Complementarity in matching markets and exchange economies

The literature has shown that complementarity places significant structure on outcomes in matching markets and exchange economies. We examine the extent to which this structure, and the economic intuition underlying it, is common across these classes of environments. We show that transferable utility matching markets can be represented as exchange economies in a way that preserves competitive equilibria and gross complementarity. Unlike canonical representations that preserve substitutability, this representation must involve the addition of brokers whenever the matching market is not two-sided. We use our representation results to uncover the relationship (or lack thereof) between existence results in the literature that rely on complementarity, and to give a new existence result for matching markets with imperfectly transferable utility and net complementarity.