Question

In: Economics

Suppose there are two inputs in the production function, labor and capital, which are substitutes. The...

  1. Suppose there are two inputs in the production function, labor and capital, which are substitutes. The current wage is $10 per hour and the current price of capital is $25 per hour.
  1. Given the following information on the marginal product of labor and the marginal product of capital, find the firm’s profit-maximizing input mix (i.e. number of workers and units of capital) in the long-run. Show your work and explain.

L

MPL

K

MPK

1

125

1

130

2

100

2

125

3

75

3

120

4

50

4

115

5

25

5

110

  1. Again, the wage is $10, the cost of capital is $25 and now firms want to spend $1000 to produce an optimal Q*. Graph the firm’s isocost line under these conditions and show the cost-minimizing level of labor and capital needed to produce Q*. (note: I am looking for a general L* and K*, you won’t be able to find specific numbers. You do need specific numbers for the isocost intercepts.)

Solutions

Expert Solution

Part a) We have the following information

Wage (w) = $10

Price of capital (r) = $25

L

MPL

K

MPK

MPL/MPK

w/r

1

125

1

130

           1.0

0.4

2

100

2

125

           0.8

0.4

3

75

3

120

           0.6

0.4

4

50

4

115

           0.4

0.4

5

25

5

110

           0.2

0.4

In the above L = Labor, K = Capital, MPL = Marginal product of labor, and MPK = Marginal product of capital.

The equilibrium condition (or the lowest cost combination) is that

MPL/MPK = w/r

So, the profit maximizing input mix is 4 labor and 4 capital.

Part b) Now it is given that the budget of the firms is $1000. Given the input prices the budget constraint will look like the following.

1000 = wL + rK

1000 = 10L + 25K

25K = 1000 – 10L

K = 40 – 0.4L (Equation of the isocost line)

When K = 0, L = 100

When L = 0, K = 40

The equilibrium looks like the following


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