MULTIDIMENSIONAL POLARIZATION

Gigliarano and Mosler (2009) constructed multivariate socioeconomic polarization indi­ces on the presumption that polarization should capture internal homogeneity, external heterogeneity, and similarity of group sizes.

To see how, consider a distribution of N individuals along C types of endowments, represented by a matrix X

where x_iq denotes the endowment q of individual i. Using inequality decomposition, the multidimensional polarization index is given by:

where GB and GW are multivariate indices that measure inequality between and within groups, respectively, and where ζ is a function that is increasing in GB and S and is decreasing in GW. S(X) is a measure of deviation from equally sized groups; it is maximal if all groups are of the same size. Particular forms of this index are given by

with constant c being positive and possibly dependent on the choice of indices GB and GW The functions φ, ψ, and τ are assumed to be continuous and strictly increasing, with φ(0) = τ(0) = 0. These forms were subsequently developed by Gigliarano and Mosler (2009) using additively decomposable multivariate inequality measures. Similar proce­dures are proposed for multiplicatively decomposable inequality measures. Equation (5.51) is similar to Zhang and Kanbur (2001) except for the explicit role of group sizes S(X) and for the fact that X is a matrix.

Two alternative multivariate polarization indices with special relevance to the case of a population split into two groups are proposed in Anderson (2010). When these two groups are made of poor and nonpoor individuals, these indices can be used as multivar­iate relative poverty measures based on distances between them. Consider then a population composed of two groups: the poor and the nonpoor. They are assumed to be distributed according to two continuous multivariate unimodal distributions: f_p(x) for the poor andf(x) for the nonpoor, where x is a 1 ? C vector of characteristics. An “overlap” measure is defined as:

This is a multidimensional extension of the overlap measure of Equation (5.45). Assessing the degree of polarization between two groups when individuals have many character­istics amounts in this formulation to capturing the degree of commonality between the two distributions.

Anderson (2010) proposed an alternative index that can be used when attributes are mutually exclusive and do not overlap. Let x_mp and x_mr be the value of the characteristic vector at the modal point of the poor and the nonpoor distributions, respectively. The index is the area of the trapezoid formed by the modal peaks of the densities and the mean normalized Euclidean distance between these two points. Let μ_q be the mean of the q th characteristic in the pooled population. When the poor and nonpoor distributions are separately identified in C dimensions, multidimensional polarization is written as:

This can be interpreted as a multivariate relative poverty index without formal poverty frontiers, the index being based on distances between two identified groups, poor and nonpoor.

A multidimensional extension of DER’s identification/alienation framework can also be designed. A natural procedure for that is found in Anderson (2011). The extension is multidimensional in the sense that both identification and alienation depend on the joint distribution of multiple socioeconomic attributes. The multivariate distribution can also be defined over a collection of both discrete and continuous variables.

To see how, let u_i and v_j∙ be stacked vectors of continuous and discrete variables, with dimension k (for the continuous variables) and h (for the discrete variables) and for indi­viduals i and j, respectively. A multivariate polarization index taking into account both continuous and discrete variables is then given by

This is the normalized Euclidean distance between individuals i and j, where u_iq is the q th variable of individual i and C = k + h.

5.9.