# WP19 — Verified literature notes, Part A: Theory Scope: housing search-and-matching theory, illiquid-housing/credit theory, search-theory foundations, real-options/transaction-cost/price-discovery theory, and the mortgage–cash premium. Companion files: `lit_part_A.csv` (73 verified rows, keys cited below), `lit_unverified_A.md`, `verification/A_*.crossref.json` and `verification/A_*.evidence.txt`. Every reference cited by key below has a verified Crossref record (DOI resolved) except `pissarides2000equilibrium` (publisher/RePEc page). Statements about a paper's content are based on the abstract retrieved from Crossref/Semantic Scholar/RePEc or on the paper itself; claims I could not confirm from a retrieved source are flagged "[unconfirmed]". Empirical papers are covered by Part B; a few that are "both" are mentioned here only for their theoretical content. --- ## 1. Housing search-and-matching theory **What is known.** The modern literature begins with `wheaton1990vacancy`, which embeds owner-occupiers in a matching model in which a mismatched household searches while carrying a vacant unit, so that the equilibrium price (Nash bargaining) depends on the vacancy rate through expected time to sell and the cost of holding two houses. `arnott1989housing` had already shown that idiosyncratic tastes in a thin market generate equilibrium vacancies, the first formalisation of the idiosyncratic match-value draw that nearly all later models (and WP19) use as the acceptance margin. `yavas1992simple` introduced two-sided search intensity with Nash bargaining into real estate, and `williams1995pricing` priced heterogeneous real assets as optimal-stopping values under Poisson buyer arrivals, giving the first explicit notion of a search-induced price discount for real assets. `krainer2001theory` is the reference theory of *liquidity* in residential markets: buyers draw match values, sellers set reservation prices that anticipate the future aggregate state, and time on market rises in downturns because sellers rationally wait; with `krainer2002equilibrium`, which values an illiquid asset relative to a liquid benchmark as a function of the arrival intensity, this gives the two objects WP19 formalises — expected selling time and the discount D = (p* − p)/p* against a frictionless price. The survey `han2015microstructure` organises this material into one-sided search, random matching, directed search, bargaining/auctions and brokerage, and is the natural taxonomy for positioning WP19 (random matching + Nash bargaining, with a posted-price robustness check). **Urgency and seller heterogeneity.** The closest antecedent to WP19's central device is `albrecht2007opportunistic`: buyers and sellers switch from a "relaxed" to a "desperate" flow value at a Poisson rate, matching is opportunistic, and the equilibrium price depends on the urgency states of the pair. The paper derives that the expected price conditional on time on market falls with duration while its conditional variance is hump-shaped — i.e., urgency produces both lower prices and dispersion. Related but distinct treatments of urgency are: `salant1991sale` (a finite-horizon seller with a moving deadline lowers price over time), `arnold1999search` (comparative statics of the optimal reservation and asking prices in the seller's discount rate and the buyer's outside option), `anenberg2020endogenous` (the cost of holding two homes, i.e., the carrying cost of the joint buyer–seller problem, varies endogenously over the cycle and amplifies volatility), `head2023default` (indebted sellers set higher asking prices and sell more slowly, so debt lowers rather than raises urgency), and `carrillo2012empirical`, whose estimated stationary equilibrium includes heterogeneity in the motivation to trade with posting plus bargaining. On the empirical side (shared with Part B), `glower1998selling` document that motivated sellers sell faster and at lower prices and explicitly call for theory with heterogeneous sellers. In all these papers urgency is either a two-state Markov process, a deadline, a debt position, or an estimated nuisance parameter; none treats it as a continuous permanent holding-cost type c whose cross-sectional distribution feeds back into tightness, hazards and the composition of transactions, which is WP19's modelling choice. **Aggregate dynamics and tightness feedback.** A second branch uses the matching framework to explain price–volume–TOM comovement and amplification: `novymarx2009hot` (feedback between the buyer–seller ratio and bargaining positions amplifies demand shocks), `diaz2013house` (random matching, Nash bargaining and entry reproduce the cyclical comovements and propagate shocks), `head2014search` (search with construction and buyer entry generates momentum because time to sell varies), `ngai2014hot` (thick-market effects on match quality turn small seasonal drivers into large seasonal swings), `ngai2020decision` and `ngai2024ins` (listing inflows versus sale outflows), `moen2021buying` (buy-first/sell-first complementarity and multiplicity), `gabrovski2019beveridge` (two-sided entry is needed for an upward-sloping housing Beveridge curve), `caplin2011trading` (trading frictions linking prices, sales and inventory), `guren2018house` (concave demand in relative list price and strategic complementarity), `anenberg2016information` (Bayesian learning by sellers generates list-price cuts and momentum), `leung2017flipping` (middlemen as liquidity providers, multiple equilibria), and `piazzesi2020segmented` (search across overlapping segments). WP19 inherits from this branch the free-entry determination of tightness and the result that matching conditions move prices without changes in housing-service values — most explicitly in `ngai2014hot`, where match quality rather than dividends moves prices — but it is stationary and partial in scope: it asks what a fall in matching efficiency A or in qualification phi does to the cross-section of hazards and prices, not what drives the cycle. **Price formation: bargaining versus posting.