Share equivalent and equitarian allocations for problems of fair division

Abstract

In this paper we begin with a given social endowment. A profile of shares (which could very well all be equal), is part of the environment. This is the planner s contribution to the economic environment, as conceived in this paper. First we formulate the concept of an envy free allocation as was done by Schmeidler and Vind [1972]. Then we propose the concept of a share equivalent allocation which is a generalization of the concept of an egalitarian equivalent allocation due to Pazner and Schmeidler [1978]. An allocation is share equivalent if every agent is indifferent between his allocation and what would result if an identical change in entitlement were affected for all the agents. (An agent s entitlement is his share of the social endowment in physical units). If the identical change is a multiple of the social endowment vector, we say that the allocation is naturally share equivalent. We prove the existence of a naturally share equivalent allocation which is also Pareto efficient and prove that such allocations correspond to maximization of the minimum utility over all feasible allocations. For two agent economies we show that naturally share equivalent allocations are envy free and all envy free allocations are share equivalent. In a final section to this paper we introduce the concept of an equitarian allocation (Yound 1993). This is a feasible allocation such that each agent is indifferent between what he/she received and a multiple of his entitlement, where the multiple is common to all the agents. The proof that a Pareto efficient equitarian allocation exists is similar to the proof which established the existence of a Pareto efficient and naturally share equivalent allocation. Hence we omit it. Subsequently, we observe that such allocations are the only ones which maximize the minimum utility over all feasible allocations, where the utility representations are once again suitable constructed. Thus, we manage to generalize an existing notion of economic equity, by incorporating possible asymmetries that may need to arise for the sake of obtaining (final) distributive justice. As observed by Moulin (1995), problems of fair division arise perpetually in managerial contexts. With these results, perhaps a new insight would be gained in resolving such problems.