1# WP19 — Formal results and proofs (Phase 2)23Notation as in `model_specification.md`. Throughout, $U$ is treated as given (partial equilibrium) unless stated; under free entry4$U=k$. $h_G(v)=g(v)/(1-G(v))$ is the hazard function of $G$, $M(v)=H(v)/(1-G(v))$ its mean residual valuation.5"IFR" = increasing failure rate = $1-G$ log-concave; a log-concave density implies IFR (Bagnoli–Bergstrom 2005).67Two elementary facts used repeatedly:8(F1) $H'(v)=-(1-G(v))$, $M'(v)=-1+h_G(v)M(v)$, hence $M'(v)>-1$ wherever $g(v)>0$ and $M'(v)\le0$ under IFR9(because under IFR $M(v)=\int_v^{w_H}\exp(-\int_v^w h_G)\,dw\le\int_v^{\infty}e^{-h_G(v)(w-v)}dw=1/h_G(v)$).10(F2) $\frac{d}{dv}E[w\mid w\ge v]=1+M'(v)=h_G(v)M(v)>0$ wherever $g(v)>0$; under IFR, $gH=g(1-G)M\le(1-G)^2$.1112## Lemma 1 (scale invariance in quality)13*Statement.* Let $w=q\varepsilon$ with $\varepsilon\sim G_\varepsilon$, $c=q\kappa$, $k=q\tilde k$, $\kappa_b=q\tilde\kappa_b$. If14$(\theta,U,V(\cdot),n(\cdot))$ is a stationary equilibrium of the economy with $q=1$ and primitives $(G_\varepsilon,\kappa,\tilde k,\tilde\kappa_b)$,15then $(\theta,qU,qV(\cdot),n(\cdot))$ is a stationary equilibrium of the economy with quality $q$; prices scale by $q$; hazards,16durations, $\theta$, $\omega$ and $D(c)$ are invariant.17*Proof.* $H_q(qv)=\int(q\varepsilon-qv)^+dG_\varepsilon=qH_1(v)$. Substituting $V_q=qV$, $U_q=qU$ into (S): $(r+\delta)qV=-q\kappa+\Lambda_s\beta qH_1(V+U)$,18which is (S) for $q=1$ divided by $q$; same for (B) and $1-G_q(q v)=1-G_\varepsilon(v)$ so hazards and stocks coincide; free entry19$qU=q\tilde k$. Prices $p=(1-\beta)qV-\beta qU+\beta q\varepsilon$. $\square$2021## Lemma 2 (seller value: existence, uniqueness, smoothness)22*Statement.* For any $U\in\mathbb R$, $\Lambda_s>0$, $c\ge0$, equation (S) has a unique solution $V(c)$; $V$ is continuously23differentiable in $(c,\Lambda_s,U)$ with24$$\partial_cV=\frac{-1}{(r+\delta)+\Lambda_s\beta(1-G(v^*))},\quad25\partial_{\Lambda_s}V=\frac{\beta H(v^*)}{(r+\delta)+\Lambda_s\beta(1-G(v^*))},\quad26\partial_UV=\frac{-\Lambda_s\beta(1-G(v^*))}{(r+\delta)+\Lambda_s\beta(1-G(v^*))}\in(-1,0].$$27*Proof.* $\Phi(V)\equiv-c+\Lambda_s\beta H(V+U)-(r+\delta)V$ is continuous, strictly decreasing ($\Phi'=-\Lambda_s\beta(1-G)-(r+\delta)\le-(r+\delta)<0$),28$\Phi(V)\to+\infty$ as $V\to-\infty$ (since $H(v)=E[w]-v$ for $v\le w_L$) and $\Phi(V)\to-\infty$ as $V\to+\infty$ ($H\ge0$). Hence a unique29root. $\Phi$ is $C^1$ in all arguments ($H'=-(1-G)$ continuous), and $\Phi_V<0$, so the implicit function theorem gives the derivatives:30$\partial_xV=-\Phi_x/\Phi_V$ with $\Phi_c=-1$, $\Phi_{\Lambda_s}=\beta H(v^*)$, $\Phi_U=-\Lambda_s\beta(1-G(v^*))$. $\square$3132## Lemma 3 (spurious negative duration dependence)33*Statement.* If $\mu(c)\equiv\eta(c)+\delta$ is non-degenerate under $F$, the population exit hazard34$\bar h(\tau)=-\frac{d}{d\tau}\ln\bar{\mathcal S}(\tau)$, $\bar{\mathcal S}(\tau)=\int e^{-\mu(c)\tau}dF(c)$, is strictly decreasing in $\tau$,35although every individual hazard is constant.36*Proof.