WP18 — Results audit (Phase 4, final version)
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:
- identity — accounting relation that holds by construction (unit-tested in
tests/); - theory — comparative static or existence result proved in
paper/appendix_proofs.texfor the benchmark (Part A ofresearch/model_specification.md) and checked here in the quantitative model (Part B); - 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); - simulation (PE, prices fixed) — household block and distribution re-solved at the baseline prices; never labelled GE;
- static expectations (A6) — temporary equilibria with stationary continuation values (Proposition 9's assumption);
- conjecture — stated as such in the paper; neither proved nor established numerically.
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.
0. What was computed, and how well it converged
| Object | Implementation | Diagnostic reported |
|---|---|---|
| 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) |
| 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) |
| 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 |
| 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 |
| 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) |
| 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 |
| 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 |
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.
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.
Experiments — hypotheses, parameters, algorithm
| # | Experiment | Hypothesis tested | Parameters / scenarios | Output files |
|---|---|---|---|---|
| 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 |
| 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 |
| 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 |
| 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 |
| 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_* |
| 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 |
| 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 |
| 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 |
| 9 | Welfare (CEV) | marginal households (unsigned in theory) | all GE scenarios vs baseline; transitions by age/tenure | welfare/*.csv, inflation/*_welfare.csv |
| 10 | Robustness | numerical stability | $\beta$, $\gamma$, $\alpha$, $\kappa$, $\chi$, grids, tolerance, sample size, seed | robustness/*.json, robustness_summary.csv, grid_sensitivity.csv |
Results by experiment (labels in bold)
1. Baseline stationary equilibrium (results/data/baseline/) — simulation (GE)
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.
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).
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.
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).
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.
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.
2. Equal-income paths (baseline/equal_income_paths.csv, Figure 5) — simulation (baseline policies)
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).
3. Down-payment sweep (theta_sweep/, Figures 3 and 14, Table 5) — simulation (GE and PE); theory check of Prop. 2
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.
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.
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
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).
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.
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.
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.
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).
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.
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).
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).
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).
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.
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).
5a. Negative house-price shocks (baseline/price_drop_accounting.csv) — identity (PE, prices fixed after the drop)
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).
5b. Real-rate shock +2 pp for 20 years (inflation/rate_shock_theta*) — simulation (GE)
$\theta = 0.20$ 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.
6. Bequests on/off, transfer tax (bequests/, Table 7, Figure 7) — simulation (GE and PE); theory check of Prop. 5
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).
7. Supply elasticity, segmentation (supply/, Figure 10) — simulation (GE)
$\varepsilon$ 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.
7b. Multiplicity (multiplicity/, Figure 13) — simulation (GE scan); conjecture closed for this calibration
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.
8. Rates, trend inflation, wedge, risk, transmission, outside asset (sensitivity/, Table 7) — simulation (GE; PE where stated)
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.
9. Welfare (welfare/welfare_all.csv, Table 8, Figure 11) — simulation (GE)
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.
10. Robustness (robustness/, grid_sensitivity.csv, Table 9) — simulation (GE)
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.
Classification of the three theory–simulation disagreements (Phase 4 decision)
| # | Disagreement | What the theory actually assumed | What the quantitative model does | Resolution in the paper |
|---|---|---|---|---|
| (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. |
| (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. |
| (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. |
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).
Synthesis — what the paper claims, with status
| Claim | Status | Where |
|---|---|---|
| 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 |
| Threshold ↑ in $\theta$ (Prop. 2) | theory under binding LTV + once-and-for-all decision; contradicted within noise outside (option to buy later) | §3 |
| 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 |
| 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 |
| Uniqueness / no ladder multiplicity | simulation (scan): monotone excess demand, one root incl. near-fixed supply; multiplicity remains a conjecture in general | §7b |
| Neutrality without segmentation (Prop. 7) | theory under unit demand; not supported with an intensive margin | §4 |
| Revaluation accounting, zero-sum, contract ordering (Prop. 8) | identity + simulation, incl. the term-vs-FRM distinction under a real-rate response | §4 |
| 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 |
| Tilt of anticipated inflation (Prop. 3) | theory; supported in PE, two-thirds offset in GE | §8, §4 |
| Welfare of marginal households (unsigned in theory) | simulation: $+0.3$ % for $\theta = 0.05$, $-0.3$ % for $\theta = 0.40$ | §9 |
| Supply elasticity × contract → mobility (contribution iv) | simulation: effect not measurable at these demand shifts | §7 |
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.
Traceability check (Phase 4)
src/tables.pygeneratesnumbers.tex(≈ 2 000 macros, all read fromresults/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 inpaper/is defined innumbers.tex.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).make all= test → simulate (skipped whenresults/data/diagnostics.jsonexists) → figures → tables → paper.