📖 寡头模型：Stackelberg、卡特尔

Economics

I. Lesson Focus

This lesson examines two important oligopoly models beyond the simultaneous-move Cournot framework: the Stackelberg sequential-move model and cartel (collusive) arrangements. Candidates must be able to derive equilibrium quantities and prices under linear demand, compare profits across models, and evaluate the conditions that make cartels stable or unstable. The material directly supports Learning Outcome Statements on identifying oligopoly equilibria and analyzing strategic interactions.

II. The Problem

An industry is served by only two firms, A and B. Firm A chooses its output first; Firm B observes A’s choice and then selects its own output. Market price is determined by total industry output. Should Firm A produce more than it would under perfect competition? If the two firms secretly agree to restrict output and raise price (a cartel), can they sustain the agreement over time? What happens if one firm secretly increases production? These questions lie at the heart of the Stackelberg leader-follower model and cartel analysis—two of the most frequently tested non-Cournot oligopoly settings on the CFA Level I exam.

III. Core Characteristics of Oligopoly

An oligopoly exists when a small number of firms (typically 2–10) control the majority of market supply and each firm’s decisions materially affect rivals’ profits. Key features include: - Few sellers - Homogeneous or differentiated products - High barriers to entry (economies of scale, patents, regulation) - Strategic interdependence: one firm’s price or quantity decision directly influences competitors’ payoffs

Because firms recognize this interdependence, equilibrium depends on assumptions about timing, information, and ability to collude. The two models covered here—Stackelberg (sequential quantity) and cartel (explicit collusion)—produce different output, price, and profit outcomes than Cournot or perfect competition.

IV. The Stackelberg Model: Leader-Follower Equilibrium

Developed by Heinrich von Stackelberg, this model assumes one firm (the leader) commits to an output level first. The second firm (the follower) observes the leader’s output and then chooses its own output to maximize profit.

Model Assumptions

• Two firms facing linear inverse demand: P = a – b(Q₁ + Q₂)
• Constant marginal cost c for both firms, with c < a
• Firm 1 (leader) selects Q₁ first; Firm 2 (follower) reacts after observing Q₁
• Both firms maximize profit

Solution by Backward Induction

Step 1 – Follower’s reaction function
Follower profit: π₂ = [a – b(Q₁ + Q₂) – c]Q₂
Differentiating with respect to Q₂ yields:
Q₂ = (a – c)/(2b) – Q₁/2

Step 2 – Leader incorporates the reaction function
Leader profit: π₁ = [a – b(Q₁ + Q₂) – c]Q₁
Substitute Q₂ and differentiate with respect to Q₁:
Q₁* = (a – c)/(2b)

Substitute back to obtain follower output:
Q₂* = (a – c)/(4b)

Comparison with Cournot

• Cournot (simultaneous): each firm produces (a – c)/(3b); total Q = 2(a – c)/(3b)
• Stackelberg: leader = (a – c)/(2b), follower = (a – c)/(4b), total Q = 3(a – c)/(4b)
• Stackelberg produces higher total output and a lower price than Cournot. The leader earns more profit than the follower (first-mover advantage), yet joint profit is lower than under Cournot.

V. Cartel Model: Collusive Oligopoly

A cartel is an explicit or implicit agreement among oligopolists to restrict output, raise price, and share monopoly profit. The classic real-world example is OPEC.

Cartel Optimum

If the two firms perfectly collude, they jointly act as a monopolist: - Monopoly output: Qₘ = (a – c)/(2b) - Monopoly price: Pₘ = (a + c)/2 - Total monopoly profit: (a – c)²/(4b)

In a symmetric cartel each firm produces half: Q = (a – c)/(4b) per firm.

Cartel Instability and the Prisoner’s Dilemma

Although joint profits are maximized by collusion, each member has a dominant incentive to cheat by secretly expanding output while the other adheres to the quota. This incentive structure creates a classic Prisoner’s Dilemma: - Both collude → each earns monopoly profit / 2 - One cheats, one colludes → cheater earns nearly monopoly profit; complier earns very low profit - Both cheat → outcome collapses toward Cournot or even lower prices, with profits below the collusive level

For a cartel to remain stable over time, the following conditions must hold: - Effective monitoring of members’ output and prices - Credible, swift, and severe punishment for deviation - Small number of participants - Stable market demand with few external shocks - Legal or practical ability to operate without detection

Worked Cases

Case 1: Stackelberg Quantity and Profit Calculation

Market demand: P = 120 – 0.5(Q₁ + Q₂). Marginal cost = 20 for both firms.

