spb/wp18_uqo

1# WP18 — Propositions: statements, status, intuition, and what the simulations found23Phase 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.45Status 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`).67| # | 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 $\varepsilon$ |2829## Proposition-by-proposition notes3031### Proposition 1 — Ownership-entry threshold32*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.3637### 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.4041### Proposition 3 — Tilt vs neutrality42Unchanged. 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.4344### Proposition 4 — Leverage with costs45Unchanged. 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.4647### Proposition 5 — Persistence48Unchanged. 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.4950### Proposition 6 and Corollary 1 — Stationary equilibrium51Unchanged. 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.5253### 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 %.5657### Proposition 8 — Revaluation accounting58Unchanged. 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).5960### 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.6364## Numbers that the numerical phase must *not* take from here65All 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`.66