A multidimensional Lorenz dominance relation

This paper considers the problem of constructing a multidimensional Lorenz dominance relation (MLDR) satisfying normatively acceptable conditions. One of the conditions, Comonotonizing Majorization (CM), is a weaker form of the condition of Correlation Increasing Majorizaton considered in the literature on multidimensional inequality indices. A condition, called Prioritization of Attributes under Comonotonicity (PAC), is also proposed: Suppose that the distributions of the attributes are comonotonic and that the distribution of attribute i Lorenz dominates that of attribute j. Consider a distribution vector y which is also comonotonic with i and j and which Lorenz dominates the distributions of both i and j. Then, the distribution matrix obtained by replacing the distribution of j by y dominates the matrix obtained by making the same replacement for i. An MLDR is constructed by considering each individual‟s well-being to be a weighted average of the attribute levels and applying the concept of unidimensional Lorenz dominance to the resulting vector of individual well-beings. The suggested weight vector is a function of the distribution of the attributes. It is shown that the proposed MLDR satisfies both CM and PAC. The existing literature does not seem to contain an example of an MLDR satisfying these two conditions simultaneously. (194 words)