Least-Cost Rule and Profit-Maximizing Hiring: MRP = MFC

A laundry that washes hotel towels has $150 left in today's budget: enough for one more worker for the day, or three more hours on a giant washer. Which one gets more towels clean for each dollar? And how would the owner know? This is Module 5.3, Profit Maximization in Factor Markets: how many workers, and which input mix.

First, the hiring rule again: the AP course files it under this topic. A firewood yard sells bundles at the market price, $8 each, and hires all the workers it wants at the market wage, $20 an hour. That's a perfectly competitive labor market: the market sets the wage, and each firm takes it. Each extra worker adds exactly $20 an hour to cost: the marginal factor cost, MFC, or marginal resource cost.

Quick refresher from Module 5.1, part one: marginal product, MP, is the extra output from one more worker. Marginal revenue product, MRP, is the extra revenue: MP times marginal revenue. For a price taker, that's MP times the price, the value of the marginal product.

With its saws and yard fixed, 1 worker makes 4 bundles an hour, 2 make 9, 3 make 13, 4 make 16, 5 make 18, 6 make 19. Third worker: 13 minus 9 is 4 more bundles, times $8 is $32. Fourth: 16 minus 13 is 3, times $8 is $24. Fifth: 18 minus 16 is 2, times $8 is $16. The rule: hire up to where MRP meets MFC. With whole workers, hire each one whose MRP beats the $20 wage. The 4th brings in $24: hire. The 5th would bring in $16: stop. The yard hires 4. Hiring the 5th would change profit by $16 minus $20: negative $4 an hour.

So what just happened? One: MRP is MP times the $8 price. Two: the 4th worker's $24 beats the $20 wage, so hire. Three: the 5th's $16 doesn't, so the yard stops at 4.

Exam trap: on two recent AP exams, fewer than two in five earned the point for explaining this choice. Total profit, MR equals MC, or the rule without numbers all missed. Give both comparisons, with numbers.

Now the yard's rent rises by $600 a month. Does it hire fewer workers?

Quick refresher from Module 3.2: a fixed cost, like rent, is the same at 4 workers or 5. Marginal cost comes only from variable cost, like the $20 per worker-hour.

So the 5th worker still adds $16 and costs $20. Neither MRP nor MFC moved: the yard still hires 4, just with less profit. Same as Module 3.5: a fixed cost leaves the MR equals MC output alone. Exam trap: explain it through what drives hiring: productivity, the output price, the wage. Rent moves none of them. On a recent AP exam, fewer than three in ten earned this point; a named mistake: arguing from total profit.

So what just happened? One: rent rose $600 a month. Two: the 5th worker still adds $16 and costs $20. Three: MRP and MFC didn't move: still 4 workers, less profit.

A firm can take the wage even when it sets its own price: a town's only movie theater picks its ticket prices but hires cashiers at the going wage every shop in town pays. Its MRP uses its own marginal revenue.

Back to the laundry. The same towels can come from different mixes of workers and machine time. Inputs can be substitutes: a folding machine can replace hand folders. Or complements: more of one raises the other's marginal product, as a rolling cart helps each worker move more towels.

The goal: the towels it needs at the lowest cost. A worker costs $150 a day; washers rent for $50 an hour each. A machine-heavy mix: 2 workers and 30 washer-hours. A worker-heavy mix: 4 workers and 22. Same towels. Machine-heavy: 2 times $150 plus 30 times $50 is $300 plus $1,500: $1,800. Worker-heavy: 4 times $150 plus 22 times $50 is $600 plus $1,100: $1,700. The worker-heavy mix saves $100.

So what just happened? One: two mixes wash the same towels. Two: we priced each one: $1,800 against $1,700. Three: the laundry picks the cheaper, worker-heavy mix.

With too many possible mixes to price, firms use marginal analysis instead.

The tool: the cost-minimization rule, or least-cost rule. MPL over W equals MPK over r, with r the rental rate, the price of using capital. The last dollar on each input adds the same output.

Quick refresher from Module 1.6: a consumer compares marginal utility per dollar: a slice with 30 utils for $3 gives 10 per dollar. A firm compares marginal product per dollar: output, not satisfaction.

