Characterizing the Equal Income Market Equilibrium Choice Correspondence: Average Envy Freeness Instead of Individual Rationality from Equal Division

In problems of fair division of a given bundle of resources, amongst a finite number of agents, individual rationality from equal division plays a significant role. In a society, where all resources are socially owned, one cannot argue in terms of equal ownership of the social endowment. One normally takes the position that each agents has the right to veto any allocation, which leaves him/her worse than equal division. Based on this premise, individual rationality from equal division has been proposed as a minimal requirement of distributive justice. In Thomson (1982), we find an equity criterion called average envy-freeness, which in the context of economies with convex preferences, implies individual rationality from equal division. Average envy-freeness says that no agent finds the average consumption of the other agents, superior to his/her own consumption. This concept has been developed in the Foley (1967) tradition of an envy-free allocation: no agent should find his/her consumption inferior to the consumption of any other agents. We show in this paper (the not too difficult result) that average envy-freeness does not automatically imply individual rationality from equal division. A solution concept which recurs with seeming regularity in the literature of fair division is the equal income market equilibrium solution concept. In a variable population framework Thomson (1988) provides an axiomatic characterization, using the axiom of consistency. Consistency basically says that the departure of some agents with their allocated consumption, should not affect the consumption of the remaining agents, provided they operate the same distribution mechanism as before, Lahiri (1997a 1997b) use this same axiom to characterize the equal income market equilibrium choice correspondence in convex and non-convex environments. Out main result reported in this paper is similar to a Lahir (1997b) result, although it may not extend to the non-convex economies considered there. It is thus a modest generalization of the Thomson (1988) result. We use consistency, replication invariance, efficiency and average envy-freeness to show that if a solution satisfies these properties, it must consist of equal income market equilibrium allocations. Subsequently we drop consistency and arrive at yet another characterization of subsolutions of equal income market equilibrium choice correspondence using the strict envy-freeness property due to Zhou (1992).