Problem:

Suppose a firm's short-run total product schedule is given in the table below. It sells its output in a competitive market for $1.50 per unit.

1. What is the marginal product of the first worker?
2. What is the marginal revenue product of the first worker?
3. Fill in the rest of the table. How many workers will this firm hire at a wage of $7?
4. If the wage rises to $9, how will the firm adjust its employment?
5. Alternatively, suppose the firm sells its output according to the following demand schedule:
   
   

   Labor	Total Product	Product Price	Total Revenue	Marginal Revenue Product
   0	0	----
   1	8	$3.50
   2	18	2.80
   3	29	2.30
   4	39	1.90
   5	47	1.65
   6	52	1.50
   7	53	1.40

   
   Fill in the remaining two columns of the table. How many workers will be hired at a wage of $7?

Answer:

1. Marginal product is the addition to total output associated with the next worker. Total output rises from 0 to 8 with the addition of the first worker, so the marginal product is 8.
2. Marginal revenue product is the increase in total revenue associated with the next worker. For a competitive firm, this is product price times marginal product. The marginal revenue product of the first worker is $1.50 x 8 = $12.
3. To determine the profit-maximizing level of employment, it is necessary to find marginal revenue product and compare it to the wage rate. The completed table is below:
   
   

   Labor	Total Product	Marginal Product	Marginal Revenue Product
   0	0
   1	8	8	$12.00
   2	18	10	15.00
   3	29	11	16.50
   4	39	10	15.00
   5	47	8	12.00
   6	52	5	7.50
   7	53	1	1.50
   8	53	0	0.00

   
   Maximum profits are obtained by hiring only those workers whose marginal revenue products exceed the wage. In this example, 6 workers are hired.
4. The 6^th worker is no longer profitable. Reducing employment by 1 worker would save $9 in wages and "costs" the firm only $7.50 in lost revenue. Maximum profits are obtained with 5 workers.
5. The completed table is shown below: Total revenue is product price times total product and marginal revenue product is the change in total revenue from hiring an additional worker.
   
   

   Labor	Total Product	Product Price	Total Revenue	Marginal Revenue Product
   0	0	----	----	----
   1	8	$3.50	$28.00	$28.00
   2	18	2.80	50.40	22.40
   3	29	2.30	66.70	16.30
   4	39	1.90	74.10	7.40
   5	47	1.65	77.55	3.45
   6	52	1.50	78.00	.45
   7	53	1.40	74.20	–3.80

   
   Comparing MRP to the wage, the firm maximizes profits by hiring 4 workers.

Problem:

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.

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?
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? Does this combination produce this output level at least cost?

Answer:

1. No. The marginal product of the fourth unit of capital is 7, and its marginal product per dollar is 7/$4 = 1.75. The marginal product of the fourth worker is 20, and her marginal product per dollar is 20/$8 = 2.50. Per dollar of expenditure, labor is more productive, implying that the same output level could be produced more cheaply using relatively more labor and less capital.
2. Find the marginal revenue products of both capital and labor by multiplying their respective marginal products by the output price. At the profit-maximizing output level, each input's marginal revenue product should be equal to its respective price. That is, MRP_K/P_K = MRP_L/P_L = 1 at the optimal combination of labor and capital. The relevant table is reproduced below:
   
   

   Capital	Labor	MRPK/PK	MRPL/PL
   0	0
   1	1	1.25	1.75
   2	2	1.125	1.875
   3	3	1	1.5
   4	4	0.875	1.25
   5	5	0.75	1
   6	6	0.625	0.75
   7	7	0.5	0.5
   8	8	0.375	0.25

   
   For example, the MRP of the first unit of capital is PxMP_K = $.50x10 = $5. With the price of capital at $4, MRP_K/P_K = 5/4 = 1.25. Since this is greater than 1, the firm will expand its use of capital. The firm should use 3 units of capital and 5 units of labor.
3. Under the assumed independence, total output can be found by accumulating the marginal products of capital and labor: Output = 10+9+8+28+30+24+20+16 = 145. Since the marginal product per dollar of each input is the same, this is the least-cost combination of resources.