 Solution

For every deflator $Y$ and $\epsilon \ge 0$, $Y^{\ast }+\epsilon Y$ is a deflator. Minimality and right differentiation at zero giveTaking $Y=Y^{\ast }$ and varying the positive scalar multiple $(1+\epsilon )Y^{\ast }$ in both directions around one gives equality.

Set ${\xi }_1=-{\hat{u}}^{\prime }(Y_1^{\ast })$ and ${\xi }_0=X_0$. The preceding inequality saysfor every deflator, with equality at $Y^{\ast }$. Part c produces $H^{\ast }$ with $H^{\ast }\cdot P_0=X_0$ and $H^{\ast }\cdot P_1={\xi }_1$. The inverse relation between conjugate derivatives gives $u^{\prime }({\xi }_1)=Y_1^{\ast }$, so equality holds in part d: