A monopolist does not read its price off a supply curve — it chooses the price, and the point it picks is the exam question. Getting there means one derivation done cleanly: why marginal revenue sits below demand, where the profit-maximising quantity lands, and how much surplus the markup destroys. This is the intermediate-micro monopoly material New York students bring to us most, built here from marginal revenue to a full welfare account.
1 · Why marginal revenue lies below price
A monopolist faces the whole downward-sloping demand curve. To sell one more unit it must lower the price — and, at a single price, it lowers the price on every unit, not just the last. So an extra unit’s revenue is the new price minus that cut applied across all the units already being sold.
That second term pushes marginal revenue below demand. Write demand as P = a − bQ; total revenue is aQ − bQ², so MR = a − 2bQ. Marginal revenue shares demand’s price intercept but has twice the slope — hitting zero at half the quantity where demand does.
Examiners reward that “twice as steep” rule on almost every monopoly diagram; drawing MR parallel to demand — a common slip — loses both the marks and the equilibrium.
2 · Marginal revenue, elasticity, and the markup
A second way to write MR carries more economics:
MR = P(1 − 1/|ε|),
where ε is the price elasticity of demand. When demand is elastic (|ε| > 1) the bracket is positive, so MR > 0 — selling more still raises revenue. When demand is inelastic (|ε| < 1) it turns negative and MR < 0 — selling more lowers revenue, because the price cut outweighs the extra unit.
That gives you an invariant examiners love: a monopolist never operates where demand is inelastic. With positive marginal cost the firm sets MR = MC > 0, forcing it onto the elastic segment. On linear demand the unit-elastic point is the midpoint — exactly where MR = 0 — so the monopolist sits on the upper half.
Rearranging gives the Lerner index, the standard measure of market power: (P − MC)/P = 1/|ε|. The markup as a fraction of price is the reciprocal of elasticity — less elastic demand, fatter markup; perfectly elastic demand drives it to zero.
3 · Deadweight loss, decomposed
Because MR lies below demand, the monopolist stops where MR = MC, short of the competitive quantity where P = MC. Between those two quantities sit units buyers value above what they cost to make — trades that never happen. Their lost surplus is the deadweight loss, the triangle between demand and marginal cost over the withheld output.
Better: track where each piece of competitive consumer surplus goes under monopoly. It splits into three parts:
- a triangle consumers keep, above the higher monopoly price;
- a rectangle transferred to the firm as profit — the markup times the quantity sold;
- the deadweight-loss triangle, which goes to no one.
The transfer is not a social loss — it is a distribution question, consumers to shareholders. The deadweight loss is the efficiency loss, and it is what “monopoly is inefficient” means. For linear demand and constant cost it works out to (a − MC)²/8b — growing with the square of the gap between choke price and cost.
Worked example — a patented migraine drug
A firm holds the patent on a weekly migraine course — a legal monopoly. Each course costs a constant MC = £6 to make (take average cost equal to marginal cost). Weekly demand, in thousands of courses, is P = 30 − Q.
Step 1 — Build marginal revenue. With P = 30 − Q, revenue is 30Q − Q², so MR = 30 − 2Q — same intercept, twice the slope, hitting zero at Q = 15.
Step 2 — Find the monopoly point. Set MR = MC: 30 − 2Q = 6, so Qm = 12, and the price off demand is Pm = 30 − 12 = £18 — three times marginal cost.
Step 3 — Check the elasticity and the markup. At (12, £18), elasticity is (dQ/dP)(P/Q) = (−1)(18/12) = −1.5 — elastic, as it must be. The Lerner index is (18 − 6)/18 = 2/3 = 1/1.5 = 1/|ε|, and MR = P(1 − 1/1.5) = £6 = MC, closing the loop.
Step 4 — Account for the welfare. A competitive market with the same £6 cost produces where P = MC: Qc = 24, with consumer surplus ½(30 − 6)(24) = £288. Under monopoly that £288 splits three ways: £72 consumers keep (½ × 12 × 12), £144 transferred to the firm as profit ((18 − 6) × 12), and a £72 deadweight loss (½ × 12 × 12). They sum back to 288.
Step 5 — Perturb it: costs rise. An active-ingredient shortage lifts marginal cost from £6 to £10. Re-solve MR = MC: 30 − 2Q = 10, so Qm = 10 and Pm = £20.
Step 6 — Read the pass-through. Marginal cost rose £4, but the price rose only £2. The monopolist passes through exactly half a cost increase — always half for linear demand, since the optimum price is (a + MC)/2, so dPm/dMC = ½.
