Full Substitutability in Trading Networks

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. In this paper, we extend earlier models' canonical definitions of substitutability to a setting in which an agent can be a buyer in some transactions and a seller in others, and show that all the different substitutability concepts are equivalent. Next, we introduce a new class of fully substitutable preferences that models the preferences of intermediaries with production capacity. We then prove that substitutability is preserved under economically important transformations such as trade endowments, mergers, and limited liability. We show that full substitutability can be recast in terms of submodularity of the indirect utility function, the single improvement property, a "no complementarities" condition, and a condition from discrete convex analysis called M♮-concavity. Finally, we show that substitutability implies two key monotonicity conditions known as the Laws of Aggregate Supply and Demand. All of our results explicitly incorporate economically important features such as indifferences, non-monotonicities, and unbounded utility functions that were not fully addressed in prior work.