spb/wp18_uqo

1% ============================================================================2\section{Equilibrium and propositions}3\label{sec:equilibrium}4% ============================================================================56This section states the analytical results of the benchmark economy. Proofs are in Appendix~\ref{app:proofs}. Each proposition is followed by the assumption that the quantitative economy relaxes and, where the simulations of Sections~\ref{sec:baseline}--\ref{sec:supply} contradict the benchmark, by a remark that identifies the responsible assumption. Three results are stated more narrowly than in a first draft of this paper precisely because the simulations contradicted their broader versions: the monotonicity of the entry threshold in the down payment (Proposition~\ref{prop:cs}), price neutrality without segmentation (Proposition~\ref{prop:noseg}) and the sign of the house-price response to a surprise (Proposition~\ref{prop:impact} and Corollary~\ref{cor:creditor}). Simulation findings are never called theorems.78\subsection{Assumptions}9\label{subsec:assumptions}1011\begin{assumption}[Preferences, demography, housing need]12\label{ass:pref}13Households have the preferences \eqref{eq:utilityA}, live three stages, need one unit of housing services at stages~1 and~2 (unit demand), and have the income process of Section~\ref{subsec:benchmark}: $y_1,y_2>0$ common and deterministic, $y_3\sim G$ with continuous density $g>0$ on $[\underline y_3,\bar y_3]$, $0<\underline y_3<\bar y_3<\infty$, independent across households and generations.14\end{assumption}1516\begin{assumption}[Prices, contracts, lender]17\label{ass:prices}18The real rate $r\ge 0$ is exogenous and $1+i_t = R(1+\pi^e_{t+1})$. A mortgage of real principal $B\le(1-\theta)p_t$ is the nominal contract of Section~\ref{subsec:benchmark} with $a\in(0,1]$; deposits pay $i_t$ and cannot be negative; the lender is competitive and risk neutral and its realised real profit accrues outside the dynasties; the rental stock is segmented, with $S(p) = \bar Hp^{\varepsilon}$, $\varepsilon\in[0,\infty)$, and $S_r(\rho) = \bar H_r\rho^{\varepsilon_r}$, $\varepsilon_r\in(0,\infty]$, per cohort.19\end{assumption}2021\begin{assumption}[Feasibility and no default]22\label{ass:feas}23For all prices considered, $c^O_2>0$, $c^R_2>0$ and $\underline y_3 + \min\{e_3p,0\}>0$.24\end{assumption}2526\begin{assumption}[Liquidity-constrained at the margin: hand-to-mouth]27\label{ass:htm}28On the wealth interval $\mathcal B = (dp-y_1,\bar b]$ relevant for the tenure decision, the strict Euler inequalities29\begin{equation}30\frac{1}{c^o_1(b)} > \beta R\,\frac{1}{c^o_2}, \qquad31\frac{1}{c^o_2} > \beta R(1+\nu)\,\mathbb E\Big[\frac{1}{W^o_3(y_3)}\Big], \qquad o\in\{O,R\},\ b\in\mathcal B,32\label{eq:euler}33\end{equation}34hold for both tenures at the corner allocation $n_1 = n_2 = 0$ (and $B = (1-\theta)p$ for owners).35\end{assumption}3637\begin{assumption}[Interior tenure margin]38\label{ass:interior}39$\Delta>0$ and $dp>\rho$, where40\begin{equation}41\Delta \equiv \chi + \beta\big[\ln c^O_2 - \ln c^R_2\big] + \beta^2(1+\nu)\,\mathbb E\big[\ln W^O_3(y_3) - \ln W^R_3(y_3)\big], \qquad \Lambda\equiv e^{-\Delta}\in(0,1).42\label{eq:Delta}43\end{equation}44\end{assumption}4546\begin{assumption}[Static expectations on impact]47\label{ass:static}48In the period of an unanticipated surprise, stage-1 households evaluate their stage-2 rent and stage-3 resale price at the pre-shock stationary values $(\rho^*,p^*)$ while transacting at the current prices.49\end{assumption}5051\begin{assumption}[Nominal positions at the shock date]52\label{ass:nominal}53At the date of a surprise the only nominal claims held by households are