We present a class of decomposable inequality indices for ordinal data (e.g. self-reported health survey). It is characterized by well-known inequality axioms (e.g. scale invariance) and a decomposability axiom which states that an index can be represented as a function of inequality values in subgroups and subgroup sizes. The only decomposable indices are strictly monotonic transformations of the weighted average of frequencies in categories. Among the indices proposed in the literature only the absolute value index (Abul Naga and Yalcin, 2008; Apouey, 2007) is decomposable. As an empirical illustration we calculate regional contributions to overall health inequality in Switzerland.

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This paper develops a normative approach to the measurement of ex-ante inequality of opportunity in a multidimensional setting, i.e., when the individual outcome is represented by a multidimensional variable. We characterize three classes of social welfare functions, all endorsing ex ante compensation but each of them reflecting a specific reward principle: (1) utilitarian, (2) agnostic and (3) averse. The first class is implemented via generalized Lorenz Dominance applied to each attribute separately. The agnostic and inequality averse classes are implemented by a welfarist Lorenz ordering, namely, of type-aggregate utilities. In the case of inequality-averse class, utility functions are submodular, hence capturing the dependence between attributes. We also develop normative inequality indices (Atkinson, 1970; Kolm 1969; Sen, 1973) for the classes of welfare functions and study their properties. Finally, we propose an empirical applications of the methods developed in the paper: by using the National Longitudinal Study of Adolescent to Adult Health (Add Health) we evaluate inequality of opportunity in U.S. for the case of three dimensions of individual outcomes, namely, education, health and income.

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We focus on a question that has been long addressed in economics, namely, of one distribution being better than another according to a normative criterion. Our criterion distinguishes between interdependence and behaviour in the margins. Many economics contexts concern interdependence only e.g. complementarities in production function, intergenerational mobility, social gradient in health. We compare bivariate discrete distributions and measure interdependence via a most general measure, namely, a copula (Schweizer and Wolff 1981). For discrete distributions we need to overcome a problem of many copulas associated with a given distribution. Drawing on a copula theory (Carley 2002, Genest andNeslehova 2007) we solve this problem, chose a method to compare copulas which together with first-order stochastic dominance of marginal distributions gives the ordering to compare distributions. We provide a type of Hardy-Littlewood-Pólya result (Hardy et al. 1934), that is, we give implementable characterizations of this ordering (Theorems 1 -3). As an application, we show how this ordering can be used to measure several phenomena that use either ordinal data (e.g. education-health gradient, bidimensional welfare) or simply discrete distributions (e.g. percentile income distributions of fathers and sons for intergenerational mobility). Welfare measures are easily decomposable into attributes and interdependence.3 Specifying the bounds of the treatment effects distribution is related to a copula theory (Frank et al. 1987). In particular, for discrete distributions of outcomes (e.g. life satisfaction and psychological indicators in assessing Moving to Opportunity (Ludwig et al. 2013)) these bounds can be improved using the results of Carley (2002) which we use a lot in the paper. 4 We use dependence/association/concordance interchangeably, although they are all different concepts (Nelsen 2006); in our setting, however, we do not need a detailed differentiation.