\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, and it is also where the solver reaches its limits. In the
baseline the share of electricity used by compute peaks near 60\% and the energy price rises eighteen-fold, but
capacity is never idle. The scenario that combines the fast compute-price decline with \emph{no} efficiency
gains (``energy-constrained'') could not be solved to tolerance: the Newton iterations stall on the kinks created
by the utilization margin (capacity switches between idle and fully used across periods), and the same happens
for the scenario in which Moore's law never ends. For these two cells the paper reports only what
Proposition~\ref{prop:energy} establishes --- a bounded energy supply with a floor on energy per unit of compute
bounds compute, keeps its price positive and keeps the labor share positive --- and the fact that in every solved
scenario the energy price rises by an order of magnitude during the boom. 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. The Cobb--Douglas
and substitutes cells ($\sigma=1$ and $\sigma=1.5$) could not be solved to tolerance on the 275-year horizon
--- with substitutes the task composite is not bounded by the physical inputs, the economy approaches the
source's additive limit, and the compute stock grows by factors that the solver cannot handle together with the
terminal conditions --- so for those cells the paper relies on Proposition~\ref{prop:regimes} and on the static
regime map of Figure~\ref{fig:regimes}: constant shares and a wage that falls at the rate of the compute price in
the Cobb--Douglas case, a compute share going to one and a positive wage in the substitutes case.
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, solved in general
equilibrium up to a maximum log excess demand for claims of \HAmaxGap{} (the quasi-Newton on the return path
stops improving at that precision; the frozen economy converges to $10^{-4}$). Three results stand out.

First, the equilibrium return path of the heterogeneous economy tracks the two-class one closely through the
boom (\rHATwoZeroFiveZero\% against \rTCTwoZeroFiveZero\% in 2050, a lower peak) and lies above it afterwards
(\rHATwoOneZeroZero\% against \rTCTwoOneZeroZero\% in 2100), because the marginal savers who must absorb the
growing supply of claims are less patient than the two-class capitalist ($\beta_3=\betaHigh$ against 0.995);
the one-year spike in 2026 is the adjustment of the initial distribution to the announced path. The pricing logic
of Section~\ref{sec:results} carries over --- every claim earns the same return, land is where value ends up ---
with a somewhat higher discount rate. The frozen economy's distribution is calibrated separately (its 2026
wealth is larger) and is flat by construction.

Second, wealth inequality is \emph{non-monotonic}. The Gini coefficient falls from \GiniBase{} in 2026 to
\GiniTwoZeroFiveZero{} in 2050 and then rises to \GiniTwoOneZeroZero{} in 2100 and above its 2026 level
afterwards; the top-10\% share goes from \TopTenBase{} to \TopTenTwoZeroFiveZero{} and back to
\TopTenTwoOneZeroZero; the bottom half's share triples by 2050 and then falls below its 2026 value. The simulation
separates the two phases. During the boom, the transitory premium of the higher skill and the high return on
savings let households in the middle of the distribution accumulate; this is Restrepo's technology-binding
transition seen from the wealth side, and it is equalizing. After the last task is automated, labor income
disappears for everybody and the distribution of wealth converges to the distribution of patience: households
with the highest discount factor hold a growing share, the bottom half holds nothing, and the Gini rises toward
the value of a pure rentier economy. The historical analogy is not the twentieth century's $r>g$; it is the
disappearance of the one asset that most households own, their labor.

Third, the citizen fund changes what households own, not only what they receive. With the fund holding one
fifth of all claims, private wealth is the remaining four fifths, every household receives the same payout, and
households at the bottom save less because the transfer substitutes for precautionary wealth: the measured
Gini of \emph{private} wealth is slightly higher than in the baseline (0.63 against 0.60 in 2050, 0.76 against
0.73 in 2100) while the share of the fund's payout in the consumption of the bottom half is, by construction, the
same as for everybody else. The right comparison is therefore in consumption, which the two-class
Table~\ref{tab:policy} reports: workers' consumption share in 2100 is three orders of magnitude higher with the
fund. The top-1\% share stays near 0.12 throughout every run: without return heterogeneity the model has no
mechanism for a Pareto tail, and the number is reported as a known miss rather than a prediction.

\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 first-order condition of production (Proposition~\ref{prop:tax}): output in 2050
differs from the baseline by less than half a percent (the difference comes from the change in who saves), 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-five percent higher in 2050 than in the baseline and output thirteen 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}.