mortgage balances: stage-3 parents and stage-1 entrants hold no deposits.54\end{assumption}5556Assumption~\ref{ass:htm} is verified, not imposed, in the numerical benchmark: the Euler slacks are computed on the support of the stationary transfer distribution and a general optimiser returns the corner (\texttt{tests/test\_toy\_model.py}). Assumption~\ref{ass:htm} implies Assumption~\ref{ass:nominal} on the hand-to-mouth region, but the two are stated separately because the sign of the inflation result depends on the second and not on the first (Corollary~\ref{cor:creditor}). $\Delta$ collects everything that happens after the purchase, the stage-2 cash-flow comparison and the stage-3 equity; the tenure decision compares the stage-1 cash sacrifice $dp-\rho$ with $\Delta$.5758\subsection{The ownership-entry threshold}59\label{subsec:threshold}6061\begin{proposition}[Ownership threshold]62\label{prop:threshold}63Under Assumptions~\ref{ass:pref}--\ref{ass:interior} there is a unique $\hat b\in\mathcal B$ such that a household with transfer $b\in\mathcal B$ owns if and only if $b\ge\hat b$, where64\begin{equation}65\hat b = \frac{d\,p - \Lambda\,\rho}{1-\Lambda} - y_1, \qquad66\hat b - (dp-y_1) = \frac{\Lambda(dp-\rho)}{1-\Lambda} > 0, \qquad67c^O_1(\hat b) = \frac{\Lambda(dp-\rho)}{1-\Lambda}.68\label{eq:bhat}69\end{equation}70If instead $dp\le\rho$, every household with $b>dp-y_1$ owns and the threshold is the feasibility bound $dp-y_1$.71\end{proposition}7273The threshold is the sum of a feasibility component, the down payment and costs net of stage-1 income, and a preference--liquidity component equal to the stage-1 consumption of the marginal buyer. The latter depends on the rent, the costs, the amortisation schedule, expected inflation and the bequest motive through $\Lambda$. The proposition is proved on the hand-to-mouth region; for households rich enough to save, the envelope derivative of each value is still $1/c^o_1$, so $V^O-V^R$ increases wherever the would-be owner consumes less than the would-be renter, but single crossing at every wealth level is not proved (Remark~\ref{rem:gap_single} in Appendix~\ref{app:proofs}). The quantitative economy finds exactly one sign change of the buy-minus-rent value along cash for every age and productivity state (Section~\ref{sec:baseline}).7475\begin{proposition}[Comparative statics of the threshold]76\label{prop:cs}77Under Assumptions~\ref{ass:pref}--\ref{ass:interior}, with the rent held fixed, the resale price moving with $p$ (stationary comparative statics), and the loan-to-value constraint binding,78\begin{gather*}79\frac{\partial\hat b}{\partial p}>0,\qquad80\frac{\partial\hat b}{\partial\rho}<0,\qquad81\frac{\partial\hat b}{\partial\theta}>0,\qquad82\frac{\partial\hat b}{\partial\kappa_b}>0,\qquad83\frac{\partial\hat b}{\partial\kappa_s}>0,\\84\frac{\partial\hat b}{\partial m}>0,\qquad85\frac{\partial\hat b}{\partial\chi}<0,\qquad86\frac{\partial\hat b}{\partial y_1}=-1,\qquad87\frac{\partial\hat b}{\partial\bar\pi}>0 .88\end{gather*}89The signs of $\partial\hat b/\partial\theta$, $\partial\hat b/\partial\kappa_s$, $\partial\hat b/\partial m$ and $\partial\hat b/\partial\bar\pi$ use the fact that, in the benchmark, a renter never buys later, so that $V^R$ does not depend on these parameters.90\end{proposition}9192All signs are liquidity effects. A higher down payment or a more front-loaded real payment schedule is present-value neutral under \eqref{eq:fisher} but moves resources from the constrained young to the unconstrained old; $r\ge 0$ guarantees that the price effect on the outlay dominates the price effect on future equity.9394\begin{remark}[Scope of Proposition~\ref{prop:cs}: the option to buy later]95\label{rem:cs_scope}96The