\section{Institutions and environment}
\label{sec:environment}

This section lays out the baseline economy (Model A). Two later sections add idiosyncratic risk and heterogeneous
ownership (Model B, Section~\ref{sec:hamodel}) and an explicit asset market with credit (Model C,
Section~\ref{sec:markets}). Throughout, we keep four roles of artificial intelligence apart
(Table~\ref{tab:roles}): a production technology, an information-processing service, a delegated decision-maker, and a
hypothetical independent economic principal. The first three exist today and have identified legal and economic
counterparties; the fourth is a counterfactual that we label as such wherever it appears.

\begin{table}[htbp]
\centering\small
\caption{Four roles of AI and where each enters the model}
\label{tab:roles}
\begin{tabularx}{\textwidth}{lXX}
\toprule
Role & Object in the model & Residual claimant and liability \\
\midrule
(i) Production technology & compute $Q^P$ performs tasks $x\le\iota$ in the task aggregator (\ref{eq:bundle}) & the firm; compute owners receive the rental $p_Q$ \\
(ii) Information service (advisory) & the human sees her own signal and the model's signal and decides herself & the household \\
(iii) Delegated decision-maker (autonomous) & the model's policy $\pi_m$ applies the household's decision rule to the model's forecast & the household (principal); the fee $\phi$ is paid to compute and energy suppliers \\
(iv) Independent AI principal & an agent with its own balance sheet and no human residual claimant (Model C only) & none; a labelled counterfactual \\
\bottomrule
\end{tabularx}
\end{table}

\subsection{Agents, goods and shocks}

Time is discrete, $t=0,1,2,\dots$, and there is one final good, the numéraire. The economy is populated by a
continuum of households $i\in[0,1]$, a representative competitive final-good firm, a competitive compute-producing
sector, an energy sector with constant marginal cost $p_E$ (goods per unit of energy), and $M$ foundation models
$m=1,\dots,M$, which are policies rather than agents. Aggregate log-productivity follows
$z_t=\rho_z z_{t-1}+\varepsilon_t$ with $\varepsilon_t\sim N(0,\sigma_\varepsilon^2)$ i.i.d.

\paragraph{An AI agent is a constrained policy.} A model-$m$ agent is a map
$\pi_m(x\mid o,\mathcal M_m,\text{mandate},\epsilon)$ from an observation $o$, the model's information
$\mathcal M_m$, a mandate (the principal's objective and constraints) and randomness $\epsilon$ into an action $x$. It
has no utility, no balance sheet and no liability. In Models A and B the mandate is ``maximise my expected utility given
your information'', so $\pi_m$ is the household's own optimal decision rule evaluated at the model's forecast. In Model C
the policy is either that rule (a Bayesian control), a rule-based policy, or a live language model whose proposed action
is validated against a schema and clipped by deterministic constraints before it touches any ledger.

\subsection{Information}
\label{sec:information}

The only friction of Model A is informational and one step ahead. After $z_t$ is known, decision-makers receive news
about $\theta_t\equiv\varepsilon_{t+1}$, the next innovation. A human receives a private signal
$s_{it}=\theta_t+e_{it}$, $e_{it}\sim N(0,\tau_{H,i}^{-1})$, independent across households and time. Model $m$ produces
a signal $s_{mt}=\theta_t+\eta_{mt}$, $\eta_{mt}\sim N(0,\tau_A^{-1})$, independent across models and time but
\emph{common to every principal that delegates to model $m$}. The prior precision is
$\tau_\theta=\sigma_\varepsilon^{-2}$ and the Bayesian weights are $\beta_H=\tau_H/(\tau_\theta+\tau_H)$ and
$\beta_A=\tau_A/(\tau_\theta+\tau_A)$. All uncertainty about $\theta_t$ is resolved at $t+1$, so the economy returns to a
full-information recursive structure every period. The assumption that model errors are common is the formal
counterpart of the observation that users of one foundation model share its ``world view'' at a point in time; the
empirical literature reviewed in Section~\ref{sec:literature} finds cross-user and even cross-vendor error correlations
far above zero, so $M$ should be read as an \emph{effective} number of independent models.

