= Solution
Let $u(x)$ be maximal expected utility without the claim. The <utility indifference price> $\pi(Y)$ is determined by
$$
\sup_\varphi\mathbb E\,U\!\left((x-\pi(Y))(1+r)+G_\varphi+Y\right)=u(x).
$$
For 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 price
$$
\lambda\pi(Y_1)+(1-\lambda)\pi(Y_2)
$$
gives expected utility at least $u(x)$. Since maximal utility decreases with purchase price,
$$
\pi(\lambda Y_1+(1-\lambda)Y_2)
\geq\lambda\pi(Y_1)+(1-\lambda)\pi(Y_2).
$$
\b[Thus $\pi$ is concave.]
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