sign of $\partial\hat b/\partial\theta$ is proved under the benchmark's structure in which the tenure decision is made once: a renter at stage~1 rents for life, so the value of renting is independent of the mortgage parameters. In the quantitative economy a renter can buy at any later age and the renter's continuation value therefore rises when $\theta$ falls, through the option of buying later with a smaller down payment. The entry threshold at a given age is then the crossing of two functions that both move with $\theta$, and its monotonicity is not implied by the proposition. Section~\ref{sec:baseline} finds the threshold at the median productivity state to be non-monotone in $\theta$ at fixed prices, within the grid sensitivity of that statistic, and finds it unchanged at the lowest state because the payment-to-income cap, not the loan-to-value cap, binds there; it rises with $\theta$ from $0.20$ to $0.40$ as the proposition predicts. The proposition should therefore be read as a statement about the once-and-for-all tenure decision of a liquidity-constrained household whose loan-to-value constraint binds.97\end{remark}9899\begin{proposition}[Anticipated inflation: tilt versus neutrality]100\label{prop:tilt}101(i) Under Assumptions~\ref{ass:pref}--\ref{ass:interior}, an increase in $\bar\pi$ raises the real stage-2 payment and lowers the real stage-3 payment of a nominal mortgage, leaves their present value unchanged, and raises $\hat b$. (ii) For a household whose optimal plan has $n_1>0$ and $n_2>0$, a small change in $\bar\pi$ or in $\theta$ leaves its value, consumption and transfer unchanged.102\end{proposition}103104\subsection{Leverage with transaction costs}105\label{subsec:leverage}106107\begin{proposition}[Leverage multiplier, break-even growth, underwater threshold]108\label{prop:leverage}109Let an owner buy at $p$, hold a mortgage $(1-\theta)p$ and resell at $(1+g)p$ at stage~3. Under Assumption~\ref{ass:prices}:110\begin{enumerate}111\item End-of-life wealth responds to the resale price with $(1-\kappa_s)$ per unit of price, i.e.\ $(1-\kappa_s)/d$ per unit of cash invested at purchase; this multiplier is decreasing in $\theta$, $\kappa_b$, $\kappa_s$ and $m$.112\item Compared with a renter who follows the same consumption path and invests the cash-flow differences at $R$, the owner has higher stage-3 wealth if and only if $g\ge g^*$, with113\begin{equation}1141+g^* = \frac{R^2(1+\kappa_b+m) + mR - (\rho/p)(R+R^2)}{1-\kappa_s},115\label{eq:gstar}116\end{equation}117increasing in $\kappa_b$, $\kappa_s$ and $m$, decreasing in $\rho/p$, and independent of $\theta$.118\item A price decline of $x$ leaves the owner with no housing equity if and only if $x\ge x^*\equiv 1-\delta_3/(1-\kappa_s)$; a renter's wealth is unaffected.119\end{enumerate}120\end{proposition}121122Leverage scales gains and losses but, with borrowing at the deposit rate, does not change the break-even: the brief's ``leverage amplification'' is a statement about the multiplier and the underwater threshold, not about the mean return. In the stationary state the owner--renter end-of-life wealth gap among equal-income hand-to-mouth households is exactly $e_3p$ and the transfer gap is $\nu e_3p/(1+\nu)$ (Proposition~\ref{prop:persist}).123124\subsection{Dynasties and persistence}125\label{subsec:persistence}126127\begin{proposition}[Tenure transitions, persistence and the equal-income wealth gap]128\label{prop:persist}129Under Assumptions~\ref{ass:pref}--\ref{ass:interior}, with stationary prices and the stationary transfer support contained in $\mathcal B$, let $\hat y_R\equiv(1+\nu)\hat b/\nu$ and $\hat y_O\equiv\hat y_R - e_3p$. Then:130\begin{enumerate}131\item The child of a renter owns