Stackelberg leadership model

Authors: Peiying Li, Xiangyu Zeng, Wen Su, Hongyan Ke, Zhiyu An (SYSEN 5800/6800 Fall 2024)

Introduction

The Stackelberg leadership model, also known as the Stackelberg game or Stackelberg competition, is a strategic game in economics and game theory where one firm (the leader) makes its decisions before other firms (the followers) in an imperfectly competitive market[1]. This sequential decision-making structure fundamentally differs from simultaneous-move games like the Cournot model, as it introduces the element of strategic advantage through first-mover position.

Heinrich von Stackelberg first introduced this concept in 1934 through his work "Market Structure and Equilibrium" (Marktform und Gleichgewicht). His pioneering research emerged during a period when economists were increasingly focused on understanding market dynamics beyond perfect competition and monopoly scenarios[2]. The model addressed a critical gap in economic theory by examining how firms might behave when they make decisions in a sequential order rather than simultaneously[3].

The motivation for studying the Stackelberg model stems from its widespread applicability in real-world markets. Many industries exhibit leader-follower dynamics, where established firms act as market leaders while others respond to their decisions. For instance, in the commercial aircraft manufacturing industry, Boeing and Airbus often demonstrate Stackelberg-like behavior in their capacity decisions and product launches. Similarly, in retail markets, large chains frequently make pricing and inventory decisions that smaller retailers must then react to.

The model serves several key purposes in modern economic analysis:

1. Understanding how sequential decision-making affects market outcomes
2. Analyzing the strategic advantages of being a first mover in a market
3. Predicting price levels and market quantities in leader-follower scenarios
4. Providing insights for regulatory policy in markets with dominant firms

Algorithm Discussion

Condition and Assumption

The Stackelberg Leadership Model consists of sequential decisions by a leader and a follower, optimizing their strategies under specific assumptions.

1. Cost function: Both leader and follower incur constant marginal costs $(C1, C2)$.
2. Rationality: Both leader and follower are profit-maximizing agents.
3. Information transparency: The follower has full knowledge of the leader’s decision.

Algorithm Description

Input:

• Inverse demand function: P(Q), where Q=q1+q2
• Cost functions: C1(q1)and C2(q2).

Definition:

• q1: leader’s quantity which maximizes the leader's profit.
• q2: follower's quantity which depends on the leader’s decision through a reaction function q2 = f(q1).
• P: market price

Step:

1. Follower’s Optimization