1\section{Limitations, falsifiable predictions and conclusion}2\label{sec:conclusion}34\subsection{Limitations}56The model is stationary and local, with exogenous listing inflow, no construction and no spillovers across segments. Urgency is a7permanent, nonnegative carrying cost, so sellers whose reservation price rises with their constraints, as in8\citet{genesove1997equity} and \citet{genesove2001lossaversion}, and mechanisms that generate positive duration dependence, such as9list-price revisions, learning \citep{anenberg2016information} and stigma \citep{taylor1999time}, are outside its scope.10Information is complete within a match, so the adverse-selection channels of \citet{kurlat2015asymmetries} and11\citet{garmaise2004asymmetries} are absent. Buyers never exit, which makes the financing extension inadmissible when the approval12probability is low. The welfare criterion is a steady-state flow, and the welfare results are numerical statements for that13criterion. Transition paths assume unanticipated shocks and free entry at every instant, which makes prices jump; anticipation or14sluggish entry would smooth them. Quality is a scale factor, so the correlation between distress and condition that confounds15measured forced-sale discounts \citep{clauretie2009foreclosure} is not modelled. Finally, no data were used: all magnitudes are16illustrative, and the paper claims nothing about any particular market.1718\subsection{Falsifiable predictions}1920Six predictions distinguish the mechanism from its alternatives. (1) Among observationally identical houses, the price21concession of urgent sellers is larger, and their sale hazard higher, in markets with fewer buyers per listing. (2) The dispersion22of prices among identical houses rises when matching efficiency or qualification falls. (3) After a liquidity shock, within-type23prices adjust at once while inventories and volumes adjust over months, and a temporary shock is followed by a rebound of sales.24(4) The cash--mortgage gap rises with seller urgency and with approval delay, and its selection component can be negative for25patient sellers, so that measured gaps understate the delay-bargaining compensation when sellers are patient. (5) The26steady-state sales-to-listings ratio falls with liquidity only in markets with non-trivial withdrawal. (6) In segments with27heavy-tailed buyer valuations, greater buyer traffic can lengthen the time on market of urgent sellers. The natural data are28listing-level records with time on market, withdrawals, financing type and a proxy for the seller's circumstances, as in29\citet{glower1998motivation} or \citet{andersen2017firesales}, with the competing-risks structure that Lemma~\ref{lem:duration}30implies.3132\subsection{Conclusion}3334The paper has formalised the housing liquidity discount as the price of urgency in a search market and has shown how it is35determined jointly with sale hazards, tightness and the composition of transactions. Four results carry the weight. The expected36transaction price falls with the seller's carrying cost by less than the reservation price does, and the gap between patient and37urgent sellers widens when buyers are scarce, provided the distribution of buyer valuations has an increasing failure rate; with38heavy-tailed valuations a more liquid market can lengthen time on market, under a condition that depends on the seller's39carrying cost but not on the arrival rate. A fall in matching efficiency or in qualification moves prices, volume, inventory and40durations together with no change in the distribution of housing-service values, and buyer entry amplifies the response. The41composition of transactions changes after such a shock in a direction that depends on the shape of the valuation distribution,42and its effect on a transaction-price index is an order of magnitude smaller than the within-type liquidity effect that no43quality adjustment can remove. The gap between what mortgage and cash buyers pay decomposes into a delay-bargaining term that44rises with urgency and whose sign depends on who bears the larger cost of waiting, an outside-option term, and a selection term45that can be negative. The frictionless benchmark against which these objects are measured is not a liquidity premium in the46usual sense, since a seller with the option to wait for a high-valuation buyer may obtain more than the Walrasian price in a47sufficiently thick market. The next step is empirical: the predictions above are stated in terms of objects that listing-level48data can measure.49