iff $y_3\ge\hat y_R$ and the child of an owner owns iff $y_3\ge\hat y_O$; the tenure of a dynasty is a Markov chain with $q_{RO} = 1-G(\hat y_R)$ and $q_{OO} = 1-G(\hat y_O)$.132\item The second eigenvalue of the chain,133\begin{equation}134\lambda = q_{OO}-q_{RO} = G(\hat y_R) - G(\hat y_R - e_3p),135\label{eq:lambda}136\end{equation}137equals the parent--child correlation of the ownership indicator and the per-generation persistence of any difference in ownership rates between groups of dynasties; $\lambda>0$ iff $e_3>0$ and $g>0$ somewhere on $(\hat y_O,\hat y_R)$; for fixed $\hat b$, $\lambda$ is non-decreasing in $e_3p$.138\item For $\lambda<1$ the chain has the unique stationary owner share $\mu^* = q_{RO}/(1-\lambda)$ and converges to it geometrically at rate $\lambda$.139\item Among dynasties with identical income processes, the expected transfer received by an owner's child exceeds that of a renter's child by $\nu e_3p/(1+\nu)$, and the end-of-life wealth gap between an owner and a renter with the same $y_3$ is $e_3p$.140\end{enumerate}141\end{proposition}142143Persistence is the probability mass of income draws for which parental housing equity is pivotal for the child's tenure. It is zero if the house is not net wealth at the end of life, and it is a half-life, $\ln 2/(-\ln\lambda)$, not a trap: without income risk the two tenure classes would be absorbing and a continuum of history-dependent steady states would exist, but any $G$ with positive density between the cutoffs makes the chain ergodic (Remark~\ref{rem:norisk} in Appendix~\ref{app:proofs}). The paper therefore does not claim path dependence or multiplicity as a generic result.144145\subsection{Stationary equilibrium}146\label{subsec:se}147148\begin{definition}[Stationary equilibrium of the benchmark]149\label{def:se}150A stationary equilibrium is $(p^*,\rho^*,\mu^*)$ with $\hat b^* = \hat b(p^*,\rho^*)$ such that $\mu^* = q_{RO}/(1-\lambda)$ evaluated at $(p^*,\rho^*)$, $S(p^*) = \mu^*$ and $S_r(\rho^*) = 1-\mu^*$ per cohort, the lender earns zero expected real profit, and the distribution of transfers is the stationary distribution of the dynasty chain.151\end{definition}152153Write $f(\mu)\equiv\mu^*\big(S^{-1}(\mu),S_r^{-1}(1-\mu)\big)$ for the owner share generated by the dynasty chain when both markets clear at a cohort owner share $\mu$, with $\mu^* = 0$ when $\Delta\le 0$ and $\mu^* = 1$ when $\hat b<0$.154155\begin{proposition}[Existence; uniqueness under a dominance condition]156\label{prop:exist}157Let Assumptions~\ref{ass:pref}--\ref{ass:feas} hold for all $(p,\rho)$ with $p\in[0,S^{-1}(1)]$, $\rho\in[0,S_r^{-1}(1)]$, with $\varepsilon>0$ and $\varepsilon_r<\infty$. Then a stationary equilibrium exists. If, in addition, Assumptions~\ref{ass:htm}--\ref{ass:interior} hold along the clearing curve and158\begin{equation}159\text{(C)}\qquad \frac{1+\nu}{\nu}\,\frac{\partial\hat b}{\partial p}\;\ge\;e_3\qquad\text{for all }\mu\in(0,1),160\label{eq:C}161\end{equation}162then $f$ is non-increasing, strictly decreasing wherever $g(\hat y_R)>0$, and the stationary equilibrium is unique.163\end{proposition}164165Condition (C) says that the entry-threshold channel, a higher price raises $\hat b$ and excludes renter-dynasty children, dominates the ladder channel, a higher price raises the equity $e_3p$ that owner-parents bequeath and includes owner-dynasty children. If (C) fails, multiple stationary equilibria of the \citet{stein1995prices} type are possible; they are not established here, and Section~\ref{sec:supply} searches for them numerically and finds none.166167\begin{corollary}[Fixed owner-occupied stock]168\label{cor:fixed}169If $\varepsilon = 0$, every stationary equilibrium has $\mu^* = \bar