\section{A search model of a local housing market}
\label{sec:model}

\subsection{Environment}

Time is continuous and the horizon infinite; all agents discount at the rate $r>0$. The market is a quality segment of a local
housing market with houses of quality $q>0$; Lemma~\ref{lem:scale} shows that $q$ scales values and prices, so the exposition
sets $q=1$ and measures prices, values and costs in units of the mean buyer valuation of the segment.

\paragraph{Sellers.} New listings arrive at the exogenous flow $s>0$ per unit time. The owner of a listing has a permanent
\emph{carrying cost} $c$, drawn at listing from a distribution $F$ with support $[c_L,c_H]$, $0\le c_L<c_H<\infty$, and pays $c$
per unit time while the house is listed. Listings are withdrawn at the exogenous rate $\delta\ge0$; a seller who withdraws keeps
the house and obtains its own-use value $w_0\ge0$. The value of being listed, $V(c)$, is therefore measured relative to this
outside option.

\paragraph{Buyers.} A buyer enters the market at the sunk cost $k>0$, pays the flow search cost $\kappa_b\ge0$ while active, and
exits upon purchase. Upon meeting a listing the buyer draws a match-specific valuation $w\sim G$, the present value of the
housing-service flow to that buyer, with density $g$ on $[w_L,w_H]$, $0\le w_L<w_H\le\infty$. A purchase at price $p$ yields
$w-p$; resale is abstracted from.

\paragraph{Matching.} With $B$ active buyers and $S$ listed sellers, meetings occur at the rate $m(B,S)=AB^{\alpha}S^{1-\alpha}$
with $\alpha\in(0,1)$ and $A>0$. Tightness is $\theta=B/S$; a seller meets buyers at the rate $\lambda_s(\theta)=A\theta^{\alpha}$ and a
buyer meets listings at $\lambda_b(\theta)=A\theta^{\alpha-1}$. Meetings are random across the stock of listings, so a buyer meets a
type-$c$ seller with probability $N(c)=n(c)/S$, where $n$ is the stock density of listed sellers. A meeting is with a qualified
buyer with probability $\phi\in(0,1]$ (a pre-approval screen); unqualified meetings are wasted. The effective contact rates are
$\Lambda_s=\phi\lambda_s(\theta)$ and $\Lambda_b=\phi\lambda_b(\theta)$. Section~\ref{sec:financing} replaces the screen by a financing
contingency with cash and mortgage buyers.

\paragraph{Bargaining.} Within a match both parties observe $(w,c)$ and split the surplus by generalised Nash bargaining with
seller share $\beta\in(0,1)$; trade occurs if and only if the match surplus is nonnegative. Section~\ref{sec:posted} replaces
bargaining by posted prices.

\paragraph{Liquidity shocks.} A liquidity shock is an unanticipated change in matching efficiency $A$ or in qualification $\phi$
with $(G,F,r,k,\kappa_b,s,\delta,w_0)$ unchanged, so that the distribution of housing-service values is the same before and after.

\begin{assumption}[Match-value distribution]
\label{ass:G}
$G$ is continuous with a density $g$ that is positive on the interior of its support and has a finite mean. Where stated, $G$ is
IFR (increasing failure rate: $1-G$ is log-concave), which holds whenever $g$ is log-concave \citep{bagnoli2005logconcave}; the
truncated normal of the baseline parameterisation satisfies this, the exponential is the boundary case, and the Pareto
violates it.
\end{assumption}

Three functions of $G$ recur. The expected surplus function $H(v)=\int(w-v)^+dG(w)=\int_v^{w_H}(1-G(w))\,dw$ is nonincreasing and
convex with $H'(v)=-(1-G(v))$; the hazard function is $h_G(v)=g(v)/(1-G(v))$; and the mean residual valuation is
$M(v)=H(v)/(1-G(v))=E[w-v\mid w\ge v]$. Two facts are used repeatedly: $M'(v)=-1+h_G(v)M(v)>-1$ wherever $g(v)>0$, with $M'\le0$
under IFR; and $gH\le(1-G)^2$ under IFR.

