# WP19 — Model specification (Phase 2)

Date: 2026-10-20. Status: formal environment, value functions, equilibrium definition, benchmark objects and the list of candidate propositions. Proofs are in research/proofs.md. Nothing here is a result until proved there or computed in Phase 4.

# 1. Environment

Continuous time, infinite horizon, discount rate $r>0$. A local market for houses of a single quality segment $q>0$ (Lemma 1 shows $q$ is a scale factor; all quantities below are expressed in units of $q$, i.e. $q=1$).

Sellers. New listings arrive at the exogenous flow $s>0$ per unit time. Each listing is owned by a seller with a permanent carrying cost (urgency) $c$ drawn at listing from a distribution $F$ with support $[c_L,c_H]$, $0\le c_L<c_H<\infty$ (mortgage interest, maintenance, double housing, relocation and psychic costs of an unsold house). The seller pays $c$ per unit time while listed. Listings are withdrawn at the exogenous rate $\delta\ge0$ (the seller keeps the house; the value of this outside option is normalised to zero, so $V(c)$ below is the net value of being listed).

Buyers. A buyer enters the market at sunk cost $k>0$, pays the flow search cost $\kappa_b\ge0$ while active, and exits upon purchase. Upon meeting a listing a buyer draws a match-specific valuation $w\sim G$ — the present value of the housing-service flow to that buyer, in units of $q$ — with continuous density $g$ on $[w_L,w_H]$, $0\le w_L<w_H\le\infty$. A buyer who purchases at price $p$ obtains $w-p$ (resale is abstracted from; Remark 1).

Matching. With $B$ active buyers and $S$ listed sellers, meetings occur at the rate $m(B,S)=A,B^{\alpha}S^{1-\alpha}$, $\alpha\in(0,1)$, $A>0$. Tightness is $\theta=B/S$; a seller meets buyers at rate $\lambda_s(\theta)=A\theta^{\alpha}$ and a buyer meets listings at rate $\lambda_b(\theta)=A\theta^{\alpha-1}$. Meetings are random across the stock of listings: a buyer meets a type-$c$ seller with probability $n(c)/S$ where $n(\cdot)$ is the stock density of listed sellers.

Financing (baseline). A meeting is with a qualified buyer with probability $\phi\in(0,1]$ (mortgage pre-approval); an unqualified meeting is wasted. Effective contact rates: $\Lambda_s=\phi\lambda_s(\theta)$, $\Lambda_b=\phi\lambda_b(\theta)$. Section 6 replaces this screen by a financing contingency with cash and mortgage buyers.

Bargaining. Complete information about $(w,c)$ within a match; generalised Nash bargaining with seller share $\beta\in(0,1)$; trade occurs iff the match surplus is nonnegative. Section 7 replaces Nash by posted prices.

Liquidity shock. $z\in{A,\phi}$: an unanticipated change in matching efficiency $A$ or qualification $\phi$ with $G$, $F$, $r$, $k$, $\kappa_b$, $s$ unchanged ("fundamental housing-service values unchanged").

Timing (continuous). Listing → repeated meetings at rate $\Lambda_s$ → within a meeting: draw $w$, bargain, trade or part → exogenous withdrawal at rate $\delta$. Listing duration $\tau$ is the time from listing to sale or withdrawal.

# 2. Value functions (stationary)

Seller of type $c$, listed: $$(r+\delta)V(c) = -c + \Lambda_s,\beta,H\big(V(c)+U\big), \qquad H(v)\equiv\int (w-v)^+,dG(w)=\int_v^{w_H}(1-G(w)),dw. \tag{S}$$ Active buyer: $$rU = -\kappa_b + \Lambda_b,(1-\beta)\int H\big(V(c)+U\big),dN(c),\qquad N(c)=n(c)/S. \tag{B}$$ $H$ is continuous, nonincreasing, convex, $H'(v)=-(1-G(v))$, $H''=g$. The mean residual valuation is $M(v)\equiv H(v)/(1-G(v))=E[w-v\mid w\ge v]$.

Match surplus and trade. $\Sigma(w,c)=w-V(c)-U$. Trade iff $w\ge v^*(c)\equiv V(c)+U$ (the reservation valuation). The seller's reservation price is $V(c)$ (the least price the seller accepts); the buyer's is $w-U$.

Price. Nash: $p(w,c)=V(c)+\beta,\Sigma(w,c)=(1-\beta)V(c)-\beta U+\beta w$.

