Question

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

2. 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.)

Answer 1

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