Oligopoly Models: Cournot, Bertrand & Stackelberg — Complete Derivations & DU Exam Numericals

Oligopoly — a market structure dominated by a small number of strategically interdependent firms — is one of the most analytically rich topics in intermediate microeconomics. Unlike perfect competition or monopoly, oligopoly models require game-theoretic reasoning because each firm's optimal decision depends on what it expects its rivals to do.

For Delhi University B.A. (Hons) Economics students studying Intermediate Microeconomics I & II, oligopoly models consistently appear as 15-mark to 20-mark compulsory numerical questions. In this 4,000-word masterclass, we derive the equilibrium outcomes for the three canonical oligopoly models — Cournot (simultaneous quantity), Bertrand (simultaneous price), and Stackelberg (sequential quantity) — and solve 3 complete step-by-step examination numericals.

1. Market Structure Spectrum

2. The Cournot Model (Simultaneous Quantity Competition)

Developed by Antoine Augustin Cournot (1838), this model assumes firms simultaneously choose output quantities, taking rivals' quantities as given.

2.1 Setup & Derivation

Consider a duopoly with inverse demand P = a - b*Q where Q = q1 + q2. Both firms have identical constant marginal cost c (no fixed costs).

Firm 1's Profit Maximization:

First-Order Condition (FOC): Take derivative with respect to q1 and set equal to 0:

Solving for q1 gives Firm 1's Best Response Function (Reaction Function):

By symmetry, Firm 2's Best Response is:

2.2 Cournot-Nash Equilibrium

At equilibrium, both reaction functions are satisfied simultaneously. By symmetry, q1* = q2*:

2.3 Generalization to N Firms

With N symmetric Cournot competitors:

As N approaches infinity, P* approaches c and Q* approaches (a - c) / b — the perfectly competitive outcome.

3. The Bertrand Model (Simultaneous Price Competition)

Joseph Bertrand (1883) argued that firms compete on price rather than quantity. With homogeneous products and equal marginal costs, Bertrand competition yields a dramatically different result.

3.1 The Bertrand Paradox

If both firms produce identical goods and consumers buy from the cheapest supplier:

• If p1 > p2: Firm 1 sells nothing, Firm 2 captures entire market.
• If p1 = p2: Market is split equally.
• If p1 < p2: Firm 1 captures entire market.

Each firm has an incentive to undercut the rival by an infinitesimal amount. This undercutting spiral continues until:

The Bertrand Paradox: With just 2 firms, the market achieves the perfectly competitive outcome (P = MC, zero economic profits). This is paradoxical because we would expect firms with market power to earn positive profits.

3.2 Escaping the Bertrand Paradox

• Product Differentiation: If goods are not perfect substitutes (e.g., Coca-Cola vs Pepsi), firms can sustain P > MC.
• Capacity Constraints: If firms cannot serve the entire market (Edgeworth model), undercutting to MC is not sustainable.
• Repeated Interaction: In infinitely repeated games, firms can sustain tacit collusion above MC.

4. The Stackelberg Model (Sequential Quantity Leadership)

Heinrich von Stackelberg (1934) modified Cournot by introducing sequential moves: Firm 1 (the Leader) moves first, and Firm 2 (the Follower) observes the leader's quantity before choosing its own.

4.1 Backward Induction Solution

Step 1: Solve for the Follower's Best Response (same as Cournot):

Step 2: Substitute into Leader's Profit Function:

FOC:

Step 3: Follower's Equilibrium Quantity:

4.2 Key Stackelberg Results

First-Mover Advantage: The Stackelberg leader earns higher profit than the follower because committing to a large quantity first forces the follower to accommodate by producing less.

5. Worked Examination Numerical Problems

Numerical 1: Cournot Duopoly with Asymmetric Costs (15 Marks)

Question: Market demand is P = 120 - Q. Firm 1 has MC = 20, Firm 2 has MC = 30. Find Cournot equilibrium quantities, price, and profits.

Solution:

Firm 1's BR: q1 = (120 - 20 - q2) / 2 = 50 - q2/2

Firm 2's BR: q2 = (120 - 30 - q1) / 2 = 45 - q1/2

Substitute q2 into q1's BR:

Numerical 2: Stackelberg with Linear Demand (15 Marks)

Question: P = 200 - 2Q, MC = 40 for both firms. Firm 1 is the Stackelberg leader. Find equilibrium.

Solution:

Follower's BR: q2 = (200 - 40)/(2*2) - q1/2 = 40 - q1/2

Leader substitutes:

6. Summary Comparison & Exam Tips

• Cournot: Firms compete on quantity simultaneously. Output between monopoly and perfect competition.
• Bertrand: Firms compete on price. With identical products, P = MC (the paradox).
• Stackelberg: Sequential quantity. Leader produces more, earns more (first-mover advantage).
• Always show complete reaction function derivation in DU exams for full marks.

For related microeconomics guides, see our Game Theory & Nash Equilibrium Guide and Consumer Theory & Indifference Curves.