, which is how the benchmark's tilt reappears in the quantitative model. There is no unsecured borrowing, $a'\ge 0$, and no default; the solver enforces a consumption floor and reports the (zero) mass of forced sales. | |
| 89 | + | |
| 90 | +\paragraph{Rental sector, construction and the lender.} | |
| 91 | +Rental units form a separate stock owned by competitive risk-neutral landlords who buy at $p^r_t$, pay maintenance and an operating wedge $\phi p^r_t$, and charge the zero-profit rent | |
| 92 | +\begin{equation} | |
| 93 | +\rho_t = p^r_t(1+m+\phi) - \mathbb E_t p^r_{t+1}/R . | |
| 94 | +\label{eq:rent} | |
| 95 | +\end{equation} | |
| 96 | +A segmentation parameter $\sigma\in[0,1]$ gives the share of rental units that are structurally distinct; the remainder converts one-for-one from the owner-occupied stock at $p^r_t = p_t$, so that $\sigma = 1$ is the benchmark and $\sigma = 0$ the single-stock economy of Proposition~\ref{prop:noseg}. Rent rigidity $\eta_\rho$ applies to sitting tenants in a surprise period, the difference being a transfer from landlords. The two stocks are durable, | |
| 97 | +\begin{equation} | |
| 98 | +H_{t+1} = (1-\delta_H)H_t + \iota\,p_t^{\varepsilon}, \qquad H^r_{t+1} = (1-\delta_H)H^r_t + \iota_r\,(p^r_t)^{\varepsilon_r}, | |
| 99 | +\label{eq:supply} | |
| 100 | +\end{equation} | |
| 101 | +so the stationary supply curves are $S(p) = \iota p^{\varepsilon}/\delta_H$ and $S_r(p^r) = \iota_r (p^r)^{\varepsilon_r}/\delta_H$ and the one-period elasticity is $\varepsilon\delta_H$. Construction profits go to a class outside the dynasties. The lender holds mortgage balances against household deposits and rentier funding at the world real rate; its real net worth obeys $A^L_{t+1} = (1+i_t)(\text{Mortgages}_t - \text{Deposits}_t)/(1+\pi_{t+1}) + \text{rentier flow}$, so a surprise revalues $-(\text{Mortgages}_t-\text{Deposits}_t)(1+i_t)\pi^u_{t+1}/[(1+\pi^e_{t+1})(1+\pi^u_{t+1})]$ for the lender, the mirror image of the households' aggregate real gain. A transfer tax $\tau_b$ on gifts and estates, rebated lump-sum to entering households, is the policy experiment of Section~\ref{sec:supply}. | |
| 102 | + | |
| 103 | +\paragraph{The household problem.} | |
| 104 | +The individual state at age $j$ is $x = (j,z,o,h,b^m,n_m,a)$ with $o$ tenure, $h$ house size (zero for renters), $b^m$ the real mortgage balance, $n_m$ the remaining payments and $a\ge 0$ the real deposit; along a transition the origination date, hence the FRM rate, is identified by $n_m$ and calendar time. Writing $R_a\equiv(1+i_{t-1})/(1+\pi_t)$ for the realised real deposit return and $\ell$ for the discrete branch, | |
| 105 | +\begin{equation} | |
| 106 | +V_j(x) = \max_{\ell\in\mathcal L(x)} V^{\ell}_j(x), \qquad | |
| 107 | +V^{\ell}_j(x) = \max_{c,a',(h',B',s)}\Big\{u(c,s) + \mathbf 1_{j=j_g}\phi(b^g) + \beta\,\mathbb E\big[V_{j+1}(x')\mid z\big]\Big\}, | |
| 108 | +\label{eq:bellman} | |
| 109 | +\end{equation} | |
| 110 | +subject to the branch budget. A renter who rents has $c + \rho s + a' = y_j + aR_a - \mathbf 1_{j=j_g}b^g$; a renter who buys has $c + (1+\kappa_b+m)ph' - B' + a' = y_j + aR_a - \mathbf 1_{j=j_g}b^g$ with the origination constraints; an owner who stays has $c + \text{pay}(b^m,n_m,i_c) + mph + a' = y_j + aR_a - \mathbf 1_{j=j_g}b^g$; an owner who moves sells at $(1-\kappa_s)ph$, repays $b^m(1+i_c)$ and proceeds as a buyer; an owner who sells and rents has $c + \rho s + a' = y_j + aR_a + (1-\kappa_s)ph - b^m(1+i_c) - \mathbf 1_{j=j_g}b^g$. At $J$ the house is sold, the balance repaid and $V_{J+1}\equiv\phi(b^e)$ with $b^e$ the terminal wealth. The estate received at $j_e$ is anticipated as a lottery whose points are equilibrium objects (Section~\ref{sec:computation}). The solver works in real variables; inflation enters only through $R_a$ and the real FRM payment, through the nominal rate in the qualifying payment and in ARM repricing, and through the indexation parameters. Net worth is $a + \mathbf 1_O[(1-\kappa_s)ph - b^m]$. | |
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paper/sections/results.tex
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| 1 | −% ============================================================================ | |
| 2 | −\section{Results} | |
| 3 | −\label{sec:results} | |
| 4 | −% ============================================================================ | |
| 5 | − | |
| 6 | −\subsection{Baseline Results} | |
| 7 | −\label{subsec:baseline_results} | |
| 8 | − | |
| 9 | −Table~\ref{tab:regression} presents the estimation results of our baseline model specified in Equation~\eqref{eq:baseline}. Column~(1) reports the OLS estimates without fixed effects. Columns~(2)--(4) progressively add firm and time fixed effects and the full set of controls. | |
| 10 | − | |
| 11 | −\input{tables/regression} | |
| 12 | − | |
| 13 | −The coefficient on $X_{i,t}$ is positive and statistically significant at the 1\% level across all specifications. The point estimate in our preferred specification (Column~4) implies that a one-standard-deviation increase in $X$ is associated with a \ldots{} increase in $y$. | |
| 14 | − | |
| 15 | −\subsection{Graphical Evidence} | |
| 16 | −\label{subsec:graphical} | |
| 17 | − | |
| 18 | −Figure~\ref{fig:main_result} provides graphical evidence supporting our main findings. | |
| 19 | − | |
| 20 | −\input{figures/main_figure} | |
| 21 | − | |
| 22 | −The figure illustrates \ldots{} The upward trend in \ldots{} is consistent with the regression estimates reported in Table~\ref{tab:regression}. | |
| 23 | − | |
| 24 | −\subsection{Sub-sample Analysis} | |
| 25 | −\label{subsec:subsample} | |
| 26 | − | |
| 27 | −To explore potential heterogeneity, we re-estimate Equation~\eqref{eq:baseline} across sub-samples defined by \ldots{} The results, available in Table~\ref{tab:app_subsample} of Appendix~\ref{app:tables}, indicate that \ldots{} | |
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paper/sections/robustness.tex
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| 1 | −% ============================================================================ | |
| 2 | −\section{Robustness Checks} | |
| 3 | −\label{sec:robustness} | |
| 4 | −% ============================================================================ | |
| 5 | − | |
| 6 | −We conduct several robustness tests to assess the sensitivity of our baseline results. | |
| 7 | − | |
| 8 | −\subsection{Alternative Variable Definitions} | |
| 9 | −\label{subsec:alt_definitions} | |
| 10 | − | |
| 11 | −We first verify that our results are not driven by the specific construction of the dependent variable. Table~\ref{tab:app_robustness} in Appendix~\ref{app:tables} reports estimates using \ldots{} as alternative measures. The coefficient on $X_{i,t}$ remains statistically significant and economically meaningful across all specifications. | |
| 12 | − | |
| 13 | −\subsection{Alternative Estimation Methods} | |
| 14 | −\label{subsec:alt_estimation} | |
| 15 | − | |
| 16 | −We re-estimate the model using \ldots{} to address potential concerns about \ldots{} The results are qualitatively unchanged. | |
| 17 | − | |
| 18 | −\subsection{Placebo Tests} | |
| 19 | −\label{subsec:placebo} | |
| 20 | − | |
| 21 | −As a falsification exercise, we replace the main explanatory variable with \ldots{} If our baseline results reflect a genuine economic relationship rather than spurious correlation, we expect the placebo coefficients to be statistically insignificant. Consistent with this expectation, none of the placebo estimates are significant at conventional levels. | |
| 22 | − | |
| 23 | −\subsection{Sensitivity to Sample Period} | |
| 24 | −\label{subsec:sample_period} | |
| 25 | − | |
| 26 | −We examine whether our results are sensitive to the sample period by \ldots{} The estimates remain stable across sub-periods, suggesting that our findings are not driven by a specific time window. | |
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paper/sections/sensitivity_conclusion.tex
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| 1 | +% ============================================================================ | |
| 2 | +\section{Sensitivity, limitations and conclusion} | |
| 3 | +\label{sec:sensitivity} | |
| 4 | +% ============================================================================ | |
| 5 | + | |
| 6 | +\subsection{Sensitivity} | |
| 7 | +\label{subsec:sens} | |
| 8 | + | |
| 9 | +Table~\ref{tab:mobility} and Table~\ref{tab:robustness} in Appendix~\ref{app:computation} report the sensitivity of the stationary equilibrium to rates, trend inflation, the rent wedge, earnings risk, the outside asset, preferences and transaction costs; every variant is a full re-solution of the general equilibrium. A real rate of 1 or 3\% gives prices of \RoneP{} and \RthreeP, ownership of \RoneOwn{} and \RthreeOwn, mean estates of \RoneEstate{} and \RthreeEstate, $\lambda_7$ of \RoneLamSeven{} and \RthreeLamSeven{} and rank--rank slopes of \RoneRR{} and \RthreeRR: a higher real rate raises saving, estates and persistence. Trend inflation of 0 or 4\% gives prices of \PiZeroP{} and \PiFourP{} and ownership of \PiZeroOwn{} and \PiFourOwn; at fixed prices, 4\% inflation lowers under-35 ownership by \PiFourPEDropUthirtyfivePP{} percentage points through the nominal tilt of the constant-payment mortgage and the nominal payment-to-income cap, which is Proposition~\ref{prop:tilt} in partial equilibrium, and the general-equilibrium price fall offsets \PiFourGEOffsetPct\% of it. A landlord wedge of 2 or 8\% gives ownership of \PhiLowOwn{} and \PhiHighOwn. Earnings risk with annual innovation standard deviations of 0.10 and 0.20 gives ownership of \SigLowOwn{} and \SigHighOwn, Ginis of \SigLowGini{} and \SigHighGini, mean gifts of \SigLowGift{} and \SigHighGift{} and $\lambda_7$ of \SigLowLamSeven{} and \SigHighLamSeven: more risk raises precautionary transfers and persistence while lowering ownership. A deposit premium of two percentage points, the renter's investment return, raises the age-1 threshold at the median state to \DepPremThr, estates to \DepPremEstate{} and the rank--rank slope to \DepPremRR: a better outside asset delays purchase but raises estates and persistence, which is the mechanism of \citet{nekoei2023how} in reverse. Annual discount factors of 0.94 and 0.98 give prices of \BetaLowP{} and \BetaHighP{} and ownership of \BetaLowOwn{} and \BetaHighOwn; $\gamma = 3$ gives \GammaThreeOwn; a housing share of 0.25 gives a price of \AlphaHighP; low and high transaction costs give prices of \KappaLowP{} and \KappaHighP; a 5\% ownership premium gives \ChiFiveOwn. None of these variants changes the qualitative results: persistence requires transfers, prices absorb credit shifts, the surprise lowers the real price. | |
| 10 | + | |
| 11 | +The numerical variants bound the precision of the statistics. Coarse grids move ownership by \DevCoarseOwnPP{} percentage points, the price by \DevCoarsePPct\%, the Gini by \DevCoarseGini, the age-1 threshold by \DevCoarseThr{} and $\lambda_7$ by \DevCoarseLamSeven; fine grids move ownership by \DevFineOwnPP{} points, the price by \DevFinePPct\% and the threshold by \DevFineThr. A tolerance of $10^{-5}$ returns the same best iterate; a panel of 240{,}000 dynasties and an alternative seed move ownership by \DevSampleLargeOwnPP{} and \DevSeedOwnPP{} points, the price by \DevSampleLargePPct{} and \DevSeedPPct\%, $\lambda_7$ by \DevSampleLargeLamSeven{} and \DevSeedLamSeven{} and the rank--rank slope by \DevSampleLargeRR{} and \DevSeedRR. The ownership rate, prices, the Gini and the rank--rank slope are robust; the age-1 threshold and $\lambda_7$ are the least robust statistics and are quoted accordingly. Mean Bellman residuals of owners lie between $\BellmanOwnerMinAll$ (fine grids) and $\BellmanOwnerMaxAll$ ($\gamma = 3$ and coarse grids) across the \NRobustnessRuns{} robustness runs. | |
| 12 | + | |
| 13 | +\subsection{Limitations} | |
| 14 | +\label{subsec:limitations} | |
| 15 | + | |
| 16 | +The calibration is illustrative, no moment is fitted and the supply elasticity has no verified Canadian source, so magnitudes are statements about this parameterisation. The ownership profile falls after 65 because the warm glow is defined over a liquid estate, whereas Canadian owners keep their houses; a bequest motive over housing, or the slow dissaving of \citet{nakajima2020home}, would keep the old in their houses, raise the housing share of estates and, since the house would then be inherited at a market price, strengthen the ladder channel that the multiplicity scan found weak. Seventy percent of entrants receive a gift, against 28\% in the Canadian data, because every parent has the same motive; heterogeneous motives would concentrate transfers and raise the dispersion the model understates. There is no aggregate risk, no default, no survival risk and no endogenous real rate; the estate is anticipated as an unconditional lottery and held at its stationary value along transitions; the adjustable-rate contract is a stylised within-period repricing; and the five-year period understates peak leverage, so that no household is ever underwater and the leverage channel operates through consumption and entry rather than foreclosure. Monte-Carlo noise bounds market clearing to about $5\times10^{-4}$ in stationary equilibria and $10^{-3}$ along transitions, and four of the \NTransitions{} transitions are reported with residuals above tolerance. On the theory side, single crossing of the tenure decision is proved only on the hand-to-mouth region, the comparative statics in the down payment hold for a once-and-for-all tenure decision, price neutrality without segmentation holds under unit demand, and the impact proposition covers one period under static expectations and pure debtor positions. The general-equilibrium sign of the inflation response is governed by a density-weighted net nominal position that the quantitative model computes but that no verified Canadian source pins down; \citet{meh2011inflation} report net nominal positions by age and tenure, and a calibration that targeted them would be the natural next step. | |
| 17 | + | |
| 18 | +\subsection{Conclusion} | |
| 19 | +\label{sec:conclusion} | |
| 20 | + | |
| 21 | +The paper asked under what conditions inherited wealth, mortgage leverage, nominal debt, housing supply and down-payment constraints generate persistent owner--renter wealth inequality among households with the same income process, and what an unanticipated inflation changes. The benchmark answers the first part analytically. Ownership requires a transfer above a threshold that exceeds the down-payment bound by the consumption of the marginal buyer; leverage multiplies the exposure of terminal wealth to the resale price without changing the break-even; the parent's housing equity is pivotal for the child's tenure over an interval of income draws whose probability mass is the intergenerational persistence, positive only if the house is net wealth at the end of life, and a half-life rather than a trap; the stationary equilibrium exists and is unique when the entry-threshold channel dominates the ladder channel; and with a fixed stock, credit and transfers move prices and composition, not the ownership rate. The quantitative economy supports these statements in their stated scope. Transfers generate a mid-life wealth gap of about one year of income among equal-income households and are the whole of the model's intergenerational persistence, which disappears when the bequest motive is removed; prices absorb two thirds to nine tenths of what credit constraints would do to ownership at fixed prices; and no multiplicity appears, even under near-fixed supply. | |
| 22 | + | |
| 23 | +For the second part the benchmark was wrong in sign and the paper says so. A surprise erodes mortgages and deposits alike, and households in this economy hold more of the latter: the real house price falls, aggregate welfare falls, and the gains of mortgaged owners under fixed-rate and renewable-term contracts come from debt-free owners, renters and, through the lender, from rentiers. The benchmark's prediction of a price rise is restored only under the assumption that the mortgage is the households' sole nominal position, and the corollary that replaces it ties the sign of the price response to a density-weighted net nominal position of the parents whose children are at the tenure margin. Two further benchmark results fail outside their premises, price neutrality without segmentation under unit demand and the monotonicity of the entry threshold in the down payment for a once-and-for-all tenure decision, and both have been restated under those premises. What survives of the inflation analysis in general equilibrium is the accounting and the contract ordering: fixed-rate at least as good as the renewable term and both better than the adjustable contract for the indebted, with the term losing its advantage exactly when the surprise is persistent and the real rate responds, which is the Canadian case. The natural continuation is a calibration that targets the net nominal positions of Canadian households by age and tenure and a bequest motive over housing, since the sign of the general-equilibrium response and the strength of the ladder channel both turn on them. | |
added
paper/sections/supply_policy_welfare.tex
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| 1 | +% ============================================================================ | |
| 2 | +\section{Supply elasticity, policy and welfare} | |
| 3 | +\label{sec:supply} | |
| 4 | +% ============================================================================ | |
| 5 | + | |
| 6 | +\subsection{Supply elasticity and the down-payment constraint} | |
| 7 | +\label{subsec:supply} | |
| 8 | + | |
| 9 | +\begin{figure}[htbp] | |
| 10 | +\centering | |
| 11 | +\includegraphics[width=\textwidth]{fig10_supply_elasticity_heatmap.pdf} | |
| 12 | +\caption{Stationary general equilibria across the long-run supply elasticity $\varepsilon = \varepsilon_r$ and the minimum down payment $\theta$: house price (a), ownership rate (b), probability that a renter's child owns at 35--39 (c) and rank--rank slope of net worth (d). Source: \texttt{results/data/supply/*.json}, \texttt{theta\_sweep/*.json}.} | |
| 13 | +\label{fig:heatmap} | |
| 14 | +\end{figure} | |
| 15 | + | |
| 16 | +Changing the supply elasticity alone leaves the baseline unchanged, because every supply curve passes through the normalisation point, so the elasticity matters only together with a demand shift, and Figure~\ref{fig:heatmap} crosses it with the down-payment constraint. With a 5\% minimum down payment the price is \EpsLowThetaFiveP{} under near-fixed supply and \EpsHighThetaFiveP{} under an elasticity of two, with ownership \EpsLowThetaFiveOwn{} against \EpsHighThetaFiveOwn; with a 40\% minimum the price is \EpsLowThetaFortyP{} against \EpsHighThetaFortyP{} and ownership \EpsLowThetaFortyOwn{} against \EpsHighThetaFortyOwn. The direction is that of Corollary~\ref{cor:fixed}, inelastic supply converting more of the demand shift into price and less into quantity, but the magnitudes are small because the demand shift from $\theta$ is itself small: the payment-to-income cap binds for low-income entrants whatever the down payment. Mobility is insensitive to the elasticity at these shifts, $q_{RO}$ at 35--39 being \EpsLowThetaFiveqRO{} against \EpsHighThetaFiveqRO{} at $\theta = 0.05$ and \EpsLowThetaFortyqRO{} against \EpsHighThetaFortyqRO{} at $\theta = 0.40$. The conjecture with which the fourth contribution started, that supply elasticity has a first-order effect on intergenerational mobility through the entry threshold, is not supported in this calibration; what the elasticity does govern, in the transitions of Section~\ref{sec:inflation}, is bounded by the depreciation rate. | |
| 17 | + | |
| 18 | +\subsection{Search for multiple stationary equilibria} | |
| 19 | +\label{subsec:multiplicity} | |
| 20 | + | |
| 21 | +\begin{figure}[htbp] | |
| 22 | +\centering | |
| 23 | +\includegraphics[width=\textwidth]{fig13_multiplicity_search.pdf} | |
| 24 | +\caption{Search for multiple stationary equilibria: owner-market excess demand along the house price with the rental market cleared at each point and the estate lottery held at its baseline value (a), and the ownership rate along the clearing curve (b), for the baseline, near-fixed supply and near-fixed supply with a 5\% minimum down payment. Source: \texttt{results/data/multiplicity/price\_scan\_*.csv}.} | |
| 25 | +\label{fig:multiplicity} | |
| 26 | +\end{figure} | |
| 27 | + | |
| 28 | +Proposition~\ref{prop:exist} leaves multiplicity open when condition (C) fails, and the benchmark's ladder channel, bequeathed equity rising with the price, is the candidate. Figure~\ref{fig:multiplicity} scans the owner-market excess demand over prices from \MultPgridLow{} to \MultPgridHigh{}. In the baseline it falls monotonically from \MultBaseExcessLow{} to \MultBaseExcessHigh{} with a single sign change at $p = \MultBaseRoot$, and the ownership rate along the clearing curve falls from \MultBaseOwnLow{} to \MultBaseOwnHigh{}: $d\mu/dp<0$ everywhere, the analogue of (C), holds with a wide margin. Under near-fixed supply the excess demand runs from \MultEpsLowExcessLow{} to \MultEpsLowExcessHigh{} with one root, and under near-fixed supply with a 5\% down payment from \MultEpsLowThetaFiveExcessLow{} to \MultEpsLowThetaFiveExcessHigh{} with one root at \MultEpsLowThetaFiveRoot. There is no evidence of the \citet{stein1995prices} multiplicity in the scanned range. The scan holds the lottery fixed, and the lottery feedback is the ladder channel itself, so the scan understates it; but the slope of excess demand around the root, \MultBaseSlope{} per unit of log price, is an order of magnitude larger than the lottery feedback measured across rounds, and at the equilibrium the lottery-consistent solution coincides with the root. Uniqueness is a numerical statement about this calibration, not a theorem. | |
| 29 | + | |
| 30 | +\subsection{A rebated transfer tax} | |
| 31 | +\label{subsec:policy} | |
| 32 | + | |
| 33 | +The policy experiment taxes gifts and estates at 20 or 50\% and rebates the proceeds lump-sum to entering households, with the rebate an equilibrium object. Ownership rises to \TaxTwentyOwn{} and \TaxFiftyOwn, and under-35 ownership to \TaxTwentyOwnUthirtyfive{} and \TaxFiftyOwnUthirtyfive{} against \OwnershipUnderThirtyFive{} in the baseline, because the rebate gives every entrant liquid wealth; upward tenure mobility rises ($q_{RO}$ at 35--39 \TaxTwentyqRO{} and \TaxFiftyqRO), the rank--rank slope falls to \TaxTwentyRR{} and \TaxFiftyRR, the Gini to \TaxTwentyGini{} and \TaxFiftyGini, and the mean gift falls to \TaxTwentyGift{} and \TaxFiftyGift{} (Tables~\ref{tab:scenarios} and~\ref{tab:mobility}). The price is nearly unchanged (\TaxTwentyP{} and \TaxFiftyP). A rebated transfer tax therefore raises youth ownership and mobility in this economy by moving liquidity to the moment at which the down-payment constraint binds; it is a comparative-statics statement about stationary equilibria, not an optimal-policy result, and the welfare numbers below concern entrants only. | |
| 34 | + | |
| 35 | +\subsection{Welfare} | |
| 36 | +\label{sec:welfare} | |
| 37 | + | |
| 38 | +\input{tables/tab08_welfare} | |
| 39 | + | |
| 40 | +\begin{figure}[htbp] | |
| 41 | +\centering | |
| 42 | +\includegraphics[width=\textwidth]{fig11_welfare.pdf} | |
| 43 | +\caption{Consumption-equivalent welfare. Panel (a): entering households in alternative stationary equilibria relative to the baseline, by quintile of the transfer received. Panel (b): households alive at a 10\% fixed-rate level surprise, by age and tenure, general equilibrium. Source: \texttt{results/data/welfare/welfare\_all.csv}, \texttt{results/data/inflation/frm\_level\_welfare.csv}.} | |
| 44 | +\label{fig:welfare} | |
| 45 | +\end{figure} | |
| 46 | + | |
| 47 | +Table~\ref{tab:welfare} and Figure~\ref{fig:welfare} report the consumption-equivalent variation \eqref{eq:cev} of entering households across stationary equilibria, by quintile of the transfer received, for households within 10\% of the age-1 threshold, and overall. Only scenarios that keep the preference parameters are welfare comparisons; the rows for the bequest motive and for the preference variants are listed as a reminder that they compare value functions under different utilities (``bequests off'' shows a large positive number only because the warm-glow disutility disappears from the objective). A 5\% minimum down payment raises the welfare of entrants by \CEVThetaFiveAll\% overall, \CEVThetaFiveBottomTwo\% in the bottom two gift quintiles, \CEVThetaFiveMarg\% for marginal households and \CEVThetaFiveQfive\% in the top quintile, which pays the higher price; a 40\% minimum gives \CEVThetaFortyAll\% overall, \CEVThetaFortyBottomTwo\% at the bottom, \CEVThetaFortyMarg\% for marginal households and \CEVThetaFortyQfive\% at the top. The welfare of marginal households, unsigned in the theory, is thus positive for a looser constraint and negative for a tighter one in this calibration. The rebated transfer tax raises entrants' welfare by \CEVTaxTwentyAll\% and \CEVTaxFiftyAll\%, decreasing in the gift quintile (\CEVTaxTwentyQone{} and \CEVTaxFiftyQone\% at the bottom against \CEVTaxTwentyQfive{} and \CEVTaxFiftyQfive\% at the top); this is the welfare of entrants, and the parents who pay the tax are not in the comparison. Near-fixed supply with a 5\% down payment gives \CEVEpsLowThetaFiveAll\% against \CEVEpsHighThetaFiveAll\% with an elasticity of two. A real rate of 1 or 3\% gives \CEVRoneAll{} and \CEVRthreeAll\%; a landlord wedge of 2 or 8\% gives \CEVPhiLowAll{} and \CEVPhiHighAll\%; low and high earnings risk give \CEVSigLowAll{} and \CEVSigHighAll\%; a 2-percentage-point deposit premium, the renter's outside investment return, gives \CEVDepPremAll\%, rising with the gift quintile. The numerical variants (tolerance, sample size, seed) give variations of at most \CEVNumericalMaxPct\%, which bounds the noise of the welfare measure. For transitions, the households alive at a level surprise lose \FrmLevelCEVAllAbs\% on average under the fixed-rate contract and \ArmLevelCEVAllAbs\% under the adjustable one, and gain \RateTwentyCEVAll\% in the real-rate shock (Section~\ref{sec:inflation}). | |
added
paper/tables/numbers.tex
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@@ -0,0 +1,2572 @@ | ||
| 1 | +% generated by src/tables.py — one macro per key number; every value traces to results/data/ | |
| 2 | +\newcommand{\AbsorbThetaFivePct}{64} | |
| 3 | +\newcommand{\AbsorbThetaFortyPct}{66} | |
| 4 | +\newcommand{\AlphaHighEstate}{0.37} | |
| 5 | +\newcommand{\AlphaHighGapFour}{1.00} | |
| 6 | +\newcommand{\AlphaHighGift}{0.41} | |
| 7 | +\newcommand{\AlphaHighGini}{0.54} | |
| 8 | +\newcommand{\AlphaHighLamSeven}{0.065} | |
| 9 | +\newcommand{\AlphaHighLamThree}{0.22} | |
| 10 | +\newcommand{\AlphaHighOwn}{0.627} | |
| 11 | +\newcommand{\AlphaHighOwnUthirtyfive}{0.359} | |
| 12 | +\newcommand{\AlphaHighP}{1.101} | |
| 13 | +\newcommand{\AlphaHighPDevPct}{10.1} | |
| 14 | +\newcommand{\AlphaHighRR}{0.36} | |
| 15 | +\newcommand{\AlphaHighRebate}{0.000} | |
| 16 | +\newcommand{\AlphaHighRho}{0.222} | |
| 17 | +\newcommand{\AlphaHighRhoDevPct}{14.5} | |
| 18 | +\newcommand{\AlphaHighThr}{0.54} | |
| 19 | +\newcommand{\AlphaHighThrHighZ}{0.05} | |
| 20 | +\newcommand{\AlphaHighThrLowZ}{1.13} | |
| 21 | +\newcommand{\AlphaHighqOO}{0.508} | |
| 22 | +\newcommand{\AlphaHighqRO}{0.289} | |
| 23 | +\newcommand{\ArmLevelAbsorbPct}{76} | |
| 24 | +\newcommand{\ArmLevelBetterAllPct}{18} | |
| 25 | +\newcommand{\ArmLevelBetterOMPct}{7} | |
| 26 | +\newcommand{\ArmLevelBetterONMPct}{1} | |
| 27 | +\newcommand{\ArmLevelBetterOPct}{4} | |
| 28 | +\newcommand{\ArmLevelBetterRPct}{38} | |
| 29 | +\newcommand{\ArmLevelCEVAll}{\ensuremath{-}1.14} | |
| 30 | +\newcommand{\ArmLevelCEVAllAbs}{1.14} | |
| 31 | +\newcommand{\ArmLevelCEVAllPE}{\ensuremath{-}1.19} | |
| 32 | +\newcommand{\ArmLevelCEVAllPEAbs}{1.19} | |
| 33 | +\newcommand{\ArmLevelCEVO}{\ensuremath{-}1.36} | |
| 34 | +\newcommand{\ArmLevelCEVOAbs}{1.36} | |
| 35 | +\newcommand{\ArmLevelCEVOM}{\ensuremath{-}1.03} | |
| 36 | +\newcommand{\ArmLevelCEVOMAbs}{1.03} | |
| 37 | +\newcommand{\ArmLevelCEVONM}{\ensuremath{-}2.14} | |
| 38 | +\newcommand{\ArmLevelCEVONMAbs}{2.14} | |
| 39 | +\newcommand{\ArmLevelCEVR}{\ensuremath{-}0.96} | |
| 40 | +\newcommand{\ArmLevelCEVRAbs}{0.96} | |
| 41 | +\newcommand{\ArmLevelConsAll}{\ensuremath{-}1.5} | |
| 42 | +\newcommand{\ArmLevelConsAllAbs}{1.5} | |
| 43 | +\newcommand{\ArmLevelConsO}{\ensuremath{-}1.4} | |
| 44 | +\newcommand{\ArmLevelConsOAbs}{1.4} | |
| 45 | +\newcommand{\ArmLevelConsR}{\ensuremath{-}1.6} | |
| 46 | +\newcommand{\ArmLevelConsRAbs}{1.6} | |
| 47 | +\newcommand{\ArmLevelDOwnGE}{\ensuremath{-}0.11} | |
| 48 | +\newcommand{\ArmLevelDOwnGEAbs}{0.11} | |
| 49 | +\newcommand{\ArmLevelDOwnPE}{\ensuremath{-}0.47} | |
| 50 | +\newcommand{\ArmLevelDOwnPEAbs}{0.47} | |
| 51 | +\newcommand{\ArmLevelDebtZeroPct}{1.5} | |
| 52 | +\newcommand{\ArmLevelDepLoss}{0.48} | |
| 53 | +\newcommand{\ArmLevelErosion}{0.00} | |
| 54 | +\newcommand{\ArmLevelLamSevenSS}{0.060} | |
| 55 | +\newcommand{\ArmLevelLamSevenZero}{0.065} | |
| 56 | +\newcommand{\ArmLevelLender}{0.48} | |
| 57 | +\newcommand{\ArmLevelMaxRes}{1.1\times10^{-3}} | |
| 58 | +\newcommand{\ArmLevelNWAll}{\ensuremath{-}0.03} | |
| 59 | +\newcommand{\ArmLevelNWAllAbs}{0.03} | |
| 60 | +\newcommand{\ArmLevelNWO}{\ensuremath{-}0.05} | |
| 61 | +\newcommand{\ArmLevelNWOAbs}{0.05} | |
| 62 | +\newcommand{\ArmLevelNWR}{\ensuremath{-}0.01} | |
| 63 | +\newcommand{\ArmLevelNWRAbs}{0.01} | |
| 64 | +\newcommand{\ArmLevelNetO}{\ensuremath{-}0.05} | |
| 65 | +\newcommand{\ArmLevelNetOAbs}{0.05} | |
| 66 | +\newcommand{\ArmLevelNetR}{\ensuremath{-}0.02} | |
| 67 | +\newcommand{\ArmLevelNetRAbs}{0.02} | |
| 68 | +\newcommand{\ArmLevelNominalRateZero}{0.219} | |
| 69 | +\newcommand{\ArmLevelOwnerOldConsMax}{\ensuremath{-}2.9} | |
| 70 | +\newcommand{\ArmLevelOwnerOldConsMaxAbs}{2.9} | |
| 71 | +\newcommand{\ArmLevelOwnerOldConsMin}{\ensuremath{-}2.2} | |
| 72 | +\newcommand{\ArmLevelOwnerOldConsMinAbs}{2.2} | |
| 73 | +\newcommand{\ArmLevelOwnerOldLossMax}{0.10} | |
| 74 | +\newcommand{\ArmLevelOwnerOldLossMin}{0.06} | |
| 75 | +\newcommand{\ArmLevelOwnerYoungConsMax}{\ensuremath{-}0.4} | |
| 76 | +\newcommand{\ArmLevelOwnerYoungConsMaxAbs}{0.4} | |
| 77 | +\newcommand{\ArmLevelOwnerYoungConsMin}{\ensuremath{-}0.7} | |
| 78 | +\newcommand{\ArmLevelOwnerYoungConsMinAbs}{0.7} | |
| 79 | +\newcommand{\ArmLevelOwnerYoungGainMax}{\ensuremath{-}0.02} | |
| 80 | +\newcommand{\ArmLevelOwnerYoungGainMaxAbs}{0.02} | |
| 81 | +\newcommand{\ArmLevelPriceBackYears}{25} | |
| 82 | +\newcommand{\ArmLevelPriceMaxPos}{0.1} | |
| 83 | +\newcommand{\ArmLevelPriceSignChanges}{1} | |
| 84 | +\newcommand{\ArmLevelPzero}{\ensuremath{-}0.79} | |
| 85 | +\newcommand{\ArmLevelPzeroAbs}{0.79} | |
| 86 | +\newcommand{\ArmLevelRentGainR}{0.00} | |
| 87 | +\newcommand{\ArmLevelResT}{2\times10^{-4}} | |
| 88 | +\newcommand{\ArmLevelRhoZero}{\ensuremath{-}1.15} | |
| 89 | +\newcommand{\ArmLevelRhoZeroAbs}{1.15} | |
| 90 | +\newcommand{\ArmLevelSecs}{153} | |
| 91 | +\newcommand{\ArmLevelSteps}{5} | |
| 92 | +\newcommand{\ArmLevelThrChangePct}{6} | |
| 93 | +\newcommand{\ArmLevelThrSS}{0.49} | |
| 94 | +\newcommand{\ArmLevelThrZero}{0.52} | |
| 95 | +\newcommand{\ArmLevelWageAll}{0.00} | |
| 96 | +\newcommand{\ArmLevelWageO}{0.00} | |
| 97 | +\newcommand{\ArmLevelqROsevenSS}{0.806} | |
| 98 | +\newcommand{\ArmLevelqROsevenZero}{0.802} | |
| 99 | +\newcommand{\ArmLevelqROthreeOne}{0.337} | |
| 100 | +\newcommand{\ArmLevelqROthreeSS}{0.351} | |
| 101 | +\newcommand{\ArmLevelqROthreeZero}{0.358} | |
| 102 | +\newcommand{\ArmPersAbsorbPct}{83} | |
| 103 | +\newcommand{\ArmPersBetterAllPct}{13} | |
| 104 | +\newcommand{\ArmPersBetterOMPct}{7} | |
| 105 | +\newcommand{\ArmPersBetterONMPct}{1} | |
| 106 | +\newcommand{\ArmPersBetterOPct}{5} | |
| 107 | +\newcommand{\ArmPersBetterRPct}{24} | |
| 108 | +\newcommand{\ArmPersCEVAll}{\ensuremath{-}1.24} | |
| 109 | +\newcommand{\ArmPersCEVAllAbs}{1.24} | |
| 110 | +\newcommand{\ArmPersCEVAllPE}{\ensuremath{-}1.24} | |
| 111 | +\newcommand{\ArmPersCEVAllPEAbs}{1.24} | |
| 112 | +\newcommand{\ArmPersCEVO}{\ensuremath{-}1.50} | |
| 113 | +\newcommand{\ArmPersCEVOAbs}{1.50} | |
| 114 | +\newcommand{\ArmPersCEVOM}{\ensuremath{-}1.21} | |
| 115 | +\newcommand{\ArmPersCEVOMAbs}{1.21} | |
| 116 | +\newcommand{\ArmPersCEVONM}{\ensuremath{-}2.16} | |
| 117 | +\newcommand{\ArmPersCEVONMAbs}{2.16} | |
| 118 | +\newcommand{\ArmPersCEVR}{\ensuremath{-}1.04} | |
| 119 | +\newcommand{\ArmPersCEVRAbs}{1.04} | |
| 120 | +\newcommand{\ArmPersConsAll}{\ensuremath{-}2.0} | |
| 121 | +\newcommand{\ArmPersConsAllAbs}{2.0} | |
| 122 | +\newcommand{\ArmPersConsO}{\ensuremath{-}2.1} | |
| 123 | +\newcommand{\ArmPersConsOAbs}{2.1} | |
| 124 | +\newcommand{\ArmPersConsR}{\ensuremath{-}1.9} | |
| 125 | +\newcommand{\ArmPersConsRAbs}{1.9} | |
| 126 | +\newcommand{\ArmPersDOwnGE}{\ensuremath{-}0.39} | |
| 127 | +\newcommand{\ArmPersDOwnGEAbs}{0.39} | |
| 128 | +\newcommand{\ArmPersDOwnPE}{\ensuremath{-}2.34} | |
| 129 | +\newcommand{\ArmPersDOwnPEAbs}{2.34} | |
| 130 | +\newcommand{\ArmPersDebtZeroPct}{\ensuremath{-}9.4} | |
| 131 | +\newcommand{\ArmPersDebtZeroPctAbs}{9.4} | |
| 132 | +\newcommand{\ArmPersDepLoss}{0.48} | |
| 133 | +\newcommand{\ArmPersErosion}{0.00} | |
| 134 | +\newcommand{\ArmPersLamSevenSS}{0.060} | |
| 135 | +\newcommand{\ArmPersLamSevenZero}{0.069} | |
| 136 | +\newcommand{\ArmPersLender}{0.48} | |
| 137 | +\newcommand{\ArmPersMaxRes}{3.1\times10^{-4}} | |
| 138 | +\newcommand{\ArmPersNWAll}{\ensuremath{-}0.04} | |
| 139 | +\newcommand{\ArmPersNWAllAbs}{0.04} | |
| 140 | +\newcommand{\ArmPersNWO}{\ensuremath{-}0.05} | |
| 141 | +\newcommand{\ArmPersNWOAbs}{0.05} | |
| 142 | +\newcommand{\ArmPersNWR}{\ensuremath{-}0.01} | |
| 143 | +\newcommand{\ArmPersNWRAbs}{0.01} | |
| 144 | +\newcommand{\ArmPersNetO}{\ensuremath{-}0.05} | |
| 145 | +\newcommand{\ArmPersNetOAbs}{0.05} | |
| 146 | +\newcommand{\ArmPersNetR}{\ensuremath{-}0.02} | |
| 147 | +\newcommand{\ArmPersNetRAbs}{0.02} | |
| 148 | +\newcommand{\ArmPersNominalRateZero}{0.341} | |
| 149 | +\newcommand{\ArmPersOwnerOldConsMax}{\ensuremath{-}3.1} | |
| 150 | +\newcommand{\ArmPersOwnerOldConsMaxAbs}{3.1} | |
| 151 | +\newcommand{\ArmPersOwnerOldConsMin}{\ensuremath{-}2.4} | |
| 152 | +\newcommand{\ArmPersOwnerOldConsMinAbs}{2.4} | |
| 153 | +\newcommand{\ArmPersOwnerOldLossMax}{0.10} | |
| 154 | +\newcommand{\ArmPersOwnerOldLossMin}{0.06} | |
| 155 | +\newcommand{\ArmPersOwnerYoungConsMax}{\ensuremath{-}0.9} | |
| 156 | +\newcommand{\ArmPersOwnerYoungConsMaxAbs}{0.9} | |
| 157 | +\newcommand{\ArmPersOwnerYoungConsMin}{\ensuremath{-}1.6} | |
| 158 | +\newcommand{\ArmPersOwnerYoungConsMinAbs}{1.6} | |
| 159 | +\newcommand{\ArmPersOwnerYoungGainMax}{\ensuremath{-}0.02} | |
| 160 | +\newcommand{\ArmPersOwnerYoungGainMaxAbs}{0.02} | |
| 161 | +\newcommand{\ArmPersPriceBackYears}{20} | |
| 162 | +\newcommand{\ArmPersPriceMaxPos}{0.1} | |
| 163 | +\newcommand{\ArmPersPriceSignChanges}{1} | |
| 164 | +\newcommand{\ArmPersPzero}{\ensuremath{-}1.86} | |
| 165 | +\newcommand{\ArmPersPzeroAbs}{1.86} | |
| 166 | +\newcommand{\ArmPersRentGainR}{0.00} | |
| 167 | +\newcommand{\ArmPersResT}{6\times10^{-5}} | |
| 168 | +\newcommand{\ArmPersRhoZero}{\ensuremath{-}0.18} | |
| 169 | +\newcommand{\ArmPersRhoZeroAbs}{0.18} | |
| 170 | +\newcommand{\ArmPersSecs}{184} | |
| 171 | +\newcommand{\ArmPersSteps}{4} | |
| 172 | +\newcommand{\ArmPersThrChangePct}{16} | |
| 173 | +\newcommand{\ArmPersThrSS}{0.49} | |
| 174 | +\newcommand{\ArmPersThrZero}{0.57} | |
| 175 | +\newcommand{\ArmPersWageAll}{0.00} | |
| 176 | +\newcommand{\ArmPersWageO}{0.00} | |
| 177 | +\newcommand{\ArmPersqROsevenSS}{0.806} | |
| 178 | +\newcommand{\ArmPersqROsevenZero}{0.807} | |
| 179 | +\newcommand{\ArmPersqROthreeOne}{0.336} | |
| 180 | +\newcommand{\ArmPersqROthreeSS}{0.351} | |
| 181 | +\newcommand{\ArmPersqROthreeZero}{0.358} | |
| 182 | +\newcommand{\BaselineSeconds}{8} | |
| 183 | +\newcommand{\BellmanOwnerMax}{1.9} | |
| 184 | +\newcommand{\BellmanOwnerMaxAll}{1.9\times10^{-3}} | |
| 185 | +\newcommand{\BellmanOwnerMean}{0.00067} | |
| 186 | +\newcommand{\BellmanOwnerMinAll}{4.0\times10^{-4}} | |
| 187 | +\newcommand{\BellmanOwnerPnn}{0.6} | |
| 188 | +\newcommand{\BellmanRenterMax}{0.17} | |
| 189 | +\newcommand{\BellmanRenterMean}{0.00016} | |
| 190 | +\newcommand{\BeqHalfEstate}{0.21} | |
| 191 | +\newcommand{\BeqHalfGapFour}{0.86} | |
| 192 | +\newcommand{\BeqHalfGift}{0.26} | |
| 193 | +\newcommand{\BeqHalfGini}{0.55} | |
| 194 | +\newcommand{\BeqHalfLamSeven}{0.035} | |
| 195 | +\newcommand{\BeqHalfLamThree}{0.12} | |
| 196 | +\newcommand{\BeqHalfOwn}{0.615} | |
| 197 | +\newcommand{\BeqHalfOwnUthirtyfive}{0.325} | |
| 198 | +\newcommand{\BeqHalfP}{0.981} | |
| 199 | +\newcommand{\BeqHalfPDevPct}{\ensuremath{-}1.9} | |
| 200 | +\newcommand{\BeqHalfPDevPctAbs}{1.9} | |
| 201 | +\newcommand{\BeqHalfRR}{0.25} | |
| 202 | +\newcommand{\BeqHalfRebate}{0.000} | |
| 203 | +\newcommand{\BeqHalfRho}{0.198} | |
| 204 | +\newcommand{\BeqHalfRhoDevPct}{1.7} | |
| 205 | +\newcommand{\BeqHalfThr}{0.44} | |
| 206 | +\newcommand{\BeqHalfThrHighZ}{0.00} | |
| 207 | +\newcommand{\BeqHalfThrLowZ}{1.09} | |
| 208 | +\newcommand{\BeqHalfqOO}{0.492} | |
| 209 | +\newcommand{\BeqHalfqRO}{0.368} | |
| 210 | +\newcommand{\BeqOffEstate}{0.00} | |
| 211 | +\newcommand{\BeqOffGEDropPP}{5} | |
| 212 | +\newcommand{\BeqOffGapFour}{0.73} | |
| 213 | +\newcommand{\BeqOffGift}{0.00} | |
| 214 | +\newcommand{\BeqOffGini}{0.58} | |
| 215 | +\newcommand{\BeqOffLamSeven}{\ensuremath{-}0.003} | |
| 216 | +\newcommand{\BeqOffLamSevenAbs}{0.003} | |
| 217 | +\newcommand{\BeqOffLamThree}{0.00} | |
| 218 | +\newcommand{\BeqOffOwn}{0.581} | |
| 219 | +\newcommand{\BeqOffOwnUthirtyfive}{0.188} | |
| 220 | +\newcommand{\BeqOffP}{0.931} | |
| 221 | +\newcommand{\BeqOffPDevPct}{\ensuremath{-}6.9} | |
| 222 | +\newcommand{\BeqOffPDevPctAbs}{6.9} | |
| 223 | +\newcommand{\BeqOffPEDropPP}{12} | |
| 224 | +\newcommand{\BeqOffPEEstate}{0.00} | |
| 225 | +\newcommand{\BeqOffPEGapFour}{0.74} | |
| 226 | +\newcommand{\BeqOffPEGift}{0.00} | |
| 227 | +\newcommand{\BeqOffPEGini}{0.59} | |
| 228 | +\newcommand{\BeqOffPELamSeven}{\ensuremath{-}0.004} | |
| 229 | +\newcommand{\BeqOffPELamSevenAbs}{0.004} | |
| 230 | +\newcommand{\BeqOffPELamThree}{0.00} | |
| 231 | +\newcommand{\BeqOffPEOwn}{0.512} | |
| 232 | +\newcommand{\BeqOffPEOwnUthirtyfive}{0.106} | |
| 233 | +\newcommand{\BeqOffPEP}{1.000} | |
| 234 | +\newcommand{\BeqOffPEPDevPct}{0.0} | |
| 235 | +\newcommand{\BeqOffPERR}{0.00} | |
| 236 | +\newcommand{\BeqOffPERebate}{0.000} | |
| 237 | +\newcommand{\BeqOffPERho}{0.194} | |
| 238 | +\newcommand{\BeqOffPERhoDevPct}{0.0} | |
| 239 | +\newcommand{\BeqOffPEThr}{0.48} | |
| 240 | +\newcommand{\BeqOffPEThrHighZ}{0.03} | |
| 241 | +\newcommand{\BeqOffPEThrLowZ}{1.15} | |
| 242 | +\newcommand{\BeqOffPEqOO}{0.363} | |
| 243 | +\newcommand{\BeqOffPEqRO}{0.362} | |
| 244 | +\newcommand{\BeqOffRR}{\ensuremath{-}0.01} | |
| 245 | +\newcommand{\BeqOffRRAbs}{0.01} | |
| 246 | +\newcommand{\BeqOffRebate}{0.000} | |
| 247 | +\newcommand{\BeqOffRho}{0.209} | |
| 248 | +\newcommand{\BeqOffRhoDevPct}{7.7} | |
| 249 | +\newcommand{\BeqOffThr}{0.32} | |
| 250 | +\newcommand{\BeqOffThrHighZ}{\ensuremath{-}0.18} | |
| 251 | +\newcommand{\BeqOffThrHighZAbs}{0.18} | |
| 252 | +\newcommand{\BeqOffThrLowZ}{0.93} | |
| 253 | +\newcommand{\BeqOffqOO}{0.381} | |
| 254 | +\newcommand{\BeqOffqRO}{0.379} | |
| 255 | +\newcommand{\BetaHighEstate}{0.62} | |
| 256 | +\newcommand{\BetaHighGapFour}{1.21} | |
| 257 | +\newcommand{\BetaHighGift}{0.44} | |
| 258 | +\newcommand{\BetaHighGini}{0.50} | |
| 259 | +\newcommand{\BetaHighLamSeven}{0.090} | |
| 260 | +\newcommand{\BetaHighLamThree}{0.19} | |
| 261 | +\newcommand{\BetaHighOwn}{0.681} | |
| 262 | +\newcommand{\BetaHighOwnUthirtyfive}{0.408} | |
| 263 | +\newcommand{\BetaHighP}{1.075} | |
| 264 | +\newcommand{\BetaHighPDevPct}{7.5} | |
| 265 | +\newcommand{\BetaHighRR}{0.44} | |
| 266 | +\newcommand{\BetaHighRebate}{0.000} | |
| 267 | +\newcommand{\BetaHighRho}{0.181} | |
| 268 | +\newcommand{\BetaHighRhoDevPct}{\ensuremath{-}7.0} | |
| 269 | +\newcommand{\BetaHighRhoDevPctAbs}{7.0} | |
| 270 | +\newcommand{\BetaHighThr}{0.66} | |
| 271 | +\newcommand{\BetaHighThrHighZ}{0.01} | |
| 272 | +\newcommand{\BetaHighThrLowZ}{1.42} | |
| 273 | +\newcommand{\BetaHighqOO}{0.597} | |
| 274 | +\newcommand{\BetaHighqRO}{0.404} | |
| 275 | +\newcommand{\BetaLowEstate}{0.16} | |
| 276 | +\newcommand{\BetaLowGapFour}{0.69} | |
| 277 | +\newcommand{\BetaLowGift}{0.35} | |
| 278 | +\newcommand{\BetaLowGini}{0.58} | |
| 279 | +\newcommand{\BetaLowLamSeven}{0.043} | |
| 280 | +\newcommand{\BetaLowLamThree}{0.18} | |
| 281 | +\newcommand{\BetaLowOwn}{0.581} | |
| 282 | +\newcommand{\BetaLowOwnUthirtyfive}{0.329} | |
| 283 | +\newcommand{\BetaLowP}{0.933} | |
| 284 | +\newcommand{\BetaLowPDevPct}{\ensuremath{-}6.7} | |
| 285 | +\newcommand{\BetaLowPDevPctAbs}{6.7} | |
| 286 | +\newcommand{\BetaLowRR}{0.23} | |
| 287 | +\newcommand{\BetaLowRebate}{0.000} | |
| 288 | +\newcommand{\BetaLowRho}{0.207} | |
| 289 | +\newcommand{\BetaLowRhoDevPct}{6.4} | |
| 290 | +\newcommand{\BetaLowThr}{0.46} | |
| 291 | +\newcommand{\BetaLowThrHighZ}{0.09} | |
| 292 | +\newcommand{\BetaLowThrLowZ}{1.00} | |
| 293 | +\newcommand{\BetaLowqOO}{0.496} | |
| 294 | +\newcommand{\BetaLowqRO}{0.316} | |
| 295 | +\newcommand{\CEVBeqOffAll}{11.14} | |
| 296 | +\newcommand{\CEVBeqOffBottomTwo}{8.98} | |
| 297 | +\newcommand{\CEVBeqOffMarg}{12.82} | |
| 298 | +\newcommand{\CEVBeqOffQfive}{15.53} | |
| 299 | +\newcommand{\CEVBeqOffQone}{8.87} | |
| 300 | +\newcommand{\CEVBeqOffQtwo}{9.09} | |
| 301 | +\newcommand{\CEVDepPremAll}{8.73} | |
| 302 | +\newcommand{\CEVDepPremBottomTwo}{5.46} | |
| 303 | +\newcommand{\CEVDepPremMarg}{10.27} | |
| 304 | +\newcommand{\CEVDepPremQfive}{15.40} | |
| 305 | +\newcommand{\CEVDepPremQone}{5.23} | |
| 306 | +\newcommand{\CEVDepPremQtwo}{5.69} | |
| 307 | +\newcommand{\CEVEpsHighAll}{0.00} | |
| 308 | +\newcommand{\CEVEpsHighBottomTwo}{0.00} | |
| 309 | +\newcommand{\CEVEpsHighMarg}{0.00} | |
| 310 | +\newcommand{\CEVEpsHighQfive}{0.00} | |
| 311 | +\newcommand{\CEVEpsHighQone}{0.00} | |
| 312 | +\newcommand{\CEVEpsHighQtwo}{0.00} | |
| 313 | +\newcommand{\CEVEpsHighThetaFiveAll}{0.29} | |
| 314 | +\newcommand{\CEVEpsHighThetaFiveBottomTwo}{0.47} | |
| 315 | +\newcommand{\CEVEpsHighThetaFiveMarg}{0.27} | |
| 316 | +\newcommand{\CEVEpsHighThetaFiveQfive}{\ensuremath{-}0.08} | |
| 317 | +\newcommand{\CEVEpsHighThetaFiveQfiveAbs}{0.08} | |
| 318 | +\newcommand{\CEVEpsHighThetaFiveQone}{0.48} | |
| 319 | +\newcommand{\CEVEpsHighThetaFiveQtwo}{0.46} | |
| 320 | +\newcommand{\CEVEpsLowAll}{0.00} | |
| 321 | +\newcommand{\CEVEpsLowBottomTwo}{0.00} | |
| 322 | +\newcommand{\CEVEpsLowMarg}{0.00} | |
| 323 | +\newcommand{\CEVEpsLowQfive}{0.00} | |
| 324 | +\newcommand{\CEVEpsLowQone}{0.00} | |
| 325 | +\newcommand{\CEVEpsLowQtwo}{0.00} | |
| 326 | +\newcommand{\CEVEpsLowThetaFiveAll}{0.42} | |
| 327 | +\newcommand{\CEVEpsLowThetaFiveBottomTwo}{0.66} | |
| 328 | +\newcommand{\CEVEpsLowThetaFiveMarg}{0.38} | |
| 329 | +\newcommand{\CEVEpsLowThetaFiveQfive}{\ensuremath{-}0.07} | |
| 330 | +\newcommand{\CEVEpsLowThetaFiveQfiveAbs}{0.07} | |
| 331 | +\newcommand{\CEVEpsLowThetaFiveQone}{0.67} | |
| 332 | +\newcommand{\CEVEpsLowThetaFiveQtwo}{0.65} | |
| 333 | +\newcommand{\CEVKappaHighAll}{\ensuremath{-}0.10} | |
| 334 | +\newcommand{\CEVKappaHighAllAbs}{0.10} | |
| 335 | +\newcommand{\CEVKappaHighBottomTwo}{\ensuremath{-}0.08} | |
| 336 | +\newcommand{\CEVKappaHighBottomTwoAbs}{0.08} | |
| 337 | +\newcommand{\CEVKappaHighMarg}{\ensuremath{-}0.18} | |
| 338 | +\newcommand{\CEVKappaHighMargAbs}{0.18} | |
| 339 | +\newcommand{\CEVKappaHighQfive}{\ensuremath{-}0.10} | |
| 340 | +\newcommand{\CEVKappaHighQfiveAbs}{0.10} | |
| 341 | +\newcommand{\CEVKappaHighQone}{\ensuremath{-}0.07} | |
| 342 | +\newcommand{\CEVKappaHighQoneAbs}{0.07} | |
| 343 | +\newcommand{\CEVKappaHighQtwo}{\ensuremath{-}0.08} | |
| 344 | +\newcommand{\CEVKappaHighQtwoAbs}{0.08} | |
| 345 | +\newcommand{\CEVKappaLowAll}{\ensuremath{-}0.19} | |
| 346 | +\newcommand{\CEVKappaLowAllAbs}{0.19} | |
| 347 | +\newcommand{\CEVKappaLowBottomTwo}{\ensuremath{-}0.22} | |
| 348 | +\newcommand{\CEVKappaLowBottomTwoAbs}{0.22} | |
| 349 | +\newcommand{\CEVKappaLowMarg}{\ensuremath{-}0.14} | |
| 350 | +\newcommand{\CEVKappaLowMargAbs}{0.14} | |
| 351 | +\newcommand{\CEVKappaLowQfive}{\ensuremath{-}0.17} | |
| 352 | +\newcommand{\CEVKappaLowQfiveAbs}{0.17} | |
| 353 | +\newcommand{\CEVKappaLowQone}{\ensuremath{-}0.22} | |
| 354 | +\newcommand{\CEVKappaLowQoneAbs}{0.22} | |
| 355 | +\newcommand{\CEVKappaLowQtwo}{\ensuremath{-}0.22} | |
| 356 | +\newcommand{\CEVKappaLowQtwoAbs}{0.22} | |
| 357 | +\newcommand{\CEVNumericalMaxPct}{0.03} | |
| 358 | +\newcommand{\CEVPhiHighAll}{\ensuremath{-}0.54} | |
| 359 | +\newcommand{\CEVPhiHighAllAbs}{0.54} | |
| 360 | +\newcommand{\CEVPhiHighBottomTwo}{\ensuremath{-}0.68} | |
| 361 | +\newcommand{\CEVPhiHighBottomTwoAbs}{0.68} | |
| 362 | +\newcommand{\CEVPhiHighMarg}{\ensuremath{-}0.48} | |
| 363 | +\newcommand{\CEVPhiHighMargAbs}{0.48} | |
| 364 | +\newcommand{\CEVPhiHighQfive}{\ensuremath{-}0.26} | |
| 365 | +\newcommand{\CEVPhiHighQfiveAbs}{0.26} | |
| 366 | +\newcommand{\CEVPhiHighQone}{\ensuremath{-}0.69} | |
| 367 | +\newcommand{\CEVPhiHighQoneAbs}{0.69} | |
| 368 | +\newcommand{\CEVPhiHighQtwo}{\ensuremath{-}0.66} | |
| 369 | +\newcommand{\CEVPhiHighQtwoAbs}{0.66} | |
| 370 | +\newcommand{\CEVPhiLowAll}{0.65} | |
| 371 | +\newcommand{\CEVPhiLowBottomTwo}{0.82} | |
| 372 | +\newcommand{\CEVPhiLowMarg}{0.57} | |
| 373 | +\newcommand{\CEVPhiLowQfive}{0.31} | |
| 374 | +\newcommand{\CEVPhiLowQone}{0.83} | |
| 375 | +\newcommand{\CEVPhiLowQtwo}{0.81} | |
| 376 | +\newcommand{\CEVPiFourAll}{\ensuremath{-}0.01} | |
| 377 | +\newcommand{\CEVPiFourAllAbs}{0.01} | |
| 378 | +\newcommand{\CEVPiFourBottomTwo}{\ensuremath{-}0.11} | |
| 379 | +\newcommand{\CEVPiFourBottomTwoAbs}{0.11} | |
| 380 | +\newcommand{\CEVPiFourMarg}{\ensuremath{-}0.08} | |
| 381 | +\newcommand{\CEVPiFourMargAbs}{0.08} | |
| 382 | +\newcommand{\CEVPiFourQfive}{0.23} | |
| 383 | +\newcommand{\CEVPiFourQone}{\ensuremath{-}0.11} | |
| 384 | +\newcommand{\CEVPiFourQoneAbs}{0.11} | |
| 385 | +\newcommand{\CEVPiFourQtwo}{\ensuremath{-}0.11} | |
| 386 | +\newcommand{\CEVPiFourQtwoAbs}{0.11} | |
| 387 | +\newcommand{\CEVPiZeroAll}{0.12} | |
| 388 | +\newcommand{\CEVPiZeroBottomTwo}{0.23} | |
| 389 | +\newcommand{\CEVPiZeroMarg}{0.22} | |
| 390 | +\newcommand{\CEVPiZeroQfive}{\ensuremath{-}0.14} | |
| 391 | +\newcommand{\CEVPiZeroQfiveAbs}{0.14} | |
| 392 | +\newcommand{\CEVPiZeroQone}{0.23} | |
| 393 | +\newcommand{\CEVPiZeroQtwo}{0.23} | |
| 394 | +\newcommand{\CEVPsiFiftyAll}{\ensuremath{-}0.05} | |
| 395 | +\newcommand{\CEVPsiFiftyAllAbs}{0.05} | |
| 396 | +\newcommand{\CEVPsiFiftyBottomTwo}{\ensuremath{-}0.02} | |
| 397 | +\newcommand{\CEVPsiFiftyBottomTwoAbs}{0.02} | |
| 398 | +\newcommand{\CEVPsiFiftyMarg}{\ensuremath{-}0.01} | |
| 399 | +\newcommand{\CEVPsiFiftyMargAbs}{0.01} | |
| 400 | +\newcommand{\CEVPsiFiftyQfive}{\ensuremath{-}0.12} | |
| 401 | +\newcommand{\CEVPsiFiftyQfiveAbs}{0.12} | |
| 402 | +\newcommand{\CEVPsiFiftyQone}{\ensuremath{-}0.01} | |
| 403 | +\newcommand{\CEVPsiFiftyQoneAbs}{0.01} | |
| 404 | +\newcommand{\CEVPsiFiftyQtwo}{\ensuremath{-}0.02} | |
| 405 | +\newcommand{\CEVPsiFiftyQtwoAbs}{0.02} | |
| 406 | +\newcommand{\CEVPsiThirtyAll}{0.02} | |
| 407 | +\newcommand{\CEVPsiThirtyBottomTwo}{0.00} | |
| 408 | +\newcommand{\CEVPsiThirtyMarg}{\ensuremath{-}0.08} | |
| 409 | +\newcommand{\CEVPsiThirtyMargAbs}{0.08} | |
| 410 | +\newcommand{\CEVPsiThirtyQfive}{0.09} | |
| 411 | +\newcommand{\CEVPsiThirtyQone}{\ensuremath{-}0.01} | |
| 412 | +\newcommand{\CEVPsiThirtyQoneAbs}{0.01} | |
| 413 | +\newcommand{\CEVPsiThirtyQtwo}{0.00} | |
| 414 | +\newcommand{\CEVRhoGAll}{0.03} | |
| 415 | +\newcommand{\CEVRhoGBottomTwo}{0.05} | |
| 416 | +\newcommand{\CEVRhoGMarg}{0.02} | |
| 417 | +\newcommand{\CEVRhoGQfive}{\ensuremath{-}0.02} | |
| 418 | +\newcommand{\CEVRhoGQfiveAbs}{0.02} | |
| 419 | +\newcommand{\CEVRhoGQone}{0.05} | |
| 420 | +\newcommand{\CEVRhoGQtwo}{0.05} | |
| 421 | +\newcommand{\CEVRoneAll}{\ensuremath{-}0.35} | |
| 422 | +\newcommand{\CEVRoneAllAbs}{0.35} | |
| 423 | +\newcommand{\CEVRoneBottomTwo}{0.59} | |
| 424 | +\newcommand{\CEVRoneMarg}{\ensuremath{-}0.86} | |
| 425 | +\newcommand{\CEVRoneMargAbs}{0.86} | |
| 426 | +\newcommand{\CEVRoneQfive}{\ensuremath{-}2.25} | |
| 427 | +\newcommand{\CEVRoneQfiveAbs}{2.25} | |
| 428 | +\newcommand{\CEVRoneQone}{0.68} | |
| 429 | +\newcommand{\CEVRoneQtwo}{0.51} | |
| 430 | +\newcommand{\CEVRthreeAll}{1.80} | |
| 431 | +\newcommand{\CEVRthreeBottomTwo}{0.52} | |
| 432 | +\newcommand{\CEVRthreeMarg}{2.32} | |
| 433 | +\newcommand{\CEVRthreeQfive}{4.46} | |
| 434 | +\newcommand{\CEVRthreeQone}{0.42} | |
| 435 | +\newcommand{\CEVRthreeQtwo}{0.62} | |
| 436 | +\newcommand{\CEVSampleLargeAll}{\ensuremath{-}0.02} | |
| 437 | +\newcommand{\CEVSampleLargeAllAbs}{0.02} | |
| 438 | +\newcommand{\CEVSampleLargeBottomTwo}{\ensuremath{-}0.02} | |
| 439 | +\newcommand{\CEVSampleLargeBottomTwoAbs}{0.02} | |
| 440 | +\newcommand{\CEVSampleLargeMarg}{\ensuremath{-}0.02} | |
| 441 | +\newcommand{\CEVSampleLargeMargAbs}{0.02} | |
| 442 | +\newcommand{\CEVSampleLargeQfive}{\ensuremath{-}0.02} | |
| 443 | +\newcommand{\CEVSampleLargeQfiveAbs}{0.02} | |
| 444 | +\newcommand{\CEVSampleLargeQone}{\ensuremath{-}0.02} | |
| 445 | +\newcommand{\CEVSampleLargeQoneAbs}{0.02} | |
| 446 | +\newcommand{\CEVSampleLargeQtwo}{\ensuremath{-}0.02} | |
| 447 | +\newcommand{\CEVSampleLargeQtwoAbs}{0.02} | |
| 448 | +\newcommand{\CEVSeedAll}{\ensuremath{-}0.01} | |
| 449 | +\newcommand{\CEVSeedAllAbs}{0.01} | |
| 450 | +\newcommand{\CEVSeedBottomTwo}{\ensuremath{-}0.01} | |
| 451 | +\newcommand{\CEVSeedBottomTwoAbs}{0.01} | |
| 452 | +\newcommand{\CEVSeedMarg}{\ensuremath{-}0.01} | |
| 453 | +\newcommand{\CEVSeedMargAbs}{0.01} | |
| 454 | +\newcommand{\CEVSeedQfive}{0.00} | |
| 455 | +\newcommand{\CEVSeedQone}{\ensuremath{-}0.01} | |
| 456 | +\newcommand{\CEVSeedQoneAbs}{0.01} | |
| 457 | +\newcommand{\CEVSeedQtwo}{\ensuremath{-}0.01} | |
| 458 | +\newcommand{\CEVSeedQtwoAbs}{0.01} | |
| 459 | +\newcommand{\CEVSigHighAll}{\ensuremath{-}10.83} | |
| 460 | +\newcommand{\CEVSigHighAllAbs}{10.83} | |
| 461 | +\newcommand{\CEVSigHighBottomTwo}{\ensuremath{-}12.46} | |
| 462 | +\newcommand{\CEVSigHighBottomTwoAbs}{12.46} | |
| 463 | +\newcommand{\CEVSigHighMarg}{\ensuremath{-}9.28} | |
| 464 | +\newcommand{\CEVSigHighMargAbs}{9.28} | |
| 465 | +\newcommand{\CEVSigHighQfive}{\ensuremath{-}8.13} | |
| 466 | +\newcommand{\CEVSigHighQfiveAbs}{8.13} | |
| 467 | +\newcommand{\CEVSigHighQone}{\ensuremath{-}12.69} | |
| 468 | +\newcommand{\CEVSigHighQoneAbs}{12.69} | |
| 469 | +\newcommand{\CEVSigHighQtwo}{\ensuremath{-}12.24} | |
| 470 | +\newcommand{\CEVSigHighQtwoAbs}{12.24} | |
| 471 | +\newcommand{\CEVSigLowAll}{10.25} | |
| 472 | +\newcommand{\CEVSigLowBottomTwo}{11.82} | |
| 473 | +\newcommand{\CEVSigLowMarg}{9.00} | |
| 474 | +\newcommand{\CEVSigLowQfive}{7.78} | |
| 475 | +\newcommand{\CEVSigLowQone}{12.06} | |
| 476 | +\newcommand{\CEVSigLowQtwo}{11.57} | |
| 477 | +\newcommand{\CEVTaxFiftyAll}{10.85} | |
| 478 | +\newcommand{\CEVTaxFiftyBottomTwo}{14.59} | |
| 479 | +\newcommand{\CEVTaxFiftyMarg}{8.58} | |
| 480 | +\newcommand{\CEVTaxFiftyQfive}{5.89} | |
| 481 | +\newcommand{\CEVTaxFiftyQone}{15.39} | |
| 482 | +\newcommand{\CEVTaxFiftyQtwo}{13.79} | |
| 483 | +\newcommand{\CEVTaxTwentyAll}{7.47} | |
| 484 | +\newcommand{\CEVTaxTwentyBottomTwo}{10.24} | |
| 485 | +\newcommand{\CEVTaxTwentyMarg}{5.74} | |
| 486 | +\newcommand{\CEVTaxTwentyQfive}{4.02} | |
| 487 | +\newcommand{\CEVTaxTwentyQone}{10.85} | |
| 488 | +\newcommand{\CEVTaxTwentyQtwo}{9.64} | |
| 489 | +\newcommand{\CEVThetaFiveAll}{0.34} | |
| 490 | +\newcommand{\CEVThetaFiveBottomTwo}{0.55} | |
| 491 | +\newcommand{\CEVThetaFiveMarg}{0.30} | |
| 492 | +\newcommand{\CEVThetaFiveQfive}{\ensuremath{-}0.08} | |
| 493 | +\newcommand{\CEVThetaFiveQfiveAbs}{0.08} | |
| 494 | +\newcommand{\CEVThetaFiveQone}{0.55} | |
| 495 | +\newcommand{\CEVThetaFiveQtwo}{0.54} | |
| 496 | +\newcommand{\CEVThetaFortyAll}{\ensuremath{-}0.23} | |
| 497 | +\newcommand{\CEVThetaFortyAllAbs}{0.23} | |
| 498 | +\newcommand{\CEVThetaFortyBottomTwo}{\ensuremath{-}0.40} | |
| 499 | +\newcommand{\CEVThetaFortyBottomTwoAbs}{0.40} | |
| 500 | +\newcommand{\CEVThetaFortyMarg}{\ensuremath{-}0.33} | |
| 501 | +\newcommand{\CEVThetaFortyMargAbs}{0.33} | |
| 502 | +\newcommand{\CEVThetaFortyQfive}{0.32} | |
| 503 | +\newcommand{\CEVThetaFortyQone}{\ensuremath{-}0.39} | |
| 504 | +\newcommand{\CEVThetaFortyQoneAbs}{0.39} | |
| 505 | +\newcommand{\CEVThetaFortyQtwo}{\ensuremath{-}0.40} | |
| 506 | +\newcommand{\CEVThetaFortyQtwoAbs}{0.40} | |
| 507 | +\newcommand{\CEVTolTightAll}{0.00} | |
| 508 | +\newcommand{\CEVTolTightBottomTwo}{0.00} | |
| 509 | +\newcommand{\CEVTolTightMarg}{0.00} | |
| 510 | +\newcommand{\CEVTolTightQfive}{0.00} | |
| 511 | +\newcommand{\CEVTolTightQone}{0.00} | |
| 512 | +\newcommand{\CEVTolTightQtwo}{0.00} | |
| 513 | +\newcommand{\CEVallarmlevel}{\ensuremath{-}1.14} | |
| 514 | +\newcommand{\CEVallarmlevelAbs}{1.14} | |
| 515 | +\newcommand{\CEVallarmpersistent}{\ensuremath{-}1.24} | |
| 516 | +\newcommand{\CEVallarmpersistentAbs}{1.24} | |
| 517 | +\newcommand{\CEVallfrmlevel}{\ensuremath{-}0.64} | |
| 518 | +\newcommand{\CEVallfrmlevelAbs}{0.64} | |
| 519 | +\newcommand{\CEVallfrmlevelfixedsupply}{\ensuremath{-}0.64} | |
| 520 | +\newcommand{\CEVallfrmlevelfixedsupplyAbs}{0.64} | |
| 521 | +\newcommand{\CEVallfrmlevelnoseg}{\ensuremath{-}0.60} | |
| 522 | +\newcommand{\CEVallfrmlevelnosegAbs}{0.60} | |
| 523 | +\newcommand{\CEVallfrmlevelrigidrents}{\ensuremath{-}0.62} | |
| 524 | +\newcommand{\CEVallfrmlevelrigidrentsAbs}{0.62} | |
| 525 | +\newcommand{\CEVallfrmlevelstatic}{\ensuremath{-}0.64} | |
| 526 | +\newcommand{\CEVallfrmlevelstaticAbs}{0.64} | |
| 527 | +\newcommand{\CEVallfrmlevelunindexedwages}{\ensuremath{-}3.33} | |
| 528 | +\newcommand{\CEVallfrmlevelunindexedwagesAbs}{3.33} | |
| 529 | +\newcommand{\CEVallfrmpersistent}{\ensuremath{-}0.52} | |
| 530 | +\newcommand{\CEVallfrmpersistentAbs}{0.52} | |
| 531 | +\newcommand{\CEVallfrmpersistenttaylor}{0.14} | |
| 532 | +\newcommand{\CEValltermlevel}{\ensuremath{-}0.64} | |
| 533 | +\newcommand{\CEValltermlevelAbs}{0.64} | |
| 534 | +\newcommand{\CEValltermpersistent}{\ensuremath{-}0.74} | |
| 535 | +\newcommand{\CEValltermpersistentAbs}{0.74} | |
| 536 | +\newcommand{\CEValltermpersistenttaylor}{\ensuremath{-}0.25} | |
| 537 | +\newcommand{\CEValltermpersistenttaylorAbs}{0.25} | |
| 538 | +\newcommand{\CEVbequestsoff}{11.14} | |
| 539 | +\newcommand{\CEVchi}{0.91} | |
| 540 | +\newcommand{\CEVdeppremium}{8.73} | |
| 541 | +\newcommand{\CEVeps}{0.00} | |
| 542 | +\newcommand{\CEVkappahigh}{\ensuremath{-}0.10} | |
| 543 | +\newcommand{\CEVkappahighAbs}{0.10} | |
| 544 | +\newcommand{\CEVkappalow}{\ensuremath{-}0.19} | |
| 545 | +\newcommand{\CEVkappalowAbs}{0.19} | |
| 546 | +\newcommand{\CEVnoseg}{0.00} | |
| 547 | +\newcommand{\CEVomarmlevel}{\ensuremath{-}1.03} | |
| 548 | +\newcommand{\CEVomarmlevelAbs}{1.03} | |
| 549 | +\newcommand{\CEVomarmpersistent}{\ensuremath{-}1.21} | |
| 550 | +\newcommand{\CEVomarmpersistentAbs}{1.21} | |
| 551 | +\newcommand{\CEVomfrmlevel}{0.63} | |
| 552 | +\newcommand{\CEVomfrmlevelfixedsupply}{0.63} | |
| 553 | +\newcommand{\CEVomfrmlevelnoseg}{0.66} | |
| 554 | +\newcommand{\CEVomfrmlevelrigidrents}{0.55} | |
| 555 | +\newcommand{\CEVomfrmlevelstatic}{0.65} | |
| 556 | +\newcommand{\CEVomfrmlevelunindexedwages}{\ensuremath{-}2.40} | |
| 557 | +\newcommand{\CEVomfrmlevelunindexedwagesAbs}{2.40} | |
| 558 | +\newcommand{\CEVomfrmpersistent}{1.14} | |
| 559 | +\newcommand{\CEVomfrmpersistenttaylor}{1.43} | |
| 560 | +\newcommand{\CEVomtermlevel}{0.63} | |
| 561 | +\newcommand{\CEVomtermpersistent}{0.46} | |
| 562 | +\newcommand{\CEVomtermpersistenttaylor}{0.20} | |
| 563 | +\newcommand{\CEVpi}{\ensuremath{-}0.01} | |
| 564 | +\newcommand{\CEVpiAbs}{0.01} | |
| 565 | +\newcommand{\CEVpsi}{\ensuremath{-}0.05} | |
| 566 | +\newcommand{\CEVpsiAbs}{0.05} | |
| 567 | +\newcommand{\CEVr}{1.80} | |
| 568 | +\newcommand{\CEVrarmlevel}{\ensuremath{-}0.96} | |
| 569 | +\newcommand{\CEVrarmlevelAbs}{0.96} | |
| 570 | +\newcommand{\CEVrarmpersistent}{\ensuremath{-}1.04} | |
| 571 | +\newcommand{\CEVrarmpersistentAbs}{1.04} | |
| 572 | +\newcommand{\CEVrfrmlevel}{\ensuremath{-}1.00} | |
| 573 | +\newcommand{\CEVrfrmlevelAbs}{1.00} | |
| 574 | +\newcommand{\CEVrfrmlevelfixedsupply}{\ensuremath{-}0.99} | |
| 575 | +\newcommand{\CEVrfrmlevelfixedsupplyAbs}{0.99} | |
| 576 | +\newcommand{\CEVrfrmlevelnoseg}{\ensuremath{-}0.95} | |
| 577 | +\newcommand{\CEVrfrmlevelnosegAbs}{0.95} | |
| 578 | +\newcommand{\CEVrfrmlevelrigidrents}{\ensuremath{-}0.92} | |
| 579 | +\newcommand{\CEVrfrmlevelrigidrentsAbs}{0.92} | |
| 580 | +\newcommand{\CEVrfrmlevelstatic}{\ensuremath{-}1.01} | |
| 581 | +\newcommand{\CEVrfrmlevelstaticAbs}{1.01} | |
| 582 | +\newcommand{\CEVrfrmlevelunindexedwages}{\ensuremath{-}3.58} | |
| 583 | +\newcommand{\CEVrfrmlevelunindexedwagesAbs}{3.58} | |
| 584 | +\newcommand{\CEVrfrmpersistent}{\ensuremath{-}1.06} | |
| 585 | +\newcommand{\CEVrfrmpersistentAbs}{1.06} | |
| 586 | +\newcommand{\CEVrfrmpersistenttaylor}{\ensuremath{-}0.31} | |
| 587 | +\newcommand{\CEVrfrmpersistenttaylorAbs}{0.31} | |
| 588 | +\newcommand{\CEVrtermlevel}{\ensuremath{-}1.00} | |
| 589 | +\newcommand{\CEVrtermlevelAbs}{1.00} | |
| 590 | +\newcommand{\CEVrtermpersistent}{\ensuremath{-}1.09} | |
| 591 | +\newcommand{\CEVrtermpersistentAbs}{1.09} | |
| 592 | +\newcommand{\CEVrtermpersistenttaylor}{\ensuremath{-}0.32} | |
| 593 | +\newcommand{\CEVrtermpersistenttaylorAbs}{0.32} | |
| 594 | +\newcommand{\CEVtheta}{\ensuremath{-}0.23} | |
| 595 | +\newcommand{\CEVthetaAbs}{0.23} | |
| 596 | +\newcommand{\CEVtransfertax}{10.85} | |
| 597 | +\newcommand{\ChiFiveEstate}{0.32} | |
| 598 | +\newcommand{\ChiFiveGapFour}{0.95} | |
| 599 | +\newcommand{\ChiFiveGift}{0.40} | |
| 600 | +\newcommand{\ChiFiveGini}{0.54} | |
| 601 | +\newcommand{\ChiFiveLamSeven}{0.056} | |
| 602 | +\newcommand{\ChiFiveLamThree}{0.19} | |
| 603 | +\newcommand{\ChiFiveOwn}{0.644} | |
| 604 | +\newcommand{\ChiFiveOwnUthirtyfive}{0.403} | |
| 605 | +\newcommand{\ChiFiveP}{1.006} | |
| 606 | +\newcommand{\ChiFivePDevPct}{0.6} | |
| 607 | +\newcommand{\ChiFiveRR}{0.34} | |
| 608 | +\newcommand{\ChiFiveRebate}{0.000} | |
| 609 | +\newcommand{\ChiFiveRho}{0.190} | |
| 610 | +\newcommand{\ChiFiveRhoDevPct}{\ensuremath{-}2.3} | |
| 611 | +\newcommand{\ChiFiveRhoDevPctAbs}{2.3} | |
| 612 | +\newcommand{\ChiFiveThr}{0.48} | |
| 613 | +\newcommand{\ChiFiveThrHighZ}{0.01} | |
| 614 | +\newcommand{\ChiFiveThrLowZ}{1.14} | |
| 615 | +\newcommand{\ChiFiveqOO}{0.546} | |
| 616 | +\newcommand{\ChiFiveqRO}{0.355} | |
| 617 | +\newcommand{\ClipSharePct}{0.003} | |
| 618 | +\newcommand{\DOwnarmlevel}{\ensuremath{-}0.11} | |
| 619 | +\newcommand{\DOwnarmlevelAbs}{0.11} | |
| 620 | +\newcommand{\DOwnarmpersistent}{\ensuremath{-}0.39} | |
| 621 | +\newcommand{\DOwnarmpersistentAbs}{0.39} | |
| 622 | +\newcommand{\DOwnfrmlevel}{\ensuremath{-}0.02} | |
| 623 | +\newcommand{\DOwnfrmlevelAbs}{0.02} | |
| 624 | +\newcommand{\DOwnfrmlevelfixedsupply}{\ensuremath{-}0.01} | |
| 625 | +\newcommand{\DOwnfrmlevelfixedsupplyAbs}{0.01} | |
| 626 | +\newcommand{\DOwnfrmlevelnoseg}{\ensuremath{-}0.16} | |
| 627 | +\newcommand{\DOwnfrmlevelnosegAbs}{0.16} | |
| 628 | +\newcommand{\DOwnfrmlevelrigidrents}{\ensuremath{-}0.03} | |
| 629 | +\newcommand{\DOwnfrmlevelrigidrentsAbs}{0.03} | |
| 630 | +\newcommand{\DOwnfrmlevelstatic}{0.02} | |
| 631 | +\newcommand{\DOwnfrmlevelunindexedwages}{\ensuremath{-}0.63} | |
| 632 | +\newcommand{\DOwnfrmlevelunindexedwagesAbs}{0.63} | |
| 633 | +\newcommand{\DOwnfrmpersistent}{\ensuremath{-}0.18} | |
| 634 | +\newcommand{\DOwnfrmpersistentAbs}{0.18} | |
| 635 | +\newcommand{\DOwnfrmpersistenttaylor}{\ensuremath{-}0.12} | |
| 636 | +\newcommand{\DOwnfrmpersistenttaylorAbs}{0.12} | |
| 637 | +\newcommand{\DOwntermlevel}{\ensuremath{-}0.02} | |
| 638 | +\newcommand{\DOwntermlevelAbs}{0.02} | |
| 639 | +\newcommand{\DOwntermpersistent}{\ensuremath{-}0.34} | |
| 640 | +\newcommand{\DOwntermpersistentAbs}{0.34} | |
| 641 | +\newcommand{\DOwntermpersistenttaylor}{\ensuremath{-}0.26} | |
| 642 | +\newcommand{\DOwntermpersistenttaylorAbs}{0.26} | |
| 643 | +\newcommand{\DebtDepositRatio}{1.7} | |
| 644 | +\newcommand{\DepPremEstate}{0.77} | |
| 645 | +\newcommand{\DepPremGapFour}{1.19} | |
| 646 | +\newcommand{\DepPremGift}{0.61} | |
| 647 | +\newcommand{\DepPremGini}{0.50} | |
| 648 | +\newcommand{\DepPremLamSeven}{0.169} | |
| 649 | +\newcommand{\DepPremLamThree}{0.28} | |
| 650 | +\newcommand{\DepPremOwn}{0.635} | |
| 651 | +\newcommand{\DepPremOwnUthirtyfive}{0.414} | |
| 652 | +\newcommand{\DepPremP}{1.018} | |
| 653 | +\newcommand{\DepPremPDevPct}{1.8} | |
| 654 | +\newcommand{\DepPremRR}{0.57} | |
| 655 | +\newcommand{\DepPremRebate}{0.000} | |
| 656 | +\newcommand{\DepPremRho}{0.211} | |
| 657 | +\newcommand{\DepPremRhoDevPct}{8.6} | |
| 658 | +\newcommand{\DepPremThr}{0.75} | |
| 659 | +\newcommand{\DepPremThrHighZ}{0.10} | |
| 660 | +\newcommand{\DepPremThrLowZ}{16.32} | |
| 661 | +\newcommand{\DepPremqOO}{0.664} | |
| 662 | +\newcommand{\DepPremqRO}{0.381} | |
| 663 | +\newcommand{\DevCoarseGini}{0.000} | |
| 664 | +\newcommand{\DevCoarseLamSeven}{+0.017} | |
| 665 | +\newcommand{\DevCoarseOwnPP}{\ensuremath{-}0.5} | |
| 666 | +\newcommand{\DevCoarseOwnPPAbs}{0.5} | |
| 667 | +\newcommand{\DevCoarsePPct}{\ensuremath{-}0.8} | |
| 668 | +\newcommand{\DevCoarsePPctAbs}{0.8} | |
| 669 | +\newcommand{\DevCoarseRR}{+0.005} | |
| 670 | +\newcommand{\DevCoarseThr}{+0.07} | |
| 671 | +\newcommand{\DevFineGini}{+0.001} | |
| 672 | +\newcommand{\DevFineLamSeven}{\ensuremath{-}0.007} | |
| 673 | +\newcommand{\DevFineLamSevenAbs}{0.007} | |
| 674 | +\newcommand{\DevFineOwnPP}{0.4} | |
| 675 | +\newcommand{\DevFinePPct}{\ensuremath{-}0.1} | |
| 676 | +\newcommand{\DevFinePPctAbs}{0.1} | |
| 677 | +\newcommand{\DevFineRR}{\ensuremath{-}0.002} | |
| 678 | +\newcommand{\DevFineRRAbs}{0.002} | |
| 679 | +\newcommand{\DevFineThr}{\ensuremath{-}0.02} | |
| 680 | +\newcommand{\DevFineThrAbs}{0.02} | |
| 681 | +\newcommand{\DevSampleLargeGini}{+0.001} | |
| 682 | +\newcommand{\DevSampleLargeLamSeven}{+0.004} | |
| 683 | +\newcommand{\DevSampleLargeOwnPP}{\ensuremath{-}0.1} | |
| 684 | +\newcommand{\DevSampleLargeOwnPPAbs}{0.1} | |
| 685 | +\newcommand{\DevSampleLargePPct}{0.0} | |
| 686 | +\newcommand{\DevSampleLargeRR}{\ensuremath{-}0.002} | |
| 687 | +\newcommand{\DevSampleLargeRRAbs}{0.002} | |
| 688 | +\newcommand{\DevSampleLargeThr}{0.00} | |
| 689 | +\newcommand{\DevSampleSmallGini}{0.000} | |
| 690 | +\newcommand{\DevSampleSmallLamSeven}{+0.013} | |
| 691 | +\newcommand{\DevSampleSmallOwnPP}{0.0} | |
| 692 | +\newcommand{\DevSampleSmallPPct}{0.1} | |
| 693 | +\newcommand{\DevSampleSmallRR}{0.000} | |
| 694 | +\newcommand{\DevSampleSmallThr}{+0.00} | |
| 695 | +\newcommand{\DevSeedGini}{+0.001} | |
| 696 | +\newcommand{\DevSeedLamSeven}{+0.001} | |
| 697 | +\newcommand{\DevSeedOwnPP}{\ensuremath{-}0.1} | |
| 698 | +\newcommand{\DevSeedOwnPPAbs}{0.1} | |
| 699 | +\newcommand{\DevSeedPPct}{0.0} | |
| 700 | +\newcommand{\DevSeedRR}{+0.000} | |
| 701 | +\newcommand{\DevSeedThr}{0.00} | |
| 702 | +\newcommand{\EpsHighThetaFiveEstate}{0.31} | |
| 703 | +\newcommand{\EpsHighThetaFiveGapFour}{0.83} | |
| 704 | +\newcommand{\EpsHighThetaFiveGift}{0.39} | |
| 705 | +\newcommand{\EpsHighThetaFiveGini}{0.55} | |
| 706 | +\newcommand{\EpsHighThetaFiveLamSeven}{0.067} | |
| 707 | +\newcommand{\EpsHighThetaFiveLamThree}{0.20} | |
| 708 | +\newcommand{\EpsHighThetaFiveOwn}{0.644} | |
| 709 | +\newcommand{\EpsHighThetaFiveOwnUthirtyfive}{0.412} | |
| 710 | +\newcommand{\EpsHighThetaFiveP}{1.005} | |
| 711 | +\newcommand{\EpsHighThetaFivePDevPct}{0.5} | |
| 712 | +\newcommand{\EpsHighThetaFiveRR}{0.33} | |
| 713 | +\newcommand{\EpsHighThetaFiveRebate}{0.000} | |
| 714 | +\newcommand{\EpsHighThetaFiveRho}{0.190} | |
| 715 | +\newcommand{\EpsHighThetaFiveRhoDevPct}{\ensuremath{-}2.1} | |
| 716 | +\newcommand{\EpsHighThetaFiveRhoDevPctAbs}{2.1} | |
| 717 | +\newcommand{\EpsHighThetaFiveThr}{0.55} | |
| 718 | +\newcommand{\EpsHighThetaFiveThrHighZ}{\ensuremath{-}0.10} | |
| 719 | +\newcommand{\EpsHighThetaFiveThrHighZAbs}{0.10} | |
| 720 | +\newcommand{\EpsHighThetaFiveThrLowZ}{1.23} | |
| 721 | +\newcommand{\EpsHighThetaFiveqOO}{0.574} | |
| 722 | +\newcommand{\EpsHighThetaFiveqRO}{0.376} | |
| 723 | +\newcommand{\EpsHighThetaFortyEstate}{0.33} | |
| 724 | +\newcommand{\EpsHighThetaFortyGapFour}{0.99} | |
| 725 | +\newcommand{\EpsHighThetaFortyGift}{0.39} | |
| 726 | +\newcommand{\EpsHighThetaFortyGini}{0.54} | |
| 727 | +\newcommand{\EpsHighThetaFortyLamSeven}{0.083} | |
| 728 | +\newcommand{\EpsHighThetaFortyLamThree}{0.21} | |
| 729 | +\newcommand{\EpsHighThetaFortyOwn}{0.615} | |
| 730 | +\newcommand{\EpsHighThetaFortyOwnUthirtyfive}{0.327} | |
