Cournot, Bertrand, and Stackelberg models explain how firms compete in oligopolies. They show how companies set prices or quantities, either at the same time or one after another, to maximize profits while considering what rivals might do.

These models help us understand real-world markets where a few big players dominate. By comparing outcomes, we can see how different competitive strategies affect prices, output, and profits in various industries.

Cournot Model of Quantity Competition

Key Assumptions and Equilibrium Concept

• Firms compete by choosing output quantities simultaneously
• Market demand determines prices based on total industry output
• Cournot equilibrium represents a Nash equilibrium where each firm maximizes profit given competitors' choices
• Reaction functions (best response curves) determine equilibrium output levels for each firm
• Results in prices above marginal cost but below monopoly price
• Industry output falls between perfect competition and monopoly levels

Model Extensions and Market Dynamics

• Number of firms affects equilibrium outcomes
  • More firms lead to lower prices and higher total output
  • As number of firms approaches infinity, price approaches marginal cost
• Model extends to asymmetric firms with different cost structures
• Accommodates product differentiation scenarios
• Can incorporate capacity constraints or investment decisions

Mathematical Representation

• Profit function for firm i: $\pi_i = P(Q)q_i - C_i(q_i)$
  • P(Q) represents inverse demand function
  • Q = total industry output
  • q_i = output of firm i
  • C_i(q_i) = cost function for firm i
• First-order condition for profit maximization: $\frac{\partial \pi_i}{\partial q_i} = P(Q) + P'(Q)q_i - C'_i(q_i) = 0$
• Solving system of equations yields Cournot-Nash equilibrium quantities

Bertrand Model of Price Competition

Core Principles and Paradox

• Firms compete by setting prices simultaneously
• Consumers purchase from firm offering lowest price
• Bertrand paradox emerges in markets with homogeneous products and identical marginal costs
  • Equilibrium price equals marginal cost
  • Firms earn zero economic profits
• Bertrand equilibrium represents Nash equilibrium where no firm can increase profit by unilaterally changing price

Model Variations and Extensions

• Product differentiation allows prices above marginal cost
  • Products become imperfect substitutes
  • Each firm faces downward-sloping demand curve
• Capacity constraints modify outcomes
  • Can lead to prices above marginal cost even with homogeneous products
  • Introduces element of quantity competition
• Repeated interactions may facilitate tacit collusion
  • Firms may sustain higher prices through threat of future punishment

Mathematical Framework

• Demand function for firm i: $q_i = D_i(p_i, p_{-i})$
  • p_i = price set by firm i
  • p_{-i} = vector of prices set by other firms
• Profit function: $\pi_i = (p_i - c_i)D_i(p_i, p_{-i})$
  • c_i = marginal cost for firm i
• First-order condition: $\frac{\partial \pi_i}{\partial p_i} = D_i(p_i, p_{-i}) + (p_i - c_i)\frac{\partial D_i}{\partial p_i} = 0$

Stackelberg Model of Sequential Competition

Sequential Decision-Making Structure

• One firm (leader) chooses output before other firm (follower)
• Leader enjoys first-mover advantage
  • Typically results in higher profits compared to follower
• Model solved using backward induction
  • First determine follower's best response function
  • Then calculate leader's optimal choice

Equilibrium Characteristics

• Stackelberg equilibrium represents subgame perfect Nash equilibrium
• Accounts for sequential nature of decisions
• Leader produces more output than in Cournot model
• Follower produces less output than in Cournot model
• Total industry output typically exceeds Cournot output

Model Extensions and Applications

• Can extend to multiple periods or multiple leaders/followers
• Alters strategic dynamics and equilibrium outcomes
• Relevant for markets with dominant firms or sequential entry decisions
• Applies to industries with clear market leaders (technology, automotive)

Cournot vs Bertrand vs Stackelberg Outcomes

Comparative Market Outcomes

• Market prices typically follow order: Bertrand < Stackelberg < Cournot < Monopoly
  • Assumes homogeneous products and identical costs
• Total industry output generally follows reverse order: Bertrand > Stackelberg > Cournot > Monopoly
• Firm profits vary across models
  • Bertrand competition often yields zero economic profits
  • Cournot and Stackelberg allow for positive profits
  • Stackelberg leader usually earns highest profit

Model Applicability and Industry Characteristics

• Bertrand model suits markets with price competition and low switching costs (retail, online marketplaces)
• Cournot model often used for quantity competition in commodity markets (oil, agriculture)
• Stackelberg model relevant for markets with dominant firms or sequential entry (technology sectors)
• Choice of model depends on specific industry characteristics
  • Nature of competition (price vs. quantity)
  • Product homogeneity
  • Timing of decisions
  • Market structure and firm dynamics

Efficiency and Welfare Implications

• Bertrand competition typically yields most efficient outcome
  • Prices closest to marginal cost
  • Highest consumer surplus
• Cournot and Stackelberg models result in deadweight loss
  • Prices above marginal cost
  • Reduced total surplus compared to perfect competition
• Monopoly outcome least efficient among oligopoly models
  • Highest prices and lowest output
  • Greatest deadweight loss