Profit Maximization: Study Guide (Microeconomics 3.5)

The question every firm must answer

A firm can produce many different quantities. Which one should it choose? In this course we assume that firms want to make as much economic profit as possible.

Economic profit = total revenue (TR) − total cost (TC)

Total cost here includes both explicit and implicit costs, as you saw in Topic 3.4. So “maximize profit” means “make the gap between TR and TC as large as possible”.

There are two ways to find that output:

1. The total approach. Work out TR − TC at every output and pick the largest.
2. The marginal approach. Compare what the next unit adds to revenue with what it adds to cost.

Both give the same answer. The marginal approach is the one you will use most, because it explains why the answer is right, and it works on a graph.

Marginal revenue and marginal cost

Marginal revenue (MR) is the extra revenue from selling one more unit:

MR = change in TR ÷ change in quantity

Marginal cost (MC) is the extra cost of producing one more unit (Topic 3.2):

MC = change in TC ÷ change in quantity

The change in profit from one more unit is simply MR − MC.

• If MR > MC, the unit adds more to revenue than to cost. Profit rises. Produce it.
• If MR < MC, the unit adds more to cost than to revenue. Profit falls. Do not produce it.

So the firm keeps adding units while MR is above MC and stops where the two meet. This gives the profit-maximizing rule:

Produce the quantity where MR = MC (where MC is rising).

This is marginal analysis again, the same idea as in Topic 1.6: keep doing an activity while its marginal benefit is at least its marginal cost. For a firm, the marginal benefit of a unit is its MR.

Using the rule with whole units

Tables usually give MR and MC for whole units. Then MR may never exactly equal MC. Use this version of the rule:

• Produce every unit whose MR is greater than or equal to its MC.
• Stop before the first unit whose MC is greater than its MR (once MC is rising).

If one unit has MR exactly equal to MC, producing it leaves profit unchanged. The usual convention is to produce that unit, so the answer is the quantity where MR = MC.

What MR looks like for different firms

Type of firm	What happens to price as it sells more	MR
A firm that takes the market price (it is too small to affect it)	price stays the same	MR = price for every unit
A firm that must lower its price to sell more	price falls	MR < price, and MR falls as output rises

Why is MR below price in the second case? To sell one more unit, the firm must cut the price on every unit it sells, not just the extra one. The revenue lost on the earlier units pulls MR below the price. You will meet these firms in detail in Unit 4. The rule MR = MC is the same for both.

Worked example 1: a firm that takes the market price

Question. Hollin Candle Co., a fictional workshop, sells boxes of candles at the market price of $15 per box. It cannot change this price. Its fixed cost is $30 per day.

Boxes per day (Q)	0	1	2	3	4	5	6	7	8
Total cost (TC, $)	30	40	47	53	60	69	81	97	116

(a) Find MR and MC for each box. (b) Use the marginal approach to find the profit-maximizing output. (c) Check your answer with the total approach, and state the profit. (d) The market price falls to $10. Find the new best output and the result.

(a) MR is $15 for every box, because each extra box adds $15 to revenue. MC is the change in TC from one row to the next:

Box	1st	2nd	3rd	4th	5th	6th	7th	8th
MR ($)	15	15	15	15	15	15	15	15
MC ($)	10	7	6	7	9	12	16	19

(b)

1. Boxes 1 to 6 each have MR ($15) greater than MC. Each adds to profit.
2. The 7th box has MC = $16 > MR = $15. It would reduce profit by $1.
3. So the profit-maximizing output is 6 boxes per day.

(c) TR = $15 × Q.

Q	0	1	2	3	4	5	6	7	8
TR ($)	0	15	30	45	60	75	90	105	120
TC ($)	30	40	47	53	60	69	81	97	116
Profit ($)	−30	−25	−17	−8	0	6	9	8	4

Profit is largest at 6 boxes: $90 − $81 = $9 per day. The two approaches agree.

(d) Now MR = $10 for every box.

1. The 1st box has MC = $10 = MR, so it neither adds nor removes profit. MC is still falling here, so do not stop: the next boxes cost less than $10 each.
2. Boxes 2 to 5 have MC of $7, $6, $7 and $9, all below $10. Produce them.
3. The 6th box has MC = $12 > $10, and MC is rising from here. Stop at 5 boxes.
4. TR = $10 × 5 = $50; TC = $69. Result = −$19 per day, a loss.

Check: at 4 boxes the result is $40 − $60 = −$20; at 6 boxes it is $60 − $81 = −$21. So 5 boxes gives the smallest loss. Producing nothing would give a loss of $30 (the fixed cost). Topic 3.6 explains how a firm decides between producing at a loss and shutting down.

Interpretation. The rule MR = MC finds the best output, but “best” can mean “smallest loss”. Notice also part (d) step 1: MR equals MC at the first box, but MC is falling there. That crossing is not the profit-maximizing point.