| 731 | +\newcommand{\EpsHighThetaFortyP}{0.989} | |
| 732 | +\newcommand{\EpsHighThetaFortyPDevPct}{\ensuremath{-}1.1} | |
| 733 | +\newcommand{\EpsHighThetaFortyPDevPctAbs}{1.1} | |
| 734 | +\newcommand{\EpsHighThetaFortyRR}{0.34} | |
| 735 | +\newcommand{\EpsHighThetaFortyRebate}{0.000} | |
| 736 | +\newcommand{\EpsHighThetaFortyRho}{0.199} | |
| 737 | +\newcommand{\EpsHighThetaFortyRhoDevPct}{2.4} | |
| 738 | +\newcommand{\EpsHighThetaFortyThr}{0.55} | |
| 739 | +\newcommand{\EpsHighThetaFortyThrHighZ}{0.20} | |
| 740 | +\newcommand{\EpsHighThetaFortyThrLowZ}{1.11} | |
| 741 | +\newcommand{\EpsHighThetaFortyqOO}{0.510} | |
| 742 | +\newcommand{\EpsHighThetaFortyqRO}{0.298} | |
| 743 | +\newcommand{\EpsLowThetaFiveEstate}{0.31} | |
| 744 | +\newcommand{\EpsLowThetaFiveGapFour}{0.84} | |
| 745 | +\newcommand{\EpsLowThetaFiveGift}{0.39} | |
| 746 | +\newcommand{\EpsLowThetaFiveGini}{0.55} | |
| 747 | +\newcommand{\EpsLowThetaFiveLamSeven}{0.068} | |
| 748 | +\newcommand{\EpsLowThetaFiveLamThree}{0.19} | |
| 749 | +\newcommand{\EpsLowThetaFiveOwn}{0.636} | |
| 750 | +\newcommand{\EpsLowThetaFiveOwnUthirtyfive}{0.402} | |
| 751 | +\newcommand{\EpsLowThetaFiveP}{1.006} | |
| 752 | +\newcommand{\EpsLowThetaFivePDevPct}{0.6} | |
| 753 | +\newcommand{\EpsLowThetaFiveRR}{0.33} | |
| 754 | +\newcommand{\EpsLowThetaFiveRebate}{0.000} | |
| 755 | +\newcommand{\EpsLowThetaFiveRho}{0.188} | |
| 756 | +\newcommand{\EpsLowThetaFiveRhoDevPct}{\ensuremath{-}3.5} | |
| 757 | +\newcommand{\EpsLowThetaFiveRhoDevPctAbs}{3.5} | |
| 758 | +\newcommand{\EpsLowThetaFiveThr}{0.56} | |
| 759 | +\newcommand{\EpsLowThetaFiveThrHighZ}{\ensuremath{-}0.09} | |
| 760 | +\newcommand{\EpsLowThetaFiveThrHighZAbs}{0.09} | |
| 761 | +\newcommand{\EpsLowThetaFiveThrLowZ}{1.26} | |
| 762 | +\newcommand{\EpsLowThetaFiveqOO}{0.566} | |
| 763 | +\newcommand{\EpsLowThetaFiveqRO}{0.374} | |
| 764 | +\newcommand{\EpsLowThetaFortyEstate}{0.33} | |
| 765 | +\newcommand{\EpsLowThetaFortyGapFour}{0.98} | |
| 766 | +\newcommand{\EpsLowThetaFortyGift}{0.39} | |
| 767 | +\newcommand{\EpsLowThetaFortyGini}{0.53} | |
| 768 | +\newcommand{\EpsLowThetaFortyLamSeven}{0.085} | |
| 769 | +\newcommand{\EpsLowThetaFortyLamThree}{0.21} | |
| 770 | +\newcommand{\EpsLowThetaFortyOwn}{0.626} | |
| 771 | +\newcommand{\EpsLowThetaFortyOwnUthirtyfive}{0.342} | |
| 772 | +\newcommand{\EpsLowThetaFortyP}{0.983} | |
| 773 | +\newcommand{\EpsLowThetaFortyPDevPct}{\ensuremath{-}1.7} | |
| 774 | +\newcommand{\EpsLowThetaFortyPDevPctAbs}{1.7} | |
| 775 | +\newcommand{\EpsLowThetaFortyRR}{0.34} | |
| 776 | +\newcommand{\EpsLowThetaFortyRebate}{0.000} | |
| 777 | +\newcommand{\EpsLowThetaFortyRho}{0.202} | |
| 778 | +\newcommand{\EpsLowThetaFortyRhoDevPct}{3.8} | |
| 779 | +\newcommand{\EpsLowThetaFortyThr}{0.54} | |
| 780 | +\newcommand{\EpsLowThetaFortyThrHighZ}{0.12} | |
| 781 | +\newcommand{\EpsLowThetaFortyThrLowZ}{1.09} | |
| 782 | +\newcommand{\EpsLowThetaFortyqOO}{0.520} | |
| 783 | +\newcommand{\EpsLowThetaFortyqRO}{0.308} | |
| 784 | +\newcommand{\EqIncAoneMedian}{0.22} | |
| 785 | +\newcommand{\EqIncAoneMinus}{0.484} | |
| 786 | +\newcommand{\EqIncAonePlus}{0.504} | |
| 787 | +\newcommand{\EqIncEndowDiffPct}{4} | |
| 788 | +\newcommand{\EqIncFirstBuyAgeLowGift}{5} | |
| 789 | +\newcommand{\EqIncFirstBuyAgeMinus}{4} | |
| 790 | +\newcommand{\EqIncFirstBuyAgePlus}{1} | |
| 791 | +\newcommand{\EqIncGapPctAgeFour}{49} | |
| 792 | +\newcommand{\EqIncGapPctAgeSeven}{12} | |
| 793 | +\newcommand{\EqIncNWMinusAgeFour}{0.33} | |
| 794 | +\newcommand{\EqIncNWMinusAgeSeven}{1.03} | |
| 795 | +\newcommand{\EqIncNWPlusAgeFour}{0.50} | |
| 796 | +\newcommand{\EqIncNWPlusAgeSeven}{1.15} | |
| 797 | +\newcommand{\Erosionarmlevel}{0.000} | |
| 798 | +\newcommand{\Erosionarmpersistent}{0.000} | |
| 799 | +\newcommand{\Erosionfrmlevel}{0.206} | |
| 800 | +\newcommand{\Erosionfrmlevelfixedsupply}{0.206} | |
| 801 | +\newcommand{\Erosionfrmlevelnoseg}{0.206} | |
| 802 | +\newcommand{\Erosionfrmlevelrigidrents}{0.206} | |
| 803 | +\newcommand{\Erosionfrmlevelstatic}{0.206} | |
| 804 | +\newcommand{\Erosionfrmlevelunindexedwages}{0.206} | |
| 805 | +\newcommand{\Erosionfrmpersistent}{0.206} | |
| 806 | +\newcommand{\Erosionfrmpersistenttaylor}{0.206} | |
| 807 | +\newcommand{\Erosiontermlevel}{0.206} | |
| 808 | +\newcommand{\Erosiontermpersistent}{0.206} | |
| 809 | +\newcommand{\Erosiontermpersistenttaylor}{0.206} | |
| 810 | +\newcommand{\ExcessOwner}{0.000004} | |
| 811 | +\newcommand{\ExcessRental}{0.000007} | |
| 812 | +\newcommand{\FixedSupplyAbsorbPct}{94} | |
| 813 | +\newcommand{\FixedSupplyBetterAllPct}{41} | |
| 814 | +\newcommand{\FixedSupplyBetterOMPct}{71} | |
| 815 | +\newcommand{\FixedSupplyBetterONMPct}{1} | |
| 816 | +\newcommand{\FixedSupplyBetterOPct}{45} | |
| 817 | +\newcommand{\FixedSupplyBetterRPct}{35} | |
| 818 | +\newcommand{\FixedSupplyCEVAll}{\ensuremath{-}0.64} | |
| 819 | +\newcommand{\FixedSupplyCEVAllAbs}{0.64} | |
| 820 | +\newcommand{\FixedSupplyCEVAllPE}{\ensuremath{-}0.68} | |
| 821 | +\newcommand{\FixedSupplyCEVAllPEAbs}{0.68} | |
| 822 | +\newcommand{\FixedSupplyCEVO}{\ensuremath{-}0.20} | |
| 823 | +\newcommand{\FixedSupplyCEVOAbs}{0.20} | |
| 824 | +\newcommand{\FixedSupplyCEVOM}{0.63} | |
| 825 | +\newcommand{\FixedSupplyCEVONM}{\ensuremath{-}2.12} | |
| 826 | +\newcommand{\FixedSupplyCEVONMAbs}{2.12} | |
| 827 | +\newcommand{\FixedSupplyCEVR}{\ensuremath{-}0.99} | |
| 828 | +\newcommand{\FixedSupplyCEVRAbs}{0.99} | |
| 829 | +\newcommand{\FixedSupplyConsAll}{\ensuremath{-}0.8} | |
| 830 | +\newcommand{\FixedSupplyConsAllAbs}{0.8} | |
| 831 | +\newcommand{\FixedSupplyConsO}{\ensuremath{-}0.6} | |
| 832 | +\newcommand{\FixedSupplyConsOAbs}{0.6} | |
| 833 | +\newcommand{\FixedSupplyConsR}{\ensuremath{-}1.4} | |
| 834 | +\newcommand{\FixedSupplyConsRAbs}{1.4} | |
| 835 | +\newcommand{\FixedSupplyDOwnGE}{\ensuremath{-}0.01} | |
| 836 | +\newcommand{\FixedSupplyDOwnGEAbs}{0.01} | |
| 837 | +\newcommand{\FixedSupplyDOwnPE}{\ensuremath{-}0.19} | |
| 838 | +\newcommand{\FixedSupplyDOwnPEAbs}{0.19} | |
| 839 | +\newcommand{\FixedSupplyDebtZeroPct}{\ensuremath{-}2.6} | |
| 840 | +\newcommand{\FixedSupplyDebtZeroPctAbs}{2.6} | |
| 841 | +\newcommand{\FixedSupplyDepLoss}{0.48} | |
| 842 | +\newcommand{\FixedSupplyErosion}{0.21} | |
| 843 | +\newcommand{\FixedSupplyLamSevenSS}{0.060} | |
| 844 | +\newcommand{\FixedSupplyLamSevenZero}{0.068} | |
| 845 | +\newcommand{\FixedSupplyLender}{0.27} | |
| 846 | +\newcommand{\FixedSupplyMaxRes}{3.1\times10^{-3}} | |
| 847 | +\newcommand{\FixedSupplyNWAll}{\ensuremath{-}0.02} | |
| 848 | +\newcommand{\FixedSupplyNWAllAbs}{0.02} | |
| 849 | +\newcommand{\FixedSupplyNWO}{\ensuremath{-}0.02} | |
| 850 | +\newcommand{\FixedSupplyNWOAbs}{0.02} | |
| 851 | +\newcommand{\FixedSupplyNWR}{\ensuremath{-}0.01} | |
| 852 | +\newcommand{\FixedSupplyNWRAbs}{0.01} | |
| 853 | +\newcommand{\FixedSupplyNetO}{\ensuremath{-}0.02} | |
| 854 | +\newcommand{\FixedSupplyNetOAbs}{0.02} | |
| 855 | +\newcommand{\FixedSupplyNetR}{\ensuremath{-}0.02} | |
| 856 | +\newcommand{\FixedSupplyNetRAbs}{0.02} | |
| 857 | +\newcommand{\FixedSupplyNominalRateZero}{0.219} | |
| 858 | +\newcommand{\FixedSupplyOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 859 | +\newcommand{\FixedSupplyOwnerOldConsMaxAbs}{2.3} | |
| 860 | +\newcommand{\FixedSupplyOwnerOldConsMin}{\ensuremath{-}1.8} | |
| 861 | +\newcommand{\FixedSupplyOwnerOldConsMinAbs}{1.8} | |
| 862 | +\newcommand{\FixedSupplyOwnerOldLossMax}{0.08} | |
| 863 | +\newcommand{\FixedSupplyOwnerOldLossMin}{0.04} | |
| 864 | +\newcommand{\FixedSupplyOwnerYoungConsMax}{1.1} | |
| 865 | +\newcommand{\FixedSupplyOwnerYoungConsMin}{0.6} | |
| 866 | +\newcommand{\FixedSupplyOwnerYoungGainMax}{0.04} | |
| 867 | +\newcommand{\FixedSupplyPriceBackYears}{20} | |
| 868 | +\newcommand{\FixedSupplyPriceMaxPos}{0.1} | |
| 869 | +\newcommand{\FixedSupplyPriceSignChanges}{1} | |
| 870 | +\newcommand{\FixedSupplyPzero}{\ensuremath{-}0.50} | |
| 871 | +\newcommand{\FixedSupplyPzeroAbs}{0.50} | |
| 872 | +\newcommand{\FixedSupplyRentGainR}{0.00} | |
| 873 | +\newcommand{\FixedSupplyResT}{2\times10^{-4}} | |
| 874 | +\newcommand{\FixedSupplyRhoZero}{\ensuremath{-}1.08} | |
| 875 | +\newcommand{\FixedSupplyRhoZeroAbs}{1.08} | |
| 876 | +\newcommand{\FixedSupplySecs}{374} | |
| 877 | +\newcommand{\FixedSupplySteps}{7} | |
| 878 | +\newcommand{\FixedSupplyThrChangePct}{8} | |
| 879 | +\newcommand{\FixedSupplyThrSS}{0.49} | |
| 880 | +\newcommand{\FixedSupplyThrZero}{0.53} | |
| 881 | +\newcommand{\FixedSupplyWageAll}{0.00} | |
| 882 | +\newcommand{\FixedSupplyWageO}{0.00} | |
| 883 | +\newcommand{\FixedSupplyqROsevenSS}{0.806} | |
| 884 | +\newcommand{\FixedSupplyqROsevenZero}{0.798} | |
| 885 | +\newcommand{\FixedSupplyqROthreeOne}{0.334} | |
| 886 | +\newcommand{\FixedSupplyqROthreeSS}{0.351} | |
| 887 | +\newcommand{\FixedSupplyqROthreeZero}{0.356} | |
| 888 | +\newcommand{\FrmLevelAbsorbPct}{92} | |
| 889 | +\newcommand{\FrmLevelBetterAllPct}{41} | |
| 890 | +\newcommand{\FrmLevelBetterOMPct}{71} | |
| 891 | +\newcommand{\FrmLevelBetterONMPct}{1} | |
| 892 | +\newcommand{\FrmLevelBetterOPct}{45} | |
| 893 | +\newcommand{\FrmLevelBetterRPct}{35} | |
| 894 | +\newcommand{\FrmLevelCEVAll}{\ensuremath{-}0.64} | |
| 895 | +\newcommand{\FrmLevelCEVAllAbs}{0.64} | |
| 896 | +\newcommand{\FrmLevelCEVAllPE}{\ensuremath{-}0.68} | |
| 897 | +\newcommand{\FrmLevelCEVAllPEAbs}{0.68} | |
| 898 | +\newcommand{\FrmLevelCEVO}{\ensuremath{-}0.20} | |
| 899 | +\newcommand{\FrmLevelCEVOAbs}{0.20} | |
| 900 | +\newcommand{\FrmLevelCEVOM}{0.63} | |
| 901 | +\newcommand{\FrmLevelCEVONM}{\ensuremath{-}2.12} | |
| 902 | +\newcommand{\FrmLevelCEVONMAbs}{2.12} | |
| 903 | +\newcommand{\FrmLevelCEVR}{\ensuremath{-}1.00} | |
| 904 | +\newcommand{\FrmLevelCEVRAbs}{1.00} | |
| 905 | +\newcommand{\FrmLevelConsAll}{\ensuremath{-}0.8} | |
| 906 | +\newcommand{\FrmLevelConsAllAbs}{0.8} | |
| 907 | +\newcommand{\FrmLevelConsO}{\ensuremath{-}0.6} | |
| 908 | +\newcommand{\FrmLevelConsOAbs}{0.6} | |
| 909 | +\newcommand{\FrmLevelConsR}{\ensuremath{-}1.3} | |
| 910 | +\newcommand{\FrmLevelConsRAbs}{1.3} | |
| 911 | +\newcommand{\FrmLevelDOwnGE}{\ensuremath{-}0.02} | |
| 912 | +\newcommand{\FrmLevelDOwnGEAbs}{0.02} | |
| 913 | +\newcommand{\FrmLevelDOwnPE}{\ensuremath{-}0.20} | |
| 914 | +\newcommand{\FrmLevelDOwnPEAbs}{0.20} | |
| 915 | +\newcommand{\FrmLevelDebtZeroPct}{\ensuremath{-}2.7} | |
| 916 | +\newcommand{\FrmLevelDebtZeroPctAbs}{2.7} | |
| 917 | +\newcommand{\FrmLevelDepLoss}{0.48} | |
| 918 | +\newcommand{\FrmLevelErosion}{0.21} | |
| 919 | +\newcommand{\FrmLevelLamSevenSS}{0.060} | |
| 920 | +\newcommand{\FrmLevelLamSevenZero}{0.069} | |
| 921 | +\newcommand{\FrmLevelLender}{0.27} | |
| 922 | +\newcommand{\FrmLevelMaxRes}{2.2\times10^{-3}} | |
| 923 | +\newcommand{\FrmLevelNWAll}{\ensuremath{-}0.02} | |
| 924 | +\newcommand{\FrmLevelNWAllAbs}{0.02} | |
| 925 | +\newcommand{\FrmLevelNWO}{\ensuremath{-}0.02} | |
| 926 | +\newcommand{\FrmLevelNWOAbs}{0.02} | |
| 927 | +\newcommand{\FrmLevelNWR}{\ensuremath{-}0.01} | |
| 928 | +\newcommand{\FrmLevelNWRAbs}{0.01} | |
| 929 | +\newcommand{\FrmLevelNetO}{\ensuremath{-}0.02} | |
| 930 | +\newcommand{\FrmLevelNetOAbs}{0.02} | |
| 931 | +\newcommand{\FrmLevelNetR}{\ensuremath{-}0.02} | |
| 932 | +\newcommand{\FrmLevelNetRAbs}{0.02} | |
| 933 | +\newcommand{\FrmLevelNominalRateZero}{0.219} | |
| 934 | +\newcommand{\FrmLevelOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 935 | +\newcommand{\FrmLevelOwnerOldConsMaxAbs}{2.3} | |
| 936 | +\newcommand{\FrmLevelOwnerOldConsMin}{\ensuremath{-}1.8} | |
| 937 | +\newcommand{\FrmLevelOwnerOldConsMinAbs}{1.8} | |
| 938 | +\newcommand{\FrmLevelOwnerOldLossMax}{0.08} | |
| 939 | +\newcommand{\FrmLevelOwnerOldLossMin}{0.04} | |
| 940 | +\newcommand{\FrmLevelOwnerYoungConsMax}{1.0} | |
| 941 | +\newcommand{\FrmLevelOwnerYoungConsMin}{0.6} | |
| 942 | +\newcommand{\FrmLevelOwnerYoungGainMax}{0.04} | |
| 943 | +\newcommand{\FrmLevelPriceBackYears}{20} | |
| 944 | +\newcommand{\FrmLevelPriceMaxPos}{0.1} | |
| 945 | +\newcommand{\FrmLevelPriceSignChanges}{1} | |
| 946 | +\newcommand{\FrmLevelPzero}{\ensuremath{-}0.45} | |
| 947 | +\newcommand{\FrmLevelPzeroAbs}{0.45} | |
| 948 | +\newcommand{\FrmLevelRentGainR}{0.00} | |
| 949 | +\newcommand{\FrmLevelResT}{3\times10^{-4}} | |
| 950 | +\newcommand{\FrmLevelRhoZero}{\ensuremath{-}1.14} | |
| 951 | +\newcommand{\FrmLevelRhoZeroAbs}{1.14} | |
| 952 | +\newcommand{\FrmLevelSecs}{471} | |
| 953 | +\newcommand{\FrmLevelSteps}{10} | |
| 954 | +\newcommand{\FrmLevelThrChangePct}{8} | |
| 955 | +\newcommand{\FrmLevelThrSS}{0.49} | |
| 956 | +\newcommand{\FrmLevelThrZero}{0.53} | |
| 957 | +\newcommand{\FrmLevelWageAll}{0.00} | |
| 958 | +\newcommand{\FrmLevelWageO}{0.00} | |
| 959 | +\newcommand{\FrmLevelqROsevenSS}{0.806} | |
| 960 | +\newcommand{\FrmLevelqROsevenZero}{0.797} | |
| 961 | +\newcommand{\FrmLevelqROthreeOne}{0.336} | |
| 962 | +\newcommand{\FrmLevelqROthreeSS}{0.351} | |
| 963 | +\newcommand{\FrmLevelqROthreeZero}{0.356} | |
| 964 | +\newcommand{\FrmPersAbsorbPct}{92} | |
| 965 | +\newcommand{\FrmPersBetterAllPct}{31} | |
| 966 | +\newcommand{\FrmPersBetterOMPct}{73} | |
| 967 | +\newcommand{\FrmPersBetterONMPct}{2} | |
| 968 | +\newcommand{\FrmPersBetterOPct}{47} | |
| 969 | +\newcommand{\FrmPersBetterRPct}{7} | |
| 970 | +\newcommand{\FrmPersCEVAll}{\ensuremath{-}0.52} | |
| 971 | +\newcommand{\FrmPersCEVAllAbs}{0.52} | |
| 972 | +\newcommand{\FrmPersCEVAllPE}{\ensuremath{-}0.53} | |
| 973 | +\newcommand{\FrmPersCEVAllPEAbs}{0.53} | |
| 974 | +\newcommand{\FrmPersCEVO}{0.15} | |
| 975 | +\newcommand{\FrmPersCEVOM}{1.14} | |
| 976 | +\newcommand{\FrmPersCEVONM}{\ensuremath{-}2.13} | |
| 977 | +\newcommand{\FrmPersCEVONMAbs}{2.13} | |
| 978 | +\newcommand{\FrmPersCEVR}{\ensuremath{-}1.06} | |
| 979 | +\newcommand{\FrmPersCEVRAbs}{1.06} | |
| 980 | +\newcommand{\FrmPersConsAll}{\ensuremath{-}1.2} | |
| 981 | +\newcommand{\FrmPersConsAllAbs}{1.2} | |
| 982 | +\newcommand{\FrmPersConsO}{\ensuremath{-}1.0} | |
| 983 | +\newcommand{\FrmPersConsOAbs}{1.0} | |
| 984 | +\newcommand{\FrmPersConsR}{\ensuremath{-}1.6} | |
| 985 | +\newcommand{\FrmPersConsRAbs}{1.6} | |
| 986 | +\newcommand{\FrmPersDOwnGE}{\ensuremath{-}0.18} | |
| 987 | +\newcommand{\FrmPersDOwnGEAbs}{0.18} | |
| 988 | +\newcommand{\FrmPersDOwnPE}{\ensuremath{-}2.17} | |
| 989 | +\newcommand{\FrmPersDOwnPEAbs}{2.17} | |
| 990 | +\newcommand{\FrmPersDebtZeroPct}{\ensuremath{-}24.7} | |
| 991 | +\newcommand{\FrmPersDebtZeroPctAbs}{24.7} | |
| 992 | +\newcommand{\FrmPersDepLoss}{0.48} | |
| 993 | +\newcommand{\FrmPersErosion}{0.21} | |
| 994 | +\newcommand{\FrmPersLamSevenSS}{0.060} | |
| 995 | +\newcommand{\FrmPersLamSevenZero}{0.065} | |
| 996 | +\newcommand{\FrmPersLender}{0.27} | |
| 997 | +\newcommand{\FrmPersMaxRes}{1.2\times10^{-3}} | |
| 998 | +\newcommand{\FrmPersNWAll}{\ensuremath{-}0.01} | |
| 999 | +\newcommand{\FrmPersNWAllAbs}{0.01} | |
| 1000 | +\newcommand{\FrmPersNWO}{\ensuremath{-}0.02} | |
| 1001 | +\newcommand{\FrmPersNWOAbs}{0.02} | |
| 1002 | +\newcommand{\FrmPersNWR}{\ensuremath{-}0.01} | |
| 1003 | +\newcommand{\FrmPersNWRAbs}{0.01} | |
| 1004 | +\newcommand{\FrmPersNetO}{\ensuremath{-}0.02} | |
| 1005 | +\newcommand{\FrmPersNetOAbs}{0.02} | |
| 1006 | +\newcommand{\FrmPersNetR}{\ensuremath{-}0.02} | |
| 1007 | +\newcommand{\FrmPersNetRAbs}{0.02} | |
| 1008 | +\newcommand{\FrmPersNominalRateZero}{0.341} | |
| 1009 | +\newcommand{\FrmPersOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 1010 | +\newcommand{\FrmPersOwnerOldConsMaxAbs}{2.3} | |
| 1011 | +\newcommand{\FrmPersOwnerOldConsMin}{\ensuremath{-}1.7} | |
| 1012 | +\newcommand{\FrmPersOwnerOldConsMinAbs}{1.7} | |
| 1013 | +\newcommand{\FrmPersOwnerOldLossMax}{0.08} | |
| 1014 | +\newcommand{\FrmPersOwnerOldLossMin}{0.04} | |
| 1015 | +\newcommand{\FrmPersOwnerYoungConsMax}{0.7} | |
| 1016 | +\newcommand{\FrmPersOwnerYoungConsMin}{0.3} | |
| 1017 | +\newcommand{\FrmPersOwnerYoungGainMax}{0.04} | |
| 1018 | +\newcommand{\FrmPersPriceBackYears}{5} | |
| 1019 | +\newcommand{\FrmPersPriceMaxPos}{0.1} | |
| 1020 | +\newcommand{\FrmPersPriceSignChanges}{3} | |
| 1021 | +\newcommand{\FrmPersPzero}{\ensuremath{-}1.88} | |
| 1022 | +\newcommand{\FrmPersPzeroAbs}{1.88} | |
| 1023 | +\newcommand{\FrmPersRentGainR}{0.00} | |
| 1024 | +\newcommand{\FrmPersResT}{4\times10^{-4}} | |
| 1025 | +\newcommand{\FrmPersRhoZero}{\ensuremath{-}0.38} | |
| 1026 | +\newcommand{\FrmPersRhoZeroAbs}{0.38} | |
| 1027 | +\newcommand{\FrmPersSecs}{167} | |
| 1028 | +\newcommand{\FrmPersSteps}{4} | |
| 1029 | +\newcommand{\FrmPersThrChangePct}{16} | |
| 1030 | +\newcommand{\FrmPersThrSS}{0.49} | |
| 1031 | +\newcommand{\FrmPersThrZero}{0.57} | |
| 1032 | +\newcommand{\FrmPersWageAll}{0.00} | |
| 1033 | +\newcommand{\FrmPersWageO}{0.00} | |
| 1034 | +\newcommand{\FrmPersqROsevenSS}{0.806} | |
| 1035 | +\newcommand{\FrmPersqROsevenZero}{0.811} | |
| 1036 | +\newcommand{\FrmPersqROthreeOne}{0.334} | |
| 1037 | +\newcommand{\FrmPersqROthreeSS}{0.351} | |
| 1038 | +\newcommand{\FrmPersqROthreeZero}{0.357} | |
| 1039 | +\newcommand{\FrmTaylorAbsorbPct}{95} | |
| 1040 | +\newcommand{\FrmTaylorBetterAllPct}{53} | |
| 1041 | +\newcommand{\FrmTaylorBetterOMPct}{70} | |
| 1042 | +\newcommand{\FrmTaylorBetterONMPct}{22} | |
| 1043 | +\newcommand{\FrmTaylorBetterOPct}{53} | |
| 1044 | +\newcommand{\FrmTaylorBetterRPct}{53} | |
| 1045 | +\newcommand{\FrmTaylorCEVAll}{0.14} | |
| 1046 | +\newcommand{\FrmTaylorCEVAllPE}{\ensuremath{-}1.32} | |
| 1047 | +\newcommand{\FrmTaylorCEVAllPEAbs}{1.32} | |
| 1048 | +\newcommand{\FrmTaylorCEVO}{0.69} | |
| 1049 | +\newcommand{\FrmTaylorCEVOM}{1.43} | |
| 1050 | +\newcommand{\FrmTaylorCEVONM}{\ensuremath{-}1.04} | |
| 1051 | +\newcommand{\FrmTaylorCEVONMAbs}{1.04} | |
| 1052 | +\newcommand{\FrmTaylorCEVR}{\ensuremath{-}0.31} | |
| 1053 | +\newcommand{\FrmTaylorCEVRAbs}{0.31} | |
| 1054 | +\newcommand{\FrmTaylorConsAll}{\ensuremath{-}3.0} | |
| 1055 | +\newcommand{\FrmTaylorConsAllAbs}{3.0} | |
| 1056 | +\newcommand{\FrmTaylorConsO}{\ensuremath{-}3.0} | |
| 1057 | +\newcommand{\FrmTaylorConsOAbs}{3.0} | |
| 1058 | +\newcommand{\FrmTaylorConsR}{\ensuremath{-}3.0} | |
| 1059 | +\newcommand{\FrmTaylorConsRAbs}{3.0} | |
| 1060 | +\newcommand{\FrmTaylorDOwnGE}{\ensuremath{-}0.12} | |
| 1061 | +\newcommand{\FrmTaylorDOwnGEAbs}{0.12} | |
| 1062 | +\newcommand{\FrmTaylorDOwnPE}{\ensuremath{-}2.64} | |
| 1063 | +\newcommand{\FrmTaylorDOwnPEAbs}{2.64} | |
| 1064 | +\newcommand{\FrmTaylorDebtZeroPct}{\ensuremath{-}26.8} | |
| 1065 | +\newcommand{\FrmTaylorDebtZeroPctAbs}{26.8} | |
| 1066 | +\newcommand{\FrmTaylorDepLoss}{0.48} | |
| 1067 | +\newcommand{\FrmTaylorErosion}{0.21} | |
| 1068 | +\newcommand{\FrmTaylorLamSevenSS}{0.060} | |
| 1069 | +\newcommand{\FrmTaylorLamSevenZero}{0.066} | |
| 1070 | +\newcommand{\FrmTaylorLender}{0.27} | |
| 1071 | +\newcommand{\FrmTaylorMaxRes}{1.3\times10^{-3}} | |
| 1072 | +\newcommand{\FrmTaylorNWAll}{\ensuremath{-}0.03} | |
| 1073 | +\newcommand{\FrmTaylorNWAllAbs}{0.03} | |
| 1074 | +\newcommand{\FrmTaylorNWO}{\ensuremath{-}0.05} | |
| 1075 | +\newcommand{\FrmTaylorNWOAbs}{0.05} | |
| 1076 | +\newcommand{\FrmTaylorNWR}{\ensuremath{-}0.01} | |
| 1077 | +\newcommand{\FrmTaylorNWRAbs}{0.01} | |
| 1078 | +\newcommand{\FrmTaylorNetO}{\ensuremath{-}0.02} | |
| 1079 | +\newcommand{\FrmTaylorNetOAbs}{0.02} | |
| 1080 | +\newcommand{\FrmTaylorNetR}{\ensuremath{-}0.02} | |
| 1081 | +\newcommand{\FrmTaylorNetRAbs}{0.02} | |
| 1082 | +\newcommand{\FrmTaylorNominalRateZero}{0.408} | |
| 1083 | +\newcommand{\FrmTaylorOwnerOldConsMax}{\ensuremath{-}4.2} | |
| 1084 | +\newcommand{\FrmTaylorOwnerOldConsMaxAbs}{4.2} | |
| 1085 | +\newcommand{\FrmTaylorOwnerOldConsMin}{\ensuremath{-}3.4} | |
| 1086 | +\newcommand{\FrmTaylorOwnerOldConsMinAbs}{3.4} | |
| 1087 | +\newcommand{\FrmTaylorOwnerOldLossMax}{0.08} | |
| 1088 | +\newcommand{\FrmTaylorOwnerOldLossMin}{0.04} | |
| 1089 | +\newcommand{\FrmTaylorOwnerYoungConsMax}{\ensuremath{-}2.0} | |
| 1090 | +\newcommand{\FrmTaylorOwnerYoungConsMaxAbs}{2.0} | |
| 1091 | +\newcommand{\FrmTaylorOwnerYoungConsMin}{\ensuremath{-}3.9} | |
| 1092 | +\newcommand{\FrmTaylorOwnerYoungConsMinAbs}{3.9} | |
| 1093 | +\newcommand{\FrmTaylorOwnerYoungGainMax}{0.04} | |
| 1094 | +\newcommand{\FrmTaylorPriceBackYears}{40} | |
| 1095 | +\newcommand{\FrmTaylorPriceMaxPos}{1.4} | |
| 1096 | +\newcommand{\FrmTaylorPriceSignChanges}{1} | |
| 1097 | +\newcommand{\FrmTaylorPzero}{\ensuremath{-}7.25} | |
| 1098 | +\newcommand{\FrmTaylorPzeroAbs}{7.25} | |
| 1099 | +\newcommand{\FrmTaylorRentGainR}{0.00} | |
| 1100 | +\newcommand{\FrmTaylorResT}{9\times10^{-5}} | |
| 1101 | +\newcommand{\FrmTaylorRhoZero}{\ensuremath{-}2.29} | |
| 1102 | +\newcommand{\FrmTaylorRhoZeroAbs}{2.29} | |
| 1103 | +\newcommand{\FrmTaylorSecs}{211} | |
| 1104 | +\newcommand{\FrmTaylorSteps}{6} | |
| 1105 | +\newcommand{\FrmTaylorThrChangePct}{12} | |
| 1106 | +\newcommand{\FrmTaylorThrSS}{0.49} | |
| 1107 | +\newcommand{\FrmTaylorThrZero}{0.55} | |
| 1108 | +\newcommand{\FrmTaylorWageAll}{0.00} | |
| 1109 | +\newcommand{\FrmTaylorWageO}{0.00} | |
| 1110 | +\newcommand{\FrmTaylorqROsevenSS}{0.806} | |
| 1111 | +\newcommand{\FrmTaylorqROsevenZero}{0.803} | |
| 1112 | +\newcommand{\FrmTaylorqROthreeOne}{0.345} | |
| 1113 | +\newcommand{\FrmTaylorqROthreeSS}{0.351} | |
| 1114 | +\newcommand{\FrmTaylorqROthreeZero}{0.362} | |
| 1115 | +\newcommand{\GammaThreeEstate}{0.26} | |
| 1116 | +\newcommand{\GammaThreeGapFour}{0.97} | |
| 1117 | +\newcommand{\GammaThreeGift}{0.28} | |
| 1118 | +\newcommand{\GammaThreeGini}{0.53} | |
| 1119 | +\newcommand{\GammaThreeLamSeven}{0.050} | |
| 1120 | +\newcommand{\GammaThreeLamThree}{0.16} | |
| 1121 | +\newcommand{\GammaThreeOwn}{0.646} | |
| 1122 | +\newcommand{\GammaThreeOwnUthirtyfive}{0.373} | |
| 1123 | +\newcommand{\GammaThreeP}{0.995} | |
| 1124 | +\newcommand{\GammaThreePDevPct}{\ensuremath{-}0.5} | |
| 1125 | +\newcommand{\GammaThreePDevPctAbs}{0.5} | |
| 1126 | +\newcommand{\GammaThreeRR}{0.28} | |
| 1127 | +\newcommand{\GammaThreeRebate}{0.000} | |
| 1128 | +\newcommand{\GammaThreeRho}{0.189} | |
| 1129 | +\newcommand{\GammaThreeRhoDevPct}{\ensuremath{-}2.6} | |
| 1130 | +\newcommand{\GammaThreeRhoDevPctAbs}{2.6} | |
| 1131 | +\newcommand{\GammaThreeThr}{0.47} | |
| 1132 | +\newcommand{\GammaThreeThrHighZ}{\ensuremath{-}0.09} | |
| 1133 | +\newcommand{\GammaThreeThrHighZAbs}{0.09} | |
| 1134 | +\newcommand{\GammaThreeThrLowZ}{1.17} | |
| 1135 | +\newcommand{\GammaThreeqOO}{0.532} | |
| 1136 | +\newcommand{\GammaThreeqRO}{0.375} | |
| 1137 | +\newcommand{\GapAgeFour}{0.95} | |
| 1138 | +\newcommand{\GapAgeSeven}{1.57} | |
| 1139 | +\newcommand{\GapAgeTen}{1.03} | |
| 1140 | +\newcommand{\GapAgeTwo}{0.62} | |
| 1141 | +\newcommand{\GapQtwoQfiveAgeOnePP}{86} | |
| 1142 | +\newcommand{\GapQtwoQfiveAgeSevenPP}{16} | |
| 1143 | +\newcommand{\GiniAgeSeven}{0.38} | |
| 1144 | +\newcommand{\GiniNW}{0.54} | |
| 1145 | +\newcommand{\GridCoarseEstate}{0.32} | |
| 1146 | +\newcommand{\GridCoarseGapFour}{0.98} | |
| 1147 | +\newcommand{\GridCoarseGift}{0.39} | |
| 1148 | +\newcommand{\GridCoarseGini}{0.54} | |
| 1149 | +\newcommand{\GridCoarseLamSeven}{0.077} | |
| 1150 | +\newcommand{\GridCoarseLamThree}{0.19} | |
| 1151 | +\newcommand{\GridCoarseOwn}{0.626} | |
| 1152 | +\newcommand{\GridCoarseOwnUthirtyfive}{0.369} | |
| 1153 | +\newcommand{\GridCoarseP}{0.992} | |
| 1154 | +\newcommand{\GridCoarsePDevPct}{\ensuremath{-}0.8} | |
| 1155 | +\newcommand{\GridCoarsePDevPctAbs}{0.8} | |
| 1156 | +\newcommand{\GridCoarseRR}{0.34} | |
| 1157 | +\newcommand{\GridCoarseRebate}{0.000} | |
| 1158 | +\newcommand{\GridCoarseRho}{0.196} | |
| 1159 | +\newcommand{\GridCoarseRhoDevPct}{0.8} | |
| 1160 | +\newcommand{\GridCoarseThr}{0.56} | |
| 1161 | +\newcommand{\GridCoarseThrHighZ}{0.02} | |
| 1162 | +\newcommand{\GridCoarseThrLowZ}{1.14} | |
| 1163 | +\newcommand{\GridCoarseqOO}{0.542} | |
| 1164 | +\newcommand{\GridCoarseqRO}{0.350} | |
| 1165 | +\newcommand{\GridFineEstate}{0.31} | |
| 1166 | +\newcommand{\GridFineGapFour}{0.94} | |
| 1167 | +\newcommand{\GridFineGift}{0.39} | |
| 1168 | +\newcommand{\GridFineGini}{0.54} | |
| 1169 | +\newcommand{\GridFineLamSeven}{0.054} | |
| 1170 | +\newcommand{\GridFineLamThree}{0.20} | |
| 1171 | +\newcommand{\GridFineOwn}{0.635} | |
| 1172 | +\newcommand{\GridFineOwnUthirtyfive}{0.382} | |
| 1173 | +\newcommand{\GridFineP}{0.999} | |
| 1174 | +\newcommand{\GridFinePDevPct}{\ensuremath{-}0.1} | |
| 1175 | +\newcommand{\GridFinePDevPctAbs}{0.1} | |
| 1176 | +\newcommand{\GridFineRR}{0.33} | |
| 1177 | +\newcommand{\GridFineRebate}{0.000} | |
| 1178 | +\newcommand{\GridFineRho}{0.193} | |
| 1179 | +\newcommand{\GridFineRhoDevPct}{\ensuremath{-}0.6} | |
| 1180 | +\newcommand{\GridFineRhoDevPctAbs}{0.6} | |
| 1181 | +\newcommand{\GridFineThr}{0.47} | |
| 1182 | +\newcommand{\GridFineThrHighZ}{0.04} | |
| 1183 | +\newcommand{\GridFineThrLowZ}{1.19} | |
| 1184 | +\newcommand{\GridFineqOO}{0.535} | |
| 1185 | +\newcommand{\GridFineqRO}{0.340} | |
| 1186 | +\newcommand{\HalfLifeGen}{0.25} | |
| 1187 | +\newcommand{\HouseValueIncome}{5.0} | |
| 1188 | +\newcommand{\HouseholdPeriods}{1{,}152{,}000} | |
| 1189 | +\newcommand{\KappaHighEstate}{0.31} | |
| 1190 | +\newcommand{\KappaHighGapFour}{0.91} | |
| 1191 | +\newcommand{\KappaHighGift}{0.39} | |
| 1192 | +\newcommand{\KappaHighGini}{0.54} | |
| 1193 | +\newcommand{\KappaHighLamSeven}{0.069} | |
| 1194 | +\newcommand{\KappaHighLamThree}{0.20} | |
| 1195 | +\newcommand{\KappaHighOwn}{0.633} | |
| 1196 | +\newcommand{\KappaHighOwnUthirtyfive}{0.366} | |
| 1197 | +\newcommand{\KappaHighP}{0.962} | |
| 1198 | +\newcommand{\KappaHighPDevPct}{\ensuremath{-}3.8} | |
| 1199 | +\newcommand{\KappaHighPDevPctAbs}{3.8} | |
| 1200 | +\newcommand{\KappaHighRR}{0.34} | |
| 1201 | +\newcommand{\KappaHighRebate}{0.000} | |
| 1202 | +\newcommand{\KappaHighRho}{0.194} | |
| 1203 | +\newcommand{\KappaHighRhoDevPct}{\ensuremath{-}0.3} | |
| 1204 | +\newcommand{\KappaHighRhoDevPctAbs}{0.3} | |
| 1205 | +\newcommand{\KappaHighThr}{0.52} | |
| 1206 | +\newcommand{\KappaHighThrHighZ}{0.06} | |
| 1207 | +\newcommand{\KappaHighThrLowZ}{1.23} | |
| 1208 | +\newcommand{\KappaHighqOO}{0.539} | |
| 1209 | +\newcommand{\KappaHighqRO}{0.335} | |
| 1210 | +\newcommand{\KappaLowEstate}{0.30} | |
| 1211 | +\newcommand{\KappaLowGapFour}{0.98} | |
| 1212 | +\newcommand{\KappaLowGift}{0.38} | |
| 1213 | +\newcommand{\KappaLowGini}{0.54} | |
| 1214 | +\newcommand{\KappaLowLamSeven}{0.068} | |
| 1215 | +\newcommand{\KappaLowLamThree}{0.17} | |
| 1216 | +\newcommand{\KappaLowOwn}{0.622} | |
| 1217 | +\newcommand{\KappaLowOwnUthirtyfive}{0.374} | |
| 1218 | +\newcommand{\KappaLowP}{1.044} | |
| 1219 | +\newcommand{\KappaLowPDevPct}{4.4} | |
| 1220 | +\newcommand{\KappaLowRR}{0.32} | |
| 1221 | +\newcommand{\KappaLowRebate}{0.000} | |
| 1222 | +\newcommand{\KappaLowRho}{0.196} | |
| 1223 | +\newcommand{\KappaLowRhoDevPct}{0.9} | |
| 1224 | +\newcommand{\KappaLowThr}{0.50} | |
| 1225 | +\newcommand{\KappaLowThrHighZ}{\ensuremath{-}0.04} | |
| 1226 | +\newcommand{\KappaLowThrHighZAbs}{0.04} | |
| 1227 | +\newcommand{\KappaLowThrLowZ}{1.06} | |
| 1228 | +\newcommand{\KappaLowqOO}{0.524} | |
| 1229 | +\newcommand{\KappaLowqRO}{0.350} | |
| 1230 | +\newcommand{\LambdaAgeSeven}{0.060} | |
| 1231 | +\newcommand{\LambdaAgeThree}{0.19} | |
| 1232 | +\newcommand{\LossNWAllTen}{12} | |
| 1233 | +\newcommand{\LossNWAllThirty}{37} | |
| 1234 | +\newcommand{\LossNWAllTwenty}{24} | |
| 1235 | +\newcommand{\LossNWNewTen}{17} | |
| 1236 | +\newcommand{\LossNWNewThirty}{52} | |
| 1237 | +\newcommand{\LossNWNewTwenty}{35} | |
| 1238 | +\newcommand{\LotteryRounds}{5} | |
| 1239 | +\newcommand{\MeanEstate}{0.32} | |
| 1240 | +\newcommand{\MeanGift}{0.39} | |
| 1241 | +\newcommand{\MeanGiftAnnual}{1.9} | |
| 1242 | +\newcommand{\MeanLTVNew}{0.60} | |
| 1243 | +\newcommand{\MeanLTVOwners}{0.27} | |
| 1244 | +\newcommand{\MedNWAgeSevenQFive}{2.31} | |
| 1245 | +\newcommand{\MedNWAgeSevenQFour}{1.54} | |
| 1246 | +\newcommand{\MedNWAgeSevenQOne}{1.02} | |
| 1247 | +\newcommand{\MedNWAgeSevenQThree}{1.16} | |
| 1248 | +\newcommand{\MedNWAgeSevenQTwo}{1.06} | |
| 1249 | +\newcommand{\MedianGift}{0.22} | |
| 1250 | +\newcommand{\MultBaseExcessHigh}{\ensuremath{-}0.43} | |
| 1251 | +\newcommand{\MultBaseExcessHighAbs}{0.43} | |
| 1252 | +\newcommand{\MultBaseExcessLow}{0.69} | |
| 1253 | +\newcommand{\MultBaseOwnHigh}{0.50} | |
| 1254 | +\newcommand{\MultBaseOwnLow}{0.73} | |
| 1255 | +\newcommand{\MultBaseRoot}{1.000} | |
| 1256 | +\newcommand{\MultBaseSlope}{\ensuremath{-}3.1} | |
| 1257 | +\newcommand{\MultBaseSlopeAbs}{3.1} | |
| 1258 | +\newcommand{\MultEpsLowExcessHigh}{\ensuremath{-}0.26} | |
| 1259 | +\newcommand{\MultEpsLowExcessHighAbs}{0.26} | |
| 1260 | +\newcommand{\MultEpsLowExcessLow}{0.33} | |
| 1261 | +\newcommand{\MultEpsLowOwnHigh}{0.54} | |
| 1262 | +\newcommand{\MultEpsLowOwnLow}{0.70} | |
| 1263 | +\newcommand{\MultEpsLowRoot}{1.000} | |
| 1264 | +\newcommand{\MultEpsLowSlope}{\ensuremath{-}2.0} | |
| 1265 | +\newcommand{\MultEpsLowSlopeAbs}{2.0} | |
| 1266 | +\newcommand{\MultEpsLowThetaFiveExcessHigh}{\ensuremath{-}0.24} | |
| 1267 | +\newcommand{\MultEpsLowThetaFiveExcessHighAbs}{0.24} | |
| 1268 | +\newcommand{\MultEpsLowThetaFiveExcessLow}{0.34} | |
| 1269 | +\newcommand{\MultEpsLowThetaFiveOwnHigh}{0.55} | |
| 1270 | +\newcommand{\MultEpsLowThetaFiveOwnLow}{0.71} | |
| 1271 | +\newcommand{\MultEpsLowThetaFiveRoot}{1.000} | |
| 1272 | +\newcommand{\MultEpsLowThetaFiveSlope}{\ensuremath{-}1.8} | |
| 1273 | +\newcommand{\MultEpsLowThetaFiveSlopeAbs}{1.8} | |
| 1274 | +\newcommand{\MultPgridHigh}{1.25} | |
| 1275 | +\newcommand{\MultPgridLow}{0.80} | |
| 1276 | +\newcommand{\NDynasties}{96{,}000} | |
| 1277 | +\newcommand{\NRobustnessRuns}{49} | |
| 1278 | +\newcommand{\NScenariosSS}{45} | |
| 1279 | +\newcommand{\NScenariosSSConverged}{44} | |
| 1280 | +\newcommand{\NTransitions}{16} | |
| 1281 | +\newcommand{\NTransitionsConverged}{12} | |
| 1282 | +\newcommand{\NWRatioYoung}{11.2} | |
| 1283 | +\newcommand{\NoSegAbsorbPct}{23} | |
| 1284 | +\newcommand{\NoSegBetterAllPct}{42} | |
| 1285 | +\newcommand{\NoSegBetterOMPct}{71} | |
| 1286 | +\newcommand{\NoSegBetterONMPct}{1} | |
| 1287 | +\newcommand{\NoSegBetterOPct}{45} | |
| 1288 | +\newcommand{\NoSegBetterRPct}{36} | |
| 1289 | +\newcommand{\NoSegCEVAll}{\ensuremath{-}0.60} | |
| 1290 | +\newcommand{\NoSegCEVAllAbs}{0.60} | |
| 1291 | +\newcommand{\NoSegCEVAllPE}{\ensuremath{-}0.68} | |
| 1292 | +\newcommand{\NoSegCEVAllPEAbs}{0.68} | |
| 1293 | +\newcommand{\NoSegCEVO}{\ensuremath{-}0.17} | |
| 1294 | +\newcommand{\NoSegCEVOAbs}{0.17} | |
| 1295 | +\newcommand{\NoSegCEVOM}{0.66} | |
| 1296 | +\newcommand{\NoSegCEVONM}{\ensuremath{-}2.10} | |
| 1297 | +\newcommand{\NoSegCEVONMAbs}{2.10} | |
| 1298 | +\newcommand{\NoSegCEVR}{\ensuremath{-}0.95} | |
| 1299 | +\newcommand{\NoSegCEVRAbs}{0.95} | |
| 1300 | +\newcommand{\NoSegConsAll}{\ensuremath{-}0.8} | |
| 1301 | +\newcommand{\NoSegConsAllAbs}{0.8} | |
| 1302 | +\newcommand{\NoSegConsO}{\ensuremath{-}0.5} | |
| 1303 | +\newcommand{\NoSegConsOAbs}{0.5} | |
| 1304 | +\newcommand{\NoSegConsR}{\ensuremath{-}1.3} | |
| 1305 | +\newcommand{\NoSegConsRAbs}{1.3} | |
| 1306 | +\newcommand{\NoSegDOwnGE}{\ensuremath{-}0.16} | |
| 1307 | +\newcommand{\NoSegDOwnGEAbs}{0.16} | |
| 1308 | +\newcommand{\NoSegDOwnPE}{\ensuremath{-}0.20} | |
| 1309 | +\newcommand{\NoSegDOwnPEAbs}{0.20} | |
| 1310 | +\newcommand{\NoSegDebtZeroPct}{\ensuremath{-}3.0} | |
| 1311 | +\newcommand{\NoSegDebtZeroPctAbs}{3.0} | |
| 1312 | +\newcommand{\NoSegDepLoss}{0.48} | |
| 1313 | +\newcommand{\NoSegErosion}{0.21} | |
| 1314 | +\newcommand{\NoSegLamSevenSS}{0.060} | |
| 1315 | +\newcommand{\NoSegLamSevenZero}{0.069} | |
| 1316 | +\newcommand{\NoSegLender}{0.27} | |
| 1317 | +\newcommand{\NoSegMaxRes}{6.5\times10^{-4}} | |
| 1318 | +\newcommand{\NoSegNWAll}{\ensuremath{-}0.02} | |
| 1319 | +\newcommand{\NoSegNWAllAbs}{0.02} | |
| 1320 | +\newcommand{\NoSegNWO}{\ensuremath{-}0.02} | |
| 1321 | +\newcommand{\NoSegNWOAbs}{0.02} | |
| 1322 | +\newcommand{\NoSegNWR}{\ensuremath{-}0.01} | |
| 1323 | +\newcommand{\NoSegNWRAbs}{0.01} | |
| 1324 | +\newcommand{\NoSegNetO}{\ensuremath{-}0.02} | |
| 1325 | +\newcommand{\NoSegNetOAbs}{0.02} | |
| 1326 | +\newcommand{\NoSegNetR}{\ensuremath{-}0.02} | |
| 1327 | +\newcommand{\NoSegNetRAbs}{0.02} | |
| 1328 | +\newcommand{\NoSegNominalRateZero}{0.219} | |
| 1329 | +\newcommand{\NoSegOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 1330 | +\newcommand{\NoSegOwnerOldConsMaxAbs}{2.3} | |
| 1331 | +\newcommand{\NoSegOwnerOldConsMin}{\ensuremath{-}1.7} | |
| 1332 | +\newcommand{\NoSegOwnerOldConsMinAbs}{1.7} | |
| 1333 | +\newcommand{\NoSegOwnerOldLossMax}{0.08} | |
| 1334 | +\newcommand{\NoSegOwnerOldLossMin}{0.04} | |
| 1335 | +\newcommand{\NoSegOwnerYoungConsMax}{1.1} | |
| 1336 | +\newcommand{\NoSegOwnerYoungConsMin}{0.7} | |
| 1337 | +\newcommand{\NoSegOwnerYoungGainMax}{0.04} | |
| 1338 | +\newcommand{\NoSegPriceBackYears}{10} | |
| 1339 | +\newcommand{\NoSegPriceMaxPos}{0.1} | |
| 1340 | +\newcommand{\NoSegPriceSignChanges}{1} | |
| 1341 | +\newcommand{\NoSegPzero}{\ensuremath{-}0.39} | |
| 1342 | +\newcommand{\NoSegPzeroAbs}{0.39} | |
| 1343 | +\newcommand{\NoSegRentGainR}{0.00} | |
| 1344 | +\newcommand{\NoSegResT}{4\times10^{-5}} | |
| 1345 | +\newcommand{\NoSegRhoZero}{\ensuremath{-}1.47} | |
| 1346 | +\newcommand{\NoSegRhoZeroAbs}{1.47} | |
| 1347 | +\newcommand{\NoSegSecs}{152} | |
| 1348 | +\newcommand{\NoSegSteps}{2} | |
| 1349 | +\newcommand{\NoSegThrChangePct}{9} | |
| 1350 | +\newcommand{\NoSegThrSS}{0.49} | |
| 1351 | +\newcommand{\NoSegThrZero}{0.54} | |
| 1352 | +\newcommand{\NoSegWageAll}{0.00} | |
| 1353 | +\newcommand{\NoSegWageO}{0.00} | |
| 1354 | +\newcommand{\NoSegqROsevenSS}{0.806} | |
| 1355 | +\newcommand{\NoSegqROsevenZero}{0.796} | |
| 1356 | +\newcommand{\NoSegqROthreeOne}{0.333} | |
| 1357 | +\newcommand{\NoSegqROthreeSS}{0.351} | |
| 1358 | +\newcommand{\NoSegqROthreeZero}{0.356} | |
| 1359 | +\newcommand{\OwnAgeOne}{0.34} | |
| 1360 | +\newcommand{\OwnAgeOneQFive}{0.89} | |
| 1361 | +\newcommand{\OwnAgeOneQFour}{0.57} | |
| 1362 | +\newcommand{\OwnAgeOneQOne}{0.02} | |
| 1363 | +\newcommand{\OwnAgeOneQThree}{0.18} | |
| 1364 | +\newcommand{\OwnAgeOneQTwo}{0.03} | |
| 1365 | +\newcommand{\OwnAgeSevenQFive}{0.97} | |
| 1366 | +\newcommand{\OwnAgeSevenQFour}{0.88} | |
| 1367 | +\newcommand{\OwnAgeSevenQOne}{0.81} | |
| 1368 | +\newcommand{\OwnAgeSevenQThree}{0.82} | |
| 1369 | +\newcommand{\OwnAgeSevenQTwo}{0.81} | |
| 1370 | +\newcommand{\OwnAgeTwelve}{0.36} | |
| 1371 | +\newcommand{\OwnBaseline}{0.631} | |
| 1372 | +\newcommand{\OwnBequestshalved}{0.615} | |
| 1373 | +\newcommand{\OwnBequestsoff}{0.581} | |
| 1374 | +\newcommand{\OwnBequestsoffpricesfixed}{0.512} | |
| 1375 | +\newcommand{\OwnDSTIcappsi}{0.634} | |
| 1376 | +\newcommand{\OwnHalfsegmentation}{0.631} | |
| 1377 | +\newcommand{\OwnNosegmentation}{0.631} | |
| 1378 | +\newcommand{\OwnPeak}{0.86} | |
| 1379 | +\newcommand{\OwnSixtyFivePlus}{0.61} | |
| 1380 | +\newcommand{\OwnThirtyFiveSixtyFour}{0.73} | |
| 1381 | +\newcommand{\OwnTransfertax}{0.639} | |
| 1382 | +\newcommand{\OwnershipAll}{0.63} | |
| 1383 | +\newcommand{\OwnershipUnderThirtyFive}{0.38} | |
| 1384 | +\newcommand{\Owntheta}{0.619} | |
| 1385 | +\newcommand{\Ownthetapricesfixed}{0.596} | |
| 1386 | +\newcommand{\Ownvarepsilon}{0.631} | |
| 1387 | +\newcommand{\PImpactarmlevel}{\ensuremath{-}0.79} | |
| 1388 | +\newcommand{\PImpactarmlevelAbs}{0.79} | |
| 1389 | +\newcommand{\PImpactarmpersistent}{\ensuremath{-}1.86} | |
| 1390 | +\newcommand{\PImpactarmpersistentAbs}{1.86} | |
| 1391 | +\newcommand{\PImpactfrmlevel}{\ensuremath{-}0.45} | |
| 1392 | +\newcommand{\PImpactfrmlevelAbs}{0.45} | |
| 1393 | +\newcommand{\PImpactfrmlevelfixedsupply}{\ensuremath{-}0.50} | |
| 1394 | +\newcommand{\PImpactfrmlevelfixedsupplyAbs}{0.50} | |
| 1395 | +\newcommand{\PImpactfrmlevelnoseg}{\ensuremath{-}0.39} | |
| 1396 | +\newcommand{\PImpactfrmlevelnosegAbs}{0.39} | |
| 1397 | +\newcommand{\PImpactfrmlevelrigidrents}{\ensuremath{-}0.46} | |
| 1398 | +\newcommand{\PImpactfrmlevelrigidrentsAbs}{0.46} | |
| 1399 | +\newcommand{\PImpactfrmlevelstatic}{\ensuremath{-}0.36} | |
| 1400 | +\newcommand{\PImpactfrmlevelstaticAbs}{0.36} | |
| 1401 | +\newcommand{\PImpactfrmlevelunindexedwages}{\ensuremath{-}3.86} | |
| 1402 | +\newcommand{\PImpactfrmlevelunindexedwagesAbs}{3.86} | |
| 1403 | +\newcommand{\PImpactfrmpersistent}{\ensuremath{-}1.88} | |
| 1404 | +\newcommand{\PImpactfrmpersistentAbs}{1.88} | |
| 1405 | +\newcommand{\PImpactfrmpersistenttaylor}{\ensuremath{-}7.25} | |
| 1406 | +\newcommand{\PImpactfrmpersistenttaylorAbs}{7.25} | |
| 1407 | +\newcommand{\PImpacttermlevel}{\ensuremath{-}0.47} | |
| 1408 | +\newcommand{\PImpacttermlevelAbs}{0.47} | |
| 1409 | +\newcommand{\PImpacttermpersistent}{\ensuremath{-}1.49} | |
| 1410 | +\newcommand{\PImpacttermpersistentAbs}{1.49} | |
| 1411 | +\newcommand{\PImpacttermpersistenttaylor}{\ensuremath{-}6.93} | |
| 1412 | +\newcommand{\PImpacttermpersistenttaylorAbs}{6.93} | |
| 1413 | +\newcommand{\PhiHighEstate}{0.32} | |
| 1414 | +\newcommand{\PhiHighGapFour}{0.94} | |
| 1415 | +\newcommand{\PhiHighGift}{0.39} | |
| 1416 | +\newcommand{\PhiHighGini}{0.54} | |
| 1417 | +\newcommand{\PhiHighLamSeven}{0.055} | |
| 1418 | +\newcommand{\PhiHighLamThree}{0.19} | |
| 1419 | +\newcommand{\PhiHighOwn}{0.650} | |
| 1420 | +\newcommand{\PhiHighOwnUthirtyfive}{0.412} | |
| 1421 | +\newcommand{\PhiHighP}{1.007} | |
| 1422 | +\newcommand{\PhiHighPDevPct}{0.7} | |
| 1423 | +\newcommand{\PhiHighRR}{0.34} | |
| 1424 | +\newcommand{\PhiHighRebate}{0.000} | |
| 1425 | +\newcommand{\PhiHighRho}{0.202} | |
| 1426 | +\newcommand{\PhiHighRhoDevPct}{4.0} | |
| 1427 | +\newcommand{\PhiHighThr}{0.47} | |
| 1428 | +\newcommand{\PhiHighThrHighZ}{0.00} | |
| 1429 | +\newcommand{\PhiHighThrLowZ}{1.13} | |
| 1430 | +\newcommand{\PhiHighqOO}{0.548} | |
| 1431 | +\newcommand{\PhiHighqRO}{0.360} | |
| 1432 | +\newcommand{\PhiLowEstate}{0.31} | |
| 1433 | +\newcommand{\PhiLowGapFour}{0.96} | |
| 1434 | +\newcommand{\PhiLowGift}{0.39} | |
| 1435 | +\newcommand{\PhiLowGini}{0.55} | |
| 1436 | +\newcommand{\PhiLowLamSeven}{0.070} | |
| 1437 | +\newcommand{\PhiLowLamThree}{0.19} | |
| 1438 | +\newcommand{\PhiLowOwn}{0.607} | |
| 1439 | +\newcommand{\PhiLowOwnUthirtyfive}{0.342} | |
| 1440 | +\newcommand{\PhiLowP}{0.991} | |
| 1441 | +\newcommand{\PhiLowPDevPct}{\ensuremath{-}0.9} | |
| 1442 | +\newcommand{\PhiLowPDevPctAbs}{0.9} | |
| 1443 | +\newcommand{\PhiLowRR}{0.33} | |
| 1444 | +\newcommand{\PhiLowRebate}{0.000} | |
| 1445 | +\newcommand{\PhiLowRho}{0.186} | |
| 1446 | +\newcommand{\PhiLowRhoDevPct}{\ensuremath{-}4.4} | |
| 1447 | +\newcommand{\PhiLowRhoDevPctAbs}{4.4} | |
| 1448 | +\newcommand{\PhiLowThr}{0.53} | |
| 1449 | +\newcommand{\PhiLowThrHighZ}{0.05} | |
| 1450 | +\newcommand{\PhiLowThrLowZ}{1.24} | |
| 1451 | +\newcommand{\PhiLowqOO}{0.524} | |
| 1452 | +\newcommand{\PhiLowqRO}{0.332} | |
| 1453 | +\newcommand{\PiFourEstate}{0.33} | |
| 1454 | +\newcommand{\PiFourGEOffsetPct}{65} | |
| 1455 | +\newcommand{\PiFourGapFour}{0.97} | |
| 1456 | +\newcommand{\PiFourGift}{0.39} | |
| 1457 | +\newcommand{\PiFourGini}{0.54} | |
| 1458 | +\newcommand{\PiFourLamSeven}{0.060} | |
| 1459 | +\newcommand{\PiFourLamThree}{0.19} | |
| 1460 | +\newcommand{\PiFourOwn}{0.625} | |
| 1461 | +\newcommand{\PiFourOwnUthirtyfive}{0.361} | |
| 1462 | +\newcommand{\PiFourP}{0.986} | |
| 1463 | +\newcommand{\PiFourPDevPct}{\ensuremath{-}1.4} | |
| 1464 | +\newcommand{\PiFourPDevPctAbs}{1.4} | |
| 1465 | +\newcommand{\PiFourPEDropUthirtyfivePP}{4.9} | |
| 1466 | +\newcommand{\PiFourPEEstate}{0.32} | |
| 1467 | +\newcommand{\PiFourPEGapFour}{0.98} | |
| 1468 | +\newcommand{\PiFourPEGift}{0.38} | |
| 1469 | +\newcommand{\PiFourPEGini}{0.55} | |
| 1470 | +\newcommand{\PiFourPELamSeven}{0.070} | |
| 1471 | +\newcommand{\PiFourPELamThree}{0.19} | |
| 1472 | +\newcommand{\PiFourPEOwn}{0.604} | |
| 1473 | +\newcommand{\PiFourPEOwnUthirtyfive}{0.330} | |
| 1474 | +\newcommand{\PiFourPEP}{1.000} | |
| 1475 | +\newcommand{\PiFourPEPDevPct}{0.0} | |
| 1476 | +\newcommand{\PiFourPERR}{0.34} | |
| 1477 | +\newcommand{\PiFourPERebate}{0.000} | |
| 1478 | +\newcommand{\PiFourPERho}{0.194} | |
| 1479 | +\newcommand{\PiFourPERhoDevPct}{0.0} | |
| 1480 | +\newcommand{\PiFourPEThr}{0.59} | |
| 1481 | +\newcommand{\PiFourPEThrHighZ}{0.09} | |
| 1482 | +\newcommand{\PiFourPEThrLowZ}{1.21} | |
| 1483 | +\newcommand{\PiFourPEqOO}{0.513} | |
| 1484 | +\newcommand{\PiFourPEqRO}{0.324} | |
| 1485 | +\newcommand{\PiFourRR}{0.34} | |
| 1486 | +\newcommand{\PiFourRebate}{0.000} | |
| 1487 | +\newcommand{\PiFourRho}{0.197} | |
| 1488 | +\newcommand{\PiFourRhoDevPct}{1.5} | |
| 1489 | +\newcommand{\PiFourThr}{0.55} | |
| 1490 | +\newcommand{\PiFourThrHighZ}{0.07} | |
| 1491 | +\newcommand{\PiFourThrLowZ}{1.13} | |
| 1492 | +\newcommand{\PiFourqOO}{0.529} | |
| 1493 | +\newcommand{\PiFourqRO}{0.338} | |
| 1494 | +\newcommand{\PiZeroEstate}{0.30} | |
| 1495 | +\newcommand{\PiZeroGapFour}{0.91} | |
| 1496 | +\newcommand{\PiZeroGift}{0.39} | |
| 1497 | +\newcommand{\PiZeroGini}{0.55} | |
| 1498 | +\newcommand{\PiZeroLamSeven}{0.058} | |
| 1499 | +\newcommand{\PiZeroLamThree}{0.19} | |
| 1500 | +\newcommand{\PiZeroOwn}{0.640} | |
| 1501 | +\newcommand{\PiZeroOwnUthirtyfive}{0.398} | |
| 1502 | +\newcommand{\PiZeroP}{1.011} | |
| 1503 | +\newcommand{\PiZeroPDevPct}{1.1} | |
| 1504 | +\newcommand{\PiZeroRR}{0.33} | |
| 1505 | +\newcommand{\PiZeroRebate}{0.000} | |
| 1506 | +\newcommand{\PiZeroRho}{0.190} | |
| 1507 | +\newcommand{\PiZeroRhoDevPct}{\ensuremath{-}2.0} | |
| 1508 | +\newcommand{\PiZeroRhoDevPctAbs}{2.0} | |
| 1509 | +\newcommand{\PiZeroThr}{0.43} | |
| 1510 | +\newcommand{\PiZeroThrHighZ}{0.01} | |
| 1511 | +\newcommand{\PiZeroThrLowZ}{1.18} | |
| 1512 | +\newcommand{\PiZeroqOO}{0.537} | |
| 1513 | +\newcommand{\PiZeroqRO}{0.346} | |
| 1514 | +\newcommand{\PriceBaseline}{1.000} | |
| 1515 | +\newcommand{\PriceBequestshalved}{0.981} | |
| 1516 | +\newcommand{\PriceBequestsoff}{0.931} | |
| 1517 | +\newcommand{\PriceBequestsoffpricesfixed}{1.000} | |
| 1518 | +\newcommand{\PriceDSTIcappsi}{1.006} | |
| 1519 | +\newcommand{\PriceHalfsegmentation}{1.000} | |
| 1520 | +\newcommand{\PriceNosegmentation}{1.000} | |
| 1521 | +\newcommand{\PriceTransfertax}{0.999} | |
| 1522 | +\newcommand{\Pricetheta}{0.987} | |
| 1523 | +\newcommand{\Pricethetapricesfixed}{1.000} | |
| 1524 | +\newcommand{\Pricevarepsilon}{1.000} | |
| 1525 | +\newcommand{\PsiFiftyEstate}{0.30} | |
| 1526 | +\newcommand{\PsiFiftyGapFour}{0.96} | |
| 1527 | +\newcommand{\PsiFiftyGift}{0.39} | |
| 1528 | +\newcommand{\PsiFiftyGini}{0.54} | |
| 1529 | +\newcommand{\PsiFiftyLamSeven}{0.056} | |
| 1530 | +\newcommand{\PsiFiftyLamThree}{0.19} | |
| 1531 | +\newcommand{\PsiFiftyOwn}{0.634} | |
| 1532 | +\newcommand{\PsiFiftyOwnUthirtyfive}{0.382} | |
| 1533 | +\newcommand{\PsiFiftyP}{1.006} | |
| 1534 | +\newcommand{\PsiFiftyPDevPct}{0.6} | |
| 1535 | +\newcommand{\PsiFiftyRR}{0.33} | |
| 1536 | +\newcommand{\PsiFiftyRebate}{0.000} | |
| 1537 | +\newcommand{\PsiFiftyRho}{0.193} | |
| 1538 | +\newcommand{\PsiFiftyRhoDevPct}{\ensuremath{-}0.7} | |
| 1539 | +\newcommand{\PsiFiftyRhoDevPctAbs}{0.7} | |
| 1540 | +\newcommand{\PsiFiftyThr}{0.46} | |
| 1541 | +\newcommand{\PsiFiftyThrHighZ}{0.04} | |
| 1542 | +\newcommand{\PsiFiftyThrLowZ}{1.13} | |
| 1543 | +\newcommand{\PsiFiftyqOO}{0.533} | |
| 1544 | +\newcommand{\PsiFiftyqRO}{0.339} | |
| 1545 | +\newcommand{\PsiThirtyEstate}{0.33} | |
| 1546 | +\newcommand{\PsiThirtyGapFour}{0.94} | |
| 1547 | +\newcommand{\PsiThirtyGift}{0.39} | |
| 1548 | +\newcommand{\PsiThirtyGini}{0.54} | |
| 1549 | +\newcommand{\PsiThirtyLamSeven}{0.053} | |
| 1550 | +\newcommand{\PsiThirtyLamThree}{0.17} | |
| 1551 | +\newcommand{\PsiThirtyOwn}{0.627} | |
| 1552 | +\newcommand{\PsiThirtyOwnUthirtyfive}{0.362} | |
| 1553 | +\newcommand{\PsiThirtyP}{0.991} | |
| 1554 | +\newcommand{\PsiThirtyPDevPct}{\ensuremath{-}0.9} | |
| 1555 | +\newcommand{\PsiThirtyPDevPctAbs}{0.9} | |
| 1556 | +\newcommand{\PsiThirtyRR}{0.33} | |
| 1557 | +\newcommand{\PsiThirtyRebate}{0.000} | |
| 1558 | +\newcommand{\PsiThirtyRho}{0.196} | |
| 1559 | +\newcommand{\PsiThirtyRhoDevPct}{0.9} | |
| 1560 | +\newcommand{\PsiThirtyThr}{0.59} | |
| 1561 | +\newcommand{\PsiThirtyThrHighZ}{0.02} | |
| 1562 | +\newcommand{\PsiThirtyThrLowZ}{1.18} | |
| 1563 | +\newcommand{\PsiThirtyqOO}{0.535} | |
| 1564 | +\newcommand{\PsiThirtyqRO}{0.362} | |
| 1565 | +\newcommand{\Pstar}{1.000} | |
| 1566 | +\newcommand{\RankRankNW}{0.34} | |
| 1567 | +\newcommand{\RateFiveAbsorbPct}{96} | |
| 1568 | +\newcommand{\RateFiveBetterAllPct}{75} | |
| 1569 | +\newcommand{\RateFiveBetterOMPct}{65} | |
| 1570 | +\newcommand{\RateFiveBetterONMPct}{69} | |
| 1571 | +\newcommand{\RateFiveBetterOPct}{67} | |
| 1572 | +\newcommand{\RateFiveBetterRPct}{87} | |
| 1573 | +\newcommand{\RateFiveCEVAll}{2.11} | |
| 1574 | +\newcommand{\RateFiveCEVAllPE}{\ensuremath{-}1.52} | |
| 1575 | +\newcommand{\RateFiveCEVAllPEAbs}{1.52} | |
| 1576 | +\newcommand{\RateFiveCEVO}{1.56} | |
| 1577 | +\newcommand{\RateFiveCEVOM}{1.00} | |
| 1578 | +\newcommand{\RateFiveCEVONM}{2.86} | |
| 1579 | +\newcommand{\RateFiveCEVR}{2.59} | |
| 1580 | +\newcommand{\RateFiveConsAll}{\ensuremath{-}5.7} | |
| 1581 | +\newcommand{\RateFiveConsAllAbs}{5.7} | |
| 1582 | +\newcommand{\RateFiveConsO}{\ensuremath{-}6.3} | |
| 1583 | +\newcommand{\RateFiveConsOAbs}{6.3} | |
| 1584 | +\newcommand{\RateFiveConsR}{\ensuremath{-}3.9} | |
| 1585 | +\newcommand{\RateFiveConsRAbs}{3.9} | |
| 1586 | +\newcommand{\RateFiveDOwnGE}{\ensuremath{-}0.20} | |
| 1587 | +\newcommand{\RateFiveDOwnGEAbs}{0.20} | |
| 1588 | +\newcommand{\RateFiveDOwnPE}{\ensuremath{-}5.56} | |
| 1589 | +\newcommand{\RateFiveDOwnPEAbs}{5.56} | |
| 1590 | +\newcommand{\RateFiveDebtZeroPct}{\ensuremath{-}13.5} | |
| 1591 | +\newcommand{\RateFiveDebtZeroPctAbs}{13.5} | |
| 1592 | +\newcommand{\RateFiveDepLoss}{0.00} | |
| 1593 | +\newcommand{\RateFiveErosion}{0.00} | |
| 1594 | +\newcommand{\RateFiveLamSevenSS}{0.069} | |
| 1595 | +\newcommand{\RateFiveLamSevenZero}{0.067} | |
| 1596 | +\newcommand{\RateFiveLender}{0.00} | |
| 1597 | +\newcommand{\RateFiveMaxRes}{8.8\times10^{-3}} | |
| 1598 | +\newcommand{\RateFiveNWAll}{\ensuremath{-}0.06} | |
| 1599 | +\newcommand{\RateFiveNWAllAbs}{0.06} | |
| 1600 | +\newcommand{\RateFiveNWO}{\ensuremath{-}0.11} | |
| 1601 | +\newcommand{\RateFiveNWOAbs}{0.11} | |
| 1602 | +\newcommand{\RateFiveNWR}{0.02} | |
| 1603 | +\newcommand{\RateFiveNetO}{0.00} | |
| 1604 | +\newcommand{\RateFiveNetR}{0.00} | |
| 1605 | +\newcommand{\RateFiveNominalRateZero}{0.346} | |
| 1606 | +\newcommand{\RateFiveOwnerOldConsMax}{\ensuremath{-}6.5} | |
| 1607 | +\newcommand{\RateFiveOwnerOldConsMaxAbs}{6.5} | |
| 1608 | +\newcommand{\RateFiveOwnerOldConsMin}{\ensuremath{-}4.9} | |
| 1609 | +\newcommand{\RateFiveOwnerOldConsMinAbs}{4.9} | |
| 1610 | +\newcommand{\RateFiveOwnerOldLossMax}{0.00} | |
| 1611 | +\newcommand{\RateFiveOwnerOldLossMin}{0.00} | |
| 1612 | +\newcommand{\RateFiveOwnerYoungConsMax}{\ensuremath{-}7.9} | |
| 1613 | +\newcommand{\RateFiveOwnerYoungConsMaxAbs}{7.9} | |
| 1614 | +\newcommand{\RateFiveOwnerYoungConsMin}{\ensuremath{-}11.0} | |
| 1615 | +\newcommand{\RateFiveOwnerYoungConsMinAbs}{11.0} | |
| 1616 | +\newcommand{\RateFiveOwnerYoungGainMax}{0.00} | |
| 1617 | +\newcommand{\RateFivePriceBackYears}{15} | |
| 1618 | +\newcommand{\RateFivePriceMaxPos}{7.6} | |
| 1619 | +\newcommand{\RateFivePriceSignChanges}{1} | |
| 1620 | +\newcommand{\RateFivePzero}{\ensuremath{-}16.16} | |
| 1621 | +\newcommand{\RateFivePzeroAbs}{16.16} | |
| 1622 | +\newcommand{\RateFiveRentGainR}{0.00} | |
| 1623 | +\newcommand{\RateFiveResT}{1\times10^{-4}} | |
| 1624 | +\newcommand{\RateFiveRhoZero}{\ensuremath{-}3.02} | |
| 1625 | +\newcommand{\RateFiveRhoZeroAbs}{3.02} | |
| 1626 | +\newcommand{\RateFiveSecs}{437} | |
| 1627 | +\newcommand{\RateFiveSteps}{4} | |
| 1628 | +\newcommand{\RateFiveThrChangePct}{\ensuremath{-}19} | |
| 1629 | +\newcommand{\RateFiveThrChangePctAbs}{19} | |
| 1630 | +\newcommand{\RateFiveThrSS}{0.56} | |
| 1631 | +\newcommand{\RateFiveThrZero}{0.45} | |
| 1632 | +\newcommand{\RateFiveWageAll}{0.00} | |
| 1633 | +\newcommand{\RateFiveWageO}{0.00} | |
| 1634 | +\newcommand{\RateFiveqROsevenSS}{0.780} | |
| 1635 | +\newcommand{\RateFiveqROsevenZero}{0.765} | |
| 1636 | +\newcommand{\RateFiveqROthreeOne}{0.372} | |
| 1637 | +\newcommand{\RateFiveqROthreeSS}{0.375} | |
| 1638 | +\newcommand{\RateFiveqROthreeZero}{0.382} | |
| 1639 | +\newcommand{\RateFortyAbsorbPct}{109} | |
| 1640 | +\newcommand{\RateFortyBetterAllPct}{80} | |
| 1641 | +\newcommand{\RateFortyBetterOMPct}{64} | |
| 1642 | +\newcommand{\RateFortyBetterONMPct}{72} | |
| 1643 | +\newcommand{\RateFortyBetterOPct}{67} | |
| 1644 | +\newcommand{\RateFortyBetterRPct}{100} | |
| 1645 | +\newcommand{\RateFortyCEVAll}{2.61} | |
| 1646 | +\newcommand{\RateFortyCEVAllPE}{\ensuremath{-}1.52} | |
| 1647 | +\newcommand{\RateFortyCEVAllPEAbs}{1.52} | |
| 1648 | +\newcommand{\RateFortyCEVO}{1.92} | |
| 1649 | +\newcommand{\RateFortyCEVOM}{1.36} | |
| 1650 | +\newcommand{\RateFortyCEVONM}{3.38} | |
| 1651 | +\newcommand{\RateFortyCEVR}{3.15} | |
| 1652 | +\newcommand{\RateFortyConsAll}{\ensuremath{-}5.1} | |
| 1653 | +\newcommand{\RateFortyConsAllAbs}{5.1} | |
| 1654 | +\newcommand{\RateFortyConsO}{\ensuremath{-}5.7} | |
| 1655 | +\newcommand{\RateFortyConsOAbs}{5.7} | |
| 1656 | +\newcommand{\RateFortyConsR}{\ensuremath{-}3.7} | |
| 1657 | +\newcommand{\RateFortyConsRAbs}{3.7} | |
| 1658 | +\newcommand{\RateFortyDOwnGE}{0.16} | |
| 1659 | +\newcommand{\RateFortyDOwnPE}{\ensuremath{-}1.77} | |
| 1660 | +\newcommand{\RateFortyDOwnPEAbs}{1.77} | |
| 1661 | +\newcommand{\RateFortyDebtZeroPct}{\ensuremath{-}17.0} | |
| 1662 | +\newcommand{\RateFortyDebtZeroPctAbs}{17.0} | |
| 1663 | +\newcommand{\RateFortyDepLoss}{0.00} | |
| 1664 | +\newcommand{\RateFortyErosion}{0.00} | |
| 1665 | +\newcommand{\RateFortyLamSevenSS}{0.083} | |
| 1666 | +\newcommand{\RateFortyLamSevenZero}{0.091} | |
| 1667 | +\newcommand{\RateFortyLender}{0.00} | |
| 1668 | +\newcommand{\RateFortyMaxRes}{1.1\times10^{-3}} | |
| 1669 | +\newcommand{\RateFortyNWAll}{\ensuremath{-}0.05} | |
| 1670 | +\newcommand{\RateFortyNWAllAbs}{0.05} | |
| 1671 | +\newcommand{\RateFortyNWO}{\ensuremath{-}0.10} | |
| 1672 | +\newcommand{\RateFortyNWOAbs}{0.10} | |
| 1673 | +\newcommand{\RateFortyNWR}{0.02} | |
| 1674 | +\newcommand{\RateFortyNetO}{0.00} | |
| 1675 | +\newcommand{\RateFortyNetR}{0.00} | |
| 1676 | +\newcommand{\RateFortyNominalRateZero}{0.346} | |
| 1677 | +\newcommand{\RateFortyOwnerOldConsMax}{\ensuremath{-}5.8} | |
| 1678 | +\newcommand{\RateFortyOwnerOldConsMaxAbs}{5.8} | |
| 1679 | +\newcommand{\RateFortyOwnerOldConsMin}{\ensuremath{-}4.1} | |
| 1680 | +\newcommand{\RateFortyOwnerOldConsMinAbs}{4.1} | |
| 1681 | +\newcommand{\RateFortyOwnerOldLossMax}{0.00} | |
| 1682 | +\newcommand{\RateFortyOwnerOldLossMin}{0.00} | |
| 1683 | +\newcommand{\RateFortyOwnerYoungConsMax}{\ensuremath{-}7.3} | |
| 1684 | +\newcommand{\RateFortyOwnerYoungConsMaxAbs}{7.3} | |
| 1685 | +\newcommand{\RateFortyOwnerYoungConsMin}{\ensuremath{-}9.6} | |
| 1686 | +\newcommand{\RateFortyOwnerYoungConsMinAbs}{9.6} | |
| 1687 | +\newcommand{\RateFortyOwnerYoungGainMax}{0.00} | |
| 1688 | +\newcommand{\RateFortyPriceBackYears}{--} | |
| 1689 | +\newcommand{\RateFortyPriceMaxPos}{7.8} | |
| 1690 | +\newcommand{\RateFortyPriceSignChanges}{1} | |
| 1691 | +\newcommand{\RateFortyPzero}{\ensuremath{-}14.61} | |
| 1692 | +\newcommand{\RateFortyPzeroAbs}{14.61} | |
| 1693 | +\newcommand{\RateFortyRentGainR}{0.00} | |
| 1694 | +\newcommand{\RateFortyResT}{3\times10^{-5}} | |
| 1695 | +\newcommand{\RateFortyRhoZero}{\ensuremath{-}5.19} | |
| 1696 | +\newcommand{\RateFortyRhoZeroAbs}{5.19} | |
| 1697 | +\newcommand{\RateFortySecs}{351} | |
| 1698 | +\newcommand{\RateFortySteps}{11} | |
| 1699 | +\newcommand{\RateFortyThrChangePct}{\ensuremath{-}10} | |
| 1700 | +\newcommand{\RateFortyThrChangePctAbs}{10} | |
| 1701 | +\newcommand{\RateFortyThrSS}{0.55} | |
| 1702 | +\newcommand{\RateFortyThrZero}{0.49} | |
| 1703 | +\newcommand{\RateFortyWageAll}{0.00} | |
| 1704 | +\newcommand{\RateFortyWageO}{0.00} | |
| 1705 | +\newcommand{\RateFortyqROsevenSS}{0.784} | |
| 1706 | +\newcommand{\RateFortyqROsevenZero}{0.744} | |
| 1707 | +\newcommand{\RateFortyqROthreeOne}{0.341} | |
| 1708 | +\newcommand{\RateFortyqROthreeSS}{0.304} | |
| 1709 | +\newcommand{\RateFortyqROthreeZero}{0.334} | |
| 1710 | +\newcommand{\RateTwentyAbsorbPct}{100} | |
| 1711 | +\newcommand{\RateTwentyBetterAllPct}{80} | |
| 1712 | +\newcommand{\RateTwentyBetterOMPct}{65} | |
| 1713 | +\newcommand{\RateTwentyBetterONMPct}{70} | |
| 1714 | +\newcommand{\RateTwentyBetterOPct}{66} | |
| 1715 | +\newcommand{\RateTwentyBetterRPct}{100} | |
| 1716 | +\newcommand{\RateTwentyCEVAll}{2.33} | |
| 1717 | +\newcommand{\RateTwentyCEVAllPE}{\ensuremath{-}1.57} | |
| 1718 | +\newcommand{\RateTwentyCEVAllPEAbs}{1.57} | |
| 1719 | +\newcommand{\RateTwentyCEVO}{1.55} | |
| 1720 | +\newcommand{\RateTwentyCEVOM}{0.97} | |
| 1721 | +\newcommand{\RateTwentyCEVONM}{2.96} | |
| 1722 | +\newcommand{\RateTwentyCEVR}{2.98} | |
| 1723 | +\newcommand{\RateTwentyConsAll}{\ensuremath{-}5.4} | |
| 1724 | +\newcommand{\RateTwentyConsAllAbs}{5.4} | |
| 1725 | +\newcommand{\RateTwentyConsO}{\ensuremath{-}6.0} | |
| 1726 | +\newcommand{\RateTwentyConsOAbs}{6.0} | |
| 1727 | +\newcommand{\RateTwentyConsR}{\ensuremath{-}3.9} | |
| 1728 | +\newcommand{\RateTwentyConsRAbs}{3.9} | |
| 1729 | +\newcommand{\RateTwentyDOwnGE}{0.01} | |
| 1730 | +\newcommand{\RateTwentyDOwnPE}{\ensuremath{-}4.11} | |
| 1731 | +\newcommand{\RateTwentyDOwnPEAbs}{4.11} | |
| 1732 | +\newcommand{\RateTwentyDebtZeroPct}{\ensuremath{-}15.0} | |
| 1733 | +\newcommand{\RateTwentyDebtZeroPctAbs}{15.0} | |
| 1734 | +\newcommand{\RateTwentyDepLoss}{0.00} | |
| 1735 | +\newcommand{\RateTwentyErosion}{0.00} | |
| 1736 | +\newcommand{\RateTwentyLamSevenSS}{0.060} | |
| 1737 | +\newcommand{\RateTwentyLamSevenZero}{0.070} | |
| 1738 | +\newcommand{\RateTwentyLender}{0.00} | |
| 1739 | +\newcommand{\RateTwentyMaxRes}{9.5\times10^{-4}} | |
| 1740 | +\newcommand{\RateTwentyNWAll}{\ensuremath{-}0.06} | |
| 1741 | +\newcommand{\RateTwentyNWAllAbs}{0.06} | |
| 1742 | +\newcommand{\RateTwentyNWO}{\ensuremath{-}0.11} | |
| 1743 | +\newcommand{\RateTwentyNWOAbs}{0.11} | |
| 1744 | +\newcommand{\RateTwentyNWR}{0.02} | |
| 1745 | +\newcommand{\RateTwentyNetO}{0.00} | |
| 1746 | +\newcommand{\RateTwentyNetR}{0.00} | |
| 1747 | +\newcommand{\RateTwentyNominalRateZero}{0.346} | |
| 1748 | +\newcommand{\RateTwentyOwnerOldConsMax}{\ensuremath{-}6.3} | |
| 1749 | +\newcommand{\RateTwentyOwnerOldConsMaxAbs}{6.3} | |
| 1750 | +\newcommand{\RateTwentyOwnerOldConsMin}{\ensuremath{-}4.7} | |
| 1751 | +\newcommand{\RateTwentyOwnerOldConsMinAbs}{4.7} | |
| 1752 | +\newcommand{\RateTwentyOwnerOldLossMax}{0.00} | |
| 1753 | +\newcommand{\RateTwentyOwnerOldLossMin}{0.00} | |
| 1754 | +\newcommand{\RateTwentyOwnerYoungConsMax}{\ensuremath{-}7.5} | |
| 1755 | +\newcommand{\RateTwentyOwnerYoungConsMaxAbs}{7.5} | |
| 1756 | +\newcommand{\RateTwentyOwnerYoungConsMin}{\ensuremath{-}10.7} | |
| 1757 | +\newcommand{\RateTwentyOwnerYoungConsMinAbs}{10.7} | |
| 1758 | +\newcommand{\RateTwentyOwnerYoungGainMax}{0.00} | |
| 1759 | +\newcommand{\RateTwentyPriceBackYears}{--} | |
| 1760 | +\newcommand{\RateTwentyPriceMaxPos}{6.9} | |
| 1761 | +\newcommand{\RateTwentyPriceSignChanges}{1} | |
| 1762 | +\newcommand{\RateTwentyPzero}{\ensuremath{-}15.54} | |
| 1763 | +\newcommand{\RateTwentyPzeroAbs}{15.54} | |
| 1764 | +\newcommand{\RateTwentyRentGainR}{0.00} | |
| 1765 | +\newcommand{\RateTwentyResT}{7\times10^{-5}} | |
| 1766 | +\newcommand{\RateTwentyRhoZero}{\ensuremath{-}5.32} | |
| 1767 | +\newcommand{\RateTwentyRhoZeroAbs}{5.32} | |
| 1768 | +\newcommand{\RateTwentySecs}{244} | |
| 1769 | +\newcommand{\RateTwentySteps}{10} | |
| 1770 | +\newcommand{\RateTwentyThrChangePct}{1} | |
| 1771 | +\newcommand{\RateTwentyThrSS}{0.49} | |
| 1772 | +\newcommand{\RateTwentyThrZero}{0.50} | |
| 1773 | +\newcommand{\RateTwentyWageAll}{0.00} | |
| 1774 | +\newcommand{\RateTwentyWageO}{0.00} | |
| 1775 | +\newcommand{\RateTwentyqROsevenSS}{0.806} | |
| 1776 | +\newcommand{\RateTwentyqROsevenZero}{0.764} | |
| 1777 | +\newcommand{\RateTwentyqROthreeOne}{0.359} | |
| 1778 | +\newcommand{\RateTwentyqROthreeSS}{0.351} | |
| 1779 | +\newcommand{\RateTwentyqROthreeZero}{0.380} | |
| 1780 | +\newcommand{\RentPriceAnnual}{0.039} | |
| 1781 | +\newcommand{\RhoGEstate}{0.32} | |
| 1782 | +\newcommand{\RhoGGapFour}{1.02} | |
| 1783 | +\newcommand{\RhoGGift}{0.40} | |
| 1784 | +\newcommand{\RhoGGini}{0.55} | |
| 1785 | +\newcommand{\RhoGLamSeven}{0.121} | |
| 1786 | +\newcommand{\RhoGLamThree}{0.32} | |
| 1787 | +\newcommand{\RhoGOwn}{0.624} | |
| 1788 | +\newcommand{\RhoGOwnUthirtyfive}{0.385} | |
| 1789 | +\newcommand{\RhoGP}{1.004} | |
| 1790 | +\newcommand{\RhoGPDevPct}{0.4} | |
| 1791 | +\newcommand{\RhoGRR}{0.41} | |
| 1792 | +\newcommand{\RhoGRebate}{0.000} | |
| 1793 | +\newcommand{\RhoGRho}{0.194} | |
| 1794 | +\newcommand{\RhoGRhoDevPct}{\ensuremath{-}0.1} | |
| 1795 | +\newcommand{\RhoGRhoDevPctAbs}{0.1} | |
| 1796 | +\newcommand{\RhoGThr}{0.50} | |
| 1797 | +\newcommand{\RhoGThrHighZ}{0.03} | |
| 1798 | +\newcommand{\RhoGThrLowZ}{1.19} | |
| 1799 | +\newcommand{\RhoGqOO}{0.545} | |
| 1800 | +\newcommand{\RhoGqRO}{0.223} | |
| 1801 | +\newcommand{\RhoImpactarmlevel}{\ensuremath{-}1.15} | |
| 1802 | +\newcommand{\RhoImpactarmlevelAbs}{1.15} | |
| 1803 | +\newcommand{\RhoImpactarmpersistent}{\ensuremath{-}0.18} | |
| 1804 | +\newcommand{\RhoImpactarmpersistentAbs}{0.18} | |
| 1805 | +\newcommand{\RhoImpactfrmlevel}{\ensuremath{-}1.14} | |
| 1806 | +\newcommand{\RhoImpactfrmlevelAbs}{1.14} | |
| 1807 | +\newcommand{\RhoImpactfrmlevelfixedsupply}{\ensuremath{-}1.08} | |
| 1808 | +\newcommand{\RhoImpactfrmlevelfixedsupplyAbs}{1.08} | |
| 1809 | +\newcommand{\RhoImpactfrmlevelnoseg}{\ensuremath{-}1.47} | |
| 1810 | +\newcommand{\RhoImpactfrmlevelnosegAbs}{1.47} | |
| 1811 | +\newcommand{\RhoImpactfrmlevelrigidrents}{7.48} | |
| 1812 | +\newcommand{\RhoImpactfrmlevelstatic}{\ensuremath{-}1.11} | |
| 1813 | +\newcommand{\RhoImpactfrmlevelstaticAbs}{1.11} | |
| 1814 | +\newcommand{\RhoImpactfrmlevelunindexedwages}{\ensuremath{-}3.21} | |
| 1815 | +\newcommand{\RhoImpactfrmlevelunindexedwagesAbs}{3.21} | |
| 1816 | +\newcommand{\RhoImpactfrmpersistent}{\ensuremath{-}0.38} | |
| 1817 | +\newcommand{\RhoImpactfrmpersistentAbs}{0.38} | |
| 1818 | +\newcommand{\RhoImpactfrmpersistenttaylor}{\ensuremath{-}2.29} | |
| 1819 | +\newcommand{\RhoImpactfrmpersistenttaylorAbs}{2.29} | |
| 1820 | +\newcommand{\RhoImpacttermlevel}{\ensuremath{-}1.03} | |
| 1821 | +\newcommand{\RhoImpacttermlevelAbs}{1.03} | |
| 1822 | +\newcommand{\RhoImpacttermpersistent}{0.04} | |
| 1823 | +\newcommand{\RhoImpacttermpersistenttaylor}{\ensuremath{-}1.91} | |
| 1824 | +\newcommand{\RhoImpacttermpersistenttaylorAbs}{1.91} | |
| 1825 | +\newcommand{\RhoStar}{0.194} | |
| 1826 | +\newcommand{\RigidRentsAbsorbPct}{99} | |
| 1827 | +\newcommand{\RigidRentsBetterAllPct}{42} | |
| 1828 | +\newcommand{\RigidRentsBetterOMPct}{70} | |
| 1829 | +\newcommand{\RigidRentsBetterONMPct}{1} | |
| 1830 | +\newcommand{\RigidRentsBetterOPct}{44} | |
| 1831 | +\newcommand{\RigidRentsBetterRPct}{38} | |
| 1832 | +\newcommand{\RigidRentsCEVAll}{\ensuremath{-}0.62} | |
| 1833 | +\newcommand{\RigidRentsCEVAllAbs}{0.62} | |
| 1834 | +\newcommand{\RigidRentsCEVAllPE}{\ensuremath{-}0.35} | |
| 1835 | +\newcommand{\RigidRentsCEVAllPEAbs}{0.35} | |
| 1836 | +\newcommand{\RigidRentsCEVO}{\ensuremath{-}0.26} | |
| 1837 | +\newcommand{\RigidRentsCEVOAbs}{0.26} | |
| 1838 | +\newcommand{\RigidRentsCEVOM}{0.55} | |
| 1839 | +\newcommand{\RigidRentsCEVONM}{\ensuremath{-}2.13} | |
| 1840 | +\newcommand{\RigidRentsCEVONMAbs}{2.13} | |
| 1841 | +\newcommand{\RigidRentsCEVR}{\ensuremath{-}0.92} | |
| 1842 | +\newcommand{\RigidRentsCEVRAbs}{0.92} | |
| 1843 | +\newcommand{\RigidRentsConsAll}{\ensuremath{-}0.8} | |
| 1844 | +\newcommand{\RigidRentsConsAllAbs}{0.8} | |
| 1845 | +\newcommand{\RigidRentsConsO}{\ensuremath{-}0.6} | |
| 1846 | +\newcommand{\RigidRentsConsOAbs}{0.6} | |
| 1847 | +\newcommand{\RigidRentsConsR}{\ensuremath{-}1.3} | |
| 1848 | +\newcommand{\RigidRentsConsRAbs}{1.3} | |
| 1849 | +\newcommand{\RigidRentsDOwnGE}{\ensuremath{-}0.03} | |
| 1850 | +\newcommand{\RigidRentsDOwnGEAbs}{0.03} | |
| 1851 | +\newcommand{\RigidRentsDOwnPE}{\ensuremath{-}2.88} | |
| 1852 | +\newcommand{\RigidRentsDOwnPEAbs}{2.88} | |
| 1853 | +\newcommand{\RigidRentsDebtZeroPct}{\ensuremath{-}3.0} | |
| 1854 | +\newcommand{\RigidRentsDebtZeroPctAbs}{3.0} | |
| 1855 | +\newcommand{\RigidRentsDepLoss}{0.48} | |
| 1856 | +\newcommand{\RigidRentsErosion}{0.21} | |
| 1857 | +\newcommand{\RigidRentsLamSevenSS}{0.060} | |
| 1858 | +\newcommand{\RigidRentsLamSevenZero}{0.068} | |
| 1859 | +\newcommand{\RigidRentsLender}{0.27} | |
| 1860 | +\newcommand{\RigidRentsMaxRes}{1.2\times10^{-3}} | |
| 1861 | +\newcommand{\RigidRentsNWAll}{\ensuremath{-}0.02} | |
| 1862 | +\newcommand{\RigidRentsNWAllAbs}{0.02} | |
| 1863 | +\newcommand{\RigidRentsNWO}{\ensuremath{-}0.02} | |
| 1864 | +\newcommand{\RigidRentsNWOAbs}{0.02} | |
| 1865 | +\newcommand{\RigidRentsNWR}{\ensuremath{-}0.01} | |
| 1866 | +\newcommand{\RigidRentsNWRAbs}{0.01} | |
| 1867 | +\newcommand{\RigidRentsNetO}{\ensuremath{-}0.02} | |
| 1868 | +\newcommand{\RigidRentsNetOAbs}{0.02} | |
| 1869 | +\newcommand{\RigidRentsNetR}{\ensuremath{-}0.02} | |
| 1870 | +\newcommand{\RigidRentsNetRAbs}{0.02} | |
| 1871 | +\newcommand{\RigidRentsNominalRateZero}{0.219} | |
| 1872 | +\newcommand{\RigidRentsOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 1873 | +\newcommand{\RigidRentsOwnerOldConsMaxAbs}{2.3} | |
| 1874 | +\newcommand{\RigidRentsOwnerOldConsMin}{\ensuremath{-}1.8} | |
| 1875 | +\newcommand{\RigidRentsOwnerOldConsMinAbs}{1.8} | |
| 1876 | +\newcommand{\RigidRentsOwnerOldLossMax}{0.08} | |
| 1877 | +\newcommand{\RigidRentsOwnerOldLossMin}{0.04} | |
| 1878 | +\newcommand{\RigidRentsOwnerYoungConsMax}{1.1} | |
| 1879 | +\newcommand{\RigidRentsOwnerYoungConsMin}{0.6} | |
| 1880 | +\newcommand{\RigidRentsOwnerYoungGainMax}{0.04} | |
| 1881 | +\newcommand{\RigidRentsPriceBackYears}{20} | |
| 1882 | +\newcommand{\RigidRentsPriceMaxPos}{0.1} | |
| 1883 | +\newcommand{\RigidRentsPriceSignChanges}{1} | |
| 1884 | +\newcommand{\RigidRentsPzero}{\ensuremath{-}0.46} | |
| 1885 | +\newcommand{\RigidRentsPzeroAbs}{0.46} | |
| 1886 | +\newcommand{\RigidRentsRentGainR}{0.01} | |
| 1887 | +\newcommand{\RigidRentsResT}{3\times10^{-4}} | |
| 1888 | +\newcommand{\RigidRentsRhoZero}{7.48} | |
| 1889 | +\newcommand{\RigidRentsSecs}{191} | |
| 1890 | +\newcommand{\RigidRentsSteps}{5} | |
| 1891 | +\newcommand{\RigidRentsThrChangePct}{9} | |
| 1892 | +\newcommand{\RigidRentsThrSS}{0.49} | |
| 1893 | +\newcommand{\RigidRentsThrZero}{0.54} | |
| 1894 | +\newcommand{\RigidRentsWageAll}{0.00} | |
| 1895 | +\newcommand{\RigidRentsWageO}{0.00} | |
| 1896 | +\newcommand{\RigidRentsqROsevenSS}{0.806} | |
| 1897 | +\newcommand{\RigidRentsqROsevenZero}{0.796} | |
| 1898 | +\newcommand{\RigidRentsqROthreeOne}{0.336} | |
| 1899 | +\newcommand{\RigidRentsqROthreeSS}{0.351} | |
| 1900 | +\newcommand{\RigidRentsqROthreeZero}{0.354} | |
| 1901 | +\newcommand{\RoneEstate}{0.24} | |
| 1902 | +\newcommand{\RoneGapFour}{0.90} | |
| 1903 | +\newcommand{\RoneGift}{0.36} | |
| 1904 | +\newcommand{\RoneGini}{0.55} | |
| 1905 | +\newcommand{\RoneLamSeven}{0.063} | |
| 1906 | +\newcommand{\RoneLamThree}{0.17} | |
| 1907 | +\newcommand{\RoneOwn}{0.601} | |
| 1908 | +\newcommand{\RoneOwnUthirtyfive}{0.323} | |
| 1909 | +\newcommand{\RoneP}{1.042} | |
| 1910 | +\newcommand{\RonePDevPct}{4.2} | |
| 1911 | +\newcommand{\RoneRR}{0.26} | |
| 1912 | +\newcommand{\RoneRebate}{0.000} | |
| 1913 | +\newcommand{\RoneRho}{0.177} | |
| 1914 | +\newcommand{\RoneRhoDevPct}{\ensuremath{-}8.9} | |
| 1915 | +\newcommand{\RoneRhoDevPctAbs}{8.9} | |
| 1916 | +\newcommand{\RoneThr}{0.48} | |
| 1917 | +\newcommand{\RoneThrHighZ}{0.08} | |
| 1918 | +\newcommand{\RoneThrLowZ}{1.12} | |
| 1919 | +\newcommand{\RoneqOO}{0.484} | |
| 1920 | +\newcommand{\RoneqRO}{0.315} | |
| 1921 | +\newcommand{\RthreeEstate}{0.48} | |
| 1922 | +\newcommand{\RthreeGapFour}{1.09} | |
| 1923 | +\newcommand{\RthreeGift}{0.46} | |
| 1924 | +\newcommand{\RthreeGini}{0.53} | |
| 1925 | +\newcommand{\RthreeLamSeven}{0.132} | |
| 1926 | +\newcommand{\RthreeLamThree}{0.24} | |
| 1927 | +\newcommand{\RthreeOwn}{0.642} | |
| 1928 | +\newcommand{\RthreeOwnUthirtyfive}{0.399} | |
| 1929 | +\newcommand{\RthreeP}{0.972} | |
| 1930 | +\newcommand{\RthreePDevPct}{\ensuremath{-}2.8} | |
| 1931 | +\newcommand{\RthreePDevPctAbs}{2.8} | |
| 1932 | +\newcommand{\RthreeRR}{0.44} | |
| 1933 | +\newcommand{\RthreeRebate}{0.000} | |
| 1934 | +\newcommand{\RthreeRho}{0.214} | |
| 1935 | +\newcommand{\RthreeRhoDevPct}{10.1} | |
| 1936 | +\newcommand{\RthreeThr}{0.60} | |
| 1937 | +\newcommand{\RthreeThrHighZ}{0.09} | |
| 1938 | +\newcommand{\RthreeThrLowZ}{1.35} | |
| 1939 | +\newcommand{\RthreeqOO}{0.596} | |
| 1940 | +\newcommand{\RthreeqRO}{0.353} | |
| 1941 | +\newcommand{\ShareEstatePositivePct}{55} | |
| 1942 | +\newcommand{\ShareFirstBuyersGiftPct}{72} | |
| 1943 | +\newcommand{\ShareGiftPositivePct}{70} | |
| 1944 | +\newcommand{\ShareMortgage}{0.41} | |
| 1945 | +\newcommand{\ShareOwnersMortgage}{0.65} | |
| 1946 | +\newcommand{\ShareParentOwnerPct}{86} | |
