Measuring Achievement and Shortfall Improvements in a Consistent Way

Abstract

In measuring improvements over time of bounded variables, one can focus on achievements or shortfalls. However, rankings of alternative social states in terms of achievements and shortfalls do not necessarily mirror one another. We characterize axiomatically different families of achievement and shortfall improvement indices, and present the necessary and sufficient conditions under which they rank social states in a consistent way. Empirical illustrations using child mortality data from South Africa suggest that consistency between achievement and shortfall improvements in standards of living is not only a matter of theoretical import but is also a problem that can be encountered in practice to a large extent.