metadata: authors: 0: author: Foster,James E.
1: author: Lokshin,Michael M.
count: World
docna: 0: docna: Multidimensional and Specific Inequalities
docty: Policy Research Working Paper
abstracts: cdata!: Despite the multitude of measures of
            multidimensional inequality, none is regularly used in
            policymaking. This paper proposes multidimensional
            inequality measures that are easily implementable and
            transparent and overcome many deficiencies of existing
            measures. The measures follow a traditional two-stage
            format, which aggregates dimensions first and then applies a
            unidimensional measure like the Gini coefficient to the
            distribution of aggregates. A novel characterization result
            identifies the precise form of aggregation needed to obtain
            axiomatically sound measures. The paper derives an additive
            decomposition formula &#x2014; breaking down
            multidimensional inequality into terms reflecting the
            average specific inequalities (within dimensions) and the
            joint distribution (across dimensions) &#x2014; for any
            measure created using a standard unidimensional measure or
            the Lorenz curve. The paper also provides an approach to
            calibrating the measure for use with data over time,
            replacing the usual ad hoc normalization of variables with
            one that accounts for a policymaker&#x2019;s normative
            weights. The technology is illustrated first using synthetic
            data to understand how the measure varies as the components
            are changed and then using data from Azerbaijan.
display_title: Multidimensional and Specific Inequalities
listing_relative_url: /research/2024/04/34295886/multidimensional-specific-inequalities
topicv3: Inequality and Shared Prosperity m1327655 700
publishtoextweb_dt: 2024-04-04T00:00:00Z
dois: 10.1596/1813-9450-10748
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
no_of_pages: 40
author_keys: 0: author_key: 1094033
1: author_key: 144523
model_extractions: Policy Research Working Paper 10748 Multidimensional and Specific Inequalities _James E. Foster Michael Lokshin_ Europe and Central Asia Region Office of the Chief Economist April 2024
Policy Research Working Paper 10748

# **Abstract**

Despite the multitude of measures of multidimensional inequality, none is regularly used in policymaking. This paper proposes multidimensional inequality measures that are easily implementable and transparent and overcome many deficiencies of existing measures. The measures follow a traditional two-stage format, which aggregates dimensions first and then applies a unidimensional measure like the Gini coefficient to the distribution of aggregates. A novel characterization result identifies the precise form of aggregation needed to obtain axiomatically sound measures. The paper derives an additive decomposition formula—breaking down multidimensional inequality into terms reflecting the average specific inequalities (within dimensions) and the joint distribution (across dimensions)—for any measure created using a standard unidimensional measure or the Lorenz curve. The paper also provides an approach to calibrating the measure for use with data over time, replacing the usual ad hoc normalization of variables with one that accounts for a policymaker’s normative weights. The technology is illustrated first using synthetic data to understand how the measure varies as the components are changed and then using data from Azerbaijan.

This paper is a product of the Office of the Chief Economist, Europe and Central Asia Region. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at mlokshin@worldbank.org and fosterje@email.gwu.edu.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Multidimensional and Specific Inequalities<sup>*</sup>

James E. Foster Oliver T. Carr Jr Professor of International Affairs and Professor of Economics The George Washington University

## Michael Lokshin

Lead Economist with the Office of the Chief Economist for Europe and Central Asia, The World Bank JEL: D30, D63 Keywords: Multidimensional Inequality, Axioms, Measures, Lorenz Curves, Decompositions

> * The authors have benefitted from conversations with Jean Dreze, Chico Ferreira, Stephen Jenkins, Ravi Kanbur, Suman Seth, Tony Shorrocks, Stephen Smith, and the participants of seminars at the LSE and the LIS. We thank Carolina Sanchez-Paramo for her support of this research project. Foster is grateful to the Bank and to the International Inequalities Institute of the LSE for hosting his sabbatical leave. This research was partially supported by the World Bank.
# **I. Introduction**

While global income inequality has declined since the 1990s, income inequality levels within countries have been rising for a wide range of developed and developing countries, capturing the attention of social activists and policymakers alike. More than half of all countries and close to 90 percent of advanced economies have seen an increase in income inequality since 2000, with the income Gini increasing by more than two points in some instances (IMF 2023). Yet inequality is present not only in the space of incomes; it inhabits other key dimensions such as health, education, and social services, whose dimension-specific inequalities may reinforce or dampen the impact of income inequality. Empirical data on income inequality offers only a partial view of what Amartya Sen has termed “economic inequality” and can limit the scope and accuracy of a country’s policy responses (Sen 1997, 1999).