| 1947 | +\newcommand{\SigHighEstate}{0.42} | |
| 1948 | +\newcommand{\SigHighGapFour}{1.35} | |
| 1949 | +\newcommand{\SigHighGift}{0.49} | |
| 1950 | +\newcommand{\SigHighGini}{0.56} | |
| 1951 | +\newcommand{\SigHighLamSeven}{0.097} | |
| 1952 | +\newcommand{\SigHighLamThree}{0.18} | |
| 1953 | +\newcommand{\SigHighOwn}{0.602} | |
| 1954 | +\newcommand{\SigHighOwnUthirtyfive}{0.416} | |
| 1955 | +\newcommand{\SigHighP}{1.033} | |
| 1956 | +\newcommand{\SigHighPDevPct}{3.3} | |
| 1957 | +\newcommand{\SigHighRR}{0.34} | |
| 1958 | +\newcommand{\SigHighRebate}{0.000} | |
| 1959 | +\newcommand{\SigHighRho}{0.189} | |
| 1960 | +\newcommand{\SigHighRhoDevPct}{\ensuremath{-}2.6} | |
| 1961 | +\newcommand{\SigHighRhoDevPctAbs}{2.6} | |
| 1962 | +\newcommand{\SigHighThr}{0.65} | |
| 1963 | +\newcommand{\SigHighThrHighZ}{\ensuremath{-}0.27} | |
| 1964 | +\newcommand{\SigHighThrHighZAbs}{0.27} | |
| 1965 | +\newcommand{\SigHighThrLowZ}{1.47} | |
| 1966 | +\newcommand{\SigHighqOO}{0.570} | |
| 1967 | +\newcommand{\SigHighqRO}{0.391} | |
| 1968 | +\newcommand{\SigLowEstate}{0.22} | |
| 1969 | +\newcommand{\SigLowGapFour}{0.59} | |
| 1970 | +\newcommand{\SigLowGift}{0.30} | |
| 1971 | +\newcommand{\SigLowGini}{0.51} | |
| 1972 | +\newcommand{\SigLowLamSeven}{0.043} | |
| 1973 | +\newcommand{\SigLowLamThree}{0.19} | |
| 1974 | +\newcommand{\SigLowOwn}{0.672} | |
| 1975 | +\newcommand{\SigLowOwnUthirtyfive}{0.318} | |
| 1976 | +\newcommand{\SigLowP}{0.963} | |
| 1977 | +\newcommand{\SigLowPDevPct}{\ensuremath{-}3.7} | |
| 1978 | +\newcommand{\SigLowPDevPctAbs}{3.7} | |
| 1979 | +\newcommand{\SigLowRR}{0.32} | |
| 1980 | +\newcommand{\SigLowRebate}{0.000} | |
| 1981 | +\newcommand{\SigLowRho}{0.197} | |
| 1982 | +\newcommand{\SigLowRhoDevPct}{1.3} | |
| 1983 | +\newcommand{\SigLowThr}{0.41} | |
| 1984 | +\newcommand{\SigLowThrHighZ}{0.27} | |
| 1985 | +\newcommand{\SigLowThrLowZ}{0.84} | |
| 1986 | +\newcommand{\SigLowqOO}{0.492} | |
| 1987 | +\newcommand{\SigLowqRO}{0.307} | |
| 1988 | +\newcommand{\SignChangesbaseline}{1} | |
| 1989 | +\newcommand{\SignChangeseps}{1} | |
| 1990 | +\newcommand{\SignChangesepstheta}{1} | |
| 1991 | +\newcommand{\SimulateMinutes}{60} | |
| 1992 | +\newcommand{\SimulateMinutesOne}{59.8} | |
| 1993 | +\newcommand{\StaticAbsorbPct}{110} | |
| 1994 | +\newcommand{\StaticBetterAllPct}{41} | |
| 1995 | +\newcommand{\StaticBetterOMPct}{71} | |
| 1996 | +\newcommand{\StaticBetterONMPct}{3} | |
| 1997 | +\newcommand{\StaticBetterOPct}{46} | |
| 1998 | +\newcommand{\StaticBetterRPct}{33} | |
| 1999 | +\newcommand{\StaticCEVAll}{\ensuremath{-}0.64} | |
| 2000 | +\newcommand{\StaticCEVAllAbs}{0.64} | |
| 2001 | +\newcommand{\StaticCEVAllPE}{\ensuremath{-}0.68} | |
| 2002 | +\newcommand{\StaticCEVAllPEAbs}{0.68} | |
| 2003 | +\newcommand{\StaticCEVO}{\ensuremath{-}0.17} | |
| 2004 | +\newcommand{\StaticCEVOAbs}{0.17} | |
| 2005 | +\newcommand{\StaticCEVOM}{0.65} | |
| 2006 | +\newcommand{\StaticCEVONM}{\ensuremath{-}2.10} | |
| 2007 | +\newcommand{\StaticCEVONMAbs}{2.10} | |
| 2008 | +\newcommand{\StaticCEVR}{\ensuremath{-}1.01} | |
| 2009 | +\newcommand{\StaticCEVRAbs}{1.01} | |
| 2010 | +\newcommand{\StaticConsAll}{\ensuremath{-}0.8} | |
| 2011 | +\newcommand{\StaticConsAllAbs}{0.8} | |
| 2012 | +\newcommand{\StaticConsO}{\ensuremath{-}0.5} | |
| 2013 | +\newcommand{\StaticConsOAbs}{0.5} | |
| 2014 | +\newcommand{\StaticConsR}{\ensuremath{-}1.4} | |
| 2015 | +\newcommand{\StaticConsRAbs}{1.4} | |
| 2016 | +\newcommand{\StaticDOwnGE}{0.02} | |
| 2017 | +\newcommand{\StaticDOwnPE}{\ensuremath{-}0.20} | |
| 2018 | +\newcommand{\StaticDOwnPEAbs}{0.20} | |
| 2019 | +\newcommand{\StaticDebtZeroPct}{\ensuremath{-}2.4} | |
| 2020 | +\newcommand{\StaticDebtZeroPctAbs}{2.4} | |
| 2021 | +\newcommand{\StaticDepLoss}{0.48} | |
| 2022 | +\newcommand{\StaticErosion}{0.21} | |
| 2023 | +\newcommand{\StaticLamSevenSS}{0.060} | |
| 2024 | +\newcommand{\StaticLamSevenZero}{0.069} | |
| 2025 | +\newcommand{\StaticLender}{0.27} | |
| 2026 | +\newcommand{\StaticMaxRes}{1.1\times10^{-3}} | |
| 2027 | +\newcommand{\StaticNWAll}{\ensuremath{-}0.02} | |
| 2028 | +\newcommand{\StaticNWAllAbs}{0.02} | |
| 2029 | +\newcommand{\StaticNWO}{\ensuremath{-}0.02} | |
| 2030 | +\newcommand{\StaticNWOAbs}{0.02} | |
| 2031 | +\newcommand{\StaticNWR}{\ensuremath{-}0.01} | |
| 2032 | +\newcommand{\StaticNWRAbs}{0.01} | |
| 2033 | +\newcommand{\StaticNetO}{\ensuremath{-}0.02} | |
| 2034 | +\newcommand{\StaticNetOAbs}{0.02} | |
| 2035 | +\newcommand{\StaticNetR}{\ensuremath{-}0.02} | |
| 2036 | +\newcommand{\StaticNetRAbs}{0.02} | |
| 2037 | +\newcommand{\StaticNominalRateZero}{0.219} | |
| 2038 | +\newcommand{\StaticOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 2039 | +\newcommand{\StaticOwnerOldConsMaxAbs}{2.3} | |
| 2040 | +\newcommand{\StaticOwnerOldConsMin}{\ensuremath{-}1.7} | |
| 2041 | +\newcommand{\StaticOwnerOldConsMinAbs}{1.7} | |
| 2042 | +\newcommand{\StaticOwnerOldLossMax}{0.08} | |
| 2043 | +\newcommand{\StaticOwnerOldLossMin}{0.04} | |
| 2044 | +\newcommand{\StaticOwnerYoungConsMax}{1.1} | |
| 2045 | +\newcommand{\StaticOwnerYoungConsMin}{0.7} | |
| 2046 | +\newcommand{\StaticOwnerYoungGainMax}{0.04} | |
| 2047 | +\newcommand{\StaticPriceBackYears}{20} | |
| 2048 | +\newcommand{\StaticPriceMaxPos}{0.1} | |
| 2049 | +\newcommand{\StaticPriceSignChanges}{1} | |
| 2050 | +\newcommand{\StaticPzero}{\ensuremath{-}0.36} | |
| 2051 | +\newcommand{\StaticPzeroAbs}{0.36} | |
| 2052 | +\newcommand{\StaticRentGainR}{0.00} | |
| 2053 | +\newcommand{\StaticResT}{8\times10^{-5}} | |
| 2054 | +\newcommand{\StaticRhoZero}{\ensuremath{-}1.11} | |
| 2055 | +\newcommand{\StaticRhoZeroAbs}{1.11} | |
| 2056 | +\newcommand{\StaticSecs}{387} | |
| 2057 | +\newcommand{\StaticSteps}{10} | |
| 2058 | +\newcommand{\StaticThrChangePct}{8} | |
| 2059 | +\newcommand{\StaticThrSS}{0.49} | |
| 2060 | +\newcommand{\StaticThrZero}{0.53} | |
| 2061 | +\newcommand{\StaticWageAll}{0.00} | |
| 2062 | +\newcommand{\StaticWageO}{0.00} | |
| 2063 | +\newcommand{\StaticqROsevenSS}{0.806} | |
| 2064 | +\newcommand{\StaticqROsevenZero}{0.797} | |
| 2065 | +\newcommand{\StaticqROthreeOne}{0.335} | |
| 2066 | +\newcommand{\StaticqROthreeSS}{0.351} | |
| 2067 | +\newcommand{\StaticqROthreeZero}{0.357} | |
| 2068 | +\newcommand{\SupplyScaleOwner}{9.41} | |
| 2069 | +\newcommand{\SupplyScaleRental}{2.90} | |
| 2070 | +\newcommand{\TaxFiftyEstate}{0.09} | |
| 2071 | +\newcommand{\TaxFiftyGapFour}{0.90} | |
| 2072 | +\newcommand{\TaxFiftyGift}{0.12} | |
| 2073 | +\newcommand{\TaxFiftyGini}{0.52} | |
| 2074 | +\newcommand{\TaxFiftyLamSeven}{0.023} | |
| 2075 | +\newcommand{\TaxFiftyLamThree}{0.12} | |
| 2076 | +\newcommand{\TaxFiftyOwn}{0.639} | |
| 2077 | +\newcommand{\TaxFiftyOwnUthirtyfive}{0.482} | |
| 2078 | +\newcommand{\TaxFiftyP}{0.999} | |
| 2079 | +\newcommand{\TaxFiftyPDevPct}{\ensuremath{-}0.1} | |
| 2080 | +\newcommand{\TaxFiftyPDevPctAbs}{0.1} | |
| 2081 | +\newcommand{\TaxFiftyRR}{0.16} | |
| 2082 | +\newcommand{\TaxFiftyRebate}{0.213} | |
| 2083 | +\newcommand{\TaxFiftyRho}{0.195} | |
| 2084 | +\newcommand{\TaxFiftyRhoDevPct}{0.4} | |
| 2085 | +\newcommand{\TaxFiftyThr}{0.48} | |
| 2086 | +\newcommand{\TaxFiftyThrHighZ}{0.02} | |
| 2087 | +\newcommand{\TaxFiftyThrLowZ}{1.15} | |
| 2088 | +\newcommand{\TaxFiftyqOO}{0.553} | |
| 2089 | +\newcommand{\TaxFiftyqRO}{0.429} | |
| 2090 | +\newcommand{\TaxTwentyEstate}{0.22} | |
| 2091 | +\newcommand{\TaxTwentyGapFour}{0.96} | |
| 2092 | +\newcommand{\TaxTwentyGift}{0.28} | |
| 2093 | +\newcommand{\TaxTwentyGini}{0.53} | |
| 2094 | +\newcommand{\TaxTwentyLamSeven}{0.054} | |
| 2095 | +\newcommand{\TaxTwentyLamThree}{0.18} | |
| 2096 | +\newcommand{\TaxTwentyOwn}{0.642} | |
| 2097 | +\newcommand{\TaxTwentyOwnUthirtyfive}{0.461} | |
| 2098 | +\newcommand{\TaxTwentyP}{1.005} | |
| 2099 | +\newcommand{\TaxTwentyPDevPct}{0.5} | |
| 2100 | +\newcommand{\TaxTwentyRR}{0.28} | |
| 2101 | +\newcommand{\TaxTwentyRebate}{0.125} | |
| 2102 | +\newcommand{\TaxTwentyRho}{0.192} | |
| 2103 | +\newcommand{\TaxTwentyRhoDevPct}{\ensuremath{-}1.1} | |
| 2104 | +\newcommand{\TaxTwentyRhoDevPctAbs}{1.1} | |
| 2105 | +\newcommand{\TaxTwentyThr}{0.51} | |
| 2106 | +\newcommand{\TaxTwentyThrHighZ}{0.04} | |
| 2107 | +\newcommand{\TaxTwentyThrLowZ}{1.21} | |
| 2108 | +\newcommand{\TaxTwentyqOO}{0.569} | |
| 2109 | +\newcommand{\TaxTwentyqRO}{0.392} | |
| 2110 | +\newcommand{\TermLevelAbsorbPct}{89} | |
| 2111 | +\newcommand{\TermLevelBetterAllPct}{41} | |
| 2112 | +\newcommand{\TermLevelBetterOMPct}{71} | |
| 2113 | +\newcommand{\TermLevelBetterONMPct}{1} | |
| 2114 | +\newcommand{\TermLevelBetterOPct}{45} | |
| 2115 | +\newcommand{\TermLevelBetterRPct}{35} | |
| 2116 | +\newcommand{\TermLevelCEVAll}{\ensuremath{-}0.64} | |
| 2117 | +\newcommand{\TermLevelCEVAllAbs}{0.64} | |
| 2118 | +\newcommand{\TermLevelCEVAllPE}{\ensuremath{-}0.68} | |
| 2119 | +\newcommand{\TermLevelCEVAllPEAbs}{0.68} | |
| 2120 | +\newcommand{\TermLevelCEVO}{\ensuremath{-}0.20} | |
| 2121 | +\newcommand{\TermLevelCEVOAbs}{0.20} | |
| 2122 | +\newcommand{\TermLevelCEVOM}{0.63} | |
| 2123 | +\newcommand{\TermLevelCEVONM}{\ensuremath{-}2.12} | |
| 2124 | +\newcommand{\TermLevelCEVONMAbs}{2.12} | |
| 2125 | +\newcommand{\TermLevelCEVR}{\ensuremath{-}1.00} | |
| 2126 | +\newcommand{\TermLevelCEVRAbs}{1.00} | |
| 2127 | +\newcommand{\TermLevelConsAll}{\ensuremath{-}0.8} | |
| 2128 | +\newcommand{\TermLevelConsAllAbs}{0.8} | |
| 2129 | +\newcommand{\TermLevelConsO}{\ensuremath{-}0.6} | |
| 2130 | +\newcommand{\TermLevelConsOAbs}{0.6} | |
| 2131 | +\newcommand{\TermLevelConsR}{\ensuremath{-}1.4} | |
| 2132 | +\newcommand{\TermLevelConsRAbs}{1.4} | |
| 2133 | +\newcommand{\TermLevelDOwnGE}{\ensuremath{-}0.02} | |
| 2134 | +\newcommand{\TermLevelDOwnGEAbs}{0.02} | |
| 2135 | +\newcommand{\TermLevelDOwnPE}{\ensuremath{-}0.20} | |
| 2136 | +\newcommand{\TermLevelDOwnPEAbs}{0.20} | |
| 2137 | +\newcommand{\TermLevelDebtZeroPct}{\ensuremath{-}2.7} | |
| 2138 | +\newcommand{\TermLevelDebtZeroPctAbs}{2.7} | |
| 2139 | +\newcommand{\TermLevelDepLoss}{0.48} | |
| 2140 | +\newcommand{\TermLevelErosion}{0.21} | |
| 2141 | +\newcommand{\TermLevelLamSevenSS}{0.060} | |
| 2142 | +\newcommand{\TermLevelLamSevenZero}{0.068} | |
| 2143 | +\newcommand{\TermLevelLender}{0.27} | |
| 2144 | +\newcommand{\TermLevelMaxRes}{6.1\times10^{-4}} | |
| 2145 | +\newcommand{\TermLevelNWAll}{\ensuremath{-}0.02} | |
| 2146 | +\newcommand{\TermLevelNWAllAbs}{0.02} | |
| 2147 | +\newcommand{\TermLevelNWO}{\ensuremath{-}0.02} | |
| 2148 | +\newcommand{\TermLevelNWOAbs}{0.02} | |
| 2149 | +\newcommand{\TermLevelNWR}{\ensuremath{-}0.01} | |
| 2150 | +\newcommand{\TermLevelNWRAbs}{0.01} | |
| 2151 | +\newcommand{\TermLevelNetO}{\ensuremath{-}0.02} | |
| 2152 | +\newcommand{\TermLevelNetOAbs}{0.02} | |
| 2153 | +\newcommand{\TermLevelNetR}{\ensuremath{-}0.02} | |
| 2154 | +\newcommand{\TermLevelNetRAbs}{0.02} | |
| 2155 | +\newcommand{\TermLevelNominalRateZero}{0.219} | |
| 2156 | +\newcommand{\TermLevelOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 2157 | +\newcommand{\TermLevelOwnerOldConsMaxAbs}{2.3} | |
| 2158 | +\newcommand{\TermLevelOwnerOldConsMin}{\ensuremath{-}1.8} | |
| 2159 | +\newcommand{\TermLevelOwnerOldConsMinAbs}{1.8} | |
| 2160 | +\newcommand{\TermLevelOwnerOldLossMax}{0.08} | |
| 2161 | +\newcommand{\TermLevelOwnerOldLossMin}{0.04} | |
| 2162 | +\newcommand{\TermLevelOwnerYoungConsMax}{1.1} | |
| 2163 | +\newcommand{\TermLevelOwnerYoungConsMin}{0.6} | |
| 2164 | +\newcommand{\TermLevelOwnerYoungGainMax}{0.04} | |
| 2165 | +\newcommand{\TermLevelPriceBackYears}{20} | |
| 2166 | +\newcommand{\TermLevelPriceMaxPos}{0.1} | |
| 2167 | +\newcommand{\TermLevelPriceSignChanges}{1} | |
| 2168 | +\newcommand{\TermLevelPzero}{\ensuremath{-}0.47} | |
| 2169 | +\newcommand{\TermLevelPzeroAbs}{0.47} | |
| 2170 | +\newcommand{\TermLevelRentGainR}{0.00} | |
| 2171 | +\newcommand{\TermLevelResT}{3\times10^{-4}} | |
| 2172 | +\newcommand{\TermLevelRhoZero}{\ensuremath{-}1.03} | |
| 2173 | +\newcommand{\TermLevelRhoZeroAbs}{1.03} | |
| 2174 | +\newcommand{\TermLevelSecs}{161} | |
| 2175 | +\newcommand{\TermLevelSteps}{3} | |
| 2176 | +\newcommand{\TermLevelThrChangePct}{8} | |
| 2177 | +\newcommand{\TermLevelThrSS}{0.49} | |
| 2178 | +\newcommand{\TermLevelThrZero}{0.53} | |
| 2179 | +\newcommand{\TermLevelWageAll}{0.00} | |
| 2180 | +\newcommand{\TermLevelWageO}{0.00} | |
| 2181 | +\newcommand{\TermLevelqROsevenSS}{0.806} | |
| 2182 | +\newcommand{\TermLevelqROsevenZero}{0.798} | |
| 2183 | +\newcommand{\TermLevelqROthreeOne}{0.335} | |
| 2184 | +\newcommand{\TermLevelqROthreeSS}{0.351} | |
| 2185 | +\newcommand{\TermLevelqROthreeZero}{0.356} | |
| 2186 | +\newcommand{\TermPersAbsorbPct}{83} | |
| 2187 | +\newcommand{\TermPersBetterAllPct}{27} | |
| 2188 | +\newcommand{\TermPersBetterOMPct}{66} | |
| 2189 | +\newcommand{\TermPersBetterONMPct}{2} | |
| 2190 | +\newcommand{\TermPersBetterOPct}{43} | |
| 2191 | +\newcommand{\TermPersBetterRPct}{4} | |
| 2192 | +\newcommand{\TermPersCEVAll}{\ensuremath{-}0.74} | |
| 2193 | +\newcommand{\TermPersCEVAllAbs}{0.74} | |
| 2194 | +\newcommand{\TermPersCEVAllPE}{\ensuremath{-}0.73} | |
| 2195 | +\newcommand{\TermPersCEVAllPEAbs}{0.73} | |
| 2196 | +\newcommand{\TermPersCEVO}{\ensuremath{-}0.32} | |
| 2197 | +\newcommand{\TermPersCEVOAbs}{0.32} | |
| 2198 | +\newcommand{\TermPersCEVOM}{0.46} | |
| 2199 | +\newcommand{\TermPersCEVONM}{\ensuremath{-}2.14} | |
| 2200 | +\newcommand{\TermPersCEVONMAbs}{2.14} | |
| 2201 | +\newcommand{\TermPersCEVR}{\ensuremath{-}1.09} | |
| 2202 | +\newcommand{\TermPersCEVRAbs}{1.09} | |
| 2203 | +\newcommand{\TermPersConsAll}{\ensuremath{-}1.4} | |
| 2204 | +\newcommand{\TermPersConsAllAbs}{1.4} | |
| 2205 | +\newcommand{\TermPersConsO}{\ensuremath{-}1.2} | |
| 2206 | +\newcommand{\TermPersConsOAbs}{1.2} | |
| 2207 | +\newcommand{\TermPersConsR}{\ensuremath{-}1.7} | |
| 2208 | +\newcommand{\TermPersConsRAbs}{1.7} | |
| 2209 | +\newcommand{\TermPersDOwnGE}{\ensuremath{-}0.34} | |
| 2210 | +\newcommand{\TermPersDOwnGEAbs}{0.34} | |
| 2211 | +\newcommand{\TermPersDOwnPE}{\ensuremath{-}2.03} | |
| 2212 | +\newcommand{\TermPersDOwnPEAbs}{2.03} | |
| 2213 | +\newcommand{\TermPersDebtZeroPct}{\ensuremath{-}13.5} | |
| 2214 | +\newcommand{\TermPersDebtZeroPctAbs}{13.5} | |
| 2215 | +\newcommand{\TermPersDepLoss}{0.48} | |
| 2216 | +\newcommand{\TermPersErosion}{0.21} | |
| 2217 | +\newcommand{\TermPersLamSevenSS}{0.060} | |
| 2218 | +\newcommand{\TermPersLamSevenZero}{0.067} | |
| 2219 | +\newcommand{\TermPersLender}{0.27} | |
| 2220 | +\newcommand{\TermPersMaxRes}{8.5\times10^{-4}} | |
| 2221 | +\newcommand{\TermPersNWAll}{\ensuremath{-}0.02} | |
| 2222 | +\newcommand{\TermPersNWAllAbs}{0.02} | |
| 2223 | +\newcommand{\TermPersNWO}{\ensuremath{-}0.03} | |
| 2224 | +\newcommand{\TermPersNWOAbs}{0.03} | |
| 2225 | +\newcommand{\TermPersNWR}{\ensuremath{-}0.01} | |
| 2226 | +\newcommand{\TermPersNWRAbs}{0.01} | |
| 2227 | +\newcommand{\TermPersNetO}{\ensuremath{-}0.02} | |
| 2228 | +\newcommand{\TermPersNetOAbs}{0.02} | |
| 2229 | +\newcommand{\TermPersNetR}{\ensuremath{-}0.02} | |
| 2230 | +\newcommand{\TermPersNetRAbs}{0.02} | |
| 2231 | +\newcommand{\TermPersNominalRateZero}{0.341} | |
| 2232 | +\newcommand{\TermPersOwnerOldConsMax}{\ensuremath{-}2.5} | |
| 2233 | +\newcommand{\TermPersOwnerOldConsMaxAbs}{2.5} | |
| 2234 | +\newcommand{\TermPersOwnerOldConsMin}{\ensuremath{-}1.9} | |
| 2235 | +\newcommand{\TermPersOwnerOldConsMinAbs}{1.9} | |
| 2236 | +\newcommand{\TermPersOwnerOldLossMax}{0.08} | |
| 2237 | +\newcommand{\TermPersOwnerOldLossMin}{0.04} | |
| 2238 | +\newcommand{\TermPersOwnerYoungConsMax}{0.4} | |
| 2239 | +\newcommand{\TermPersOwnerYoungConsMin}{0.2} | |
| 2240 | +\newcommand{\TermPersOwnerYoungGainMax}{0.04} | |
| 2241 | +\newcommand{\TermPersPriceBackYears}{5} | |
| 2242 | +\newcommand{\TermPersPriceMaxPos}{0.1} | |
| 2243 | +\newcommand{\TermPersPriceSignChanges}{3} | |
| 2244 | +\newcommand{\TermPersPzero}{\ensuremath{-}1.49} | |
| 2245 | +\newcommand{\TermPersPzeroAbs}{1.49} | |
| 2246 | +\newcommand{\TermPersRentGainR}{0.00} | |
| 2247 | +\newcommand{\TermPersResT}{2\times10^{-4}} | |
| 2248 | +\newcommand{\TermPersRhoZero}{0.04} | |
| 2249 | +\newcommand{\TermPersSecs}{213} | |
| 2250 | +\newcommand{\TermPersSteps}{8} | |
| 2251 | +\newcommand{\TermPersThrChangePct}{18} | |
| 2252 | +\newcommand{\TermPersThrSS}{0.49} | |
| 2253 | +\newcommand{\TermPersThrZero}{0.58} | |
| 2254 | +\newcommand{\TermPersWageAll}{0.00} | |
| 2255 | +\newcommand{\TermPersWageO}{0.00} | |
| 2256 | +\newcommand{\TermPersqROsevenSS}{0.806} | |
| 2257 | +\newcommand{\TermPersqROsevenZero}{0.807} | |
| 2258 | +\newcommand{\TermPersqROthreeOne}{0.333} | |
| 2259 | +\newcommand{\TermPersqROthreeSS}{0.351} | |
| 2260 | +\newcommand{\TermPersqROthreeZero}{0.356} | |
| 2261 | +\newcommand{\TermTaylorAbsorbPct}{91} | |
| 2262 | +\newcommand{\TermTaylorBetterAllPct}{44} | |
| 2263 | +\newcommand{\TermTaylorBetterOMPct}{56} | |
| 2264 | +\newcommand{\TermTaylorBetterONMPct}{22} | |
| 2265 | +\newcommand{\TermTaylorBetterOPct}{43} | |
| 2266 | +\newcommand{\TermTaylorBetterRPct}{46} | |
| 2267 | +\newcommand{\TermTaylorCEVAll}{\ensuremath{-}0.25} | |
| 2268 | +\newcommand{\TermTaylorCEVAllAbs}{0.25} | |
| 2269 | +\newcommand{\TermTaylorCEVAllPE}{\ensuremath{-}1.67} | |
| 2270 | +\newcommand{\TermTaylorCEVAllPEAbs}{1.67} | |
| 2271 | +\newcommand{\TermTaylorCEVO}{\ensuremath{-}0.17} | |
| 2272 | +\newcommand{\TermTaylorCEVOAbs}{0.17} | |
| 2273 | +\newcommand{\TermTaylorCEVOM}{0.20} | |
| 2274 | +\newcommand{\TermTaylorCEVONM}{\ensuremath{-}1.05} | |
| 2275 | +\newcommand{\TermTaylorCEVONMAbs}{1.05} | |
| 2276 | +\newcommand{\TermTaylorCEVR}{\ensuremath{-}0.32} | |
| 2277 | +\newcommand{\TermTaylorCEVRAbs}{0.32} | |
| 2278 | +\newcommand{\TermTaylorConsAll}{\ensuremath{-}3.4} | |
| 2279 | +\newcommand{\TermTaylorConsAllAbs}{3.4} | |
| 2280 | +\newcommand{\TermTaylorConsO}{\ensuremath{-}3.6} | |
| 2281 | +\newcommand{\TermTaylorConsOAbs}{3.6} | |
| 2282 | +\newcommand{\TermTaylorConsR}{\ensuremath{-}3.2} | |
| 2283 | +\newcommand{\TermTaylorConsRAbs}{3.2} | |
| 2284 | +\newcommand{\TermTaylorDOwnGE}{\ensuremath{-}0.26} | |
| 2285 | +\newcommand{\TermTaylorDOwnGEAbs}{0.26} | |
| 2286 | +\newcommand{\TermTaylorDOwnPE}{\ensuremath{-}2.94} | |
| 2287 | +\newcommand{\TermTaylorDOwnPEAbs}{2.94} | |
| 2288 | +\newcommand{\TermTaylorDebtZeroPct}{\ensuremath{-}19.4} | |
| 2289 | +\newcommand{\TermTaylorDebtZeroPctAbs}{19.4} | |
| 2290 | +\newcommand{\TermTaylorDepLoss}{0.48} | |
| 2291 | +\newcommand{\TermTaylorErosion}{0.21} | |
| 2292 | +\newcommand{\TermTaylorLamSevenSS}{0.060} | |
| 2293 | +\newcommand{\TermTaylorLamSevenZero}{0.068} | |
| 2294 | +\newcommand{\TermTaylorLender}{0.27} | |
| 2295 | +\newcommand{\TermTaylorMaxRes}{7.7\times10^{-4}} | |
| 2296 | +\newcommand{\TermTaylorNWAll}{\ensuremath{-}0.04} | |
| 2297 | +\newcommand{\TermTaylorNWAllAbs}{0.04} | |
| 2298 | +\newcommand{\TermTaylorNWO}{\ensuremath{-}0.07} | |
| 2299 | +\newcommand{\TermTaylorNWOAbs}{0.07} | |
| 2300 | +\newcommand{\TermTaylorNWR}{\ensuremath{-}0.01} | |
| 2301 | +\newcommand{\TermTaylorNWRAbs}{0.01} | |
| 2302 | +\newcommand{\TermTaylorNetO}{\ensuremath{-}0.02} | |
| 2303 | +\newcommand{\TermTaylorNetOAbs}{0.02} | |
| 2304 | +\newcommand{\TermTaylorNetR}{\ensuremath{-}0.02} | |
| 2305 | +\newcommand{\TermTaylorNetRAbs}{0.02} | |
| 2306 | +\newcommand{\TermTaylorNominalRateZero}{0.408} | |
| 2307 | +\newcommand{\TermTaylorOwnerOldConsMax}{\ensuremath{-}4.6} | |
| 2308 | +\newcommand{\TermTaylorOwnerOldConsMaxAbs}{4.6} | |
| 2309 | +\newcommand{\TermTaylorOwnerOldConsMin}{\ensuremath{-}3.8} | |
| 2310 | +\newcommand{\TermTaylorOwnerOldConsMinAbs}{3.8} | |
| 2311 | +\newcommand{\TermTaylorOwnerOldLossMax}{0.08} | |
| 2312 | +\newcommand{\TermTaylorOwnerOldLossMin}{0.04} | |
| 2313 | +\newcommand{\TermTaylorOwnerYoungConsMax}{\ensuremath{-}3.1} | |
| 2314 | +\newcommand{\TermTaylorOwnerYoungConsMaxAbs}{3.1} | |
| 2315 | +\newcommand{\TermTaylorOwnerYoungConsMin}{\ensuremath{-}3.6} | |
| 2316 | +\newcommand{\TermTaylorOwnerYoungConsMinAbs}{3.6} | |
| 2317 | +\newcommand{\TermTaylorOwnerYoungGainMax}{0.04} | |
| 2318 | +\newcommand{\TermTaylorPriceBackYears}{40} | |
| 2319 | +\newcommand{\TermTaylorPriceMaxPos}{1.4} | |
| 2320 | +\newcommand{\TermTaylorPriceSignChanges}{1} | |
| 2321 | +\newcommand{\TermTaylorPzero}{\ensuremath{-}6.93} | |
| 2322 | +\newcommand{\TermTaylorPzeroAbs}{6.93} | |
| 2323 | +\newcommand{\TermTaylorRentGainR}{0.00} | |
| 2324 | +\newcommand{\TermTaylorResT}{8\times10^{-4}} | |
| 2325 | +\newcommand{\TermTaylorRhoZero}{\ensuremath{-}1.91} | |
| 2326 | +\newcommand{\TermTaylorRhoZeroAbs}{1.91} | |
| 2327 | +\newcommand{\TermTaylorSecs}{213} | |
| 2328 | +\newcommand{\TermTaylorSteps}{7} | |
| 2329 | +\newcommand{\TermTaylorThrChangePct}{13} | |
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| 2331 | +\newcommand{\TermTaylorThrZero}{0.56} | |
| 2332 | +\newcommand{\TermTaylorWageAll}{0.00} | |
| 2333 | +\newcommand{\TermTaylorWageO}{0.00} | |
| 2334 | +\newcommand{\TermTaylorqROsevenSS}{0.806} | |
| 2335 | +\newcommand{\TermTaylorqROsevenZero}{0.799} | |
| 2336 | +\newcommand{\TermTaylorqROthreeOne}{0.349} | |
| 2337 | +\newcommand{\TermTaylorqROthreeSS}{0.351} | |
| 2338 | +\newcommand{\TermTaylorqROthreeZero}{0.361} | |
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| 2340 | +\newcommand{\ThetaFiveGapFour}{0.84} | |
| 2341 | +\newcommand{\ThetaFiveGift}{0.39} | |
| 2342 | +\newcommand{\ThetaFiveGini}{0.55} | |
| 2343 | +\newcommand{\ThetaFiveLamSeven}{0.069} | |
| 2344 | +\newcommand{\ThetaFiveLamThree}{0.20} | |
| 2345 | +\newcommand{\ThetaFiveOwn}{0.640} | |
| 2346 | +\newcommand{\ThetaFiveOwnUthirtyfive}{0.409} | |
| 2347 | +\newcommand{\ThetaFiveP}{1.005} | |
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| 2349 | +\newcommand{\ThetaFivePEEstate}{0.32} | |
| 2350 | +\newcommand{\ThetaFivePEGapFour}{0.83} | |
| 2351 | +\newcommand{\ThetaFivePEGift}{0.39} | |
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| 2353 | +\newcommand{\ThetaFivePELamSeven}{0.061} | |
| 2354 | +\newcommand{\ThetaFivePELamThree}{0.21} | |
| 2355 | +\newcommand{\ThetaFivePEOwn}{0.657} | |
| 2356 | +\newcommand{\ThetaFivePEOwnUthirtyfive}{0.429} | |
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| 2360 | +\newcommand{\ThetaFivePERebate}{0.000} | |
| 2361 | +\newcommand{\ThetaFivePERho}{0.194} | |
| 2362 | +\newcommand{\ThetaFivePERhoDevPct}{0.0} | |
| 2363 | +\newcommand{\ThetaFivePEThr}{0.52} | |
| 2364 | +\newcommand{\ThetaFivePEThrHighZ}{\ensuremath{-}0.13} | |
| 2365 | +\newcommand{\ThetaFivePEThrHighZAbs}{0.13} | |
| 2366 | +\newcommand{\ThetaFivePEThrLowZ}{1.18} | |
| 2367 | +\newcommand{\ThetaFivePEqOO}{0.585} | |
| 2368 | +\newcommand{\ThetaFivePEqRO}{0.379} | |
| 2369 | +\newcommand{\ThetaFiveRR}{0.33} | |
| 2370 | +\newcommand{\ThetaFiveRebate}{0.000} | |
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| 2376 | +\newcommand{\ThetaFiveThrHighZAbs}{0.10} | |
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| 2379 | +\newcommand{\ThetaFiveqRO}{0.375} | |
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| 2390 | +\newcommand{\ThetaFortyPDevPctAbs}{1.3} | |
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| 2392 | +\newcommand{\ThetaFortyPEGapFour}{0.99} | |
| 2393 | +\newcommand{\ThetaFortyPEGift}{0.39} | |
| 2394 | +\newcommand{\ThetaFortyPEGini}{0.54} | |
| 2395 | +\newcommand{\ThetaFortyPELamSeven}{0.079} | |
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| 2397 | +\newcommand{\ThetaFortyPEOwn}{0.596} | |
| 2398 | +\newcommand{\ThetaFortyPEOwnUthirtyfive}{0.304} | |
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| 2402 | +\newcommand{\ThetaFortyPERebate}{0.000} | |
| 2403 | +\newcommand{\ThetaFortyPERho}{0.194} | |
| 2404 | +\newcommand{\ThetaFortyPERhoDevPct}{0.0} | |
| 2405 | +\newcommand{\ThetaFortyPEThr}{0.59} | |
| 2406 | +\newcommand{\ThetaFortyPEThrHighZ}{0.24} | |
| 2407 | +\newcommand{\ThetaFortyPEThrLowZ}{1.18} | |
| 2408 | +\newcommand{\ThetaFortyPEqOO}{0.488} | |
| 2409 | +\newcommand{\ThetaFortyPEqRO}{0.279} | |
| 2410 | +\newcommand{\ThetaFortyRR}{0.34} | |
| 2411 | +\newcommand{\ThetaFortyRebate}{0.000} | |
| 2412 | +\newcommand{\ThetaFortyRho}{0.200} | |
| 2413 | +\newcommand{\ThetaFortyRhoDevPct}{3.0} | |
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| 2415 | +\newcommand{\ThetaFortyThrHighZ}{0.18} | |
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| 2418 | +\newcommand{\ThetaFortyqRO}{0.304} | |
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| 2426 | +\newcommand{\ThetaTenOwnUthirtyfive}{0.401} | |
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| 2429 | +\newcommand{\ThetaTenRR}{0.33} | |
| 2430 | +\newcommand{\ThetaTenRebate}{0.000} | |
| 2431 | +\newcommand{\ThetaTenRho}{0.190} | |
| 2432 | +\newcommand{\ThetaTenRhoDevPct}{\ensuremath{-}2.1} | |
| 2433 | +\newcommand{\ThetaTenRhoDevPctAbs}{2.1} | |
| 2434 | +\newcommand{\ThetaTenThr}{0.58} | |
| 2435 | +\newcommand{\ThetaTenThrHighZ}{\ensuremath{-}0.05} | |
| 2436 | +\newcommand{\ThetaTenThrHighZAbs}{0.05} | |
| 2437 | +\newcommand{\ThetaTenThrLowZ}{1.24} | |
| 2438 | +\newcommand{\ThetaTenqOO}{0.560} | |
| 2439 | +\newcommand{\ThetaTenqRO}{0.372} | |
| 2440 | +\newcommand{\ThetaThirtyEstate}{0.32} | |
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| 2442 | +\newcommand{\ThetaThirtyGift}{0.39} | |
| 2443 | +\newcommand{\ThetaThirtyGini}{0.54} | |
| 2444 | +\newcommand{\ThetaThirtyLamSeven}{0.066} | |
| 2445 | +\newcommand{\ThetaThirtyLamThree}{0.20} | |
| 2446 | +\newcommand{\ThetaThirtyOwn}{0.625} | |
| 2447 | +\newcommand{\ThetaThirtyOwnUthirtyfive}{0.353} | |
| 2448 | +\newcommand{\ThetaThirtyP}{0.992} | |
| 2449 | +\newcommand{\ThetaThirtyPDevPct}{\ensuremath{-}0.8} | |
| 2450 | +\newcommand{\ThetaThirtyPDevPctAbs}{0.8} | |
| 2451 | +\newcommand{\ThetaThirtyRR}{0.34} | |
| 2452 | +\newcommand{\ThetaThirtyRebate}{0.000} | |
| 2453 | +\newcommand{\ThetaThirtyRho}{0.197} | |
| 2454 | +\newcommand{\ThetaThirtyRhoDevPct}{1.6} | |
| 2455 | +\newcommand{\ThetaThirtyThr}{0.49} | |
| 2456 | +\newcommand{\ThetaThirtyThrHighZ}{0.11} | |
| 2457 | +\newcommand{\ThetaThirtyThrLowZ}{1.12} | |
| 2458 | +\newcommand{\ThetaThirtyqOO}{0.523} | |
| 2459 | +\newcommand{\ThetaThirtyqRO}{0.321} | |
| 2460 | +\newcommand{\ThrBaseline}{0.49} | |
| 2461 | +\newcommand{\ThrBequestshalved}{0.44} | |
| 2462 | +\newcommand{\ThrBequestsoff}{0.32} | |
| 2463 | +\newcommand{\ThrBequestsoffpricesfixed}{0.48} | |
| 2464 | +\newcommand{\ThrDSTIcappsi}{0.46} | |
| 2465 | +\newcommand{\ThrHalfsegmentation}{0.49} | |
| 2466 | +\newcommand{\ThrNosegmentation}{0.49} | |
| 2467 | +\newcommand{\ThrTransfertax}{0.48} | |
| 2468 | +\newcommand{\ThresholdAgeOneHighZ}{0.03} | |
| 2469 | +\newcommand{\ThresholdAgeOneLowZ}{1.18} | |
| 2470 | +\newcommand{\ThresholdAgeOneMidZ}{0.49} | |
| 2471 | +\newcommand{\ThresholdAgeOneZfour}{0.24} | |
| 2472 | +\newcommand{\ThresholdAgeOneZtwo}{0.88} | |
| 2473 | +\newcommand{\ThresholdAgeThreeHighZ}{\ensuremath{-}0.61} | |
| 2474 | +\newcommand{\ThresholdAgeThreeHighZAbs}{0.61} | |
| 2475 | +\newcommand{\ThresholdAgeThreeMidZ}{0.19} | |
| 2476 | +\newcommand{\ThresholdAgeTwoHighZ}{\ensuremath{-}0.31} | |
| 2477 | +\newcommand{\ThresholdAgeTwoHighZAbs}{0.31} | |
| 2478 | +\newcommand{\Thrtheta}{0.55} | |
| 2479 | +\newcommand{\Thrthetapricesfixed}{0.59} | |
| 2480 | +\newcommand{\Thrvarepsilon}{0.49} | |
| 2481 | +\newcommand{\TotalDebt}{2.47} | |
| 2482 | +\newcommand{\TotalDeposits}{4.12} | |
| 2483 | +\newcommand{\UnderwaterAllTen}{0} | |
| 2484 | +\newcommand{\UnderwaterAllThirty}{0} | |
| 2485 | +\newcommand{\UnderwaterAllTwenty}{0} | |
| 2486 | +\newcommand{\UnderwaterNewTen}{0} | |
| 2487 | +\newcommand{\UnderwaterNewThirty}{48} | |
| 2488 | +\newcommand{\UnderwaterNewTwenty}{40} | |
| 2489 | +\newcommand{\UnindexedAbsorbPct}{80} | |
| 2490 | +\newcommand{\UnindexedBetterAllPct}{0} | |
| 2491 | +\newcommand{\UnindexedBetterOMPct}{0} | |
| 2492 | +\newcommand{\UnindexedBetterONMPct}{0} | |
| 2493 | +\newcommand{\UnindexedBetterOPct}{0} | |
| 2494 | +\newcommand{\UnindexedBetterRPct}{0} | |
| 2495 | +\newcommand{\UnindexedCEVAll}{\ensuremath{-}3.33} | |
| 2496 | +\newcommand{\UnindexedCEVAllAbs}{3.33} | |
| 2497 | +\newcommand{\UnindexedCEVAllPE}{\ensuremath{-}3.45} | |
| 2498 | +\newcommand{\UnindexedCEVAllPEAbs}{3.45} | |
| 2499 | +\newcommand{\UnindexedCEVO}{\ensuremath{-}3.02} | |
| 2500 | +\newcommand{\UnindexedCEVOAbs}{3.02} | |
| 2501 | +\newcommand{\UnindexedCEVOM}{\ensuremath{-}2.40} | |
| 2502 | +\newcommand{\UnindexedCEVOMAbs}{2.40} | |
| 2503 | +\newcommand{\UnindexedCEVONM}{\ensuremath{-}4.48} | |
| 2504 | +\newcommand{\UnindexedCEVONMAbs}{4.48} | |
| 2505 | +\newcommand{\UnindexedCEVR}{\ensuremath{-}3.58} | |
| 2506 | +\newcommand{\UnindexedCEVRAbs}{3.58} | |
| 2507 | +\newcommand{\UnindexedConsAll}{\ensuremath{-}4.4} | |
| 2508 | +\newcommand{\UnindexedConsAllAbs}{4.4} | |
| 2509 | +\newcommand{\UnindexedConsO}{\ensuremath{-}3.5} | |
| 2510 | +\newcommand{\UnindexedConsOAbs}{3.5} | |
| 2511 | +\newcommand{\UnindexedConsR}{\ensuremath{-}6.3} | |
| 2512 | +\newcommand{\UnindexedConsRAbs}{6.3} | |
| 2513 | +\newcommand{\UnindexedDOwnGE}{\ensuremath{-}0.63} | |
| 2514 | +\newcommand{\UnindexedDOwnGEAbs}{0.63} | |
| 2515 | +\newcommand{\UnindexedDOwnPE}{\ensuremath{-}3.18} | |
| 2516 | +\newcommand{\UnindexedDOwnPEAbs}{3.18} | |
| 2517 | +\newcommand{\UnindexedDebtZeroPct}{\ensuremath{-}1.6} | |
| 2518 | +\newcommand{\UnindexedDebtZeroPctAbs}{1.6} | |
| 2519 | +\newcommand{\UnindexedDepLoss}{0.48} | |
| 2520 | +\newcommand{\UnindexedErosion}{0.21} | |
| 2521 | +\newcommand{\UnindexedLamSevenSS}{0.060} | |
| 2522 | +\newcommand{\UnindexedLamSevenZero}{0.074} | |
| 2523 | +\newcommand{\UnindexedLender}{0.27} | |
| 2524 | +\newcommand{\UnindexedMaxRes}{2.0\times10^{-3}} | |
| 2525 | +\newcommand{\UnindexedNWAll}{\ensuremath{-}0.09} | |
| 2526 | +\newcommand{\UnindexedNWAllAbs}{0.09} | |
| 2527 | +\newcommand{\UnindexedNWO}{\ensuremath{-}0.12} | |
| 2528 | +\newcommand{\UnindexedNWOAbs}{0.12} | |
| 2529 | +\newcommand{\UnindexedNWR}{\ensuremath{-}0.04} | |
| 2530 | +\newcommand{\UnindexedNWRAbs}{0.04} | |
| 2531 | +\newcommand{\UnindexedNetO}{\ensuremath{-}0.02} | |
| 2532 | +\newcommand{\UnindexedNetOAbs}{0.02} | |
| 2533 | +\newcommand{\UnindexedNetR}{\ensuremath{-}0.02} | |
| 2534 | +\newcommand{\UnindexedNetRAbs}{0.02} | |
| 2535 | +\newcommand{\UnindexedNominalRateZero}{0.219} | |
| 2536 | +\newcommand{\UnindexedOwnerOldConsMax}{\ensuremath{-}4.8} | |
| 2537 | +\newcommand{\UnindexedOwnerOldConsMaxAbs}{4.8} | |
| 2538 | +\newcommand{\UnindexedOwnerOldConsMin}{\ensuremath{-}4.3} | |
| 2539 | +\newcommand{\UnindexedOwnerOldConsMinAbs}{4.3} | |
| 2540 | +\newcommand{\UnindexedOwnerOldLossMax}{0.08} | |
| 2541 | +\newcommand{\UnindexedOwnerOldLossMin}{0.04} | |
| 2542 | +\newcommand{\UnindexedOwnerYoungConsMax}{\ensuremath{-}1.8} | |
| 2543 | +\newcommand{\UnindexedOwnerYoungConsMaxAbs}{1.8} | |
| 2544 | +\newcommand{\UnindexedOwnerYoungConsMin}{\ensuremath{-}2.8} | |
| 2545 | +\newcommand{\UnindexedOwnerYoungConsMinAbs}{2.8} | |
| 2546 | +\newcommand{\UnindexedOwnerYoungGainMax}{0.04} | |
| 2547 | +\newcommand{\UnindexedPriceBackYears}{40} | |
| 2548 | +\newcommand{\UnindexedPriceMaxPos}{0.1} | |
| 2549 | +\newcommand{\UnindexedPriceSignChanges}{1} | |
| 2550 | +\newcommand{\UnindexedPzero}{\ensuremath{-}3.86} | |
| 2551 | +\newcommand{\UnindexedPzeroAbs}{3.86} | |
| 2552 | +\newcommand{\UnindexedRentGainR}{0.00} | |
| 2553 | +\newcommand{\UnindexedResT}{9\times10^{-5}} | |
| 2554 | +\newcommand{\UnindexedRhoZero}{\ensuremath{-}3.21} | |
| 2555 | +\newcommand{\UnindexedRhoZeroAbs}{3.21} | |
| 2556 | +\newcommand{\UnindexedSecs}{487} | |
| 2557 | +\newcommand{\UnindexedSteps}{11} | |
| 2558 | +\newcommand{\UnindexedThrChangePct}{10} | |
| 2559 | +\newcommand{\UnindexedThrSS}{0.49} | |
| 2560 | +\newcommand{\UnindexedThrZero}{0.55} | |
| 2561 | +\newcommand{\UnindexedWageAll}{0.08} | |
| 2562 | +\newcommand{\UnindexedWageO}{0.09} | |
| 2563 | +\newcommand{\UnindexedqROsevenSS}{0.806} | |
| 2564 | +\newcommand{\UnindexedqROsevenZero}{0.807} | |
| 2565 | +\newcommand{\UnindexedqROthreeOne}{0.308} | |
| 2566 | +\newcommand{\UnindexedqROthreeSS}{0.351} | |
| 2567 | +\newcommand{\UnindexedqROthreeZero}{0.352} | |
| 2568 | +\newcommand{\WelfareCheckMaxPct}{4.9} | |
| 2569 | +\newcommand{\qOOageSeven}{0.87} | |
| 2570 | +\newcommand{\qOOageThree}{0.54} | |
| 2571 | +\newcommand{\qROageSeven}{0.81} | |
| 2572 | +\newcommand{\qROageThree}{0.35} | |
added
paper/tables/tab01_notation.tex
+37 −0
@@ -0,0 +1,37 @@ | ||
| 1 | +\begin{table}[htbp] | |
| 2 | +\centering | |
| 3 | +\small | |
| 4 | +\begin{threeparttable} | |
| 5 | +\caption{Notation of the quantitative model} | |
| 6 | +\label{tab:notation} | |
| 7 | +\begin{tabular}{lp{9.3cm}l} | |
| 8 | +\toprule | |
| 9 | +Symbol & Meaning & Units \\ | |
| 10 | +\midrule | |
| 11 | +$j$ & age index, $j = 1,\dots,J=12$ (5-year periods, 25--84) & -- \\ | |
| 12 | +$z$ & idiosyncratic productivity (Rouwenhorst, $n_z = 5$), frozen in retirement & -- \\ | |
| 13 | +$h \in \mathcal H$ & owner-occupied house size (four sizes, $h_1 = \underline h$) & housing units \\ | |
| 14 | +$s$ & rented housing services ($s \ge \underline s$) & housing units \\ | |
| 15 | +$a$ & liquid real asset (nominal deposit), $a \ge 0$ & goods \\ | |
| 16 | +$b^m$ & real mortgage balance at the start of the period ($B^{nom}/P$) & goods \\ | |
| 17 | +$n_m$ & remaining number of mortgage payments ($N = 5$ at origination) & -- \\ | |
| 18 | +$p$, $p^r$, $\rho$ & real house price, rental-unit price, rent per period & goods / unit \\ | |
| 19 | +$P$, $\pi$, $\pi^u$ & CPI level, inflation, unexpected inflation & -- \\ | |
| 20 | +$R$, $i$, $i_q$ & gross real rate, nominal rate ($1+i = R(1+\bar\pi)$), qualifying rate & -- \\ | |
| 21 | +$\theta$, $\psi$ & minimum down payment, payment-to-income cap & -- \\ | |
| 22 | +$\kappa_b$, $\kappa_s$, $\kappa_f$, $m$ & buying, selling, refinancing and maintenance costs & share of value \\ | |
| 23 | +$b^g$, $b^e$ & inter-vivos gift (parent's age $j_g = 7$), estate (parent's age $J$) & goods \\ | |
| 24 | +$\nu$, $\underline x$ & warm-glow weight and luxury shift & -- \\ | |
| 25 | +$\varepsilon$, $\varepsilon_r$, $\delta_H$ & supply elasticities, depreciation of the stocks & -- \\ | |
| 26 | +$\sigma$, $\eta_\rho$, $\eta_w$ & segmentation, rent and wage indexation to a surprise & -- \\ | |
| 27 | +$\hat a_j(z)$ & ownership-entry threshold: smallest liquid wealth at which the age-$j$ renter buys & goods \\ | |
| 28 | +$q_{RO}$, $q_{OO}$, $\lambda$ & child ownership given parent renter / owner; persistence $\lambda = q_{OO} - q_{RO}$ & -- \\ | |
| 29 | +$\lambda^{CE}$ & consumption-equivalent variation (Section~\ref{sec:welfare}) & -- \\ | |
| 30 | +\bottomrule | |
| 31 | +\end{tabular} | |
| 32 | +\begin{tablenotes}[flushleft] | |
| 33 | +\footnotesize | |
| 34 | +\item Units: ``goods'' = units of the consumption good; the numeraire of the simulations is the average 5-year labour income of a working-age household (E$[z]=1$). Source: research/model\_specification.md, Part B. | |
| 35 | +\end{tablenotes} | |
| 36 | +\end{threeparttable} | |
| 37 | +\end{table} | |
added
paper/tables/tab02_parameters.tex
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| 1 | +\begin{table}[htbp] | |
| 2 | +\centering | |
| 3 | +\footnotesize | |
| 4 | +\begin{threeparttable} | |
| 5 | +\caption{Parameters of the baseline (period = 5 years)} | |
| 6 | +\label{tab:parameters} | |
| 7 | +\begin{tabular}{lp{2.2cm}p{2.3cm}p{6cm}} | |
| 8 | +\toprule | |
| 9 | +Parameter & Value & Unit & Source / status \\ | |
| 10 | +\midrule | |
| 11 | +J & 12 & periods & spec B.1 \\ | |
| 12 | +j\_retire & 9 & age index & spec B.1 \\ | |
| 13 | +j\_gift = j\_estate & 7 & age index & spec B.1 \\ | |
| 14 | +beta (period) & 0.8154 & - & 0.96 annual, illustrative \\ | |
| 15 | +gamma & 2 & - & illustrative \\ | |
| 16 | +alpha & 0.2 & - & illustrative \\ | |
| 17 | +chi & 0 & - & spec B.2 baseline 0 \\ | |
| 18 | +nu & 2 & - & illustrative \\ | |
| 19 | +x\_shift & 1 & income units & illustrative \\ | |
| 20 | +rho\_z (period) & 0.7738 & - & 0.95\textasciicircum{}5, illustrative \\ | |
| 21 | +sigma\_z (period) & 0.3043 & - & same unconditional variance as annual, illustrative \\ | |
| 22 | +n\_z & 5 & - & Rouwenhorst \\ | |
| 23 | +replacement rate & 0.5 & - & illustrative \\ | |
| 24 | +sigma\_transitory (period, log) & 0.03 & - & illustrative; unanticipated, breaks cash atoms \\ | |
| 25 | +h sizes & [1.0, 1.5, 2.25, 3.4] & housing units & spec B.12 \\ | |
| 26 | +s\_min & 0.3 & housing units & illustrative \\ | |
| 27 | +m (period) & 0.05 & share of value & 0.01 annual, illustrative \\ | |
| 28 | +kappa\_b & 0.025 & share of price & illustrative \\ | |
| 29 | +kappa\_s & 0.05 & share of price & illustrative \\ | |
| 30 | +kappa\_f & 0.01 & share of value & illustrative \\ | |
| 31 | +delta\_H (period) & 0.04901 & - & 0.01 annual, illustrative \\ | |
| 32 | +N & 5 & periods & 25-year amortisation (literature\_review §7) \\ | |
| 33 | +theta & 0.2 & - & illustrative (uninsured threshold) \\ | |
| 34 | +psi & 0.39 & - & illustrative \\ | |
| 35 | +stress add-on (period) & 0.1218 & - & 0.02 annual, illustrative \\ | |
| 36 | +r (period) & 0.1041 & - & 0.02 annual, illustrative \\ | |
| 37 | +pibar (period) & 0.1041 & - & 0.02 annual (2 \% target) \\ | |
| 38 | +i (period, Fisher) & 0.219 & - & derived \\ | |
| 39 | +phi (landlord wedge) & 0.05 & share of value & illustrative \\ | |
| 40 | +sigma\_seg & 1 & - & spec B.6 \\ | |
| 41 | +eps & 1 & - & ILLUSTRATIVE (no verified Canadian estimate) \\ | |
| 42 | +eps\_r & 1 & - & illustrative \\ | |
| 43 | +tau\_b & 0 & - & policy experiment \\ | |
| 44 | +n\_a / a\_max & 50 / 15.0 & - & grid \\ | |
| 45 | +n\_b & 20 & - & grid \\ | |
| 46 | +n\_x & 120 & - & grid \\ | |
| 47 | +n\_B & 11 & - & grid \\ | |
| 48 | +K\_estate & 5 & - & lottery \\ | |
| 49 | +n\_dynasties & 96{,}000 & - & simulation \\ | |
| 50 | +seed & 20{,}261{,}020 & - & simulation \\ | |
| 51 | +\bottomrule | |
| 52 | +\end{tabular} | |
| 53 | +\begin{tablenotes}[flushleft] | |
| 54 | +\footnotesize | |
| 55 | +\item Values in period units after the conversions recorded in src/model.py (annual rates compounded over 5 years; the 5-year AR(1) keeps the unconditional variance of the annual process). ``Illustrative'' = no verified Canadian estimate in the literature matrix; the supply elasticity in particular has no verified Canadian source (research/literature\_review.md, \S7). Supply scales $\bar H$, $\bar H_r$ are set by the normalisation $p^* = p^{r*} = 1$. Source file: results/data/parameters.csv. | |
| 56 | +\end{tablenotes} | |
| 57 | +\end{threeparttable} | |
| 58 | +\end{table} | |
added
paper/tables/tab03_baseline_moments.tex
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| 1 | +\begin{table}[htbp] | |
| 2 | +\centering | |
| 3 | +\footnotesize | |
| 4 | +\begin{threeparttable} | |
| 5 | +\caption{Baseline stationary equilibrium: model moments and verified comparison points} | |
| 6 | +\label{tab:moments} | |
| 7 | +\begin{tabular}{p{6.3cm}cp{6cm}} | |
| 8 | +\toprule | |
| 9 | +Moment & Model & Comparison point (source) \\ | |
| 10 | +\midrule | |
| 11 | +Ownership rate, all households & 0.631 & about 0.66--0.69 (not a verified target); plateau $\approx$ 0.75 near 65 (Hou 2010) \\ | |
| 12 | +Ownership rate, under 35 / 35--64 / 65+ & 0.38 / 0.73 / 0.61 & steep before 40, plateau near 65 (Hou 2010) \\ | |
| 13 | +Share of households with a mortgage & 0.411 & 0.39 (SFS 2023) \\ | |
| 14 | +Share of owners with a mortgage & 0.651 & -- \\ | |
| 15 | +Mean net worth owners / renters, under 35 & 11.2 & median 457{,}100 / 44{,}000 $\approx$ 10.4 (SFS 2023); model median of renters $= 0$ \\ | |
| 16 | +Mean gift received at entry (5-year income units) & 0.388 & about \$80{,}000 (CIBC, 2021, via Mirdamadi--Khalid 2023) \\ | |
| 17 | +Share of entrants receiving a positive gift & 0.695 & 28\% of first-time buyers received gifts (same source) \\ | |
| 18 | +Mean estate received at age 55--59 & 0.316 & -- \\ | |
| 19 | +Median house value / annual income (owners) & 5.0 & -- \\ | |
| 20 | +Mean LTV of new loans & 0.605 & -- \\ | |
| 21 | +Annual rent--price ratio & 0.039 & illustrative (landlord wedge $\phi$) \\ | |
| 22 | +Gini of net worth (all ages) & 0.542 & -- \\ | |
| 23 | +P(child owns at 35--39 $\mid$ parent renter / owner) & 0.35 / 0.54 & children of owners more likely to own (Mirdamadi--Khalid 2023) \\ | |
| 24 | +Rank--rank slope of net worth (child at 55--59 vs parent at 55--59) & 0.34 & -- \\ | |
| 25 | +Aggregate mortgage debt / deposits (per age mass) & 2.47 / 4.12 & -- \\ | |
| 26 | +\bottomrule | |
| 27 | +\end{tabular} | |
| 28 | +\begin{tablenotes}[flushleft] | |
| 29 | +\footnotesize | |
| 30 | +\item Model moments from results/data/baseline/summary.json (pooled cross-section of 6 periods, 1,152,000 household-periods). Comparison points are quoted from the verified sources listed in research/literature\_matrix.csv (research/literature\_review.md \S1, \S7) and are not fitted: the calibration is illustrative (Table~\ref{tab:parameters}). The young renters' median net worth is zero in the model (no liquid saving before purchase), so the median ratio is undefined; the mean ratio is reported. | |
| 31 | +\end{tablenotes} | |
| 32 | +\end{threeparttable} | |
| 33 | +\end{table} | |
added
paper/tables/tab04_convergence.tex
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| 1 | +\begin{table}[htbp] | |
| 2 | +\centering | |
| 3 | +\footnotesize | |
| 4 | +\begin{threeparttable} | |
| 5 | +\caption{Convergence and numerical diagnostics} | |
| 6 | +\label{tab:convergence} | |
| 7 | +\begin{tabular}{p{8.5cm}p{6cm}} | |
| 8 | +\toprule | |
| 9 | +Diagnostic & Value \\ | |
| 10 | +\midrule | |
| 11 | +Owner-occupied market, relative excess demand & 0.000004 \\ | |
| 12 | +Rental market, relative excess demand & \ensuremath{-}0.000007 \\ | |
| 13 | +Tolerance (relative excess demand) & 2.5e-4 \\ | |
| 14 | +Estate-lottery fixed point: max.\ change (last round) & 0.000009 \\ | |
| 15 | +Bellman residual, owners: mean / 99th pct.\ / max & 0.00067 / 0.0062 / 0.0193 \\ | |
| 16 | +Bellman residual, renters: mean / 99th pct.\ / max & 0.00016 / 0.0008 / 0.0017 \\ | |
| 17 | +Share of households at the forced-sale rule & 0.00000 \\ | |
| 18 | +Share of households with a clipped (infeasible interpolated) policy & 0.00003 \\ | |
| 19 | +Outer (lottery) rounds / household-block evaluations & 1 / 5 (calibration) \\ | |
| 20 | +Wall-clock time of the baseline (s) & 8 \\ | |
| 21 | +Total wall-clock time of \texttt{make simulate} (min) & 59.8 \\ | |
| 22 | +Transitions solved (GE converged / total) & 12 / 16 \\ | |
| 23 | +\midrule | |
| 24 | +\multicolumn{2}{l}{\emph{Numerical variants (deviation from baseline, in percent of the baseline value)}} \\ | |
| 25 | +grid\_coarse & own. -0.83, $p^*$ -0.85, Gini -0.06, gift +0.11, rank--rank +1.55 \\ | |
| 26 | +grid\_fine & own. +0.60, $p^*$ -0.11, Gini +0.13, gift +0.10, rank--rank -0.52 \\ | |
| 27 | +tol\_tight & own. +0.00, $p^*$ +0.00, Gini +0.00, gift -0.00, rank--rank +0.00 \\ | |
| 28 | +sample\_small & own. +0.06, $p^*$ +0.09, Gini -0.06, gift +0.47, rank--rank -0.13 \\ | |
| 29 | +sample\_large & own. -0.11, $p^*$ +0.00, Gini +0.13, gift -0.08, rank--rank -0.64 \\ | |
| 30 | +seed\_alt & own. -0.10, $p^*$ -0.01, Gini +0.11, gift -0.53, rank--rank +0.09 \\ | |
| 31 | +\bottomrule | |
| 32 | +\end{tabular} | |
| 33 | +\begin{tablenotes}[flushleft] | |
| 34 | +\footnotesize | |
| 35 | +\item Bellman residuals are relative differences between the interpolated value function and a re-optimisation on a 400-point fine asset grid at 400 random off-grid states (gift age excluded). Numerical variants: grids (coarse: $n_a=30,n_b=12,n_x=80$; fine: $80,30,180$), tighter tolerance ($10^{-5}$), panel of 24{,}000 and 240{,}000 dynasties (baseline 96{,}000), alternative seed. Source files: results/data/baseline/summary.json, results/data/grid\_sensitivity.csv, results/data/diagnostics.json. | |
| 36 | +\end{tablenotes} | |
| 37 | +\end{threeparttable} | |
| 38 | +\end{table} | |
added
paper/tables/tab05_scenarios.tex
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| 1 | +\begin{table}[htbp] | |
| 2 | +\centering | |
| 3 | +\footnotesize | |
| 4 | +\caption{Stationary equilibria across credit, supply, segmentation and bequest scenarios} | |
| 5 | +\label{tab:scenarios} | |
| 6 | +\resizebox{\textwidth}{!}{% | |
| 7 | +\begin{tabular}{lcccccccccccc} | |
| 8 | +\toprule | |
| 9 | +Scenario & Eq. & $p^*$ & $\rho/p$ & Own. & Own.\ $<$35 & $\hat a_1$ & $q_{RO}$ & $q_{OO}$ & rank--rank & Gini & gift \\ | |
| 10 | +\midrule | |
| 11 | +Baseline ($\theta=0.20$, $\varepsilon=1$) & GE & 1.000 & 0.039 & 0.631 & 0.378 & 0.49 & 0.35 & 0.54 & 0.34 & 0.54 & 0.39 \\ | |
| 12 | +$\theta = 0.05$ & GE & 1.005 & 0.039 & 0.640 & 0.409 & 0.56 & 0.38 & 0.57 & 0.33 & 0.55 & 0.39 \\ | |
| 13 | +$\theta = 0.10$ & GE & 1.005 & 0.039 & 0.639 & 0.401 & 0.58 & 0.37 & 0.56 & 0.33 & 0.54 & 0.39 \\ | |
| 14 | +$\theta = 0.30$ & GE & 0.992 & 0.039 & 0.625 & 0.353 & 0.49 & 0.32 & 0.52 & 0.34 & 0.54 & 0.39 \\ | |
| 15 | +$\theta = 0.40$ & GE & 0.987 & 0.039 & 0.619 & 0.333 & 0.55 & 0.30 & 0.51 & 0.34 & 0.54 & 0.39 \\ | |
| 16 | +$\theta = 0.05$, prices fixed & PE & 1.000 & 0.039 & 0.657 & 0.429 & 0.52 & 0.38 & 0.58 & 0.33 & 0.54 & 0.39 \\ | |
| 17 | +$\theta = 0.40$, prices fixed & PE & 1.000 & 0.039 & 0.596 & 0.304 & 0.59 & 0.28 & 0.49 & 0.33 & 0.54 & 0.39 \\ | |
| 18 | +DSTI cap $\psi = 0.30$ & GE & 0.991 & 0.039 & 0.627 & 0.362 & 0.59 & 0.36 & 0.53 & 0.33 & 0.54 & 0.39 \\ | |
| 19 | +DSTI cap $\psi = 0.50$ & GE & 1.006 & 0.039 & 0.634 & 0.382 & 0.46 & 0.34 & 0.53 & 0.33 & 0.54 & 0.39 \\ | |
| 20 | +$\varepsilon = 0.05$ (near-fixed supply) & GE & 1.000 & 0.039 & 0.631 & 0.378 & 0.49 & 0.35 & 0.54 & 0.34 & 0.54 & 0.39 \\ | |
| 21 | +$\varepsilon = 0.25$ & GE & 1.000 & 0.039 & 0.631 & 0.378 & 0.49 & 0.35 & 0.54 & 0.34 & 0.54 & 0.39 \\ | |
| 22 | +$\varepsilon = 0.5$ & GE & 1.000 & 0.039 & 0.631 & 0.378 & 0.49 & 0.35 & 0.54 & 0.34 & 0.54 & 0.39 \\ | |
| 23 | +$\varepsilon = 2$ & GE & 1.000 & 0.039 & 0.631 & 0.378 & 0.49 & 0.35 & 0.54 & 0.34 & 0.54 & 0.39 \\ | |
| 24 | +No segmentation ($\sigma = 0$) & GE & 1.000 & 0.039 & 0.631 & 0.378 & 0.49 & 0.35 & 0.54 & 0.34 & 0.54 & 0.39 \\ | |
| 25 | +Half segmentation ($\sigma = 0.5$) & GE & 1.000 & 0.039 & 0.631 & 0.378 & 0.49 & 0.35 & 0.54 & 0.34 & 0.54 & 0.39 \\ | |
| 26 | +Bequests off ($\nu = 0$) & GE & 0.931 & 0.039 & 0.581 & 0.188 & 0.32 & 0.38 & 0.38 & \ensuremath{-}0.01 & 0.58 & 0.00 \\ | |
| 27 | +Bequests halved ($\nu = 1$) & GE & 0.981 & 0.039 & 0.615 & 0.325 & 0.44 & 0.37 & 0.49 & 0.25 & 0.55 & 0.26 \\ | |
| 28 | +Transfer tax 20\% (rebated) & GE & 1.005 & 0.039 & 0.642 & 0.461 & 0.51 & 0.39 & 0.57 & 0.28 & 0.53 & 0.28 \\ | |
| 29 | +Transfer tax 50\% (rebated) & GE & 0.999 & 0.039 & 0.639 & 0.482 & 0.48 & 0.43 & 0.55 & 0.16 & 0.52 & 0.12 \\ | |
| 30 | +Bequests off, prices fixed & PE & 1.000 & 0.039 & 0.512 & 0.106 & 0.48 & 0.36 & 0.36 & 0.00 & 0.59 & 0.00 \\ | |
| 31 | +\bottomrule | |
| 32 | +\end{tabular}} | |
| 33 | +\begin{minipage}{\textwidth}\footnotesize\vspace{2pt} | |
| 34 | +GE = general equilibrium (prices, estate lottery and rebate re-solved; supply scales fixed at the baseline normalisation); PE = partial equilibrium with prices fixed at the baseline (household block and distribution re-solved). $^{\dagger}$ = not converged to tolerance (residuals in results/data/diagnostics.csv). $\rho/p$ = annual rent--price ratio; $\hat a_1$ = age-1 ownership-entry threshold at the median productivity state (5-year income units); $q_{RO}$, $q_{OO}$ = probability that the child owns at 35--39 given the parent's tenure at the gift; rank--rank = slope of the child's net-worth rank on the parent's (both at 55--59); gift = mean transfer received at entry. Source files: results/data/group/name.json. | |
| 35 | +\end{minipage} | |
| 36 | +\end{table} | |
added
paper/tables/tab06_inflation.tex
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| 1 | +\begin{table}[htbp] | |
| 2 | +\centering | |
| 3 | +\scriptsize | |
| 4 | +\caption{Unanticipated 10\% price-level surprise: impact effects by mortgage contract and persistence} | |
| 5 | +\label{tab:inflation} | |
| 6 | +\resizebox{\textwidth}{!}{% | |
| 7 | +\begin{tabular}{lccccccccccc} | |
| 8 | +\toprule | |
| 9 | +Scenario & Eq. & $p_0$ & $\rho_0$ & $\Delta$own & $\Delta$own PE & erosion & dep.\ loss & CEV o/m & CEV r & CEV all & CEV PE \\ | |
| 10 | +\midrule | |
| 11 | +FRM, level surprise & GE$^{\dagger}$ & \ensuremath{-}0.45 & \ensuremath{-}1.14 & \ensuremath{-}0.02 & \ensuremath{-}0.20 & 0.206 & 0.481 & 0.63 & \ensuremath{-}1.00 & \ensuremath{-}0.64 & \ensuremath{-}0.68 \\ | |
| 12 | +5-year term, level & GE & \ensuremath{-}0.47 & \ensuremath{-}1.03 & \ensuremath{-}0.02 & \ensuremath{-}0.20 & 0.206 & 0.481 & 0.63 & \ensuremath{-}1.00 & \ensuremath{-}0.64 & \ensuremath{-}0.68 \\ | |
| 13 | +ARM, level & GE & \ensuremath{-}0.79 & \ensuremath{-}1.15 & \ensuremath{-}0.11 & \ensuremath{-}0.47 & 0.000 & 0.481 & \ensuremath{-}1.03 & \ensuremath{-}0.96 & \ensuremath{-}1.14 & \ensuremath{-}1.19 \\ | |
| 14 | +FRM, persistent (2 periods) & GE & \ensuremath{-}1.88 & \ensuremath{-}0.38 & \ensuremath{-}0.18 & \ensuremath{-}2.17 & 0.206 & 0.481 & 1.14 & \ensuremath{-}1.06 & \ensuremath{-}0.52 & \ensuremath{-}0.53 \\ | |
| 15 | +5-year term, persistent & GE & \ensuremath{-}1.49 & 0.04 & \ensuremath{-}0.34 & \ensuremath{-}2.03 & 0.206 & 0.481 & 0.46 & \ensuremath{-}1.09 & \ensuremath{-}0.74 & \ensuremath{-}0.73 \\ | |
| 16 | +ARM, persistent & GE & \ensuremath{-}1.86 & \ensuremath{-}0.18 & \ensuremath{-}0.39 & \ensuremath{-}2.34 & 0.000 & 0.481 & \ensuremath{-}1.21 & \ensuremath{-}1.04 & \ensuremath{-}1.24 & \ensuremath{-}1.24 \\ | |
| 17 | +FRM, level, rigid rents ($\eta_\rho=0$) & GE & \ensuremath{-}0.46 & 7.48 & \ensuremath{-}0.03 & \ensuremath{-}2.88 & 0.206 & 0.481 & 0.55 & \ensuremath{-}0.92 & \ensuremath{-}0.62 & \ensuremath{-}0.35 \\ | |
| 18 | +FRM, level, unindexed wages ($\eta_w=0$) & GE$^{\dagger}$ & \ensuremath{-}3.86 & \ensuremath{-}3.21 & \ensuremath{-}0.63 & \ensuremath{-}3.18 & 0.206 & 0.481 & \ensuremath{-}2.40 & \ensuremath{-}3.58 & \ensuremath{-}3.33 & \ensuremath{-}3.45 \\ | |
| 19 | +FRM, persistent, real-rate response & GE & \ensuremath{-}7.25 & \ensuremath{-}2.29 & \ensuremath{-}0.12 & \ensuremath{-}2.64 & 0.206 & 0.481 & 1.43 & \ensuremath{-}0.31 & 0.14 & \ensuremath{-}1.32 \\ | |
| 20 | +5-year term, persistent, real-rate response & GE & \ensuremath{-}6.93 & \ensuremath{-}1.91 & \ensuremath{-}0.26 & \ensuremath{-}2.94 & 0.206 & 0.481 & 0.20 & \ensuremath{-}0.32 & \ensuremath{-}0.25 & \ensuremath{-}1.67 \\ | |
| 21 | +FRM, level, $\varepsilon=0.05$ & GE$^{\dagger}$ & \ensuremath{-}0.50 & \ensuremath{-}1.08 & \ensuremath{-}0.01 & \ensuremath{-}0.19 & 0.206 & 0.481 & 0.63 & \ensuremath{-}0.99 & \ensuremath{-}0.64 & \ensuremath{-}0.68 \\ | |
| 22 | +FRM, level, no segmentation & GE & \ensuremath{-}0.39 & \ensuremath{-}1.47 & \ensuremath{-}0.16 & \ensuremath{-}0.20 & 0.206 & 0.481 & 0.66 & \ensuremath{-}0.95 & \ensuremath{-}0.60 & \ensuremath{-}0.68 \\ | |
| 23 | +FRM, level, static expectations (A6) & A6 & \ensuremath{-}0.36 & \ensuremath{-}1.11 & 0.02 & \ensuremath{-}0.20 & 0.206 & 0.481 & 0.65 & \ensuremath{-}1.01 & \ensuremath{-}0.64 & \ensuremath{-}0.68 \\ | |
| 24 | +\bottomrule | |
| 25 | +\end{tabular}} | |
| 26 | +\begin{minipage}{\textwidth}\footnotesize\vspace{2pt} | |
| 27 | +Perfect-foresight transitions (GE: prices clear every period; $^{\dagger}$ = not converged to the $1.5\times10^{-3}$ per-period tolerance (stalled at the Monte-Carlo noise floor; maximum residual in results/data/inflation/name.json); A6 = static expectations, stationary continuation values). Columns: real house price on impact (\%; the impact is also the peak deviation in every run), real rent on impact (\%), change of the ownership rate on impact (pp) in GE and with prices fixed (PE), aggregate real debt erosion and real deposit loss per unit mass of one age (the lender's revaluation is their difference with the opposite sign), consumption-equivalent welfare of households alive at the shock (owners with a mortgage, renters, all; \%), and the all-household CEV with prices fixed. Level surprise: CPI 10\% above expectation in the shock period, inflation back to target afterwards; persistent: two periods. Source: results/data/inflation/*.json and *\_welfare.csv. | |
| 28 | +\end{minipage} | |
| 29 | +\end{table} | |
added
paper/tables/tab07_mobility_inequality.tex
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| 1 | +\begin{table}[htbp] | |
| 2 | +\centering | |
| 3 | +\scriptsize | |
| 4 | +\caption{Intergenerational persistence and inequality across stationary equilibria} | |
| 5 | +\label{tab:mobility} | |
| 6 | +\resizebox{\textwidth}{!}{% | |
| 7 | +\begin{tabular}{lcccccccccccc} | |
| 8 | +\toprule | |
| 9 | +Scenario & Eq. & $p^*$ & Own. & $q_{RO}$ & $q_{OO}$ & $\lambda_3$ & $\lambda_7$ & rank--rank & Gini & gap$_4$ & gift & estate \\ | |
| 10 | +\midrule | |
| 11 | +Baseline & GE & 1.000 & 0.631 & 0.35 & 0.54 & 0.19 & 0.060 & 0.34 & 0.54 & 0.95 & 0.39 & 0.32 \\ | |
| 12 | +Bequests off & GE & 0.931 & 0.581 & 0.38 & 0.38 & 0.00 & \ensuremath{-}0.003 & \ensuremath{-}0.01 & 0.58 & 0.73 & 0.00 & 0.00 \\ | |
| 13 | +Bequests halved & GE & 0.981 & 0.615 & 0.37 & 0.49 & 0.12 & 0.035 & 0.25 & 0.55 & 0.86 & 0.26 & 0.21 \\ | |
| 14 | +Transfer tax 20\% & GE & 1.005 & 0.642 & 0.39 & 0.57 & 0.18 & 0.054 & 0.28 & 0.53 & 0.96 & 0.28 & 0.22 \\ | |
| 15 | +Transfer tax 50\% & GE & 0.999 & 0.639 & 0.43 & 0.55 & 0.12 & 0.023 & 0.16 & 0.52 & 0.90 & 0.12 & 0.09 \\ | |
| 16 | +Intergenerational income correlation 0.4 & GE & 1.004 & 0.624 & 0.22 & 0.54 & 0.32 & 0.121 & 0.41 & 0.55 & 1.02 & 0.40 & 0.32 \\ | |
| 17 | +Earnings risk $\sigma = 0.10$ & GE & 0.963 & 0.672 & 0.31 & 0.49 & 0.19 & 0.043 & 0.32 & 0.51 & 0.59 & 0.30 & 0.22 \\ | |
| 18 | +Earnings risk $\sigma = 0.20$ & GE & 1.033 & 0.602 & 0.39 & 0.57 & 0.18 & 0.097 & 0.34 & 0.56 & 1.35 & 0.49 & 0.42 \\ | |
| 19 | +$\theta = 0.05$ & GE & 1.005 & 0.640 & 0.38 & 0.57 & 0.20 & 0.069 & 0.33 & 0.55 & 0.84 & 0.39 & 0.31 \\ | |
| 20 | +$\theta = 0.40$ & GE & 0.987 & 0.619 & 0.30 & 0.51 & 0.21 & 0.084 & 0.34 & 0.54 & 0.98 & 0.39 & 0.33 \\ | |
| 21 | +$\varepsilon = 0.05$ & GE & 1.000 & 0.631 & 0.35 & 0.54 & 0.19 & 0.060 & 0.34 & 0.54 & 0.95 & 0.39 & 0.32 \\ | |
| 22 | +$\varepsilon = 2$ & GE & 1.000 & 0.631 & 0.35 & 0.54 & 0.19 & 0.060 & 0.34 & 0.54 & 0.95 & 0.39 & 0.32 \\ | |
| 23 | +Deposit return $+2$ pp (renter investment) & GE & 1.018 & 0.635 & 0.38 & 0.66 & 0.28 & 0.169 & 0.57 & 0.50 & 1.19 & 0.61 & 0.77 \\ | |
| 24 | +Real rate 1\% & GE & 1.042 & 0.601 & 0.31 & 0.48 & 0.17 & 0.063 & 0.26 & 0.55 & 0.90 & 0.36 & 0.24 \\ | |
| 25 | +Real rate 3\% & GE & 0.972 & 0.642 & 0.35 & 0.60 & 0.24 & 0.132 & 0.44 & 0.53 & 1.09 & 0.46 & 0.48 \\ | |
| 26 | +Trend inflation 0\% & GE & 1.011 & 0.640 & 0.35 & 0.54 & 0.19 & 0.058 & 0.33 & 0.55 & 0.91 & 0.39 & 0.30 \\ | |
| 27 | +Trend inflation 4\% & GE & 0.986 & 0.625 & 0.34 & 0.53 & 0.19 & 0.060 & 0.34 & 0.54 & 0.97 & 0.39 & 0.33 \\ | |
| 28 | +Landlord wedge $\phi = 0.02$ & GE & 0.991 & 0.607 & 0.33 & 0.52 & 0.19 & 0.070 & 0.33 & 0.55 & 0.96 & 0.39 & 0.31 \\ | |
| 29 | +Landlord wedge $\phi = 0.08$ & GE & 1.007 & 0.650 & 0.36 & 0.55 & 0.19 & 0.055 & 0.34 & 0.54 & 0.94 & 0.39 & 0.32 \\ | |
| 30 | +\bottomrule | |
| 31 | +\end{tabular}} | |
| 32 | +\begin{minipage}{\textwidth}\footnotesize\vspace{2pt} | |
| 33 | +$q_{RO}$, $q_{OO}$: probability that the child owns at 35--39 given the parent's tenure when the gift is made; $\lambda_3 = q_{OO} - q_{RO}$; $\lambda_7$: the same difference measured at 55--59 (the Part A persistence statistic); rank--rank: slope of the child's net-worth rank on the parent's at the same age (55--59); gap$_4$: mean net-worth difference between owners and renters at 40--44 (equal income process by design); gift, estate: mean transfers received. $^{\dagger}$ = not converged to tolerance. Source: results/data/group/name.json. | |
| 34 | +\end{minipage} | |
| 35 | +\end{table} | |
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paper/tables/tab08_welfare.tex
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| 1 | +\begin{table}[htbp] | |
| 2 | +\centering | |
| 3 | +\footnotesize | |
| 4 | +\caption{Welfare of entering households across stationary equilibria (consumption-equivalent variation, \%)} | |
| 5 | +\label{tab:welfare} | |
| 6 | +\resizebox{\textwidth}{!}{% | |
| 7 | +\begin{tabular}{lccccccccc} | |
| 8 | +\toprule | |
| 9 | +Scenario & Q1 & Q2 & Q3 & Q4 & Q5 & marginal & low $z_1$ & high $z_1$ & all \\ | |
| 10 | +\midrule | |
| 11 | +$\theta = 0.05$ & 0.55 & 0.54 & 0.43 & 0.26 & \ensuremath{-}0.08 & 0.30 & 0.54 & 0.20 & 0.34 \\ | |
| 12 | +$\theta = 0.10$ & 0.41 & 0.40 & 0.33 & 0.19 & \ensuremath{-}0.05 & 0.24 & 0.41 & 0.14 & 0.26 \\ | |
| 13 | +$\theta = 0.30$ & \ensuremath{-}0.20 & \ensuremath{-}0.21 & \ensuremath{-}0.22 & \ensuremath{-}0.07 & 0.21 & \ensuremath{-}0.13 & \ensuremath{-}0.23 & 0.08 & \ensuremath{-}0.10 \\ | |
| 14 | +$\theta = 0.40$ & \ensuremath{-}0.39 & \ensuremath{-}0.40 & \ensuremath{-}0.44 & \ensuremath{-}0.25 & 0.32 & \ensuremath{-}0.33 & \ensuremath{-}0.44 & 0.07 & \ensuremath{-}0.23 \\ | |
| 15 | +DSTI cap $\psi = 0.30$ & \ensuremath{-}0.01 & 0.00 & 0.01 & \ensuremath{-}0.02 & 0.09 & \ensuremath{-}0.08 & \ensuremath{-}0.08 & 0.21 & 0.02 \\ | |
| 16 | +DSTI cap $\psi = 0.50$ & \ensuremath{-}0.01 & \ensuremath{-}0.02 & \ensuremath{-}0.05 & \ensuremath{-}0.06 & \ensuremath{-}0.12 & \ensuremath{-}0.01 & 0.03 & \ensuremath{-}0.16 & \ensuremath{-}0.05 \\ | |
| 17 | +Bequests off ($\nu = 0$)$^{\ddagger}$ & 8.87 & 9.09 & 10.11 & 12.08 & 15.53 & 12.82 & 4.41 & 20.16 & 11.14 \\ | |
| 18 | +Transfer tax 20\% (rebated) & 10.85 & 9.64 & 7.20 & 5.64 & 4.02 & 5.74 & 16.92 & 2.58 & 7.47 \\ | |
| 19 | +Transfer tax 50\% (rebated) & 15.39 & 13.79 & 10.68 & 8.52 & 5.89 & 8.58 & 24.44 & 2.87 & 10.85 \\ | |
| 20 | +$\varepsilon = 0.05$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ | |
| 21 | +$\varepsilon = 2$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ | |
| 22 | +No segmentation & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ | |
| 23 | +Real rate 1\% & 0.68 & 0.51 & \ensuremath{-}0.03 & \ensuremath{-}0.68 & \ensuremath{-}2.25 & \ensuremath{-}0.86 & \ensuremath{-}0.31 & \ensuremath{-}1.18 & \ensuremath{-}0.35 \\ | |
| 24 | +Real rate 3\% & 0.42 & 0.62 & 1.25 & 2.23 & 4.46 & 2.32 & 1.46 & 3.06 & 1.80 \\ | |
| 25 | +Trend inflation 4\% & \ensuremath{-}0.11 & \ensuremath{-}0.11 & \ensuremath{-}0.08 & \ensuremath{-}0.01 & 0.23 & \ensuremath{-}0.08 & \ensuremath{-}0.16 & 0.21 & \ensuremath{-}0.01 \\ | |
| 26 | +Deposit return $+2$ pp & 5.23 & 5.69 & 7.34 & 9.97 & 15.40 & 10.27 & 7.76 & 11.92 & 8.73 \\ | |
| 27 | +Ownership premium $\chi = 0.05$$^{\ddagger}$ & 0.81 & 0.82 & 0.86 & 0.95 & 1.08 & 0.94 & 0.77 & 1.09 & 0.91 \\ | |
| 28 | +Transaction costs low & \ensuremath{-}0.22 & \ensuremath{-}0.22 & \ensuremath{-}0.21 & \ensuremath{-}0.14 & \ensuremath{-}0.17 & \ensuremath{-}0.14 & \ensuremath{-}0.29 & \ensuremath{-}0.08 & \ensuremath{-}0.19 \\ | |
| 29 | +Transaction costs high & \ensuremath{-}0.07 & \ensuremath{-}0.08 & \ensuremath{-}0.11 & \ensuremath{-}0.16 & \ensuremath{-}0.10 & \ensuremath{-}0.18 & \ensuremath{-}0.03 & \ensuremath{-}0.12 & \ensuremath{-}0.10 \\ | |
| 30 | +\bottomrule | |
| 31 | +\end{tabular}} | |
| 32 | +\begin{minipage}{\textwidth}\footnotesize\vspace{2pt} | |
| 33 | +$\lambda^{CE}$ such that scaling non-housing consumption by $1+\lambda^{CE}$ in every period of the baseline makes an age-1 household with state $(z_1, a_1)$ as well off as in the alternative stationary equilibrium: $V^{alt}_1 = (1+\lambda^{CE})^{(1-\alpha)(1-\gamma)} U^{base} + \Phi^{base}$, where $U^{base}$ (discounted flow utility) and $\Phi^{base}$ (warm glow) come from cohort simulations under baseline policies. Columns: quintiles of the transfer received at entry (averaged over $z_1$ with stationary weights), households within $\pm10\%$ of the age-1 ownership threshold (``marginal''), lowest and highest $z_1$, all. Comparisons are between stationary equilibria, not along transitions. $^{\ddagger}$ = the scenario changes a preference parameter, so the number compares value functions under different utilities and is not a welfare statement. Source: results/data/welfare/welfare\_all.csv. | |
| 34 | +\end{minipage} | |
| 35 | +\end{table} | |
added
paper/tables/tab09_robustness.tex
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| 1 | +\begin{table}[htbp] | |
| 2 | +\centering | |
| 3 | +\scriptsize | |
| 4 | +\caption{Robustness: preferences, transaction costs, grids, tolerance, sample size and seed} | |
| 5 | +\label{tab:robustness} | |
| 6 | +\resizebox{\textwidth}{!}{% | |
| 7 | +\begin{tabular}{lccccccccccc} | |
| 8 | +\toprule | |
| 9 | +Variant & Eq. & $p^*$ & Own. & Own.\ $<$35 & $\hat a_1$ & Gini & gift & $q_{RO}$ & $q_{OO}$ & rank--rank & min \\ | |
| 10 | +\midrule | |
| 11 | +baseline & GE & 1.000 & 0.631 & 0.378 & 0.49 & 0.542 & 0.388 & 0.351 & 0.538 & 0.335 & 0.1 \\ | |
| 12 | +beta\_094 & GE & 0.933 & 0.581 & 0.329 & 0.46 & 0.576 & 0.355 & 0.316 & 0.496 & 0.225 & 2.7 \\ | |
| 13 | +beta\_098 & GE & 1.075 & 0.681 & 0.408 & 0.66 & 0.498 & 0.435 & 0.404 & 0.597 & 0.443 & 3.2 \\ | |
| 14 | +gamma\_3 & GE & 0.995 & 0.646 & 0.373 & 0.47 & 0.527 & 0.276 & 0.375 & 0.532 & 0.283 & 1.8 \\ | |
| 15 | +alpha\_025 & GE & 1.101 & 0.627 & 0.359 & 0.54 & 0.536 & 0.412 & 0.289 & 0.508 & 0.362 & 2.4 \\ | |
| 16 | +kappa\_low & GE & 1.044 & 0.622 & 0.374 & 0.50 & 0.543 & 0.378 & 0.350 & 0.524 & 0.324 & 1.6 \\ | |
| 17 | +kappa\_high & GE & 0.962 & 0.633 & 0.366 & 0.52 & 0.544 & 0.388 & 0.335 & 0.539 & 0.339 & 1.4 \\ | |
| 18 | +chi\_005 & GE & 1.006 & 0.644 & 0.403 & 0.48 & 0.537 & 0.396 & 0.355 & 0.546 & 0.339 & 1.0 \\ | |
| 19 | +sigma\_low & GE & 0.963 & 0.672 & 0.318 & 0.41 & 0.507 & 0.304 & 0.307 & 0.492 & 0.321 & 4.0 \\ | |
| 20 | +sigma\_high & GE & 1.033 & 0.602 & 0.416 & 0.65 & 0.565 & 0.490 & 0.391 & 0.570 & 0.344 & 1.9 \\ | |
| 21 | +grid\_coarse & GE & 0.992 & 0.626 & 0.369 & 0.56 & 0.542 & 0.388 & 0.350 & 0.542 & 0.341 & 0.6 \\ | |
| 22 | +grid\_fine & GE & 0.999 & 0.635 & 0.382 & 0.47 & 0.543 & 0.388 & 0.340 & 0.535 & 0.334 & 1.5 \\ | |
| 23 | +tol\_tight & GE & 1.000 & 0.631 & 0.378 & 0.49 & 0.542 & 0.388 & 0.351 & 0.538 & 0.335 & 0.2 \\ | |
| 24 | +sample\_small & GE$^{\dagger}$ & 1.001 & 0.632 & 0.379 & 0.50 & 0.542 & 0.390 & 0.340 & 0.542 & 0.335 & 9.9 \\ | |
| 25 | +sample\_large & GE & 1.000 & 0.631 & 0.379 & 0.49 & 0.543 & 0.388 & 0.349 & 0.539 & 0.333 & 0.6 \\ | |
| 26 | +seed\_alt & GE & 1.000 & 0.631 & 0.378 & 0.49 & 0.543 & 0.386 & 0.347 & 0.536 & 0.336 & 0.3 \\ | |
| 27 | +\bottomrule | |
| 28 | +\end{tabular}} | |
| 29 | +\begin{minipage}{\textwidth}\footnotesize\vspace{2pt} | |
| 30 | +Each row is a full re-solution of the stationary general equilibrium (prices, lottery). beta\_094/098: annual discount factor 0.94/0.98; gamma\_3: $\gamma=3$; alpha\_025: housing share 0.25; kappa\_low/high: transaction costs (1\%, 2\%) / (5\%, 8\%); chi\_005: ownership utility premium 5\%; grid\_coarse/fine, tol\_tight, sample\_small/large, seed\_alt as in Table~\ref{tab:convergence}. Source: results/data/robustness\_summary.csv. | |
| 31 | +\end{minipage} | |
| 32 | +\end{table} | |
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research/failed_hypotheses.md
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@@ -35,3 +35,30 @@ Recorded 2026-10-20. Each entry gives the hypothesis as it appeared in the brief | ||
| 35 | 35 | |
| 36 | 36 | ## 9. Two-period version — abandoned in favour of three stages |
| 37 | 37 | A two-period lifecycle (buy, then sell) makes the whole mortgage a balloon repaid with compound interest at the end of life, so the house is net wealth only if $\theta > 1 - (1-\kappa_s)/R$, which fails for generation-length periods; persistence through housing equity then has the wrong sign. The three-stage structure with amortisation at stage 2 restores $e_3 > 0$ for reasonable parameters and is what the propositions use. |
| 38 | + | |
| 39 | +--- | |
| 40 | + | |
| 41 | +# Phase 4 additions (2026-10-20): hypotheses tested against the quantitative model | |
| 42 | + | |
| 43 | +## 10. "Surprise inflation raises the real house price in general equilibrium" (Proposition 9) — reversed in the simulation | |
| 44 | +*Hypothesis.* Under segmentation, the erosion of the parents' mortgages raises transfers, the demand of owner-dynasty children and the price (Prop. 9, $dp \ge 0$). | |
| 45 | +*Finding.* In the quantitative model the real price **falls** after a 10 % surprise ($-0.45$ % level FRM, $-1.88$ % persistent) because households are net nominal creditors (deposits 4.12 > debt 2.47 per unit age mass): the deposit loss exceeds the debt erosion, the lender gains, household wealth and housing demand fall. | |
| 46 | +*Diagnosis.* Prop. 9 relied on an unstated premise — the mortgage is the only nominal position — implied by A4 on the hand-to-mouth region but false in Part B. | |
| 47 | +*Replacement.* Assumption 7 (nominal positions) made explicit; Prop. 9 restated under it; **Corollary 2**: the sign of $dp$ is the sign of $\Xi = \mu^* g(\hat y_O)(\delta_3 p^* - Rn^O) - (1-\mu^*) g(\hat y_R) Rn^R$. The simulated fall is the $\Xi < 0$ case. Not reconciled by the corollary and reported as results: the rent falls too (intensive margin), and the entry threshold rises (entrants' own deposits are eroded). | |
