Interpreting inequality measures and changes in inequality

Abstract

This paper explores, in the context of the Atkinson inequality measure, attempts to make interpretations of orders of magnitude transparent. One suggestion is that the analogy of sharing a cake among a very small number of people provides a useful intuitive description for people who want some idea of what an inequality measure ‘actually means’. In contrast with the Gini measure, for which a simple ‘cake-sharing’ result is available, the Atkinson measure requires a nonlinear equation to be solved. Comparisons of ‘excess shares’ (the share obtained by the richer person in excess of the arithmetic mean) for a range of assumptions are provided. The implications for the ‘leaky bucket’ experiments are also examined. An additional approach is to obtain the ‘pivotal income’, above which a small increase for any individual increases inequality. The properties of this measure for the Atkinson index are also explored.

JEL Classifications:

• D331
• D63

Keywords:

• inequality
• Atkinson measure
• excess share

Acknowledgements

I have benefited from detailed comments by Anita King, Justin van de Ven and two referees on an earlier draft. I am grateful to Nicolas Hérault and Stephen Jenkins for drawing my attention to literature on the pivotal income.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Furthermore, its relationship with value judgements, such as the principle of transfers, has been well explored. The concepts of Lorenz dominance and generalised Lorenz dominance play a central role in inequality and welfare comparisons; see, for example, Lambert (2001).

2. Two-person cake sharing in the context of the extended Gini measure is examined by Tibiletti and Subramaniam (2015).

3. He also reported values of the relative poverty line for five regions of Brazil.

4. They also considered the role of the benchmark income in the leaky bucket experiment, in the context of effects on inequality measures, rather than the usual reference to constant ‘social welfare’ evaluations.

5. For a critique of the axiom of replication invariance, see Subramanian (2010)

6. This is demonstrated in Appendix below.

7. Where ϵ = 1, , or the geometric mean.

8. Subramanian (2002) related the Gini measure to a welfare function based on rank-order weights, using a corresponding equally distributed equivalent income level; see the Appendix below.

9. With just two income shares, there is only one degree of freedom, and hence one equation in one unknown, x, to be solved. This is found to admit of just one feasible root (if n is sufficiently large). An attempt to extend the comparison to three income shares would result in more than one feasible solution.

10. Curiously, Shorrocks reports (2005, Table 3) values of the excess share which, for n = 10, increase as ϵ is increased.