\section{Numerical solution, parameterisation and validation}
\label{sec:numerical}

\subsection{Algorithms}

The seller block \eqref{eq:S} is solved type by type as the unique root of the strictly decreasing function
$\Phi(V)=-c+\delta w_0+\Lambda_s\beta H(V+U)-(r+\delta)V$ (Lemma~\ref{lem:value}) by Brent's method with a bracket that is widened
automatically when the support of $G$ is unbounded. The distribution of carrying costs $F$ is discretised into $\nTypes$ types at
the conditional means of equiprobable-mass bins of a Beta distribution; the expected surplus $H$, the mean residual valuation $M$
and the conditional variance of $G$ are computed in closed form for the truncated normal, exponential, Pareto and Weibull
families. Equilibrium tightness is found by scanning the free-entry residual on a logarithmic grid of 60 points in $[0.02,60]$
and applying Brent's method on every sign change; the number of roots is recorded, so that multiplicity would be reported rather
than hidden. The posted-price protocol nests the seller's first-order condition inside the value root. The financing extension
is solved as a three-equation root problem in $(\ln\theta,U_c,\operatorname{logit}\tilde\pi)$ with $U_m$ imposed by ex-ante free entry;
a damped fixed-point iteration on the buyer values was tried first and diverged, because the buyer map has a slope of order
$\lambda_b/r$, and was replaced. The planner's steady-state problem maximises \eqref{eq:welfare} over tightness and the vector of
acceptance thresholds by derivative-free optimisation started at the decentralised allocation. Transition paths solve the
backward equation \eqref{eq:HJBdyn} by a semi-implicit scheme, the forward stock equation by implicit Euler, and the pointwise
free-entry condition by damped iteration on the tightness path until the maximum update falls below $10^{-7}$. Tolerances,
iteration counts and residuals are stored with every result file; Appendix~\ref{app:computation} gives details.

\subsection{Parameterisation}

The parameterisation is illustrative. No moment is estimated and no data are used; the values in Table~\ref{tab:parameters}
are chosen so that the baseline equilibrium displays durations, discounts and dispersion of the order of magnitude reported in
the literature reviewed in Section~\ref{sec:literature}, and every experiment reports how the results move when they change.
Time is measured in years and values in units of the mean buyer valuation of the segment. The discount rate is 4 percent. Match
values are normally distributed with a 10 percent standard deviation and truncated at three standard deviations, which makes
$G$ log-concave. Carrying costs range from 2 percent of the house value per year (maintenance and property taxes for an owner
who still lives in the house) to 30 percent (a seller carrying two homes, a bridge loan or a relocation deadline) with a
right-skewed Beta$(2,4)$ shape, so that most sellers are patient and a minority urgent. Withdrawals occur at the rate 0.5 per
year, which at baseline hazards removes about a tenth of listings, and a withdrawing seller keeps a house worth 0.8 to her. A
buyer pays an entry cost of 2 percent of the house value and a flow search cost of 5 percent per year, a figure that includes
temporary housing and that must be comparable to sellers' carrying costs for the free-entry condition to deliver a tightness of
a few buyers per listing; 85 percent of meetings are with qualified buyers. The matching function is Cobb--Douglas with
elasticity one half and efficiency 4.5 meetings per year per side at unit tightness: a meeting is a serious offer, not a visit.
The seller's Nash share is one half, the Hosios value for $\alpha=1/2$. The financing extension uses a 30 percent share of cash
buyers, a one-month expected financing decision and an approval probability of 0.85.

\begin{table}[H]
\centering
\caption{Baseline parameterisation (illustrative; units of the mean buyer valuation, time in years)}
\label{tab:parameters}
\footnotesize
\input{tables/tab01_parameters}
\end{table}

\subsection{Validation}

Twenty-four unit tests cover the closed forms of the distributions, the residuals and uniqueness of the seller value, the
analytic derivatives of Lemma~\ref{lem:value} against finite differences, each part of Propositions~\ref{prop:urgency}
to~\ref{prop:posted} on the baseline parameterisation, the Pareto counterexample in both regions of \eqref{eq:pareto}, the
exponential elasticity formula of Proposition~\ref{prop:selection}, boundary cases (zero urgency and no withdrawal, a frozen market
with $\Lambda_s\to0$, a very liquid market with $\Lambda_s=10^7$ in which $v^*\to w_H$ and $\bar p\to w_H-k$), the equilibrium stock
equations and probability sums, and the convergence of the equilibrium in the type grid and the root tolerance. All tests pass.
Independently of the analytic solution, the listing process was simulated listing by listing (type draw, Poisson meetings,
exponential withdrawal, match draws and the acceptance rule) for \mcN{} listings and three seeds; the simulated sold share,
durations, meetings per listing, mean and standard deviation of prices agree with the analytic moments to within
\mcMaxRelErr{} percent, and the per-type mean prices to within \mcTypesRelErr{} percent. With 9 seller types the mean price differs
from the solution with \convTypesMax{} types by \convTypesRelErr{} percent (Figure~\ref{fig:convergence}); tightness is invariant to
the root tolerance between $10^{-6}$ and $10^{-12}$; and the transition solver converges at first order in the time step.
Appendix~\ref{app:computation} reports these diagnostics in full.