Answer to Question #280413 in Microeconomics for Geoffroy

Question

The production for a productive is given by q=100KL. If the Price of capital is 120€ per day and the Price of labor 30€ per day, what is the minimum cost of producing 1000 units of output

Answer 1

The cost-minimizing combination of capital and labor is,

MRTS = MPL/MPK = w/r

The marginal product of labor is dq/dL = 100K. The marginal product of capital is dq/dK =100L. Therefore, the marginal rate of technical substitution is 100K/100L = K/L.

K/L = 30/120 (or) L = 4K

Substitute for L in the production function and solve where K yields an output of 1,000 units:

1,000 = (100)(K)(4K), or K = 1.58.

L equals 4K, this means L equals 6.32(4*1.58)

With these levels of the two inputs, total cost is:

TC = wL + rK

= (30)(6.32) + (120)(1.58) = 379.20€

Next, consider small changes in K and L. At K = 1.6 and L = 6.32, total cost would be 381.60€, and at K = 1.58 and L = 6.4, total cost would be 381.6€, both values are greater than 379.20€. Hence, we have found the cost-minimizing levels of K and L.