* $\bar h(\tau)=\int\mu e^{-\mu\tau}dF/\int e^{-\mu\tau}dF=E_\tau[\mu]$ where $E_\tau$ is the expectation under the tilted37measure $dF_\tau\propto e^{-\mu\tau}dF$. Differentiating, $\bar h'(\tau)=-\mathrm{Var}_\tau(\mu)<0$. (Classical mixture result;38cf. Lancaster 1979.) $\square$3940## Proposition 1 (seller urgency)41*Statement.* Fix $U$ and $\Lambda_s>0$. (i) $V'(c)=-1/[(r+\delta)+\Lambda_s\beta(1-G(v^*(c)))]\in(-1/(r+\delta),0)$: the reservation42price falls with the carrying cost, by less than one-for-one in present-value terms. (ii) The sale hazard $\eta(c)$ is nondecreasing in $c$43and strictly increasing wherever $g(v^*(c))>0$; expected listing duration is nonincreasing. (iii) The expected transaction price44$\bar p(c)=V(c)+\beta M(v^*(c))$ is strictly decreasing in $c$, with $\bar p'(c)=V'(c)[1+\beta M'(v^*(c))]$ and $1+\beta M'>1-\beta>0$; under45IFR, $|\bar p'(c)|\le|V'(c)|$ (the price falls by less than the reservation price). Hence the urgency discount $D(c)$ is zero at $c_L$,46strictly increasing in $c$ and nonnegative. (iv) Under free entry ($U=k$), a type-$c$ seller has $V(c)\ge0$ iff $c\le\bar c\equiv\Lambda_s\beta H(k)$.47*Proof.* (i) is Lemma 2. (ii) $\eta(c)=\Lambda_s[1-G(V(c)+U)]$, so $\eta'(c)=-\Lambda_sg(v^*)V'(c)\ge0$, strict iff $g(v^*)>0$. (iii) Conditional on48sale, $w\ge v^*$ and $p=V+\beta(w-v^*)$, so $\bar p=V+\beta E[w-v^*\mid w\ge v^*]=V+\beta M(v^*)$. Differentiate: $\bar p'=V'+\beta M'(v^*)v^{*\prime}=V'[1+\beta M']$49because $v^{*\prime}=V'$ ($U$ fixed). By (F1) $M'>-1$ so $1+\beta M'>1-\beta>0$, and $V'<0$ gives $\bar p'<0$. Under IFR $M'\le0$ so50$0<1+\beta M'\le1$. $D(c)=1-\bar p(c)/\bar p(c_L)$ inherits monotonicity provided $\bar p(c_L)>0$ (true when $V(c_L)\ge0$).51(iv) $\Phi$ is decreasing, so the root is $\ge0$ iff $\Phi(0)\ge0$, i.e. $-c+\Lambda_s\beta H(k)\ge0$. $\square$5253## Proposition 2 (effective arrival rate: partial equilibrium)54*Statement.* Fix $U$. Let $\Lambda_s=\phi\lambda_s$ (so $A$, $\phi$ and $\theta$ act only through $\Lambda_s$).55(i) $\partial V/\partial\Lambda_s>0$ and $\partial\bar p/\partial\Lambda_s>0$ whenever $v^*<w_H$.56(ii) If $G$ is IFR, $\partial\eta/\partial\Lambda_s>0$: a decline in arrivals or in qualification lengthens expected time on market.57Without IFR the sign can reverse: for Pareto $G(w)=1-(w_L/w)^a$, $a>1$, $\partial\eta/\partial\Lambda_s<0$ iff $\beta\eta(c)>(a-1)(r+\delta)$,58which, using (S), is equivalent to $c>\delta w_0+(r+\delta)U$ — a condition independent of $\Lambda_s$: for a sufficiently urgent seller59facing heavy-tailed valuations, a more liquid market *lengthens* time on market because the seller becomes pickier faster than60buyers arrive.61(iii) Under IFR, the reservation-value gap $V(c_L)-V(c)$, $c>c_L$, is strictly decreasing in $\Lambda_s$: lower liquidity widens the62cross-sectional dispersion of reservation prices.63(iv) Limits (patient type $c=0$, $U=k\le w_L$, $w_H<\infty$): $\lim_{\Lambda_s\to0}\bar p=\beta(E[w]-k)$ and64$\lim_{\Lambda_s\to\infty}\bar p=w_H-k$. If $\beta(E[w]-k)<p^*<w_H-k$ (the second inequality holds iff $H(w_H-k)<rk$), the market65liquidity gap $\Gamma$ of the patient type is positive in thin markets and negative in thick markets.66*Proof.