Persistent Link:
http://hdl.handle.net/10150/290535
Title:
Stability of equilibria in dynamic oligopolies
Author:
Li, Weiye
Issue Date:
2001
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
The Lyapunov function method is used in proving stability, asymptotic or globally asymptotic stability of discrete dynamic systems. We show that the slightly relaxed versions of the well known sufficient conditions are also necessary. The stability of the equilibria of time-invariant nonlinear dynamical systems with discrete time scale is investigated. We present an elementary proof showing that in the case of a stable equilibrium and continuously differentiable state transition function, all eigenvalues of the Jacobian computed at the equilibrium must be inside or on the unit circle. We also demonstrate via numerical examples that if some eigenvalues are on the unit circle and all other eigenvalues are inside the unit circle, then the equilibrium maybe unstable, or stable, or even asymptotically stable, which show that the necessary condition cannot be further restricted in general. In addition, the necessary condition is given in terms of spectral radius and matrix norms. The asymptotic stability of equilibria in a number of discrete dynamic oligopolies is analyzed. First the equivalence of the equilibrium problem of a large class of nonlinear games and the equilibrium problem of a class of discrete dynamic systems is verified. Stability conditions are then derived for a certain class of dynamic models, and these results are finally applied to single-product oligopolies, multiproduct oligopolies, and labor-managed oligopolies. The economic interpretation of the stability conditions are also presented. The stability properties of a special class of homogeneous dynamic economic systems are examined. The nonlinearity of the models and the presence of eigenvalues with zero real parts in a normally hyperbolic invariant set make the application of the classical theory impossible. Some principles of the modern theory of dynamical systems and invariant manifolds are applied. The local and global strong attractivity of the set of equilibria is verified under mild conditions. As an application, special labor-managed oligopolies are investigated.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Rychlik, Marek

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleStability of equilibria in dynamic oligopoliesen_US
dc.creatorLi, Weiyeen_US
dc.contributor.authorLi, Weiyeen_US
dc.date.issued2001en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThe Lyapunov function method is used in proving stability, asymptotic or globally asymptotic stability of discrete dynamic systems. We show that the slightly relaxed versions of the well known sufficient conditions are also necessary. The stability of the equilibria of time-invariant nonlinear dynamical systems with discrete time scale is investigated. We present an elementary proof showing that in the case of a stable equilibrium and continuously differentiable state transition function, all eigenvalues of the Jacobian computed at the equilibrium must be inside or on the unit circle. We also demonstrate via numerical examples that if some eigenvalues are on the unit circle and all other eigenvalues are inside the unit circle, then the equilibrium maybe unstable, or stable, or even asymptotically stable, which show that the necessary condition cannot be further restricted in general. In addition, the necessary condition is given in terms of spectral radius and matrix norms. The asymptotic stability of equilibria in a number of discrete dynamic oligopolies is analyzed. First the equivalence of the equilibrium problem of a large class of nonlinear games and the equilibrium problem of a class of discrete dynamic systems is verified. Stability conditions are then derived for a certain class of dynamic models, and these results are finally applied to single-product oligopolies, multiproduct oligopolies, and labor-managed oligopolies. The economic interpretation of the stability conditions are also presented. The stability properties of a special class of homogeneous dynamic economic systems are examined. The nonlinearity of the models and the presence of eigenvalues with zero real parts in a normally hyperbolic invariant set make the application of the classical theory impossible. Some principles of the modern theory of dynamical systems and invariant manifolds are applied. The local and global strong attractivity of the set of equilibria is verified under mild conditions. As an application, special labor-managed oligopolies are investigated.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorRychlik, Mareken_US
dc.identifier.proquest3023503en_US
dc.identifier.bibrecord.b4195760xen_US
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