Two types of inequality: Inequality between persons and inequality between subgroups

Social scientists study two kinds of inequality: inequality between persons (as in income inequality) and inequality between subgroups (as in racial inequality). This paper analyzes the mathematical connections between the two kinds of inequality. The paper proceeds by exploring a set of two-parameter continuous probability distributions widely used in economic and sociological applications. We define a general inequality parameter, which governs all measures of personal inequality (such as the Gini coefficient), and we link this parameter to the gap (difference or ratio) between the means of subdistributions. In this way we establish that, at least in the two-parameter distributions analyzed here, and for the case of two nonoverlapping subgroups, as personal inequality increases, so does inequality between subgroups. This general inequality parameter also governs Lorenz dominance. Further, we explore the connection between subgroup inequality (in particular, the ratio of the bottom subgroup mean to the top subgroup mean) and decomposition of personal inequality into between-subgroup and within-subgroup components, focusing on an important decomposable measure, Theil's MLD, and its operation in the Pareto case. This allows us to establish that all the quantities in the decomposition are monotonic functions of the general inequality parameter. Thus, the general inequality parameter captures the deep structure of inequality. We also introduce a whole-distribution graphical tool for assessing personal and subgroup inequality. Substantively, this work suggests that in at least some societies, characterized by special income distributions, whenever inequality disrupts social harmony and social cohesion, it attacks on two fronts, via subgroup inequality as well as personal inequality.