Multidimensional Inequality and Dependence between its Dimensions

Well-being and its inequality are inherently multidimensional concepts (Tobin, 1970; Sen, 1992). Multidimensional measures of inequality allow taking this multidimensionality explicitly into account. One of the important added values of such an approach compared to the standard unidimensional ones is its sensitivity to the dependence between the dimensions. Intuitively, we say that a multidimensional distribution is more dependent than another one, when its dimensions tend to more ”large” or ”small” together. The purpose of the paper is to look in detail at the notion of dependence between the dimensions of a multidimensional distribution by proposing and comparing some partial and complete dependence orderings. These orderings can be useful in the field of measuring multidimensional inequality, horizontal inequality or income mobility. In the second section we start by investigating the notion of dependence by comparing the dependence of two multidimensional distributions with the same marginal distributions. Two partial orderings are defined. The first partial ordering, the supermodular ordering is introduced by Epstein and Tanny (1980) and reintroduced in the multidimensional inequality literature by Tsui (1999). A m-dimensional distribution is said to be more dependent according to the supermodular ordering, when it can be obtained from another one by a finite series of correlation increasing transfers, defined as follows. ∗