\section{Introduction}
\label{sec:introduction}

Houses do not trade at a single price. Two identical houses listed in the same street in the same month can sell weeks apart at
prices several percent apart, and the difference is systematically related to the circumstances of the sellers: those who
have already moved, who carry two mortgages, or who face a deadline sell faster and for less \citep{glower1998motivation,
springer1996motivation,campbell2011forced,andersen2017firesales}. The size of this concession varies with the state of the market,
larger when buyers are scarce \citep{mayer1998auctions}, and it varies with the way the sale is financed, since cash buyers pay
less than mortgage buyers for the same house \citep{han2024cash,reher2024mortgage}. At the aggregate level, transaction-price
indices and sales volumes move together over the cycle and inventories move against them \citep{clayton2010pricevolume,
leamer2007businesscycle}, and the houses that transact in a downturn are not the houses that transact in a boom
\citep{gatzlaff1997selection,goetzmann2006reservation}. These facts are usually discussed separately, as a liquidity discount, a cash
discount, a volume--price correlation and a selection bias. This paper argues that they are four faces of one object: the price
of urgency in a market where buyers arrive slowly, sellers differ in what waiting costs them, and the two sides bargain.

The paper asks how market tightness, heterogeneous seller reservation values, financing frictions, time-on-market costs and
bargaining power jointly determine an endogenous liquidity discount, transaction prices, sale hazards and price dispersion, and
under which conditions a liquidity shock propagates through prices and quantities when the value of housing services has not
changed. It answers with a continuous-time search model of a local housing market that is deliberately simple. New listings
arrive at a constant rate and their owners pay a permanent, heterogeneous carrying cost while the house is listed; buyers enter
freely at a sunk cost, meet listings through a constant-returns matching technology, and draw a match-specific valuation at each
meeting; a meeting is with a qualified buyer with some probability; the surplus is split by Nash bargaining; listings are
withdrawn at an exogenous rate. Free entry pins the buyer's value at the entry cost, which isolates the seller side of the
market and makes tightness the only general-equilibrium channel. Two controlled extensions add a financing contingency, in which
a mortgage agreement spends time in a pending state and fails with some probability while cash agreements close at once, and a
posted-price protocol. A dynamic version tracks the economy after an unanticipated change in matching efficiency or
qualification.

The model yields a set of analytical results and a program of numerical experiments. On the analytical side, the reservation
price of a seller falls with her carrying cost by less than one-for-one, her sale hazard rises, and her expected transaction
price falls by less than her reservation price, because urgent sellers accept lower valuations but capture a share of the larger
surplus of the matches they accept; the urgency discount defined relative to the most patient seller is therefore a discount
without qualification. A fall in buyer arrivals or in qualification lowers every seller's reservation price and expected price; it
lengthens time on market and widens the dispersion of reservation prices when the distribution of buyer valuations has an
increasing failure rate, and the paper gives an exact counterexample: with Pareto valuations, a more liquid market lengthens the
time on market of every seller whose carrying cost exceeds a threshold that does not depend on the arrival rate. With free entry
and homogeneous sellers, tightness is unique and amplifies qualification and matching shocks, which are observationally
equivalent in the baseline. A liquidity shock moves prices, volume, inventory and durations together with no change in
fundamentals, and the steady-state sales flow falls only if listings are withdrawn. Price dispersion among identical houses
has a between-type component that is positive whenever sellers are heterogeneous, bounded by the dispersion of carrying costs
scaled by the effective discount rate, and widening when liquidity falls. The composition of transactions shifts after a
liquidity shock in proportion to the elasticity of each seller type's hazard, and the resulting bias in a transaction-price index
has a sign that depends on whether the product of the hazard function and the mean residual valuation of the match-value
distribution rises or falls: exponential and Pareto valuations make selection amplify measured price changes, light-tailed
valuations make it dampen them. In the financing extension the gap between mortgage and cash prices for the same house
decomposes exactly into a delay-bargaining term whose sign depends on which party bears the larger bargaining-weighted cost of
waiting and which rises with the seller's urgency, an outside-option term, and a selection term that is positive only when the
pending wedge exceeds the cash buyers' advantage in continuation value. Finally, the Walrasian free-entry price that would seem
the natural benchmark for a liquidity premium is shown to lie \emph{below} the search price in thick markets, because a seller
with the option to wait for a high-valuation buyer captures match-specific rents that a uniform price does not; the paper
therefore measures the liquidity discount among sellers in the same market and reports the gap to the Walrasian price only as a
decomposition.

