Nonlinear Oligopolies: Stability and Bifurcations

✍ Scribed by Gian-Italo Bischi, Carl Chiarella, Michael Kopel, Ferenc Szidarovszky

• Publisher: Springer
• Year: 2009
• Tongue: English
• Leaves: 340
• Category: Library

✦ Synopsis

Oligopoly theory is one of the most intensively studied areas of mathematical economics. On the basis of the pioneering works of Cournot (1838), many res- rchers have developed and extensively examined the different variants of oligopoly models. Initially, the existence and uniqueness of the equilibrium of the different types of oligopolies was the main concern, and later the dynamic extensions of these models became the focus. The classical result of Theocharis (1960) asserts that under discrete time scales and static expectations, the equilibrium of a sing- product oligopoly without product differentiation and with linear price and cost functions is asymptotically stable if and only if it is a duopoly. In the continuous time case, asymptotic stability is guaranteed for any number of ?rms. In these cases the resulting dynamical systems are also linear, where local and global asymptotic stability are equivalent to each other. The classical book of Okuguchi (1976) gives a comprehensive summary of the earlier results and developments. The multipr- uct extensionshave been discussed in Okuguchiand Szidarovszky(1999);however, nonlinear features were barely touched upon in these contributions. WiththedevelopmentofthecriticalcurvemethodbyGumowskiandMira(1980) (see also Mira et al. (1996))fordiscrete time systemsand the introductionof cont- uously distributed information lags by Invernizzi and Medio (1991) in continuous time systems, increasing attention has been given to the global dynamics of n- linear oligopolies. The authors of this book have devoted a great deal of research effort to this area.

✦ Table of Contents

Nonlinear Oligopolies
1 The Classical Cournot Model
1.1 Introduction
1.2 Dynamic Adjustment Processes
1.3 An Introduction to the Analysis of Global Dynamics
1.3.1 A Cournot Duopoly Game
1.3.2 A Cournot Oligopoly Game
1.3.3 Cournot Duopoly Revisited: A GradientType Adjustment Process
1.3.4 Simple Basins and Critical Curves
1.3.5 Disconnected Basins
2 Concave Oligopolies
2.1 Introduction
2.2 Discrete Time Models and Local Stability
2.3 Discrete Time Oligopolies and Global Stability
2.4 Gradient Adjustments
2.5 Continuous Time Oligopolies and Local Stability
2.6 Oligopolies with Continuously DistributedTime Lags
3 General Oligopolies
3.1 Isoelastic Price Functions
3.1.1 Discrete Time Models and Local Stability
3.1.2 Global Dynamics of Discrete TimeModels
3.1.3 Continuous Time Models and LocalStability
3.2 Cost Externalities and Multiple Interior Nash Equilibria
3.2.1 Identical Speeds of Adjustment
3.2.2 Non-Identical Speeds of Adjustment
4 Modified and Extended Oligopolies
4.1 Market Share Attraction Games
4.1.1 Local Stability
4.1.2 The Feasible Set and Global Stability
4.2 Labor-Managed Oligopolies
4.2.1 Discrete Time Models and Local Stability
4.2.2 Discrete Time Models and Global Dynamics
4.2.3 Continuous Time Models
4.3 Oligopolies with Intertemporal Demand Interaction
4.3.1 Discrete Time Models and Local Stability
4.3.2 Discrete Time Models and Global Stability
4.3.3 Continuous Time Models
4.4 Models with Production Adjustment Costs
4.5 Oligopolies with Partial Cooperation
4.5.1 Local Stability Analysis
4.5.2 Global Dynamics
5 Misspecified and uncertain price functions
5.1 Misspecified Price Functions
5.1.1 Discrete Time Models and Local Stability
5.1.2 Discrete Time Models and Global Dynamics
5.1.3 Continuous Time Models
5.2 Cournot Oligopolies with Local Monopolistic Approximation
5.2.1 Adjustments with Local Monopolistic Approximation
5.2.2 Dynamics Under Adaptive Adjustment
5.3 Other Learning Processes
5.3.1 Unknown Slope with Known Market Saturation Point
5.3.2 Unknown Reservation Price with Known Slope
5.3.3 Unknown Slope with Known Reservation Price
5.4 Uncertain Price Functions
6 Overview and Directions for Future Research
A Elements of Lyapunov Theory
B Local Linearization
C Noninvertible Maps and Critical Sets
C.1 Definitions and Simple Examples
C.2 Discrete Time Dynamical Systems as Iterated Maps
C.3 Critical Sets and the Delineation of Trapping Regions
C.4 Critical Sets and the Creation of Disconnected Basins
D Continuously Distributed Time Lags
E A Determinantal Identity
F Stable Quadratic Polynomials
References
Index