Measuring group disadvantage with inter-distributional inequality indices: A critical review and some amendments to existing indices

Yalonetzky, Gaston Working Paper Measuring group disadvantage with inter-distributional inequality indices: A critical review and some amendments to existing indices Economics Discussion Papers, No. 2011-46 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Yalonetzky, Gaston (2011) : Measuring group disadvantage with interdistributional inequality indices: A critical review and some amendments to existing indices, Economics Discussion Papers, No. 2011-46, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/52227 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en Measuring Group Disadvantage with Interdistributional Inequality Indices: A Critical Review and Some Amendments to Existing Indices Gaston Yalonetzky University of Leeds Abstract A long literature on inter-distributional inequality (IDI) has developed statistical tools for measuring the extent of inequality between two groups (e.g. men versus women). Firstly, I introduce the property of group-specific disadvantage focus (GDF). Indices satisfying this property are only sensitive to inequalities that are disadvantageous to one specific group. Then the paper reviews some of the most prominents IDI indices proposed in the last four decades. The assessment focuses on whether these indices satisfy GDF and, if not, how they react to inequalities that are disadvantageous to different groups. I also discuss whether these indices are informative, or not, regarding other interesting features related to IDI comparisons, e.g. distributional equality, absence of distributional overlap and presence of first-order stochastic dominance. Finally, I propose amendments to several of these indices in order to render them in fulfillment of GDF and more informative on the mentioned distributional features. Paper submitted to the special issue The Measurement of Inequality and Well-Being: New Perspectives JEL D30, J71 Keywords Inter-distributional inequality Correspondence Gaston Yalonetzky, Maurice Keyworth Building, Leeds University Business School, Leeds LS2 9JT, United Kingdom; e-mail: [email protected] I would like to thank the co-editor Satya Chakravarty, Carlos Gradin, and seminar participants at the Human Development Report Seminar Series of the UNDP and the Economics Network Meetings of the Inter-American Development Bank for their very helpful comments and suggestions. © Author(s) 2011. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany Discussion Paper No. 2011-46 | November 2, 2011 | http://www.economics-ejournal.org/economics/discussionpapers/2011-46 conomics Discussion Paper Introduction The concern for differences in the distribution of wellbeing characteristics among groups within societies has earned a long-standing interest in the Social Sciences and Political Philosophy. This concern has often emphasized the potential presence of socio-economic discrimination of different natures (e.g. Becker, 1971; Phelps, 1972; Arrow, 1973). In general, it has been associated with concepts of inequality of opportunities.1The normative view for between-groups differences related to ethnicity or gender states that they are intrinsically unfair (particularly when the groups are dened over characteristics beyond their members' control), and instrumentally detrimental to individuals and societies (e.g. Arneson, 1989; Cohen, 1989; Nussbaum and Glover, 1995; Roemer, 1998; Fleurbaey, 2001; Sen, 2001). From a quantitative perspective, one way of measuring the extent of differences in wellbeing between groups is to use indices that capture between-group inequalities and that declare the total absence of between-group inequality if and only if the conditional distributions of wellbeing are identical across groups.2 There is also an interest in quantifying between-group inequalities with a focus on capturing inequality if and when it is (more) detrimental to one specic group as opposed to other(s), i.e. a concept of relative economic disadvantage. Even though several authors have focused on inequalities detrimental to one group,3 only recently formal denitions of the concept have been put forward, with a concern for censoring inequalities when they are not detrimental to the group of concern. The most recent and neat denition by del Rio et al. (2011), based on the work of Jenkins (1994), applies to comparisons of actual distributions against counterfactuals. This approach effectively deals with distributions of the same population size. In this paper I rst propose a property of a(n index's) sensitivity to inequality that is detrimental exclusively to one specic group and that is applicable to indices of inter-distributional inequality that deal with populations of different size. 1For a good review of the literature on inequality of opportunity see Fleurbaey (2008). Also Roemer (1998). 2This condition is consistent with a literalist denition of inequality of opportunity by Roemer (1998, p. 15-6) as well as with Van De Gaer's rule (Ooghe et al., 2007). It is also consistent with Fleurbaey's concept of circumstance neutralization (Fleurbaey, 2008, p. 25). There are alternative ways of measuring between-group inequality. For instance, it could be measured as the residual inequality after within-group inequality has been suppressed (e.g. by replacing individual's wellbeing values with those of their group mean). Such approach has been followed, among others, by Roemer (2006); Elbers et al. (2008); ?); Lanjouw and Rao (2008). 3This literature is abundant. Some important examples are Gastwirth (1975), Butler and McDonald (1987), Dagum (1987), Jenkins (1994), van Krem (2009), Gradin et al. (2010) and del Rio et al. (2011). www.economics-ejournal.org 2 conomics Discussion Paper I call this property Group-Specic Disadvantage Focus (GDF). An advantage of this denition is that it can be related to indices that measure inequality on quantile space, or on probability space. Secondly, I explore how we can measure inequalities with metrics satisfying (GDF), i.e. with an exclusive focus on one specic group's disadvantage. Since there are several indices of inter-distributional inequality (IDI) already available, I propose some ways of measuring this focused inequality by suggesting some amendments to existing indices which do not measure IDI with a focus on specic disadvantages in their current forms. Thirdly, I take the opportunity to extend this review of existing indices in order to evalute whether these indices are informative, or not, regarding other interesting features related to IDI comparisons. For instance, I assess whether these indices are able to pinpoint situations in which two distributions are identical. Remarkably most of them are not. I also assess whether they are informative as to the absence of distributional overlap and/or presence of rst-order stochastic dominance. I propose some further amendments that improve the indices' informative content on these features. Since there are several indices of distributional change, or IDI, in this paper I focus on indices that are characterized by: i) being useful especically for twogroup comparisons, ii) being more informative than just comparing two means, and iii) being useful when the two distributions have different sample sizes.4I rst review the PROB index by Gastwirth (1975), followed by the closely related indices of relative distributions by Le Breton et al. (2008). I show that these indices do not fulll GDF because in some cases they compensate inequalities detrimental to one group with inequalities detrimental to the other group, while in other cases they just add up the two forms of inequalities together. In relation to that, most of these indices do not distinguish situations of equal distributions from other situations wherein there is inequality among distributions. Finally, while several of these indices are helpful to pinpoint situations of lack of distributional overlap, they are not informative to the presence of rst-order stochastic dominance. I propose some simple amendments to these indices that render them more informative about the abovementioned features; chiey, the extent of groupspecic disadvantage. I then review the family of percentile-based indices of Ebert (1984) and Vinod (1985). I show that, again, these indices do not fulll GDF either because they 4When sample sizes are identical the literature on counterfactual comparisons, e.g. del Rio et al. (2011), provides the relevant indices. However, even without the explicit purpose, mobility indices may also be amendable to render them suitable for the analysis of between-group inequalities with GDF and identical populations. Good examples of such indices are provided by Cowell's measures of distributional change (Cowell, 1985), by Fields and Ok (1996, 1999) and by Schluter and van de Gaer (2011). www.economics-ejournal.org 3 conomics Discussion Paper compensate group-specic detrimental inequalities or because they add them up indiscriminately. As for other features, while Ebert's index does differentiate between distributional