Optimal marginal utility as a one-period pricing density ($Z=U^{\prime }({\theta }_{\ast }X)/\mathbb{E}[U^{\prime }({\theta }_{\ast }X)]$)

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 $|X|U^{\prime }({\theta }_{\ast }X)$; boundedness of $U^{\prime }$ on the remaining compact set gives integrability of the positive marginal utility itself. Its normalization has expectation one and prices the gain at zero.

At a nonzero optimum, the derivative of expected utility of a proportional transaction cost payoff is zero. Normalized marginal utility therefore prices $X$ at $\epsilon$ for a positive optimum or $-\epsilon$ for a negative optimum. At a zero optimum the one-sided derivative inequalities put $\mathbb{E}X$ inside that interval, so density one works. The competing one-sided-payoff alternative prevents the coercive-maximizer argument from being assumed when a net gain has only one sign.