WP19 — Formal results and proofs (Phase 2)
Notation as in model_specification.md. Throughout, $U$ is treated as given (partial equilibrium) unless stated; under free entry
$U=k$. $h_G(v)=g(v)/(1-G(v))$ is the hazard function of $G$, $M(v)=H(v)/(1-G(v))$ its mean residual valuation.
"IFR" = increasing failure rate = $1-G$ log-concave; a log-concave density implies IFR (Bagnoli–Bergstrom 2005).
Two elementary facts used repeatedly: (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 IFR (because under IFR $M(v)=\int_v^{w_H}\exp(-\int_v^w h_G),dw\le\int_v^{\infty}e^{-h_G(v)(w-v)}dw=1/h_G(v)$). (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, $gH=g(1-G)M\le(1-G)^2$.
Lemma 1 (scale invariance in quality)
Statement. Let $w=q\varepsilon$ with $\varepsilon\sim G_\varepsilon$, $c=q\kappa$, $k=q\tilde k$, $\kappa_b=q\tilde\kappa_b$. If $(\theta,U,V(\cdot),n(\cdot))$ is a stationary equilibrium of the economy with $q=1$ and primitives $(G_\varepsilon,\kappa,\tilde k,\tilde\kappa_b)$, then $(\theta,qU,qV(\cdot),n(\cdot))$ is a stationary equilibrium of the economy with quality $q$; prices scale by $q$; hazards, durations, $\theta$, $\omega$ and $D(c)$ are invariant. Proof. $H_q(qv)=\int(q\varepsilon-qv)^+dG_\varepsilon=qH_1(v)$. Substituting $V_q=qV$, $U_q=qU$ into (S): $(r+\delta)qV=-q\kappa+\Lambda_s\beta qH_1(V+U)$, which 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 entry $qU=q\tilde k$. Prices $p=(1-\beta)qV-\beta qU+\beta q\varepsilon$. $\square$
Lemma 2 (seller value: existence, uniqueness, smoothness)
Statement. For any $U\in\mathbb R$, $\Lambda_s>0$, $c\ge0$, equation (S) has a unique solution $V(c)$; $V$ is continuously differentiable in $(c,\Lambda_s,U)$ with $$\partial_cV=\frac{-1}{(r+\delta)+\Lambda_s\beta(1-G(v^))},\quad \partial_{\Lambda_s}V=\frac{\beta H(v^)}{(r+\delta)+\Lambda_s\beta(1-G(v^))},\quad \partial_UV=\frac{-\Lambda_s\beta(1-G(v^))}{(r+\delta)+\Lambda_s\beta(1-G(v^))}\in(-1,0].$$ Proof. $\Phi(V)\equiv-c+\Lambda_s\beta H(V+U)-(r+\delta)V$ is continuous, strictly decreasing ($\Phi'=-\Lambda_s\beta(1-G)-(r+\delta)\le-(r+\delta)<0$), $\Phi(V)\to+\infty$ 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 unique root. $\Phi$ is $C^1$ in all arguments ($H'=-(1-G)$ continuous), and $\Phi_V<0$, so the implicit function theorem gives the derivatives: $\partial_xV=-\Phi_x/\Phi_V$ with $\Phi_c=-1$, $\Phi_{\Lambda_s}=\beta H(v^)$, $\Phi_U=-\Lambda_s\beta(1-G(v^*))$. $\square$
Lemma 3 (spurious negative duration dependence)
Statement. If $\mu(c)\equiv\eta(c)+\delta$ is non-degenerate under $F$, the population exit hazard $\bar h(\tau)=-\frac{d}{d\tau}\ln\bar{\mathcal S}(\tau)$, $\bar{\mathcal S}(\tau)=\int e^{-\mu(c)\tau}dF(c)$, is strictly decreasing in $\tau$, although every individual hazard is constant. 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 tilted measure $dF_\tau\propto e^{-\mu\tau}dF$. Differentiating, $\bar h'(\tau)=-\mathrm{Var}_\tau(\mu)<0$. (Classical mixture result; cf. Lancaster 1979.) $\square$
Proposition 1 (seller urgency)