Dashboards and weighted averages of dimension-specific inequalities can help paint a broader picture of economic inequality within a country, and its evolution through time. However, they completely ignore the joint distribution of dimensional variables, which conveys important information on how people in the country are experiencing inequality. For example, it could be the case that most people with lower levels of one variable have lower levels of the others, resulting in a rigid hierarchy of achievement vectors and, arguably, greater economic inequality. Alternatively, people could exhibit mixed levels of achievements, dampening positive association and its impact on inequality. Measurement tools sensitive to the joint distribution can distinguish between these situations, and better gauge the extent of economic inequality.<sup>1</sup> Following the pioneering work of Kolm (1977), Atkinson and Bourguignon (1982), and Maasoumi (1986), there have been significant advances in the range of tools available for measuring inequality when there are multiple dimensions, including new classes of measures, axioms for discerning among measures, and dominance methods for ensuring comparisons are robust.<sup>2</sup> In their survey on multidimensional poverty and inequality, Aaberge and Brandolini (2015, p. 201) note a strong demand for multidimensional analyses by policymakers and other stakeholders, and indeed, empirical and policy applications of multidimensional poverty indices

> 1 For Aaberge and Brandolini (2015, p. 146) this sensitivity is “the single feature that distinguishes multidimensional from unidimensional analysis.” Note that it also places additional requirements on the data that can be used. 2 See the recent surveys of Aalberg and Brandolini (2015), Andreoli and Zoli (2020), Glassman (2019), and Seth and Santos (2018).

2
(MPIs) are numerous.<sup>3</sup> However, for multidimensional _inequality_ measures, the policy impact has been more muted.

Why is this? One possibility might be the complexity of existing measures.<sup>4</sup> To be effective in policy analysis, a measure needs to be easily understood and communicated. But this desirable characteristic is unlikely to be present unless it has been prioritized and intentionally built into the measure along with the traditional axiomatic requirements.<sup>5</sup> In addition, the process of bringing a particular multidimensional measure to data can be daunting, requiring many consequential choices not addressed in most theoretical presentations (Alkire and Foster 2010). On what basis should a given cardinalization of a variable be selected? How can the variables be made comparable to one another? Where should the relative importance of variables be reflected? The structure of a measure might facilitate or hinder the answers to such questions, impacting its dependability and ease of use in policy analyses.

The aim of this paper is to identify axiomatically-sound multidimensional inequality measures having attributes well-suited for policy.<sup>6</sup> Our focus is the two-stage approach of Maasoumi (1986), which yields intuitive measures frequently used in empirical analyses.<sup>7</sup> In this approach, a multidimensional inequality measure is constructed using two components: an _aggregation function_ converting each person’s dimensional achievements into an aggregate indicator; and an _inequality measure_ evaluating the resulting vector of indicators. His original presentation used general means and generalized entropy measures; ours considers a general set of aggregation functions based on Bosmans et al. (2015) and Lorenz-consistent inequality measures.<sup>8</sup> Two-stage measures inherit several of the standard axioms for multidimensional inequality measures from the properties of their components. However, Dardanoni (1995) has shown that at

> 3 Google Scholar and the site www.mppn.org list hundreds of empirical studies using MPIs along with many policy applications. The relative infrequency for multidimensional inequality indices is noted in Hong (2009), Seth and Santos (2018) and IPSS (2018).

> 4 For example, Lugo (2007) notes how the parameters of a measure can obscure its meaning, hampering applications.

> 5 See Foster (2024) who discusses an approach to “intentional measurement”.

> 6 As we shall see below, the required properties include anonymity, scale invariance and replication invariance, as well as the two generalizations of the transfer principle to multidimensional measures.

