1\section{Benchmark equilibrium: the delegation game}2\label{sec:theory}34This section contains the analytical core. We first reduce the within-period decision problem to a linear-quadratic5game with dispersed information, then characterise delegation, aggregate ``model noise'', and the private and social6values of delegation. Proofs are in Appendix~\ref{app:proofs}; every closed form below is also implemented in7\texttt{age.equilibrium.lq\_game} and cross-checked against Monte Carlo simulations and against a direct computation of8expected utility in the test suite.910\subsection{From the economy to a quadratic game}1112Let $x_i$ denote household $i$'s log-deviation of its payoff-relevant action from the full-information steady-state13path and $X=\int x_i\,\dd i$ the aggregate action. Expanding the household's objective to second order around the14symmetric steady state gives15\begin{equation}16U(x_i,X,\theta)=\tfrac12U_{xx}x_i^2+U_{xX}x_iX+\tfrac12U_{XX}X^2+U_{x\theta}x_i\theta+U_{X\theta}X\theta+\ldots,17\label{eq:quadratic}18\end{equation}19where the omitted terms do not involve $x_i$. Normalising by $|U_{xx}|$, the best response is20$x_i=\E_i[\kappa_0\theta+\alpha X]$ with $\kappa_0=-U_{x\theta}/U_{xx}$ (the partial-equilibrium response to a known21$\theta$) and $\alpha=-U_{xX}/U_{xx}$ (the degree of strategic complementarity). In capital accumulation22$U_{xX}=\beta u'(c')F_{KK}<0$: actions are strategic substitutes, $\alpha<0$. The class is that of23\citet{angeletos2007efficient}; what is new is the information structure, which is chosen in a market.2425Two mappings from the economy of Section~\ref{sec:environment} to (\ref{eq:quadratic}) are used. In the26\emph{savings mapping}, $x_i$ is the log deviation of savings $a_{i,t+1}$ and $\theta$ the TFP news; $\kappa_0$ and27$\alpha$ are computed from the log-linearised household problem (Appendix~\ref{app:algorithms}). In the28\emph{portfolio mapping}, households allocate wealth between risky capital claims and a safe store of value with29mean-variance preferences; $\theta$ is the predictable component of next period's return,30\begin{equation}31\kappa_0=\frac{1}{\gamma\sigma_R^2},\qquad \alpha=-(1-\alpha_K)\bar R^K\kappa_0,\qquad32|U_{xx}|=\gamma\sigma_R^2\bar K\ \text{(goods)},33\label{eq:portfolio_map}34\end{equation}35where $\sigma_R$ is the return standard deviation and $\alpha_K$ the capital share. Section~\ref{sec:results_a}36reports that the savings mapping yields a negligible stake ($\kappa_0\approx-0.0015$): wealth and substitution effects37of productivity news on saving nearly cancel. The portfolio mapping is therefore our baseline. The lesson is itself a38result: \emph{which} decision is delegated determines whether delegation has aggregate consequences.3940\begin{lemma}[Automation threshold]41\label{lem:iota}42For $Q^P>0$ equation (\ref{eq:threshold}) has a unique solution $\iota^*(Q^P/L)\in(0,1)$, increasing in $Q^P/L$;43the automated share is $\iota=\min\{\bar\iota,\iota^*\}$, and factor payments exhaust output.44\end{lemma}4546\begin{lemma}[Constant-returns restriction]47\label{lem:crs}48With constant returns, price-taking and symmetric ownership, the marginal effect of the aggregate action on an49individual's payoff vanishes at symmetric allocations for every state, $U_X\equiv0$; hence50$U_{X\theta}=0$ and $U_{xX}+U_{XX}=0$. The social curvature of welfare in the aggregate action is $(1-\alpha)|U_{xx}|$.51\end{lemma}5253The lemma is Euler's theorem applied to factor payments: a change in aggregate capital redistributes income between54capital, labour and compute, but the first-order sum of these transfers is zero when each household receives all three55incomes in the same proportions. We write $U_{XX}=-\upsilon\,U_{xX}$ in general and call $\upsilon=1$ the56constant-returns benchmark.5758\subsection{Linear equilibrium and aggregate model noise}5960A share $D$ of households delegates, split evenly across the $M$ models. We look for equilibria in which61non-delegators play $x_i=a_Hs_i$ and the users of model $m$ play $x_i=a_As_m$.6263\begin{proposition}[Linear equilibrium]64\label{prop:linear}65For every $\alpha<1$, $D\in[0,1]$ and $M\ge1$ there is a unique linear equilibrium,66\begin{equation}67a_H=\beta_Hc,\qquad a_A=\beta_A\xi