\section{Scenarios, heterogeneous households, policy and welfare}
\label{sec:robustness}

Table~\ref{tab:scenarios} summarizes the scenario catalogue in 2050 and 2100; Figures~\ref{fig:tech}--\ref{fig:policy}
show the paths. Every scenario re-calibrates the 2026 base point to the same shares, so that differences are
differences in the anticipated future, not in the starting balance sheet.

\begin{table}[htbp]
\centering\scriptsize
\caption{Scenario catalogue: selected outcomes in 2050 and 2100 (two-class economy). $w_L$ per worker relative to
2026; GDP relative to 2026; $W$/GDP aggregate wealth to GDP; land P/R price--rent ratio of production land;
$V^Q$/GDP market value of compute capital; $C_W$/GDP workers' consumption share.}
\label{tab:scenarios}
\resizebox{\textwidth}{!}{\input{tables/tab04_scenarios}}
\end{table}

\subsection{Technology: how fast, how cheap, how constrained}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fig05_technology_scenarios.pdf}
\caption{Technology scenarios. GDP relative to trend; labor share; wage of the lower skill; real return;
wealth-to-GDP; land price--rent ratio.}
\label{fig:tech}
\end{figure}

The \emph{timing} of the transition is governed by the rate of capability progress and by the compute price
path; the \emph{destination} by the floors. With capability progress at 20\% a year (fast) the labor share is
below 0.07 by 2050 and the real return peaks above 50\%; at 5\% a year (slow) the labor share is still 0.27 in
2050 and the wage of the lower skill is three times higher than in the baseline at that date, but by 2100 the
two paths are within a factor of two of each other. The rapid-AGI scenario, in which every task reaches its
terminal compute requirement in 2032, is the source's technology-binding transition in its purest form: output
rises forty-fold by 2050, the real return exceeds 100\% during the 2030s, the labor share is one in a thousand by
2050, and the expert skill's premium never materializes because nothing is the last task to be automated.
The frozen-capability scenario (``no capability'') --- 2026 algorithms with hardware that keeps getting cheaper
--- is instructive: Moore's law alone, at 2026 capability, automates three of the four task classes by 2060 and
halves the labor share, because the compute requirements of 2026 are finite and a hundredfold cheaper compute
eventually meets them. Only the fully frozen economy (``AI frozen'') stays on its balanced path.

The compute price floor is the parameter the long run depends on. With Moore's law ending in 2040 (compute
only eight times cheaper than today) the wage of the lower skill in 2100 is \wLMooreEndsTwoOneZeroZero{} of its
2026 level against \wLTwoOneZeroZero{} in the baseline; with a floor of one thousandth it is lower still, and when
the terminal compute requirements are five times higher (``$\alpha^\infty\times5$'') wages are five times higher
than in the baseline at every date after 2070, exactly as Proposition~\ref{prop:bgp} says. The level of wages in
the AGI limit is the product of two unidentified numbers; the paper can only report the dependence.

Energy is where the physical world pushes back. With a fast compute-price decline and no efficiency gains
(``energy-constrained'') the energy price rises far more than in the baseline, utilization of compute capacity
falls below one for several decades, the share of electricity used by compute exceeds one half, and the labor
share converges to a positive limit (Proposition~\ref{prop:energy}); with fast efficiency gains (``cheap
energy'') the same compute-price path produces the fastest collapse of the labor share in the catalogue. Energy
efficiency, not energy capacity, is the margin that matters, because capacity is reproducible and the resource
share in energy is modest.

