\section{Counterfactuals: arrivals, financing, heterogeneity, dynamics and selection}
\label{sec:counterfactuals}

Each experiment of this section was pre-registered with a hypothesis, run from the stored configuration, and saved with its
diagnostics before any text was written; the hypotheses and outcomes are listed in Appendix~\ref{app:extra}.

\subsection{A collapse in buyer arrivals}
\label{sec:arrival}

Table~\ref{tab:compstat} and Figure~\ref{fig:discount} vary matching efficiency between 20 and 300 percent of its baseline
value, in general equilibrium (tightness re-solved) and in partial equilibrium (tightness fixed). Halving $A$ lowers the
transaction-weighted price by \priceDropHalfA{} percent in general equilibrium against \priceDropHalfAPE{} percent with tightness
fixed, lengthens the time on market of sales from \tomBase{} to \tomHalfA{} months, raises the inventory of listings from
\inventoryBase{} to \inventoryHalfA, lowers tightness from \thetaBase{} to \thetaHalfA, and raises the urgency discount of the
most urgent type from \DurgentBase{} to \DurgentHalfA{} percent. The amplification through entry is modest but systematic: the
elasticity of the effective contact rate with respect to matching efficiency is \elAGELam{} in general equilibrium against
exactly one in partial equilibrium, and the elasticities of prices and durations are correspondingly larger
(\elAGEPrice{} against \elAPEPrice{} for prices, \elAGETom{} against \elAPETom{} for time on market). Because $A$ and $\phi$ are
observationally equivalent in the baseline (Proposition~\ref{prop:entry}), the same table describes a tightening of
qualification; Table~\ref{tab:screen} in Appendix~\ref{app:extra} reports the $\phi$ experiment directly.

\begin{table}[H]
\centering
\caption{Comparative statics of matching efficiency in general (GE) and partial (PE) equilibrium (model-generated)}
\label{tab:compstat}
\footnotesize
\resizebox{\textwidth}{!}{\input{tables/tab03_comparative_statics}}
\begin{tablenotes}
\footnotesize
\item Note: PE holds tightness at its baseline value. $D_{urg}$ is the urgency discount of the most urgent type, $\bar D$ the
transaction-weighted discount, $\Gamma$ the gap to the Walrasian free-entry price.
\end{tablenotes}
\end{table}

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig04_liquidity_discount_vs_arrivals.pdf}
\caption{Liquidity and the urgency discount (model-generated). Panel (a): urgency discount of the most urgent type and
transaction-weighted discount as functions of matching efficiency, in general (solid) and partial (dashed) equilibrium. Panel
(b): transaction-weighted mean price. Panel (c): market liquidity gap $\Gamma$ of the patient type relative to the Walrasian
free-entry price as a function of the effective arrival rate, partial equilibrium with $\delta=0$.}
\label{fig:discount}
\end{figure}

Figure~\ref{fig:volume} plots the comovement that the experiment generates. Across liquidity states the steady-state sales flow
and the mean price move together and inventory and time on market move inversely, all with the same distribution of buyer
valuations. The movement of the steady-state sales flow is entirely due to withdrawals: with $\delta=0$ every listing eventually
sells and the sales flow equals the listing inflow at every level of liquidity (Proposition~\ref{prop:shock}(iv)), as verified in
the $\delta=0$ experiment of Section~\ref{sec:dynamics}.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig06_volume_vs_price.pdf}
\caption{Volume, price, inventory and time on market across liquidity states (model-generated, general equilibrium). Each point
is a stationary equilibrium for a different matching efficiency, from 20 to 300 percent of the baseline. Panel (a): sales flow
and mean price. Panel (b): inventory of listings and mean time on market of sales.}
\label{fig:volume}
\end{figure}

\subsection{Financing: qualification, approval delay and the cash--mortgage gap}
\label{sec:financing_results}

