The gini index, the dual decomposition of aggregation functions, and the consistent measurement of inequality

Abstract

In several economic fields, such as those related to health, education or poverty, the individuals' characteristics are measured by bounded variables. Accordingly, these characteristics may be indistinctly represented by achievements or shortfalls. A difficulty arises when inequality needs to be assessed. One may focus either on achievements or on shortfalls but the respective inequality rankings may lead to contradictory results. Specifically, this paper concentrates on the poverty measure proposed by Sen. According to this measure the inequality among the poor is captured by the Gini index. However, the rankings obtained by the Gini index applied to either the achievements or the shortfalls do not coincide in general. To overcome this drawback, we show that an OWA operator is underlying in the definition of the Sen measure. The dual decomposition of the OWA operators into a self-dual core and anti-self-dual remainder allows us to propose an inequality component which measures consistently the achievement and shortfall inequality among the poor. Keywords AbstractIn several economic fields, such as those related to health, education or poverty, the individuals' characteristics are measured by bounded variables. Accordingly, these characteristics may be indistinctly represented by achievements or shortfalls. A difficulty arises when inequality needs to be assessed. One may focus either on achievements or on shortfalls but the respective inequality rankings may lead to contradictory results. Specifically, this paper concentrates on the poverty measure proposed by Sen. According to this measure the inequality among the poor is captured by the Gini index. However, the rankings obtained by the Gini index applied to either the achievements or the shortfalls do not coincide in general. To overcome this drawback, we show that an OWA operator is underlying in the definition of the Sen measure. The dual decomposition of the OWA operators into a self-dual core and anti-self-dual remainder allows us to propose an inequality component which measures consistently the achievement and shortfall inequality among the poor.

The Generalized Gini Welfare Function in the Framework of Symmetric Choquet Integration

“…which can be easily shown to be equivalent to (1). In fact, the double summation expression for n 2 G c A (x) as in (2) corresponds to…”

“…The proof follows straightforwardly from (44) - (45), associated to the constraints (43), and the definition of the classical Gini inequality index (1).…”

“…, x n ) represents the 1 income distribution of a population of n ≥ 2 individuals. Recently, the extended interpretation of this formula in terms of the dual decomposition [19] of aggregation functions has been discussed in [20,1].…”

The dual decomposition of aggregation functions and its application in welfare economics

“…, n}, and by x > y we mean x ≥ y and x = y. Given x ∈ [0, 1] n , the increasing and decreasing reorderings of the coordinates of x are indicated as x (1) ≤ · · · ≤ x (n) and x [1] ≥ · · · ≥ x [n] , respectively. In particular, x (1) = min{x 1 , .…”

“…We now discuss an application of the dual decomposition in the context of a well-known poverty measure proposed by Sen, following Aristondo et al [1].…”