WP18 — Failed, weakened or abandoned hypotheses (Phase 2)
Recorded 2026-10-20. Each entry gives the hypothesis as it appeared in the brief or in the Phase 1 novelty matrix, what the analysis found, and what replaced it.
1. "Persistence through multiple equilibria / path dependence" — not established; knife-edge
Hypothesis (contribution ii). Inherited wealth and the down-payment constraint generate path-dependent, possibly multiple, long-run ownership distributions among equal-income dynasties. Finding. With no income risk, the benchmark has absorbing tenure classes and a continuum of history-dependent stationary owner shares (Remark after Proposition 5). This disappears for any income distribution with positive density between the two cutoffs: the dynasty chain is then ergodic with a unique stationary distribution given prices. Multiplicity of equilibrium prices can only come from the ladder channel (bequeathed equity rising with the price) dominating the threshold channel, i.e. failure of condition (C); in the benchmark (C) holds by a factor of 50. Replacement. Persistence is stated as the intergenerational tenure correlation $\lambda = G(\hat y_R) - G(\hat y_R - e_3p)$ and its half-life (Proposition 5). Multiplicity is kept as an open conjecture to be searched numerically under near-fixed supply.
2. "Surprise inflation raises real house prices" — not a theorem
Finding. Without segmentation the price does not move at all (Proposition 7). With segmentation the impact sign is positive only through a specific channel — owner-parents with eroded debt whose children are at the margin ($\mu^* g(\hat y_O) > 0$) — and the magnitude is small in the benchmark (0.03 % for a 10 % surprise). Nothing in the model links the surprise to real prices directly; a monetary tightening ($\phi_r > 0$) lowers demand through $\Delta$ and could reverse the sign (not analysed formally). Replacement. Proposition 9 states conditions and signs; the quantitative claim is left to Phase 3.
3. "Inflation enriches owners" — false in general
Finding. Four classes of owners lose: ARM borrowers when the real rate rises by more than $R$ per unit of surprise (Proposition 8.1); owners with imperfectly indexed wages when $y_2(1-\eta_w)$ exceeds the eroded payments; owners holding nominal deposits larger than their mortgage; and new buyers in inelastic markets, whose transfer rises by less than the price-induced outlay (Proposition 9.4). Stage-3 sellers always gain. Replacement. Condition (8.4) and the list of losers.
4. "Leverage raises the return hurdle / lowers it" — leverage is neutral for the break-even
Finding. With borrowing at the deposit rate, the break-even real price growth $g^*$ at which owning beats renting-and-investing is independent of the LTV (Proposition 4.2); leverage only scales gains and losses. The brief's "leverage amplification with costs" is therefore a statement about the multiplier $(1-\kappa_s)/d$ and the underwater threshold, not about the mean return.
5. "Ownership threshold is a feasibility (down-payment) threshold" — only partly
Finding. The down payment gives the feasibility bound $dp - y_1$; the actual threshold is higher by the marginal buyer's stage-1 consumption $\Lambda(dp-\rho)/(1-\Lambda)$, a preference/liquidity component that depends on rent, costs, the amortisation schedule, expected inflation and the bequest motive. When owning costs less cash than renting ($dp \le \rho$) the preference component vanishes.
6. Single crossing of the tenure decision for savers — proof gap
Finding. Proposition 1 is proved on the hand-to-mouth region. For households that save, the envelope derivative argument shows $V^O - V^R$ increases where the would-be owner consumes less than the would-be renter, but we could not show this at every crossing. Numerically the crossing is unique on the tested grid. Left open (Remark in the appendix); Phase 3 must report the ownership policy as a function of $a_1$ and flag non-monotonicity.
7. Static expectations in the impact proposition — strong assumption
Finding. Proposition 9 needs the stage-1 cohort to expect pre-shock future prices; with forward-looking prices the threshold depends on $p_{t+3}$ and the dominance condition changes to (C). The myopic simulation also shows a cohort-crowding oscillation (units held two periods) that a durable-stock, forward-looking model may not have. Labelled P-strong; the transition is a numerical object.
8. Comparative statics that are ambiguous in theory (not failures, but not signed)
- $\partial\lambda/\partial\theta$: a higher down payment raises $e_3$ (more persistence) and $\hat b$ (moves both cutoffs); sign depends on $g$.
- $\partial\mu^*/\partial\theta$ in GE with fixed stock: zero by construction; with elastic supply negative under (C).
- Welfare of renter-dynasty children priced out on impact: they lose the option to own at the old price but rent at a lower rent; unsigned.
- $\partial\hat b/\partial p$ with an intensive housing margin (Part B): not covered by the unit-demand proof.
9. Two-period version — abandoned in favour of three stages
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.
Phase 4 additions (2026-10-20): hypotheses tested against the quantitative model
10. "Surprise inflation raises the real house price in general equilibrium" (Proposition 9) — reversed in the simulation
Hypothesis. Under segmentation, the erosion of the parents' mortgages raises transfers, the demand of owner-dynasty children and the price (Prop. 9, $dp \ge 0$). 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. 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. 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).
11. "Price neutrality without segmentation" (Proposition 7) — fails with an intensive margin
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. 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 %.
12. "The entry threshold is increasing in the down payment" (Proposition 2) — not monotone once households can buy later
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. 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).
13. "Multiple stationary equilibria under near-fixed supply" — none found
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.
14. "Supply elasticity has a first-order effect on intergenerational mobility" (contribution iv) — not measurable
$q_{RO}$ 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.
15. Two calibration moments not supported by the comparison points (not hypotheses, but recorded)
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.