Measuring Achievement and Shortfall Improvements in a Consistent Way
Iñaki Permanyer, Universitat Autònoma de Barcelona
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 a family of achievement and shortfall improvement indices and present the necessary and sufficient conditions under which they rank social states in a consistent way. We also suggest collecting information at small levels of aggregation to uncover unobserved heterogeneity, therefore complementing traditional country-level analysis and taking into account the entire distribution of local-level improvements over time. Empirical illustrations using census data from five African countries suggest that consistency between achievement and shortfall improvements in standards of living is not only a matter of theoretical import but it is also a problem that can be encountered in practice to a large extent.