📖 寡头垄断导论

Economics

I. Lesson Focus

This lesson introduces oligopoly as a market structure characterized by a small number of interdependent firms. Candidates must be able to distinguish oligopoly from perfect competition, monopolistic competition, and monopoly; derive equilibrium outcomes under the Cournot, Bertrand, Stackelberg, and cartel frameworks; and explain why cartels are inherently unstable. The material emphasizes strategic interaction, reaction functions, and the application of basic game theory (especially the prisoner’s dilemma) to oligopolistic behavior.

II. The Problem

An industry is dominated by only three large firms, each holding a significant market share with partially differentiated products. If one firm decides to increase output, how will the other two respond—by also increasing output, decreasing output, or holding output constant? Does this interdependence cause the market price to lie between the perfectly competitive and monopoly levels? CFA candidates frequently fail to correctly determine equilibrium quantity and price under oligopoly or to recognize the inherent instability of cartels. This lesson solves these core real-world and exam problems by providing clear models, numerical derivations, and common traps.

III. Characteristics of Oligopoly

Oligopoly is a market structure in which a few large firms account for the majority of industry output. Its defining features are: - Small number of firms (typically 2–10) - Strong interdependence: one firm’s pricing or output decision materially affects rivals’ profits - Products may be homogeneous (e.g., steel) or differentiated (e.g., automobiles) - High barriers to entry: economies of scale, patents, capital requirements, government licenses - Possibility of positive economic profit in both short and long run

Comparison with other market structures:

IV. The Central Challenge: Interdependence and Game Theory

Because of interdependence, oligopolists must anticipate rivals’ reactions. Traditional MR = MC analysis is insufficient; game-theoretic concepts such as reaction functions and Nash equilibrium become essential. Oligopoly situations can be modeled as non-cooperative games (Cournot, Bertrand, Stackelberg) or cooperative games (cartels).

V. The Cournot Model

The Cournot duopoly assumes two firms simultaneously choose quantities, each treating the rival’s output as fixed. Market demand is linear: $P = a - bQ$, where $Q = q_1 + q_2$. Marginal cost is constant at $c$.

Firm 1’s profit: $\pi_1 = (a - b(q_1 + q_2))q_1 - c q_1$.
Taking the derivative with respect to $q_1$ and setting to zero yields the reaction function: $$ q_1 = \frac{a - c}{2b} - \frac{1}{2}q_2 $$ The symmetric reaction function for firm 2 is identical. Solving simultaneously gives the Nash equilibrium: $$ q_1^ = q_2^ = \frac{a - c}{3b},\quad Q^ = \frac{2(a - c)}{3b},\quad P^ = \frac{a + 2c}{3} $$ Cournot total output lies between the competitive output $(a-c)/b$ and the monopoly output $(a-c)/(2b)$.

VI. The Bertrand Model

In the Bertrand model, firms simultaneously set prices rather than quantities, with homogeneous products and identical constant marginal costs. Price competition drives the equilibrium to $P_1 = P_2 = MC$, resulting in zero economic profit—identical to perfect competition. This counter-intuitive result is called the Bertrand paradox. In practice, product differentiation, capacity constraints, or repeated interaction soften the outcome.

VII. The Stackelberg Model

A leader-follower sequential game. The Stackelberg leader chooses quantity first; the follower observes and then optimizes. The leader’s reaction function is substituted into the follower’s best-response function. Equilibrium quantities are: $$ q_L = \frac{a - c}{2b},\quad q_F = \frac{a - c}{4b},\quad Q = \frac{3(a - c)}{4b} $$ The leader produces and earns more than in the simultaneous-move Cournot equilibrium, demonstrating a first-mover advantage.

VIII. Cartel Model

A cartel is an explicit or tacit agreement among oligopolists to restrict total output to the monopoly level and share the resulting profit. The cartel as a whole solves $MR = MC$, so industry output equals the monopoly quantity $Q_m = (a - c)/(2b)$. However, each member has a strong incentive to cheat by secretly expanding output while others adhere to the quota. This incentive structure is a classic prisoner’s dilemma, explaining why cartels are unstable (e.g., OPEC’s historical quota violations).

