1\section{Welfare and policy}2\label{sec:welfare}34Table~\ref{tab:welfare} and Figure~\ref{fig:welfare} report the steady-state flow surplus \eqref{eq:welfare} across bargaining5shares and matching elasticities, for the baseline economy with heterogeneous sellers and for an economy with a single seller6type at the mean carrying cost. Welfare is hump-shaped in the seller's share. Too low a share crowds the market with buyers who7search long and transact at low prices; too high a share deters entry and leaves sellers waiting. The welfare-maximising share is8$\beta=\betaStarHet$ in both economies, slightly above the Hosios value $1-\alpha=0.5$; the welfare loss at the Hosios value is9\welfareLossHet{} percent and the loss at $\beta=0.2$ is \welfareLossBetaTwenty{} percent. Across matching elasticities the10welfare-maximising share tracks the Hosios line with a small upward displacement (Figure~\ref{fig:welfare}(a)). The displacement11reflects the criterion, not heterogeneity: the flow surplus does not discount across cohorts, whereas the Hosios condition12characterises the discounted problem, and the homogeneous economy displays the same displacement. Heterogeneity in carrying costs13adds a wedge that is numerically negligible at the baseline dispersion: the planner's flow optimum exceeds the decentralised14one by \plannerGainHet{} percent with heterogeneous sellers and by \plannerGainHom{} percent with homogeneous sellers, with lower15tightness in both cases (\plannerThetaHet{} against \thetaBase{} buyers per listing), and the planner's acceptance thresholds are16flatter across urgency types than the decentralised ones (Table~\ref{tab:planner}, Appendix~\ref{app:extra}). These are numerical17statements for the flow criterion, not theorems, and they do not extend to economies with endogenous listing, where18\citet{gabrovski2025note} show that two-sided entry breaks the Hosios condition.1920\begin{table}[H]21\centering22\caption{Bargaining power and steady-state flow welfare, heterogeneous and homogeneous sellers (model-generated, $\alpha=0.5$)}23\label{tab:welfare}24\footnotesize25\resizebox{\textwidth}{!}{\input{tables/tab08_welfare}}26\begin{tablenotes}27\footnotesize28\item Note: ``welfare'' is the flow surplus \eqref{eq:welfare}; ``sellers' value'' the ex-ante value of a listing cohort net of the29own-use value; ``holding costs'' the flow of carrying costs paid by the stock of listings.30\end{tablenotes}31\end{table}3233\begin{figure}[H]34\centering35\includegraphics[width=\textwidth]{fig11_welfare_friction_contours.pdf}36\caption{Welfare and market frictions (model-generated). Panel (a): steady-state flow surplus over the seller's bargaining share37and the matching elasticity, heterogeneous sellers; the dashed line is the Hosios locus $\beta=1-\alpha$ and the dots the38welfare-maximising share at each elasticity. Panel (b): welfare as a function of the bargaining share at $\alpha=0.5$ for39heterogeneous and homogeneous sellers.}40\label{fig:welfare}41\end{figure}4243Three policy readings follow, each conditional on the model's assumptions. First, measures that raise matching efficiency or44qualification, such as faster or more reliable mortgage decisions, raise prices, shorten durations and narrow the urgency45discount, and part of their effect works through buyer entry; the general-equilibrium elasticities of Section~\ref{sec:arrival}46exceed the partial-equilibrium ones by about ten percent. Second, the welfare cost of a liquidity shock is borne by sellers in47proportion to their urgency: the holding-cost flow and the reservation-value gap both rise when arrivals fall, so that policies48that reduce the carrying cost of urgent sellers (bridge financing, forbearance) have a distributional rationale even when total49surplus is little affected. Third, the welfare of the market is insensitive to the split of the surplus in a wide neighbourhood of50the Hosios share but falls steeply outside it, and institutions that shift bargaining power toward one side, such as the bidding51wars documented by \citet{han2014biddingwars} or the brokerage arrangements studied by \citet{levitt2008agents}, matter for52welfare through their effect on entry rather than through the price level.53