The Measurement of Intergroup Income Inequality: A Conceptual Review

This paper reviews conceptual differences between popular measures of relative intergroup income inequality. To organize our review we introduce a set of guidelines identifying desirable characteristics of measures of relative intergroup income inequality and then evaluate different measures in terms of these guidelines. Significantly, we find that many popular measures (e.g., the ratio of median incomes, the Index of Dissimilarity, and the Index of Net Difference) have one or more logical weaknesses which prevent them from fully satisfying our guidelines. On the other hand, comparisons of group mean incomes (e.g., the ratio of mean incomes or the difference of the means for the logarithm of income) satisfy most of the guidelines we set forth. Additionally, we introduce a new measure of relative intergroup income inequality, the Index of Average Relative Advantage, which meets our guidelines and has an appealing interpretation. Social scientists have long been interested in two basic forms of income inequality; that between persons in a single population (intragroup income inequality), and that between people in different groups (intergroup income inequality). While the measurement of income inequality within a single population has received a good deal of scholarly attention (e.g., Allison; Atkinson; Blau; Schwartz and Winship), surprisingly little has been written about the conceptualization and measurement of income inequality between groups. (Exceptions include Lieberson; Palmore and Whittington; Villemez and Rowe.) As a result, measures of intergroup inequality are often selected on the basis of convention or convenience and the conceptual differences between different measures are not taken into account. To remedy this, this paper reviews the conceptual differences between a num*We thank Omer R. Galle, Aage B. Sorensen, W Parker Frisbie, and two anonymous referees for helpful comments on earlier versions of this paper. We also acknowledge the support of NSF Grant SES 80-07901 and NICHD Grant P30 HD06160. An earlier version of this paper was presented at the 1981 meetings of the American Sociological Association. ? 1983 The University of North Carolina Press. 0037-7732/83/030855-71$01.70 855 This content downloaded from 207.46.13.48 on Wed, 12 Oct 2016 05:43:34 UTC All use subject to http://about.jstor.org/terms 856 / Social Forces Volume 61:3, March 1983 ber of popular measures of intergroup inequality, giving particular attention to their capacity to summarize the relative income differences between two groups. The organization of the paper is as follows. The first section introduces six guidelines for measures of relative intergroup income inequality. These guidelines clarify our thinking about the conceptualization of relative income inequality between groups, and provide a framework for comparing different measures of inequality. The middle section of the paper discusses a number of specific measures of relative intergroup income inequality, the extent of their conformity to the guidelines we have suggested, and the implications of failure to meet specific guidelines. This section also introduces the Index of Average Relative Advantage, a new measure of inequality. The final section comments briefly on our findings, the goals and problems involved in measuring relative intergroup income inequality, and the potential directions of future research on this topic. Guidelines for Measures of Relative Intergroup Income Inequality The Principle of Directionality Since the notion of inequality implies that group income differences may favor one or the other of two groups, it is crucial that a measure of inequality exhibit directionality. Thus, there should be some value representing the point of equality for the measure, with values above this point reflecting advantage to one group, and values below this point reflecting advantage to the other group.1 If a measure fails to satisfy this principle, it will be impossible to determine the direction of group advantage from the value of the measure alone. The Principle of Transitivity We next require that inequality scores be transitive. Thus, given three groups, A, B, and C, where the measure shows A to enjoy an advantage over B and B to enjoy an advantage over C, two things should follow. First, the measure should also show A to enjoy an advantage over C. Second, the measure should indicate that A's advantage over C is greater than A's advantage over B. If a measure fails to satisfy these requirements, it will not yield an unambiguous ordering of inequality comparisons. The Principle of Transfers Our third requirement is that, if income is transferred from an individual in one group to an individual in another group, the measure must change from its initial value to one that is more favorable to the receiving group after the transfer, and the change in the measure must increase as the size of the transfer increases.2 If a measure fails to satisfy this requirement, it This content downloaded from 207.46.13.48 on Wed, 12 Oct 2016 05:43:34 UTC All use subject to http://about.jstor.org/terms Income Inequality / 857 will not always be sensitive to the clear-cut changes in inequality that occur when one group gains at another's expense. The Principle of Scale Invariance Our fourth requirement is that the measure should remain unchanged when incomes are multiplied by a positive, nonzero constant. Failure to satisfy this requirement opens the possibility that inequality will be a function of the arbitrary metric of income (e.g., dollars, cents, yen) or the impact of inflation. The Principle of Equal Additions To insure that a measure is sensitive to relative inequality, we require the measure to show greater equality between groups when a positive constant is added to all incomes. Conversely, the measure should show greater inequality when a negative constant is added to all incomes.3 This guideline insures that any given absolute difference in income becomes less important as the overall level of income rises, and more important as the level of income falls. The Principle of Symmetry Our final requirement is that the logical range of a measure's values be symmetrically distributed around zero with an upper limit of one and a lower limit of negative one. The result of this requirement is that, if two groups were to exchange income distributions, the absolute value of the measure would not change, but the sign of the measure would be reversed. If this guideline is satisfied, three desirable qualities result. First, the sign of the measure will indicate the direction of group advantage. Second, the value of the measure will indicate how far inequality is from its maximum. For example, a score of .5 would indicate that the advantage one group enjoys over another is one-half of the maximum possible advantage it could enjoy. Third, it is possible to compare scores above and below the point of equality to determine whether they are equally unequal. Taken together, these guidelines define an incomplete conceptualization of intergroup income inequality. This conceptualization is incomplete because adherence to these guidelines does not yield an unambiguous ordering of inequality comparisons. Different measures can satisfy each of the guidelines we have suggested, yet not rank group inequality comparisons in the same way. This possibility arises because we have not completely restricted the manner in which a measure of inequality must respond to different kinds of transfers of income. Our principle of transfers requires that the measure shift to show a greater advantage (lesser disadvantage) to the receiving group after a transfer, and that this shift be larger as the size of the transfer increases. It says nothing, however, of whether the impact of the transfer should (or should not) vary depending This content downloaded from 207.46.13.48 on Wed, 12 Oct 2016 05:43:34 UTC All use subject to http://about.jstor.org/terms 858 / Social Forces Volume 61:3, March 1983 on the initial income of donor and recipient. For example, it says nothing of whether the impact of a transfer of a fixed amount of income from one group to another should vary depending on whether the transfer is taken from a rich or poor donor, or whether the transfer is given to a rich or poor recipient. This ambiguity is unfortunate, but it reflects the fact that social scientists do not agree on this aspect of the conceptualization of relative inequality.4 We turn now to consider how various popular measures stand in relation to these guidelines, and the problems that result when they fail to satisfy one or more of the guidelines. Measures of Relative Intergroup Income Inequality