Solution
Follower reaction: Q₂ = 100 – 0.5Q₁
Leader profit: π₁ = (50 – 0.25Q₁)Q₁
First-order condition: 50 – 0.5Q₁ = 0 → Q₁ = 100
Q₂ = 50
Total Q = 150, P = 45
π₁ = (45 – 20) × 100 = 2,500
π₂ = (45 – 20) × 50 = 1,250

The leader produces and earns exactly twice as much as the follower.

Case 2: Cartel versus Non-Cooperative Equilibria

Using the same demand and cost parameters:
- Perfect collusion (cartel): monopoly Qₘ = 100, Pₘ = 70. Each firm produces 50 and earns 2,500; joint profit = 5,000.
- Cournot: each firm produces ≈26.67, total Q = 80, P ≈ 60, each profit ≈1,066.67.
- Stackelberg (from Case 1): total profit = 3,750.

Cartel profit is highest but extremely fragile.

Case 3: Cartel Breakdown Scenario

Both firms agree to produce 50 each. Firm A secretly raises output to 80 while B still produces 50. Total Q = 130, P = 55.
- A’s profit = (55 – 20) × 80 = 2,800 (> 2,500 collusive profit)
- B’s profit = (55 – 20) × 50 = 1,750 (< 2,500)

If B retaliates by also producing 80, total Q = 160, P = 40, each earns only 1,600. The example illustrates how one deviation can trigger a destructive price war.

Traps

Key Formulas

• Linear demand P = a – bQ, constant MC = c:
• Stackelberg leader: $Q_1^* = \frac{a-c}{2b}$
• Stackelberg follower: $Q_2^* = \frac{a-c}{4b}$
• Stackelberg total: $Q = \frac{3(a-c)}{4b}$
• Cartel (monopoly) total: $Q_m = \frac{a-c}{2b}$
• Cartel output per firm (symmetric): $\frac{a-c}{4b}$
• Cartel stability conditions: effective monitoring + credible punishment + small number of firms + stable demand
• Profit ranking: Cartel joint profit > Stackelberg joint profit > Cournot joint profit

Practice Questions

Q1. In the Stackelberg model, the leader’s optimal output is typically:
A. Equal to the follower’s output
B. Twice the follower’s output
C. Half the follower’s output
D. Dependent on whether marginal costs differ

Q2. Market demand is P = 100 – 2Q and MC = 10. The Stackelberg leader’s output is:
A. 22.5
B. 11.25
C. 45
D. 15

Q3. Which factor is most likely to increase cartel stability?
A. Increasing the number of participants to ten
B. Effective monitoring of each member’s output
C. Highly volatile market demand
D. Strict legal prohibition of collusion

Q4. Compared with the Cournot model, total output under Stackelberg is:
A. Lower
B. Higher
C. The same
D. Not comparable

Q5. A cartel member’s strongest incentive is to:
A. Strictly adhere to the quota
B. Secretly increase output while others comply
C. Publicly announce price cuts
D. Exit the cartel immediately

Q6. When two symmetric firms perfectly collude, each should produce what fraction of the monopoly output?
A. One-quarter
B. One-half
C. Two-thirds
D. The entire monopoly output

Q7. Under linear demand, Stackelberg total output relative to perfect competition output is:
A. Equal
B. Greater
C. Smaller
D. Indeterminate

Q8. Which statement about OPEC is most accurate?
A. OPEC is best described by the Bertrand price-competition model
B. OPEC attempts to operate as a cartel but frequently suffers from member overproduction
C. OPEC members choose output simultaneously, consistent with Cournot
D. Every OPEC member acts as a Stackelberg leader

Answers

Takeaways

• Stackelberg is a sequential quantity game in which the leader enjoys a first-mover advantage, producing twice as much as the follower under linear demand.
• Stackelberg industry output lies between Cournot and perfect competition, resulting in an intermediate price and lower joint profit than Cournot.
• A cartel seeks to replicate monopoly profit but is undermined by strong individual incentives to cheat, creating a prisoner’s dilemma.
• Cartel durability requires effective monitoring, credible punishment, few participants, and stable demand.
• Exam questions frequently require numerical calculation of Stackelberg or cartel quantities and qualitative assessment of cartel-breakdown scenarios.
• Memorize the output relationships: monopoly < Stackelberg leader < Cournot < perfect competition.