Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 380530, 22 pages
http://dx.doi.org/10.1155/2011/380530
Research Article

Stability, Bifurcation, and Chaos in 𝑁 -Firm Nonlinear Cournot Games

Akio Matsumoto1 and Ferenc Szidarovszky2

1Department of Economics, Chuo University, 742-1 Higashi-Nakano, Hachioji, Tokyo 192-0393, Japan
2Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721-0020, USA

Received 10 June 2010; Revised 10 November 2010; Accepted 6 January 2011

Academic Editor: Weihong Huang

Copyright © 2011 Akio Matsumoto and Ferenc Szidarovszky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An 𝑁 -firm production game known as oligopoly will be examined with isoelastic price function and linear cost under al Cournot competition. After the best responses of the firms are determined, a dynamic system with adaptive expectations is introduced. It is first shown that the local asymptotic behavior of the system is identical with that of the adaptive adjustment process in which the firms cautiously determine their outputs. Dynamic analysis is confined to two special cases, one in which 𝑁 is divided into two groups and the other in which 𝑁 is divided into three groups. Then stability conditions will be derived and the global behavior of the equilibria will be illustrated including chaos control. Lastly the two- and three-group models are compared with two-firm (duopoly) and three-firm (triopoly) models to shed light on roles of the number of the firms.