* (i) Lemma 2 gives $\partial V/\partial\Lambda_s=\beta H(v^*)/[(r+\delta)+\Lambda_s\beta(1-G)]>0$ when $H(v^*)>0$; and67$\partial\bar p/\partial\Lambda_s=\partial V/\partial\Lambda_s\,[1+\beta M'(v^*)]>0$ by (F1).68(ii) $\partial\eta/\partial\Lambda_s=(1-G)-\Lambda_sg\,\partial V/\partial\Lambda_s$. Multiplying by the positive denominator, the sign equals that of69$(1-G)(r+\delta)+\Lambda_s\beta[(1-G)^2-gH]$, which is positive under IFR by (F2). Pareto: $1-G=(w_L/v)^a$, $g=a w_L^av^{-a-1}$,70$M=v/(a-1)$, so $gH=\frac{a}{a-1}(w_L/v)^{2a}$ and $(1-G)^2-gH=-\frac{(1-G)^2}{a-1}$; the sign becomes that of71$(1-G)\{(r+\delta)-\beta\Lambda_s(1-G)/(a-1)\}=(1-G)\{(r+\delta)-\beta\eta/(a-1)\}$. For Pareto, $H(v^*)=(1-G(v^*))v^*/(a-1)$, so72(S) reads $(r+\delta)V+c-\delta w_0=\beta\eta\,v^*/(a-1)$ with $v^*=V+U$, i.e. $\beta\eta=(a-1)(r+\delta)+(a-1)[c-\delta w_0-(r+\delta)U]/v^*$,73whence $\beta\eta>(a-1)(r+\delta)$ iff $c>\delta w_0+(r+\delta)U$.74(iii) $V(c_L)-V(c)=\int_{c_L}^{c}\frac{du}{(r+\delta)+\beta\eta(u)}$ by (i) of Proposition 1, and $\eta(u)$ is increasing in $\Lambda_s$ by (ii).75(iv) As $\Lambda_s\to0$, (S) gives $V\to-c/(r+\delta)=0$ for $c=0$, so $v^*\to k\le w_L$, $M(k)=E[w]-k$ and $\bar p\to\beta(E[w]-k)$. As76$\Lambda_s\to\infty$, $H(v^*)=[(r+\delta)V+c]/(\Lambda_s\beta)\to0$ forces $v^*\to w_H$, so $V\to w_H-k$ and $M(v^*)\to0$. $p^*<w_H-k$ iff77$H(p^*)=rk>H(w_H-k)$ since $H$ is decreasing. $\square$7879## Proposition 3 (free entry and tightness)80*Statement.* (a) Homogeneous sellers ($F$ degenerate at $c$). The free-entry residual81$R(\theta)\equiv-\kappa_b+\phi\lambda_b(\theta)(1-\beta)H(V(\theta)+k)-rk$ is continuous and strictly decreasing with $R(0^+)=+\infty$,82$R(\infty)=-\kappa_b-rk<0$; hence a unique equilibrium tightness $\theta^*$. Moreover $d\theta^*/d\phi>0$ and $d\theta^*/dA>0$, and83$\frac{d\ln\Lambda_s}{d\ln\phi}=1+\alpha\frac{d\ln\theta^*}{d\ln\phi}>1$ (tightness amplifies a qualification shock). In the baseline, $A$84and $\phi$ enter all equilibrium conditions only through the product $\phi A$ (observational equivalence).85(b) Heterogeneous sellers: an equilibrium exists; uniqueness is not guaranteed (the stock composition $N$ depends on $\theta$) and is86verified numerically by checking the sign pattern of $R$ on a fine grid.87*Proof.* (a) $\lambda_b=A\theta^{\alpha-1}$ is strictly decreasing; $V(\theta)$ is increasing in $\theta$ through $\Lambda_s=\phi A\theta^\alpha$88(Proposition 2(i)), so $H(V+k)$ is nonincreasing; the product of a strictly decreasing positive function and a nonincreasing positive89function is strictly decreasing. Limits: $\theta\to\infty$ gives $\lambda_b\to0$; $\theta\to0$ gives $\lambda_b\to\infty$ and90$H(V+k)\to H(k-c/(r+\delta))>0$. Comparative statics: $\partial R/\partial\phi=(1-\beta)\lambda_b[H-\phi(1-G)\partial V/\partial\phi]$ with91$\partial V/\partial\phi=\lambda_s\beta H/[(r+\delta)+\Lambda_s\beta(1-G)]$, so $\partial R/\partial\phi=(1-\beta)\lambda_bH\,(r+\delta)/[(r+\delta)+\Lambda_s\beta(1-G)]>0$92and $d\theta^*/d\phi=-R_\phi/R_\theta>0$; identical algebra for $A$. Observational equivalence: $\Lambda_s=\phi A\theta^\alpha$,93$\Lambda_b=\phi A\theta^{\alpha-1}$. (b) $R$ is continuous in $\theta$ because $V(c;\theta)$, $\eta$, $n$ and $N$ are; the limits are as94in (a) (bounded $H$, positive weights), so the intermediate value theorem applies. $\square$9596## Proposition 4 (a liquidity shock without a fundamental shock)97*Statement.