Business Economics · Profit maximisation under perfect competition and monopoly

Profit Maximisation Rule (MR = MC) Explained

Q: Why must MC cut MR from below?

If MC cuts MR from above, output beyond that point has MC below MR, so producing more adds profit. That point is a profit minimum, not a maximum. A maximum needs MC above MR just after the intersection, which is what cutting from below means.

Q: Is MR = MC the same as the TR and TC approach?

Yes. Profit is largest where the gap between TR and TC is greatest. At that point the two curves have equal slopes. The slope of TR is MR and the slope of TC is MC.

Q: Does the profit-maximising firm always make a profit?

No. MR = MC gives the best output, not a guarantee of positive profit. If price is below average cost at that output, the firm makes a loss, but it is the smallest loss it can get at that output.

Q: How do I find the profit-maximising output from equations?

Find MR and MC by differentiating TR and TC. Set them equal and solve for Q. Then check that d²π/dQ² is negative, and use the demand curve to get price.

Updated 11 October 2026 · Fact-checked

A firm maximises profit at the output where marginal revenue equals marginal cost, provided marginal cost is rising and cuts marginal revenue from below. Up to that output each extra unit adds more revenue than cost. Beyond it, each extra unit adds more cost than revenue. Total revenue minus total cost is then largest.

Understand Profit Maximisation Rule (MR = MC)

A firm wants the biggest gap between total revenue (TR) and total cost (TC). That gap is profit: π = TR − TC. The question is how many units to produce.

Think one unit at a time. Marginal revenue (MR) is the extra revenue from selling one more unit. Marginal cost (MC) is the extra cost of making it. If MR > MC, the unit adds to profit, so make it. If MR < MC, the unit reduces profit, so do not make it. The firm keeps expanding while MR > MC and stops when they meet.

That gives the first order condition: MR = MC. It is only a necessary condition. MR = MC can also hold at an output where profit is at a minimum. The second order condition tells you which case you have. Profit is at a maximum if the slope of MC is greater than the slope of MR at that output. In plain words, MC must cut MR from below. For a price taker, MR is flat, so MC must be rising. For a monopoly, MR slopes down, so MC may be falling only if it falls less steeply than MR.

The same answer comes from the total revenue and total cost approach. Profit is largest where the vertical gap between the TR curve and the TC curve is largest. At that point the slopes of TR and TC are equal. The slope of TR is MR and the slope of TC is MC, so this is the same MR = MC rule. Where TR lies below TC, the firm makes a loss at that output.

MR = MC finds the best output, not whether profit is positive. After finding the output, compare price (or average revenue) with average cost. If price is below average cost at the best output, the firm makes the smallest possible loss there. Whether it should then keep producing is a separate shutdown question.

Key rules to remember

Profit: π = TR − TC; TR = P × Q. TC includes all costs, including normal profit in the usual economic treatment.
Marginal revenue and marginal cost: MR = dTR/dQ; MC = dTC/dQ; With discrete data, use the change in total divided by the change in quantity.
First order condition: dπ/dQ = 0, so MR = MC; Necessary, not sufficient. It also holds at a profit minimum.
Second order condition: d²π/dQ² < 0, so slope of MC > slope of MR; MC cuts MR from below. For a price taker, this means MC is rising.
Profit per unit: Average profit = P − AC; total profit = (P − AC) × Q; Use AC at the profit-maximising output, not at any other output.
Price taker: MR = P; So the rule becomes P = MC.

How to solve Profit Maximisation Rule (MR = MC) questions

Use this method for numerical and diagram questions on profit maximisation.

1Write down or derive TR as a function of Q. For a price taker, TR = P × Q. For a firm facing a demand curve, put P in terms of Q first, then TR = P × Q.
2Differentiate TR to get MR, and differentiate TC to get MC. With a table, take the change in total divided by the change in output.
3Set MR = MC and solve for Q. If there are several solutions, keep a list.
4Check the second order condition: the slope of MC must exceed the slope of MR at that Q. Equivalently, d²π/dQ² < 0. Reject any Q that fails this test.
5Substitute Q into the demand curve or price to find P, then compute TR and TC.
6Calculate profit π = TR − TC, or (P − AC) × Q. State clearly whether it is a profit or a loss.
7If the question uses a diagram, read Q where MC cuts MR from below, go up to the demand or AR curve for price, and compare with the AC curve at that Q.

Quickest way: Differentiate, equate, check the sign

When to use it: Use it when TR and TC (or P and TC) are given as simple polynomials and the question wants output or maximum profit.

Form π = TR − TC and write the first and second derivatives mentally.
Solve dπ/dQ = 0. If it is a quadratic, expect two roots.
Pick the root where d²π/dQ² < 0. The other root is usually a profit minimum.
Compute profit only at that root and check that the answer makes sense, for example Q ≥ 0 and P > 0.

