\section{Analytical results}
\label{sec:propositions}

Proofs are in Appendix~\ref{app:proofs}. Unless stated otherwise, the buyer value $U$ is held fixed (under free entry $U=k$) and
$\Lambda_s>0$ is given; results that involve the response of tightness are stated separately.

\begin{lemma}[Scale invariance in quality]
\label{lem:scale}
Let $w=q\varepsilon$ with $\varepsilon\sim G_\varepsilon$, $c=q\kappa$, $k=q\tilde k$ and $\kappa_b=q\tilde\kappa_b$. If $(\theta,U,V,n)$
is a stationary equilibrium of the economy with $q=1$, then $(\theta,qU,qV,n)$ is a stationary equilibrium of the economy with
quality $q$; prices scale by $q$, while hazards, durations, tightness, the transaction composition and the discounts $D(c)$ are
invariant.
\end{lemma}

\begin{lemma}[Seller value]
\label{lem:value}
For any $U$, $\Lambda_s>0$ and $c\ge0$, equation \eqref{eq:S} has a unique solution $V(c)$, continuously differentiable in
$(c,\Lambda_s,U)$, with
\begin{equation}
\frac{\partial V}{\partial c}=\frac{-1}{(r+\delta)+\Lambda_s\beta(1-G(v^*))},\qquad
\frac{\partial V}{\partial\Lambda_s}=\frac{\beta H(v^*)}{(r+\delta)+\Lambda_s\beta(1-G(v^*))},\qquad
\frac{\partial V}{\partial U}\in(-1,0].
\label{eq:Vderiv}
\end{equation}
\end{lemma}

\begin{lemma}[Spurious negative duration dependence]
\label{lem:duration}
If the exit hazard $\mu(c)=\eta(c)+\delta$ is non-degenerate under $F$, the population exit hazard
$\bar h(\tau)=-\frac{d}{d\tau}\ln\int e^{-\mu(c)\tau}dF(c)$ is strictly decreasing in listing duration $\tau$, although every
individual hazard is constant.
\end{lemma}

\begin{proposition}[Seller urgency]
\label{prop:urgency}
(i) The reservation price falls with the carrying cost, by less than one-for-one in present-value terms:
$V'(c)\in(-1/(r+\delta),0)$. (ii) The sale hazard $\eta(c)$ is nondecreasing in $c$, strictly where $g(v^*(c))>0$, and the expected
listing duration is nonincreasing. (iii) The expected transaction price is strictly decreasing in $c$,
$\bar p'(c)=V'(c)\,[1+\beta M'(v^*(c))]$ with $1+\beta M'>1-\beta>0$, and under IFR $|\bar p'(c)|\le|V'(c)|$: the price falls by less
than the reservation price. The urgency discount $D(c)$ is therefore zero at $c_L$, nonnegative and strictly increasing.
(iv) Under free entry a type-$c$ seller prefers listing to keeping the house iff $c\le\bar c\equiv\Lambda_s\beta H(w_0+k)-rw_0$.
\end{proposition}

Part (i) is the classical reservation-value result of \citet{lippman1976economics} in the present setting and the
continuous-type counterpart of the desperate-seller effect of \citet{albrecht2007opportunistic}; it is stated for completeness
and because the magnitude of the slope, $1/[(r+\delta)+\beta\eta(c)]$, is what later results build on. Part (iii) contains the
less obvious point that transaction prices respond to urgency \emph{less} than reservation prices. An urgent seller accepts
lower matches, but conditional on acceptance the buyer's surplus is larger on average, and the seller captures the share $\beta$
of it. No log-concavity is needed for the sign; IFR is needed only for the attenuation.

\begin{proposition}[Effective arrival rate]
\label{prop:arrival}
(i) $\partial V/\partial\Lambda_s>0$ and $\partial\bar p/\partial\Lambda_s>0$ whenever $v^*<w_H$. (ii) If $G$ is IFR,
$\partial\eta/\partial\Lambda_s>0$: a decline in buyer arrivals or in qualification lengthens expected time on market. The sign can
reverse without IFR: for the Pareto distribution $G(w)=1-(w_L/w)^a$, $a>1$,
\begin{equation}
\frac{\partial\eta}{\partial\Lambda_s}<0\quad\Longleftrightarrow\quad \beta\eta(c)>(a-1)(r+\delta)\quad\Longleftrightarrow\quad
c>\delta w_0+(r+\delta)U,
\label{eq:pareto}
\end{equation}
a condition that does not depend on $\Lambda_s$. (iii) Under IFR the reservation-value gap $V(c_L)-V(c)$ is strictly decreasing in
$\Lambda_s$ for every $c>c_L$: lower liquidity widens the cross-sectional dispersion of reservation prices. (iv) For the patient
type $c=0$ with $U=k\le w_L$ and $w_H<\infty$, $\bar p\to\beta(E[w]-k)$ as $\Lambda_s\to0$ and $\bar p\to w_H-k$ as $\Lambda_s\to\infty$; if
$\beta(E[w]-k)<p^*<w_H-k$, which holds iff $H(w_H-k)<rk$, the market liquidity gap $\Gamma$ is positive in thin markets and
negative in thick ones.
\end{proposition}

