\section{Limitations, falsifiable predictions and conclusion}
\label{sec:conclusion}

\subsection{Limitations}

The model is stationary and local, with exogenous listing inflow, no construction and no spillovers across segments. Urgency is a
permanent, nonnegative carrying cost, so sellers whose reservation price rises with their constraints, as in
\citet{genesove1997equity} and \citet{genesove2001lossaversion}, and mechanisms that generate positive duration dependence, such as
list-price revisions, learning \citep{anenberg2016information} and stigma \citep{taylor1999time}, are outside its scope.
Information is complete within a match, so the adverse-selection channels of \citet{kurlat2015asymmetries} and
\citet{garmaise2004asymmetries} are absent. Buyers never exit, which makes the financing extension inadmissible when the approval
probability is low. The welfare criterion is a steady-state flow, and the welfare results are numerical statements for that
criterion. Transition paths assume unanticipated shocks and free entry at every instant, which makes prices jump; anticipation or
sluggish entry would smooth them. Quality is a scale factor, so the correlation between distress and condition that confounds
measured forced-sale discounts \citep{clauretie2009foreclosure} is not modelled. Finally, no data were used: all magnitudes are
illustrative, and the paper claims nothing about any particular market.

\subsection{Falsifiable predictions}

Six predictions distinguish the mechanism from its alternatives. (1) Among observationally identical houses, the price
concession of urgent sellers is larger, and their sale hazard higher, in markets with fewer buyers per listing. (2) The dispersion
of prices among identical houses rises when matching efficiency or qualification falls. (3) After a liquidity shock, within-type
prices adjust at once while inventories and volumes adjust over months, and a temporary shock is followed by a rebound of sales.
(4) The cash--mortgage gap rises with seller urgency and with approval delay, and its selection component can be negative for
patient sellers, so that measured gaps understate the delay-bargaining compensation when sellers are patient. (5) The
steady-state sales-to-listings ratio falls with liquidity only in markets with non-trivial withdrawal. (6) In segments with
heavy-tailed buyer valuations, greater buyer traffic can lengthen the time on market of urgent sellers. The natural data are
listing-level records with time on market, withdrawals, financing type and a proxy for the seller's circumstances, as in
\citet{glower1998motivation} or \citet{andersen2017firesales}, with the competing-risks structure that Lemma~\ref{lem:duration}
implies.

\subsection{Conclusion}

The paper has formalised the housing liquidity discount as the price of urgency in a search market and has shown how it is
determined jointly with sale hazards, tightness and the composition of transactions. Four results carry the weight. The expected
transaction price falls with the seller's carrying cost by less than the reservation price does, and the gap between patient and
urgent sellers widens when buyers are scarce, provided the distribution of buyer valuations has an increasing failure rate; with
heavy-tailed valuations a more liquid market can lengthen time on market, under a condition that depends on the seller's
carrying cost but not on the arrival rate. A fall in matching efficiency or in qualification moves prices, volume, inventory and
durations together with no change in the distribution of housing-service values, and buyer entry amplifies the response. The
composition of transactions changes after such a shock in a direction that depends on the shape of the valuation distribution,
and its effect on a transaction-price index is an order of magnitude smaller than the within-type liquidity effect that no
quality adjustment can remove. The gap between what mortgage and cash buyers pay decomposes into a delay-bargaining term that
rises with urgency and whose sign depends on who bears the larger cost of waiting, an outside-option term, and a selection term
that can be negative. The frictionless benchmark against which these objects are measured is not a liquidity premium in the
usual sense, since a seller with the option to wait for a high-valuation buyer may obtain more than the Walrasian price in a
sufficiently thick market. The next step is empirical: the predictions above are stated in terms of objects that listing-level
data can measure.