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 reservation price falls with the carrying cost, by less than one-for-one in present-value terms. (ii) The sale hazard $\eta(c)$ is nondecreasing in $c$ and strictly increasing wherever $g(v^(c))>0$; expected listing duration is nonincreasing. (iii) The expected transaction price $\bar p(c)=V(c)+\beta M(v^(c))$ is strictly decreasing in $c$, with $\bar p'(c)=V'(c)[1+\beta M'(v^(c))]$ and $1+\beta M'>1-\beta>0$; under IFR, $|\bar p'(c)|\le|V'(c)|$ (the price falls by less than the reservation price). Hence the urgency discount $D(c)$ is zero at $c_L$, strictly increasing in $c$ and nonnegative. (iv) Under free entry ($U=k$), a type-$c$ seller has $V(c)\ge0$ iff $c\le\bar c\equiv\Lambda_s\beta H(k)$. 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 on sale, $w\ge v^$ and $p=V+\beta(w-v^)$, so $\bar p=V+\beta E[w-v^\mid w\ge v^]=V+\beta M(v^)$. Differentiate: $\bar p'=V'+\beta M'(v^)v^{\prime}=V'[1+\beta M']$ because $v^{\prime}=V'$ ($U$ fixed). By (F1) $M'>-1$ so $1+\beta M'>1-\beta>0$, and $V'<0$ gives $\bar p'<0$. Under IFR $M'\le0$ so $0<1+\beta M'\le1$. $D(c)=1-\bar p(c)/\bar p(c_L)$ inherits monotonicity provided $\bar p(c_L)>0$ (true when $V(c_L)\ge0$). (iv) $\Phi$ is decreasing, so the root is $\ge0$ iff $\Phi(0)\ge0$, i.e. $-c+\Lambda_s\beta H(k)\ge0$. $\square$
Proposition 2 (effective arrival rate: partial equilibrium)
Statement. Fix $U$. Let $\Lambda_s=\phi\lambda_s$ (so $A$, $\phi$ and $\theta$ act only through $\Lambda_s$). (i) $\partial V/\partial\Lambda_s>0$ and $\partial\bar p/\partial\Lambda_s>0$ whenever $v^<w_H$. (ii) If $G$ is IFR, $\partial\eta/\partial\Lambda_s>0$: a decline in arrivals or in qualification lengthens expected time on market. Without IFR the sign can reverse: for Pareto $G(w)=1-(w_L/w)^a$, $a>1$, $\partial\eta/\partial\Lambda_s<0$ iff $\beta\eta(c)>(a-1)(r+\delta)$, which, using (S), is equivalent to $c>\delta w_0+(r+\delta)U$ — a condition independent of $\Lambda_s$: for a sufficiently urgent seller facing heavy-tailed valuations, a more liquid market lengthens time on market because the seller becomes pickier faster than buyers arrive. (iii) Under IFR, the reservation-value gap $V(c_L)-V(c)$, $c>c_L$, is strictly decreasing in $\Lambda_s$: lower liquidity widens the cross-sectional dispersion of reservation prices. (iv) Limits (patient type $c=0$, $U=k\le w_L$, $w_H<\infty$): $\lim_{\Lambda_s\to0}\bar p=\beta(E[w]-k)$ and $\lim_{\Lambda_s\to\infty}\bar p=w_H-k$. If $\beta(E[w]-k)<p^<w_H-k$ (the second inequality holds iff $H(w_H-k)<rk$), the market liquidity gap $\Gamma$ of the patient type is positive in thin markets and negative in thick markets. Proof. (i) Lemma 2 gives $\partial V/\partial\Lambda_s=\beta H(v^)/[(r+\delta)+\Lambda_s\beta(1-G)]>0$ when $H(v^)>0$; and $\partial\bar p/\partial\Lambda_s=\partial V/\partial\Lambda_s,[1+\beta M'(v^)]>0$ by (F1). (ii) $\partial\eta/\partial\Lambda_s=(1-G)-\Lambda_sg,\partial V/\partial\Lambda_s$. Multiplying by the positive denominator, the sign equals that of $(1-G)(r+\delta)+\Lambda_s\beta[(1-G)^2-gH]$, which is positive under IFR by (F2). Pareto: $1-G=(w_L/v)^a$, $g=a w_L^av^{-a-1}$, $M=v/(a-1)$, 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 of $(1-G){(r+\delta)-\beta\Lambda_s(1-G)/(a-1)}=(1-G){(r+\delta)-\beta\eta/(a-1)}$. For Pareto, $H(v^)=(1-G(v^))v^/(a-1)$, so (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) $V(c_L)-V(c)=\int_{c_L}^{c}\frac{du}{(r+\delta)+\beta\eta(u)}$ by (i) of Proposition 1, and $\eta(u)$ is increasing in $\Lambda_s$ by (ii). (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 $\bar p\to\beta(E[w]-k)$. As $\Lambda_s\to\infty$, $H(v^)=[(r+\delta)V+c]/(\Lambda_s\beta)\to0$ forces $v^\to w_H$, so $V\to w_H-k$ and $M(v^)\to0$. $p^<w_H-k$ iff $H(p^*)=rk>H(w_H-k)$ since $H$ is decreasing. $\square$