The numerical part solves the stationary equilibrium, the posted-price and financing variants, a steady-state planner's problem
and transition paths for an illustrative parameterisation in which a listing sells in about two months, a buyer searches for
about eleven months, and the most urgent seller concedes about six percent of the price of the most patient. Twelve
pre-registered experiments trace the comparative statics of arrivals, qualification, approval delay, bargaining power, entry
costs and seller heterogeneity, the transition after permanent and temporary shocks, the decomposition of the observed price index
across four match-value distributions, the posted-price protocol, and the sensitivity of every result to the parameters.
Three numerical findings stand out. Entry amplifies arrival shocks by about ten percent in the baseline. Prices adjust on impact
while inventories and sales adjust over months, and a temporary shock is followed by a burst of sales. The composition term in
the observed price index is at most a few basis points against a within-type liquidity effect of one to two percent, in every
distribution and at every urgency dispersion considered, so that the bias from sale selection is second order under the present
mechanism while the liquidity effect that quality adjustment cannot remove is first order. In the financing extension mortgage
buyers pay 1.7 percent more than cash buyers on average, between 1.1 and 3.0 percent depending on the seller's urgency, with a
negative selection term for patient sellers; tighter approval and longer delays widen the gap. All code, tests, results and the
manuscript are reproducible from a single configuration file; no data are used and no empirical claim is made.

The paper is related to four literatures, reviewed in Section~\ref{sec:literature}. It builds on the search-and-matching theory
of housing markets \citep{wheaton1990vacancy,krainer2001theory,albrecht2007opportunistic,diaz2013house,ngai2014hot,head2014search,
han2015microstructure}, from which it takes the acceptance margin, the matching technology and free entry, and to which it adds a
continuous distribution of seller urgency as a state of the market. It is related to the theory of illiquid housing and credit
\citep{hedlund2016illiquidity,eerola2018borrowing,gabrovski2021search,guren2020foreclosures}, to which it adds an explicit financing
contingency and the decomposition of the cash--mortgage gap that the evidence of \citet{han2024cash} and \citet{reher2024mortgage}
calls for. It is related to the measurement literature on constant-liquidity values and sale selection
\citep{fisher2003controlling,lin2007illiquidity,goetzmann2006reservation}, to which it adds the sign condition for the selection
bias in a search equilibrium. And it rests on the foundations of \citet{diamond1982aggregate}, \citet{mortensen1994job},
\citet{hosios1990efficiency}, \citet{lippman1976economics} and \citet{bagnoli2005logconcave}. The first proposition restates a
known comparative static in a new setting and is presented as a benchmark; the contributions that survive the novelty audit
documented with the replication material are the joint treatment of continuous urgency, free entry and pending financing, the
exact condition under which liquidity lengthens time on market, the sign condition for the index-selection bias, the three-term
decomposition of the cash--mortgage gap, and the welfare analysis with heterogeneous sellers.

Section~\ref{sec:literature} reviews the literature. Section~\ref{sec:benchmark} sets out the institutional intuition and the two
benchmarks. Section~\ref{sec:model} presents the model, its dynamic version and the two extensions. Section~\ref{sec:propositions}
states the analytical results. Section~\ref{sec:numerical} describes the numerical solution, the parameterisation and its
validation. Section~\ref{sec:baseline} presents the baseline equilibrium, Section~\ref{sec:counterfactuals} the counterfactual
experiments, and Section~\ref{sec:welfare} the welfare analysis. Section~\ref{sec:conclusion} states limitations, falsifiable
predictions and conclusions. Proofs, computational details and additional tables are in the appendices.