AP® Microeconomics review sheet from Aim for Five (aimforfive.com/micro/units/5/5-3)
Unit 5 · Topic 5.3
5.3 Profit-Maximizing Behavior in Perfectly Competitive Factor Markets
A firm hiring in a perfectly competitive labor market is a wage taker: it can hire all the workers it wants at the market wage. It maximizes profit by hiring until marginal revenue product equals the wage. When it uses more than one input, it chooses the mix that gets the most output per dollar, and the amounts that make each input's MRP equal its price.
Key terms
- wage taker
- marginal factor cost
- MRP = MFC
- least-cost rule
- profit-maximizing input combination
The wage taker: market and firm side by side
In a perfectly competitive labor market, there are many firms hiring the same kind of worker, and no single firm's hiring changes the wage. The market wage is set by market supply and demand.
Describe two graphs side by side. On the left, the market: wage on the vertical axis, quantity of labor on the horizontal, an upward-sloping market supply curve and a downward-sloping market demand curve crossing at the market wage W* and quantity L*. On the right, the firm: the same wage axis, the firm's quantity of labor on the horizontal. Draw a horizontal line at W*, labeled S = MFC. The firm can hire as many workers as it wants at W*, so the cost of each extra worker (MFC) is exactly the wage.
The firm's downward-sloping MRP curve (its labor demand) crosses the horizontal wage line at the quantity it hires.
The hiring rule: MRP = MFC
Hire one more worker if they add more to revenue than to cost. Keep going until MRP = MFC. For a wage taker, MFC = the wage, so the rule is MRP = wage.
With a table, hire every worker whose MRP is at least the wage, and stop before the first one whose MRP is below it.
Being a wage taker in the labor market and being a price taker in the product market are separate questions. A firm with market power where it sells can still be a wage taker where it hires. Then its MRP uses MR (5.1). If it also sells in a perfectly competitive product market, MR = P, so MRP = MP × P.
What changes the number of workers hired
- Market wage rises (for example, market labor supply decreases): the firm's horizontal S = MFC line shifts up, and it hires fewer workers.
- Market wage falls: the line shifts down, and it hires more.
- Product price or productivity rises: the firm's MRP curve shifts right, and it hires more at the same wage.
Using more than one input
Firms use labor and capital together. Two rules tell them how much of each to use.
The least-cost rule: to make any given amount of output as cheaply as possible, the last dollar spent on each input should bring in the same extra output: MP of labor ÷ wage = MP of capital ÷ price of capital. If labor gives more output per dollar, shift spending from capital to labor, and the other way around. As you use more of an input, its MP falls, which brings the ratios together.
The profit-maximizing rule: to make the most profit, also hire each input until its MRP equals its price: MRP of labor ÷ wage = MRP of capital ÷ price of capital = 1. If a ratio is above 1, that input adds more revenue than it costs, so use more of it. If it's below 1, use less.
Any input mix that maximizes profit also minimizes cost, but not the other way around. A firm can produce the wrong amount of output very cheaply.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
How many workers to hire
A car wash sells washes in a perfectly competitive market for $5 each and hires workers in a perfectly competitive labor market. With 1 worker it does 20 washes a day; 2 workers, 38; 3 workers, 52; 4 workers, 62; 5 workers, 68; 6 workers, 70. (a) If the market wage is $50 a day, how many workers does it hire? (b) What if the wage rises to $60?
Show the solutionHide the solution
- Step 1: MP: 20, 18, 14, 10, 6, 2 washes.
- Step 2: MRP = MP × P = MP × $5: $100, $90, $70, $50, $30, $10.
- Step 3: (a) The firm is a wage taker, so MFC = wage = $50. Hire while MRP ≥ $50: workers 1 to 4 (the 4th worker's MRP is exactly $50). The 5th worker's MRP is $30, below $50, so stop at 4.
- Step 4: (b) At $60: workers 1 to 3 have MRP of at least $60. The 4th worker's MRP ($50) is now below the wage, so the firm hires 3.
Answer: (a) 4 workers. (b) 3 workers.
- Example 2
Least-cost and profit-maximizing checks (classic trap)
A firm sells its product for $2 in a perfectly competitive market. Right now, the marginal product of labor is 30 units and the wage is $15. The marginal product of capital is 40 units and capital rents for $10 per unit. (a) Is the firm producing its current output at the lowest cost? If not, what should it change? (b) Is it maximizing profit?
Show the solutionHide the solution
- Step 1: (a) Compare MP per dollar, not MP alone. Labor: 30 ÷ 15 = 2 units per dollar. Capital: 40 ÷ 10 = 4 units per dollar.
- Step 2: The last dollar spent on capital brings in twice as much output as the last dollar spent on labor. So the firm is not minimizing cost. It should use more capital and less labor until the ratios are equal.
- Step 3: The trap: capital's MP (40) is higher than labor's (30), but that alone doesn't settle it. You have to divide by each input's price.
- Step 4: (b) MRP = MP × P. Labor: 30 × $2 = $60, and 60 ÷ 15 = 4. Capital: 40 × $2 = $80, and 80 ÷ 10 = 8. Both ratios are above 1, so each input adds more revenue than it costs. The firm should use more of both, and it is not maximizing profit.
Answer: (a) No: capital gives 4 units per dollar and labor only 2, so shift toward more capital and less labor. (b) No: MRP ÷ price is 4 for labor and 8 for capital, both above 1, so it should use more of both inputs.
Common mistakes
- Drawing the firm's labor supply curve as upward-sloping in a perfectly competitive labor market. For a wage taker, it's horizontal at the market wage, and it equals MFC.
- Comparing MP of labor with MP of capital without dividing by their prices.
- Thinking that a firm that minimizes cost must be maximizing profit. Profit maximization also needs MRP ÷ price = 1 for each input.
- Hiring the first worker whose MRP is below the wage. Stop at the last worker whose MRP is at least the wage.
On the exam
- Free-response questions often show the market and firm side by side, then change something in the market (like market labor supply) and ask what happens to the firm's wage and the number it hires. Shift the market curve first, then move the firm's horizontal wage line.
- In a table question, write out the MRP of each worker before deciding. If the question gives inputs with prices, check MP ÷ price for each input.
Connected topics
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Check yourself
4 questions on 5.3 Profit-Maximizing Behavior in Perfectly Competitive Factor Markets. Pick an answer to see if you got it, and why.
| Input | Price of one unit of the input | Marginal product of the last unit used |
|---|---|---|
| Labor (worker-hours) | $20 | 15 units |
| Capital (machine-hours) | $40 | 20 units |
Hypothetical data for a firm that sells its output in a perfectly competitive market for $2 per unit and buys both inputs in perfectly competitive markets
Is the firm producing its current output at the lowest possible cost?
What is the marginal revenue product of the last worker-hour of labor?
To maximize profit, what should the firm do next?
A small firm hires workers in a perfectly competitive labor market. The labor supply curve facing this firm is
0 of 4 answered