H$, and under (C) the price $p^*$ is the unique solution of $\mu^*(p,S_r^{-1}(1-\bar H)) = \bar H$. A parameter change that raises $\mu^*$ at given prices (a lower $\theta$, lower transaction costs, a higher $\chi$ or $y_1$) raises $p^*$ and leaves the ownership rate unchanged; it changes the composition of owners and, through $e_3p^*$ and $\hat b^*$, the persistence $\lambda$.170\end{corollary}171172\begin{proposition}[Price neutrality without segmentation, under unit demand]173\label{prop:noseg}174Suppose, instead of the segmented stocks of Assumption~\ref{ass:prices}, that owner-occupied and rental units are the same stock, converted costlessly by competitive risk-neutral landlords with maintenance $m$ and operating wedge $\phi\ge 0$, and that Assumption~\ref{ass:pref} (unit demand) holds. Then in any stationary equilibrium $p^*$ solves $S(p^*) = 1$ per cohort and $\rho^* = p^*(1+m+\phi-1/R)$: the price and the rent are independent of $\theta$, $\kappa_b$, $\kappa_s$, $\chi$, $\nu$, $G$, the distribution of transfers and of inflation surprises; tenure shares, $\hat b^*$ and $\lambda$ adjust to these prices without feeding back on them.175\end{proposition}176177\begin{remark}[Scope of Proposition~\ref{prop:noseg}: the intensive margin]178\label{rem:noseg_scope}179The neutrality is a consequence of unit demand: every household occupies one unit whatever its tenure, so credit, transfers and inflation can only reshuffle who owns, not how many units are demanded. With an intensive margin, house sizes and rented services that respond to wealth and prices, total housing demand depends on the wealth distribution, and a shock that lowers household wealth lowers the single-stock price. Section~\ref{sec:inflation} finds that the price response to a surprise in the unsegmented quantitative economy ($\PImpactfrmlevelnoseg\%$) is almost as large as in the segmented one ($\PImpactfrmlevel\%$): in the quantitative model the price feedback is mostly a wealth effect on total housing demand, not the tenure-composition effect that the benchmark isolates. The \citet{kaplan2020housing} neutrality is thus recovered only in its unit-demand form.180\end{remark}181182\subsection{Surprise inflation}183\label{subsec:surprise}184185Two experiments are considered from a stationary equilibrium. (E1) A \emph{level} surprise at $t+1$: $1+\pi_{t+1} = (1+\bar\pi)(1+\pi^u)$ and expectations for $t+2$ onward unchanged. (E2) A \emph{persistent} surprise: $\pi_{t+1} = \pi_{t+2} = (1+\bar\pi)(1+\pi^u)-1$, learned at $t+1$, so that ARMs reset to the higher nominal rate while FRMs do not, with an optional real-rate response $r' = r+\phi_r\pi^u$ for $(t+1,t+2)$.186187\begin{proposition}[Direct revaluation and its incidence]188\label{prop:accounting}189Holding real prices fixed:190\begin{enumerate}191\item For the cohort aged~2 at $t+1$, the real stage-2 payment is $\delta_2p/(1+\pi^u)$ for every contract. The real stage-3 payment is $\delta_3p/(1+\pi^u)$ for every contract under (E1); under (E2) it is $\delta_3p/(1+\pi^u)^2$ for an FRM and $\delta_3p\,(R'/R)/(1+\pi^u)$ for an ARM, $R' = 1+r'$. The FRM borrower gains at least as much as the ARM borrower, strictly under (E2), and the ARM stage-3 payment rises in real terms iff $\phi_r>R$.192\item For the cohort aged~3 at $t+1$, the real stage-3 payment is $\delta_3p/(1+\pi^u)$ for every contract.193\item A household holding nominal deposits $n$ loses $nR\pi^u/(1+\pi^u)$ in real terms; a household whose nominal wage is indexed with elasticity $\eta_w$ loses $y[1-(1+\pi^u)^{-(1-\eta_w)}]$ of real income in the surprise period; a sitting tenant whose rent is indexed with elasticity $\eta_\rho$ gains $\rho[1-(1+\pi^u)^{-(1-\eta_\rho)}]$ at the landlord's expense.194\item