Marginal Revenue Product and Labor Demand: How Many Workers to Hire

A bakery could hire one more baker, and she would definitely bake more bread. So how could hiring her lose the owner money? She adds loaves, and she adds a wage. Which one is worth more? This is Module 5.1, part one: factor demand and marginal revenue product.

A baker's work is a factor of production: a resource used to make goods and services, bought and sold in factor markets at factor prices. Your textbook names four factors: labor, or human work; land, what nature provides; capital, both physical, like ovens, and human, the skill gained from training; and entrepreneurship, which combines them and takes the risk. Labor earns a wage. The College Board pairs capital with interest and land with rent; your textbook prices both with a rental rate. Those prices steer resources. When bakeries need more bakers, a higher wage draws workers to them.

Nobody wants bakers for their own sake; bakeries want them because people buy bread. That's derived demand: demand for a factor comes from demand for the product it makes. If bread demand rises, its price and marginal revenue rise, each baker brings in more revenue, and the bakery wants more bakers. Module 5.2 works that chain.

Most factor markets are treated as perfectly competitive, so one more worker costs just the wage, her marginal factor cost, or MFC. Our bakery is a price taker in both markets: loaves sell for $4, and bakers earn $130 a day. With 1 to 7 bakers, it bakes 50, 120, 180, 220, 250, 270 and 280 loaves a day.

Quick refresher from Module 3.1: marginal product, MP, is the change in total output from one more worker. Baker one: 50 minus 0 is 50 loaves. Baker two: 120 minus 50 is 70.

Baker three: 180 minus 120, or 60. Then 40, 30, 20 and 10. Diminishing returns begin with baker three, the first whose MP is smaller than the one before.

Now turn loaves into dollars. Marginal revenue product, MRP, is the change in total revenue from one more unit of a factor: MP times marginal revenue.

Quick refresher from Module 3.5: a price taker sells every loaf at the market price, so each extra loaf adds exactly $4: MR equals price.

So a price taker's MRP is MP times price, also called the value of the marginal product of labor, VMPL. Baker one: 50 loaves times $4 is $200. Baker two: 70 times $4 is $280. Baker three: 60 times $4 is $240. The rest follow the same way: $160, $120, $80 and $40 a day.

So what just happened? One: each MP is the change in total loaves. Two: MRP is MP times the $4 price. Three: baker three adds 60 loaves, worth $240.

Exam trap: MRP is in dollars. On a recent AP exam, giving the MP as the MRP, or subtracting the wage, missed the point; only about half earned it. Never write MR times P.

Which bakers should the bakery hire? Three things decide: each baker's productivity, the bread's price, and the wage.

Quick refresher from Unit 1 and Module 3.5: do more while marginal benefit beats marginal cost; for output, MR equals MC. In hiring, compare a worker's MRP with the wage: $160 in for $130 out adds $30 to profit.

Hire every baker whose MRP is above the $130 wage. Baker one, $200: hire. Two, $280: hire. Three, $240: hire. Four, $160: hire. Baker five: her MRP is only $120, below the $130 wage. So stop at four bakers. That answers the opening question. Baker five would bake 30 more loaves, worth $120, but she costs $130. Hiring her changes profit by $120 minus $130: negative $10 a day.

So what just happened? One: baker four adds $160 for a $130 wage: hire. Two: baker five adds only $120. Three: hiring her changes profit by negative $10, so stop at four.

Exam trap: give both sides, with numbers: the last worker hired has an MRP above the wage, the next one an MRP below it, so hiring it lowers profit. Total profit, the highest MP, or the bare rule all missed.

Plot each MRP against the number of bakers. Past baker two, each is lower than the one before, so the curve slopes down. The flat $130 wage line crosses it between four and five bakers. That falling MRP curve is the bakery's labor demand curve. At a $170 wage, baker three's $240 clears it but baker four's $160 does not: it hires three. At $100, baker five's $120 clears it but baker six's $80 does not: it hires five. Other things equal, a higher wage means a lower quantity of labor demanded: a move along the curve.

