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1% ============================================================================2\section{Supply elasticity, policy and welfare}3\label{sec:supply}4% ============================================================================56\subsection{Supply elasticity and the down-payment constraint}7\label{subsec:supply}89\begin{figure}[htbp]10\centering11\includegraphics[width=\textwidth]{fig10_supply_elasticity_heatmap.pdf}12\caption{Stationary general equilibria across the long-run supply elasticity $\varepsilon = \varepsilon_r$ and the minimum down payment $\theta$: house price (a), ownership rate (b), probability that a renter's child owns at 35--39 (c) and rank--rank slope of net worth (d). Source: \texttt{results/data/supply/*.json}, \texttt{theta\_sweep/*.json}.}13\label{fig:heatmap}14\end{figure}1516Changing the supply elasticity alone leaves the baseline unchanged, because every supply curve passes through the normalisation point, so the elasticity matters only together with a demand shift, and Figure~\ref{fig:heatmap} crosses it with the down-payment constraint. With a 5\% minimum down payment the price is \EpsLowThetaFiveP{} under near-fixed supply and \EpsHighThetaFiveP{} under an elasticity of two, with ownership \EpsLowThetaFiveOwn{} against \EpsHighThetaFiveOwn; with a 40\% minimum the price is \EpsLowThetaFortyP{} against \EpsHighThetaFortyP{} and ownership \EpsLowThetaFortyOwn{} against \EpsHighThetaFortyOwn. The direction is that of Corollary~\ref{cor:fixed}, inelastic supply converting more of the demand shift into price and less into quantity, but the magnitudes are small because the demand shift from $\theta$ is itself small: the payment-to-income cap binds for low-income entrants whatever the down payment. Mobility is insensitive to the elasticity at these shifts, $q_{RO}$ at 35--39 being \EpsLowThetaFiveqRO{} against \EpsHighThetaFiveqRO{} at $\theta = 0.05$ and \EpsLowThetaFortyqRO{} against \EpsHighThetaFortyqRO{} at $\theta = 0.40$. The conjecture with which the fourth contribution started, that supply elasticity has a first-order effect on intergenerational mobility through the entry threshold, is not supported in this calibration; what the elasticity does govern, in the transitions of Section~\ref{sec:inflation}, is bounded by the depreciation rate.1718\subsection{Search for multiple stationary equilibria}19\label{subsec:multiplicity}2021\begin{figure}[htbp]22\centering23\includegraphics[width=\textwidth]{fig13_multiplicity_search.pdf}24\caption{Search for multiple stationary equilibria: owner-market excess demand along the house price with the rental market cleared at each point and the estate lottery held at its baseline value (a), and the ownership rate along the clearing curve (b), for the baseline, near-fixed supply and near-fixed supply with a 5\% minimum down payment. Source: \texttt{results/data/multiplicity/price\_scan\_*.csv}.}25\label{fig:multiplicity}26\end{figure}2728Proposition~\ref{prop:exist} leaves multiplicity open when condition (C) fails, and the benchmark's ladder channel, bequeathed equity rising with the price, is the candidate. Figure~\ref{fig:multiplicity} scans the owner-market excess demand over prices from \MultPgridLow{} to \MultPgridHigh{}. In the baseline it falls monotonically from \MultBaseExcessLow{} to \MultBaseExcessHigh{} with a single sign change at $p = \MultBaseRoot$, and the ownership rate along the clearing curve falls from \MultBaseOwnLow{} to \MultBaseOwnHigh{}: $d\mu/dp<0$ everywhere, the analogue of (C), holds with a wide margin. Under near-fixed supply the excess demand runs from \MultEpsLowExcessLow{} to \MultEpsLowExcessHigh{} with one root, and under near-fixed supply with a 5\% down payment from \MultEpsLowThetaFiveExcessLow{} to \MultEpsLowThetaFiveExcessHigh{} with one root at \MultEpsLowThetaFiveRoot. There is no evidence of the \citet{stein1995prices} multiplicity in the scanned range. The scan holds the lottery fixed, and the lottery feedback is the ladder channel itself, so the scan understates it; but the slope of excess demand around the root, \MultBaseSlope{} per unit of log price, is an order of magnitude larger than the lottery feedback measured across rounds, and at the equilibrium the