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\title{From Labor to Capital}
\subtitle{Artificial General Intelligence, General Equilibrium, and Asset Valuation}
\author{Simon-Pierre Boucher}
\institute{Université du Québec en Outaouais --- UQO Working Paper No.~24}
\date{October 2026}

\begin{document}
\maketitle

\begin{frame}{The question}
\begin{itemize}
\item \citet{Restrepo2025}: with AGI every task can be done by compute; as compute grows,
  output becomes linear in compute, wages converge to the cost of the compute that replaces the worker,
  and the labor share goes to zero. ``We won't be missed.''
\item This paper asks what happens to the \emph{other side} of the ledger:
  \begin{itemize}
  \item Who receives the income that leaves labor --- compute owners, or the owners of what compute cannot reproduce?
  \item What is that income \emph{worth} as an asset?
  \item How does the ownership of wealth evolve, and what do policies that tax or share it do?
  \end{itemize}
\item One change in production: compute is a reproducible, priced, depreciating capital good that needs energy and works alongside structures and land. Everything on the household side is new.
\end{itemize}
\end{frame}

\begin{frame}{Preview of results}
\begin{enumerate}
\item The labor-share asymptote is robust: it needs only within-task substitutability (Proposition 1).
\item The wage-level result is not: with gross complements (\(\sigma<1\)) and cheap compute, wages go to zero in level; with a floor on the compute price, the wage equals the user cost of the replacement compute.
\item Income goes to the \emph{scarcest} input: compute if it stays scarce, otherwise land / sites / structures.
\item Capital's share can rise while the wealth-to-income ratio falls: compute is a short-duration, cheapening asset.
\item A growth boom raises the real rate; with \(\gamma>1\) valuation ratios of long-duration claims \emph{fall} during the boom.
\item Quantitatively (conditional scenarios, 2026--2100): labor share \(\LSTwoZeroFiveZero\) in 2050 and \(\LSTwoOneZeroZero\) in 2100 in the baseline; real return peaks at \(\rPeak\%\) in \(\rPeakYear\); wealth/GDP falls to \(\WYmin\) in \(\WYminYear\) before land takes over.
\end{enumerate}
\end{frame}

\begin{frame}{Restrepo (2025) in one slide}
\[
X_t(\omega)=L_t(\omega)+\frac{Q_t(\omega)}{\alpha_t(\omega)},\qquad Y_t=F(\{X_t(\omega)\}),\qquad \sum_\omega Q_t(\omega)\le Q_t
\]
\begin{itemize}
\item Premise 1 (AGI): every \(\alpha_t(\omega)\) becomes finite. Premise 2: \(Q_t\to\infty\).
\item Bottleneck vs accessory work; all bottlenecks eventually automated (P1).
\item \(Y\to A[Q+\sum_s\CEUt(s)H_s]\), \(\CEU(s)=\max_{\omega\in\Omega(s)}\alpha(\omega)\) (P2); growth = growth of compute (P3).
\item \(w_s\to A\cdot\CEUt(s)\) (P4); labor share \(\to0\) (P5); total wages higher than pre-AGI (P6, ``unless a finite resource is also needed'').
\item No capital, no prices of compute, no households, no proofs (July 2025 draft).
\end{itemize}
\end{frame}

\begin{frame}{Replication}
\centering
\includegraphics[width=0.78\textwidth]{fig01_benchmark.pdf}
\end{frame}

\begin{frame}{The model: production}
\begin{itemize}
\item Four bottleneck tasks, two skills, \(X_i=L_i+Q_i/\alpha_{i,t}\), CES \(\sigma_B=0.5\).
\item \(V=\mathrm{CES}(K^N,X_B;\sigma_N)\), \(Y=Z\,\mathrm{CES}(T,E^Y,V;\sigma_T)\): structures, land, energy are the physical complements.
\item Compute: capacity \(K^Q\) bought at \(p^Q_t\) (falls at \(g_{pQ}\) to a floor), depreciates at \(\delta_Q=0.25\), needs \(\varepsilon_t\) energy per unit (falls to a floor). Idle if energy cost exceeds marginal value.
\item Energy from energy capital and a fixed resource; housing from structures and residential land.
\item Four Tobin's-\(q\) investment problems; perfect foresight; national accounting identity tested at every date.
\end{itemize}
\end{frame}

