\section{Proofs}
\label{app:proofs}

Throughout, $U$ is given, $h_G=g/(1-G)$ and $M=H/(1-G)$. Two facts are used repeatedly. (F1) $H'(v)=-(1-G(v))$ and
$M'(v)=-1+h_G(v)M(v)$, so $M'>-1$ wherever $g>0$; under IFR $M(v)=\int_v^{w_H}\exp(-\int_v^wh_G)\,dw\le\int_v^\infty e^{-h_G(v)(w-v)}dw
=1/h_G(v)$, hence $M'\le0$. (F2) $\frac{d}{dv}E[w\mid w\ge v]=1+M'(v)=h_G(v)M(v)>0$ wherever $g>0$; under IFR $gH=g(1-G)M\le(1-G)^2$.

\paragraph{Lemma~\ref{lem:scale}.}
$H_q(qv)=\int(q\varepsilon-qv)^+dG_\varepsilon=qH_1(v)$ and $1-G_q(qv)=1-G_\varepsilon(v)$. Substituting $V_q=qV$, $U_q=qU$ into \eqref{eq:S}
and \eqref{eq:B} gives the $q=1$ equations multiplied by $q$; hazards, stocks and $\theta$ are unchanged; free entry reads
$qU=q\tilde k$; prices scale by \eqref{eq:price}. \qed

\paragraph{Lemma~\ref{lem:value}.}
$\Phi(V)=-c+\delta w_0+\Lambda_s\beta H(V+U)-(r+\delta)V$ is continuous and strictly decreasing, $\Phi'=-\Lambda_s\beta(1-G)-(r+\delta)\le-(r+\delta)$;
$\Phi\to+\infty$ as $V\to-\infty$ because $H(v)=E[w]-v$ for $v\le w_L$, and $\Phi\to-\infty$ as $V\to+\infty$ because $H\ge0$. Hence a unique
root. $\Phi$ is $C^1$ in all arguments and $\Phi_V<0$, so the implicit function theorem applies with $\Phi_c=-1$,
$\Phi_{\Lambda_s}=\beta H(v^*)$ and $\Phi_U=-\Lambda_s\beta(1-G(v^*))$. \qed

\paragraph{Lemma~\ref{lem:duration}.}
$\bar h(\tau)=\int\mu e^{-\mu\tau}dF/\int e^{-\mu\tau}dF=E_\tau[\mu]$ under the tilted measure $dF_\tau\propto e^{-\mu\tau}dF$, and
$\bar h'(\tau)=-\mathrm{Var}_\tau(\mu)<0$. \qed

\paragraph{Proposition~\ref{prop:urgency}.}
(i) is Lemma~\ref{lem:value}. (ii) $\eta'(c)=-\Lambda_sg(v^*)V'(c)\ge0$. (iii) Conditional on sale $p=V+\beta(w-v^*)$, so
$\bar p=V+\beta M(v^*)$ and $\bar p'=V'[1+\beta M'(v^*)]$ since $v^{*\prime}=V'$; by (F1) $1+\beta M'>1-\beta>0$, and under IFR $1+\beta M'\le1$.
$D(c)=1-\bar p(c)/\bar p(c_L)$ inherits monotonicity when $\bar p(c_L)>0$. (iv) The root of the decreasing $\Phi$ is at least $w_0$ iff
$\Phi(w_0)\ge0$, i.e. $-c+\delta w_0+\Lambda_s\beta H(w_0+k)-(r+\delta)w_0\ge0$. \qed