In the extension with cash and mortgage buyers, the baseline equilibrium has tightness close to the baseline screen, a share
$\tilde\pi=\piTildeBase$ of meetings with cash buyers (below their 30 percent entry share, because mortgage buyers spend longer
searching), and buyer values $U_c=\UcBase$ against $U_m=\UmBase$: a cash buyer's search is worth four times a mortgage buyer's.
Table~\ref{tab:financing} and Figure~\ref{fig:financing} report the decomposition of Proposition~\ref{prop:financing}. For the
same house, mortgage buyers pay \gapMeanBase{} percent more than cash buyers on average, \gapPatientBase{} percent when the seller
is the most patient and \gapUrgentBase{} percent when she is the most urgent. The delay-bargaining term rises from
\deltaBPatientBase{} to \deltaBUrgentBase{} percent across types, as part (i) of the proposition requires; the outside-option term
is \outsideBase{} percent for every type; and the selection term is negative for patient sellers, because cash buyers' better
outside option raises the valuation at which a cash deal is struck above the valuation at which a mortgage deal is struck, and
turns positive only for the most urgent sellers, for whom the pending wedge dominates. On average the selection term is
\selMeanBase{} percent. The measured gap is therefore smaller than the sum of its bargaining and outside-option components, and
its cross-sectional gradient in urgency comes entirely from the delay-bargaining term.

\begin{table}[H]
\centering
\caption{Financing extension: the cash--mortgage gap and its components as the approval probability varies (model-generated)}
\label{tab:financing}
\footnotesize
\resizebox{\textwidth}{!}{\input{tables/tab04b_financing_extension}}
\begin{tablenotes}
\footnotesize
\item Note: components are listing-weighted averages across seller types, in percent of the house value; ``listed'' is the
expected duration in the listed state and ``to closing'' the expected total duration including pending spells (Little's law).
Rows with $U_m<0$ violate the mortgage buyers' participation constraint and are not used in the text.
\end{tablenotes}
\end{table}

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig05_financing_counterfactual.pdf}
\caption{The cash--mortgage gap (model-generated, financing extension). Panel (a): decomposition of the gap by seller type at the
baseline into the delay-bargaining term $\Delta_b(c)$, the outside-option term and the selection term. Panel (b): listing-weighted
gap as a function of the approval probability $\phi_m$ (values below 0.70 violate buyer participation). Panel (c): gap as a
function of the expected approval delay.}
\label{fig:financing}
\end{figure}

Tighter approval widens the gap: at $\phi_m=0.70$ the average gap is \gapPhiSeventy{} percent and at $\phi_m=0.60$ the value of searching as a
mortgage buyer turns negative, so that the model's assumption that buyers never exit is violated and the economy is no longer
admissible (Table~\ref{tab:financing}, last column); results are reported only for $\phi_m\ge\phiMinParticipation$. A longer
approval delay widens the gap almost linearly, from \gapNuWeek{} percent with a one-week decision to \gapNuFourMonths{} percent with a four-month
decision (Table~\ref{tab:fin_delay}, Appendix~\ref{app:extra}), while the baseline screen corresponds to the instantaneous limit in
which the gap collapses to the outside-option term. Bargaining power shifts the sign of the delay term: with $\beta=0.8$ the delay
term is negative for patient sellers, who then compensate mortgage buyers for the risk the buyers bear (Table~\ref{tab:fin_beta}).
The magnitudes are in the range of the 2--3.9 percent structural discount of \citet{han2024cash} and well below the 11 percent
reduced-form premium of \citet{reher2024mortgage}, which is the pattern the decomposition leads one to expect when the selection
term is small or negative.

