1\section{Asset prices}2\label{sec:assetpricing}34\subsection{The discount factor and the return on claims}56Under perfect foresight the capitalists' Euler equation \eqref{eq:euler} defines a one-period discount factor7$M_{t,t+1}=\beta\,(x^K_{t+1}/x^K_t)^{-\gamma}(P_{t+1}/P_t)^{\gamma-1}$ with $M_{t,t+1}(1+r_{t+1})=1$. Every claim8is priced by $V_t=M_{t,t+1}(D_{t+1}+V_{t+1})$, and because there is no aggregate risk every claim earns $r_{t+1}$:9the model has a term structure (the path of $r_{t+1}$) but no risk premia. The marginal investor is an unconstrained10saver in both layers of Section~\ref{sec:households}: in the two-class economy by construction, in the11heterogeneous economy because households with positive savings satisfy the Euler equation with equality and the12return is the same for all of them. Borrowing constraints change who saves, not the pricing kernel13\citep{KrusellSmith1997}.1415\subsection{Capital goods}1617For each capital type $k$ with purchase price $p^k_t$ the firm maximizes the present value of net payouts18$D^k_t=R^k_tK^k_t-p^k_tI^k_t-\tfrac{\phi_k}{2}(x^k_t-\delta_k)^2K^k_tp^k_t$. The first-order conditions are19$q^k_t=p^k_t[1+\phi_k(x^k_t-\delta_k)]$ and20\begin{equation}21q^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},22\label{eq:capital_euler}23\end{equation}24and 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 net25issuance of shares; there is no dilution problem because $V^k_t$ is the value of all shares after the issuance and26the fund buys the new shares. Three valuation objects are reported: the ratio of market value to27\emph{replacement cost}, $q^k_t/p^k_t$, which exceeds one in investment booms; the price--earnings ratio28$V^k_t/(R^k_tK^k_t)$, which for compute is dominated by depreciation and the expected price decline29(Proposition~\ref{prop:wealth}); and the price--payout ratio $V^k_t/D^k_t$, which is undefined while the sector30issues shares. Market-to-\emph{book} ratios at historical cost are higher than market-to-replacement for compute,31because the book value of a stock bought at yesterday's price overstates its replacement cost when prices fall;32we report replacement-cost ratios throughout.3334\subsection{Land, housing and the aggregate balance sheet}3536Production 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 the37terminal value given by the Gordon formula at the terminal growth rate of rents, and the transversality38condition is verified by the horizon check of Appendix~\ref{app:algorithms}. A house is a bundle of a structure39and a plot: its price is $q^H_tK^H_{t+1}+P^{TH}_tT^H$ and its price--rent ratio is that bundle divided by40$R^H_tH^S_t$. The structure part is bounded by replacement cost and the land part is not, which is why land is41where the value of an AI boom goes when the top nest has gross complements (Proposition~\ref{prop:regimes}c).4243Aggregate wealth $W_t$ is the ex-dividend value of all claims. It differs from the replacement cost of the44physical 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 the45value of land, and from national wealth in the foreign-ownership scenario by the claims held abroad. We never add46$V^Q_t$ to $p^Q_tK^Q_{t+1}$: the first is the market value of the claim on the second.4748\subsection{Decomposing valuation changes}4950Along a deterministic path, the log price--payout ratio of any claim satisfies the Campbell--Shiller identity51exactly, so the difference between two scenarios in $\log(V_t/D_t)$ can be written as the present value of52differences in payout growth minus the present value of differences in returns \citep{CampbellShiller1988a}.53We use this to separate, for each claim and each scenario, the \emph{cash-flow} channel (AI raises or lowers54what the claim pays) from the \emph{discount-rate} channel (AI changes the return the marginal investor55requires, Proposition~\ref{prop:discount}), and we add the \emph{supply} channel for capital goods (new capital56is produced at replacement cost, so the value of existing claims is diluted by entry at $q^k=p^k$) and the57\emph{competition} channel in the AI-monopoly scenario (markup rents). Because the paths are deterministic there58is no risk channel; the stochastic AGI-arrival extension of Section~\ref{sec:robustness} shows what a hazard of59arrival adds.6061\subsection{Bubbles}6263We impose the transversality condition on every claim, so there are no bubbles in the baseline. A rational64bubble on compute equity is inconsistent with free entry at replacement cost (new capacity can always be65produced at $p^Q_t$); the only claims that could carry a bubble in this economy are land and non-reproducible66intangibles, and only if the interest rate falls to the growth rate. We check along every path that $r_t>g_t$67asymptotically, which is the condition under which the land price recursion converges.68