Answer to Question #169062 in Microeconomics for Riza

Question

A cob Douglas production function for a firm is given as Q=4L ½K½. The firm has also established that wage rate and interest paid on capital are $3 and $5 respectively for a production period. The firm intents to spend $200 million for the period on production cost. Compute the levels of capital and labor that will maximize output. What is the maximum output?

Answer 1

Solution:

Cost function: 200,000,000 = 3L + 5K

MP_L = dQ/dL = 2L^0.5-1K^0.5 = 2L^-0.5K^0.5

MP_K = dQ/dK = 0.5L^0.5K^0.5-1 = 0.5L^0.5K^-0.5

MP_L/MP_K = w/r

MP_L/w = MP_K /r

2L^-0.5K^0.5 / 3 = 0.5L^0.5K^-0.5 / 5

K = 15L

Plug into cost constraint:

200,000,000 = 3L + 5K

200,000,000 = 3L + (5x15L)

200,000,000 = 3L + 75L

200,000,000 = 78L

L = 2,564,102

K = 15L = (15x2564102) = 38,461,530

Maximum output = K + L = 38,461,530 + 2,564,102 = 41,025,632