A Unified Approach to Measuring Poverty and Inequality
To detect the difference between these two situations, we need to use a measure that is sensitive to association between dimensions. Seth (2009) extended the method of Foster and others to a class of multidimensional standards that are sensitive to both forms of inequality: the welfare indicators of each person are first aggregated using a general mean of order b < 1; then these personal aggregates are aggregated using a general mean of order a < 1 to obtain the overall measure. Note that when a is equal to b, the measure belongs to the Foster and others class and is neutral to the second form of inequality. When a is not equal to b, the measure is sensitive to association among dimensions. For the detailed methodology, see Seth (2012). This second form of inequality has also been discussed in the poverty measurement literature (see Tsui 2002; Bourguignon and Chakravarty 2003; Alkire and Foster 2007, 2011). Given a multidimensional standard s incorporating one or both notions of inequality, it is then straightforward to define a multidimensional inequality measure as the percentage shortfall of s from the overall mean achievement, namely, I = (m − s)/m. It should be noted, though, that many assumptions are needed to construct s, which can make multidimensional inequality I hard to measure in practice. Key among these are assumptions pertaining to the cardinalization and comparability of the component indicators; changing the way a variable is measured and altering its value vis-à-vis other variables can change the rankings provided by s and the inequality measure. Particularly vexing is the case where one or more of the variables are ordinal, so that the cardinal form of each variable must, by definition, be arbitrary. One way forward is to restrict consideration to multidimensional versions of stochastic dominance (see Atkinson and Bourguignon 1982). However, the case that addresses this issue—first-order dominance—is precisely the case where the first form of inequality must be ignored. Further work is needed to construct robust multidimensional standards and practical indicators of multidimensional inequality.
Inequality of Opportunity The previous section examined the general case where several welfare indicators contribute to a person’s well-being. We now return to the simpler case of a single welfare indicator, but where other variables provide information on relevant characteristics or “identities” of the individuals. Recent
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