= Marginal utility pricing with proportional transaction costs
{title2=$\mathbb EZ=1,\quad\mathbb E[XZ]\in[-\varepsilon,\varepsilon]$}
At a nonzero optimum, the derivative of expected utility of a <proportional transaction cost> payoff is zero. Normalized marginal utility therefore prices $X$ at $\varepsilon$ for a positive optimum or $-\varepsilon$ for a negative optimum. At a zero optimum the one-sided derivative inequalities put $\mathbb EX$ 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.
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