** The list-price literature (`horowitz1992role`: list price as a ceiling distinct from the reservation price; `yavas1995strategic`: list price trades off arrival rate against bargained price; `merlo2015home`: optimal stopping with tiny menu costs rationalises sticky list prices) and the directed-search literature (`albrecht2016directed`: asking prices with limited commitment signal seller reservation types and the equilibrium is constrained efficient; `diaz2024housing`: competitive search with credit constraints where wealthier buyers pay to transact faster) provide the two benchmarks between which WP19 sits. Dispersion for observationally identical houses has been derived from buyer search strategies (`maury2014search`) and from search-based asset pricing with transaction risk (`sagi2021asset`); seller-urgency-driven dispersion appears only in `albrecht2007opportunistic`. Non-search theories of volume–price comovement through credit constraints (`stein1995prices`, `ortalomagne2006housing`) are the relevant frictionless-with-credit benchmarks; belief-based cycle models (Burnside–Eichenbaum–Rebelo 2016; Kaplan–Mitman–Violante 2020, both verified but not retained, see `lit_unverified_A.md`) do not contain a liquidity margin. **What is open.** (i) No stationary model combines a continuous seller urgency distribution, free entry of buyers and a definition of the liquidity discount against a frictionless benchmark. (ii) The mapping from a *pure* matching-efficiency shock (fundamentals fixed) to the joint distribution of hazards and prices, including which sellers transact, is implicit at best in `ngai2014hot` and `novymarx2009hot`. (iii) None of the housing search papers above derives the bias in a transaction-based price index induced by the urgency composition of sellers after a liquidity shock. ## 2. Illiquid housing, credit and foreclosure theory **What is known.** `hedlund2016illiquidity` and `hedlund2016cyclical` make search-based housing illiquidity a state variable in a model with long-term defaultable mortgages: longer selling times raise foreclosure risk, foreclosures tighten credit, and the resulting "liquidity spiral" amplifies busts; `garriga2020mortgage` embeds endogenously illiquid housing in a heterogeneous-agent macro model and shows that the joint deterioration of prices and liquidity, transmitted through balance sheets, is central to the Great Recession patterns. `guren2020foreclosures` show in a dynamic search model that foreclosures freeze the non-distressed market through choosier buyers, ruined credit and lender losses, which is the extreme form of a composition/selection effect: distressed sellers are a high-urgency type. On the credit–search interaction without default, `eerola2018borrowing` find that a moderate tightening of borrowing constraints raises average time on market and idiosyncratic price dispersion substantially, and `gabrovski2021search` show that search and credit frictions together open a channel from the financial sector to liquidity and prices (with multiplicity) and that credit shocks move liquidity more than prices. `head2012housing` first used "liquidity = how fast owners can sell" as the state variable linking housing and labour markets. **Competing mechanisms.** Two different routes connect credit to liquidity: through buyers (fewer qualified or wealthy buyers → lower effective arrival rate, lower bargained prices, longer TOM: `eerola2018borrowing`, `gabrovski2021search`, `diaz2024housing`) and through sellers (debt overhang raises asking prices and lengthens TOM: `head2023default`; foreclosure externalities: `guren2020foreclosures`). WP19's financing extension belongs to the buyer route but is more specific than any of these: it introduces a *pending* state (approval at rate nu, success with probability phi) that converts a mortgage match into a delayed and uncertain sale, so that the cash–mortgage price gap is derived rather than assumed. **What is open.