Summary of Cournot Model of Duopoly Lecture
In this lecture, the instructor explained the Cournot model of duopoly, focusing on how two firms independently decide their production quantities which then determine the market price. The main concepts covered include the determination of residual demand, profit maximizing problems for firms, and the derivation of reaction functions leading to the determination of the Nash equilibrium.
Key Notes on Cournot Model of Duopoly
Market Demand
- General Demand Function: Market demand is linear, represented by ( P = 10 - Q ).
Firm Decisions and Output Determination
- Each firm decides its production quantity independently.
- Total output ( Q ) is the sum of outputs from both firms (( Q = Q_1 + Q_2 )).
- Market price is determined by ( P = 10 - (Q_1 + Q_2) ).
Residual Demand Function
- Each firm perceives a residual demand depending on the estimated output of the competitor.
- Example: If Firm 1 assumes Firm 2 produces 2 units, the residual demand for Firm 1 becomes ( P = 8 - Q_1 ).
Profit Maximization
- Each firm maximizes profit, ( \pi = TR - TC ).
- Profit function for Firm 1, considering linear costs (e.g., ( TC = 2Q_1 )), becomes ( \pi = (8 - Q_1)Q_1 - 2Q_1 ).
- Deriving the maximization condition leads to ( Q_1 = 3 ) when ( Q_2 = 2 ), assuming those values maximize its profit.
General Case Analysis
- If general market demand function is ( P = a - bQ ), and cost functions are similarly structured for both firms, both firms face a competitive setup.
- Reaction Functions: Establish equations considering each firm’s belief about the other's production, leading to individual output decisions.
Nash Equilibrium
- Achieved when neither firm has the incentive to unilaterally deviate from their production decision.
- Equilibrium quantities for both firms typically expressed as ( Q_1 = Q_2 = \frac{a - C}{3B} ).
- Total output and market price in equilibrium can be deduced accordingly.
Implications and Conclusion
- The Cournot Nash equilibrium points to a specific production level where both firms find it optimal, given their mutual best response strategies.
- Instructor emphasized the importance of understanding these dynamics, especially in comparison to other models like Stackelberg competition, which will be discussed in future sessions.
Key Formulae and Equations
- Residual Demand: ( P = a - b(Q_1 + Q_2) )
- Profit Function: ( \pi = (a - b(Q_1 + Q_2))Q_1 - CQ_1 )
- First-Order Condition: ( a - b(Q_1 + Q_2) - bQ_1 - C = 0 )
- Nash Equilibrium Output:
- ( Q_1 = Q_2 = \frac{a - C}{3B} )
- Total Output = ( \frac{2(a - C)}{3B} )
- Equilibrium Price = ( \frac{1}{3}a + \frac{2}{3}C )
As the course progresses, the comparison between Cournot and other competitive strategies will provide deeper understanding on strategic business decisions in oligopolistic markets.