So what just happened? One: the MRP curve is labor demand. Two: at $170, three bakers; at $130, four; at $100, five. Three: higher wage, fewer workers.

Exam trap: why does a price taker's labor demand slope down? Diminishing marginal returns: each extra worker adds less output, so less revenue. Not economies of scale, and not market power.

Now picture a bakery with market power, the only one in town. To sell more loaves, it must cut its price.

Quick refresher from Module 4.2: a seller with market power cuts its price on every unit to sell one more. Two loaves at $5 bring in $10; three at $4 bring in $12. The third loaf adds just $2, below its $4 price.

So its MRP is MP times MR, and MR is below price. Say MR is $3.50 on baker one's loaves, then 50 cents less per baker. Baker one: 50 times $3.50 is $175. Baker two: 70 times $3 is $210. Every row is lower than the price taker's, and MRP falls for two reasons: diminishing returns and falling MR. At the $130 market wage, baker three's $150 clears it but baker four's $80 does not: it hires only three.

So what just happened? One: with market power, MR is below price. Two: MRP is MP times MR, so every row is lower. Three: it hires three bakers, not four.

Land and capital follow the same rule. Rent more while each unit's MRP is above its rental rate, the cost of using it for a set period.

Even an owned oven has that cost. Quick refresher from Module 3.4: an implicit cost is income given up. An oven that could rent out for $70 a day costs $70 a day to use.

Say a second oven adds 25 loaves a day: 25 times $4 is $100, above the $70 rental rate, so use it. A third would add 15 loaves, $60: stop at two.

So what just happened? One: owned or rented, an oven costs $70 a day. Two: oven two adds $100, so use it. Three: oven three adds $60, so stop at two.

Exam moment: two questions the way the AP exam asks them, then one free-response part.

Question 1. A greenhouse, a price taker in its product and labor markets, sells herbs at $3 a pot and pays $14 an hour. With 1, 2, 3 and 4 workers, it grows 8, 15, 21 and 26 pots an hour. What is the third worker's marginal revenue product? A) $4 B) $6 C) $9 D) $18 E) $63

The answer is D. The third worker's MP is 21 minus 15, or 6 pots, times the $3 price: $18. A: the MRP minus the wage. B: the MP alone. C multiplies MR by price. E multiplies total output by price.

Question 2. The only ferry line to an island, a monopolist, hires deckhands in a perfectly competitive labor market. Why does its marginal revenue product of labor fall as it hires more? A) Economies of scale lower its average cost. B) Each adds less output; MR equals price. C) Each adds less output; MR falls. D) Each adds more output; MR falls. E) It must raise every deckhand's wage.

The answer is C. Each deckhand adds less output, and a monopolist's MR falls as it sells more: both pull MRP down. A: a long-run cost idea. B is the price taker's story; here MR falls. D reverses diminishing returns. E describes a monopsony: Module 5.4.

Free response, 1 point. Pinecrest Candles, a price taker in its product and labor markets, sells candles for $7 and pays $50 a shift. With 1 to 6 workers, it makes 12, 28, 41, 50, 56 and 60 candles a shift. Determine the profit-maximizing number of workers. Explain using marginal analysis.

What earns the point: 4 workers, and both comparisons. Worker 4's MRP, 9 times $7, is $63, above the $50 wage, its MFC. Worker 5's MRP, 6 times $7, is $42, below $50, so hiring it lowers profit. The costly mistakes: total profit, the highest MP, or the rule with no numbers. On a recent AP exam, fewer than one student in three earned this point.

So, which answer earned the point? One: 4 workers. Two: worker 4's $63 beats the $50 wage. Three: worker 5's $42 falls short, so hiring it lowers profit.

Recap: factor demand is derived from product demand. MRP is MP times MR, or MP times price for a price taker. Hire while MRP beats the wage, so the MRP curve is labor demand. Next: Module 5.1, part two, labor supply and the labor market.