= Solution
In a price-taking equilibrium, market clearing fixes aggregate consumption at the dividend $\delta_t$, so $c_t^i=p_t^i\delta_t$ and $\sum_i p_t^i=1$. Each agent takes this aggregate benchmark as given when varying their own consumption. Let $\xi_t$ be a common <state-price density> and $y_i>0$ their budget multipliers. Marginal pricing gives
$$
e^{-\rho_it}\frac{U_i'(p_t^i)}{\delta_t}=y_i\xi_t.
$$
Write $I_i=(U_i')^{-1}$ and $b_t=\xi_t\delta_t$. Then
$$
p_t^i=I_i(y_ie^{\rho_it}b_t),\qquad\sum_i I_i(y_ie^{\rho_it}b_t)=1.
$$
For fixed $t$, the second left side is continuous and strictly decreasing from infinity to zero, so it determines a unique positive number $b(t)$. Every coefficient in that equation is deterministic. Consequently
$$
\boxed{p_t^i=I_i(y_ie^{\rho_it}b(t))\text{ is deterministic},\qquad\xi_t=\frac{b(t)}{\delta_t}.}
$$
This is the <relative-consumption share equilibrium>. The absolute consumption streams remain random through $\delta_t$.
Assuming finite fundamental prices, the ex-dividend price of the productive asset follows from conditional pricing:
$$
\boxed{S_t=\xi_t^{-1}\mathbb E_t\int_t^\infty\xi_s\delta_s\,ds
=\frac{\delta_t}{b(t)}\int_t^\infty b(s)\,ds.}
$$
The initial <state-price budget constraints> determine the multipliers up to a common normalization:
$$
\int_0^\infty b(t)I_i(y_ie^{\rho_it}b(t))\,dt
=\pi_0^i\int_0^\infty b(t)\,dt,\qquad i=1,\ldots,J.
$$
Together with the scalar market-clearing equation these characterize the equilibrium. A common scaling of all $y_i$ is offset by the reciprocal scaling of $b$. At least one redundant budget equation can be removed, since the initial fractions sum to one.
For completeness, define $B(t)=\int_t^\infty b(s)ds$ and $B_i(t)=\int_t^\infty b(s)p_s^i ds$. The consumption-financing wealth is $w_t^i=\delta_tB_i(t)/b(t)$. Its stochastic exposure and that of $S_t$ are both proportional to $\delta_t$. Thus holding $B_i(t)/B(t)$ units of the productive asset, with no bank-account balance, finances this wealth and consumption. Indeed, the holding's deterministic changes and dividend consumption balance because $B_i'=-bp^i$ and $B'=-b$.
If all $\rho_i=\rho$ and $U_i=U$, market clearing gives $b(t)=ke^{-\rho t}$ for a constant $k>0$, and $p_t^i=I(y_ik)$ is constant. The budget equations then imply
$$
\boxed{p_t^i=\pi_0^i,\qquad c_t^i=\pi_0^i\delta_t,\qquad S_t=\delta_t/\rho.}
$$
For differentiable $b$, the general equilibrium has short rate $r_t=\mu-\sigma^2-b'(t)/b(t)$ and risky-asset total expected return $r_t+\sigma^2$; its volatility is $\sigma$. These follow directly from $\xi_t=b(t)/\delta_t$ and the <Itô formula>. In the identical-agent case, $r=\mu-\sigma^2+\rho$.
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