Proposition 3 (free entry and tightness)
Statement. (a) Homogeneous sellers ($F$ degenerate at $c$). The free-entry residual $R(\theta)\equiv-\kappa_b+\phi\lambda_b(\theta)(1-\beta)H(V(\theta)+k)-rk$ is continuous and strictly decreasing with $R(0^+)=+\infty$, $R(\infty)=-\kappa_b-rk<0$; hence a unique equilibrium tightness $\theta^$. Moreover $d\theta^/d\phi>0$ and $d\theta^/dA>0$, and $\frac{d\ln\Lambda_s}{d\ln\phi}=1+\alpha\frac{d\ln\theta^}{d\ln\phi}>1$ (tightness amplifies a qualification shock). In the baseline, $A$ and $\phi$ enter all equilibrium conditions only through the product $\phi A$ (observational equivalence). (b) Heterogeneous sellers: an equilibrium exists; uniqueness is not guaranteed (the stock composition $N$ depends on $\theta$) and is verified numerically by checking the sign pattern of $R$ on a fine grid. Proof. (a) $\lambda_b=A\theta^{\alpha-1}$ is strictly decreasing; $V(\theta)$ is increasing in $\theta$ through $\Lambda_s=\phi A\theta^\alpha$ (Proposition 2(i)), so $H(V+k)$ is nonincreasing; the product of a strictly decreasing positive function and a nonincreasing positive function is strictly decreasing. Limits: $\theta\to\infty$ gives $\lambda_b\to0$; $\theta\to0$ gives $\lambda_b\to\infty$ and $H(V+k)\to H(k-c/(r+\delta))>0$. Comparative statics: $\partial R/\partial\phi=(1-\beta)\lambda_b[H-\phi(1-G)\partial V/\partial\phi]$ 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 $d\theta^*/d\phi=-R_\phi/R_\theta>0$; identical algebra for $A$. Observational equivalence: $\Lambda_s=\phi A\theta^\alpha$, $\Lambda_b=\phi A\theta^{\alpha-1}$. (b) $R$ is continuous in $\theta$ because $V(c;\theta)$, $\eta$, $n$ and $N$ are; the limits are as in (a) (bounded $H$, positive weights), so the intermediate value theorem applies. $\square$
Proposition 4 (a liquidity shock without a fundamental shock)
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 every $c$ (partial and, by Proposition 3(a), general equilibrium, where $\theta$ falls as well); (ii) lowers every sale hazard and the sales flow on impact (stock of listings predetermined); (iii) raises the steady-state inventory $S$ and mean listing duration; (iv) lowers the steady-state sales flow iff $\delta>0$ (with $\delta=0$ the steady-state sales flow equals the listing inflow $s$ regardless of liquidity). Transaction prices and volume therefore co-move, and inventory and time on market move inversely, with no change in $G$. Proof. (i) Proposition 2(i); in general equilibrium $\theta^*$ falls (Proposition 3(a)), reinforcing the fall in $\Lambda_s$. (ii) Proposition 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 $c$; $E[\tau\mid c]=1/(\eta+\delta)$ rises. (iv) $Q_{ss}=s\int\frac{\eta(c)}{\eta(c)+\delta}dF(c)$, strictly increasing in each $\eta$ iff $\delta>0$. $\square$
Proposition 5 (price dispersion among identical houses)