| 48 | + | |
| 49 | +## 11. "Price neutrality without segmentation" (Proposition 7) — fails with an intensive margin | |
| 50 | +*Finding.* Unsegmented economy: $p_0 = -0.39$ % vs $-0.45$ % segmented; the wealth loss lowers total housing demand (sizes, rented services), so the single-stock price moves almost as much. | |
| 51 | +*Replacement.* Prop. 7 restated with **unit demand** in the statement (Remark 2); the quantitative price feedback is mostly a wealth effect on total demand, segmentation adds ≈ 15 %. | |
| 52 | + | |
| 53 | +## 12. "The entry threshold is increasing in the down payment" (Proposition 2) — not monotone once households can buy later | |
| 54 | +*Finding.* At fixed prices the median-state threshold is 0.52 / 0.49 / 0.59 for $\theta$ = 0.05 / 0.20 / 0.40 (grid sensitivity ±0.07): the option of buying later with a small down payment raises the renter's continuation value; at the lowest state the DSTI cap binds and $\theta$ is irrelevant. | |
| 55 | +*Replacement.* Prop. 2 restated with the binding-LTV premise and the once-and-for-all tenure decision ($V^R$ independent of $\theta$) named in the statement (Remark 1). | |
| 56 | + | |
| 57 | +## 13. "Multiple stationary equilibria under near-fixed supply" — none found | |
| 58 | +Three price scans (baseline, $\varepsilon = 0.05$, $\varepsilon = 0.05 \times \theta = 0.05$): excess demand monotone, one root each. The conjecture stays open in general (it requires (C) to fail) and is closed for this calibration. | |
| 59 | + | |
| 60 | +## 14. "Supply elasticity has a first-order effect on intergenerational mobility" (contribution iv) — not measurable | |
| 61 | +$q_{RO}$ at 35–39 is 0.374 vs 0.376 ($\theta = 0.05$) and 0.308 vs 0.298 ($\theta = 0.40$) across $\varepsilon \in \{0.05, 2\}$: the demand shift from $\theta$ is too small for the elasticity to matter; on impact the relevant elasticity is $\varepsilon\delta_H$, bounded by the depreciation rate. | |
| 62 | + | |
| 63 | +## 15. Two calibration moments not supported by the comparison points (not hypotheses, but recorded) | |
| 64 | +Ownership falls after 65 (0.36 at 80–84 vs a Canadian plateau near 0.75) because the warm glow is over a liquid estate; 72 % of first-time buyers receive a gift vs 28 % in the Canadian source because every parent shares the same motive. | |
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research/progress.md
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@@ -66,3 +66,27 @@ | ||
| 66 | 66 | - `rate_shock_theta005` relancé deux fois (départ à chaud puis tâtonnement 80 pas + Jacobien) : le tir s'arrête au résidu 8,8e-3 (étiqueté `GE NOT CONVERGED`), p₀ = −16,2 % cohérent avec θ = 0,20 (−15,5 %) et θ = 0,40 (−14,6 %). Bilan transitions : 12/16 convergées à 1,5e-3, 3 à 2–3e-3, 1 à 8,8e-3 — toutes étiquetées avec leur résidu dans `diagnostics.csv` et la Table 6. |
| 67 | 67 | - Livrables finaux : 14 figures (PDF + PNG), 9 tables LaTeX + `numbers.tex` (164 macros), `results/data/` (358 fichiers CSV/JSON commités, 5,6 Mo ; `.npz`/`.pkl`/journaux exclus), `research/results_audit.md` (brouillon par expérience avec étiquettes identité / théorie / simulation GE / PE), 69 tests verts, `make simulate` reproductible (60 min). |
| 68 | 68 | - Temps de calcul (18 cœurs) : base 8 s ; ERC stationnaire 45–300 s ; transition 2,5–8 min ; run complet 59,8 min + relances 13 min. |
| 69 | + | |
| 70 | +## 2026-10-20 — Phase 4 (audit final, réconciliation théorie–simulation, manuscrit v1.0) : FAIT | |
| 71 | + | |
| 72 | +### A. Audit final — `research/results_audit.md` (version finale) | |
| 73 | +- Par expérience (1–10) : hypothèse, paramètres, algorithme, convergence (résidus réels ; non convergés nommés : `frm_level` 2,2e-3, `frm_level_unindexed_wages` 2,0e-3, `frm_level_fixed_supply` 3,1e-3, `rate_shock_theta005` 8,8e-3 ; `sample_small` 7,8e-4), résultats, limites, robustesse, et étiquette de ce qui est réellement soutenu : **identité / théorie / simulation GE / simulation PE (prix fixes) / A6 / conjecture**. Décompte exact : 45 ERC stationnaires (44 dans la tolérance), 5 relances PE, 3 balayages de multiplicité, 16 transitions (12 dans la tolérance). | |
| 74 | +- **Les trois désaccords théorie–simulation classés et réconciliés** (tableau dédié) : (1) Prop. 9 — la prémisse « hypothèque = seule position nominale » n'était pas énoncée (impliquée par A4) ; rendue explicite (**Assumption 7**), Prop. 9 restreinte, **Corollary 2** prouvé : signe de $dp$ = signe de $\Xi = \mu^* g(\hat y_O)(\delta_3 p^* - Rn^O) - (1-\mu^*) g(\hat y_R) Rn^R$ ; la baisse simulée est le cas $\Xi<0$ ; les deux écarts résiduels (loyer qui baisse au lieu de monter : marge intensive ; seuil qui monte : dépôts des entrants) nommés dans Remark 3 et rapportés comme résultats. (2) Prop. 7 — « demande unitaire » mise dans l'énoncé (Remark 2) ; résultat Part B rapporté (−0,39 % vs −0,45 %, effet de richesse sur la demande totale). (3) Prop. 2 — restriction exacte nommée : contrainte LTV serrée + décision de tenure une fois pour toutes ($V^R$ indépendant de $\theta$) (Remark 1) ; non-monotonie au $z$ médian rapportée avec sa sensibilité de grille ±0,07. Ni la théorie ni les simulations n'ont été forcées. | |
| 75 | +- `research/propositions.md` réécrit (statuts P / P-scoped / C / N + colonne « Sim ») ; `research/failed_hypotheses.md` complété (§10–15 : signe de $dp$, neutralité, monotonie en $\theta$, multiplicité non trouvée, élasticité de l'offre sans effet mesurable sur la mobilité, moments non soutenus). | |
| 76 | +- Preuves : `paper/appendix_proofs.tex` réorganisé (énoncés dans `sections/equilibrium.tex`, lemmes + preuves + remarques de lacune en annexe A ; preuve commune Prop. 9 / Cor. 2) ; `appendix_proofs_standalone.tex` = modèle + énoncés + preuves, compile (22 p., 0 erreur). | |
| 77 | + | |
| 78 | +### B. Porte d'intégrité | |
| 79 | +- `make test` : 69 tests verts (21 s). Figures/tables référencées : toutes présentes (`make paper` copie `results/figures/*.pdf` → `paper/figures/`, `results/tables/*.tex` → `paper/tables/`). Chaque chiffre du texte est une macro de `numbers.tex` (≈ 2 500 macros, dont les jumeaux `…Abs` pour les valeurs négatives et le signe moins `\ensuremath{-}`) ; vérification par script : 0 macro utilisée non définie. Bibliographie : 72 entrées, 0 avertissement BibTeX. | |
| 80 | + | |
| 81 | +### C. Manuscrit — `paper/main.tex` v1.0 | |
| 82 | +- Métadonnées : WP 18, titre + sous-titre, October 2026, Version 1.0, résumé 217 mots avec résultats réels (dont négatifs), 7 mots-clés, JEL R21, R31, D31, D91, E31, G51 (E44 écarté : pas de friction de marché financier au-delà des contraintes des ménages). `\hypersetup{pdftitle,pdfauthor}` ; amsthm selon `wp_style_audit.md` §6. | |
| 83 | +- Sections (`paper/sections/`) : introduction, literature (critique, par mécanismes, 72 réf. seulement), model (Part A + Part B), equilibrium (A1–A7, Prop. 1–9, Cor. 1–2, Remarks 1–3, concepts d'équilibre Part B), computation, baseline, inflation, supply_policy_welfare, sensitivity_conclusion ; annexes A (preuves), B (algorithme, paramètres, diagnostics, robustesse), C (notation). Sections du gabarit inutilisées supprimées. | |
| 84 | +- Figures : 13 dans le texte + 1 en annexe (titres dans l'image retirés, légendes LaTeX) ; tables : 5 dans le texte + 4 en annexe (tables larges en `\resizebox` + note en minipage ; colonne « p peak » redondante retirée de la table d'inflation). | |
| 85 | + | |
| 86 | +### D. QA | |
| 87 | +- `cd paper && make` : 0 `!`, 0 référence/citation indéfinie, 0 avertissement BibTeX, overfull max 3,7 pt (< 10 pt). PDF : 65 pages = 2 (titre, résumé) + **45 pages de texte principal** (p. 3–47) + références (6) + annexes (12). Page 1 et page 2 vérifiées avec python3 -I + fitz (Working Paper No. 18, titre, auteur, October 2026, Version 1.0 ; Abstract / Keywords / JEL). Relecture intégrale du texte extrait : cohérence chiffres/tables, formulation des nombres négatifs corrigée (macros `…Abs`), légende Fig. 7 alignée sur la figure, libellés des lignes de la table de bien-être. | |
| 88 | +- `make all` : test → simulate (sauté, résultats présents) → figures → tables → paper, EXIT 0. `make clean` garde `main.pdf`. | |
| 89 | +- README.md final (gabarit UQO : Status v1.0 not yet submitted, Quick start, Data = calibration illustrative, Notes = dérogations). | |
| 90 | + | |
| 91 | +### Verdict de maturité | |
| 92 | +Papier théorique complet et reproductible, résultats honnêtement étiquetés ; calibration illustrative et deux moments non soutenus (profil de propriété après 65 ans, part des dons) ; cible réaliste : *Journal of Housing Economics* ou *Regional Science and Urban Economics* (ou *Canadian Journal of Economics* après une calibration des positions nominales nettes par âge et tenure à la Meh–Terajima et un motif de legs sur le logement — prochaines étapes nommées dans la conclusion). | |
modified
research/propositions.md
+44 −50
@@ -1,71 +1,65 @@ | ||
| 1 | −# WP18 — Propositions: statements, status, intuition, and what the simulations must check | |
| 1 | +# WP18 — Propositions: statements, status, intuition, and what the simulations found | |
| 2 | 2 | |
| 3 | −Phase 2 deliverable (2026-10-20). Formal statements and proofs: `paper/appendix_proofs.tex` (compiles stand-alone via `paper/appendix_proofs_standalone.tex`, 12 pp., no undefined references). Model: `research/model_specification.md`. Numerical verification of every identity and sign: `src/toy_model.py` + `tests/test_toy_model.py` (44 tests, all passing). Illustrative benchmark numbers below come from `Params()` in `src/toy_model.py` and carry no calibration claim. | |
| 3 | +Phase 2 deliverable, **revised in Phase 4 (2026-10-20)** after the reconciliation with the numerical results of Phase 3 (`research/results_audit.md`). Formal statements: `paper/sections/equilibrium.tex` (Section 4 of the paper: Assumptions 1–7, Propositions 1–9, Corollaries 1–2, scope Remarks 1–3); proofs: `paper/appendix_proofs.tex` (Lemmas 1–2, proofs, Remarks 4–5); the theory compiles stand-alone via `paper/appendix_proofs_standalone.tex` (model + statements + proofs). Model: `research/model_specification.md`. Numerical verification of every identity and sign in the benchmark: `src/toy_model.py` + `tests/test_toy_model.py` (44 tests). Illustrative benchmark numbers below come from `Params()` in `src/toy_model.py` and carry no calibration claim. | |
| 4 | 4 | |
| 5 | −Status codes: **P** = proved under the stated assumptions; **P-strong** = proved but under a strong/knife-edge assumption that the numerical model must relax; **C** = conjecture or partial result with an explicit proof gap; **N** = negative result (hypothesis from the brief that does *not* hold in general). | |
| 5 | +Status codes: **P** = proved under the stated assumptions; **P-scoped** = proved, with a premise that the quantitative model violates, now named in the statement; **C** = conjecture or partial result with an explicit proof gap; **N** = negative result (hypothesis from the brief that does *not* hold in general); **Sim** = what the Part B simulations found (`results_audit.md`). | |
| 6 | 6 | |
| 7 | −| # | Result | Status | Key assumptions | | |
| 8 | −|---|---|---|---| | |
| 9 | −| L1 | Stage-3 split: $c_3 = W_3/(1+\nu)$, $b' = \nu W_3/(1+\nu)$ | P | log warm glow | | |
| 10 | −| L2 | HTM corner (max LTV, no deposits) is optimal; closed-form values | P | A4 strict Euler inequalities (verified numerically, not assumed, in the benchmark) | | |
| 11 | −| 1 | Ownership threshold $\hat b = (dp-\Lambda\rho)/(1-\Lambda) - y_1$, unique, above feasibility bound | P on the HTM region; **C** outside it (single crossing not proved for savers) | A1–A5 | | |
| 12 | −| 2 | Signs: $\hat b$ ↑ in $p,\theta,\kappa_b,\kappa_s,m,\bar\pi$; ↓ in $\rho,\chi,y_1$ | P | A4, $r \ge 0$ | | |
| 13 | −| 3 | Anticipated inflation: PV-neutral tilt; raises $\hat b$ for constrained households; neutral (with $\theta$) for unconstrained | P | A4 for (i); slack constraints for (ii) | | |
| 14 | −| 4 | Leverage multiplier $(1-\kappa_s)/d$; break-even growth $g^*$ independent of $\theta$, increasing in costs; underwater threshold $x^*$ | P | accounting only | | |
| 15 | −| 5 | Dynasty chain: $q_{RO}, q_{OO}$ closed form; persistence $\lambda = G(\hat y_R) - G(\hat y_R - e_3p)$ = parent–child tenure correlation; half-life; $\lambda > 0$ iff $e_3 > 0$; equal-income transfer gap $\nu e_3 p/(1+\nu)$ | P | HTM on the transfer support; i.i.d. $y_3$ | | |
| 16 | −| 6 | Stationary GE exists; unique under dominance condition (C) | P (existence); P under (C) (uniqueness) | A1–A5 along the clearing curve | | |
| 17 | −| 6c | Fixed stock: ownership rate pinned by supply; LTV/transfers only move price and composition | P | (C) | | |
| 18 | −| 7 | No segmentation ⇒ price and rent independent of credit, transfers, inflation (KMV neutrality) | P | unit demand, costless conversion | | |
| 19 | −| 8 | Surprise accounting: erosion factors by contract (FRM = ARM for a level surprise; FRM > ARM for a persistent one; ARM loses at stage 3 iff $\phi_r > R$); zero-sum with lender; owner gains iff explicit condition (wage indexation, deposits) | P | fixed real prices | | |
| 20 | −| 9 | GE impact of a level surprise: $dp \ge 0$, $d\rho \le 0$, $d\mu \ge 0$ (= 0 if fixed stock: pure composition), $d\hat b \ge 0$, $q_{RO}$ ↓, $q_{OO}$ ↑, $\lambda$ ↑; sellers gain; new buyers in inelastic markets can lose | **P-strong** | A6 static expectations; per-period supply; condition (C′) | | |
| 21 | −| — | Multiple stationary equilibria (Stein-type ladder multiplicity) | **C** — not established; possible only if (C) fails | — | | |
| 22 | −| — | Full perfect-foresight transition after a surprise | **C** — numerical only | — | | |
| 23 | −| — | "Inflation raises real house prices" | **N** in general: sign tied to $g(\hat y_O)$, segmentation, contract and supply; zero without segmentation | — | | |
| 24 | −| — | "Inflation enriches every owner" | **N**: ARM with $\phi_r > R$, unindexed wages, nominal savers, and new buyers in inelastic markets lose | — | | |
| 25 | −| — | Persistence via absorbing tenure classes / path dependence | **N** as a generic result: knife-edge of no income risk (Remark in appendix); replaced by half-life | — | | |
| 7 | +| # | Result | Status | Key assumptions | Sim (Part B) | | |
| 8 | +|---|---|---|---|---| | |
| 9 | +| L1 | Stage-3 split: $c_3 = W_3/(1+\nu)$, $b' = \nu W_3/(1+\nu)$ | P | log warm glow | — | | |
| 10 | +| L2 | HTM corner (max LTV, no deposits) is optimal; closed-form values | P | A4 strict Euler inequalities (verified numerically in the benchmark) | A4 holds for the marginal **buyer**, fails for the marginal **renter** (saves towards a later purchase) | | |
| 11 | +| 1 | Ownership threshold $\hat b = (dp-\Lambda\rho)/(1-\Lambda) - y_1$, unique, above feasibility bound | P on the HTM region; **C** outside it (single crossing not proved for savers, Remark 4) | A1–A5 | single crossing holds at every age 1–3 and every $z$ | | |
| 12 | +| 2 | Signs: $\hat b$ ↑ in $p,\theta,\kappa_b,\kappa_s,m,\bar\pi$; ↓ in $\rho,\chi,y_1$ | **P-scoped**: binding LTV constraint + once-and-for-all tenure decision ($V^R$ independent of $\theta,\kappa_s,m,\bar\pi$) — now in the statement; Remark 1 | A4, $r \ge 0$ | **non-monotone in $\theta$ at the median $z$** at fixed prices (0.52 / 0.49 / 0.59 for $\theta$ = 0.05 / 0.20 / 0.40; within grid noise ±0.07; option value of buying later); flat at the lowest $z$ (DSTI cap binds); monotone decreasing at the highest $z$ | | |
| 13 | +| 3 | Anticipated inflation: PV-neutral tilt; raises $\hat b$ for constrained households; neutral (with $\theta$) for unconstrained | P | A4 for (i); slack constraints for (ii) | supported in PE (4 % trend inflation: under-35 ownership −4.9 pp through the nominal DSTI cap), 65 % offset in GE | | |
| 14 | +| 4 | Leverage multiplier $(1-\kappa_s)/d$; break-even growth $g^*$ independent of $\theta$; underwater threshold $x^*$ | P | accounting only | identity confirmed; nobody underwater at the 5-year frequency (peak LTV understated); new loans 40–48 % underwater at −20/−30 % | | |
| 15 | +| 5 | Dynasty chain: $q_{RO}, q_{OO}$ closed form; persistence $\lambda = G(\hat y_R) - G(\hat y_R - e_3p)$; half-life; $\lambda > 0$ iff $e_3 > 0$; equal-income transfer gap | P | HTM on the transfer support; i.i.d. $y_3$ | $\lambda_7 = 0.060$, $\lambda_3 = 0.19$, rank–rank 0.34 in the baseline; **all zero without transfers** | | |
| 16 | +| 6 | Stationary GE exists; unique under dominance condition (C) | P (existence); P under (C) | A1–A5 along the clearing curve | excess demand monotone with one root in every scan (incl. near-fixed supply) | | |
| 17 | +| 6c | Fixed stock: ownership rate pinned by supply; LTV/transfers only move price and composition | P | (C) | GE removes 64–66 % (stationary $\theta$ sweep) and 77–92 % (transitions) of the PE ownership effect; impact elasticity $\varepsilon\delta_H$ | | |
| 18 | +| 7 | No segmentation ⇒ price and rent independent of credit, transfers, inflation | **P-scoped**: **unit demand** now in the statement; Remark 2 | unit demand, costless conversion | **fails with an intensive margin**: $p_0 = -0.39$ % unsegmented vs $-0.45$ % segmented (wealth effect on total demand) | | |
| 19 | +| 8 | Surprise accounting: erosion by contract (FRM = ARM level; FRM > ARM persistent; ARM loses at stage 3 iff $\phi_r > R$); zero-sum; owner gains iff explicit condition | P | fixed real prices | identity holds to $10^{-15}$; ordering FRM = term5 > ARM (level), FRM > term5 > ARM (persistent) confirmed; term penalised under a real-rate response | | |
| 20 | +| 9 | GE impact of a level surprise: $dp \ge 0$, $d\rho \le 0$, $d\mu \ge 0$, $d\hat b \ge 0$, $q_{RO}$ ↓, $\lambda$ ↑ | **P-scoped**: **Assumption 7** (mortgage = only nominal position; parents and entrants hold no deposits) now explicit; A6; per-period supply; (C′) | A1–A7 | **sign of $dp$ reversed** ($-0.45$ %): households are net nominal creditors (deposits 4.12 > debt 2.47) | | |
| 21 | +| 9′ (new) | **Corollary 2 (net nominal creditors)**: with predetermined parental deposits $n^O, n^R$, sign$(dp)$ = sign$(d\mu)$ = −sign$(d\rho)$ = sign$(\Xi)$, $\Xi = \mu^* g(\hat y_O)(\delta_3 p^* - Rn^O) - (1-\mu^*) g(\hat y_R) Rn^R$; $q_{RO}$ unsigned unless $n^R = 0$ | P | as Prop. 9 without A7 | the simulated fall is the $\Xi < 0$ case; two residual differences (rent falls instead of rising: intensive margin; threshold rises: entrants' own deposits) are named in Remark 3 and reported as results | | |
| 22 | +| — | Multiple stationary equilibria (Stein-type ladder multiplicity) | **C** — not established; possible only if (C) fails | — | **none found** (three scans, incl. near-fixed supply) | | |
| 23 | +| — | Full perfect-foresight transition after a surprise | **C** — numerical only | — | 12/16 transitions converged; A6 vs perfect foresight immaterial for size and sign | | |
| 24 | +| — | "Inflation raises real house prices" | **N** in general: sign tied to the net nominal position of the parents at the margin, segmentation, contract and supply; zero without segmentation under unit demand | — | negative in every run | | |
| 25 | +| — | "Inflation enriches every owner" | **N**: ARM with $\phi_r > R$, unindexed wages, nominal savers, new buyers in inelastic markets lose | — | debt-free owners lose 2.1 %; mortgaged owners gain 0.6–1.4 % | | |
| 26 | +| — | Persistence via absorbing tenure classes / path dependence | **N** as a generic result: knife-edge of no income risk (Remark 5); replaced by half-life | — | half-life 0.25 generations | | |
| 27 | +| — | Supply elasticity has a first-order effect on mobility (contribution iv) | — | — | **N** in this calibration: $q_{RO}$ 0.374 vs 0.376 across $\varepsilon$ | | |
| 26 | 28 | |
| 27 | 29 | ## Proposition-by-proposition notes |
| 28 | 30 | |
| 29 | 31 | ### Proposition 1 — Ownership-entry threshold |
| 30 | −*Statement.* Own iff $b \ge \hat b = (dp - \Lambda\rho)/(1-\Lambda) - y_1$, with $\Lambda = e^{-\Delta}$ and $\Delta$ the utility premium of ownership after the purchase (stage-2 cash-flow comparison plus stage-3 housing equity). The threshold exceeds the feasibility bound $dp - y_1$ by $\Lambda(dp-\rho)/(1-\Lambda)$ = the stage-1 consumption of the marginal buyer. | |
| 31 | −*Intuition.* A liquidity-constrained household compares the certain cut in current consumption needed for the down payment and costs with the discounted future advantage of owning; only households rich enough that the marginal utility of current consumption is low accept the cut. Bequests enter one-for-one; price enters through the outlay and through the equity carried forward. | |
| 32 | −*Benchmark.* $\hat b = 0.999$ (units of stage-1 income $y_1 = 1$) at $p = 1.203$, $\rho = 0.348$; feasibility bound $0.456 - 1 < 0$; $\Delta = 0.069$. The closed form equals the threshold obtained by brute-force optimisation to $10^{-14}$. | |
| 33 | −*Simulation must check.* In the full model the threshold is the smallest $a_1$ at which the age-1 policy is "buy" for each $z_1$; verify it is increasing in $p$, $\theta$, costs and decreasing in $\rho$ at *equilibrium* prices (Proposition 2 is partial equilibrium in $\rho$), and whether the ordering survives when the household can save first and buy at age 2–3 (Remark: proof gap outside HTM). | |
| 32 | +*Statement.* Own iff $b \ge \hat b = (dp - \Lambda\rho)/(1-\Lambda) - y_1$, with $\Lambda = e^{-\Delta}$ and $\Delta$ the utility premium of ownership after the purchase. The threshold exceeds the feasibility bound $dp - y_1$ by $\Lambda(dp-\rho)/(1-\Lambda)$ = the stage-1 consumption of the marginal buyer. | |
| 33 | +*Intuition.* A liquidity-constrained household compares the certain cut in current consumption needed for the down payment and costs with the discounted future advantage of owning; only households rich enough that the marginal utility of current consumption is low accept the cut. | |
| 34 | +*Benchmark.* $\hat b = 0.999$ at $p = 1.203$, $\rho = 0.348$; closed form = brute-force optimum to $10^{-14}$. | |
| 35 | +*Part B.* $\hat a_1(z)$ = 1.18 / 0.88 / 0.49 / 0.24 / 0.03 (units of average 5-year income); single sign change of $W_B - W_R$ along cash at every age and state; the marginal buyer is hand-to-mouth, the marginal renter is not. | |
| 34 | 36 | |
| 35 | −### Proposition 2 — Comparative statics | |
| 36 | −*Intuition.* All signs are liquidity effects: a higher down payment or more front-loaded real payments (higher $\bar\pi$) are present-value neutral but shift resources from the constrained young to the unconstrained old. The proof bounds the stage-3 equity term by the Euler inequalities; $r \ge 0$ guarantees that the price effect on the outlay dominates the price effect on future equity. | |
| 37 | −*Simulation must check.* Magnitudes (benchmark: $\partial\hat b/\partial\theta = 9.8$, $\partial\hat b/\partial p = 7.3$, $\partial \hat b/\partial\bar\pi = 1.2$ per unit); whether $\partial\hat b/\partial p$ stays positive with an intensive housing margin (bigger houses at higher $p$ could change the ladder term). | |
| 37 | +### Proposition 2 — Comparative statics (scoped in Phase 4) | |
| 38 | +*What is proved.* All signs are liquidity effects under A4 and $r \ge 0$. The signs in $\theta$, $\kappa_s$, $m$, $\bar\pi$ additionally use the benchmark's once-and-for-all tenure decision: a renter never buys later, so $V^R$ does not depend on these parameters, and only $V^O$ moves. The LTV constraint must bind for $\theta$ to enter at all. | |
| 39 | +*What Part B found.* With the option to buy later, the renter's continuation value rises when $\theta$ falls, so the threshold is the crossing of two functions that both move with $\theta$. At fixed prices the median-state threshold is 0.52 / 0.49 / 0.59 ($\theta$ = 0.05 / 0.20 / 0.40): monotone from 0.20 to 0.40, not below 0.20 (difference 0.03 within the ±0.07 grid sensitivity); flat at the lowest $z$ (DSTI cap binds, LTV does not); monotone decreasing at the highest $z$ (−0.13 / 0.03 / 0.24). *Decision:* the premise is now named in the statement and in Remark 1; the proposition is not weakened further and the simulation is reported with its noise. | |
| 38 | 40 | |
| 39 | 41 | ### Proposition 3 — Tilt vs neutrality |
| 40 | −*Intuition.* Nominal contracts front-load real payments when expected inflation is higher; this is irrelevant to a household that can re-time consumption with deposits, and costly to one that cannot. The same holds for the down payment. | |
| 41 | −*Simulation must check.* In Part B the tilt operates through the payment-to-income constraint and the constant-nominal-payment schedule; verify that raising $\bar\pi$ lowers ownership at ages 1–3 in partial equilibrium, and trace the GE offset through $p$ and $\rho$. | |
| 42 | +Unchanged. Part B: trend inflation of 4 % lowers under-35 ownership by 4.9 pp at fixed prices through the nominal DSTI cap; the GE price fall offsets 65 %; along the persistent-surprise transition the nominal rate expected for the second period (0.219 → 0.341) tightens the cap and is why the persistent surprise lowers the price four times more than the level surprise. | |
| 42 | 43 | |
| 43 | 44 | ### Proposition 4 — Leverage with costs |
| 44 | −*Intuition.* Leverage multiplies the exposure of terminal wealth to the resale price (multiplier $(1-\kappa_s)/d = 2.5$ in the benchmark) but does not change the break-even growth rate, which depends on costs and the rent–price ratio only. In the benchmark $g^* = -0.27$ (rent is high relative to user cost) and the underwater threshold is a 63 % price fall. The owner–renter wealth gap among equal-income HTM households is $e_3 p = 0.72$ and the transfer gap $0.17$. | |
| 45 | −*Simulation must check.* Experiment 5 (negative price shocks, LTV sweeps): the dispersion of terminal wealth across inheritance quantiles should scale with the multiplier; the sign of the mean owner–renter gap should flip at $g^*$ when the renter is allowed to invest the down payment (experiment 2, "renter investment return"). | |
| 45 | +Unchanged. Part B: identity confirmed in the price-drop accounting; the PE ownership response to a real-rate shock ranks by leverage exposure ($-5.6 / -4.1 / -1.8$ pp for $\theta$ = 0.05 / 0.20 / 0.40) and GE removes the ranking. | |
| 46 | 46 | |
| 47 | 47 | ### Proposition 5 — Persistence |
| 48 | −*Intuition.* Owner-parents bequeath $e_3 p$ more; their children clear the threshold for a wider range of income draws. Persistence is the probability mass of income draws for which the parental housing equity is pivotal. It is zero if the house is not net wealth at the end of life (no amortisation, high selling costs, low down payment). | |
| 49 | −*Benchmark.* $q_{RO} = 0.56$, $q_{OO} = 0.78$, $\lambda = 0.23$, half-life 0.47 generations, $\mu^* = 0.72$. | |
| 50 | −*Simulation must check.* Parent–child tenure correlation and rank–rank slope of net worth with and without bequests (experiment 6); monotonicity of $\lambda$ in amortisation $a$, $\theta$ (ambiguous in theory: $\theta$ raises both $e_3$ and $\hat b$), and $p$; whether persistence is much higher when income shocks are persistent across generations ($\rho_g > 0$, experiment 8). | |
| 48 | +Unchanged. Part B: $q_{OO} = 0.87$, $q_{RO} = 0.81$ at the gift age ($\lambda_7 = 0.060$, half-life 0.25 generations; weak because 86 % of parents own by then), $\lambda_3 = 0.19$ at 35–39, rank–rank 0.34; without bequests $q_{RO} = q_{OO} = 0.38$ and rank–rank $-0.01$. With $\rho_g = 0.4$ income transmission adds more persistence ($\lambda_3 = 0.32$) than housing does. | |
| 51 | 49 | |
| 52 | −### Proposition 6 and Corollary — Stationary equilibrium | |
| 53 | −*Intuition.* Higher ownership raises the price and lowers the rent, both of which raise the threshold and reduce the next generation's ownership (the stabilising channel); but a higher price also raises the equity incumbents bequeath (the ladder channel). Uniqueness holds if the first dominates (condition (C), satisfied by a wide margin in the benchmark: 31 vs 0). With a fixed stock, credit relaxation only raises the price and reshuffles who owns. | |
| 54 | −*Simulation must check.* Experiment 3 and 7: search for multiple equilibria by starting the price iteration from several initial prices under near-fixed supply and high $e_3$; report the sign of $d\mu^*/dp$ along the clearing curve. | |
| 50 | +### Proposition 6 and Corollary 1 — Stationary equilibrium | |
| 51 | +Unchanged. Part B: (C)'s analogue $d\mu/dp < 0$ holds with slope $-0.48$ per unit of $p$ along the clearing curve; excess demand slope $-3.1$ per unit of $\log p$ at the root; no multiplicity found. | |
| 55 | 52 | |
| 56 | −### Proposition 7 — No segmentation | |
| 57 | −*Intuition.* With a single stock and costless conversion, tenure is a pure portfolio choice that does not change the number of units demanded, so credit, bequests and inflation cannot move the price. This is the Kaplan–Mitman–Violante / Greenwald–Guren lesson restated; the feedback results of WP18 are statements about segmentation and supply. | |
| 58 | −*Simulation must check.* Run the baseline at $\sigma = 0$ and $\sigma = 1$ (B.6); with an intensive margin the neutrality is only approximate — quantify. | |
| 53 | +### Proposition 7 — No segmentation (scoped in Phase 4) | |
| 54 | +*What is proved.* With one stock, costless conversion **and unit demand**, total demand for units is fixed, so credit, transfers and inflation only reshuffle who owns. | |
| 55 | +*What Part B found.* With an intensive margin, total housing demand depends on wealth; the unsegmented price falls by 0.39 % after the surprise against 0.45 % segmented. *Decision:* unit demand is now in the statement (Remark 2); the Part B result is reported as a finding — the price feedback in the quantitative model is mostly a wealth effect on total demand, segmentation adds ≈ 15 %. | |
| 59 | 56 | |
| 60 | 57 | ### Proposition 8 — Revaluation accounting |
| 61 | −*Intuition.* A surprise erodes every nominal payment fixed before it. A level surprise erodes all remaining FRM and ARM payments equally (the ARM reset rate is unchanged); a persistent surprise is recovered by the lender on ARM payments after the reset; a real-rate response can make ARM borrowers lose. Deposits lose symmetrically; unindexed wages lose; rent rigidity transfers from landlords to sitting tenants. The lender bears the mirror image. | |
| 62 | −*Benchmark (10 % surprise).* Age-2 owner gains 0.049 (stage 2) + 0.038 (stage 3, level) or 0.073 (FRM, persistent) in units of $y_1$; lender loses 0.126 (level) / 0.161 (FRM persistent); ARM with $\phi_r = 2 > R = 1.1$ loses 0.031 at stage 3; with $\eta_w = 0$ the owner's net is $-0.185$. | |
| 63 | −*Simulation must check.* Experiment 4 with the three contracts (long FRM, five-year term, ARM) and two rent regimes; the zero-sum identity state by state; the distribution of gains by age and tenure (compare with Meh–Terajima). | |
| 58 | +Unchanged. Part B: zero-sum to $10^{-15}$; FRM = term5 (level), FRM > term5 > ARM (persistent); with a real-rate response the term's gain is clawed back (CEV of mortgaged owners +1.43 % FRM vs +0.20 % term); with unindexed wages every indebted owner loses (condition 8.4). | |
| 64 | 59 | |
| 65 | −### Proposition 9 — GE impact | |
| 66 | −*Intuition.* Eroded debt of stage-3 owners becomes larger transfers, more owner-dynasty children qualify, the price rises and the rent falls, the threshold rises, and renter-dynasty children at the margin are displaced: inflation lowers upward tenure mobility and raises persistence, with no change in the ownership rate if the stock is fixed. Sellers gain twice; new buyers can lose. | |
| 67 | −*Benchmark (10 % surprise, $\varepsilon = \varepsilon_r = 1$).* $dp = +0.03$ %, $d\rho = -0.06$ %, $d\mu = +0.03$ pp, $d\hat b = +0.6$ %, $q_{RO}$ 0.559 → 0.550, $\lambda$ 0.226 → 0.238; with $\varepsilon = 0$: $d\mu = 0$, $dp = +0.05$ %. The effects are small because the eroded payment is small relative to $W_3$ and the price absorbs most of the shift (demand is very price-elastic under static expectations). | |
| 68 | −*Simulation must check.* The full perfect-foresight transition with a durable stock (A6 removed); second-round effects through the cohort aged 2 (contract-dependent); the alternating cohort-crowding dynamics found in the myopic simulation; welfare of marginal households (not signed in theory); whether the mobility effect survives when prices are forward-looking. | |
| 60 | +### Proposition 9 and Corollary 2 — GE impact (scoped and extended in Phase 4) | |
| 61 | +*What is proved.* Under Assumption 7 (mortgage = only nominal position), static expectations, per-period supply and (C′): $dp \ge 0$, $d\rho \le 0$, $d\mu \ge 0$, $d\hat b \ge 0$, $q_{RO}$ ↓, $\lambda$ ↑. **Corollary 2** drops A7: with predetermined parental deposits, the signs of $dp$, $d\mu$ and $-d\rho$ are the sign of $\Xi = \mu^* g(\hat y_O)(\delta_3 p^* - Rn^O) - (1-\mu^*) g(\hat y_R) Rn^R$ (density-weighted net nominal debtor position of the parents whose children are at the margin); Prop. 9 is $n^O = n^R = 0$; the change in $q_{RO}$ is unsigned unless $n^R = 0$. | |
| 62 | +*What Part B found.* Net nominal creditors (deposits 4.12 > debt 2.47) ⇒ $p_0 = -0.45$ % (FRM level), $-1.88$ % (persistent); rent also falls ($-1.14$ %); threshold rises (+8 %) through entrants' own deposit losses; $q_{RO}$ at the gift age falls and $\lambda_7$ rises as in Prop. 9.3 (direction), by < 0.01. *Decision:* the premise is explicit (A7), the corollary covers the creditor case, and the two residual differences (rent sign: intensive margin; threshold: entrants' deposits) are attributed to benchmark reductions in Remark 3 and reported as results of the paper. The theory is not bent to the simulation; the simulation is not discarded. | |
| 69 | 63 | |
| 70 | 64 | ## Numbers that the numerical phase must *not* take from here |
| 71 | −All numbers above are from illustrative parameters (stage lengths unequal in spirit, $R = 1.1$ per stage, Beta-distributed $y_3$). They verify signs and identities only. | |
| 65 | +All benchmark numbers are from illustrative parameters ($R = 1.1$ per stage, Beta-distributed $y_3$). They verify signs and identities only. Every quantitative claim in the paper comes from `results/data/` through `results/tables/numbers.tex`. | |
modified
research/results_audit.md
+96 −93
@@ -1,144 +1,147 @@ | ||
| 1 | −# WP18 — Results audit (Phase 3 draft; final version in Phase 4) | |
| 1 | +# WP18 — Results audit (Phase 4, final version) | |
| 2 | 2 | |
| 3 | −Status: **draft written during Phase 3 (2026-10-20)**. Every number below is read from a file under `results/data/` (named in each entry); figures and tables are regenerated by `make figures tables`. Labels used throughout: | |
| 3 | +Status: **final (2026-10-20, Phase 4)**. This document is the integrity gate between the numerical work (Phase 3) and the manuscript (`paper/main.tex`). Every number below is read from a file under `results/data/` (named in each entry) and, in the manuscript, is injected through a macro of `results/tables/numbers.tex` (generated by `src/tables.py`; `paper/tables/numbers.tex` is the copy used by LaTeX). Figures and tables are regenerated by `make figures tables`; the paper by `make paper`. Labels used throughout: | |
| 4 | 4 | |
| 5 | −* **identity** — accounting relation that holds by construction (tested in `tests/`); | |
| 6 | −* **theory** — comparative static proved in `paper/appendix_proofs.tex` (Part A) and *checked* here in the Part B model; | |
| 7 | −* **simulation (GE)** — stationary general equilibrium or perfect-foresight transition in which the owner-occupied and rental markets clear to the stated tolerance; | |
| 5 | +* **identity** — accounting relation that holds by construction (unit-tested in `tests/`); | |
| 6 | +* **theory** — comparative static or existence result proved in `paper/appendix_proofs.tex` for the benchmark (Part A of `research/model_specification.md`) and *checked* here in the quantitative model (Part B); | |
| 7 | +* **simulation (GE)** — stationary general equilibrium or perfect-foresight transition in which the owner-occupied and rental markets clear to the stated tolerance (residuals in `results/data/diagnostics.csv`); | |
| 8 | 8 | * **simulation (PE, prices fixed)** — household block and distribution re-solved at the baseline prices; never labelled GE; |
| 9 | −* **static expectations (A6)** — temporary equilibria with stationary continuation values (Proposition 9's assumption). | |
| 9 | +* **static expectations (A6)** — temporary equilibria with stationary continuation values (Proposition 9's assumption); | |
| 10 | +* **conjecture** — stated as such in the paper; neither proved nor established numerically. | |
| 10 | 11 | |
| 11 | −The calibration is **illustrative** (no moment is fitted; verified Canadian comparison points are listed in `results/data/calibration_targets.csv`); all magnitudes are therefore qualitative statements about *this* parameterisation, not estimates for Canada. | |
| 12 | +The calibration is **illustrative** (no moment is fitted; verified Canadian comparison points are listed in `results/data/calibration_targets.csv` and Table 1 of the paper); all magnitudes are qualitative statements about *this* parameterisation, not estimates for Canada. | |
| 12 | 13 | |
| 13 | −## 0. Algorithm, tolerances and what was actually computed | |
| 14 | +## 0. What was computed, and how well it converged | |
| 14 | 15 | |
| 15 | 16 | | Object | Implementation | Diagnostic reported | |
| 16 | 17 | |---|---|---| |
| 17 | −| Household problem | backward VFI over 12 ages; discrete branches (stay / refinance / move / sell-and-rent / rent / buy) on a cash-on-hand grid ($n_x = 120$) except *stay* on the full state grid ($n_z = 5$, $n_h = 4$, $n_b = 20$, $N = 5$, $n_a = 50$); exhaustive search over $a'$ on the grid + golden-section refinement; loans on 11 fractions of the LTV/DSTI cap; gift on 12 fractions of cash; estate anticipated as a 5-point lottery; annuity-due nominal mortgages | relative Bellman residuals at 400 random off-grid states vs a 400-point re-optimisation (`baseline/summary.json: bellman`) | | |
| 18 | −| Distribution | Monte-Carlo panel of 96 000 dynasty slots (young + old member each, phases equidistributed), common random numbers seeded by (seed, period), 60 burn-in periods (10 generations) from an arbitrary state, 6 recorded periods pooled (1 152 000 household-periods) | sample-size (24 000 / 240 000) and seed sensitivity (`grid_sensitivity.csv`) | | |
| 19 | −| Stationary equilibrium | nested Illinois regula falsi (rental market in $\tilde p_r$ inside the owner market in $p$), then 2-D Broyden rounds; damped fixed point (0.7) on the estate lottery and the transfer rebate; tolerance $2.5\times10^{-4}$ on relative excess demand, $10^{-3}$ on the lottery | residuals of every run in `diagnostics.csv` | | |
| 20 | −| Supply normalisation | $\bar H$, $\bar H_r$ set so that the baseline clears at $p = \tilde p_r = 1$; all counterfactuals keep these scales | `baseline/supply_scales.json` | | |
| 21 | −| Transition (MIT) | sequence-space shooting on $(\log p_t, \log \tilde p_{r,t})_{t<14}$, durable stocks $H_t = (1-\delta_H)H_{t-1} + \delta_H \bar H p_t^{\varepsilon}$, finite-difference Jacobian (step 0.01, parallel), Broyden with refresh, tolerance $10^{-3}$ per period; terminal condition = stationary values at $T = 14$ (70 years) | max residual and residual at $T$ in `inflation/<name>.json` | | |
| 22 | −| Welfare | consumption-equivalent variation with the decomposition $V = (1+\lambda)^{(1-\alpha)(1-\gamma)}U + \Phi$ ($U$ flow utility, $\Phi$ warm glow); $\Phi$ from cohort / remaining-lifetime simulations under baseline policies, $U = V - \Phi$ (consistent at the baseline; the simulated $U$ is reported as a check — it differs from $V-\Phi$ by 1–5 % because simulated policies use nearest-grid loan/size choices, which would inflate the CEV if used directly) | `welfare/*.csv` (`U_base_simulated` vs `U_base`, `V_check_sim_vs_fn`) | | |
| 18 | +| Household problem | backward VFI over 12 ages; discrete branches (stay / refinance / move / sell-and-rent / rent / buy) on a cash-on-hand grid ($n_x = 120$) except *stay* on the full state grid ($n_z = 5$, $n_h = 4$, $n_b = 20$, $N = 5$, $n_a = 50$); exhaustive search over $a'$ on the grid + golden-section refinement; loans on 11 fractions of the LTV/DSTI cap; gift on 12 fractions of cash; estate anticipated as a 5-point lottery; annuity-due nominal mortgages | relative Bellman residuals at 400 random off-grid states vs a 400-point re-optimisation: owners mean $6.7\times10^{-4}$, p99 0.6 %, max 1.9 %; renters mean $1.6\times10^{-4}$, max 0.17 % (`baseline/summary.json: bellman`) | | |
| 19 | +| Distribution | Monte-Carlo panel of 96 000 dynasty slots (young + old member each, phases equidistributed), common random numbers seeded by (seed, period), 60 burn-in periods, 6 recorded periods pooled (1 152 000 household-periods); unanticipated transitory income component $\sigma = 0.03$ that breaks the atoms of the cash distribution | sample-size (24 000 / 240 000) and seed sensitivity (`grid_sensitivity.csv`) | | |
| 20 | +| Stationary equilibrium | nested Illinois regula falsi (rental market in $\tilde p_r$ inside the owner market in $p$); damped fixed point (0.7) on the estate lottery and the transfer rebate; tolerance $5\times10^{-4}$ on relative excess demand, $10^{-3}$ on the lottery | **45 stationary GE runs, 44 within tolerance** (`sample_small` stopped at the noise floor $7.8\times10^{-4}$, labelled); 5 PE re-solutions; residuals of every run in `diagnostics.csv` | | |
| 21 | +| Supply normalisation | $\bar H = 9.41$, $\bar H_r = 2.90$ set so that the baseline clears at $p = \tilde p_r = 1$; all counterfactuals keep these scales | `baseline/supply_scales.json` | | |
| 22 | +| Transition (MIT) | sequence-space shooting on $(\log p_t, \log \tilde p_{r,t})_{t<14}$, durable stocks $H_t = (1-\delta_H)H_{t-1} + \delta_H \bar H p_t^{\varepsilon}$, finite-difference Jacobian (step 0.01, parallel), Broyden with refresh, damped tâtonnement pre-phase for large shocks; tolerance $1.5\times10^{-3}$ per period; terminal condition = stationary values at $T = 14$ (70 years) | **16 transition experiments, 12 within tolerance**; not converged and labelled: `frm_level` ($2.2\times10^{-3}$), `frm_level_unindexed_wages` ($2.0\times10^{-3}$), `frm_level_fixed_supply` ($3.1\times10^{-3}$), `rate_shock_theta005` ($8.8\times10^{-3}$); residual at $T$ of the converged runs $10^{-4}$–$4\times10^{-4}$ (`inflation/<name>.json`) | | |
| 23 | +| Multiplicity | owner-market excess demand scanned on $p \in [0.80, 1.25]$ (step 0.025) with the rental market cleared at each point and the lottery held fixed; baseline, $\varepsilon = 0.05$, $\varepsilon = 0.05 \times \theta = 0.05$ | `multiplicity/multiplicity_summary.json` | | |
| 24 | +| Welfare | CEV with $V = (1+\lambda)^{(1-\alpha)(1-\gamma)}U + \Phi$; $\Phi$ from cohort simulations under baseline policies, $U = V - \Phi$ (exact at the baseline); simulated $U$ kept as a check (differs from $V-\Phi$ by at most 4.9 % because simulated policies use nearest-grid loan/size choices) | `welfare/*.csv` | | |
| 23 | 25 | |
| 24 | −Two approximations are deliberate and documented: (i) the estate is anticipated as the unconditional 5-point lottery (the realised estate in the panel is exactly linked to the parent; the expectation ignores the information in one's own gift); (ii) along a transition the lottery is held at its stationary value (realised estates are correct). | |
| 26 | +Deliberate approximations: (i) the estate is anticipated as the unconditional 5-point lottery (the realised estate in the panel is exactly linked to the parent); (ii) along a transition the lottery is held at its stationary value; (iii) the "ARM" is a stylised within-period repricing (no erosion); (iv) no EGM, no taste shocks, no survival risk, no default, no aggregate risk, no endogenous real rate. | |
| 25 | 27 | |
| 26 | −What was *not* done: no EGM (justified in `household_solver.py`), no taste shocks, no survival risk, no default, no aggregate risk, no endogenous real rate (small open economy), no within-period monthly structure (the "ARM" contract is a stylised within-period repricing). | |
| 28 | +Design change recorded during Phase 3: a first full run showed a genuine discontinuity in rental excess demand (1.5 % jump between $\tilde p_r = 0.9730$ and $0.9735$ in `eps_200_theta005`) caused by atoms of the cash distribution; the transitory income component removed it; budget identities were re-tested with the realised income. | |
| 27 | 29 | |
| 28 | −## Experiments — fixed design (hypotheses, parameters, algorithm) | |
| 30 | +## Experiments — hypotheses, parameters, algorithm | |
| 29 | 31 | |
| 30 | −| # | Experiment | Hypothesis tested (from `propositions.md`) | Parameters / scenarios | Output files | | |
| 32 | +| # | Experiment | Hypothesis tested | Parameters / scenarios | Output files | | |
| 31 | 33 | |---|---|---|---|---| |
| 32 | −| 1 | Baseline by age, income, inheritance quantile | model generates owner–renter gaps among equal-income households; ownership profile; A4 (HTM at the threshold); single crossing for savers (proof gap 6) | `baseline.yaml` | `baseline/summary.json`, `profiles_by_age.csv`, `by_gift_quintile.csv`, `thresholds.csv`, `policy_age*.csv` | | |
| 34 | +| 1 | Baseline by age, income, inheritance quantile | equal-income owner–renter gaps; ownership profile; A4 at the threshold; single crossing (proof gap) | `baseline.yaml` | `baseline/summary.json`, `profiles_by_age.csv`, `by_gift_quintile.csv`, `thresholds.csv`, `policy_age*.csv` | | |
| 33 | 35 | | 2 | Equal-income paths with different endowments | Prop. 4/5: equal-income gap = housing equity carried; divergence of two households straddling the threshold | median $z$ path, $a_1$ at gift quantiles and $\hat a \pm 0.01$ | `baseline/equal_income_paths.csv` | |
| 34 | −| 3 | Down-payment / threshold sweep | Prop. 1–2: $\hat a$ increasing in $\theta$; GE offset through $p$, $\rho$; Corollary 6c (ownership vs composition) | $\theta \in \{0.05, 0.10, 0.20, 0.30, 0.40\}$ GE and PE; $\psi \in \{0.30, 0.50\}$ | `theta_sweep/*.json` | | |
| 35 | −| 4 | Surprise inflation, contracts, persistence, rent rigidity | Prop. 8 (ordering FRM = term5 > ARM for a level surprise; FRM > term5 for a persistent one; zero-sum), Prop. 9 (signs of $dp$, $d\rho$, $d\mu$, $d\hat a$, $q_{RO}$) | $\pi^u = 10\%$, FRM / term5 / ARM × level / persistent; $\eta_\rho = 0$; $\eta_w = 0$; $\phi_r = 0.5$; $\varepsilon = 0.05$; $\sigma = 0$; static expectations | `inflation/*.json`, `*_path.csv`, `*_impact.csv`, `*_welfare.csv` | | |
| 36 | −| 5 | Leverage / LTV, negative price shocks | Prop. 4 (multiplier, underwater threshold $x^*$ independent of $\theta$ in break-even terms) | price drops 10/20/30 % on the cross-section (accounting, PE); real-rate shock +2 pp for 20 years under $\theta \in \{0.05, 0.20, 0.40\}$ (GE) | `baseline/price_drop_accounting.csv`, `inflation/rate_shock_*` | | |
| 37 | −| 6 | Bequests on/off, transfer tax | Prop. 5: persistence requires bequeathed housing equity; tax with rebate | $\nu \in \{0, 1, 2\}$, $\tau_b \in \{0.2, 0.5\}$ (GE and PE) | `bequests/*.json` | | |
| 38 | −| 7 | Supply elasticity incl.\ near-fixed | Prop. 6c / 7: fixed stock → price and composition absorb credit; neutrality without segmentation; multiplicity search (condition (C)) | $\varepsilon \in \{0.05, 0.25, 0.5, 1, 2\}$ × $\theta$; $\sigma \in \{0, 0.5, 1\}$; price scans | `supply/*.json`, `multiplicity/*.csv` | | |
| 39 | −| 8 | Rates, rent growth, earnings risk, expectations | Prop. 3 (tilt), Remark on $\rho_g$, A6 vs perfect foresight | $r \in \{1, 3\}\%$, $\bar\pi \in \{0, 4\}\%$ (GE, PE), $\phi \in \{0.02, 0.08\}$, $\sigma_z \in \{0.10, 0.20\}$, $\rho_g = 0.4$, deposit premium 2 pp, rent growth 1 %/yr (PE), static expectations | `sensitivity/*.json` | | |
| 36 | +| 3 | Down-payment / threshold sweep | Prop. 1–2: $\hat a$ in $\theta$; GE offset through $p$, $\rho$; Cor. 6c | $\theta \in \{0.05, 0.10, 0.20, 0.30, 0.40\}$ GE and PE; $\psi \in \{0.30, 0.50\}$ | `theta_sweep/*.json` | | |
| 37 | +| 4 | Surprise inflation, contracts, persistence, rent rigidity | Prop. 8 (ordering, zero-sum), Prop. 9 (signs) | $\pi^u = 10\%$, FRM / term5 / ARM × level / persistent; $\eta_\rho = 0$; $\eta_w = 0$; $\phi_r = 0.5$; $\varepsilon = 0.05$; $\sigma = 0$; A6 | `inflation/*.json`, `*_path.csv`, `*_impact.csv`, `*_welfare.csv` | | |
| 38 | +| 5 | Leverage / LTV, negative price shocks, real-rate shock | Prop. 4 (multiplier, $x^*$) | price drops 10/20/30 % (accounting, PE); real rate +2 pp/yr for 20 years under $\theta \in \{0.05, 0.20, 0.40\}$ (GE) | `baseline/price_drop_accounting.csv`, `inflation/rate_shock_*` | | |
| 39 | +| 6 | Bequests on/off, transfer tax | Prop. 5: persistence requires bequeathed equity; rebated tax | $\nu \in \{0, 1, 2\}$, $\tau_b \in \{0.2, 0.5\}$ (GE and PE) | `bequests/*.json` | | |
| 40 | +| 7 | Supply elasticity incl. near-fixed; segmentation; multiplicity | Cor. 6c / Prop. 7; condition (C) | $\varepsilon \in \{0.05, 0.25, 0.5, 1, 2\}$ × $\theta$; $\sigma \in \{0, 0.5, 1\}$; price scans | `supply/*.json`, `multiplicity/*.csv` | | |
| 41 | +| 8 | Rates, trend inflation, rent wedge, earnings risk, expectations | Prop. 3 (tilt); income transmission; A6 vs perfect foresight | $r \in \{1, 3\}\%$, $\bar\pi \in \{0, 4\}\%$ (GE, PE), $\phi \in \{0.02, 0.08\}$, $\sigma_z \in \{0.10, 0.20\}$, $\rho_g = 0.4$, deposit premium 2 pp, rent growth (PE), static expectations | `sensitivity/*.json` | | |
| 40 | 42 | | 9 | Welfare (CEV) | marginal households (unsigned in theory) | all GE scenarios vs baseline; transitions by age/tenure | `welfare/*.csv`, `inflation/*_welfare.csv` | |
| 41 | −| 10 | Robustness | numerical stability of the above | $\beta$, $\gamma$, $\alpha$, $\kappa$, $\chi$, grids, tolerance, sample size, seed | `robustness/*.json`, `robustness_summary.csv` | | |
| 43 | +| 10 | Robustness | numerical stability | $\beta$, $\gamma$, $\alpha$, $\kappa$, $\chi$, grids, tolerance, sample size, seed | `robustness/*.json`, `robustness_summary.csv`, `grid_sensitivity.csv` | | |
| 42 | 44 | |
| 43 | −## Results by experiment | |
| 45 | +## Results by experiment (labels in bold) | |
| 44 | 46 | |
| 45 | −### 1. Baseline stationary equilibrium (`results/data/baseline/`) | |
| 46 | −*Convergence.* GE: $p^* = 1.0000$, $\tilde p_r^* = 1.0000$, $\rho^* = 0.1943$ (annual rent–price ratio 3.9 %); relative excess demands $4\times10^{-6}$ (owner) and $-7\times10^{-6}$ (rental); lottery change $< 10^{-5}$; Bellman residuals mean $6.7\times10^{-4}$ (owners; p99 0.6 %, max 1.9 %) and $1.6\times10^{-4}$ (renters; max 0.17 %); no forced sale; $3\times10^{-5}$ of household-periods with a clipped interpolated policy. Supply normalisation in 5 lottery rounds ($\bar H = 9.406$, $\bar H_r = 2.902$), GE in 5 evaluations; 8 s wall-clock. All 48 stationary scenarios of the run converged (`diagnostics.csv`; max relative excess demand $4.8\times10^{-4}$, i.e. within the tolerance $5\times10^{-4}$). | |
| 47 | +### 1. Baseline stationary equilibrium (`results/data/baseline/`) — **simulation (GE)** | |
| 48 | +*Convergence.* $p^* = 1.0000$, $\tilde p_r^* = 1.0000$, $\rho^* = 0.1943$ (annual rent–price ratio 3.9 %); relative excess demands $4\times10^{-6}$ (owner) and $-7\times10^{-6}$ (rental); lottery change $< 10^{-5}$; 5 lottery rounds; 8 s wall-clock; no forced sale; $3\times10^{-5}$ of household-periods with a clipped interpolated policy. | |
| 47 | 49 | |
| 48 | −*Results (simulation, GE).* Ownership 0.63 overall, 0.38 under 35, 0.73 at 35–64, 0.61 at 65+ (profile: 0.34 at 25–29 → 0.86 at 55–64 → 0.36 at 80–84: households sell late in life to consume and to leave a liquid estate — a feature of the warm-glow-over-liquid-estate specification that contradicts the Canadian plateau near 3/4 after 65 reported by Hou 2010; flagged for the paper). Share with a mortgage 0.41 (SFS 2023: 0.39). Mean net worth of owners / renters under 35 = 11.2 (SFS medians 10.4; the model's young renters have zero median net worth). Mean gift at entry 0.39 (1.9 years of average income; 70 % of entrants and 72 % of first-time buyers receive a positive gift — the latter well above the 28 % quoted by Mirdamadi–Khalid 2023), mean estate 0.32 at 55–59 (55 % positive). Gini of net worth 0.54 (0.38 at 55–59). Aggregate real mortgage debt 2.47 vs deposits 4.12 per unit mass of one age: **households are net nominal creditors in aggregate** (this drives the sign of the inflation results in §4). | |
| 50 | +*Results.* Ownership 0.63 overall, 0.38 under 35, 0.73 at 35–64, 0.61 at 65+ (profile 0.34 at 25–29 → 0.86 at 55–64 → 0.36 at 80–84: households sell late in life to consume and to leave a liquid estate — contradicts the Canadian plateau near 3/4 after 65 of Hou 2010; **flagged in the paper**). Share with a mortgage 0.41 (SFS 2023: 0.39). Mean net worth of owners / renters under 35 = 11.2 (SFS medians ≈ 10.4; the model's young renters have zero median net worth). Mean gift 0.39 (1.9 years of average income; 70 % of entrants and 72 % of first-time buyers receive a positive gift — the latter well above the 28 % quoted by Mirdamadi–Khalid 2023; **flagged**), mean estate 0.32 (55 % positive). Gini of net worth 0.54 (0.38 at 55–59). **Aggregate real mortgage debt 2.47 vs deposits 4.12 per unit mass of one age: households are net nominal creditors** (this drives the sign of §4). | |
| 49 | 51 | |
| 50 | −*Ownership-entry threshold (Proposition 1 analogue).* $\hat a_1(z)$ = 1.18, 0.88, 0.49, 0.24, 0.03 for the five productivity states (units of average 5-year income); at ages 2–3 it falls and is negative for the two highest states. **Single crossing holds** for every age 1–3 and every $z$ (exactly one sign change of $W_B - W_R$ along cash: proof gap 6 of `failed_hypotheses.md` is not a problem numerically). **Assumption A4 holds for the buyer and fails for the renter**: the marginal buyer sets $a' \approx 0$ for all $z$, the marginal renter saves — renters accumulate towards a later purchase, which Part A rules out. The threshold is numerically the least robust statistic: $\pm 0.07$ (median $z$) between the coarse and fine grids (§10). | |
| 52 | +*Threshold (Prop. 1 analogue).* $\hat a_1(z)$ = 1.18, 0.88, 0.49, 0.24, 0.03 for the five productivity states; negative at ages 2–3 for the two highest states. **Single crossing holds** at every age 1–3 and every $z$ (exactly one sign change of $W_B - W_R$ along cash): proof gap 6 is not a problem numerically. **A4 holds for the buyer and fails for the renter**: the marginal buyer sets $a' \approx 0$, the marginal renter saves towards a later purchase (the option that Part A rules out — origin of the Prop. 2 scope restriction). The threshold is the least robust statistic: $\pm 0.07$ between coarse and fine grids. | |
| 51 | 53 | |
| 52 | −*Equal-income gaps (identity + simulation).* Owners minus renters mean net worth, same income process by design: 0.63 at 30–34, 0.95 at 40–44, 1.57 at 55–59, 1.03 at 70–74. By quintile of the transfer received: ownership at 25–29 is 0.02/0.03/0.18/0.57/0.89 from the bottom to the top quintile (the bottom two quintiles receive nothing); the gap between quintiles 2 and 5 narrows from 86 pp at 25–29 to 16 pp at 55–59, and median net worth at 55–59 is 1.06 (Q2) vs 2.31 (Q5) (`by_gift_quintile.csv`). | |
| 54 | +*Equal-income gaps.* Owners minus renters mean net worth: 0.62 at 30–34, 0.95 at 40–44, 1.57 at 55–59, 1.03 at 70–74. By gift quintile: ownership at 25–29 is 0.02/0.03/0.18/0.57/0.89; the Q2–Q5 gap narrows from 86 pp at 25–29 to 16 pp at 55–59; median net worth at 55–59 is 1.06 (Q2) vs 2.31 (Q5). | |
| 53 | 55 | |
| 54 | −*Mobility (Proposition 5 analogue).* At the gift age (55–59): $q_{OO} = 0.866$, $q_{RO} = 0.806$, $\lambda_7 = 0.060$ — weak because 86 % own by then. At 35–39: $q_{OO} = 0.538$ vs $q_{RO} = 0.351$ ($\lambda_3 = 0.19$). Rank–rank slope of net worth (child vs parent, both at 55–59) 0.335. | |
| 56 | +*Mobility (Prop. 5 analogue).* At 55–59: $q_{OO} = 0.87$, $q_{RO} = 0.81$, $\lambda_7 = 0.060$ (half-life 0.25 generations) — weak because 86 % of parents own by then. At 35–39: $q_{OO} = 0.54$ vs $q_{RO} = 0.35$ ($\lambda_3 = 0.19$). Rank–rank slope of net worth 0.34. | |
| 55 | 57 | |
| 56 | −*Supported claims.* Existence of a cleared stationary equilibrium with an interior ownership rate (simulation); single crossing and buyer-side HTM at the threshold (numerical check of the proof gap); equal-income wealth gaps of about one year's income at mid-life (simulation, illustrative). *Not supported:* the old-age ownership profile; the share of gift recipients. | |
| 58 | +*Supported.* Existence of a cleared stationary equilibrium with an interior ownership rate (**simulation**); single crossing and buyer-side HTM at the threshold (**numerical check of a proof gap**); equal-income gaps of about one year's income at mid-life (**simulation, illustrative**). *Not supported by the comparison points:* the old-age ownership profile; the share of gift recipients. | |
| 57 | 59 | |
| 58 | −### 2. Equal-income paths with different initial transfers (`baseline/equal_income_paths.csv`, Figure 5) | |
| 59 | −Deterministic median-income path, baseline policies, estate at 55–59 = median lottery point. Entrants with gifts at the 10th–50th percentile ($a_1 \le 0.22$) rent until 45–49 and then buy the smallest house with the maximum loan; the household just above the age-1 threshold ($a_1 = 0.504$) buys at 25–29, the one just below ($a_1 = 0.484$) at 40–44. Net worth of the two straddling households: 0.50 vs 0.33 at 40–44, 1.15 vs 1.03 at 55–59 (a 12 % gap from a 4 % difference in endowment), converging after retirement when both sell (75–79). *Supported:* the threshold converts small endowment differences into two-decade wealth gaps through earlier leverage and forced amortisation (simulation; sign consistent with Proposition 4's multiplier; magnitude illustrative). | |
| 60 | +### 2. Equal-income paths (`baseline/equal_income_paths.csv`, Figure 5) — **simulation (baseline policies)** | |
| 61 | +Entrants with gifts at or below the median ($a_1 \le 0.22$) rent until 45–49 and then buy the smallest house with the maximum loan; the household just above the age-1 threshold ($a_1 = 0.504$) buys at 25–29, the one just below ($a_1 = 0.484$) at 40–44. Net worth 0.50 vs 0.33 at 40–44 (+49 %) and 1.15 vs 1.03 at 55–59 (+12 %) from a 4 % endowment difference; convergence after retirement when both sell. *Supported:* the threshold converts small endowment differences into two-decade wealth gaps through earlier leverage and forced amortisation (sign of Prop. 4's multiplier; magnitude illustrative). | |
| 60 | 62 | |
| 61 | −### 3. Down-payment constraint and threshold sweep (`theta_sweep/`, Figures 3 and 14, Table 5) | |
| 62 | −GE: $\theta = 0.05 / 0.10 / 0.20 / 0.30 / 0.40 \Rightarrow p^* = 1.0055 / 1.0045 / 1.000 / 0.992 / 0.987$; ownership 0.640 / 0.639 / 0.631 / 0.625 / 0.619; under-35 ownership 0.409 / 0.401 / 0.378 / 0.353 / 0.333. PE (prices fixed): $\theta = 0.05$ gives ownership 0.657 (under 35: 0.429), $\theta = 0.40$ gives 0.596 (0.304): **the price response absorbs about half of the partial-equilibrium ownership effect** (Corollary 6c direction; with $\varepsilon = 1$ the absorption is partial, not total). Mobility: $q_{RO}$ at 35–39 falls from 0.375 ($\theta = 0.05$) to 0.304 ($\theta = 0.40$), $\lambda_3$ rises from 0.20 to 0.21, $\lambda_7$ from 0.07 to 0.08; the rank–rank slope is flat (0.33–0.34). DSTI cap: $\psi = 0.30 / 0.50 \Rightarrow$ ownership 0.627 / 0.634, $p^* = 0.991 / 1.006$. | |
| 63 | +### 3. Down-payment sweep (`theta_sweep/`, Figures 3 and 14, Table 5) — **simulation (GE and PE)**; **theory check of Prop. 2** | |
| 64 | +GE: $\theta = 0.05 / 0.10 / 0.20 / 0.30 / 0.40 \Rightarrow p^* = 1.005 / 1.005 / 1.000 / 0.992 / 0.987$; ownership 0.640 / 0.639 / 0.631 / 0.625 / 0.619; under-35 0.409 / 0.401 / 0.378 / 0.353 / 0.333. PE: $\theta = 0.05$ gives 0.657 (under 35: 0.429), $\theta = 0.40$ gives 0.596 (0.304): **the price response absorbs 64 % ($\theta = 0.05$) and 66 % ($\theta = 0.40$) of the PE ownership effect** (Cor. 6c direction; partial, not total, with $\varepsilon = 1$). Mobility: $q_{RO}$ at 35–39 0.375 → 0.304, $\lambda_7$ 0.069 → 0.084, rank–rank flat (0.33–0.34). DSTI cap: $\psi = 0.30 / 0.50 \Rightarrow$ ownership 0.627 / 0.634. | |
| 63 | 65 | |
| 64 | −*Threshold (Proposition 2 check).* In PE the age-1 threshold at the median $z$ is 0.52 ($\theta = 0.05$), 0.49 (0.20), 0.59 (0.40): it rises with $\theta$ from 0.20 to 0.40 as Proposition 2 predicts, but is *not lower* at $\theta = 0.05$ — the difference (0.03) is within the grid sensitivity of the threshold ($\pm 0.07$) and, economically, the option value of buying later with a small down payment raises the renter's continuation value (a channel absent from Part A, where there is no later purchase). For the highest $z$ the threshold falls monotonically with $\theta$ ($-0.13 / 0.03 / 0.24$); for the lowest $z$ it is unchanged in PE (1.18) because the DSTI cap, not the LTV cap, binds. **Contrary to Proposition 2 in the strict sense** (monotonicity for every state) — supported for the states where the LTV cap binds. *Label:* simulation (GE and PE), theory check. | |
| 66 | +*Prop. 2 check — disagreement (3).* PE age-1 threshold at the median $z$: 0.52 ($\theta = 0.05$), 0.49 (0.20), 0.59 (0.40): rises from 0.20 to 0.40 as predicted but is *not lower* at 0.05 — difference 0.03, within grid sensitivity ($\pm 0.07$); economic source: option value of buying later with a small down payment raises the renter's continuation value (absent from Part A). Highest $z$: monotone decreasing in $\theta$ ($-0.13 / 0.03 / 0.24$); lowest $z$: unchanged (1.18) because the DSTI cap, not the LTV cap, binds. **Classification: Prop. 2 is supported where its premises hold (binding LTV constraint, once-and-for-all tenure decision) and contradicted, within noise, where they do not; the proposition is restated with the premise explicit (Remark on scope in the paper). Not forced either way.** | |
| 65 | 67 | |
| 66 | −### 4. Unanticipated 10 % price-level surprise (`inflation/`, Figures 8, 9, 11b, Table 6) | |
| 67 | −*Design.* MIT shock at $t = 0$ from the baseline SRCE: CPI 10 % above expectation in the shock period (level), or in two consecutive periods (persistent, the second known at $t = 0$); perfect foresight afterwards; durable stocks; $T = 14$ (70 years), terminal condition = stationary values; prices $(p_t, \tilde p_{r,t})$ solved by shooting with a parallel finite-difference Jacobian (28 columns) and Broyden updates; tolerance $1.5\times10^{-3}$ per period on relative excess demand (single-period cross-sections of 192 000 households, noise floor about $10^{-3}$). Contracts: FRM (nominal payment fixed for the remaining term), 5-year term (reset at the next period boundary), ARM (stylised within-period repricing: no erosion of the balance). PE counterpart = same shock with prices fixed at the SRCE, always reported separately. | |
| 68 | +### 4. Unanticipated 10 % price-level surprise (`inflation/`, Figures 8, 9, 11b, Table 6) — **simulation (GE transitions)**; **identity** for the accounting; **theory check of Props. 8–9** | |
| 69 | +*Convergence.* `term5_level` $6.1\times10^{-4}$ (3 Broyden steps, 161 s); `arm_level` $1.1\times10^{-3}$; `frm_persistent` $1.2\times10^{-3}$; `term5_persistent` $8.5\times10^{-4}$; `arm_persistent` $3.1\times10^{-4}$; `frm_level` **stalled at $2.2\times10^{-3}$** (10 steps, 471 s; labelled). FRM and term5 are identical for a level surprise at the 5-year frequency and agree to two decimals ($p_0$: $-0.45$ vs $-0.47$ %): the pair bounds the solver's price noise at ≈ 0.02 pp. Price paths back within 0.1 % after 20 years; residual at $T$ $3\times10^{-4}$. Zero-sum check $< 10^{-15}$ (**identity**). | |
| 68 | 70 | |
| 69 | −*Convergence.* `term5_level`: converged, max residual $6.1\times10^{-4}$ (3 Broyden steps, 161 s); `arm_level`: $1.1\times10^{-3}$ (5 steps); `frm_persistent`: $1.2\times10^{-3}$ (4 steps); `frm_level`: **stalled at $2.2\times10^{-3}$** after a Jacobian refresh (10 steps, 471 s) and is labelled `GE NOT CONVERGED (max residual 2.24e-03)`. For a *level* surprise the FRM and 5-year-term economies are identical at the 5-year frequency (same balance erosion, same post-shock nominal rate), and indeed every impact statistic of the two runs agrees to 3 decimals ($p_0$: $-0.448$ vs $-0.469$ %, CEV $-0.64$ % both): the pair bounds the solver's price noise at about 0.02 pp of $p$. Residual at $T$: $2$–$4\times10^{-4}$ (the economy is back within noise of the SRCE after 70 years). Zero-sum revaluation check $< 10^{-15}$ (identity). | |
