6.1 Cournot and Bertrand Competition Models

Cournot and Bertrand models are key frameworks for understanding market competition. Cournot focuses on quantity-based competition, while Bertrand emphasizes price-based rivalry. These models help explain how firms make strategic decisions and reach equilibrium in different market structures.

The outcomes of these models vary significantly. Cournot competition often leads to higher prices and lower quantities than perfect competition. In contrast, Bertrand competition with homogeneous products results in prices equal to marginal cost, mirroring perfect competition.

Cournot and Bertrand Competition Models

Cournot vs Bertrand competition

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• Cournot model assumes firms compete based on the quantity of output they produce (oil, steel)
  • Firms make simultaneous decisions about their production levels
  • Each firm takes its competitor's output as fixed when deciding its own production quantity
• Bertrand model assumes firms compete based on the prices they set for their products (retailers, online sellers)
  • Firms make simultaneous decisions about the prices they will charge
  • Each firm takes its competitor's price as fixed when deciding its own price
• Equilibrium outcomes differ between the two models
  • Cournot competition generally results in higher prices and lower quantities compared to perfect competition (OPEC)
  • Bertrand competition with homogeneous products leads to the same outcome as perfect competition, where price equals marginal cost (generic medications)

Cournot duopoly equilibrium

• Assumes two firms producing a homogeneous product (Coke and Pepsi)
  • Both firms have the same constant marginal cost of production ($c$)
  • The firms face a linear inverse demand function: $P = a - bQ$, where $Q = q_1 + q_2$ represents the total market quantity
• To find Firm 1's profit-maximizing quantity
  • Firm 1 maximizes its profit function: $\max_{q_1} \pi_1 = (a - b(q_1 + q_2))q_1 - cq_1$
  • The first-order condition for profit maximization is: $\frac{\partial \pi_1}{\partial q_1} = a - 2bq_1 - bq_2 - c = 0$
  • Solving the first-order condition yields Firm 1's best response function: $q_1 = \frac{a - c}{2b} - \frac{q_2}{2}$, which shows how Firm 1's optimal quantity depends on Firm 2's quantity
• Firm 2's profit maximization problem and best response function are symmetric to Firm 1's
• To find the Nash equilibrium quantities
  • Solve the system of best response functions for both firms simultaneously
  • The equilibrium quantities are: $q_1^* = q_2^* = \frac{a - c}{3b}$
• To find the equilibrium price
  • Substitute the equilibrium quantities into the inverse demand function
  • The equilibrium price is: $P^* = a - b(\frac{a - c}{3b} + \frac{a - c}{3b}) = \frac{a + 2c}{3}$

Nash equilibrium in Bertrand models

• Assumes two firms producing a homogeneous product (generic drugs)
  • Both firms have the same constant marginal cost of production ($c$)
  • Consumers purchase from the firm offering the lowest price
• In the Nash equilibrium
  • Both firms set their prices equal to marginal cost: $p_1^* = p_2^* = c$
  • If one firm sets a price above marginal cost, its competitor can slightly undercut that price and capture the entire market
  • If a firm sets a price below marginal cost, it will incur losses on each unit sold
• Equilibrium quantities
  • At the equilibrium price, the firms split the market demand equally
  • Each firm's equilibrium quantity is: $q_1^* = q_2^* = \frac{Q(c)}{2}$, where $Q(c)$ represents the total market demand at the price equal to marginal cost

Product differentiation in Bertrand competition

• Product differentiation occurs when firms produce goods that are imperfect substitutes (Coke and Pepsi)
  • Consumers have distinct preferences for the different products
• With differentiated products, the demand functions for each firm are
  • Firm 1's demand: $q_1 = a - bp_1 + dp_2$
  • Firm 2's demand: $q_2 = a - bp_2 + dp_1$
  • $b > d > 0$, where $b$ represents the own-price effect and $d$ represents the cross-price effect
• Firm 1's profit maximization problem
  • Firm 1 maximizes its profit function: $\max_{p_1} \pi_1 = (p_1 - c)(a - bp_1 + dp_2)$
  • The first-order condition is: $\frac{\partial \pi_1}{\partial p_1} = a - 2bp_1 + dp_2 + bc = 0$
  • Solving the first-order condition yields Firm 1's best response function: $p_1 = \frac{a + bc + dp_2}{2b}$
• Firm 2's profit maximization problem and best response function are symmetric to Firm 1's
• To find the Nash equilibrium prices with differentiated products
  • Solve the system of best response functions for both firms simultaneously
  • The equilibrium prices are: $p_1^* = p_2^* = \frac{a + bc}{2b - d}$
• The equilibrium prices with differentiated products are higher than marginal cost
  • Product differentiation gives firms some market power, allowing them to set prices above marginal cost
  • As products become more differentiated ($d$ decreases), equilibrium prices increase, reflecting greater market power