ECON5001 · Microeconomic Theory
Oligopoly: Cournot, Stackelberg & Bertrand
An oligopoly has a handful of firms, each large enough that its own choice moves the market price — so every firm must reason strategically about its rivals. ECON5001 examines four canonical duopoly models, and the assessable skill is solving them, not reciting them. Cournot firms pick quantities simultaneously and best-respond through a downward-sloping reaction function; the Cournot-Nash equilibrium is where the reaction functions cross. Collusion mimics monopoly for higher joint profit but is unstable (a Prisoner's Dilemma). A Stackelberg leader moves first and substitutes the follower's reaction into its own profit, gaining a first-mover advantage. Bertrand firms set prices: identical goods collapse to price = marginal cost (the Bertrand paradox), while differentiated goods restore a positive markup. The reliable payoff is the welfare ranking — output runs Bertrand > Stackelberg > Cournot > monopoly, and price runs the other way.
What this chapter covers
- 011. Oligopoly & strategic interdependence — few firms, each a price-mover, game theory applied to firms
- 022. Cournot model — simultaneous quantity choice, profit πᵢ = [P(q₁+q₂) − c]·qᵢ taking the rival as given
- 033. Reaction (best-response) function — the FOC ∂πᵢ/∂qᵢ = 0 gives qᵢ = Rᵢ(qⱼ); quantities are strategic substitutes
- 044. Cournot-Nash equilibrium — intersect the reaction functions; output sits between monopoly and perfect competition
- 055. Collusion & the cartel — joint monopoly raises profit but each firm has an incentive to cheat (unstable)
- 066. Stackelberg leadership — leader substitutes the follower's reaction in and commits first → first-mover advantage
- 077. Bertrand price competition — homogeneous goods drive p → MC (Bertrand paradox); differentiated goods keep p > MC
- 088. Welfare ranking — output Bertrand > Stackelberg > Cournot > monopoly; price runs the opposite way
Cournot duopoly: reaction functions, equilibrium, price and profit
- +2(a) Write firm 1's profit, treating q₂ as fixed: π₁ = (120 − q₁ − q₂)·q₁ − 30·q₁ = (90 − q₁ − q₂)·q₁.
- +2(a) Take the first-order condition ∂π₁/∂q₁ = 90 − 2q₁ − q₂ = 0, which rearranges to the reaction function q₁ = 45 − ½q₂ (and by symmetry q₂ = 45 − ½q₁).
- +2(b) Impose symmetry q₁ = q₂ = q: q = 45 − ½q ⇒ 1.5q = 45 ⇒ q = 30. So q₁ = q₂ = 30 and Q = 60.
- +1(b) Price: P = 120 − 60 = 60. Each firm's profit: π = (60 − 30) × 30 = 900.
- +1(b) Check the band: a monopolist would set MR = 120 − 2Q = 30 ⇒ Q = 45; perfect competition gives P = MC ⇒ Q = 90. The Cournot Q = 60 sits between 45 and 90. ✓
Key terms
- Oligopoly
- A market with a small number of firms, each large enough that its own output or price choice noticeably affects the market price — so firms must act strategically, anticipating rivals' responses.
- Reaction (best-response) function
- A firm's profit-maximising choice as a function of its rival's choice, qᵢ = Rᵢ(qⱼ), obtained from the first-order condition ∂πᵢ/∂qᵢ = 0. In Cournot it slopes down (quantities are strategic substitutes); in differentiated Bertrand the price reaction slopes up (strategic complements).
- Cournot-Nash equilibrium
- The output pair where both reaction functions hold simultaneously — each firm best-responds to the other, so no firm can raise profit by unilaterally changing its quantity. It is the intersection of the reaction curves.
- Strategic substitutes vs complements
- Choices are strategic substitutes when more of one makes the other want less (Cournot quantities — reaction slopes down), and strategic complements when more of one makes the other want more (Bertrand prices — reaction slopes up).
- Collusion / cartel
- Firms jointly set output to maximise total profit, acting as a single monopolist. Joint profit beats Cournot, but each firm has a private incentive to expand and cheat, so collusion is unstable — a Prisoner's Dilemma.
