= Solution
Use the buyer's <utility indifference price>: $\pi(Y)$ is the cash payment satisfying
$$
\sup_{X\in V}\mathbb E U(w_0+X)
=\sup_{X\in V}\mathbb E U(w_0-\pi(Y)+X+Y).
$$
Assume 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 $U$ gives
$$
\mathbb E U\big(w_0-[\theta p_1+(1-\theta)p_2]+X+\theta Y_1+(1-\theta)Y_2\big)\geq\theta M+(1-\theta)M=M.
$$
The 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 $\theta p_1+(1-\theta)p_2$. Thus
$$
\boxed{\pi(\theta Y_1+(1-\theta)Y_2)\geq\theta\pi(Y_1)+(1-\theta)\pi(Y_2).}
$$
The sign corresponds to buying the payoff. A seller's price uses a different convention.
Back to article page