Worked Cases

Case 1: Cournot Duopoly Equilibrium

Market demand: $P = 200 - 2Q$. Both firms have $MC = 20$.

Solution steps: 1. Reaction function for firm 1: $q_1 = 45 - 0.5q_2$ 2. Symmetric reaction for firm 2. 3. Solve: $q_1 = 45 - 0.5(45 - 0.5q_1)$ → $q_1^ = q_2^ = 30$ 4. Total $Q^ = 60$, $P^ = 200 - 120 = 80$ 5. Each firm’s profit = $(80 - 20) \times 30 = 1,800$

Comparison: Perfect competition gives $Q = 90$, $P = 20$; monopoly gives $Q = 45$, $P = 110$. Cournot lies in between.

Case 2: Stackelberg First-Mover Advantage

Same demand and costs; firm 1 is the leader.

Leader anticipates follower’s reaction $q_2 = 45 - 0.5q_1$. Substituting into its profit function and maximizing yields $q_1^ = 45$, $q_2^ = 22.5$, $Q = 67.5$, $P = 65$. Leader profit = 2,025; follower profit = 1,012.5. The leader clearly benefits from moving first.

Case 3: Cartel Instability (Prisoner’s Dilemma)

Two firms can cooperate (each produce 22.5, monopoly price 110) or cheat (produce 45). Payoff matrix (profits):

The Nash equilibrium is mutual cheating, yielding far lower profits than sustained cooperation. This explains why cartels tend to collapse.

Traps

Key Formulas

• Cournot duopoly (linear demand): $q_1^ = q_2^ = \frac{a-c}{3b}$
• Cournot industry output: $Q^* = \frac{2(a-c)}{3b}$
• Stackelberg leader output: $q_L = \frac{a-c}{2b}$
• Cartel industry output = monopoly output: $Q_m = \frac{a-c}{2b}$
• Bertrand (homogeneous goods): $P_1 = P_2 = MC$
• Price ordering: $P_{monopoly} > P_{Cournot} > P_{Stackelberg} > P_{perfect\ competition}$

Practice Questions

Q1. In a Cournot duopoly with demand $P=120-3Q$ and constant $MC=10$ for both firms, each firm’s equilibrium output is closest to:
A. 12.2
B. 18.3
C. 36.7
D. 55

Q2. In the Bertrand model with homogeneous products and identical marginal costs, the equilibrium outcome is most likely:
A. Price equals average cost
B. Price equals marginal cost
C. Price equals the monopoly price
D. Price lies between Cournot and monopoly levels

Q3. Which feature best illustrates oligopolistic interdependence?
A. One firm raises price and rivals follow
B. Extremely large number of firms
C. Very low barriers to entry
D. Perfectly homogeneous products with no fixed costs

Q4. Compared with the Cournot model, the Stackelberg leader’s output is usually:
A. Lower
B. Higher
C. The same
D. Dependent on demand elasticity

Q5. The fundamental reason cartels are unstable is:
A. Government antitrust laws
B. Each member has an incentive to secretly increase output
C. Inelastic market demand
D. Low barriers to entry

Q6. If two oligopolists successfully form and maintain a cartel, the market outcome most closely resembles:
A. Perfect competition
B. Monopoly
C. Monopolistic competition
D. Bertrand equilibrium

Q7. In a linear-demand Cournot model, industry output as a fraction of the perfectly competitive output is:
A. 1/3
B. 1/2
C. 2/3
D. 3/4

Q8. Which statement about oligopoly is correct?
A. Long-run economic profit must be zero
B. Firms need not consider rivals’ reactions
C. High barriers to entry are essential for long-run positive profit
D. Bertrand firms typically earn positive economic profit

Answers

Takeaways

• Oligopoly is defined by few firms, high barriers, and strong strategic interdependence.
• Cournot quantity competition yields industry output equal to two-thirds of the competitive level.
• Bertrand price competition with homogeneous goods produces $P = MC$.
• Stackelberg first-mover (leader) enjoys higher output and profit than the follower.
• Cartels aim for monopoly profit but collapse because of cheating incentives (prisoner’s dilemma).
• Equilibrium prices typically satisfy: monopoly > Cournot > Stackelberg > perfect competition.