\section{A task economy with priced compute and scarce complements}
\label{sec:model}

The quantitative model keeps the work equation and the skill partition of Section~\ref{sec:benchmark} and
adds four capital stocks, energy, land and housing. Time is discrete and annual; $t=0$ is 2026. All quantities
are real in units of the 2026 final good. Notation is collected in Table~\ref{tab:notation} and in the model
dictionary of the replication package.

\subsection{Tasks and automation}

There are four bottleneck task classes $i\in B=\{1,2,3,4\}$ --- routine, lower cognitive, higher cognitive,
expert --- and two skills with $\Omega(L)=\{1,2\}$ and $\Omega(H)=\{3,4\}$. Workers supply $H_s$ efficiency
units inelastically; efficiency units, land and the energy resource grow at the trend rate $g_Z$ (Harrod-neutral
non-AI technical progress), so that the frozen-technology economy has an exact balanced growth path and all
statements about wages below are per efficiency unit. Task output is
\begin{equation}
X_{i,t} = L_{i,t}+\frac{Q_{i,t}}{\alpha_{i,t}},\qquad
X_{B,t}=\Big(\sum_{i\in B}\nu_iX_{i,t}^{\rho_B}\Big)^{1/\rho_B},\quad \sigma_B=\tfrac{1}{1-\rho_B}<1,
\label{eq:tasks}
\end{equation}
where $\alpha_{i,t}$ is the compute needed to replicate one efficiency unit of labor in task $i$. Automation
feasibility improves along a scenario path: in the baseline $\log\alpha_{i,t}$ declines linearly at rate
$g_\alpha$ from its 2026 value to a terminal value $\alpha_i^\infty$, ordered $(1,2,4,8)$ across tasks so that
the expert task is the last to become cheap to replicate; in the ``rapid AGI'' scenario all $\alpha_{i,t}$ jump
to their terminal values at a date. Whether a task is actually automated is an equilibrium outcome: firms use
compute in task $i$ if and only if $A_t\alpha_{i,t}\le w_{s(i),t}$, and labor of skill $s$ flows to the task in
$\Omega(s)$ with the highest marginal value. Automation can therefore be feasible but unprofitable, exactly as in
the source.

\subsection{Final goods, structures, land and energy}

Final output is a nested CES in the task composite, structures and machinery $K^N$, production land $T$ and
energy $E^Y$:
\begin{equation}
V_t = \big[\nu_N (K^N_t)^{\rho_N}+(1-\nu_N)X_{B,t}^{\rho_N}\big]^{1/\rho_N},\qquad
Y_t = Z\big[\nu_T T_t^{\rho_T}+\nu_E (E^Y_t)^{\rho_T}+\nu_V V_t^{\rho_T}\big]^{1/\rho_T},
\label{eq:nests}
\end{equation}
with $\sigma_N=1/(1-\rho_N)$ the elasticity between tasks and structures and $\sigma_T=1/(1-\rho_T)$ the
elasticity between the task--structure bundle and the two physical inputs that cannot be reproduced at will. The
baseline uses $\sigma_N=0.7$ and $\sigma_T=0.6$; Section~\ref{sec:robustness} varies both across the regimes of
Table~\ref{tab:regimes}. Competitive final-goods firms rent structures at $R^N_t$, buy compute services at $A_t$,
hire labor at $w_{s,t}$, rent land at $R^T_t$ and buy energy at $P^E_t$; prices equal marginal products.

\subsection{Compute as capital}

Installed compute capacity $K^Q_t$ delivers $Q_t=u_tK^Q_t$ units of services, $u_t\le1$ the utilization rate, and
each unit of service requires $\varepsilon_t$ units of energy. Capacity is bought one period ahead at price
$p^Q_t$ in units of the final good and depreciates at $\delta_Q$:
\begin{equation}
K^Q_{t+1}=(1-\delta_Q)K^Q_t+I^Q_t,\qquad p^Q_t = p^Q_0\max\{e^{-g_{pQ}t},\underline{p}/p^Q_0\},\qquad
\varepsilon_t=\varepsilon_0\max\{e^{-g_{\varepsilon}t},\underline{\varepsilon}/\varepsilon_0\}.
\end{equation}
The price decline is Moore's law for installed capacity, and the floor $\underline{p}$ is the date at which it
ends; the energy-efficiency decline is Koomey's law with its own floor. The owner of a unit of capacity earns the
net rental $R^Q_t = u_t(A_t-\varepsilon_tP^E_t)$; capacity is idle ($u_t<1$) only when its energy cost exceeds its
marginal value, which never happens on the baseline path and happens in the energy-constrained scenarios.

