Question

Suppose the Isoquant and Isocost curves of a manufacturing company are given as Q=4K^0.5 L^0.5 and 4K + 8L= 320 respectively, where r is the price of capital and w refers to the price of labour. How many units of L and K the firm should employ to maximize profit?

Answer 1

Solution:

The profit-maximizing rule is the price of input resource must be equal to its marginal revenue product.

Q = 4K^0.5L^0.5

$\frac{MP_{L} }{w} = \frac{MP_{K} }{r} = \frac{MP_{L} }{MP_{K}} = \frac{w }{r}$

MP_L = $\frac{\partial L} {\partial Q}$ = 2K^0.5L^-0.5

MP_K = $\frac{\partial K} {\partial Q}$ = 2K^-0.5 L^0.5

$\frac{MP_{L} }{MP_{K}} = \frac{w }{r}$

= $\frac{2K^{0.5} L^{0.5}}{2K^{-0.5} L^{0.5}} = \frac{8 }{4}$

=$\frac{K }{L} = \frac{8 }{4}$

K = 2L

Plug into isocost line:

320 = 4K + 8L

320 = 4(2L) + 8L

320 = 8L + 8L

320 = 16L

L = 20

K = 2L = 2$\times$20 = 40

 

The units of L and K the firm should employ to maximize profit are = 20 and 40