📖 寡头模型：Cournot、Bertrand

Economics

I. Lesson Focus

This lesson examines the two classic oligopoly models tested on the CFA Level I exam: the Cournot (quantity) model and the Bertrand (price) model. Candidates must be able to derive reaction functions, solve for Nash equilibria, calculate equilibrium price, quantity, and profit for each model, and compare outcomes with monopoly and perfect competition. The lesson also addresses the Bertrand paradox and the effect of product differentiation.

II. The Problem

Suppose two airlines, A and B, dominate a popular route. Each must decide how many daily flights to offer. If both fly many flights, fares collapse and profits shrink; if both restrict output, fares stay high but each risks losing market share. How will they behave? The Cournot model assumes firms simultaneously choose quantities; the Bertrand model assumes they simultaneously choose prices. What are the resulting equilibrium price, output, and profit compared with monopoly or perfect competition? These strategic-interaction outcomes are frequent CFA exam topics.

III. Core Characteristics of Oligopoly

An oligopoly is a market structure in which a small number of firms (typically 2–10) control the majority of market share. Key features include: - Few sellers leading to strategic interdependence: each firm’s decision affects and is affected by rivals. - Products may be homogeneous (steel, cement) or differentiated (cars, airline seats). - High barriers to entry (economies of scale, patents, regulation). - Behavior is inherently game-theoretic; firms must anticipate rivals’ reactions.

Unlike perfect competitors (price takers) or a monopolist (sole price setter), oligopolists are interdependent price/quantity setters.

IV. The Cournot Model (Quantity Competition)

Developed by Augustin Cournot in 1838, the model assumes: - Firms simultaneously and independently choose output quantities. - Products are homogeneous; market price is determined by total quantity: $P = a - b(Q_1 + Q_2)$. - Constant and identical marginal cost $MC = c$ for both firms. - Each firm maximizes its own profit, treating the rival’s output as fixed (an early version of Nash equilibrium).

Reaction Functions

Firm 1’s profit: $\pi_1 = (a - b(Q_1 + Q_2))Q_1 - cQ_1$.
Taking the derivative with respect to $Q_1$ and setting it to zero yields Firm 1’s reaction function:
$Q_1 = \frac{a - c}{2b} - \frac{1}{2}Q_2$.

Firm 2’s reaction function is symmetric:
$Q_2 = \frac{a - c}{2b} - \frac{1}{2}Q_1$.

Cournot-Nash Equilibrium

Solving the two reaction functions simultaneously gives the symmetric equilibrium:
$Q_1^ = Q_2^ = \frac{a - c}{3b}$,
Total quantity $Q^ = \frac{2(a - c)}{3b}$,
Equilibrium price $P^ = \frac{a + 2c}{3}$,
Profit per firm $\pi^* = \frac{(a - c)^2}{9b}$.

Compared with monopoly, Cournot output is higher, price is lower, and profit is smaller. Compared with perfect competition, Cournot output is lower, price is higher, and positive economic profit exists.

V. The Bertrand Model (Price Competition)

The Bertrand model assumes firms simultaneously set prices, capacity is unlimited, products are homogeneous, and consumers buy only from the cheapest seller.

Key assumptions: - Perfectly homogeneous products. - Identical constant marginal cost $MC = c$. - “Winner-takes-all” demand: the firm with the lowest price captures the entire market; equal prices split the market; a higher price yields zero sales.

Bertrand Equilibrium

With homogeneous products, identical costs, and no capacity constraints, the unique Nash equilibrium is $P_1^ = P_2^ = MC = c$. Both firms earn zero economic profit.

This outcome is known as the Bertrand paradox: even with only two firms, price competition drives the market to the perfectly competitive outcome (price equals marginal cost).

When products are differentiated, equilibrium prices lie above marginal cost ($P^* > c$); the greater the differentiation, the higher the equilibrium price.

VI. Cournot versus Bertrand: A Comparison

Worked Cases

Case 1: Cournot Duopoly Equilibrium

Market demand: $P = 100 - 2(Q_1 + Q_2)$, $MC_1 = MC_2 = 10$.

