\section{Analytical results: when does income leave labor, and where does it go?}
\label{sec:theory}

This section isolates the mechanisms in a version of the model that can be solved by hand. It keeps the task
block of Section~\ref{sec:benchmark}, adds one complementary input $S$ and a price for compute, and asks three
questions: whether the labor-share asymptote of \citet{Restrepo2025} survives; whether the \emph{level} of
wages survives; and what the valuation of the claims to the income that leaves labor looks like. Proofs are in
Appendix~\ref{app:proofs}. Every statement is conditional on the stated assumptions; the quantitative model of
Sections~\ref{sec:calibration}--\ref{sec:results} relaxes several of them numerically.

\subsection{The limit economy}

\begin{assumption}[Limit economy]\label{ass:limit}
Final output is $Y = Z\,F(X_B, S)$ with $F(X_B,S)=\big[\nu X_B^{\rho} + (1-\nu) S^{\rho}\big]^{1/\rho}$,
$\rho = (\sigma-1)/\sigma$, $\sigma\in(0,\infty)$, $\nu\in(0,1)$, and the Cobb--Douglas limit at $\sigma=1$.
The bottleneck composite is $X_B = (\sum_{i\in B}\nu_i X_i^{\rho_B})^{1/\rho_B}$ with $\sigma_B<1$ and
$X_i = L_i + Q_i/\alpha_i$, $\alpha_i\in(0,\infty)$ for all $i$ (Premise~1 has been reached), skills
$s\in\{L,H\}$ with inelastic supplies $H_s$ and a partition $\Omega(s)$ of $B$. $S>0$ is either fixed or
reproducible at constant price one with depreciation $\delta_S$.
\end{assumption}

\begin{lemma}[Compute-equivalent labor]\label{lem:ceu}
Under Assumption~\ref{ass:limit} there is a finite $\bar Q$ such that for all $Q\ge\bar Q$ the output-maximizing
allocation uses compute in every bottleneck task, and
\begin{equation}
X_B = \mathcal{B}\,\Qt, \qquad \Qt \equiv Q + \sum_s \CEUt(s)H_s, \qquad
\mathcal{B} = \Big[\sum_{i\in B}\nu_i^{\sigma_B}\alpha_i^{1-\sigma_B}\Big]^{-1/(1-\sigma_B)},
\label{eq:XB}
\end{equation}
where $\CEUt(s)=\max_{i\in\Omega(s)}\alpha_i$ is the compute-equivalent unit of skill $s$ among its tasks.
The price of compute services is $A = Z F_X \mathcal{B}$ and $w_s = A\,\CEUt(s)$.
\end{lemma}

Lemma~\ref{lem:ceu} is Restrepo's Proposition~2 applied to the composite instead of to final output: once
every bottleneck is automated, the composite is linear in effective compute. The constant $\mathcal{B}$ is the
unit cost of the composite in compute units; it falls when any $\alpha_i$ falls. With the accessory block of
Section~\ref{sec:benchmark} the same statement holds with $\CEUt(s)$ replaced by the discounted value of
footnote~6 of the source; the quantitative model has no accessory tasks (Section~\ref{sec:model}).

\subsection{The labor share result is robust; the wage result is not}

\begin{proposition}[Labor share]\label{prop:laborshare}
Under Assumption~\ref{ass:limit}, for any $Q\ge\bar Q$ and any $S$,
\begin{equation}
\frac{\sum_s w_s H_s}{Y} \;\le\; \frac{\sum_s \CEUt(s) H_s}{Q+\sum_s\CEUt(s)H_s}\;\xrightarrow[Q\to\infty]{}\;0 .
\end{equation}
The bound holds for every constant-returns $F$ that is increasing in $X_B$, with or without complementary
inputs, and whether compute is an endowment or a priced capital good.
\end{proposition}

The proposition says that the only ingredient behind Restrepo's Proposition~5 is the \emph{within-task} perfect
substitutability of labor and compute: labor is paid the value of the compute it saves, and the value of all
compute cannot exceed output. Nothing about the rest of production matters. What the rest of production
determines is the \emph{level} at which wages settle, and who receives the income that labor loses.

