1\section{Heterogeneous agents and ownership}2\label{sec:hamodel}34Model A says nothing about who delegates and who owns the compute. Model B is a stationary heterogeneous-agent economy5in the tradition of \citet{aiyagari1994uninsured} in which the decision channel of Section~\ref{sec:theory} appears in6the cross-section as a \emph{decision-quality} return term, and compute ownership is either broad or concentrated.78\subsection{Households}910A continuum of households has CRRA utility $u(c)=c^{1-\gamma}/(1-\gamma)$, discount factor $\beta$, and earnings11$w\exp(\epsilon_{it})$ with $\epsilon'=\rho_e\epsilon+\sigma_e\nu$ discretised by the Rouwenhorst method on seven12states. Savings $a'\ge\underline a$ earn the fund return $R$ plus an idiosyncratic term that reflects the quality of the13household's own financial decisions:14\begin{equation}15x_{i,t+1}=(R+\nu^r_{i,t+1})\,a_{i,t+1}+w\exp(\epsilon_{i,t+1}),\qquad16\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}17\label{eq:hb_return}18\end{equation}19at a fee $\phi_0+\phi_1a'$ paid when delegating. The mean cost of mistakes $m_H$ is a resource cost (it is destroyed,20like trading costs); the dispersion is a zero-sum redistribution across households (it averages out). The household21chooses $(c,a',d)$ every period:22\begin{equation}23V(x,\epsilon)=\max_{d\in\{0,1\}}\ \max_{a'\ge\underline a}\ u\big(x-a'-d(\phi_0+\phi_1a')\big)24+\beta\,\E\big[V(x',\epsilon')\mid\epsilon,d\big].25\label{eq:hb_bellman}26\end{equation}27Because the gain from a better decision is proportional to the stake $a'$ while part of the fee is fixed, the rich28delegate first: this is Proposition~\ref{prop:threshold} in the cross-section, and it feeds back into accumulation.29The specification is deliberately reduced-form---it stands in for the portfolio-choice stake of30Section~\ref{sec:theory} in an economy without aggregate risk---and the size of $m_H$ and $\sigma_{rH}$ is informed by31the household-finance evidence on investment mistakes and return heterogeneity discussed in32Section~\ref{sec:calibration}.3334\subsection{Firms, compute and ownership}3536The firm uses the task technology (\ref{eq:production})--(\ref{eq:threshold}) with a compute stock supplied along37$K_Q=K_{Q,0}(p_Q/u_Q)^{\epsilon_Q}$ ($\epsilon_Q=0$: fixed stock; $\epsilon_Q\to\infty$: perfectly elastic at user38cost $u_Q$). Inference for delegated decisions uses $\bar qD$ units of compute, so the compute market (\ref{eq:compute_clearing})39links the adoption share $D=\int d\,\dd\lambda$ to production automation. Physical capital and the user-cost part of40compute are held through a mutual fund that pays $R=1+r$ (plus, under broad ownership, the scarcity rents41$(p_Q-u_Q)K_Q$ in proportion to wealth). Under \emph{concentrated} ownership the scarcity rents accrue to a measure-$n_K$42owner class outside the household block, whose members hold the capitalised rent $(p_Q-u_Q)K_Q/(n_Kr)$ and are added43to the wealth distribution as a point mass---the two-class economy of Proposition~\ref{prop:ownership}.4445\subsection{Equilibrium and computation}4647\begin{definition}[Stationary recursive competitive equilibrium]48A value function $V$, policies $(c,a',d)$, an invariant distribution $\lambda$ over $(x,\epsilon)$, prices49$(r,w,p_Q)$ and quantities $(K,K_Q,Q^P,D)$ such that (i) $V$ solves (\ref{eq:hb_bellman}); (ii) $\lambda$ is50invariant under the policies and the shocks; (iii) $K+p^K_QK_Q=\int a'\,\dd\lambda$ (asset market),51$L=\int\exp(\epsilon)\,\dd\lambda=1$, $Q^P+\bar qD=K_Q$ with $K_Q$ on its supply curve, and factor prices are the52marginal products of Section~\ref{sec:technology}.53\end{definition}5455The household problem is solved by value-function iteration with a discrete delegation choice and Howard56acceleration (numba kernels), the distribution by non-stochastic simulation on the cash-on-hand grid, and the interest57rate by a scan-and-Brent root search on excess assets (Appendix~\ref{app:algorithms}). The zero-AI limit is a standard58Aiyagari economy and is used as a control; the channel decomposition switches on, one at a time, production compute59(productivity and displacement), delegation (information), ownership and the borrowing limit.60