Industrial 4: Cournot Model

Oligopoly Model

• Definition: A market structure with a small number of large firms.
  • Characterized by strategic interactions where each firm's actions impact others and consumer choices.

Strategy Profile

• Definition: The combination of strategies made by the players.

Payoff

• Definition: For each strategy vector, the utility for each payoff. The impact on other players is denoted by s-i.

Nash Equilibrium

• Definition: A situation where no benefit is derived from unilateral deviation. No one wants to deviate from this strategy.

Strategy Example

• Decisions like advertisement levels, quantity, and pricing strategies.

Game Theory vs. Duopoly

• Game Theory: Often a two-by-two scenario to check all options.
• Duopoly: Infinite options available.

Best Payoffs in Duopoly

• Use the Best Response Function.
  • Ri(aj) = argmax ai (ai, aj)

Cournot Duopoly Model Setup

• Firms: 2
• Output Levels: Each firm chooses an output level qi (i=1,2).
• Aggregate Output: Q=q1+q2
• Cost Function: C(q1)
• Inverse Market Demand Function: P(Q)=P(q1+q2)
• Payoffs: Profits

Payoffs Equation

• 1(q1,q2) = p(q1+q2)q1 - C(q1)

Assumptions of Cournot Duopoly Model

• No further entry of firms.
• Completely homogeneous goods.
• Production decisions are the main strategic variable.
• Output levels chosen simultaneously.
• Decreasing inverse market demand function (dP/dQ < 0).

Nash Equilibrium in Cournot Duopoly Model

• Pair of strategies (q1 and q2) where neither firm can increase profits by changing its output, given the other firm's output choice.

Best Response Function in Cournot Duopoly

• q1 = R1(q2) and q2 = R2(q1)

Maximizing Profits in Cournot Duopoly

• Take the FOC of the profit function to find marginal revenue and set it equal to marginal cost.
• Rearrange to find qi and qj.

Slope of MR and Importance

• Downward sloping: If qj increases, MR(qi) and qi decrease.

Cournot Model with N Firms

• Use i(q1,Q-i) = (A-B(qi+Q-1)-c)qi
  • i is specific to the firm, Q-1 is the output of all other firms.
• Best response: Symmetry implies Q-1 = (N-1)qi

Large N (Tending to Infinity)

• Total output converges to competitive level (A-c)/B.
• 1/(N+1) tends to 0, N/(N+1) tends to 1.

Inefficiency in N

• Occurs when N is small (causing deadweight loss).

Market Power in Oligopoly

• Assumptions: All firms have the same marginal cost.
• Equation: (P(Q)-c) * qi, take FOC to find MR=MC.
• Use qi/Q for firm's market share.
• Elasticity = partial of Q with respect to P times P/Q.*

Market Power Summary

• Exercised but limited by elasticity.
  • Less elastic demand equates to more market power.

Profit Function for Different Costs

• Details vary based on cost differences.

Herfindahl-Hirschman Index (HHI)

• Definition: Measure of market concentration to assess competitiveness.
• Calculated by multiplying both sides of the elasticity equation by si.

Greater Market Power

• Achieved with greater concentration of firms.