The laundry's choice: one more worker adds 900 towels a day for $150, and 900 over 150 is 6 towels per dollar. One more washer-hour adds 200 towels for $50: 4 per dollar. 6 beats 4, so a dollar does more on labor. That answers the opening question: the worker. And keep shifting: more workers, fewer washer-hours.

Quick refresher from Module 3.1: diminishing marginal returns. Past some point, each extra unit of an input adds less than the one before, like the yard's 4, then 3, then 2 bundles.

As workers are added, the newest adds less, say 750 towels; with fewer washer-hours, the last one adds more, say 250. Now 750 over 150 is 5, and 250 over 50 is 5. Equal: the cheapest mix.

So what just happened? One: the worker gave 6 towels per dollar, the washer 4. Two: so the laundry shifted toward workers. Three: diminishing returns brought both to 5: the cheapest mix.

Exam trap: three wrong versions. Marginal utility per dollar is the consumer's rule, a named error on a recent AP exam. Equal marginal products is wrong too: at the cheapest mix, a worker adds 750 towels, a washer-hour 250. And the bigger MP isn't automatically the better buy: at $300 a day, that 900-towel worker would give just 3 per dollar, less than the washer's 4. How the rules fit: the least-cost rule says how to make an amount cheaply, not how much. For maximum profit, also use each input until its MRP meets its price.

The AP exam has run it backward. Another laundry uses its cheapest mix. Its last washer-hour adds 240 towels and rents for $60; its last worker adds 840 towels a day. What's the daily wage? Step one, write the rule: MPL over W equals MPK over r. Step two, plug in: 840 over W equals 240 over $60. Step three: 240 divided by 60 is 4 towels per dollar. Step four: 840 over W equals 4, so W is 840 divided by 4: $210 a day. Check: 840 over 210 is 4.

So what just happened? One: we wrote the rule. Two: the washer side gave 4 towels per dollar. Three: 840 over W equals 4, so the wage is $210 a day.

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

Question 1. A snack maker pays $16 per worker-hour and $40 per fryer-hour. Its last worker-hour adds 48 bags; its last fryer-hour adds 80. To make the same output for less, it should use more A) fryers, until MPs are equal. B) fryers: their MP is larger. C) workers, until MU per dollar is equal. D) workers, until MP per dollar is equal. E) of neither: both MPs are positive.

The answer is D. Workers give 48 over 16, 3 bags per dollar; fryers, 80 over 40, just 2. So use more workers. B compares MPs, not MP per dollar. A wants equal MPs. C is the consumer's rule. E's reason is true but irrelevant.

Question 2. A workshop sells chairs at the market price and hires carpenters at the market wage, $110 a day. Its 6th carpenter's MRP is $130; a 7th's would be $95. Rent rises $900 a month. It will hire A) 5: total cost rose. B) 5: MFC rose. C) 6: MRP and MFC are unchanged. D) 6: profit is unchanged. E) 7: to spread the rent.

The answer is C. Rent is fixed, so MRP and MFC stay put: the 6th's $130 still beats $110, a 7th's $95 doesn't. D is false: profit falls. A, like D, reasons from totals. B treats rent as a marginal cost. E: a 7th carpenter loses $15 a day.

Free response, 1 point. A print shop uses its cost-minimizing mix of labor and capital. Printers rent for $30 an hour. The last printer-hour adds 360 flyers; the last worker-hour, 240. Calculate the wage rate. Show your work.

What earns the point: setup, numbers, answer. MPL over W equals MPK over r. 240 over W equals 360 over $30, or 12 per dollar. So W is 240 over 12: $20 an hour. The costly mistakes: the consumer's marginal utility formula, or the right setup with a math slip. On a recent AP exam, about half earned this point.

So what earned the point? One: write the rule. Two: 360 over 30 is 12 flyers per dollar. Three: 240 divided by 12 is $20 an hour.

Recap: hire while MRP beats MFC; fixed costs don't change that. Pick the mix where each input's last dollar adds the same output. Run the rule backward for a missing price. Next: Module 5.4, Monopsony.