| 71 | +*Impact — disagreement (1): the sign of the price response is the opposite of Prop. 9.* Level surprise, FRM / term5: $p_0 = -0.45 / -0.47$ %, $\rho_0 = -1.14 / -1.03$ %; ARM: $-0.79$ %, $-1.15$ %; persistent FRM / term5 / ARM: $p_0 = -1.88 / -1.49 / -1.86$ %. Reason (**identity**): households are net nominal creditors — real deposit loss 0.48 vs real debt erosion 0.21 per unit mass of one age (0 under the ARM), so the lender gains 0.27 (0.48 under the ARM) and aggregate household wealth falls; housing demand falls on both margins. **Classification: Prop. 9 assumed (implicitly, through A4 on the HTM region) that the mortgage is the only nominal position; that premise fails in the quantitative model. Reconciliation: the premise is now an explicit Assumption 7 in the paper, Prop. 9 is stated under it, and a new Corollary (net nominal creditors) proves that the sign of $dp$, $d\mu$ and $-d\rho$ is the sign of $\Xi = \mu^* g(\hat y_O)(\delta_3 p^* - R n^O) - (1-\mu^*) g(\hat y_R) R n^R$, so the simulated fall is the corollary's case $\Xi < 0$. Two residual differences remain and are stated as such: in Part B the rent falls (the corollary's unit demand would make it rise) because the wealth loss lowers total demand for rented services; and the entry threshold rises (0.49 → 0.53, +8 %) because would-be buyers are themselves nominal creditors — a channel absent from the benchmark.** Ownership on impact: $-0.02$ pp (FRM/term, GE) vs $-0.20$ pp (PE); ARM $-0.11$ vs $-0.47$; persistent FRM $-0.18$ vs $-2.17$: **GE absorbs 77–92 % of the PE ownership response**. One sign change of the price path (level), three (persistent), all $\le 0.1$ %; no cohort-crowding oscillation. | |
| 70 | 72 | |
| 71 | −*Impact — the sign of the price response is the opposite of Proposition 9.* Level surprise, FRM / 5-year term: $p_0 = -0.47$ %, $\rho_0 = -1.03$ %; ARM: $p_0 = -0.79$ %, $\rho_0 = -1.15$ %; persistent FRM: $p_0 = -1.88$ %, $\rho_0 = -0.38$ %. Reason (accounting, identity): households are net nominal creditors in aggregate — real deposit loss 0.48 vs real debt erosion 0.21 per unit mass of one age (0 under the ARM), so the lender gains 0.27 (0.48 under the ARM) and aggregate household wealth falls; housing demand falls with it. Proposition 9 assumes that the only nominal position is the mortgage; with deposits of this size its premise fails. Ownership on impact: $-0.02$ pp (FRM/term, GE) vs $-0.20$ pp (PE); ARM $-0.11$ vs $-0.47$ pp; persistent FRM $-0.18$ vs $-2.17$ pp: **general equilibrium absorbs 80–90 % of the partial-equilibrium ownership response**. Price paths return to the SRCE within 30–40 years with one sign change (level) and three (persistent): no cohort-crowding oscillation of the kind seen in the Part A myopic simulation. | |
| 73 | +*Redistribution (Prop. 8 accounting, **identity** + **simulation**).* Owners 30–44 gain up to $+0.04$ of a 5-year income and raise impact consumption by 0.6–1.0 %; owners 60–74 lose 0.04–0.08 and cut consumption 1.8–2.3 %; renters lose 0.02 and cut consumption 1.3 %. CEV (GE, alive at the shock): all $-0.64$ % (level FRM/term), $-1.14$ % (ARM), $-0.52$ % (persistent FRM); mortgaged owners $+0.63$ % (71 % better off) under FRM/term, $-1.03$ % (7 %) under ARM, $+1.14$ % (73 %) under persistent FRM, $+0.46$ % (66 %) under persistent term, $-1.21$ % under persistent ARM; debt-free owners $-2.1$ %; renters $-1.0$ to $-1.1$ %. **Contract ordering of Prop. 8 confirmed**: FRM = term (level), FRM > term > ARM (persistent). PE welfare differs from GE by < 0.05 pp. | |
| 72 | 74 | |
| 73 | −*Redistribution (Proposition 8 accounting + GE).* Owners aged 30–44 gain on their nominal position (up to $+0.04$ of a 5-year income) and raise impact consumption by 0.3–1.1 %; owners aged 60–74 (deposit-rich, debt-free) lose 0.07–0.08 and cut consumption by 1.8–2.3 %; renters lose 0.02 on average and cut consumption by 1.4 %. Welfare (CEV, GE, households alive at the shock): all $-0.64$ % (level FRM/term), $-1.14$ % (ARM), $-0.52$ % (persistent FRM); owners with a mortgage $+0.63$ % (71 % better off) under FRM/term, $-1.03$ % (7 % better off) under the ARM, $+1.14$ % (73 %) under the persistent FRM; owners without a mortgage $-2.1$ % in every case; renters $-1.0$ %. **Contract ordering of Proposition 8 holds**: FRM = term for a level surprise, FRM > term for a persistent one (term: see below), ARM last. The PE welfare numbers differ from GE by less than 0.05 pp: the price response barely changes the welfare incidence because prices move little. | |
| 75 | +*Mobility and threshold (Prop. 9.3).* $q_{RO}$ at 55–59 falls (0.806 → 0.797), $\lambda_7$ rises (0.060 → 0.069), as predicted in direction; at 35–39 $q_{RO}$ rises slightly on impact (0.351 → 0.356) and falls one period later (0.336). Composition effects of a fraction of a percentage point. | |
| 74 | 76 | |
| 75 | −*Mobility and threshold (Proposition 9.3).* $q_{RO}$ at 55–59 falls on impact (0.806 → 0.798) and $\lambda_7$ rises (0.060 → 0.068), as predicted; at 35–39 $q_{RO}$ rises slightly on impact (0.351 → 0.356) and falls below the SRCE one period later (0.335). The age-1 threshold at the median $z$ rises from 0.49 to 0.53 (+8 %), but through the erosion of the real return on the savings of would-be buyers, not through a higher price. These are **composition effects of a fraction of a percentage point**; their sign is as in Proposition 9.3 for the gift-age statistic and ambiguous one period later. | |
| 77 | +*Persistent surprise.* Real debt on impact: FRM $-24.7$ %, term5 $-13.5$ %, ARM $-9.4$ % (lower origination, not erosion: nominal rate quoted at $t=0$ jumps from 0.219 to 0.341 and the nominal DSTI cap tightens); ownership $-2.0$ to $-2.3$ pp (PE), $-0.2$ to $-0.4$ pp (GE): **Prop. 3's tilt operates through the nominal DSTI cap along the transition** (**simulation**, consistent with theory). | |
| 76 | 78 | |
| 77 | −*Supported.* Revaluation accounting and zero-sum (identity); contract ordering (Proposition 8, confirmed); GE absorption of the ownership response (simulation); *contrary to Proposition 9*: the real house price *falls* on impact because households' nominal deposits exceed their mortgages in this calibration — the proposition's sign requires net nominal debtors. *Limitation:* the estate lottery is held at its SRCE value along the path; the `frm_level` run is reported with its residual. | |
| 79 | +*Rent rigidity ($\eta_\rho = 0$).* Sitting tenants gain 0.013 (one period); market rent for new tenants $+7.5$ %; ownership $-2.9$ pp (PE) vs $-0.03$ pp (GE); renters' CEV $-1.00 \to -0.92$ %. *Supported:* the Pallotti et al. hedge, small at the 5-year frequency. | |
| 78 | 80 | |
| 79 | −*Persistent surprise (two periods, second known at $t=0$).* Real debt on impact: FRM $-24.7$ % (nominal payments fixed through both inflation periods), 5-year term $-13.5$ %, ARM $-9.4$ % (no erosion; the fall is lower origination because the nominal rate quoted at $t=0$ jumps from 0.219 to 0.341 and the nominal DSTI cap tightens). CEV of mortgaged owners: FRM $+1.14$ % (73 % better off), term $+0.46$ % (66 %), ARM $-1.21$ % (7 %) — **the ordering FRM > 5-year term > ARM of Proposition 8 is confirmed**; debt-free owners $-2.1$ % and renters $-1.1$ % in all three. Prices: $p_0 = -1.88$ % (FRM), $-1.49$ % (term), $-1.86$ % (ARM), $\rho_0$ between $-0.4$ and $+0.04$ %. The larger price fall than under the level surprise comes from the nominal rate expected for the next period: the qualifying payment rises, the DSTI cap binds for more entrants, ownership falls by 2.0–2.3 pp with prices fixed (PE) and by 0.2–0.4 pp in GE — **Proposition 3's tilt operates through the nominal DSTI cap along the transition** (simulation, consistent with the theory). Price paths show three sign changes (small, $\le 0.1$ %) after the impact. | |
| 81 | +*Unindexed wages ($\eta_w = 0$; stalled at $2.0\times10^{-3}$, labelled).* Real wage loss 0.075 per household (0.085 per owner) dwarfs the revaluation: $p_0 = -3.9$ %, $\rho_0 = -3.2$ %, consumption $-4.4$ %, CEV all $-3.3$ %, mortgaged owners $-2.4$ % (0 % better off). *Supported:* Prop. 8.4's condition (**identity** + **simulation**). | |
| 80 | 82 | |
| 81 | −*Rent rigidity ($\eta_\rho = 0$, level FRM).* Sitting tenants pay the pre-shock nominal rent (real rent $-9$ % for one period, a transfer of 0.013 per renter from landlords); the market rent for new tenants rises by 7.5 % on impact because incumbents stay put (ownership $-2.9$ pp with prices fixed, $-0.03$ pp in GE); $p_0 = -0.46$ % as without rigidity; renters' CEV improves only from $-1.00$ to $-0.92$ % (the rigidity lasts one period). *Supported:* rent rigidity protects sitting tenants (Proposition 8's landlord transfer, identity) but is quantitatively small at the 5-year frequency (simulation). | |
| 83 | +*Real-rate response ($\phi_r = 0.5$, persistent).* $p_0 = -7.3$ % (FRM) / $-6.9$ % (term5); mortgaged owners $+1.43$ % (70 % better off) vs $+0.20$ % (56 %): the renewal at the higher real rate claws back most of the term borrowers' gain (Prop. 8.1); all households $+0.14$ % (FRM) / $-0.25$ % (term). *Supported:* the FRM-vs-term distinction matters when the surprise is persistent and policy reacts (**simulation**). | |
| 82 | 84 | |
| 83 | −*Unindexed wages ($\eta_w = 0$, level FRM; run stalled at residual $2.0\times10^{-3}$, labelled).* A 10 % one-period real wage loss (0.075 per household, 0.085 per owner) dwarfs the nominal revaluation (0.023): $p_0 = -3.9$ %, $\rho_0 = -3.2$ %, ownership $-0.63$ pp (GE) vs $-3.2$ pp (PE), consumption $-4.4$ % on impact, CEV $-3.3$ % for all, $-2.4$ % for mortgaged owners (0 % better off), $-3.6$ % for renters. *Supported:* Proposition 8.4's condition — an indebted owner loses from the surprise when the wage-indexation loss exceeds the debt erosion (identity + simulation). | |
| 85 | +*No segmentation ($\sigma = 0$) — disagreement (2).* $p_0 = -0.39$ % (vs $-0.45/-0.47$ % segmented), $\rho_0 = -1.47$ %, ownership $-0.16$ pp (GE) vs $-0.20$ pp (PE), CEV identical within 0.05 pp. **Classification: Prop. 7's neutrality holds under unit demand (proved) and fails with an intensive margin (sizes, rented services): the wealth loss of net creditors lowers *total* housing demand, so the single-stock price falls almost as much as the segmented owner-occupied price; segmentation adds only ≈ 15 % to the impact response. Reconciliation: Prop. 7 is restated with unit demand in the statement and a scope remark; the Part B result is reported as a result of the paper (the price feedback in the quantitative model is mostly a wealth effect on total demand, not the composition effect of Part A).** | |
| 84 | 86 | |
| 85 | −*Real-rate response ($\phi_r = 0.5$: real rate $+5$ pp per period in the two surprise periods, persistent surprise).* The rate response dominates: $p_0 = -7.3$ % (FRM) / $-6.9$ % (5-year term), $\rho_0 = -2.3$ / $-1.9$ %, ownership $-0.13$ / $-0.26$ pp (GE) vs $-2.6$ / $-2.9$ pp (PE). CEV of mortgaged owners: FRM $+1.43$ % (70 % better off) vs 5-year term $+0.20$ % (56 %) — the renewal at the higher real rate claws back most of the term borrowers' gain, as Proposition 8.1 states for $\phi_r > 0$; debt-free owners $-1.0$ % and renters $-0.3$ % (the higher deposit return cushions savers), all households $+0.14$ % (FRM) / $-0.25$ % (term). *Supported:* the FRM-vs-renewable-term distinction of the Canadian institutional setting matters precisely when the surprise is persistent and monetary policy reacts (simulation; sign as in Proposition 8). | |
| 87 | +*Near-fixed supply ($\varepsilon = 0.05$; stalled at $3.1\times10^{-3}$, labelled).* $p_0 = -0.50$ % vs $-0.45/-0.47$ %; ownership $-0.01$ pp (GE) vs $-0.19$ pp (PE). The impact elasticity is $\varepsilon\delta_H$ (0.05 vs 0.0025): shock-period supply is essentially fixed in both cases (Cor. 6c composition result; **simulation**). *Not supported:* a quantitatively important role of the long-run elasticity in the inflation experiment. | |
| 86 | 88 | |
| 87 | −*No segmentation ($\sigma = 0$, one stock, level FRM).* $p_0 = -0.39$ % (vs $-0.45/-0.47$ % with segmented stocks), $\rho_0 = -1.47$ %, ownership $-0.16$ pp (GE) vs $-0.20$ pp (PE), CEV identical to the segmented case within 0.05 pp. **Proposition 7's neutrality does not carry over to Part B**: with an intensive housing margin (house sizes, rented services), the wealth loss of net nominal creditors lowers *total* housing demand, so the single-stock price falls almost as much as the segmented owner-occupied price; segmentation itself adds only about 15 % to the impact response. The price feedback in the quantitative model is therefore mostly a wealth effect on total housing demand, not the tenure-composition effect of Part A (simulation; a documented limitation of the unit-demand proposition). | |
| 89 | +*Static expectations (A6).* $p_0 = -0.36$ % (vs $-0.45/-0.47$ %), ownership $+0.02$ pp, CEV identical to two decimals, $q_{RO}$ at 55–59 0.806 → 0.797 in both: **A6 is innocuous for the size of the effects and the sign is negative under both expectation schemes** (**simulation**). | |
| 88 | 90 | |
| 89 | −*Near-fixed supply ($\varepsilon = \varepsilon_r = 0.05$, level FRM; run stalled at residual $3.1\times10^{-3}$, labelled).* $p_0 = -0.50$ % vs $-0.45/-0.47$ % with $\varepsilon = 1$; ownership $-0.01$ pp (GE) vs $-0.19$ pp (PE); CEV identical to the elastic case. The long-run elasticity barely matters on impact because the stock is durable: the impact elasticity is $\varepsilon\delta_H$ (0.05 vs 0.0025), so in both cases the shock-period supply is essentially fixed and the price absorbs the demand shift; Corollary 6c's prediction (fixed stock → pure composition effect, no ownership change) is what the GE numbers show in both cases. *Supported:* the composition result (simulation); *not supported:* a quantitatively important role for the long-run supply elasticity in the inflation experiment. | |
| 91 | +### 5a. Negative house-price shocks (`baseline/price_drop_accounting.csv`) — **identity (PE, prices fixed after the drop)** | |
| 92 | +Existing owners: mean LTV 0.27; **nobody underwater after a 10/20/30 % fall** (first annuity payment, 20 % of the loan, made within the purchase period; the 5-year period understates peak LTV — limitation). New borrowers at origination (mean LTV 0.60): 0 / 40 / 48 % underwater ($x^* = 1 - 0.8/0.95 \approx 0.16$ at the 80 % cap). Loss / net worth 12 / 24 / 37 % (all), 17 / 35 / 52 % (new). | |
| 90 | 93 | |
| 91 | −*Static expectations (Assumption A6: stationary continuation values, prices clear period by period; level FRM).* $p_0 = -0.36$ % (vs $-0.45/-0.47$ % with perfect foresight), $\rho_0 = -1.11$ %, ownership $+0.02$ pp on impact (vs $-0.02$ pp), CEV identical to 2 decimals. **Perfect foresight vs. static expectations makes almost no difference here**: the price response is small in both cases and the threshold/mobility statistics agree to 3 decimals ($q_{RO}$ at 55–59: 0.806 → 0.797 on impact in both). The A6 assumption of Proposition 9 is therefore innocuous *for the size* of the effects in this calibration; the *sign* of the price response is negative under both expectation schemes (simulation). | |
| 94 | +### 5b. Real-rate shock +2 pp for 20 years (`inflation/rate_shock_theta*`) — **simulation (GE)** | |
| 95 | +$\theta = 0.20$ converged ($1.0\times10^{-3}$), $\theta = 0.40$ converged ($1.1\times10^{-3}$), $\theta = 0.05$ **not converged ($8.8\times10^{-3}$ after warm start + 80-step tâtonnement; indicative only)**. $p_0 = -15.5 / -14.6$ % ($-16.2$ % unconverged), overshoot $+7$ % at 20 years, $\rho_0 = -5.3 / -5.2$ %. Ownership on impact $+0.01 / +0.16$ pp (GE) vs $-4.1 / -1.8$ pp (PE; $-5.6$ pp at $\theta = 0.05$): the PE ranking is Prop. 4's leverage-exposure ordering, removed in GE. Owners lose 0.11 / 0.10 of net worth, consumption $-6.0 / -5.7$ %; renters' consumption $-3.9 / -3.7$ %. CEV all $+2.3 / +2.6$ %; mortgaged owners $+1.0 / +1.4$ % (65 % better off); debt-free owners $+3.0 / +3.4$ %; renters $+3.0 / +3.2$ %. Threshold 0.546 → 0.494 ($\theta = 0.40$), $q_{RO}$ 0.351 → 0.380 ($\theta = 0.20$). *Supported:* leverage amplifies the housing-equity loss (**identity**); with FRM contracts and net creditor positions a real-rate shock is on average welfare-improving here (**simulation, illustrative**); no default ⇒ the channel runs through consumption and entry. | |
| 92 | 96 | |
| 93 | −### 5a. Negative house-price shocks — accounting (`baseline/price_drop_accounting.csv`; PE, prices fixed after the drop) | |
| 94 | −Existing owners: mean LTV 0.27; **no household is underwater after a 10, 20 or 30 % price fall** (the maximum LTV carried across 5-year periods is 0.63 because the first annuity payment, 20 % of the loan, is made within the purchase period; the underwater threshold $x^* = 1 - b/((1-\kappa_s)ph)$ exceeds 30 % for everybody). New borrowers evaluated at origination (mean LTV 0.60): 0 % underwater at $-10$ %, 40 % at $-20$ %, 48 % at $-30$ % (identity: Proposition 4's $x^* = 16$ % at the 80 % cap with $\kappa_s = 0.05$). Mean loss relative to net worth 12 / 24 / 37 % for all owners, 17 / 35 / 52 % for new borrowers. *Supported:* the leverage-multiplier accounting (identity); the time-aggregation caveat that the 5-year period understates peak LTV exposure (limitation). The GE part (real-rate shock under different $\theta$) is in §5b. | |
| 97 | +### 6. Bequests on/off, transfer tax (`bequests/`, Table 7, Figure 7) — **simulation (GE and PE)**; **theory check of Prop. 5** | |
| 98 | +Bequests off: $p^* = 0.931$, $\rho^* = 0.209$ ($+7.7$ %), ownership 0.581 (under 35: 0.188 vs 0.378), **$q_{RO} = 0.379 \approx q_{OO} = 0.381$, rank–rank $-0.01$: persistence disappears entirely** (Prop. 5; here it is the whole of it because incomes are independent across generations). Gini 0.58; equal-income gap at 40–44 0.95 → 0.73. PE: ownership 0.512 (a 12 pp fall, of which GE offsets all but 5). Halved: $p^* = 0.981$, ownership 0.615, $q_{OO} = 0.49$, rank–rank 0.25. Transfer tax 20 / 50 % with rebate: ownership 0.642 / 0.639, under 35 0.461 / 0.482, $q_{RO}$ 0.39 / 0.43, rank–rank 0.28 / 0.16, Gini 0.53 / 0.52, mean gift 0.28 / 0.12; rebate an equilibrium object. *Supported:* persistence is generated by transfers (**simulation**, consistent with Prop. 5); a rebated transfer tax raises youth ownership and mobility (**simulation, comparative statics, not optimal policy**). | |
| 95 | 99 | |
| 96 | −### 5b. Leverage and an unanticipated real-rate increase (`inflation/rate_shock_theta*`, GE) | |
| 97 | −*Design.* Real rate $+2$ pp per year for 20 years (four periods, perfect foresight after the surprise; $R_t = 1.219$ instead of 1.104), from the stationary equilibrium of each $\theta \in \{0.05, 0.20, 0.40\}$ (the $\theta$ variants are re-solved first); FRM contracts keep their origination rate. This is the general-equilibrium counterpart of the price-drop accounting of §5a: a persistent rise in the real rate is the fundamental that produces a large house-price fall. | |
| 100 | +### 7. Supply elasticity, segmentation (`supply/`, Figure 10) — **simulation (GE)** | |
| 101 | +$\varepsilon$ alone leaves the baseline unchanged (normalisation). $\theta = 0.05$: $p^* = 1.006$ ($\varepsilon = 0.05$) vs 1.005 ($\varepsilon = 2$), ownership 0.636 vs 0.644; $\theta = 0.40$: 0.983 vs 0.989, 0.626 vs 0.615. Direction of Cor. 6c, magnitudes small (the demand shift from $\theta$ is small because the DSTI cap binds for low-income entrants). Mobility: $q_{RO}$ at 35–39 0.374 vs 0.376 ($\theta = 0.05$), 0.308 vs 0.298 ($\theta = 0.40$): **no measurable supply-elasticity effect on mobility** (the contribution-(iv) conjecture is not supported here). $\sigma = 0$ and $0.5$ reproduce the baseline at $p = 1$ by construction. | |
| 98 | 102 | |
| 99 | −*Convergence.* $\theta = 0.20$: converged ($1.0\times10^{-3}$, 244 s); $\theta = 0.40$: converged ($1.1\times10^{-3}$, 351 s); $\theta = 0.05$: the cold-start shooting diverged (initial residual 0.17, Broyden steps rejected); two re-runs (warm start from the $\theta = 0.20$ path, then an 80-step damped tâtonnement pre-phase followed by a fresh Jacobian) brought the residual to $8.8\times10^{-3}$, where Broyden steps are again rejected — the run is **labelled `GE NOT CONVERGED (max residual 8.82e-03)`** and its numbers are indicative only: $p_0 = -16.2$ % (vs $-15.5$ % at $\theta = 0.20$ and $-14.6$ % at $\theta = 0.40$), $\rho_0 = -3.0$ %, ownership $-0.20$ pp (GE) vs $-5.6$ pp (PE), CEV all $+2.1$ %, mortgaged owners $+1.0$ %, renters $+2.6$ %, $q_{RO}$ at 35–39 0.375 → 0.382, age-1 threshold 0.56 → 0.45. The ranking of the PE ownership response ($-5.6 / -4.1 / -1.8$ pp for $\theta = 0.05 / 0.20 / 0.40$) is the leverage-exposure ordering of Proposition 4; the GE responses are all within $\pm 0.2$ pp. | |
| 103 | +### 7b. Multiplicity (`multiplicity/`, Figure 13) — **simulation (GE scan)**; **conjecture closed for this calibration** | |
| 104 | +Baseline: excess demand monotone from $+0.69$ ($p = 0.80$) to $-0.43$ ($p = 1.25$), one root at $p = 1.000$, slope $-3.1$ per unit of $\log p$ at the root, ownership along the clearing curve 0.73 → 0.50 ($d\mu/dp < 0$, analogue of (C)). $\varepsilon = 0.05$: $+0.33 \to -0.26$, one root; $\varepsilon = 0.05$, $\theta = 0.05$: $+0.34 \to -0.24$, one root at $p \approx 1.006$. *Supported:* uniqueness in the scanned range (**simulation**); no Stein-type ladder multiplicity. *Limitation:* the lottery is held fixed in the scan (understates the ladder channel), but the slope of excess demand is an order of magnitude larger than the lottery feedback. | |
| 100 | 105 | |
| 101 | −*Results (converged runs).* $p_0 = -15.5$ % ($\theta = 0.20$) and $-14.6$ % ($\theta = 0.40$), with an overshoot of $+7$ % twenty years later when the rate returns to 2 % (one sign change); $\rho_0 = -5.3 / -5.2$ %. Ownership on impact: $+0.01 / +0.16$ pp in GE vs $-4.1 / -1.8$ pp with prices fixed — the price fall fully offsets the credit tightening. Owners lose 0.11 / 0.10 of a 5-year income of net worth on impact (about 9 % of mean owner net worth) and cut consumption by 6.0 / 5.7 %; renters' net worth rises slightly (0.02) and their consumption falls 3.9 / 3.7 %. Welfare (CEV, households alive at the shock): all $+2.3 / +2.6$ % — net nominal creditors gain from the higher real return, which dominates the housing-wealth loss; mortgaged owners $+1.0 / +1.4$ % (64 % better off: the FRM shields them from the rate rise while their deposits earn more); debt-free owners $+3.0 / +3.4$ %; renters $+3.0 / +3.2$ %. Mobility: $q_{RO}$ at 35–39 rises on impact (0.351 → 0.380 at $\theta = 0.20$; 0.304 → 0.334 at $\theta = 0.40$) because the price fall lowers the entry threshold (0.546 → 0.494 at $\theta = 0.40$, median $z$). | |
| 106 | +### 8. Rates, trend inflation, wedge, risk, transmission, outside asset (`sensitivity/`, Table 7) — **simulation (GE; PE where stated)** | |
| 107 | +Real rate 1 / 3 %: $p^* = 1.042 / 0.972$, ownership 0.601 / 0.642, estate 0.24 / 0.48, $\lambda_7 = 0.063 / 0.132$, rank–rank 0.26 / 0.44. Trend inflation 0 / 4 %: $p^* = 1.011 / 0.986$, ownership 0.640 / 0.625; **PE** 4 %: under-35 ownership $-4.9$ pp (nominal tilt through the constant-payment schedule and the nominal DSTI cap — **Prop. 3 supported in PE**); GE price fall offsets 65 %. Landlord wedge 0.02 / 0.08: ownership 0.607 / 0.650. Earnings risk 0.10 / 0.20: ownership 0.672 / 0.602, Gini 0.51 / 0.57, gift 0.30 / 0.49, $\lambda_7 = 0.043 / 0.097$. $\rho_g = 0.4$: $q_{RO} = 0.22$, $\lambda_3 = 0.32$, rank–rank 0.41: **most of the measured tenure persistence beyond the baseline comes from income transmission, not housing** (Fagereng et al. caveat; supported). Deposit premium +2 pp: threshold 0.75, estate 0.77, rank–rank 0.57. | |
| 102 | 108 | |
| 103 | −*Supported.* Leverage amplifies the *housing-equity* loss (identity) but, with FRM contracts and net nominal creditor positions, a real-rate shock is on average welfare-improving in this calibration; the down-payment constraint has a second-order effect on the price response ($-15.5$ vs $-14.6$ %) and a first-order effect on the PE ownership response ($-4.1$ vs $-1.8$ pp) that general equilibrium removes (simulation; illustrative). *Limitation:* no default — with LTVs below 0.63 after the first payment nobody is underwater, so the leverage channel operates through consumption and entry, not through foreclosure. | |
| 109 | +### 9. Welfare (`welfare/welfare_all.csv`, Table 8, Figure 11) — **simulation (GE)** | |
| 110 | +Only scenarios that keep the preference parameters are welfare comparisons (bequests off shows $+11$ % purely because the warm-glow disutility disappears). Entrants: $\theta = 0.05$: $+0.34$ % overall, $+0.55$ % bottom two quintiles, $+0.30$ % marginal, $-0.08$ % top; $\theta = 0.40$: $-0.23$ %, $-0.40$ %, $-0.33$ %, $+0.32$ %. Transfer tax with rebate: $+7.5 / +10.9$ % (entrants only; the taxed parents are not in the comparison). $\varepsilon = 0.05 \times \theta = 0.05$: $+0.42$ % vs $+0.29$ % ($\varepsilon = 2$). Numerical variants: $|\lambda^{CE}| \le 0.03$ %. *Supported:* marginal households gain from a looser constraint and lose from a tighter one (answers the unsigned question for this calibration). *Not meaningful:* CEVs across preference parameters. | |
| 104 | 111 | |
| 105 | −### 6. Bequests on/off, transfer tax (`bequests/`, Table 7, Figure 7) | |
| 106 | −Bequests off ($\nu = 0$; GE): $p^* = 0.931$ ($-6.9$ %), $\rho^* = 0.209$ ($+7.6$ %), ownership 0.581 (under 35: 0.188 vs 0.378), **$q_{RO} = 0.379 \approx q_{OO} = 0.381$ at 35–39 and rank–rank slope $-0.005$: intergenerational persistence disappears entirely without transfers** (Proposition 5: persistence requires bequeathed wealth; here it is the whole of it, since incomes are independent across generations by design). Gini of net worth 0.577 (higher than 0.542: transfers lift the bottom of the young distribution more than the top). Equal-income gap at 40–44 falls from 0.95 to 0.73. PE (prices fixed) bequests off: ownership 0.512 — the GE price fall offsets 8 of the 12 pp. Bequests halved ($\nu = 1$): $p^* = 0.981$, ownership 0.615, $q_{RO} = 0.368$, $q_{OO} = 0.492$, rank–rank 0.25. Transfer tax with lump-sum rebate to entrants: $\tau_b = 0.2 / 0.5 \Rightarrow$ ownership 0.642 / 0.639 (under 35: 0.461 / 0.482, *higher* than the baseline 0.378 because the rebate gives every entrant liquid wealth), $q_{RO}$ 0.39 / 0.43, $q_{OO}$ 0.57 / 0.55, rank–rank 0.28 / 0.16, Gini 0.53 / 0.52, mean gift 0.28 / 0.12. *Supported:* persistence is generated by transfers (simulation, consistent with Proposition 5); a rebated transfer tax raises youth ownership and mobility in this calibration (simulation, illustrative). The rebate is an equilibrium object (fixed point converged, `rebate` field). | |
| 112 | +### 10. Robustness (`robustness/`, `grid_sensitivity.csv`, Table 9) — **simulation (GE)** | |
| 113 | +Coarse grids: ownership $-0.5$ pp, $p^*$ $-0.8$ %, threshold $+0.07$, $\lambda_7$ $+0.017$; fine grids: $+0.4$ pp, $-0.1$ %, $-0.02$; tolerance $10^{-5}$: identical; 240 000 dynasties / alternative seed: ownership within 0.1 pp, $p^*$ within 0.01 %, $\lambda_7$ within 0.004, rank–rank within 0.002. Economic variants: $\beta = 0.94 / 0.98$: $p^* = 0.933 / 1.075$, ownership 0.58 / 0.68; $\gamma = 3$: 0.646; $\alpha = 0.25$: $p^* = 1.10$; transaction costs low / high: $p^* = 1.044 / 0.962$; $\chi = 0.05$: 0.644. Mean owner Bellman residuals $4\times10^{-4}$ to $1.9\times10^{-3}$ across 16 runs. | |
| 107 | 114 | |
| 108 | −### 7. Supply elasticity, segmentation, multiplicity (`supply/`, `multiplicity/`, Figures 10 and 13) | |
| 109 | −Changing $\varepsilon$ alone leaves the baseline equilibrium unchanged (every supply curve passes through the normalisation point $(p, H) = (1, \bar H)$): the elasticity matters only in combination with a demand shift, which is why the $\varepsilon \times \theta$ cells are reported. $\theta = 0.05$: $p^* = 1.0060$ with $\varepsilon = 0.05$ vs 1.0046 with $\varepsilon = 2$; ownership 0.636 vs 0.644; $\theta = 0.40$: $p^* = 0.983$ vs 0.989, ownership 0.626 vs 0.615. **Direction consistent with Corollary 6c** (inelastic supply → larger price response, smaller quantity response) **but the magnitudes are small** because the demand shift from $\theta$ is itself small (the DSTI cap binds for low-income entrants whatever $\theta$). Mobility: $q_{RO}$ at 35–39 is 0.374 ($\varepsilon=0.05$) vs 0.376 ($\varepsilon=2$) at $\theta = 0.05$ and 0.308 vs 0.298 at $\theta = 0.40$ — no measurable supply-elasticity effect on mobility at these demand shifts. No segmentation ($\sigma = 0$) and half segmentation reproduce the baseline exactly at $p = 1$ (by construction of the normalisation); the segmentation test is therefore in the inflation transition (`frm_level_noseg`, §4) and in the multiplicity scan. Multiplicity scan: §7b. | |
| 115 | +## Classification of the three theory–simulation disagreements (Phase 4 decision) | |
| 110 | 116 | |
| 111 | −### 7b. Search for multiple stationary equilibria (`multiplicity/price_scan_*.csv`, Figure 13) | |
| 112 | −Owner-market excess demand along $p \in [0.80, 1.25]$ (step 0.025) with the rental market cleared at each point and the estate lottery held at its baseline value. Baseline ($\varepsilon = 1$): excess demand falls monotonically from $+0.69$ at $p = 0.80$ to $-0.43$ at $p = 1.25$, with a single sign change at $p = 1.000$; ownership along the clearing curve falls monotonically from 0.73 to 0.50, i.e. $d\mu/dp < 0$ everywhere — the analogue of condition (C) (threshold channel dominating the ladder channel) holds with a wide margin. Near-fixed supply ($\varepsilon = 0.05$): excess demand monotone from $+0.33$ to $-0.26$, single root at $p = 1.000$, ownership 0.70 → 0.54 along the curve; near-fixed supply with $\theta = 0.05$: monotone from $+0.34$ to $-0.24$, single root at $p \approx 1.006$, ownership 0.71 → 0.55. `multiplicity_summary.json`: one sign change and monotone decreasing excess demand on all three curves. *Supported:* uniqueness of the stationary equilibrium in the scanned range for this calibration (simulation); no evidence for the Stein-type ladder multiplicity conjectured in `failed_hypotheses.md` §1. *Limitation:* the scan holds the lottery fixed; the lottery feedback (bequeathed equity rising with $p$) is the ladder channel itself, so the scan understates it — but at the equilibrium the lottery-consistent solution coincides with the scan root, and the slope of excess demand ($-3$ per unit of $\log p$ around the root) is an order of magnitude larger than the lottery feedback measured across rounds. | |
| 113 | − | |
| 114 | −### 8. Rates, trend inflation, rent wedge, earnings risk, expectations (`sensitivity/`, Table 7) | |
| 115 | −Real rate 1 % / 3 % (annual): $p^* = 1.042 / 0.972$, ownership 0.601 / 0.642, mean estate 0.24 / 0.48, $\lambda_3 = 0.17 / 0.24$, $\lambda_7 = 0.06 / 0.13$, rank–rank 0.27 / 0.44: a higher real rate raises saving, estates and persistence. Trend inflation 0 % / 4 %: $p^* = 1.011 / 0.986$, ownership 0.640 / 0.625, under 35 0.398 / 0.361; **at fixed prices** 4 % inflation lowers ownership to 0.604 and under-35 ownership to 0.330 ($-4.8$ pp) through the nominal tilt of the constant-payment mortgage and the nominal DSTI cap (Proposition 3, supported in PE); the GE price fall ($-1.4$ %) offsets about two thirds. Landlord wedge $\phi = 0.02 / 0.08$ (rent–price 3.7 / 4.1 %/yr): ownership 0.607 / 0.650. Earnings risk $\sigma = 0.10 / 0.20$: ownership 0.672 / 0.602, Gini 0.51 / 0.57, mean gift 0.30 / 0.49, $\lambda_7 = 0.04 / 0.10$. Intergenerational income correlation $\rho_g = 0.4$: $q_{RO} = 0.223$ vs $q_{OO} = 0.545$ ($\lambda_3 = 0.32$ vs 0.19), rank–rank 0.41 (vs 0.335): **most of the measured tenure persistence beyond the baseline comes from income transmission, not housing** (Remark in the appendix, supported). Deposit premium +2 pp (renter investment return): $p^* = 1.018$, threshold 0.75 (vs 0.49), estates 0.77, $\lambda_3 = 0.28$, rank–rank 0.57: a better outside asset delays purchase but raises estates and persistence. Rent growth (PE, `rent_growth_pe`) and static expectations (`frm_level_static`): §4. | |
| 116 | − | |
| 117 | −### 9. Welfare — consumption-equivalent variations (`welfare/welfare_all.csv`, Table 8, Figure 11) | |
| 118 | −*Definition.* $\lambda^{CE}$ such that scaling non-housing consumption by $1+\lambda^{CE}$ in every period of the baseline makes an age-1 household with state $(z_1, a_1)$ as well off as in the alternative stationary equilibrium; $\Phi$ (warm glow) from cohort simulation under baseline policies, $U = V - \Phi$. Reported by quintile of the transfer received at entry, for households within $\pm10$ % of the age-1 threshold ("marginal"), and overall (stationary $z_1$ weights). **Only scenarios that keep the preference parameters are comparable**: the CEVs listed for $\nu$ (bequests off/half), $\beta$, $\gamma$, $\alpha$ and $\chi$ variants compare value functions under different utilities and are reported in Table 8 only as a reminder of this caveat (e.g. "bequests off" shows $+11$ % purely because the warm-glow disutility disappears from the objective). | |
| 119 | − | |
| 120 | −*Results (entrants, GE vs baseline).* Down payment $\theta = 0.05$: $+0.34$ % overall, $+0.55$ % in the bottom two gift quintiles, $-0.08$ % in the top quintile (which pays the higher price), $+0.30$ % for marginal households; $\theta = 0.40$: $-0.23$ % overall, $-0.40$ % in the bottom quintiles, $+0.32$ % at the top (lower price), $-0.33$ % for marginal households. DSTI cap: $\psi = 0.30 / 0.50$: $+0.02 / -0.05$ %. Supply × credit: $\varepsilon = 0.05$ with $\theta = 0.05$: $+0.42$ % (vs $+0.29$ % with $\varepsilon = 2$). Transfer tax with lump-sum rebate to entrants: $+7.5$ % (20 %) and $+10.9$ % (50 %), decreasing in the gift quintile (bottom $+10.9/+15.4$ %, top $+4.0/+5.9$ %) — the rebate relaxes the entry constraint of the liquidity-constrained young; this is the welfare of *entrants* only, the parents who pay the tax are not in the comparison. Real rate 1 % / 3 %: $-0.35 / +1.80$ % (top quintile $-2.3 / +4.5$ %); trend inflation 0 % / 4 %: $+0.12 / -0.01$ %; landlord wedge 0.02 / 0.08: $+0.65 / -0.54$ %; earnings risk low / high: $+10.3 / -10.8$ %; intergenerational income correlation 0.4: $+0.03$ %; deposit premium +2 pp: $+8.7$ % (rising to $+15$ % in the top quintile); transaction costs low / high: $-0.19 / -0.10$ %. Numerical variants: $|\lambda^{CE}| \le 0.02$ % (tolerance, sample size, seed). The decomposition check ($V$ simulated vs value function) is within 4.9 % for every cell; the CEV uses the consistent $U = V - \Phi$. | |
| 121 | − | |
| 122 | −*Transitions (households alive at the shock; §4 and §5b).* 10 % level surprise: $-0.64$ % overall, $+0.63$ % mortgaged owners, $-2.1$ % debt-free owners, $-1.0$ % renters (FRM / 5-year term); ARM: $-1.14$ % overall. Real-rate shock: $+2.3$ % overall. | |
| 123 | − | |
| 124 | −*Supported.* Signs and ordering of the credit-constraint CEVs across gift quintiles (simulation; marginal households gain from a lower $\theta$ and lose from a higher one, which answers the unsigned question of `failed_hypotheses.md` §8 for this calibration). *Not supported / not meaningful:* CEVs across preference parameters. | |
| 117 | +| # | Disagreement | What the theory actually assumed | What the quantitative model does | Resolution in the paper | | |
| 118 | +|---|---|---|---|---| | |
| 119 | +| (1) | Prop. 9: $dp > 0$ after a surprise; simulation: $dp < 0$ | mortgage = only nominal position of parents and entrants (implied by A4 on the HTM region, never stated) | households are net nominal creditors (deposits 4.12 > debt 2.47) | **Assumption 7 (nominal positions) made explicit; Prop. 9 restated under it; new Corollary (net nominal creditors): sign of $dp$ = sign of $\Xi$; the simulation is the $\Xi < 0$ case. Residual differences (rent sign, threshold rising through entrants' own deposit losses) attributed to the intensive margin and to entrants' nominal saving, and reported as results.** Theory not bent to the simulation; simulation not discarded. | | |
| 120 | +| (2) | Prop. 7: price neutrality without segmentation; simulation: $p_0 = -0.39$ % vs $-0.45$ % | unit demand (one unit per household regardless of wealth) | intensive margin (sizes, rented services) | **Prop. 7 restated "under unit demand" with a scope remark; the Part B result (price feedback = wealth effect on total demand, segmentation adds ≈ 15 %) reported as a finding.** | | |
| 121 | +| (3) | Prop. 2: $\partial\hat b/\partial\theta > 0$; simulation: non-monotone at the median $z$ (within grid noise), flat at the lowest $z$ | binding LTV constraint; once-and-for-all tenure decision ($V^R$ independent of $\theta$) | renters can buy later (option value); DSTI cap binds for low $z$ | **Prop. 2 restated with the binding-LTV premise and the no-later-purchase restriction named in the statement and in a scope remark; the simulation reported with its grid sensitivity.** | | |
| 125 | 122 | |
| 126 | −### 10. Robustness (`robustness/`, `grid_sensitivity.csv`, Table 9) | |
| 127 | −All variants converged in GE. Numerical variants (deviation from baseline): coarse grids ($n_a=30, n_b=12, n_x=80$): ownership $-0.5$ pp, $p^*$ $-0.9$ %, Gini $+0.000$, threshold $+0.07$, $\lambda_7$ $+0.017$; fine grids ($80/30/180$): ownership $+0.4$ pp, $p^*$ $-0.1$ %, threshold $-0.02$; tolerance $10^{-5}$: identical to the baseline (same best iterate); panel of 240 000 dynasties and alternative seed: ownership identical to 3 decimals, $p^*$ within $10^{-4}$, $\lambda_7$ within 0.004, rank–rank within 0.002. **The ownership rate, prices, Gini and the rank–rank slope are robust to grids, tolerance, sample size and seed; the age-1 threshold and $\lambda_7$ are the least robust statistics** (grid-dependent at the $\pm 0.07$ and $\pm 0.02$ level). Economic variants: $\beta_{annual} = 0.94 / 0.98$: $p^* = 0.933 / 1.075$, ownership 0.58 / 0.68; $\gamma = 3$: 0.646; $\alpha = 0.25$: $p^* = 1.10$, ownership 0.627; transaction costs low / high: $p^* = 1.044 / 0.962$, ownership 0.622 / 0.633; ownership premium $\chi = 0.05$: 0.644. Bellman residuals (owners, mean) between $4\times10^{-4}$ (fine grids) and $1.9\times10^{-3}$ ($\gamma = 3$, coarse grids). | |
| 123 | +Two further hypotheses from the brief are reported as negative results: multiple equilibria (none found, §7b) and a first-order supply-elasticity effect on mobility (not found, §7). | |
| 128 | 124 | |
| 129 | −## Synthesis — what the Phase 3 simulations support (for Phase 4) | |
| 125 | +## Synthesis — what the paper claims, with status | |
| 130 | 126 | |
| 131 | 127 | | Claim | Status | Where | |
| 132 | 128 | |---|---|---| |
| 133 | −| Ownership-entry threshold exists, single crossing, buyer hand-to-mouth at the threshold | supported (numerical check of proof gap 6; A4 holds for buyers, fails for renters) | §1 | | |
| 134 | −| Threshold increasing in $\theta$ (Prop. 2) | supported for states where the LTV cap binds; **not monotone** at the median state in PE (option value of waiting; within grid sensitivity) | §3 | | |
| 135 | −| Price absorbs credit shifts; fixed stock → composition only (Cor. 6c) | supported: GE removes 50–100 % of the PE ownership effect; $\varepsilon$ matters little on impact (impact elasticity $\varepsilon\delta_H$) | §3, §4, §5b, §7 | | |
| 136 | −| Persistence requires bequeathed wealth (Prop. 5) | supported: $\lambda \to 0$, rank–rank $\to 0$ without transfers; $\rho_g$ adds more than housing | §6, §8 | | |
| 137 | −| Uniqueness / no ladder multiplicity (Prop. 6, (C)) | supported: monotone excess demand, one root, incl. near-fixed supply | §7b | | |
| 138 | −| Neutrality without segmentation (Prop. 7) | **not supported** in Part B (intensive margin: wealth effect on total demand) | §4 | | |
| 139 | −| Revaluation accounting, zero-sum, contract ordering (Prop. 8) | supported (identity + simulation), incl. the term-vs-FRM distinction under a real-rate response | §4 | | |
| 140 | −| Surprise inflation raises $p$, lowers $\rho$, raises $\lambda$ (Prop. 9) | **sign of $dp$ reversed** (households net nominal creditors); $q_{RO}$ at the gift age falls and $\lambda_7$ rises on impact as predicted, but by < 0.01; A6 vs perfect foresight immaterial | §4 | | |
| 141 | −| Tilt of anticipated inflation (Prop. 3) | supported in PE (nominal DSTI cap), two-thirds offset in GE | §8, §4 | | |
| 142 | −| Welfare of marginal households (unsigned in theory) | gain from lower $\theta$ ($+0.3$ %), lose from higher ($-0.3$ %) | §9 | | |
| 143 | − | |
| 144 | −Numerical caveats to carry into the paper: Monte-Carlo noise floor ($5\times10^{-4}$ stationary, $\sim10^{-3}$ transitions) and the four transitions reported with residuals above tolerance; the transitory income component and the unconditional estate lottery as modelling approximations; the illustrative calibration (no fitted moment; supply elasticity without a verified Canadian source); the old-age sale of houses; the 5-year period's understatement of peak LTV. | |
| 129 | +| Ownership-entry threshold = feasibility + liquidity component; single crossing; buyer HTM at the threshold | **theory** (Prop. 1 on the HTM region) + **numerical check** of the proof gap | §1 | | |
| 130 | +| Threshold ↑ in $\theta$ (Prop. 2) | **theory** under binding LTV + once-and-for-all decision; **contradicted within noise** outside (option to buy later) | §3 | | |
| 131 | +| Prices absorb credit shifts; fixed stock → composition only (Cor. 6c) | **simulation**: GE removes 64–66 % (stationary $\theta$) and 77–92 % (transitions) of the PE ownership effect; $\varepsilon$ matters little on impact | §3, §4, §5b, §7 | | |
| 132 | +| Persistence requires bequeathed wealth (Prop. 5); half-life, not trap | **theory** + **simulation**: $\lambda$, rank–rank → 0 without transfers; $\rho_g$ adds more than housing | §6, §8 | | |
| 133 | +| Uniqueness / no ladder multiplicity | **simulation** (scan): monotone excess demand, one root incl. near-fixed supply; multiplicity remains a **conjecture** in general | §7b | | |
| 134 | +| Neutrality without segmentation (Prop. 7) | **theory under unit demand**; **not supported** with an intensive margin | §4 | | |
| 135 | +| Revaluation accounting, zero-sum, contract ordering (Prop. 8) | **identity** + **simulation**, incl. the term-vs-FRM distinction under a real-rate response | §4 | | |
| 136 | +| Surprise inflation raises $p$ (Prop. 9) | **theory under Assumption 7**; **sign reversed in the simulation** (net creditors), as the new Corollary allows; mobility effects in the predicted direction but < 0.01 | §4 | | |
| 137 | +| Tilt of anticipated inflation (Prop. 3) | **theory**; **supported in PE**, two-thirds offset in GE | §8, §4 | | |
| 138 | +| Welfare of marginal households (unsigned in theory) | **simulation**: $+0.3$ % for $\theta = 0.05$, $-0.3$ % for $\theta = 0.40$ | §9 | | |
| 139 | +| Supply elasticity × contract → mobility (contribution iv) | **simulation**: effect not measurable at these demand shifts | §7 | | |
| 140 | + | |
| 141 | +Numerical caveats carried into the paper: Monte-Carlo noise floor ($5\times10^{-4}$ stationary, $\sim10^{-3}$ transitions) and the four transitions reported with residuals above tolerance; the transitory income component and the unconditional estate lottery; the illustrative calibration (no fitted moment; supply elasticity without a verified Canadian source); the old-age sale of houses; the 5-year period's understatement of peak LTV. | |
| 142 | + | |
| 143 | +## Traceability check (Phase 4) | |
| 144 | + | |
| 145 | +* `src/tables.py` generates `numbers.tex` (≈ 2 000 macros, all read from `results/data/`); every number in the manuscript's prose is a macro (`grep -c '\\\\[A-Z][A-Za-z]*' paper/sections/*.tex`); a script check confirms that every macro used in `paper/` is defined in `numbers.tex`. | |
| 146 | +* `make test`: 69 tests pass (identities, debt law, feasibility, terminal condition, dynastic link, lottery, common random numbers, monotonicity, single crossing, Bellman residuals, two slow GE tests). | |
| 147 | +* `make all` = test → simulate (skipped when `results/data/diagnostics.json` exists) → figures → tables → paper. | |
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@@ -1,70 +1,799 @@ | ||
| 1 | 1 | % generated by src/tables.py — one macro per key number; every value traces to results/data/ |
| 2 | +\newcommand{\AbsorbThetaFivePct}{64} | |
| 3 | +\newcommand{\AbsorbThetaFortyPct}{66} | |
| 4 | +\newcommand{\AlphaHighEstate}{0.37} | |
| 5 | +\newcommand{\AlphaHighGapFour}{1.00} | |
| 6 | +\newcommand{\AlphaHighGift}{0.41} | |
| 7 | +\newcommand{\AlphaHighGini}{0.54} | |
| 8 | +\newcommand{\AlphaHighLamSeven}{0.065} | |
| 9 | +\newcommand{\AlphaHighLamThree}{0.22} | |
| 10 | +\newcommand{\AlphaHighOwn}{0.627} | |
| 11 | +\newcommand{\AlphaHighOwnUthirtyfive}{0.359} | |
| 12 | +\newcommand{\AlphaHighP}{1.101} | |
| 13 | +\newcommand{\AlphaHighPDevPct}{10.1} | |
| 14 | +\newcommand{\AlphaHighRR}{0.36} | |
| 15 | +\newcommand{\AlphaHighRebate}{0.000} | |
| 16 | +\newcommand{\AlphaHighRho}{0.222} | |
| 17 | +\newcommand{\AlphaHighRhoDevPct}{14.5} | |
| 18 | +\newcommand{\AlphaHighThr}{0.54} | |
| 19 | +\newcommand{\AlphaHighThrHighZ}{0.05} | |
| 20 | +\newcommand{\AlphaHighThrLowZ}{1.13} | |
| 21 | +\newcommand{\AlphaHighqOO}{0.508} | |
| 22 | +\newcommand{\AlphaHighqRO}{0.289} | |
| 23 | +\newcommand{\ArmLevelAbsorbPct}{76} | |
| 24 | +\newcommand{\ArmLevelBetterAllPct}{18} | |
| 25 | +\newcommand{\ArmLevelBetterOMPct}{7} | |
| 26 | +\newcommand{\ArmLevelBetterONMPct}{1} | |
| 27 | +\newcommand{\ArmLevelBetterOPct}{4} | |
| 28 | +\newcommand{\ArmLevelBetterRPct}{38} | |
| 29 | +\newcommand{\ArmLevelCEVAll}{\ensuremath{-}1.14} | |
| 30 | +\newcommand{\ArmLevelCEVAllAbs}{1.14} | |
| 31 | +\newcommand{\ArmLevelCEVAllPE}{\ensuremath{-}1.19} | |
| 32 | +\newcommand{\ArmLevelCEVAllPEAbs}{1.19} | |
| 33 | +\newcommand{\ArmLevelCEVO}{\ensuremath{-}1.36} | |
| 34 | +\newcommand{\ArmLevelCEVOAbs}{1.36} | |
| 35 | +\newcommand{\ArmLevelCEVOM}{\ensuremath{-}1.03} | |
| 36 | +\newcommand{\ArmLevelCEVOMAbs}{1.03} | |
| 37 | +\newcommand{\ArmLevelCEVONM}{\ensuremath{-}2.14} | |
| 38 | +\newcommand{\ArmLevelCEVONMAbs}{2.14} | |
| 39 | +\newcommand{\ArmLevelCEVR}{\ensuremath{-}0.96} | |
| 40 | +\newcommand{\ArmLevelCEVRAbs}{0.96} | |
| 41 | +\newcommand{\ArmLevelConsAll}{\ensuremath{-}1.5} | |
| 42 | +\newcommand{\ArmLevelConsAllAbs}{1.5} | |
| 43 | +\newcommand{\ArmLevelConsO}{\ensuremath{-}1.4} | |
| 44 | +\newcommand{\ArmLevelConsOAbs}{1.4} | |
| 45 | +\newcommand{\ArmLevelConsR}{\ensuremath{-}1.6} | |
| 46 | +\newcommand{\ArmLevelConsRAbs}{1.6} | |
| 47 | +\newcommand{\ArmLevelDOwnGE}{\ensuremath{-}0.11} | |
| 48 | +\newcommand{\ArmLevelDOwnGEAbs}{0.11} | |
| 49 | +\newcommand{\ArmLevelDOwnPE}{\ensuremath{-}0.47} | |
| 50 | +\newcommand{\ArmLevelDOwnPEAbs}{0.47} | |
| 51 | +\newcommand{\ArmLevelDebtZeroPct}{1.5} | |
| 52 | +\newcommand{\ArmLevelDepLoss}{0.48} | |
| 53 | +\newcommand{\ArmLevelErosion}{0.00} | |
| 54 | +\newcommand{\ArmLevelLamSevenSS}{0.060} | |
| 55 | +\newcommand{\ArmLevelLamSevenZero}{0.065} | |
| 56 | +\newcommand{\ArmLevelLender}{0.48} | |
| 57 | +\newcommand{\ArmLevelMaxRes}{1.1\times10^{-3}} | |
| 58 | +\newcommand{\ArmLevelNWAll}{\ensuremath{-}0.03} | |
| 59 | +\newcommand{\ArmLevelNWAllAbs}{0.03} | |
| 60 | +\newcommand{\ArmLevelNWO}{\ensuremath{-}0.05} | |
| 61 | +\newcommand{\ArmLevelNWOAbs}{0.05} | |
| 62 | +\newcommand{\ArmLevelNWR}{\ensuremath{-}0.01} | |
| 63 | +\newcommand{\ArmLevelNWRAbs}{0.01} | |
| 64 | +\newcommand{\ArmLevelNetO}{\ensuremath{-}0.05} | |
| 65 | +\newcommand{\ArmLevelNetOAbs}{0.05} | |
| 66 | +\newcommand{\ArmLevelNetR}{\ensuremath{-}0.02} | |
| 67 | +\newcommand{\ArmLevelNetRAbs}{0.02} | |
| 68 | +\newcommand{\ArmLevelNominalRateZero}{0.219} | |
| 69 | +\newcommand{\ArmLevelOwnerOldConsMax}{\ensuremath{-}2.9} | |
| 70 | +\newcommand{\ArmLevelOwnerOldConsMaxAbs}{2.9} | |
| 71 | +\newcommand{\ArmLevelOwnerOldConsMin}{\ensuremath{-}2.2} | |
| 72 | +\newcommand{\ArmLevelOwnerOldConsMinAbs}{2.2} | |
| 73 | +\newcommand{\ArmLevelOwnerOldLossMax}{0.10} | |
| 74 | +\newcommand{\ArmLevelOwnerOldLossMin}{0.06} | |
| 75 | +\newcommand{\ArmLevelOwnerYoungConsMax}{\ensuremath{-}0.4} | |
| 76 | +\newcommand{\ArmLevelOwnerYoungConsMaxAbs}{0.4} | |
| 77 | +\newcommand{\ArmLevelOwnerYoungConsMin}{\ensuremath{-}0.7} | |
| 78 | +\newcommand{\ArmLevelOwnerYoungConsMinAbs}{0.7} | |
| 79 | +\newcommand{\ArmLevelOwnerYoungGainMax}{\ensuremath{-}0.02} | |
| 80 | +\newcommand{\ArmLevelOwnerYoungGainMaxAbs}{0.02} | |
| 81 | +\newcommand{\ArmLevelPriceBackYears}{25} | |
| 82 | +\newcommand{\ArmLevelPriceMaxPos}{0.1} | |
| 83 | +\newcommand{\ArmLevelPriceSignChanges}{1} | |
| 84 | +\newcommand{\ArmLevelPzero}{\ensuremath{-}0.79} | |
| 85 | +\newcommand{\ArmLevelPzeroAbs}{0.79} | |
| 86 | +\newcommand{\ArmLevelRentGainR}{0.00} | |
| 87 | +\newcommand{\ArmLevelResT}{2\times10^{-4}} | |
| 88 | +\newcommand{\ArmLevelRhoZero}{\ensuremath{-}1.15} | |
| 89 | +\newcommand{\ArmLevelRhoZeroAbs}{1.15} | |
| 90 | +\newcommand{\ArmLevelSecs}{153} | |
| 91 | +\newcommand{\ArmLevelSteps}{5} | |
| 92 | +\newcommand{\ArmLevelThrChangePct}{6} | |
| 93 | +\newcommand{\ArmLevelThrSS}{0.49} | |
| 94 | +\newcommand{\ArmLevelThrZero}{0.52} | |
| 95 | +\newcommand{\ArmLevelWageAll}{0.00} | |
| 96 | +\newcommand{\ArmLevelWageO}{0.00} | |
| 97 | +\newcommand{\ArmLevelqROsevenSS}{0.806} | |
| 98 | +\newcommand{\ArmLevelqROsevenZero}{0.802} | |
| 99 | +\newcommand{\ArmLevelqROthreeOne}{0.337} | |
| 100 | +\newcommand{\ArmLevelqROthreeSS}{0.351} | |
| 101 | +\newcommand{\ArmLevelqROthreeZero}{0.358} | |
| 102 | +\newcommand{\ArmPersAbsorbPct}{83} | |
| 103 | +\newcommand{\ArmPersBetterAllPct}{13} | |
| 104 | +\newcommand{\ArmPersBetterOMPct}{7} | |
| 105 | +\newcommand{\ArmPersBetterONMPct}{1} | |
| 106 | +\newcommand{\ArmPersBetterOPct}{5} | |
| 107 | +\newcommand{\ArmPersBetterRPct}{24} | |
| 108 | +\newcommand{\ArmPersCEVAll}{\ensuremath{-}1.24} | |
| 109 | +\newcommand{\ArmPersCEVAllAbs}{1.24} | |
| 110 | +\newcommand{\ArmPersCEVAllPE}{\ensuremath{-}1.24} | |
| 111 | +\newcommand{\ArmPersCEVAllPEAbs}{1.24} | |
| 112 | +\newcommand{\ArmPersCEVO}{\ensuremath{-}1.50} | |
| 113 | +\newcommand{\ArmPersCEVOAbs}{1.50} | |
| 114 | +\newcommand{\ArmPersCEVOM}{\ensuremath{-}1.21} | |
| 115 | +\newcommand{\ArmPersCEVOMAbs}{1.21} | |
| 116 | +\newcommand{\ArmPersCEVONM}{\ensuremath{-}2.16} | |
| 117 | +\newcommand{\ArmPersCEVONMAbs}{2.16} | |
| 118 | +\newcommand{\ArmPersCEVR}{\ensuremath{-}1.04} | |
| 119 | +\newcommand{\ArmPersCEVRAbs}{1.04} | |
| 120 | +\newcommand{\ArmPersConsAll}{\ensuremath{-}2.0} | |
| 121 | +\newcommand{\ArmPersConsAllAbs}{2.0} | |
| 122 | +\newcommand{\ArmPersConsO}{\ensuremath{-}2.1} | |
| 123 | +\newcommand{\ArmPersConsOAbs}{2.1} | |
| 124 | +\newcommand{\ArmPersConsR}{\ensuremath{-}1.9} | |
| 125 | +\newcommand{\ArmPersConsRAbs}{1.9} | |
| 126 | +\newcommand{\ArmPersDOwnGE}{\ensuremath{-}0.39} | |
| 127 | +\newcommand{\ArmPersDOwnGEAbs}{0.39} | |
| 128 | +\newcommand{\ArmPersDOwnPE}{\ensuremath{-}2.34} | |
| 129 | +\newcommand{\ArmPersDOwnPEAbs}{2.34} | |
| 130 | +\newcommand{\ArmPersDebtZeroPct}{\ensuremath{-}9.4} | |
| 131 | +\newcommand{\ArmPersDebtZeroPctAbs}{9.4} | |
| 132 | +\newcommand{\ArmPersDepLoss}{0.48} | |
| 133 | +\newcommand{\ArmPersErosion}{0.00} | |
| 134 | +\newcommand{\ArmPersLamSevenSS}{0.060} | |
| 135 | +\newcommand{\ArmPersLamSevenZero}{0.069} | |
| 136 | +\newcommand{\ArmPersLender}{0.48} | |
| 137 | +\newcommand{\ArmPersMaxRes}{3.1\times10^{-4}} | |
| 138 | +\newcommand{\ArmPersNWAll}{\ensuremath{-}0.04} | |
| 139 | +\newcommand{\ArmPersNWAllAbs}{0.04} | |
| 140 | +\newcommand{\ArmPersNWO}{\ensuremath{-}0.05} | |
| 141 | +\newcommand{\ArmPersNWOAbs}{0.05} | |
| 142 | +\newcommand{\ArmPersNWR}{\ensuremath{-}0.01} | |
| 143 | +\newcommand{\ArmPersNWRAbs}{0.01} | |
| 144 | +\newcommand{\ArmPersNetO}{\ensuremath{-}0.05} | |
| 145 | +\newcommand{\ArmPersNetOAbs}{0.05} | |
| 146 | +\newcommand{\ArmPersNetR}{\ensuremath{-}0.02} | |
| 147 | +\newcommand{\ArmPersNetRAbs}{0.02} | |
| 148 | +\newcommand{\ArmPersNominalRateZero}{0.341} | |
| 149 | +\newcommand{\ArmPersOwnerOldConsMax}{\ensuremath{-}3.1} | |
| 150 | +\newcommand{\ArmPersOwnerOldConsMaxAbs}{3.1} | |
| 151 | +\newcommand{\ArmPersOwnerOldConsMin}{\ensuremath{-}2.4} | |
| 152 | +\newcommand{\ArmPersOwnerOldConsMinAbs}{2.4} | |
| 153 | +\newcommand{\ArmPersOwnerOldLossMax}{0.10} | |
| 154 | +\newcommand{\ArmPersOwnerOldLossMin}{0.06} | |
| 155 | +\newcommand{\ArmPersOwnerYoungConsMax}{\ensuremath{-}0.9} | |
| 156 | +\newcommand{\ArmPersOwnerYoungConsMaxAbs}{0.9} | |
| 157 | +\newcommand{\ArmPersOwnerYoungConsMin}{\ensuremath{-}1.6} | |
| 158 | +\newcommand{\ArmPersOwnerYoungConsMinAbs}{1.6} | |
| 159 | +\newcommand{\ArmPersOwnerYoungGainMax}{\ensuremath{-}0.02} | |
| 160 | +\newcommand{\ArmPersOwnerYoungGainMaxAbs}{0.02} | |
| 161 | +\newcommand{\ArmPersPriceBackYears}{20} | |
| 162 | +\newcommand{\ArmPersPriceMaxPos}{0.1} | |
| 163 | +\newcommand{\ArmPersPriceSignChanges}{1} | |
| 164 | +\newcommand{\ArmPersPzero}{\ensuremath{-}1.86} | |
| 165 | +\newcommand{\ArmPersPzeroAbs}{1.86} | |
| 166 | +\newcommand{\ArmPersRentGainR}{0.00} | |
| 167 | +\newcommand{\ArmPersResT}{6\times10^{-5}} | |
| 168 | +\newcommand{\ArmPersRhoZero}{\ensuremath{-}0.18} | |
| 169 | +\newcommand{\ArmPersRhoZeroAbs}{0.18} | |
| 170 | +\newcommand{\ArmPersSecs}{184} | |
| 171 | +\newcommand{\ArmPersSteps}{4} | |
| 172 | +\newcommand{\ArmPersThrChangePct}{16} | |
| 173 | +\newcommand{\ArmPersThrSS}{0.49} | |
| 174 | +\newcommand{\ArmPersThrZero}{0.57} | |
| 175 | +\newcommand{\ArmPersWageAll}{0.00} | |
| 176 | +\newcommand{\ArmPersWageO}{0.00} | |
| 177 | +\newcommand{\ArmPersqROsevenSS}{0.806} | |
| 178 | +\newcommand{\ArmPersqROsevenZero}{0.807} | |
| 179 | +\newcommand{\ArmPersqROthreeOne}{0.336} | |
| 180 | +\newcommand{\ArmPersqROthreeSS}{0.351} | |
| 181 | +\newcommand{\ArmPersqROthreeZero}{0.358} | |
| 182 | +\newcommand{\BaselineSeconds}{8} | |
| 183 | +\newcommand{\BellmanOwnerMax}{1.9} | |
| 184 | +\newcommand{\BellmanOwnerMaxAll}{1.9\times10^{-3}} | |
| 2 | 185 | \newcommand{\BellmanOwnerMean}{0.00067} |
| 186 | +\newcommand{\BellmanOwnerMinAll}{4.0\times10^{-4}} | |
| 187 | +\newcommand{\BellmanOwnerPnn}{0.6} | |
| 188 | +\newcommand{\BellmanRenterMax}{0.17} | |
| 3 | 189 | \newcommand{\BellmanRenterMean}{0.00016} |
| 4 | −\newcommand{\CEVallarmlevel}{-1.14} | |
| 5 | −\newcommand{\CEVallarmpersistent}{-1.24} | |
| 6 | −\newcommand{\CEVallfrmlevel}{-0.64} | |
| 7 | −\newcommand{\CEVallfrmlevelfixedsupply}{-0.64} | |
| 8 | −\newcommand{\CEVallfrmlevelnoseg}{-0.60} | |
| 9 | −\newcommand{\CEVallfrmlevelrigidrents}{-0.62} | |
| 10 | −\newcommand{\CEVallfrmlevelstatic}{-0.64} | |
| 11 | −\newcommand{\CEVallfrmlevelunindexedwages}{-3.33} | |
| 12 | −\newcommand{\CEVallfrmpersistent}{-0.52} | |
| 190 | +\newcommand{\BeqHalfEstate}{0.21} | |
| 191 | +\newcommand{\BeqHalfGapFour}{0.86} | |
| 192 | +\newcommand{\BeqHalfGift}{0.26} | |
| 193 | +\newcommand{\BeqHalfGini}{0.55} | |
| 194 | +\newcommand{\BeqHalfLamSeven}{0.035} | |
| 195 | +\newcommand{\BeqHalfLamThree}{0.12} | |
| 196 | +\newcommand{\BeqHalfOwn}{0.615} | |
| 197 | +\newcommand{\BeqHalfOwnUthirtyfive}{0.325} | |
| 198 | +\newcommand{\BeqHalfP}{0.981} | |
| 199 | +\newcommand{\BeqHalfPDevPct}{\ensuremath{-}1.9} | |
| 200 | +\newcommand{\BeqHalfPDevPctAbs}{1.9} | |
| 201 | +\newcommand{\BeqHalfRR}{0.25} | |
| 202 | +\newcommand{\BeqHalfRebate}{0.000} | |
| 203 | +\newcommand{\BeqHalfRho}{0.198} | |
| 204 | +\newcommand{\BeqHalfRhoDevPct}{1.7} | |
| 205 | +\newcommand{\BeqHalfThr}{0.44} | |
| 206 | +\newcommand{\BeqHalfThrHighZ}{0.00} | |
| 207 | +\newcommand{\BeqHalfThrLowZ}{1.09} | |
| 208 | +\newcommand{\BeqHalfqOO}{0.492} | |
| 209 | +\newcommand{\BeqHalfqRO}{0.368} | |
| 210 | +\newcommand{\BeqOffEstate}{0.00} | |
| 211 | +\newcommand{\BeqOffGEDropPP}{5} | |
| 212 | +\newcommand{\BeqOffGapFour}{0.73} | |
| 213 | +\newcommand{\BeqOffGift}{0.00} | |
| 214 | +\newcommand{\BeqOffGini}{0.58} | |
| 215 | +\newcommand{\BeqOffLamSeven}{\ensuremath{-}0.003} | |
| 216 | +\newcommand{\BeqOffLamSevenAbs}{0.003} | |
| 217 | +\newcommand{\BeqOffLamThree}{0.00} | |
| 218 | +\newcommand{\BeqOffOwn}{0.581} | |
| 219 | +\newcommand{\BeqOffOwnUthirtyfive}{0.188} | |
| 220 | +\newcommand{\BeqOffP}{0.931} | |
| 221 | +\newcommand{\BeqOffPDevPct}{\ensuremath{-}6.9} | |
| 222 | +\newcommand{\BeqOffPDevPctAbs}{6.9} | |
| 223 | +\newcommand{\BeqOffPEDropPP}{12} | |
| 224 | +\newcommand{\BeqOffPEEstate}{0.00} | |
| 225 | +\newcommand{\BeqOffPEGapFour}{0.74} | |
| 226 | +\newcommand{\BeqOffPEGift}{0.00} | |
| 227 | +\newcommand{\BeqOffPEGini}{0.59} | |
| 228 | +\newcommand{\BeqOffPELamSeven}{\ensuremath{-}0.004} | |
| 229 | +\newcommand{\BeqOffPELamSevenAbs}{0.004} | |
| 230 | +\newcommand{\BeqOffPELamThree}{0.00} | |
| 231 | +\newcommand{\BeqOffPEOwn}{0.512} | |
| 232 | +\newcommand{\BeqOffPEOwnUthirtyfive}{0.106} | |
| 233 | +\newcommand{\BeqOffPEP}{1.000} | |
| 234 | +\newcommand{\BeqOffPEPDevPct}{0.0} | |
| 235 | +\newcommand{\BeqOffPERR}{0.00} | |
| 236 | +\newcommand{\BeqOffPERebate}{0.000} | |
| 237 | +\newcommand{\BeqOffPERho}{0.194} | |
| 238 | +\newcommand{\BeqOffPERhoDevPct}{0.0} | |
| 239 | +\newcommand{\BeqOffPEThr}{0.48} | |
| 240 | +\newcommand{\BeqOffPEThrHighZ}{0.03} | |
| 241 | +\newcommand{\BeqOffPEThrLowZ}{1.15} | |
| 242 | +\newcommand{\BeqOffPEqOO}{0.363} | |
| 243 | +\newcommand{\BeqOffPEqRO}{0.362} | |
| 244 | +\newcommand{\BeqOffRR}{\ensuremath{-}0.01} | |
| 245 | +\newcommand{\BeqOffRRAbs}{0.01} | |
| 246 | +\newcommand{\BeqOffRebate}{0.000} | |
| 247 | +\newcommand{\BeqOffRho}{0.209} | |
| 248 | +\newcommand{\BeqOffRhoDevPct}{7.7} | |
| 249 | +\newcommand{\BeqOffThr}{0.32} | |
| 250 | +\newcommand{\BeqOffThrHighZ}{\ensuremath{-}0.18} | |
| 251 | +\newcommand{\BeqOffThrHighZAbs}{0.18} | |
| 252 | +\newcommand{\BeqOffThrLowZ}{0.93} | |
| 253 | +\newcommand{\BeqOffqOO}{0.381} | |
| 254 | +\newcommand{\BeqOffqRO}{0.379} | |
| 255 | +\newcommand{\BetaHighEstate}{0.62} | |
| 256 | +\newcommand{\BetaHighGapFour}{1.21} | |
| 257 | +\newcommand{\BetaHighGift}{0.44} | |
| 258 | +\newcommand{\BetaHighGini}{0.50} | |
| 259 | +\newcommand{\BetaHighLamSeven}{0.090} | |
| 260 | +\newcommand{\BetaHighLamThree}{0.19} | |
| 261 | +\newcommand{\BetaHighOwn}{0.681} | |
| 262 | +\newcommand{\BetaHighOwnUthirtyfive}{0.408} | |
| 263 | +\newcommand{\BetaHighP}{1.075} | |
| 264 | +\newcommand{\BetaHighPDevPct}{7.5} | |
| 265 | +\newcommand{\BetaHighRR}{0.44} | |
| 266 | +\newcommand{\BetaHighRebate}{0.000} | |
| 267 | +\newcommand{\BetaHighRho}{0.181} | |
| 268 | +\newcommand{\BetaHighRhoDevPct}{\ensuremath{-}7.0} | |
| 269 | +\newcommand{\BetaHighRhoDevPctAbs}{7.0} | |
| 270 | +\newcommand{\BetaHighThr}{0.66} | |
| 271 | +\newcommand{\BetaHighThrHighZ}{0.01} | |
| 272 | +\newcommand{\BetaHighThrLowZ}{1.42} | |
| 273 | +\newcommand{\BetaHighqOO}{0.597} | |
| 274 | +\newcommand{\BetaHighqRO}{0.404} | |
| 275 | +\newcommand{\BetaLowEstate}{0.16} | |
| 276 | +\newcommand{\BetaLowGapFour}{0.69} | |
| 277 | +\newcommand{\BetaLowGift}{0.35} | |
| 278 | +\newcommand{\BetaLowGini}{0.58} | |
| 279 | +\newcommand{\BetaLowLamSeven}{0.043} | |
| 280 | +\newcommand{\BetaLowLamThree}{0.18} | |
| 281 | +\newcommand{\BetaLowOwn}{0.581} | |
| 282 | +\newcommand{\BetaLowOwnUthirtyfive}{0.329} | |
| 283 | +\newcommand{\BetaLowP}{0.933} | |
| 284 | +\newcommand{\BetaLowPDevPct}{\ensuremath{-}6.7} | |
| 285 | +\newcommand{\BetaLowPDevPctAbs}{6.7} | |
| 286 | +\newcommand{\BetaLowRR}{0.23} | |
| 287 | +\newcommand{\BetaLowRebate}{0.000} | |
| 288 | +\newcommand{\BetaLowRho}{0.207} | |
| 289 | +\newcommand{\BetaLowRhoDevPct}{6.4} | |
| 290 | +\newcommand{\BetaLowThr}{0.46} | |
| 291 | +\newcommand{\BetaLowThrHighZ}{0.09} | |
| 292 | +\newcommand{\BetaLowThrLowZ}{1.00} | |
| 293 | +\newcommand{\BetaLowqOO}{0.496} | |
| 294 | +\newcommand{\BetaLowqRO}{0.316} | |
| 295 | +\newcommand{\CEVBeqOffAll}{11.14} | |
| 296 | +\newcommand{\CEVBeqOffBottomTwo}{8.98} | |
| 297 | +\newcommand{\CEVBeqOffMarg}{12.82} | |
| 298 | +\newcommand{\CEVBeqOffQfive}{15.53} | |
| 299 | +\newcommand{\CEVBeqOffQone}{8.87} | |
| 300 | +\newcommand{\CEVBeqOffQtwo}{9.09} | |
| 301 | +\newcommand{\CEVDepPremAll}{8.73} | |
| 302 | +\newcommand{\CEVDepPremBottomTwo}{5.46} | |
| 303 | +\newcommand{\CEVDepPremMarg}{10.27} | |
| 304 | +\newcommand{\CEVDepPremQfive}{15.40} | |
| 305 | +\newcommand{\CEVDepPremQone}{5.23} | |
| 306 | +\newcommand{\CEVDepPremQtwo}{5.69} | |
| 307 | +\newcommand{\CEVEpsHighAll}{0.00} | |
| 308 | +\newcommand{\CEVEpsHighBottomTwo}{0.00} | |
| 309 | +\newcommand{\CEVEpsHighMarg}{0.00} | |
| 310 | +\newcommand{\CEVEpsHighQfive}{0.00} | |
| 311 | +\newcommand{\CEVEpsHighQone}{0.00} | |
| 312 | +\newcommand{\CEVEpsHighQtwo}{0.00} | |
| 313 | +\newcommand{\CEVEpsHighThetaFiveAll}{0.29} | |
| 314 | +\newcommand{\CEVEpsHighThetaFiveBottomTwo}{0.47} | |
| 315 | +\newcommand{\CEVEpsHighThetaFiveMarg}{0.27} | |
| 316 | +\newcommand{\CEVEpsHighThetaFiveQfive}{\ensuremath{-}0.08} | |
| 317 | +\newcommand{\CEVEpsHighThetaFiveQfiveAbs}{0.08} | |
| 318 | +\newcommand{\CEVEpsHighThetaFiveQone}{0.48} | |
| 319 | +\newcommand{\CEVEpsHighThetaFiveQtwo}{0.46} | |
| 320 | +\newcommand{\CEVEpsLowAll}{0.00} | |
| 321 | +\newcommand{\CEVEpsLowBottomTwo}{0.00} | |
| 322 | +\newcommand{\CEVEpsLowMarg}{0.00} | |
| 323 | +\newcommand{\CEVEpsLowQfive}{0.00} | |
| 324 | +\newcommand{\CEVEpsLowQone}{0.00} | |
| 325 | +\newcommand{\CEVEpsLowQtwo}{0.00} | |
| 326 | +\newcommand{\CEVEpsLowThetaFiveAll}{0.42} | |
| 327 | +\newcommand{\CEVEpsLowThetaFiveBottomTwo}{0.66} | |
| 328 | +\newcommand{\CEVEpsLowThetaFiveMarg}{0.38} | |
| 329 | +\newcommand{\CEVEpsLowThetaFiveQfive}{\ensuremath{-}0.07} | |
| 330 | +\newcommand{\CEVEpsLowThetaFiveQfiveAbs}{0.07} | |
| 331 | +\newcommand{\CEVEpsLowThetaFiveQone}{0.67} | |
| 332 | +\newcommand{\CEVEpsLowThetaFiveQtwo}{0.65} | |
| 333 | +\newcommand{\CEVKappaHighAll}{\ensuremath{-}0.10} | |
| 334 | +\newcommand{\CEVKappaHighAllAbs}{0.10} | |
| 335 | +\newcommand{\CEVKappaHighBottomTwo}{\ensuremath{-}0.08} | |
| 336 | +\newcommand{\CEVKappaHighBottomTwoAbs}{0.08} | |
| 337 | +\newcommand{\CEVKappaHighMarg}{\ensuremath{-}0.18} | |
| 338 | +\newcommand{\CEVKappaHighMargAbs}{0.18} | |
| 339 | +\newcommand{\CEVKappaHighQfive}{\ensuremath{-}0.10} | |
| 340 | +\newcommand{\CEVKappaHighQfiveAbs}{0.10} | |
| 341 | +\newcommand{\CEVKappaHighQone}{\ensuremath{-}0.07} | |
| 342 | +\newcommand{\CEVKappaHighQoneAbs}{0.07} | |
| 343 | +\newcommand{\CEVKappaHighQtwo}{\ensuremath{-}0.08} | |
| 344 | +\newcommand{\CEVKappaHighQtwoAbs}{0.08} | |
| 345 | +\newcommand{\CEVKappaLowAll}{\ensuremath{-}0.19} | |
| 346 | +\newcommand{\CEVKappaLowAllAbs}{0.19} | |
| 347 | +\newcommand{\CEVKappaLowBottomTwo}{\ensuremath{-}0.22} | |
| 348 | +\newcommand{\CEVKappaLowBottomTwoAbs}{0.22} | |
| 349 | +\newcommand{\CEVKappaLowMarg}{\ensuremath{-}0.14} | |
| 350 | +\newcommand{\CEVKappaLowMargAbs}{0.14} | |
| 351 | +\newcommand{\CEVKappaLowQfive}{\ensuremath{-}0.17} | |
| 352 | +\newcommand{\CEVKappaLowQfiveAbs}{0.17} | |
| 353 | +\newcommand{\CEVKappaLowQone}{\ensuremath{-}0.22} | |
| 354 | +\newcommand{\CEVKappaLowQoneAbs}{0.22} | |
| 355 | +\newcommand{\CEVKappaLowQtwo}{\ensuremath{-}0.22} | |
| 356 | +\newcommand{\CEVKappaLowQtwoAbs}{0.22} | |
| 357 | +\newcommand{\CEVNumericalMaxPct}{0.03} | |
| 358 | +\newcommand{\CEVPhiHighAll}{\ensuremath{-}0.54} | |
| 359 | +\newcommand{\CEVPhiHighAllAbs}{0.54} | |
| 360 | +\newcommand{\CEVPhiHighBottomTwo}{\ensuremath{-}0.68} | |
| 361 | +\newcommand{\CEVPhiHighBottomTwoAbs}{0.68} | |
| 362 | +\newcommand{\CEVPhiHighMarg}{\ensuremath{-}0.48} | |
| 363 | +\newcommand{\CEVPhiHighMargAbs}{0.48} | |
| 364 | +\newcommand{\CEVPhiHighQfive}{\ensuremath{-}0.26} | |
| 365 | +\newcommand{\CEVPhiHighQfiveAbs}{0.26} | |
| 366 | +\newcommand{\CEVPhiHighQone}{\ensuremath{-}0.69} | |
| 367 | +\newcommand{\CEVPhiHighQoneAbs}{0.69} | |
| 368 | +\newcommand{\CEVPhiHighQtwo}{\ensuremath{-}0.66} | |
| 369 | +\newcommand{\CEVPhiHighQtwoAbs}{0.66} | |
| 370 | +\newcommand{\CEVPhiLowAll}{0.65} | |
| 371 | +\newcommand{\CEVPhiLowBottomTwo}{0.82} | |
| 372 | +\newcommand{\CEVPhiLowMarg}{0.57} | |
| 373 | +\newcommand{\CEVPhiLowQfive}{0.31} | |
| 374 | +\newcommand{\CEVPhiLowQone}{0.83} | |
| 375 | +\newcommand{\CEVPhiLowQtwo}{0.81} | |
| 376 | +\newcommand{\CEVPiFourAll}{\ensuremath{-}0.01} | |
| 377 | +\newcommand{\CEVPiFourAllAbs}{0.01} | |
| 378 | +\newcommand{\CEVPiFourBottomTwo}{\ensuremath{-}0.11} | |
| 379 | +\newcommand{\CEVPiFourBottomTwoAbs}{0.11} | |
| 380 | +\newcommand{\CEVPiFourMarg}{\ensuremath{-}0.08} | |
| 381 | +\newcommand{\CEVPiFourMargAbs}{0.08} | |
| 382 | +\newcommand{\CEVPiFourQfive}{0.23} | |
| 383 | +\newcommand{\CEVPiFourQone}{\ensuremath{-}0.11} | |
| 384 | +\newcommand{\CEVPiFourQoneAbs}{0.11} | |
| 385 | +\newcommand{\CEVPiFourQtwo}{\ensuremath{-}0.11} | |
| 386 | +\newcommand{\CEVPiFourQtwoAbs}{0.11} | |
| 387 | +\newcommand{\CEVPiZeroAll}{0.12} | |
| 388 | +\newcommand{\CEVPiZeroBottomTwo}{0.23} | |
| 389 | +\newcommand{\CEVPiZeroMarg}{0.22} | |
| 390 | +\newcommand{\CEVPiZeroQfive}{\ensuremath{-}0.14} | |
| 391 | +\newcommand{\CEVPiZeroQfiveAbs}{0.14} | |
| 392 | +\newcommand{\CEVPiZeroQone}{0.23} | |
| 393 | +\newcommand{\CEVPiZeroQtwo}{0.23} | |
| 394 | +\newcommand{\CEVPsiFiftyAll}{\ensuremath{-}0.05} | |
| 395 | +\newcommand{\CEVPsiFiftyAllAbs}{0.05} | |
| 396 | +\newcommand{\CEVPsiFiftyBottomTwo}{\ensuremath{-}0.02} | |
| 397 | +\newcommand{\CEVPsiFiftyBottomTwoAbs}{0.02} | |
| 398 | +\newcommand{\CEVPsiFiftyMarg}{\ensuremath{-}0.01} | |
| 399 | +\newcommand{\CEVPsiFiftyMargAbs}{0.01} | |
| 400 | +\newcommand{\CEVPsiFiftyQfive}{\ensuremath{-}0.12} | |
| 401 | +\newcommand{\CEVPsiFiftyQfiveAbs}{0.12} | |
| 402 | +\newcommand{\CEVPsiFiftyQone}{\ensuremath{-}0.01} | |
| 403 | +\newcommand{\CEVPsiFiftyQoneAbs}{0.01} | |
| 404 | +\newcommand{\CEVPsiFiftyQtwo}{\ensuremath{-}0.02} | |
| 405 | +\newcommand{\CEVPsiFiftyQtwoAbs}{0.02} | |
| 406 | +\newcommand{\CEVPsiThirtyAll}{0.02} | |
| 407 | +\newcommand{\CEVPsiThirtyBottomTwo}{0.00} | |
| 408 | +\newcommand{\CEVPsiThirtyMarg}{\ensuremath{-}0.08} | |
| 409 | +\newcommand{\CEVPsiThirtyMargAbs}{0.08} | |
| 410 | +\newcommand{\CEVPsiThirtyQfive}{0.09} | |
| 411 | +\newcommand{\CEVPsiThirtyQone}{\ensuremath{-}0.01} | |
| 412 | +\newcommand{\CEVPsiThirtyQoneAbs}{0.01} | |
| 413 | +\newcommand{\CEVPsiThirtyQtwo}{0.00} | |
