Model Answer
0 min readIntroduction
Duopoly, a market structure characterized by two firms, is a fundamental concept in industrial organization. A common model used to analyze duopolies is the Stackelberg model, a dynamic game where one firm (the leader) moves first, and the other firm (the follower) observes the leader’s action and then chooses its own output. This sequential decision-making creates a distinct asymmetry in the problems faced by each firm. Understanding the 'follower's problem' and how it differs from the 'leader's problem' is crucial for comprehending the strategic interactions within a duopoly and predicting market outcomes. This analysis is particularly relevant in industries like telecommunications and aviation, where a few dominant firms often compete.
The Stackelberg Duopoly Model: A Foundation
The Stackelberg model assumes that the follower firm believes the leader’s output is fixed when making its own production decision. This is a key distinction from the Cournot model, where firms simultaneously choose their output levels. The leader, knowing this, strategically chooses its output to influence the follower’s behavior and maximize its own profits.
The Follower’s Problem
The follower’s problem is a standard profit maximization problem, conditional on the leader’s output choice. The follower takes the leader’s output (QL) as given and determines its own optimal output (QF) to maximize its profit.
- Profit Function: The follower’s profit (πF) is given by: πF = P(QL + QF) * QF - CF(QF), where P is the market price, and CF is the follower’s cost function.
- Reaction Function: To find the optimal QF, the follower takes the derivative of its profit function with respect to QF, sets it equal to zero, and solves for QF. This results in the follower’s reaction function, which expresses QF as a function of QL: QF = f(QL). The reaction function shows how the follower will react to any given output level chosen by the leader.
- Example: If both firms have linear demand P = a - b(QL + QF) and constant marginal cost c, the follower’s reaction function will be QF = (a - c - bQL) / 2b.
The Leader’s Problem
The leader’s problem is more complex. The leader must anticipate the follower’s reaction to its own output choice. The leader doesn’t just maximize profit given a fixed follower output; it maximizes profit taking into account how the follower will respond according to its reaction function.
- Profit Function: The leader’s profit (πL) is given by: πL = P(QL + f(QL)) * QL - CL(QL), where f(QL) is the follower’s reaction function.
- Strategic Substitution: The leader substitutes the follower’s reaction function into its own profit function. This effectively transforms the leader’s problem into an unconstrained optimization problem, where the leader chooses QL to maximize its profit, knowing how the follower will respond.
- First-Order Condition: The leader then takes the derivative of the substituted profit function with respect to QL, sets it equal to zero, and solves for QL. This yields the leader’s optimal output level.
- Example: Continuing with the linear demand and constant marginal cost example, the leader’s optimal output will be QL = (a - c) / 2b, which is higher than the Cournot equilibrium output.
Key Differences: A Comparative Analysis
| Feature | Follower | Leader |
|---|---|---|
| Decision Timing | Second mover | First mover |
| Information | Observes leader’s output | Anticipates follower’s reaction |
| Optimization | Maximizes profit given leader’s output | Maximizes profit considering follower’s reaction function |
| Output Level | Determined by reaction function | Strategically chosen to influence follower |
| Complexity | Relatively simpler | More complex, requires anticipating opponent’s behavior |
The leader enjoys a first-mover advantage, allowing it to influence the market outcome. The follower, however, has the benefit of observing the leader’s action before making its own decision, reducing its uncertainty.
Conclusion
In conclusion, the follower’s problem in a Stackelberg duopoly is a standard profit maximization exercise conditional on the leader’s output, resulting in a reaction function. The leader’s problem, however, is significantly more complex, requiring the anticipation of the follower’s response and strategic substitution of the reaction function into its own profit maximization problem. This difference in decision-making processes highlights the importance of first-mover advantages and strategic interaction in duopoly markets. Understanding these dynamics is crucial for firms operating in oligopolistic industries and for policymakers seeking to regulate such markets.
Answer Length
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