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Business Economics · Profit maximisation under imperfect competition

Revenue Curves and Marginal Revenue Under Downward-Sloping Demand

Updated 11 October 2026 · Fact-checked

Marginal revenue (MR) is the extra revenue from selling one more unit. For a firm facing a downward-sloping demand curve, price must fall to sell more, so MR lies below price (AR). For linear demand P = a − bQ, MR = a − 2bQ. Also MR = P(1 − 1/|e|), where e is price elasticity of demand.

Understand Revenue Curves and Marginal Revenue

A firm that faces a downward-sloping demand curve (a monopolist or any imperfect competitor) cannot sell more without cutting its price. This is the starting point for everything in this topic.

Total revenue (TR) is price times quantity: TR = P × Q. Average revenue (AR) is TR ÷ Q. Since TR = P × Q, AR = P at every output. So the demand curve is the AR curve.

Marginal revenue (MR) is the change in TR from selling one more unit. When the firm cuts the price to sell one more unit, it gains the price on that extra unit. But it also loses revenue on all the units it could have sold at the higher price. So MR is less than price. That is why the MR curve lies below the demand curve.

For a straight-line demand curve, the MR curve starts at the same price intercept but is twice as steep. It hits the quantity axis at half the quantity where demand does. TR rises while MR is positive, peaks where MR = 0, and falls when MR is negative.

Elasticity links it all. Where demand is elastic (|e| > 1), MR is positive and cutting price raises TR. Where demand is unit elastic (|e| = 1), MR = 0 and TR is at its maximum. Where demand is inelastic (|e| < 1), MR is negative. A profit-maximising firm therefore never chooses an output on the inelastic part of demand, provided marginal cost is positive. Under perfect competition the firm is a price taker, so demand is horizontal and MR = AR = P.

Key rules to remember

Total revenue
TR = P × Q
P is read from the demand curve at quantity Q.
Average revenue
AR = TR ÷ Q = P
The demand curve is the AR curve.
Marginal revenue (definition)
MR = dTR/dQ (or ΔTR ÷ ΔQ for discrete changes)
Use the derivative for a continuous demand function.
Linear demand and MR
If P = a − bQ, then TR = aQ − bQ² and MR = a − 2bQ
MR has the same intercept a and twice the slope. MR = 0 at Q = a ÷ (2b).
MR and elasticity
MR = P × (1 + 1/e) = P × (1 − 1/|e|)
e is the price elasticity of demand, which is negative on a downward-sloping curve.
Point elasticity (linear demand)
|e| = (P ÷ Q) × (1 ÷ b) = P ÷ (bQ)
Equals 1 at the midpoint of the demand line, where Q = a ÷ (2b).
Perfect competition
MR = AR = P
Demand facing the firm is horizontal.

How to solve Revenue Curves and Marginal Revenue questions

Use this method for any question on revenue curves, marginal revenue or the elasticity link.

  1. 1Write the demand function in the form P = f(Q). If it is given as Q = f(P), invert it first.
  2. 2Find TR = P × Q by substituting P = f(Q).
  3. 3Differentiate TR with respect to Q to get MR. For a table of data, use ΔTR ÷ ΔQ.
  4. 4Check the shape: for linear demand, MR has the same intercept and double the slope. Use this as a quick check on your algebra.
  5. 5If asked about elasticity, use |e| = P ÷ (bQ) for linear demand, or use MR = P(1 − 1/|e|).
  6. 6Find key points: set MR = 0 for maximum TR, and compare |e| with 1 to say whether MR is positive or negative.
  7. 7If profit is asked, set MR = MC, find Q, then read P from the demand curve (not from MR).
  8. 8State the answer with units and a short interpretation.

Quickest way: Double the slope, then test the elasticity

When to use it: For linear demand questions in the multiple-choice section, or to check a written answer quickly.

  1. Write demand as P = a − bQ.
  2. Write MR directly as a − 2bQ. No integration or long working needed.
  3. For the quantity where MR = 0, use Q = a ÷ (2b). This is also where |e| = 1 and TR is maximum.
  4. For any other point, compute |e| = P ÷ (bQ) and compare with 1.
  5. If given P and |e|, use MR = P(1 − 1/|e|) directly.

Common mistakes in Revenue Curves and Marginal Revenue

  • Drawing the MR curve above the demand curve, or equal to it, for an imperfect competitor.

    Students remember MR = AR from perfect competition and apply it everywhere.

    Fix: MR = AR only when price does not fall with output. With downward-sloping demand, MR is always below AR.

  • Writing MR = a − bQ by differentiating P instead of TR.

    Students confuse the demand slope with the slope of total revenue.

    Fix: Always multiply by Q first to get TR, then differentiate. Check that the MR slope is 2b.

  • Using the sign of e wrongly in MR = P(1 + 1/e).

    Elasticity is negative on a demand curve, and students plug in its absolute value.

    Fix: Use P(1 + 1/e) with e negative, or P(1 − 1/|e|) with the absolute value. Never mix the two.