\begin{proposition}[Scarce complement: factor shares and wage levels]\label{prop:regimes}
Under Assumption~\ref{ass:limit} with $S$ fixed, as $Q\to\infty$:
\begin{enumerate}[label=(\alph*)]
\item if $\sigma>1$: $A\to Z\nu^{1/\rho}\mathcal{B}>0$, $w_s\to A\,\CEUt(s)>0$, the compute share
      $s_Q\equiv AQ/Y\to1$, $Y/Q\to A$; Propositions~2--6 of \citet{Restrepo2025} hold;
\item if $\sigma=1$: $Y = Z(\mathcal{B}\Qt)^{\nu}S^{1-\nu}$, $A=\nu Y/\Qt\to0$, $w_s\to0$, factor shares are
      constant at $(\nu,1-\nu)$ and output grows at $\nu$ times the growth rate of compute;
\item if $\sigma<1$: $Y\to Z(1-\nu)^{1/\rho}S<\infty$, $A\to0$ at rate $\Qt^{\rho-1}$, $w_s\to0$, and the
      share of $S$ converges to one.
\end{enumerate}
In cases (b) and (c) the total wage bill converges to zero, so the ``no-AGI zone'' argument behind
Restrepo's Proposition~6 fails for $Q$ large enough, as the source anticipates for ``a finite resource that is
also needed by human labor'' (p.~11).
\end{proposition}

The economics is simple. Labor is paid its compute-equivalent, and the compute-equivalent is worth
whatever compute is worth. With gross complementarity, abundant compute is worth nothing at the margin: a
worker whose work can be replicated by compute that costs nothing is paid nothing, however skilled. The
source's optimistic reading (``humans can still work \ldots but their contribution becomes economically
negligible'') therefore depends on compute remaining scarce, which in the source is guaranteed by treating
$Q_t$ as an endowment that grows exogenously.

\subsection{Compute as a reproducible capital good}

Treating compute as capital changes which object is scarce. Let installed capacity $K^Q$ be bought at price
$p^Q_t$ with depreciation $\delta_Q$, no adjustment costs, and let the capitalists' Euler equation hold with
discount factor $\beta$ and curvature $\gamma$.

\begin{proposition}[Balanced AGI growth with a floor on the compute price]\label{prop:bgp}
Let $S$ be reproducible at price one and let $p^Q_t\to\bar p>0$. Under Assumption~\ref{ass:limit} with
$\sigma<1$ there exists a balanced growth path on which $Y$, $S$, $K^Q$ and capitalists' consumption grow at a
common rate $g$, the interest rate is $r = R_S-\delta_S$ with $R_S$ the marginal product of $S$, and
\begin{equation}
A = (r+\delta_Q)\bar p, \qquad w_s = (r+\delta_Q)\,\bar p\,\CEUt(s) >0, \qquad
s_Q = \frac{\nu (\mathcal{B}\Qt/S)^{\rho}}{\nu(\mathcal{B}\Qt/S)^{\rho}+(1-\nu)}\in(0,1).
\label{eq:bgp}
\end{equation}
Along this path the labor share converges to zero (Proposition~\ref{prop:laborshare}), the compute share is
constant and strictly below one, and the wage equals the user cost of the compute that replaces the worker.
\end{proposition}

\begin{proposition}[Compute price converging to zero]\label{prop:moore}
Let $p^Q_t\to0$ (Moore's law never ends) with $S$ reproducible at price one.
\begin{enumerate}[label=(\alph*)]
\item If $\sigma>1$, no balanced growth path exists: the growth rate of compute and of output increases without
      bound (the economy is an $AK$ economy whose capital price falls to zero).
\item If $\sigma\le1$, the economy converges to an $AK$ path in $S$ with $Y/S\to Z(1-\nu)^{1/\rho}$
      (Cobb--Douglas: $Y/S$ grows without bound but $S$'s share stays $1-\nu$); $A\to0$, $w_s\to0$, and the
      income share and the market value of compute capital converge to zero.
\end{enumerate}
\end{proposition}