\subsection{Matching efficiency versus approval}
\label{sec:matching_vs_approval}

The extension distinguishes the two shocks that the baseline cannot. Table~\ref{tab:match_vs_approval} compares a 30 percent fall
in matching efficiency with a fall in the approval probability from 0.85 to 0.70. The first lowers prices by \expSixPriceA{}
percent and raises the expected time to closing from \expSixClosingBase{} to \expSixClosingA{} months; the second lowers prices by
\expSixPricePhi{} percent, raises the time to closing to \expSixClosingPhi{} months, and widens the cash--mortgage gap from \gapMeanBase{} to
\expSixGapPhi{} percent, whereas the matching shock leaves the gap almost unchanged (\expSixGapA{} percent). An approval shock also
\emph{raises} the hazard of agreements in the listed state, because mortgage buyers whose search is worth less accept more
readily, while lowering the hazard of completed sales. The two shocks therefore have different signatures: a fall in matching
efficiency shows up in durations and prices alike, a fall in approval shows up in the cash--mortgage gap and in the ratio of
agreements to closings.

\begin{table}[H]
\centering
\caption{Matching-efficiency versus approval shocks in the financing extension (model-generated)}
\label{tab:match_vs_approval}
\footnotesize
\resizebox{\textwidth}{!}{\input{tables/tab04e_matching_vs_approval}}
\end{table}

\subsection{Bargaining power and entry costs}
\label{sec:bargaining}

A higher seller share $\beta$ raises prices, lowers tightness sharply because buyers' expected surplus per meeting falls, and
lengthens time on market (Table~\ref{tab:welfare}): between $\beta=0.2$ and $\beta=0.8$ tightness falls from \thetaBetaTwenty{} to \thetaBetaEighty{}
buyers per listing and the time on market of sales rises from \tomBetaTwenty{} to \tomBetaEighty{} months. The urgency discount falls with $\beta$, since
the carrying cost enters the price with weight $1-\beta$ in \eqref{eq:price}, but it does not vanish as $\beta\to1$, because urgency
still governs which matches are accepted. Entry and search costs work through the same channel: halving the buyer's search cost
raises tightness and prices and shortens sellers' time on market (Table~\ref{tab:entry}, Appendix~\ref{app:extra}).

\subsection{Heterogeneity and dispersion}
\label{sec:heterogeneity}

Table~\ref{tab:heterogeneity} varies the upper bound of the carrying-cost distribution, which scales its dispersion, and the
seller's bargaining share. Widening the support from $c_H=0.3$ to $0.6$ raises the between-type share of price variance from
\betweenShareCmaxThree{} to \betweenShareCmaxSix{} percent and the discount of the most urgent type from \DurgentBase{} to
\DurgentCmaxSix{} percent; lowering matching efficiency to 70 percent of baseline raises the between-type share from
\betweenShareBase{} to \betweenShareLowA{} percent and the total standard deviation from \sdPriceBase{} to \sdPriceLowA{} percent,
the widening of Proposition~\ref{prop:dispersion}(iii). As $\beta\to0$ the within-type component vanishes and all dispersion is
urgency-driven; as $\beta\to1$ the between-type component shrinks but, as noted, does not vanish (Table~\ref{tab:disp_beta},
Appendix~\ref{app:extra}).

\begin{table}[H]
\centering
\caption{Seller heterogeneity and price dispersion among identical houses (model-generated)}
\label{tab:heterogeneity}
\footnotesize
\resizebox{\textwidth}{!}{\input{tables/tab05_heterogeneity}}
\begin{tablenotes}
\footnotesize
\item Note: $c_H$ is the upper bound of the carrying-cost support (Beta$(2,4)$ shape rescaled), ``between'' the share of the
transaction-price variance between seller types, ``$\bar p$ range'' the gap in expected price between the most patient and the
most urgent type.
\end{tablenotes}
\end{table}

