 Solution

For fixed $\gamma$, differentiation under the integral sign is justified by bounded $P$ and $X$. At the unconstrained minimizer $H_{\gamma }$, the first-order condition isDefine the strictly positive, bounded pricing kernelThe first-order condition says $\mathbb{E}[Z_{\gamma }P]=p$, so $Z_{\gamma }\in \mathcal{Z}$. When taking the requested partial derivative, hold $H$ fixed and only afterwards set $H=H_{\gamma }$. Writing $A_{\gamma }=H_{\gamma }\cdot p-x$, we obtainThe assumed dual bound applies to this particular pricing kernel. ThusNo derivative of the map $\gamma \mapsto H_{\gamma }$ is needed; confusing a partial derivative with a derivative along the minimizing path would add an unnecessary hypothesis.