Inequality Measurement with Subgroup Decomposability and Level-Sensitivity

Subgroup Decomposability is a very useful property in an inequality measure, and level-sensitivity, which requires a given level of inequality to acquire a greater significance the poorer a population is, is a distributionally appealing axiom for an inequality index to satisfy. In this paper, which is largely in the nature of a recollection of important results on the characterization of subgroup decomposable inequality measures, the mutual compatibility of subgroup decomposability and level-sensitivity is examined, with specific reference to a classification of inequality measures into relative, absolute, centrist, and unit-consistent types. Arguably, the most appealing combination of properties for a symmetric, continuous, normalized, transfer-preferring and replication-invariant (S-C-N-T-R) inequality measure to satisfy is that of subgroup decomposability, centrism, unit-consistency and level-sensitivity. The existence of such an inequality index is (as far as this author is aware) yet to be established. However, it can be shown, as is done in this paper, that there does exist an S-C-N-T-R measure satisfying the (plausibly) next-best combination of properties - those of decomposability, centrism, unit-consistency and level-neutrality.


Introduction
In the axiomatic approach to the measurement of inequality, a number of desirable properties of inequality indices have been advanced. In this article, we consider two specific properties -those of 'decomposability' and 'level-sensitivity' -and check for their mutual compatibility in the presence of other specified properties. The points made in this essay draw on a number of important results which have already been established in the literature: it is then mainly a matter of putting these results together in order to present a set of observations on the prospects of simultaneously meeting the requirements of decomposability and level-sensitivity. The outcome is arguably useful, insofar as taxonomies (in this case of inequality measures) are generally useful; the outcome is also inarguably dependent on a great deal of important prior work that has been done on the subject of decomposable inequality measures.
Subgroup decomposability (see Bourguignon 1979, Cowell 1980, Cowell and Kuga 1981, Shorrocks 1980, 1984, 1988 is the property that an inequality measure be expressible as an exact sum of a 'between-group component' (obtained by imagining that each person in any subgroup receives the subgroup's mean income) and a 'within-group component' (obtained as a weighted sum of subgroup inequality levels, the weights depending on the subgroups' income shares or population shares or some combination of the two shares).
Level-sensitivity can be thought of as a group-related egalitarian requirement that arises when a population is partitioned into non-overlapping income groups of the same size: it postulates that in this circumstance, and other things remaining the same, a given increase in subgroup inequality should cause overall inequality to rise by more the poorer (in terms of subgroup mean income) the subgroup is. This property has a strong affinity to a concern expressed in an early contribution by Amartya Sen (1973), and relating to the question of how our view on inequality ought to vary with the general level of a society's prosperity. As observed by Sen (1973: 36): Can it be asserted that our judgment of the extent of inequality will not vary according to whether the people involved are generally poor or generally rich? Some have taken the view that our concern with inequality increases as a society gets prosperous since the society can 'afford' to be inequalityconscious. Others have asserted that the poorer an economy, the more 'disastrous' the consequences of inequality, so that inequality measures should be sharper for low average income. This is a fairly complex question and is bedeviled by a mixture of positive and normative considerations. The view that for poorer economies inequality measures must be themselves sharper can be contrasted with the view that greater importance must be attached to any given inequality measure if the economy is poorer. The former incorporates the value in question into the measure of inequality itself, while the latter brings it in through the evaluation of the relative importance of a given measure at different levels of average income.
It is the former of the two views asserted by Sen at the conclusion of the preceding quote that is upheld by the level-sensitivity axiom.
In this essay, we examine the mutual compatibility of subgroup decomposability and level-sensitivity for certain broad classes of inequality measures, taxonomised according to their invariance to multiplicative or additive transformations of an income distribution. In terms of this classification, inequality measures can be relative or absolute (see Blackorby and Donaldson 1980). A relative inequality measure is 'scale-invariant', while an absolute inequality measure is 'translation-invariant'. Scale-invariance is the property that the value of an inequality measure should remain unchanged if all persons' incomes were to be uniformly multiplied by any positive scalar, while translation-invariance requires such constancy in the value of an inequality measure when all persons' incomes are increased (or decreased) by the addition (or subtraction) of a fixed amount.