equality and other situations, Vinod's does not. Neither index is helpful to detect absolute lack of distributional overlap. I propose simple amendments to these indices that render them both in fulllment of GDF and more informative in terms of lack of distributional overlap. The I turn to an assessment of the P(1;1)measure of Butler and McDonald (1987) and the measures of Dagum (1980, 1987). Notwithstanding these indices' merits and usefulness in other situations, I show that they do not fulll GDF and that they do not distinguish a situation of distributional equality from other cases of inequality. Finally, I complete the review with an appraisal of the family of ethical distance functions proposed by Shorrocks (1982) and Chakravarty and Dutta (1987). Ethical distance indices are different from the previous ones in that they compare equally-distributed-equivalent (EDE) standards from the distributions.5 This requires a rst aggregation step in which each distribution is mapped into its respective EDE standard. Then two such standards are compared. Despite this difference, I include these indices in the review because they have been proposed as alternatives to, and contrasted with, some IDI indices (see Shorrocks (1982)). I show that, notwithstanding their merit and appeal, this family of indices does not fulll GDF. The indices are also of little help for pinpointing situations of distributional equality, rst-order stochastic dominance and/or absolute absence of distributional overlap. The next section denes the property of group-specic disadvantage focus. Then the review and proposal of new amendments is done in subsequent sections: one for the PROB measure and indices based on relative distributions; followed by a section on the percentile indices of Ebert and Vinod; then followed by a section on the P(1;1)measure of Butler and McDonald, a section on the REA measures of Dagum and a section on ethical distance indices. Finally the paper ends with some concluding remarks. A focus on group-specic disadvantage Two distributions may be different in many ways. For instance, they may have different means. Or even if they have equal means, they may differ in their variance, skewness or kurtosis. More importantly, from a wellbeing perspective, these inter-distributional differences may render one distribution more desirable than the other one as a "lottery". The stochastic dominance literature discusses this type of partial-ordering comparisons. But even when stochastic dominance rela5EDE standards were introduced by Atkinson (1970). www.economics-ejournal.org 4 conomics Discussion Paper tionships do not hold over the whole admissible range of a wellbeing variable, one may be able to make statements about whether certain parts of a distribution are more advantageous for one group vis-a-vis another one. For instance, consider income distributions A and B. Both are symmetric and have equal means, but people in A are closely clustered around the mean, whereas people in B exhibits signicantly higher variance. In that case, one may nd that the poorest people in B are poorer than the poorest people in A whereas the richest people in B are richer than the richest people in A. In such situations, one may be interested in measuring only the amount of inequality that is detrimental to, say, A. If that is the purpose then one may want to have an index that is sensitive to the fact that the richest people in A are poorer than the richest people in B, while being insensitive to the fact that the poorest people in A are better-off than the poorest people in B. The purpose of such a focused approach is served by comparing the percentiles of the two groups, i.e. people who are in the same relative wellbeing position within their own group (i.e. the same position in the Pen's Parade). 6Let y(p); p2[0;1];be the ppercentile of distribution Y. Then I propose the following denition of Group-specic Disadvantage Focus for an index that is meant to capture only inequalities that are detrimental to a distribution Xwhen compared to a distribution Y: Denition 1 An index measuring inter-distributional inequality between Y and X satises the property of group-specic disadvantage focus (GDF) if and only if it is sensitive to the gap y(p)x(p)8p2[0;1]jy(p)x(p)and it is insensitive to the gap y(p)x(p)8p2[0;1]jy(p)x(p). In particular, the index does not decrease (increase) if the gap y(p)x(p)increases (decreases) given that initially y(p)x(p)and the index does not react to changes in y(p)x(p)as long as y(p)x(p)before and after the changes. The