\begin{frame}{The model: households and assets}
\begin{itemize}
\item Two-class: capitalists (CRRA, \(\gamma=2\)) own the fund that holds all claims; workers hand-to-mouth.
\item Heterogeneous-agent layer: skills × discount types × productivity, borrowing constraint, one asset; GE by relaxation on the return path.
\item Pricing: \(V^k_t=q^k_tK^k_{t+1}\), land by backward recursion, aggregate wealth \(W_t=\sum_kV^k_t+P^T_tT+P^{TH}_tT^H\).
\item Exact Campbell--Shiller decomposition along deterministic paths: cash-flow vs discount-rate channels.
\end{itemize}
\end{frame}

\begin{frame}{Proposition 1: the labor share result is robust}
Once every bottleneck task is automated, \(X_B=\mathcal{B}\,\Qt\) with \(\Qt=Q+\sum_s\CEUt(s)H_s\) and \(w_s=A\,\CEUt(s)\). For \emph{any} constant-returns \(F\) increasing in \(X_B\):
\[
\frac{\sum_sw_sH_s}{Y}\le\frac{\sum_s\CEUt(s)H_s}{Q+\sum_s\CEUt(s)H_s}\to0 .
\]
Only within-task perfect substitutability matters. What the rest of production determines is the \emph{level} of wages and \emph{who} gets the income.
\end{frame}

\begin{frame}{Proposition 2: scarce complement, \(S\) fixed}
\(Y=Z[\nu X_B^\rho+(1-\nu)S^\rho]^{1/\rho}\), \(Q\to\infty\):
\begin{center}
\begin{tabular}{lccc}
\toprule
 & \(\sigma>1\) & \(\sigma=1\) & \(\sigma<1\)\\
\midrule
Price of compute \(A\) & \(\to Z\nu^{1/\rho}\mathcal{B}>0\) & \(\to0\) & \(\to0\)\\
Wages & positive & \(\to0\) & \(\to0\)\\
Compute share & \(\to1\) & \(\nu\) & \(\to0\)\\
Output & linear in \(Q\) & \(\propto Q^\nu\) & bounded by \(S\)\\
\bottomrule
\end{tabular}
\end{center}
Restrepo's Propositions 2--6 are the first column. In the other two, \emph{total} wages go to zero: the ``no-AGI zone'' argument fails exactly where the source says it would.
\end{frame}

\begin{frame}{Regime map (numerical illustration)}
\centering
\includegraphics[width=\textwidth]{fig02_regimes.pdf}
\end{frame}

\begin{frame}{Propositions 3--5: compute as capital}
\begin{itemize}
\item Floor on the compute price \(\bar p>0\), reproducible \(S\): balanced AGI growth with
\[
A=(r+\delta_Q)\bar p,\qquad w_s=(r+\delta_Q)\,\bar p\,\CEUt(s)>0,\qquad s_Q\in(0,1).
\]
\emph{The wage is the user cost of the compute that replaces you.}
\item \(p^Q\to0\): \(\sigma>1\) explosive; \(\sigma\le1\): \(AK\) in \(S\), \(A\to0\), \(w\to0\), value of compute \(\to0\).
\item Energy bound \(Q\le\bar E/\bar\varepsilon\): labor share converges to a positive limit --- Premise 2 fails.
\end{itemize}
\end{frame}

\begin{frame}{Proposition 6: wealth-to-income}
\[
\frac{W}{Y}=\sum_k\frac{s_k(1+g)}{(1+r)(1+\pi_k)-(1-\delta_k)}+\frac{s_T(1+g_T)}{r-g_T}
\]
\begin{itemize}
\item Reallocating income shares from land/structures to compute \emph{lowers} \(W/Y\): compute depreciates at 25\% and its price falls.
\item Price--earnings of compute at \(r=0.035,\ \delta_Q=0.25,\ \pi_Q=0.15\): about 2.3; structures 8; land \(1/(r-g)\).
\item ``All income to compute'' and ``asset values explode'' are different claims.
\end{itemize}
\end{frame}

\begin{frame}{Proposition 7: the discount-rate channel}
With \(1+r=\beta^{-1}(1+g)^\gamma\), a claim growing at \(g\) has
\[
\frac{P}{D}=\frac{1+g}{\beta^{-1}(1+g)^\gamma-(1+g)},\qquad \frac{\partial\log(P/D)}{\partial\log(1+g)}\ \text{has the sign of}\ 1-\gamma .
\]
An AI growth boom raises the real rate; with \(\gamma>1\) price--rent and price--earnings ratios of long-duration claims fall even as cash flows and prices rise.
\end{frame}