\paragraph{Proposition~\ref{prop:arrival}.}
(i) Lemma~\ref{lem:value} and $\partial\bar p/\partial\Lambda_s=\partial V/\partial\Lambda_s[1+\beta M']$. (ii)
$\partial\eta/\partial\Lambda_s=(1-G)-\Lambda_sg\,\partial V/\partial\Lambda_s$; multiplying by the positive denominator of \eqref{eq:Vderiv}, the sign
is that of $(1-G)(r+\delta)+\Lambda_s\beta[(1-G)^2-gH]$, positive under IFR by (F2). Pareto: $1-G=(w_L/v)^a$, $g=aw_L^av^{-a-1}$,
$M=v/(a-1)$, so $gH=\frac{a}{a-1}(w_L/v)^{2a}$ and $(1-G)^2-gH=-(1-G)^2/(a-1)$; the sign becomes that of $(r+\delta)-\beta\eta/(a-1)$. Since
$H(v^*)=(1-G(v^*))v^*/(a-1)$, equation \eqref{eq:S} reads $(r+\delta)V+c-\delta w_0=\beta\eta v^*/(a-1)$ with $v^*=V+U$, i.e.
$\beta\eta=(a-1)(r+\delta)+(a-1)[c-\delta w_0-(r+\delta)U]/v^*$, whence $\beta\eta>(a-1)(r+\delta)$ iff $c>\delta w_0+(r+\delta)U$.
(iii) $V(c_L)-V(c)=\int_{c_L}^c du/[(r+\delta)+\beta\eta(u)]$ by (i) of Proposition~\ref{prop:urgency}, and $\eta(u)$ is increasing in
$\Lambda_s$ by (ii). (iv) As $\Lambda_s\to0$, \eqref{eq:S} gives $V\to(-c+\delta w_0)/(r+\delta)$, which is $0$ for $c=0$, $w_0=0$; then $v^*\to k\le w_L$,
$M(k)=E[w]-k$ and $\bar p\to\beta(E[w]-k)$. As $\Lambda_s\to\infty$, $H(v^*)=[(r+\delta)V+c-\delta w_0]/(\Lambda_s\beta)\to0$ forces $v^*\to w_H$, so
$V\to w_H-k$ and $M\to0$. $p^*<w_H-k$ iff $H(p^*)=rk>H(w_H-k)$ since $H$ is decreasing. \qed

\paragraph{Proposition~\ref{prop:entry}.}
(a) $\lambda_b=A\theta^{\alpha-1}$ is strictly decreasing; $V$ is increasing in $\theta$ through $\Lambda_s=\phi A\theta^\alpha$, so $H(V+k)$ is
nonincreasing; the product of a strictly decreasing positive function and a nonincreasing positive function is strictly decreasing.
As $\theta\to\infty$, $\lambda_b\to0$ and $R\to-\kappa_b-rk$; as $\theta\to0$, $\lambda_b\to\infty$ and $H(V+k)\to H(k+(\delta w_0-c)/(r+\delta))>0$.
$\partial R/\partial\phi=(1-\beta)\lambda_b[H-\phi(1-G)\partial V/\partial\phi]$ with $\partial V/\partial\phi=\lambda_s\beta H/[(r+\delta)+\Lambda_s\beta(1-G)]$, so
$\partial R/\partial\phi=(1-\beta)\lambda_bH(r+\delta)/[(r+\delta)+\Lambda_s\beta(1-G)]>0$ and $d\theta^*/d\phi=-R_\phi/R_\theta>0$; the same algebra applies to
$A$. Observational equivalence: $\Lambda_s=\phi A\theta^\alpha$ and $\Lambda_b=\phi A\theta^{\alpha-1}$. (b) $R$ is continuous in $\theta$ because
$V(c;\theta)$, $\eta$, $n$ and $N$ are, with the same limits, so the intermediate value theorem applies. \qed

\paragraph{Proposition~\ref{prop:shock}.}
(i) Proposition~\ref{prop:arrival}(i), and Proposition~\ref{prop:entry}(a) for the fall in $\theta^*$. (ii)
Proposition~\ref{prop:arrival}(ii) with $n$ fixed in $Q=\int\eta n$. (iii) $n(c)=sf(c)/(\eta(c)+\delta)$ and $1/(\eta+\delta)$ rise.
(iv) $Q_{ss}=s\int\eta/(\eta+\delta)\,dF$ is strictly increasing in each $\eta$ iff $\delta>0$. \qed