sensitivity part of Denition 1 is similar to the monotonicity axiom of del Rio et al. (2011) for counterfactual comparisons, while the insensitivity part is similar to their focus axiom. Now Denition 1 can be expressed also in terms of cumulative probabilities. This dual expression is useful for applications based on ordinal variables. It stems from the fact that, if it is true that y(p)x(p)over the 6A similar approach was advocated in the inequality-of-opportunity literature by Roemer (1998). He proposed that in order to measure inequality of opportunity between different groups of people (dened in terms of their specic sets of life circumstances), people in a given percentile within their own group should be compared against people from the same percentile in a different group. The percentile is used as a measure of relative effort within the group, under certain assumptions. www.economics-ejournal.org 5 conomics Discussion Paper interval p2[p;p], and is also the case that yp=xpand y(p) = x(p);then the following equation holds: Zx(p) x(p)[FX(z)F Y(z)]+dz =Zp p [y(p)x(p)]+dp;(1) where FX(z)is the cumulative density function (cdf) of Xand [m]+ maxfm;0g:In words, (1) says that the sum of positive gaps, y(p)x(p);over the interval [p;p], is equal to the sum of positive gaps of cdfs, FX(z)F Y(z);in the interval xp;x(p)(or yp;y(p)) dened by [p;p]. Hence a dual for denition 1 can be proposed: Denition 2 An index measuring inter-distributional inequality between Y and X satises the property of group-specic disadvantage focus (GDF) if and only if it is sensitive to the gap FX(z)F Y(z)jFX(z)F Y(z)and it is insensitive to the gap FX(z)F Y(z)jFX(z)F Y(z). In particular, the index does not decrease (increase) if the gap FX(z)F Y(z)increases (decreases) given that initially FX(z)F Y(z)and the index does not react to changes in FX(z)F Y(z)as long as FX(z)F Y(z)before and after the changes. Denition 2 is useful for indices that map from probability space like the PROB index and those based on relative distributions. It can also be considered for applications with ordinal variables. Notice also the connection between the two denitions and rst-order stochastic dominance. The following three statements are identical: (i) Distribution Y(weakly) rst-order dominantes X: (ii) y(p)x(p)8p2[0;1]: (iii) FX(z)F Y(z)8z: Hence indices that satisfy GDF are expected to be informative about the presence of rst-order stochastic dominance, especially in its weak form, as is shown below. The PROB measure and relative distributions: review and amendments The PROB measure of Gastwirth (1975) is dened as: PROB Z∞ ∞[1FX(z)] fY(z)dz. It measures the probability of nding an individual in Xhaving at least as much of zas a random individual in Y(hence Yis the reference distribution and Xis the compared distribution). PROB does not fulll GDF because it pits inequalities www.economics-ejournal.org 6 conomics Discussion Paper that are detrimental to Xagainst inequalities that are detrimental to Y. To see this notice the following simple decomposition stemming from adding and subtracting Z∞ ∞F Y(z)fY(z)dz and considering that Z∞ 0 F Y(z)fY(z)dz =0:5 : PROB =Z∞ ∞[F Y(z)FX(z)]+fY(z)dzZ∞ ∞[FX(z)F Y(z)]+fY(z)dz+0:5 (2) Hence it is clear from (2) that inequalities detrimental to Y[F Y(z)FX(z)]+ are compensated with inequalities detrimental to X[FX(z)F Y(z)]+. For this reason PROB cannot distinguish a situation of distributional equality from others of distributional inequality. Whenever fX=fY,PROB =0:5.7However the reverse is not true, as is clear from (2). As it stands, PROB does not take any specic value that signals rst-order stochastic dominance. By contrast, PROB is useful to pinpoint absences of distributional overlap. For instance: PROB =0$ FXzY min=1, where zY min is the minimum value for which Yhas support. When PROB =0 the richest person in Xis not better off than the poorest person in Y (whose value of zis zY min). On the other extreme: PROB =1$FXzY max=0. When PROB =1 the poorest person in Xis richer than the richest person in Y. In summary: PROB does not satisfy GDF, does not exclusively identify distributional equality or rst-order stochastic dominance, but it does identify lack of distributional overlap. However, some simple measures based on PROB can be used in conjunction with it in order to provide more information on the abovementioned distributional features. I propose the following: PROBα Y(YX)(α+1)Z∞ ∞[F Y(z)FX(z)]α +fY(z)dz;(3) PROBα Y(XY)(α+1)Z∞ ∞[FX(z)F Y(z)]α +fY(z)dz;(4) where αis a parameter and the subindex Yin PROBα Y(YX)denotes that