Cournot Duopoly
Two firms compete on quantity. The cleanest oligopoly model that's still useful in real markets.
Built and reviewed by Stephen Omukoko Okoth
Mathematical Economist · ex-Morgan Stanley FI · Equilar
Theory
What the model says, and why
Two firms simultaneously choose how much to produce. Each takes the rival's output as given. Inverse demand pins down price as a function of total quantity:
Each firm's profit:
Maximising w.r.t. qᵢ holding q_(j) fixed gives firm i's best-response function:
The Cournot-Nash equilibrium is the simultaneous solution of both best responses. With symmetric costs c it simplifies to qᵢ* = (a − c) / (3b). Total Cournot output sits between perfect competition and monopoly — closer to competition as the number of firms grows.
What the model is for. It captures imperfect competition without requiring collusion: even fully strategic firms taking each other's actions as given fall short of monopoly profit, but earn more than zero. It's the right starting point for thinking about telecoms, airlines, and any market with two-or-three dominant players.
Interactive playground
Move the parameters, watch the equilibrium move
Parameters
Demand and costs
Comparison benchmarks
Monopoly
Q = 40.0, P = 60.0
π = 1600
Perfect competition
Q = 80.0, P = 20.0
π = 0
Cournot-Nash
q₁* = 26.7, q₂* = 26.7, P* = 46.7
Firm 1 output (q₁*)
26.7
Firm 2 output (q₂*)
26.7
Market price
46.7
Total Q = 53.3
Profits π₁ / π₂
711 / 711
Best-response curves: BR₁ shows firm 1's optimal q given firm 2's choice; BR₂ vice versa. The equilibrium is the intersection.
In the classroom
How to teach it well
Convergence by iteration. Cournot's original story was iterative: firm 1 picks q assuming firm 2 produces zero; firm 2 then best-responds; firm 1 revises; etc. The process converges to the Nash equilibrium under standard conditions. Modern treatment skips the iteration and solves simultaneously, but the iterative intuition is what makes it click for students.
Asymmetric costs. Drop c₁ below c₂. The lower-cost firm produces more — but the higher-cost firm doesn't disappear. Both still earn positive profit in equilibrium. That's why inefficient incumbents survive in oligopolistic markets without explicit protection.
Cournot vs Bertrand. If firms compete on price instead of quantity (Bertrand), even two firms drive price to marginal cost. Quantity competition is far more forgiving. The choice between Cournot and Bertrand is one of the most important industry-specific calls — it depends on capacity constraints and product differentiation.
Adding firms. Generalising to n symmetric firms gives Q = n·(a−c)/((n+1)·b), P = (a + n·c)/(n+1). As n → ∞, P → c. That's the convergence to perfect competition, plotted on a single parameter.