Unambiguous Comparison of Intersecting Distribution Functions∗

Comparing distribution functions is a key task in descriptive research and policy evaluation. In many applications, a reasonable refinement of second-degree stochastic dominance criterion is necessary to attain unambigous ranking of intersecting distribution functions. Although the literature offers dominance criteria of third or higher degree, these ranking criteria are rarely used because they are viewed as difficult to interpret and hard to justify. We provide a general framework that unambigously ranks any set of distributions functions and quantifies the social welfare level of a dominating distribution as compared to a dominated distribution. Our framework is based on two complementary sequences of nested inverse stochastic dominance criteria: One places more emphasis on differences that occur in the lower part of the distributions, while the other places more emphasis on differences that occur in the upper part of the distribution. The two sequences coincide at second-degree dominance, and thus both satisfy the Pigou-Dalton transfer principle. For each sequence, we show equivalence in the ranking of distributions according to the dominance criteria and a general family of rank dependent social welfare functions. Because the sequences of dominance criteria are hierarchical and nested, our equivalence results allow us to uniquely identify the largest subfamily of welfare functions – and thus the least restrictive social preferences – that give an unambiguous ranking of any set of distribution functions. We also provide a characterization of the largest subfamily of social welfare functions that rank consistently with dominance of any given degree. Because of the equivalence result, this characterization provides interpretation and justification not only for the social welfare functions, but also for the use of higher degree dominance criteria in comparison of distribution functions. We show the usefulness of our framework through two empirical applications; we assess the welfare implications of changes in household income distributions over the business cycle, and evaluate the actual and counterfactual outcome distributions from a policy experiment.