* Let $G$ be IFR. A fall in $A$ or in $\phi$ with $(G,F,r,k,\kappa_b,s)$ unchanged: (i) lowers $V(c)$ and $\bar p(c)$ for every98$c$ (partial and, by Proposition 3(a), general equilibrium, where $\theta$ falls as well); (ii) lowers every sale hazard and the sales99flow on impact (stock of listings predetermined); (iii) raises the steady-state inventory $S$ and mean listing duration; (iv) lowers the100steady-state sales flow iff $\delta>0$ (with $\delta=0$ the steady-state sales flow equals the listing inflow $s$ regardless of liquidity).101Transaction prices and volume therefore co-move, and inventory and time on market move inversely, with no change in $G$.102*Proof.* (i) Proposition 2(i); in general equilibrium $\theta^*$ falls (Proposition 3(a)), reinforcing the fall in $\Lambda_s$. (ii)103Proposition 2(ii); $Q=\int\eta(c)n(c)dc$ with $n$ fixed on impact. (iii) $n(c)=sf(c)/(\eta(c)+\delta)$ rises for every $c$;104$E[\tau\mid c]=1/(\eta+\delta)$ rises. (iv) $Q_{ss}=s\int\frac{\eta(c)}{\eta(c)+\delta}dF(c)$, strictly increasing in each $\eta$ iff $\delta>0$. $\square$105106## Proposition 5 (price dispersion among identical houses)107*Statement.* Conditional on a sale, $\mathrm{Var}_\omega(p)=\beta^2E_\omega[\mathrm{Var}(w\mid w\ge v^*(c))]+\mathrm{Var}_\omega[\bar p(c)]$.108(i) The between-type component is strictly positive iff the transaction composition $\omega$ is non-degenerate.109(ii) With $\beta=0$ all dispersion is urgency-driven: $p=V(c)$ and $\mathrm{Var}_\omega(p)\le\mathrm{Var}_\omega(c)/(r+\delta)^2$;110more generally, under IFR, the range of expected prices satisfies $\bar p(c_L)-\bar p(c_H)\le(c_H-c_L)/(r+\delta)$.111(iii) Under IFR the between-type reservation-value gap widens when $\Lambda_s$ falls (Proposition 2(iii)).112*Proof.* Law of total variance with $p=(1-\beta)V(c)-\beta U+\beta w$ and $\mathrm{Var}(p\mid c)=\beta^2\mathrm{Var}(w\mid w\ge v^*(c))$. (i)113$\bar p$ is strictly monotone (Proposition 1(iii)). (ii) $V$ is Lipschitz with constant $1/(r+\delta)$ (Proposition 1(i)); for any114Lipschitz $f$ with constant $L$, $\mathrm{Var}f(X)=\tfrac12E[(f(X)-f(X'))^2]\le L^2\mathrm{Var}X$. Under IFR $|\bar p'|\le|V'|\le1/(r+\delta)$. $\square$115116## Proposition 6 (sale selection and transaction-price indices)117*Statement.* (i) If $\delta=0$, the steady-state transaction composition is $\omega(c)=f(c)$ for every $\Lambda_s$: comparisons across118steady states contain no composition effect, and the entire cross-steady-state price change is a within-type liquidity effect.119(ii) *Impact effect.* Consider an unanticipated permanent change $d\ln\Lambda_s$ with the stock $n(\cdot)$ predetermined. Let120$\epsilon(c)\equiv\partial\ln\eta(c)/\partial\ln\Lambda_s=1-\beta\eta(c)\,h_G(v^*)M(v^*)/[(r+\delta)+\beta\eta(c)]$. Then121$d\ln\omega(c)=[\epsilon(c)-E_\omega\epsilon]\,d\ln\Lambda_s$ and122$$d\ln P=\underbrace{E_\omega[d\ln\bar p(c)]}_{\text{within-type (liquidity)}}+\underbrace{\mathrm{Cov}_\omega\big(\bar p(c),\epsilon(c)\big)\,d\ln\Lambda_s/P}_{\text{composition (selection)}}.