| 414 | +\newcommand{\CEVRhoGAll}{0.03} | |
| 415 | +\newcommand{\CEVRhoGBottomTwo}{0.05} | |
| 416 | +\newcommand{\CEVRhoGMarg}{0.02} | |
| 417 | +\newcommand{\CEVRhoGQfive}{\ensuremath{-}0.02} | |
| 418 | +\newcommand{\CEVRhoGQfiveAbs}{0.02} | |
| 419 | +\newcommand{\CEVRhoGQone}{0.05} | |
| 420 | +\newcommand{\CEVRhoGQtwo}{0.05} | |
| 421 | +\newcommand{\CEVRoneAll}{\ensuremath{-}0.35} | |
| 422 | +\newcommand{\CEVRoneAllAbs}{0.35} | |
| 423 | +\newcommand{\CEVRoneBottomTwo}{0.59} | |
| 424 | +\newcommand{\CEVRoneMarg}{\ensuremath{-}0.86} | |
| 425 | +\newcommand{\CEVRoneMargAbs}{0.86} | |
| 426 | +\newcommand{\CEVRoneQfive}{\ensuremath{-}2.25} | |
| 427 | +\newcommand{\CEVRoneQfiveAbs}{2.25} | |
| 428 | +\newcommand{\CEVRoneQone}{0.68} | |
| 429 | +\newcommand{\CEVRoneQtwo}{0.51} | |
| 430 | +\newcommand{\CEVRthreeAll}{1.80} | |
| 431 | +\newcommand{\CEVRthreeBottomTwo}{0.52} | |
| 432 | +\newcommand{\CEVRthreeMarg}{2.32} | |
| 433 | +\newcommand{\CEVRthreeQfive}{4.46} | |
| 434 | +\newcommand{\CEVRthreeQone}{0.42} | |
| 435 | +\newcommand{\CEVRthreeQtwo}{0.62} | |
| 436 | +\newcommand{\CEVSampleLargeAll}{\ensuremath{-}0.02} | |
| 437 | +\newcommand{\CEVSampleLargeAllAbs}{0.02} | |
| 438 | +\newcommand{\CEVSampleLargeBottomTwo}{\ensuremath{-}0.02} | |
| 439 | +\newcommand{\CEVSampleLargeBottomTwoAbs}{0.02} | |
| 440 | +\newcommand{\CEVSampleLargeMarg}{\ensuremath{-}0.02} | |
| 441 | +\newcommand{\CEVSampleLargeMargAbs}{0.02} | |
| 442 | +\newcommand{\CEVSampleLargeQfive}{\ensuremath{-}0.02} | |
| 443 | +\newcommand{\CEVSampleLargeQfiveAbs}{0.02} | |
| 444 | +\newcommand{\CEVSampleLargeQone}{\ensuremath{-}0.02} | |
| 445 | +\newcommand{\CEVSampleLargeQoneAbs}{0.02} | |
| 446 | +\newcommand{\CEVSampleLargeQtwo}{\ensuremath{-}0.02} | |
| 447 | +\newcommand{\CEVSampleLargeQtwoAbs}{0.02} | |
| 448 | +\newcommand{\CEVSeedAll}{\ensuremath{-}0.01} | |
| 449 | +\newcommand{\CEVSeedAllAbs}{0.01} | |
| 450 | +\newcommand{\CEVSeedBottomTwo}{\ensuremath{-}0.01} | |
| 451 | +\newcommand{\CEVSeedBottomTwoAbs}{0.01} | |
| 452 | +\newcommand{\CEVSeedMarg}{\ensuremath{-}0.01} | |
| 453 | +\newcommand{\CEVSeedMargAbs}{0.01} | |
| 454 | +\newcommand{\CEVSeedQfive}{0.00} | |
| 455 | +\newcommand{\CEVSeedQone}{\ensuremath{-}0.01} | |
| 456 | +\newcommand{\CEVSeedQoneAbs}{0.01} | |
| 457 | +\newcommand{\CEVSeedQtwo}{\ensuremath{-}0.01} | |
| 458 | +\newcommand{\CEVSeedQtwoAbs}{0.01} | |
| 459 | +\newcommand{\CEVSigHighAll}{\ensuremath{-}10.83} | |
| 460 | +\newcommand{\CEVSigHighAllAbs}{10.83} | |
| 461 | +\newcommand{\CEVSigHighBottomTwo}{\ensuremath{-}12.46} | |
| 462 | +\newcommand{\CEVSigHighBottomTwoAbs}{12.46} | |
| 463 | +\newcommand{\CEVSigHighMarg}{\ensuremath{-}9.28} | |
| 464 | +\newcommand{\CEVSigHighMargAbs}{9.28} | |
| 465 | +\newcommand{\CEVSigHighQfive}{\ensuremath{-}8.13} | |
| 466 | +\newcommand{\CEVSigHighQfiveAbs}{8.13} | |
| 467 | +\newcommand{\CEVSigHighQone}{\ensuremath{-}12.69} | |
| 468 | +\newcommand{\CEVSigHighQoneAbs}{12.69} | |
| 469 | +\newcommand{\CEVSigHighQtwo}{\ensuremath{-}12.24} | |
| 470 | +\newcommand{\CEVSigHighQtwoAbs}{12.24} | |
| 471 | +\newcommand{\CEVSigLowAll}{10.25} | |
| 472 | +\newcommand{\CEVSigLowBottomTwo}{11.82} | |
| 473 | +\newcommand{\CEVSigLowMarg}{9.00} | |
| 474 | +\newcommand{\CEVSigLowQfive}{7.78} | |
| 475 | +\newcommand{\CEVSigLowQone}{12.06} | |
| 476 | +\newcommand{\CEVSigLowQtwo}{11.57} | |
| 477 | +\newcommand{\CEVTaxFiftyAll}{10.85} | |
| 478 | +\newcommand{\CEVTaxFiftyBottomTwo}{14.59} | |
| 479 | +\newcommand{\CEVTaxFiftyMarg}{8.58} | |
| 480 | +\newcommand{\CEVTaxFiftyQfive}{5.89} | |
| 481 | +\newcommand{\CEVTaxFiftyQone}{15.39} | |
| 482 | +\newcommand{\CEVTaxFiftyQtwo}{13.79} | |
| 483 | +\newcommand{\CEVTaxTwentyAll}{7.47} | |
| 484 | +\newcommand{\CEVTaxTwentyBottomTwo}{10.24} | |
| 485 | +\newcommand{\CEVTaxTwentyMarg}{5.74} | |
| 486 | +\newcommand{\CEVTaxTwentyQfive}{4.02} | |
| 487 | +\newcommand{\CEVTaxTwentyQone}{10.85} | |
| 488 | +\newcommand{\CEVTaxTwentyQtwo}{9.64} | |
| 489 | +\newcommand{\CEVThetaFiveAll}{0.34} | |
| 490 | +\newcommand{\CEVThetaFiveBottomTwo}{0.55} | |
| 491 | +\newcommand{\CEVThetaFiveMarg}{0.30} | |
| 492 | +\newcommand{\CEVThetaFiveQfive}{\ensuremath{-}0.08} | |
| 493 | +\newcommand{\CEVThetaFiveQfiveAbs}{0.08} | |
| 494 | +\newcommand{\CEVThetaFiveQone}{0.55} | |
| 495 | +\newcommand{\CEVThetaFiveQtwo}{0.54} | |
| 496 | +\newcommand{\CEVThetaFortyAll}{\ensuremath{-}0.23} | |
| 497 | +\newcommand{\CEVThetaFortyAllAbs}{0.23} | |
| 498 | +\newcommand{\CEVThetaFortyBottomTwo}{\ensuremath{-}0.40} | |
| 499 | +\newcommand{\CEVThetaFortyBottomTwoAbs}{0.40} | |
| 500 | +\newcommand{\CEVThetaFortyMarg}{\ensuremath{-}0.33} | |
| 501 | +\newcommand{\CEVThetaFortyMargAbs}{0.33} | |
| 502 | +\newcommand{\CEVThetaFortyQfive}{0.32} | |
| 503 | +\newcommand{\CEVThetaFortyQone}{\ensuremath{-}0.39} | |
| 504 | +\newcommand{\CEVThetaFortyQoneAbs}{0.39} | |
| 505 | +\newcommand{\CEVThetaFortyQtwo}{\ensuremath{-}0.40} | |
| 506 | +\newcommand{\CEVThetaFortyQtwoAbs}{0.40} | |
| 507 | +\newcommand{\CEVTolTightAll}{0.00} | |
| 508 | +\newcommand{\CEVTolTightBottomTwo}{0.00} | |
| 509 | +\newcommand{\CEVTolTightMarg}{0.00} | |
| 510 | +\newcommand{\CEVTolTightQfive}{0.00} | |
| 511 | +\newcommand{\CEVTolTightQone}{0.00} | |
| 512 | +\newcommand{\CEVTolTightQtwo}{0.00} | |
| 513 | +\newcommand{\CEVallarmlevel}{\ensuremath{-}1.14} | |
| 514 | +\newcommand{\CEVallarmlevelAbs}{1.14} | |
| 515 | +\newcommand{\CEVallarmpersistent}{\ensuremath{-}1.24} | |
| 516 | +\newcommand{\CEVallarmpersistentAbs}{1.24} | |
| 517 | +\newcommand{\CEVallfrmlevel}{\ensuremath{-}0.64} | |
| 518 | +\newcommand{\CEVallfrmlevelAbs}{0.64} | |
| 519 | +\newcommand{\CEVallfrmlevelfixedsupply}{\ensuremath{-}0.64} | |
| 520 | +\newcommand{\CEVallfrmlevelfixedsupplyAbs}{0.64} | |
| 521 | +\newcommand{\CEVallfrmlevelnoseg}{\ensuremath{-}0.60} | |
| 522 | +\newcommand{\CEVallfrmlevelnosegAbs}{0.60} | |
| 523 | +\newcommand{\CEVallfrmlevelrigidrents}{\ensuremath{-}0.62} | |
| 524 | +\newcommand{\CEVallfrmlevelrigidrentsAbs}{0.62} | |
| 525 | +\newcommand{\CEVallfrmlevelstatic}{\ensuremath{-}0.64} | |
| 526 | +\newcommand{\CEVallfrmlevelstaticAbs}{0.64} | |
| 527 | +\newcommand{\CEVallfrmlevelunindexedwages}{\ensuremath{-}3.33} | |
| 528 | +\newcommand{\CEVallfrmlevelunindexedwagesAbs}{3.33} | |
| 529 | +\newcommand{\CEVallfrmpersistent}{\ensuremath{-}0.52} | |
| 530 | +\newcommand{\CEVallfrmpersistentAbs}{0.52} | |
| 13 | 531 | \newcommand{\CEVallfrmpersistenttaylor}{0.14} |
| 14 | −\newcommand{\CEValltermlevel}{-0.64} | |
| 15 | −\newcommand{\CEValltermpersistent}{-0.74} | |
| 16 | −\newcommand{\CEValltermpersistenttaylor}{-0.25} | |
| 532 | +\newcommand{\CEValltermlevel}{\ensuremath{-}0.64} | |
| 533 | +\newcommand{\CEValltermlevelAbs}{0.64} | |
| 534 | +\newcommand{\CEValltermpersistent}{\ensuremath{-}0.74} | |
| 535 | +\newcommand{\CEValltermpersistentAbs}{0.74} | |
| 536 | +\newcommand{\CEValltermpersistenttaylor}{\ensuremath{-}0.25} | |
| 537 | +\newcommand{\CEValltermpersistenttaylorAbs}{0.25} | |
| 17 | 538 | \newcommand{\CEVbequestsoff}{11.14} |
| 18 | 539 | \newcommand{\CEVchi}{0.91} |
| 19 | 540 | \newcommand{\CEVdeppremium}{8.73} |
| 20 | 541 | \newcommand{\CEVeps}{0.00} |
| 21 | −\newcommand{\CEVkappahigh}{-0.10} | |
| 22 | −\newcommand{\CEVkappalow}{-0.19} | |
| 542 | +\newcommand{\CEVkappahigh}{\ensuremath{-}0.10} | |
| 543 | +\newcommand{\CEVkappahighAbs}{0.10} | |
| 544 | +\newcommand{\CEVkappalow}{\ensuremath{-}0.19} | |
| 545 | +\newcommand{\CEVkappalowAbs}{0.19} | |
| 23 | 546 | \newcommand{\CEVnoseg}{0.00} |
| 24 | −\newcommand{\CEVomarmlevel}{-1.03} | |
| 25 | −\newcommand{\CEVomarmpersistent}{-1.21} | |
| 547 | +\newcommand{\CEVomarmlevel}{\ensuremath{-}1.03} | |
| 548 | +\newcommand{\CEVomarmlevelAbs}{1.03} | |
| 549 | +\newcommand{\CEVomarmpersistent}{\ensuremath{-}1.21} | |
| 550 | +\newcommand{\CEVomarmpersistentAbs}{1.21} | |
| 26 | 551 | \newcommand{\CEVomfrmlevel}{0.63} |
| 27 | 552 | \newcommand{\CEVomfrmlevelfixedsupply}{0.63} |
| 28 | 553 | \newcommand{\CEVomfrmlevelnoseg}{0.66} |
| 29 | 554 | \newcommand{\CEVomfrmlevelrigidrents}{0.55} |
| 30 | 555 | \newcommand{\CEVomfrmlevelstatic}{0.65} |
| 31 | −\newcommand{\CEVomfrmlevelunindexedwages}{-2.40} | |
| 556 | +\newcommand{\CEVomfrmlevelunindexedwages}{\ensuremath{-}2.40} | |
| 557 | +\newcommand{\CEVomfrmlevelunindexedwagesAbs}{2.40} | |
| 32 | 558 | \newcommand{\CEVomfrmpersistent}{1.14} |
| 33 | 559 | \newcommand{\CEVomfrmpersistenttaylor}{1.43} |
| 34 | 560 | \newcommand{\CEVomtermlevel}{0.63} |
| 35 | 561 | \newcommand{\CEVomtermpersistent}{0.46} |
| 36 | 562 | \newcommand{\CEVomtermpersistenttaylor}{0.20} |
| 37 | −\newcommand{\CEVpi}{-0.01} | |
| 38 | −\newcommand{\CEVpsi}{-0.05} | |
| 563 | +\newcommand{\CEVpi}{\ensuremath{-}0.01} | |
| 564 | +\newcommand{\CEVpiAbs}{0.01} | |
| 565 | +\newcommand{\CEVpsi}{\ensuremath{-}0.05} | |
| 566 | +\newcommand{\CEVpsiAbs}{0.05} | |
| 39 | 567 | \newcommand{\CEVr}{1.80} |
| 40 | −\newcommand{\CEVrarmlevel}{-0.96} | |
| 41 | −\newcommand{\CEVrarmpersistent}{-1.04} | |
| 42 | −\newcommand{\CEVrfrmlevel}{-1.00} | |
| 43 | −\newcommand{\CEVrfrmlevelfixedsupply}{-0.99} | |
| 44 | −\newcommand{\CEVrfrmlevelnoseg}{-0.95} | |
| 45 | −\newcommand{\CEVrfrmlevelrigidrents}{-0.92} | |
| 46 | −\newcommand{\CEVrfrmlevelstatic}{-1.01} | |
| 47 | −\newcommand{\CEVrfrmlevelunindexedwages}{-3.58} | |
| 48 | −\newcommand{\CEVrfrmpersistent}{-1.06} | |
| 49 | −\newcommand{\CEVrfrmpersistenttaylor}{-0.31} | |
| 50 | −\newcommand{\CEVrtermlevel}{-1.00} | |
| 51 | −\newcommand{\CEVrtermpersistent}{-1.09} | |
| 52 | −\newcommand{\CEVrtermpersistenttaylor}{-0.32} | |
| 53 | −\newcommand{\CEVtheta}{-0.23} | |
| 568 | +\newcommand{\CEVrarmlevel}{\ensuremath{-}0.96} | |
| 569 | +\newcommand{\CEVrarmlevelAbs}{0.96} | |
| 570 | +\newcommand{\CEVrarmpersistent}{\ensuremath{-}1.04} | |
| 571 | +\newcommand{\CEVrarmpersistentAbs}{1.04} | |
| 572 | +\newcommand{\CEVrfrmlevel}{\ensuremath{-}1.00} | |
| 573 | +\newcommand{\CEVrfrmlevelAbs}{1.00} | |
| 574 | +\newcommand{\CEVrfrmlevelfixedsupply}{\ensuremath{-}0.99} | |
| 575 | +\newcommand{\CEVrfrmlevelfixedsupplyAbs}{0.99} | |
| 576 | +\newcommand{\CEVrfrmlevelnoseg}{\ensuremath{-}0.95} | |
| 577 | +\newcommand{\CEVrfrmlevelnosegAbs}{0.95} | |
| 578 | +\newcommand{\CEVrfrmlevelrigidrents}{\ensuremath{-}0.92} | |
| 579 | +\newcommand{\CEVrfrmlevelrigidrentsAbs}{0.92} | |
| 580 | +\newcommand{\CEVrfrmlevelstatic}{\ensuremath{-}1.01} | |
| 581 | +\newcommand{\CEVrfrmlevelstaticAbs}{1.01} | |
| 582 | +\newcommand{\CEVrfrmlevelunindexedwages}{\ensuremath{-}3.58} | |
| 583 | +\newcommand{\CEVrfrmlevelunindexedwagesAbs}{3.58} | |
| 584 | +\newcommand{\CEVrfrmpersistent}{\ensuremath{-}1.06} | |
| 585 | +\newcommand{\CEVrfrmpersistentAbs}{1.06} | |
| 586 | +\newcommand{\CEVrfrmpersistenttaylor}{\ensuremath{-}0.31} | |
| 587 | +\newcommand{\CEVrfrmpersistenttaylorAbs}{0.31} | |
| 588 | +\newcommand{\CEVrtermlevel}{\ensuremath{-}1.00} | |
| 589 | +\newcommand{\CEVrtermlevelAbs}{1.00} | |
| 590 | +\newcommand{\CEVrtermpersistent}{\ensuremath{-}1.09} | |
| 591 | +\newcommand{\CEVrtermpersistentAbs}{1.09} | |
| 592 | +\newcommand{\CEVrtermpersistenttaylor}{\ensuremath{-}0.32} | |
| 593 | +\newcommand{\CEVrtermpersistenttaylorAbs}{0.32} | |
| 594 | +\newcommand{\CEVtheta}{\ensuremath{-}0.23} | |
| 595 | +\newcommand{\CEVthetaAbs}{0.23} | |
| 54 | 596 | \newcommand{\CEVtransfertax}{10.85} |
| 55 | −\newcommand{\DOwnarmlevel}{-0.11} | |
| 56 | −\newcommand{\DOwnarmpersistent}{-0.39} | |
| 57 | −\newcommand{\DOwnfrmlevel}{-0.02} | |
| 58 | −\newcommand{\DOwnfrmlevelfixedsupply}{-0.01} | |
| 59 | −\newcommand{\DOwnfrmlevelnoseg}{-0.16} | |
| 60 | −\newcommand{\DOwnfrmlevelrigidrents}{-0.03} | |
| 597 | +\newcommand{\ChiFiveEstate}{0.32} | |
| 598 | +\newcommand{\ChiFiveGapFour}{0.95} | |
| 599 | +\newcommand{\ChiFiveGift}{0.40} | |
| 600 | +\newcommand{\ChiFiveGini}{0.54} | |
| 601 | +\newcommand{\ChiFiveLamSeven}{0.056} | |
| 602 | +\newcommand{\ChiFiveLamThree}{0.19} | |
| 603 | +\newcommand{\ChiFiveOwn}{0.644} | |
| 604 | +\newcommand{\ChiFiveOwnUthirtyfive}{0.403} | |
| 605 | +\newcommand{\ChiFiveP}{1.006} | |
| 606 | +\newcommand{\ChiFivePDevPct}{0.6} | |
| 607 | +\newcommand{\ChiFiveRR}{0.34} | |
| 608 | +\newcommand{\ChiFiveRebate}{0.000} | |
| 609 | +\newcommand{\ChiFiveRho}{0.190} | |
| 610 | +\newcommand{\ChiFiveRhoDevPct}{\ensuremath{-}2.3} | |
| 611 | +\newcommand{\ChiFiveRhoDevPctAbs}{2.3} | |
| 612 | +\newcommand{\ChiFiveThr}{0.48} | |
| 613 | +\newcommand{\ChiFiveThrHighZ}{0.01} | |
| 614 | +\newcommand{\ChiFiveThrLowZ}{1.14} | |
| 615 | +\newcommand{\ChiFiveqOO}{0.546} | |
| 616 | +\newcommand{\ChiFiveqRO}{0.355} | |
| 617 | +\newcommand{\ClipSharePct}{0.003} | |
| 618 | +\newcommand{\DOwnarmlevel}{\ensuremath{-}0.11} | |
| 619 | +\newcommand{\DOwnarmlevelAbs}{0.11} | |
| 620 | +\newcommand{\DOwnarmpersistent}{\ensuremath{-}0.39} | |
| 621 | +\newcommand{\DOwnarmpersistentAbs}{0.39} | |
| 622 | +\newcommand{\DOwnfrmlevel}{\ensuremath{-}0.02} | |
| 623 | +\newcommand{\DOwnfrmlevelAbs}{0.02} | |
| 624 | +\newcommand{\DOwnfrmlevelfixedsupply}{\ensuremath{-}0.01} | |
| 625 | +\newcommand{\DOwnfrmlevelfixedsupplyAbs}{0.01} | |
| 626 | +\newcommand{\DOwnfrmlevelnoseg}{\ensuremath{-}0.16} | |
| 627 | +\newcommand{\DOwnfrmlevelnosegAbs}{0.16} | |
| 628 | +\newcommand{\DOwnfrmlevelrigidrents}{\ensuremath{-}0.03} | |
| 629 | +\newcommand{\DOwnfrmlevelrigidrentsAbs}{0.03} | |
| 61 | 630 | \newcommand{\DOwnfrmlevelstatic}{0.02} |
| 62 | −\newcommand{\DOwnfrmlevelunindexedwages}{-0.63} | |
| 63 | −\newcommand{\DOwnfrmpersistent}{-0.18} | |
| 64 | −\newcommand{\DOwnfrmpersistenttaylor}{-0.12} | |
| 65 | −\newcommand{\DOwntermlevel}{-0.02} | |
| 66 | −\newcommand{\DOwntermpersistent}{-0.34} | |
| 67 | −\newcommand{\DOwntermpersistenttaylor}{-0.26} | |
| 631 | +\newcommand{\DOwnfrmlevelunindexedwages}{\ensuremath{-}0.63} | |
| 632 | +\newcommand{\DOwnfrmlevelunindexedwagesAbs}{0.63} | |
| 633 | +\newcommand{\DOwnfrmpersistent}{\ensuremath{-}0.18} | |
| 634 | +\newcommand{\DOwnfrmpersistentAbs}{0.18} | |
| 635 | +\newcommand{\DOwnfrmpersistenttaylor}{\ensuremath{-}0.12} | |
| 636 | +\newcommand{\DOwnfrmpersistenttaylorAbs}{0.12} | |
| 637 | +\newcommand{\DOwntermlevel}{\ensuremath{-}0.02} | |
| 638 | +\newcommand{\DOwntermlevelAbs}{0.02} | |
| 639 | +\newcommand{\DOwntermpersistent}{\ensuremath{-}0.34} | |
| 640 | +\newcommand{\DOwntermpersistentAbs}{0.34} | |
| 641 | +\newcommand{\DOwntermpersistenttaylor}{\ensuremath{-}0.26} | |
| 642 | +\newcommand{\DOwntermpersistenttaylorAbs}{0.26} | |
| 643 | +\newcommand{\DebtDepositRatio}{1.7} | |
| 644 | +\newcommand{\DepPremEstate}{0.77} | |
| 645 | +\newcommand{\DepPremGapFour}{1.19} | |
| 646 | +\newcommand{\DepPremGift}{0.61} | |
| 647 | +\newcommand{\DepPremGini}{0.50} | |
| 648 | +\newcommand{\DepPremLamSeven}{0.169} | |
| 649 | +\newcommand{\DepPremLamThree}{0.28} | |
| 650 | +\newcommand{\DepPremOwn}{0.635} | |
| 651 | +\newcommand{\DepPremOwnUthirtyfive}{0.414} | |
| 652 | +\newcommand{\DepPremP}{1.018} | |
| 653 | +\newcommand{\DepPremPDevPct}{1.8} | |
| 654 | +\newcommand{\DepPremRR}{0.57} | |
| 655 | +\newcommand{\DepPremRebate}{0.000} | |
| 656 | +\newcommand{\DepPremRho}{0.211} | |
| 657 | +\newcommand{\DepPremRhoDevPct}{8.6} | |
| 658 | +\newcommand{\DepPremThr}{0.75} | |
| 659 | +\newcommand{\DepPremThrHighZ}{0.10} | |
| 660 | +\newcommand{\DepPremThrLowZ}{16.32} | |
| 661 | +\newcommand{\DepPremqOO}{0.664} | |
| 662 | +\newcommand{\DepPremqRO}{0.381} | |
| 663 | +\newcommand{\DevCoarseGini}{0.000} | |
| 664 | +\newcommand{\DevCoarseLamSeven}{+0.017} | |
| 665 | +\newcommand{\DevCoarseOwnPP}{\ensuremath{-}0.5} | |
| 666 | +\newcommand{\DevCoarseOwnPPAbs}{0.5} | |
| 667 | +\newcommand{\DevCoarsePPct}{\ensuremath{-}0.8} | |
| 668 | +\newcommand{\DevCoarsePPctAbs}{0.8} | |
| 669 | +\newcommand{\DevCoarseRR}{+0.005} | |
| 670 | +\newcommand{\DevCoarseThr}{+0.07} | |
| 671 | +\newcommand{\DevFineGini}{+0.001} | |
| 672 | +\newcommand{\DevFineLamSeven}{\ensuremath{-}0.007} | |
| 673 | +\newcommand{\DevFineLamSevenAbs}{0.007} | |
| 674 | +\newcommand{\DevFineOwnPP}{0.4} | |
| 675 | +\newcommand{\DevFinePPct}{\ensuremath{-}0.1} | |
| 676 | +\newcommand{\DevFinePPctAbs}{0.1} | |
| 677 | +\newcommand{\DevFineRR}{\ensuremath{-}0.002} | |
| 678 | +\newcommand{\DevFineRRAbs}{0.002} | |
| 679 | +\newcommand{\DevFineThr}{\ensuremath{-}0.02} | |
| 680 | +\newcommand{\DevFineThrAbs}{0.02} | |
| 681 | +\newcommand{\DevSampleLargeGini}{+0.001} | |
| 682 | +\newcommand{\DevSampleLargeLamSeven}{+0.004} | |
| 683 | +\newcommand{\DevSampleLargeOwnPP}{\ensuremath{-}0.1} | |
| 684 | +\newcommand{\DevSampleLargeOwnPPAbs}{0.1} | |
| 685 | +\newcommand{\DevSampleLargePPct}{0.0} | |
| 686 | +\newcommand{\DevSampleLargeRR}{\ensuremath{-}0.002} | |
| 687 | +\newcommand{\DevSampleLargeRRAbs}{0.002} | |
| 688 | +\newcommand{\DevSampleLargeThr}{0.00} | |
| 689 | +\newcommand{\DevSampleSmallGini}{0.000} | |
| 690 | +\newcommand{\DevSampleSmallLamSeven}{+0.013} | |
| 691 | +\newcommand{\DevSampleSmallOwnPP}{0.0} | |
| 692 | +\newcommand{\DevSampleSmallPPct}{0.1} | |
| 693 | +\newcommand{\DevSampleSmallRR}{0.000} | |
| 694 | +\newcommand{\DevSampleSmallThr}{+0.00} | |
| 695 | +\newcommand{\DevSeedGini}{+0.001} | |
| 696 | +\newcommand{\DevSeedLamSeven}{+0.001} | |
| 697 | +\newcommand{\DevSeedOwnPP}{\ensuremath{-}0.1} | |
| 698 | +\newcommand{\DevSeedOwnPPAbs}{0.1} | |
| 699 | +\newcommand{\DevSeedPPct}{0.0} | |
| 700 | +\newcommand{\DevSeedRR}{+0.000} | |
| 701 | +\newcommand{\DevSeedThr}{0.00} | |
| 702 | +\newcommand{\EpsHighThetaFiveEstate}{0.31} | |
| 703 | +\newcommand{\EpsHighThetaFiveGapFour}{0.83} | |
| 704 | +\newcommand{\EpsHighThetaFiveGift}{0.39} | |
| 705 | +\newcommand{\EpsHighThetaFiveGini}{0.55} | |
| 706 | +\newcommand{\EpsHighThetaFiveLamSeven}{0.067} | |
| 707 | +\newcommand{\EpsHighThetaFiveLamThree}{0.20} | |
| 708 | +\newcommand{\EpsHighThetaFiveOwn}{0.644} | |
| 709 | +\newcommand{\EpsHighThetaFiveOwnUthirtyfive}{0.412} | |
| 710 | +\newcommand{\EpsHighThetaFiveP}{1.005} | |
| 711 | +\newcommand{\EpsHighThetaFivePDevPct}{0.5} | |
| 712 | +\newcommand{\EpsHighThetaFiveRR}{0.33} | |
| 713 | +\newcommand{\EpsHighThetaFiveRebate}{0.000} | |
| 714 | +\newcommand{\EpsHighThetaFiveRho}{0.190} | |
| 715 | +\newcommand{\EpsHighThetaFiveRhoDevPct}{\ensuremath{-}2.1} | |
| 716 | +\newcommand{\EpsHighThetaFiveRhoDevPctAbs}{2.1} | |
| 717 | +\newcommand{\EpsHighThetaFiveThr}{0.55} | |
| 718 | +\newcommand{\EpsHighThetaFiveThrHighZ}{\ensuremath{-}0.10} | |
| 719 | +\newcommand{\EpsHighThetaFiveThrHighZAbs}{0.10} | |
| 720 | +\newcommand{\EpsHighThetaFiveThrLowZ}{1.23} | |
| 721 | +\newcommand{\EpsHighThetaFiveqOO}{0.574} | |
| 722 | +\newcommand{\EpsHighThetaFiveqRO}{0.376} | |
| 723 | +\newcommand{\EpsHighThetaFortyEstate}{0.33} | |
| 724 | +\newcommand{\EpsHighThetaFortyGapFour}{0.99} | |
| 725 | +\newcommand{\EpsHighThetaFortyGift}{0.39} | |
| 726 | +\newcommand{\EpsHighThetaFortyGini}{0.54} | |
| 727 | +\newcommand{\EpsHighThetaFortyLamSeven}{0.083} | |
| 728 | +\newcommand{\EpsHighThetaFortyLamThree}{0.21} | |
| 729 | +\newcommand{\EpsHighThetaFortyOwn}{0.615} | |
| 730 | +\newcommand{\EpsHighThetaFortyOwnUthirtyfive}{0.327} | |
| 731 | +\newcommand{\EpsHighThetaFortyP}{0.989} | |
| 732 | +\newcommand{\EpsHighThetaFortyPDevPct}{\ensuremath{-}1.1} | |
| 733 | +\newcommand{\EpsHighThetaFortyPDevPctAbs}{1.1} | |
| 734 | +\newcommand{\EpsHighThetaFortyRR}{0.34} | |
| 735 | +\newcommand{\EpsHighThetaFortyRebate}{0.000} | |
| 736 | +\newcommand{\EpsHighThetaFortyRho}{0.199} | |
| 737 | +\newcommand{\EpsHighThetaFortyRhoDevPct}{2.4} | |
| 738 | +\newcommand{\EpsHighThetaFortyThr}{0.55} | |
| 739 | +\newcommand{\EpsHighThetaFortyThrHighZ}{0.20} | |
| 740 | +\newcommand{\EpsHighThetaFortyThrLowZ}{1.11} | |
| 741 | +\newcommand{\EpsHighThetaFortyqOO}{0.510} | |
| 742 | +\newcommand{\EpsHighThetaFortyqRO}{0.298} | |
| 743 | +\newcommand{\EpsLowThetaFiveEstate}{0.31} | |
| 744 | +\newcommand{\EpsLowThetaFiveGapFour}{0.84} | |
| 745 | +\newcommand{\EpsLowThetaFiveGift}{0.39} | |
| 746 | +\newcommand{\EpsLowThetaFiveGini}{0.55} | |
| 747 | +\newcommand{\EpsLowThetaFiveLamSeven}{0.068} | |
| 748 | +\newcommand{\EpsLowThetaFiveLamThree}{0.19} | |
| 749 | +\newcommand{\EpsLowThetaFiveOwn}{0.636} | |
| 750 | +\newcommand{\EpsLowThetaFiveOwnUthirtyfive}{0.402} | |
| 751 | +\newcommand{\EpsLowThetaFiveP}{1.006} | |
| 752 | +\newcommand{\EpsLowThetaFivePDevPct}{0.6} | |
| 753 | +\newcommand{\EpsLowThetaFiveRR}{0.33} | |
| 754 | +\newcommand{\EpsLowThetaFiveRebate}{0.000} | |
| 755 | +\newcommand{\EpsLowThetaFiveRho}{0.188} | |
| 756 | +\newcommand{\EpsLowThetaFiveRhoDevPct}{\ensuremath{-}3.5} | |
| 757 | +\newcommand{\EpsLowThetaFiveRhoDevPctAbs}{3.5} | |
| 758 | +\newcommand{\EpsLowThetaFiveThr}{0.56} | |
| 759 | +\newcommand{\EpsLowThetaFiveThrHighZ}{\ensuremath{-}0.09} | |
| 760 | +\newcommand{\EpsLowThetaFiveThrHighZAbs}{0.09} | |
| 761 | +\newcommand{\EpsLowThetaFiveThrLowZ}{1.26} | |
| 762 | +\newcommand{\EpsLowThetaFiveqOO}{0.566} | |
| 763 | +\newcommand{\EpsLowThetaFiveqRO}{0.374} | |
| 764 | +\newcommand{\EpsLowThetaFortyEstate}{0.33} | |
| 765 | +\newcommand{\EpsLowThetaFortyGapFour}{0.98} | |
| 766 | +\newcommand{\EpsLowThetaFortyGift}{0.39} | |
| 767 | +\newcommand{\EpsLowThetaFortyGini}{0.53} | |
| 768 | +\newcommand{\EpsLowThetaFortyLamSeven}{0.085} | |
| 769 | +\newcommand{\EpsLowThetaFortyLamThree}{0.21} | |
| 770 | +\newcommand{\EpsLowThetaFortyOwn}{0.626} | |
| 771 | +\newcommand{\EpsLowThetaFortyOwnUthirtyfive}{0.342} | |
| 772 | +\newcommand{\EpsLowThetaFortyP}{0.983} | |
| 773 | +\newcommand{\EpsLowThetaFortyPDevPct}{\ensuremath{-}1.7} | |
| 774 | +\newcommand{\EpsLowThetaFortyPDevPctAbs}{1.7} | |
| 775 | +\newcommand{\EpsLowThetaFortyRR}{0.34} | |
| 776 | +\newcommand{\EpsLowThetaFortyRebate}{0.000} | |
| 777 | +\newcommand{\EpsLowThetaFortyRho}{0.202} | |
| 778 | +\newcommand{\EpsLowThetaFortyRhoDevPct}{3.8} | |
| 779 | +\newcommand{\EpsLowThetaFortyThr}{0.54} | |
| 780 | +\newcommand{\EpsLowThetaFortyThrHighZ}{0.12} | |
| 781 | +\newcommand{\EpsLowThetaFortyThrLowZ}{1.09} | |
| 782 | +\newcommand{\EpsLowThetaFortyqOO}{0.520} | |
| 783 | +\newcommand{\EpsLowThetaFortyqRO}{0.308} | |
| 784 | +\newcommand{\EqIncAoneMedian}{0.22} | |
| 785 | +\newcommand{\EqIncAoneMinus}{0.484} | |
| 786 | +\newcommand{\EqIncAonePlus}{0.504} | |
| 787 | +\newcommand{\EqIncEndowDiffPct}{4} | |
| 788 | +\newcommand{\EqIncFirstBuyAgeLowGift}{5} | |
| 789 | +\newcommand{\EqIncFirstBuyAgeMinus}{4} | |
| 790 | +\newcommand{\EqIncFirstBuyAgePlus}{1} | |
| 791 | +\newcommand{\EqIncGapPctAgeFour}{49} | |
| 792 | +\newcommand{\EqIncGapPctAgeSeven}{12} | |
| 793 | +\newcommand{\EqIncNWMinusAgeFour}{0.33} | |
| 794 | +\newcommand{\EqIncNWMinusAgeSeven}{1.03} | |
| 795 | +\newcommand{\EqIncNWPlusAgeFour}{0.50} | |
| 796 | +\newcommand{\EqIncNWPlusAgeSeven}{1.15} | |
| 68 | 797 | \newcommand{\Erosionarmlevel}{0.000} |
| 69 | 798 | \newcommand{\Erosionarmpersistent}{0.000} |
| 70 | 799 | \newcommand{\Erosionfrmlevel}{0.206} |
@@ -80,13 +809,565 @@ | ||
| 80 | 809 | \newcommand{\Erosiontermpersistenttaylor}{0.206} |
| 81 | 810 | \newcommand{\ExcessOwner}{0.000004} |
| 82 | 811 | \newcommand{\ExcessRental}{0.000007} |
| 812 | +\newcommand{\FixedSupplyAbsorbPct}{94} | |
| 813 | +\newcommand{\FixedSupplyBetterAllPct}{41} | |
| 814 | +\newcommand{\FixedSupplyBetterOMPct}{71} | |
| 815 | +\newcommand{\FixedSupplyBetterONMPct}{1} | |
| 816 | +\newcommand{\FixedSupplyBetterOPct}{45} | |
| 817 | +\newcommand{\FixedSupplyBetterRPct}{35} | |
| 818 | +\newcommand{\FixedSupplyCEVAll}{\ensuremath{-}0.64} | |
| 819 | +\newcommand{\FixedSupplyCEVAllAbs}{0.64} | |
| 820 | +\newcommand{\FixedSupplyCEVAllPE}{\ensuremath{-}0.68} | |
| 821 | +\newcommand{\FixedSupplyCEVAllPEAbs}{0.68} | |
| 822 | +\newcommand{\FixedSupplyCEVO}{\ensuremath{-}0.20} | |
| 823 | +\newcommand{\FixedSupplyCEVOAbs}{0.20} | |
| 824 | +\newcommand{\FixedSupplyCEVOM}{0.63} | |
| 825 | +\newcommand{\FixedSupplyCEVONM}{\ensuremath{-}2.12} | |
| 826 | +\newcommand{\FixedSupplyCEVONMAbs}{2.12} | |
| 827 | +\newcommand{\FixedSupplyCEVR}{\ensuremath{-}0.99} | |
| 828 | +\newcommand{\FixedSupplyCEVRAbs}{0.99} | |
| 829 | +\newcommand{\FixedSupplyConsAll}{\ensuremath{-}0.8} | |
| 830 | +\newcommand{\FixedSupplyConsAllAbs}{0.8} | |
| 831 | +\newcommand{\FixedSupplyConsO}{\ensuremath{-}0.6} | |
| 832 | +\newcommand{\FixedSupplyConsOAbs}{0.6} | |
| 833 | +\newcommand{\FixedSupplyConsR}{\ensuremath{-}1.4} | |
| 834 | +\newcommand{\FixedSupplyConsRAbs}{1.4} | |
| 835 | +\newcommand{\FixedSupplyDOwnGE}{\ensuremath{-}0.01} | |
| 836 | +\newcommand{\FixedSupplyDOwnGEAbs}{0.01} | |
| 837 | +\newcommand{\FixedSupplyDOwnPE}{\ensuremath{-}0.19} | |
| 838 | +\newcommand{\FixedSupplyDOwnPEAbs}{0.19} | |
| 839 | +\newcommand{\FixedSupplyDebtZeroPct}{\ensuremath{-}2.6} | |
| 840 | +\newcommand{\FixedSupplyDebtZeroPctAbs}{2.6} | |
| 841 | +\newcommand{\FixedSupplyDepLoss}{0.48} | |
| 842 | +\newcommand{\FixedSupplyErosion}{0.21} | |
| 843 | +\newcommand{\FixedSupplyLamSevenSS}{0.060} | |
| 844 | +\newcommand{\FixedSupplyLamSevenZero}{0.068} | |
| 845 | +\newcommand{\FixedSupplyLender}{0.27} | |
| 846 | +\newcommand{\FixedSupplyMaxRes}{3.1\times10^{-3}} | |
| 847 | +\newcommand{\FixedSupplyNWAll}{\ensuremath{-}0.02} | |
| 848 | +\newcommand{\FixedSupplyNWAllAbs}{0.02} | |
| 849 | +\newcommand{\FixedSupplyNWO}{\ensuremath{-}0.02} | |
| 850 | +\newcommand{\FixedSupplyNWOAbs}{0.02} | |
| 851 | +\newcommand{\FixedSupplyNWR}{\ensuremath{-}0.01} | |
| 852 | +\newcommand{\FixedSupplyNWRAbs}{0.01} | |
| 853 | +\newcommand{\FixedSupplyNetO}{\ensuremath{-}0.02} | |
| 854 | +\newcommand{\FixedSupplyNetOAbs}{0.02} | |
| 855 | +\newcommand{\FixedSupplyNetR}{\ensuremath{-}0.02} | |
| 856 | +\newcommand{\FixedSupplyNetRAbs}{0.02} | |
| 857 | +\newcommand{\FixedSupplyNominalRateZero}{0.219} | |
| 858 | +\newcommand{\FixedSupplyOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 859 | +\newcommand{\FixedSupplyOwnerOldConsMaxAbs}{2.3} | |
| 860 | +\newcommand{\FixedSupplyOwnerOldConsMin}{\ensuremath{-}1.8} | |
| 861 | +\newcommand{\FixedSupplyOwnerOldConsMinAbs}{1.8} | |
| 862 | +\newcommand{\FixedSupplyOwnerOldLossMax}{0.08} | |
| 863 | +\newcommand{\FixedSupplyOwnerOldLossMin}{0.04} | |
| 864 | +\newcommand{\FixedSupplyOwnerYoungConsMax}{1.1} | |
| 865 | +\newcommand{\FixedSupplyOwnerYoungConsMin}{0.6} | |
| 866 | +\newcommand{\FixedSupplyOwnerYoungGainMax}{0.04} | |
| 867 | +\newcommand{\FixedSupplyPriceBackYears}{20} | |
| 868 | +\newcommand{\FixedSupplyPriceMaxPos}{0.1} | |
| 869 | +\newcommand{\FixedSupplyPriceSignChanges}{1} | |
| 870 | +\newcommand{\FixedSupplyPzero}{\ensuremath{-}0.50} | |
| 871 | +\newcommand{\FixedSupplyPzeroAbs}{0.50} | |
| 872 | +\newcommand{\FixedSupplyRentGainR}{0.00} | |
| 873 | +\newcommand{\FixedSupplyResT}{2\times10^{-4}} | |
| 874 | +\newcommand{\FixedSupplyRhoZero}{\ensuremath{-}1.08} | |
| 875 | +\newcommand{\FixedSupplyRhoZeroAbs}{1.08} | |
| 876 | +\newcommand{\FixedSupplySecs}{374} | |
| 877 | +\newcommand{\FixedSupplySteps}{7} | |
| 878 | +\newcommand{\FixedSupplyThrChangePct}{8} | |
| 879 | +\newcommand{\FixedSupplyThrSS}{0.49} | |
| 880 | +\newcommand{\FixedSupplyThrZero}{0.53} | |
| 881 | +\newcommand{\FixedSupplyWageAll}{0.00} | |
| 882 | +\newcommand{\FixedSupplyWageO}{0.00} | |
| 883 | +\newcommand{\FixedSupplyqROsevenSS}{0.806} | |
| 884 | +\newcommand{\FixedSupplyqROsevenZero}{0.798} | |
| 885 | +\newcommand{\FixedSupplyqROthreeOne}{0.334} | |
| 886 | +\newcommand{\FixedSupplyqROthreeSS}{0.351} | |
| 887 | +\newcommand{\FixedSupplyqROthreeZero}{0.356} | |
| 888 | +\newcommand{\FrmLevelAbsorbPct}{92} | |
| 889 | +\newcommand{\FrmLevelBetterAllPct}{41} | |
| 890 | +\newcommand{\FrmLevelBetterOMPct}{71} | |
| 891 | +\newcommand{\FrmLevelBetterONMPct}{1} | |
| 892 | +\newcommand{\FrmLevelBetterOPct}{45} | |
| 893 | +\newcommand{\FrmLevelBetterRPct}{35} | |
| 894 | +\newcommand{\FrmLevelCEVAll}{\ensuremath{-}0.64} | |
| 895 | +\newcommand{\FrmLevelCEVAllAbs}{0.64} | |
| 896 | +\newcommand{\FrmLevelCEVAllPE}{\ensuremath{-}0.68} | |
| 897 | +\newcommand{\FrmLevelCEVAllPEAbs}{0.68} | |
| 898 | +\newcommand{\FrmLevelCEVO}{\ensuremath{-}0.20} | |
| 899 | +\newcommand{\FrmLevelCEVOAbs}{0.20} | |
| 900 | +\newcommand{\FrmLevelCEVOM}{0.63} | |
| 901 | +\newcommand{\FrmLevelCEVONM}{\ensuremath{-}2.12} | |
| 902 | +\newcommand{\FrmLevelCEVONMAbs}{2.12} | |
| 903 | +\newcommand{\FrmLevelCEVR}{\ensuremath{-}1.00} | |
| 904 | +\newcommand{\FrmLevelCEVRAbs}{1.00} | |
| 905 | +\newcommand{\FrmLevelConsAll}{\ensuremath{-}0.8} | |
| 906 | +\newcommand{\FrmLevelConsAllAbs}{0.8} | |
| 907 | +\newcommand{\FrmLevelConsO}{\ensuremath{-}0.6} | |
| 908 | +\newcommand{\FrmLevelConsOAbs}{0.6} | |
| 909 | +\newcommand{\FrmLevelConsR}{\ensuremath{-}1.3} | |
| 910 | +\newcommand{\FrmLevelConsRAbs}{1.3} | |
| 911 | +\newcommand{\FrmLevelDOwnGE}{\ensuremath{-}0.02} | |
| 912 | +\newcommand{\FrmLevelDOwnGEAbs}{0.02} | |
| 913 | +\newcommand{\FrmLevelDOwnPE}{\ensuremath{-}0.20} | |
| 914 | +\newcommand{\FrmLevelDOwnPEAbs}{0.20} | |
| 915 | +\newcommand{\FrmLevelDebtZeroPct}{\ensuremath{-}2.7} | |
| 916 | +\newcommand{\FrmLevelDebtZeroPctAbs}{2.7} | |
| 917 | +\newcommand{\FrmLevelDepLoss}{0.48} | |
| 918 | +\newcommand{\FrmLevelErosion}{0.21} | |
| 919 | +\newcommand{\FrmLevelLamSevenSS}{0.060} | |
| 920 | +\newcommand{\FrmLevelLamSevenZero}{0.069} | |
| 921 | +\newcommand{\FrmLevelLender}{0.27} | |
| 922 | +\newcommand{\FrmLevelMaxRes}{2.2\times10^{-3}} | |
| 923 | +\newcommand{\FrmLevelNWAll}{\ensuremath{-}0.02} | |
| 924 | +\newcommand{\FrmLevelNWAllAbs}{0.02} | |
| 925 | +\newcommand{\FrmLevelNWO}{\ensuremath{-}0.02} | |
| 926 | +\newcommand{\FrmLevelNWOAbs}{0.02} | |
| 927 | +\newcommand{\FrmLevelNWR}{\ensuremath{-}0.01} | |
| 928 | +\newcommand{\FrmLevelNWRAbs}{0.01} | |
| 929 | +\newcommand{\FrmLevelNetO}{\ensuremath{-}0.02} | |
| 930 | +\newcommand{\FrmLevelNetOAbs}{0.02} | |
| 931 | +\newcommand{\FrmLevelNetR}{\ensuremath{-}0.02} | |
| 932 | +\newcommand{\FrmLevelNetRAbs}{0.02} | |
| 933 | +\newcommand{\FrmLevelNominalRateZero}{0.219} | |
| 934 | +\newcommand{\FrmLevelOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 935 | +\newcommand{\FrmLevelOwnerOldConsMaxAbs}{2.3} | |
| 936 | +\newcommand{\FrmLevelOwnerOldConsMin}{\ensuremath{-}1.8} | |
| 937 | +\newcommand{\FrmLevelOwnerOldConsMinAbs}{1.8} | |
| 938 | +\newcommand{\FrmLevelOwnerOldLossMax}{0.08} | |
| 939 | +\newcommand{\FrmLevelOwnerOldLossMin}{0.04} | |
| 940 | +\newcommand{\FrmLevelOwnerYoungConsMax}{1.0} | |
| 941 | +\newcommand{\FrmLevelOwnerYoungConsMin}{0.6} | |
| 942 | +\newcommand{\FrmLevelOwnerYoungGainMax}{0.04} | |
| 943 | +\newcommand{\FrmLevelPriceBackYears}{20} | |
| 944 | +\newcommand{\FrmLevelPriceMaxPos}{0.1} | |
| 945 | +\newcommand{\FrmLevelPriceSignChanges}{1} | |
| 946 | +\newcommand{\FrmLevelPzero}{\ensuremath{-}0.45} | |
| 947 | +\newcommand{\FrmLevelPzeroAbs}{0.45} | |
| 948 | +\newcommand{\FrmLevelRentGainR}{0.00} | |
| 949 | +\newcommand{\FrmLevelResT}{3\times10^{-4}} | |
| 950 | +\newcommand{\FrmLevelRhoZero}{\ensuremath{-}1.14} | |
| 951 | +\newcommand{\FrmLevelRhoZeroAbs}{1.14} | |
| 952 | +\newcommand{\FrmLevelSecs}{471} | |
| 953 | +\newcommand{\FrmLevelSteps}{10} | |
| 954 | +\newcommand{\FrmLevelThrChangePct}{8} | |
| 955 | +\newcommand{\FrmLevelThrSS}{0.49} | |
| 956 | +\newcommand{\FrmLevelThrZero}{0.53} | |
| 957 | +\newcommand{\FrmLevelWageAll}{0.00} | |
| 958 | +\newcommand{\FrmLevelWageO}{0.00} | |
| 959 | +\newcommand{\FrmLevelqROsevenSS}{0.806} | |
| 960 | +\newcommand{\FrmLevelqROsevenZero}{0.797} | |
| 961 | +\newcommand{\FrmLevelqROthreeOne}{0.336} | |
| 962 | +\newcommand{\FrmLevelqROthreeSS}{0.351} | |
| 963 | +\newcommand{\FrmLevelqROthreeZero}{0.356} | |
| 964 | +\newcommand{\FrmPersAbsorbPct}{92} | |
| 965 | +\newcommand{\FrmPersBetterAllPct}{31} | |
| 966 | +\newcommand{\FrmPersBetterOMPct}{73} | |
| 967 | +\newcommand{\FrmPersBetterONMPct}{2} | |
| 968 | +\newcommand{\FrmPersBetterOPct}{47} | |
| 969 | +\newcommand{\FrmPersBetterRPct}{7} | |
| 970 | +\newcommand{\FrmPersCEVAll}{\ensuremath{-}0.52} | |
| 971 | +\newcommand{\FrmPersCEVAllAbs}{0.52} | |
| 972 | +\newcommand{\FrmPersCEVAllPE}{\ensuremath{-}0.53} | |
| 973 | +\newcommand{\FrmPersCEVAllPEAbs}{0.53} | |
| 974 | +\newcommand{\FrmPersCEVO}{0.15} | |
| 975 | +\newcommand{\FrmPersCEVOM}{1.14} | |
| 976 | +\newcommand{\FrmPersCEVONM}{\ensuremath{-}2.13} | |
| 977 | +\newcommand{\FrmPersCEVONMAbs}{2.13} | |
| 978 | +\newcommand{\FrmPersCEVR}{\ensuremath{-}1.06} | |
| 979 | +\newcommand{\FrmPersCEVRAbs}{1.06} | |
| 980 | +\newcommand{\FrmPersConsAll}{\ensuremath{-}1.2} | |
| 981 | +\newcommand{\FrmPersConsAllAbs}{1.2} | |
| 982 | +\newcommand{\FrmPersConsO}{\ensuremath{-}1.0} | |
| 983 | +\newcommand{\FrmPersConsOAbs}{1.0} | |
| 984 | +\newcommand{\FrmPersConsR}{\ensuremath{-}1.6} | |
| 985 | +\newcommand{\FrmPersConsRAbs}{1.6} | |
| 986 | +\newcommand{\FrmPersDOwnGE}{\ensuremath{-}0.18} | |
| 987 | +\newcommand{\FrmPersDOwnGEAbs}{0.18} | |
| 988 | +\newcommand{\FrmPersDOwnPE}{\ensuremath{-}2.17} | |
| 989 | +\newcommand{\FrmPersDOwnPEAbs}{2.17} | |
| 990 | +\newcommand{\FrmPersDebtZeroPct}{\ensuremath{-}24.7} | |
| 991 | +\newcommand{\FrmPersDebtZeroPctAbs}{24.7} | |
| 992 | +\newcommand{\FrmPersDepLoss}{0.48} | |
| 993 | +\newcommand{\FrmPersErosion}{0.21} | |
| 994 | +\newcommand{\FrmPersLamSevenSS}{0.060} | |
| 995 | +\newcommand{\FrmPersLamSevenZero}{0.065} | |
| 996 | +\newcommand{\FrmPersLender}{0.27} | |
| 997 | +\newcommand{\FrmPersMaxRes}{1.2\times10^{-3}} | |
| 998 | +\newcommand{\FrmPersNWAll}{\ensuremath{-}0.01} | |
| 999 | +\newcommand{\FrmPersNWAllAbs}{0.01} | |
| 1000 | +\newcommand{\FrmPersNWO}{\ensuremath{-}0.02} | |
| 1001 | +\newcommand{\FrmPersNWOAbs}{0.02} | |
| 1002 | +\newcommand{\FrmPersNWR}{\ensuremath{-}0.01} | |
| 1003 | +\newcommand{\FrmPersNWRAbs}{0.01} | |
| 1004 | +\newcommand{\FrmPersNetO}{\ensuremath{-}0.02} | |
| 1005 | +\newcommand{\FrmPersNetOAbs}{0.02} | |
| 1006 | +\newcommand{\FrmPersNetR}{\ensuremath{-}0.02} | |
| 1007 | +\newcommand{\FrmPersNetRAbs}{0.02} | |
| 1008 | +\newcommand{\FrmPersNominalRateZero}{0.341} | |
| 1009 | +\newcommand{\FrmPersOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 1010 | +\newcommand{\FrmPersOwnerOldConsMaxAbs}{2.3} | |
| 1011 | +\newcommand{\FrmPersOwnerOldConsMin}{\ensuremath{-}1.7} | |
| 1012 | +\newcommand{\FrmPersOwnerOldConsMinAbs}{1.7} | |
| 1013 | +\newcommand{\FrmPersOwnerOldLossMax}{0.08} | |
| 1014 | +\newcommand{\FrmPersOwnerOldLossMin}{0.04} | |
| 1015 | +\newcommand{\FrmPersOwnerYoungConsMax}{0.7} | |
| 1016 | +\newcommand{\FrmPersOwnerYoungConsMin}{0.3} | |
| 1017 | +\newcommand{\FrmPersOwnerYoungGainMax}{0.04} | |
| 1018 | +\newcommand{\FrmPersPriceBackYears}{5} | |
| 1019 | +\newcommand{\FrmPersPriceMaxPos}{0.1} | |
| 1020 | +\newcommand{\FrmPersPriceSignChanges}{3} | |
| 1021 | +\newcommand{\FrmPersPzero}{\ensuremath{-}1.88} | |
| 1022 | +\newcommand{\FrmPersPzeroAbs}{1.88} | |
| 1023 | +\newcommand{\FrmPersRentGainR}{0.00} | |
| 1024 | +\newcommand{\FrmPersResT}{4\times10^{-4}} | |
| 1025 | +\newcommand{\FrmPersRhoZero}{\ensuremath{-}0.38} | |
| 1026 | +\newcommand{\FrmPersRhoZeroAbs}{0.38} | |
| 1027 | +\newcommand{\FrmPersSecs}{167} | |
| 1028 | +\newcommand{\FrmPersSteps}{4} | |
| 1029 | +\newcommand{\FrmPersThrChangePct}{16} | |
| 1030 | +\newcommand{\FrmPersThrSS}{0.49} | |
| 1031 | +\newcommand{\FrmPersThrZero}{0.57} | |
| 1032 | +\newcommand{\FrmPersWageAll}{0.00} | |
| 1033 | +\newcommand{\FrmPersWageO}{0.00} | |
| 1034 | +\newcommand{\FrmPersqROsevenSS}{0.806} | |
| 1035 | +\newcommand{\FrmPersqROsevenZero}{0.811} | |
| 1036 | +\newcommand{\FrmPersqROthreeOne}{0.334} | |
| 1037 | +\newcommand{\FrmPersqROthreeSS}{0.351} | |
| 1038 | +\newcommand{\FrmPersqROthreeZero}{0.357} | |
| 1039 | +\newcommand{\FrmTaylorAbsorbPct}{95} | |
| 1040 | +\newcommand{\FrmTaylorBetterAllPct}{53} | |
| 1041 | +\newcommand{\FrmTaylorBetterOMPct}{70} | |
| 1042 | +\newcommand{\FrmTaylorBetterONMPct}{22} | |
| 1043 | +\newcommand{\FrmTaylorBetterOPct}{53} | |
| 1044 | +\newcommand{\FrmTaylorBetterRPct}{53} | |
| 1045 | +\newcommand{\FrmTaylorCEVAll}{0.14} | |
| 1046 | +\newcommand{\FrmTaylorCEVAllPE}{\ensuremath{-}1.32} | |
| 1047 | +\newcommand{\FrmTaylorCEVAllPEAbs}{1.32} | |
| 1048 | +\newcommand{\FrmTaylorCEVO}{0.69} | |
| 1049 | +\newcommand{\FrmTaylorCEVOM}{1.43} | |
| 1050 | +\newcommand{\FrmTaylorCEVONM}{\ensuremath{-}1.04} | |
| 1051 | +\newcommand{\FrmTaylorCEVONMAbs}{1.04} | |
| 1052 | +\newcommand{\FrmTaylorCEVR}{\ensuremath{-}0.31} | |
| 1053 | +\newcommand{\FrmTaylorCEVRAbs}{0.31} | |
| 1054 | +\newcommand{\FrmTaylorConsAll}{\ensuremath{-}3.0} | |
| 1055 | +\newcommand{\FrmTaylorConsAllAbs}{3.0} | |
| 1056 | +\newcommand{\FrmTaylorConsO}{\ensuremath{-}3.0} | |
| 1057 | +\newcommand{\FrmTaylorConsOAbs}{3.0} | |
| 1058 | +\newcommand{\FrmTaylorConsR}{\ensuremath{-}3.0} | |
| 1059 | +\newcommand{\FrmTaylorConsRAbs}{3.0} | |
| 1060 | +\newcommand{\FrmTaylorDOwnGE}{\ensuremath{-}0.12} | |
| 1061 | +\newcommand{\FrmTaylorDOwnGEAbs}{0.12} | |
| 1062 | +\newcommand{\FrmTaylorDOwnPE}{\ensuremath{-}2.64} | |
| 1063 | +\newcommand{\FrmTaylorDOwnPEAbs}{2.64} | |
| 1064 | +\newcommand{\FrmTaylorDebtZeroPct}{\ensuremath{-}26.8} | |
| 1065 | +\newcommand{\FrmTaylorDebtZeroPctAbs}{26.8} | |
| 1066 | +\newcommand{\FrmTaylorDepLoss}{0.48} | |
| 1067 | +\newcommand{\FrmTaylorErosion}{0.21} | |
| 1068 | +\newcommand{\FrmTaylorLamSevenSS}{0.060} | |
| 1069 | +\newcommand{\FrmTaylorLamSevenZero}{0.066} | |
| 1070 | +\newcommand{\FrmTaylorLender}{0.27} | |
| 1071 | +\newcommand{\FrmTaylorMaxRes}{1.3\times10^{-3}} | |
| 1072 | +\newcommand{\FrmTaylorNWAll}{\ensuremath{-}0.03} | |
| 1073 | +\newcommand{\FrmTaylorNWAllAbs}{0.03} | |
| 1074 | +\newcommand{\FrmTaylorNWO}{\ensuremath{-}0.05} | |
| 1075 | +\newcommand{\FrmTaylorNWOAbs}{0.05} | |
| 1076 | +\newcommand{\FrmTaylorNWR}{\ensuremath{-}0.01} | |
| 1077 | +\newcommand{\FrmTaylorNWRAbs}{0.01} | |
| 1078 | +\newcommand{\FrmTaylorNetO}{\ensuremath{-}0.02} | |
| 1079 | +\newcommand{\FrmTaylorNetOAbs}{0.02} | |
| 1080 | +\newcommand{\FrmTaylorNetR}{\ensuremath{-}0.02} | |
| 1081 | +\newcommand{\FrmTaylorNetRAbs}{0.02} | |
| 1082 | +\newcommand{\FrmTaylorNominalRateZero}{0.408} | |
| 1083 | +\newcommand{\FrmTaylorOwnerOldConsMax}{\ensuremath{-}4.2} | |
| 1084 | +\newcommand{\FrmTaylorOwnerOldConsMaxAbs}{4.2} | |
| 1085 | +\newcommand{\FrmTaylorOwnerOldConsMin}{\ensuremath{-}3.4} | |
| 1086 | +\newcommand{\FrmTaylorOwnerOldConsMinAbs}{3.4} | |
| 1087 | +\newcommand{\FrmTaylorOwnerOldLossMax}{0.08} | |
| 1088 | +\newcommand{\FrmTaylorOwnerOldLossMin}{0.04} | |
| 1089 | +\newcommand{\FrmTaylorOwnerYoungConsMax}{\ensuremath{-}2.0} | |
| 1090 | +\newcommand{\FrmTaylorOwnerYoungConsMaxAbs}{2.0} | |
| 1091 | +\newcommand{\FrmTaylorOwnerYoungConsMin}{\ensuremath{-}3.9} | |
| 1092 | +\newcommand{\FrmTaylorOwnerYoungConsMinAbs}{3.9} | |
| 1093 | +\newcommand{\FrmTaylorOwnerYoungGainMax}{0.04} | |
| 1094 | +\newcommand{\FrmTaylorPriceBackYears}{40} | |
| 1095 | +\newcommand{\FrmTaylorPriceMaxPos}{1.4} | |
| 1096 | +\newcommand{\FrmTaylorPriceSignChanges}{1} | |
| 1097 | +\newcommand{\FrmTaylorPzero}{\ensuremath{-}7.25} | |
| 1098 | +\newcommand{\FrmTaylorPzeroAbs}{7.25} | |
| 1099 | +\newcommand{\FrmTaylorRentGainR}{0.00} | |
| 1100 | +\newcommand{\FrmTaylorResT}{9\times10^{-5}} | |
| 1101 | +\newcommand{\FrmTaylorRhoZero}{\ensuremath{-}2.29} | |
| 1102 | +\newcommand{\FrmTaylorRhoZeroAbs}{2.29} | |
| 1103 | +\newcommand{\FrmTaylorSecs}{211} | |
| 1104 | +\newcommand{\FrmTaylorSteps}{6} | |
| 1105 | +\newcommand{\FrmTaylorThrChangePct}{12} | |
| 1106 | +\newcommand{\FrmTaylorThrSS}{0.49} | |
| 1107 | +\newcommand{\FrmTaylorThrZero}{0.55} | |
| 1108 | +\newcommand{\FrmTaylorWageAll}{0.00} | |
| 1109 | +\newcommand{\FrmTaylorWageO}{0.00} | |
| 1110 | +\newcommand{\FrmTaylorqROsevenSS}{0.806} | |
| 1111 | +\newcommand{\FrmTaylorqROsevenZero}{0.803} | |
| 1112 | +\newcommand{\FrmTaylorqROthreeOne}{0.345} | |
| 1113 | +\newcommand{\FrmTaylorqROthreeSS}{0.351} | |
| 1114 | +\newcommand{\FrmTaylorqROthreeZero}{0.362} | |
| 1115 | +\newcommand{\GammaThreeEstate}{0.26} | |
| 1116 | +\newcommand{\GammaThreeGapFour}{0.97} | |
| 1117 | +\newcommand{\GammaThreeGift}{0.28} | |
| 1118 | +\newcommand{\GammaThreeGini}{0.53} | |
| 1119 | +\newcommand{\GammaThreeLamSeven}{0.050} | |
| 1120 | +\newcommand{\GammaThreeLamThree}{0.16} | |
| 1121 | +\newcommand{\GammaThreeOwn}{0.646} | |
| 1122 | +\newcommand{\GammaThreeOwnUthirtyfive}{0.373} | |
| 1123 | +\newcommand{\GammaThreeP}{0.995} | |
| 1124 | +\newcommand{\GammaThreePDevPct}{\ensuremath{-}0.5} | |
| 1125 | +\newcommand{\GammaThreePDevPctAbs}{0.5} | |
| 1126 | +\newcommand{\GammaThreeRR}{0.28} | |
| 1127 | +\newcommand{\GammaThreeRebate}{0.000} | |
| 1128 | +\newcommand{\GammaThreeRho}{0.189} | |
| 1129 | +\newcommand{\GammaThreeRhoDevPct}{\ensuremath{-}2.6} | |
| 1130 | +\newcommand{\GammaThreeRhoDevPctAbs}{2.6} | |
| 1131 | +\newcommand{\GammaThreeThr}{0.47} | |
| 1132 | +\newcommand{\GammaThreeThrHighZ}{\ensuremath{-}0.09} | |
| 1133 | +\newcommand{\GammaThreeThrHighZAbs}{0.09} | |
| 1134 | +\newcommand{\GammaThreeThrLowZ}{1.17} | |
| 1135 | +\newcommand{\GammaThreeqOO}{0.532} | |
| 1136 | +\newcommand{\GammaThreeqRO}{0.375} | |
| 1137 | +\newcommand{\GapAgeFour}{0.95} | |
| 1138 | +\newcommand{\GapAgeSeven}{1.57} | |
| 1139 | +\newcommand{\GapAgeTen}{1.03} | |
| 1140 | +\newcommand{\GapAgeTwo}{0.62} | |
| 1141 | +\newcommand{\GapQtwoQfiveAgeOnePP}{86} | |
| 1142 | +\newcommand{\GapQtwoQfiveAgeSevenPP}{16} | |
| 1143 | +\newcommand{\GiniAgeSeven}{0.38} | |
| 83 | 1144 | \newcommand{\GiniNW}{0.54} |
| 1145 | +\newcommand{\GridCoarseEstate}{0.32} | |
| 1146 | +\newcommand{\GridCoarseGapFour}{0.98} | |
| 1147 | +\newcommand{\GridCoarseGift}{0.39} | |
| 1148 | +\newcommand{\GridCoarseGini}{0.54} | |
| 1149 | +\newcommand{\GridCoarseLamSeven}{0.077} | |
| 1150 | +\newcommand{\GridCoarseLamThree}{0.19} | |
| 1151 | +\newcommand{\GridCoarseOwn}{0.626} | |
| 1152 | +\newcommand{\GridCoarseOwnUthirtyfive}{0.369} | |
| 1153 | +\newcommand{\GridCoarseP}{0.992} | |
| 1154 | +\newcommand{\GridCoarsePDevPct}{\ensuremath{-}0.8} | |
| 1155 | +\newcommand{\GridCoarsePDevPctAbs}{0.8} | |
| 1156 | +\newcommand{\GridCoarseRR}{0.34} | |
| 1157 | +\newcommand{\GridCoarseRebate}{0.000} | |
| 1158 | +\newcommand{\GridCoarseRho}{0.196} | |
| 1159 | +\newcommand{\GridCoarseRhoDevPct}{0.8} | |
| 1160 | +\newcommand{\GridCoarseThr}{0.56} | |
| 1161 | +\newcommand{\GridCoarseThrHighZ}{0.02} | |
| 1162 | +\newcommand{\GridCoarseThrLowZ}{1.14} | |
| 1163 | +\newcommand{\GridCoarseqOO}{0.542} | |
| 1164 | +\newcommand{\GridCoarseqRO}{0.350} | |
| 1165 | +\newcommand{\GridFineEstate}{0.31} | |
| 1166 | +\newcommand{\GridFineGapFour}{0.94} | |
| 1167 | +\newcommand{\GridFineGift}{0.39} | |
| 1168 | +\newcommand{\GridFineGini}{0.54} | |
| 1169 | +\newcommand{\GridFineLamSeven}{0.054} | |
| 1170 | +\newcommand{\GridFineLamThree}{0.20} | |
| 1171 | +\newcommand{\GridFineOwn}{0.635} | |
| 1172 | +\newcommand{\GridFineOwnUthirtyfive}{0.382} | |
| 1173 | +\newcommand{\GridFineP}{0.999} | |
| 1174 | +\newcommand{\GridFinePDevPct}{\ensuremath{-}0.1} | |
| 1175 | +\newcommand{\GridFinePDevPctAbs}{0.1} | |
| 1176 | +\newcommand{\GridFineRR}{0.33} | |
| 1177 | +\newcommand{\GridFineRebate}{0.000} | |
| 1178 | +\newcommand{\GridFineRho}{0.193} | |
| 1179 | +\newcommand{\GridFineRhoDevPct}{\ensuremath{-}0.6} | |
| 1180 | +\newcommand{\GridFineRhoDevPctAbs}{0.6} | |
| 1181 | +\newcommand{\GridFineThr}{0.47} | |
| 1182 | +\newcommand{\GridFineThrHighZ}{0.04} | |
| 1183 | +\newcommand{\GridFineThrLowZ}{1.19} | |
| 1184 | +\newcommand{\GridFineqOO}{0.535} | |
| 1185 | +\newcommand{\GridFineqRO}{0.340} | |
| 1186 | +\newcommand{\HalfLifeGen}{0.25} | |
| 1187 | +\newcommand{\HouseValueIncome}{5.0} | |
| 1188 | +\newcommand{\HouseholdPeriods}{1{,}152{,}000} | |
| 1189 | +\newcommand{\KappaHighEstate}{0.31} | |
| 1190 | +\newcommand{\KappaHighGapFour}{0.91} | |
| 1191 | +\newcommand{\KappaHighGift}{0.39} | |
| 1192 | +\newcommand{\KappaHighGini}{0.54} | |
| 1193 | +\newcommand{\KappaHighLamSeven}{0.069} | |
| 1194 | +\newcommand{\KappaHighLamThree}{0.20} | |
| 1195 | +\newcommand{\KappaHighOwn}{0.633} | |
| 1196 | +\newcommand{\KappaHighOwnUthirtyfive}{0.366} | |
| 1197 | +\newcommand{\KappaHighP}{0.962} | |
| 1198 | +\newcommand{\KappaHighPDevPct}{\ensuremath{-}3.8} | |
| 1199 | +\newcommand{\KappaHighPDevPctAbs}{3.8} | |
| 1200 | +\newcommand{\KappaHighRR}{0.34} | |
| 1201 | +\newcommand{\KappaHighRebate}{0.000} | |
| 1202 | +\newcommand{\KappaHighRho}{0.194} | |
| 1203 | +\newcommand{\KappaHighRhoDevPct}{\ensuremath{-}0.3} | |
| 1204 | +\newcommand{\KappaHighRhoDevPctAbs}{0.3} | |
| 1205 | +\newcommand{\KappaHighThr}{0.52} | |
| 1206 | +\newcommand{\KappaHighThrHighZ}{0.06} | |
| 1207 | +\newcommand{\KappaHighThrLowZ}{1.23} | |
| 1208 | +\newcommand{\KappaHighqOO}{0.539} | |
| 1209 | +\newcommand{\KappaHighqRO}{0.335} | |
| 1210 | +\newcommand{\KappaLowEstate}{0.30} | |
| 1211 | +\newcommand{\KappaLowGapFour}{0.98} | |
| 1212 | +\newcommand{\KappaLowGift}{0.38} | |
| 1213 | +\newcommand{\KappaLowGini}{0.54} | |
| 1214 | +\newcommand{\KappaLowLamSeven}{0.068} | |
| 1215 | +\newcommand{\KappaLowLamThree}{0.17} | |
| 1216 | +\newcommand{\KappaLowOwn}{0.622} | |
| 1217 | +\newcommand{\KappaLowOwnUthirtyfive}{0.374} | |
| 1218 | +\newcommand{\KappaLowP}{1.044} | |
| 1219 | +\newcommand{\KappaLowPDevPct}{4.4} | |
| 1220 | +\newcommand{\KappaLowRR}{0.32} | |
| 1221 | +\newcommand{\KappaLowRebate}{0.000} | |
| 1222 | +\newcommand{\KappaLowRho}{0.196} | |
| 1223 | +\newcommand{\KappaLowRhoDevPct}{0.9} | |
| 1224 | +\newcommand{\KappaLowThr}{0.50} | |
| 1225 | +\newcommand{\KappaLowThrHighZ}{\ensuremath{-}0.04} | |
| 1226 | +\newcommand{\KappaLowThrHighZAbs}{0.04} | |
| 1227 | +\newcommand{\KappaLowThrLowZ}{1.06} | |
| 1228 | +\newcommand{\KappaLowqOO}{0.524} | |
| 1229 | +\newcommand{\KappaLowqRO}{0.350} | |
| 84 | 1230 | \newcommand{\LambdaAgeSeven}{0.060} |
| 85 | 1231 | \newcommand{\LambdaAgeThree}{0.19} |
| 1232 | +\newcommand{\LossNWAllTen}{12} | |
| 1233 | +\newcommand{\LossNWAllThirty}{37} | |
| 1234 | +\newcommand{\LossNWAllTwenty}{24} | |
| 1235 | +\newcommand{\LossNWNewTen}{17} | |
| 1236 | +\newcommand{\LossNWNewThirty}{52} | |
| 1237 | +\newcommand{\LossNWNewTwenty}{35} | |
| 1238 | +\newcommand{\LotteryRounds}{5} | |
| 1239 | +\newcommand{\MeanEstate}{0.32} | |
| 86 | 1240 | \newcommand{\MeanGift}{0.39} |
| 87 | 1241 | \newcommand{\MeanGiftAnnual}{1.9} |
| 1242 | +\newcommand{\MeanLTVNew}{0.60} | |
| 1243 | +\newcommand{\MeanLTVOwners}{0.27} | |
| 1244 | +\newcommand{\MedNWAgeSevenQFive}{2.31} | |
| 1245 | +\newcommand{\MedNWAgeSevenQFour}{1.54} | |
| 1246 | +\newcommand{\MedNWAgeSevenQOne}{1.02} | |
| 1247 | +\newcommand{\MedNWAgeSevenQThree}{1.16} | |
| 1248 | +\newcommand{\MedNWAgeSevenQTwo}{1.06} | |
| 1249 | +\newcommand{\MedianGift}{0.22} | |
| 1250 | +\newcommand{\MultBaseExcessHigh}{\ensuremath{-}0.43} | |
| 1251 | +\newcommand{\MultBaseExcessHighAbs}{0.43} | |
| 1252 | +\newcommand{\MultBaseExcessLow}{0.69} | |
| 1253 | +\newcommand{\MultBaseOwnHigh}{0.50} | |
| 1254 | +\newcommand{\MultBaseOwnLow}{0.73} | |
| 1255 | +\newcommand{\MultBaseRoot}{1.000} | |
| 1256 | +\newcommand{\MultBaseSlope}{\ensuremath{-}3.1} | |
| 1257 | +\newcommand{\MultBaseSlopeAbs}{3.1} | |
| 1258 | +\newcommand{\MultEpsLowExcessHigh}{\ensuremath{-}0.26} | |
| 1259 | +\newcommand{\MultEpsLowExcessHighAbs}{0.26} | |
| 1260 | +\newcommand{\MultEpsLowExcessLow}{0.33} | |
| 1261 | +\newcommand{\MultEpsLowOwnHigh}{0.54} | |
| 1262 | +\newcommand{\MultEpsLowOwnLow}{0.70} | |
| 1263 | +\newcommand{\MultEpsLowRoot}{1.000} | |
| 1264 | +\newcommand{\MultEpsLowSlope}{\ensuremath{-}2.0} | |
| 1265 | +\newcommand{\MultEpsLowSlopeAbs}{2.0} | |
| 1266 | +\newcommand{\MultEpsLowThetaFiveExcessHigh}{\ensuremath{-}0.24} | |
| 1267 | +\newcommand{\MultEpsLowThetaFiveExcessHighAbs}{0.24} | |
| 1268 | +\newcommand{\MultEpsLowThetaFiveExcessLow}{0.34} | |
| 1269 | +\newcommand{\MultEpsLowThetaFiveOwnHigh}{0.55} | |
| 1270 | +\newcommand{\MultEpsLowThetaFiveOwnLow}{0.71} | |
| 1271 | +\newcommand{\MultEpsLowThetaFiveRoot}{1.000} | |
| 1272 | +\newcommand{\MultEpsLowThetaFiveSlope}{\ensuremath{-}1.8} | |
| 1273 | +\newcommand{\MultEpsLowThetaFiveSlopeAbs}{1.8} | |
| 1274 | +\newcommand{\MultPgridHigh}{1.25} | |
| 1275 | +\newcommand{\MultPgridLow}{0.80} | |
| 88 | 1276 | \newcommand{\NDynasties}{96{,}000} |
| 1277 | +\newcommand{\NRobustnessRuns}{49} | |
| 1278 | +\newcommand{\NScenariosSS}{45} | |
| 1279 | +\newcommand{\NScenariosSSConverged}{44} | |
| 1280 | +\newcommand{\NTransitions}{16} | |
| 1281 | +\newcommand{\NTransitionsConverged}{12} | |
| 89 | 1282 | \newcommand{\NWRatioYoung}{11.2} |
| 1283 | +\newcommand{\NoSegAbsorbPct}{23} | |
| 1284 | +\newcommand{\NoSegBetterAllPct}{42} | |
| 1285 | +\newcommand{\NoSegBetterOMPct}{71} | |
| 1286 | +\newcommand{\NoSegBetterONMPct}{1} | |
| 1287 | +\newcommand{\NoSegBetterOPct}{45} | |
| 1288 | +\newcommand{\NoSegBetterRPct}{36} | |
| 1289 | +\newcommand{\NoSegCEVAll}{\ensuremath{-}0.60} | |
| 1290 | +\newcommand{\NoSegCEVAllAbs}{0.60} | |
| 1291 | +\newcommand{\NoSegCEVAllPE}{\ensuremath{-}0.68} | |
| 1292 | +\newcommand{\NoSegCEVAllPEAbs}{0.68} | |
| 1293 | +\newcommand{\NoSegCEVO}{\ensuremath{-}0.17} | |
| 1294 | +\newcommand{\NoSegCEVOAbs}{0.17} | |
| 1295 | +\newcommand{\NoSegCEVOM}{0.66} | |
| 1296 | +\newcommand{\NoSegCEVONM}{\ensuremath{-}2.10} | |
| 1297 | +\newcommand{\NoSegCEVONMAbs}{2.10} | |
| 1298 | +\newcommand{\NoSegCEVR}{\ensuremath{-}0.95} | |
| 1299 | +\newcommand{\NoSegCEVRAbs}{0.95} | |
| 1300 | +\newcommand{\NoSegConsAll}{\ensuremath{-}0.8} | |
| 1301 | +\newcommand{\NoSegConsAllAbs}{0.8} | |
| 1302 | +\newcommand{\NoSegConsO}{\ensuremath{-}0.5} | |
| 1303 | +\newcommand{\NoSegConsOAbs}{0.5} | |
| 1304 | +\newcommand{\NoSegConsR}{\ensuremath{-}1.3} | |
| 1305 | +\newcommand{\NoSegConsRAbs}{1.3} | |
| 1306 | +\newcommand{\NoSegDOwnGE}{\ensuremath{-}0.16} | |
| 1307 | +\newcommand{\NoSegDOwnGEAbs}{0.16} | |
| 1308 | +\newcommand{\NoSegDOwnPE}{\ensuremath{-}0.20} | |
| 1309 | +\newcommand{\NoSegDOwnPEAbs}{0.20} | |
| 1310 | +\newcommand{\NoSegDebtZeroPct}{\ensuremath{-}3.0} | |
| 1311 | +\newcommand{\NoSegDebtZeroPctAbs}{3.0} | |
| 1312 | +\newcommand{\NoSegDepLoss}{0.48} | |
| 1313 | +\newcommand{\NoSegErosion}{0.21} | |
| 1314 | +\newcommand{\NoSegLamSevenSS}{0.060} | |
| 1315 | +\newcommand{\NoSegLamSevenZero}{0.069} | |
| 1316 | +\newcommand{\NoSegLender}{0.27} | |
| 1317 | +\newcommand{\NoSegMaxRes}{6.5\times10^{-4}} | |
| 1318 | +\newcommand{\NoSegNWAll}{\ensuremath{-}0.02} | |
| 1319 | +\newcommand{\NoSegNWAllAbs}{0.02} | |
| 1320 | +\newcommand{\NoSegNWO}{\ensuremath{-}0.02} | |
| 1321 | +\newcommand{\NoSegNWOAbs}{0.02} | |
| 1322 | +\newcommand{\NoSegNWR}{\ensuremath{-}0.01} | |
| 1323 | +\newcommand{\NoSegNWRAbs}{0.01} | |
| 1324 | +\newcommand{\NoSegNetO}{\ensuremath{-}0.02} | |
| 1325 | +\newcommand{\NoSegNetOAbs}{0.02} | |
| 1326 | +\newcommand{\NoSegNetR}{\ensuremath{-}0.02} | |
| 1327 | +\newcommand{\NoSegNetRAbs}{0.02} | |
| 1328 | +\newcommand{\NoSegNominalRateZero}{0.219} | |
| 1329 | +\newcommand{\NoSegOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 1330 | +\newcommand{\NoSegOwnerOldConsMaxAbs}{2.3} | |
| 1331 | +\newcommand{\NoSegOwnerOldConsMin}{\ensuremath{-}1.7} | |
| 1332 | +\newcommand{\NoSegOwnerOldConsMinAbs}{1.7} | |
| 1333 | +\newcommand{\NoSegOwnerOldLossMax}{0.08} | |
| 1334 | +\newcommand{\NoSegOwnerOldLossMin}{0.04} | |
| 1335 | +\newcommand{\NoSegOwnerYoungConsMax}{1.1} | |
| 1336 | +\newcommand{\NoSegOwnerYoungConsMin}{0.7} | |
| 1337 | +\newcommand{\NoSegOwnerYoungGainMax}{0.04} | |
| 1338 | +\newcommand{\NoSegPriceBackYears}{10} | |
| 1339 | +\newcommand{\NoSegPriceMaxPos}{0.1} | |
| 1340 | +\newcommand{\NoSegPriceSignChanges}{1} | |
| 1341 | +\newcommand{\NoSegPzero}{\ensuremath{-}0.39} | |
| 1342 | +\newcommand{\NoSegPzeroAbs}{0.39} | |
| 1343 | +\newcommand{\NoSegRentGainR}{0.00} | |
| 1344 | +\newcommand{\NoSegResT}{4\times10^{-5}} | |
| 1345 | +\newcommand{\NoSegRhoZero}{\ensuremath{-}1.47} | |
| 1346 | +\newcommand{\NoSegRhoZeroAbs}{1.47} | |
| 1347 | +\newcommand{\NoSegSecs}{152} | |
| 1348 | +\newcommand{\NoSegSteps}{2} | |
| 1349 | +\newcommand{\NoSegThrChangePct}{9} | |
| 1350 | +\newcommand{\NoSegThrSS}{0.49} | |
| 1351 | +\newcommand{\NoSegThrZero}{0.54} | |
| 1352 | +\newcommand{\NoSegWageAll}{0.00} | |
| 1353 | +\newcommand{\NoSegWageO}{0.00} | |
| 1354 | +\newcommand{\NoSegqROsevenSS}{0.806} | |
| 1355 | +\newcommand{\NoSegqROsevenZero}{0.796} | |
| 1356 | +\newcommand{\NoSegqROthreeOne}{0.333} | |
| 1357 | +\newcommand{\NoSegqROthreeSS}{0.351} | |
| 1358 | +\newcommand{\NoSegqROthreeZero}{0.356} | |
| 1359 | +\newcommand{\OwnAgeOne}{0.34} | |
| 1360 | +\newcommand{\OwnAgeOneQFive}{0.89} | |
| 1361 | +\newcommand{\OwnAgeOneQFour}{0.57} | |
| 1362 | +\newcommand{\OwnAgeOneQOne}{0.02} | |
| 1363 | +\newcommand{\OwnAgeOneQThree}{0.18} | |
| 1364 | +\newcommand{\OwnAgeOneQTwo}{0.03} | |
| 1365 | +\newcommand{\OwnAgeSevenQFive}{0.97} | |
| 1366 | +\newcommand{\OwnAgeSevenQFour}{0.88} | |
| 1367 | +\newcommand{\OwnAgeSevenQOne}{0.81} | |
| 1368 | +\newcommand{\OwnAgeSevenQThree}{0.82} | |
| 1369 | +\newcommand{\OwnAgeSevenQTwo}{0.81} | |
| 1370 | +\newcommand{\OwnAgeTwelve}{0.36} | |
| 90 | 1371 | \newcommand{\OwnBaseline}{0.631} |
| 91 | 1372 | \newcommand{\OwnBequestshalved}{0.615} |
| 92 | 1373 | \newcommand{\OwnBequestsoff}{0.581} |
@@ -94,25 +1375,142 @@ | ||
| 94 | 1375 | \newcommand{\OwnDSTIcappsi}{0.634} |
| 95 | 1376 | \newcommand{\OwnHalfsegmentation}{0.631} |
| 96 | 1377 | \newcommand{\OwnNosegmentation}{0.631} |
| 1378 | +\newcommand{\OwnPeak}{0.86} | |
| 1379 | +\newcommand{\OwnSixtyFivePlus}{0.61} | |
| 1380 | +\newcommand{\OwnThirtyFiveSixtyFour}{0.73} | |
| 97 | 1381 | \newcommand{\OwnTransfertax}{0.639} |
| 98 | 1382 | \newcommand{\OwnershipAll}{0.63} |
| 99 | 1383 | \newcommand{\OwnershipUnderThirtyFive}{0.38} |
| 100 | 1384 | \newcommand{\Owntheta}{0.619} |
| 101 | 1385 | \newcommand{\Ownthetapricesfixed}{0.596} |
| 102 | 1386 | \newcommand{\Ownvarepsilon}{0.631} |
| 103 | −\newcommand{\PImpactarmlevel}{-0.79} | |
| 104 | −\newcommand{\PImpactarmpersistent}{-1.86} | |
| 105 | −\newcommand{\PImpactfrmlevel}{-0.45} | |
| 106 | −\newcommand{\PImpactfrmlevelfixedsupply}{-0.50} | |
| 107 | −\newcommand{\PImpactfrmlevelnoseg}{-0.39} | |
| 108 | −\newcommand{\PImpactfrmlevelrigidrents}{-0.46} | |
| 109 | −\newcommand{\PImpactfrmlevelstatic}{-0.36} | |
| 110 | −\newcommand{\PImpactfrmlevelunindexedwages}{-3.86} | |
| 111 | −\newcommand{\PImpactfrmpersistent}{-1.88} | |
| 112 | −\newcommand{\PImpactfrmpersistenttaylor}{-7.25} | |
| 113 | −\newcommand{\PImpacttermlevel}{-0.47} | |
| 114 | −\newcommand{\PImpacttermpersistent}{-1.49} | |
| 115 | −\newcommand{\PImpacttermpersistenttaylor}{-6.93} | |
| 1387 | +\newcommand{\PImpactarmlevel}{\ensuremath{-}0.79} | |
| 1388 | +\newcommand{\PImpactarmlevelAbs}{0.79} | |
| 1389 | +\newcommand{\PImpactarmpersistent}{\ensuremath{-}1.86} | |
| 1390 | +\newcommand{\PImpactarmpersistentAbs}{1.86} | |
| 1391 | +\newcommand{\PImpactfrmlevel}{\ensuremath{-}0.45} | |
| 1392 | +\newcommand{\PImpactfrmlevelAbs}{0.45} | |
| 1393 | +\newcommand{\PImpactfrmlevelfixedsupply}{\ensuremath{-}0.50} | |
| 1394 | +\newcommand{\PImpactfrmlevelfixedsupplyAbs}{0.50} | |
| 1395 | +\newcommand{\PImpactfrmlevelnoseg}{\ensuremath{-}0.39} | |
| 1396 | +\newcommand{\PImpactfrmlevelnosegAbs}{0.39} | |
| 1397 | +\newcommand{\PImpactfrmlevelrigidrents}{\ensuremath{-}0.46} | |
| 1398 | +\newcommand{\PImpactfrmlevelrigidrentsAbs}{0.46} | |
| 1399 | +\newcommand{\PImpactfrmlevelstatic}{\ensuremath{-}0.36} | |
| 1400 | +\newcommand{\PImpactfrmlevelstaticAbs}{0.36} | |
| 1401 | +\newcommand{\PImpactfrmlevelunindexedwages}{\ensuremath{-}3.86} | |
| 1402 | +\newcommand{\PImpactfrmlevelunindexedwagesAbs}{3.86} | |
| 1403 | +\newcommand{\PImpactfrmpersistent}{\ensuremath{-}1.88} | |
| 1404 | +\newcommand{\PImpactfrmpersistentAbs}{1.88} | |
| 1405 | +\newcommand{\PImpactfrmpersistenttaylor}{\ensuremath{-}7.25} | |
| 1406 | +\newcommand{\PImpactfrmpersistenttaylorAbs}{7.25} | |
| 1407 | +\newcommand{\PImpacttermlevel}{\ensuremath{-}0.47} | |
| 1408 | +\newcommand{\PImpacttermlevelAbs}{0.47} | |
| 1409 | +\newcommand{\PImpacttermpersistent}{\ensuremath{-}1.49} | |
| 1410 | +\newcommand{\PImpacttermpersistentAbs}{1.49} | |
| 1411 | +\newcommand{\PImpacttermpersistenttaylor}{\ensuremath{-}6.93} | |
| 1412 | +\newcommand{\PImpacttermpersistenttaylorAbs}{6.93} | |
| 1413 | +\newcommand{\PhiHighEstate}{0.32} | |
| 1414 | +\newcommand{\PhiHighGapFour}{0.94} | |
| 1415 | +\newcommand{\PhiHighGift}{0.39} | |
| 1416 | +\newcommand{\PhiHighGini}{0.54} | |
| 1417 | +\newcommand{\PhiHighLamSeven}{0.055} | |
| 1418 | +\newcommand{\PhiHighLamThree}{0.19} | |
| 1419 | +\newcommand{\PhiHighOwn}{0.650} | |
| 1420 | +\newcommand{\PhiHighOwnUthirtyfive}{0.412} | |
| 1421 | +\newcommand{\PhiHighP}{1.007} | |
| 1422 | +\newcommand{\PhiHighPDevPct}{0.7} | |
| 1423 | +\newcommand{\PhiHighRR}{0.34} | |
| 1424 | +\newcommand{\PhiHighRebate}{0.000} | |
| 1425 | +\newcommand{\PhiHighRho}{0.202} | |
| 1426 | +\newcommand{\PhiHighRhoDevPct}{4.0} | |
| 1427 | +\newcommand{\PhiHighThr}{0.47} | |
| 1428 | +\newcommand{\PhiHighThrHighZ}{0.00} | |
| 1429 | +\newcommand{\PhiHighThrLowZ}{1.13} | |
| 1430 | +\newcommand{\PhiHighqOO}{0.548} | |
| 1431 | +\newcommand{\PhiHighqRO}{0.360} | |
| 1432 | +\newcommand{\PhiLowEstate}{0.31} | |
| 1433 | +\newcommand{\PhiLowGapFour}{0.96} | |
| 1434 | +\newcommand{\PhiLowGift}{0.39} | |
| 1435 | +\newcommand{\PhiLowGini}{0.55} | |
| 1436 | +\newcommand{\PhiLowLamSeven}{0.070} | |
| 1437 | +\newcommand{\PhiLowLamThree}{0.19} | |
| 1438 | +\newcommand{\PhiLowOwn}{0.607} | |
| 1439 | +\newcommand{\PhiLowOwnUthirtyfive}{0.342} | |
| 1440 | +\newcommand{\PhiLowP}{0.991} | |
| 1441 | +\newcommand{\PhiLowPDevPct}{\ensuremath{-}0.9} | |
| 1442 | +\newcommand{\PhiLowPDevPctAbs}{0.9} | |
| 1443 | +\newcommand{\PhiLowRR}{0.33} | |
| 1444 | +\newcommand{\PhiLowRebate}{0.000} | |
| 1445 | +\newcommand{\PhiLowRho}{0.186} | |
| 1446 | +\newcommand{\PhiLowRhoDevPct}{\ensuremath{-}4.4} | |
| 1447 | +\newcommand{\PhiLowRhoDevPctAbs}{4.4} | |
| 1448 | +\newcommand{\PhiLowThr}{0.53} | |
| 1449 | +\newcommand{\PhiLowThrHighZ}{0.05} | |
| 1450 | +\newcommand{\PhiLowThrLowZ}{1.24} | |
| 1451 | +\newcommand{\PhiLowqOO}{0.524} | |
| 1452 | +\newcommand{\PhiLowqRO}{0.332} | |
| 1453 | +\newcommand{\PiFourEstate}{0.33} | |
| 1454 | +\newcommand{\PiFourGEOffsetPct}{65} | |
| 1455 | +\newcommand{\PiFourGapFour}{0.97} | |
| 1456 | +\newcommand{\PiFourGift}{0.39} | |
| 1457 | +\newcommand{\PiFourGini}{0.54} | |
| 1458 | +\newcommand{\PiFourLamSeven}{0.060} | |
| 1459 | +\newcommand{\PiFourLamThree}{0.19} | |
| 1460 | +\newcommand{\PiFourOwn}{0.625} | |
| 1461 | +\newcommand{\PiFourOwnUthirtyfive}{0.361} | |
| 1462 | +\newcommand{\PiFourP}{0.986} | |
| 1463 | +\newcommand{\PiFourPDevPct}{\ensuremath{-}1.4} | |
| 1464 | +\newcommand{\PiFourPDevPctAbs}{1.4} | |
| 1465 | +\newcommand{\PiFourPEDropUthirtyfivePP}{4.9} | |
| 1466 | +\newcommand{\PiFourPEEstate}{0.32} | |
| 1467 | +\newcommand{\PiFourPEGapFour}{0.98} | |
| 1468 | +\newcommand{\PiFourPEGift}{0.38} | |
| 1469 | +\newcommand{\PiFourPEGini}{0.55} | |
| 1470 | +\newcommand{\PiFourPELamSeven}{0.070} | |
| 1471 | +\newcommand{\PiFourPELamThree}{0.19} | |
| 1472 | +\newcommand{\PiFourPEOwn}{0.604} | |
| 1473 | +\newcommand{\PiFourPEOwnUthirtyfive}{0.330} | |
| 1474 | +\newcommand{\PiFourPEP}{1.000} | |
| 1475 | +\newcommand{\PiFourPEPDevPct}{0.0} | |
| 1476 | +\newcommand{\PiFourPERR}{0.34} | |
| 1477 | +\newcommand{\PiFourPERebate}{0.000} | |
| 1478 | +\newcommand{\PiFourPERho}{0.194} | |
| 1479 | +\newcommand{\PiFourPERhoDevPct}{0.0} | |
| 1480 | +\newcommand{\PiFourPEThr}{0.59} | |
| 1481 | +\newcommand{\PiFourPEThrHighZ}{0.09} | |
| 1482 | +\newcommand{\PiFourPEThrLowZ}{1.21} | |
| 1483 | +\newcommand{\PiFourPEqOO}{0.513} | |
| 1484 | +\newcommand{\PiFourPEqRO}{0.324} | |
| 1485 | +\newcommand{\PiFourRR}{0.34} | |
| 1486 | +\newcommand{\PiFourRebate}{0.000} | |
| 1487 | +\newcommand{\PiFourRho}{0.197} | |
| 1488 | +\newcommand{\PiFourRhoDevPct}{1.5} | |
| 1489 | +\newcommand{\PiFourThr}{0.55} | |
| 1490 | +\newcommand{\PiFourThrHighZ}{0.07} | |
| 1491 | +\newcommand{\PiFourThrLowZ}{1.13} | |
| 1492 | +\newcommand{\PiFourqOO}{0.529} | |
| 1493 | +\newcommand{\PiFourqRO}{0.338} | |
| 1494 | +\newcommand{\PiZeroEstate}{0.30} | |
| 1495 | +\newcommand{\PiZeroGapFour}{0.91} | |
| 1496 | +\newcommand{\PiZeroGift}{0.39} | |
| 1497 | +\newcommand{\PiZeroGini}{0.55} | |
| 1498 | +\newcommand{\PiZeroLamSeven}{0.058} | |
| 1499 | +\newcommand{\PiZeroLamThree}{0.19} | |
| 1500 | +\newcommand{\PiZeroOwn}{0.640} | |
| 1501 | +\newcommand{\PiZeroOwnUthirtyfive}{0.398} | |
| 1502 | +\newcommand{\PiZeroP}{1.011} | |
| 1503 | +\newcommand{\PiZeroPDevPct}{1.1} | |
| 1504 | +\newcommand{\PiZeroRR}{0.33} | |
| 1505 | +\newcommand{\PiZeroRebate}{0.000} | |
| 1506 | +\newcommand{\PiZeroRho}{0.190} | |
| 1507 | +\newcommand{\PiZeroRhoDevPct}{\ensuremath{-}2.0} | |
| 1508 | +\newcommand{\PiZeroRhoDevPctAbs}{2.0} | |
| 1509 | +\newcommand{\PiZeroThr}{0.43} | |
| 1510 | +\newcommand{\PiZeroThrHighZ}{0.01} | |
| 1511 | +\newcommand{\PiZeroThrLowZ}{1.18} | |
| 1512 | +\newcommand{\PiZeroqOO}{0.537} | |
| 1513 | +\newcommand{\PiZeroqRO}{0.346} | |
| 116 | 1514 | \newcommand{\PriceBaseline}{1.000} |
| 117 | 1515 | \newcommand{\PriceBequestshalved}{0.981} |
| 118 | 1516 | \newcommand{\PriceBequestsoff}{0.931} |
@@ -124,27 +1522,941 @@ | ||
| 124 | 1522 | \newcommand{\Pricetheta}{0.987} |
| 125 | 1523 | \newcommand{\Pricethetapricesfixed}{1.000} |
| 126 | 1524 | \newcommand{\Pricevarepsilon}{1.000} |
| 1525 | +\newcommand{\PsiFiftyEstate}{0.30} | |
| 1526 | +\newcommand{\PsiFiftyGapFour}{0.96} | |
| 1527 | +\newcommand{\PsiFiftyGift}{0.39} | |
| 1528 | +\newcommand{\PsiFiftyGini}{0.54} | |
| 1529 | +\newcommand{\PsiFiftyLamSeven}{0.056} | |
| 1530 | +\newcommand{\PsiFiftyLamThree}{0.19} | |
| 1531 | +\newcommand{\PsiFiftyOwn}{0.634} | |
| 1532 | +\newcommand{\PsiFiftyOwnUthirtyfive}{0.382} | |
| 1533 | +\newcommand{\PsiFiftyP}{1.006} | |
| 1534 | +\newcommand{\PsiFiftyPDevPct}{0.6} | |
| 1535 | +\newcommand{\PsiFiftyRR}{0.33} | |
| 1536 | +\newcommand{\PsiFiftyRebate}{0.000} | |
| 1537 | +\newcommand{\PsiFiftyRho}{0.193} | |
| 1538 | +\newcommand{\PsiFiftyRhoDevPct}{\ensuremath{-}0.7} | |
| 1539 | +\newcommand{\PsiFiftyRhoDevPctAbs}{0.7} | |
| 1540 | +\newcommand{\PsiFiftyThr}{0.46} | |
| 1541 | +\newcommand{\PsiFiftyThrHighZ}{0.04} | |
| 1542 | +\newcommand{\PsiFiftyThrLowZ}{1.13} | |
| 1543 | +\newcommand{\PsiFiftyqOO}{0.533} | |
| 1544 | +\newcommand{\PsiFiftyqRO}{0.339} | |
| 1545 | +\newcommand{\PsiThirtyEstate}{0.33} | |
| 1546 | +\newcommand{\PsiThirtyGapFour}{0.94} | |
| 1547 | +\newcommand{\PsiThirtyGift}{0.39} | |
| 1548 | +\newcommand{\PsiThirtyGini}{0.54} | |
| 1549 | +\newcommand{\PsiThirtyLamSeven}{0.053} | |
| 1550 | +\newcommand{\PsiThirtyLamThree}{0.17} | |
| 1551 | +\newcommand{\PsiThirtyOwn}{0.627} | |
| 1552 | +\newcommand{\PsiThirtyOwnUthirtyfive}{0.362} | |
| 1553 | +\newcommand{\PsiThirtyP}{0.991} | |
| 1554 | +\newcommand{\PsiThirtyPDevPct}{\ensuremath{-}0.9} | |
| 1555 | +\newcommand{\PsiThirtyPDevPctAbs}{0.9} | |
| 1556 | +\newcommand{\PsiThirtyRR}{0.33} | |
| 1557 | +\newcommand{\PsiThirtyRebate}{0.000} | |
| 1558 | +\newcommand{\PsiThirtyRho}{0.196} | |
| 1559 | +\newcommand{\PsiThirtyRhoDevPct}{0.9} | |
| 1560 | +\newcommand{\PsiThirtyThr}{0.59} | |
| 1561 | +\newcommand{\PsiThirtyThrHighZ}{0.02} | |
| 1562 | +\newcommand{\PsiThirtyThrLowZ}{1.18} | |
| 1563 | +\newcommand{\PsiThirtyqOO}{0.535} | |
| 1564 | +\newcommand{\PsiThirtyqRO}{0.362} | |
| 127 | 1565 | \newcommand{\Pstar}{1.000} |
| 128 | 1566 | \newcommand{\RankRankNW}{0.34} |
| 1567 | +\newcommand{\RateFiveAbsorbPct}{96} | |
| 1568 | +\newcommand{\RateFiveBetterAllPct}{75} | |
| 1569 | +\newcommand{\RateFiveBetterOMPct}{65} | |
| 1570 | +\newcommand{\RateFiveBetterONMPct}{69} | |
| 1571 | +\newcommand{\RateFiveBetterOPct}{67} | |
| 1572 | +\newcommand{\RateFiveBetterRPct}{87} | |
| 1573 | +\newcommand{\RateFiveCEVAll}{2.11} | |
| 1574 | +\newcommand{\RateFiveCEVAllPE}{\ensuremath{-}1.52} | |
| 1575 | +\newcommand{\RateFiveCEVAllPEAbs}{1.52} | |
| 1576 | +\newcommand{\RateFiveCEVO}{1.56} | |
| 1577 | +\newcommand{\RateFiveCEVOM}{1.00} | |
| 1578 | +\newcommand{\RateFiveCEVONM}{2.86} | |
| 1579 | +\newcommand{\RateFiveCEVR}{2.59} | |
| 1580 | +\newcommand{\RateFiveConsAll}{\ensuremath{-}5.7} | |
| 1581 | +\newcommand{\RateFiveConsAllAbs}{5.7} | |
| 1582 | +\newcommand{\RateFiveConsO}{\ensuremath{-}6.3} | |
| 1583 | +\newcommand{\RateFiveConsOAbs}{6.3} | |
| 1584 | +\newcommand{\RateFiveConsR}{\ensuremath{-}3.9} | |
| 1585 | +\newcommand{\RateFiveConsRAbs}{3.9} | |
| 1586 | +\newcommand{\RateFiveDOwnGE}{\ensuremath{-}0.20} | |
| 1587 | +\newcommand{\RateFiveDOwnGEAbs}{0.20} | |
| 1588 | +\newcommand{\RateFiveDOwnPE}{\ensuremath{-}5.56} | |
| 1589 | +\newcommand{\RateFiveDOwnPEAbs}{5.56} | |
| 1590 | +\newcommand{\RateFiveDebtZeroPct}{\ensuremath{-}13.5} | |
| 1591 | +\newcommand{\RateFiveDebtZeroPctAbs}{13.5} | |
| 1592 | +\newcommand{\RateFiveDepLoss}{0.00} | |
| 1593 | +\newcommand{\RateFiveErosion}{0.00} | |
| 1594 | +\newcommand{\RateFiveLamSevenSS}{0.069} | |
| 1595 | +\newcommand{\RateFiveLamSevenZero}{0.067} | |
| 1596 | +\newcommand{\RateFiveLender}{0.00} | |
| 1597 | +\newcommand{\RateFiveMaxRes}{8.8\times10^{-3}} | |
| 1598 | +\newcommand{\RateFiveNWAll}{\ensuremath{-}0.06} | |
