\section{Asset prices}
\label{sec:assetpricing}

\subsection{The discount factor and the return on claims}

Under perfect foresight the capitalists' Euler equation \eqref{eq:euler} defines a one-period discount factor
$M_{t,t+1}=\beta\,(x^K_{t+1}/x^K_t)^{-\gamma}(P_{t+1}/P_t)^{\gamma-1}$ with $M_{t,t+1}(1+r_{t+1})=1$. Every claim
is priced by $V_t=M_{t,t+1}(D_{t+1}+V_{t+1})$, and because there is no aggregate risk every claim earns $r_{t+1}$:
the model has a term structure (the path of $r_{t+1}$) but no risk premia. The marginal investor is an unconstrained
saver in both layers of Section~\ref{sec:households}: in the two-class economy by construction, in the
heterogeneous economy because households with positive savings satisfy the Euler equation with equality and the
return is the same for all of them. Borrowing constraints change who saves, not the pricing kernel
\citep{KrusellSmith1997}.

\subsection{Capital goods}

For each capital type $k$ with purchase price $p^k_t$ the firm maximizes the present value of net payouts
$D^k_t=R^k_tK^k_t-p^k_tI^k_t-\tfrac{\phi_k}{2}(x^k_t-\delta_k)^2K^k_tp^k_t$. The first-order conditions are
$q^k_t=p^k_t[1+\phi_k(x^k_t-\delta_k)]$ and
\begin{equation}
q^k_t(1+r_{t+1})=R^k_{t+1}+p^k_{t+1}\Big[\phi_k(x^k_{t+1}-\delta_k)x^k_{t+1}-\frac{\phi_k}{2}(x^k_{t+1}-\delta_k)^2\Big]+(1-\delta_k)q^k_{t+1},
\label{eq:capital_euler}
\end{equation}
and by \citet{Hayashi1982} the ex-dividend value of the firm is $V^k_t=q^k_tK^k_{t+1}$. A negative payout is a net
issuance of shares; there is no dilution problem because $V^k_t$ is the value of all shares after the issuance and
the fund buys the new shares. Three valuation objects are reported: the ratio of market value to
\emph{replacement cost}, $q^k_t/p^k_t$, which exceeds one in investment booms; the price--earnings ratio
$V^k_t/(R^k_tK^k_t)$, which for compute is dominated by depreciation and the expected price decline
(Proposition~\ref{prop:wealth}); and the price--payout ratio $V^k_t/D^k_t$, which is undefined while the sector
issues shares. Market-to-\emph{book} ratios at historical cost are higher than market-to-replacement for compute,
because the book value of a stock bought at yesterday's price overstates its replacement cost when prices fall;
we report replacement-cost ratios throughout.

\subsection{Land, housing and the aggregate balance sheet}

Production land and residential land are priced by $P^T_t(1+r_{t+1})=(1-\tau_T)R^T_{t+1}+P^T_{t+1}$ with the
terminal value given by the Gordon formula at the terminal growth rate of rents, and the transversality
condition is verified by the horizon check of Appendix~\ref{app:algorithms}. A house is a bundle of a structure
and a plot: its price is $q^H_tK^H_{t+1}+P^{TH}_tT^H$ and its price--rent ratio is that bundle divided by
$R^H_tH^S_t$. The structure part is bounded by replacement cost and the land part is not, which is why land is
where the value of an AI boom goes when the top nest has gross complements (Proposition~\ref{prop:regimes}c).

Aggregate wealth $W_t$ is the ex-dividend value of all claims. It differs from the replacement cost of the
physical stocks $\sum_kp^k_tK^k_{t+1}$ by the adjustment-cost premia $\sum_k(q^k_t-p^k_t)K^k_{t+1}$ and by the
value of land, and from national wealth in the foreign-ownership scenario by the claims held abroad. We never add
$V^Q_t$ to $p^Q_tK^Q_{t+1}$: the first is the market value of the claim on the second.

\subsection{Decomposing valuation changes}

Along a deterministic path, the log price--payout ratio of any claim satisfies the Campbell--Shiller identity
exactly, so the difference between two scenarios in $\log(V_t/D_t)$ can be written as the present value of
differences in payout growth minus the present value of differences in returns \citep{CampbellShiller1988a}.
We use this to separate, for each claim and each scenario, the \emph{cash-flow} channel (AI raises or lowers
what the claim pays) from the \emph{discount-rate} channel (AI changes the return the marginal investor
requires, Proposition~\ref{prop:discount}), and we add the \emph{supply} channel for capital goods (new capital
is produced at replacement cost, so the value of existing claims is diluted by entry at $q^k=p^k$) and the
\emph{competition} channel in the AI-monopoly scenario (markup rents). Because the paths are deterministic there
is no risk channel; the stochastic AGI-arrival extension of Section~\ref{sec:robustness} shows what a hazard of
arrival adds.

\subsection{Bubbles}

We impose the transversality condition on every claim, so there are no bubbles in the baseline. A rational
bubble on compute equity is inconsistent with free entry at replacement cost (new capacity can always be
produced at $p^Q_t$); the only claims that could carry a bubble in this economy are land and non-reproducible
intangibles, and only if the interest rate falls to the growth rate. We check along every path that $r_t>g_t$
asymptotically, which is the condition under which the land price recursion converges.