Capturing Income Distributions and Inequality Indices Using NETs (Negative Extremal Transfers)

We introduce the concept of negative extremal transfers (NETs), which are transfers from the poorest individuals to the richest individuals. This family of transfers alone is rich enough to describe the entire space of income distributions: our first result is that any income distribution can be obtained as an expansion from the uniform distribution by applying a sequence of NETs. In other words, NETs constitute a mathematical basis of the space of income distributions. Our second representation theorem establishes that one can describe any given inequality index based on the weight it attaches to all possible NETs. These results allow one to observe how much importance a given inequality index attaches to poverty concerns in addition to inequality concerns. Anecdotally, we find that indices used in practice lie in a relatively small region of the index space: our NET representation theorem can serve as a guide to proposing new inequality indices. Practitioners will find this representation result useful to quantify the contribution of a given quantile or subgroup to the population's inequality level as well as to guide policy toward the most effective transfers to lower the inequality measure.