the marginal product of labour function for International trading Inc. is given by the equation MPL= 10K^0.5/L^0.5 Currently the firm is using 100 units of capital and 121 units of labour. Given the very specialized nature of the capital equipment, it takes six to nine months to increase the capital stock, but the rate of labour input can be varied daily. If the price of labour is $10 per unit and the price of output is $2 per unit, is the firm operating efficiently in the short run? if not why and determine the optimal rate of labour input.
As MPL = Q'(L), so Q = 20K^0.5*L^0.5, MPK = Q'(K) = 10L^0.5/K^0.5. If K = 100 units, L = 121 units, then: MPL = 10*100^0.5/121^0.5 = 100/11 = 9.09, MPK = 10*121^0.5/100^0.5 = 121/10 = 12.1. Firm is operating efficiently, if MPK/MPL = k/w. If w = $10 per unit, then 12.1/9.09 = k/10, k = 13.31 and the firm is operating efficiently in the short run.