Lexical Measures of Social Inequality: From Pigou‐Dalton to Hammond

• Published date: 01 September 2019
• Author: Kui Ou‐Yang
• Date: 01 September 2019
• DOI: http://doi.org/10.1111/roiw.12402

© 2018 Internationa l Association for Res earch in Incom e and Wealth

657

LEXICAL MEASURES OF SOCIAL INEQUALITY: FROM

PIGOU-DALTON TO H AMMOND

by Kui Ou-yang*

School of Eco nomics and Manage ment,Northwest Universi ty, China

A measure of social inequality is essentially a rational ordering over a space of social distributions.

However, different measures, including the most popular ones, may provide very different rankings over

the same set of typical distributions. We thus propose an axiomatic approach to inequality measure-

ment mainly based on the Hammond principle, a natural generalization of the Pigou-Dalton principle,

attempting to clarify the true nature of social inequality: the rich get richer and the poor get poorer.

Under the standard assumptions of anonymity and scale independence, we show that a social inequality

ordering is the leximinimax measure if and only if it satisfies the first Hammond principle, and it is the

leximaximin measure if and only if it satisfies the second Hammond principle.

JEL Codes: D63

Keywords: inequality measurement, the Pigou-Dalton principle, the Hammond principle, leximinimax,

leximaximin

1. intrOductiOn

A measure of social inequality is essentially a rational ordering over a space of

social distributions. However, different measures, including the most popular ones,

may provide very different rankings over the same set of typical distributions. For

example, consider three income distributions a = (10, 40, 50), b = (20, 20, 60), and

c = (100/3, 100/3, 100/3). The three famous measures of inequality, i.e., the Gini

index (Gini, 1912), the Theil index (Theil, 1967), and the coefficient of variation,

are calculated in Table 1.

While all indices consider c = (100/3, 100/3, 100/3) the lest unequal, the Gini

index considers a = (10, 40, 50) as unequal as b = (20, 20, 60), the Theil index

considers b less unequal than a, and the coefficient of variation considers a less

unequal than b. Which index should we believe in the end? Or perhaps naively,

what is inequality and how to measure it?

As pointed out by Foster (1985), there are two common approaches to deal

with the paradox of measuring inequality. The first approach is to find a par-

tial ranking that some large class of measures all would be consistent with; for

Note: The author would like to thank the editor and two anonymous referees for their helpful

suggestions and comments. The support from the Key Research Foundation of Humanities and Social

Sciences of the Ministry of Education of China (16JJD790046) is gratefully acknowledged. Any re-

maining errors are the responsibility of the author.

*Correspondence to: Kui Ou-Yang, School of Economics and Management, Northwest University,

Chang’an District, Xi’an, Shaanxi, China (byloyk@gmail.com).

Review of Inc ome and Wealth

Series 65, Numb er 3, September 2019

DOI : 10.1111 /roi w.124 02

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Review of Income and Wealth, Series 65, Number 3, September 2019

658

© 2018 Internationa l Association for Res earch in Incom e and Wealth

example, the Lorenz criterion, an incomplete ranking, is generally accepted as an

unambiguous principle for inequality comparisons and satisfied by most types of

reasonable inequality measures, including the Gini index, the Theil index, and the

coefficient of variation. The second approach is to axiomatically characterize a

reasonable inequality measure by a relevant set of axioms elucidating the nature of

measuring inequality. Those axioms, of course, should better be intuitive, ethically

justifiable, mathematically tractable, and empirically implementable (Cowell, 2000,

2016). The following three principles are fairly standard in the theory of inequality

measurement.

Anonymity: Permutation of income distribution should not change the degree

of inequality, i.e. inequality measures should be symmetrical between social mem-

bers. For example, a = (10, 40, 50) should be as unequal as both (40, 50, 10) and

(50, 10, 40).

Scale Independence: Equally proportionate changes in individual incomes

should not change the degree of inequality. Thus d = (6, 24, 30) should be as

unequal as a = (10, 40, 50), and e = (10, 10, 30) should be as unequal as b = (20,

20, 60).

Pigou-Dalton Principle (Pigou, 1912; Dalton, 1920): A mean-preserving trans-

fer of income from a person to a richer one should increase the degree of inequal-

ity. Hence (10, 30, 60) should be more unequal than both a = (10, 40, 50) and

b = (20, 20, 60).

The Pigou-Dalton principle elucidates the true nature of inequality: it gets

more unequal when the rich get richer and the poor get poorer; but it only applies

when two distributions have the same mean. When two distributions have unequal

means, the Pigou-Dalton principle can be extended into the following principle.

Hammond Principle (Hammond, 1976): Increasing the income of a richer per-

son and decreasing the income of a poorer person should increase the degree of

inequality. Therefore, e = (10, 10, 30) must be less unequal than d = (6, 24, 30),

which cannot be derived from the Pigou-Dalton principle.

Now, we claim that b = (20, 20, 60) is less unequal than a = (10, 40, 50). By

the Hammond principle, e = (10, 10, 30) must be less unequal than d = (6, 24, 30).

By scale independence, d is as unequal as a, and e is as unequal as b; that is, e is

less unequal than d if and only if b is less unequal than a. Therefore, b must be

less unequal than a, which means that both the Gini index and the coefficient of

variation violate the Hammond principle. In fact, the Theil index must also violate

the Hammond principle, since, as we will show, there is no continuous inequality

measure satisfying the Hammond principle.

TABLE 1

Difficulty in Measuring Inequality

Measure s Form ulas abcRankings

Gini index

∑n

i=1

∑n

j=1

�

xi−xj

�

2n(n

−

1)𝜇

.4000 .4000 0c ≺ a ∼ b

Theil index

1

nlnn∑n

i=1

x

i

𝜇

ln

x

i

𝜇

.1413 .1350 0c ≺ b ≺ a

Coeff icient of

variation



1

n(n−1)



n

i=1



1−xi

𝜇

2

.3606 .4000 0c ≺ a ≺ b