- Stackelberg leadership
- Sequential quantity competition: a leader chooses output first and commits, having substituted the follower's reaction function into its own profit. Committing to a large output yields a first-mover advantage — the leader earns more than under Cournot.
- Bertrand paradox
- With homogeneous products and equal constant marginal cost, price undercutting drives the equilibrium price down to marginal cost with zero profit — just two firms reproduce the perfectly competitive outcome.
- Differentiated Bertrand
- When products are differentiated, each firm faces its own demand in both prices and keeps some loyal buyers even if dearer. Solving the price reaction functions gives equilibrium prices above marginal cost and positive profit.
Oligopoly: Cournot, Stackelberg & Bertrand FAQ
What's the difference between Cournot, Stackelberg and Bertrand?
They differ on two questions: what firms choose and when. Cournot = quantities chosen simultaneously. Stackelberg = quantities chosen sequentially (a leader moves first). Bertrand = prices chosen simultaneously. The solution machinery is the same every time — write profit as a function of both firms' choices, take the first-order condition for a reaction function, then solve the reaction functions together. The model name only fixes the choice variable and the timing.
Why does the Cournot reaction function slope down?
If your rival produces more, total output is higher and the market price is already lower, so your profit-maximising response is to produce less. More rival output ⇒ less of your own output — that is what 'strategic substitutes' means, and it makes each reaction curve downward-sloping. The two curves cross once, at the Cournot-Nash equilibrium.
How is Stackelberg different from Cournot if the demand and costs are the same?
In Cournot the leader takes the rival's output as given. In Stackelberg the leader knows the follower will best-respond, so it substitutes the follower's reaction function into its own profit before optimising. That commitment lets the leader produce more (q₁ = 45 on the standard P = 120 − Q, c = 30 economy) and earn more (1012.5) than under Cournot (900), while the follower earns less — the first-mover advantage. Total output is also higher, so welfare improves and deadweight loss falls.
Why is profit zero in Bertrand with identical products?
If both firms sell the same good and set prices, buyers all go to the cheaper seller. So whenever price exceeds marginal cost, each firm can profitably undercut the other by a cent and capture the whole market. Undercutting continues until price equals marginal cost, where there is nothing left to undercut. The unique equilibrium is p₁ = p₂ = MC with zero profit — the Bertrand paradox. It dissolves once you add product differentiation, capacity limits or repeated play.
What is the welfare ranking I should memorise?
For homogeneous goods with identical costs on linear demand: output runs Bertrand > Stackelberg > Cournot > monopoly/cartel, and price runs the opposite way. More firms, or sequential leadership, push output up, price down and deadweight loss down. On the P = 120 − Q, c = 30 economy that is Q = 90 / 67.5 / 60 / 45 and P = 30 / 52.5 / 60 / 75.
How do I know which model a question wants?
Read the cues. Homogeneous product and firms set quantities → Cournot (simultaneous) or Stackelberg (a clear first-mover or dominant lead firm). Differentiated product, or firms set prices → Bertrand. 'Identical product, price war' is the Bertrand-paradox signal; 'a dominant firm announces capacity first' is the Stackelberg signal.
Exam move
Drill the five-step solution loop until it is automatic, because every oligopoly item reuses it: (1) name the game — quantity or price, simultaneous or sequential; (2) write profit πᵢ as a function of both firms' choices with the rival's as a letter; (3) take the first-order condition to get a reaction function — and for Stackelberg substitute the follower's reaction in first; (4) solve the reaction functions together, imposing symmetry if the firms are identical; (5) back out price and profit, then sanity-check against the welfare ranking. Practise on a fresh economy such as P = 120 − Q with c = 30: get q = 30 / P = 60 / π = 900 for Cournot, q₁ = 45 / q₂ = 22.5 for Stackelberg, and p = MC = 30 / π = 0 for homogeneous Bertrand. Memorise two anchors — the n-firm shortcut q = (a − c)/(n + 1) for instant Cournot checks, and the output ranking Bertrand > Stackelberg > Cournot > monopoly — and watch the four recurring traps: differentiating the rival's output in Cournot, forgetting to substitute the follower's reaction in Stackelberg, expecting positive profit in homogeneous Bertrand, and confusing strategic substitutes (Cournot, reaction down) with strategic complements (Bertrand, reaction up).
