Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities

Abstract

An Equivalence Theorem between geometric structures and utility functions allows new methods for understanding preferences. Our classification of valuations into “Demand Types” incorporates existing definitions (substitutes, complements, “strong substitutes,” etc.) and permits new ones. Our Unimodularity Theorem generalizes previous results about when competitive equilibrium exists for any set of agents whose valuations are all of a “demand type.” Contrary to popular belief, equilibrium is guaranteed for more classes of purely‐complements than of purely‐substitutes, preferences. Our Intersection Count Theorem checks equilibrium existence for combinations of agents with specific valuations by counting the intersection points of geometric objects. Applications include matching and coalition‐formation, and the “Product‐Mix Auction” introduced by the Bank of England in response to the financial crisis.

Full substitutability

“…Sun and Yang () extended Gul and Stacchetti's equivalence result to their setting. Baldwin and Klemperer () showed that in their setting the single improvement property is equivalent to requiring that agents have complete preferences.…”

“…Various forms of substitutability are essential for establishing the existence of equilibria and other useful properties in diverse settings such as matching, auctions, exchange economies with indivisible goods, and trading networks (Kelso and Crawford 1982, Roth 1984, Bikhchandani and Mamer 1997, Gul and Stacchetti 1999, , Milgrom 2000, Ausubel and Milgrom 2005, Hatfield and Milgrom 2005, Sun and Yang 2006, , Ostrovsky 2008, Hatfield et al 2013, Fleiner et al 2017). Substitutability arises in a number of important applications, including matching with distributional constraints (Abdulkadiroğlu and Sönmez 2003, Hafalir et al 2013, Ehlers et al 2014, Echenique and Yenmez 2015), supply chains (Ostrovsky 2008), markets with horizontal subcontracting (Hatfield et al 2013), “swap” deals in exchange markets (Milgrom 2009), and combinatorial auctions for bank securities (Klemperer 2010, Baldwin and Klemperer 2019).…”

“…Subsequent to our work, Baldwin and Klemperer () obtained additional insights on the underlying mathematical structure of fully substitutable preferences using techniques from tropical geometry. Baldwin and Klemperer () studied the set of price vectors for which the demand correspondence is multi‐valued, and associated them with convex‐geometric objects called tropical hypersurfaces. Then, using the normal vectors that determine agents' tropical hypersurfaces, they distinguish among preferences that are strongly substitutable, are gross substitutable, or have complementarities.…”

“…Then, using the normal vectors that determine agents' tropical hypersurfaces, they distinguish among preferences that are strongly substitutable, are gross substitutable, or have complementarities. Full substitutability corresponds to the “strong substitutes demand” condition of Baldwin and Klemperer ()…”

“… Baldwin and Klemperer () also consider several other economically important classes of preferences for which the existence of competitive equilibria is guaranteed. …”