1\section{Related literature}2\label{sec:literature}34This paper belongs to the search-and-matching literature on housing markets and borrows its tools from the5Diamond--Mortensen--Pissarides framework \citep{diamond1982aggregate,mortensen1994job,pissarides2000equilibrium,rogerson2005search}.6It is organised around four themes: the theory of housing liquidity and seller urgency, the credit and financing7channel, the measurement of prices and liquidity, and the foundations on which the formal results rest. The review8draws on a verified bibliography of 132 references; the closest antecedents of each proposition are identified explicitly so9that what is restated and what is new can be told apart.1011\subsection{Search, liquidity and seller urgency}1213The matching approach to housing begins with \citet{arnott1989housing}, who shows that idiosyncratic tastes generate14equilibrium vacancies in a thin market, and \citet{wheaton1990vacancy}, in which a mismatched owner searches while carrying a15vacant unit, so that the Nash-bargained price depends on the vacancy rate through the expected time to sell and the cost of16holding two houses. \citet{yavas1992simple} adds two-sided search intensity, and \citet{williams1995pricing} prices heterogeneous17real assets as optimal-stopping values under Poisson buyer arrivals, which yields the first explicit search-induced price18discount. \citet{krainer2001theory} is the reference theory of residential \emph{liquidity}: buyers draw match-specific values,19sellers set reservation prices that anticipate the aggregate state, and time on market lengthens in downturns because waiting20is rational; \citet{krainer2002equilibrium} value an illiquid asset relative to a liquid benchmark as a function of the21arrival intensity. \citet{han2015microstructure} organise the field into one-sided search, random matching, directed search,22bargaining and brokerage. The present model is a random-matching, Nash-bargaining model with a posted-price robustness check,23and it inherits from this strand the two objects it formalises: the sale hazard and a discount measured against a benchmark24price. A distinction that matters for what follows is that the ``liquidity shock'' of \citet{krainer2001theory} is a shock to25the dividend from housing services, whereas the shocks studied here change matching efficiency or buyer qualification and leave26the distribution of housing-service values unchanged. \citet{ngai2014hot} provide the closest precedent for a non-fundamental27driver: thick-market effects on match quality turn small seasonal shifts into large seasonal swings in prices and volume.28\citet{novymarx2009hot} shows that the feedback between the buyer--seller ratio and bargaining positions amplifies demand29shocks; \citet{diaz2013house} and \citet{head2014search} reproduce the comovement of prices, sales and time on market with random30matching, bargaining and entry; \citet{ngai2020decision}, \citet{ngai2024ins}, \citet{moen2021buying}, \citet{gabrovski2019beveridge},31\citet{caplin2011trading}, \citet{guren2018house}, \citet{anenberg2016information}, \citet{leung2017flipping} and32\citet{piazzesi2020segmented} develop the aggregate-dynamics branch. The model below is stationary and local in scope: it asks33what a fall in matching efficiency or qualification does to the cross-section of hazards and prices and to the composition of34transactions, not what drives the cycle.3536The nearest antecedent of the central device of this paper is \citet{albrecht2007opportunistic}, in which buyers and sellers switch37from a ``relaxed'' to a ``desperate'' flow value at a Poisson rate and the bargained price depends on the urgency states of the38pair; they derive that the expected price conditional on time on market falls with duration while its variance is hump-shaped.39Other formal treatments of urgency are deadlines \citep{salant1991sale}, the seller's discount rate \citep{arnold1999search}, the40endogenous cost of holding two homes \citep{anenberg2020endogenous}, debt overhang that \emph{raises} asking prices41\citep{head2023default}, and an estimated motivation parameter \citep{carrillo2012empirical}. The empirical call for42heterogeneous-seller theory is explicit in \citet{glower1998motivation}: sellers with a planned move sell faster and motivation43lowers the sale price but not the list markup. \citet{springer1996motivation} finds lower prices for eager, relocated, vacant and44foreclosed sellers; \citet{harding2003bargainingeffects} find weaker bargaining power for vacant sellers; \citet{levitt2008agents}45estimate