The invariance requirements just considered have both purely 'analytical' and 'normative' implications. At the analytical level, scale-invariance ensures that the value of an inequality index does not change with the units in which income is measured, while translation-invariance violates this property of neutrality with respect to the units of measurement. From this 'analytical' perspective, scaleinvariance would appear to possess an attractive advantage over translationinvariance. However, from a 'normative' perspective, scale-invariance can be seen to uphold a 'right-wing' view of inequality and translation-invariance to uphold a 'left-wing' view, as pointed out by Serge-Christophe Kolm (1976aKolm ( , 1976b. Notice that, given a two-person ordered income distribution x = (1,100), a doubling of each person's income would lead to the distribution y = (2,200): a scale-invariant index would uphold the (typically right-wing) judgment that the extent of inequality is the same in both distributions, despite the fact that out of the additional total income of 101 units in y vis-à-vis x, 100 units of income have gone to the richer person and only 1 unit to the poorer person. In contrast, if z were to be derived from x by the addition of 100 units of income to each person, so that z = (101, 200), a translation-invariant index would uphold the (typically left-wing) judgment that the extent of inequality is the same in both distributions, despite the fact that in the transition from x to z, the poorer person's income has risen by a factor of 10,000 per cent and the richer person's income by a factor of just 100 per cent.
One can see now that one can have inequality measures which are a 'compromise' between absolute and relative measures. The compromise we effect would depend on whether we take a purely analytical or a normative view of the two classes of measures. Under a purely analytical interpretation, a compromise class of measures would be unit-consistent measures (Zheng 2007), namely inequality measures which satisfy the requirement that the inequality-ranking of distributions is invariant with respect to the choice of units in which income is measured. As it happens, all right-wing measures and some left-wing measures are unit-consistent. A different type of compromise is the normative one between right-and left-wing measures, which leads to a class of centrist or intermediate measures (see, for example, Zheng 2007): an intermediate measure is one which satisfies the property that (i) a uniform scaling-up of every individual's income should increase inequality and (ii) the addition of any given income to every person's income should reduce inequality. It should be noted that the two types of compromise we have just considered are mutually independent: unit-consistent inequality measures are not necessarily centrist measures, and similarly centrist inequality measures are not necessarily unit-consistent. In examining subgroup decomposability and level-sensitivity of inequality measures for a classification of measures according to their disposition toward distributional values and unit-consistency, this article proceeds as follows. The following section introduces concepts and notation. This is followed by a section which advances a set of observations on subgroup decomposability and levelsensitivity for alternative types of inequality measures. The final section offers a summary and conclusions.

Basic Concepts
N is the set of positive integers, and R is the real line. For every , n is the set of positive n-vectors , and each is to be interpreted as an income vector whose typical element is the set , and an inequality index is a mapping such that, for every supposed to indicate the amount of inequality associated with the distribution x.
For every income vector , is the set of individuals represented in x, and is the dimensionality of is the dimensionality of j , is the mean income of j , and is the extent of inequality associated with j ( Where there is no ambiguity, we shall also write I for , for , for , for , and so on.
Let I* be the set of inequality measures such that a typical member of this set, , satisfies the following properties: is any appropriately dimensioned permutation matrix (so measured inequality is impervious to the personal identities of individuals); , which is the requirement that for all , , where is the vector obtained from by setting x(so that inequality is taken to be zero when all incomes are equalized); Continuity (Axiom C), which is the requirement that I be continuous on n for all (so that 'similar income distributions have similar inequality values'); where B is any appropriately dimensioned bistochastic matrix which is not a permutation matrix ( so that any movement toward equalization of the incomes in a distribution causes measured inequality to decline); (RI), which is the requirement that for all , whenever is a q-replication of , that is, , x , and q is any positive integer greater than 1 (so that inequality values depend only on the relative, not the absolute, frequency distribution of incomes); and Differentiability (D), which is the requirement that for all X x ∈ , I should have continuous first and second partial derivatives   Zheng 2007).
