Question

Aamir's company has the production function Q=8K^0.75L^0.25, where Q measures output, K measures machine hours, and L measures labor hours. Suppose that the rental rate of capital is R=$120, the wage rate is W=$20, and the firm wants to produce 800 units of output. Use the Lagrange method to find the optimal input mix. What the optimal level of K?

Answer 1

Cost minimizing problem will look like

Minimize 120*K+20^L

Subject to 8K^0.75L^0.25=800

Lagrange will look like

$L=120K+20L-\lambda (800-8K^{0.75}L^{0.25})$

First order conditions give us

$dL/dK=120-6K^{-0.25}L^{0.25}*\lambda =0$

or

120 = 6K-0.25 0.25* -------(1)

$dL/dK=20-2K^{0.75}L^{-0.75}*\lambda =0$

or

$20=2K^{0.75}L^{-0.25}*\lambda -------(2)$

$dL/d\lambda =(800-8K^{0.75}L^{0.25})=0$

or

$800=8K^{0.75}L^{0.25}-------(3)$​​​​​​

Divide equation 1 by equation 2, we get

3L/K=120/20

L/K=2

L=2K

Set L=2K in equation 3, we get

800=8K0.75(2*K)0.25=8*2^0.25K

K=100/2^0.25=84.0896 (Optimal capital)

L=2K=2*84.0896=168.1792 (Optimal labor)