Reading the rule on a graph

On a graph, find the profit-maximizing output in three steps:

1. Draw MR and MC with dollars per unit on the vertical axis and quantity on the horizontal axis.
2. Find where MC rises through MR.
3. Drop a line down to the quantity axis. That quantity is the profit-maximizing output.

To the left of that point, MR is above MC, so each extra unit adds profit. To the right, MC is above MR, so each extra unit cuts profit. Profit is highest exactly at the crossing.

Notice what the graph does not show directly: the size of the profit. To find it you need average total cost or TR and TC. You will do this on a full cost-curve graph in Topics 3.6 and 3.7.

Worked example 2: a firm that must lower its price to sell more

Question. Larchmont Glass Studio, a fictional workshop, sells hand-blown glass bowls. To sell more bowls per day it must lower its price. Its fixed cost is $20 per day.

Bowls per day (Q)	1	2	3	4	5	6	7
Price ($)	30	28	26	24	22	20	18
MC of that bowl ($)	8	10	12	14	16	18	20

(a) Calculate TR and MR. (b) Find the profit-maximizing output and price. (c) Calculate the maximum profit. (d) The owner says, “Sell 7 bowls. More sales means more revenue.” Evaluate this.

(a) TR = price × quantity. MR = change in TR.

Q	1	2	3	4	5	6	7
TR ($)	30	56	78	96	110	120	126
MR ($)	30	26	22	18	14	10	6

MR is below the price from the 2nd bowl on. For example, the 4th bowl sells for $24, but to sell it the studio cuts the price of the first 3 bowls by $2 each. So MR = $24 − $6 = $18.

(b)

1. Bowls 1 to 4: MR ($30, $26, $22, $18) is above MC ($8, $10, $12, $14). Each adds profit.
2. The 5th bowl: MR = $14 < MC = $16. It would reduce profit by $2.
3. So the profit-maximizing output is 4 bowls per day.
4. The price comes from the price row at that quantity: $24 per bowl. Do not use MR as the price.

(c) TC = fixed cost + the sum of the MCs: $20 + $8 + $10 + $12 + $14 = $64. TR = $24 × 4 = $96. Profit = $96 − $64 = $32 per day.

Check with the totals: profit is $28 at 3 bowls and $30 at 5 bowls, both less than $32.

(d) The owner is right that revenue rises: TR is $126 at 7 bowls, more than $96. But the 5th, 6th and 7th bowls each cost more to make than they add to revenue (MR of $14, $10 and $6 against MC of $16, $18 and $20). TC rises to $20 + $98 = $118, so profit falls to $126 − $118 = $8. Maximizing revenue is not the same as maximizing profit.

Check. The units of profit are dollars per day, the same as TR and TC. The price ($24) is above MR ($18) and above MC ($14) at the chosen output. That is normal for a firm that must cut its price to sell more.

Why fixed costs do not change the output

Suppose Hollin Candle Co.’s rent rises, so its fixed cost goes up from $30 to $50 per day. Fixed costs do not change with output, so MC does not change. The 7th box still costs $16 more than the 6th. The firm still makes 6 boxes, but its profit falls by $20, from $9 to −$11.

A change that does affect MC changes the output. If the cost of wax rises, every box’s MC rises. MC now crosses MR at a lower quantity, so the firm makes fewer boxes. In the same way, a higher market price raises MR for a price-taking firm, so it makes more.

Change	Effect on MR or MC	Profit-maximizing output
Fixed cost rises (rent, a yearly licence)	neither changes	no change; profit falls
Cost of a variable input rises (materials, hourly wages)	MC rises	falls
Market price rises (price-taking firm)	MR rises	rises

Common misconceptions

• “Firms maximize profit where profit per unit is highest.” Profit per unit (price − ATC) is not the target. A firm may gain total profit by selling more units at a lower margin, as long as MR ≥ MC.
• “Firms maximize revenue.” Revenue keeps rising after profit has peaked (Worked example 2(d)). The goal is the biggest gap between TR and TC.
• “Produce where price equals MC.” That is the rule only when price = MR, for a price-taking firm. For a firm that must cut its price to sell more, use MR, not price.
• “The price is where MR and MC cross.” The crossing gives the quantity. A price-setting firm reads its price from the demand curve at that quantity.
• “MR = MC means the firm makes a profit.” It gives the largest profit or the smallest loss (Worked example 1(d)).
• “Any point where MR = MC will do.” If MC crosses MR while MC is falling, producing more still raises profit. Use the crossing where MC is rising.
• “Higher fixed costs make a firm cut output.” Fixed costs do not change MC, so the best output is unchanged in the short run.

Where this leads

The MR = MC rule is the decision rule for every firm in Units 3 and 4. Next, Topic 3.6 asks what a firm does when even its best output makes a loss: keep producing or shut down in the short run, and stay or leave in the long run. Read Topic 3.6: short-run and long-run decisions. First, test yourself with the practice questions, then use the revision notes and the checklist to consolidate. If you need a reminder of economic profit, go back to Topic 3.4: types of profit.