One-period marginal-utility certificate of optimality ($U^{\prime }(W^{\ast })=yZ$)

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