Monopoly and perfect competition are the easy cases: one firm has no rivals to watch, and a competitive firm is too small for rivals to notice. Oligopoly is the hard middle, where each firm’s best choice depends on what the other does. Cournot competition is the version intermediate papers test most — and a staple request brought to any microeconomics tutor in London.
1 · Two firms choosing quantities
The setup is deliberately spare. Two firms sell an identical product, and each chooses its quantity — how much to produce — once, at the same time as its rival. With inverse demand P = a − b(q₁ + q₂), the market sells whatever the two firms bring at the price that clears it.
The tension is immediate. Every extra unit earns a margin, but it drags the price down on everything you — and your rival — already sell. Your best output depends on your rival’s. That dependence is the whole subject.
2 · The reaction function
Take firm 1. Treat q₂ as fixed, and maximise profit:
π₁ = [a − b(q₁ + q₂) − c] q₁
Differentiate with respect to q₁ and set the result to zero. The first-order condition solves to
q₁ = (a − c)/(2b) − q₂/2
This is firm 1’s reaction function (or best-response function): the profit-maximising q₁ for every possible q₂. It slopes down — the more your rival produces, the less you want to. Quantities are strategic substitutes: rival output depresses the price, shrinks the margin on your marginal unit, and makes you pull back.
Its two ends are old friends. A rival producing nothing leaves you the monopoly quantity, (a − c)/(2b). A rival flooding the market with the competitive quantity, (a − c)/b, drives price to marginal cost, and your best response is zero.
3 · Where the two lines cross
Firm 2 faces the mirror-image problem and gets the mirror-image reaction function. Plot both in (q₁, q₂) space and they cross exactly once. That crossing is the Cournot–Nash equilibrium: each firm’s output is the best response to the other’s, simultaneously. It is a Nash equilibrium in the familiar sense — neither firm can raise its profit by changing its own output alone.
Why nowhere else? At any other point, at least one firm is off its reaction function and would change its output if given the chance. Only the intersection survives. Examiners reward this argument stated in one clean sentence.
4 · Between monopoly and competition
With symmetric firms and linear demand, each firm produces (a − c)/(3b), so the duopoly’s total output is two-thirds of the competitive quantity — comfortably above the monopoly’s half. Price lands strictly between marginal cost and the monopoly price.
The logic generalises: with n symmetric firms, total output is n/(n + 1) times the competitive quantity. One firm gives the monopoly outcome, two give Cournot, and as n grows the market slides toward P = MC — which is why concentrated industries price above cost without any collusion story.
Worked example — a bottled spring-water duopoly
Two producers bottle water from neighbouring springs and sell into the same market. Inverse demand is P = 120 − Q, with P in pence per litre and Q = q₁ + q₂ in thousands of litres per week. Both firms bottle at a constant marginal cost of 30p per litre.
Step 1 — Firm 1’s profit. π₁ = (120 − q₁ − q₂ − 30) q₁ = (90 − q₁ − q₂) q₁.
Step 2 — Firm 1’s reaction function. The first-order condition is 90 − 2q₁ − q₂ = 0, so q₁ = 45 − q₂/2. Alone, firm 1 would bottle the monopoly quantity, 45.
Step 3 — Firm 2’s reaction function. The problem is symmetric: q₂ = 45 − q₁/2.
Step 4 — Solve the system. Substitute: q₁ = 45 − (45 − q₁/2)/2, which gives (3/4)q₁ = 22.5, so q₁* = 30 and, by symmetry, q₂* = 30 — the crossing in the diagram below.
Step 5 — Price and profits. Q = 60, so P* = 120 − 60 = 60p. Each firm earns a 30p margin on 30,000 litres: £9,000 per week. Check the benchmarks: a monopolist would sell 45 at 75p; competition would force 90 at 30p. Cournot lands between both.
Step 6 — A cost shock. Firm 2 signs a cheaper filtration contract and its marginal cost falls to 15p. Its reaction function shifts out to q₂ = 52.5 − q₁/2; firm 1’s is unchanged.
