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Consider a price-taking firm that produces widgets with only labour input. Let the relation between widget...

Consider a price-taking firm that produces widgets with only labour input. Let the relation between widget output and labour input be the function f(z), where z is labour input. Denote the price of widgets by p and the wage rate (the price of labour) by w; assume both p and w are positive and beyond the firm’s control. Assume the firm chooses labour input to maximize profits. Denote the labour demand function by z(w, p), the output supply function by y(w, p), and the value function for this problem, that is, the highest level of profits possible with prices (w, p) — by Π(w, p) = py(w, p) − wz(w, p). Prove Π(w, p) is convex in w and p.

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