c,\qquad68\xi=\frac{1}{1-\alpha\frac{D}{M}(1-\beta_A)},\qquad69c=\frac{\kappa_0}{1-\alpha\big[(1-D)\beta_H+D\beta_A\xi\big]},70\label{eq:strategies}71\end{equation}72and the aggregate action is73\begin{equation}74X=\bar A\theta+\frac{Da_A}{M}\sum_{m=1}^M\eta_m,\qquad \bar A=(1-D)a_H+Da_A,\qquad75\Var\Big(\frac{Da_A}{M}\sum_m\eta_m\Big)=\frac{D^2a_A^2}{M\tau_A}.76\label{eq:aggregate}77\end{equation}78When $\alpha<0$, $\xi<1$: a delegator, knowing that a mass $D/M$ of others act on the same error and that actions are79substitutes, dampens its response. The law of large numbers removes the non-delegators' errors from $X$ and leaves80the delegators' errors in it with weight $D$ and variance $\propto D^2/M$.81\end{proposition}8283Two aggregate consequences follow. First, $X$ tracks the fundamental more closely as $D$ rises ($\bar A$ moves toward84the full-information response $\kappa_0/(1-\alpha)$ because the model is better informed than the median human).85Second, $X$ acquires a component that has nothing to do with fundamentals. Wages, the rental rate and asset prices86inherit this component, which we call \emph{model noise}. Figure~\ref{fig:A_pv_noise}(b) plots its variance.8788\subsection{Private value of delegation and the adoption threshold}8990A measure-zero household compares the precision of its target $\hat x_i=\kappa_0\theta+\alpha X$ under its own signal91and under the model's signal.9293\begin{proposition}[Private value and adoption threshold]94\label{prop:threshold}95The private value of delegation (gross of fees, in units of $|U_{xx}|$) is96\begin{equation}97\PV(D)=\tfrac12\big[\Var(\hat x_i\mid s_i)-\Var(\hat x_i\mid s_m)\big],98\label{eq:pv}99\end{equation}100which at $\alpha=0$ reduces to $\PV=\frac{\kappa_0^2}{2}\frac{\tau_A-\tau_H}{(\tau_A+\tau_\theta)(\tau_H+\tau_\theta)}$;101it is increasing in $\tau_A$, decreasing in $\tau_H$, and strictly positive if and only if $\tau_A>\tau_H$.102Household $i$ delegates if and only if103\begin{equation}104\PV(D)\,|U_{xx}|\ \ge\ u'(\bar c)\phi+\kappa_i+\tau_s\bar wu'(\bar c),105\label{eq:threshold_rule}106\end{equation}107so the delegation share solves the fixed point $D=F_\kappa\big(\PV(D)|U_{xx}|-u'(\bar c)\phi-\tau_s\bar wu'(\bar c)\big)$.108A solution exists; it is increasing in $\tau_A$ and decreasing in the fee $\phi$ and in supervision time $\tau_s$.109\end{proposition}110111The value of the shared signal is not constant in $D$: it includes a ``knowing what others know'' component in the112sense of \citet{hellwig2009knowing}, because the model's error is also informative about the aggregate action. With113substitutes this component makes $\PV$ \emph{decrease} in $D$ (Figure~\ref{fig:A_pv_noise}(a)): the more people act on114the same model, the less a marginal user gains, and a model only slightly better than the median human can be worth115less than nothing once widely adopted. Adoption is self-limiting without any congestion in compute.116117\subsection{The efficiency benchmark}118119Welfare per capita, relative to full information and in units of $|U_{xx}|$, is120$W(D)=\E\,U(x_i,X,\theta)-D\tilde\phi$, which under Lemma~\ref{lem:crs} decomposes as121\begin{equation}122W(D)=-\tfrac12\Big[(1-\alpha)\,\E(X-X^*)^2+\E(x_i-X)^2\Big]-D\tilde\phi,\qquad X^*=\frac{\kappa_0}{1-\alpha}\theta,123\label{eq:welfare}124\end{equation}125the sum of an aggregate-volatility loss and a dispersion loss \citep{angeletos2007efficient}.126127\begin{proposition}[Efficiency benchmark]128\label{prop:benchmark}129Under Lemma~\ref{lem:crs} ($\upsilon=1$, $U_{X\theta}=0$), for every $D\in[0,1]$, $M\ge1$, $\alpha<1$ and130precisions $(\tau_\theta,\tau_H,\tau_A)$,131\begin{equation}132\PV(D)=\frac{\partial}{\partial D}\big[W(D)+D\tilde\phi\big].133\end{equation}134The competitive delegation share is constrained efficient: the planner who can choose $D$ (but cannot pool signals)135chooses the same $D$ as the market, for any number of models $M$.136\end{proposition}137138This is the first organising result of the paper, and it is a negative one for the most common policy narrative. Model139noise is real---wages, returns and capital fluctuate with the foundation model's mistakes---but in a price-taking140economy with constant returns and complete markets it is not an inefficiency. Delegators already dampen their response141to the common signal exactly as a planner would (strategic substitutability), and the