\section{Benchmark equilibrium: the delegation game}
\label{sec:theory}

This section contains the analytical core. We first reduce the within-period decision problem to a linear-quadratic
game with dispersed information, then characterise delegation, aggregate ``model noise'', and the private and social
values of delegation. Proofs are in Appendix~\ref{app:proofs}; every closed form below is also implemented in
\texttt{age.equilibrium.lq\_game} and cross-checked against Monte Carlo simulations and against a direct computation of
expected utility in the test suite.

\subsection{From the economy to a quadratic game}

Let $x_i$ denote household $i$'s log-deviation of its payoff-relevant action from the full-information steady-state
path and $X=\int x_i\,\dd i$ the aggregate action. Expanding the household's objective to second order around the
symmetric steady state gives
\begin{equation}
U(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,
\label{eq:quadratic}
\end{equation}
where the omitted terms do not involve $x_i$. Normalising by $|U_{xx}|$, the best response is
$x_i=\E_i[\kappa_0\theta+\alpha X]$ with $\kappa_0=-U_{x\theta}/U_{xx}$ (the partial-equilibrium response to a known
$\theta$) and $\alpha=-U_{xX}/U_{xx}$ (the degree of strategic complementarity). In capital accumulation
$U_{xX}=\beta u'(c')F_{KK}<0$: actions are strategic substitutes, $\alpha<0$. The class is that of
\citet{angeletos2007efficient}; what is new is the information structure, which is chosen in a market.

Two mappings from the economy of Section~\ref{sec:environment} to (\ref{eq:quadratic}) are used. In the
\emph{savings mapping}, $x_i$ is the log deviation of savings $a_{i,t+1}$ and $\theta$ the TFP news; $\kappa_0$ and
$\alpha$ are computed from the log-linearised household problem (Appendix~\ref{app:algorithms}). In the
\emph{portfolio mapping}, households allocate wealth between risky capital claims and a safe store of value with
mean-variance preferences; $\theta$ is the predictable component of next period's return,
\begin{equation}
\kappa_0=\frac{1}{\gamma\sigma_R^2},\qquad \alpha=-(1-\alpha_K)\bar R^K\kappa_0,\qquad
|U_{xx}|=\gamma\sigma_R^2\bar K\ \text{(goods)},
\label{eq:portfolio_map}
\end{equation}
where $\sigma_R$ is the return standard deviation and $\alpha_K$ the capital share. Section~\ref{sec:results_a}
reports that the savings mapping yields a negligible stake ($\kappa_0\approx-0.0015$): wealth and substitution effects
of productivity news on saving nearly cancel. The portfolio mapping is therefore our baseline. The lesson is itself a
result: \emph{which} decision is delegated determines whether delegation has aggregate consequences.

\begin{lemma}[Automation threshold]
\label{lem:iota}
For $Q^P>0$ equation (\ref{eq:threshold}) has a unique solution $\iota^*(Q^P/L)\in(0,1)$, increasing in $Q^P/L$;
the automated share is $\iota=\min\{\bar\iota,\iota^*\}$, and factor payments exhaust output.
\end{lemma}

\begin{lemma}[Constant-returns restriction]
\label{lem:crs}
With constant returns, price-taking and symmetric ownership, the marginal effect of the aggregate action on an
individual's payoff vanishes at symmetric allocations for every state, $U_X\equiv0$; hence
$U_{X\theta}=0$ and $U_{xX}+U_{XX}=0$. The social curvature of welfare in the aggregate action is $(1-\alpha)|U_{xx}|$.
\end{lemma}

The lemma is Euler's theorem applied to factor payments: a change in aggregate capital redistributes income between
capital, labour and compute, but the first-order sum of these transfers is zero when each household receives all three
incomes in the same proportions. We write $U_{XX}=-\upsilon\,U_{xX}$ in general and call $\upsilon=1$ the
constant-returns benchmark.

\subsection{Linear equilibrium and aggregate model noise}

A share $D$ of households delegates, split evenly across the $M$ models. We look for equilibria in which
non-delegators play $x_i=a_Hs_i$ and the users of model $m$ play $x_i=a_As_m$.