$$123If $\epsilon(c)$ is decreasing in $c$, the covariance is positive (both $\bar p$ and $\epsilon$ decrease in $c$), so the index falls by124more than every within-type price after a negative liquidity shock and rises by more after a positive one: sale selection amplifies125measured price responses to liquidity. (iii) For exponential $G$, $\epsilon(c)=(r+\delta)/[(r+\delta)+\beta\eta(c)]\in(0,1)$, strictly126decreasing in $c$ (amplification). For Pareto $G$, $h_GM=a/(a-1)$ is constant and $\epsilon(c)$ is again decreasing in $c$. In general,127a sufficient condition for amplification is that $v\mapsto h_G(v)M(v)$ be nonincreasing; when $h_GM$ is *increasing* in $v$ (light-tailed128$G$ such as the truncated normal, where $h_GM\uparrow1$ in the upper tail) the two factors in $\epsilon$ move in opposite directions and129$\epsilon(c)$ can be increasing in $c$, in which case $\mathrm{Cov}_\omega(\bar p,\epsilon)<0$ and selection *dampens* the measured response. The130sign of the selection term is therefore a property of the match-value distribution, not a general law; it is reported numerically in131Experiment 10 for the truncated normal, exponential, Weibull and Pareto cases.132(iv) With $\delta=0$ the composition returns to $f$ in the new steady state, so the impact composition effect is transitory: the observed133index overshoots its long-run response.134*Proof.* (i) $\omega(c)=\eta(c)n(c)/Q$ with $n=sf/\eta$ and $Q=s$. (ii) $\omega^+\propto\eta^+n$ with $n$ fixed, so135$d\ln\omega=d\ln\eta-d\ln Q=(\epsilon-E_\omega\epsilon)d\ln\Lambda_s$. Then $dP=\int\bar p\,d\omega+\int\omega\,d\bar p$ and136$\int\bar p\,d\omega=\int\bar p\,\omega(\epsilon-E_\omega\epsilon)\,d\ln\Lambda_s=\mathrm{Cov}_\omega(\bar p,\epsilon)d\ln\Lambda_s$. Two functions137that are both decreasing in $c$ are comonotone, hence positively correlated under any $\omega$ (Chebyshev's sum inequality), strictly138if both are strictly monotone and $\omega$ non-degenerate. Formula for $\epsilon$: $\eta=\Lambda_s(1-G(v^*))$,139$\partial\ln\eta/\partial\ln\Lambda_s=1-h_G(v^*)\Lambda_s\partial V/\partial\Lambda_s=1-h_G\Lambda_s\beta H/[(r+\delta)+\Lambda_s\beta(1-G)]$ and140$\Lambda_sH=\eta M$. (iii) Exponential: $h_GM=1$. If $h_GM$ is nonincreasing in $v$, then as $c$ rises $v^*$ falls, $h_GM$ rises and141$\beta\eta/(r+\delta+\beta\eta)$ rises, so $\epsilon$ falls. (iv) is (i). $\square$142143## Proposition 7 (financing contingency: cash versus mortgage buyers)144*Statement.* In the extension of §6, for a seller of type $c$ and a given match $w$ acceptable to both buyer types,145$$p_m(w,c)-p_c(w,c)=\underbrace{\frac{(1-\beta)[c+rV(c)]-\beta[\kappa_b+rU_m]}{\nu\phi}}_{\Delta_b(c)\ \text{(delay-bargaining term)}}146+\underbrace{\beta(U_c-U_m)}_{\text{outside-option term}}.