Hazards. Meeting hazard $\Lambda_s$; conditional sale probability $1-G(v^(c))$; sale hazard $\eta(c)=\Lambda_s[1-G(v^(c))]$; exit hazard $\eta(c)+\delta$; survival of a listing $\mathcal S(\tau\mid c)=e^{-(\eta(c)+\delta)\tau}$; expected listing duration $E[\tau\mid c]=1/(\eta(c)+\delta)$; probability of eventual sale $\eta(c)/(\eta(c)+\delta)$; population survival $\bar{\mathcal S}(\tau)=\int e^{-(\eta(c)+\delta)\tau}dF(c)$ (a mixture; its hazard is decreasing in $\tau$ — Lemma 3).

Expected transaction price by type. $\bar p(c)\equiv E[p\mid \text{sale},c]=V(c)+\beta M(v^*(c))$.

# 3. Stationary equilibrium

Stocks: inflow $sf(c)$ = outflow $(\eta(c)+\delta)n(c)$, so $n(c)=sf(c)/(\eta(c)+\delta)$, $S=\int n$, $B=\theta S$. Sales flow $Q=\int\eta(c)n(c),dc$; withdrawals $\delta S$; transaction composition $\omega(c)=\eta(c)n(c)/Q$. Buyer entry flow $b=Q$ (buyers exit only by purchase).

Definition (stationary equilibrium with free entry). Given $(r,\delta,s,F,G,A,\alpha,\phi,\beta,k,\kappa_b)$, a stationary equilibrium is $(\theta,U,V(\cdot),n(\cdot))$ such that (S) holds for every $c$ (with the withdrawal term $\delta w_0$ added to the right-hand side: $(r+\delta)V(c)=-c+\delta w_0+\Lambda_s\beta H(V(c)+U)$, where $w_0\ge0$ is the seller's own-use value upon withdrawal), (B) holds, free entry $U=k$, and the stock equation holds. Partial equilibrium fixes $\theta$ and $U=k$ and drops (B).

Participation. Listing dominates keeping the house ($V(c)\ge w_0$) iff $c\le\bar c\equiv\Lambda_s\beta H(w_0+k)-rw_0$ (Proposition 1(iv)); types above $\bar c$ would not list. Parameter restriction: $c_H\le\bar c$ in every reported equilibrium (checked numerically, reported if violated).

# 4. Benchmarks and liquidity-premium definitions (sign conventions)

1. Urgency (forced-sale) discount. $D(c)\equiv[\bar p(c_L)-\bar p(c)]/\bar p(c_L)$, the expected price concession of a type-$c$ seller relative to the most patient seller in the same market; $D(c)\ge0$ (Proposition 1). Transaction-weighted market discount $\bar D=\int D(c),\omega(c),dc$. This is the model counterpart of the forced-sale discount of the empirical literature.
2. Frictionless (Walrasian, free-entry) benchmark. $p^$ solves $H(p^)=rk$: buyers enter until the expected gain from an immediate purchase at a uniform price equals the entry cost; sellers trade immediately at $p^$ and pay no carrying cost. $p^$ depends only on $(G,r,k)$. The market liquidity gap $\Gamma\equiv(p^-\bar p_T)/p^$ with $\bar p_T=\int\bar p(c)\omega(c)dc$ can take either sign (Proposition 2(iv)): search delay and carrying costs push transaction prices below $p^*$, while the seller's option to wait for a high-valuation buyer (rent extraction through bargaining) pushes them above. $\Gamma$ is therefore reported as a decomposition, never called "the liquidity premium".
3. Seller's liquidity cost. $L(c)\equiv c/(r+\delta+\eta(c))$, the expected discounted carrying cost paid over the listing spell; and $V(c_L)-V(c)$, the reservation-value gap.
4. Buyer's liquidity premium (what a buyer would pay to skip search) is $U$-based and distinct from the seller's objects; it is not reported as a price discount.