> 7 According to Bosmans, Decancq, and Ooghe (2015 p. 95) the two-stage approach “dominates the empirical literature”. Recent applications include Nilson (2010), Justino (2012), Rhode and Guest (2013, 2018), Bartels and Stockhausen (2016).

> 8 Aggregation functions are assumed to satisfy continuity, concavity, and linear homogeneity; Lorenz-consistent measures follow the Lorenz criterion when it applies.

3
least one property is not assured: Kolm’s (1977) weak majorization axiom, which requires the level of multidimensional inequality not to rise when each dimensional distribution is “smoothed” using the same bistochastic matrix. This is a fundamental axiom that generalizes the Pigou-Dalton transfer principle to the multidimensional context. Consequently, additional restrictions on the two components may be needed to ensure that the resulting measures are axiomatically sound.

Our first results characterize the subset of measures satisfying the standard multidimensional axioms, including weak uniform majorization. We show that, while any Lorenz-consistent inequality measure can be used at the second stage, the only form of aggregation that can be used in the first is linear. Given this specification, we show that the two-stage measures satisfy all the basic axioms, including, perhaps surprisingly, the unfair rearrangement axiom, which ensures that a multidimensional inequality measure is appropriately sensitive to positive association among the variables. The Lorenz curve can also be applied in the second stage to obtain a graphical depiction of multidimensional inequality and a dominance criterion that indicates when all two-stage measures with the same aggregation would agree on a comparison.

The next series of results explores the link between multidimensional inequality and the (dimension-)specific inequalities. Following Shorrocks (1978), we consider Lorenz consistent measures satisfying a basic convexity assumption.<sup>9</sup> We show that multidimensional inequality can be expressed as a weighted average of specific inequalities minus a non-negative term reflecting the relevant aspects of the joint distribution across dimensions. In the case of the Gini coefficient (or the Lorenz curve) this final term is particularly intuitive: it is the extent to which multidimensional inequality would rise if achievements were completely aligned.

To implement the measure, we provide a calibration approach based on data in an initial period and normative policy weights. An average Lorenz curve is constructed by weighting and summing up the specific Lorenz curves for the country in the initial period. Then, working back through the Lorenz formula, coefficients for the linear aggregation function are extracted to reflect the normative weights. In essence, dimensions are rendered comparable using the measuring rod of Lorenz (or Gini) inequality.<sup>10</sup> Once the multidimensional inequality measure

> 9 The property of constant-sum convexity is satisfied by virtually all traditional measures.

> 10 This is analogous to the role of deprivations in multidimensional poverty (Alkire et al 2015, p. 50).

4
has been calibrated using the initial period’s data, it can then be used to gauge the country’s multidimensional inequality through time and, with additional assumptions, through space.

We illustrate the methodology using simulated data to allow specific inequalities, distributional means, and correlation levels to vary freely. Examples show the impact of each factor on multidimensional inequality for measures based on the Lorenz curve, the Gini coefficient, and other Lorenz-consistent measures. A second illustration using data from Azerbaijan examines the evolution of multidimensional inequality during a period of rapid income growth.

Previous studies have considered linear aggregation as one among several options, but to our knowledge this is the first paper that selects this structure based on the axiomatic properties of its associated measures. Likewise, a number of authors have linked multidimensional inequality to specific inequalities or to positive association, but none has the elegant simplicity of our decomposition, which highlights the fundamental role of Shorrocks mobility in representing a positive association in multidimensional inequality. In addition, the approach is unique in its use of normative weights and Lorenz curves to calibrate the measure in a base year, and then to judge subsequent changes accordingly. Finally, the paper is unusual in its focus on identifying multidimensional inequality measures that are especially useful in conducting policy.

Section II introduces the definitions and notation used in the paper. Section III presents different approaches to multidimensional inequality measurement and then establishes our main characterization result, identifying the subset of measures from Maasoumi’s (1986) two-stage approach that are axiomatically sound. Section IV provides expressions linking members of this class of multidimensional inequality measures to specific inequalities and Shorrocks mobility. Our calibration method is described in Section V, selecting initial parameters of the measure based on normative weights and Lorenz curves. Section VI illustrates our methods, while the final section concludes.

