Abstract
In this chapter, we begin to take cognisance of the multidimensional nature of development. However, we approach the matter in steps. Since we desire an inequality-sensitive measure of development, we first consider the problem of how to measure multidimensional inequality. This is the subject matter of this chapter. As in the unidimensional case, here again, inequality can be measured in various ways. Again, however, there is no guarantee that if inequality as measured by a particular multidimensional inequality index is seen to decrease, the same will be true of inequality measured by a different index. This suggests that a more appropriate procedure would be to look for a way to extend the concept of Lorenz dominance from the unidimensional context to the multidimensional one. The chapter starts by stating a number of conditions or properties that one would intuitively expect any notion of multidimensional Lorenz dominance to satisfy and using these conditions to formulate a definition of a multidimensional Lorenz dominance relation. It then examines a number of “candidate” relations that have been proposed in the literature and shows that all of these fail to satisfy the definition formulated here. The question, therefore, arises as to whether there exists a Lorenz dominance relation as defined here. The chapter gives an affirmative answer to the question.
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Notes
- 1.
It is possible to formulate weaker versions of the condition of UM. For instance, we may require that if X and Y are such that X = BY for some bistochastic B but X is neither equal to nor a row permutation of Y, then X D Y (rather than X DP Y). Our claim in the text that some of the dominance relations proposed in the literature violate UM is based on the version that has been formulated in this chapter, i.e. the one that is based on the original Kolm (1977) formulation. A major reason for preferring UM over the weaker condition referred to above is that, as can be easily seen, if m = 1, the weaker condition would not reduce to the unidimensional PD majorisation principle. In other words, it is not a multidimensional version of that unidimensional principle in the proper sense of the term.
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Banerjee, A.K. (2020). Multidimensional Lorenz Dominance. In: Measuring Development. Themes in Economics. Springer, Singapore. https://doi.org/10.1007/978-981-15-6161-0_5
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