On the Stability of a Retarded Functional Differential Equation Edgeworth Process

Abstract

Following Smale's notations, let n and 1 denote respectively the number of agents and commodities in a pure exchange barter economy. Let u₁,u₂, ... u_n denote n real valued functions representing the individuals preferences and which depend on the past as well as present holdings. We assume that the mechanism which describes the adjustment in the quantities traded at any point in time be given by an Edgeworth process in the form of an autonomous retarded functional differential equation type. The aim of this study is first to develop necessary conditions for this type of mechanism to converge to the set of Pareto-points. Second, we examine the conditions under which positive results may be obtained as solutions of the Eggeworth dynamic approach the boundary of the smooth submanifold of feasible allocations W. Third, necessary conditions are developped to assure local stabiltyof a Pareto point. Finally, we investigate the conditions under which a Pareto point is unstable. The result on asymptotic behavior uses the ideas of LaSalle'Theorem and those on stability are proved by means of Liapunov'theory, using Hale'stability Theorem on retarded functional differential equations.

Keywords:

• Pareto-points
• Edgeworth process
• asymptotic behavior
• wedge function
• stability
• retarded functional differential equation

Keywords:

• AMS(MOS)

^*This work is based on the author's math thesis. I am grateful to Professor K.MEYER for his supervision. I would like to thank the Department of mathematical Sciences of the University of Cincinnati for their kind hospitality during my stay. I am pleased to acknowledge conversations with A.Lazer, A.Leung, and E.P.Merkes. I would like also to thank, Professors George sell and Hans Weinberger for their kind invitation to the 1983-84 I.M.A. Meeting. I am grateful to Professor J.K. Hale for advicing me this Journal.

^*This work is based on the author's math thesis. I am grateful to Professor K.MEYER for his supervision. I would like to thank the Department of mathematical Sciences of the University of Cincinnati for their kind hospitality during my stay. I am pleased to acknowledge conversations with A.Lazer, A.Leung, and E.P.Merkes. I would like also to thank, Professors George sell and Hans Weinberger for their kind invitation to the 1983-84 I.M.A. Meeting. I am grateful to Professor J.K. Hale for advicing me this Journal.

Notes

^*This work is based on the author's math thesis. I am grateful to Professor K.MEYER for his supervision. I would like to thank the Department of mathematical Sciences of the University of Cincinnati for their kind hospitality during my stay. I am pleased to acknowledge conversations with A.Lazer, A.Leung, and E.P.Merkes. I would like also to thank, Professors George sell and Hans Weinberger for their kind invitation to the 1983-84 I.M.A. Meeting. I am grateful to Professor J.K. Hale for advicing me this Journal.