Model Answer
0 min readIntroduction
The Bertrand model, a cornerstone of game theory in economics, analyzes competition between firms offering homogenous products. Developed by Joseph Bertrand in 1883 as a critique of Cournot’s model, it posits that firms compete on price rather than quantity. This leads to a significantly different market outcome than other oligopoly models. In a Bertrand duopoly, where two firms compete, the intense price competition drives the market price down to the marginal cost of production, resulting in zero economic profits. Understanding this model is crucial for analyzing industries with close substitutes and price-sensitive consumers.
Understanding the Bertrand Duopoly
The Bertrand model rests on several key assumptions:
- Homogenous Products: Both firms produce identical products.
- Simultaneous Price Setting: Firms choose their prices simultaneously.
- Perfect Information: Consumers know the prices offered by both firms.
- Sufficient Capacity: Firms have the capacity to meet all market demand at any price.
- Constant Marginal Cost: Each firm has a constant marginal cost of production (denoted as 'c').
Deriving Market Equilibrium Price and Quantity
The core logic of the Bertrand model is based on the following:
- If Firm 1 sets a price above the marginal cost (P > c), Firm 2 can undercut that price by setting P - ε (where ε is a small positive number) and capture the entire market demand.
- If Firm 1 sets a price equal to the marginal cost (P = c), Firm 2 has no incentive to undercut, as it would result in zero profits.
- If Firm 1 sets a price below the marginal cost (P < c), it will incur losses.
This leads to a Nash Equilibrium where both firms set their price equal to the marginal cost (P = c). This is because, given the other firm’s price is 'c', neither firm can improve its profits by deviating and setting a different price.
Calculating Market Equilibrium Quantity
The market equilibrium quantity depends on the market demand function. Let's denote the market demand function as Q = a - bP, where 'a' and 'b' are positive constants. Since the equilibrium price is P = c, the market equilibrium quantity (Q*) is:
Q* = a - bc
However, because both firms charge the same price (equal to marginal cost), they split the market demand equally. Therefore, each firm produces Q*/2.
Determining Firms’ Profits
Since the price equals the marginal cost (P = c), each firm earns zero economic profit. This is because:
Profit = (Price - Marginal Cost) * Quantity
Profit = (c - c) * (Q*/2) = 0
Therefore, in a Bertrand duopoly, both firms earn zero economic profits in equilibrium.
Illustrative Example
Suppose the market demand function is Q = 100 - 2P, and the marginal cost for both firms is c = 10. Then:
- Equilibrium Price (P) = 10
- Market Equilibrium Quantity (Q*) = 100 - 2(10) = 80
- Each Firm’s Quantity = 80/2 = 40
- Each Firm’s Profit = (10 - 10) * 40 = 0
Limitations of the Bertrand Model
The Bertrand model’s strong assumptions often don’t hold in the real world. Product differentiation, capacity constraints, and switching costs can mitigate the intense price competition and allow firms to earn positive profits. Repeated interactions and the possibility of collusion can also alter the outcome.
Conclusion
In conclusion, the Bertrand model demonstrates that even with only two firms, intense price competition can drive prices down to marginal cost, resulting in zero economic profits. While the model’s assumptions are often unrealistic, it provides a valuable benchmark for understanding competitive dynamics. The model highlights the importance of factors like product differentiation and capacity constraints in determining market outcomes. Further research and more complex models are needed to accurately represent real-world oligopolistic markets.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.