1% ============================================================================2% Economic welfare and market design: a stylised buyer–seller–lender model (WP 26)3% Author: Simon-Pierre Boucher — simon-pierre.boucher@uqo.ca4% Code: src/agentic_valuation/theory/welfare.py, scripts/40_welfare_sim.py5% Proofs: appendix/proofs.tex6% ============================================================================7\makeatletter8\@ifundefined{proposition}{\theoremstyle{plain}\newtheorem{proposition}{Proposition}}{}9\@ifundefined{assumption}{\theoremstyle{definition}\newtheorem{assumption}{Assumption}}{}10\makeatother11\theoremstyle{plain}12\providecommand{\figdir}{../figures}13\providecommand{\E}{\mathbb{E}}14\providecommand{\Var}{\operatorname{Var}}15\providecommand{\Cov}{\operatorname{Cov}}1617\section{Economic Welfare and Market Design}18\label{sec:welfare}1920The accuracy metrics of the preceding sections measure the distance between an estimate and the selected market value. Whether a more accurate valuation is worth having depends on how valuations enter decisions, and the mapping is not monotone: a valuation that is unbiased on average can still ration credit, shift bargaining surplus, and, when it is shared by every lender, concentrate losses in the same states of the world. This section sets out a deliberately stylised model of one transaction with a buyer, a seller and a lender, derives three comparative statics that hold in general, and illustrates them numerically. Only the loan-to-value cap and the form of the haircut rule are transparent calibrations; every other parameter is illustrative and the numbers that follow should be read as orders of magnitude under stated assumptions, not as estimates.2122\subsection{The model}23\label{subsec:welfare-model}2425\begin{assumption}[Transaction stage]26\label{ass:welfare}27All quantities are expressed as fractions of the market-consistent value $V$ of the dwelling, normalised to one. The seller's reservation value is $S=1+u_S$ and the buyer's willingness to pay is $B=1+u_B$, with $(u_S,u_B)$ drawn from a continuous joint distribution; a match is mutually beneficial when $B\ge S$. The lender observes an appraisal $\hat A=1+b+sZ$, $Z\sim N(0,1)$ independent of $(u_S,u_B)$, with bias $b$ and standard deviation $s$, and lends $L=\ell(1-h)\hat A$, where $\ell$ is the loan-to-value cap and $h=h_0+h_1s$ is a haircut that may rise with valuation uncertainty. The buyer holds cash $W=(1-\ell)\kappa$ with $\kappa\ge1$. The transaction completes if and only if $B\ge S$ and $S\le W+L$, at the price $P=\min\{S+\theta(B-S),\,W+L\}$, where $\theta\in(0,1)$ is the seller's bargaining share.28\end{assumption}2930The structure follows the down-payment mechanism of \citet{stein1995prices}: the buyer's capacity to pay is the sum of own cash and what the lender will advance against the collateral, and the lender's advance is tied to a valuation rather than to the negotiated price. The financing cap $W+L$ truncates the price from above; when it binds, the match either fails or completes at a price below the bargaining outcome, which transfers surplus from the seller to the buyer. The collateral channel through which valuations propagate to credit is the one formalised by \citet{kiyotaki1997credit}.3132\begin{assumption}[Default stage]33\label{ass:default}34The loan defaults with exogenous probability $p_d$. At default the lender recovers $(1-c_\ell)V_1$, where $c_\ell$ is a liquidation cost and $V_1=1+g$ with $g\sim N(\mu_g,\sigma_g^{2})$ independent of $Z$; the loss is $\max\{L-(1-c_\ell)V_1,0\}$ and the expected loss at origination is $\mathrm{EL}(b,s)=p_d\,\E[\max\{L-(1-c_\ell)V_1,0\}]$. There are $N$ lenders whose appraisal errors are exchangeable with correlation $\rho$, $Z_n=\sqrt{\rho}\,C+\sqrt{1-\rho}\,E_n$, where $C,E_1,\dots,E_N$ are independent standard normal; $\rho$ measures common-model dependence across lenders' valuation models.35\end{assumption}3637The independence of $V_1$ from $Z$ is restrictive: it rules out the case in which the valuation error is itself informative about the subsequent price path, as when a model overvalues precisely the dwellings that later underperform. The results below are therefore lower bounds on the cost of correlated mispricing in that sense.3839\subsection{Three comparative statics}40\label{subsec:welfare-results}4142\begin{proposition}[Credit rationing]43\label{prop:rationing}44Conditional on the seller's draw, let $a(u_S)=(1+u_S)-(1-\ell)\kappa-\ell(1-h)(1+b)$ denote the financing gap at the error-free appraisal, and suppose financing is needed, $(1+u_S)>(1-\ell)\kappa$. The probability that a mutually beneficial match fails for lack of financing is45\begin{equation}46F(u_S;b,h,s)=\Phi\!