Part (ii) is the result that a more liquid market shortens time on market, together with the condition under which it fails.
A higher arrival rate raises the seller's reservation valuation (part (i)); whether the hazard rises depends on whether the
direct effect of more meetings dominates the indirect effect of greater selectivity. Under IFR the mean residual valuation is
bounded by the reciprocal hazard, $M\le1/h_G$, which is exactly what is needed. With heavy-tailed valuations the seller's option to
wait for an exceptional buyer becomes more valuable faster than buyers arrive, and the condition in \eqref{eq:pareto} shows that
this happens for every sufficiently urgent seller, at every arrival rate. The empirical counterpart is a segment of unique or
luxury properties in which greater buyer traffic does not shorten marketing time. Part (iv) is the formal reason for not calling
$\Gamma$ a liquidity premium.

\begin{proposition}[Free entry and tightness]
\label{prop:entry}
(a) With homogeneous sellers the free-entry residual $R(\theta)=-\kappa_b+\phi\lambda_b(\theta)(1-\beta)H(V(\theta)+k)-rk$ is continuous
and strictly decreasing from $+\infty$ to $-\kappa_b-rk$, so equilibrium tightness $\theta^*$ is unique; $d\theta^*/d\phi>0$,
$d\theta^*/dA>0$, and $d\ln\Lambda_s/d\ln\phi=1+\alpha\,d\ln\theta^*/d\ln\phi>1$. In the baseline, $A$ and $\phi$ enter all equilibrium
conditions only through the product $\phi A$. (b) With heterogeneous sellers an equilibrium exists; uniqueness is not guaranteed,
because the stock composition $N$ depends on $\theta$, and is verified numerically.
\end{proposition}

\begin{proposition}[A liquidity shock without a fundamental shock]
\label{prop:shock}
Let $G$ be IFR. A fall in $A$ or in $\phi$ with all other primitives unchanged (i) lowers $V(c)$ and $\bar p(c)$ for every $c$, in
partial equilibrium and, since $\theta^*$ falls as well, in general equilibrium with homogeneous sellers; (ii) lowers every sale
hazard and the sales flow on impact, when the stock of listings is predetermined; (iii) raises the steady-state inventory $S$
and mean listing duration; (iv) lowers the steady-state sales flow if and only if $\delta>0$: with $\delta=0$ every listing
eventually sells and the steady-state sales flow equals the listing inflow whatever the level of liquidity. Transaction prices and
volume therefore co-move, and inventory and time on market move inversely, with no change in the distribution of housing-service
values.
\end{proposition}

Part (iv) is a diagnostic rather than a curiosity. In a market where listings are rarely withdrawn, a permanent fall in
liquidity lowers volume only during the transition; a persistent fall in the sales-to-listings ratio across steady states
requires withdrawals or relistings, which the duration literature finds to be common \citep{anglin2003tradeoff,
benefield2015measurement}.

\begin{proposition}[Price dispersion among identical houses]
\label{prop:dispersion}
Conditional on sale,
\begin{equation}
\mathrm{Var}_\omega(p)=\beta^2E_\omega\big[\mathrm{Var}(w\mid w\ge v^*(c))\big]+\mathrm{Var}_\omega\big[\bar p(c)\big].
\label{eq:vardecomp}
\end{equation}
(i) The
between-type component is strictly positive iff the transaction composition is non-degenerate. (ii) With $\beta=0$ all dispersion
is urgency-driven and $\mathrm{Var}_\omega(p)\le\mathrm{Var}_\omega(c)/(r+\delta)^2$; under IFR the range of expected prices satisfies
$\bar p(c_L)-\bar p(c_H)\le(c_H-c_L)/(r+\delta)$. (iii) Under IFR the between-type reservation-value gap widens when $\Lambda_s$ falls.
\end{proposition}

\begin{proposition}[Sale selection and transaction-price indices]
\label{prop:selection}
(i) If $\delta=0$, the steady-state transaction composition equals the listing composition, $\omega=f$, for every $\Lambda_s$:
comparisons across steady states contain no composition effect. (ii) On impact of an unanticipated change $d\ln\Lambda_s$ with the
stock predetermined, let $\epsilon(c)=\partial\ln\eta(c)/\partial\ln\Lambda_s=1-\beta\eta(c)\,h_G(v^*)M(v^*)/[(r+\delta)+\beta\eta(c)]$.
Then $d\ln\omega(c)=[\epsilon(c)-E_\omega\epsilon]\,d\ln\Lambda_s$ and
\begin{equation}
d\ln P=E_\omega\big[d\ln\bar p(c)\big]+\frac{\mathrm{Cov}_\omega\big(\bar p(c),\epsilon(c)\big)}{P}\,d\ln\Lambda_s .
\label{eq:impact}
\end{equation}
If $\epsilon$ is decreasing in $c$ the covariance is positive and the index moves by more than every within-type price
(selection amplifies); if $\epsilon$ is increasing in $c$ the covariance is negative and selection dampens. (iii) For exponential
$G$, $\epsilon(c)=(r+\delta)/[(r+\delta)+\beta\eta(c)]$ is strictly decreasing in $c$, and for Pareto $G$ likewise; a sufficient condition
for amplification is that $v\mapsto h_G(v)M(v)$ be nonincreasing. When $h_GM$ is increasing, as for light-tailed distributions such
as the truncated normal, $\epsilon$ can be increasing in $c$ and selection dampens. (iv) With $\delta=0$ the composition returns to $f$
in the new steady state, so the impact composition effect is transitory.
\end{proposition}