\subsection{Energy, housing and the other capital stocks}

Energy is produced from energy capital and a fixed resource (sites, fuel, grid rights),
$E_t = Z^E(K^E_t)^{1-\eta}(R^E_t)^{\eta}$, and allocated between compute and final production,
$E_t=\varepsilon_tQ_t+E^Y_t$. The resource earns the rent $\eta P^E_tE_t$. Housing services
$H^S_t=(K^H_t)^{1-\nu_H}(T^H_t)^{\nu_H}$ are produced from residential structures and residential land and rented to
households at $R^H_t$; households spend a Cobb--Douglas share $\theta_H$ of total expenditure on housing, so
$R^H_tH^S_t=\theta_H/(1-\theta_H)\,C_t$ with $C_t$ aggregate goods consumption. Structures, energy capital and
housing structures are bought at price one, depreciate at $\delta_N,\delta_E,\delta_H$ and, like compute, are
subject to quadratic adjustment costs $\tfrac{\phi_k}{2}(x^k_t-\delta_k)^2K^k_tp^k_t$, $x^k_t=I^k_t/K^k_t$, so that
Tobin's marginal $q$ is $q^k_t=p^k_t[1+\phi_k(x^k_t-\delta_k)]$ \citep{Hayashi1982}.

\subsection{National accounting}

Compute services and energy are intermediate inputs. Value added is
$\mathrm{GDP}_t = Y_t + R^H_tH^S_t$, and the income identity
\begin{equation}
\mathrm{GDP}_t=\sum_sw_{s,t}H_{s,t}+R^N_tK^N_t+R^Q_tK^Q_t+R^E_tK^E_t+\eta P^E_tE_t+R^T_tT_t+R^{H}_{K,t}K^H_t+R^{TH}_tT^H_t
\label{eq:identity}
\end{equation}
holds exactly (Euler's theorem on each constant-returns block), with $R^H_{K}$ and $R^{TH}$ the rentals of housing
structures and residential land. Equation \eqref{eq:identity} is tested numerically at every date of every
scenario (Appendix~\ref{app:replication}); a deliberately broken version of the accounting --- counting compute
services as final output --- fails the test, as it should. There is no double counting of compute services and
compute capital: $A_tQ_t$ is the gross revenue of the compute sector, $R^Q_tK^Q_t$ its value added net of energy.
Profits are zero under perfect competition; the AI-monopoly scenario adds a markup wedge on compute services
that appears as a separate profit line accruing to the owners of compute firms.

\subsection{Who owns what}

Every stock --- the four capital types, production land, residential land --- is held through a single fund whose
shares are the only asset households trade. Under perfect foresight all claims earn the same return $r_{t+1}$
between $t$ and $t+1$ (no arbitrage), and the fund's ex-dividend value is
\begin{equation}
W_t=\sum_{k\in\{N,Q,E,H\}}q^k_tK^k_{t+1}+P^T_tT+P^{TH}_tT^H .
\end{equation}
Who holds the fund's shares is the subject of Section~\ref{sec:households}: capitalists alone in the analytical
two-class economy, a continuum of households in the heterogeneous-agent economy. The citizen-fund scenario gives a
public fund a fixed share $\theta_F$ of the claims and distributes its net payout equally; the foreign-ownership
scenario lets a share of compute equity be held abroad, so that compute payouts leave the economy and foreign
investors finance the corresponding share of compute investment.

\begin{table}[htbp]
\centering\small
\caption{Notation.}
\label{tab:notation}
\begin{tabular}{@{}llll@{}}
\toprule
Symbol & Meaning & Symbol & Meaning\\
\midrule
$X_{i,t}, L_{i,t}, Q_{i,t}$ & task output, labor, compute in task $i$ & $K^N,K^Q,K^E,K^H$ & structures, compute, energy capital, housing\\
$\alpha_{i,t}$ & compute per worker-equivalent in task $i$ & $p^Q_t,\varepsilon_t$ & compute price, energy per compute unit\\
$\CEUt(s)$ & compute-equivalent unit of skill $s$ & $T,T^H,R^E$ & production land, residential land, energy resource\\
$w_{s,t}$ & wage per efficiency unit of skill $s$ & $q^k_t,x^k_t$ & Tobin's $q$, investment rate of capital $k$\\
$A_t$ & price of compute services & $P^T_t,P^{TH}_t$ & land prices\\
$R^k_t$ & rental of capital $k$ & $W_t$ & aggregate wealth (ex-dividend value of all claims)\\
$P^E_t,R^H_t$ & energy price, housing rent & $r_{t+1}$ & real return on all claims between $t$ and $t+1$\\
$\sigma_B,\sigma_N,\sigma_T$ & elasticities (tasks; tasks--structures; top nest) & $\beta,\gamma$ & discount factor, curvature of utility\\
\bottomrule
\end{tabular}
\end{table}