Question

Suppose a firm's marginal product of capital and marginal product of labor schedules are as shown in the table below. The firm hires both capital and labor competitively for $4 and $8, respectively. Its output is sold in a competitive market for $.50 per unit.

Capital	MP of Capital	Labor	MP of Labor
0		0
1	10	1	28
2	9	2	30
3	8	3	24
4	7	4	20
5	6	5	16
6	5	6	12
7	4	7	8
8	3	8	4

1. Suppose the firm is currently using 4 units of capital and 4 units of labor. Is the corresponding output being produced at least cost? How do you know?
   
   
   
   
2. What combination of labor and capital should the firm use to maximize its profit? Show.
   
   
   
   
3. If the marginal product of each resource does not depend on the quantity of the other resource, what output level corresponds to the input combination you just found? Show how.
   
   
   
4. Does this combination produce this output level at least cost?

Answer 1

a. The corresponding output is not least cost because for least cost the factors should paid according to their marginal product at that output level. Here at this output level K=4 units, L=4 the marginal product of capital and labour is 7 and 20. But factor price is 4 and 8. So factors are paid less than their marginal product. As there is two factors ratio of marginal product must be same with ratio of factor price. At this output ratio of MP = MP of K/ MP of L = 7/20. Ratio of their prices 4/8. These two are not equal. Therefore it is not least cost.

b. For maximise profit firm should use combination of capital and labour so that their marginal product is just equal to their factor price. This situation take place at the use of K= 7 and L =7. Because at this level of capital and labour ratio of marginal product is 4/8. The ratio of price paid to factors also 4/8. So the combination K =7 and L =7 is the profit maximising.

c. The output level corresponds to the input combination K =7 and L =7 will be the total output produced at this level of capital and labour use TP of capital upto K =7 is 10+9+8+7+6+5+4 =49. Total product of labour at this level of labour use i.e L =7 is 28+30+24+20+16+12+8 = 138. But if output level is made by combination of capital and labour this will be not be the output. It depends on technology. If output function is homogeneous production function of degree 1 then output will be 7. It means 1 unit K and 1 unit of L produce 1 unit of Q. Actually it depends on production function.

d. Yes this combination is produced at least cost. Because here factors are paid according to their marginal product.