\section{Welfare and policy}
\label{sec:welfare}

Table~\ref{tab:welfare} and Figure~\ref{fig:welfare} report the steady-state flow surplus \eqref{eq:welfare} across bargaining
shares and matching elasticities, for the baseline economy with heterogeneous sellers and for an economy with a single seller
type at the mean carrying cost. Welfare is hump-shaped in the seller's share. Too low a share crowds the market with buyers who
search long and transact at low prices; too high a share deters entry and leaves sellers waiting. The welfare-maximising share is
$\beta=\betaStarHet$ in both economies, slightly above the Hosios value $1-\alpha=0.5$; the welfare loss at the Hosios value is
\welfareLossHet{} percent and the loss at $\beta=0.2$ is \welfareLossBetaTwenty{} percent. Across matching elasticities the
welfare-maximising share tracks the Hosios line with a small upward displacement (Figure~\ref{fig:welfare}(a)). The displacement
reflects the criterion, not heterogeneity: the flow surplus does not discount across cohorts, whereas the Hosios condition
characterises the discounted problem, and the homogeneous economy displays the same displacement. Heterogeneity in carrying costs
adds a wedge that is numerically negligible at the baseline dispersion: the planner's flow optimum exceeds the decentralised
one by \plannerGainHet{} percent with heterogeneous sellers and by \plannerGainHom{} percent with homogeneous sellers, with lower
tightness in both cases (\plannerThetaHet{} against \thetaBase{} buyers per listing), and the planner's acceptance thresholds are
flatter across urgency types than the decentralised ones (Table~\ref{tab:planner}, Appendix~\ref{app:extra}). These are numerical
statements for the flow criterion, not theorems, and they do not extend to economies with endogenous listing, where
\citet{gabrovski2025note} show that two-sided entry breaks the Hosios condition.

\begin{table}[H]
\centering
\caption{Bargaining power and steady-state flow welfare, heterogeneous and homogeneous sellers (model-generated, $\alpha=0.5$)}
\label{tab:welfare}
\footnotesize
\resizebox{\textwidth}{!}{\input{tables/tab08_welfare}}
\begin{tablenotes}
\footnotesize
\item Note: ``welfare'' is the flow surplus \eqref{eq:welfare}; ``sellers' value'' the ex-ante value of a listing cohort net of the
own-use value; ``holding costs'' the flow of carrying costs paid by the stock of listings.
\end{tablenotes}
\end{table}

\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{fig11_welfare_friction_contours.pdf}
\caption{Welfare and market frictions (model-generated). Panel (a): steady-state flow surplus over the seller's bargaining share
and the matching elasticity, heterogeneous sellers; the dashed line is the Hosios locus $\beta=1-\alpha$ and the dots the
welfare-maximising share at each elasticity. Panel (b): welfare as a function of the bargaining share at $\alpha=0.5$ for
heterogeneous and homogeneous sellers.}
\label{fig:welfare}
\end{figure}

Three policy readings follow, each conditional on the model's assumptions. First, measures that raise matching efficiency or
qualification, such as faster or more reliable mortgage decisions, raise prices, shorten durations and narrow the urgency
discount, and part of their effect works through buyer entry; the general-equilibrium elasticities of Section~\ref{sec:arrival}
exceed the partial-equilibrium ones by about ten percent. Second, the welfare cost of a liquidity shock is borne by sellers in
proportion to their urgency: the holding-cost flow and the reservation-value gap both rise when arrivals fall, so that policies
that reduce the carrying cost of urgent sellers (bridge financing, forbearance) have a distributional rationale even when total
surplus is little affected. Third, the welfare of the market is insensitive to the split of the surplus in a wide neighbourhood of
the Hosios share but falls steeply outside it, and institutions that shift bargaining power toward one side, such as the bidding
wars documented by \citet{han2014biddingwars} or the brokerage arrangements studied by \citet{levitt2008agents}, matter for
welfare through their effect on entry rather than through the price level.