\subsection{Propagation and recovery}
\label{sec:dynamics}

Figure~\ref{fig:irf} and Table~\ref{tab:dynamics} show transition paths after a permanent 20 percent fall in matching
efficiency, a permanent 20 percent rise, and a temporary 20 percent fall that lasts one year. Prices adjust immediately: the
observed index falls by \dynPriceImpact{} percent on impact and stays there, because the seller value is forward-looking and
tightness, pinned by free entry at every instant, barely moves along the path. Quantities adjust slowly: the sales flow falls by
\dynQImpact{} percent on impact and recovers to \dynQEnd{} percent below its initial level as the stock of listings builds up,
inventory rises by \dynSEnd{} percent with a half-life of \dynSHalfLifeMonths{} months, and the time on market of completed sales
rises and stays up. After the temporary shock, the end of the shock releases the accumulated inventory in a burst of sales
\dynTempQRebound{} percent above the initial flow before the economy returns to its original steady state. A permanent fall in
qualification produces the same path as the fall in $A$, by the observational equivalence of the baseline.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig10_shock_impulse_responses.pdf}
\caption{Transition paths after liquidity shocks at date zero (model-generated, free entry at every instant). Permanent and
temporary 20 percent changes in matching efficiency. Panel (a): tightness. Panel (b): observed transaction-price index relative
to the initial steady state. Panel (c): sales flow. Panel (d): inventory of listings. Panel (e): mean time on market of completed
sales. Panel (f): composition term $\ln(P_t/P^{fix}_t)$ in basis points, where $P^{fix}_t$ uses the pre-shock transaction weights.}
\label{fig:irf}
\end{figure}

\begin{table}[H]
\centering
\caption{Transition experiments: impact and long-run values (model-generated)}
\label{tab:dynamics}
\footnotesize
\resizebox{\textwidth}{!}{\input{tables/tab06b_dynamics}}
\begin{tablenotes}
\footnotesize
\item Note: ``comp.'' is the composition term $\ln(P_t/P^{fix}_t)$ in basis points; ``conv.'' and ``iter.'' report the convergence
flag and the number of iterations on the tightness path.
\end{tablenotes}
\end{table}

\subsection{Sale selection and the observed price index}
\label{sec:selection}

The composition term of \eqref{eq:decomp} is at most \dynCompMaxBp{} basis points along every transition path in
Table~\ref{tab:dynamics}, against a within-type price change of about 150 basis points: in the baseline, selection is an order of
magnitude smaller than the liquidity effect it accompanies, and it dampens rather than amplifies. Table~\ref{tab:selection} and
Figure~\ref{fig:index} apply the impact decomposition of Proposition~\ref{prop:selection} to four match-value distributions and
three widths of the carrying-cost distribution, for 30 percent falls and rises in matching efficiency. The sign condition works
as the proposition states. For the truncated normal and the Weibull with shape two, both light-tailed with $h_GM$ increasing over
the relevant range, the hazard elasticity rises with urgency, the sales mix shifts toward patient sellers after a negative shock,
and the index falls by \emph{less} than within-type prices (ratio below one). For the exponential, where $h_GM=1$, and for the
Pareto, where $h_GM$ is a constant above one, the elasticity falls with urgency and the index falls by more. The magnitudes are
small in every case: the largest composition term is \compMaxAbs{} log-percent (\compMaxG, $c_H=\compMaxCmax$, $A\times\compMaxA$),
and the ratio of the observed to the within-type change lies between \ampRatioMin{} and \ampRatioMax. In the baseline
distribution with $c_H=0.3$ the within-type change after a 30 percent fall in $A$ is \withinTN{} percent and the composition term
\compTN{} percent; with exponential valuations the figures are \withinExp{} and \compExp{} percent. Steady-state composition
effects, which exist only through withdrawals, are smaller still.

\begin{table}[H]
\centering
\caption{Impact decomposition of the observed price index by match-value distribution (model-generated, general equilibrium,
$\pm30$ percent matching-efficiency shocks)}
\label{tab:selection}
\footnotesize
\resizebox{\textwidth}{!}{\input{tables/tab06_selection_bias}}
\begin{tablenotes}
\footnotesize
\item Note: ``within'' is $\ln(P^{fix}/P_0)$, ``comp.\ impact'' is $\ln(P^{obs}/P^{fix})$ with the pre-shock stock, ``comp.\ s.s.''
the same across steady states, ``ratio'' the observed over the within-type log change, $\epsilon(c)$ the hazard elasticity of the
most patient and most urgent type.
\end{tablenotes}
\end{table}