** The cited models are quantitative and dynamic; none isolates analytically (a) how a fall in phi alone changes the hazard and the discount, (b) how the induced cash discount decomposes into a bargaining component (compensation for delay/failure risk) and a selection component (which sellers accept cash), or (c) the implied bias in observed price indices. These are the gaps WP19's Propositions on qualification and the cash discount target. ## 3. Search-theory foundations and tools The equilibrium architecture of WP19 — CRS matching function, tightness theta, free entry pinning the entrant's value, Nash surplus sharing, continuous-time Bellman equations — is the Diamond–Mortensen–Pissarides framework (`diamond1982aggregate`, `mortensen1994job`, `pissarides2000equilibrium`, surveyed in `mortensen1999new` and `rogerson2005search`). The efficiency benchmark is the Hosios condition (`hosios1990efficiency`); `gabrovski2025note` is the housing-specific caveat that with *two-sided* entry the Hosios–Mortensen–Pissarides condition no longer delivers efficiency because of congestion and participation externalities, so WP19's welfare proposition must state clearly that listing inflow is exogenous (one-sided entry) and that urgency heterogeneity adds a composition wedge of its own. The reservation-value characterisation and its comparative statics (a higher flow cost of search lowers the reservation value) go back to `lippman1976economics`; WP19's "reservation price decreasing in c" is an application of this classical result inside a market equilibrium. The monotone-hazard result "sale hazard increasing in the effective arrival rate under log-concave G" uses the properties catalogued in `bagnoli2005logconcave`. The CRS matching assumption and the interpretation of A as matching efficiency are supported by `petrongolo2001looking`; the volatility-of-tightness benchmark for calibration discussions is `shimer2005cyclical`. Bargaining: `nash1950bargaining` (axiomatic split, WP19's baseline), `rubinstein1982perfect` (impatience determines the split, which gives a strategic reading of why urgent sellers obtain lower prices). Posted prices and directed search: `moen1997competitive`, surveyed in `wright2021directed`. Price dispersion with homogeneous agents: `burdett1983equilibrium`. The asset-market search literature supplies the finance notion of an endogenous liquidity premium: `duffie2005over` (spreads and premia decreasing in search intensity and shaped by bargaining power), `lagos2009liquidity` (with unrestricted holdings, investors offset frictions by adjusting positions), surveyed in `rocheteau2011liquidity`. WP19's housing unit is indivisible and heterogeneous in match value, so the holdings margin is absent and the premium is driven by seller urgency and tightness instead; this distinction should be made explicit in the paper. ## 4. Real options to wait, transaction costs, marketing-period risk and price discovery **What is known.** `lin2007illiquidity` argue that the defining features of real estate illiquidity are the time required to sell and the *uncertainty* of the marketing period, and that valuation based only on sold properties with implicit immediate execution biases estimated returns upward and risks downward. `bond2007marketing` price marketing-period risk in a portfolio context; `cheng2010illiquidity` derive an optimal holding period from illiquidity and transaction costs; `lin2008real` show that sellers with different degrees of financial distress face different marketing-period risk and realised return distributions, so "return and risk are investor-specific" — the closest statement in this strand to WP19's urgency gradient, but in partial equilibrium without bargaining or tightness. On indices, `fisher2003controlling` define a "constant-liquidity value" and show that transaction-based indices embed time-varying ease of sale, with constant-liquidity indices leading and showing larger amplitude (both/theory-informed econometrics; see also Part B). `taylor1999time` provides the competing explanation of duration dependence in which time on the market signals low quality (stigma). **What is open.** This literature treats the marketing period as exogenous risk to be priced or corrected; it does not derive the distribution of time on market and the selection of transacting sellers from an equilibrium with heterogeneous urgency, and it does not sign the index bias after a liquidity shock from primitives. WP19's "dynamic selection bias" proposition is the equilibrium counterpart of `fisher2003controlling` and `lin2007illiquidity`, and its duration-dependence result must be distinguished from `taylor1999time` (stigma) and `anenberg2016information` (learning). ## 5. The mortgage–cash premium **What is known.** `reher2024mortgage` estimate that mortgaged buyers pay roughly 11% more than all-cash buyers; a dynamic representative-seller model with mortgage transaction frictions delivers only about 3%; heterogeneity in selling conditions explains half of the remaining gap and belief distortions the rest in high-risk states. `han2024cash` estimate a 2–3.9% cash discount and 29% shorter time to close in Los Angeles and show, with a calibrated search model, that closing risk alone is insufficient: the discount turns on the possibility that a property returned to the market fails to sell, so that the discount is smaller in booms and in larger markets — "the inseparability of financial frictions in the mortgage market and search frictions in the housing market". `bian2018bargaining` is the only formal bargaining theory with financing I could verify: embedding mortgage terms in bilateral bargaining changes the gain from trade and allows trade even when the buyer's valuation is below the seller's reservation. No other formal theory of financing contingencies in housing transactions was found (see `lit_unverified_A.md`, section A). **Competing mechanisms and what is open.