Full substitutability

Abstract

Various forms of substitutability are essential for establishing the existence of equilibria and other useful properties in diverse settings such as matching, auctions, and exchange economies with indivisible goods. We extend earlier models' definitions of substitutability to settings in which each agent can be both a buyer in some transactions and a seller in others, and show that all these definitions are equivalent. We then introduce a new class of substitutable preferences that allows us to model intermediaries with production capacity. We also prove that substitutability is preserved under economically important transformations such as trade endowments, mergers, and limited liability.

A Compact, Logical Approach to Large-Market Analysis

“…Thus our exercise here is in some sense similar to that of Blume and Zame (1994), who unified our understanding of perfect and sequential equilibria by way of the Tarski-Seidenberg Theorem. But at the same time, we use our methods to prove completely new results: our results for trading networks-all of which are novel to this work-generalize the existence result of Hatfield et al (2013) to infinite trading networks (see also Hatfield et al, 2019Hatfield et al, , 2018Fleiner et al, forthcoming). Meanwhile, we show a way that Logical Compactness can be used to convert a dynamic game with a finite start time into an "ongoing" dynamic game with neither start nor end-connecting our work to the broad literature on infinite-horizon games (see, e.g., Fudenberg and Levine, 2009) .…”

The structure of equilibria in trading networks with frictions

“…The following notion of full substitutability is due to Hatfield et al (2019). 7 Precursors of the full substitutability notion were introduced for exchange economies (Sun and Yang (2006)) and for trading networks without 7 Hatfield et al, 2019 call this version of full substitutability the "demand-language expansion" version of full substitutability (cf.…”

“…Hatfield et al (2019) show that for transferable utility various ways of defining full substitutability are equivalent, and hence one can work with either of the definitions discussed in their paper. Going beyond transferable utility makes issues more subtle: Not all equivalence results of Hatfield et al (2019) generalize and it matters which of the full substitutability conditions are used. More specifically, it matters how the full substitutes condition is defined in instances where indifferences matter, that is, when the demand is multivalued.…”

“…The following notion of full substitutability is due to Hatfield et al (2019). 7 Precursors of the full substitutability notion were introduced for exchange economies (Sun and Yang (2006)) and for trading networks without transfers (Ostrovsky (2008)).…”

“…77 Hatfield et al, 2019 call this version of full substitutability the “demand‐language expansion” version of full substitutability (cf. Definition A.3 in Hatfield et al, 2019).…”