Investigating equality: The Rényi spectrum

An equality index is a score quantifying the socioeconomic egalitarianism of the distribution of wealth in human societies; the score takes values in the unit interval, with the unit upper bound characterizing purely communist societies. In this paper we explore the Rényi spectrum, a continuum of equality indices that: (i) is based on the moments of the societies' distributions of wealth; (ii) unifies various measures of socioeconomic inequality, including the Theil and Atkinson indices; (iii) displays a collection of amicable analytic properties; (iv) admits multiple Rényi-divergence representations; and (v) provides a high-resolution gauging of egalitarianism that is way beyond what can be offered by the common-practice measures of socioeconomic inequality, the Gini and Pietra indices. At large, the Rényi spectrum is applicable in the context of any distribution of non-negative sizes with a positive mean-yielding a high-resolution gauging of the distribution's inherent statistical heterogeneity.