Propositions~\ref{prop:regimes}--\ref{prop:moore} give the regime map summarized in
Table~\ref{tab:regimes}. The vanishing labor share is common to all cells. The destination of the income that
leaves labor is not: it is compute owners only when compute stays scarce (an endowment with $\sigma>1$, or a
priced capital good whose price has a floor), and it is the owners of the scarcest complement (structures, land,
energy sites) otherwise. The level of wages is positive exactly in the cells in which compute keeps a positive
price.

\begin{table}[htbp]
\centering\small
\caption{Asymptotic regimes of the limit economy (all cells: labor share $\to0$).}
\label{tab:regimes}
\begin{threeparttable}
\begin{tabular}{@{}llll@{}}
\toprule
Compute & $\sigma>1$ (substitutes) & $\sigma=1$ & $\sigma<1$ (complements) \\
\midrule
Endowment (source) & $w>0$, $s_Q\to1$, $g_Y=g_Q$ & $w\to0$, shares constant, $g_Y=\nu g_Q$ & $w\to0$, $Y$ bounded by $S$ \\
Priced, floor $\bar p>0$ & $AK$ in compute, $w>0$ & BGP, $w>0$, shares constant & BGP, $w>0$, shares constant \\
Priced, $p^Q\to0$ & explosive & $AK$ in $S$, $w\to0$ & $AK$ in $S$, $w\to0$, $s_Q\to0$ \\
Energy-bounded ($Q\le\bar E/\bar\varepsilon$) & \multicolumn{3}{c}{labor share $\to$ positive limit (Proposition~\ref{prop:energy})} \\
\bottomrule
\end{tabular}
\begin{tablenotes}\footnotesize
\item $S$ fixed in the first row, reproducible at constant price in the next two. $w$ is the wage per efficiency unit; with trend labor-augmenting progress at $g_Z$ all wage statements hold per efficiency unit.
\end{tablenotes}
\end{threeparttable}
\end{table}

\begin{proposition}[Energy bottleneck]\label{prop:energy}
Suppose compute services require $\varepsilon_t\ge\bar\varepsilon>0$ units of energy per unit and energy
available to compute is bounded by $\bar E$. Then $Q_t\le\bar E/\bar\varepsilon$ for all $t$, the price of
compute is bounded below by a positive constant whenever $F_X>0$, and the labor share converges to a positive
limit no smaller than $s_Q^{\infty}\sum_s\CEUt(s)H_s/(\bar E/\bar\varepsilon+\sum_s\CEUt(s)H_s)$, where
$s_Q^\infty$ is the limiting share of the task composite. Restrepo's Premise~2 fails and so does his Proposition~5.
\end{proposition}

\subsection{Valuation: capital's share rises, wealth-to-income can fall}

Let $s_k$ denote the gross income share of capital type $k$ and $\pi_k\ge0$ the rate at which its purchase price
declines ($\pi_Q = g_{pQ}$ until the floor, zero for structures). On a balanced growth path with growth $g$ and
interest rate $r$, no adjustment costs and competitive rentals:

\begin{proposition}[Wealth-to-income ratio]\label{prop:wealth}
\begin{equation}
\frac{W}{Y} = \sum_{k\in\{N,Q,E,H\}}\frac{s_k\,(1+g)}{(1+r)(1+\pi_k)-(1-\delta_k)} + \frac{s_T(1+g_T)}{r-g_T},
\label{eq:WY}
\end{equation}
where $s_T$ and $g_T$ are the share and growth rate of land rents. A reallocation of income shares from a
long-duration claim (land, structures) to compute capital lowers $W/Y$ whenever
$(1+r)(1+\pi_Q)-(1-\delta_Q) > (1+r)-(1-\delta_N)$, i.e.\ whenever compute depreciates faster or its price
falls faster than that of the claim it displaces. The price--earnings ratio of compute capital is
$(1+g)/[(1+r)(1+\pi_Q)-(1-\delta_Q)]$, which at $r=0.035$, $\delta_Q=0.25$, $\pi_Q=0.15$ is about $2.3$ against
$8$ for structures and $1/(r-g_T)$ for land.
\end{proposition}

