 Solution

Let $u(x)$ be maximal expected utility without the claim. The utility indifference price $\pi (Y)$ is determined byFor claims $Y_1,Y_2$, mix optimal portfolios for their indifference problems with weights $\lambda$ and $1-\lambda$. Concavity of $U$ shows that buying $\lambda Y_1+(1-\lambda )Y_2$ at pricegives expected utility at least $u(x)$. Since maximal utility decreases with purchase price,Thus $\pi$ is concave.