**II. Notation, Axioms, and Other Fundamentals** The data for measuring inequality are given in an array with rows and columns. The row lists data for person ; the column lists data for dimension ; xx nn dd ii<sup>th</sup> and each entry ii ≥0 is person ’s achievement in dimension ∙jj . The population size can vary, xx ii= 1, ... , nn jj<sup>th</sup> xx jj= 1, ... , dd but is fixed in any given application; hence, the set iijj of possible arrays is a subset of xx ii jj nn dd XX

5
∪∞ Rn+ . Two alternatives for will be considered: 1 containing arrays that are strictly positive, and nn=1 2 containing nonnegative arrays having at least one positive quantity in each XX XX= XX column.<sup>11</sup> A _multidimensional inequality measure_ is a mapping associating a real XX= XX number to each array, interpreted as its level of multidimensional inequality. MM: XX→RR When developing a measurement tool like , it is important to be clear about its intended purpose, its desired characteristics, and the axioms it should satisfy (Foster 2024). The purpose MM of the measure we seek is to monitor multidimensional inequality in a country over time. A useful list of desired characteristics (or desiderata) can be found in Szekely (2006); of special relevance to the present paper is one that calls for a measure to be understandable and easy to describe.<sup>12</sup> The list of axioms to be satisfied by a multidimensional inequality measure will be given below.

We make use of unidimensional inequality measures in producing and understanding multidimensional measures. Let denote the set of column vectors of arbitrary length associated with . A _unidimensional inequality measure_ is a mapping associating a VV<sup>⋮</sup> real number to each vector in interpreted as its inequality level. The _Lorenz curve_ XX II: VV<sup>⋮</sup> →RR : [0,1] →[0,1] associated with depicts its level of equality (and inequality), where is II(vv) vv VV<sup>⋮</sup> the share of the income received by the lowest vv share of the population. Distributions vv LL vv LL (pp) have the same level of inequality by the Lorenz criterion if ′ for all ; distribution pp vv, vv<sup>′</sup> εε VV<sup>⋮</sup> has no less inequality than if ′ for all , in which case we say that vv vv _weakly_ LL (pp) = LL (pp) pp _Lorenz dominates_ ; and has greater inequality than vv vv if ′ for all , and ′ ≠ vv′ vv LL (pp) ≤LL (pp) pp vv , in which case we say that _(strictly) Lorenz dominates_ vv. A measure vv is said to be _Lorenz_ vv vv′ vv′ vv LL (pp) ≤LL (pp) pp LL _consistent_ vv if it renders the same judgment as the Lorenz criterion when the criterion applies; LL vv vv′ II equivalently, satisfies the axioms of anonymity, scale invariance, replication invariance, and the (Pigou-Dalton) transfer principle (Foster 1985). A measure is said to be _constant-sum_ II _convex_ if it is convex over slices of having the same population size and the same total.<sup>13</sup> II VV<sup>⋮</sup>

> 11 In symbols, 1 = ∪∞ Rn++ while 2 = ∪∞ (R+\0) , where R+\0 denotes the nonnegative orthant in excluding its origin. nn nn nn nn 12 nn=1 nn=1 Other desiderata include: conforming to a common-sense notion of what is being measured; fitting the stated XX XX RR purpose; being technically solid; being operationally viable; and being easily replicable. See also Alkire et al (2015) and Foster (2024). Such “proto-axioms” are less precise than formal axioms but help ensure a measure is fit for purpose.

> 13 See Kolm (1976, p. 93). Like decomposability, this property constrains the cardinal values of a measure.

