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On Economic Inequality$

Amartya Sen

Print publication date: 1973

Print ISBN-13: 9780198281931

Published to British Academy Scholarship Online: November 2003

DOI: 10.1093/0198281935.001.0001

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A.4.1 The Class of Relative Inequality Measures

A.4.1 The Class of Relative Inequality Measures

Source:
On Economic Inequality
Publisher:
Oxford University Press

In this section, we are concerned with the properties of inequality measures and the quasi‐orderings that can be obtained from their congruence. We focus particularly on the relative aspect of inequality comparisons. Most of the commonly used numerical measures of inequality are replication invariant and mean independent; that is, they are invariant to changes in population size or mean income which leave the relative distribution unchanged. Lorenz comparisons also have these invariance properties. For example, if the income levels in distribution x were replicated arbitrarily to obtain the distribution (x, . . . , x), or if they were rescaled by a positive k to obtain the distribution k x, the Lorenz curve would be unaltered.41

Inequality measures that satisfy (1) symmetry, (2) replication invariance, and (3) mean independence (these three stand, respectively, for invariance under permutations, population replications, and scalar multiplication), and also (4) the Pigou–Dalton condition (inequality increases as a result of a regressive transfer), are called measures of relative inequality (p.140) (or simply relative measures).42 Prominent examples include the coefficient of variation C, the Gini coefficient G, and the Theil measure T, each described in OEI‐1973 (pp. 27–36).

Two other families are also worth considering, which are generalizations of the Theil and Gini measures respectively. The first is the generalized entropy class of measures, defined for values α other than 0 and 1 by:43

I α ( x ) = 1 α ( 1 α ) 1 n i = 1 n [ 1 ( x i μ ) α ]
with I 1 being the Theil measure:
I 1 ( x ) = T ( x ) = 1 n i = 1 n x i μ ln  ( x i μ )
and I 0 being Theil's ‘second’ measure, also known as the mean logarithmic deviation:
I 0 ( x ) = D ( x ) = 1 n i = 1 n ln  ( μ x i )
Note that I 2 is a multiple of the squared coefficient of variation C 2.

It may seem odd to generalize the Theil measure which itself is ‘not exactly overflowing with intuitive sense’ (OEI‐1973, p. 36). The primary justification for I α relates to the decomposition properties to be considered in the next section, but there are also some other merits. For example, the measures in the range α < 1 are seen to be Dalton indices—measuring the percentage social welfare loss due to inequality—where social welfare is utilitarian and the individual utility function takes a particular form with constant relative risk aversion (or ‘isoelastic’, of the type discussed by (p.141) Atkinson 1970a).44 Indeed, each I α in this range is a monotonic transformation of an Atkinson measure, and the parameter α can be seen as an indicator of ‘inequality aversion’ (more averse as α falls).45 The parameter also indicates the measure's sensitivity to transfers at different parts of the distribution. For each I α, the effect of a small regressive transfer depends not only on the incomes of the giver and receiver, and on the mean income, but also on the parameter α (the specific relations are identified in a formula characterized by Cowell 1995). I 2, for example, exhibits ‘transfer neutrality’, since a given size of transfer between two persons who are a fixed income distance apart has the same effect at high and low incomes. T, D, and all measures with α < 2 (including those satisfying Atkinson's condition of α < 1) favour transfers at the lower end of the distribution.46

The second class of measures, the generalized Gini measures, also have the merit of being able to exhibit different amounts of transfer sensitivity to transfers along the distribution. To understand what is involved, it is important to recall that the unmodified Gini has the property that the effect of a transfer depends on the relative ranks or positions of the two persons between whom the transfer takes place, and not on the actual incomes. In fact, since the effect on the Gini depends only on the difference in ranks, or equivalently, on the number of people who have intermediate incomes in between the two persons, and not on their specific ranks, the Gini exhibits a special type of ‘positional transfer neutrality’.

One can retain the focus on position without requiring the (p.142) Gini's strict neutrality. For example, to emphasize transfers at the lower end, one might alter the weights on incomes in the definition of the Gini (see equation (2.8.3) in OEI‐1973). Alternatively, the Lorenz distance [pL x (p)] used to calculate the Gini area could receive different (positive) weights θ (p) at different p, yielding the generalized Gini class defined by Shorrocks and Slottje (1995) as:

G θ ( x ) = 0 1 [ p L x ( p ) ] θ ( p ) d p / 0 1 p θ ( p ) d p
Notice that the numerator is the weighted area between the 45° line and the Lorenz curve, while the denominator is the weighted area below the 45° line, so that when θ (p) is a constant, G θ (x) reduces to the normal Gini. By choosing a decreasing weighting function θ (p), for example, one can ensure that transfers between persons who are a given ‘distance’ apart (measured by the number of people occupying intermediate positions) have greater effect at the lower part of the income distribution (i.e., when the people involved in the transfer are poorer). Consequently, G θ (x) can be made to conform with ‘positional transfer sensitivity’ and other desired forms of positional sensitivities through the specification of the weighting function.47 It should also be noted that the effect of a small transfer between two persons is independent of the distribution of income among the remaining persons—so long as the respective rankings of the two remain the same. This is a restriction on the type of information that is allowed to count, but of a different category from the informational invariance imposed by the generalized entropy measures.

Notes:

(41) This relies on the standard definition of Gastwirth (1971), which, for discrete distributions like x, amounts to plotting the income share of the poorest l persons against their population share (for each l = 0, 1, . . . , n x) and connecting the points with line segments. More generally, where F is any cumulative distribution function (indicating the proportion of the population F(s) with income no greater than s), and F −1 is its inverse (or ‘generalized’ inverse if F has jumps), the Lorenz curve of F is defined for 0 ≤ p ≤ 1 as:

L ( p ) = 0 p F 1 ( q ) d q / μ , where  μ = 0 1 F 1 ( q ) d q is the mean of  F

(42) Anand (1983), who investigated the shared properties of these measures, called them the Lorenz class of inequality indices (pp. 339–40).

(43) See Shorrocks (1980), Cowell (1980), Cowell and Kuga (1981b), who defined the family as such, and Bourguignon (1979) who did nearly the same.

(44) See Bourguignon (1979, p. 913) who sketched this argument. We must be careful to take absolute values when needed, since utility and hence social welfare can be negative as the formulae stand. Cowell (1995) interprets I α as measuring the distance from complete equality.

(45) See Atkinson (1983) for an illuminating discussion of this interpretation.

(46) Properties of ‘transfer sensitivity’ are discussed in section A.4.3. The measures beyond α = 2 stress transfers at higher incomes in a kind of ‘reverse sensitivity’—which calls into question I α's usefulness for this range. Note that all the generalized entropy measures have the property that the effect of a transfer between two persons is independent of the distribution of income among the remaining persons—a rather strong restriction on the information used in judging distributional changes (on this more later).

(47) Papers with generalizations of the Gini include Mehran (1976), Pyatt (1976, 1987), Donaldson and Weymark (1980), Kakwani (1980a, 1981), Weymark (1981), Nygård and Sandström (1982), Yitzhaki (1983), among others, although not all of the indices considered are ‘relative’ in the sense defined earlier.