Consistent comparisons of attainment and shortfall inequality: a critical examination

An inequality measure is ‘consistent’ if it ranks distributions the same irrespective of whether health quantities are represented in terms of attainments or shortfalls. This consistency property severely restricts the set of admissible inequality measures. We show that, within a more general setting of separate measures for attainments and shortfalls, the consistency property is a combination of two conditions. The first is a compelling rationality condition that says that the attainment measure should rank attainment distributions as the shortfall measure ranks shortfall distributions. The second is an overly demanding condition that says that the attainment measure and the shortfall measure should be identical. By dropping the latter condition, the restrictions on the admissible inequality measures disappear.