\left(\frac{a(u_S)}{\ell(1-h)s}\right).47\label{eq:failprob}48\end{equation}49(i) $F$ is strictly decreasing in $b$ and strictly increasing in $h$. (ii) If $a(u_S)<0$, the match is financeable at the error-free appraisal and $F$ is strictly increasing in $s$; if $a(u_S)>0$, $F$ is strictly decreasing in $s$. (iii) The failure rate among beneficial matches is increasing in $s$ whenever $a(u_S)<0$ almost surely on $\{B\ge S\}$; in general it is the sum of a rationing effect on financeable matches and a lucky-financing effect on non-financeable ones. (iv) A haircut rule with $h_1>0$ reinforces the rationing effect.50\end{proposition}5152The proposition separates two things that discussions of valuation accuracy often conflate. For the matches that an accurate valuation would finance, which are the bulk of beneficial matches under any reasonable distribution of reservation values, valuation noise is a pure cost: each unit of $s$ raises the chance that the appraisal falls below what the buyer needs. For matches that an accurate valuation would not finance, noise occasionally lets the transaction through; whether that is a welfare gain depends on whether the lender's rule was right to exclude them, which the model does not adjudicate. The direction of bias matters in the opposite way to its effect on the lender: undervaluation rations credit, overvaluation expands it.5354\begin{proposition}[Lender's expected loss]55\label{prop:lenderloss}56Hold the haircut $h$ fixed. (i) $\mathrm{EL}$ is strictly increasing in the bias $b$. (ii) $\mathrm{EL}$ is strictly increasing in the standard deviation $s$: a mean-preserving spread of the appraisal raises the expected loss because the loss is a strictly convex function of the loan amount. (iii) Under the haircut rule $h=h_0+h_1s$, the loan's mean falls with $s$ and the sign of $\partial\,\mathrm{EL}/\partial s$ is not determined; for every $s$ there is a haircut that holds $\mathrm{EL}$ at its error-free level, and by Proposition~\ref{prop:rationing}(i) it raises the failure probability of every financeable match.57\end{proposition}5859Part (iii) is the sense in which valuation uncertainty has a price that someone must pay. A lender that targets a constant expected loss responds to a noisier valuation technology by widening the haircut; the cost of the noise is then borne by borrowers in the form of rationed credit rather than by the lender in the form of losses. The empirical interval widths of Section~\ref{sec:results} map directly into $s$ for a lender that uses the committee's interval as its measure of uncertainty, and the coverage results determine whether that mapping is honest.6061\begin{proposition}[Common-model dependence across lenders]62\label{prop:systemic}63Let $\ell_n=g(Z_n)$ be lender $n$'s expected loss conditional on its appraisal, $g(z)=p_d\,\E[\max\{\ell(1-h)(1+b+sz)-(1-c_\ell)V_1,0\}]$. (i) The expected system loss $\E[\sum_n\ell_n]=N\E[g(Z)]$ does not depend on $\rho$. (ii) The variance of the system loss, $N\Var(g(Z))+N(N-1)\Cov(g(Z_1),g(Z_2))$, is strictly increasing in $\rho$ on $[0,1]$, rising from $N\Var(g(Z))$ at $\rho=0$ to $N^{2}\Var(g(Z))$ at $\rho=1$. (iii) Replacing $N$ distinct models with loss variance $v_0$ and loss correlation $\tilde\rho_0$ by a single common model with loss variance $v_1<v_0$ raises the variance of the system loss if and only if $v_1/v_0>[1+(N-1)\tilde\rho_0]/N$.64\end{proposition}6566Part (i) is the reason that the welfare case for accurate valuation cannot be made at the level of expected losses alone: correlation does not change what lenders lose on average, it changes when they lose it. Part (ii) is the collateral-market counterpart of Proposition~\ref{prop:nested}; it formalises the monoculture risk discussed by \citet{kleinberg2021algorithmic} and \citet{bommasani2022picking} for a market in which lenders converge on the same valuation model. Part (iii) is the trade-off that a market with a single dominant provider faces: a common model that is more accurate for each lender can nevertheless raise the dispersion of aggregate losses, and the condition shows that the threshold is easy to cross when $N$ is large and the existing models are only moderately correlated. Within the paper's design, the provider-balanced committees are the within-committee analogue of this proposition; the market-level implication is that the diversity that helps a committee is also what a lending system loses when every participant adopts the best available committee.6768\subsection{Illustrative simulation}69\label{subsec:welfare-sim}7071Figure~\ref{fig:welfare-failure} and Figure~\ref{fig:welfare-systemic} illustrate the three propositions with $\ell=0.80$, the conventional cap for an uninsured mortgage in Canada, two haircut rules ($h_1=0$ and $h_1=0.5$), and illustrative