\begin{proposition}[Linear equilibrium]
\label{prop:linear}
For every $\alpha<1$, $D\in[0,1]$ and $M\ge1$ there is a unique linear equilibrium,
\begin{equation}
a_H=\beta_Hc,\qquad a_A=\beta_A\xi c,\qquad
\xi=\frac{1}{1-\alpha\frac{D}{M}(1-\beta_A)},\qquad
c=\frac{\kappa_0}{1-\alpha\big[(1-D)\beta_H+D\beta_A\xi\big]},
\label{eq:strategies}
\end{equation}
and the aggregate action is
\begin{equation}
X=\bar A\theta+\frac{Da_A}{M}\sum_{m=1}^M\eta_m,\qquad \bar A=(1-D)a_H+Da_A,\qquad
\Var\Big(\frac{Da_A}{M}\sum_m\eta_m\Big)=\frac{D^2a_A^2}{M\tau_A}.
\label{eq:aggregate}
\end{equation}
When $\alpha<0$, $\xi<1$: a delegator, knowing that a mass $D/M$ of others act on the same error and that actions are
substitutes, dampens its response. The law of large numbers removes the non-delegators' errors from $X$ and leaves
the delegators' errors in it with weight $D$ and variance $\propto D^2/M$.
\end{proposition}

Two aggregate consequences follow. First, $X$ tracks the fundamental more closely as $D$ rises ($\bar A$ moves toward
the full-information response $\kappa_0/(1-\alpha)$ because the model is better informed than the median human).
Second, $X$ acquires a component that has nothing to do with fundamentals. Wages, the rental rate and asset prices
inherit this component, which we call \emph{model noise}. Figure~\ref{fig:A_pv_noise}(b) plots its variance.

\subsection{Private value of delegation and the adoption threshold}

A measure-zero household compares the precision of its target $\hat x_i=\kappa_0\theta+\alpha X$ under its own signal
and under the model's signal.

\begin{proposition}[Private value and adoption threshold]
\label{prop:threshold}
The private value of delegation (gross of fees, in units of $|U_{xx}|$) is
\begin{equation}
\PV(D)=\tfrac12\big[\Var(\hat x_i\mid s_i)-\Var(\hat x_i\mid s_m)\big],
\label{eq:pv}
\end{equation}
which at $\alpha=0$ reduces to $\PV=\frac{\kappa_0^2}{2}\frac{\tau_A-\tau_H}{(\tau_A+\tau_\theta)(\tau_H+\tau_\theta)}$;
it is increasing in $\tau_A$, decreasing in $\tau_H$, and strictly positive if and only if $\tau_A>\tau_H$.
Household $i$ delegates if and only if
\begin{equation}
\PV(D)\,|U_{xx}|\ \ge\ u'(\bar c)\phi+\kappa_i+\tau_s\bar wu'(\bar c),
\label{eq:threshold_rule}
\end{equation}
so the delegation share solves the fixed point $D=F_\kappa\big(\PV(D)|U_{xx}|-u'(\bar c)\phi-\tau_s\bar wu'(\bar c)\big)$.
A solution exists; it is increasing in $\tau_A$ and decreasing in the fee $\phi$ and in supervision time $\tau_s$.
\end{proposition}

The value of the shared signal is not constant in $D$: it includes a ``knowing what others know'' component in the
sense of \citet{hellwig2009knowing}, because the model's error is also informative about the aggregate action. With
substitutes this component makes $\PV$ \emph{decrease} in $D$ (Figure~\ref{fig:A_pv_noise}(a)): the more people act on
the same model, the less a marginal user gains, and a model only slightly better than the median human can be worth
less than nothing once widely adopted. Adoption is self-limiting without any congestion in compute.

\subsection{The efficiency benchmark}

Welfare per capita, relative to full information and in units of $|U_{xx}|$, is
$W(D)=\E\,U(x_i,X,\theta)-D\tilde\phi$, which under Lemma~\ref{lem:crs} decomposes as
\begin{equation}
W(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,
\label{eq:welfare}
\end{equation}
the sum of an aggregate-volatility loss and a dispersion loss \citep{angeletos2007efficient}.

\begin{proposition}[Efficiency benchmark]
\label{prop:benchmark}
Under Lemma~\ref{lem:crs} ($\upsilon=1$, $U_{X\theta}=0$), for every $D\in[0,1]$, $M\ge1$, $\alpha<1$ and
precisions $(\tau_\theta,\tau_H,\tau_A)$,
\begin{equation}
\PV(D)=\frac{\partial}{\partial D}\big[W(D)+D\tilde\phi\big].
\end{equation}
The competitive delegation share is constrained efficient: the planner who can choose $D$ (but cannot pool signals)
chooses the same $D$ as the market, for any number of models $M$.
\end{proposition}

This is the first organising result of the paper, and it is a negative one for the most common policy narrative. Model
noise is real---wages, returns and capital fluctuate with the foundation model's mistakes---but in a price-taking
economy with constant returns and complete markets it is not an inefficiency. Delegators already dampen their response
to the common signal exactly as a planner would (strategic substitutability), and the pecuniary effects of their common
error on everyone else are zero-sum. The result is the counterpart, for the \emph{acquisition} of a correlated signal, of
the efficiency of the use of information in neoclassical economies \citep{angeletos2007efficient,colombo2014information};
it was derived symbolically before any numerical work, and the identity ``wedge $=0$'' is a property-based test of the
replication package.

