Sorting and Decentralized Price Competition

We investigate under which conditions price competition in a market with matching frictions leads to sorting of buyers and sellers. Positive assortative matching obtains only if there is a high enough degree of complementarity between buyer and seller types. The relevant condition is root-supermodularity; i.e., the square root of the match value function is supermodular. It is a necessary and sufficient condition for positive assortative matching under any distribution of buyer and seller types, and does not depend on the details of the underlying matching function that describes the search process. The condition is weaker than log-supermodularity, a condition required for positive assortative matching in markets with random search. This highlights the role competition plays in matching heterogeneous agents. Negative assortative matching obtains whenever the match value function is weakly submodular.