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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.

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.