I have to show that the following constrained profit function, $\pi$, is convex.
A firm choses non-negative quantity, $x$, of inputs with price $p$ and non-negative quantity, $y$, of outputs with price $q$ subject to a production possibility constraint $G(x,y) \leq 0$.
$$ \pi (q, p) = \max_{(x, y) \in \mathbb{R}^2_+}{qy - px}$$
subject to the following production possibility constraint: $$ G(x,y) \leq 0 $$
How would you proceed?
Let: \begin{align} \pi&=f\circ g \\ f(x,y)&=\max\{g(x,y)\} \\ g(x,y)&=g_1(x)+g_2(y) \\ g_1(y)&=qy \\ g_2(x)&=-px \end{align} Then $g_1,g_2$ are linear and therefore convex. $g(x,y)$ is convex as the sum of convex functions.
Finally, $\pi(q,p)$ is convex as the maximum of a collection of convex functions is convex. In our case this collection is a single function.