| 1599 | +\newcommand{\RateFiveNWAllAbs}{0.06} | |
| 1600 | +\newcommand{\RateFiveNWO}{\ensuremath{-}0.11} | |
| 1601 | +\newcommand{\RateFiveNWOAbs}{0.11} | |
| 1602 | +\newcommand{\RateFiveNWR}{0.02} | |
| 1603 | +\newcommand{\RateFiveNetO}{0.00} | |
| 1604 | +\newcommand{\RateFiveNetR}{0.00} | |
| 1605 | +\newcommand{\RateFiveNominalRateZero}{0.346} | |
| 1606 | +\newcommand{\RateFiveOwnerOldConsMax}{\ensuremath{-}6.5} | |
| 1607 | +\newcommand{\RateFiveOwnerOldConsMaxAbs}{6.5} | |
| 1608 | +\newcommand{\RateFiveOwnerOldConsMin}{\ensuremath{-}4.9} | |
| 1609 | +\newcommand{\RateFiveOwnerOldConsMinAbs}{4.9} | |
| 1610 | +\newcommand{\RateFiveOwnerOldLossMax}{0.00} | |
| 1611 | +\newcommand{\RateFiveOwnerOldLossMin}{0.00} | |
| 1612 | +\newcommand{\RateFiveOwnerYoungConsMax}{\ensuremath{-}7.9} | |
| 1613 | +\newcommand{\RateFiveOwnerYoungConsMaxAbs}{7.9} | |
| 1614 | +\newcommand{\RateFiveOwnerYoungConsMin}{\ensuremath{-}11.0} | |
| 1615 | +\newcommand{\RateFiveOwnerYoungConsMinAbs}{11.0} | |
| 1616 | +\newcommand{\RateFiveOwnerYoungGainMax}{0.00} | |
| 1617 | +\newcommand{\RateFivePriceBackYears}{15} | |
| 1618 | +\newcommand{\RateFivePriceMaxPos}{7.6} | |
| 1619 | +\newcommand{\RateFivePriceSignChanges}{1} | |
| 1620 | +\newcommand{\RateFivePzero}{\ensuremath{-}16.16} | |
| 1621 | +\newcommand{\RateFivePzeroAbs}{16.16} | |
| 1622 | +\newcommand{\RateFiveRentGainR}{0.00} | |
| 1623 | +\newcommand{\RateFiveResT}{1\times10^{-4}} | |
| 1624 | +\newcommand{\RateFiveRhoZero}{\ensuremath{-}3.02} | |
| 1625 | +\newcommand{\RateFiveRhoZeroAbs}{3.02} | |
| 1626 | +\newcommand{\RateFiveSecs}{437} | |
| 1627 | +\newcommand{\RateFiveSteps}{4} | |
| 1628 | +\newcommand{\RateFiveThrChangePct}{\ensuremath{-}19} | |
| 1629 | +\newcommand{\RateFiveThrChangePctAbs}{19} | |
| 1630 | +\newcommand{\RateFiveThrSS}{0.56} | |
| 1631 | +\newcommand{\RateFiveThrZero}{0.45} | |
| 1632 | +\newcommand{\RateFiveWageAll}{0.00} | |
| 1633 | +\newcommand{\RateFiveWageO}{0.00} | |
| 1634 | +\newcommand{\RateFiveqROsevenSS}{0.780} | |
| 1635 | +\newcommand{\RateFiveqROsevenZero}{0.765} | |
| 1636 | +\newcommand{\RateFiveqROthreeOne}{0.372} | |
| 1637 | +\newcommand{\RateFiveqROthreeSS}{0.375} | |
| 1638 | +\newcommand{\RateFiveqROthreeZero}{0.382} | |
| 1639 | +\newcommand{\RateFortyAbsorbPct}{109} | |
| 1640 | +\newcommand{\RateFortyBetterAllPct}{80} | |
| 1641 | +\newcommand{\RateFortyBetterOMPct}{64} | |
| 1642 | +\newcommand{\RateFortyBetterONMPct}{72} | |
| 1643 | +\newcommand{\RateFortyBetterOPct}{67} | |
| 1644 | +\newcommand{\RateFortyBetterRPct}{100} | |
| 1645 | +\newcommand{\RateFortyCEVAll}{2.61} | |
| 1646 | +\newcommand{\RateFortyCEVAllPE}{\ensuremath{-}1.52} | |
| 1647 | +\newcommand{\RateFortyCEVAllPEAbs}{1.52} | |
| 1648 | +\newcommand{\RateFortyCEVO}{1.92} | |
| 1649 | +\newcommand{\RateFortyCEVOM}{1.36} | |
| 1650 | +\newcommand{\RateFortyCEVONM}{3.38} | |
| 1651 | +\newcommand{\RateFortyCEVR}{3.15} | |
| 1652 | +\newcommand{\RateFortyConsAll}{\ensuremath{-}5.1} | |
| 1653 | +\newcommand{\RateFortyConsAllAbs}{5.1} | |
| 1654 | +\newcommand{\RateFortyConsO}{\ensuremath{-}5.7} | |
| 1655 | +\newcommand{\RateFortyConsOAbs}{5.7} | |
| 1656 | +\newcommand{\RateFortyConsR}{\ensuremath{-}3.7} | |
| 1657 | +\newcommand{\RateFortyConsRAbs}{3.7} | |
| 1658 | +\newcommand{\RateFortyDOwnGE}{0.16} | |
| 1659 | +\newcommand{\RateFortyDOwnPE}{\ensuremath{-}1.77} | |
| 1660 | +\newcommand{\RateFortyDOwnPEAbs}{1.77} | |
| 1661 | +\newcommand{\RateFortyDebtZeroPct}{\ensuremath{-}17.0} | |
| 1662 | +\newcommand{\RateFortyDebtZeroPctAbs}{17.0} | |
| 1663 | +\newcommand{\RateFortyDepLoss}{0.00} | |
| 1664 | +\newcommand{\RateFortyErosion}{0.00} | |
| 1665 | +\newcommand{\RateFortyLamSevenSS}{0.083} | |
| 1666 | +\newcommand{\RateFortyLamSevenZero}{0.091} | |
| 1667 | +\newcommand{\RateFortyLender}{0.00} | |
| 1668 | +\newcommand{\RateFortyMaxRes}{1.1\times10^{-3}} | |
| 1669 | +\newcommand{\RateFortyNWAll}{\ensuremath{-}0.05} | |
| 1670 | +\newcommand{\RateFortyNWAllAbs}{0.05} | |
| 1671 | +\newcommand{\RateFortyNWO}{\ensuremath{-}0.10} | |
| 1672 | +\newcommand{\RateFortyNWOAbs}{0.10} | |
| 1673 | +\newcommand{\RateFortyNWR}{0.02} | |
| 1674 | +\newcommand{\RateFortyNetO}{0.00} | |
| 1675 | +\newcommand{\RateFortyNetR}{0.00} | |
| 1676 | +\newcommand{\RateFortyNominalRateZero}{0.346} | |
| 1677 | +\newcommand{\RateFortyOwnerOldConsMax}{\ensuremath{-}5.8} | |
| 1678 | +\newcommand{\RateFortyOwnerOldConsMaxAbs}{5.8} | |
| 1679 | +\newcommand{\RateFortyOwnerOldConsMin}{\ensuremath{-}4.1} | |
| 1680 | +\newcommand{\RateFortyOwnerOldConsMinAbs}{4.1} | |
| 1681 | +\newcommand{\RateFortyOwnerOldLossMax}{0.00} | |
| 1682 | +\newcommand{\RateFortyOwnerOldLossMin}{0.00} | |
| 1683 | +\newcommand{\RateFortyOwnerYoungConsMax}{\ensuremath{-}7.3} | |
| 1684 | +\newcommand{\RateFortyOwnerYoungConsMaxAbs}{7.3} | |
| 1685 | +\newcommand{\RateFortyOwnerYoungConsMin}{\ensuremath{-}9.6} | |
| 1686 | +\newcommand{\RateFortyOwnerYoungConsMinAbs}{9.6} | |
| 1687 | +\newcommand{\RateFortyOwnerYoungGainMax}{0.00} | |
| 1688 | +\newcommand{\RateFortyPriceBackYears}{--} | |
| 1689 | +\newcommand{\RateFortyPriceMaxPos}{7.8} | |
| 1690 | +\newcommand{\RateFortyPriceSignChanges}{1} | |
| 1691 | +\newcommand{\RateFortyPzero}{\ensuremath{-}14.61} | |
| 1692 | +\newcommand{\RateFortyPzeroAbs}{14.61} | |
| 1693 | +\newcommand{\RateFortyRentGainR}{0.00} | |
| 1694 | +\newcommand{\RateFortyResT}{3\times10^{-5}} | |
| 1695 | +\newcommand{\RateFortyRhoZero}{\ensuremath{-}5.19} | |
| 1696 | +\newcommand{\RateFortyRhoZeroAbs}{5.19} | |
| 1697 | +\newcommand{\RateFortySecs}{351} | |
| 1698 | +\newcommand{\RateFortySteps}{11} | |
| 1699 | +\newcommand{\RateFortyThrChangePct}{\ensuremath{-}10} | |
| 1700 | +\newcommand{\RateFortyThrChangePctAbs}{10} | |
| 1701 | +\newcommand{\RateFortyThrSS}{0.55} | |
| 1702 | +\newcommand{\RateFortyThrZero}{0.49} | |
| 1703 | +\newcommand{\RateFortyWageAll}{0.00} | |
| 1704 | +\newcommand{\RateFortyWageO}{0.00} | |
| 1705 | +\newcommand{\RateFortyqROsevenSS}{0.784} | |
| 1706 | +\newcommand{\RateFortyqROsevenZero}{0.744} | |
| 1707 | +\newcommand{\RateFortyqROthreeOne}{0.341} | |
| 1708 | +\newcommand{\RateFortyqROthreeSS}{0.304} | |
| 1709 | +\newcommand{\RateFortyqROthreeZero}{0.334} | |
| 1710 | +\newcommand{\RateTwentyAbsorbPct}{100} | |
| 1711 | +\newcommand{\RateTwentyBetterAllPct}{80} | |
| 1712 | +\newcommand{\RateTwentyBetterOMPct}{65} | |
| 1713 | +\newcommand{\RateTwentyBetterONMPct}{70} | |
| 1714 | +\newcommand{\RateTwentyBetterOPct}{66} | |
| 1715 | +\newcommand{\RateTwentyBetterRPct}{100} | |
| 1716 | +\newcommand{\RateTwentyCEVAll}{2.33} | |
| 1717 | +\newcommand{\RateTwentyCEVAllPE}{\ensuremath{-}1.57} | |
| 1718 | +\newcommand{\RateTwentyCEVAllPEAbs}{1.57} | |
| 1719 | +\newcommand{\RateTwentyCEVO}{1.55} | |
| 1720 | +\newcommand{\RateTwentyCEVOM}{0.97} | |
| 1721 | +\newcommand{\RateTwentyCEVONM}{2.96} | |
| 1722 | +\newcommand{\RateTwentyCEVR}{2.98} | |
| 1723 | +\newcommand{\RateTwentyConsAll}{\ensuremath{-}5.4} | |
| 1724 | +\newcommand{\RateTwentyConsAllAbs}{5.4} | |
| 1725 | +\newcommand{\RateTwentyConsO}{\ensuremath{-}6.0} | |
| 1726 | +\newcommand{\RateTwentyConsOAbs}{6.0} | |
| 1727 | +\newcommand{\RateTwentyConsR}{\ensuremath{-}3.9} | |
| 1728 | +\newcommand{\RateTwentyConsRAbs}{3.9} | |
| 1729 | +\newcommand{\RateTwentyDOwnGE}{0.01} | |
| 1730 | +\newcommand{\RateTwentyDOwnPE}{\ensuremath{-}4.11} | |
| 1731 | +\newcommand{\RateTwentyDOwnPEAbs}{4.11} | |
| 1732 | +\newcommand{\RateTwentyDebtZeroPct}{\ensuremath{-}15.0} | |
| 1733 | +\newcommand{\RateTwentyDebtZeroPctAbs}{15.0} | |
| 1734 | +\newcommand{\RateTwentyDepLoss}{0.00} | |
| 1735 | +\newcommand{\RateTwentyErosion}{0.00} | |
| 1736 | +\newcommand{\RateTwentyLamSevenSS}{0.060} | |
| 1737 | +\newcommand{\RateTwentyLamSevenZero}{0.070} | |
| 1738 | +\newcommand{\RateTwentyLender}{0.00} | |
| 1739 | +\newcommand{\RateTwentyMaxRes}{9.5\times10^{-4}} | |
| 1740 | +\newcommand{\RateTwentyNWAll}{\ensuremath{-}0.06} | |
| 1741 | +\newcommand{\RateTwentyNWAllAbs}{0.06} | |
| 1742 | +\newcommand{\RateTwentyNWO}{\ensuremath{-}0.11} | |
| 1743 | +\newcommand{\RateTwentyNWOAbs}{0.11} | |
| 1744 | +\newcommand{\RateTwentyNWR}{0.02} | |
| 1745 | +\newcommand{\RateTwentyNetO}{0.00} | |
| 1746 | +\newcommand{\RateTwentyNetR}{0.00} | |
| 1747 | +\newcommand{\RateTwentyNominalRateZero}{0.346} | |
| 1748 | +\newcommand{\RateTwentyOwnerOldConsMax}{\ensuremath{-}6.3} | |
| 1749 | +\newcommand{\RateTwentyOwnerOldConsMaxAbs}{6.3} | |
| 1750 | +\newcommand{\RateTwentyOwnerOldConsMin}{\ensuremath{-}4.7} | |
| 1751 | +\newcommand{\RateTwentyOwnerOldConsMinAbs}{4.7} | |
| 1752 | +\newcommand{\RateTwentyOwnerOldLossMax}{0.00} | |
| 1753 | +\newcommand{\RateTwentyOwnerOldLossMin}{0.00} | |
| 1754 | +\newcommand{\RateTwentyOwnerYoungConsMax}{\ensuremath{-}7.5} | |
| 1755 | +\newcommand{\RateTwentyOwnerYoungConsMaxAbs}{7.5} | |
| 1756 | +\newcommand{\RateTwentyOwnerYoungConsMin}{\ensuremath{-}10.7} | |
| 1757 | +\newcommand{\RateTwentyOwnerYoungConsMinAbs}{10.7} | |
| 1758 | +\newcommand{\RateTwentyOwnerYoungGainMax}{0.00} | |
| 1759 | +\newcommand{\RateTwentyPriceBackYears}{--} | |
| 1760 | +\newcommand{\RateTwentyPriceMaxPos}{6.9} | |
| 1761 | +\newcommand{\RateTwentyPriceSignChanges}{1} | |
| 1762 | +\newcommand{\RateTwentyPzero}{\ensuremath{-}15.54} | |
| 1763 | +\newcommand{\RateTwentyPzeroAbs}{15.54} | |
| 1764 | +\newcommand{\RateTwentyRentGainR}{0.00} | |
| 1765 | +\newcommand{\RateTwentyResT}{7\times10^{-5}} | |
| 1766 | +\newcommand{\RateTwentyRhoZero}{\ensuremath{-}5.32} | |
| 1767 | +\newcommand{\RateTwentyRhoZeroAbs}{5.32} | |
| 1768 | +\newcommand{\RateTwentySecs}{244} | |
| 1769 | +\newcommand{\RateTwentySteps}{10} | |
| 1770 | +\newcommand{\RateTwentyThrChangePct}{1} | |
| 1771 | +\newcommand{\RateTwentyThrSS}{0.49} | |
| 1772 | +\newcommand{\RateTwentyThrZero}{0.50} | |
| 1773 | +\newcommand{\RateTwentyWageAll}{0.00} | |
| 1774 | +\newcommand{\RateTwentyWageO}{0.00} | |
| 1775 | +\newcommand{\RateTwentyqROsevenSS}{0.806} | |
| 1776 | +\newcommand{\RateTwentyqROsevenZero}{0.764} | |
| 1777 | +\newcommand{\RateTwentyqROthreeOne}{0.359} | |
| 1778 | +\newcommand{\RateTwentyqROthreeSS}{0.351} | |
| 1779 | +\newcommand{\RateTwentyqROthreeZero}{0.380} | |
| 129 | 1780 | \newcommand{\RentPriceAnnual}{0.039} |
| 130 | −\newcommand{\RhoImpactarmlevel}{-1.15} | |
| 131 | −\newcommand{\RhoImpactarmpersistent}{-0.18} | |
| 132 | −\newcommand{\RhoImpactfrmlevel}{-1.14} | |
| 133 | −\newcommand{\RhoImpactfrmlevelfixedsupply}{-1.08} | |
| 134 | −\newcommand{\RhoImpactfrmlevelnoseg}{-1.47} | |
| 1781 | +\newcommand{\RhoGEstate}{0.32} | |
| 1782 | +\newcommand{\RhoGGapFour}{1.02} | |
| 1783 | +\newcommand{\RhoGGift}{0.40} | |
| 1784 | +\newcommand{\RhoGGini}{0.55} | |
| 1785 | +\newcommand{\RhoGLamSeven}{0.121} | |
| 1786 | +\newcommand{\RhoGLamThree}{0.32} | |
| 1787 | +\newcommand{\RhoGOwn}{0.624} | |
| 1788 | +\newcommand{\RhoGOwnUthirtyfive}{0.385} | |
| 1789 | +\newcommand{\RhoGP}{1.004} | |
| 1790 | +\newcommand{\RhoGPDevPct}{0.4} | |
| 1791 | +\newcommand{\RhoGRR}{0.41} | |
| 1792 | +\newcommand{\RhoGRebate}{0.000} | |
| 1793 | +\newcommand{\RhoGRho}{0.194} | |
| 1794 | +\newcommand{\RhoGRhoDevPct}{\ensuremath{-}0.1} | |
| 1795 | +\newcommand{\RhoGRhoDevPctAbs}{0.1} | |
| 1796 | +\newcommand{\RhoGThr}{0.50} | |
| 1797 | +\newcommand{\RhoGThrHighZ}{0.03} | |
| 1798 | +\newcommand{\RhoGThrLowZ}{1.19} | |
| 1799 | +\newcommand{\RhoGqOO}{0.545} | |
| 1800 | +\newcommand{\RhoGqRO}{0.223} | |
| 1801 | +\newcommand{\RhoImpactarmlevel}{\ensuremath{-}1.15} | |
| 1802 | +\newcommand{\RhoImpactarmlevelAbs}{1.15} | |
| 1803 | +\newcommand{\RhoImpactarmpersistent}{\ensuremath{-}0.18} | |
| 1804 | +\newcommand{\RhoImpactarmpersistentAbs}{0.18} | |
| 1805 | +\newcommand{\RhoImpactfrmlevel}{\ensuremath{-}1.14} | |
| 1806 | +\newcommand{\RhoImpactfrmlevelAbs}{1.14} | |
| 1807 | +\newcommand{\RhoImpactfrmlevelfixedsupply}{\ensuremath{-}1.08} | |
| 1808 | +\newcommand{\RhoImpactfrmlevelfixedsupplyAbs}{1.08} | |
| 1809 | +\newcommand{\RhoImpactfrmlevelnoseg}{\ensuremath{-}1.47} | |
| 1810 | +\newcommand{\RhoImpactfrmlevelnosegAbs}{1.47} | |
| 135 | 1811 | \newcommand{\RhoImpactfrmlevelrigidrents}{7.48} |
| 136 | −\newcommand{\RhoImpactfrmlevelstatic}{-1.11} | |
| 137 | −\newcommand{\RhoImpactfrmlevelunindexedwages}{-3.21} | |
| 138 | −\newcommand{\RhoImpactfrmpersistent}{-0.38} | |
| 139 | −\newcommand{\RhoImpactfrmpersistenttaylor}{-2.29} | |
| 140 | −\newcommand{\RhoImpacttermlevel}{-1.03} | |
| 1812 | +\newcommand{\RhoImpactfrmlevelstatic}{\ensuremath{-}1.11} | |
| 1813 | +\newcommand{\RhoImpactfrmlevelstaticAbs}{1.11} | |
| 1814 | +\newcommand{\RhoImpactfrmlevelunindexedwages}{\ensuremath{-}3.21} | |
| 1815 | +\newcommand{\RhoImpactfrmlevelunindexedwagesAbs}{3.21} | |
| 1816 | +\newcommand{\RhoImpactfrmpersistent}{\ensuremath{-}0.38} | |
| 1817 | +\newcommand{\RhoImpactfrmpersistentAbs}{0.38} | |
| 1818 | +\newcommand{\RhoImpactfrmpersistenttaylor}{\ensuremath{-}2.29} | |
| 1819 | +\newcommand{\RhoImpactfrmpersistenttaylorAbs}{2.29} | |
| 1820 | +\newcommand{\RhoImpacttermlevel}{\ensuremath{-}1.03} | |
| 1821 | +\newcommand{\RhoImpacttermlevelAbs}{1.03} | |
| 141 | 1822 | \newcommand{\RhoImpacttermpersistent}{0.04} |
| 142 | −\newcommand{\RhoImpacttermpersistenttaylor}{-1.91} | |
| 1823 | +\newcommand{\RhoImpacttermpersistenttaylor}{\ensuremath{-}1.91} | |
| 1824 | +\newcommand{\RhoImpacttermpersistenttaylorAbs}{1.91} | |
| 1825 | +\newcommand{\RhoStar}{0.194} | |
| 1826 | +\newcommand{\RigidRentsAbsorbPct}{99} | |
| 1827 | +\newcommand{\RigidRentsBetterAllPct}{42} | |
| 1828 | +\newcommand{\RigidRentsBetterOMPct}{70} | |
| 1829 | +\newcommand{\RigidRentsBetterONMPct}{1} | |
| 1830 | +\newcommand{\RigidRentsBetterOPct}{44} | |
| 1831 | +\newcommand{\RigidRentsBetterRPct}{38} | |
| 1832 | +\newcommand{\RigidRentsCEVAll}{\ensuremath{-}0.62} | |
| 1833 | +\newcommand{\RigidRentsCEVAllAbs}{0.62} | |
| 1834 | +\newcommand{\RigidRentsCEVAllPE}{\ensuremath{-}0.35} | |
| 1835 | +\newcommand{\RigidRentsCEVAllPEAbs}{0.35} | |
| 1836 | +\newcommand{\RigidRentsCEVO}{\ensuremath{-}0.26} | |
| 1837 | +\newcommand{\RigidRentsCEVOAbs}{0.26} | |
| 1838 | +\newcommand{\RigidRentsCEVOM}{0.55} | |
| 1839 | +\newcommand{\RigidRentsCEVONM}{\ensuremath{-}2.13} | |
| 1840 | +\newcommand{\RigidRentsCEVONMAbs}{2.13} | |
| 1841 | +\newcommand{\RigidRentsCEVR}{\ensuremath{-}0.92} | |
| 1842 | +\newcommand{\RigidRentsCEVRAbs}{0.92} | |
| 1843 | +\newcommand{\RigidRentsConsAll}{\ensuremath{-}0.8} | |
| 1844 | +\newcommand{\RigidRentsConsAllAbs}{0.8} | |
| 1845 | +\newcommand{\RigidRentsConsO}{\ensuremath{-}0.6} | |
| 1846 | +\newcommand{\RigidRentsConsOAbs}{0.6} | |
| 1847 | +\newcommand{\RigidRentsConsR}{\ensuremath{-}1.3} | |
| 1848 | +\newcommand{\RigidRentsConsRAbs}{1.3} | |
| 1849 | +\newcommand{\RigidRentsDOwnGE}{\ensuremath{-}0.03} | |
| 1850 | +\newcommand{\RigidRentsDOwnGEAbs}{0.03} | |
| 1851 | +\newcommand{\RigidRentsDOwnPE}{\ensuremath{-}2.88} | |
| 1852 | +\newcommand{\RigidRentsDOwnPEAbs}{2.88} | |
| 1853 | +\newcommand{\RigidRentsDebtZeroPct}{\ensuremath{-}3.0} | |
| 1854 | +\newcommand{\RigidRentsDebtZeroPctAbs}{3.0} | |
| 1855 | +\newcommand{\RigidRentsDepLoss}{0.48} | |
| 1856 | +\newcommand{\RigidRentsErosion}{0.21} | |
| 1857 | +\newcommand{\RigidRentsLamSevenSS}{0.060} | |
| 1858 | +\newcommand{\RigidRentsLamSevenZero}{0.068} | |
| 1859 | +\newcommand{\RigidRentsLender}{0.27} | |
| 1860 | +\newcommand{\RigidRentsMaxRes}{1.2\times10^{-3}} | |
| 1861 | +\newcommand{\RigidRentsNWAll}{\ensuremath{-}0.02} | |
| 1862 | +\newcommand{\RigidRentsNWAllAbs}{0.02} | |
| 1863 | +\newcommand{\RigidRentsNWO}{\ensuremath{-}0.02} | |
| 1864 | +\newcommand{\RigidRentsNWOAbs}{0.02} | |
| 1865 | +\newcommand{\RigidRentsNWR}{\ensuremath{-}0.01} | |
| 1866 | +\newcommand{\RigidRentsNWRAbs}{0.01} | |
| 1867 | +\newcommand{\RigidRentsNetO}{\ensuremath{-}0.02} | |
| 1868 | +\newcommand{\RigidRentsNetOAbs}{0.02} | |
| 1869 | +\newcommand{\RigidRentsNetR}{\ensuremath{-}0.02} | |
| 1870 | +\newcommand{\RigidRentsNetRAbs}{0.02} | |
| 1871 | +\newcommand{\RigidRentsNominalRateZero}{0.219} | |
| 1872 | +\newcommand{\RigidRentsOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 1873 | +\newcommand{\RigidRentsOwnerOldConsMaxAbs}{2.3} | |
| 1874 | +\newcommand{\RigidRentsOwnerOldConsMin}{\ensuremath{-}1.8} | |
| 1875 | +\newcommand{\RigidRentsOwnerOldConsMinAbs}{1.8} | |
| 1876 | +\newcommand{\RigidRentsOwnerOldLossMax}{0.08} | |
| 1877 | +\newcommand{\RigidRentsOwnerOldLossMin}{0.04} | |
| 1878 | +\newcommand{\RigidRentsOwnerYoungConsMax}{1.1} | |
| 1879 | +\newcommand{\RigidRentsOwnerYoungConsMin}{0.6} | |
| 1880 | +\newcommand{\RigidRentsOwnerYoungGainMax}{0.04} | |
| 1881 | +\newcommand{\RigidRentsPriceBackYears}{20} | |
| 1882 | +\newcommand{\RigidRentsPriceMaxPos}{0.1} | |
| 1883 | +\newcommand{\RigidRentsPriceSignChanges}{1} | |
| 1884 | +\newcommand{\RigidRentsPzero}{\ensuremath{-}0.46} | |
| 1885 | +\newcommand{\RigidRentsPzeroAbs}{0.46} | |
| 1886 | +\newcommand{\RigidRentsRentGainR}{0.01} | |
| 1887 | +\newcommand{\RigidRentsResT}{3\times10^{-4}} | |
| 1888 | +\newcommand{\RigidRentsRhoZero}{7.48} | |
| 1889 | +\newcommand{\RigidRentsSecs}{191} | |
| 1890 | +\newcommand{\RigidRentsSteps}{5} | |
| 1891 | +\newcommand{\RigidRentsThrChangePct}{9} | |
| 1892 | +\newcommand{\RigidRentsThrSS}{0.49} | |
| 1893 | +\newcommand{\RigidRentsThrZero}{0.54} | |
| 1894 | +\newcommand{\RigidRentsWageAll}{0.00} | |
| 1895 | +\newcommand{\RigidRentsWageO}{0.00} | |
| 1896 | +\newcommand{\RigidRentsqROsevenSS}{0.806} | |
| 1897 | +\newcommand{\RigidRentsqROsevenZero}{0.796} | |
| 1898 | +\newcommand{\RigidRentsqROthreeOne}{0.336} | |
| 1899 | +\newcommand{\RigidRentsqROthreeSS}{0.351} | |
| 1900 | +\newcommand{\RigidRentsqROthreeZero}{0.354} | |
| 1901 | +\newcommand{\RoneEstate}{0.24} | |
| 1902 | +\newcommand{\RoneGapFour}{0.90} | |
| 1903 | +\newcommand{\RoneGift}{0.36} | |
| 1904 | +\newcommand{\RoneGini}{0.55} | |
| 1905 | +\newcommand{\RoneLamSeven}{0.063} | |
| 1906 | +\newcommand{\RoneLamThree}{0.17} | |
| 1907 | +\newcommand{\RoneOwn}{0.601} | |
| 1908 | +\newcommand{\RoneOwnUthirtyfive}{0.323} | |
| 1909 | +\newcommand{\RoneP}{1.042} | |
| 1910 | +\newcommand{\RonePDevPct}{4.2} | |
| 1911 | +\newcommand{\RoneRR}{0.26} | |
| 1912 | +\newcommand{\RoneRebate}{0.000} | |
| 1913 | +\newcommand{\RoneRho}{0.177} | |
| 1914 | +\newcommand{\RoneRhoDevPct}{\ensuremath{-}8.9} | |
| 1915 | +\newcommand{\RoneRhoDevPctAbs}{8.9} | |
| 1916 | +\newcommand{\RoneThr}{0.48} | |
| 1917 | +\newcommand{\RoneThrHighZ}{0.08} | |
| 1918 | +\newcommand{\RoneThrLowZ}{1.12} | |
| 1919 | +\newcommand{\RoneqOO}{0.484} | |
| 1920 | +\newcommand{\RoneqRO}{0.315} | |
| 1921 | +\newcommand{\RthreeEstate}{0.48} | |
| 1922 | +\newcommand{\RthreeGapFour}{1.09} | |
| 1923 | +\newcommand{\RthreeGift}{0.46} | |
| 1924 | +\newcommand{\RthreeGini}{0.53} | |
| 1925 | +\newcommand{\RthreeLamSeven}{0.132} | |
| 1926 | +\newcommand{\RthreeLamThree}{0.24} | |
| 1927 | +\newcommand{\RthreeOwn}{0.642} | |
| 1928 | +\newcommand{\RthreeOwnUthirtyfive}{0.399} | |
| 1929 | +\newcommand{\RthreeP}{0.972} | |
| 1930 | +\newcommand{\RthreePDevPct}{\ensuremath{-}2.8} | |
| 1931 | +\newcommand{\RthreePDevPctAbs}{2.8} | |
| 1932 | +\newcommand{\RthreeRR}{0.44} | |
| 1933 | +\newcommand{\RthreeRebate}{0.000} | |
| 1934 | +\newcommand{\RthreeRho}{0.214} | |
| 1935 | +\newcommand{\RthreeRhoDevPct}{10.1} | |
| 1936 | +\newcommand{\RthreeThr}{0.60} | |
| 1937 | +\newcommand{\RthreeThrHighZ}{0.09} | |
| 1938 | +\newcommand{\RthreeThrLowZ}{1.35} | |
| 1939 | +\newcommand{\RthreeqOO}{0.596} | |
| 1940 | +\newcommand{\RthreeqRO}{0.353} | |
| 1941 | +\newcommand{\ShareEstatePositivePct}{55} | |
| 1942 | +\newcommand{\ShareFirstBuyersGiftPct}{72} | |
| 1943 | +\newcommand{\ShareGiftPositivePct}{70} | |
| 143 | 1944 | \newcommand{\ShareMortgage}{0.41} |
| 1945 | +\newcommand{\ShareOwnersMortgage}{0.65} | |
| 1946 | +\newcommand{\ShareParentOwnerPct}{86} | |
| 1947 | +\newcommand{\SigHighEstate}{0.42} | |
| 1948 | +\newcommand{\SigHighGapFour}{1.35} | |
| 1949 | +\newcommand{\SigHighGift}{0.49} | |
| 1950 | +\newcommand{\SigHighGini}{0.56} | |
| 1951 | +\newcommand{\SigHighLamSeven}{0.097} | |
| 1952 | +\newcommand{\SigHighLamThree}{0.18} | |
| 1953 | +\newcommand{\SigHighOwn}{0.602} | |
| 1954 | +\newcommand{\SigHighOwnUthirtyfive}{0.416} | |
| 1955 | +\newcommand{\SigHighP}{1.033} | |
| 1956 | +\newcommand{\SigHighPDevPct}{3.3} | |
| 1957 | +\newcommand{\SigHighRR}{0.34} | |
| 1958 | +\newcommand{\SigHighRebate}{0.000} | |
| 1959 | +\newcommand{\SigHighRho}{0.189} | |
| 1960 | +\newcommand{\SigHighRhoDevPct}{\ensuremath{-}2.6} | |
| 1961 | +\newcommand{\SigHighRhoDevPctAbs}{2.6} | |
| 1962 | +\newcommand{\SigHighThr}{0.65} | |
| 1963 | +\newcommand{\SigHighThrHighZ}{\ensuremath{-}0.27} | |
| 1964 | +\newcommand{\SigHighThrHighZAbs}{0.27} | |
| 1965 | +\newcommand{\SigHighThrLowZ}{1.47} | |
| 1966 | +\newcommand{\SigHighqOO}{0.570} | |
| 1967 | +\newcommand{\SigHighqRO}{0.391} | |
| 1968 | +\newcommand{\SigLowEstate}{0.22} | |
| 1969 | +\newcommand{\SigLowGapFour}{0.59} | |
| 1970 | +\newcommand{\SigLowGift}{0.30} | |
| 1971 | +\newcommand{\SigLowGini}{0.51} | |
| 1972 | +\newcommand{\SigLowLamSeven}{0.043} | |
| 1973 | +\newcommand{\SigLowLamThree}{0.19} | |
| 1974 | +\newcommand{\SigLowOwn}{0.672} | |
| 1975 | +\newcommand{\SigLowOwnUthirtyfive}{0.318} | |
| 1976 | +\newcommand{\SigLowP}{0.963} | |
| 1977 | +\newcommand{\SigLowPDevPct}{\ensuremath{-}3.7} | |
| 1978 | +\newcommand{\SigLowPDevPctAbs}{3.7} | |
| 1979 | +\newcommand{\SigLowRR}{0.32} | |
| 1980 | +\newcommand{\SigLowRebate}{0.000} | |
| 1981 | +\newcommand{\SigLowRho}{0.197} | |
| 1982 | +\newcommand{\SigLowRhoDevPct}{1.3} | |
| 1983 | +\newcommand{\SigLowThr}{0.41} | |
| 1984 | +\newcommand{\SigLowThrHighZ}{0.27} | |
| 1985 | +\newcommand{\SigLowThrLowZ}{0.84} | |
| 1986 | +\newcommand{\SigLowqOO}{0.492} | |
| 1987 | +\newcommand{\SigLowqRO}{0.307} | |
| 144 | 1988 | \newcommand{\SignChangesbaseline}{1} |
| 145 | 1989 | \newcommand{\SignChangeseps}{1} |
| 146 | 1990 | \newcommand{\SignChangesepstheta}{1} |
| 147 | 1991 | \newcommand{\SimulateMinutes}{60} |
| 1992 | +\newcommand{\SimulateMinutesOne}{59.8} | |
| 1993 | +\newcommand{\StaticAbsorbPct}{110} | |
| 1994 | +\newcommand{\StaticBetterAllPct}{41} | |
| 1995 | +\newcommand{\StaticBetterOMPct}{71} | |
| 1996 | +\newcommand{\StaticBetterONMPct}{3} | |
| 1997 | +\newcommand{\StaticBetterOPct}{46} | |
| 1998 | +\newcommand{\StaticBetterRPct}{33} | |
| 1999 | +\newcommand{\StaticCEVAll}{\ensuremath{-}0.64} | |
| 2000 | +\newcommand{\StaticCEVAllAbs}{0.64} | |
| 2001 | +\newcommand{\StaticCEVAllPE}{\ensuremath{-}0.68} | |
| 2002 | +\newcommand{\StaticCEVAllPEAbs}{0.68} | |
| 2003 | +\newcommand{\StaticCEVO}{\ensuremath{-}0.17} | |
| 2004 | +\newcommand{\StaticCEVOAbs}{0.17} | |
| 2005 | +\newcommand{\StaticCEVOM}{0.65} | |
| 2006 | +\newcommand{\StaticCEVONM}{\ensuremath{-}2.10} | |
| 2007 | +\newcommand{\StaticCEVONMAbs}{2.10} | |
| 2008 | +\newcommand{\StaticCEVR}{\ensuremath{-}1.01} | |
| 2009 | +\newcommand{\StaticCEVRAbs}{1.01} | |
| 2010 | +\newcommand{\StaticConsAll}{\ensuremath{-}0.8} | |
| 2011 | +\newcommand{\StaticConsAllAbs}{0.8} | |
| 2012 | +\newcommand{\StaticConsO}{\ensuremath{-}0.5} | |
| 2013 | +\newcommand{\StaticConsOAbs}{0.5} | |
| 2014 | +\newcommand{\StaticConsR}{\ensuremath{-}1.4} | |
| 2015 | +\newcommand{\StaticConsRAbs}{1.4} | |
| 2016 | +\newcommand{\StaticDOwnGE}{0.02} | |
| 2017 | +\newcommand{\StaticDOwnPE}{\ensuremath{-}0.20} | |
| 2018 | +\newcommand{\StaticDOwnPEAbs}{0.20} | |
| 2019 | +\newcommand{\StaticDebtZeroPct}{\ensuremath{-}2.4} | |
| 2020 | +\newcommand{\StaticDebtZeroPctAbs}{2.4} | |
| 2021 | +\newcommand{\StaticDepLoss}{0.48} | |
| 2022 | +\newcommand{\StaticErosion}{0.21} | |
| 2023 | +\newcommand{\StaticLamSevenSS}{0.060} | |
| 2024 | +\newcommand{\StaticLamSevenZero}{0.069} | |
| 2025 | +\newcommand{\StaticLender}{0.27} | |
| 2026 | +\newcommand{\StaticMaxRes}{1.1\times10^{-3}} | |
| 2027 | +\newcommand{\StaticNWAll}{\ensuremath{-}0.02} | |
| 2028 | +\newcommand{\StaticNWAllAbs}{0.02} | |
| 2029 | +\newcommand{\StaticNWO}{\ensuremath{-}0.02} | |
| 2030 | +\newcommand{\StaticNWOAbs}{0.02} | |
| 2031 | +\newcommand{\StaticNWR}{\ensuremath{-}0.01} | |
| 2032 | +\newcommand{\StaticNWRAbs}{0.01} | |
| 2033 | +\newcommand{\StaticNetO}{\ensuremath{-}0.02} | |
| 2034 | +\newcommand{\StaticNetOAbs}{0.02} | |
| 2035 | +\newcommand{\StaticNetR}{\ensuremath{-}0.02} | |
| 2036 | +\newcommand{\StaticNetRAbs}{0.02} | |
| 2037 | +\newcommand{\StaticNominalRateZero}{0.219} | |
| 2038 | +\newcommand{\StaticOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 2039 | +\newcommand{\StaticOwnerOldConsMaxAbs}{2.3} | |
| 2040 | +\newcommand{\StaticOwnerOldConsMin}{\ensuremath{-}1.7} | |
| 2041 | +\newcommand{\StaticOwnerOldConsMinAbs}{1.7} | |
| 2042 | +\newcommand{\StaticOwnerOldLossMax}{0.08} | |
| 2043 | +\newcommand{\StaticOwnerOldLossMin}{0.04} | |
| 2044 | +\newcommand{\StaticOwnerYoungConsMax}{1.1} | |
| 2045 | +\newcommand{\StaticOwnerYoungConsMin}{0.7} | |
| 2046 | +\newcommand{\StaticOwnerYoungGainMax}{0.04} | |
| 2047 | +\newcommand{\StaticPriceBackYears}{20} | |
| 2048 | +\newcommand{\StaticPriceMaxPos}{0.1} | |
| 2049 | +\newcommand{\StaticPriceSignChanges}{1} | |
| 2050 | +\newcommand{\StaticPzero}{\ensuremath{-}0.36} | |
| 2051 | +\newcommand{\StaticPzeroAbs}{0.36} | |
| 2052 | +\newcommand{\StaticRentGainR}{0.00} | |
| 2053 | +\newcommand{\StaticResT}{8\times10^{-5}} | |
| 2054 | +\newcommand{\StaticRhoZero}{\ensuremath{-}1.11} | |
| 2055 | +\newcommand{\StaticRhoZeroAbs}{1.11} | |
| 2056 | +\newcommand{\StaticSecs}{387} | |
| 2057 | +\newcommand{\StaticSteps}{10} | |
| 2058 | +\newcommand{\StaticThrChangePct}{8} | |
| 2059 | +\newcommand{\StaticThrSS}{0.49} | |
| 2060 | +\newcommand{\StaticThrZero}{0.53} | |
| 2061 | +\newcommand{\StaticWageAll}{0.00} | |
| 2062 | +\newcommand{\StaticWageO}{0.00} | |
| 2063 | +\newcommand{\StaticqROsevenSS}{0.806} | |
| 2064 | +\newcommand{\StaticqROsevenZero}{0.797} | |
| 2065 | +\newcommand{\StaticqROthreeOne}{0.335} | |
| 2066 | +\newcommand{\StaticqROthreeSS}{0.351} | |
| 2067 | +\newcommand{\StaticqROthreeZero}{0.357} | |
| 2068 | +\newcommand{\SupplyScaleOwner}{9.41} | |
| 2069 | +\newcommand{\SupplyScaleRental}{2.90} | |
| 2070 | +\newcommand{\TaxFiftyEstate}{0.09} | |
| 2071 | +\newcommand{\TaxFiftyGapFour}{0.90} | |
| 2072 | +\newcommand{\TaxFiftyGift}{0.12} | |
| 2073 | +\newcommand{\TaxFiftyGini}{0.52} | |
| 2074 | +\newcommand{\TaxFiftyLamSeven}{0.023} | |
| 2075 | +\newcommand{\TaxFiftyLamThree}{0.12} | |
| 2076 | +\newcommand{\TaxFiftyOwn}{0.639} | |
| 2077 | +\newcommand{\TaxFiftyOwnUthirtyfive}{0.482} | |
| 2078 | +\newcommand{\TaxFiftyP}{0.999} | |
| 2079 | +\newcommand{\TaxFiftyPDevPct}{\ensuremath{-}0.1} | |
| 2080 | +\newcommand{\TaxFiftyPDevPctAbs}{0.1} | |
| 2081 | +\newcommand{\TaxFiftyRR}{0.16} | |
| 2082 | +\newcommand{\TaxFiftyRebate}{0.213} | |
| 2083 | +\newcommand{\TaxFiftyRho}{0.195} | |
| 2084 | +\newcommand{\TaxFiftyRhoDevPct}{0.4} | |
| 2085 | +\newcommand{\TaxFiftyThr}{0.48} | |
| 2086 | +\newcommand{\TaxFiftyThrHighZ}{0.02} | |
| 2087 | +\newcommand{\TaxFiftyThrLowZ}{1.15} | |
| 2088 | +\newcommand{\TaxFiftyqOO}{0.553} | |
| 2089 | +\newcommand{\TaxFiftyqRO}{0.429} | |
| 2090 | +\newcommand{\TaxTwentyEstate}{0.22} | |
| 2091 | +\newcommand{\TaxTwentyGapFour}{0.96} | |
| 2092 | +\newcommand{\TaxTwentyGift}{0.28} | |
| 2093 | +\newcommand{\TaxTwentyGini}{0.53} | |
| 2094 | +\newcommand{\TaxTwentyLamSeven}{0.054} | |
| 2095 | +\newcommand{\TaxTwentyLamThree}{0.18} | |
| 2096 | +\newcommand{\TaxTwentyOwn}{0.642} | |
| 2097 | +\newcommand{\TaxTwentyOwnUthirtyfive}{0.461} | |
| 2098 | +\newcommand{\TaxTwentyP}{1.005} | |
| 2099 | +\newcommand{\TaxTwentyPDevPct}{0.5} | |
| 2100 | +\newcommand{\TaxTwentyRR}{0.28} | |
| 2101 | +\newcommand{\TaxTwentyRebate}{0.125} | |
| 2102 | +\newcommand{\TaxTwentyRho}{0.192} | |
| 2103 | +\newcommand{\TaxTwentyRhoDevPct}{\ensuremath{-}1.1} | |
| 2104 | +\newcommand{\TaxTwentyRhoDevPctAbs}{1.1} | |
| 2105 | +\newcommand{\TaxTwentyThr}{0.51} | |
| 2106 | +\newcommand{\TaxTwentyThrHighZ}{0.04} | |
| 2107 | +\newcommand{\TaxTwentyThrLowZ}{1.21} | |
| 2108 | +\newcommand{\TaxTwentyqOO}{0.569} | |
| 2109 | +\newcommand{\TaxTwentyqRO}{0.392} | |
| 2110 | +\newcommand{\TermLevelAbsorbPct}{89} | |
| 2111 | +\newcommand{\TermLevelBetterAllPct}{41} | |
| 2112 | +\newcommand{\TermLevelBetterOMPct}{71} | |
| 2113 | +\newcommand{\TermLevelBetterONMPct}{1} | |
| 2114 | +\newcommand{\TermLevelBetterOPct}{45} | |
| 2115 | +\newcommand{\TermLevelBetterRPct}{35} | |
| 2116 | +\newcommand{\TermLevelCEVAll}{\ensuremath{-}0.64} | |
| 2117 | +\newcommand{\TermLevelCEVAllAbs}{0.64} | |
| 2118 | +\newcommand{\TermLevelCEVAllPE}{\ensuremath{-}0.68} | |
| 2119 | +\newcommand{\TermLevelCEVAllPEAbs}{0.68} | |
| 2120 | +\newcommand{\TermLevelCEVO}{\ensuremath{-}0.20} | |
| 2121 | +\newcommand{\TermLevelCEVOAbs}{0.20} | |
| 2122 | +\newcommand{\TermLevelCEVOM}{0.63} | |
| 2123 | +\newcommand{\TermLevelCEVONM}{\ensuremath{-}2.12} | |
| 2124 | +\newcommand{\TermLevelCEVONMAbs}{2.12} | |
| 2125 | +\newcommand{\TermLevelCEVR}{\ensuremath{-}1.00} | |
| 2126 | +\newcommand{\TermLevelCEVRAbs}{1.00} | |
| 2127 | +\newcommand{\TermLevelConsAll}{\ensuremath{-}0.8} | |
| 2128 | +\newcommand{\TermLevelConsAllAbs}{0.8} | |
| 2129 | +\newcommand{\TermLevelConsO}{\ensuremath{-}0.6} | |
| 2130 | +\newcommand{\TermLevelConsOAbs}{0.6} | |
| 2131 | +\newcommand{\TermLevelConsR}{\ensuremath{-}1.4} | |
| 2132 | +\newcommand{\TermLevelConsRAbs}{1.4} | |
| 2133 | +\newcommand{\TermLevelDOwnGE}{\ensuremath{-}0.02} | |
| 2134 | +\newcommand{\TermLevelDOwnGEAbs}{0.02} | |
| 2135 | +\newcommand{\TermLevelDOwnPE}{\ensuremath{-}0.20} | |
| 2136 | +\newcommand{\TermLevelDOwnPEAbs}{0.20} | |
| 2137 | +\newcommand{\TermLevelDebtZeroPct}{\ensuremath{-}2.7} | |
| 2138 | +\newcommand{\TermLevelDebtZeroPctAbs}{2.7} | |
| 2139 | +\newcommand{\TermLevelDepLoss}{0.48} | |
| 2140 | +\newcommand{\TermLevelErosion}{0.21} | |
| 2141 | +\newcommand{\TermLevelLamSevenSS}{0.060} | |
| 2142 | +\newcommand{\TermLevelLamSevenZero}{0.068} | |
| 2143 | +\newcommand{\TermLevelLender}{0.27} | |
| 2144 | +\newcommand{\TermLevelMaxRes}{6.1\times10^{-4}} | |
| 2145 | +\newcommand{\TermLevelNWAll}{\ensuremath{-}0.02} | |
| 2146 | +\newcommand{\TermLevelNWAllAbs}{0.02} | |
| 2147 | +\newcommand{\TermLevelNWO}{\ensuremath{-}0.02} | |
| 2148 | +\newcommand{\TermLevelNWOAbs}{0.02} | |
| 2149 | +\newcommand{\TermLevelNWR}{\ensuremath{-}0.01} | |
| 2150 | +\newcommand{\TermLevelNWRAbs}{0.01} | |
| 2151 | +\newcommand{\TermLevelNetO}{\ensuremath{-}0.02} | |
| 2152 | +\newcommand{\TermLevelNetOAbs}{0.02} | |
| 2153 | +\newcommand{\TermLevelNetR}{\ensuremath{-}0.02} | |
| 2154 | +\newcommand{\TermLevelNetRAbs}{0.02} | |
| 2155 | +\newcommand{\TermLevelNominalRateZero}{0.219} | |
| 2156 | +\newcommand{\TermLevelOwnerOldConsMax}{\ensuremath{-}2.3} | |
| 2157 | +\newcommand{\TermLevelOwnerOldConsMaxAbs}{2.3} | |
| 2158 | +\newcommand{\TermLevelOwnerOldConsMin}{\ensuremath{-}1.8} | |
| 2159 | +\newcommand{\TermLevelOwnerOldConsMinAbs}{1.8} | |
| 2160 | +\newcommand{\TermLevelOwnerOldLossMax}{0.08} | |
| 2161 | +\newcommand{\TermLevelOwnerOldLossMin}{0.04} | |
| 2162 | +\newcommand{\TermLevelOwnerYoungConsMax}{1.1} | |
| 2163 | +\newcommand{\TermLevelOwnerYoungConsMin}{0.6} | |
| 2164 | +\newcommand{\TermLevelOwnerYoungGainMax}{0.04} | |
| 2165 | +\newcommand{\TermLevelPriceBackYears}{20} | |
| 2166 | +\newcommand{\TermLevelPriceMaxPos}{0.1} | |
| 2167 | +\newcommand{\TermLevelPriceSignChanges}{1} | |
| 2168 | +\newcommand{\TermLevelPzero}{\ensuremath{-}0.47} | |
| 2169 | +\newcommand{\TermLevelPzeroAbs}{0.47} | |
| 2170 | +\newcommand{\TermLevelRentGainR}{0.00} | |
| 2171 | +\newcommand{\TermLevelResT}{3\times10^{-4}} | |
| 2172 | +\newcommand{\TermLevelRhoZero}{\ensuremath{-}1.03} | |
| 2173 | +\newcommand{\TermLevelRhoZeroAbs}{1.03} | |
| 2174 | +\newcommand{\TermLevelSecs}{161} | |
| 2175 | +\newcommand{\TermLevelSteps}{3} | |
| 2176 | +\newcommand{\TermLevelThrChangePct}{8} | |
| 2177 | +\newcommand{\TermLevelThrSS}{0.49} | |
| 2178 | +\newcommand{\TermLevelThrZero}{0.53} | |
| 2179 | +\newcommand{\TermLevelWageAll}{0.00} | |
| 2180 | +\newcommand{\TermLevelWageO}{0.00} | |
| 2181 | +\newcommand{\TermLevelqROsevenSS}{0.806} | |
| 2182 | +\newcommand{\TermLevelqROsevenZero}{0.798} | |
| 2183 | +\newcommand{\TermLevelqROthreeOne}{0.335} | |
| 2184 | +\newcommand{\TermLevelqROthreeSS}{0.351} | |
| 2185 | +\newcommand{\TermLevelqROthreeZero}{0.356} | |
| 2186 | +\newcommand{\TermPersAbsorbPct}{83} | |
| 2187 | +\newcommand{\TermPersBetterAllPct}{27} | |
| 2188 | +\newcommand{\TermPersBetterOMPct}{66} | |
| 2189 | +\newcommand{\TermPersBetterONMPct}{2} | |
| 2190 | +\newcommand{\TermPersBetterOPct}{43} | |
| 2191 | +\newcommand{\TermPersBetterRPct}{4} | |
| 2192 | +\newcommand{\TermPersCEVAll}{\ensuremath{-}0.74} | |
| 2193 | +\newcommand{\TermPersCEVAllAbs}{0.74} | |
| 2194 | +\newcommand{\TermPersCEVAllPE}{\ensuremath{-}0.73} | |
| 2195 | +\newcommand{\TermPersCEVAllPEAbs}{0.73} | |
| 2196 | +\newcommand{\TermPersCEVO}{\ensuremath{-}0.32} | |
| 2197 | +\newcommand{\TermPersCEVOAbs}{0.32} | |
| 2198 | +\newcommand{\TermPersCEVOM}{0.46} | |
| 2199 | +\newcommand{\TermPersCEVONM}{\ensuremath{-}2.14} | |
| 2200 | +\newcommand{\TermPersCEVONMAbs}{2.14} | |
| 2201 | +\newcommand{\TermPersCEVR}{\ensuremath{-}1.09} | |
| 2202 | +\newcommand{\TermPersCEVRAbs}{1.09} | |
| 2203 | +\newcommand{\TermPersConsAll}{\ensuremath{-}1.4} | |
| 2204 | +\newcommand{\TermPersConsAllAbs}{1.4} | |
| 2205 | +\newcommand{\TermPersConsO}{\ensuremath{-}1.2} | |
| 2206 | +\newcommand{\TermPersConsOAbs}{1.2} | |
| 2207 | +\newcommand{\TermPersConsR}{\ensuremath{-}1.7} | |
| 2208 | +\newcommand{\TermPersConsRAbs}{1.7} | |
| 2209 | +\newcommand{\TermPersDOwnGE}{\ensuremath{-}0.34} | |
| 2210 | +\newcommand{\TermPersDOwnGEAbs}{0.34} | |
| 2211 | +\newcommand{\TermPersDOwnPE}{\ensuremath{-}2.03} | |
| 2212 | +\newcommand{\TermPersDOwnPEAbs}{2.03} | |
| 2213 | +\newcommand{\TermPersDebtZeroPct}{\ensuremath{-}13.5} | |
| 2214 | +\newcommand{\TermPersDebtZeroPctAbs}{13.5} | |
| 2215 | +\newcommand{\TermPersDepLoss}{0.48} | |
| 2216 | +\newcommand{\TermPersErosion}{0.21} | |
| 2217 | +\newcommand{\TermPersLamSevenSS}{0.060} | |
| 2218 | +\newcommand{\TermPersLamSevenZero}{0.067} | |
| 2219 | +\newcommand{\TermPersLender}{0.27} | |
| 2220 | +\newcommand{\TermPersMaxRes}{8.5\times10^{-4}} | |
| 2221 | +\newcommand{\TermPersNWAll}{\ensuremath{-}0.02} | |
| 2222 | +\newcommand{\TermPersNWAllAbs}{0.02} | |
| 2223 | +\newcommand{\TermPersNWO}{\ensuremath{-}0.03} | |
| 2224 | +\newcommand{\TermPersNWOAbs}{0.03} | |
| 2225 | +\newcommand{\TermPersNWR}{\ensuremath{-}0.01} | |
| 2226 | +\newcommand{\TermPersNWRAbs}{0.01} | |
| 2227 | +\newcommand{\TermPersNetO}{\ensuremath{-}0.02} | |
| 2228 | +\newcommand{\TermPersNetOAbs}{0.02} | |
| 2229 | +\newcommand{\TermPersNetR}{\ensuremath{-}0.02} | |
| 2230 | +\newcommand{\TermPersNetRAbs}{0.02} | |
| 2231 | +\newcommand{\TermPersNominalRateZero}{0.341} | |
| 2232 | +\newcommand{\TermPersOwnerOldConsMax}{\ensuremath{-}2.5} | |
| 2233 | +\newcommand{\TermPersOwnerOldConsMaxAbs}{2.5} | |
| 2234 | +\newcommand{\TermPersOwnerOldConsMin}{\ensuremath{-}1.9} | |
| 2235 | +\newcommand{\TermPersOwnerOldConsMinAbs}{1.9} | |
| 2236 | +\newcommand{\TermPersOwnerOldLossMax}{0.08} | |
| 2237 | +\newcommand{\TermPersOwnerOldLossMin}{0.04} | |
| 2238 | +\newcommand{\TermPersOwnerYoungConsMax}{0.4} | |
| 2239 | +\newcommand{\TermPersOwnerYoungConsMin}{0.2} | |
| 2240 | +\newcommand{\TermPersOwnerYoungGainMax}{0.04} | |
| 2241 | +\newcommand{\TermPersPriceBackYears}{5} | |
| 2242 | +\newcommand{\TermPersPriceMaxPos}{0.1} | |
| 2243 | +\newcommand{\TermPersPriceSignChanges}{3} | |
| 2244 | +\newcommand{\TermPersPzero}{\ensuremath{-}1.49} | |
| 2245 | +\newcommand{\TermPersPzeroAbs}{1.49} | |
| 2246 | +\newcommand{\TermPersRentGainR}{0.00} | |
| 2247 | +\newcommand{\TermPersResT}{2\times10^{-4}} | |
| 2248 | +\newcommand{\TermPersRhoZero}{0.04} | |
| 2249 | +\newcommand{\TermPersSecs}{213} | |
| 2250 | +\newcommand{\TermPersSteps}{8} | |
| 2251 | +\newcommand{\TermPersThrChangePct}{18} | |
| 2252 | +\newcommand{\TermPersThrSS}{0.49} | |
| 2253 | +\newcommand{\TermPersThrZero}{0.58} | |
| 2254 | +\newcommand{\TermPersWageAll}{0.00} | |
| 2255 | +\newcommand{\TermPersWageO}{0.00} | |
| 2256 | +\newcommand{\TermPersqROsevenSS}{0.806} | |
| 2257 | +\newcommand{\TermPersqROsevenZero}{0.807} | |
| 2258 | +\newcommand{\TermPersqROthreeOne}{0.333} | |
| 2259 | +\newcommand{\TermPersqROthreeSS}{0.351} | |
| 2260 | +\newcommand{\TermPersqROthreeZero}{0.356} | |
| 2261 | +\newcommand{\TermTaylorAbsorbPct}{91} | |
| 2262 | +\newcommand{\TermTaylorBetterAllPct}{44} | |
| 2263 | +\newcommand{\TermTaylorBetterOMPct}{56} | |
| 2264 | +\newcommand{\TermTaylorBetterONMPct}{22} | |
| 2265 | +\newcommand{\TermTaylorBetterOPct}{43} | |
| 2266 | +\newcommand{\TermTaylorBetterRPct}{46} | |
| 2267 | +\newcommand{\TermTaylorCEVAll}{\ensuremath{-}0.25} | |
| 2268 | +\newcommand{\TermTaylorCEVAllAbs}{0.25} | |
| 2269 | +\newcommand{\TermTaylorCEVAllPE}{\ensuremath{-}1.67} | |
| 2270 | +\newcommand{\TermTaylorCEVAllPEAbs}{1.67} | |
| 2271 | +\newcommand{\TermTaylorCEVO}{\ensuremath{-}0.17} | |
| 2272 | +\newcommand{\TermTaylorCEVOAbs}{0.17} | |
| 2273 | +\newcommand{\TermTaylorCEVOM}{0.20} | |
| 2274 | +\newcommand{\TermTaylorCEVONM}{\ensuremath{-}1.05} | |
| 2275 | +\newcommand{\TermTaylorCEVONMAbs}{1.05} | |
| 2276 | +\newcommand{\TermTaylorCEVR}{\ensuremath{-}0.32} | |
| 2277 | +\newcommand{\TermTaylorCEVRAbs}{0.32} | |
| 2278 | +\newcommand{\TermTaylorConsAll}{\ensuremath{-}3.4} | |
| 2279 | +\newcommand{\TermTaylorConsAllAbs}{3.4} | |
| 2280 | +\newcommand{\TermTaylorConsO}{\ensuremath{-}3.6} | |
| 2281 | +\newcommand{\TermTaylorConsOAbs}{3.6} | |
| 2282 | +\newcommand{\TermTaylorConsR}{\ensuremath{-}3.2} | |
| 2283 | +\newcommand{\TermTaylorConsRAbs}{3.2} | |
| 2284 | +\newcommand{\TermTaylorDOwnGE}{\ensuremath{-}0.26} | |
| 2285 | +\newcommand{\TermTaylorDOwnGEAbs}{0.26} | |
| 2286 | +\newcommand{\TermTaylorDOwnPE}{\ensuremath{-}2.94} | |
| 2287 | +\newcommand{\TermTaylorDOwnPEAbs}{2.94} | |
| 2288 | +\newcommand{\TermTaylorDebtZeroPct}{\ensuremath{-}19.4} | |
| 2289 | +\newcommand{\TermTaylorDebtZeroPctAbs}{19.4} | |
| 2290 | +\newcommand{\TermTaylorDepLoss}{0.48} | |
| 2291 | +\newcommand{\TermTaylorErosion}{0.21} | |
| 2292 | +\newcommand{\TermTaylorLamSevenSS}{0.060} | |
| 2293 | +\newcommand{\TermTaylorLamSevenZero}{0.068} | |
| 2294 | +\newcommand{\TermTaylorLender}{0.27} | |
| 2295 | +\newcommand{\TermTaylorMaxRes}{7.7\times10^{-4}} | |
| 2296 | +\newcommand{\TermTaylorNWAll}{\ensuremath{-}0.04} | |
| 2297 | +\newcommand{\TermTaylorNWAllAbs}{0.04} | |
| 2298 | +\newcommand{\TermTaylorNWO}{\ensuremath{-}0.07} | |
| 2299 | +\newcommand{\TermTaylorNWOAbs}{0.07} | |
| 2300 | +\newcommand{\TermTaylorNWR}{\ensuremath{-}0.01} | |
| 2301 | +\newcommand{\TermTaylorNWRAbs}{0.01} | |
| 2302 | +\newcommand{\TermTaylorNetO}{\ensuremath{-}0.02} | |
| 2303 | +\newcommand{\TermTaylorNetOAbs}{0.02} | |
| 2304 | +\newcommand{\TermTaylorNetR}{\ensuremath{-}0.02} | |
| 2305 | +\newcommand{\TermTaylorNetRAbs}{0.02} | |
| 2306 | +\newcommand{\TermTaylorNominalRateZero}{0.408} | |
| 2307 | +\newcommand{\TermTaylorOwnerOldConsMax}{\ensuremath{-}4.6} | |
| 2308 | +\newcommand{\TermTaylorOwnerOldConsMaxAbs}{4.6} | |
| 2309 | +\newcommand{\TermTaylorOwnerOldConsMin}{\ensuremath{-}3.8} | |
| 2310 | +\newcommand{\TermTaylorOwnerOldConsMinAbs}{3.8} | |
| 2311 | +\newcommand{\TermTaylorOwnerOldLossMax}{0.08} | |
| 2312 | +\newcommand{\TermTaylorOwnerOldLossMin}{0.04} | |
| 2313 | +\newcommand{\TermTaylorOwnerYoungConsMax}{\ensuremath{-}3.1} | |
| 2314 | +\newcommand{\TermTaylorOwnerYoungConsMaxAbs}{3.1} | |
| 2315 | +\newcommand{\TermTaylorOwnerYoungConsMin}{\ensuremath{-}3.6} | |
| 2316 | +\newcommand{\TermTaylorOwnerYoungConsMinAbs}{3.6} | |
| 2317 | +\newcommand{\TermTaylorOwnerYoungGainMax}{0.04} | |
| 2318 | +\newcommand{\TermTaylorPriceBackYears}{40} | |
| 2319 | +\newcommand{\TermTaylorPriceMaxPos}{1.4} | |
| 2320 | +\newcommand{\TermTaylorPriceSignChanges}{1} | |
| 2321 | +\newcommand{\TermTaylorPzero}{\ensuremath{-}6.93} | |
| 2322 | +\newcommand{\TermTaylorPzeroAbs}{6.93} | |
| 2323 | +\newcommand{\TermTaylorRentGainR}{0.00} | |
| 2324 | +\newcommand{\TermTaylorResT}{8\times10^{-4}} | |
| 2325 | +\newcommand{\TermTaylorRhoZero}{\ensuremath{-}1.91} | |
| 2326 | +\newcommand{\TermTaylorRhoZeroAbs}{1.91} | |
| 2327 | +\newcommand{\TermTaylorSecs}{213} | |
| 2328 | +\newcommand{\TermTaylorSteps}{7} | |
| 2329 | +\newcommand{\TermTaylorThrChangePct}{13} | |
| 2330 | +\newcommand{\TermTaylorThrSS}{0.49} | |
| 2331 | +\newcommand{\TermTaylorThrZero}{0.56} | |
| 2332 | +\newcommand{\TermTaylorWageAll}{0.00} | |
| 2333 | +\newcommand{\TermTaylorWageO}{0.00} | |
| 2334 | +\newcommand{\TermTaylorqROsevenSS}{0.806} | |
| 2335 | +\newcommand{\TermTaylorqROsevenZero}{0.799} | |
| 2336 | +\newcommand{\TermTaylorqROthreeOne}{0.349} | |
| 2337 | +\newcommand{\TermTaylorqROthreeSS}{0.351} | |
| 2338 | +\newcommand{\TermTaylorqROthreeZero}{0.361} | |
| 2339 | +\newcommand{\ThetaFiveEstate}{0.31} | |
| 2340 | +\newcommand{\ThetaFiveGapFour}{0.84} | |
| 2341 | +\newcommand{\ThetaFiveGift}{0.39} | |
| 2342 | +\newcommand{\ThetaFiveGini}{0.55} | |
| 2343 | +\newcommand{\ThetaFiveLamSeven}{0.069} | |
| 2344 | +\newcommand{\ThetaFiveLamThree}{0.20} | |
| 2345 | +\newcommand{\ThetaFiveOwn}{0.640} | |
| 2346 | +\newcommand{\ThetaFiveOwnUthirtyfive}{0.409} | |
| 2347 | +\newcommand{\ThetaFiveP}{1.005} | |
| 2348 | +\newcommand{\ThetaFivePDevPct}{0.5} | |
| 2349 | +\newcommand{\ThetaFivePEEstate}{0.32} | |
| 2350 | +\newcommand{\ThetaFivePEGapFour}{0.83} | |
| 2351 | +\newcommand{\ThetaFivePEGift}{0.39} | |
| 2352 | +\newcommand{\ThetaFivePEGini}{0.54} | |
| 2353 | +\newcommand{\ThetaFivePELamSeven}{0.061} | |
| 2354 | +\newcommand{\ThetaFivePELamThree}{0.21} | |
| 2355 | +\newcommand{\ThetaFivePEOwn}{0.657} | |
| 2356 | +\newcommand{\ThetaFivePEOwnUthirtyfive}{0.429} | |
| 2357 | +\newcommand{\ThetaFivePEP}{1.000} | |
| 2358 | +\newcommand{\ThetaFivePEPDevPct}{0.0} | |
| 2359 | +\newcommand{\ThetaFivePERR}{0.33} | |
| 2360 | +\newcommand{\ThetaFivePERebate}{0.000} | |
| 2361 | +\newcommand{\ThetaFivePERho}{0.194} | |
| 2362 | +\newcommand{\ThetaFivePERhoDevPct}{0.0} | |
| 2363 | +\newcommand{\ThetaFivePEThr}{0.52} | |
| 2364 | +\newcommand{\ThetaFivePEThrHighZ}{\ensuremath{-}0.13} | |
| 2365 | +\newcommand{\ThetaFivePEThrHighZAbs}{0.13} | |
| 2366 | +\newcommand{\ThetaFivePEThrLowZ}{1.18} | |
| 2367 | +\newcommand{\ThetaFivePEqOO}{0.585} | |
| 2368 | +\newcommand{\ThetaFivePEqRO}{0.379} | |
| 2369 | +\newcommand{\ThetaFiveRR}{0.33} | |
| 2370 | +\newcommand{\ThetaFiveRebate}{0.000} | |
| 2371 | +\newcommand{\ThetaFiveRho}{0.189} | |
| 2372 | +\newcommand{\ThetaFiveRhoDevPct}{\ensuremath{-}2.7} | |
| 2373 | +\newcommand{\ThetaFiveRhoDevPctAbs}{2.7} | |
| 2374 | +\newcommand{\ThetaFiveThr}{0.56} | |
| 2375 | +\newcommand{\ThetaFiveThrHighZ}{\ensuremath{-}0.10} | |
| 2376 | +\newcommand{\ThetaFiveThrHighZAbs}{0.10} | |
| 2377 | +\newcommand{\ThetaFiveThrLowZ}{1.25} | |
| 2378 | +\newcommand{\ThetaFiveqOO}{0.572} | |
| 2379 | +\newcommand{\ThetaFiveqRO}{0.375} | |
| 2380 | +\newcommand{\ThetaFortyEstate}{0.33} | |
| 2381 | +\newcommand{\ThetaFortyGapFour}{0.98} | |
| 2382 | +\newcommand{\ThetaFortyGift}{0.39} | |
| 2383 | +\newcommand{\ThetaFortyGini}{0.54} | |
| 2384 | +\newcommand{\ThetaFortyLamSeven}{0.084} | |
| 2385 | +\newcommand{\ThetaFortyLamThree}{0.21} | |
| 2386 | +\newcommand{\ThetaFortyOwn}{0.619} | |
| 2387 | +\newcommand{\ThetaFortyOwnUthirtyfive}{0.333} | |
| 2388 | +\newcommand{\ThetaFortyP}{0.987} | |
| 2389 | +\newcommand{\ThetaFortyPDevPct}{\ensuremath{-}1.3} | |
| 2390 | +\newcommand{\ThetaFortyPDevPctAbs}{1.3} | |
| 2391 | +\newcommand{\ThetaFortyPEEstate}{0.32} | |
| 2392 | +\newcommand{\ThetaFortyPEGapFour}{0.99} | |
| 2393 | +\newcommand{\ThetaFortyPEGift}{0.39} | |
| 2394 | +\newcommand{\ThetaFortyPEGini}{0.54} | |
| 2395 | +\newcommand{\ThetaFortyPELamSeven}{0.079} | |
| 2396 | +\newcommand{\ThetaFortyPELamThree}{0.21} | |
| 2397 | +\newcommand{\ThetaFortyPEOwn}{0.596} | |
| 2398 | +\newcommand{\ThetaFortyPEOwnUthirtyfive}{0.304} | |
| 2399 | +\newcommand{\ThetaFortyPEP}{1.000} | |
| 2400 | +\newcommand{\ThetaFortyPEPDevPct}{0.0} | |
| 2401 | +\newcommand{\ThetaFortyPERR}{0.33} | |
| 2402 | +\newcommand{\ThetaFortyPERebate}{0.000} | |
| 2403 | +\newcommand{\ThetaFortyPERho}{0.194} | |
| 2404 | +\newcommand{\ThetaFortyPERhoDevPct}{0.0} | |
| 2405 | +\newcommand{\ThetaFortyPEThr}{0.59} | |
| 2406 | +\newcommand{\ThetaFortyPEThrHighZ}{0.24} | |
| 2407 | +\newcommand{\ThetaFortyPEThrLowZ}{1.18} | |
| 2408 | +\newcommand{\ThetaFortyPEqOO}{0.488} | |
| 2409 | +\newcommand{\ThetaFortyPEqRO}{0.279} | |
| 2410 | +\newcommand{\ThetaFortyRR}{0.34} | |
| 2411 | +\newcommand{\ThetaFortyRebate}{0.000} | |
| 2412 | +\newcommand{\ThetaFortyRho}{0.200} | |
| 2413 | +\newcommand{\ThetaFortyRhoDevPct}{3.0} | |
| 2414 | +\newcommand{\ThetaFortyThr}{0.55} | |
| 2415 | +\newcommand{\ThetaFortyThrHighZ}{0.18} | |
| 2416 | +\newcommand{\ThetaFortyThrLowZ}{1.10} | |
| 2417 | +\newcommand{\ThetaFortyqOO}{0.515} | |
| 2418 | +\newcommand{\ThetaFortyqRO}{0.304} | |
| 2419 | +\newcommand{\ThetaTenEstate}{0.31} | |
| 2420 | +\newcommand{\ThetaTenGapFour}{0.87} | |
| 2421 | +\newcommand{\ThetaTenGift}{0.39} | |
| 2422 | +\newcommand{\ThetaTenGini}{0.54} | |
| 2423 | +\newcommand{\ThetaTenLamSeven}{0.063} | |
| 2424 | +\newcommand{\ThetaTenLamThree}{0.19} | |
| 2425 | +\newcommand{\ThetaTenOwn}{0.639} | |
| 2426 | +\newcommand{\ThetaTenOwnUthirtyfive}{0.401} | |
| 2427 | +\newcommand{\ThetaTenP}{1.005} | |
| 2428 | +\newcommand{\ThetaTenPDevPct}{0.5} | |
| 2429 | +\newcommand{\ThetaTenRR}{0.33} | |
| 2430 | +\newcommand{\ThetaTenRebate}{0.000} | |
| 2431 | +\newcommand{\ThetaTenRho}{0.190} | |
| 2432 | +\newcommand{\ThetaTenRhoDevPct}{\ensuremath{-}2.1} | |
| 2433 | +\newcommand{\ThetaTenRhoDevPctAbs}{2.1} | |
| 2434 | +\newcommand{\ThetaTenThr}{0.58} | |
| 2435 | +\newcommand{\ThetaTenThrHighZ}{\ensuremath{-}0.05} | |
| 2436 | +\newcommand{\ThetaTenThrHighZAbs}{0.05} | |
| 2437 | +\newcommand{\ThetaTenThrLowZ}{1.24} | |
| 2438 | +\newcommand{\ThetaTenqOO}{0.560} | |
| 2439 | +\newcommand{\ThetaTenqRO}{0.372} | |
| 2440 | +\newcommand{\ThetaThirtyEstate}{0.32} | |
| 2441 | +\newcommand{\ThetaThirtyGapFour}{0.97} | |
| 2442 | +\newcommand{\ThetaThirtyGift}{0.39} | |
| 2443 | +\newcommand{\ThetaThirtyGini}{0.54} | |
| 2444 | +\newcommand{\ThetaThirtyLamSeven}{0.066} | |
| 2445 | +\newcommand{\ThetaThirtyLamThree}{0.20} | |
| 2446 | +\newcommand{\ThetaThirtyOwn}{0.625} | |
| 2447 | +\newcommand{\ThetaThirtyOwnUthirtyfive}{0.353} | |
| 2448 | +\newcommand{\ThetaThirtyP}{0.992} | |
| 2449 | +\newcommand{\ThetaThirtyPDevPct}{\ensuremath{-}0.8} | |
| 2450 | +\newcommand{\ThetaThirtyPDevPctAbs}{0.8} | |
| 2451 | +\newcommand{\ThetaThirtyRR}{0.34} | |
| 2452 | +\newcommand{\ThetaThirtyRebate}{0.000} | |
| 2453 | +\newcommand{\ThetaThirtyRho}{0.197} | |
| 2454 | +\newcommand{\ThetaThirtyRhoDevPct}{1.6} | |
| 2455 | +\newcommand{\ThetaThirtyThr}{0.49} | |
| 2456 | +\newcommand{\ThetaThirtyThrHighZ}{0.11} | |
| 2457 | +\newcommand{\ThetaThirtyThrLowZ}{1.12} | |
| 2458 | +\newcommand{\ThetaThirtyqOO}{0.523} | |
| 2459 | +\newcommand{\ThetaThirtyqRO}{0.321} | |
| 148 | 2460 | \newcommand{\ThrBaseline}{0.49} |
| 149 | 2461 | \newcommand{\ThrBequestshalved}{0.44} |
| 150 | 2462 | \newcommand{\ThrBequestsoff}{0.32} |
@@ -156,10 +2468,105 @@ | ||
| 156 | 2468 | \newcommand{\ThresholdAgeOneHighZ}{0.03} |
| 157 | 2469 | \newcommand{\ThresholdAgeOneLowZ}{1.18} |
| 158 | 2470 | \newcommand{\ThresholdAgeOneMidZ}{0.49} |
| 2471 | +\newcommand{\ThresholdAgeOneZfour}{0.24} | |
| 2472 | +\newcommand{\ThresholdAgeOneZtwo}{0.88} | |
| 2473 | +\newcommand{\ThresholdAgeThreeHighZ}{\ensuremath{-}0.61} | |
| 2474 | +\newcommand{\ThresholdAgeThreeHighZAbs}{0.61} | |
| 2475 | +\newcommand{\ThresholdAgeThreeMidZ}{0.19} | |
| 2476 | +\newcommand{\ThresholdAgeTwoHighZ}{\ensuremath{-}0.31} | |
| 2477 | +\newcommand{\ThresholdAgeTwoHighZAbs}{0.31} | |
| 159 | 2478 | \newcommand{\Thrtheta}{0.55} |
| 160 | 2479 | \newcommand{\Thrthetapricesfixed}{0.59} |
| 161 | 2480 | \newcommand{\Thrvarepsilon}{0.49} |
| 162 | 2481 | \newcommand{\TotalDebt}{2.47} |
| 163 | 2482 | \newcommand{\TotalDeposits}{4.12} |
| 2483 | +\newcommand{\UnderwaterAllTen}{0} | |
| 2484 | +\newcommand{\UnderwaterAllThirty}{0} | |
| 2485 | +\newcommand{\UnderwaterAllTwenty}{0} | |
| 2486 | +\newcommand{\UnderwaterNewTen}{0} | |
| 2487 | +\newcommand{\UnderwaterNewThirty}{48} | |
| 2488 | +\newcommand{\UnderwaterNewTwenty}{40} | |
| 2489 | +\newcommand{\UnindexedAbsorbPct}{80} | |
| 2490 | +\newcommand{\UnindexedBetterAllPct}{0} | |
| 2491 | +\newcommand{\UnindexedBetterOMPct}{0} | |
| 2492 | +\newcommand{\UnindexedBetterONMPct}{0} | |
| 2493 | +\newcommand{\UnindexedBetterOPct}{0} | |
| 2494 | +\newcommand{\UnindexedBetterRPct}{0} | |
| 2495 | +\newcommand{\UnindexedCEVAll}{\ensuremath{-}3.33} | |
| 2496 | +\newcommand{\UnindexedCEVAllAbs}{3.33} | |
| 2497 | +\newcommand{\UnindexedCEVAllPE}{\ensuremath{-}3.45} | |
| 2498 | +\newcommand{\UnindexedCEVAllPEAbs}{3.45} | |
| 2499 | +\newcommand{\UnindexedCEVO}{\ensuremath{-}3.02} | |
| 2500 | +\newcommand{\UnindexedCEVOAbs}{3.02} | |
| 2501 | +\newcommand{\UnindexedCEVOM}{\ensuremath{-}2.40} | |
| 2502 | +\newcommand{\UnindexedCEVOMAbs}{2.40} | |
| 2503 | +\newcommand{\UnindexedCEVONM}{\ensuremath{-}4.48} | |
| 2504 | +\newcommand{\UnindexedCEVONMAbs}{4.48} | |
| 2505 | +\newcommand{\UnindexedCEVR}{\ensuremath{-}3.58} | |
| 2506 | +\newcommand{\UnindexedCEVRAbs}{3.58} | |
| 2507 | +\newcommand{\UnindexedConsAll}{\ensuremath{-}4.4} | |
| 2508 | +\newcommand{\UnindexedConsAllAbs}{4.4} | |
| 2509 | +\newcommand{\UnindexedConsO}{\ensuremath{-}3.5} | |
| 2510 | +\newcommand{\UnindexedConsOAbs}{3.5} | |
| 2511 | +\newcommand{\UnindexedConsR}{\ensuremath{-}6.3} | |
| 2512 | +\newcommand{\UnindexedConsRAbs}{6.3} | |
| 2513 | +\newcommand{\UnindexedDOwnGE}{\ensuremath{-}0.63} | |
| 2514 | +\newcommand{\UnindexedDOwnGEAbs}{0.63} | |
| 2515 | +\newcommand{\UnindexedDOwnPE}{\ensuremath{-}3.18} | |
| 2516 | +\newcommand{\UnindexedDOwnPEAbs}{3.18} | |
| 2517 | +\newcommand{\UnindexedDebtZeroPct}{\ensuremath{-}1.6} | |
| 2518 | +\newcommand{\UnindexedDebtZeroPctAbs}{1.6} | |
| 2519 | +\newcommand{\UnindexedDepLoss}{0.48} | |
| 2520 | +\newcommand{\UnindexedErosion}{0.21} | |
| 2521 | +\newcommand{\UnindexedLamSevenSS}{0.060} | |
| 2522 | +\newcommand{\UnindexedLamSevenZero}{0.074} | |
| 2523 | +\newcommand{\UnindexedLender}{0.27} | |
| 2524 | +\newcommand{\UnindexedMaxRes}{2.0\times10^{-3}} | |
| 2525 | +\newcommand{\UnindexedNWAll}{\ensuremath{-}0.09} | |
| 2526 | +\newcommand{\UnindexedNWAllAbs}{0.09} | |
| 2527 | +\newcommand{\UnindexedNWO}{\ensuremath{-}0.12} | |
| 2528 | +\newcommand{\UnindexedNWOAbs}{0.12} | |
| 2529 | +\newcommand{\UnindexedNWR}{\ensuremath{-}0.04} | |
| 2530 | +\newcommand{\UnindexedNWRAbs}{0.04} | |
| 2531 | +\newcommand{\UnindexedNetO}{\ensuremath{-}0.02} | |
| 2532 | +\newcommand{\UnindexedNetOAbs}{0.02} | |
| 2533 | +\newcommand{\UnindexedNetR}{\ensuremath{-}0.02} | |
| 2534 | +\newcommand{\UnindexedNetRAbs}{0.02} | |
| 2535 | +\newcommand{\UnindexedNominalRateZero}{0.219} | |
| 2536 | +\newcommand{\UnindexedOwnerOldConsMax}{\ensuremath{-}4.8} | |
| 2537 | +\newcommand{\UnindexedOwnerOldConsMaxAbs}{4.8} | |
| 2538 | +\newcommand{\UnindexedOwnerOldConsMin}{\ensuremath{-}4.3} | |
| 2539 | +\newcommand{\UnindexedOwnerOldConsMinAbs}{4.3} | |
| 2540 | +\newcommand{\UnindexedOwnerOldLossMax}{0.08} | |
| 2541 | +\newcommand{\UnindexedOwnerOldLossMin}{0.04} | |
| 2542 | +\newcommand{\UnindexedOwnerYoungConsMax}{\ensuremath{-}1.8} | |
| 2543 | +\newcommand{\UnindexedOwnerYoungConsMaxAbs}{1.8} | |
| 2544 | +\newcommand{\UnindexedOwnerYoungConsMin}{\ensuremath{-}2.8} | |
| 2545 | +\newcommand{\UnindexedOwnerYoungConsMinAbs}{2.8} | |
| 2546 | +\newcommand{\UnindexedOwnerYoungGainMax}{0.04} | |
| 2547 | +\newcommand{\UnindexedPriceBackYears}{40} | |
| 2548 | +\newcommand{\UnindexedPriceMaxPos}{0.1} | |
| 2549 | +\newcommand{\UnindexedPriceSignChanges}{1} | |
| 2550 | +\newcommand{\UnindexedPzero}{\ensuremath{-}3.86} | |
| 2551 | +\newcommand{\UnindexedPzeroAbs}{3.86} | |
| 2552 | +\newcommand{\UnindexedRentGainR}{0.00} | |
| 2553 | +\newcommand{\UnindexedResT}{9\times10^{-5}} | |
| 2554 | +\newcommand{\UnindexedRhoZero}{\ensuremath{-}3.21} | |
| 2555 | +\newcommand{\UnindexedRhoZeroAbs}{3.21} | |
| 2556 | +\newcommand{\UnindexedSecs}{487} | |
| 2557 | +\newcommand{\UnindexedSteps}{11} | |
| 2558 | +\newcommand{\UnindexedThrChangePct}{10} | |
| 2559 | +\newcommand{\UnindexedThrSS}{0.49} | |
| 2560 | +\newcommand{\UnindexedThrZero}{0.55} | |
| 2561 | +\newcommand{\UnindexedWageAll}{0.08} | |
| 2562 | +\newcommand{\UnindexedWageO}{0.09} | |
| 2563 | +\newcommand{\UnindexedqROsevenSS}{0.806} | |
| 2564 | +\newcommand{\UnindexedqROsevenZero}{0.807} | |
| 2565 | +\newcommand{\UnindexedqROthreeOne}{0.308} | |
| 2566 | +\newcommand{\UnindexedqROthreeSS}{0.351} | |
| 2567 | +\newcommand{\UnindexedqROthreeZero}{0.352} | |
| 2568 | +\newcommand{\WelfareCheckMaxPct}{4.9} | |
| 2569 | +\newcommand{\qOOageSeven}{0.87} | |
| 164 | 2570 | \newcommand{\qOOageThree}{0.54} |
| 2571 | +\newcommand{\qROageSeven}{0.81} | |
| 165 | 2572 | \newcommand{\qROageThree}{0.35} |
modified
results/tables/tab01_notation.tex
+1 −1
@@ -4,7 +4,7 @@ | ||
| 4 | 4 | \begin{threeparttable} |
| 5 | 5 | \caption{Notation of the quantitative model} |
| 6 | 6 | \label{tab:notation} |
| 7 | −\begin{tabular}{lll} | |
| 7 | +\begin{tabular}{lp{9.3cm}l} | |
| 8 | 8 | \toprule |
| 9 | 9 | Symbol & Meaning & Units \\ |
| 10 | 10 | \midrule |
modified
results/tables/tab02_parameters.tex
+2 −2
@@ -46,8 +46,8 @@ n\_b & 20 & - & grid \\ | ||
| 46 | 46 | n\_x & 120 & - & grid \\ |
| 47 | 47 | n\_B & 11 & - & grid \\ |
| 48 | 48 | K\_estate & 5 & - & lottery \\ |
| 49 | −n\_dynasties & 9.6e+04 & - & simulation \\ | |
| 50 | −seed & 2.026e+07 & - & simulation \\ | |
| 49 | +n\_dynasties & 96{,}000 & - & simulation \\ | |
| 50 | +seed & 20{,}261{,}020 & - & simulation \\ | |
| 51 | 51 | \bottomrule |
| 52 | 52 | \end{tabular} |
| 53 | 53 | \begin{tablenotes}[flushleft] |
modified
results/tables/tab04_convergence.tex
+1 −1
@@ -9,7 +9,7 @@ | ||
| 9 | 9 | Diagnostic & Value \\ |
| 10 | 10 | \midrule |
| 11 | 11 | Owner-occupied market, relative excess demand & 0.000004 \\ |
| 12 | −Rental market, relative excess demand & -0.000007 \\ | |
| 12 | +Rental market, relative excess demand & \ensuremath{-}0.000007 \\ | |
| 13 | 13 | Tolerance (relative excess demand) & 2.5e-4 \\ |
| 14 | 14 | Estate-lottery fixed point: max.\ change (last round) & 0.000009 \\ |
| 15 | 15 | Bellman residual, owners: mean / 99th pct.\ / max & 0.00067 / 0.0062 / 0.0193 \\ |
modified
results/tables/tab05_scenarios.tex
+7 −9
@@ -1,9 +1,9 @@ | ||
| 1 | 1 | \begin{table}[htbp] |
| 2 | 2 | \centering |
| 3 | 3 | \footnotesize |
| 4 | −\begin{threeparttable} | |
| 5 | 4 | \caption{Stationary equilibria across credit, supply, segmentation and bequest scenarios} |
| 6 | 5 | \label{tab:scenarios} |
| 6 | +\resizebox{\textwidth}{!}{% | |
| 7 | 7 | \begin{tabular}{lcccccccccccc} |
| 8 | 8 | \toprule |
| 9 | 9 | Scenario & Eq. & $p^*$ & $\rho/p$ & Own. & Own.\ $<$35 & $\hat a_1$ & $q_{RO}$ & $q_{OO}$ & rank--rank & Gini & gift \\ |
@@ -23,16 +23,14 @@ $\varepsilon = 0.5$ & GE & 1.000 & 0.039 & 0.631 & 0.378 & 0.49 & 0.35 & 0.54 & | ||
| 23 | 23 | $\varepsilon = 2$ & GE & 1.000 & 0.039 & 0.631 & 0.378 & 0.49 & 0.35 & 0.54 & 0.34 & 0.54 & 0.39 \\ |
| 24 | 24 | No segmentation ($\sigma = 0$) & GE & 1.000 & 0.039 & 0.631 & 0.378 & 0.49 & 0.35 & 0.54 & 0.34 & 0.54 & 0.39 \\ |
| 25 | 25 | Half segmentation ($\sigma = 0.5$) & GE & 1.000 & 0.039 & 0.631 & 0.378 & 0.49 & 0.35 & 0.54 & 0.34 & 0.54 & 0.39 \\ |
| 26 | −Bequests off ($\nu = 0$) & GE & 0.931 & 0.039 & 0.581 & 0.188 & 0.32 & 0.38 & 0.38 & -0.01 & 0.58 & 0.00 \\ | |
| 26 | +Bequests off ($\nu = 0$) & GE & 0.931 & 0.039 & 0.581 & 0.188 & 0.32 & 0.38 & 0.38 & \ensuremath{-}0.01 & 0.58 & 0.00 \\ | |
| 27 | 27 | Bequests halved ($\nu = 1$) & GE & 0.981 & 0.039 & 0.615 & 0.325 & 0.44 & 0.37 & 0.49 & 0.25 & 0.55 & 0.26 \\ |
| 28 | 28 | Transfer tax 20\% (rebated) & GE & 1.005 & 0.039 & 0.642 & 0.461 & 0.51 & 0.39 & 0.57 & 0.28 & 0.53 & 0.28 \\ |
| 29 | 29 | Transfer tax 50\% (rebated) & GE & 0.999 & 0.039 & 0.639 & 0.482 & 0.48 & 0.43 & 0.55 & 0.16 & 0.52 & 0.12 \\ |
| 30 | −Bequests off, prices fixed & PE & 1.000 & 0.039 & 0.512 & 0.106 & 0.48 & 0.36 & 0.36 & -0.00 & 0.59 & 0.00 \\ | |
| 30 | +Bequests off, prices fixed & PE & 1.000 & 0.039 & 0.512 & 0.106 & 0.48 & 0.36 & 0.36 & 0.00 & 0.59 & 0.00 \\ | |
| 31 | 31 | \bottomrule |
| 32 | −\end{tabular} | |
| 33 | −\begin{tablenotes}[flushleft] | |
| 34 | −\footnotesize | |
| 35 | −\item GE = general equilibrium (prices, estate lottery and rebate re-solved; supply scales fixed at the baseline normalisation); PE = partial equilibrium with prices fixed at the baseline (household block and distribution re-solved). $^{\dagger}$ = not converged to tolerance (residuals in results/data/diagnostics.csv). $\rho/p$ = annual rent--price ratio; $\hat a_1$ = age-1 ownership-entry threshold at the median productivity state (5-year income units); $q_{RO}$, $q_{OO}$ = probability that the child owns at 35--39 given the parent's tenure at the gift; rank--rank = slope of the child's net-worth rank on the parent's (both at 55--59); gift = mean transfer received at entry. Source files: results/data/group/name.json. | |
| 36 | −\end{tablenotes} | |
| 37 | −\end{threeparttable} | |
| 32 | +\end{tabular}} | |
| 33 | +\begin{minipage}{\textwidth}\footnotesize\vspace{2pt} | |
| 34 | +GE = general equilibrium (prices, estate lottery and rebate re-solved; supply scales fixed at the baseline normalisation); PE = partial equilibrium with prices fixed at the baseline (household block and distribution re-solved). $^{\dagger}$ = not converged to tolerance (residuals in results/data/diagnostics.csv). $\rho/p$ = annual rent--price ratio; $\hat a_1$ = age-1 ownership-entry threshold at the median productivity state (5-year income units); $q_{RO}$, $q_{OO}$ = probability that the child owns at 35--39 given the parent's tenure at the gift; rank--rank = slope of the child's net-worth rank on the parent's (both at 55--59); gift = mean transfer received at entry. Source files: results/data/group/name.json. | |
| 35 | +\end{minipage} | |
| 38 | 36 | \end{table} |
modified
results/tables/tab06_inflation.tex
+20 −22
@@ -1,31 +1,29 @@ | ||
| 1 | 1 | \begin{table}[htbp] |
| 2 | 2 | \centering |
| 3 | 3 | \scriptsize |
| 4 | −\begin{threeparttable} | |
| 5 | 4 | \caption{Unanticipated 10\% price-level surprise: impact effects by mortgage contract and persistence} |
| 6 | 5 | \label{tab:inflation} |
| 7 | −\begin{tabular}{lcccccccccccc} | |
| 6 | +\resizebox{\textwidth}{!}{% | |
| 7 | +\begin{tabular}{lccccccccccc} | |
| 8 | 8 | \toprule |
| 9 | −Scenario & Eq. & $p_0$ & $p$ peak & $\rho_0$ & $\Delta$own & $\Delta$own PE & erosion & dep.\ loss & CEV o/m & CEV r & CEV all & CEV PE \\ | |
| 9 | +Scenario & Eq. & $p_0$ & $\rho_0$ & $\Delta$own & $\Delta$own PE & erosion & dep.\ loss & CEV o/m & CEV r & CEV all & CEV PE \\ | |
| 10 | 10 | \midrule |
| 11 | −FRM, level surprise & GE$^{\dagger}$ & -0.45 & -0.45 & -1.14 & -0.02 & -0.20 & 0.206 & 0.481 & 0.63 & -1.00 & -0.64 & -0.68 \\ | |
| 12 | −5-year term, level & GE & -0.47 & -0.47 & -1.03 & -0.02 & -0.20 & 0.206 & 0.481 & 0.63 & -1.00 & -0.64 & -0.68 \\ | |
| 13 | −ARM, level & GE & -0.79 & -0.79 & -1.15 & -0.11 & -0.47 & 0.000 & 0.481 & -1.03 & -0.96 & -1.14 & -1.19 \\ | |
| 14 | −FRM, persistent (2 periods) & GE & -1.88 & -1.88 & -0.38 & -0.18 & -2.17 & 0.206 & 0.481 & 1.14 & -1.06 & -0.52 & -0.53 \\ | |
| 15 | −5-year term, persistent & GE & -1.49 & -1.49 & 0.04 & -0.34 & -2.03 & 0.206 & 0.481 & 0.46 & -1.09 & -0.74 & -0.73 \\ | |
| 16 | −ARM, persistent & GE & -1.86 & -1.86 & -0.18 & -0.39 & -2.34 & 0.000 & 0.481 & -1.21 & -1.04 & -1.24 & -1.24 \\ | |
| 17 | −FRM, level, rigid rents ($\eta_\rho=0$) & GE & -0.46 & -0.46 & 7.48 & -0.03 & -2.88 & 0.206 & 0.481 & 0.55 & -0.92 & -0.62 & -0.35 \\ | |
| 18 | −FRM, level, unindexed wages ($\eta_w=0$) & GE$^{\dagger}$ & -3.86 & -3.86 & -3.21 & -0.63 & -3.18 & 0.206 & 0.481 & -2.40 & -3.58 & -3.33 & -3.45 \\ | |
| 19 | −FRM, persistent, real-rate response & GE & -7.25 & -7.25 & -2.29 & -0.12 & -2.64 & 0.206 & 0.481 & 1.43 & -0.31 & 0.14 & -1.32 \\ | |
| 20 | −5-year term, persistent, real-rate response & GE & -6.93 & -6.93 & -1.91 & -0.26 & -2.94 & 0.206 & 0.481 & 0.20 & -0.32 & -0.25 & -1.67 \\ | |
| 21 | −FRM, level, $\varepsilon=0.05$ & GE$^{\dagger}$ & -0.50 & -0.50 & -1.08 & -0.01 & -0.19 & 0.206 & 0.481 & 0.63 & -0.99 & -0.64 & -0.68 \\ | |
| 22 | −FRM, level, no segmentation & GE & -0.39 & -0.39 & -1.47 & -0.16 & -0.20 & 0.206 & 0.481 & 0.66 & -0.95 & -0.60 & -0.68 \\ | |
| 23 | −FRM, level, static expectations (A6) & A6 & -0.36 & -0.36 & -1.11 & 0.02 & -0.20 & 0.206 & 0.481 & 0.65 & -1.01 & -0.64 & -0.68 \\ | |
| 11 | +FRM, level surprise & GE$^{\dagger}$ & \ensuremath{-}0.45 & \ensuremath{-}1.14 & \ensuremath{-}0.02 & \ensuremath{-}0.20 & 0.206 & 0.481 & 0.63 & \ensuremath{-}1.00 & \ensuremath{-}0.64 & \ensuremath{-}0.68 \\ | |
| 12 | +5-year term, level & GE & \ensuremath{-}0.47 & \ensuremath{-}1.03 & \ensuremath{-}0.02 & \ensuremath{-}0.20 & 0.206 & 0.481 & 0.63 & \ensuremath{-}1.00 & \ensuremath{-}0.64 & \ensuremath{-}0.68 \\ | |
| 13 | +ARM, level & GE & \ensuremath{-}0.79 & \ensuremath{-}1.15 & \ensuremath{-}0.11 & \ensuremath{-}0.47 & 0.000 & 0.481 & \ensuremath{-}1.03 & \ensuremath{-}0.96 & \ensuremath{-}1.14 & \ensuremath{-}1.19 \\ | |
| 14 | +FRM, persistent (2 periods) & GE & \ensuremath{-}1.88 & \ensuremath{-}0.38 & \ensuremath{-}0.18 & \ensuremath{-}2.17 & 0.206 & 0.481 & 1.14 & \ensuremath{-}1.06 & \ensuremath{-}0.52 & \ensuremath{-}0.53 \\ | |
| 15 | +5-year term, persistent & GE & \ensuremath{-}1.49 & 0.04 & \ensuremath{-}0.34 & \ensuremath{-}2.03 & 0.206 & 0.481 & 0.46 & \ensuremath{-}1.09 & \ensuremath{-}0.74 & \ensuremath{-}0.73 \\ | |
| 16 | +ARM, persistent & GE & \ensuremath{-}1.86 & \ensuremath{-}0.18 & \ensuremath{-}0.39 & \ensuremath{-}2.34 & 0.000 & 0.481 & \ensuremath{-}1.21 & \ensuremath{-}1.04 & \ensuremath{-}1.24 & \ensuremath{-}1.24 \\ | |
| 17 | +FRM, level, rigid rents ($\eta_\rho=0$) & GE & \ensuremath{-}0.46 & 7.48 & \ensuremath{-}0.03 & \ensuremath{-}2.88 & 0.206 & 0.481 & 0.55 & \ensuremath{-}0.92 & \ensuremath{-}0.62 & \ensuremath{-}0.35 \\ | |
| 18 | +FRM, level, unindexed wages ($\eta_w=0$) & GE$^{\dagger}$ & \ensuremath{-}3.86 & \ensuremath{-}3.21 & \ensuremath{-}0.63 & \ensuremath{-}3.18 & 0.206 & 0.481 & \ensuremath{-}2.40 & \ensuremath{-}3.58 & \ensuremath{-}3.33 & \ensuremath{-}3.45 \\ | |
| 19 | +FRM, persistent, real-rate response & GE & \ensuremath{-}7.25 & \ensuremath{-}2.29 & \ensuremath{-}0.12 & \ensuremath{-}2.64 & 0.206 & 0.481 & 1.43 & \ensuremath{-}0.31 & 0.14 & \ensuremath{-}1.32 \\ | |
| 20 | +5-year term, persistent, real-rate response & GE & \ensuremath{-}6.93 & \ensuremath{-}1.91 & \ensuremath{-}0.26 & \ensuremath{-}2.94 & 0.206 & 0.481 & 0.20 & \ensuremath{-}0.32 & \ensuremath{-}0.25 & \ensuremath{-}1.67 \\ | |
| 21 | +FRM, level, $\varepsilon=0.05$ & GE$^{\dagger}$ & \ensuremath{-}0.50 & \ensuremath{-}1.08 & \ensuremath{-}0.01 & \ensuremath{-}0.19 & 0.206 & 0.481 & 0.63 & \ensuremath{-}0.99 & \ensuremath{-}0.64 & \ensuremath{-}0.68 \\ | |
| 22 | +FRM, level, no segmentation & GE & \ensuremath{-}0.39 & \ensuremath{-}1.47 & \ensuremath{-}0.16 & \ensuremath{-}0.20 & 0.206 & 0.481 & 0.66 & \ensuremath{-}0.95 & \ensuremath{-}0.60 & \ensuremath{-}0.68 \\ | |
| 23 | +FRM, level, static expectations (A6) & A6 & \ensuremath{-}0.36 & \ensuremath{-}1.11 & 0.02 & \ensuremath{-}0.20 & 0.206 & 0.481 & 0.65 & \ensuremath{-}1.01 & \ensuremath{-}0.64 & \ensuremath{-}0.68 \\ | |
| 24 | 24 | \bottomrule |
| 25 | −\end{tabular} | |
| 26 | −\begin{tablenotes}[flushleft] | |
| 27 | −\footnotesize | |
| 28 | −\item Perfect-foresight transitions (GE: prices clear every period; $^{\dagger}$ = not converged to the $1.5\times10^{-3}$ per-period tolerance (stalled at the Monte-Carlo noise floor; maximum residual in results/data/inflation/name.json); A6 = static expectations, stationary continuation values). Columns: real house price on impact and at its peak deviation (\%), real rent on impact (\%), change of the ownership rate on impact (pp) in GE and with prices fixed (PE), aggregate real debt erosion and real deposit loss per unit mass of one age (the lender's revaluation is their difference with the opposite sign), consumption-equivalent welfare of households alive at the shock (owners with a mortgage, renters, all; \%), and the all-household CEV with prices fixed. Level surprise: CPI 10\% above expectation in the shock period, inflation back to target afterwards; persistent: two periods. Source: results/data/inflation/*.json and *\_welfare.csv. | |
| 29 | −\end{tablenotes} | |
| 30 | −\end{threeparttable} | |
| 25 | +\end{tabular}} | |
| 26 | +\begin{minipage}{\textwidth}\footnotesize\vspace{2pt} | |