values for everything else: $\kappa=1.25$, $\theta=0.5$, $u_S\sim N(-0.05,0.05^{2})$, $u_B\sim N(0.05,0.08^{2})$, $p_d=0.02$, $c_\ell=0.15$, $g\sim N(-0.10,0.10^{2})$ and $N=50$ lenders. These values are not estimated from Quebec data; they are chosen so that most beneficial matches are financeable at the error-free appraisal, which is the regime in which Proposition~\ref{prop:rationing}(iii) applies. All quantities are produced by \texttt{scripts/40\_welfare\_sim.py} and stored in \texttt{results/theory/welfare\_sim.csv}.7273\begin{figure}[t]74\centering75\includegraphics[width=\textwidth]{\figdir/welfare_failure_vs_s.pdf}76\caption{Share of mutually beneficial matches that fail for lack of financing as a function of the valuation error standard deviation $s$, for three values of the bias $b$, without an uncertainty haircut (left) and with $h=0.5s$ (right). Equation~\eqref{eq:failprob} integrated over the seller draws, $200{,}000$ draws. Illustrative parameters; see text. Source: \texttt{scripts/40\_welfare\_sim.py}.}77\label{fig:welfare-failure}78\end{figure}7980\begin{figure}[t]81\centering82\includegraphics[width=\textwidth]{\figdir/welfare_systemic_vs_rho.pdf}83\caption{System loss of $N=50$ lenders whose appraisal errors have common-model correlation $\rho$, for three values of the per-lender valuation error $s$. Left: mean (dashed, flat in $\rho$ by Proposition~\ref{prop:systemic}(i)) and 95 percent expected shortfall. Right: standard deviation. Losses are expected losses conditional on the appraisal, as fractions of $V$, summed over lenders; $20{,}000$ draws with common random numbers across $\rho$. Illustrative parameters; see text. Source: \texttt{scripts/40\_welfare\_sim.py}.}84\label{fig:welfare-systemic}85\end{figure}8687In the left panel of Figure~\ref{fig:welfare-failure}, with no bias and no uncertainty haircut, the failure rate rises from about $1$ percent at $s=0$ to $4$ percent at $s=0.05$, $13$ percent at $s=0.10$ and $26$ percent at $s=0.20$; with the haircut rule the same path runs through $8$, $23$ and $43$ percent. A negative bias of three percent shifts the curve up and a positive bias shifts it down, as Proposition~\ref{prop:rationing}(i) states. Among completed transactions the financing cap also moves the split of surplus: the seller's average surplus falls and the buyer's rises as $s$ grows, because a binding cap truncates the price from above. The lender's expected loss, with the haircut fixed, rises from $0.109$ to $0.182$ percent of $V$ between $s=0$ and $s=0.20$ (Proposition~\ref{prop:lenderloss}(ii)); with $h=0.5s$ it falls below its error-free value, which is the transfer of the cost of uncertainty from the lender to rationed borrowers described in part (iii). In Figure~\ref{fig:welfare-systemic} the mean system loss is flat in $\rho$ while its standard deviation at $s=0.10$ rises sevenfold between $\rho=0$ and $\rho=1$ and the expected shortfall rises from $0.082$ to $0.202$. The concentration case of Proposition~\ref{prop:systemic}(iii) is computed in the same file: moving from fifty lenders with $s=0.15$ and $\rho=0.2$ to a single shared model with $s=0.10$ and $\rho=1$ lowers the expected system loss by $15$ percent and raises its standard deviation by $53$ percent and its expected shortfall by $26$ percent.8889\subsection{What accuracy does and does not establish}90\label{subsec:welfare-caveat}9192The empirical sections ask whether a committee of agents reduces the error $s$ and the bias $b$ of a valuation relative to the alternatives. This section shows that the welfare consequences of such a reduction are mediated by three objects that the accuracy metrics do not contain: the distribution of reservation values and cash, which determines how many beneficial matches sit on the financing margin; the lender's haircut rule, which determines who bears the cost of uncertainty; and the correlation of valuation errors across lenders, which determines the dispersion of aggregate losses without affecting their mean. A reduction in $s$ that is achieved by every lender adopting the same committee lowers the failure rate and the per-lender expected loss while raising $\rho$, and Proposition~\ref{prop:systemic}(iii) shows that the net effect on the dispersion of system losses can have either sign. Predictive accuracy is therefore necessary for a welfare gain in this model but not sufficient for one, and the paper does not claim a welfare gain on the strength of the accuracy results. What it can claim, if the empirical results support it, is a measured reduction in $s$ and $b$ under a stated information set, together with the mapping above from which a reader with a view on the distribution of matches, the haircut rule and the market structure can draw the welfare implication.93