Step 7 — The new equilibrium. Solving the new system: q₁ = 25, q₂ = 40, so Q = 65 and P = 55p. Firm 2 earns (55 − 15) × 40,000 = £16,000 a week; firm 1’s profit falls to (55 − 30) × 25,000 = £6,250.
Step 8 — Interpretation. The low-cost firm expands; its rival — costs unchanged — contracts, squeezed through the market price. Total output rises, price falls, and consumers pocket part of the saving. A cost advantage in Cournot buys market share, not just margin.
Could you derive the reaction function from the profit function — not just quote it — and say in one sentence why only the crossing survives? That derivation carries the marks, and the equilibrium argument is where careless answers lose them. Rehearsing both on past papers is exactly what a one-on-one microeconomics tutor does with you. Book a trial session.
Practice
Q1. Two rival coffee-cart operators face market demand P = 100 − 2Q and each has marginal cost 16. Find each cart’s Cournot output, the market price, and each operator’s profit.
Q2. A third bottler with marginal cost 30p enters the spring-water market above (P = 120 − Q). Find each firm’s output, total output, the price, and each firm’s weekly profit.
Q3. A market has inverse demand P = 80 − Q and constant marginal cost 20. Compute total output and price under monopoly, Cournot duopoly, and perfect competition, and confirm the output ranking.
Answers. Q1: each produces 14, so Q = 28 and P = 100 − 56 = 44; profit each = (44 − 16) × 14 = 392. Q2: each of the three produces 22.5; Q = 67.5, P = 52.5p; profit each = 22.5p × 22,500 litres = £5,062.50. Q3: monopoly 30 at 50; Cournot 40 at 40; competition 60 at 20. Ranking: 30 < 40 < 60, with prices ordered the opposite way.
Key takeaways
- Cournot firms choose quantities simultaneously; price clears whatever total output the market receives.
- A reaction function is a solved first-order condition: q₁ = (a − c)/(2b) − q₂/2 for linear demand. It slopes down — quantities are strategic substitutes.
- The Cournot–Nash equilibrium is the crossing point of the reaction functions: each output a best response to the other, so neither firm regrets its choice.
- Cournot sits strictly between the poles: duopoly delivers 2/3 of competitive output; with n firms the fraction n/(n + 1) climbs toward competition.
Why London students choose our microeconomics tutoring
- Matched to your syllabus: whether your module sets Cournot with calculus or with the shortcut formulas, sessions use your lecture notes’ notation and your problem sets.
- Derivations drilled, not memorised: the reaction-function derivation carries the marks, and it is rehearsed until it comes out cleanly under exam pressure.
- One-on-one and online: tutors work through your past papers over a shared screen, so each session lands on the questions your exam will ask.
FAQ
Q: What is Cournot competition in simple terms?
A: Firms selling the same product each decide how much to produce, at the same time, and the price adjusts to sell the total. Each firm’s best output depends on its rivals’, and equilibrium is where all the best responses hold at once.
Q: How do I derive a reaction function?
A: Write one firm’s profit treating rivals’ quantities as fixed, differentiate with respect to its own quantity, set the derivative to zero, and solve. The result — best output as a function of rivals’ outputs — is the reaction function.
Q: Why is the Cournot price between the monopoly and competitive prices?
A: Each firm restricts output to protect the price, but ignores the damage its extra units do to rivals’ margins. Total output therefore lands above the monopoly level and below the competitive level, and price strictly between the two.
Q: What is the difference between Cournot and Bertrand competition?
A: In Cournot firms set quantities; in Bertrand they set prices, which produces a much fiercer outcome — a topic we cover on its own page.
Q: What happens to the Cournot price as more firms enter?
A: It falls toward marginal cost: with n symmetric firms output is n/(n + 1) of the competitive quantity, so each entrant closes part of the gap. Concentration is what keeps price above cost.
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Cournot questions reward students who can derive the reaction function, not just quote it. One-on-one sessions build that fluency on your course’s own past papers. Tell us your module and exam date, and we’ll match you with the right tutor this week.
