Question

Suppose you have the following production function: Q =f(L,K)=10L^1/2K^1/2 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.). The optimal amount of labor and capital is where: $\frac{MP_{L} }{MP_{K} } = \frac{w}{r}$

Q = 10L^0.5K^0.5

MP_L = 5L^-0.5K^0.5

MP_K = 5L^0.5K^-0.5

w = 1

r = 4

$\frac{5L^{-0.5}K^{0.5} }{5L^{0.5}K^{-0.5}} = \frac{1}{4}$

L = 4K

20 = 10L^0.5K^0.5 = 10(4K^0.5) K^0.5

K = 1

L = 4K = 4 $\times$ 1 = 4

Labor = 4 units

Capital = 1 unit

 

b.). Minimum cost is where: C = wL + rK

C = (1 $\times$ 4) + (4 $\times$ 1) = 4 + 4 = 8

The minimum cost = 8