Multidimensional Gini Indices, Weak Pigou-Dalton Bundle Dominance and Comonotonizing Majorization

This paper considers the problem of constructing a normatively significant multidimensional Gini index of relative inequality. The social evaluation relation (SER) from which the index is derived is required to satisfy a weak version of the Pigou-Dalton Bundle Principle (WPDBP) (rather than Uniform Majorization or similar conditions). It is also desired to satisfy a weak form of the condition of Correlation Increasing Majorization called Comonotonizing Majorization (CM). The problem of measuring multidimensional inequality is here interpreted to be essentially a problem of setting weights on the different attributes. It is argued that determination of these weights is linked to the problem of determining the weights of the individuals. A number of conditions on the two sets of weights and on their interrelationships are proposed. By combining these conditions with a social evaluation function which is decomposable between equality and efficiency components we obtain a specific SER. The Kolm index derived from this relation is then suggested as the multidimensional inequality index. It is shown that the proposed index is a multidimensional Gini index satisfying the (inequality index versions of the) properties of WPDBP and CM. The index does not seem to have appeared in the literature before. Moreover, the literature does not seem to contain any other normatively significant multidimensional Gini index that would satisfy both of these properties if the allocation matrices are not restricted to be strictly positive. In this paper this restriction has been relaxed on grounds of potential empirical applicability of the index.