Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 30K / a / Solution

Let $\mathcal X$ be a vector space of random variables and suppose
$$
F(y)=\sup_{X\in\mathcal X}\mathbb E[U(X+y)]
$$
has an optimizer for every $y$. If $U$ is increasing and concave, then $F$ is increasing and concave. For concavity, combine optimizers at two values using the same convex coefficient as the values themselves.


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 Incoming links: Optimized affine shift of concave utility