$$147(i) $\Delta_b(c)$ is positive iff the seller's bargaining-weighted flow cost of waiting, $(1-\beta)[c+rV(c)]$, exceeds the buyer's,148$\beta[\kappa_b+rU_m]$; it is nondecreasing in $c$ ($d\Delta_b/dc=(1-\beta)(1+rV')/(\nu\phi)\ge0$) and vanishes as $\nu\to\infty$.149(ii) The acceptance thresholds differ by $v^*_m(c)-v^*_c(c)=\underbrace{[c+\kappa_b+r(V+U_m)]/(\nu\phi)}_{\text{pending wedge}}-(U_c-U_m)$:150mortgage agreements require a better match than cash agreements iff the pending wedge exceeds the cash buyers' outside-option151advantage $U_c-U_m$ (the wedge is increasing in $c$, so this is more likely for urgent sellers). The observed mean price gap conditional152on sale adds the selection term $\beta\{E[w\mid w\ge v^*_m]-E[w\mid w\ge v^*_c]\}$, which has the sign of $v^*_m-v^*_c$ (by (F2)) and can153therefore be negative for patient sellers when cash buyers have a sufficiently better outside option.154(iii) The baseline screen is the limit $\nu\to\infty$: $v^*_m\to V+U_m$, $\Delta_b\to0$, and the seller's effective contact rate is155$\lambda_s[\tilde\pi+(1-\tilde\pi)\phi]$.156*Proof.* Pending values are linear in $p$ with slopes $\pm\nu\phi/(r+\nu)$, so the Nash product is maximised where157$P_s-V=\beta\Sigma_m$. Writing $\Sigma_m=[\nu\phi(w-V-U_m)-c-\kappa_b-r(V+U_m)]/(r+\nu)$ and solving $P_s-V=\beta\Sigma_m$ for $p$ gives158$p_m=V+\beta(w-V-U_m)+\{(1-\beta)[c+rV]-\beta[\kappa_b+rU_m]\}/(\nu\phi)$; subtracting $p_c=V+\beta(w-V-U_c)$ yields the display.159(i) $1+rV'=[\delta+\lambda_s\beta(\cdot)]/[(r+\delta)+\lambda_s\beta(\cdot)]\ge0$ where $(\cdot)$ is the positive acceptance mass. (ii)160$\Sigma_m\ge0$ iff $w\ge v^*_m$; subtract $v^*_c=V+U_c$; $E[w\mid w\ge v]$ is strictly increasing by (F2). (iii) Direct from the formulas. $\square$161162## Proposition 8 (posted prices)163*Statement.* Let $g$ be log-concave. (i) The seller's problem $\max_y[1-G(y)](y-U-V)$ has a unique interior solution $y(c)$ characterised by164$y-U-V(c)=1/h_G(y)$; (ii) $V(c)$ is unique, with $V'(c)=-1/[(r+\delta)+\Lambda_s(1-G(y(c)))]<0$, and the posted price $p(c)=y(c)-U$ and the165threshold $y(c)$ are strictly decreasing in $c$; the hazard is increasing in $c$; (iii) at equal $(V,U)$, $y(c)>v^*_{Nash}(c)=V+U$: posted-price166sellers hold out for better matches, and all price dispersion among identical houses is urgency-driven.167*Proof.* (i) $\ln[(1-G(y))(y-a)]$ is strictly concave on $(a,w_H)$ (log-concave $g$ implies log-concave $1-G$), zero at both ends; the FOC168$-g(y)(y-a)+(1-G(y))=0$ gives the characterisation. (ii) By the envelope theorem the maximised value is decreasing in $V$ with slope169$-(1-G(y))$, so $\Phi_P(V)=-c+\Lambda_s\max_y(\cdot)-(r+\delta)V$ is strictly decreasing and has a unique root; the implicit function theorem gives170$V'$. $\psi(y)\equiv y-1/h_G(y)$ is strictly increasing under IFR, so $y(c)=\psi^{-1}(U+V(c))$ is strictly increasing in $V$, hence decreasing in $c$.171(iii) $1/h_G(y)>0$. $\square$172173## Results stated as numerical (no proof attempted)174- Uniqueness of $\theta^*$ with heterogeneous sellers (checked on a grid; reported in `validation.md`).175- Efficiency: the welfare-maximising $\beta$ relative to the Hosios value $1-\alpha$, with and without heterogeneity.176- Monotonicity of $h_GM$ for the truncated normal (checked numerically to justify Proposition 6(iii) in the baseline calibration).177- Full transition paths and the size of the index overshoot (Experiment 9–10).178