Answer to Question #319992 in Microeconomics for Pat

Question

The marginal product of labour function for International Trading Inc. is given by the equation

MP_L = 10 K^0.5/ L^0.5

Currently, the firm is using 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, explain why, and determine the optimal rate of labour input.

Answer 1

As $MPL = Q'(L)$ ,

So $Q = 20K^{0.5}\times L^{0.5}$

$MPK = Q'(K) = 10L^{0.5}K^{0.5}$

If K = 100 units, L = 121 units, then;

$MPL = \frac{10\times100^{0.5}}{121^{0.5}}= 100/11 = 9.09$ ,

$MPK = \frac{10\times121^{0.5}}{100^{0.5}} = 121/10 = 12.1.$

For the firm to operate efficiently, $\frac {MPK}{MPL} = \frac{k}{w}$ .

If w = $10 per unit, then

$\frac{12.1}{9.09} = \frac{k}{10}$ ,

$k = 13.31$

The firm is therefore operating efficiently in the short run