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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 givesWrite $I_i=(U_i^{\prime }{)}^{-1}$ and $b_t={\xi }_t{\delta }_t$. ThenFor 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. ConsequentlyThis 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:The initial state-price budget constraints determine the multipliers up to a common normalization: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 $w_t^i={\delta }_t{B}_i(t)/b(t)$. 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 $B_i^{\prime }=-b{p}^i$ and $B^{\prime }=-b$.

If all ${\rho }_i=\rho$ and $U_i=U$, market clearing gives $b(t)=k{e}^{-\rho t}$ for a constant $k>0$, and $p_t^i=I(y_{i}k)$ is constant. The budget equations then implyFor differentiable $b$, the general equilibrium has short rate $r_t=\mu -{\sigma }^2-b^{\prime }(t)/b(t)$ 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, $r=\mu -{\sigma }^2+\rho$.