Answer to Question #280371 in Economics of Enterprise for mari

Question

Suppose you have the following production function: Q = f (L, K) = 10L ^½ K^½. In addition, the price of labor is $1 and the price of capital is $4

a. What is the optimal amount of labor and capital if you want to produce 20 units?

b. What is the level of minimum cost ?( Ans L=4 and K=1,Min C=$8)

Answer 1

Solution:

a.). Cost minimization:

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

TC = 10L ^½ K^½

MP_L = $\frac{\partial Q} {\partial L}$ = 5L^-0.5 K^0.5

MP_K = $\frac{\partial Q} {\partial K}$ = 5L^0.5 K^-0.5

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

w = 20

r = 80

$\frac{5L^{-0.5} K^{0.5} }{5L^{0.5} K^{-0.5}} = \frac{20 }{80}$

$\frac{K }{L} = 0.25$

K = 0.25L

Q = 10L ^½ K^½

20 = 10(L^0.5) (0.25L^0.5)

L = 8

K = 0.25L = 0.25(8) = 2

The optimal amount of labor and capital to produce 2000 units (L,K) = (8, 2)

Labor = 8

Capital = 2

b.). Minimum cost:

C = wL + rK

C = (20 $\times$ 8) + (80 $\times$ 2) = 160 + 160 = 320

Minimum cost = 320