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Multidimensional inequality indices for ordinal data

Western governments have recently been putting increasing emphasis on using nonincome dimensions in measuring society's well-being (e.g. the UK, France). It is stressed that well-being consists of many indicators such as happiness, health or education and that not only averages, but the characteristics of the whole distribution (e.g. inequalities) should be acknowledged. Yet widely used data such as subjective life satisfaction, declared health status or level of education completed are, opposite to income, ordinal in nature and there is not yet any literature on how to measure inequality when we have several ordinal variables. We address this problem. As a main result we characterize the class of multidimensional inequality indices by standard inequality axioms and based on the characterization theorem we propose a bidimensional inequality measure. A novel feature of our approach is the use of copula orderings to capture association between variables. Our method ensures that the index is also attribute decomposable; that is, we can say exactly how much of the overall inequality comes from inequality in marginals and how much from association between dimensions. We illustrate our methodology with the US data for the period 1972-2010. Except for year 1985 inequality in perceived happiness contributes more to overall inequality than health inequality; association's contribution reaches its peak (close to 30 percent) in year 2004.