Marginal utility pricing with proportional transaction costs ($\mathbb{E}Z=1,\mathbb{E}[XZ]\in [-\epsilon ,\epsilon ]$)

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.