Monopoly: maximising profit
A monopolist picks the quantity where marginal revenue equals marginal cost and reads the price off the demand curve. It sells less and charges more than a competitive market.
Formulas and rules
- Revenue R(Q) = p(Q) × Q
- With p = α − β·Q: marginal revenue MR = α − 2β·Q (twice as steep)
- Profit maximum: MR = MC → Q_M
- Price: put Q_M into the inverse demand function
- Competition: p = MC; deadweight loss = triangle between the two quantities
Your turn
Three problems on this topic. Type numbers like 12.5 or 1,250 and elasticities with their sign.
Task
A monopolist faces the inverse demand p = 272 − 4Q (p in €). Marginal cost is constant at €8.00 and there are no fixed costs. Find the profit-maximising quantity, the monopoly price, the profit and the deadweight loss compared with perfect competition.
Profit-maximising quantity
Monopoly price in €
Profit in €
Deadweight loss in €
Numbers like 12.5 or 1,250; elasticities with their sign (−0.75); percentages as a number (12.5). Round to 2 decimal places.
Worked example
A monopolist faces the inverse demand p = 45 − 2Q (p in €). Marginal cost is constant at €25.00 and there are no fixed costs. Find the profit-maximising quantity, the monopoly price, the profit and the deadweight loss compared with perfect competition.
Worked solution
- 1.Revenue R = p × Q = 45Q − 2Q² → marginal revenue MR = 45 − 4Q
- 2.MR = MC: 45 − 4Q = 25 → Q = 5
- 3.Price from demand: p = 45 − 2 × 5 = 35
- 4.Profit = (35 − 25) × 5 = €50.00
- 5.Competition: p = MC → Q = 10; deadweight loss = ½ × (35 − 25) × (10 − 5) = €25.00
Typical mistake: Setting price equal to marginal cost (that is the competitive outcome), or putting the quantity from MR = MC into the marginal revenue function instead of the demand function.
Common questions
Why is marginal revenue twice as steep as demand?
To sell one more unit the monopolist has to cut the price on every unit, not just the last one. With linear demand that doubles the slope.