Inequality measurement for bounded variables

Many health indicators are bounded, that is, their values lie between a lower and an upper bound. Inequality measurement with bounded variables faces two normative challenges well-known in the health inequality literature. One is that inequality rankings may or may not be consistent across admissible attainment and shortfall representations of the variable. The other is that the set of maximum-inequality distributions for bounded variables is different from the respective set for variables with no upper bound. Therefore, the ethical criteria for ranking maximum-inequality distributions with unbounded variables may not be appropriate for bounded variables. In a novel proposal, we justify an axiom requiring maximum-inequality distributions of bounded variables to be ranked equally, irrespective of their means. Then, our axiomatic characterization naturally leads to indices that measure inequality as an increasing function of the observed proportion of maximum attainable inequality for a given mean. Additionally, our inequality indices rank distributions consistently when switching between attainment and shortfall representations. In our empirical illustration with three health indicators, a starkly different picture of cross-country inter-temporal inequality emerges when traditional inequality indices give way to our proposed normalized inequality indices.