Unit 3 · Topic 3.5

3.5 Profit Maximization

Every firm, in any market structure, makes the most profit by producing the quantity where marginal revenue equals marginal cost. Profit is then total revenue minus total cost, which you can also find as (price − ATC) × quantity. This rule drives almost every firm question on the exam.

Key terms

total revenue
marginal revenue
marginal cost
profit-maximizing rule (MR = MC)
economic profit

Revenue terms

Total revenue (TR) = price × quantity. Marginal revenue (MR) is the extra revenue from selling one more unit: change in TR ÷ change in Q. Average revenue (AR) = TR ÷ Q, which equals the price.

For a firm in perfect competition, which can sell as much as it wants at the market price, MR equals the price. For firms that must lower the price to sell more (Unit 4), MR is less than the price.

The MR = MC rule

Think about each unit one at a time. If the next unit brings in more revenue than it costs (MR > MC), making it adds to profit, so make it. If it brings in less than it costs (MR < MC), making it lowers profit, so don't. Profit is maximized where MR = MC.

With a table of whole units, produce every unit where MR ≥ MC and stop before the first unit where MR < MC. On a graph, find where the MC curve crosses the MR curve while MC is rising, and drop straight down to the quantity axis.

This is just marginal analysis from topic 1.6, with revenue as the benefit.

Finding profit on a graph

Once you know the profit-maximizing quantity Q*, go up from Q* to the ATC curve to find average cost, and to the demand (or price) line to find the price.

Profit = TR − TC = (P − ATC) × Q*. On the graph, it's a rectangle: its width is Q* and its height is the gap between the price and ATC at Q*. If price is above ATC, the rectangle is economic profit; if price is below ATC, it's an economic loss.

Saying it in a free response

A full answer names the rule, gives the number and explains it. For example: 'The firm produces 5 units, where marginal revenue equals marginal cost. Each of the first 5 units adds at least as much to revenue as to cost, and a sixth unit would add more to cost ($70) than to revenue ($55), lowering profit.'

If the question asks for profit, show the calculation, either TR − TC or (P − ATC) × Q, with the numbers filled in. A bare number with no setup can lose a point when the work is asked for.

What the rule doesn't say

It doesn't say to maximize revenue. Selling more can raise revenue while lowering profit.
It doesn't say to produce at the lowest ATC. The highest profit per unit isn't the same as the highest total profit.
It doesn't promise a positive profit. If price is below ATC at Q*, MR = MC gives the smallest possible loss (topic 3.6 covers when to shut down instead).

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
Profit-maximizing output from a table
A perfectly competitive firm sells at a market price of $55. Its total cost for 0 to 6 units is $60, $100, $130, $150, $180, $230, $300. How many units should it produce, and what is its profit?
Show the solutionHide the solution
1. Step 1: In perfect competition, MR = P = $55 for every unit.
2. Step 2: MC for units 1–6 is $40, $30, $20, $30, $50, $70.
3. Step 3: Units 1–5 each have MC ≤ $55, so make them. Unit 6 costs $70 > $55, so stop at 5.
4. Step 4: TR = $55 × 5 = $275. TC = $230. Profit = $275 − $230 = $45.
5. Step 5: Check with the rectangle: ATC at 5 units = $230 ÷ 5 = $46. Profit = ($55 − $46) × 5 = $45.
Answer: 5 units, for an economic profit of $45.
Example 2Calculator allowed
Lowest ATC isn't highest profit (classic trap)
With the same firm and price, a student says the firm should produce 4 units, because ATC is lowest there ($45). Compare profit at 4 and 5 units.
Show the solutionHide the solution
1. Step 1: At 4 units: TR = $55 × 4 = $220; TC = $180; profit = $40. Profit per unit = $55 − $45 = $10.
2. Step 2: At 5 units: profit = $45, as before. Profit per unit = $55 − $46 = $9.
3. Step 3: The fifth unit adds $55 in revenue and only $50 in cost, so it raises total profit by $5, even though profit per unit falls.
4. Step 4: The trap is maximizing profit per unit instead of total profit.
Answer: 5 units ($45 profit) beats 4 units ($40 profit); follow MR = MC, not the lowest ATC.

Common mistakes

Producing where price equals ATC or where ATC is lowest. Output is set where MR = MC.
Finding profit as (P − MC) × Q. The height of the profit rectangle is P − ATC, not P − MC.
Producing past the point where MC exceeds MR to 'sell more'. Every unit past MR = MC lowers profit.
Reading price off the MC curve for a firm with market power. Price comes from the demand curve at Q* (Unit 4).

On the exam

Almost every firm free-response question asks for the profit-maximizing quantity; state 'where MR = MC' and give the number.
When asked to identify profit or loss on a graph, describe the rectangle: width Q*, height between price and ATC at Q*.

Connected topics

Videos

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