6
Applying to column in yields the inequality level ) for dimension , which will be termed its _specific_ ∙jj inequality level; analogously (or ∙jj when is understood) II xx xx II(xx jj= 1, ... , dd xx.jj jj graphically depicts its specific inequality level. LL LL xx <u>i</u> Let be an element of . We denote the _mean_ of by or, more succinctly, by nn vv if vv . The _ordered_ VV<sup>⋮</sup> vector of is the permutation of vv μμ(vv) = ∑ for which ii=1 nn 1 2 . Given array jj .jj , the _completely aligned array_ of is the array whose columns are nn ; μμ vv= xx vv� vv vv vv� ≤vv� ≤⋯≤vv� it takes the same dimensional achievements and reorders them so that person 1 has the lowest .jj .jj xx εε XX xx̿ xx xx̿ = xx� entry for each dimension, person 2 has the next lowest, and so forth. An array is said to be _aligned_ if it has the same rows as its completely aligned version, but potentially in a different order. Some measurement approaches use a function to tally up the dimensional achievements of a given person. Let denote the set of row vectors of length associated with . An _aggregation function_ is a mapping associating a real number to each vector in ; VV<sup>⋯</sup> dd XX applying h to in yields person i’s _aggregate level_ ). Given vectors , we define h: VV<sup>⋯</sup> →RR VV<sup>⋯</sup> the _vector dominance_ ii relations as follows: iff ii ′ for all ; iff ′ for all xx xx h(xx vv, vv<sup>′</sup> εε VV<sup>⋯</sup> ; and iff and not . The rows of an aligned array can be ranked by vector jj jj jj jj vv≫vv<sup>′</sup> vv > vv jj vv≥vv′ vv ≥vv dominance; and in the completely aligned array the later rows vector dominate earlier rows, so jj vv> vv<sup>′</sup> vv≥vv<sup>′</sup> vv<sup>′</sup> ≥vv that for ii′ ii We consider two categories of axioms for multidimensional inequality measures – invariance xx̿ ≥xx̿ ii<sup>′</sup> > ii.

axioms and dominance axioms. _Invariance_ axioms specify the transformations of an array that leave the measure unchanged. They include _anonymity_ , _scale invariance_ , and _replication invariance_ and are entirely analogous to the unidimensional versions. _Dominance_ axioms specify transformations that cause multidimensional inequality to move in a certain direction. They include the two multidimensional generalizations of the Pigou-Dalton transfer principle associated with Kolm (1977) and Atkinson and Bourguignon (1982), respectively.

Intuitively, Kolm’s (1977) weak uniform majorization axiom requires multidimensional inequality not to increase when each dimension is “smoothed” in the same way. More precisely, we say that is obtained from by a _uniform smoothing_ if for some bistochastic xx′ xx xx<sup>′</sup> = Βxx 7
matrix Β.<sup>14</sup> Under a uniform smoothing, person _i_ ’s vector in is a weighted average of all persons’ initial vectors in , where the weights are the elements of the _i_<sup>th</sup> row of Β. This creates a xx<sup>′</sup> new array whose columns weakly Lorenz dominate the respective columns of the original array. xx The first dominance axiom for is given by the following. **Weak Uniform Majorization Axiom** MM . If is obtained from by a uniform smoothing, then . xx′ xx MM(xxThis axiom specifies that multidimensional inequality should not increase as a result of uniform<sup>′</sup> ) ≤MM(xx) smoothing.

When might we expect multidimensional inequality to strictly fall as a result of a uniform smoothing? When will it stay the same? We say that is obtained from by a _permutation_ if for some permutation matrix Π.<sup>15</sup> Note that a given uniform smoothing could also be a xx′ xx permutation if, say, Β were itself a permutation matrix or if the rows averaged by Β happened to xx<sup>′</sup> = Πxx be identical. Since anonymous multidimensional inequality measures are unchanged by a permutation (and anonymity is typically assumed), the strict version of the axiom accounts for this possibility.