lottery-consistent solution coincides with the root. Uniqueness is a numerical statement about this calibration, not a theorem.2930\subsection{A rebated transfer tax}31\label{subsec:policy}3233The policy experiment taxes gifts and estates at 20 or 50\% and rebates the proceeds lump-sum to entering households, with the rebate an equilibrium object. Ownership rises to \TaxTwentyOwn{} and \TaxFiftyOwn, and under-35 ownership to \TaxTwentyOwnUthirtyfive{} and \TaxFiftyOwnUthirtyfive{} against \OwnershipUnderThirtyFive{} in the baseline, because the rebate gives every entrant liquid wealth; upward tenure mobility rises ($q_{RO}$ at 35--39 \TaxTwentyqRO{} and \TaxFiftyqRO), the rank--rank slope falls to \TaxTwentyRR{} and \TaxFiftyRR, the Gini to \TaxTwentyGini{} and \TaxFiftyGini, and the mean gift falls to \TaxTwentyGift{} and \TaxFiftyGift{} (Tables~\ref{tab:scenarios} and~\ref{tab:mobility}). The price is nearly unchanged (\TaxTwentyP{} and \TaxFiftyP). A rebated transfer tax therefore raises youth ownership and mobility in this economy by moving liquidity to the moment at which the down-payment constraint binds; it is a comparative-statics statement about stationary equilibria, not an optimal-policy result, and the welfare numbers below concern entrants only.3435\subsection{Welfare}36\label{sec:welfare}3738\input{tables/tab08_welfare}3940\begin{figure}[htbp]41\centering42\includegraphics[width=\textwidth]{fig11_welfare.pdf}43\caption{Consumption-equivalent welfare. Panel (a): entering households in alternative stationary equilibria relative to the baseline, by quintile of the transfer received. Panel (b): households alive at a 10\% fixed-rate level surprise, by age and tenure, general equilibrium. Source: \texttt{results/data/welfare/welfare\_all.csv}, \texttt{results/data/inflation/frm\_level\_welfare.csv}.}44\label{fig:welfare}45\end{figure}4647Table~\ref{tab:welfare} and Figure~\ref{fig:welfare} report the consumption-equivalent variation \eqref{eq:cev} of entering households across stationary equilibria, by quintile of the transfer received, for households within 10\% of the age-1 threshold, and overall. Only scenarios that keep the preference parameters are welfare comparisons; the rows for the bequest motive and for the preference variants are listed as a reminder that they compare value functions under different utilities (``bequests off'' shows a large positive number only because the warm-glow disutility disappears from the objective). A 5\% minimum down payment raises the welfare of entrants by \CEVThetaFiveAll\% overall, \CEVThetaFiveBottomTwo\% in the bottom two gift quintiles, \CEVThetaFiveMarg\% for marginal households and \CEVThetaFiveQfive\% in the top quintile, which pays the higher price; a 40\% minimum gives \CEVThetaFortyAll\% overall, \CEVThetaFortyBottomTwo\% at the bottom, \CEVThetaFortyMarg\% for marginal households and \CEVThetaFortyQfive\% at the top. The welfare of marginal households, unsigned in the theory, is thus positive for a looser constraint and negative for a tighter one in this calibration. The rebated transfer tax raises entrants' welfare by \CEVTaxTwentyAll\% and \CEVTaxFiftyAll\%, decreasing in the gift quintile (\CEVTaxTwentyQone{} and \CEVTaxFiftyQone\% at the bottom against \CEVTaxTwentyQfive{} and \CEVTaxFiftyQfive\% at the top); this is the welfare of entrants, and the parents who pay the tax are not in the comparison. Near-fixed supply with a 5\% down payment gives \CEVEpsLowThetaFiveAll\% against \CEVEpsHighThetaFiveAll\% with an elasticity of two. A real rate of 1 or 3\% gives \CEVRoneAll{} and \CEVRthreeAll\%; a landlord wedge of 2 or 8\% gives \CEVPhiLowAll{} and \CEVPhiHighAll\%; low and high earnings risk give \CEVSigLowAll{} and \CEVSigHighAll\%; a 2-percentage-point deposit premium, the renter's outside investment return, gives \CEVDepPremAll\%, rising with the gift quintile. The numerical variants (tolerance, sample size, seed) give variations of at most \CEVNumericalMaxPct\%, which bounds the noise of the welfare measure. For transitions, the households alive at a level surprise lose \FrmLevelCEVAllAbs\% on average under the fixed-rate contract and \ArmLevelCEVAllAbs\% under the adjustable one, and gain \RateTwentyCEVAll\% in the real-rate shock (Section~\ref{sec:inflation}).48