\section{Related literature}
\label{sec:literature}

This paper belongs to the search-and-matching literature on housing markets and borrows its tools from the
Diamond--Mortensen--Pissarides framework \citep{diamond1982aggregate,mortensen1994job,pissarides2000equilibrium,rogerson2005search}.
It is organised around four themes: the theory of housing liquidity and seller urgency, the credit and financing
channel, the measurement of prices and liquidity, and the foundations on which the formal results rest. The review
draws on a verified bibliography of 132 references; the closest antecedents of each proposition are identified explicitly so
that what is restated and what is new can be told apart.

\subsection{Search, liquidity and seller urgency}

The matching approach to housing begins with \citet{arnott1989housing}, who shows that idiosyncratic tastes generate
equilibrium vacancies in a thin market, and \citet{wheaton1990vacancy}, in which a mismatched owner searches while carrying a
vacant unit, so that the Nash-bargained price depends on the vacancy rate through the expected time to sell and the cost of
holding two houses. \citet{yavas1992simple} adds two-sided search intensity, and \citet{williams1995pricing} prices heterogeneous
real assets as optimal-stopping values under Poisson buyer arrivals, which yields the first explicit search-induced price
discount. \citet{krainer2001theory} is the reference theory of residential \emph{liquidity}: buyers draw match-specific values,
sellers set reservation prices that anticipate the aggregate state, and time on market lengthens in downturns because waiting
is rational; \citet{krainer2002equilibrium} value an illiquid asset relative to a liquid benchmark as a function of the
arrival intensity. \citet{han2015microstructure} organise the field into one-sided search, random matching, directed search,
bargaining and brokerage. The present model is a random-matching, Nash-bargaining model with a posted-price robustness check,
and it inherits from this strand the two objects it formalises: the sale hazard and a discount measured against a benchmark
price. A distinction that matters for what follows is that the ``liquidity shock'' of \citet{krainer2001theory} is a shock to
the dividend from housing services, whereas the shocks studied here change matching efficiency or buyer qualification and leave
the distribution of housing-service values unchanged. \citet{ngai2014hot} provide the closest precedent for a non-fundamental
driver: thick-market effects on match quality turn small seasonal shifts into large seasonal swings in prices and volume.
\citet{novymarx2009hot} shows that the feedback between the buyer--seller ratio and bargaining positions amplifies demand
shocks; \citet{diaz2013house} and \citet{head2014search} reproduce the comovement of prices, sales and time on market with random
matching, bargaining and entry; \citet{ngai2020decision}, \citet{ngai2024ins}, \citet{moen2021buying}, \citet{gabrovski2019beveridge},
\citet{caplin2011trading}, \citet{guren2018house}, \citet{anenberg2016information}, \citet{leung2017flipping} and
\citet{piazzesi2020segmented} develop the aggregate-dynamics branch. The model below is stationary and local in scope: it asks
what a fall in matching efficiency or qualification does to the cross-section of hazards and prices and to the composition of
transactions, not what drives the cycle.

The nearest antecedent of the central device of this paper is \citet{albrecht2007opportunistic}, in which buyers and sellers switch
from a ``relaxed'' to a ``desperate'' flow value at a Poisson rate and the bargained price depends on the urgency states of the
pair; they derive that the expected price conditional on time on market falls with duration while its variance is hump-shaped.
Other formal treatments of urgency are deadlines \citep{salant1991sale}, the seller's discount rate \citep{arnold1999search}, the
endogenous cost of holding two homes \citep{anenberg2020endogenous}, debt overhang that \emph{raises} asking prices
\citep{head2023default}, and an estimated motivation parameter \citep{carrillo2012empirical}. The empirical call for
heterogeneous-seller theory is explicit in \citet{glower1998motivation}: sellers with a planned move sell faster and motivation
lowers the sale price but not the list markup. \citet{springer1996motivation} finds lower prices for eager, relocated, vacant and
foreclosed sellers; \citet{harding2003bargainingeffects} find weaker bargaining power for vacant sellers; \citet{levitt2008agents}
estimate that agents selling their own homes obtain about 3.7 percent more after about 9.5 more days on the market. Equity and
reference dependence work in the opposite direction: high loan-to-value and loss-facing sellers set higher asking prices and sell
more slowly \citep{genesove1997equity,genesove2001lossaversion}, and patient sellers sort into for-sale-by-owner platforms
\citep{hendel2009fsbo}. In every theoretical treatment cited, urgency is a two-state process, a deadline, a debt position or a
nuisance parameter. None models a continuous, permanent carrying-cost type whose cross-sectional distribution determines the stock
of listings, the composition of transactions and, through free entry, market tightness. That is the modelling choice made here,
and Proposition~\ref{prop:urgency} should accordingly be read as the continuous-type version of a known comparative static
\citep{lippman1976economics} rather than as a new result.

