Cournot vs Bertrand vs Stackelberg: The Three Oligopoly Models Compared

The Cournot, Bertrand and Stackelberg models are the three classic models of competition between a few firms, and they differ only in what each firm chooses and when. In Cournot, firms choose quantities at the same time; in Bertrand, firms choose prices at the same time; in Stackelberg, a leader chooses its quantity first and a follower responds.

This guide solves all three on one linear demand curve, compares them with monopoly and perfect competition in one table, and explains why two price-setting firms can earn zero profit (the Bertrand paradox). If you want the broader picture of oligopoly first (interdependence, cartels and payoff matrices), start with Oligopoly Explained.

The setup: one market, five outcomes

Every model below uses the same ingredients:

• Inverse market demand: P = a - bQ, where Q is total industry output.
• Two identical firms (except in the monopoly and competitive benchmarks), each with constant marginal cost c and no fixed cost.
• Identical products, so buyers care only about price.

For the numbers, use P = 130 - Q (so a = 130 and b = 1) and marginal cost c = $10 at every firm. The gap a - c = 120 drives every answer.

### Benchmark 1: perfect competition

Competition pushes price down to marginal cost. Set 130 - Q = 10, so Q = 120 and P = $10. Every unit sells at cost, so profit is $0.

### Benchmark 2: monopoly

A single seller sets marginal revenue equal to marginal cost. With linear demand, marginal revenue has twice the slope of demand: MR = 130 - 2Q.

• Set MR = MC: 130 - 2Q = 10, so 2Q = 120 and Q = 60.
• Price from demand: P = 130 - 60 = $70.
• Profit: (70 - 10) × 60 = 60 × 60 = $3,600.

This is also what two firms would earn together if they formed a perfect cartel and split the monopoly output.

Cournot: quantity competition, simultaneous moves

In Cournot competition, each firm picks an output level while treating its rival's output as fixed. The market price then adjusts so that the combined output sells.

### Step 1: write one firm's profit

Firm 1 earns price times its own output, minus cost:

profit₁ = (130 - q₁ - q₂) × q₁ - 10q₁

### Step 2: find the reaction function

Holding q₂ fixed, firm 1's marginal revenue is MR₁ = 130 - 2q₁ - q₂. Set it equal to marginal cost:

• 130 - 2q₁ - q₂ = 10
• 2q₁ = 120 - q₂
• q₁ = 60 - q₂/2

This is firm 1's reaction function. Firm 2's is the mirror image: q₂ = 60 - q₁/2.

### Step 3: solve where the reaction functions cross

The firms are identical, so in equilibrium q₁ = q₂ = q:

• q = 60 - q/2
• 1.5q = 60
• q = 40

Check: plug q₂ = 40 into firm 1's rule: 60 - 40/2 = 60 - 20 = 40. Each firm's best reply to 40 is 40, so neither wants to move. That mutual best response is exactly what a Nash equilibrium is.

### Step 4: price and profit

• Total output: Q = 40 + 40 = 80.
• Price: P = 130 - 80 = $50.
• Profit per firm: (50 - 10) × 40 = 40 × 40 = $1,600.
• Industry profit: 2 × 1,600 = $3,200.

The general duopoly formulas are q = (a - c)/3b per firm and P = (a + 2c)/3. Check: (130 + 2 × 10)/3 = 150/3 = $50. For any number of firms, the Cournot equilibrium calculator uses q = (a - c)/[(n + 1)b].

### Why Cournot firms do not stay at the cartel output

Suppose the two firms agree to split the monopoly output, 30 units each. Price would be 130 - 60 = $70, and each would earn (70 - 10) × 30 = $1,800, more than the $1,600 Cournot profit. But look at firm 1's reaction function: if firm 2 sticks to 30, firm 1's best reply is 60 - 30/2 = 45.

• Total output: 45 + 30 = 75, so P = 130 - 75 = $55.
• Firm 1 (cheater): (55 - 10) × 45 = 45 × 45 = $2,025.
• Firm 2 (loyal): (55 - 10) × 30 = 45 × 30 = $1,350.

Each firm gains by cheating, the prisoner's dilemma structure covered in Game Theory for AP Economics.

Bertrand: price competition, simultaneous moves

In Bertrand competition, each firm posts a price. Buyers purchase from whichever firm is cheaper, and the firms split the market if their prices tie.

### Step 1: test a price above cost

Suppose both firms charge the Cournot price of $50. Demand is Q = 130 - 50 = 80, so each sells 40 units and earns (50 - 10) × 40 = $1,600.

Now firm 1 cuts its price to $49. It takes the whole market: Q = 130 - 49 = 81 units, and its profit becomes (49 - 10) × 81 = 39 × 81 = $3,159. A one dollar cut nearly doubles its profit, so $50 cannot be an equilibrium, and the same logic rules out any common price above $10.