Definition 5 (Bossert-Pfingsten Restriction). A centrist inequality measure will be said to obey the Bossert-Pfingsten restriction (see Bossert and Pfingsten 1990) if and only if, for all , and for any and ) ,..., [The restriction stated above provides a particular operationalization of the notion of a centrist inequality measure by specifying a plausible condition under which the measure should remain unchanged for some combination of a uniform scale increase and a uniform incremental increase in all incomes of a distribution.] Next, the notion of 'level-sensitivity' is defined. Level-sensitivity essentially demands that, when a population is partitioned into equi-dimensional nonoverlapping income groups, then, other things equal, a given increase in subgroup inequality should cause aggregate inequality to rise by more the poorer (in terms of mean income) the subgroup is. More formally: Level-Sensitivity (Axiom LS). An inequality measure is levelsensitive if and only if, for all for some subgroups and such that The level-sensitivity axiom is kindred in spirit to a property of poverty measures which Nanak Kakwani (1980) has called Monotonicity-Sensitivity, namely the requirement that 'if represents the increase in the poverty measure due to a small reduction in the income of the i th poor, then given that incomes are arranged in non-descending order]' (Kakwani 1980: 438). Indeed, Kakwani (1993) where, for all , is the vector obtained by setting Notice that if a population is partitioned into non-overlapping income groups, then as long as the group-specific inequality levels and population shares remain unchanged, it is reasonable -even if the group-specific income shares should change -to expect the within-group component of a decomposable inequality index to also remain unchanged. Decomposability subjected to this reasonable restriction can be called 'proper decomposability', and it is easy to see that proper decomposability implies the requirement that the group-specific weights ( ) Result 2 (Chakravarty andTyagarupananda 1998, Bosmans andCowell 2010). For all ∈ x X, an absolute inequality measure I belongs to the set I* and satisfies subgroup decomposability if and only if it is a continuous and strictly increasing function of the following class of measures: (Chakravarty 2000). For all ∈ x X, an absolute inequality measure I belongs to the set I* and satisfies proper subgroup decomposability if and only if it is a positive multiple of the variance, given by: Result 4 (Chakravarty and Tyagarupananda 2009 , and depends on both and c a π . Result 5 (Zheng 2007). For all , a unit-consistent inequality measure ∈ x X I belongs to the set I* and satisfies subgroup decomposability if and only if it is a positive multiple of a member of the following class of measures: [In the interests of formal accuracy, it should be pointed out that in the Bosmans-Cowell 2010 version of Result 2, the axioms of normalization and differentiability are dispensed with, and Result 3 (Chakravarty 2000) does not really invoke the replication invariance property.] Result 3 relates to the characterization of a properly subgroup decomposable inequality measure which is absolute, while Results 1, 2, 4 and 5 relate to the characterization of subgroup decomposable measures which are, respectively, relative, absolute, centrist, and unit-consistent. How do these measures fare in relation to level-sensitivity? This issue is examined in the following section.

Some Observations on Subgroup Decomposability and Level-Sensitivity
While both subgroup decomposability and level-sensitivity appear to be attractive properties of an inequality index, it may not always be possible for an inequality measure to satisfy both properties. We illustrate this proposition by considering the Gini coefficient of inequality which, though it is not a subgroup decomposable (nor even subgroup consistent) measure, does lend itself to decomposability in the special case in which the population is partitioned into non-overlapping income groups (see Anand 1983). Specifically, it can be shown that if a population is divided into, say, non-overlapping income groups of the same size, so that with [so that ] , then one can write: Of interest is the fact that in the expression for the within-group component of aggregate inequality, the weight on the jth subgroup's inequality level is : if the groups are indexed in ascending order of mean-income, then it is clear that when , a given increase in inequality will raise aggregate inequality by more the richer (in terms of mean income) the subgroup is, since the weight on j G , , is an increasing function of j : this precisely reverses what the axiom of level-sensitivity demands.
