 Marginal utility pricing in a dividend economy

At an interior consumption and share-holding optimum, the Lagrange multiplier on the discrete dividend budget equation is ${\Lambda }_t={\beta }^t{U}^{\prime }(C_t)$. Varying the next holding by a variable measurable at date $t$ gives the displayed Euler equation. Thus the positive normalized process ${\zeta }_t={\Lambda }_t/{\Lambda }_0$ is a state-price density for traded dividend-inclusive gains. In an incomplete market, different investors can have different such densities; equality of their marginal-utility densities is an additional risk-sharing condition, not a consequence of trading one asset alone.