= Marginal utility pricing in a dividend economy
{title2=$S_t\beta^tU'(C_t)=\mathbb E_t[\beta^{t+1}U'(C_{t+1})(S_{t+1}+d_{t+1})]$}
At an interior consumption and share-holding optimum, the <Lagrange multiplier> on the <discrete dividend budget equation> is $\Lambda_t=\beta^tU'(C_t)$. 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.
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