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© 1976

The Computation of Fixed Points and Applications

Book

Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 124)

Table of contents

  1. Front Matter
    Pages I-VII
  2. Michael J. Todd
    Pages 1-16
  3. Michael J. Todd
    Pages 17-23
  4. Michael J. Todd
    Pages 24-39
  5. Michael J. Todd
    Pages 54-63
  6. Michael J. Todd
    Pages 73-81
  7. Michael J. Todd
    Pages 82-93
  8. Michael J. Todd
    Pages 94-100
  9. Michael J. Todd
    Pages 116-125
  10. Back Matter
    Pages 126-132

About this book

Introduction

Fixed-point algorithms have diverse applications in economics, optimization, game theory and the numerical solution of boundary-value problems. Since Scarf's pioneering work [56,57] on obtaining approximate fixed points of continuous mappings, a great deal of research has been done in extending the applicability and improving the efficiency of fixed-point methods. Much of this work is available only in research papers, although Scarf's book [58] gives a remarkably clear exposition of the power of fixed-point methods. However, the algorithms described by Scarf have been super~eded by the more sophisticated restart and homotopy techniques of Merrill [~8,~9] and Eaves and Saigal [1~,16]. To understand the more efficient algorithms one must become familiar with the notions of triangulation and simplicial approxi- tion, whereas Scarf stresses the concept of primitive set. These notes are intended to introduce to a wider audience the most recent fixed-point methods and their applications. Our approach is therefore via triangu- tions. For this reason, Scarf is cited less in this manuscript than his contri- tions would otherwise warrant. We have also confined our treatment of applications to the computation of economic equilibria and the solution of optimization problems. Hansen and Koopmans [28] apply fixed-point methods to the computation of an invariant optimal capital stock in an economic growth model. Applications to game theory are discussed in Scarf [56,58], Shapley [59], and Garcia, Lemke and Luethi [24]. Allgower [1] and Jeppson [31] use fixed-point algorithms to find many solutions to boundary-value problems.

Keywords

Computation Invariant economic growth economics game theory

Authors and affiliations

  1. 1.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA

Bibliographic information

  • Book Title The Computation of Fixed Points and Applications
  • Authors M. J. Todd
  • Series Title Lecture Notes in Economics and Mathematical Systems
  • DOI https://doi.org/10.1007/978-3-642-50327-6
  • Copyright Information Springer-Verlag Berlin Heidelberg 1976
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Softcover ISBN 978-3-540-07685-8
  • eBook ISBN 978-3-642-50327-6
  • Series ISSN 0075-8442
  • Edition Number 1
  • Number of Pages VII, 132
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Topology
    Business and Management, general
  • Buy this book on publisher's site