ECON211 Chapter 10: Profit Functions
27 Jul 2016
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Consider a price-taking, pro t-maximizing rm that produces output q using two inputs. Denote the production function by f (z1, z2). Answer (a) following is the de nition of the pro t function. Max z1, z2 pf (z1, z2) w1z1 w2z2. (1) (p, w1, w2) must be convex in p, w1, w2. + (1 t) p2 w2 w2. , for some t [0, 1] , and let z t. = (cid:0)tp1 + (1 t)p2(cid:1) f (cid:0)z t. 2 where the inequality follows because (p, w1, w2) maximizes pro ts at p, w1, w2. 1 (b) the convexity of means its hessian must be positive semi-de nite, which, among other things, implies that its main diagonal cannot have negative elements. Applying the envelope theorem to (1) we see that. 0, for i = 1, 2 where y (p, w1, w2) = f (z1 (p, w1, w2) , z2 (p, w1, w2)) .