The size and state dependence of liquidity discounts are documented by \citet{campbell2011forced} (foreclosure sales about 27
percent below comparable houses in Massachusetts, smaller discounts for death- and bankruptcy-related sales, and a spillover of
about one percent per nearby foreclosure), \citet{clauretie2009foreclosure} (naive discounts above 20 percent overstated by about
one third once condition, endogenous marketing time and spatial dependence are controlled), \citet{penningtoncross2006reo},
\citet{mayer1998auctions} (a price of immediacy of 0--9 percent in a boom against 9--21 percent in a bust) and
\citet{andersen2017firesales} (forced sales after sudden death: discounts rising with urgency, poor market conditions and seller
constraints, with demand held fixed). Spillovers from urgent sellers \citep{harding2009contagion,anenberg2014foreclosures,
mian2015foreclosures} show that a larger mass of urgent sellers lowers prices for all sellers without a change in fundamentals,
which is the empirical content of Proposition~\ref{prop:shock}. What these studies do not provide is a decomposition of the
discount into bargaining under a high carrying cost and the selection of which houses transact.

\subsection{Credit, qualification and the cash--mortgage gap}

\citet{hedlund2016illiquidity}, \citet{hedlund2016cyclical} and \citet{garriga2020mortgage} make search-based illiquidity a state
variable that interacts with long-term defaultable mortgages; \citet{guren2020foreclosures} show how foreclosures freeze the
non-distressed market; \citet{eerola2018borrowing} find that a moderate tightening of borrowing constraints raises time on market
and idiosyncratic price dispersion; \citet{gabrovski2021search} open a channel from credit to liquidity with multiplicity;
\citet{diaz2024housing} have wealthier buyers paying to transact faster under competitive search; and \citet{stein1995prices} and
\citet{ortalomagne2006housing} are the frictionless benchmarks with credit constraints. On the evidence side, qualification is
elastic and delays are measurable \citep{anenberg2019creditavailability,bhutta2021insurance,defusco2020leverage,fuster2019technology};
\citet{allen2019searchfrictions} show that the Canadian insured-mortgage market is itself a negotiated-price search market.

The cash--mortgage gap is the empirical object to which the financing extension of Section~\ref{sec:model} speaks.
\citet{reher2024mortgage} estimate a mortgage premium of about 11 percent over 1980--2017, against about 3 percent from a
representative-seller model, and find that heterogeneity in selling conditions closes half of the gap. \citet{han2024cash}
estimate a cash discount of 2--3.9 percent with 29 percent faster closing in Los Angeles and show, with a calibrated search model,
that closing risk alone is insufficient: the risk that a relisted property fails to sell is needed, which makes the discount smaller
in booms and in larger markets. \citet{asabere1992cash}, \citet{jauregui2019fundingprice}, \citet{aroul2023cashdiscount},
\citet{lin2025contingencies} and \citet{buchak2026ibuyers} add magnitudes and selection caveats. The only formal bargaining theory
with financing that could be verified is \citet{bian2018bargaining}. No paper decomposes the measured gap into a bargaining
component that depends on seller urgency, an outside-option component, and a selection component arising from which matches are
accepted; Proposition~\ref{prop:financing} does so.

