Equilibria and Indivisibilities: Gross Substitutes and Complements

Abstract

This paper examines an exchange economy with heterogeneous indivisible objects that can be substitutable or complementary. We show that a competitive equilibrium exists in such economies, provided that all the objects can be partitioned into two groups, and from the viewpoint of each agent, objects in the same group are substitutes and objects across the two groups are complements. This condition generalizes the well-known Kelso-Crawford gross substitutes condition and is called gross substitutes and complements. We also provide practical and typical examples from which substitutes and complements are both jointly observed. Copyright The Econometric Society 2006.

A model of partnership formation

“…They prove through a salary adjustment process that there exists an equilibrium, if every firm views all workers as substitutes. This latter condition is called gross substitutes and has been widely used in auction, matching and housing models, see for instance, Roth and Sotomayor (1990), Stacchetti (1999, 2000), Milgrom (2000), Ausubel (2006), Sun and Yang (2006), and Ostrovsky (2008). This paper presents a new model, the partnership formation problem that subsumes and extends the assignment models studied by Koopmans and Beckmann (1957), Shapley and Shubik (1972), and Crawford and Knoer (1981).…”

Gross substitutability: An algorithmic survey

“…To the present date, the original paper by Kelso and Crawford [23] has more then 800 citations and the term "gross substitutes" appears in more then 3500 publications according in Google Scholar. A comprehensive exposition on all the work is out of the scope of this survey, but we point the reader to some key generalizations and applications of gross substitutes in economics that will serve as starting point to the reader: Roth [40], Hatfield and Milgrom [1], Sun and Yang [43,44], Gul and Stachetti [19,20] and Bikhchandani and Mamer [6].…”

Market structure and matching with contracts

“…Another special case is a discrete version of the many-to-one matching model with substitutes and complements studied by Sun and Yang [15]: A set of workers can be decomposed into two sets W 1 and W 2 such that all firms view two workers from the same set as substitutes (in the sense of SSS) and two workers from different sets as complements (in the sense of CSC). Workers can work for at most one firm, while firms can hire an arbitrary number of workers.…”