# 5. Dynamic (transition) version — Experiment 9

$U_t=k$ for all $t$ (free entry at every instant). Backward HJB $\dot V_t(c)=(r+\delta)V_t(c)+c-\Lambda_s(\theta_t)\beta H(V_t(c)+k)$; static free entry $rk=-\kappa_b+\Lambda_b(\theta_t)(1-\beta)\int H(V_t(c)+k),dN_t(c)$; forward KFE $\dot n_t(c)=sf(c)-(\eta_t(c)+\delta)n_t(c)$. Observed index $P_t=\int\bar p_t(c),\omega_t(c),dc$ with $\omega_t(c)=\eta_t(c)n_t(c)/Q_t$; fixed-composition index $P^{fix}_t=\int\bar p_t(c),\omega_0(c),dc$; fundamentals ($G$) constant, so the fundamental index is flat. Decomposition: $\ln P_t-\ln P_0=[\ln P^{fix}_t-\ln P_0]+[\ln P_t-\ln P^{fix}_t]$ = within-type (liquidity) + composition (selection).

# 6. Financing-contingency extension — Experiment 4 (cash vs mortgage buyers)

A buyer is a cash buyer with probability $\pi$ (type drawn at entry) and a mortgage buyer otherwise. A cash agreement closes immediately. A mortgage agreement at price $p$ enters a pending state: approval arrives at rate $\nu$ and succeeds with probability $\phi$; on failure the house returns to the market and the buyer to search. Pending values: $P_s(c,p)=\dfrac{-c+\nu\phi p+\nu(1-\phi)V(c)}{r+\nu}$, $P_b(w,p)=\dfrac{-\kappa_b+\nu\phi(w-p)+\nu(1-\phi)U_m}{r+\nu}$. Joint surplus $\Sigma_m(w,c)=\dfrac{\nu\phi,(w-V-U_m)-c-\kappa_b-r(V+U_m)}{r+\nu}$; threshold $v^m(c)=V(c)+U_m+\dfrac{c+\kappa_b+r(V(c)+U_m)}{\nu\phi}$; Nash price $p_m(w,c)=p{cash}(w,c)+\Delta_b(c)$ with $\Delta_b(c)=\dfrac{(1-\beta),[c+rV(c)]-\beta,[\kappa_b+rU_m]}{\nu\phi}$ evaluated at the cash price $p_{cash}(w,c)=V+\beta(w-V-U_m)$. Seller: $(r+\delta)V=-c+\lambda_s\beta\big[\tilde\pi H(V+U_c)+(1-\tilde\pi)\tfrac{\nu\phi}{r+\nu}H(v^_m)\big]$, $\tilde\pi$ = share of meetings with cash buyers (stock-consistent). Buyers: $rU_c=-\kappa_b+\lambda_b(1-\beta)\int H(V+U_c)dN$, $rU_m=-\kappa_b+\lambda_b(1-\beta)\tfrac{\nu\phi}{r+\nu}\int H(v^_m)dN$; free entry $\pi U_c+(1-\pi)U_m=k$. Stocks: $n(c)=sf(c)/[\eta_c(c)+\phi\eta_m(c)+\delta]$ with $\eta_c=\lambda_s\tilde\pi(1-G(V+U_c))$, $\eta_m=\lambda_s(1-\tilde\pi)(1-G(v^_m))$; pending $m(c)=\eta_m n/\nu$; cash buyers $B_c=\pi b/\xi_c$, searching mortgage buyers $B_m=(1-\pi)b/(\phi\xi_m)$ with $\xi_c=\lambda_b\int(1-G(V+U_c))dN$, $\xi_m=\lambda_b\int(1-G(v^_m))dN$; $\tilde\pi=B_c/(B_c+B_m)$; $\theta=(B_c+B_m)/S$; $b=Q=\int(\eta_c+\phi\eta_m)n$. The baseline is the limit $\nu\to\infty$ (then $v^_m\to V+U$, $\Delta_b\to0$ and the seller's effective contact rate is $\lambda_s[\tilde\pi+(1-\tilde\pi)\phi]$). Observed mortgage–cash price gap by type: $E[p_m\mid c]-E[p_c\mid c]=\Delta_b(c)+\beta{E[w\mid w\ge v^_m]-E[w\mid w\ge v^_c]}$ = bargaining term + selection term (Proposition 6).

# 7. Posted-price protocol — Experiment 11

The seller commits to a price $p$ before the buyer's $w$ is drawn; the buyer accepts iff $w-p\ge U$. Seller: $(r+\delta)V(c)=-c+\Lambda_s\max_y,[1-G(y)],(y-U-V(c))$ with $y=p+U$; interior optimum $y(c)-U-V(c)=[1-G(y(c))]/g(y(c))$ (unique under log-concave $g$). Prices are deterministic within type: all dispersion among identical houses comes from $c$.