that agents selling their own homes obtain about 3.7 percent more after about 9.5 more days on the market. Equity and46reference dependence work in the opposite direction: high loan-to-value and loss-facing sellers set higher asking prices and sell47more slowly \citep{genesove1997equity,genesove2001lossaversion}, and patient sellers sort into for-sale-by-owner platforms48\citep{hendel2009fsbo}. In every theoretical treatment cited, urgency is a two-state process, a deadline, a debt position or a49nuisance parameter. None models a continuous, permanent carrying-cost type whose cross-sectional distribution determines the stock50of listings, the composition of transactions and, through free entry, market tightness. That is the modelling choice made here,51and Proposition~\ref{prop:urgency} should accordingly be read as the continuous-type version of a known comparative static52\citep{lippman1976economics} rather than as a new result.5354The size and state dependence of liquidity discounts are documented by \citet{campbell2011forced} (foreclosure sales about 2755percent below comparable houses in Massachusetts, smaller discounts for death- and bankruptcy-related sales, and a spillover of56about one percent per nearby foreclosure), \citet{clauretie2009foreclosure} (naive discounts above 20 percent overstated by about57one third once condition, endogenous marketing time and spatial dependence are controlled), \citet{penningtoncross2006reo},58\citet{mayer1998auctions} (a price of immediacy of 0--9 percent in a boom against 9--21 percent in a bust) and59\citet{andersen2017firesales} (forced sales after sudden death: discounts rising with urgency, poor market conditions and seller60constraints, with demand held fixed). Spillovers from urgent sellers \citep{harding2009contagion,anenberg2014foreclosures,61mian2015foreclosures} show that a larger mass of urgent sellers lowers prices for all sellers without a change in fundamentals,62which is the empirical content of Proposition~\ref{prop:shock}. What these studies do not provide is a decomposition of the63discount into bargaining under a high carrying cost and the selection of which houses transact.6465\subsection{Credit, qualification and the cash--mortgage gap}6667\citet{hedlund2016illiquidity}, \citet{hedlund2016cyclical} and \citet{garriga2020mortgage} make search-based illiquidity a state68variable that interacts with long-term defaultable mortgages; \citet{guren2020foreclosures} show how foreclosures freeze the69non-distressed market; \citet{eerola2018borrowing} find that a moderate tightening of borrowing constraints raises time on market70and idiosyncratic price dispersion; \citet{gabrovski2021search} open a channel from credit to liquidity with multiplicity;71\citet{diaz2024housing} have wealthier buyers paying to transact faster under competitive search; and \citet{stein1995prices} and72\citet{ortalomagne2006housing} are the frictionless benchmarks with credit constraints. On the evidence side, qualification is73elastic and delays are measurable \citep{anenberg2019creditavailability,bhutta2021insurance,defusco2020leverage,fuster2019technology};74\citet{allen2019searchfrictions} show that the Canadian insured-mortgage market is itself a negotiated-price search market.7576The cash--mortgage gap is the empirical object to which the financing extension of Section~\ref{sec:model} speaks.77\citet{reher2024mortgage} estimate a mortgage premium of about 11 percent over 1980--2017, against about 3 percent from a78representative-seller model, and find that heterogeneity in selling conditions closes half of the gap. \citet{han2024cash}79estimate a cash discount of 2--3.9 percent with 29 percent faster closing in Los Angeles and show, with a calibrated search model,80that closing risk alone is insufficient: the risk that a relisted property fails to sell is needed, which makes the discount smaller81in booms and in larger markets. \citet{asabere1992cash}, \citet{jauregui2019fundingprice}, \citet{aroul2023cashdiscount},82\citet{lin2025contingencies} and \citet{buchak2026ibuyers} add magnitudes and selection caveats. The only formal bargaining theory83with financing that could be verified is \citet{bian2018bargaining}. No paper decomposes the measured gap into a bargaining84component that depends on seller urgency, an outside-option component, and a selection component arising from which matches are85accepted; Proposition~\ref{prop:financing} does so.8687\subsection{Selection, indices and comovement}8889The measurement literature establishes that