Question #12070

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.

Expert's answer

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

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

40=16K

K=2.5

L=10

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