\subsection{Value functions, trade and prices}

Let $U$ be the value of an active buyer. A listed seller of type $c$ solves
\begin{equation}
(r+\delta)V(c)=-c+\delta w_0+\Lambda_s\,\beta\,H\big(V(c)+U\big),
\label{eq:S}
\end{equation}
and an active buyer solves
\begin{equation}
rU=-\kappa_b+\Lambda_b\,(1-\beta)\int H\big(V(c)+U\big)\,dN(c).
\label{eq:B}
\end{equation}
The match surplus is $\Sigma(w,c)=w-V(c)-U$, so trade occurs iff $w\ge v^*(c)\equiv V(c)+U$, the \emph{reservation valuation} of a
type-$c$ seller; $V(c)$ is the seller's reservation price and $w-U$ the buyer's. The Nash price is
\begin{equation}
p(w,c)=V(c)+\beta\,\Sigma(w,c)=(1-\beta)V(c)-\beta U+\beta w .
\label{eq:price}
\end{equation}
The \emph{sale hazard} of a type-$c$ listing is $\eta(c)=\Lambda_s[1-G(v^*(c))]$, the product of the meeting hazard and the
conditional sale probability; the exit hazard is $\eta(c)+\delta$; the survival of a listing is $e^{-(\eta(c)+\delta)\tau}$; the
expected listing duration is $1/(\eta(c)+\delta)$ and the probability of eventual sale is $\eta(c)/(\eta(c)+\delta)$. The expected
transaction price of type $c$ is
\begin{equation}
\bar p(c)\equiv E[p\mid\text{sale},c]=V(c)+\beta M(v^*(c)),
\label{eq:pbar}
\end{equation}
the reservation price plus the seller's share of the mean surplus of accepted matches.

\subsection{Stocks, composition and equilibrium}

In a stationary state the inflow of type-$c$ listings equals their outflow, so that $sf(c)=(\eta(c)+\delta)n(c)$ and
\begin{equation}
n(c)=\frac{sf(c)}{\eta(c)+\delta},\qquad S=\int n(c)\,dc,\qquad B=\theta S,\qquad Q=\int\eta(c)n(c)\,dc,\qquad
\omega(c)=\frac{\eta(c)n(c)}{Q},
\label{eq:stocks}
\end{equation}
where $Q$ is the sales flow and $\omega$ the \emph{transaction composition}, the distribution of carrying costs among houses that
sell. Three weightings are therefore distinguished: the listing cohort $f$, the stock $N$, and the transactions $\omega$. Buyers
exit only by purchase, so the buyer entry flow equals $Q$.

\begin{definition}[Stationary equilibrium]
\label{def:eq}
Given primitives $(r,\delta,w_0,s,F,G,A,\alpha,\phi,\beta,k,\kappa_b)$, a stationary equilibrium with free entry is a tuple
$(\theta,U,V(\cdot),n(\cdot))$ such that \eqref{eq:S} holds for every $c$, \eqref{eq:B} holds, buyers enter until $U=k$, and the
stock equations \eqref{eq:stocks} hold. A \emph{partial equilibrium} fixes $\theta$, sets $U=k$, and drops \eqref{eq:B}.
\end{definition}

Free entry pins the buyer's value at the entry cost at every tightness. This has two consequences that organise the analysis.
First, the seller block \eqref{eq:S} can be studied at a given $\Lambda_s$ with $U=k$, and all comparative statics of reservation
prices, hazards and prices with respect to arrival rates are partial-equilibrium statements that hold conditionally on
tightness. Second, tightness itself responds: a fall in $\phi$ or $A$ lowers the surplus buyers expect per meeting, buyers exit,
$\theta$ falls, and sellers meet buyers even less often. Proposition~\ref{prop:entry} characterises this feedback.

Listing is worthwhile for a type-$c$ seller iff $V(c)\ge w_0$. Proposition~\ref{prop:urgency}(iv) gives the threshold carrying cost
$\bar c=\Lambda_s\beta H(w_0+k)-rw_0$ above which a seller would not list; the baseline parameterisation satisfies $c_H<\bar c$, and the
condition is checked in every reported equilibrium.

\subsection{Transition dynamics}
\label{sec:dynamics_model}

An unanticipated permanent or temporary shock to $A$ or $\phi$ at date zero leaves the stock of listings predetermined and
moves the economy along a path $(\theta_t,V_t(\cdot),n_t(\cdot))$. Buyers enter or exit freely at every instant, so $U_t=k$
throughout. The seller value obeys the backward equation
\begin{equation}
\dot V_t(c)=(r+\delta)V_t(c)+c-\delta w_0-\Lambda_s(\theta_t)\,\beta\,H\big(V_t(c)+k\big),
\label{eq:HJBdyn}
\end{equation}
free entry holds pointwise and the stock evolves forward,
\begin{equation}
rk=-\kappa_b+\Lambda_b(\theta_t)(1-\beta)\int H\big(V_t(c)+k\big)\,dN_t(c),\qquad
\dot n_t(c)=sf(c)-(\eta_t(c)+\delta)\,n_t(c).
\label{eq:dynsystem}
\end{equation}
The observed transaction-price index is $P_t=\int\bar p_t(c)\,\omega_t(c)\,dc$; a
fixed-composition index $P^{fix}_t=\int\bar p_t(c)\,\omega_0(c)\,dc$ holds the pre-shock transaction weights fixed, so that
\begin{equation}
\ln P_t-\ln P_0=\underbrace{\big[\ln P^{fix}_t-\ln P_0\big]}_{\text{within-type (liquidity)}}
+\underbrace{\big[\ln P_t-\ln P^{fix}_t\big]}_{\text{composition (selection)}} .
\label{eq:decomp}
\end{equation}
Because $G$ is unchanged, a fundamental index is flat and the whole movement of $P_t$ is non-fundamental. Section~\ref{sec:selection}
signs the composition term.

