Suppose a firm has the production function given by the Cobb-Douglas function: q=ALK (Where ,>0 ), and that the company can hire all the labor and buy all the capital it wants in a competitive market at costs " w " and " r ", respectively. a) Prove using Lagrangian optimization analysis, that cost minimization requires to hold: wL=rK What is the shape of the expansion path for this company? b) Now assume that A=2,==1/2, and that capital is fixed at K=KO in the short run. Compute the firm's total cost as a function of q,w,r, and KO. c) Given q,r, and w, how should work (L) be chosen to minimize total cost? And the capital (K) ? (Hint: It is about determining the conditional demands of the productive factors). d) Use your results from part (c) to calculate the total cost of production in the long run