Assume $U(x,y,a,c )= - c x + B(x,y,a)$, with $\frac{\partial B(x,y,a)}{\partial c }=0$, and with $a$ and $c\geq 0$ being parameters, and with $x$ and $y$ being variables. Further, $B(x,y,a)$ is continuous and twice continuously differentiable in all its inputs. Further, the $(x,y)$ maximizing $U(x,y,a,c )$ is unique everywhere and it is continuous in $a$ and $c$.

Suppose that $(x_1,y_1)$ maximizes $U(x,y,a_1,c )$, and $(x_2,y_2)$ maximizes $U(x,y,a_2,c )$, and that $0\leq x_1<x_2$.

Let $U^*(a):=\underset{x,y}{\max} U(x,y,a,c)$.

To show: $\frac{\partial U^*(a_1)}{\partial c } > \frac{\partial U^*(a_2)}{\partial c } $