Consider the production function f(l, k) = Ala kB where A > 0, a > 0, B > 0 and a + B

Questions: 1. E. [12 marks] Consider the producon function Hi, k) = Al\" 1:3 where A > 0, or > 0.. )3 > D and a + ,6' c: 1 and where l and k denote labour and capital, respectively. Let w, r and p denote the prices of labour. capital and output, respectively, and let x denote output. a. Solve the prot maximization problem to show that the factor demand and supply functions are: 1 Algal-13,313 W {{W. Til-1) = [W] Apaal-a MMP) = [w] b. Solve the cost minimization problem to show that the conditional factor demand functions are: 1 mm = (gringo a 1 oars) = (iii)m (i)m or A Show that all cost-minimizing production plans line on a common ray from the origin in k4 space. Show that the cost function is: 1 xwar3]m C(w,r,x) = (a: +3) Aaa Derive the average cost function Aw, r. x) and marginal cost function MC(w, r, 2:). Use the results ofPart [c] to derive the supply function x(w. rap), and then use this along with the results of Part [b] to derive the factor demand functions i(w. r. p) and MW, no). {Hint To verify correctness, check that the functions derived in this part are the same as those derived in Part [a]}