Statement. Conditional on a sale, $\mathrm{Var}\omega(p)=\beta^2E\omega[\mathrm{Var}(w\mid w\ge v^(c))]+\mathrm{Var}\omega[\bar p(c)]$. (i) The between-type component is strictly positive iff the transaction composition $\omega$ is non-degenerate. (ii) With $\beta=0$ all dispersion is urgency-driven: $p=V(c)$ and $\mathrm{Var}\omega(p)\le\mathrm{Var}_\omega(c)/(r+\delta)^2$; more generally, under IFR, the range of expected prices satisfies $\bar p(c_L)-\bar p(c_H)\le(c_H-c_L)/(r+\delta)$. (iii) Under IFR the between-type reservation-value gap widens when $\Lambda_s$ falls (Proposition 2(iii)). 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) $\bar p$ is strictly monotone (Proposition 1(iii)). (ii) $V$ is Lipschitz with constant $1/(r+\delta)$ (Proposition 1(i)); for any Lipschitz $f$ with constant $L$, $\mathrm{Var}f(X)=\tfrac12E[(f(X)-f(X'))^2]\le L^2\mathrm{Var}X$. Under IFR $|\bar p'|\le|V'|\le1/(r+\delta)$. $\square$
Proposition 6 (sale selection and transaction-price indices)
Statement. (i) If $\delta=0$, the steady-state transaction composition is $\omega(c)=f(c)$ for every $\Lambda_s$: comparisons across steady states contain no composition effect, and the entire cross-steady-state price change is a within-type liquidity effect. (ii) Impact effect. Consider an unanticipated permanent change $d\ln\Lambda_s$ with the stock $n(\cdot)$ predetermined. Let $\epsilon(c)\equiv\partial\ln\eta(c)/\partial\ln\Lambda_s=1-\beta\eta(c),h_G(v^)M(v^)/[(r+\delta)+\beta\eta(c)]$. Then $d\ln\omega(c)=[\epsilon(c)-E_\omega\epsilon],d\ln\Lambda_s$ and $$d\ln P=\underbrace{E_\omega[d\ln\bar p(c)]}{\text{within-type (liquidity)}}+\underbrace{\mathrm{Cov}\omega\big(\bar p(c),\epsilon(c)\big),d\ln\Lambda_s/P}{\text{composition (selection)}}.$$ If $\epsilon(c)$ is decreasing in $c$, the covariance is positive (both $\bar p$ and $\epsilon$ decrease in $c$), so the index falls by more than every within-type price after a negative liquidity shock and rises by more after a positive one: sale selection amplifies measured price responses to liquidity. (iii) For exponential $G$, $\epsilon(c)=(r+\delta)/[(r+\delta)+\beta\eta(c)]\in(0,1)$, strictly decreasing in $c$ (amplification). For Pareto $G$, $h_GM=a/(a-1)$ is constant and $\epsilon(c)$ is again decreasing in $c$. In general, a sufficient condition for amplification is that $v\mapsto h_G(v)M(v)$ be nonincreasing; when $h_GM$ is increasing in $v$ (light-tailed $G$ such as the truncated normal, where $h_GM\uparrow1$ in the upper tail) the two factors in $\epsilon$ move in opposite directions and $\epsilon(c)$ can be increasing in $c$, in which case $\mathrm{Cov}\omega(\bar p,\epsilon)<0$ and selection dampens the measured response. The sign of the selection term is therefore a property of the match-value distribution, not a general law; it is reported numerically in Experiment 10 for the truncated normal, exponential, Weibull and Pareto cases. (iv) With $\delta=0$ the composition returns to $f$ in the new steady state, so the impact composition effect is transitory: the observed index overshoots its long-run response. Proof. (i) $\omega(c)=\eta(c)n(c)/Q$ 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 $\int\bar p,d\omega=\int\bar p,\omega(\epsilon-E_\omega\epsilon),d\ln\Lambda_s=\mathrm{Cov}_\omega(\bar p,\epsilon)d\ln\Lambda_s$. Two functions that are both decreasing in $c$ are comonotone, hence positively correlated under any $\omega$ (Chebyshev's sum inequality), strictly if both are strictly monotone and $\omega$ non-degenerate. Formula for $\epsilon$: $\eta=\Lambda_s(1-G(v^))$, $\partial\ln\eta/\partial\ln\Lambda_s=1-h_G(v^)\Lambda_s\partial V/\partial\Lambda_s=1-h_G\Lambda_s\beta H/[(r+\delta)+\Lambda_s\beta(1-G)]$ and $\Lambda_sH=\eta M$. (iii) Exponential: $h_GM=1$. 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. (iv) is (i). $\square$
