b1-cournot-A

Cournot Duopoly Overview

  • Reference: Martin, Chapter 5

  • Key Characteristics:

    • Two identical firms that are present in the same market.

    • Both firms produce homogeneous goods.

    • Firms set their output levels simultaneously.

    • The concept of Nash equilibrium is utilized for analysis.

Cournot Equations

  • Firms:

    • Firm i (where i = 1, 2) has output denoted as q_i.

  • Industry Output:

    • Overall industry output (Q):

      • Q = q1 + q2

  • Demand Market:

    • Demand is represented as a linear equation:

      • P = a − bQ

  • Cost Structure:

    • Total cost for firm i:

      • TCi(qi) = F + c * qi

    • where F = fixed costs and c = constant marginal costs.

Cournot Equilibrium

  • Definition:

  • The Cournot Equilibrium is positioned as a Nash equilibrium in the Cournot game.

  • Best Response Functions:

    • Each firm's optimal output is determined simultaneously.

    • Reaction function for Firm 1 is denoted as qr1(q2).

  • Profit Maximization:

    • To maximize profits, firms adjust their output:

      • Set q1 = qr1(q2) leading to the condition MR1 = mc1,

        • where P + dP/dq1 * q1 = mc1.

    • The equilibrium outputs are defined by (q1, q2) such that:

      • q1 = qr1(q2) and q2 = qr2(q1).

Graphical Intuition of Reaction Functions

  • Functionality:

  • Reaction function qr1(q2) maximizes profit under the condition MR1 = mc1.

  • Various graphical representations elucidate:

    • qr1(0) = QM (Monopoly output)

    • Link between adjustments in q2 and corresponding changes in q1 and market price.

Algebra of Reaction Functions and Equilibrium

  • Equations:

    • Q = q1 + q2

    • P = a − bQ

    • Reaction Functions:

      • qr1(q2) = (a - c - bq2) / (2b)

      • qr2(q1) = (a - c - bq1) / (2b)

    • Equilibrium output where:

      • q1 = q2 = (a - c) / (3b).

Stackelberg Leadership in Cournot

  • Overview:

  • Focus on strategic behavior and leadership dynamics.

  • Steps Involved:

    • Firm 1 acts as a leader, deciding on quantities first.

    • Firm 1's choice is vital as it impacts Firm 2's response given by qr2(q1).

    • Potential entrant firm dynamics can also be analysed under this model.

Calculus of Strategic Benefit

  • Firm 1's Objective:

    • Maximize profit using: π1 = [P(q1 + q2) - c] * q1 - F, subject to constraints from Firm 2.

    • Derivation leads to conditions for maximizing expected revenue against marginal costs.

Static Limit Pricing Revisited

  • Timing Insights:

  • Sequence of moves: Firm 1 sets q1, Firm 2 observes and reacts.

  • Reaction functions operationalized as qE2(q1).

  • Decisions based on thresholds where firms either enter or stay out of the market.

Diagrams in Static Limit Pricing

  • Utilization of various graphs to elucidate on reaction functions and the slope of profitability under different conditions.

  • Understanding limits (qL) for monopoly and competitive structures alongside profit range analysis.