\subsection{Technology}
\label{sec:technology}

Tasks $x\in[0,1]$ are combined with unit elasticity into a task bundle $T_t$, which is combined with capital:
\begin{equation}
Y_t=e^{z_t}K_t^{\alpha_K}T_t^{1-\alpha_K},\qquad \ln T_t=\int_0^1\ln y_t(x)\,\dd x .
\label{eq:production}
\end{equation}
Task $x$ is produced either by labour, $y(x)=l(x)$, or---if $x\le\bar\iota$, the technological frontier---by compute,
$y(x)=\psi(x)q(x)$ with $\psi(x)=\psi_0e^{-\psi_1x}$ decreasing in the task index (easy tasks are automated first).
Cost minimisation automates $x\le\iota_t=\min\{\bar\iota,\iota_t^*\}$, where $\iota_t^*$ solves
$p_{Q,t}/\psi(\iota_t^*)=w_t$. With unit elasticity across tasks every automated task uses $Q^P_t/\iota_t$ units of
compute and every manual task $L_t/(1-\iota_t)$ units of labour, so
\begin{equation}
T_t=\Psi(\iota_t)\Big(\frac{Q^P_t}{\iota_t}\Big)^{\iota_t}\Big(\frac{L_t}{1-\iota_t}\Big)^{1-\iota_t},\qquad
\Psi(\iota)=\exp\Big(\int_0^\iota\ln\psi(x)\,\dd x\Big)=\psi_0^{\iota}e^{-\psi_1\iota^2/2}.
\label{eq:bundle}
\end{equation}
Competitive factor prices are $R^K_t=\alpha_KY_t/K_t$, $w_t=(1-\alpha_K)(1-\iota_t)Y_t/L_t$ and
$p_{Q,t}=(1-\alpha_K)\iota_tY_t/Q^P_t$. Combining the automation threshold with the two last expressions gives the
interior condition
\begin{equation}
\psi(\iota_t)=\frac{p_{Q,t}}{w_t}=\frac{\iota_t}{1-\iota_t}\,\frac{L_t}{Q^P_t},
\label{eq:threshold}
\end{equation}
whose left side decreases and right side increases in $\iota$ (Lemma~\ref{lem:iota}). Factor shares are $\alpha_K$ for
capital, $(1-\alpha_K)\iota_t$ for compute and $(1-\alpha_K)(1-\iota_t)$ for labour. This is the standard task-based
structure of \citet{acemoglu2018race}; it is not the contribution of the paper, but it is needed so that the
decision channel and the production channel compete for the same compute.

\subsection{Compute}
\label{sec:compute}

Compute is a stock $K_{Q,t}$, predetermined at $t$, rented at $p_{Q,t}$ per unit and period. It has two uses: production
($Q^P_t$) and inference for delegated decisions, $\bar q D_t$, where $\bar q$ is the compute consumed by one delegated
household-decision per period and $D_t\in[0,1]$ is the delegation share. The compute market clears,
\begin{equation}
Q^P_t+\bar q D_t=K_{Q,t},\qquad p_{Q,t}\ge0 .
\label{eq:compute_clearing}
\end{equation}
The stock accumulates according to $K_{Q,t+1}=(1-\delta_Q)K_{Q,t}+I_{Q,t}/p^K_{Q,t}$, where
$p^K_{Q,t}=p^K_{Q,0}(1-g_Q)^t$ is the (declining) price of compute capital in goods. Compute owners are households
with shares $\omega_i$, $\int\omega_i\,\dd i=1$; they receive $\Pi_{Q,t}=p_{Q,t}K_{Q,t}$ and finance $I_{Q,t}$. In the
short run $K_{Q,t}$ is fixed and $p_{Q,t}$ carries scarcity rents; in the long run, with elastic supply, $p_Q$ converges
to the user cost $u_{Q,t}\approx(r_t+\delta_Q-g_Q)p^K_{Q,t}$. Model B interpolates with a supply elasticity $\epsilon_Q$.

