We develop the measurement theory of polarization for the case in which income distributions can be described using density functions. The main theorem uniquely characterizes a class of polarization measures that fit into what we call the "identity-alienation" framework, and simultaneously satisfies a set of axioms. Second, we provide sample estimators of population polarization indices that can be used to compare polarization across time or entities.Distribution-free statistical inference results are also used in order to ensure that the orderings of polarization across entities are not simply due to samplingnoise. An illustration of the use of these tools using data from 21 countries shows that polarization and inequality orderings can often differ in practice.

When comparing poverty across distributions, an analyst must select a poverty line to identify the poor, an equivalence scale to compare individuals from households of di erent compositions and sizes, and a poverty index to aggregate individual deprivation into an index of total poverty. A di erent c hoice of poverty line, poverty index or equivalence scale can of course reverse an initial poverty ordering. This paper develops sequential stochastic dominance conditions that throw light on the robustness of poverty comparisons to these important measurement issues. These general conditions extend well-known results to any order of dominance, to the choice of individual versus family based aggregation, and to the estimation of critical sets of measurement assumptions. Our theoretical results are brie y illustrated using data for four countries drawn from the Luxembourg Income Study data bases.

Abstract:The paper investigates how comparisons of multivariate inequality can be made robust to varying the intensity of focus on the share of the population that are more relatively deprived. It follows the dominance approach to making inequality comparisons, as developed for instance by Atkinson (1970), Foster andShorrocks (1988) and Formby, Smith, and Zheng (1999) in the unidimensional context, and Atkinson and Bourguignon (1982) in the multidimensional context. By focusing on those below a multidimensional inequality "frontier", we are able to reconcile the literature on multivariate relative poverty and multivariate inequality. Some existing approaches to multivariate inequality actually reduce the distributional analysis to a univariate problem, either by using a utility function first to aggregate an individual's multiple dimensions of well-being, or by applying a univariate inequality analysis to each dimension independently. One of our innovations is that unlike previous approaches, the distribution of relative well-being in one dimension is allowed to affect how other dimensions influence overall inequality. We apply our approach to data from India and Mexico using monetary and non-monetary indicators of well-being.