\section{Households, saving and the ownership of compute}
\label{sec:households}

Income can flow from labor to capital without anything happening to the distribution of wealth if everybody
owns capital in proportion to their consumption, and it can concentrate wealth without any change in factor
shares if only some households save. The model therefore tracks ownership explicitly, in two layers.

\subsection{The two-class economy}

The analytical layer has two types of households. Capitalists, of unit mass, own all the claims of
Section~\ref{sec:model}, have no labor income, and choose expenditure $x^K_t$ on goods and housing to maximize
$\sum_t\beta^tu(c_t,h_t)$ with $u=(c^{1-\theta_H}h^{\theta_H})^{1-\gamma}/(1-\gamma)$. With the Cobb--Douglas
inner aggregator the problem reduces to one over total expenditure with the ideal price index
$P_t\propto(R^H_t)^{\theta_H}$, and the Euler equation is
\begin{equation}
\Big(\frac{x^K_{t+1}}{x^K_t}\Big)^{\gamma}\Big(\frac{P_{t+1}}{P_t}\Big)^{1-\gamma}=\beta(1+r_{t+1}).
\label{eq:euler}
\end{equation}
Workers of skill $s$, of mass $H_s$, consume their wage income plus transfers each period. The two-class economy
is the one in which Propositions~\ref{prop:bgp}--\ref{prop:tax} are proved and in which all scenarios of
Section~\ref{sec:results} are solved; it is also the limit of the heterogeneous economy in which a group of
patient households holds almost all wealth. It has two purposes: the capitalists' Euler equation
\eqref{eq:euler} is the stochastic discount factor that prices every claim, and the hand-to-mouth workers make
the ``demand compression'' channel of Section~\ref{sec:results} visible, because any fall in the wage bill is a
one-for-one fall in workers' expenditure.

\subsection{The heterogeneous-agent economy}

The quantitative distributional layer replaces the two types by a continuum. A household is characterized by a
permanent skill $s\in\{L,H\}$, a permanent discount factor $\beta_j\in\{\beta_1,\beta_2,\beta_3\}$ with masses
$(0.5,0.3,0.2)$ within each skill \citep{KrusellSmith1998,Carroll2017}, an idiosyncratic productivity
$e_t$ following a five-state Rouwenhorst approximation to an AR(1) in logs with persistence $0.95$ and
innovation standard deviation $0.20$, and assets $a_t\ge0$ invested in the fund. Income is
$w_{s,t}e_t+\tau_t$ with $\tau_t$ the lump-sum transfer of the policy scenarios; expenditure $x_t$ and savings
$a_{t+1}$ satisfy $x_t+a_{t+1}=a_t(1+r_t)+w_{s,t}e_t+\tau_t$, and the Euler equation holds with equality
whenever the borrowing constraint is slack. The problem is solved by the endogenous grid method
\citep{Carroll2006} on a 300-point grid and the distribution by non-stochastic simulation
(Appendix~\ref{app:algorithms}). The discount factors are set so that stationary savings at the 2026 prices equal
the two-class wealth $W_0$, so that both layers start from the same balance sheet; the resulting wealth Gini and
top-10\% share are reported against the Distributional Financial Accounts in Section~\ref{sec:calibration}
\citep{FedDFA2026}. The model has no return heterogeneity and cannot reproduce the top-1\% share
\citep{Gabaix2016,HubmerKrusellSmith2021}; we report it as an untargeted moment.

General equilibrium in the heterogeneous economy is the requirement that the fund's shares are willingly held:
$\int a_{t+1}\,\dd\mu_t=W_t$ at every date, where $W_t$ is the ex-dividend value of all claims. We solve for the
path of returns by relaxation: given a path $\{r_{t+1}\}$, the production side (four capital Euler equations and
the goods market) is solved with the banded Newton method of Appendix~\ref{app:algorithms}, the household block
is solved backward and simulated forward, and the return is lowered where households demand more claims than
exist. The two-class solution is the starting point.

\subsection{Wealth accounting}

For both layers we decompose the change in aggregate wealth between any two dates into net saving, net payouts
received and reinvested, and revaluation:
\begin{equation}
W_{t+1}-W_t=\underbrace{\big[(1+r_{t+1})W_t-W_t\big]}_{\text{return}}
-\underbrace{\big[\textstyle\sum_kD^k_{t+1}+R^T_{t+1}T+R^{TH}_{t+1}T^H\big]}_{\text{payouts}}+\ldots,
\end{equation}
where the return splits further into the payout yield and the capital gain $\sum_k(q^k_{t+1}K^k_{t+2}-q^k_tK^k_{t+1})$
and the land-price change. In the heterogeneous economy the same decomposition is applied household by
household in simulation, which is what allows the paper to say whether wealth concentrates because the rich save
more or because what they already own is revalued (Section~\ref{sec:results}).