sold houses are not a random sample of the stock and that the resulting bias90co-moves with the cycle \citep{jud1994selection,gatzlaff1997selection,gatzlaff1998selection,hwang2004selectivity}. The closest91antecedent of Proposition~\ref{prop:selection} is \citet{goetzmann2006reservation}: with heterogeneous reservation prices, falling92volume biases index returns upward and rising volume biases them downward. In commercial real estate, \citet{fisher2003controlling},93\citet{fisher2007tbi} and \citet{clayton2008liquidity} build constant-liquidity indices that lead transaction indices and are more94volatile; \citet{lin2007illiquidity}, \citet{bond2007marketing}, \citet{cheng2010illiquidity}, \citet{lin2008real} and95\citet{cheng2013liquidityrisk} show that ignoring the uncertain marketing period biases returns upward and risk downward, and that96the risk is seller-specific. The classical index tools \citep{bailey1963repeatsales,case1989efficiency,clapp1992assessed} assume97random sampling. Bargaining and information confounders are documented by \citet{harding2003bargainingpower},98\citet{kurlat2015asymmetries} and \citet{garmaise2004asymmetries}; volume leads prices in \citet{berkovec1996turnover},99\citet{hort2000turnover}, \citet{clayton2010pricevolume}, \citet{andrew2003transactions} and \citet{leamer2007businesscycle};100\citet{dachis2012transfertax} provide a Canadian transfer-tax experiment. The literature recognises the bias; no verified paper101derives its sign from primitives in a search equilibrium, and the result below is that the sign is not a general law.102103The duration literature \citep{zuehlke1987duration,haurin1988duration,jud1996brokerage,anglin2003tradeoff,knight2002listing,104haurin2010listprices,huang2001withdrawal,benefield2015measurement,carrillo2012potatoes,anglin2006liquidity} documents the105price--time trade-off, positive duration dependence in parametric hazards without unobserved heterogeneity, and the sensitivity of106results to withdrawals and relistings; \citet{taylor1999time} (stigma) and \citet{anenberg2016information} (learning) are competing107explanations of duration effects. The observed hazard in the model is a mixture over urgency types with a competing withdrawal108risk and predicts spurious negative duration dependence (Lemma~\ref{lem:duration}); positive dependence in the data requires109mechanisms that are outside the model.110111\subsection{Foundations and tools}112113Efficiency is assessed against the Hosios condition \citep{hosios1990efficiency}, with the caveat of \citet{gabrovski2025note} that114two-sided entry breaks it in housing markets; the model keeps listing inflow exogenous so that heterogeneity is the only115additional wedge. The reservation-value characterisation follows \citet{lippman1976economics}; the monotone-hazard results use116the log-concavity properties catalogued by \citet{bagnoli2005logconcave}; the matching function and the interpretation of its117efficiency parameter follow \citet{petrongolo2001looking}, with the contact-hazard elasticity of 0.84 estimated by118\citet{genesove2012search} and the tightness measures of \citet{carrillo2013heat}, \citet{carrillo2015tightness} and119\citet{han2014biddingwars} as empirical counterparts; \citet{shimer2005cyclical} and \citet{mortensen1999new} provide the120labour-market benchmarks. Bargaining follows \citet{nash1950bargaining}, with \citet{rubinstein1982perfect} as its strategic121foundation; posted prices and directed search follow \citet{moen1997competitive}, \citet{wright2021directed} and122\citet{albrecht2016directed}, and the list-price literature \citep{horowitz1992role,yavas1995strategic,merlo2004bargaining,123merlo2015home,han2016askingprice} provides the empirical counterpart of the posted-price protocol. Price dispersion among124homogeneous agents goes back to \citet{burdett1983equilibrium}, with \citet{maury2014search} and \citet{sagi2021asset} as housing125and asset-market applications. The asset-market search literature \citep{duffie2005over,lagos2009liquidity,rocheteau2011liquidity}126supplies the finance notion of an endogenous liquidity premium; the housing unit here is indivisible and match-specific, so the127holdings margin that offsets frictions in those models is absent and the premium is driven by seller urgency and tightness128instead. \citet{head2012housing} first used ``liquidity as how fast owners can sell'' as a state variable linking housing and129labour markets, and \citet{piazzesi2009momentum} note that the composition of marginal traders moves prices.130