\paragraph{Proposition~\ref{prop:dispersion}.}
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)
$\bar p$ is strictly monotone. (ii) $V$ is Lipschitz with constant $1/(r+\delta)$; for Lipschitz $f$ with constant $L$,
$\mathrm{Var}f(X)=\tfrac12E[(f(X)-f(X'))^2]\le L^2\mathrm{Var}X$; under IFR $|\bar p'|\le|V'|\le1/(r+\delta)$. (iii) Proposition~\ref{prop:arrival}(iii). \qed

\paragraph{Proposition~\ref{prop:selection}.}
(i) $\omega=\eta n/Q$ with $n=sf/\eta$ and $Q=s$. (ii) $\omega^+\propto\eta^+n$ with $n$ fixed, so $d\ln\omega=d\ln\eta-d\ln Q=(\epsilon-E_\omega\epsilon)\,d\ln\Lambda_s$.
Then $dP=\int\bar p\,d\omega+\int\omega\,d\bar p$ and $\int\bar p\,d\omega=\mathrm{Cov}_\omega(\bar p,\epsilon)\,d\ln\Lambda_s$. Two functions that are both
decreasing in $c$ are comonotone and positively correlated under any $\omega$ (Chebyshev's sum inequality), strictly if both are
strictly monotone and $\omega$ is non-degenerate; two functions of opposite monotonicity are negatively correlated. The formula for
$\epsilon$: $\partial\ln\eta/\partial\ln\Lambda_s=1-h_G(v^*)\Lambda_s\partial V/\partial\Lambda_s=1-h_G\Lambda_s\beta H/[(r+\delta)+\Lambda_s\beta(1-G)]$ and
$\Lambda_sH=\eta M$. (iii) Exponential: $h_GM=1$. Pareto: $h_GM=a/(a-1)$. If $h_GM$ is nonincreasing in $v$, then as $c$ rises $v^*$
falls, $h_GM$ rises and $\beta\eta/(r+\delta+\beta\eta)$ rises, so $\epsilon$ falls; if $h_GM$ is increasing the two factors move in opposite
directions and no general sign obtains. (iv) is (i). \qed

\paragraph{Proposition~\ref{prop:financing}.}
Pending values are linear in $p$ with slopes $\pm\nu\phi_m/(r+\nu)$, so the Nash product is maximised where $P_s-V=\beta\Sigma_m$.
Writing $\Sigma_m=[\nu\phi_m(w-V-U_m)-c-\kappa_b-r(V+U_m)]/(r+\nu)$ and solving $P_s-V=\beta\Sigma_m$ for $p$ gives
$p_m=V+\beta(w-V-U_m)+\{(1-\beta)[c+rV]-\beta[\kappa_b+rU_m]\}/(\nu\phi_m)$; subtracting $p_c=V+\beta(w-V-U_c)$ yields \eqref{eq:gap}. (i)
$1+rV'=[\delta+\lambda_s\beta(\cdot)]/[(r+\delta)+\lambda_s\beta(\cdot)]\ge0$, where $(\cdot)$ is the positive acceptance mass of the two buyer types.
(ii) $\Sigma_m\ge0$ iff $w\ge v^*_m$; subtract $v^*_c=V+U_c$; $E[w\mid w\ge v]$ is strictly increasing by (F2). (iii) Direct. \qed

\paragraph{Proposition~\ref{prop:posted}.}
(i) $\ln[(1-G(y))(y-a)]$ is strictly concave on $(a,w_H)$ because log-concave $g$ implies log-concave $1-G$, and the objective
vanishes at both ends; the first-order condition $-g(y)(y-a)+(1-G(y))=0$ gives the characterisation. (ii) By the envelope theorem
the maximised value has slope $-(1-G(y))$ in $V$, so $\Phi_P(V)=-c+\delta w_0+\Lambda_s\max_y(\cdot)-(r+\delta)V$ is strictly decreasing with a
unique root, and the implicit function theorem gives $V'$. $\psi(y)=y-1/h_G(y)$ is strictly increasing under IFR, so
$y(c)=\psi^{-1}(U+V(c))$ is increasing in $V$ and decreasing in $c$. (iii) $1/h_G(y)>0$. \qed