this possibility. **Uniform Majorization Axiom** . If is obtained from by a uniform smoothing, then ) ≤ ; if, in addition, is not obtained from by a permutation, then . xx′ xx MM(xx<sup>′</sup> Many multidimensional measures have MM(xx) xx′ _two stages_ xx : the first employs an aggregation function MM(xx<sup>′</sup> ) < MM(xx) and a second applies some symmetric function to the aggregate distribution 1 ii )ii.<sup>16</sup> It could be argued that when dealing with this form of measure, the strict form of h(xx ) = ss ss= the uniform majorization axiom needs to be modified further. To be sure, if nn is a permutation (ss , ... , ss of , then will be a permutation of . However, the converse need not be true. And if it were xx<sup>′</sup> not, then the axiom would require (since is not a permutation of ) at the same xx ss<sup>′</sup> ss time that anonymity would be requiring (since is a permutation of ). The MM(xx<sup>′</sup> ) < MM(xx) xx′ xx following modification accounts for this issue. MM(xx<sup>′</sup> ) = MM(xx) ss<sup>′</sup> ss

> 14 A bistochastic matrix is a square nonnegative matrix whose rows and columns sum to 1.

> 15 A permutation matrix is a square matrix containing 0’s and 1’s whose rows and columns sum to 1. 16 Examples of papers that aggregate rows first include Tsui (1995, 1999), Bourguignon (1999), Diez et al. (2007), Decanq and Lugo (2012), and Seth (2013).

8
**Limited Uniform Majorization Axiom** . If is obtained from by a uniform smoothing, then ; if, in addition, is not obtained from by a permutation, then . xx′ xx This revised axiom only requires strict inequality to hold when the aggregate vectors are not MM(xx<sup>′</sup> ) ≤MM(xx) ss′ ss MM(xx<sup>′</sup> ) < MM(xx) permutations of one another. It should be noted that this axiom is tailor-made for two-stage multidimensional inequality measures, and it is a joint restriction on and h.<sup>17</sup> The second type of dominance axiom, based on Atkinson and Bourguignon (1982), takes into MM account the association among variables.<sup>18</sup> We say that is obtained from by an _unfair rearrangement_ if , where is the completely aligned version of . An unfair xx<sup>′</sup> xx rearrangement reassigns achievement levels to people so that person 1 has the lowest level in xx<sup>′</sup> = xx̿ xx̿ xx

rearrangement reassigns achievement levels to people so that person 1 has the lowest level in each dimension, person 2 has the next lowest levels, and so forth, thereby maximizing positive association among variables. The following is a second multidimensional generalization of the transfer axiom.

transfer axiom. **Weak Unfair Rearrangement Axiom** . If is obtained from by an unfair rearrangement, then . xx′ xx MM(xxAccording to this axiom, reallocating achievements so as to maximize positive association<sup>′</sup> ) ≥MM(xx) should not decrease multidimensional inequality.

Note that as before, this transformation may yield an array that is a permutation of the original array, as would happen if had the same rows as , but in a different order across people. The strict version of this axiom accounts for this possibility. xx xx̿ **Unfair Rearrangement Axiom** . If is obtained from by an unfair rearrangement, then ; if, in addition, is not obtained from by a permutation, then . xx<sup>′</sup> xx This axiom goes beyond the weaker version by requiring multidimensional inequality to strictly MM(xx<sup>′</sup> ) ≥MM(xx) xx<sup>′</sup> xx MM(xx<sup>′</sup> ) > MM(xx) increase when the unfair rearrangement is not a permutation of the original array.

# **III. Multidimensional Inequality Measures**

> 17 Thus, its use is limited to the class of two-stage measures. 18

> Our presentation follows Dardanoni (1995). For other versions of the axiom see, for example, Tsui (1995).