  • Reading the price off the MR curve when finding the profit-maximising price.

    MR = MC gives the quantity, and students then stop at the MR curve.

    Fix: Take Q from MR = MC, then go up to the demand curve to find the price the market will pay.

  • Saying a firm may produce where demand is inelastic.

    Students do not connect negative MR with the MR = MC rule.

    Fix: With positive MC, MR = MC needs MR > 0, so the firm operates where |e| > 1.

  • Assuming MR = 0 where price is zero.

    Students mix up the price intercept and the MR intercept on the quantity axis.

    Fix: MR = 0 at half the quantity where demand meets the axis, i.e. Q = a ÷ (2b).

Worked examples

Example 1

A firm faces the demand curve P = 120 − 2Q, where P is in ₹ per unit. (a) Find the MR function. (b) Find the output at which TR is maximum and the maximum TR. (c) Find the price elasticity of demand at Q = 20.

Show the solution
  1. TR = P × Q = (120 − 2Q)Q = 120Q − 2Q².
  2. MR = dTR/dQ = 120 − 4Q. This has the same intercept as demand and double the slope, as expected.
  3. TR is maximum where MR = 0: 120 − 4Q = 0, so Q = 30.
  4. Price at Q = 30: P = 120 − 60 = ₹60. Maximum TR = 60 × 30 = ₹1,800.
  5. Check with the formula TR = 120Q − 2Q²: 3,600 − 1,800 = ₹1,800. This matches.
  6. At Q = 20: P = 120 − 40 = ₹80. Here b = 2.
  7. |e| = P ÷ (bQ) = 80 ÷ (2 × 20) = 80 ÷ 40 = 2.
  8. Cross-check with MR = P(1 − 1/|e|) = 80 × (1 − 0.5) = 40. Direct MR = 120 − 4 × 20 = 40. This matches.

Answer: (a) MR = 120 − 4Q. (b) TR is maximum at Q = 30 with TR = ₹1,800. (c) |e| = 2 at Q = 20, so demand is elastic and MR (₹40) is positive.

Example 2

A monopolist has the demand function Q = 50 − 0.5P and constant marginal cost of ₹30 per unit. (a) Find the profit-maximising output and price. (b) Find the elasticity of demand at that point and show it agrees with MR = P(1 − 1/|e|).

Show the solution
  1. Invert demand: 0.5P = 50 − Q, so P = 100 − 2Q.
  2. TR = (100 − 2Q)Q = 100Q − 2Q². MR = 100 − 4Q.
  3. Set MR = MC: 100 − 4Q = 30, so 4Q = 70 and Q = 17.5.
  4. Price from the demand curve: P = 100 − 2 × 17.5 = 100 − 35 = ₹65.
  5. Elasticity: |e| = P ÷ (bQ) with b = 2, so |e| = 65 ÷ (2 × 17.5) = 65 ÷ 35 = 13/7 ≈ 1.857.
  6. Check: MR = 65 × (1 − 7/13) = 65 × 6/13 = ₹30. This equals MC, so the result is consistent.
  7. Since |e| > 1, the firm is on the elastic part of demand, as theory requires.

Answer: Profit-maximising output is 17.5 units at a price of ₹65. Elasticity is 13/7 ≈ 1.86 at that point, and MR = ₹30 = MC.

Exam tips

  • For a written question, always show TR first, then differentiate. Marks are usually given for the method, not just the final MR.
  • Use the elasticity check MR = P(1 − 1/|e|) to verify your answer. It takes seconds and catches algebra errors.
  • In diagram questions, label demand (AR), MR, the MR = 0 quantity, and where |e| = 1. Mark the elastic and inelastic sections clearly.
  • In multiple-choice questions, a negative MR option is usually a trap for the inelastic region. Compare |e| with 1 before choosing.
  • If the demand function is given as Q in terms of P, invert it before finding MR. Differentiating Q = f(P) gives the wrong curve.

Practice questions from Profit maximisation under imperfect competition

Revenue Curves and Marginal Revenue in other exams

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

Revenue Curves and Marginal Revenue: frequently asked questions

Why is marginal revenue less than price for a monopolist?

To sell one more unit, the firm must lower the price on all units, not just the extra one. The gain is the new price on the extra unit. The loss is the price cut on every unit already sold. So MR is below price.

How do you derive marginal revenue from a linear demand function?

Write demand as P = a − bQ. Multiply by Q to get TR = aQ − bQ². Differentiate to get MR = a − 2bQ. The MR curve has the same intercept as demand and twice the slope.

What is the relationship between marginal revenue and price elasticity of demand?

MR = P(1 − 1/|e|). If |e| > 1, MR is positive. If |e| = 1, MR is zero. If |e| < 1, MR is negative. This is why a firm with positive marginal cost avoids the inelastic part of its demand curve.

Is the demand curve the same as the average revenue curve?

Yes, for a firm that sells all units at a single price. Since AR = TR ÷ Q = P, each point on the demand curve shows the average revenue at that quantity.