Supermodularity and Complementarity

Supermodularity and Complementarity

Supermodularity and Complementarity

Supermodularity and Complementarity

Synopsis

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.

Excerpt

This monograph has had a remarkably long gestation period. My involvement in this area spans three decades, commencing with my doctoral dissertation (Topkis [1968]), which studied the problem of maximizing a supermodular function on a lattice with its implications for monotone comparative statics and with applications to various decision problems. The study of complementarity, of course, goes back much further. Yet a research monograph on the present subject—covering supermodular functions on a lattice, their connection with complementarity, their role for monotone comparative statics, and their use in diverse economic models—is certainly timely. Supermodularity has received considerable attention in the economics literature in recent years, with important extensions of the theory and significant new applications. However, much of the previously published theory and its applications were developed primarily in the applied mathematics and operations research literature, and the basic concepts and results are relatively new to the economics lexicon. This monograph is intended to present 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 objectives are to provide an integrated and accessible view of the relevant theory pertaining to supermodularity, complementarity, and monotone comparative statics, and to demonstrate the utility of the methodology through diverse applications of that theory to economic problems. I am grateful to Kyle Bagwell, Herv´e Moulin, Tom Sargent, and Bill Sharkey for helpful comments on earlier versions of this monograph.

Search by... Author
Show... All Results Primary Sources Peer-reviewed

Oops!

An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.