Question

The firm’s production function is given by q = 4K^0.5L^0.5, where q denotes the output (measured as the number of research reports per month). The firm hires people (“labor”, measured in hours of work) and rents office space (“capital”, measured in sq. feet). The marginal product of labor is given by MP_L = 2K^0.5L^–0.5 and the marginal product of capital is given by MP_K = 2L^0.5K^–0.5.

^1. Find the cost-minimizing levels of capital (K*) and labor (L*) required to produce the target level of output of q_o = 480 if the cost of capital is r = $40/sq foot and the wage rate is w = $10/hour.

2.

Find the cost-minimizing levels of capital (K*) and labor (L*) required to produce the target level of output of q_o = 480 if the cost of capital is r = $40/sq foot and the wage rate is w = $10/hour. What is the firm’s total cost of producing 480 units of output?

3.

Find the cost-minimizing levels of capital (K*) and labor (L*) required to produce the target level of output of q_o = 480 if the cost of capital is r = $40/sq foot and the wage rate is w = $10/hour. What is the firm’s average cost of producing 480 units of output?

Answer 1

1)

q = 4K^0.5L^0.5  

MP_L =  2K^0.5L^–0.5

MP_K = 2L^0.5K^–0.5  

MRTS =  2K^0.5L^–0.5 / 2L^0.5K^–0.5  

= K/L

cost minimising condition is

MRTS = w/r

K/L = 10/40

K/L = 1/4  

L = 4K

Put L = 4K in production function

q = 4K^0.5L^0.5  

q_o =   4K^0.5L^0.5  

q_o = 4K^0.5(4K)^0.5

q_o = 4K^0.5(2K^0.5)  

q_o = 8K

q_o/8 = K

K = q_o/8

L = 4K

L = 4(q_o/8)

L = q_o/2

q_o =  480  

K = 480/8 = 60  

L = 480/2 = 240

Thus K* = 60

L* = 240

2)  

Total cost C = wL + rK  

= 10(240) + 40(60)  

= 2400 + 2400

= 4800

3)

Average cost AC = C/q  

= 4800/q  

= 4800/480  

= 10