\subsection{Substitutability and scarcity}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fig06_structure_scenarios.pdf}
\caption{Substitutability between tasks and physical complements, scarce land, and chip depreciation.}
\label{fig:structure}
\end{figure}

The regime map of Section~\ref{sec:theory} is visible in Figure~\ref{fig:structure}. With strong complementarity
($\sigma_N=\sigma_T=0.4$) output rises only twenty-fold by 2100 (against \GDPTwoOneZeroZero{} in the baseline), the
land price--rent ratio stays high throughout because the real rate never rises much, and the wealth-to-income
ratio does not fall during the boom: there is not enough of a boom to raise the discount rate. With substitutes
($\sigma=1.5$) the economy approaches the source's additive limit, compute captures a large share of income, the
value of compute capital is correspondingly larger, and land captures little. The Cobb--Douglas case gives
constant shares and a wage that falls at the rate at which the price of compute falls. Scarce land (a rent
share of 8\% and $\sigma_T=0.4$) is the Piketty-like cell: land's share of income exceeds two thirds by 2100 and
the wealth-to-income ratio is the highest in the catalogue. Severe chip depreciation ($\delta_Q=0.45$) raises the
user cost of compute by nearly half, raises the wage of the lower skill in 2050 by a fifth relative to the
baseline, as Proposition~\ref{prop:bgp} implies, and lowers the price--earnings ratio of compute further (1.4
against 1.8 in 2050); it does not change the destination of income.
The reinstatement scenario, in which the expert task is never automated, keeps the higher skill's premium
permanently: its wage rises with output, the labor share converges to the expert task's share of income rather
than to zero, and the lower skill's wage path is unchanged. This is the one scenario in which labor as a whole is
missed, and it requires that some bottleneck task stay beyond compute forever.

\subsection{Preferences and the discount-rate channel}

With log utility ($\gamma=1$) the real return peaks at 26\% instead of \rPeak\%, the land price--rent ratio
does not dip during the boom (it starts far higher, because a lower $\gamma$ makes the anticipated growth path
raise the 2026 price of land, and declines slowly as rent growth slows), and the wealth-to-income ratio rises
monotonically from 9.5 in 2026; with $\gamma=4$ the return peaks at 67\%, the price--rent ratio of land falls
below three in the mid-2050s and the wealth-to-income ratio falls from 3.5 to 1.4 before recovering
(Figure~\ref{fig:pref}). The sign of the valuation response to the boom is a preference parameter, as the
proposition says; its size in the data would depend on how much of the boom the marginal investor anticipates.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fig09_preference_scenarios.pdf}
\caption{Risk aversion, reinstatement of a human task, and the end of Moore's law.}
\label{fig:pref}
\end{figure}

\subsection{Heterogeneous households}

Figure~\ref{fig:distribution} and Table~\ref{tab:distribution} report the heterogeneous-agent economy of
Section~\ref{sec:households} for the baseline, the frozen economy and the citizen fund, in general equilibrium
(maximum log excess demand for claims \HAmaxGap). Three results stand out. First, the return path of the
heterogeneous economy is close to the two-class one (\rHATwoZeroFiveZero\% against \rTCTwoZeroFiveZero\% in 2050):
the marginal investors are the patient households whose Euler equation is the capitalists', so the pricing
results of Section~\ref{sec:results} carry over. Second, the wealth Gini rises from \GiniBase{} in 2026 to
\GiniTwoZeroFiveZero{} in 2050 and \GiniTwoOneZeroZero{} in 2100, and the top-10\% share from \TopTenBase{} to
\TopTenTwoOneZeroZero, through two channels that the simulation separates: households with little wealth lose their
only source of income and dissave, while households with wealth see it revalued and compounded at a high return.
Third, the distribution of \emph{consumption} is less unequal than that of wealth because labor income remains
positive and because the citizen fund, where present, gives every household the same payout; the fund lowers
the 2100 Gini and raises the bottom half's consumption share by an order of magnitude at a cost to the top
decile that is small relative to its gains from the boom.

