Solution
= Solution
Under a <proportional transaction cost>, the sample gain $h_x(\theta)=\theta x-\varepsilon|\theta|$ is concave when $\varepsilon\geq0$. The <utility function> is increasing and concave, so
$$
U(h_x(s\theta+(1-s)\eta))\geq U(sh_x(\theta)+(1-s)h_x(\eta))\geq sU(h_x(\theta))+(1-s)U(h_x(\eta)).
$$
Take finite <expectations>. \b[Thus $\boxed{G\text{ is concave}}$.] The monotonicity of $U$ is essential to composing it with the concave transaction-cost payoff.