Answer to Question #12070 in Microeconomics for Temesgen

Question

the production function of the firm is 411/2k1/2. the cost of labor is $40 per unit and the cost of using capital is $ 10 per unit determine the amount of labor and capital that should be used in order to minimize the cost to produce 40 units of output.

Answer 1

The cost-minimizing combination of capital and labor is the one where
MRTS=MPl/MPk
Marginal product of labor is 4K
The marginal product of capital is 4L
The marginal rate of technical substitution is
4K/4L= K/L
To determine the optimal capital-labor ratio set the marginal rate of technical substitution equal to the ratio of wage rate to rental rate of capital
K/L=10/40
L=4K
Substitute for L in the production function and solve where K yields an output of 40 units
40 = 4*4K^1/2*K^1/2
40=16K
K=2.5
L=10