Answer to Question #252601 in Microeconomics for West

Question

 Suppose a firm produces according to the production function Q = 2L0.6K0.2, and faces wage rate ₵10, a rental cost of capital ₵5, and sells output at a price of ₵20.

a. Obtain and expression for the factor demand functions.

b. Compute the profit-maximizing factor demands for capital and labour.

Answer 1

Solution:

a.). Expressions for the factor demand functions are as follows:

TC = wL + rK = 10L + 5K

TR = P $\times$ Q = 20(2L^0.6K^0.2)

Profit = PQ – wL – rK = 20(2L^0.6K^0.2) - 10L - 5K

b.). Profit = PQ – wL – rK = 20(2L^0.6K^0.2) - 10L - 5K

$\frac{\partial \pi } {\partial L} =$ 24L^-0.4K^0.2 – 10 = 0         (i)

 

$\frac{\partial \pi } {\partial K} =$ = 8L^0.6K^-0.8 – 5 = 0           (ii)

First, solve for equation (i) for L:

24L^-0.4K^0.2 = 10

$\frac{24K^{0.2} }{L^{0.4} } = 10$

$24K^{0.2} = 10L^{0.4}$

$2.4K^{0.2} = L^{0.4}$

L = 8.9K^0.5

Plug into equation (ii):

8(8.9K^0.5)^0.6K^-0.8 = 5

29.7K^0.3K^-0.8 = 5

29.7K^-0.5 = 5

K = 35.28

Substitute to derive L:

L = 8.9K^0.5

L = 8.9(35.28^0.5) = 52.86

L = 52.86

Q = 2L^0.6K^0.2

Q = 2(52.86^0.6) (35.28^0.2) = 21.62 $\times$ 2.04 = 44.10

Q = 44

Derive Profit:

Profit = 20(44) – 10(52.86) – 5(35.28)

Profit = 880 – 528.6 – 176.4

Profit = 175