\subsection{When delegation becomes a policy problem}

\begin{proposition}[Sources of a delegation externality]
\label{prop:wedges}
Let $\Delta(D)\equiv\PV(D)-\partial[W+D\tilde\phi]/\partial D$ denote the wedge between the private and social marginal
value of delegation (positive: over-delegation).
\begin{enumerate}[label=(\alph*)]
\item \emph{Non-pecuniary aggregate effects.} With $U_{XX}=-\upsilon U_{xX}$ and $U_{X\theta}=0$,
$\Delta(D)=(\upsilon-1)\,\frac{\alpha}{2}\,\frac{\partial\E[X^2]}{\partial D}$. With strategic substitutes
($\alpha<0$) and $\partial\E[X^2]/\partial D>0$, there is over-delegation if $\upsilon<1$ and under-delegation if
$\upsilon>1$.
\item \emph{Uninsurable price risk.} If a mass $m_w$ of hand-to-mouth workers consume their wage and the log wage loads
on the aggregate action with elasticity $\alpha_K$, then
$\Delta_b(D)=m_w\,\tfrac12\gamma\bar w\alpha_K^2\,\partial\Var(X)/\partial D>0$ in goods per unit of $D$, and the
Pigouvian delegation tax equals $\Delta_b$. Since $\partial\Var(X)/\partial D$ contains the model-noise term
$2Da_A^2/(M\tau_A)$, the tax falls with model diversity at rate $1/M$.
\item \emph{Collateral constraints.} When prices enter borrowing constraints, the price impact of correlated orders
creates a pecuniary externality of the \citet{davila2018pecuniary} type (Proposition~\ref{prop:collateral}).
\end{enumerate}
In the benchmark of Proposition~\ref{prop:benchmark} model diversity $M$ is welfare-irrelevant; under (b) and (c) it is
a substitute for the tax.
\end{proposition}

\subsection{Compute market and ownership}

\begin{proposition}[Compute market in the short run]
\label{prop:compute}
With $K_Q$ fixed, equations (\ref{eq:threshold}), (\ref{eq:compute_clearing}) and the threshold rule
(\ref{eq:threshold_rule}) have a unique equilibrium $(p_Q,Q^P,D)$. An improvement in model quality (higher $\tau_A$)
raises $D$ and $p_Q$ and lowers $Q^P$ and $\iota$; the labour share $(1-\alpha_K)(1-\iota)$ rises. With compute supplied
elastically at user cost, $Q^P$ is unaffected by delegation and $\iota$ and $D$ move together.
\end{proposition}

\begin{proposition}[Ownership of algorithmic capital]
\label{prop:ownership}
Compute income $(1-\alpha_K)\iota Y+p_Q\bar qD$ accrues to households in proportion to $\omega_i$. In a two-class
economy in which a mass $n$ of owners holds all compute and a pro-rata share of physical capital, the owners' income
share is $n\alpha_K+(1-\alpha_K)\iota$ plus scarcity rents; it increases with automation $\iota$ and with the rents,
while broad ownership ($\omega_i=1$) leaves the income distribution unchanged by $\iota$ at given factor prices.
\end{proposition}

\subsection{Collateral amplification}

\begin{proposition}[Margin spiral on model noise]
\label{prop:collateral}
Consider a mass $D$ of leveraged delegators holding $h$ units each at the margin limit $\ell=1-m$, a liquidity provider
with relative price impact $\lambda_r$ per unit sold, and a common negative model error that moves the price by
$\epsilon_0=\lambda_rDc\,\bar\eta<0$ on impact. Forced sales to restore the margin constraint multiply the price change,
\begin{equation}
\epsilon=\mu\,\epsilon_0,\qquad \mu=\frac{1}{1-\lambda_rhD\,\frac{1-m}{m}},\qquad
\Var(\epsilon)=\mu^2\lambda_r^2D^2c^2\frac{\sigma_A^2}{M},
\label{eq:multiplier}
\end{equation}
provided $\lambda_rhD(1-m)/m<1$; the multiplier increases in leverage $(1-m)/m$, in the mass of leveraged delegators
$D$ and in price impact, and the spiral is unbounded as $\lambda_rhD(1-m)/m\to1$. Model diversity lowers the variance
at rate $1/M$ but leaves the multiplier unchanged.
\end{proposition}

The proposition is static and deliberately minimal; Model C (Section~\ref{sec:markets}) checks it with explicit
balance sheets, defaults and three market designs.