Inequality of ratios

Some socioeconomic indicators can be represented in alternative ways. For example, as either attainments (e.g. child survival rates) or shortfalls (e.g. child mortality rates) if the variables are bounded. The literature has long been concerned with the consistency of inequality comparisons across such alternative representations. The case of bounded variables and their two alternative representations (attainments versus shortfalls) has been largely settled. This paper addresses the extent of the consistency problem in inequality comparisons involving ratio indicators which also have two potential representations (e.g. number of people per room or number of rooms per capita in the case of overcrowding). First, we probe welfare comparisons based on the generalised Lorenz curve and find that consistency can be secured in the presence of rank dominance. Second, we show that robust inequality comparisons based on all possible Zoli partial orderings (which include all relative and absolute inequality partial orderings, among others) are inconsistent across alternative representations of ratios. Third, with the identification of a class of inequality indices based on the ratio of the harmonic to the arithmetic mean, we show that complete orderings consistent across alternative representations of ratios do exist. Then we consider and ponder the pros and cons of three alternative solutions: defending one representation, using inequality indices that combine both representations, and functional transformations of the ratio variable. We find that the costs of these alternatives render them inferior to the class of indices based on the harmonic and arithmetic means. Both the consistency problem and its preferred proposed solution are illustrated with an empirical study of intergenerational changes in overcrowding inequality in Mexico.