Let f(z) be the production function for a price-taking firm that produces one output with only one variable input.

Let f(z) be the production function for a price-taking firm that produces one output with only one variable input. Denote the price of output by p and the input price by w, so firm profits are pf(z) − wz. Assume f 0 (z) > 0, f 00(z) < 0 and profits attain a maximum at some positive and finite value of z for any positive values of w and p. For any pair (w, p) denote the highest level of profits possible by Π(w, p). Prove Π(w, p) is convex in (w, p).