Substitutes, Complements, and Equilibrium in Two-Sided Market Models

Abstract

There are two sets of agents: buyers B and sellers S. Each type of agent is allowed to trade with as many agents on the opposite side it wishes. Agents’ decision process is determined by a market price system p, where p=(p(s, b), (s, b) ∈ S × B. Namely, a seller s solves the task max _{B ⊂ B} [p(s, B)-c(s, B)] where c(s, B) is the cost incurred by seller s when he contracts with a set B of buyers. A buyer b, similarly, will solve for max _s⊂s [u(S, b) - p(S,b)], where u(S, b) is the utility of buyer b after contracting with S sellers.

We examine the existence of competitive equilibrium in this market. We show that equilibria exist in those markets for which there are appropriate combinations of substitutes and complements among the goods on sale. In particular, equilibria exist if all the goods on sale are pure substitutes, or if they are all pure complements. We also establish results about the structure of the sets of equilibrium prices and allocations. We show that the substitution and complementarity requirements are intimately related to the discrete convexity (or concavity) requirements imposed on the corresponding cost and utility functions of the market agents.

The authors wish to thank two anonymous referees for their remarks.

The financial support of RFBR grant # 00–15–98873 is gratefully acknowledged.

The financial support # 8210–053488 of the Swiss National Science Foundation is gratefully acknowledged.