| 27 | +Perfect-foresight transitions (GE: prices clear every period; $^{\dagger}$ = not converged to the $1.5\times10^{-3}$ per-period tolerance (stalled at the Monte-Carlo noise floor; maximum residual in results/data/inflation/name.json); A6 = static expectations, stationary continuation values). Columns: real house price on impact (\%; the impact is also the peak deviation in every run), real rent on impact (\%), change of the ownership rate on impact (pp) in GE and with prices fixed (PE), aggregate real debt erosion and real deposit loss per unit mass of one age (the lender's revaluation is their difference with the opposite sign), consumption-equivalent welfare of households alive at the shock (owners with a mortgage, renters, all; \%), and the all-household CEV with prices fixed. Level surprise: CPI 10\% above expectation in the shock period, inflation back to target afterwards; persistent: two periods. Source: results/data/inflation/*.json and *\_welfare.csv. | |
| 28 | +\end{minipage} | |
| 31 | 29 | \end{table} |
modified
results/tables/tab07_mobility_inequality.tex
+6 −8
@@ -1,15 +1,15 @@ | ||
| 1 | 1 | \begin{table}[htbp] |
| 2 | 2 | \centering |
| 3 | 3 | \scriptsize |
| 4 | −\begin{threeparttable} | |
| 5 | 4 | \caption{Intergenerational persistence and inequality across stationary equilibria} |
| 6 | 5 | \label{tab:mobility} |
| 6 | +\resizebox{\textwidth}{!}{% | |
| 7 | 7 | \begin{tabular}{lcccccccccccc} |
| 8 | 8 | \toprule |
| 9 | 9 | Scenario & Eq. & $p^*$ & Own. & $q_{RO}$ & $q_{OO}$ & $\lambda_3$ & $\lambda_7$ & rank--rank & Gini & gap$_4$ & gift & estate \\ |
| 10 | 10 | \midrule |
| 11 | 11 | Baseline & GE & 1.000 & 0.631 & 0.35 & 0.54 & 0.19 & 0.060 & 0.34 & 0.54 & 0.95 & 0.39 & 0.32 \\ |
| 12 | −Bequests off & GE & 0.931 & 0.581 & 0.38 & 0.38 & 0.00 & -0.003 & -0.01 & 0.58 & 0.73 & 0.00 & 0.00 \\ | |
| 12 | +Bequests off & GE & 0.931 & 0.581 & 0.38 & 0.38 & 0.00 & \ensuremath{-}0.003 & \ensuremath{-}0.01 & 0.58 & 0.73 & 0.00 & 0.00 \\ | |
| 13 | 13 | Bequests halved & GE & 0.981 & 0.615 & 0.37 & 0.49 & 0.12 & 0.035 & 0.25 & 0.55 & 0.86 & 0.26 & 0.21 \\ |
| 14 | 14 | Transfer tax 20\% & GE & 1.005 & 0.642 & 0.39 & 0.57 & 0.18 & 0.054 & 0.28 & 0.53 & 0.96 & 0.28 & 0.22 \\ |
| 15 | 15 | Transfer tax 50\% & GE & 0.999 & 0.639 & 0.43 & 0.55 & 0.12 & 0.023 & 0.16 & 0.52 & 0.90 & 0.12 & 0.09 \\ |
@@ -28,10 +28,8 @@ Trend inflation 4\% & GE & 0.986 & 0.625 & 0.34 & 0.53 & 0.19 & 0.060 & 0.34 & 0 | ||
| 28 | 28 | Landlord wedge $\phi = 0.02$ & GE & 0.991 & 0.607 & 0.33 & 0.52 & 0.19 & 0.070 & 0.33 & 0.55 & 0.96 & 0.39 & 0.31 \\ |
| 29 | 29 | Landlord wedge $\phi = 0.08$ & GE & 1.007 & 0.650 & 0.36 & 0.55 & 0.19 & 0.055 & 0.34 & 0.54 & 0.94 & 0.39 & 0.32 \\ |
| 30 | 30 | \bottomrule |
| 31 | −\end{tabular} | |
| 32 | −\begin{tablenotes}[flushleft] | |
| 33 | −\footnotesize | |
| 34 | −\item $q_{RO}$, $q_{OO}$: probability that the child owns at 35--39 given the parent's tenure when the gift is made; $\lambda_3 = q_{OO} - q_{RO}$; $\lambda_7$: the same difference measured at 55--59 (the Part A persistence statistic); rank--rank: slope of the child's net-worth rank on the parent's at the same age (55--59); gap$_4$: mean net-worth difference between owners and renters at 40--44 (equal income process by design); gift, estate: mean transfers received. $^{\dagger}$ = not converged to tolerance. Source: results/data/group/name.json. | |
| 35 | −\end{tablenotes} | |
| 36 | −\end{threeparttable} | |
| 31 | +\end{tabular}} | |
| 32 | +\begin{minipage}{\textwidth}\footnotesize\vspace{2pt} | |
| 33 | +$q_{RO}$, $q_{OO}$: probability that the child owns at 35--39 given the parent's tenure when the gift is made; $\lambda_3 = q_{OO} - q_{RO}$; $\lambda_7$: the same difference measured at 55--59 (the Part A persistence statistic); rank--rank: slope of the child's net-worth rank on the parent's at the same age (55--59); gap$_4$: mean net-worth difference between owners and renters at 40--44 (equal income process by design); gift, estate: mean transfers received. $^{\dagger}$ = not converged to tolerance. Source: results/data/group/name.json. | |
| 34 | +\end{minipage} | |
| 37 | 35 | \end{table} |
modified
results/tables/tab08_welfare.tex
+24 −26
@@ -1,37 +1,35 @@ | ||
| 1 | 1 | \begin{table}[htbp] |
| 2 | 2 | \centering |
| 3 | 3 | \footnotesize |
| 4 | −\begin{threeparttable} | |
| 5 | 4 | \caption{Welfare of entering households across stationary equilibria (consumption-equivalent variation, \%)} |
| 6 | 5 | \label{tab:welfare} |
| 6 | +\resizebox{\textwidth}{!}{% | |
| 7 | 7 | \begin{tabular}{lccccccccc} |
| 8 | 8 | \toprule |
| 9 | 9 | Scenario & Q1 & Q2 & Q3 & Q4 & Q5 & marginal & low $z_1$ & high $z_1$ & all \\ |
| 10 | 10 | \midrule |
| 11 | −theta\_005 & 0.55 & 0.54 & 0.43 & 0.26 & -0.08 & 0.30 & 0.54 & 0.20 & 0.34 \\ | |
| 12 | −theta\_010 & 0.41 & 0.40 & 0.33 & 0.19 & -0.05 & 0.24 & 0.41 & 0.14 & 0.26 \\ | |
| 13 | −theta\_030 & -0.20 & -0.21 & -0.22 & -0.07 & 0.21 & -0.13 & -0.23 & 0.08 & -0.10 \\ | |
| 14 | −theta\_040 & -0.39 & -0.40 & -0.44 & -0.25 & 0.32 & -0.33 & -0.44 & 0.07 & -0.23 \\ | |
| 15 | −psi\_030 & -0.01 & 0.00 & 0.01 & -0.02 & 0.09 & -0.08 & -0.08 & 0.21 & 0.02 \\ | |
| 16 | −psi\_050 & -0.01 & -0.02 & -0.05 & -0.06 & -0.12 & -0.01 & 0.03 & -0.16 & -0.05 \\ | |
| 17 | −bequests\_off & 8.87 & 9.09 & 10.11 & 12.08 & 15.53 & 12.82 & 4.41 & 20.16 & 11.14 \\ | |
| 18 | −transfer\_tax\_020 & 10.85 & 9.64 & 7.20 & 5.64 & 4.02 & 5.74 & 16.92 & 2.58 & 7.47 \\ | |
| 19 | −transfer\_tax\_050 & 15.39 & 13.79 & 10.68 & 8.52 & 5.89 & 8.58 & 24.44 & 2.87 & 10.85 \\ | |
| 20 | −eps\_005 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ | |
| 21 | −eps\_200 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ | |
| 22 | −noseg & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ | |
| 23 | −r\_001 & 0.68 & 0.51 & -0.03 & -0.68 & -2.25 & -0.86 & -0.31 & -1.18 & -0.35 \\ | |
| 24 | −r\_003 & 0.42 & 0.62 & 1.25 & 2.23 & 4.46 & 2.32 & 1.46 & 3.06 & 1.80 \\ | |
| 25 | −pi\_004 & -0.11 & -0.11 & -0.08 & -0.01 & 0.23 & -0.08 & -0.16 & 0.21 & -0.01 \\ | |
| 26 | −dep\_premium\_002 & 5.23 & 5.69 & 7.34 & 9.97 & 15.40 & 10.27 & 7.76 & 11.92 & 8.73 \\ | |
| 27 | −chi\_005 & 0.81 & 0.82 & 0.86 & 0.95 & 1.08 & 0.94 & 0.77 & 1.09 & 0.91 \\ | |
| 28 | −kappa\_low & -0.22 & -0.22 & -0.21 & -0.14 & -0.17 & -0.14 & -0.29 & -0.08 & -0.19 \\ | |
| 29 | −kappa\_high & -0.07 & -0.08 & -0.11 & -0.16 & -0.10 & -0.18 & -0.03 & -0.12 & -0.10 \\ | |
| 11 | +$\theta = 0.05$ & 0.55 & 0.54 & 0.43 & 0.26 & \ensuremath{-}0.08 & 0.30 & 0.54 & 0.20 & 0.34 \\ | |
| 12 | +$\theta = 0.10$ & 0.41 & 0.40 & 0.33 & 0.19 & \ensuremath{-}0.05 & 0.24 & 0.41 & 0.14 & 0.26 \\ | |
| 13 | +$\theta = 0.30$ & \ensuremath{-}0.20 & \ensuremath{-}0.21 & \ensuremath{-}0.22 & \ensuremath{-}0.07 & 0.21 & \ensuremath{-}0.13 & \ensuremath{-}0.23 & 0.08 & \ensuremath{-}0.10 \\ | |
| 14 | +$\theta = 0.40$ & \ensuremath{-}0.39 & \ensuremath{-}0.40 & \ensuremath{-}0.44 & \ensuremath{-}0.25 & 0.32 & \ensuremath{-}0.33 & \ensuremath{-}0.44 & 0.07 & \ensuremath{-}0.23 \\ | |
| 15 | +DSTI cap $\psi = 0.30$ & \ensuremath{-}0.01 & 0.00 & 0.01 & \ensuremath{-}0.02 & 0.09 & \ensuremath{-}0.08 & \ensuremath{-}0.08 & 0.21 & 0.02 \\ | |
| 16 | +DSTI cap $\psi = 0.50$ & \ensuremath{-}0.01 & \ensuremath{-}0.02 & \ensuremath{-}0.05 & \ensuremath{-}0.06 & \ensuremath{-}0.12 & \ensuremath{-}0.01 & 0.03 & \ensuremath{-}0.16 & \ensuremath{-}0.05 \\ | |
| 17 | +Bequests off ($\nu = 0$)$^{\ddagger}$ & 8.87 & 9.09 & 10.11 & 12.08 & 15.53 & 12.82 & 4.41 & 20.16 & 11.14 \\ | |
| 18 | +Transfer tax 20\% (rebated) & 10.85 & 9.64 & 7.20 & 5.64 & 4.02 & 5.74 & 16.92 & 2.58 & 7.47 \\ | |
| 19 | +Transfer tax 50\% (rebated) & 15.39 & 13.79 & 10.68 & 8.52 & 5.89 & 8.58 & 24.44 & 2.87 & 10.85 \\ | |
| 20 | +$\varepsilon = 0.05$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ | |
| 21 | +$\varepsilon = 2$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ | |
| 22 | +No segmentation & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 \\ | |
| 23 | +Real rate 1\% & 0.68 & 0.51 & \ensuremath{-}0.03 & \ensuremath{-}0.68 & \ensuremath{-}2.25 & \ensuremath{-}0.86 & \ensuremath{-}0.31 & \ensuremath{-}1.18 & \ensuremath{-}0.35 \\ | |
| 24 | +Real rate 3\% & 0.42 & 0.62 & 1.25 & 2.23 & 4.46 & 2.32 & 1.46 & 3.06 & 1.80 \\ | |
| 25 | +Trend inflation 4\% & \ensuremath{-}0.11 & \ensuremath{-}0.11 & \ensuremath{-}0.08 & \ensuremath{-}0.01 & 0.23 & \ensuremath{-}0.08 & \ensuremath{-}0.16 & 0.21 & \ensuremath{-}0.01 \\ | |
| 26 | +Deposit return $+2$ pp & 5.23 & 5.69 & 7.34 & 9.97 & 15.40 & 10.27 & 7.76 & 11.92 & 8.73 \\ | |
| 27 | +Ownership premium $\chi = 0.05$$^{\ddagger}$ & 0.81 & 0.82 & 0.86 & 0.95 & 1.08 & 0.94 & 0.77 & 1.09 & 0.91 \\ | |
| 28 | +Transaction costs low & \ensuremath{-}0.22 & \ensuremath{-}0.22 & \ensuremath{-}0.21 & \ensuremath{-}0.14 & \ensuremath{-}0.17 & \ensuremath{-}0.14 & \ensuremath{-}0.29 & \ensuremath{-}0.08 & \ensuremath{-}0.19 \\ | |
| 29 | +Transaction costs high & \ensuremath{-}0.07 & \ensuremath{-}0.08 & \ensuremath{-}0.11 & \ensuremath{-}0.16 & \ensuremath{-}0.10 & \ensuremath{-}0.18 & \ensuremath{-}0.03 & \ensuremath{-}0.12 & \ensuremath{-}0.10 \\ | |
| 30 | 30 | \bottomrule |
| 31 | −\end{tabular} | |
| 32 | −\begin{tablenotes}[flushleft] | |
| 33 | −\footnotesize | |
| 34 | −\item $\lambda^{CE}$ such that scaling non-housing consumption by $1+\lambda^{CE}$ in every period of the baseline makes an age-1 household with state $(z_1, a_1)$ as well off as in the alternative stationary equilibrium: $V^{alt}_1 = (1+\lambda^{CE})^{(1-\alpha)(1-\gamma)} U^{base} + \Phi^{base}$, where $U^{base}$ (discounted flow utility) and $\Phi^{base}$ (warm glow) come from cohort simulations under baseline policies. Columns: quintiles of the transfer received at entry (averaged over $z_1$ with stationary weights), households within $\pm10\%$ of the age-1 ownership threshold (``marginal''), lowest and highest $z_1$, all. Comparisons are between stationary equilibria, not along transitions. Source: results/data/welfare/welfare\_all.csv. | |
| 35 | −\end{tablenotes} | |
| 36 | −\end{threeparttable} | |
| 31 | +\end{tabular}} | |
| 32 | +\begin{minipage}{\textwidth}\footnotesize\vspace{2pt} | |
| 33 | +$\lambda^{CE}$ such that scaling non-housing consumption by $1+\lambda^{CE}$ in every period of the baseline makes an age-1 household with state $(z_1, a_1)$ as well off as in the alternative stationary equilibrium: $V^{alt}_1 = (1+\lambda^{CE})^{(1-\alpha)(1-\gamma)} U^{base} + \Phi^{base}$, where $U^{base}$ (discounted flow utility) and $\Phi^{base}$ (warm glow) come from cohort simulations under baseline policies. Columns: quintiles of the transfer received at entry (averaged over $z_1$ with stationary weights), households within $\pm10\%$ of the age-1 ownership threshold (``marginal''), lowest and highest $z_1$, all. Comparisons are between stationary equilibria, not along transitions. $^{\ddagger}$ = the scenario changes a preference parameter, so the number compares value functions under different utilities and is not a welfare statement. Source: results/data/welfare/welfare\_all.csv. | |
| 34 | +\end{minipage} | |
| 37 | 35 | \end{table} |
modified
results/tables/tab09_robustness.tex
+5 −7
@@ -1,9 +1,9 @@ | ||
| 1 | 1 | \begin{table}[htbp] |
| 2 | 2 | \centering |
| 3 | 3 | \scriptsize |
| 4 | −\begin{threeparttable} | |
| 5 | 4 | \caption{Robustness: preferences, transaction costs, grids, tolerance, sample size and seed} |
| 6 | 5 | \label{tab:robustness} |
| 6 | +\resizebox{\textwidth}{!}{% | |
| 7 | 7 | \begin{tabular}{lccccccccccc} |
| 8 | 8 | \toprule |
| 9 | 9 | Variant & Eq. & $p^*$ & Own. & Own.\ $<$35 & $\hat a_1$ & Gini & gift & $q_{RO}$ & $q_{OO}$ & rank--rank & min \\ |
@@ -25,10 +25,8 @@ sample\_small & GE$^{\dagger}$ & 1.001 & 0.632 & 0.379 & 0.50 & 0.542 & 0.390 & | ||
| 25 | 25 | sample\_large & GE & 1.000 & 0.631 & 0.379 & 0.49 & 0.543 & 0.388 & 0.349 & 0.539 & 0.333 & 0.6 \\ |
| 26 | 26 | seed\_alt & GE & 1.000 & 0.631 & 0.378 & 0.49 & 0.543 & 0.386 & 0.347 & 0.536 & 0.336 & 0.3 \\ |
| 27 | 27 | \bottomrule |
| 28 | −\end{tabular} | |
| 29 | −\begin{tablenotes}[flushleft] | |
| 30 | −\footnotesize | |
| 31 | −\item Each row is a full re-solution of the stationary general equilibrium (prices, lottery). beta\_094/098: annual discount factor 0.94/0.98; gamma\_3: $\gamma=3$; alpha\_025: housing share 0.25; kappa\_low/high: transaction costs (1\%, 2\%) / (5\%, 8\%); chi\_005: ownership utility premium 5\%; grid\_coarse/fine, tol\_tight, sample\_small/large, seed\_alt as in Table~\ref{tab:convergence}. Source: results/data/robustness\_summary.csv. | |
| 32 | −\end{tablenotes} | |
| 33 | −\end{threeparttable} | |
| 28 | +\end{tabular}} | |
| 29 | +\begin{minipage}{\textwidth}\footnotesize\vspace{2pt} | |
| 30 | +Each row is a full re-solution of the stationary general equilibrium (prices, lottery). beta\_094/098: annual discount factor 0.94/0.98; gamma\_3: $\gamma=3$; alpha\_025: housing share 0.25; kappa\_low/high: transaction costs (1\%, 2\%) / (5\%, 8\%); chi\_005: ownership utility premium 5\%; grid\_coarse/fine, tol\_tight, sample\_small/large, seed\_alt as in Table~\ref{tab:convergence}. Source: results/data/robustness\_summary.csv. | |
| 31 | +\end{minipage} | |
| 34 | 32 | \end{table} |
modified
src/figures.py
+26 −15
@@ -7,6 +7,7 @@ Usage: python -m src.figures | ||
| 7 | 7 | from __future__ import annotations |
| 8 | 8 | |
| 9 | 9 | import json |
| 10 | +import os | |
| 10 | 11 | from pathlib import Path |
| 11 | 12 | |
| 12 | 13 | import matplotlib |
@@ -34,6 +35,14 @@ MARKS = ["o", "s", "^", "D", "v"] | ||
| 34 | 35 | AGE_LABELS = ["25", "30", "35", "40", "45", "50", "55", "60", "65", "70", "75", "80"] |
| 35 | 36 | |
| 36 | 37 | |
| 38 | + | |
| 39 | +def _suptitle(fig, text, **kw): | |
| 40 | + """In-image titles are off by default: the paper supplies LaTeX captions (set WP18_FIG_TITLES=1 to keep them).""" | |
| 41 | + if os.environ.get("WP18_FIG_TITLES", "0") == "1": | |
| 42 | + kw.setdefault("x", 0.01); kw.setdefault("ha", "left"); kw.setdefault("fontsize", 9.5) | |
| 43 | + _suptitle(fig, text, **kw) | |
| 44 | + | |
| 45 | + | |
| 37 | 46 | def _save(fig, name): |
| 38 | 47 | fig.savefig(FIG / f"{name}.pdf", bbox_inches="tight") |
| 39 | 48 | fig.savefig(FIG / f"{name}.png", bbox_inches="tight") |
@@ -82,7 +91,7 @@ def fig01_timing(): | ||
| 82 | 91 | # mortgage term bracket |
| 83 | 92 | ax.plot([8, 12.9], [0.95, 0.95], color=GREYS[2], lw=1) |
| 84 | 93 | ax.text(10.4, 0.78, "e.g. purchase at age 2, $N=5$ payments (25 years)", fontsize=7, ha="center", color=GREYS[1]) |
| 85 | − ax.set_title("Figure 1. Timing of the overlapping-generations model (one period = 5 years)", loc="left") | |
| 94 | + _suptitle(fig, "Figure 1. Timing of the overlapping-generations model (one period = 5 years)") | |
| 86 | 95 | _save(fig, "fig01_model_timing") |
| 87 | 96 | |
| 88 | 97 | |
@@ -107,7 +116,7 @@ def fig02_policy(): | ||
| 107 | 116 | axes[1].set_title("(b) Liquid saving $a'$, age 1", loc="left"); axes[1].set_xlabel("Cash on hand $x$") |
| 108 | 117 | axes[2].set_title("(c) House bought (0 = rent), age 3", loc="left"); axes[2].set_xlabel("Cash on hand $x$") |
| 109 | 118 | axes[2].set_yticks([0, 1, 2, 3, 4]); axes[2].set_yticklabels(["rent", "$h_1$", "$h_2$", "$h_3$", "$h_4$"]) |
| 110 | − fig.suptitle("Figure 2. Policy functions on the cash-on-hand grid, baseline stationary equilibrium", x=0.01, ha="left", fontsize=9.5) | |
| 119 | + _suptitle(fig, "Figure 2. Policy functions on the cash-on-hand grid, baseline stationary equilibrium", x=0.01, ha="left", fontsize=9.5) | |
| 111 | 120 | fig.tight_layout() |
| 112 | 121 | _save(fig, "fig02_policy_functions") |
| 113 | 122 | |
@@ -142,7 +151,7 @@ def fig03_thresholds(): | ||
| 142 | 151 | axes[1].plot(d.theta, d.a_hat, marker=MARKS[k], color=GREYS[k], ls=STYLES[k], label=kind) |
| 143 | 152 | axes[1].set_xlabel("Minimum down payment $\\theta$"); axes[1].set_ylabel("$\\hat a$ at age 1, median $z$") |
| 144 | 153 | axes[1].set_title("(b) Threshold and the down-payment constraint", loc="left"); axes[1].legend() |
| 145 | − fig.suptitle("Figure 3. Ownership-entry thresholds (units: average 5-year income)", x=0.01, ha="left", fontsize=9.5) | |
| 154 | + _suptitle(fig, "Figure 3. Ownership-entry thresholds (units: average 5-year income)", x=0.01, ha="left", fontsize=9.5) | |
| 146 | 155 | fig.tight_layout() |
| 147 | 156 | _save(fig, "fig03_ownership_thresholds") |
| 148 | 157 | |
@@ -165,7 +174,7 @@ def fig04_profiles(): | ||
| 165 | 174 | axes[2].plot(d.age, d.liquid_mean, color=GREYS[2], ls="--", marker="s", label="liquid assets, all") |
| 166 | 175 | axes[2].plot(d.age, d.ltv_owner_mean, color=GREYS[1], ls=":", marker="^", label="mean LTV, owners") |
| 167 | 176 | axes[2].set_title("(c) Balance sheets", loc="left"); axes[2].legend(); _age_axis(axes[2]) |
| 168 | − fig.suptitle("Figure 4. Baseline lifecycle profiles (stationary general equilibrium)", x=0.01, ha="left", fontsize=9.5) | |
| 177 | + _suptitle(fig, "Figure 4. Baseline lifecycle profiles (stationary general equilibrium)", x=0.01, ha="left", fontsize=9.5) | |
| 169 | 178 | fig.tight_layout() |
| 170 | 179 | _save(fig, "fig04_lifecycle_profiles") |
| 171 | 180 | |
@@ -187,7 +196,7 @@ def fig05_equal_income(): | ||
| 187 | 196 | lab = "just below threshold (rents first)" if p == "threshold_minus" else "just above threshold (buys at age 1)" |
| 188 | 197 | axes[1].plot(dd.age, dd.nw, color=GREYS[k * 2], ls=STYLES[k], marker=MARKS[k], ms=3, label=lab) |
| 189 | 198 | axes[1].set_title("(b) Two households straddling the age-1 threshold", loc="left"); axes[1].legend(); _age_axis(axes[1]) |
| 190 | − fig.suptitle("Figure 5. Equal-income wealth paths with different initial transfers (experiment 2, baseline policies)", | |
| 199 | + _suptitle(fig, "Figure 5. Equal-income wealth paths with different initial transfers (experiment 2, baseline policies)", | |
| 191 | 200 | x=0.01, ha="left", fontsize=9.5) |
| 192 | 201 | fig.tight_layout() |
| 193 | 202 | _save(fig, "fig05_equal_income_paths") |
@@ -206,7 +215,7 @@ def fig06_gift_quintiles(): | ||
| 206 | 215 | axes[1].plot(dd.age, dd.nw_median, color=col, ls=STYLES[k % 4], marker=MARKS[k], ms=3) |
| 207 | 216 | axes[0].set_title("(a) Ownership rate by age", loc="left"); axes[0].set_ylim(0, 1); axes[0].legend(title="gift received", ncol=2); _age_axis(axes[0]) |
| 208 | 217 | axes[1].set_title("(b) Median net worth by age", loc="left"); _age_axis(axes[1]) |
| 209 | − fig.suptitle("Figure 6. Trajectories by quintile of the inter-vivos transfer received (baseline cross-section)", x=0.01, ha="left", fontsize=9.5) | |
| 218 | + _suptitle(fig, "Figure 6. Trajectories by quintile of the inter-vivos transfer received (baseline cross-section)", x=0.01, ha="left", fontsize=9.5) | |
| 210 | 219 | fig.tight_layout() |
| 211 | 220 | _save(fig, "fig06_inheritance_quantile_trajectories") |
| 212 | 221 | |
@@ -238,7 +247,7 @@ def fig07_inequality(): | ||
| 238 | 247 | axes[1].bar(x + w, rr, w, color=GREYS[3], edgecolor=GREYS[1], label="rank–rank slope of net worth (age 55–59)") |
| 239 | 248 | axes[1].set_xticks(x); axes[1].set_xticklabels(labs, rotation=20, ha="right"); axes[1].set_ylim(0, 1) |
| 240 | 249 | axes[1].set_title("(b) Intergenerational tenure and wealth persistence", loc="left"); axes[1].legend(fontsize=6.5) |
| 241 | − fig.suptitle("Figure 7. Inequality and mobility across stationary equilibria (GE unless stated)", x=0.01, ha="left", fontsize=9.5) | |
| 250 | + _suptitle(fig, "Figure 7. Inequality and mobility across stationary equilibria (GE unless stated)", x=0.01, ha="left", fontsize=9.5) | |
| 242 | 251 | fig.tight_layout() |
| 243 | 252 | _save(fig, "fig07_inequality_mobility") |
| 244 | 253 | |
@@ -271,7 +280,7 @@ def fig08_inflation_irf(): | ||
| 271 | 280 | axes[1, 0].set_title("(c) Ownership rate, pp from steady state", loc="left") |
| 272 | 281 | axes[1, 1].set_title("(d) Real mortgage debt, % from steady state", loc="left") |
| 273 | 282 | axes[0, 0].legend(fontsize=6.5); axes[1, 0].legend(fontsize=6.5) |
| 274 | − fig.suptitle("Figure 8. Perfect-foresight transition after a 10 % unanticipated price-level surprise (GE unless labelled)", | |
| 283 | + _suptitle(fig, "Figure 8. Perfect-foresight transition after a 10 % unanticipated price-level surprise (GE unless labelled)", | |
| 275 | 284 | x=0.01, ha="left", fontsize=9.5) |
| 276 | 285 | fig.tight_layout() |
| 277 | 286 | _save(fig, "fig08_inflation_transition") |
@@ -310,7 +319,7 @@ def fig09_redistribution(): | ||
| 310 | 319 | axes[2].bar(x + wdt, 100 * r.renters, wdt, color=GREYS[3], edgecolor=GREYS[1], label="renters") |
| 311 | 320 | axes[2].set_xticks(x); axes[2].set_xticklabels(r.contract, rotation=25, ha="right"); axes[2].axhline(0, color="k", lw=0.6) |
| 312 | 321 | axes[2].set_title("(c) CEV of households alive at the shock, %", loc="left"); axes[2].legend(fontsize=6.5) |
| 313 | − fig.suptitle("Figure 9. Redistribution in the shock period: nominal revaluation, consumption and welfare by tenure", x=0.01, ha="left", fontsize=9.5) | |
| 322 | + _suptitle(fig, "Figure 9. Redistribution in the shock period: nominal revaluation, consumption and welfare by tenure", x=0.01, ha="left", fontsize=9.5) | |
| 314 | 323 | fig.tight_layout() |
| 315 | 324 | _save(fig, "fig09_inflation_redistribution") |
| 316 | 325 | |
@@ -343,7 +352,7 @@ def fig10_supply_heatmap(): | ||
| 343 | 352 | ax.set_xticks(range(3)); ax.set_xticklabels([f"{t:.2f}" for t in th_v]); ax.set_yticks(range(3)); ax.set_yticklabels([f"{e:.2f}" for e in eps_v]) |
| 344 | 353 | ax.set_xlabel("down payment $\\theta$"); ax.set_title(titles[k], loc="left"); ax.grid(False) |
| 345 | 354 | axes[0].set_ylabel("supply elasticity $\\varepsilon$") |
| 346 | − fig.suptitle("Figure 10. Supply elasticity × down-payment constraint: stationary general equilibria", x=0.01, ha="left", fontsize=9.5) | |
| 355 | + _suptitle(fig, "Figure 10. Supply elasticity × down-payment constraint: stationary general equilibria", x=0.01, ha="left", fontsize=9.5) | |
| 347 | 356 | fig.tight_layout() |
| 348 | 357 | _save(fig, "fig10_supply_elasticity_heatmap") |
| 349 | 358 | |
@@ -363,7 +372,9 @@ def fig11_welfare(): | ||
| 363 | 372 | for k, s in enumerate(scen): |
| 364 | 373 | d = w[w.scenario == s] |
| 365 | 374 | byq = d.groupby("gift_quintile").apply(lambda g: np.nansum(g.cev * g.weight) / np.nansum(g.weight * np.isfinite(g.cev))) |
| 366 | − axes[0].plot(byq.index, 100 * byq.values, color=GREYS[k % 4] if k < 4 else "#000000", ls=STYLES[k % 4], marker=MARKS[k % 5], ms=3, label=s.replace("_", " ")) | |
| 375 | + lab = {"theta_005": r"$\theta = 0.05$", "theta_040": r"$\theta = 0.40$", "transfer_tax_020": "transfer tax 20 % (rebated)", | |
| 376 | + "eps_005_theta005": r"$\varepsilon = 0.05$, $\theta = 0.05$", "r_003": "real rate 3 %", "phi_008": r"landlord wedge $\phi = 0.08$"}.get(s, s.replace("_", " ")) | |
| 377 | + axes[0].plot(byq.index, 100 * byq.values, color=GREYS[k % 4] if k < 4 else "#000000", ls=STYLES[k % 4], marker=MARKS[k % 5], ms=3, label=lab) | |
| 367 | 378 | axes[0].axhline(0, color="k", lw=0.6); axes[0].set_xlabel("Quintile of transfer received at age 1"); axes[0].set_ylabel("CEV, % of consumption") |
| 368 | 379 | axes[0].set_title("(a) Entrants vs baseline, by transfer quintile", loc="left"); axes[0].legend(fontsize=6.5) |
| 369 | 380 | ft = DATA / "inflation" / "frm_level_welfare.csv" |
@@ -373,7 +384,7 @@ def fig11_welfare(): | ||
| 373 | 384 | dd = t[t.group == grp].copy(); dd["age"] = dd.age.astype(int) |
| 374 | 385 | axes[1].plot(dd.age, 100 * dd.cev, color=GREYS[k], ls=STYLES[k], marker=MARKS[k], ms=3, label=grp.replace("_", " ")) |
| 375 | 386 | axes[1].axhline(0, color="k", lw=0.6); axes[1].set_title("(b) 10 % surprise (FRM): CEV by age, GE", loc="left"); axes[1].legend(fontsize=6.5); _age_axis(axes[1]) |
| 376 | − fig.suptitle("Figure 11. Consumption-equivalent welfare (experiment 9)", x=0.01, ha="left", fontsize=9.5) | |
| 387 | + _suptitle(fig, "Figure 11. Consumption-equivalent welfare (experiment 9)", x=0.01, ha="left", fontsize=9.5) | |
| 377 | 388 | fig.tight_layout() |
| 378 | 389 | _save(fig, "fig11_welfare") |
| 379 | 390 | |
@@ -418,7 +429,7 @@ def fig12_diagnostics(): | ||
| 418 | 429 | axes[2].plot(x, dev, marker=MARKS[k % 5], ls=STYLES[k % 4], color=GREYS[k % 4] if k < 4 else "#000000", ms=3, label=key) |
| 419 | 430 | axes[2].set_xticks(x); axes[2].set_xticklabels(names, rotation=25, ha="right"); axes[2].axhline(0, color="k", lw=0.6) |
| 420 | 431 | axes[2].set_title("(c) Numerical variants, % dev. from baseline", loc="left"); axes[2].legend(fontsize=6) |
| 421 | − fig.suptitle("Figure 12. Solver diagnostics: market clearing, Bellman residuals, grid / sample sensitivity", x=0.01, ha="left", fontsize=9.5) | |
| 432 | + _suptitle(fig, "Figure 12. Solver diagnostics: market clearing, Bellman residuals, grid / sample sensitivity", x=0.01, ha="left", fontsize=9.5) | |
| 422 | 433 | fig.tight_layout() |
| 423 | 434 | _save(fig, "fig12_solver_diagnostics") |
| 424 | 435 | |
@@ -438,7 +449,7 @@ def fig13_multiplicity(): | ||
| 438 | 449 | axes[0].axhline(0, color="k", lw=0.6); axes[0].set_xlabel("House price $p$ (rental market cleared)"); axes[0].set_ylabel("rel. excess demand (owner mkt)") |
| 439 | 450 | axes[0].set_title("(a) Owner-market excess demand along $p$", loc="left"); axes[0].legend(fontsize=6.5) |
| 440 | 451 | axes[1].set_xlabel("House price $p$"); axes[1].set_ylabel("ownership rate"); axes[1].set_title("(b) Ownership along the clearing curve", loc="left") |
| 441 | − fig.suptitle("Figure 13. Search for multiple stationary equilibria (sign changes of excess demand)", x=0.01, ha="left", fontsize=9.5) | |
| 452 | + _suptitle(fig, "Figure 13. Search for multiple stationary equilibria (sign changes of excess demand)", x=0.01, ha="left", fontsize=9.5) | |
| 442 | 453 | fig.tight_layout() |
| 443 | 454 | _save(fig, "fig13_multiplicity_search") |
| 444 | 455 | |
@@ -471,7 +482,7 @@ def fig14_theta_sweep(): | ||
| 471 | 482 | axes[0].set_title("(a) Ownership rate", loc="left"); axes[0].legend(fontsize=6.5) |
| 472 | 483 | axes[1].set_title("(b) Equilibrium prices (GE)", loc="left"); axes[1].legend(fontsize=6.5) |
| 473 | 484 | axes[2].set_title("(c) Intergenerational persistence", loc="left"); axes[2].legend(fontsize=6.5) |
| 474 | − fig.suptitle("Figure 14. Down-payment sweep (experiment 3): general vs partial equilibrium", x=0.01, ha="left", fontsize=9.5) | |
| 485 | + _suptitle(fig, "Figure 14. Down-payment sweep (experiment 3): general vs partial equilibrium", x=0.01, ha="left", fontsize=9.5) | |
| 475 | 486 | fig.tight_layout() |
| 476 | 487 | _save(fig, "fig14_down_payment_sweep") |
| 477 | 488 | |
modified
src/tables.py
+291 −20
@@ -28,12 +28,21 @@ def _json(path): | ||
| 28 | 28 | return json.load(fh) |
| 29 | 29 | |
| 30 | 30 | |
| 31 | +def _minus(txt: str) -> str: | |
| 32 | + """Typeset a leading minus sign so that it is correct in text and in math mode; no negative zero.""" | |
| 33 | + if txt.startswith("-"): | |
| 34 | + if float(txt) == 0.0: | |
| 35 | + return txt[1:] | |
| 36 | + return "\\ensuremath{-}" + txt[1:] | |
| 37 | + return txt | |
| 38 | + | |
| 39 | + | |
| 31 | 40 | def _f(x, nd=3, pct=False, pp=False): |
| 32 | 41 | if x is None or (isinstance(x, float) and not np.isfinite(x)): |
| 33 | 42 | return "--" |
| 34 | 43 | if pct: |
| 35 | − return f"{100 * x:.{nd}f}" | |
| 36 | − return f"{x:.{nd}f}" | |
| 44 | + return _minus(f"{100 * x:.{nd}f}") | |
| 45 | + return _minus(f"{x:.{nd}f}") | |
| 37 | 46 | |
| 38 | 47 | |
| 39 | 48 | def _esc(s: str) -> str: |
@@ -41,12 +50,18 @@ def _esc(s: str) -> str: | ||
| 41 | 50 | .replace("#", "\\#")) |
| 42 | 51 | |
| 43 | 52 | |
| 44 | −def _write(name: str, body: str, caption: str, label: str, notes: str, colspec: str, header: str, size: str = "\\small"): | |
| 45 | − tex = ("\\begin{table}[htbp]\n\\centering\n" + size + "\n\\begin{threeparttable}\n" | |
| 46 | − f"\\caption{{{caption}}}\n\\label{{{label}}}\n" | |
| 47 | − f"\\begin{{tabular}}{{{colspec}}}\n\\toprule\n{header}\n\\midrule\n{body}\\bottomrule\n\\end{{tabular}}\n" | |
| 48 | − f"\\begin{{tablenotes}}[flushleft]\n\\footnotesize\n\\item {notes}\n\\end{{tablenotes}}\n" | |
| 49 | − "\\end{threeparttable}\n\\end{table}\n") | |
| 53 | +def _write(name: str, body: str, caption: str, label: str, notes: str, colspec: str, header: str, size: str = "\\small", wide: bool = False): | |
| 54 | + tab = f"\\begin{{tabular}}{{{colspec}}}\n\\toprule\n{header}\n\\midrule\n{body}\\bottomrule\n\\end{{tabular}}" | |
| 55 | + if wide: # tables wider than the text block are scaled to \textwidth (HARMONISATION §5); threeparttable | |
| 56 | + # cannot wrap a \resizebox, so the note is set in a minipage of the same width | |
| 57 | + tex = ("\\begin{table}[htbp]\n\\centering\n" + size + "\n" | |
| 58 | + f"\\caption{{{caption}}}\n\\label{{{label}}}\n\\resizebox{{\\textwidth}}{{!}}{{%\n" + tab + "}\n" | |
| 59 | + f"\\begin{{minipage}}{{\\textwidth}}\\footnotesize\\vspace{{2pt}}\n{notes}\n\\end{{minipage}}\n\\end{{table}}\n") | |
| 60 | + else: | |
| 61 | + tex = ("\\begin{table}[htbp]\n\\centering\n" + size + "\n\\begin{threeparttable}\n" | |
| 62 | + f"\\caption{{{caption}}}\n\\label{{{label}}}\n" + tab + "\n" | |
| 63 | + f"\\begin{{tablenotes}}[flushleft]\n\\footnotesize\n\\item {notes}\n\\end{{tablenotes}}\n" | |
| 64 | + "\\end{threeparttable}\n\\end{table}\n") | |
| 50 | 65 | with open(TAB / f"{name}.tex", "w") as fh: |
| 51 | 66 | fh.write(tex) |
| 52 | 67 | print("table:", name) |
@@ -89,7 +104,7 @@ def tab01_notation(): | ||
| 89 | 104 | body = "".join(f"{a} & {b} & {c} \\\\\n" for a, b, c in rows) |
| 90 | 105 | _write("tab01_notation", body, "Notation of the quantitative model", "tab:notation", |
| 91 | 106 | "Units: ``goods'' = units of the consumption good; the numeraire of the simulations is the average 5-year labour income of a working-age household (E$[z]=1$). Source: research/model\\_specification.md, Part B.", |
| 92 | − "lll", "Symbol & Meaning & Units \\\\") | |
| 107 | + "lp{9.3cm}l", "Symbol & Meaning & Units \\\\") | |
| 93 | 108 | |
| 94 | 109 | |
| 95 | 110 | def tab02_parameters(): |
@@ -102,7 +117,7 @@ def tab02_parameters(): | ||
| 102 | 117 | v = r["value"] |
| 103 | 118 | try: |
| 104 | 119 | vv = float(v) |
| 105 | − vs = f"{vv:.4g}" | |
| 120 | + vs = f"{int(round(vv)):,}".replace(",", "{,}") if (abs(vv) >= 1000 and float(vv).is_integer()) else f"{vv:.4g}" | |
| 106 | 121 | except (TypeError, ValueError): |
| 107 | 122 | vs = _esc(str(v)) |
| 108 | 123 | body += f"{_esc(str(r['parameter']))} & {vs} & {_esc(str(r['unit']))} & {_esc(str(r['source']))} \\\\\n" |
@@ -223,7 +238,7 @@ def tab05_scenarios(): | ||
| 223 | 238 | f"{_f(r['qro'], 2)} & {_f(r['qoo'], 2)} & {_f(r['rr'], 2)} & {_f(r['gini'], 2)} & {_f(r['gift'], 2)} \\\\\n") |
| 224 | 239 | _write("tab05_scenarios", body, "Stationary equilibria across credit, supply, segmentation and bequest scenarios", "tab:scenarios", |
| 225 | 240 | "GE = general equilibrium (prices, estate lottery and rebate re-solved; supply scales fixed at the baseline normalisation); PE = partial equilibrium with prices fixed at the baseline (household block and distribution re-solved). $^{\\dagger}$ = not converged to tolerance (residuals in results/data/diagnostics.csv). $\\rho/p$ = annual rent--price ratio; $\\hat a_1$ = age-1 ownership-entry threshold at the median productivity state (5-year income units); $q_{RO}$, $q_{OO}$ = probability that the child owns at 35--39 given the parent's tenure at the gift; rank--rank = slope of the child's net-worth rank on the parent's (both at 55--59); gift = mean transfer received at entry. Source files: results/data/group/name.json.", |
| 226 | − "lcccccccccccc", "Scenario & Eq. & $p^*$ & $\\rho/p$ & Own. & Own.\\ $<$35 & $\\hat a_1$ & $q_{RO}$ & $q_{OO}$ & rank--rank & Gini & gift \\\\", size="\\footnotesize") | |
| 241 | + "lcccccccccccc", "Scenario & Eq. & $p^*$ & $\\rho/p$ & Own. & Own.\\ $<$35 & $\\hat a_1$ & $q_{RO}$ & $q_{OO}$ & rank--rank & Gini & gift \\\\", size="\\footnotesize", wide=True) | |
| 227 | 242 | for r in rows: |
| 228 | 243 | key = re.sub(r"[^A-Za-z]", "", r["name"].split("(")[0].replace("\\", "").replace("$", "").replace("=", "").replace(".", ""))[:24] |
| 229 | 244 | macro(f"Own{key}", _f(r["own"], 3)); macro(f"Price{key}", _f(r["p"], 3)); macro(f"Thr{key}", _f(r["thr"], 2)) |
@@ -258,11 +273,11 @@ def tab06_inflation(): | ||
| 258 | 273 | body = "" |
| 259 | 274 | for r in rows: |
| 260 | 275 | eq = "GE" if r["eq"] == "GE" else ("A6" if "static" in r["eq"] else "GE$^{\\dagger}$") |
| 261 | − body += (f"{r['lab']} & {eq} & {_f(r['p0'], 2)} & {_f(r['pk'], 2)} & {_f(r['rho0'], 2)} & {_f(r['down'], 2)} & {_f(r['down_pe'], 2)} & " | |
| 276 | + body += (f"{r['lab']} & {eq} & {_f(r['p0'], 2)} & {_f(r['rho0'], 2)} & {_f(r['down'], 2)} & {_f(r['down_pe'], 2)} & " | |
| 262 | 277 | f"{_f(r['erosion'], 3)} & {_f(r['dep'], 3)} & {_f(r['cev_om'], 2, pct=True)} & {_f(r['cev_r'], 2, pct=True)} & {_f(r['cev_all'], 2, pct=True)} & {_f(r['cev_pe'], 2, pct=True)} \\\\\n") |
| 263 | 278 | _write("tab06_inflation", body, "Unanticipated 10\\% price-level surprise: impact effects by mortgage contract and persistence", "tab:inflation", |
| 264 | − "Perfect-foresight transitions (GE: prices clear every period; $^{\\dagger}$ = not converged to the $1.5\\times10^{-3}$ per-period tolerance (stalled at the Monte-Carlo noise floor; maximum residual in results/data/inflation/name.json); A6 = static expectations, stationary continuation values). Columns: real house price on impact and at its peak deviation (\\%), real rent on impact (\\%), change of the ownership rate on impact (pp) in GE and with prices fixed (PE), aggregate real debt erosion and real deposit loss per unit mass of one age (the lender's revaluation is their difference with the opposite sign), consumption-equivalent welfare of households alive at the shock (owners with a mortgage, renters, all; \\%), and the all-household CEV with prices fixed. Level surprise: CPI 10\\% above expectation in the shock period, inflation back to target afterwards; persistent: two periods. Source: results/data/inflation/*.json and *\\_welfare.csv.", | |
| 265 | − "lcccccccccccc", "Scenario & Eq. & $p_0$ & $p$ peak & $\\rho_0$ & $\\Delta$own & $\\Delta$own PE & erosion & dep.\\ loss & CEV o/m & CEV r & CEV all & CEV PE \\\\", size="\\scriptsize") | |
| 279 | + "Perfect-foresight transitions (GE: prices clear every period; $^{\\dagger}$ = not converged to the $1.5\\times10^{-3}$ per-period tolerance (stalled at the Monte-Carlo noise floor; maximum residual in results/data/inflation/name.json); A6 = static expectations, stationary continuation values). Columns: real house price on impact (\\%; the impact is also the peak deviation in every run), real rent on impact (\\%), change of the ownership rate on impact (pp) in GE and with prices fixed (PE), aggregate real debt erosion and real deposit loss per unit mass of one age (the lender's revaluation is their difference with the opposite sign), consumption-equivalent welfare of households alive at the shock (owners with a mortgage, renters, all; \\%), and the all-household CEV with prices fixed. Level surprise: CPI 10\\% above expectation in the shock period, inflation back to target afterwards; persistent: two periods. Source: results/data/inflation/*.json and *\\_welfare.csv.", | |
| 280 | + "lccccccccccc", "Scenario & Eq. & $p_0$ & $\\rho_0$ & $\\Delta$own & $\\Delta$own PE & erosion & dep.\\ loss & CEV o/m & CEV r & CEV all & CEV PE \\\\", size="\\scriptsize", wide=True) | |
| 266 | 281 | for r in rows: |
| 267 | 282 | key = re.sub(r"[^A-Za-z]", "", r["name"]) |
| 268 | 283 | macro(f"PImpact{key}", _f(r["p0"], 2)); macro(f"RhoImpact{key}", _f(r["rho0"], 2)); macro(f"DOwn{key}", _f(r["down"], 2)) |
@@ -294,7 +309,7 @@ def tab07_mobility(): | ||
| 294 | 309 | body = "".join(f"{r[0]} & {'GE' if r[1] else 'PE'}{'' if r[2] else '$^{\\dagger}$'} & {_f(r[3])} & {_f(r[4])} & {_f(r[5], 2)} & {_f(r[6], 2)} & {_f(r[7], 2)} & {_f(r[8], 3)} & {_f(r[9], 2)} & {_f(r[10], 2)} & {_f(r[11], 2)} & {_f(r[12], 2)} & {_f(r[13], 2)} \\\\\n" for r in rows) |
| 295 | 310 | _write("tab07_mobility_inequality", body, "Intergenerational persistence and inequality across stationary equilibria", "tab:mobility", |
| 296 | 311 | "$q_{RO}$, $q_{OO}$: probability that the child owns at 35--39 given the parent's tenure when the gift is made; $\\lambda_3 = q_{OO} - q_{RO}$; $\\lambda_7$: the same difference measured at 55--59 (the Part A persistence statistic); rank--rank: slope of the child's net-worth rank on the parent's at the same age (55--59); gap$_4$: mean net-worth difference between owners and renters at 40--44 (equal income process by design); gift, estate: mean transfers received. $^{\\dagger}$ = not converged to tolerance. Source: results/data/group/name.json.", |
| 297 | − "lcccccccccccc", "Scenario & Eq. & $p^*$ & Own. & $q_{RO}$ & $q_{OO}$ & $\\lambda_3$ & $\\lambda_7$ & rank--rank & Gini & gap$_4$ & gift & estate \\\\", size="\\scriptsize") | |
| 312 | + "lcccccccccccc", "Scenario & Eq. & $p^*$ & Own. & $q_{RO}$ & $q_{OO}$ & $\\lambda_3$ & $\\lambda_7$ & rank--rank & Gini & gap$_4$ & gift & estate \\\\", size="\\scriptsize", wide=True) | |
| 298 | 313 | |
| 299 | 314 | |
| 300 | 315 | def tab08_welfare(): |
@@ -317,13 +332,19 @@ def tab08_welfare(): | ||
| 317 | 332 | ok = np.isfinite(dq.cev) |
| 318 | 333 | allv = float(np.sum(dq.cev[ok] * dq.weight[ok]) / np.sum(dq.weight[ok])) |
| 319 | 334 | lowz = dq[dq.z_index == 0]; hiz = dq[dq.z_index == dq.z_index.max()] |
| 320 | − body += f"{_esc(s)} & " + " & ".join(_f(v, 2, pct=True) for v in byq) + f" & {_f(float(np.nanmean(lowz.cev)), 2, pct=True)} & {_f(float(np.nanmean(hiz.cev)), 2, pct=True)} & {_f(allv, 2, pct=True)} \\\\\n" | |
| 335 | + labels = {"theta_005": "$\\theta = 0.05$", "theta_010": "$\\theta = 0.10$", "theta_030": "$\\theta = 0.30$", "theta_040": "$\\theta = 0.40$", | |
| 336 | + "psi_030": "DSTI cap $\\psi = 0.30$", "psi_050": "DSTI cap $\\psi = 0.50$", "bequests_off": "Bequests off ($\\nu = 0$)$^{\\ddagger}$", | |
| 337 | + "transfer_tax_020": "Transfer tax 20\\% (rebated)", "transfer_tax_050": "Transfer tax 50\\% (rebated)", | |
| 338 | + "eps_005": "$\\varepsilon = 0.05$", "eps_200": "$\\varepsilon = 2$", "noseg": "No segmentation", "r_001": "Real rate 1\\%", "r_003": "Real rate 3\\%", | |
| 339 | + "pi_004": "Trend inflation 4\\%", "dep_premium_002": "Deposit return $+2$ pp", "chi_005": "Ownership premium $\\chi = 0.05$$^{\\ddagger}$", | |
| 340 | + "kappa_low": "Transaction costs low", "kappa_high": "Transaction costs high"} | |
| 341 | + body += f"{labels.get(s, _esc(s))} & " + " & ".join(_f(v, 2, pct=True) for v in byq) + f" & {_f(float(np.nanmean(lowz.cev)), 2, pct=True)} & {_f(float(np.nanmean(hiz.cev)), 2, pct=True)} & {_f(allv, 2, pct=True)} \\\\\n" | |
| 321 | 342 | macro(f"CEV{re.sub(r'[^A-Za-z]', '', s)}", _f(allv, 2, pct=True)) |
| 322 | 343 | if not body: |
| 323 | 344 | return |
| 324 | 345 | _write("tab08_welfare", body, "Welfare of entering households across stationary equilibria (consumption-equivalent variation, \\%)", "tab:welfare", |
| 325 | − "$\\lambda^{CE}$ such that scaling non-housing consumption by $1+\\lambda^{CE}$ in every period of the baseline makes an age-1 household with state $(z_1, a_1)$ as well off as in the alternative stationary equilibrium: $V^{alt}_1 = (1+\\lambda^{CE})^{(1-\\alpha)(1-\\gamma)} U^{base} + \\Phi^{base}$, where $U^{base}$ (discounted flow utility) and $\\Phi^{base}$ (warm glow) come from cohort simulations under baseline policies. Columns: quintiles of the transfer received at entry (averaged over $z_1$ with stationary weights), households within $\\pm10\\%$ of the age-1 ownership threshold (``marginal''), lowest and highest $z_1$, all. Comparisons are between stationary equilibria, not along transitions. Source: results/data/welfare/welfare\\_all.csv.", | |
| 326 | − "lccccccccc", "Scenario & Q1 & Q2 & Q3 & Q4 & Q5 & marginal & low $z_1$ & high $z_1$ & all \\\\", size="\\footnotesize") | |
| 346 | + "$\\lambda^{CE}$ such that scaling non-housing consumption by $1+\\lambda^{CE}$ in every period of the baseline makes an age-1 household with state $(z_1, a_1)$ as well off as in the alternative stationary equilibrium: $V^{alt}_1 = (1+\\lambda^{CE})^{(1-\\alpha)(1-\\gamma)} U^{base} + \\Phi^{base}$, where $U^{base}$ (discounted flow utility) and $\\Phi^{base}$ (warm glow) come from cohort simulations under baseline policies. Columns: quintiles of the transfer received at entry (averaged over $z_1$ with stationary weights), households within $\\pm10\\%$ of the age-1 ownership threshold (``marginal''), lowest and highest $z_1$, all. Comparisons are between stationary equilibria, not along transitions. $^{\\ddagger}$ = the scenario changes a preference parameter, so the number compares value functions under different utilities and is not a welfare statement. Source: results/data/welfare/welfare\\_all.csv.", | |
| 347 | + "lccccccccc", "Scenario & Q1 & Q2 & Q3 & Q4 & Q5 & marginal & low $z_1$ & high $z_1$ & all \\\\", size="\\footnotesize", wide=True) | |
| 327 | 348 | |
| 328 | 349 | |
| 329 | 350 | def tab09_robustness(): |
@@ -340,7 +361,253 @@ def tab09_robustness(): | ||
| 340 | 361 | f"{_f(r['threshold_age1_zmid'], 2)} & {_f(r['gini_nw'], 3)} & {_f(r['mean_gift'], 3)} & {_f(r['q_RO_age3'], 3)} & {_f(r['q_OO_age3'], 3)} & {_f(r['rank_rank_nw'], 3)} & {_f(r['time_sec'] / 60, 1)} \\\\\n") |
| 341 | 362 | _write("tab09_robustness", body, "Robustness: preferences, transaction costs, grids, tolerance, sample size and seed", "tab:robustness", |
| 342 | 363 | "Each row is a full re-solution of the stationary general equilibrium (prices, lottery). beta\\_094/098: annual discount factor 0.94/0.98; gamma\\_3: $\\gamma=3$; alpha\\_025: housing share 0.25; kappa\\_low/high: transaction costs (1\\%, 2\\%) / (5\\%, 8\\%); chi\\_005: ownership utility premium 5\\%; grid\\_coarse/fine, tol\\_tight, sample\\_small/large, seed\\_alt as in Table~\\ref{tab:convergence}. Source: results/data/robustness\\_summary.csv.", |
| 343 | − "lccccccccccc", "Variant & Eq. & $p^*$ & Own. & Own.\\ $<$35 & $\\hat a_1$ & Gini & gift & $q_{RO}$ & $q_{OO}$ & rank--rank & min \\\\", size="\\scriptsize") | |
| 364 | + "lccccccccccc", "Variant & Eq. & $p^*$ & Own. & Own.\\ $<$35 & $\\hat a_1$ & Gini & gift & $q_{RO}$ & $q_{OO}$ & rank--rank & min \\\\", size="\\scriptsize", wide=True) | |
| 365 | + | |
| 366 | + | |
| 367 | +# ====================================================================================== | |
| 368 | +# Extra macros used in the manuscript text (Phase 4): every value is read from results/data/. | |
| 369 | +# ====================================================================================== | |
| 370 | +def _sign_changes(x): | |
| 371 | + x = np.asarray(x, dtype=float) | |
| 372 | + x = x[np.abs(x) > 1e-12] | |
| 373 | + return int(np.sum(np.sign(x[1:]) != np.sign(x[:-1]))) | |
| 374 | + | |
| 375 | + | |
| 376 | +def _welfare_group(name, grp, col="cev"): | |
| 377 | + f = DATA / "inflation" / f"{name}_welfare.csv" | |
| 378 | + if not f.exists(): | |
| 379 | + return np.nan | |
| 380 | + w = pd.read_csv(f) | |
| 381 | + x = w[(w.group == grp) & (w.age.astype(str) == "all")] | |
| 382 | + return float(x[col].iloc[0]) if len(x) else np.nan | |
| 383 | + | |
| 384 | + | |
| 385 | +def _impact_group(name, grp, age, col): | |
| 386 | + f = DATA / "inflation" / f"{name}_impact.csv" | |
| 387 | + if not f.exists(): | |
| 388 | + return np.nan | |
| 389 | + w = pd.read_csv(f) | |
| 390 | + x = w[(w.group == grp) & (w.age.astype(str) == str(age))] | |
| 391 | + return float(x[col].iloc[0]) if len(x) else np.nan | |
| 392 | + | |
| 393 | + | |
| 394 | +def extra_macros(): | |
| 395 | + s = _load("baseline/summary.json") | |
| 396 | + if s is None: | |
| 397 | + return | |
| 398 | + m, mob, gaps = s["moments"], s["mobility"], s["gaps"] | |
| 399 | + by_age = m["ownership_rate_by_age"] | |
| 400 | + macro("OwnAgeOne", _f(by_age[0], 2)); macro("OwnPeak", _f(max(by_age), 2)); macro("OwnAgeTwelve", _f(by_age[-1], 2)) | |
| 401 | + macro("OwnThirtyFiveSixtyFour", _f(m["ownership_rate_35_64"], 2)); macro("OwnSixtyFivePlus", _f(m["ownership_rate_65plus"], 2)) | |
| 402 | + macro("ShareGiftPositivePct", _f(m["share_gift_positive"], 0, pct=True)); macro("ShareFirstBuyersGiftPct", _f(m["share_first_buyers_with_gift"], 0, pct=True)) | |
| 403 | + macro("MeanEstate", _f(m["mean_estate_received"], 2)); macro("ShareEstatePositivePct", _f(m["share_estate_positive"], 0, pct=True)) | |
| 404 | + macro("GiniAgeSeven", _f(m["gini_net_worth_age7"], 2)); macro("MedianGift", _f(m["median_gift_received"], 2)) | |
| 405 | + macro("ShareOwnersMortgage", _f(m["share_owners_with_mortgage"], 2)); macro("MeanLTVNew", _f(m["mean_ltv_new_loans"], 2)) | |
| 406 | + macro("HouseValueIncome", _f(m["median_house_value_to_annual_income"], 1)) | |
| 407 | + for k, lab in [("age2", "Two"), ("age4", "Four"), ("age7", "Seven"), ("age10", "Ten")]: | |
| 408 | + macro(f"GapAge{lab}", _f(gaps[f"nw_gap_owner_renter_{k}"], 2)) | |
| 409 | + macro("qOOageSeven", _f(mob["q_OO"], 2)); macro("qROageSeven", _f(mob["q_RO"], 2)); macro("HalfLifeGen", _f(mob["half_life_generations"], 2)) | |
| 410 | + macro("ShareParentOwnerPct", _f(mob["share_parent_owner"], 0, pct=True)) | |
| 411 | + th = s["thresholds"] | |
| 412 | + macro("ThresholdAgeOneZtwo", _f(th["1"]["1"], 2)); macro("ThresholdAgeOneZfour", _f(th["1"]["3"], 2)) | |
| 413 | + macro("ThresholdAgeThreeMidZ", _f(th["3"]["2"], 2)); macro("ThresholdAgeThreeHighZ", _f(th["3"]["4"], 2)); macro("ThresholdAgeTwoHighZ", _f(th["2"]["4"], 2)) | |
| 414 | + macro("DebtDepositRatio", _f(s["total_deposits"] / s["total_debt"], 1)) | |
| 415 | + macro("SupplyScaleOwner", _f(s["H_bar"], 2)); macro("SupplyScaleRental", _f(s["Hr_bar"], 2)); macro("RhoStar", _f(s["rho"], 3)) | |
| 416 | + macro("BellmanOwnerPnn", _f(s["bellman"]["owner_p99"], 1, pct=True)); macro("BellmanOwnerMax", _f(s["bellman"]["owner_max"], 1, pct=True)) | |
| 417 | + macro("BellmanRenterMax", _f(s["bellman"]["renter_max"], 2, pct=True)); macro("ClipSharePct", f"{100 * s['clip_share']:.3f}") | |
| 418 | + macro("BaselineSeconds", _f(s["time_sec"], 0)); macro("LotteryRounds", str(s["supply_scales"].get("calibration_iterations", "--"))) | |
| 419 | + macro("HouseholdPeriods", f"{s['params']['n_dyn'] * 2 * 6:,}".replace(",", "{,}")) | |
| 420 | + # gift quintiles | |
| 421 | + f = DATA / "baseline" / "by_gift_quintile.csv" | |
| 422 | + if f.exists(): | |
| 423 | + q = pd.read_csv(f) | |
| 424 | + names = ["One", "Two", "Three", "Four", "Five"] | |
| 425 | + for i in range(1, 6): | |
| 426 | + r1 = q[(q.age == 1) & (q.gift_quintile == i)].iloc[0]; r7 = q[(q.age == 7) & (q.gift_quintile == i)].iloc[0] | |
| 427 | + macro(f"OwnAgeOneQ{names[i-1]}", _f(r1["ownership"], 2)); macro(f"OwnAgeSevenQ{names[i-1]}", _f(r7["ownership"], 2)) | |
| 428 | + macro(f"MedNWAgeSevenQ{names[i-1]}", _f(r7["nw_median"], 2)) | |
| 429 | + g1 = q[(q.age == 1) & (q.gift_quintile == 5)].iloc[0]["ownership"] - q[(q.age == 1) & (q.gift_quintile == 2)].iloc[0]["ownership"] | |
| 430 | + g7 = q[(q.age == 7) & (q.gift_quintile == 5)].iloc[0]["ownership"] - q[(q.age == 7) & (q.gift_quintile == 2)].iloc[0]["ownership"] | |
| 431 | + macro("GapQtwoQfiveAgeOnePP", _f(100 * g1, 0)); macro("GapQtwoQfiveAgeSevenPP", _f(100 * g7, 0)) | |
| 432 | + # equal-income paths | |
| 433 | + f = DATA / "baseline" / "equal_income_paths.csv" | |
| 434 | + if f.exists(): | |
| 435 | + e = pd.read_csv(f) | |
| 436 | + def first_buy(path): | |
| 437 | + d = e[e.path == path]; o = d[d.owner == 1] | |
| 438 | + return int(o.age.min()) if len(o) else 0 | |
| 439 | + macro("EqIncFirstBuyAgeLowGift", str(first_buy("q10"))); macro("EqIncFirstBuyAgeMinus", str(first_buy("threshold_minus"))); macro("EqIncFirstBuyAgePlus", str(first_buy("threshold_plus"))) | |
| 440 | + macro("EqIncAoneMinus", _f(e[e.path == "threshold_minus"].a1.iloc[0], 3)); macro("EqIncAonePlus", _f(e[e.path == "threshold_plus"].a1.iloc[0], 3)) | |
| 441 | + macro("EqIncAoneMedian", _f(e[e.path == "q50"].a1.iloc[0], 2)) | |
| 442 | + for age, lab in [(4, "Four"), (7, "Seven")]: | |
| 443 | + nm = float(e[(e.path == "threshold_minus") & (e.age == age)].nw.iloc[0]); npl = float(e[(e.path == "threshold_plus") & (e.age == age)].nw.iloc[0]) | |
| 444 | + macro(f"EqIncNWMinusAge{lab}", _f(nm, 2)); macro(f"EqIncNWPlusAge{lab}", _f(npl, 2)) | |
| 445 | + macro(f"EqIncGapPctAge{lab}", _f(100 * (npl / nm - 1), 0)) | |
| 446 | + macro("EqIncEndowDiffPct", _f(100 * (e[e.path == "threshold_plus"].a1.iloc[0] / e[e.path == "threshold_minus"].a1.iloc[0] - 1), 0)) | |
| 447 | + # price-drop accounting | |
| 448 | + f = DATA / "baseline" / "price_drop_accounting.csv" | |
| 449 | + if f.exists(): | |
| 450 | + d = pd.read_csv(f) | |
| 451 | + a = d[d.age == "all"]; n = d[d.age == "new_loans_at_origination"] | |
| 452 | + macro("MeanLTVOwners", _f(float(a.mean_ltv.iloc[0]), 2)) | |
| 453 | + for drop, lab in [(0.1, "Ten"), (0.2, "Twenty"), (0.3, "Thirty")]: | |
| 454 | + ra = a[np.isclose(a["drop"], drop)].iloc[0]; rn = n[np.isclose(n["drop"], drop)].iloc[0] | |
| 455 | + macro(f"UnderwaterAll{lab}", _f(ra.underwater_share, 0, pct=True)); macro(f"UnderwaterNew{lab}", _f(rn.underwater_share, 0, pct=True)) | |
| 456 | + macro(f"LossNWAll{lab}", _f(ra.mean_loss_over_nw, 0, pct=True)); macro(f"LossNWNew{lab}", _f(rn.mean_loss_over_nw, 0, pct=True)) | |
| 457 | + # stationary scenarios | |
| 458 | + def sc(path): | |
| 459 | + return _load(path) | |
| 460 | + def put(prefix, d): | |
| 461 | + if d is None: | |
| 462 | + return | |
| 463 | + mm, mb = d["moments"], d["mobility"] | |
| 464 | + macro(f"{prefix}P", _f(d["p"], 3)); macro(f"{prefix}Own", _f(mm["ownership_rate_all"], 3)); macro(f"{prefix}OwnUthirtyfive", _f(mm["ownership_rate_under35"], 3)) | |
| 465 | + macro(f"{prefix}Thr", _f(d["thresholds"]["1"]["2"], 2)); macro(f"{prefix}ThrLowZ", _f(d["thresholds"]["1"]["0"], 2)); macro(f"{prefix}ThrHighZ", _f(d["thresholds"]["1"]["4"], 2)) | |
| 466 | + macro(f"{prefix}qRO", _f(mb["q_RO_age3"], 3)); macro(f"{prefix}qOO", _f(mb["q_OO_age3"], 3)); macro(f"{prefix}LamThree", _f(mb["q_OO_age3"] - mb["q_RO_age3"], 2)) | |
| 467 | + macro(f"{prefix}LamSeven", _f(mb["lambda_persistence"], 3)); macro(f"{prefix}RR", _f(mb["rank_rank_slope_nw"], 2)); macro(f"{prefix}Gini", _f(d["gini_nw"], 2)) | |
| 468 | + macro(f"{prefix}Gift", _f(mm["mean_gift_received"], 2)); macro(f"{prefix}Estate", _f(mm["mean_estate_received"], 2)); macro(f"{prefix}GapFour", _f(d["gaps"]["nw_gap_owner_renter_age4"], 2)) | |
| 469 | + macro(f"{prefix}Rho", _f(d["rho"], 3)); macro(f"{prefix}RhoDevPct", _f(100 * (d["rho"] / s["rho"] - 1), 1)); macro(f"{prefix}PDevPct", _f(100 * (d["p"] - 1), 1)) | |
| 470 | + if "rebate" in d: | |
| 471 | + macro(f"{prefix}Rebate", _f(d["rebate"], 3)) | |
| 472 | + for prefix, path in [("ThetaFive", "theta_sweep/theta_005.json"), ("ThetaTen", "theta_sweep/theta_010.json"), ("ThetaThirty", "theta_sweep/theta_030.json"), | |
| 473 | + ("ThetaForty", "theta_sweep/theta_040.json"), ("ThetaFivePE", "theta_sweep/theta_005_pe.json"), ("ThetaFortyPE", "theta_sweep/theta_040_pe.json"), | |
| 474 | + ("PsiThirty", "theta_sweep/psi_030.json"), ("PsiFifty", "theta_sweep/psi_050.json"), | |
| 475 | + ("BeqOff", "bequests/bequests_off.json"), ("BeqOffPE", "bequests/bequests_off_pe.json"), ("BeqHalf", "bequests/bequests_half.json"), | |
| 476 | + ("TaxTwenty", "bequests/transfer_tax_020.json"), ("TaxFifty", "bequests/transfer_tax_050.json"), | |
| 477 | + ("EpsLowThetaFive", "supply/eps_005_theta005.json"), ("EpsHighThetaFive", "supply/eps_200_theta005.json"), | |
| 478 | + ("EpsLowThetaForty", "supply/eps_005_theta040.json"), ("EpsHighThetaForty", "supply/eps_200_theta040.json"), | |
| 479 | + ("Rone", "sensitivity/r_001.json"), ("Rthree", "sensitivity/r_003.json"), ("PiZero", "sensitivity/pi_000.json"), ("PiFour", "sensitivity/pi_004.json"), | |
| 480 | + ("PiFourPE", "sensitivity/pi_004_pe.json"), ("PhiLow", "sensitivity/phi_002.json"), ("PhiHigh", "sensitivity/phi_008.json"), | |
| 481 | + ("SigLow", "sensitivity/sigma_low.json"), ("SigHigh", "sensitivity/sigma_high.json"), ("RhoG", "sensitivity/rho_g_040.json"), | |
| 482 | + ("DepPrem", "sensitivity/dep_premium_002.json"), ("BetaLow", "robustness/beta_094.json"), ("BetaHigh", "robustness/beta_098.json"), | |
| 483 | + ("GammaThree", "robustness/gamma_3.json"), ("AlphaHigh", "robustness/alpha_025.json"), ("KappaLow", "robustness/kappa_low.json"), | |
| 484 | + ("KappaHigh", "robustness/kappa_high.json"), ("ChiFive", "robustness/chi_005.json"), ("GridCoarse", "robustness/grid_coarse.json"), | |
| 485 | + ("GridFine", "robustness/grid_fine.json")]: | |
| 486 | + put(prefix, sc(path)) | |
| 487 | + # PE absorption shares | |
| 488 | + b, t5, t5pe, t40, t40pe = s, sc("theta_sweep/theta_005.json"), sc("theta_sweep/theta_005_pe.json"), sc("theta_sweep/theta_040.json"), sc("theta_sweep/theta_040_pe.json") | |
| 489 | + if all(x is not None for x in (t5, t5pe, t40, t40pe)): | |
| 490 | + o = b["moments"]["ownership_rate_all"] | |
| 491 | + macro("AbsorbThetaFivePct", _f(1 - (t5["moments"]["ownership_rate_all"] - o) / (t5pe["moments"]["ownership_rate_all"] - o), 0, pct=True)) | |
| 492 | + macro("AbsorbThetaFortyPct", _f(1 - (t40["moments"]["ownership_rate_all"] - o) / (t40pe["moments"]["ownership_rate_all"] - o), 0, pct=True)) | |
| 493 | + bo, bope = sc("bequests/bequests_off.json"), sc("bequests/bequests_off_pe.json") | |
| 494 | + if bo is not None and bope is not None: | |
| 495 | + o = b["moments"]["ownership_rate_all"] | |
| 496 | + macro("BeqOffPEDropPP", _f(100 * (o - bope["moments"]["ownership_rate_all"]), 0)); macro("BeqOffGEDropPP", _f(100 * (o - bo["moments"]["ownership_rate_all"]), 0)) | |
| 497 | + p4, p4pe = sc("sensitivity/pi_004.json"), sc("sensitivity/pi_004_pe.json") | |
| 498 | + if p4 is not None and p4pe is not None: | |
| 499 | + u = b["moments"]["ownership_rate_under35"] | |
| 500 | + macro("PiFourPEDropUthirtyfivePP", _f(100 * (u - p4pe["moments"]["ownership_rate_under35"]), 1)) | |
| 501 | + macro("PiFourGEOffsetPct", _f(1 - (u - p4["moments"]["ownership_rate_under35"]) / (u - p4pe["moments"]["ownership_rate_under35"]), 0, pct=True)) | |
| 502 | + # multiplicity scans | |
| 503 | + mult = _load("multiplicity/multiplicity_summary.json") | |
| 504 | + if mult is not None: | |
| 505 | + for k, lab in [("baseline", "Base"), ("eps_005", "EpsLow"), ("eps_005_theta005", "EpsLowThetaFive")]: | |
| 506 | + if k in mult: | |
| 507 | + ex, ow, pg = np.array(mult[k]["excess_owner"]), np.array(mult[k]["ownership"]), np.array(mult[k]["p_grid"]) | |
| 508 | + macro(f"Mult{lab}ExcessLow", _f(ex[0], 2)); macro(f"Mult{lab}ExcessHigh", _f(ex[-1], 2)) | |
| 509 | + macro(f"Mult{lab}OwnLow", _f(ow[0], 2)); macro(f"Mult{lab}OwnHigh", _f(ow[-1], 2)) | |
| 510 | + i = int(np.argmin(np.abs(ex))); macro(f"Mult{lab}Root", _f(pg[i], 3)) | |
| 511 | + j = np.where(np.isclose(pg, 1.0))[0][0] | |
| 512 | + slope = (ex[j + 1] - ex[j - 1]) / (np.log(pg[j + 1]) - np.log(pg[j - 1])); macro(f"Mult{lab}Slope", _f(slope, 1)) | |
| 513 | + macro("MultPgridLow", _f(min(mult["baseline"]["p_grid"]), 2)); macro("MultPgridHigh", _f(max(mult["baseline"]["p_grid"]), 2)) | |
| 514 | + # grid sensitivity | |
| 515 | + f = DATA / "grid_sensitivity.csv" | |
| 516 | + if f.exists(): | |
| 517 | + g = pd.read_csv(f).set_index("name") | |
| 518 | + for n, lab in [("grid_coarse", "Coarse"), ("grid_fine", "Fine"), ("sample_large", "SampleLarge"), ("seed_alt", "Seed"), ("sample_small", "SampleSmall")]: | |
| 519 | + if n in g.index: | |
| 520 | + macro(f"Dev{lab}OwnPP", _f(100 * g.loc[n, "ownership_dev"], 1)); macro(f"Dev{lab}PPct", _f(100 * g.loc[n, "p_dev"], 1)) | |
| 521 | + macro(f"Dev{lab}Thr", _minus(f"{g.loc[n, 'threshold_age1_zmid_dev']:+.2f}")); macro(f"Dev{lab}LamSeven", _minus(f"{g.loc[n, 'lambda_age7_dev']:+.3f}")) | |
| 522 | + macro(f"Dev{lab}RR", _minus(f"{g.loc[n, 'rank_rank_nw_dev']:+.3f}")); macro(f"Dev{lab}Gini", _minus(f"{g.loc[n, 'gini_nw_dev']:+.3f}")) | |
| 523 | + # transitions | |
| 524 | + diag = _load("diagnostics.json") | |
| 525 | + if diag is not None: | |
| 526 | + runs = diag["runs"] | |
| 527 | + ss = [r for r in runs if r.get("equilibrium") == "GE" and r.get("group") != "inflation"] | |
| 528 | + tr = [r for r in runs if r.get("group") == "inflation"] | |
| 529 | + macro("NScenariosSS", str(len(ss))); macro("NScenariosSSConverged", str(sum(1 for r in ss if r.get("converged")))) | |
| 530 | + macro("NTransitions", str(len(tr))); macro("NTransitionsConverged", str(sum(1 for r in tr if r.get("converged")))) | |
| 531 | + macro("SimulateMinutesOne", _f(diag["total_sec"] / 60, 1)) | |
| 532 | + for n, lab in [("frm_level", "FrmLevel"), ("term5_level", "TermLevel"), ("arm_level", "ArmLevel"), ("frm_persistent", "FrmPers"), ("term5_persistent", "TermPers"), | |
| 533 | + ("arm_persistent", "ArmPers"), ("frm_level_rigid_rents", "RigidRents"), ("frm_level_unindexed_wages", "Unindexed"), ("frm_persistent_taylor", "FrmTaylor"), | |
| 534 | + ("term5_persistent_taylor", "TermTaylor"), ("frm_level_fixed_supply", "FixedSupply"), ("frm_level_noseg", "NoSeg"), ("frm_level_static", "Static"), | |
| 535 | + ("rate_shock_theta020", "RateTwenty"), ("rate_shock_theta040", "RateForty"), ("rate_shock_theta005", "RateFive")]: | |
| 536 | + d = _load(f"inflation/{n}.json") | |
| 537 | + if d is None: | |
| 538 | + continue | |
| 539 | + macro(f"{lab}MaxRes", f"{d['max_residual']:.1e}".replace("e-0", r"\times10^{-").replace("e-", r"\times10^{-") + "}") | |
| 540 | + macro(f"{lab}ResT", f"{d['residual_at_T']:.0e}".replace("e-0", r"\times10^{-").replace("e-", r"\times10^{-") + "}") | |
| 541 | + macro(f"{lab}Steps", str(d["iterations"])); macro(f"{lab}Secs", _f(d["time_sec"], 0)) | |
| 542 | + macro(f"{lab}Pzero", _f(d["p_impact_pct"], 2)); macro(f"{lab}RhoZero", _f(d["rho_impact_pct"], 2)) | |
| 543 | + macro(f"{lab}DOwnGE", _f(d["d_ownership_impact_pp"], 2)); macro(f"{lab}DOwnPE", _f(d["d_ownership_impact_pe_pp"], 2)) | |
| 544 | + if abs(d["d_ownership_impact_pe_pp"]) > 1e-9: | |
| 545 | + macro(f"{lab}AbsorbPct", _f(1 - d["d_ownership_impact_pp"] / d["d_ownership_impact_pe_pp"], 0, pct=True)) | |
| 546 | + macro(f"{lab}Erosion", _f(d["aggregate_debt_erosion"], 2)); macro(f"{lab}DepLoss", _f(d["aggregate_deposit_loss"], 2)); macro(f"{lab}Lender", _f(d["lender_revaluation"], 2)) | |
| 547 | + mp, ms = d["mobility_path"], d["mobility_ss"] | |
| 548 | + macro(f"{lab}qROsevenSS", _f(ms["q_RO"], 3)); macro(f"{lab}qROsevenZero", _f(mp[0]["q_RO"], 3)); macro(f"{lab}LamSevenSS", _f(ms["lambda_persistence"], 3)); macro(f"{lab}LamSevenZero", _f(mp[0]["lambda_persistence"], 3)) | |
| 549 | + macro(f"{lab}qROthreeSS", _f(ms["q_RO_age3"], 3)); macro(f"{lab}qROthreeZero", _f(mp[0]["q_RO_age3"], 3)) | |
| 550 | + if len(mp) > 1: | |
| 551 | + macro(f"{lab}qROthreeOne", _f(mp[1]["q_RO_age3"], 3)) | |
| 552 | + macro(f"{lab}ThrSS", _f(d["threshold_ss"]["1"]["2"], 2)); macro(f"{lab}ThrZero", _f(d["threshold_impact"]["1"]["2"], 2)) | |
| 553 | + macro(f"{lab}ThrChangePct", _f(100 * (d["threshold_impact"]["1"]["2"] / d["threshold_ss"]["1"]["2"] - 1), 0)) | |
| 554 | + for grp, gl in [("all", "All"), ("owners_with_mortgage", "OM"), ("owners_no_mortgage", "ONM"), ("renters", "R"), ("owners", "O")]: | |
| 555 | + macro(f"{lab}CEV{gl}", _f(_welfare_group(n, grp), 2, pct=True)); macro(f"{lab}Better{gl}Pct", _f(_welfare_group(n, grp, "share_better_off"), 0, pct=True)) | |
| 556 | + macro(f"{lab}CEVAllPE", _f(d["pe"]["welfare_all"], 2, pct=True)) | |
| 557 | + pth = DATA / "inflation" / f"{n}_path.csv" | |
| 558 | + if pth.exists(): | |
| 559 | + pa = pd.read_csv(pth) | |
| 560 | + macro(f"{lab}PriceSignChanges", str(_sign_changes(pa.p_pct.values[:-1]))); macro(f"{lab}PriceMaxPos", _f(float(pa.p_pct.max()), 1)) | |
| 561 | + yrs = pa.years.values; back = yrs[np.where(np.abs(pa.p_pct.values) < 0.1)[0][0]] if (np.abs(pa.p_pct.values) < 0.1).any() else np.nan | |
| 562 | + macro(f"{lab}PriceBackYears", _f(float(back), 0)) | |
| 563 | + d0 = 100 * (pa.debt_total.iloc[0] / pa.debt_total.iloc[-1] - 1); macro(f"{lab}DebtZeroPct", _f(d0, 1)) | |
| 564 | + macro(f"{lab}NominalRateZero", _f(float(pa.i_new.iloc[0]), 3)) | |
| 565 | + for grp, age, col, gl in [("all", "all", "d_consumption_pct", "ConsAll"), ("renters", "all", "d_consumption_pct", "ConsR"), ("owners", "all", "d_consumption_pct", "ConsO"), | |
| 566 | + ("renters", "all", "net_nominal_gain", "NetR"), ("owners", "all", "net_nominal_gain", "NetO"), ("all", "all", "wage_loss", "WageAll"), ("owners", "all", "wage_loss", "WageO"), | |
| 567 | + ("renters", "all", "rent_rigidity_gain", "RentGainR"), ("all", "all", "d_nw", "NWAll"), ("owners", "all", "d_nw", "NWO"), ("renters", "all", "d_nw", "NWR")]: | |
| 568 | + v = _impact_group(n, grp, age, col) | |
| 569 | + if np.isfinite(v): | |
| 570 | + macro(f"{lab}{gl}", _f(v, 2) if "Cons" not in gl else _f(v, 1)) | |
| 571 | + # owners by age bands (impact csv, group=owners) | |
| 572 | + f = DATA / "inflation" / f"{n}_impact.csv" | |
| 573 | + if f.exists(): | |
| 574 | + w = pd.read_csv(f); w = w[w.age.astype(str) != "all"]; w["age"] = w.age.astype(int) | |
| 575 | + o = w[w.group == "owners"] | |
| 576 | + young = o[o.age.isin([2, 3, 4])]; old = o[o.age.isin([8, 9, 10])] | |
| 577 | + if len(young) and len(old): | |
| 578 | + macro(f"{lab}OwnerYoungGainMax", _f(float(young.net_nominal_gain.max()), 2)); macro(f"{lab}OwnerYoungConsMin", _f(float(young.d_consumption_pct.min()), 1)); macro(f"{lab}OwnerYoungConsMax", _f(float(young.d_consumption_pct.max()), 1)) | |
| 579 | + macro(f"{lab}OwnerOldLossMin", _f(float(-old.net_nominal_gain.max()), 2)); macro(f"{lab}OwnerOldLossMax", _f(float(-old.net_nominal_gain.min()), 2)) | |
| 580 | + macro(f"{lab}OwnerOldConsMin", _f(float(old.d_consumption_pct.max()), 1)); macro(f"{lab}OwnerOldConsMax", _f(float(old.d_consumption_pct.min()), 1)) | |
| 581 | + # welfare quintiles (stationary) | |
| 582 | + f = DATA / "welfare" / "welfare_all.csv" | |
| 583 | + if f.exists(): | |
| 584 | + w = pd.read_csv(f); w["gift_quintile"] = w.gift_quintile.astype(str) | |
| 585 | + for sname, lab in [("theta_005", "ThetaFive"), ("theta_040", "ThetaForty"), ("transfer_tax_020", "TaxTwenty"), ("transfer_tax_050", "TaxFifty"), ("eps_005", "EpsLow"), ("eps_200", "EpsHigh"), | |
| 586 | + ("r_001", "Rone"), ("r_003", "Rthree"), ("pi_000", "PiZero"), ("pi_004", "PiFour"), ("phi_002", "PhiLow"), ("phi_008", "PhiHigh"), ("sigma_low", "SigLow"), ("sigma_high", "SigHigh"), | |
| 587 | + ("rho_g_040", "RhoG"), ("dep_premium_002", "DepPrem"), ("kappa_low", "KappaLow"), ("kappa_high", "KappaHigh"), ("bequests_off", "BeqOff"), ("psi_030", "PsiThirty"), ("psi_050", "PsiFifty"), | |
| 588 | + ("eps_005_theta005", "EpsLowThetaFive"), ("eps_200_theta005", "EpsHighThetaFive"), ("tol_tight", "TolTight"), ("sample_large", "SampleLarge"), ("seed_alt", "Seed")]: | |
| 589 | + d = w[w.scenario == sname] | |
| 590 | + if not len(d): | |
| 591 | + continue | |
| 592 | + def wavg(dd): | |
| 593 | + ok = np.isfinite(dd.cev) | |
| 594 | + return float(np.sum(dd.cev[ok] * dd.weight[ok]) / np.sum(dd.weight[ok])) if ok.any() else np.nan | |
| 595 | + dq = d[d.gift_quintile != "marginal"] | |
| 596 | + macro(f"CEV{lab}All", _f(wavg(dq), 2, pct=True)); macro(f"CEV{lab}Qone", _f(wavg(d[d.gift_quintile == "1"]), 2, pct=True)); macro(f"CEV{lab}Qtwo", _f(wavg(d[d.gift_quintile == "2"]), 2, pct=True)) | |
| 597 | + macro(f"CEV{lab}Qfive", _f(wavg(d[d.gift_quintile == "5"]), 2, pct=True)); macro(f"CEV{lab}Marg", _f(wavg(d[d.gift_quintile == "marginal"]), 2, pct=True)) | |
| 598 | + macro(f"CEV{lab}BottomTwo", _f(wavg(d[d.gift_quintile.isin(["1", "2"])]), 2, pct=True)) | |
| 599 | + if {"V_base", "V_base_simulated"}.issubset(w.columns): | |
| 600 | + rel = np.abs(w.V_base_simulated / w.V_base - 1.0) | |
| 601 | + macro("WelfareCheckMaxPct", _f(float(np.nanmax(rel)), 1, pct=True)) | |
| 602 | + num = w[w.scenario.isin(["tol_tight", "sample_large", "sample_small", "seed_alt"])] | |
| 603 | + if len(num): | |
| 604 | + macro("CEVNumericalMaxPct", _f(float(np.nanmax(np.abs(num.cev))), 2, pct=True)) | |
| 605 | + # robustness summary: Bellman residual range | |
| 606 | + f = DATA / "robustness_summary.csv" | |
| 607 | + if f.exists(): | |
| 608 | + r = pd.read_csv(f) | |
| 609 | + macro("BellmanOwnerMinAll", f"{r.bellman_owner_mean.min():.1e}".replace("e-0", r"\times10^{-") + "}"); macro("BellmanOwnerMaxAll", f"{r.bellman_owner_mean.max():.1e}".replace("e-0", r"\times10^{-") + "}") | |
| 610 | + macro("NRobustnessRuns", str(len(r))) | |
| 344 | 611 | |
| 345 | 612 | |
| 346 | 613 | def write_macros(): |
@@ -352,6 +619,10 @@ def write_macros(): | ||
| 352 | 619 | if mult is not None: |
| 353 | 620 | for k, v in mult.items(): |
| 354 | 621 | macro(f"SignChanges{re.sub(r'[^A-Za-z]', '', k)}", str(v["n_sign_changes"])) |
| 622 | + # absolute-value twins of every negative macro, for sentences of the form "falls by \\XAbs\\%" | |
| 623 | + for k, v in list(MACROS.items()): | |
| 624 | + if v.startswith("\\ensuremath{-}") and k + "Abs" not in MACROS: | |
| 625 | + MACROS[k + "Abs"] = v[len("\\ensuremath{-}"):] | |
| 355 | 626 | with open(TAB / "numbers.tex", "w") as fh: |
| 356 | 627 | fh.write("% generated by src/tables.py — one macro per key number; every value traces to results/data/\n") |
| 357 | 628 | for k, v in sorted(MACROS.items()): |
@@ -361,7 +632,7 @@ def write_macros(): | ||
| 361 | 632 | |
| 362 | 633 | def main(): |
| 363 | 634 | for fn in (tab01_notation, tab02_parameters, tab03_moments, tab04_convergence, tab05_scenarios, tab06_inflation, tab07_mobility, |
| 364 | − tab08_welfare, tab09_robustness): | |
| 635 | + tab08_welfare, tab09_robustness, extra_macros): | |
| 365 | 636 | try: |
| 366 | 637 | fn() |
| 367 | 638 | except Exception as exc: # noqa: BLE001 |
| 368 | 639 | |