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abstract

I take a decentralized approach to fractional matching with and without money. For the model with money I define and show the existence of competitive equilibria. For the model without money, while competitive equilibria may not exist, I define a version of approximate equilibrium and show existence. From a welfare standpoint, I show that equilibrium allocations are in the core. Similarly, approximate equilibrium allocations are in the approximate version of the core.

6. Discussion

I have kept the model in this paper simple in the interest of clarity. The analysis, however, is robust to several extensions. A general model that includes the following features along with a generalization of the definition of DIP equilibrium and a proof of existence are in Appendix E. Large markets and fairness As with Lindahl equilibria (Lindahl, 1958), personalized prices undermine the assumption that agents are price-takers. However, the model that I have presented here can be extended in the following way: each agent is associated with a “kind” which describes his external characteristics (as opposed to a “type” which would also encapsulate internal characteristics such as the agent’s preferences over others). That is, his kind determines how other agents feel about him, but not necessarily how he feels about them. For instance, for a labor market application, the workers could be partitioned into various kinds based on their training, tenure, achievements, and so on, while managers could be partitioned based on the kind of work they are asking workers to do.23 In such a model, prices need only be indexed by kinds. Rather than saying what agent i pays to be with agent j, the prices would specify what an agent of kind k pays to be with an agent of kind l. Thus, every agent of the same kind faces the same prices. Beyond justifying the price-taking assumption, this also ensures that agents are treated “fairly” in the sense that agents of the same kind face the same prices. In classical exchange economies, given private endowments, a competitive equilibrium allocation is not necessarily fair in the sense of “envy-freeness.” For instance, if one agent is endowed with more of each good than another, then he has a larger budget set at every price vector and the agent with the smaller endowment may envy him in equilibrium. However, a competitive equilibrium allocation does satisfy the following criterion of fairness: if two agents have the same endowment, then neither envies the other. In the model that I have presented, an agent’s time is his endowment. If two agents are of the same kind so that everyone is indifferent between them, then these two agents have the same endowment of time. Since prices are indexed by kinds rather than identities of agents, at a DIP equilibrium allocation, no agent envies another agent of the same kind who has the same endowment of money