Intermediate Micro Question.

1. The production function of a rm is given by q = ALO'K, where q denotes output produced and K and L denote capital and labour used per period. Further, the wage and rental rates are denoted by w and 9", respectively. For this Cobb Douglas production function, the marginal product of labour is M PL = aAL"_1K, and the marginal product of capital is M PK 2 ALaK_1. Suppose oz 2 = %, and that in the short run the rm's capital is xed at the level K0. In this problem, think of w, r, A and K0 as given constants or numbers. Using this information, answer the following questions for a cost-minimizing rm. Show your work. (i) In the short run, how many units of L will this rm use in order to produce q units of output? (ii) Derive the short run total cost function of this rm (i.e. express short run total cost as a function of q). (iii) In the long run, how many units of L and K will this rm use in order to produce q units of output? (iv) Derive the long run total cost function of this rm (i.e. express long run total cost as a function of q)