Profit Maximization: The MR = MC Rule, Worked with Numbers

A grocery chain will buy every batch of bread a small bakery bakes, at $30 a batch. So why doesn't the bakery bake a million batches? This is Module 3.5, Profit Maximization. By the end, you'll know where a firm should stop, and how to prove it.

Start with costs. Rent is $50 a day, owed even at zero batches: a fixed cost. Flour and labor grow with every batch. Total cost runs from $50 at zero batches to $227 at eight. It counts every cost, even the owner's time, so profit here is economic profit.

Quick refresher: total revenue is price times quantity; profit is total revenue minus total cost. At 6 batches, $30 times 6 is $180, minus $157 of total cost: $23 of profit.

Do that for every row. Profit climbs from negative $50 at zero to a peak of $23 at 6 batches, then falls: $20 at 7, $13 at 8. So the best output is 6 batches: not the most bread, but the most profit.

So what just happened at the bakery? One: revenue is $30 times batches. Two: profit is revenue minus total cost. Three: profit peaks at $23, at 6 batches.

Totals work, but the exam wants a faster tool: one batch at a time.

Quick refresher from Unit 1: marginal analysis. Keep going while the next step's benefit is at least its cost. A pizza slice worth $4 to you, priced at $3? Buy it. One worth $2? Stop.

For a firm, the benefit of one more unit is marginal revenue, or MR: the change in total revenue from one more unit. Each batch adds $30. Every batch sells at one price, so for a firm like this, marginal revenue equals the price. More in Module 3.7.

Quick refresher: marginal cost, or MC, is the change in total cost divided by the change in quantity. From 5 to 6 batches, total cost goes from $130 to $157: $27 divided by 1, or $27.

Now line them up. Batch 1 adds $30 of revenue and $10 of cost, so profit grows $20. Batch 2: $30 against $12. Through batch 6, revenue beats cost every time. Batch 6: $30 in, $27 of cost. It still adds $3, so bake it. Batch 7: $30 in, but $33 of cost: it would cut profit by $3. Stop. That answers the million batches. With one oven, extra workers have less of it to use, so each batch costs more than the last, until marginal cost passes the price.

So what just happened? One: each batch adds $30, its MR. Two: batches 1 to 6 each cost less than $30, so each raises profit. Three: batch 7 costs $33, so stop at 6.

That's the profit-maximizing rule: produce the quantity where marginal revenue equals marginal cost. MR = MC. If no row has MR exactly equal to MC, make every unit where MR is greater than MC, and stop before the first where MC is greater. Check with totals: 5 batches earn $20, and so do 7. Both are less than the $23 at 6.

Now the graph: price and cost up the side, quantity along the bottom. MR is a flat line at $30. MC rises from left to right. Where it crosses the MR line is the profit-maximizing quantity, labeled QF. Here, that's 6 batches.

So what just happened on the graph? One: MR is flat at the $30 price. Two: MC rises. Three: they cross at QF, 6 batches, the most profit.

Exam trap: with ATC on the graph, a common mistake is stopping where MC crosses ATC. That's the lowest cost per batch, not the most profit.

Quick refresher: average total cost is total cost divided by quantity. At 5 batches, that's $130 divided by 5: $26 a batch, the bakery's lowest.

But stopping at 5 throws away batch 6 and its $3 of profit. And read the price off the MR line, $30, never off ATC. Exam trap: "Explain using marginal analysis" needs two comparisons, with numbers. First, the last unit made: for batch 6, MR of $30 is greater than MC of $27. Second, the next unit refused: for batch 7, MR of $30 is less than MC of $33. Leave either one out, and the explanation is incomplete.

Now costs change. Which changes move the best quantity? Say rent rises $10, to $60 a day. Every total cost rises $10, so every profit falls $10. The peak drops from $23 to $13, but stays at 6 batches.

Quick refresher: marginal cost is the change in total cost divided by the change in quantity. From 5 to 6 batches, total cost now goes from $140 to $167: still $27. Rent is in both totals, so it cancels.

MR is still the $30 price. Neither MC nor MR changed, so their crossing can't move. On the graph, the higher fixed cost lifts ATC, but QF stays put.

So what just happened when rent rose? One: every total cost rose $10, and peak profit fell to $13. Two: MC and MR stayed the same. Three: output stayed at 6.

Exam trap: on a recent AP exam, only about three in ten students earned this point. Many knew ATC rises; naming only MR still lost it. Name both: a fixed cost changes neither MC nor MR. A lump-sum tax works the same way: one fixed amount, not a per-unit charge, so MC and output stay put.

Now a per-unit change: a government subsidy of $5 for every batch. It cuts the cost of each extra batch by $5, so marginal cost falls. Batch 7 now costs $33 minus $5: $28, below the $30 price. Bake it. Batch 8: $37 minus $5 is $32, above $30. Stop. Output rises from 6 to 7. On the graph, MC shifts down, so it meets the flat MR line at a larger quantity.

So what just happened with the subsidy? One: it cut each batch's cost by $5, so MC fell. Two: batch 7 now costs $28, under $30. Three: output rose from 6 to 7.

Exam trap: "output increases" alone is not enough. On a recent AP exam, most students missed this point, often by giving no mechanism. The fix: the subsidy lowers MC, so MR = MC at a larger quantity. A higher wage works the other way: each batch's labor costs more, so MC shifts up, meets MR at a smaller quantity, and output falls. Ask one question: does the change touch MC or MR? Rent, other fixed costs, a lump-sum tax: no, so output stays. Per-unit costs and subsidies: yes, so output moves.

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

A firm can sell as many units as it wants at $40 each. Its marginal cost for units 1 to 6: $30, $20, $16, $24, $35, $49. Its average total cost: $60, $40, $32, $30, $31, $34. Which output maximizes profit? A) 2 units B) 3 units C) 4 units D) 5 units E) 6 units.

The answer is D. Compare MC with MR, the $40 price: unit 5 costs $35, so make it; unit 6 costs $49, so stop at 5. C is the lowest ATC, where MC crosses ATC. A is where price equals ATC. B is the lowest MC. E adds unit 6, which loses $9.

A profit-maximizing firm's yearly insurance premium, a fixed cost, rises. In the short run, what happens to its MC, ATC, and output, in that order? A) Rises, rises, falls B) Same, rises, falls C) Same, rises, same D) Same, rises, rises E) Same, same, same.

The answer is C. Insurance adds the same amount to every total cost, so MC and MR don't change, and output stays put. It does raise ATC. A treats insurance as a per-unit cost. B and D move output, but MC and MR never moved. E forgets that ATC includes fixed cost.

Free response. A beekeeper sells as many jars of honey as she wants at $9 each; her marginal cost of jars 1 to 6 is $4, $5, $6, $7, $8 and $11. Identify the profit-maximizing quantity. Explain using marginal analysis and numbers.

To earn the point: 5 jars, plus both comparisons. The 5th jar adds profit: MR, the $9 price, is greater than MC, $8. The 6th would lower profit: MR, $9, is less than MC, $11. The costly mistake: quoting total profit, or writing "MR = MC" with no numbers when no jar has MR = MC. On a recent AP exam, in a version about hiring workers, 29% of students earned the point.

So what earned the point? One: 5 jars. Two: the last jar made, MR $9 above MC $8. Three: the next jar refused, MR $9 below MC $11.

Recap: produce where MR = MC; in a table, the last unit where MR beats MC. Explain with two comparisons. Fixed costs don't move output; per-unit costs do. One question comes before "how much": should a firm produce at all? Next: Module 3.6, Firms' Entry and Exit Decisions.