Proposition 7 (financing contingency: cash versus mortgage buyers)
Statement. In the extension of §6, for a seller of type $c$ and a given match $w$ acceptable to both buyer types, $$p_m(w,c)-p_c(w,c)=\underbrace{\frac{(1-\beta)[c+rV(c)]-\beta[\kappa_b+rU_m]}{\nu\phi}}{\Delta_b(c)\ \text{(delay-bargaining term)}} +\underbrace{\beta(U_c-U_m)}{\text{outside-option term}}.$$ (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, $\beta[\kappa_b+rU_m]$; it is nondecreasing in $c$ ($d\Delta_b/dc=(1-\beta)(1+rV')/(\nu\phi)\ge0$) and vanishes as $\nu\to\infty$. (ii) The acceptance thresholds differ by $v^_m(c)-v^c(c)=\underbrace{[c+\kappa_b+r(V+U_m)]/(\nu\phi)}{\text{pending wedge}}-(U_c-U_m)$: mortgage agreements require a better match than cash agreements iff the pending wedge exceeds the cash buyers' outside-option advantage $U_c-U_m$ (the wedge is increasing in $c$, so this is more likely for urgent sellers). The observed mean price gap conditional on 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 can therefore be negative for patient sellers when cash buyers have a sufficiently better outside option. (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 is $\lambda_s[\tilde\pi+(1-\tilde\pi)\phi]$. Proof. Pending values are linear in $p$ with slopes $\pm\nu\phi/(r+\nu)$, so the Nash product is maximised where $P_s-V=\beta\Sigma_m$. 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 $p$ gives $p_m=V+\beta(w-V-U_m)+{(1-\beta)[c+rV]-\beta[\kappa_b+rU_m]}/(\nu\phi)$; subtracting $p_c=V+\beta(w-V-U_c)$ yields the display. (i) $1+rV'=[\delta+\lambda_s\beta(\cdot)]/[(r+\delta)+\lambda_s\beta(\cdot)]\ge0$ where $(\cdot)$ is the positive acceptance mass. (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 from the formulas. $\square$
Proposition 8 (posted prices)
Statement. Let $g$ be log-concave. (i) The seller's problem $\max_y1-G(y)$ has a unique interior solution $y(c)$ characterised by $y-U-V(c)=1/h_G(y)$; (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 the threshold $y(c)$ are strictly decreasing in $c$; the hazard is increasing in $c$; (iii) at equal $(V,U)$, $y(c)>v^*_{Nash}(c)=V+U$: posted-price sellers hold out for better matches, and all price dispersion among identical houses is urgency-driven. 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 FOC $-g(y)(y-a)+(1-G(y))=0$ gives the characterisation. (ii) By the envelope theorem the maximised value is decreasing in $V$ with slope $-(1-G(y))$, 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 gives $V'$. $\psi(y)\equiv y-1/h_G(y)$ is strictly increasing under IFR, so $y(c)=\psi^{-1}(U+V(c))$ is strictly increasing in $V$, hence decreasing in $c$. (iii) $1/h_G(y)>0$. $\square$
Results stated as numerical (no proof attempted)
- Uniqueness of $\theta^*$ with heterogeneous sellers (checked on a grid; reported in
validation.md). - Efficiency: the welfare-maximising $\beta$ relative to the Hosios value $1-\alpha$, with and without heterogeneity.
- Monotonicity of $h_GM$ for the truncated normal (checked numerically to justify Proposition 6(iii) in the baseline calibration).
- Full transition paths and the size of the index overshoot (Experiment 9–10).