A Multidimensional Lorenz Dominance Relation

Multiple attributes of standard of living Hence, methods of measurement of inequality need to be extended to the multidimensional context. As in unidimensional theory, two approaches to comparing the degrees of inequality of alternative distributions: (1) complete ordering by means of inequality indices.[See, for instance, Kolm (1977) and Tsui (1995, 1999).] Different indices may lead to different orderings. (2) partial ordering; not necessarily complete but intuitively more acceptable; dominance criteria. 2

In unidimensional theory the most widely used dominance relation is Lorenz dominance. This paper considers the problem of obtaining a multidimensional Lorenz dominance relation (MLDR). If relative weights of the different attributes are known, the problem is trivial. Otherwise, no obvious or unique answer. The literature contains several suggestions. A general definition of an MLDR is developed in this paper. The paper proposes two axioms which an MLDR may reasonably be expected to satisfy, notes that the existing literature does not seem to contain an example of an MLDR that satisfies both the axioms and constructs one that does. 3

We shall be concerned with relative inequality. Confine attention to discrete distributions. n individuals; m attributes; X = distribution matrix = ((x p j )) ; x p j = amount of j-th attribute allocated to the p-th individual; x p and x j are, resp.,the p-th row and the j-th column of X. X is non-negative; at least one positive entry in each column. For technical reasons, assume that no attribute is equally distributed. X = the set of all such X’s. A binary relation D on X is called a multidimensional inequality dominance relation (MIDR) if it satisfies the following conditions: 4

Continuity (CONT), Quasi-ordering (QORD), Ratio Scale Invariance (RSI), Invariance w.r.t. Row Permutations (IRP) and Uniform Lorenz Majorization (ULM) [If X and Y in X are such that X = BY for some positive, symmetric and bistochastic matrix B, then X D Y]. ULM is similar to (but technically independent of) Kolm’s (1977) axiom of Uniform Majorization [If X = BY for some bistochastic matrix B, then society prefers X to Y.] The antecedent of ULM implies that as we move from Y to X, the Lorenz curve for each attribute moves “upward” in such a way that the area between the line of equality and the Lorenz curve decreases in the same proportion for each attribute. 5

For all X and Y in X, X D Y is interpreted to mean that the over-all degree of inequality in the distribution matrix X is no more than that in the distribution matrix Y, whatever the method of measuring inequality may be. Let L be the unidimensional Lorenz dominance relation on the set of non-negative distribution vectors. A multidimensional Lorenz dominance relation (MLDR), L M, on X is an MIDR such that if m = 1, then L M = L. 6

The literature contains several specific MLDR’s. Two examples: (1) For all X and Y in X, X L M Y if and only if (Xw) L (Yw) for all w in R ++ m. (2) For all X and Y in X, X L M Y if and only if x j L y j for all j = 1,2,…,m. We impose the following two axioms on any MLDR, L M. Axiom 1[ Comonotizing Majorization (CM)]: For all X and Y in X such that (i) X is mixed monotonic and (ii) Y is a comonotonization of X, X P M Y where P M is the asymmetric component of L M. 7

[A real vector whose components are in non- increasing (resp. non-decreasing) order is non- increasing (non-decreasing) monotonic. A matrix is comonotonic if either all its columns are non- increasing monotonic or all of them are non- decreasing monotonic. It is mixed monotonic if all its columns are non-increasing or non-decreasing comonotonic but it is not comonotonic. For any matrix, its comonotonization is a comonotonic matrix obtained by rearranging the entries in each column, if necessary. Obviously, a matrix has two comonotonizations (non-increasing and non-decreasing).] 8

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Axiom 2 [Prioritization of Attributes under Comonotonicity (PAC)]: For all comonotonic X in X, and for all i, j = 1,2,…,m such that x j L x i, if y in R + n is such that (i) 0 ≠ y ≠ k1 n for any scalar k, (ii) y is comonotonic with the columns of X and (iii) yLx i for all i=1,2,…,m, then [X − i,y L M X − j,y ] where X − i,y and X − j,y are the matrices obtained by replacing the i-th and the j-th columns of X, resp., by y. Intuitively, PAC means that if x j Lorenz dominates x i and y lorenz dominates all columns of X, then by replacing x i by y in X get a better distribution (in terms of L M ) than that obtained by making the same replacement for x j i.e. it is “more important” to reduce the more acute inequalities. 10

CM and PAC are independent. The existing literature does not seem to contain an example of an MLDR which satisfies both CM and PAC. E.g., Example (1) above [Dominance by all positive weights] satisfies CM but violates PAC. Example (2) [Column-wise Dominance] satisfies PAC but violates CM. Does there exist an MLDR satisfying CM and PAC? The answer is in the affirmative. For all X in X, let X^ denote the ‘scaled’ version of X obtained by dividing each entry by the arithmetic mean of the column containing it. 11

Let Com(X^) be a comonotonization of X^ and C(X^) the matrix of covariances of the columns of Com(X^). For all X in X, let w(X^) be the first eigen vector (i.e. the eigen vector associated with the maximal eigen value) of C(X^). Consider the following relation L* on X: For all X and Y in X, X L*Y if and only if [X^w(X^)] L [Y^w(Y^)]. Proposition: L* is an MLDR satisfying CM and PAC. 12