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Problem 13.1 - Resource Demand
Problem: Suppose a firm's short-run total product schedule is given in the table below. It sells its output competitively for $1.50 each. Labor | Total Product | Marginal Product | Marginal Revenue Product | 0 | 0 | | | 1 | 8 | | | 2 | 18 | | | 3 | 29 | | | 4 | 39 | | | 5 | 47 | | | 6 | 52 | | | 7 | 53 | | | 8 | 53 | | |
- What is the marginal product of the 1st worker?
- What is the marginal revenue product of the first worker?
- Suppose the wage is $7. How many workers will this firm hire?
- If the wage rises to $9, how will the firm adjust its employment?
- Alternatively, suppose the firm sells its output according to the following demand schedule:
Labor | Total Product | Product Price | Marginal Product | 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: - 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.
- Marginal revenue product is the increase in total revenue associated with the next worker. For a competitive, this is product price times marginal product. The marginal revenue product of the first worker is $12. $12 = $1.50 x 8.
- 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. - The 6th worker is no longer profitable. Reducing employment revenue. Maximum profits requires 5 workers.
- The completed table is shown below:
Labor | Total Product | Product Price | Marginal Product | 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.
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Problem 25.2 - Optimal combination of resources 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 competitively for $.50 each. 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 | 9 | 4 |
- 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?
- What is the profit-maximizing combination of labor and capital the firm should use?
- What output level corresponds to the input combination you just found? Is this the least-costly combination of labor and capital to produce this level of output?
| Answer: - No. The marginal product of the fourth unit of capital is 7, and its marginal product per dollar is 1.75 = 7/$4. The marginal product of the fourth worker is 20, and her marginal product per dollar is 2.50 = 20/$8. Labor is more productive per dollar, implying that the same output level could be produced more cheaply using relatively more labor and less capital.
- 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 of these should be equal to their respective prices (MRPK/PK = MRPL/PL = 1). 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 | 9 | 0.375 | 0.25 |
For example, the MRP of the first unit of capital is PxMPK = $.50x10 = $5. Since the price of capital is $4, MRPK/PK = 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. - Total output is found by accumulating the marginal products of capital and labor: Output = 7+9+8+28+30+24+20+16 = 142. Since the marginal product per dollar of each input is the same, this is the least cost combination of resources.
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