\subsection{Selection, indices and comovement}

The measurement literature establishes that sold houses are not a random sample of the stock and that the resulting bias
co-moves with the cycle \citep{jud1994selection,gatzlaff1997selection,gatzlaff1998selection,hwang2004selectivity}. The closest
antecedent of Proposition~\ref{prop:selection} is \citet{goetzmann2006reservation}: with heterogeneous reservation prices, falling
volume biases index returns upward and rising volume biases them downward. In commercial real estate, \citet{fisher2003controlling},
\citet{fisher2007tbi} and \citet{clayton2008liquidity} build constant-liquidity indices that lead transaction indices and are more
volatile; \citet{lin2007illiquidity}, \citet{bond2007marketing}, \citet{cheng2010illiquidity}, \citet{lin2008real} and
\citet{cheng2013liquidityrisk} show that ignoring the uncertain marketing period biases returns upward and risk downward, and that
the risk is seller-specific. The classical index tools \citep{bailey1963repeatsales,case1989efficiency,clapp1992assessed} assume
random sampling. Bargaining and information confounders are documented by \citet{harding2003bargainingpower},
\citet{kurlat2015asymmetries} and \citet{garmaise2004asymmetries}; volume leads prices in \citet{berkovec1996turnover},
\citet{hort2000turnover}, \citet{clayton2010pricevolume}, \citet{andrew2003transactions} and \citet{leamer2007businesscycle};
\citet{dachis2012transfertax} provide a Canadian transfer-tax experiment. The literature recognises the bias; no verified paper
derives its sign from primitives in a search equilibrium, and the result below is that the sign is not a general law.

The duration literature \citep{zuehlke1987duration,haurin1988duration,jud1996brokerage,anglin2003tradeoff,knight2002listing,
haurin2010listprices,huang2001withdrawal,benefield2015measurement,carrillo2012potatoes,anglin2006liquidity} documents the
price--time trade-off, positive duration dependence in parametric hazards without unobserved heterogeneity, and the sensitivity of
results to withdrawals and relistings; \citet{taylor1999time} (stigma) and \citet{anenberg2016information} (learning) are competing
explanations of duration effects. The observed hazard in the model is a mixture over urgency types with a competing withdrawal
risk and predicts spurious negative duration dependence (Lemma~\ref{lem:duration}); positive dependence in the data requires
mechanisms that are outside the model.

\subsection{Foundations and tools}

Efficiency is assessed against the Hosios condition \citep{hosios1990efficiency}, with the caveat of \citet{gabrovski2025note} that
two-sided entry breaks it in housing markets; the model keeps listing inflow exogenous so that heterogeneity is the only
additional wedge. The reservation-value characterisation follows \citet{lippman1976economics}; the monotone-hazard results use
the log-concavity properties catalogued by \citet{bagnoli2005logconcave}; the matching function and the interpretation of its
efficiency parameter follow \citet{petrongolo2001looking}, with the contact-hazard elasticity of 0.84 estimated by
\citet{genesove2012search} and the tightness measures of \citet{carrillo2013heat}, \citet{carrillo2015tightness} and
\citet{han2014biddingwars} as empirical counterparts; \citet{shimer2005cyclical} and \citet{mortensen1999new} provide the
labour-market benchmarks. Bargaining follows \citet{nash1950bargaining}, with \citet{rubinstein1982perfect} as its strategic
foundation; posted prices and directed search follow \citet{moen1997competitive}, \citet{wright2021directed} and
\citet{albrecht2016directed}, and the list-price literature \citep{horowitz1992role,yavas1995strategic,merlo2004bargaining,
merlo2015home,han2016askingprice} provides the empirical counterpart of the posted-price protocol. Price dispersion among
homogeneous agents goes back to \citet{burdett1983equilibrium}, with \citet{maury2014search} and \citet{sagi2021asset} as housing
and asset-market applications. The asset-market search literature \citep{duffie2005over,lagos2009liquidity,rocheteau2011liquidity}
supplies the finance notion of an endogenous liquidity premium; the housing unit here is indivisible and match-specific, so the
holdings margin that offsets frictions in those models is absent and the premium is driven by seller urgency and tightness
instead. \citet{head2012housing} first used ``liquidity as how fast owners can sell'' as a state variable linking housing and
labour markets, and \citet{piazzesi2009momentum} note that the composition of marginal traders moves prices.