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig09_observed_vs_latent_index.pdf}
\caption{Sale selection and the transaction-price index (model-generated, $c_H=0.3$). Panel (a): within-type (liquidity) term of
the impact decomposition for four match-value distributions. Panel (b): composition (selection) term in basis points. Panel (c):
the sign condition of Proposition~\ref{prop:selection}: the product of the hazard function and the mean residual valuation,
$h_G(v)M(v)$, is constant for the exponential and the Pareto and increasing for the truncated normal and the Weibull.}
\label{fig:index}
\end{figure}

The conclusion for measurement is twofold. A transaction-price index can move by one to two percent, in either direction, with
no change in the distribution of housing-service values, and the movement is almost entirely a within-type liquidity effect
that a hedonic or repeat-sales index cannot remove, because the carrying cost of the seller is not a characteristic of the house.
The additional bias from the changing composition of sellers who transact, the object of \citet{gatzlaff1997selection} and
\citet{goetzmann2006reservation}, is of second order under the present mechanism and has a sign that depends on the shape of the
distribution of buyer valuations. Larger composition effects would require a wider dispersion of urgency than the one assumed, or
mechanisms outside the model, such as the foreclosure externalities of \citet{guren2020foreclosures}.

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

Under posted prices sellers hold out for better matches: tightness falls to \postedTheta, the time on market of sales rises to
\postedTom{} months, the expected price conditional on sale rises to \postedPrice, and the sales flow falls (Table~\ref{tab:protocols},
Figure~\ref{fig:posted}). All dispersion is between types by construction, and the total standard deviation of prices falls to
\postedSd{} percent. The urgency gradient is preserved, with a discount of \postedDurg{} percent for the most urgent type, and
flow welfare is \postedWelfareGap{} percent lower than under bargaining, because the monopoly mark-up that each seller applies to
her own listing delays trade without creating surplus. The qualitative results of the previous subsections do not depend on the
price-setting protocol.

\begin{table}[H]
\centering
\caption{Nash bargaining versus posted prices at three levels of matching efficiency (model-generated)}
\label{tab:protocols}
\footnotesize
\resizebox{\textwidth}{!}{\input{tables/tab09_protocols}}
\end{table}

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig14_posted_vs_nash.pdf}
\caption{Price-setting protocols (model-generated, baseline). Panel (a): expected price by seller type under Nash bargaining and
the posted price under commitment. Panel (b): sale hazard by seller type.}
\label{fig:posted}
\end{figure}

\subsection{Sensitivity and robustness}
\label{sec:robustness}

Figure~\ref{fig:tornado} and Table~\ref{tab:sensitivity} (Appendix~\ref{app:extra}) report one-at-a-time changes of $\pm25$
percent in every parameter, together with the dispersion of match values and the width of the carrying-cost distribution. The
urgency discount is most sensitive to the width of the carrying-cost distribution, to the own-use value of withdrawal and to the
dispersion of valuations; time on market is most sensitive to the own-use value, the dispersion of valuations and bargaining
power. In \globalN{} random draws of all parameters within wide ranges, an equilibrium exists and is unique in every draw, the
urgency discount is positive and expected prices are decreasing in urgency in \globalSignD{} percent of draws, and hazards are
increasing in urgency in \globalSignEta{} percent (the exceptions are draws in which hazards are flat to numerical precision, not
reversals); time on market ranges from \globalTomMin{} to \globalTomMax{} months and the discount of the most urgent type from
\globalDurgMin{} to \globalDurgMax{} percent. Alternative match-value families (Table~\ref{tab:distributions}, Appendix~\ref{app:extra})
preserve every sign; heavy-tailed families produce equilibria with prices above the mean valuation and a between-type share of
dispersion above 90 percent, which is why they are used only to illustrate the sign conditions.

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig12_sensitivity_tornado.pdf}
\caption{One-at-a-time sensitivity (model-generated). Change in the urgency discount of the most urgent type (left) and in the
mean time on market of sales (right) when each parameter is set to 75 and 125 percent of its baseline value ($\sigma_w$: 0.05 and
0.20; $c_{max}$: 0.2 and 0.45).}
\label{fig:tornado}
\end{figure}