= Optimal marginal utility as a one-period pricing density
{title2=$Z=U'(\theta_*X)/\mathbb E[U'(\theta_*X)]$}
For a negative, strictly increasing, concave differentiable utility with finite expected utility at every holding, a gain having both signs makes the objective coercive at both ends. At a finite optimum its derivative is zero. <Secant domination for expected utility derivatives> gives <integrability> of $|X|U'(\theta_*X)$; boundedness of $U'$ on the remaining compact set gives <integrability> of the positive marginal utility itself. Its normalization has <expectation> one and prices the gain at zero.
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