Common mistakes in Profit Maximisation Rule (MR = MC)

Stopping at MR = MC without checking that MC cuts MR from below.
Students treat the first order condition as the whole rule.
Fix: Always test the slope of MC against the slope of MR, or check that d²π/dQ² < 0. Write the check as a line in your answer.
Using price as MR for a monopoly or any firm facing a downward-sloping demand curve.
MR = P is true only for a price taker, and students overgeneralise it.
Fix: Derive TR = P × Q first, then differentiate. For a linear demand P = a − bQ, MR = a − 2bQ.
Reading the profit-maximising price from the MC curve.
Students find Q at the MR = MC point and then stop on the MC curve.
Fix: Go up from Q to the demand (AR) curve. The price is read from demand, not from MC.
Thinking profit is always positive at MR = MC.
The rule is called profit maximisation, so students assume profit exists.
Fix: Compare P with AC at that Q. If P < AC, the firm makes a loss, the smallest loss available at that output.
Confusing AC and MC when calculating profit.
Both are cost per unit measures and sit on the same diagram.
Fix: Total profit = (P − AC) × Q. MC is only used to locate Q.
Using total cost with fixed cost left out of the derivative or treating fixed cost as part of MC.
Students differentiate sloppily or add the fixed cost back into MC.
Fix: A constant in TC disappears on differentiation, so MC ignores fixed cost. Put it back when you compute TC and profit.

Worked examples

Example 1

A price-taking firm sells at ₹40 per unit. Its total cost is TC = 100 + 10Q + 0.5Q². Find the profit-maximising output and the maximum profit.

Show the solution

Price taker, so MR = P = 40.
MC = dTC/dQ = 10 + Q.
Set MR = MC: 40 = 10 + Q, so Q = 30.
Second order check: slope of MC is 1 and slope of MR is 0, so MC cuts MR from below. Equivalently d²π/dQ² = −1 < 0.
TR = 40 × 30 = ₹1,200.
TC = 100 + 10 × 30 + 0.5 × 900 = 100 + 300 + 450 = ₹850.
Profit = 1,200 − 850 = ₹350.

Answer: Q = 30 units; maximum profit = ₹350.

Example 2

A monopolist faces demand P = 100 − 2Q and has TC = 200 + 20Q + Q². Find the profit-maximising output, price and profit, and confirm it is a maximum.

Show the solution

TR = P × Q = 100Q − 2Q², so MR = 100 − 4Q.
MC = dTC/dQ = 20 + 2Q.
Set MR = MC: 100 − 4Q = 20 + 2Q, so 80 = 6Q and Q = 40/3 ≈ 13.33.
Second order check: slope of MR is −4 and slope of MC is +2. MC has the greater slope, so MC cuts MR from below. Also d²π/dQ² = −4 − 2 = −6 < 0.
Price: P = 100 − 2 × (40/3) = 100 − 80/3 = 220/3 ≈ 73.33.
TR = (220/3) × (40/3) = 8,800/9 ≈ 977.78.
TC = 200 + 20 × (40/3) + (40/3)² = 200 + 266.67 + 177.78 = 644.44.
Profit = 977.78 − 644.44 = 333.33, which is 3,000/9.

Answer: Q = 40/3 ≈ 13.33; P = 220/3 ≈ 73.33; maximum profit = 1,000/3 ≈ 333.33 (in the same currency units as the demand curve).

Exam tips

Write the rule in full: MR = MC with MC cutting MR from below. Examiners give marks for the second condition.
In diagram questions, label Q at the MR = MC intersection, then draw a vertical line to the demand curve for price and to AC for cost per unit. Shade the profit rectangle.
For MCQs, check whether the firm is a price taker. If so, MR = P and the rule is P = MC. If not, derive MR from the demand curve.
For quadratic problems, you may get two roots. Use the second derivative to reject the one that gives minimum profit.
In written answers, link the rule to the TR-TC approach: maximum gap between TR and TC occurs where their slopes are equal.

Practice questions from Profit maximisation under perfect competition and monopoly

Profit Maximisation Rule (MR = MC) in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Profit Maximisation Rule (MR = MC): frequently asked questions

Why must MC cut MR from below?

If MC cuts MR from above, output beyond that point has MC below MR, so producing more adds profit. That point is a profit minimum, not a maximum. A maximum needs MC above MR just after the intersection, which is what cutting from below means.

Is MR = MC the same as the TR and TC approach?

Yes. Profit is largest where the gap between TR and TC is greatest. At that point the two curves have equal slopes. The slope of TR is MR and the slope of TC is MC.

Does the profit-maximising firm always make a profit?

No. MR = MC gives the best output, not a guarantee of positive profit. If price is below average cost at that output, the firm makes a loss, but it is the smallest loss it can get at that output.

How do I find the profit-maximising output from equations?

Find MR and MC by differentiating TR and TC. Set them equal and solve for Q. Then check that d²π/dQ² is negative, and use the demand curve to get price.