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If one is an egalitarian, should one try to equalize resources available to agents, or to equalize their welfares? With a suitably general conception of what resources are, these two conceptions cannot be distinguished. An allocation mechanism is defined on a space of economies, and is required to satisfy various axioms that are necessary conditions for "equalizing resources" among a population. The unique mechanism satisfying these axioms on a large domain of economies is the one that allocates commodities so as to equalize utilities of the agents. Methodologically, the theorem is a generalization of bargaining theory because the mechanisms are defined on commodity space.
John E. Roemer (1986) studied this question.