\subsection{Financing contingency: cash and mortgage buyers}
\label{sec:financing}

In the baseline the qualification screen $\phi$ multiplies both contact rates, so that $A$ and $\phi$ enter every equilibrium
condition only through the product $\phi A$ (Proposition~\ref{prop:entry}): a qualification shock and a matching-efficiency
shock are observationally equivalent. The extension of this subsection breaks the equivalence by modelling the financing step
explicitly. A buyer is a cash buyer with probability $\pi$ and a mortgage buyer otherwise, the type being drawn at entry. A cash
agreement closes immediately. A mortgage agreement at price $p$ enters a \emph{pending} state: the lender's decision arrives at
rate $\nu$ and is favourable with probability $\phi_m$; on failure the house returns to the market and the buyer to search. During
the pending spell the seller keeps paying $c$ and the buyer $\kappa_b$. The pending values are
\begin{equation}
P_s(c,p)=\frac{-c+\nu\phi_m p+\nu(1-\phi_m)V(c)}{r+\nu},\qquad
P_b(w,p)=\frac{-\kappa_b+\nu\phi_m(w-p)+\nu(1-\phi_m)U_m}{r+\nu},
\label{eq:pending}
\end{equation}
where $U_m$ is the value of a searching mortgage buyer and $U_c$ that of a cash buyer. The joint surplus of a mortgage agreement
is $\Sigma_m(w,c)=[\nu\phi_m(w-V-U_m)-c-\kappa_b-r(V+U_m)]/(r+\nu)$, so a mortgage agreement requires
\begin{equation}
w\ge v^*_m(c)\equiv V(c)+U_m+\frac{c+\kappa_b+r\,(V(c)+U_m)}{\nu\phi_m},
\label{eq:vm}
\end{equation}
whereas a cash agreement requires $w\ge v^*_c(c)=V(c)+U_c$. Sellers meet cash buyers in a share $\tilde\pi$ of meetings that is
determined by the stocks of searching buyers of each type; buyers of each type solve their own version of \eqref{eq:B}; ex-ante
free entry requires $\pi U_c+(1-\pi)U_m=k$; and the stock equations account for pending houses and for houses returning to the
market. Appendix~\ref{app:computation} states the full system. The baseline is the limit $\nu\to\infty$ with $\phi=\tilde\pi+(1-\tilde\pi)\phi_m$.

\subsection{Posted prices}
\label{sec:posted}

As a robustness protocol the seller commits to a price $p$ before the buyer's valuation is drawn, and the buyer accepts iff
$w-p\ge U$. Writing $y=p+U$, the seller solves
\begin{equation}
(r+\delta)V(c)=-c+\delta w_0+\Lambda_s\max_{y}\,[1-G(y)]\,\big(y-U-V(c)\big),
\label{eq:posted}
\end{equation}
whose interior optimum satisfies $y(c)-U-V(c)=1/h_G(y(c))$. Prices are deterministic within a type, so all dispersion among
identical houses is urgency-driven under this protocol.

\subsection{Welfare}
\label{sec:welfare_model}

The steady-state flow of surplus generated by the market is
\begin{equation}
\Omega=\int\eta(c)n(c)\,\big(E[w\mid w\ge v^*(c)]-w_0\big)\,dc-\int c\,n(c)\,dc-\kappa_bB-kQ,
\label{eq:welfare}
\end{equation}
the value created by sales net of the own-use value forgone, carrying costs, search costs and entry costs. Under free entry
buyers earn zero net of entry, so $\Omega$ accrues to sellers in equilibrium; the ex-ante value of a listing cohort is
$s\int(V(c)-w_0)\,dF(c)$. This flow criterion ignores discounting across cohorts and transition costs; the Hosios condition
$\beta=1-\alpha$ of the discounted problem \citep{hosios1990efficiency} is used as a reference point, not as a theorem about
\eqref{eq:welfare}.