 Solution

Use the buyer's utility indifference price: $\pi (Y)$ is the cash payment satisfyingAssume the utilities are well-defined, the finite indifference price exists, and the suprema are attained as allowed. Write the common reference value as $M$. For claims $Y_1,Y_2$, choose optimizers $X_1,X_2$ at their indifference prices $p_1,p_2$. Because $V$ is a vector space, $X=\theta X_1+(1-\theta )X_2$ is admissible. The concavity of $U$ givesThe optimized utility decreases with the payment; strict increase of $U$ and attainment make the comparison strict whenever a smaller payment is substituted into an attained optimizer. Hence the payment restoring utility $M$ is at least $\theta p_1+(1-\theta )p_2$. ThusThe sign corresponds to buying the payoff. A seller's price uses a different convention.