Question

1. The production function of a firm is f(1,k) = Vlk where l is labor and k is capital/machinery. a. In the short run, if the quantity of capital is fixed at 64, derive the short run total cost SC(q), average cost SAC(q), and marginal cost SMC(q) of this firm. Assume each input costs $1 per unit. At what output does the minimum of SAC(q) occur? b. If labor and capital cost r and w respectively, and the quantity of capital is fixed at k, derive the short run total cost SC(q, r, w,k) of this firm.- C. Holding fixed r and w, if the firm was free to choose k, what k* would minimize the SC(q,r,w,k) of producing an output of q?- d. In the long-run the firm can vary k and set it to the k* (q, r, w) from (c). What is the long run total cost function, C(q, r, w) of the firm? e. Plot C(q,9,4) against q.- f. Prove that SC(q, r,w,k) > C(q,r, w) for a > 0. Hint: use the fact that a + b > 2 Vab for a, b 20.-

Answer 1

va) production function;, qu- veli In short rum, kE.GA ..y = 811 Then short run total cost: scca) = we trk. - 1 tk (as War:17 + sac case also you smc (9) = a/ will b) In short run: k=k 19. veka Alis = q² :. SCCa.) ano atrás

4 c) for minimization Foc requires olsc:69)}. bingolo in ok. wq² = rek? w q = fra 1 d) The long run cost function: cca, v, w) = win - wig + r w .a. = adwria e) The prot (Ca, 9 4) = 12 q depicted in (fig. 1)

Figure 1

C(q)|400 LRTC 300- 200- -100- 20 30 50 Output

Y SC rk sccaja nice jerk · Wa? * r ** wraza. ?*? - [cca, r, w)]*+ rmanente - [<caorg m) In tā 22 rh X secair, w; e) > .cca, r, w) (proved)