1\section{Equilibrium computation}2\label{sec:computation}34Three solution methods correspond to the three models; each is tested against a known case before it is used.56\paragraph{Model A.} The deterministic steady state solves (\ref{eq:threshold})--(\ref{eq:clearing}) with7$R=1/\beta$ for given $(K_Q,D)$. The linear-quadratic game is solved in closed form (\ref{eq:strategies}); its moments,8welfare, private value and wedge are implemented in numpy and cross-checked by Monte Carlo (20{,}000 agents,94{,}000 draws) and by a direct computation of expected utility from the full quadratic payoff. The mapping from the10economy to $(\kappa_0,\alpha,|U_{xx}|)$ is computed either from the log-linearised household problem in sequence space11(savings mapping; a $2T+2$ linear system with $T=400$, terminal condition $\hat a_{T+1}=\hat a_T$ because wealth has a12unit root when $\beta R=1$) or from the mean-variance closed form (\ref{eq:portfolio_map}). The delegation share is a13fixed point solved by Brent's method, and the short-run compute market is a one-dimensional root in $Q^P$ (monotone14excess demand, Proposition~\ref{prop:compute}). Tests: zero-AI limit against Brock--Mirman; Euler's theorem on factor15payments; marginal products against finite differences; the identity ``wedge $=0$'' on random parameters.1617\paragraph{Model B.} Value-function iteration on a $160\times7$ cash-on-hand/earnings grid with a discrete delegation18choice, five Gauss--Hermite nodes for the return term, Howard acceleration (20 policy steps) and numba kernels; the19invariant distribution by non-stochastic simulation with lotteries onto the grid; the equilibrium $r$ by a coarse scan20followed by Brent's method on excess assets, with an inner fixed point on $D$ for the compute market. One equilibrium21takes 4--30 seconds on a laptop. Tests: Rouwenhorst moments; distribution mass; asset-market clearing within221\%; goods-market identity within 2\% (grid lotteries); delegation increasing in wealth.2324\paragraph{Model C.} A discrete-event simulation with common random numbers: all dividend innovations, private noises25and model errors are drawn up-front from the seed, so treatment cells differ only in the treatment. Twenty seeds per26cell and bootstrap intervals in the full runs; five seeds in the quick runs reported in the logs. Tests: the three27designs match feasibly on hand-built order sets; runs are bit-for-bit deterministic; ledger identities hold under28100 random operation sequences (property-based testing); the qualitative predictions of H2 and H4 hold in the Bayesian29control.3031\paragraph{Reproducibility.} All scripts are deterministic given the seeds in \texttt{configs/}; LLM calls are cached32content-addressed, so a run replays from the cache without inference; the git revision, package versions and33hardware are recorded in the experiment manifests (Appendix~\ref{app:replication}).34