A note on inequality-preserving distributional changes

ABSTRACT

This note considers the problem of distributing a fixed amount of money (‘income’) among a given number of people, such that inequality (measured by either the Gini or Atkinson measure) takes a specified value. It is well known that simultaneous equations admit of many solutions where the number of variables exceeds that of equations (constraints). However, the approach examines cases where there are just one or two degrees of freedom, clarifying the resulting range of distributions. The properties of simultaneous disequalising and equalising transfers are discussed.

KEYWORDS:

• Inequality
• Gini measure
• Atkinson measure

Acknowledgment

In preparing this note I have benefited from encouragement by Peter Lambert and discussions with Chris Ball, Norman Gemmell and Justin van de Ven.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. The question of the accounting period, the unit of analysis and the precise ‘welfare metric’ are rarely given the attention they deserve.

2. Writing the reverse-ranked-weighted mean as , it can be seen that, for large n, the Gini is effectively 1 − . Hence, is the equally distributed equivalent income for the Gini welfare function.

3. This form of Gini is not scale invariant, though this matters only for small n. In this case of n = 2, the maximum value that can be taken by G is 0.5. The following expression for G with n = 2 is given by Shorrocks (2005). An alternative version is given by Subramanian (2002) for the form of Gini that, for small n, lies between 0 and 1.

4. For these two distributions [1, 2, 3] and [0.8, 2.4, 2.8], the Atkinson values are quite different. For example, for ϵ of 0.2 and 1.2, the Atkinson values for the first distribution are 0.018 and 0.110, respectively, while for the second distribution they are 0.022 and 0.152.

5. Extending the requirement to include a fixed third moment would thus be rather cumbersome as it would require the solution to a cubic in addition to a quadratic similar to Equation (16(16) ).

6. Furthermore, using dx₁ = −(dx₂ + dx₃) and setting the total differential, dA_ϵ, equal to zero gives dx₂ = −dx₃(x₃ − x₁)/(x₂ − x₁). Since dx₂/dx₃ is not constant, this expression cannot be used to examine discrete changes. This contrasts with the Gini case discussed above where only the arithmetic mean and inequality are constant and there are three individuals.

7. For this case of dx⁻ = 0.05, the expression in Equation (24(24) ) gives a value of dx⁺ = 0.43, showing the extent of the nonlinearity of the constraint, since for the discrete change examined, it is necessary only to transfer 0.2 from x₃ to x₄.

8. To give just one example using numerical methods, the distributions [1, 2, 4, 5, 10] and [0.8, 2.6, 6.3, 5.2, 10] have the same Gini and the first three moments. The second can be obtained from the first by using two disequalising transfers and one equalising transfer. Small variations in the top income (of 10) generate two solutions for each imposed value of x₅.