The Computation of Fixed Points and Applications

  • Michael J. Todd

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

  • Michael J. Todd
    • 1
  1. 1.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA

Bibliographic information

  • 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
  • Print ISBN 978-3-540-07685-8
  • Online ISBN 978-3-642-50327-6
  • Series Print ISSN 0075-8442
  • About this book