Lorenz domination, utilitarian deprivation rule and equal sacrifice principle

Of two income distributions x and y, over a given population size, we say that the former dominates the latter by the utilitarian deprivation rule if, for any person, the aggregate utility shortfall from richer persons under x is at least as large as that under y. In this paper we show that if the relative risk aversion associated with the utility function does not exceed unity, then any Lorenz-consistent income tax function will make the post-tax distribution no more deprived than the pre-tax distribution according to the utilitarian deprivation rule. The converse of this proposition holds if the risk-aversion measure is not less than one. It then follows that if the utility function is of logarithmic type then consistency between the two criteria holds. Finally, we relate our results to the equal sacrifice principle.