pecuniary effects of their common142error on everyone else are zero-sum. The result is the counterpart, for the \emph{acquisition} of a correlated signal, of143the efficiency of the use of information in neoclassical economies \citep{angeletos2007efficient,colombo2014information};144it was derived symbolically before any numerical work, and the identity ``wedge $=0$'' is a property-based test of the145replication package.146147\subsection{When delegation becomes a policy problem}148149\begin{proposition}[Sources of a delegation externality]150\label{prop:wedges}151Let $\Delta(D)\equiv\PV(D)-\partial[W+D\tilde\phi]/\partial D$ denote the wedge between the private and social marginal152value of delegation (positive: over-delegation).153\begin{enumerate}[label=(\alph*)]154\item \emph{Non-pecuniary aggregate effects.} With $U_{XX}=-\upsilon U_{xX}$ and $U_{X\theta}=0$,155$\Delta(D)=(\upsilon-1)\,\frac{\alpha}{2}\,\frac{\partial\E[X^2]}{\partial D}$. With strategic substitutes156($\alpha<0$) and $\partial\E[X^2]/\partial D>0$, there is over-delegation if $\upsilon<1$ and under-delegation if157$\upsilon>1$.158\item \emph{Uninsurable price risk.} If a mass $m_w$ of hand-to-mouth workers consume their wage and the log wage loads159on the aggregate action with elasticity $\alpha_K$, then160$\Delta_b(D)=m_w\,\tfrac12\gamma\bar w\alpha_K^2\,\partial\Var(X)/\partial D>0$ in goods per unit of $D$, and the161Pigouvian delegation tax equals $\Delta_b$. Since $\partial\Var(X)/\partial D$ contains the model-noise term162$2Da_A^2/(M\tau_A)$, the tax falls with model diversity at rate $1/M$.163\item \emph{Collateral constraints.} When prices enter borrowing constraints, the price impact of correlated orders164creates a pecuniary externality of the \citet{davila2018pecuniary} type (Proposition~\ref{prop:collateral}).165\end{enumerate}166In the benchmark of Proposition~\ref{prop:benchmark} model diversity $M$ is welfare-irrelevant; under (b) and (c) it is167a substitute for the tax.168\end{proposition}169170\subsection{Compute market and ownership}171172\begin{proposition}[Compute market in the short run]173\label{prop:compute}174With $K_Q$ fixed, equations (\ref{eq:threshold}), (\ref{eq:compute_clearing}) and the threshold rule175(\ref{eq:threshold_rule}) have a unique equilibrium $(p_Q,Q^P,D)$. An improvement in model quality (higher $\tau_A$)176raises $D$ and $p_Q$ and lowers $Q^P$ and $\iota$; the labour share $(1-\alpha_K)(1-\iota)$ rises. With compute supplied177elastically at user cost, $Q^P$ is unaffected by delegation and $\iota$ and $D$ move together.178\end{proposition}179180\begin{proposition}[Ownership of algorithmic capital]181\label{prop:ownership}182Compute income $(1-\alpha_K)\iota Y+p_Q\bar qD$ accrues to households in proportion to $\omega_i$. In a two-class183economy in which a mass $n$ of owners holds all compute and a pro-rata share of physical capital, the owners' income184share is $n\alpha_K+(1-\alpha_K)\iota$ plus scarcity rents; it increases with automation $\iota$ and with the rents,185while broad ownership ($\omega_i=1$) leaves the income distribution unchanged by $\iota$ at given factor prices.186\end{proposition}187188\subsection{Collateral amplification}189190\begin{proposition}[Margin spiral on model noise]191\label{prop:collateral}192Consider a mass $D$ of leveraged delegators holding $h$ units each at the margin limit $\ell=1-m$, a liquidity provider193with relative price impact $\lambda_r$ per unit sold, and a common negative model error that moves the price by194$\epsilon_0=\lambda_rDc\,\bar\eta<0$ on impact. Forced sales to restore the margin constraint multiply the price change,195\begin{equation}196\epsilon=\mu\,\epsilon_0,\qquad \mu=\frac{1}{1-\lambda_rhD\,\frac{1-m}{m}},\qquad197\Var(\epsilon)=\mu^2\lambda_r^2D^2c^2\frac{\sigma_A^2}{M},198\label{eq:multiplier}199\end{equation}200provided $\lambda_rhD(1-m)/m<1$; the multiplier increases in leverage $(1-m)/m$, in the mass of leveraged delegators201$D$ and in price impact, and the spiral is unbounded as $\lambda_rhD(1-m)/m\to1$. Model diversity lowers the variance202at rate $1/M$ but leaves the multiplier unchanged.203\end{proposition}204205The proposition is static and deliberately minimal; Model C (Section~\ref{sec:markets}) checks it with explicit206balance sheets, defaults and three market designs.207