### Step 2: test a price at cost

At P = $10, demand is 130 - 10 = 120, so each firm sells 60 units and earns (10 - 10) × 60 = $0.

• Raising the price above $10 loses every customer to the rival, so profit stays $0.
• Cutting below $10 wins the market but loses money on every unit.

No firm can do better by changing its price, so both pricing at marginal cost is the Nash equilibrium. Output is 120, the same as perfect competition.

### The Bertrand paradox

The Bertrand paradox is the result that just two firms, selling identical goods at identical constant cost, compete price all the way down to marginal cost and earn nothing. It feels wrong because two sellers should have some market power. Relaxing any one of its four assumptions can lift price above cost:

The last row is easy to check. If firm 1's marginal cost is $10 and firm 2's is $20, firm 1 prices just below $20. Firm 2 cannot match without losing money, so firm 1 serves the market at a margin of almost $10 per unit. For a side by side of the two simultaneous models, see Cournot vs Bertrand.

Stackelberg: quantity leadership

In the Stackelberg model, the leader chooses its output first, the follower observes it, and then the follower chooses. You solve it by backward induction: work out what the follower will do, then let the leader plan around that.

### Step 1: the follower's best response

The follower faces the same problem as a Cournot firm with the leader's output already fixed, so its reaction function is the same: q₂ = 60 - q₁/2.

### Step 2: the leader plans around that response

Substitute the follower's reaction into the demand curve:

• P = 130 - q₁ - (60 - q₁/2)
• P = 70 - q₁/2

The leader's profit is (70 - q₁/2 - 10) × q₁ = 60q₁ - q₁²/2. Its marginal profit is 60 - q₁, which equals zero at q₁ = 60.

### Step 3: fill in the rest

• Follower: q₂ = 60 - 60/2 = 30.
• Total output: Q = 60 + 30 = 90.
• Price: P = 130 - 90 = $40.
• Leader profit: (40 - 10) × 60 = 30 × 60 = $1,800.
• Follower profit: (40 - 10) × 30 = 30 × 30 = $900.
• Industry profit: 1,800 + 900 = $2,700.

Check with the general formulas: the leader makes (a - c)/2b = 120/2 = 60, the follower makes (a - c)/4b = 120/4 = 30, and price is (a + 3c)/4 = (130 + 30)/4 = $40. All three match. The Stackelberg model calculator runs the same steps for other numbers.

The leader earns $1,800 instead of its $1,600 Cournot profit, and the follower drops from $1,600 to $900. That gap is the first mover advantage. It only exists if the leader's output is a credible commitment, such as capacity already built; otherwise the game falls back to Cournot. For the two quantity models side by side, see Cournot vs Stackelberg.

The comparison table

All five outcomes on P = 130 - Q with marginal cost $10:

The same ranking in general form, for P = a - bQ and marginal cost c:

### What buyers lose

Consumer surplus is the triangle under demand and above price: (130 - P) × Q/2. Deadweight loss is the triangle between the demand curve and marginal cost over the units that are never produced: (P - 10) × (120 - Q)/2.

Every row adds back to $7,200, the total surplus available when all 120 efficient units are produced. That is a useful check on the arithmetic. The deadweight loss column measures the gains from trade left on the table, which is the same idea as a missed Pareto improvement in Pareto Efficiency Explained.

The intuition behind the ranking

The ranking comes down to how much a firm gains by being aggressive.

• Bertrand: a tiny price cut steals the whole market. The reward for undercutting is enormous, so price falls until there is nothing left to steal.
• Cournot: an extra unit lowers the market price only a little, and the firm counts that loss only on its own units, not its rival's. Each firm expands beyond the cartel level, but stops well short of the competitive level.
• Stackelberg: the leader knows every extra unit it commits to makes the follower cut back by half a unit, so expanding is cheaper for the leader than for a Cournot firm. Industry output rises, price falls, and profit shifts from the follower to the leader.
• Monopoly: one firm counts the price drop on every unit sold, so it restricts output the most.

AP Microeconomics tests oligopoly mainly through payoff matrices, dominant strategies and Nash equilibrium rather than these named models. The named models appear in intermediate microeconomics and industrial organization courses, and they explain why the cooperative cell in an AP payoff matrix is so hard to sustain.

Common mistakes

1. Using market marginal revenue for a Cournot firm. Setting MR = 130 - 2Q equal to $10 gives Q = 60, which is the cartel answer. Each Cournot firm has its own marginal revenue, 130 - 2q₁ - q₂.

2. Forgetting to use total output in the demand curve. Price depends on Q = q₁ + q₂, not on one firm's output.

3. Thinking the Stackelberg leader wins just by acting sooner. The advantage needs a commitment the follower believes.

4. Assuming higher leader profit means higher industry profit. Stackelberg industry profit ($2,700) is lower than Cournot ($3,200), even though the leader does better.

5. Taking the Bertrand result as a prediction for real oligopolies. It depends on identical goods, equal costs, no capacity limits and one-shot play.