# 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.

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# 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.