# 8. Welfare (steady-state flow)

$\Omega=\int\eta(c)n(c),E[w\mid w\ge v^*(c)],dc-\int c,n(c),dc-\kappa_b B-k,Q$ (value created by sales, net of carrying costs, search costs and entry costs, per unit time). Seller surplus $s\int V(c)dF(c)$; buyer surplus is zero net of entry under free entry. The Hosios benchmark $\beta=1-\alpha$ is tested numerically (homogeneous and heterogeneous sellers).

# 9. Candidate propositions (to be proved in proofs.md)

• Lemma 1 (scale). With $w=q\varepsilon$, $c=q\kappa$, $k=q\tilde k$, $\kappa_b=q\tilde\kappa_b$, all values and prices are homogeneous of degree one in $q$; hazards and $\theta$ are invariant.
• Lemma 2 (existence, uniqueness of $V$). For any $U$, $\Lambda_s>0$: (S) has a unique solution; $V$ is $C^1$ in $c$ and in the parameters.
• Lemma 3 (spurious duration dependence). The population exit hazard $-,d\ln\bar{\mathcal S}/d\tau$ is decreasing in $\tau$ whenever $F$ is non-degenerate.
• Proposition 1 (urgency). $V'(c)=-1/[(r+\delta)+\Lambda_s\beta(1-G(v^))]\in(-1/(r+\delta),0)$; $\eta'(c)>0$ where $g(v^)>0$; $\bar p'(c)=V'(c)[1+\beta M'(v^*)]<0$ (no log-concavity needed); participation threshold $\bar c=\Lambda_s\beta H(k)$ under $U=k$.
• Proposition 2 (arrival and qualification, partial equilibrium). (i) $\partial V/\partial\Lambda_s>0$, $\partial\bar p/\partial\Lambda_s>0$; (ii) $\partial\eta/\partial\Lambda_s>0$ if $G$ is IFR (log-concave survival), and a Pareto counterexample where $\partial\eta/\partial\Lambda_s<0$; (iii) the reservation-value gap $V(c_L)-V(c)$ is decreasing in $\Lambda_s$ under IFR; (iv) sign of $\Gamma$: $\bar p(c)\to$ depends on $\Lambda_s$ with limits $-c/(r+\delta)$ as $\Lambda_s\to0$ and $w_H-k$ as $\Lambda_s\to\infty$, so $\Gamma$ changes sign.
• Proposition 3 (free entry, homogeneous sellers). Unique $\theta$; $d\theta/d\phi>0$, $d\theta/dA$ (sign determined), amplification of a $\phi$ shock through $\theta$; heterogeneous case: existence and a numerical uniqueness check.
• Proposition 4 (liquidity shock without fundamentals). A fall in $A$ or $\phi$ lowers $V(c)$, $\bar p(c)$ for all $c$, lowers the sales flow on impact and (if $\delta>0$) in steady state, raises inventory $S$ and mean duration.
• Proposition 5 (dispersion). Between-type variance $\mathrm{Var}\omega[\bar p(c)]>0$ iff $F$ non-degenerate; with $\beta=0$ all dispersion among identical houses is urgency-driven and bounded by $\mathrm{Var}\omega(c)/(r+\delta)^2$; the between-type reservation gap widens when $\Lambda_s$ falls (IFR).
• Proposition 6 (selection in indices). (i) $\delta=0$: steady-state transaction composition equals $f$ for any $\Lambda_s$; (ii) impact effect of a permanent $\Lambda_s$ shock: composition shifts toward types with smaller hazard elasticity $\epsilon(c)=\partial\ln\eta/\partial\ln\Lambda_s$; for exponential $G$, $\epsilon(c)=(r+\delta)/(r+\delta+\beta\eta(c))$ is decreasing in $c$, so the index falls (rises) by more than every within-type price after a negative (positive) shock.
• Proposition 7 (financing contingency). Bargaining term $\Delta_b(c)$ has the sign of $(1-\beta)[c+rV]-\beta[\kappa_b+rU_m]$ and is increasing in $c$; the selection term is strictly positive; the baseline screen is the $\nu\to\infty$ limit.
• Proposition 8 (posted prices). Under log-concave $g$, $y(c)$ is unique, decreasing in $c$, and $y(c)>v^*_{Nash}(c)$ at equal $(V,U)$: posted-price sellers are pickier.