= Secant domination for expected utility derivatives
If a negative differentiable concave <utility function> satisfies $\mathbb EU(\theta X)>-\infty$ on an open interval, outer secants at four points bound the sample derivative on a smaller compact interval. Negativity makes the endpoint values absolutely integrable. The <dominated convergence theorem> then gives $d\mathbb EU(\theta X)/d\theta=\mathbb E[XU'(\theta X)]$ and continuous derivative when $U'$ is continuous. This is a justification of differentiation, rather than an assumption of <integrability> of marginal utility.
Back to article page