The sum over households of real gains on nominal positions equals the lender's real loss: the revaluation is zero-sum. With $\Gamma_3$ the real stage-3 gain of the cohort aged~2 ($\Gamma_3 = \delta_3p\,\pi^u/(1+\pi^u)$ under (E1) for both contracts, $\delta_3p[1-(1+\pi^u)^{-2}]$ for an FRM and $\delta_3p[1-(R'/R)/(1+\pi^u)]$ for an ARM under (E2)), an owner of that cohort with deposits $n$ gains on net over its remaining life iff195\begin{equation}196\delta_2p\,\frac{\pi^u}{1+\pi^u} + \Gamma_3 \;>\; y_2\Big[1-(1+\pi^u)^{-(1-\eta_w)}\Big] + nR\,\frac{\pi^u}{1+\pi^u}.197\label{eq:gaincond}198\end{equation}199\end{enumerate}200\end{proposition}201202Four classes of owners lose from a surprise: ARM borrowers when the real rate rises by more than $R$ per unit of surprise, owners with imperfectly indexed wages when the real-income loss exceeds the debt erosion, owners holding nominal deposits larger than their mortgage, and, in general equilibrium, new buyers in inelastic markets (Proposition~\ref{prop:impact}, item~4). The claim that inflation enriches owners is false in general; condition \eqref{eq:gaincond} is what replaces it.203204\begin{proposition}[General-equilibrium impact of a level surprise under pure debtor positions]205\label{prop:impact}206Let the economy be in a stationary equilibrium satisfying Assumptions~\ref{ass:pref}--\ref{ass:interior} and \ref{ass:nominal} with $g(\hat y_O)>0$, and let a level surprise $\pi^u>0$ occur at $t+1$ (any contract). Under Assumption~\ref{ass:static}, with stage-2 owners not transacting, owner-market clearing $2S(p_{t+1}) = \mu^*+\mu_{t+1}$, rental clearing $2S_r(\rho_{t+1}) = 2-\mu^*-\mu_{t+1}$, and the static-expectations dominance condition207\begin{equation}208\text{(C$'$)}\qquad \frac{1+\nu}{\nu}\,\frac{\partial\hat b}{\partial p}\Big|_{p''=p^*}\;\ge\;1-\kappa_s,209\label{eq:Cprime}210\end{equation}211the impact responses satisfy:212\begin{enumerate}213\item $dp_{t+1}/d\pi^u\ge 0$ and $d\rho_{t+1}/d\pi^u\le 0$, both strict if $\varepsilon<\infty$ (respectively $\varepsilon_r<\infty$); if $\varepsilon = 0$ the price response is strictly positive and $\mu_{t+1} = \mu^*$.214\item $d\mu_{t+1}/d\pi^u\ge 0$, strictly if $\varepsilon>0$.215\item $d\hat b_{t+1}/d\pi^u\ge 0$; $q_{RO}$ falls, $q_{OO}$ rises, and $\lambda$ rises.216\item Stage-3 owner-sellers gain $\delta_3p^*\pi^u/(1+\pi^u) + (1-\kappa_s)dp_{t+1}\ge 0$; the lender loses the nominal erosion; the stage-1 cash of an inframarginal owner-dynasty child changes by $\frac{\nu}{1+\nu}\big[\delta_3p^*\frac{\pi^u}{1+\pi^u}+(1-\kappa_s)dp_{t+1}\big] - d\,dp_{t+1}$, which is negative when $dp_{t+1}$ is large relative to the erosion and $\nu/(1+\nu)<d/(1-\kappa_s)$; renter-dynasty children who rent in both situations gain iff $\rho_{t+1}<\rho^*$.217\item Without segmentation (Proposition~\ref{prop:noseg}) items 1--3 are zero and only Proposition~\ref{prop:accounting} applies.218\end{enumerate}219\end{proposition}220221The mechanism runs entirely through the parents whose debt is eroded: their larger transfers let more owner-dynasty children qualify, the price rises and the rent falls, the threshold rises, and renter-dynasty children at the margin are displaced. Assumption~\ref{ass:nominal} is what makes the parents' revaluation a gain. The next result drops it.222223\begin{corollary}[Net nominal creditors reverse the sign]224\label{cor:creditor}225Replace Assumption~\ref{ass:nominal} by predetermined nominal deposits $n^O\ge 0$ and $n^R\ge 0$ held at the shock date by owner-parents and renter-parents of the cohort aged~3 at $t+1$, chosen before the surprise, and keep the other assumptions of Proposition~\ref{prop:impact}. Define226\begin{equation}227\Xi \;\equiv\; \mu^*\,g(\hat