Egalitarian Equivalent Allocations: A New Concept of Economic Equity

Elisha A. Pazner, David Schmeidler
1978 Quarterly Journal of Economics  
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more » ... pt of Pareto-efficient-egalitarian-equivalent-allocations (PEEEA), 674.-III. PEEEA as a fair arbitration scheme for allocations, 676.-IV. Maximin properties of PEEEA, 678.-V. PEEEA in economies with production, 680.-Mathematical appendices, 682. on such functions, a very powerful analytical concept would indeed have become available. Significant insights to normative economics have nevertheless been provided by the social welfare function approach. But the explicit underlying (ordinal) interpersonal welfare comparisons (effectively ruled out by Arrow's result) imply that a robust equity criterion, normatively compelling for any possible economy, cannot be derived from this particular approach. The question that suggests itself therefore is whether any reasonable equity criterion, the normative significance of which is equally valid in any particular society (economy), can be advanced. A few years ago Foley (1967) advanced the concept of fair (or envy-free) allocations as a reasonable equity criterion. An allocation is said to be fair if nobody prefers anybody else's bundle over his own. The concept of fairness is appealing from an equity viewpoint in that it treats economic agents symmetrically, is ordinal in nature, and is free of interpersonal comparisons of utility. However, as shown by Pazner and Schmeidler (1974), standard Arrow-Debreu production economies may display the disturbing feature that among all Pareto-efficient allocations none can be found that is fair. In light of the general acceptance of the Pareto criterion, it would be desirable to have a concept of equity that never conflicts with Pareto efficiency (under the standard assumptions on the economic environment). Since the fairness criterion does not possess this property and since there is also the question of whether an equity concept based on envy can be morally acceptable in the first place (see Rawls, 1971), the issue of defining an adequate criterion is still open. The present paper introduces a concept of economic equity that, as fairness, possesses an appealing symmetry property, is ordinal in nature, and is free of interpersonal welfare comparisons. Specifically, an allocation is said to be egalitarian-equivalent if there exists a fixed commodity bundle (the same for each agent) that is considered by each agent to be indifferent to the bundle that he actually gets in the allocation under consideration. In other words, an egalitarianequivalent allocation has the special property that its underlying welfare distribution could have been generated by an egalitarian economy. It is shown that Pareto-efficient and egalitarian-equivalent allocations always exist under (even weaker than) the standard conditions on the economic environment. When supplemented by the egalitarian-equivalence criterion, the set of Pareto-efficient allocations is thus restricted to those allocations having the property that there exists an egalitarian economy (i.e., an economy in which everybody gets an identical bundle) in which every agent enjoys precisely the This content downloaded from 14.139.45.244 on Tue, 14 Oct 2014 09:45:32 AM All use subject to JSTOR Terms and Conditions 683 PROPOSITION 3. Let there be given a convergent sequence (n) in RI with a limit x-> 0 and suppose also that Yn > 0 for all n. Denote by T (and T) the real number corresponding to Yn (and x) via Proposition 1. Then one has n -rT. Denote by P the set {x R' Ix >0 and E xi and denote by RP the set of egalitarian-reference-bundles: {rx E R+' I XY e P and rcorresponds to x-via Proposition 1}. As a simple consequence of Proposition 3, one has that RP is homeomorphic to P. Furthermore, if RP is a bounded set, then the homeomorphism, as well as Propositions I and 3, can be extended to P, the closure of P in R , and RP correspondingly. Next denote by ARP the set of allocations, each of them being Pareto-efficient and egalitarian-equivalent to some bundle in RP, By Propositions 1, 2, and 3 we have PROPOSITION 4. Under the strict convexity assumption the correspondence that applies Pareto-efficient x-equivalent allocations to each x-in RP is a well-defined continuous function from RP onto ARP.
doi:10.2307/1883182 fatcat:25solbj2jfeldagz5gjn3fjauy