An elementary characterization of the Gini index

The Gini index is one of the most used indicators of social and economic inequality. In this paper we characterize the Gini index as the unique function that satis…es the properties of scale independence, symmetry, standardization and separability. Furthermore, we propose a simpler way to compute it. Keywords: Gini index; income inequality; axiomatization. JEL Classi…cation: D31, D63, I31. 1 Introduction In this article we o¤er a novel characterization of the Gini index1 , exploiting the geometry of the contour lines of this index restricted to the unit simplex. In the case of three agents, these contour lines are hexagons whose edges are line segments on distribution sets that maintain a similar ordering (see Foster and Sen [4]). There are six permutations of the agents, which lead to the six edges of any contour line of the Gini index. The basic idea is to …nd appropriate axioms that take advantage of this fact. The treatment of extreme Corresponding author. Facultad de Economía, UASLP; Av. Pintores s/n, Col. B. del Estado 78213, San Luis Potosí, México. Tel. +52 (444) 8131238 Ext. 120, Fax. +52 (444) 8174705 1The coe¢ cient attributed to Gini [5] has been presented and analyzed by Dalton [2], Atkinson [1] and Sen [6], among others. The discrete formula of the Gini coe¢ cient as a standardized average of income di¤erences is presented in several equivalent versions, a good explanation is available in the classic paper of Sen [6] or in the survey of Dutta [3]. Atkinson [1] provides the basis to discuss the relation between inequality and social welfare orders.