The proposition separates three objects that the source and much commentary conflate: the share of income
accruing to compute, the market value of compute capital, and aggregate wealth. Income can flow
overwhelmingly to compute while the \emph{capitalized} value of that income is small, because compute is a
short-duration, rapidly cheapening asset. Whether the wealth-to-income ratio rises with AI depends on whether
the income that labor loses is captured by claims that are long-lived (land, scarce sites, intangible moats) or
short-lived (chips).

\begin{proposition}[Discount-rate channel]\label{prop:discount}
With the capitalists' Euler equation and constant growth $g$ of their expenditure, $1+r = \beta^{-1}(1+g)^{\gamma}$.
For a claim whose cash flow grows at the common rate $g$ (land under Cobb--Douglas, structures on the BGP), the
price--cash-flow ratio is $(1+g)/[\beta^{-1}(1+g)^{\gamma}-(1+g)]$, which is decreasing in $g$ if and only if
$\gamma>1$. An AI-driven increase in the growth rate therefore \emph{lowers} price--rent and price--earnings
ratios of long-duration claims when the elasticity of intertemporal substitution is below one, and raises them
otherwise; the level of prices rises in both cases if cash flows rise.
\end{proposition}

Proposition~\ref{prop:discount} is the general-equilibrium reason why ``higher profits'' need not mean ``higher
valuations'': the same boom that raises cash flows raises the rate at which the marginal investor discounts
them. The quantitative section decomposes the valuation paths into these two channels.

\subsection{Ownership}

\begin{proposition}[Two-class distribution]\label{prop:distribution}
In the two-class economy, workers' expenditure equals $\sum_s w_s H_s + \tau_t$ where $\tau_t$ are transfers.
With $\tau_t=0$ and in any regime in which $Y_t\to\infty$, the consumption share of workers converges to zero;
their consumption \emph{level} converges to $A\sum_s\CEUt(s)H_s$, positive in the cells of
Table~\ref{tab:regimes} with $w>0$ and zero otherwise. With a citizen fund paying workers a share $\theta_F$ of
aggregate net payouts, the workers' consumption share is bounded below by $\theta_F\,\bar d\,\mu_W$, where
$\bar d$ is the net-payout-to-GDP ratio on the balanced path and $\mu_W$ the workers' population share.
\end{proposition}

\begin{proposition}[Wages as the tax-inclusive rental price of the replacement]\label{prop:tax}
On the balanced path of Proposition~\ref{prop:bgp} with a tax $\tau_Q$ on compute rental income rebated lump-sum,
and an energy cost $\varepsilon P_E$ per unit of compute,
\begin{equation}
w_s = \CEUt(s)\Big[\frac{(r+\delta_Q)\bar p}{1-\tau_Q}+\varepsilon P_E\Big].
\end{equation}
A compute tax raises wages one-for-one with the tax-inclusive user cost, at the cost of a lower compute stock and
lower output; a land tax leaves all wages, allocations and prices except the land price unchanged.
\end{proposition}

The wage expression is the paper's one-line summary: in the AGI limit a worker earns what it costs,
inclusive of taxes and energy, to rent the compute that would do the worker's job. Policies that make compute
dearer raise wages and lower output; policies that tax the scarce complement (land) redistribute without changing
production.

\subsection{What the propositions do not say}

They are statements about limits. They do not say how fast the labor share falls, whether wages rise before
they fall (they can: Restrepo's Propositions~7--8 describe the transitory premium of the last-automated skills),
how much of the wealth that households hold in 2026 is revalued, or how a heterogeneous population shares the
income of the fund. Those are the questions for the quantitative model. The propositions also treat $\sigma$ as a
structural constant; if physical complements can themselves be economized over time (land through density,
energy through efficiency), the relevant elasticity is the long-run one, and the quantitative model lets
$\varepsilon_t$ fall to a floor for that reason.