(Prot maximization and Hotelling Lemma) Consider the problem of maximizing prots for the Cobb-Douglas production function in the form: q = f (z) = z 13 . (a) What returns to scale does this technology exhibit? Derive the inputdemand function z (p; w) and the supply function q (p; w). Represent the solution (q ; z) graphically at some given price vector (p ; w). (b) Calculate explicitly the prot function (p; w) and verify that it is homogeneous of degree one and convex in (p; w). (c) Represent graphically (p; w) a "passive prot function" (p) = pq wz resulting from varying the output price p but maintaining xed the input price w and the rms optimal production plan (q ; z) at (p ; q). Using the obtained gure, argue that the Hotellings Lemma holds. Show what happens if the output price increases from p to (p)0