Question

A production function has the following form Q = 4KL. Price of labor is PL = 8 and price of capital is PK = 4. How many units of labor and capital the firm hires, if it minimizes its costs and its goal is to produce 800 units of output? Calculate total costs.

Answer 1

Solution:

Cost minimization is where: MP_L/MP_K = w/r or MP_L/w = MP_K/r

Q = 4KL

MP_L = $\frac{\partial Q} {\partial L} = 4K$

MP_K = $\frac{\partial Q} {\partial K} = 4L$

w = 8

r = 4

$\frac{4K} {4L} = \frac{8} {4}$

K = 2L

Substitute in the production function:

Q = 4KL

800 = 4(2L) L

800 = 8L²

L² = 100

L = 10

K = 2L = 2$\times$10 = 20

K = 20

The firm will hire 10 units of labor and 20 units of capital.

 

TC = wL + rK

TC = 8(10) + 4(20) = 80 + 80 = 160

Total Costs = 160