A New Generalized Variance Approach for Measuring Multidimensional Inequality and Poverty

The paper suggests a new generalized variance concept for measuring multidimensional inequality of a stratified society, based on multivariate statistical methods, where the members of society form a cloud in the oblique space of dimensions of inequality, such as income, expenditure and property. The cloud presents the multidimensional inequality capsulized in the cloud. The goal is to condense all the inequality information embodied by the cloud into a composite compact metric characterizing both the shape and the inner structure of the cloud. Contrary to the conventional literature that considers multidimensionality as a unidimensional weighted combination of the dimensions, our new composite index measures the inequality of the configuration of the points in the cloud. Our aim is twofold. First, we introduce the Inequality Covariance Matrix (ICM) assigned to the cloud, with elements measuring the correlations among dimensions. Having ICM, we propose the Generalized Variance (GV) of ICM to measure the composite Generalized Variance Inequality (GVI) level. Second, to evaluate the stratum-specific structure of the overall inequality, we suggest a new two-stage procedure. In the first stage, we divide the total GVI into between-groups and within-groups effects. Then, in the second stage the contributions of the strata to the within-groups inequality and, the contributions of the dimensions to the between-groups inequality are calculated. This GVI approach is sensitive to the correlation system, decomposable into stratum effects and, the number of dimensions is not limited. Moreover, including the log-dimensions in the analysis, GVI yields an Entropy Covariance Matrix giving a new Generalized Variance Entropy index. Finally, the GVI of censored poverty indicators means multidimensional poverty measurement. This special complex task is not yet solved in the traditional literature so far.

Lorenz Surfaces Based on the Sarmanov–Lee Distribution with Applications to Multidimensional Inequality in Well-Being

The purpose of this paper is to derive analytic expressions for the multivariate Lorenz surface for a relevant type of models based on the class of distributions with given marginals described by Sarmanov and Lee. The expression of the bivariate Lorenz surface can be conveniently interpreted as the convex linear combination of products of classical and concentrated univariate Lorenz curves. Thus, the generalized Gini index associated with this surface is expressed as a function of marginal Gini indices and concentration indices. This measure is additively decomposable in two factors, corresponding to inequality within and between variables. We present different parametric models using several marginal distributions including the classical Beta, the GB1, the Gamma, the lognormal distributions and others. We illustrate the use of these models to measure multidimensional inequality using data on two dimensions of well-being, wealth and health, in five developing countries.

Multidimensional Inequality in Vietnam, 2002–2012

We investigate the evolution of multidimensional inequality of well-being in Vietnam in the period 2002–2012 using household survey data. Our study focuses on four crucial dimensions of human welfare: consumption, education, health and housing. We measure inequality by means of the multidimensional Atkinson index, which belongs to the Atkinson family of relative inequality indices. The choice of the values of two crucial parameters, with respect to the aversion to inequality on the one hand and the degree of substitutability between dimensions on the other hand, has a significant influence on the perceived trends of inequality. We consider different combinations of dimensions (two, three and four dimensions) and a wide variety of values of the parameters, with the aim of arriving at a robust understanding of the extent of inequality in Vietnam. Our results suggest that the level of multidimensional inequality in Vietnam has decreased, albeit that this is not the case for all combinations of the parameter values. Our study shows that looking at multidimensional rather than one-dimensional inequality leads to a richer understanding of the evolution of inequality, and indicates that it is important to be aware of the influence of value judgments on the assessment of inequality.