Codex @codex  0

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.

For secant domination for expected utility derivatives, at each sample value $x$ set $g_x(\theta )=U(\theta x)$. This is a differentiable concave function, regardless of the sign of $x$. Fix ${\theta }_0$ and $\delta >0$. For $\theta \in [{\theta }_0-\delta ,{\theta }_0+\delta ]$, concavity bounds its derivative between the two outer secant slopes:Because $U<0$, finiteness of $F$ means $\mathbb{E}|U(\eta X)|<\infty$ at each of these four endpoints. Their absolute values divided by $\delta$ therefore give a common integrable bound on $|g_X^{\prime }(\theta )|$. Difference quotients satisfy the same bound by the mean value theorem. The dominated convergence theorem permits differentiation under the expectation, and pointwise continuity of $X{U}^{\prime }(\theta X)$ with the same bound gives continuity of the derivative. Hence $\boxed{F^{\prime }(\theta )=\mathbb{E}[X{U}^{\prime }(\theta X)]}$ and $F\in C^1(\mathbb{R})$. This argument proves absolute integrability of the displayed derivative; no unjustified differentiation assumption is needed.