\subsection{Households}
\label{sec:households}

Preferences are $\E_0\sum_t\beta^t\big[u(c_{it})-v(l_{it})-\kappa_i d_{it}\big]$ with $u(c)=c^{1-\gamma}/(1-\gamma)$,
$v(l)=\chi l^{1+1/\nu}/(1+1/\nu)$, and $\kappa_i\ge0$ a delegation disutility (trust, agency loss) distributed with
c.d.f.\ $F_\kappa$. The delegation indicator $d_{it}\in\{0,1\}$ is chosen at the start of the period, before signals
arrive. Time is divided between work and supervision: $l_{it}+\tau_sd_{it}\le1$. The budget constraint is
\begin{equation}
c_{it}+a_{i,t+1}+\phi_td_{it}=w_tl_{it}+R_ta_{it}+\omega_i\big(\Pi_{Q,t}-I_{Q,t}\big),\qquad R_t=1-\delta+R^K_t,
\label{eq:budget}
\end{equation}
with the delegation fee $\phi_t=p_{Q,t}\bar q+p_E\bar e$ (compute rental paid to compute owners, energy paid to the
energy sector). A borrowing limit $a_{i,t+1}\ge\underline a$ is slack in Model A and binds for some households in
Model B. The transversality condition is $\lim_T\beta^T\E[u'(c_{iT})a_{i,T+1}]=0$.

\paragraph{Decision rule and delegation modes.} Let $a^*(\cdot)$ denote the household's optimal savings rule under its
information. A self-managing household applies it at $\E[\theta_t\mid s_{it}]$; a delegating household's agent applies
the same rule at $\E[\theta_t\mid s_{mt}]$ (the mandate is the principal's objective). Advisory AI (role ii) applies it
at $\E[\theta_t\mid s_{it},s_{mt}]$ at the cost of a larger supervision time $\tau_s^{adv}>\tau_s$. The difference
between modes is therefore the weight on the common error, not the decision rule itself.

\subsection{Market clearing, flow of funds and the equilibrium concept}
\label{sec:equilibrium_def}

Labour, capital, compute and goods markets clear:
\begin{equation}
L_t=\int l_{it}\,\dd i,\quad K_{t+1}=\int a_{i,t+1}\,\dd i,\quad (\ref{eq:compute_clearing}),\quad
C_t+I_t+I_{Q,t}+p_E\bar eD_t=Y_t,\quad I_t=K_{t+1}-(1-\delta)K_t .
\label{eq:clearing}
\end{equation}
Every flow has a payer and a payee: fees $p_Q\bar qD$ go from delegating households to compute owners; energy
$p_E\bar eD$ goes to the energy sector and is a resource use; factor payments flow from the firm to households; compute
investment flows from owners to the compute-producing sector. These identities are asserted in the code after every
operation (Appendix~\ref{app:replication}).

\begin{definition}[Recursive competitive equilibrium with dispersed information]
\label{def:rce}
Let $S_t=(K_t,K_{Q,t},z_t)$. An equilibrium consists of price functions $(w,R^K,p_Q)$, household policies
$(c,a',l,d)$ measurable with respect to each household's information, firm policies, an aggregate law of motion
$K_{t+1}=\mathcal K(S_t,\theta_t,\eta_{1t},\dots,\eta_{Mt})$ and a delegation share $D(S_t)$ such that
(a) policies are optimal given beliefs; (b) beliefs are Bayesian given signals and the law of motion; (c) the markets in
(\ref{eq:compute_clearing}) and (\ref{eq:clearing}) clear; (d) $D(S_t)=\int d_{it}\,\dd i$ is consistent with
individual delegation choices.
\end{definition}

\begin{definition}[LQ-recursive equilibrium]
\label{def:lq}
The second-order approximation of Definition~\ref{def:rce} around the deterministic steady state, in which the
within-period decision is the linear-quadratic game of Section~\ref{sec:theory} and the between-period dynamics are the
log-linearised economy with the aggregate action of the game entering the capital law of motion.
\end{definition}

Propositions are stated for Definition~\ref{def:lq}; the numerical sections check their accuracy. In the
\emph{zero-AI limit} ($K_Q=0$, hence $\iota=0$ and $D=0$) equations (\ref{eq:production})--(\ref{eq:clearing})
collapse to the stochastic growth model $Y=e^zK^{\alpha_K}L^{1-\alpha_K}$, which the code reproduces against the
Brock--Mirman closed form when $\delta=1$ and $\gamma=1$ (Appendix~\ref{app:algorithms}).