y_O)\,\big(\delta_3p^* - Rn^O\big) \;-\; (1-\mu^*)\,g(\hat y_R)\,Rn^R .228\label{eq:Xi}229\end{equation}230Then $dp_{t+1}/d\pi^u$, $d\mu_{t+1}/d\pi^u$ and $-d\rho_{t+1}/d\pi^u$ have the sign of $\Xi$, with the same strictness conditions as in Proposition~\ref{prop:impact}; Proposition~\ref{prop:impact} is the case $n^O = n^R = 0$. If $\Xi<0$, the real house price and the stage-1 ownership rate fall and the rent rises on impact, and $\hat b$ falls; the change in $q_{RO}$ is the sum of a price term, positive when $\Xi<0$, and a direct deposit-loss term $-g(\hat y_R)Rn^R/(1+\pi^u)^2\le 0$, so it is unsigned unless $n^R = 0$, in which case $q_{RO}$ rises and $\lambda$ falls.231\end{corollary}232233\begin{remark}[Scope of Proposition~\ref{prop:impact}: the sign is a statement about net nominal positions]234\label{rem:impact_scope}235$\Xi$ is a density-weighted net nominal debtor position of the parents whose children are at the tenure margin. In the quantitative economy households are net nominal creditors in aggregate (real deposits \TotalDeposits{} against mortgage debt \TotalDebt{} per unit age mass), so the premise of Proposition~\ref{prop:impact} fails and Section~\ref{sec:inflation} finds, as the corollary indicates, that the real house price \emph{falls} on impact. Two further differences from the corollary are also traceable to benchmark reductions. The rent falls rather than rises, because with an intensive margin the wealth loss of nominal creditors lowers the total demand for housing services, and the two markets are no longer mirror images of a fixed unit demand. And the entry threshold rises rather than falls, because would-be buyers are themselves nominal creditors whose saving for the down payment is eroded, a channel absent from a benchmark in which entrants hold nothing before the transfer arrives. The proposition also covers only the first period under per-period supply and static expectations; the perfect-foresight transition with durable stocks, the second-round effects through the cohort aged~2, which are contract-dependent, and the welfare of marginal households are numerical objects.236\end{remark}237238\subsection{Equilibrium concepts for the quantitative economy}239\label{subsec:quant_eq}240241A \emph{stationary recursive competitive equilibrium} with constant inflation $\bar\pi$ and exogenous $r$ consists of value functions and policies solving \eqref{eq:bellman} at prices $(p^*,p^{r*},\rho^*)$ with $i^* = R(1+\bar\pi)-1$, stocks $(H^*,H^{r*})$ and a distribution $\Phi^*$ over states such that: the owner-occupied market clears, $\int h\,\mathbf 1\{o = O\}\,d\Phi^* = H^* = \iota(p^*)^{\varepsilon}/\delta_H$; the rental market clears, $\int s\,\mathbf 1\{o = R\}\,d\Phi^* = H^{r*}$ with \eqref{eq:rent} and $H^{r*} = \iota_r(p^{r*})^{\varepsilon_r}/\delta_H$ on the segmented margin; the distribution of gifts received at age~1 equals the distribution of gifts chosen at $j_g$, and likewise for estates at $j_e$, with the estate lottery used by households equal to the distribution of estates in $\Phi^*$; $\Phi^*$ is invariant under the policies, the income chain and the dynasty link; and the lender earns zero expected real profit. A \emph{transition equilibrium} after an unanticipated shock at $t = 0$ is a path of prices, stocks and distributions such that households optimise with perfect foresight of the post-shock path, markets clear every period, pre-shock FRM cohorts keep their origination rate, and the economy is within tolerance of the stationary equilibrium at a terminal date $T$. Partial-equilibrium counterparts, which re-solve the household block and the distribution at the stationary prices, are reported separately and are never labelled general equilibrium.242