## Abstract

This paper rigorously demonstrates that for any unequal income distribution, the well-known Gini index of inequality is bounded above by the recently revived Bonferroni inequality index. The bound is exactly attained if and only if out of *n* incomes in the society, *n* − 1 poor incomes are identical. The boundedness theorem is shown to possess a duality-type inequality implication. These two inequality metrics, two popular members of a general class of inequality indices generated by Aaberge’s (J Econ Inequal 5:305–322, 2007) ‘scaled conditional mean curve’, may lead to different directional rankings of alternative income distributions because of some important differences between them. We then explicitly examine their sensitivity to Weymark’s (Math Soc Sci 1:409–430, 1981) ‘comonotonic additivity’ postulate.

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## Notes

A common characteristic of these welfare functions is that they are linear homogenous and when all incomes increase by a constant amount, they increase by the amount itself. A welfare function possessing this characteristic becomes helpful in measuring ethical distance between two income distributions (see Chakravarty and Dutta 1987).

Strictly speaking, Weymark (1981) considered non-increasing ordering of incomes. All his arguments apply as well to the framework that relies on non-decreasing income ordering. Multidimensional extensions of the (non-increasing) comonotonic additivity postulate were considered by Gajdos and Weymark (2005) to characterize variants of the multidimensional generalized Gini social welfare functions.

A social welfare function is called S-concave if for all \(\mathbf {x}\in D_n^{+}\), \(W(\mathbf {x}B)\ge W(\mathbf {x})\), where

*B*is any \(n\times n\) bistochastic matrix, a non-negative square matrix of order*n*each of whose of rows and columns sums to 1.*W*is called strictly S-concave, if the weak inequality is replaced by a strict inequality whenever \(\mathbf {x}B\) is not a permutation of \(\mathbf {x}\). All S-concave functions are symmetric.Dutta 2002 made an extensive discussion on this generalized index.

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We thank the reviewers for their kind comments which have helped in improving the paper.

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Chakravarty, S.R., Sarkar, P. New perspectives on the Gini and Bonferroni indices of inequality.
*Soc Choice Welf* (2021). https://doi.org/10.1007/s00355-021-01311-4

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DOI: https://doi.org/10.1007/s00355-021-01311-4