Step 7 — Interpret it. The competitive quantity is now Qc = 20, so the deadweight loss falls to ½ × 10 × 10 = £50. A higher cost narrows the gap between choke price and marginal cost, so the market-power wedge — and the surplus it destroys — shrinks.
Could you draw demand, MR, MC and read the profit rectangle and the deadweight-loss triangle straight off — and say which one is a loss to society? That diagram, plus the transfer-versus-deadweight-loss distinction, is exactly what monopoly questions reward. Building it until it comes from memory is what a one-on-one microeconomics tutor does with you. Book a trial session.
Practice
Q1. A monopolist faces P = 200 − 4Q with constant marginal cost £40. Find its output, price, and the competitive quantity, then the deadweight loss and the Lerner index.
Q2. Demand is P = 90 − 3Q. (a) At what quantity is demand unit-elastic? (b) Show that a monopolist with MC = £30 produces on the elastic segment, and give the elasticity there.
Q3. For the monopoly in Q1, split the surplus a competitive market would create into the part consumers keep, the part transferred to the firm, and the deadweight loss.
Answers. Q1: MR = 200 − 8Q = 40 gives Qm = 20, Pm = £120; P = MC gives Qc = 40. Deadweight loss = ½ × 80 × 20 = £800; Lerner = 2/3 = 1/|ε|, |ε| = 1.5. Q2: (a) MR = 90 − 6Q = 0 at Q = 15, elasticity exactly −1. (b) MR = MC gives Qm = 10, below 15, so elastic; there P = £60 and elasticity is −2. Q3: competitive surplus £3,200 splits into £800 kept, £1,600 transferred, and the £800 deadweight loss.
Key takeaways
- Marginal revenue lies below demand because a price cut applies to every unit; for linear demand MR shares the intercept and has twice the slope.
- A monopolist never prices on the inelastic segment. Since MR = P(1 − 1/|ε|) = MC > 0, the optimum has |ε| > 1, and the Lerner markup (P − MC)/P equals 1/|ε|.
- Deadweight loss is the efficiency cost, distinct from the consumer-to-firm transfer; competitive surplus splits into kept surplus, transfer, and the lost triangle.
- A monopolist passes through only half of a linear-demand cost increase, and the deadweight loss shrinks with the market-power wedge.
Why New York students choose our microeconomics tutoring
- Course-matched derivations: whether your module works monopoly through calculus or the MR = MC diagram, sessions use your professor’s notation and the surplus decomposition your problem sets ask for.
- The marks live in the diagram: you leave able to draw demand, MR, MC, the profit rectangle and the deadweight-loss triangle from memory, and read every area straight off.
- One-on-one problem sets: a private tutor works the numerical questions with you until MR = MC, the Lerner index and the welfare split are second nature — not a scramble in the exam hall.
FAQ
Q: Why is marginal revenue below the demand curve for a monopoly?
A: Because a monopolist must cut the price on every unit to sell one more, not just the last. Marginal revenue is the new price minus that cut spread over existing sales, so it sits below demand — and for linear demand it falls twice as fast.
Q: Why does a monopolist never produce where demand is inelastic?
A: On the inelastic segment marginal revenue is negative, so selling more lowers total revenue. With positive marginal cost the firm sets MR = MC > 0, which can only happen on the elastic segment — the upper half of linear demand.
Q: What is the difference between the transfer and the deadweight loss?
A: The transfer is the profit rectangle — surplus moving from consumers to the firm, not a loss to society. The deadweight loss is the triangle of trades that never happen once output is restricted; that surplus goes to no one.
Q: How do I calculate monopoly deadweight loss?
A: Find the monopoly quantity from MR = MC and the competitive quantity from P = MC. The loss is the triangle between demand and marginal cost over the gap: ½ × (Pm − MC) × (Qc − Qm).
Q: What does the Lerner index measure?
A: Market power: the markup of price over marginal cost divided by price, (P − MC)/P. It runs from zero under perfect competition upward and equals the reciprocal of demand elasticity at the chosen quantity.
Book a university microeconomics tutor in New York
Monopoly questions reward the student who can derive marginal revenue, land the MR = MC point, and read the profit and deadweight loss straight off the diagram. A one-on-one session builds exactly that — the markup rule, the elasticity condition, and the full welfare account — on the questions your course sets. Tell us your university and module, and we will match you with the right tutor this week, in person or online.