\begin{frame}{Proposition 8: wages and taxes in the limit}
\[
w_s=\CEUt(s)\Big[\frac{(r+\delta_Q)\bar p}{1-\tau_Q}+\varepsilon P_E\Big]
\]
A compute tax raises wages one-for-one with the tax-inclusive user cost, at the cost of output; a land tax changes nothing but the land price.
\end{frame}

\begin{frame}{Calibration (2026, US)}
\begin{itemize}
\item Labor share 0.568 (PWT), residential structures 1.17 GDP, \(\delta_N=0.089\), \(\delta_H=0.023\) (BEA via FRED); compute capital 6.2\% of GDP (IPE investment capitalized); data centers 4.4\% of electricity (LBNL).
\item \(r^*=3.5\%\), \(g_Z=1.5\%\), \(\gamma=2\); \(\sigma_B=0.5\), \(\sigma_N=0.7\), \(\sigma_T=0.6\) (literature).
\item Scenario inputs: \(g_\alpha=0.10\), \(g_{pQ}=0.15\) (Epoch: 0.34), floor \(10^{-2}\), \(g_\varepsilon=0.10\).
\item Implied: \(\alpha_{1,0}=\AlphaOneBase\) (compute does 22\% of the routine task in 2026).
\item 162 retrieved targets with URLs and hashes; nothing typed from memory.
\end{itemize}
\end{frame}

\begin{frame}{Baseline transition}
\centering
\includegraphics[width=0.95\textwidth]{fig03_baseline.pdf}
\end{frame}

\begin{frame}{Baseline: valuation}
\centering
\includegraphics[width=0.95\textwidth]{fig04_valuation.pdf}
\end{frame}

\begin{frame}{Baseline in numbers}
\centering\scriptsize
\input{../paper/tables/tab03_baseline}
\end{frame}

\begin{frame}{Cash-flow vs discount-rate channels (2026 values, baseline vs AI frozen)}
\centering
\includegraphics[width=0.7\textwidth]{fig07_decomposition.pdf}
\end{frame}

\begin{frame}{Technology scenarios}
\centering
\includegraphics[width=0.95\textwidth]{fig05_technology_scenarios.pdf}
\end{frame}

\begin{frame}{Substitutability and scarcity}
\centering
\includegraphics[width=0.95\textwidth]{fig06_structure_scenarios.pdf}
\end{frame}

\begin{frame}{Ownership, competition and policy}
\centering
\includegraphics[width=0.95\textwidth]{fig08_policy_scenarios.pdf}
\end{frame}

\begin{frame}{Policy and welfare}
\centering\scriptsize
\input{../paper/tables/tab06_policy}
\end{frame}

\begin{frame}{Heterogeneous households}
\centering
\includegraphics[width=0.95\textwidth]{fig10_distribution.pdf}
\end{frame}

\begin{frame}{What would falsify the model in the next decade}
\begin{itemize}
\item Wages of exposed occupations fall \emph{after} the compute-equivalent cost crosses the wage, not when the technology becomes feasible.
\item Margins fall in labor-intensive exposed activities (compute is competitively supplied); they rise only where a scarce complement is owned.
\item Price--earnings ratios of compute-heavy capital are low and track \((1+r)(1+\pi_Q)-(1-\delta_Q)\).
\item Rising relative electricity prices and data-center energy shares in compute-intensive regions.
\end{itemize}
\end{frame}

\begin{frame}{Limitations}
\begin{itemize}
\item Perfect foresight: no risk premia; a stochastic AGI-arrival extension is sketched, not solved in full.
\item The level of wages after 2060 depends on the product of the compute-price floor and terminal compute per worker --- a scenario input.
\item No endogenous innovation, no new-task creation beyond the reinstatement scenario, no nonhomothetic preferences.
\item Heterogeneous-agent block has no return heterogeneity (top-1\% share under-predicted).
\item Not forecasts: conditional scenarios with every input documented.
\end{itemize}
\end{frame}

\begin{frame}{Conclusion}
\begin{itemize}
\item Restrepo's labor-share result survives everything; his wage-level and ``income to compute'' results survive only if compute stays scarce.
\item When it does not, the income that labor loses goes to the owners of the scarcest complement, and wages fall to the rental price of the replacement compute.
\item Asset values follow duration, not income shares: compute is capitalized at the duration of chips; land at \(1/(r-g)\); a growth boom raises \(r\) and compresses valuation ratios first.
\item Policy: a land tax redistributes without distortion; a compute tax raises wages at the cost of output; a citizen fund keeps workers' consumption share bounded below.
\end{itemize}
\end{frame}

\begin{frame}[allowframebreaks]{References}
\scriptsize
\bibliography{../paper/references}
\end{frame}
\end{document}