What can be said at a more general level about subgroup decomposability and level-sensitivity? A first and immediately obvious conclusion that emerges from a consideration of the concepts and definitions discussed in the preceding section is that there is a mutual incompatibility between the properties of proper decomposability and level-sensitivity of an inequality measure. This follows from noting that when a population is partitioned into non-overlapping income groups of equal size, any properly decomposable inequality measure I belonging to the set I* will (by definition) have a within-group inequality component which is a weighted sum of subgroup inequality levels where the weights depend only on the subgroup population shares -which must all be equal since the subgroups are of equal size: a given increase in subgroup inequality will therefore cause overall inequality to rise by the same extent, irrespective of the average level of prosperity of the subgroup. The outcome is that level-sensitivity is a casualty. This leads to our first observation: Observation 1. There exists no properly decomposable inequality measure I ∈I* which is level-sensitive. Observation 1 suggests that if level-sensitivity is a desired normative property of an inequality index, then insistence on proper decomposability may have to be sacrificed. Indeed, the following observation, it can be shown, is true: this latter index, however, is not decomposable.) What is relevant to note is that the decomposition of is defined by:  Notice now that since all relative inequality indices are also unit-consistent, we are assured by Observation 2 that there exists a unit-consistent relative inequality measure belonging to the set I* which is also level-sensitive. Unfortunately, we have no such assurance regarding absolute inequality measures from Observation 3, since absolute measures may or may not be unit-consistent. Result 2 confines our attention to those absolute indices which are either exponential indices or the variance. Zheng (2007) points out that the family of exponential indices is not unit-consistent. The variance, however, is a unit-consistent measure, but Result 4 (Chakravarty 2000) asserts that the only absolute inequality measure in the set I* which is properly decomposable is the variance; and from Observation 1 we know that no properly decomposable index belonging to the set I* is level-sensitive. This leads to the following negative observation: Observation 4. There exists no absolute unit-consistent inequality measure I ∈I* which is level-sensitive.
Observation 4 is a harsh verdict for those who would value both subgroup decomposability and level-sensitivity but whose distributional judgments favour only left-wing inequality indices. For those who are happy to settle for centrist measures, the present state of knowledge may be inadequate to arrive at a definitive conclusion on the prospects of meeting the requirements of both subgroup decomposability and level-sensitivity, as reflected in the following observation.
Observation 5. Since (to the best of this author's awareness) there is no characterization available of unit-consistent, centrist inequality measures which are subgroup decomposable, it is not known if there exists a unit-consistent and centrist measure which is both decomposable and level-sensitive.
It may be added that the available evidence on this question is not encouraging. Result 4 (Chakravarty and Tyagarupananda 2009) presents a class ˆc I of centrist inequality measures belonging to the set I* which are decomposable, but, as pointed out by Zheng (2007), none of these indices is unit-consistent.
Result 5 (Zheng, 2007) presents a class c I % of unit-consistent inequality measures www.economics-ejournal.org

Summary and Conclusion
This article has been mainly a quick review of a set of important results on the characterization of decomposable inequality measures, classified into relative, absolute, centrist, and unit-consistent indices, and an examination of the mutual compatibility of the properties of subgroup decomposability and level-sensitivity.
For inequality measurement to be coherent, it appears that inequality measures must be unit-consistent. For inequality measurement to be informed by nonextreme distributional values, it also seems to be desirable that inequality measures be centrist. Thus, in the interests of both coherence and normative appeal, there would appear to be a strong case to confine attention to the set of unit-consistent and centrist measures. Decomposability is an extremely convenient property for an inequality index to possess, though it is not clear that this property is imbued with any particularly striking normative values (except in so far as what the philosopher Derek Parfit 1997 has called 'prioritarianism' is compatible with the strong separability underlying additively decomposable inequality indices). Levelsensitivity is a fairly compelling property of an inequality measure, requiring as it does that inequality be regarded as a more severe problem the poorer the population experiencing it is. Level-neutrality is a weaker requirement, demanding only that inequality should be regarded as a problem whose severity does not diminish as a population becomes poorer. In an 'ideal' situation, one may wish to have inequality measures which are centrist, unit-consistent, subgroup decomposable and level-sensitive. Whether such measures exist is still (as far as the present author is aware) an open question. What can, however, be asserted is that there does exist a symmetric, normalized, continuous, differentiable, strictly Schur-convex and replication-invariant measure which is unit-consistent, centrist, subgroup decomposable and level-neutral. This is the index, or rather family of indices (see Zheng 2007), given by