Relative-consumption share equilibrium ($p_t^i=I_i(y_i{e}^{{\rho }_{i}t}b(t))$)

Suppose a productive asset pays total consumption ${\delta }_t$, while price-taking agents value their consumption shares $p_t^i=c_t^i/{\delta }_t$ through discounted utility functions. Marginal pricing by a common state-price density ${\xi }_t$ gives $e^{-{\rho }_{i}t}{U}_i^{\prime }(p_t^i)/{\delta }_t=y_i{\xi }_t$. Writing $b(t)={\xi }_t{\delta }_t$, market clearing reduces to the scalar equation ${\sum }_i{I}_i(y_i{e}^{{\rho }_{i}t}b(t))=1$, where $I_i$ is the inverse marginal utility. Its unique positive solution is deterministic. Therefore the consumption shares are deterministic even when dividend levels are random.

With finite fundamental prices, the asset's price is $S_t={\delta }_{t}b(t{)}^{-1}{\int }_t^{\infty }b(s)ds$. Initial budget equations determine the multipliers. Identical preferences and discount rates make the shares constant and equal to initial ownership fractions. The aggregate benchmark is taken as given in the price-taking first-order conditions.