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fig10_distribution.pdf}
\caption{Heterogeneous-agent economy: wealth Gini, top-10\%, top-1\% and bottom-50\% shares, the equilibrium
return path against the two-class one, and household expenditure relative to GDP.}
\label{fig:distribution}
\end{figure}

\begin{table}[htbp]
\centering\small
\caption{Wealth distribution along the transition (heterogeneous-agent economy, general equilibrium).}
\label{tab:distribution}
\resizebox{\textwidth}{!}{\input{tables/tab07_distribution}}
\end{table}

\subsection{Ownership, competition and policy}

\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{fig08_policy_scenarios.pdf}
\caption{Ownership, competition and policy scenarios.}
\label{fig:policy}
\end{figure}

\begin{table}[htbp]
\centering\small
\caption{Policy and ownership scenarios: output, wages, workers' consumption and welfare (consumption-equivalent
variations over 2026--2065 relative to the baseline, two-class economy).}
\label{tab:policy}
\resizebox{\textwidth}{!}{\input{tables/tab06_policy}}
\end{table}

Table~\ref{tab:policy} compares five instruments with the baseline. A \emph{land tax} of 50\% of rents, rebated
lump-sum to workers, changes no allocation and no price except the price of land (Proposition~\ref{prop:tax}):
output is the same to three digits, and because land captures \LandShareTwoOneZeroZero{} of income in 2100 the
rebate gives workers a quarter of GDP in the long run; capitalists lose \CEVKlandtaxFiveZero\% of consumption over
2026--2065, and the wealth-to-income ratio in 2100 is halved because land is worth half as much. A
\emph{compute tax} of 30\% raises the wage of the lower skill by about thirty percent in 2050 --- the tax-inclusive
user cost of the replacement compute is higher --- and lowers output by eight percent at that date; it is the
instrument that changes what workers \emph{earn} rather than what they are \emph{given}. A \emph{capital-income
tax} of 20\% has the largest output cost in the catalogue (output in 2050 sixteen percent below the baseline)
because it taxes the accumulation of every complement as well as of compute. An \emph{AI monopoly} with a
markup of 1.5 on compute services behaves like a compute tax whose proceeds go to the monopolist: the wage of the
lower skill is fifty percent higher in 2050 than in the baseline and output eleven percent lower, and workers
gain while capitalists, who own the monopolist, are nearly indifferent. \emph{Foreign ownership} of half of
compute equity leaves every real path unchanged and sends half of compute payouts abroad; since compute payouts
are small relative to the rents that stay at home, the effect on national income is modest, and the result
reverses only in the substitutes scenario where compute captures most of income. The \emph{citizen fund} holding
one fifth of all claims does what Proposition~\ref{prop:distribution} says: workers' consumption share is bounded
below by a fifth of the net payout share, it is three orders of magnitude higher than in the baseline by 2100,
and capitalists give up \CEVKcitizenfundTwoZero\% of consumption over 2026--2065 for it.

The welfare numbers for workers in Table~\ref{tab:policy} are large because their baseline consumption falls
toward zero: any instrument that gives them a claim on the scarce complements is worth a great deal to them. The
comparison that the table allows is across instruments at a given cost to capitalists: per point of capitalists'
consumption-equivalent loss, the land tax and the citizen fund deliver far more to workers than the compute or
capital-income taxes, because they do not reduce output. This is the general-equilibrium content of the source's
suggestion to treat compute ``akin to land'': in our economy it is land itself, not compute, that should be
treated as land.

\subsection{AGI-arrival risk: a one-period illustration}

The paths above are deterministic. To see the direction in which uncertainty about the arrival of AGI would move
asset prices, we take the 2026 stocks as given and let the economy learn in 2027, with probability $\pi$, that
the baseline path is realized and otherwise the frozen path, pricing each claim with the capitalists' marginal
utility in each branch (Section~\ref{sec:assetpricing}). The exercise is not an equilibrium under uncertainty ---
2026 decisions ignore the lottery --- and the magnitudes are not meaningful, because capitalists' consumption
jumps in the baseline branch. The signs are: compute capital carries a large positive premium (it pays in the
state in which the marginal investor is rich), land and housing land carry negative premia (they pay relatively
more in the state in which the marginal investor is poor, and are therefore hedges), and structures are in
between. A full stochastic solution, in which the premium of the last-automated skill is itself a lottery, is
the first item on the agenda of Section~\ref{sec:conclusion}.