the reference distribution is Y.8It is straightforward to notice that both (3) and (4) fulll GDF. It is also the case that: fX=fY$ (PROBα Y(YX) = 0^PROBα Y(XY) = 0). Hence, used together, both indices (for any positive value of α) identify distributional equality. Two interesting 7When PROB <0:5 the distribution of Yhas some advantage over X's such that the probability of nding someone in Xhaving at least as much of zas a randomly chosen person from Yis lower than the probability that would ensue from identical distributions. A similar interpretation, favouring X's distribution over Y's, ensues when PROB >0:5: 8Analogue indices can be dened using Xas the reference distribution. www.economics-ejournal.org 7 conomics Discussion Paper sets of indices are related to the cases when α=0 and α=1:When α=0 the indices help to pinpoint situations of rst-order stochastic dominance since: PROB0 Y(YX) = 1$XFD Y, where FD reads "weakly rst-order dominates".9When α=1;both PROB1 Y(YX)and PROB1 Y(XY)are sensitive to changes in the percentile gaps, and they are helpful to detect absence of distributional overlap because: PROB1 Y(XY) = 1$FXzY max=0:When α=1 the following relationship holds: 2PROB =PROB1 Y(YX)PROB1 Y(XY)+1 (5) Since these indices map from probability space, it is easy to show that they fulll properties of population replication invariance and ratio scale invariance. Relative distributions The PROB index and this paper's amendments are closely related to indices stemming from discrimination curves based on cumulative relative distributions. A cumulative relative distribution function maps the cumulative distribution of a reference distribution, F Y(z), into the interval [0;1]:Specically, the cumulative distribution function is: GX=Y(F Y)FX[y(F Y)] and the discrimination curve is the drawing of GX=Y(F Y)on an horizontal axis of F Y.10 Le Breton et al. (2008) studied dominance conditions for the discrimination curve and proposed some indices based on the area between the discrimination curve and the 45 degree line. Two of their measures are relevant for this paper: AAD =Z1 0GX=Y(F Y)F YdF Y(6) C=Z1 0GX=Y(F Y)F YdF Y;(7) where AAD is the average absolute deviation between the discrimination curve and the distributional equality line (45 degree). 11 Now notice that: 2AAD =PROB1 Y(YX) + PROB1 Y(XY)and 2C=PROB1 Y(XY) PROB1 Y(YX):Hence it is easy to see that both AAD and Cdo not fulll GDF. In the rst case both types of inequalities, i.e those detrimental to Xand those detrimental to Y, are added up; while in the second case they compensate each other. As for other distributional features, Cdoes not distinguish between distributional 9Likewise: PROB0 Y(XY) = 0$XFD Y: 10 Hence when: fX=fYthe discrimination curve is a 45 degree line. 11 Le Breton et al. (2008) use different names for these indices. www.economics-ejournal.org 8 conomics Discussion Paper that most of the reviewed IDI indices map from either percentiles or probabilities. Hence in this paper, I rst set out to dene GDF in the context of IDI measurement with different population sizes. Neither of the indices reviewed satises GDF. Several are also limited in the information they provide on some interesting distributional features including distributional equality, presence of rst-order stochastic dominance and absence of distributional overlap. However, as the paper shows, in most cases it is straightforward to amend these indices in order to render them in fulllment of GDF and, often, more informative in terms of the additional distributional features mentioned above. The examination of several indices suggests that fulllment of GDF by IDI indices, for two distributions with different population sizes, may require that the indices map explicitly from either percentiles or probabilities. References Arneson, R. (1989). Equality and equal opportunity for welfare. Philosophical studies, 56: 77–93. Arrow, K. (1973). 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Journal of Business and Economic Statistics, 3(1): 78–88. www.economics-ejournal.org 17 conomics Discussion Paper Figure 1: Left panel: the two CDFs overlap in part of their common support. Right panel: the two CDFs cross once    www.economics-ejournal.org 18 Please note: You are most sincerely encouraged to participate in the open assessment of this discussion paper. You can do so by either recommending the paper or by posting your comments. Please go to: http://www.economics-ejournal.org/economics/discussionpapers/2011-46 The Editor © Author(s) 2011. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany