Supermodularity and Complementarity

Supermodularity and Complementarity

Copyright Date: 1998
Pages: 312
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  • Book Info
    Supermodularity and Complementarity
    Book Description:

    The economics literature is replete with examples of monotone comparative statics; that is, scenarios where optimal decisions or equilibria in a parameterized collection of models vary monotonically with the parameter. Most of these examples are manifestations of complementarity, with a common explicit or implicit theoretical basis in properties of a super-modular function on a lattice. Supermodular functions yield a characterization for complementarity and extend the notion of complementarity to a general setting that is a natural mathematical context for studying complementarity and monotone comparative statics. Concepts and results related to supermodularity and monotone comparative statics constitute a new and important formal step in the long line of economics literature on complementarity.

    This monograph links complementarity to powerful concepts and results involving supermodular functions on lattices and focuses on analyses and issues related to monotone comparative statics. Don Topkis, who is known for his seminal contributions to this area, here presents a self-contained and up-to-date view of this field, including many new results, to scholars interested in economic theory and its applications as well as to those in related disciplines. The emphasis is on methodology. The book systematically develops a comprehensive, integrated theory pertaining to supermodularity, complementarity, and monotone comparative statics. It then applies that theory in the analysis of many diverse economic models formulated as decision problems, noncooperative games, and cooperative games.

    eISBN: 978-1-4008-2253-9
    Subjects: Economics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-x)
  3. Preface
    (pp. xi-2)
  4. 1 Introduction
    (pp. 3-6)

    One may sometimes conclude too readily that an old, familiar, and simple concept, exemplified in myriad common situations, has little new to offer. In particular, how much novelty could be expected from the descriptive notion of complementarity, whereby two products are considered complements if having more of one product increases the marginal value derived from having more of the other product? Indeed, a half century ago, Samuelson [1947] declared:

    In my opinion the problem of complementarity has received more attention than is merited by its intrinsic importance.

    Yet, a quarter century later, Samuelson [1974] came to assert:

    The time is...

  5. 2 Lattices, Supermodular Functions, and Related Topics
    (pp. 7-93)

    This chapter includes general concepts and results relevant for the present perspective on supermodularity and complementarity. The theory in this chapter is used in the applications that follow in the three subsequent chapters. For a collection of optimization problems where the objective function and the constraint set depend on a parameter, comparative statics is concerned with the dependence of optimal solutions on the parameter and monotone comparative statics is concerned with optimal solutions varying monotonically with the parameter. (These definitions extend from optimization problems to game problems, as in Chapter 4 and Chapter 5.) Monotone comparative statics is the primary...

  6. 3 Optimal Decision Models
    (pp. 94-174)

    This chapter applies the results of Chapter 2 to optimal decision models. The primary focus is on establishing increasing optimal solutions and complementarity properties. The various sections of this chapter can be read independently, except that Section 3.7 should be read before Subsection 3.8.2 and Subsection 3.9.1 should be read before Section 3.10. The sections are organized as follows.

    Section 3.2 examines matching problems, where workers of different types are to be assigned among multiple firms. Conditions are given for optimal matchings to be increasing or ordered with respect to the qualities of the workers and the efficiencies of the...

  7. 4 Noncooperative Games
    (pp. 175-206)

    This chapter considers the role of supermodularity and complementarity in noncooperative games. The results primarily involve supermodular games, where the payoff function of each player has properties of supermodularity and increasing differences. This introductory section briefly summarizes subsequent sections of this chapter and gives basic definitions and notation used in later sections of this chapter.

    Section 4.2 studies general properties of supermodular games. Under modest regularity conditions, an equilibrium point exists. Certain parameterized collections of supermodular games have the property that equilibrium points for each game increase with the parameter. A consequence of the latter result is that for certain...

  8. 5 Cooperative Games
    (pp. 207-262)

    This chapter considers the role of supermodularity and complementarity in cooperative games. The results primarily involve convex games, where the net return to any subset of players acting together (that is, the characteristic function evaluated for any coalition) is a supermodular function of the set of players. This introductory section proceeds first by briefly summarizing the content of the subsequent sections in the chapter and then by providing basic definitions and notation for the remaining sections.

    Section 5.2 presents general properties of convex games. Much of the analysis is constructive and involves the remarkably simple, effective, and efficient greedy algorithm....

  9. Bibliography
    (pp. 263-268)
  10. Index
    (pp. 269-272)