1\section{The baseline equilibrium}2\label{sec:baseline}34\subsection{Tightness, durations and prices}56Figure~\ref{fig:equilibrium} shows the free-entry residual and the matching technology. The residual is strictly decreasing in7tightness and crosses zero once, at $\theta^*=\thetaBase$ buyers per listing, so the equilibrium is unique in the baseline; the8same sign pattern obtains in all \uniqDraws{} random parameter draws of Section~\ref{sec:robustness}. At this tightness a9listing meets a qualified buyer \LamSBase{} times per year and a buyer meets a listing \LamBBase{} times per year.10Table~\ref{tab:baseline} reports the baseline moments and Table~\ref{tab:types} the cross-section by seller type.1112\begin{figure}[H]13\centering14\includegraphics[width=\textwidth]{fig01_equilibrium_matching.pdf}15\caption{Free entry and matching in the baseline. Panel (a): the free-entry residual $R(\theta)$ (model-generated, baseline16parameterisation of Table~\ref{tab:parameters}) and its unique root. Panel (b): seller and buyer meeting rates implied by the17Cobb--Douglas matching function; the dashed line marks equilibrium tightness.}18\label{fig:equilibrium}19\end{figure}2021\begin{table}[H]22\centering23\caption{Baseline stationary equilibrium: aggregate moments (model-generated)}24\label{tab:baseline}25\small26\input{tables/tab02_baseline_moments}27\end{table}2829The most patient sellers accept \accPatientBase{} percent of qualified offers and sell at the hazard \etaPatientBase{} per year;30the most urgent accept \accUrgentBase{} percent and sell at \etaUrgentBase{} per year. The mean time on market of houses that31sell is \tomBase{} months and \soldShareBase{} percent of listings eventually sell; the remainder are withdrawn. A buyer searches32for \buyerDurBase{} months on average, longer than a seller because free entry crowds the buyer side of the market until the33surplus a buyer expects per meeting just covers the cost of searching. The transaction-weighted mean price is \priceBase, close to34the mean buyer valuation by construction of the units, with reservation prices between \VUrgentBase{} and \VPatientBase.3536\begin{table}[H]37\centering38\caption{Baseline cross-section by seller type (model-generated; $c$ in units of value per year, durations in months)}39\label{tab:types}40\footnotesize41\resizebox{\textwidth}{!}{\input{tables/tab02b_baseline_types}}42\begin{tablenotes}43\footnotesize44\item Note: $f(c)$ is the listing composition, ``stock share'' the share of type $c$ in the stock of listings and ``sales share''45its share in transactions. $D(c)$ is the urgency discount \eqref{eq:Ddef}.46\end{tablenotes}47\end{table}4849\subsection{The urgency discount and its attenuation}5051The most urgent sellers obtain \pbarUrgentBase{} against \pbarPatientBase{} for the most patient, an urgency discount of52\DurgentBase{} percent; weighted by transactions the discount is \DmeanBase{} percent because most sellers are patient. The53reservation-price gap between the two extreme types is larger than the gap in expected prices, as Proposition~\ref{prop:urgency}(iii)54predicts: urgent sellers accept lower valuations, but conditional on acceptance the buyer's surplus is larger, and the seller's55share of it recovers part of the concession. Figures~\ref{fig:urgency_price} and~\ref{fig:urgency_hazard} trace the reservation56value, the expected price, the sale hazard and the expected duration along a fine grid of carrying costs for five levels of57matching efficiency, holding tightness at its baseline value. Three properties of Proposition~\ref{prop:urgency} and58\ref{prop:arrival} are visible. The schedules are monotone in $c$; they shift down (prices) and up (hazards) when matching59efficiency falls; and they become steeper, so that the gap between patient and urgent sellers widens when the market is less60liquid. The slope of the reservation value equals the analytic expression $-1/[(r+\delta)+\beta\eta(c)]$ along the whole grid.6162\begin{figure}[H]63\centering64\includegraphics[width=\textwidth]{fig02_urgency_reservation_price.pdf}65\caption{Urgency, reservation values and expected prices (model-generated, partial equilibrium with tightness fixed at its66baseline value). Each curve corresponds to a multiple of baseline matching efficiency; the effective arrival rate $\Lambda_s$ is67given in the legend. Panel (a): reservation value $V(c)$. Panel (b): expected transaction price $\bar p(c)$ conditional on sale.}68\label{fig:urgency_price}69\end{figure}7071\begin{figure}[H]72\centering73\includegraphics[width=\textwidth]{fig03_urgency_sale_hazard.pdf}74\caption{Urgency and the sale hazard (model-generated, partial equilibrium). Panel (a): sale hazard $\eta(c)$ per year for five75levels of matching efficiency; dots mark the nine baseline types. Panel (b): expected listing duration in months.}76\label{fig:urgency_hazard}77\end{figure}7879\subsection{Dispersion, survival and duration dependence}8081Among identical houses the standard deviation of transaction prices is \sdPriceBase{} percent of the mean, of which82\betweenShareBase{} percent of the variance is between seller types and the remainder reflects the match-specific component of83the Nash price. Figure~\ref{fig:price_dist} shows the mixture density of prices by seller type: urgent sellers populate the left84tail, and a fall in matching efficiency shifts and widens the whole distribution. Figure~\ref{fig:survival} shows the survival85functions of listings for three types and for the population, and the population exit hazard, which declines with duration as86Lemma~\ref{lem:duration} requires although every individual hazard is constant. The decline is modest in the baseline because87the dispersion of hazards across types is modest; it would be invisible in a sample of the size usually available, which is why88the duration literature's finding of positive dependence \citep{zuehlke1987duration,jud1996brokerage} is not evidence against89heterogeneity but evidence that other mechanisms, such as list-price revisions and learning \citep{knight2002listing,90anenberg2016information}, dominate.9192\begin{figure}[H]93\centering94\includegraphics[width=\textwidth]{fig08_seller_type_price_distributions.pdf}95\caption{Transaction-price distributions among identical houses (model-generated). Panel (a): mixture density of transaction96prices in the baseline, stacked by seller type from the most patient (light) to the most urgent (dark). Panel (b): the same97density at baseline matching efficiency and at 60 percent of it.}98\label{fig:price_dist}99\end{figure}100101\begin{figure}[H]102\centering103\includegraphics[width=\textwidth]{fig07_tom_survival.pdf}104\caption{Survival of listings and duration dependence (model-generated). Panel (a): survival functions of three seller types and105of the population mixture. Panel (b): population exit hazard (sale or withdrawal) as a function of elapsed duration; individual106hazards are constant.}107\label{fig:survival}108\end{figure}109110\subsection{The Walrasian benchmark}111112The frictionless free-entry price is $p^*=\pstarBase$ and the market liquidity gap is $\Gamma=\GammaBase$ percent in the baseline.113Figure~\ref{fig:discount}(c) traces $\Gamma$ for the patient type across effective arrival rates from 0.05 to 2{,}000 per year: the114gap is 43 percent in a frozen market, falls monotonically, and turns negative beyond about \GammaCrossLam{} contacts per year, as115Proposition~\ref{prop:arrival}(iv) predicts. At that point the seller's option to wait for a high-valuation buyer is worth more116than the delay costs, and the search price exceeds the uniform Walrasian price. The baseline is far from this region, so the117gap is positive and of the same order as the discounts reported for distressed sales, but the sign change is the reason the paper118does not interpret $\Gamma$ as a liquidity premium.119