Question

Suppose that a firm had a production function given by: q=L025K0 75 The wage rate ( $10 and the rental rate (r) is $5 Calculate the amount of labor the firm would hire when it produces 400 units of output in a cost-minimizing way (Round to the nearest 2 decimal places if necessary.) Answer:

Answer 1

A firm produces output in the cost minimizing way when it used that quantity of two inputs (Labor and capital) at which following condition is fulfilled -

MP_L/MP_K = P_L/P_K

Production function is as follows -

q = L^0.25K^0.75

Calculate MPL -

MPL = dq/dL

MPL = d(L^0.25K^0.75)/dL

MPL = 0.25K^0.75/L^0.25

MPK = dq/dK

MPK = d(L^0.25K^0.75)/dK

MPK = 0.75L^0.25/K^0.25

Cost minimizing condition is as follows -

MP_L/MP_K = P_L/P_K

(0.25K^0.75/L^0.25)/(0.75L^0.25/K^0.25) = 10/5

0.25K/0.75L = 2

K/3L = 2

K = 6L

Firm is producing 400 units of output.

So,

q = L^0.25K^0.75

400 = L^0.25K^0.75

400 = L^0.25(6L)^0.75

400 = 6^0.75 L^0.25 L^0.75

400 = 3.83 * L

L = 400/3.83 = 104.44

Thus,

The firm will use 104.44 units of labor.