WP18 — Propositions: statements, status, intuition, and what the simulations found
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.
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).
| # | Result | Status | Key assumptions | Sim (Part B) |
|---|---|---|---|---|
| L1 | Stage-3 split: $c_3 = W_3/(1+\nu)$, $b' = \nu W_3/(1+\nu)$ | P | log warm glow | — |
| 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) |
| 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 $z$ |
| 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 $z$ |
| 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 |
| 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 % |
| 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 |
| 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) |
| 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 $\varepsilon\delta_H$ |
| 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) |
| 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 |
| 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) |
| 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 |
| — | Multiple stationary equilibria (Stein-type ladder multiplicity) | C — not established; possible only if (C) fails | — | none found (three scans, incl. near-fixed supply) |
| — | Full perfect-foresight transition after a surprise | C — numerical only | — | 12/16 transitions converged; A6 vs perfect foresight immaterial for size and sign |
| — | "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 |
| — | "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 % |
| — | 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 |
| — | Supply elasticity has a first-order effect on mobility (contribution iv) | — | — | N in this calibration: $q_{RO}$ 0.374 vs 0.376 across $\varepsilon$ |
Proposition-by-proposition notes
Proposition 1 — Ownership-entry threshold
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. 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. Benchmark. $\hat b = 0.999$ at $p = 1.203$, $\rho = 0.348$; closed form = brute-force optimum to $10^{-14}$. 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.
Proposition 2 — Comparative statics (scoped in Phase 4)
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. 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.
Proposition 3 — Tilt vs neutrality
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.
Proposition 4 — Leverage with costs
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.
Proposition 5 — Persistence
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.
Proposition 6 and Corollary 1 — Stationary equilibrium
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.
Proposition 7 — No segmentation (scoped in Phase 4)
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. 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 %.
Proposition 8 — Revaluation accounting
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).
Proposition 9 and Corollary 2 — GE impact (scoped and extended in Phase 4)
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 $n^R = 0$. 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.
Numbers that the numerical phase must not take from here
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.