= One-period marginal-utility certificate of optimality
{title2=$U'(W^*)=yZ$}
For a positive <state-price density> $Z$ pricing every attainable payoff, a feasible positive terminal payoff $W^*$ is optimal at its budget if its marginal utility is $yZ$ for some scalar $y>0$. The pointwise conjugate inequality gives $\mathbb EU(W)\leq\mathbb E\widehat U(yZ)+yx$ for every competitor; the marginal-utility identity makes equality hold for $W^*$. Finite states make integrability straightforward. In a <complete market> an interior optimum also necessarily has this proportionality, by variations preserving the state-price budget.
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