The economics of part (ii) is a race between two forces. After a fall in arrivals every seller lowers the valuation she accepts,
and the share of sales made by each type changes in proportion to the elasticity of its hazard. Urgent sellers already accept
most offers, so their hazard is less elastic to arrivals through the acceptance margin, which pushes the sales mix toward them and
pulls the index down by more than the fall in any individual price; this is the mechanism of \citet{goetzmann2006reservation} seen
from the supply side. But urgent sellers also sit lower in the valuation distribution, where the product of the hazard function and
the mean residual valuation of $G$ can be smaller, which makes their acceptance threshold more responsive. The exponential
distribution, for which $h_GM=1$, is the knife-edge where only the first force operates; for lighter tails the second force can
dominate. The sign of the selection bias in a transaction index is therefore a property of the distribution of buyer
valuations, not a general law; Section~\ref{sec:selection} quantifies both cases.

\begin{proposition}[Financing contingency]
\label{prop:financing}
In the extension of Section~\ref{sec:financing}, for a seller of type $c$ and a match $w$ acceptable to both buyer types,
\begin{equation}
p_m(w,c)-p_c(w,c)=\underbrace{\frac{(1-\beta)\,[c+rV(c)]-\beta\,[\kappa_b+rU_m]}{\nu\phi_m}}_{\Delta_b(c)\ \text{(delay-bargaining term)}}
+\underbrace{\beta\,(U_c-U_m)}_{\text{outside-option term}} .
\label{eq:gap}
\end{equation}
(i) $\Delta_b(c)$ is positive iff the seller's bargaining-weighted flow cost of waiting, $(1-\beta)[c+rV(c)]$, exceeds the buyer's,
$\beta[\kappa_b+rU_m]$; it is nondecreasing in $c$ and vanishes as $\nu\to\infty$. (ii) The acceptance thresholds differ by
$v^*_m(c)-v^*_c(c)=[c+\kappa_b+r(V+U_m)]/(\nu\phi_m)-(U_c-U_m)$, a pending wedge increasing in $c$ net of the cash buyers'
outside-option advantage; the observed mean price gap conditional on sale adds the selection term
$\beta\{E[w\mid w\ge v^*_m]-E[w\mid w\ge v^*_c]\}$, which has the sign of $v^*_m-v^*_c$ and can be negative for patient sellers.
(iii) The baseline screen is the limit $\nu\to\infty$.
\end{proposition}

Equation \eqref{eq:gap} says that the party with the larger bargaining-weighted cost of delay pays for the financing contingency.
When sellers are urgent and buyers patient, mortgage buyers pay more than cash buyers for the same house: this is the cash
discount of \citet{han2024cash} and \citet{reher2024mortgage}, derived rather than assumed. When buyers bear the larger cost of
waiting, or when sellers have most of the bargaining power and extract less from a buyer whose own waiting cost reduces the
surplus, the sign reverses. The measured gap adds two further terms. Cash buyers have a better continuation value, which lowers
the price they pay by $\beta(U_c-U_m)$, and the two types of agreement are selected from different parts of the valuation
distribution. The third term is the one that \citet{reher2024mortgage} attribute to ``heterogeneity in selling conditions''; the
proposition shows that it need not be positive.

\begin{proposition}[Posted prices]
\label{prop:posted}
Let $g$ be log-concave. (i) The posted-price problem \eqref{eq:posted} has a unique interior solution $y(c)$ with
$y(c)-U-V(c)=1/h_G(y(c))$. (ii) $V(c)$ is unique with $V'(c)=-1/[(r+\delta)+\Lambda_s(1-G(y(c)))]<0$; the posted price and the
acceptance threshold are strictly decreasing in $c$ and the hazard is increasing in $c$. (iii) At equal $(V,U)$, $y(c)>V(c)+U$:
posted-price sellers hold out for better matches than bargaining sellers.
\end{proposition}

Two statements are made numerically rather than proved: the uniqueness of tightness with heterogeneous sellers
(Proposition~\ref{prop:entry}(b)), and the location of the welfare-maximising bargaining share relative to the Hosios value.
Both are reported in Section~\ref{sec:welfare} with the parameter regions in which they were checked.