A Class of Acceptable and Practical Social Welfare Orderings with Variable Population: Stepwise Social Welfare Orderings and Their Applications

This study proposes a new class of social welfare orderings, a stepwise rank-dependent social welfare ordering, which naturally generalizes a rank-dependent utilitarianism in the setting of social choice with variable population size. In fact, a stepwise social welfare ordering is simply designed to have the same value for each proportion of the population, with the obvious advantage that allows functional form to be freely chosen for assessing well-being inequality. We show that a stepwise rank-dependent social welfare ordering is easily characterized by standard axioms: strong Pareto, anonymity, Pigou-Dalton transfer equity, continuity, rank-separability, and consistency for population replication. If additional requirements on standard invariance are imposed on it, its functional form is specified as a well-known rank-weighted social welfare ordering. As some practical applications to measure social welfare, we propose three simple methods such as a quantile mean comparison method, which evaluates social welfare by comparing each quantile’s average income in an approximate lexicographic ordering or quantile-dependent weighted summation. These applications have obvious advantages in that they can see the whole picture of income distributions compared with standard tools such as the traditional GDP per capita, median, range, and top/bottom ratios. Furthermore, we show a representation theorem that generalizes stepwise social welfare orderings for the problem of optimal population size.