Steps: 1. Firm 1 reaction function: $Q_1 = (100 - 10)/4 - 0.5Q_2 = 22.5 - 0.5Q_2$. 2. Firm 2 reaction function is symmetric. 3. Solving simultaneously: $Q_1^ = Q_2^ = 15$. 4. Total $Q = 30$, $P = 100 - 2 \times 30 = 40$. 5. Profit per firm = $(40 - 10) \times 15 = 450$.

Conclusion: Cournot price of 40 lies well above MC = 10 but below the monopoly price of 55.

Case 2: Bertrand Equilibrium with Homogeneous Products

Using the same demand and cost as Case 1. If both firms set prices simultaneously: - Any price above 10 invites the rival to undercut and capture the entire market. - Equilibrium: $P_1^ = P_2^ = 10$ (equal to MC), profit = 0 for both. - Total quantity = $100 - 2 \times 10 = 80$ (perfectly competitive level).

Case 3: Bertrand with Differentiated Products

Demand: $Q_1 = 60 - 2P_1 + P_2$, $Q_2 = 60 - 2P_2 + P_1$, $MC = 5$ for both.

Best-response price functions:
$P_1 = 17.5 + 0.25P_2$,
$P_2 = 17.5 + 0.25P_1$.

Equilibrium: $P_1^ = P_2^ = 23.33$, $Q_1 = Q_2 \approx 36.67$, profit per firm ≈ 673.61.

Conclusion: Differentiation raises equilibrium price substantially above marginal cost.

Traps

Key Formulas

• Cournot reaction function (symmetric MC = c): $Q_1 = \frac{a-c}{2b} - \frac{1}{2}Q_2$
• Cournot equilibrium output per firm: $Q_i^* = \frac{a-c}{3b}$
• Cournot equilibrium price: $P^* = \frac{a + 2c}{3}$
• Bertrand homogeneous-goods equilibrium: $P_1^ = P_2^ = MC$
• With product differentiation, Bertrand price > MC; greater differentiation raises price further
• Profit ranking: Monopoly > Cournot > differentiated Bertrand > homogeneous Bertrand = perfect competition

Practice Questions

Q1. In a Cournot duopoly, market demand is $P = 120 - 3Q$ and both firms have $MC = 20$. Each firm’s equilibrium output is closest to:
A. 10
B. 11.11
C. 13.33
D. 20

Q2. In the Bertrand model with homogeneous products and identical marginal costs and no capacity constraints, the equilibrium price equals:
A. The monopoly price
B. Marginal cost
C. Average cost
D. The Cournot equilibrium price

Q3. Relative to perfect competition, a Cournot oligopoly produces:
A. Higher output and lower price
B. Lower output and higher price
C. The same output and price
D. Higher profit but zero economic profit

Q4. Which industry is best described by the Cournot model?
A. Supermarket retailing (frequent price changes)
B. Cement manufacturing (slow capacity adjustment)
C. Airlines using real-time pricing systems
D. E-commerce flash-sale platforms

Q5. When the two oligopolists’ products are significantly differentiated, the Bertrand equilibrium price will be:
A. Equal to marginal cost
B. Above marginal cost
C. Equal to the competitive price
D. Equal to the monopoly price

Q6. If the two firms in Case 1 merge into a monopolist, the new equilibrium price becomes:
A. 10
B. 40
C. 55
D. 70

Q7. In the Cournot model, a firm’s reaction function has a negative slope. This implies:
A. When the rival increases output, the firm increases its own output
B. When the rival increases output, the firm reduces its own output
C. Output and price are positively related
D. Profit rises when the rival increases output

Q8. The central economic intuition of the Bertrand paradox is that:
A. Oligopolists always collude
B. Price competition is far more intense than quantity competition
C. Capacity constraints always raise price
D. Differentiated products necessarily produce zero profit

Answers

Takeaways

• Cournot firms compete on quantity; each produces one-third of the competitive output, price lies between monopoly and competitive levels.
• Bertrand firms compete on price; with homogeneous goods, equilibrium price equals MC and profit is zero (Bertrand paradox).
• Product differentiation softens price competition and raises equilibrium price above MC.
• Reaction functions are the essential tool for finding Nash equilibria; negative slope indicates strategic substitutes.
• The speed of capacity adjustment in an industry determines whether Cournot or Bertrand is the more appropriate framework.
• Profit ranking in oligopoly: monopoly > Cournot > differentiated Bertrand > homogeneous Bertrand = perfect competition.