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Correspondences

  • Charalambos D. Aliprantis
  • Kim C. Border
Part of the Studies in Economic Theory book series (ECON.THEORY, volume 4)

Abstract

A correspondence is a set-valued function. That is, a correspondence associates to each point in one set a set of points in another set. As such, it can be viewed simply as a subset of the Cartesian product of the two sets. It may seem a bit silly to dedicate a chapter to such a topic, but correspondences arise naturally in many applications. For instance, the budget correspondence in economic theory associates the set of affordable consumption bundles to each price—income combination; the excess demand correspondence is a useful tool in studying economic equilibria; and the best-reply correspondence is the key to analyzing noncooperative games. The theory of “differential inclusions” deals with set-valued differential equations and plays an important role in control theory.

Keywords

Topological Space Compact Convex Topological Vector Space Polish Space Metrizable Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    J. C. Moore [173] identifies five slightly different definitions of upper semicontinuity in use by economists, and points out some of the differences for compositions, etc.Google Scholar
  2. 2.
    The terms “multifunction” and “set-valued function” are used by some authors in place of “correspondence.”Google Scholar
  3. 3.
    If Y is Hausdorff, then the converse is true. That is, if φ is continuous at x 0, then each f i is continuous at x 0 . We leave the proof as an exercise.Google Scholar
  4. 4.
    N.B. Weak measurability has nothing to do with weak topologies.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Charalambos D. Aliprantis
    • 1
  • Kim C. Border
    • 2
  1. 1.Department of MathematicsIndiana University-Purdue University Indianapolis (IUPUI)IndianapolisUSA
  2. 2.Division of the Humanities and Social SciencesCALTECHPasadenaUSA

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