1\section{Numerical solution, parameterisation and validation}2\label{sec:numerical}34\subsection{Algorithms}56The seller block \eqref{eq:S} is solved type by type as the unique root of the strictly decreasing function7$\Phi(V)=-c+\delta w_0+\Lambda_s\beta H(V+U)-(r+\delta)V$ (Lemma~\ref{lem:value}) by Brent's method with a bracket that is widened8automatically when the support of $G$ is unbounded. The distribution of carrying costs $F$ is discretised into $\nTypes$ types at9the conditional means of equiprobable-mass bins of a Beta distribution; the expected surplus $H$, the mean residual valuation $M$10and the conditional variance of $G$ are computed in closed form for the truncated normal, exponential, Pareto and Weibull11families. Equilibrium tightness is found by scanning the free-entry residual on a logarithmic grid of 60 points in $[0.02,60]$12and applying Brent's method on every sign change; the number of roots is recorded, so that multiplicity would be reported rather13than hidden. The posted-price protocol nests the seller's first-order condition inside the value root. The financing extension14is 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;15a damped fixed-point iteration on the buyer values was tried first and diverged, because the buyer map has a slope of order16$\lambda_b/r$, and was replaced. The planner's steady-state problem maximises \eqref{eq:welfare} over tightness and the vector of17acceptance thresholds by derivative-free optimisation started at the decentralised allocation. Transition paths solve the18backward equation \eqref{eq:HJBdyn} by a semi-implicit scheme, the forward stock equation by implicit Euler, and the pointwise19free-entry condition by damped iteration on the tightness path until the maximum update falls below $10^{-7}$. Tolerances,20iteration counts and residuals are stored with every result file; Appendix~\ref{app:computation} gives details.2122\subsection{Parameterisation}2324The parameterisation is illustrative. No moment is estimated and no data are used; the values in Table~\ref{tab:parameters}25are chosen so that the baseline equilibrium displays durations, discounts and dispersion of the order of magnitude reported in26the literature reviewed in Section~\ref{sec:literature}, and every experiment reports how the results move when they change.27Time is measured in years and values in units of the mean buyer valuation of the segment. The discount rate is 4 percent. Match28values are normally distributed with a 10 percent standard deviation and truncated at three standard deviations, which makes29$G$ log-concave. Carrying costs range from 2 percent of the house value per year (maintenance and property taxes for an owner30who still lives in the house) to 30 percent (a seller carrying two homes, a bridge loan or a relocation deadline) with a31right-skewed Beta$(2,4)$ shape, so that most sellers are patient and a minority urgent. Withdrawals occur at the rate 0.5 per32year, which at baseline hazards removes about a tenth of listings, and a withdrawing seller keeps a house worth 0.8 to her. A33buyer pays an entry cost of 2 percent of the house value and a flow search cost of 5 percent per year, a figure that includes34temporary housing and that must be comparable to sellers' carrying costs for the free-entry condition to deliver a tightness of35a few buyers per listing; 85 percent of meetings are with qualified buyers. The matching function is Cobb--Douglas with36elasticity one half and efficiency 4.5 meetings per year per side at unit tightness: a meeting is a serious offer, not a visit.37The seller's Nash share is one half, the Hosios value for $\alpha=1/2$. The financing extension uses a 30 percent share of cash38buyers, a one-month expected financing decision and an approval probability of 0.85.3940\begin{table}[H]41\centering42\caption{Baseline parameterisation (illustrative; units of the mean buyer valuation, time in years)}43\label{tab:parameters}44\footnotesize45\input{tables/tab01_parameters}46\end{table}4748\subsection{Validation}4950Twenty-four unit tests cover the closed forms of the distributions, the residuals and uniqueness of the seller value, the51analytic derivatives of Lemma~\ref{lem:value} against finite differences, each part of Propositions~\ref{prop:urgency}52to~\ref{prop:posted} on the baseline parameterisation, the Pareto counterexample in both regions of \eqref{eq:pareto}, the53exponential elasticity formula of Proposition~\ref{prop:selection}, boundary cases (zero urgency and no withdrawal, a frozen market54with $\Lambda_s\to0$, a very liquid market with $\Lambda_s=10^7$ in which $v^*\to w_H$ and $\bar p\to w_H-k$), the equilibrium stock55equations and probability sums, and the convergence of the equilibrium in the type grid and the root tolerance. All tests pass.56Independently of the analytic solution, the listing process was simulated listing by listing (type draw, Poisson meetings,57exponential withdrawal, match draws and the acceptance rule) for \mcN{} listings and three seeds; the simulated sold share,58durations, meetings per listing, mean and standard deviation of prices agree with the analytic moments to within59\mcMaxRelErr{} percent, and the per-type mean prices to within \mcTypesRelErr{} percent. With 9 seller types the mean price differs60from the solution with \convTypesMax{} types by \convTypesRelErr{} percent (Figure~\ref{fig:convergence}); tightness is invariant to61the root tolerance between $10^{-6}$ and $10^{-12}$; and the transition solver converges at first order in the time step.62Appendix~\ref{app:computation} reports these diagnostics in full.63