\section{Heterogeneous agents and ownership}
\label{sec:hamodel}

Model A says nothing about who delegates and who owns the compute. Model B is a stationary heterogeneous-agent economy
in the tradition of \citet{aiyagari1994uninsured} in which the decision channel of Section~\ref{sec:theory} appears in
the cross-section as a \emph{decision-quality} return term, and compute ownership is either broad or concentrated.

\subsection{Households}

A continuum of households has CRRA utility $u(c)=c^{1-\gamma}/(1-\gamma)$, discount factor $\beta$, and earnings
$w\exp(\epsilon_{it})$ with $\epsilon'=\rho_e\epsilon+\sigma_e\nu$ discretised by the Rouwenhorst method on seven
states. Savings $a'\ge\underline a$ earn the fund return $R$ plus an idiosyncratic term that reflects the quality of the
household's own financial decisions:
\begin{equation}
x_{i,t+1}=(R+\nu^r_{i,t+1})\,a_{i,t+1}+w\exp(\epsilon_{i,t+1}),\qquad
\nu^r\sim\begin{cases}N(-m_H,\sigma_{rH}^2)& d=0\ \text{(self-managed)}\\ N(0,\sigma_{rA}^2)& d=1\ \text{(delegated)},\end{cases}
\label{eq:hb_return}
\end{equation}
at a fee $\phi_0+\phi_1a'$ paid when delegating. The mean cost of mistakes $m_H$ is a resource cost (it is destroyed,
like trading costs); the dispersion is a zero-sum redistribution across households (it averages out). The household
chooses $(c,a',d)$ every period:
\begin{equation}
V(x,\epsilon)=\max_{d\in\{0,1\}}\ \max_{a'\ge\underline a}\ u\big(x-a'-d(\phi_0+\phi_1a')\big)
+\beta\,\E\big[V(x',\epsilon')\mid\epsilon,d\big].
\label{eq:hb_bellman}
\end{equation}
Because the gain from a better decision is proportional to the stake $a'$ while part of the fee is fixed, the rich
delegate first: this is Proposition~\ref{prop:threshold} in the cross-section, and it feeds back into accumulation.
The specification is deliberately reduced-form---it stands in for the portfolio-choice stake of
Section~\ref{sec:theory} in an economy without aggregate risk---and the size of $m_H$ and $\sigma_{rH}$ is informed by
the household-finance evidence on investment mistakes and return heterogeneity discussed in
Section~\ref{sec:calibration}.

\subsection{Firms, compute and ownership}

The firm uses the task technology (\ref{eq:production})--(\ref{eq:threshold}) with a compute stock supplied along
$K_Q=K_{Q,0}(p_Q/u_Q)^{\epsilon_Q}$ ($\epsilon_Q=0$: fixed stock; $\epsilon_Q\to\infty$: perfectly elastic at user
cost $u_Q$). Inference for delegated decisions uses $\bar qD$ units of compute, so the compute market (\ref{eq:compute_clearing})
links the adoption share $D=\int d\,\dd\lambda$ to production automation. Physical capital and the user-cost part of
compute are held through a mutual fund that pays $R=1+r$ (plus, under broad ownership, the scarcity rents
$(p_Q-u_Q)K_Q$ in proportion to wealth). Under \emph{concentrated} ownership the scarcity rents accrue to a measure-$n_K$
owner class outside the household block, whose members hold the capitalised rent $(p_Q-u_Q)K_Q/(n_Kr)$ and are added
to the wealth distribution as a point mass---the two-class economy of Proposition~\ref{prop:ownership}.

\subsection{Equilibrium and computation}

\begin{definition}[Stationary recursive competitive equilibrium]
A value function $V$, policies $(c,a',d)$, an invariant distribution $\lambda$ over $(x,\epsilon)$, prices
$(r,w,p_Q)$ and quantities $(K,K_Q,Q^P,D)$ such that (i) $V$ solves (\ref{eq:hb_bellman}); (ii) $\lambda$ is
invariant under the policies and the shocks; (iii) $K+p^K_QK_Q=\int a'\,\dd\lambda$ (asset market),
$L=\int\exp(\epsilon)\,\dd\lambda=1$, $Q^P+\bar qD=K_Q$ with $K_Q$ on its supply curve, and factor prices are the
marginal products of Section~\ref{sec:technology}.
\end{definition}

The household problem is solved by value-function iteration with a discrete delegation choice and Howard
acceleration (numba kernels), the distribution by non-stochastic simulation on the cash-on-hand grid, and the interest
rate by a scan-and-Brent root search on excess assets (Appendix~\ref{app:algorithms}). The zero-AI limit is a standard
Aiyagari economy and is used as a control; the channel decomposition switches on, one at a time, production compute
(productivity and displacement), delegation (information), ownership and the borrowing limit.