A Class of Acceptable and Practical Social Welfare Functions with Variable Populations: A Stepwise Rank-Dependent Utilitarianism and Its Application

This study proposes a new class of social welfare orderings, stepwise rank-dependent utilitarianisms, which generalize rank-dependent utilitarianisms in the setting of social choice with variable population size. It is shown that the stepwise rank-dependent utilitarianism is equivalent to a social welfare ordering that satisfies the desirable axioms: strong Pareto, anonymity, Pigou-Dalton transfer equity, continuity, rank-separability, cardinal full comparability, and consistency for population replication. This social welfare ordering is a kind of rank-dependent utilitarianisms designed to have the same weight for each proportion of the population, with the obvious advantage that allows weights to be freely chosen for assessing well-being inequality. As a practical application of the stepwise rank-dependent utilitarianism, we propose a k-quantile mean comparison method, which evaluates social welfare by comparing each quantile’s average income in an approximate lexicographic manner. The method has a methodological advantage in that it makes flexible consideration on distributive justice possible compared with the traditional GDP per capita. Furthermore, we show a representation theorem of a social welfare function that generalizes the stepwise rank-dependent utilitarianism for the problem of optimal population size. In addition, the paper reexamines and reformulates several impossibility theorems with a variable population and shows that they are not so serious in the context of cardinal full comparability. Combining the previous results with our findings, theoretical correspondence between interpersonal comparability of individual well-being and acceptable social welfare orderings would be clarified. If cardinal partial comparability were admissible, stepwise rank-dependent Kolm-Pollack or Atkinson-Blackorby-Donaldson social welfare functions should be recommended. If ordinal full comparability were admissible, a stepwise leximin should be recommended. The stepwise rank-dependent utilitarianism should be used if cardinal full comparability were admissible.