** The literature attributes the cash discount to (i) compensation for delay and failure risk given relisting costs (`han2024cash`), (ii) seller beliefs about mortgage frictions (`reher2024mortgage`), and (iii) financing-induced changes in the bargaining surplus (`bian2018bargaining`). What is missing is an equilibrium statement of which sellers accept cash and how that selection contributes to the measured discount; `reher2024mortgage`'s finding that "heterogeneity in selling conditions" closes half the puzzle is exactly the margin WP19's decomposition (bargaining term + selection term by urgency type) formalises, with tightness feedback through free entry. WP19 should also note that both empirical papers measure a discount conditional on sale, i.e., subject to the same selection its index proposition analyses. --- ## Closest prior work for the WP19 propositions Judgment scale: **established** (result exists in essentially the same form), **partially established** (mechanism present but in a different setting, informal, or without the WP19 object), **open** (no verified prior statement). 1. **Urgency (holding cost c) → lower reservation price and lower bargained price.** Closest: `albrecht2007opportunistic` (desperate state lowers price), `lippman1976economics` (reservation value decreasing in search cost), `arnold1999search` (reservation price decreasing in seller discount rate), `salant1991sale` (deadline lowers price path). Judgment: **established** as a comparative static; WP19's contribution is the continuous-type equilibrium version with tightness feedback, which should be presented as a benchmark result, not as new. 2. **Lower effective arrival/qualification rate → lower hazard, longer TOM, wider discount (log-concave G).** Closest: `krainer2001theory` (TOM rises when sellers wait), `eerola2018borrowing` (credit tightening raises TOM and dispersion), `gabrovski2021search` (credit shocks move liquidity), `williams1995pricing` / `krainer2002equilibrium` (discount decreasing in arrival intensity), tools in `bagnoli2005logconcave`. Judgment: **partially established** — the direction is known; the joint hazard–discount statement with an explicit pending/qualification stage and the log-concavity condition for monotone hazards is not in the housing literature. 3. **Liquidity shock (fall in A or phi) moves prices and volume with housing-service values unchanged.** Closest: `ngai2014hot` (thick-market match quality, not dividends, moves prices), `novymarx2009hot` (tightness feedback amplifies), `diaz2013house`, `duffie2005over` (search intensity moves asset prices in OTC markets), `hedlund2016cyclical` (liquidity spirals). Judgment: **partially established** — qualitatively present, but no paper isolates a pure matching-efficiency or qualification shock in a stationary model and traces it to the cross-section of sellers; `krainer2001theory`'s liquidity shock is itself a dividend shock, which is the distinction WP19 must stress. 4. **Price dispersion from seller urgency for identical houses.** Closest: `albrecht2007opportunistic` (hump-shaped conditional variance), `eerola2018borrowing` (dispersion from credit), `maury2014search` (buyer strategies), `burdett1983equilibrium` (homogeneous-agent dispersion), `sagi2021asset`. Judgment: **partially established** — dispersion from a two-state urgency process exists; the continuous-type mapping from the distribution of c to the transaction-price distribution, and its response to A and phi, is open. 5. **Dynamic selection bias in transaction indices after a liquidity shock (urgent sellers' hazards less elastic → composition shift).** Closest: `fisher2003controlling` (constant-liquidity value), `lin2007illiquidity` (bias from ignoring marketing period), `guren2020foreclosures` (distressed composition), `piazzesi2009momentum` (composition of marginal traders moves prices). Judgment: **open** as a theoretical proposition from primitives — the empirical/econometric literature recognises the bias but no verified paper derives its sign from heterogeneous hazard elasticities in a search equilibrium. This is WP19's strongest novelty claim and should be audited carefully against Part B's empirical index literature (Gatzlaff–Haurin selection papers belong there). 6. **Cash discount = bargaining term + selection term.** Closest: `han2024cash` (closing risk plus failure-to-resell risk, calibrated search model), `reher2024mortgage` (representative-seller model, heterogeneity in selling conditions closes half the puzzle), `bian2018bargaining` (financing in bargaining), `diaz2024housing` (buyers pay to transact faster). Judgment: **partially established** — the bargaining/compensation term is established and quantified; the selection term by seller urgency type and its interaction with tightness is open and directly addresses the residual in `reher2024mortgage`. **Overall novelty assessment for Part A.** Propositions 1 and (the direction of) 2–3 are re-derivations in a new setting and must be labelled as such; the paper's genuine contributions, as far as the theoretical literature verified here is concerned, are the joint treatment (continuous urgency × free entry × pending financing), the decomposition of the cash discount, and the index-selection proposition (5). The novelty gate in CLAUDE.md is therefore passed provided the manuscript explicitly credits `albrecht2007opportunistic`, `krainer2001theory`, `eerola2018borrowing`, `han2024cash` and `fisher2003controlling` as the nearest antecedents.