9
We now present several intuitive approaches to measuring multidimensional inequality and the properties they satisfy. A _dashboard_ ∙1 )� is a vector of specific inequalities, which can be interpreted as a multidimensional inequality measure (or rather a ∙nn DD(xx) = �II(xx ), ... , II(xx quasiordering on ) when used with vector dominance.<sup>19</sup> Multidimensional inequality is then judged to be higher when one specific inequality level is higher, and the rest are no lower. So XX long as is Lorenz-consistent, satisfies all but one of the general axioms required of a multidimensional inequality measure.<sup>20</sup> The unfair rearrangement axiom fails, since ignores II DD information on positive association and considers and to be identical. Of course, the practical DD utility of is also hampered by its inability to make comparisons when one specific inequality xx̿ xx rises and another falls.<sup>21</sup> DD A simple way of moving from a dashboard to a multidimensional inequality measure is to take a weighted average of specific inequalities using positive weights 1 that sum to 1, resulting in an _average specific inequality_ measure nn ωω , ... , ωω 1 ∙1 ) (1) nn ∙nn Gajdos and Weymark (2005, p. 489), for example, use the Gini coefficient and weights AA(xx) = ωω II(xx ) + ⋯+ ωω II(xx = to obtain a measure of this form, which they contrast to Koshevoy and Mosler (1997) jj ωω who consider fixed weights. Whether weights depend on means or are fixed, jj kk kk satisfies the same μμ / ∑μμ list of axioms as when is Lorenz-consistent.<sup>22</sup> Unlike a dashboard, an average specific AA inequality measure can make comparisons between any two arrays, but it also ignores the DD II association between dimensions. In particular, it views and as identical and violates the unfair rearrangement axiom. xx̿ xx

> 19 Dashboards can also be populated by distinct inequality measures or applied to unrelated sample populations. The above case fits best in the present context. A quasiordering is a reflexive and transitive relation, that is not necessarily complete (Sen 1997 Ch 3).

> 20 In particular, dashboard inequality levels are unchanged by permutations, by a scalar multiple, by a population replication, and by an unfair rearrangement; they do not rise, and can fall, as a result of a uniform smoothing. Consequently, the quasiordering generated by satisfies the three invariance axioms, the uniform majorization axiom, and the weak unfair rearrangement axiom. 21 Even when _can_ compare two arrays, the comparison might go against judgments that take into account DD information on dimensional means (and dependence); the conclusions rendered by dashboards are partial and not unambiguous. DD

> 22 In particular, satisfies the three invariance axioms, the weak unfair rearrangement axiom, and the uniform dominance axiom, where the latter property holds since a uniform smoothing leaves weights unchanged. AA

10
Maasoumi (1986) constructs a multidimensional inequality measure that reverses the order of aggregation by first combining a person’s dimensional achievements into a single aggregate indicator ) and then applying a unidimensional inequality measure to the aggregate distribution ii ii1 )<sup>′</sup> . The resulting _two-stage_ measure is intuitive in ss = h(xx II structure, with components nn and h that can be readily understood and applied.<sup>23</sup> However, ss= (ss , ... , ss εε VV<sup>⋮</sup> MM(xx) = II(ss) questions about the axiomatic suitability of the approach have been raised. Dardanoni (1995) II shows how a two-stage measure can violate the weak uniform majorization axiom, which leads him to critique the axiom; Weymark (2006) reinterprets this finding as a critique of Maasoumi’s two-stage approach.

Bosmans et al. (2015) provide a novel justification of two-stage measures in the context of normative inequality measurement, which views multidimensional inequality as the welfare loss from falling short of an optimal allocation.<sup>24</sup> They divide each normative multidimensional inequality measure into two distinct terms: one that evaluates inefficiency and another that evaluates inequity and show that the latter term is, in fact, a two-stage inequality measure. The authors conclude: “If one would insist that inequality measures should be concerned with inequity alone, and not with inefficiency, then we arrive at the striking conclusion that the normative approach itself pushes two-stage measures to the forefront.” This is a remarkable observation, which sheds light on the structure of normative multidimensional inequality measures as well as the suitability of the two-stage class. Note, though, that it justifies the twostage measures not as independent multidimensional inequality measures but as useful “partial” indices that focus on one aspect of multidimensional inequality.<sup>25</sup> The broader suitability of the two-stage measures depends on the axioms they satisfy, which in turn depends on the range of components h and being considered. For the first component, we consider all aggregation functions that are continuous, concave, linear homogenous, II and strictly increasing (as in Bosmans et al., 2015), and denote the resulting set by H. For the h: VV<sup>⋯</sup> →RR second, we consider all Lorenz-consistent unidimensional inequality measures , and II: VV<sup>⋮</sup> →RR

> 23 While many, including Maasoumi (1986) and Bosemans et al (2015), interpret the aggregation function as utility, here we are “making no use of information on individual relative valuations” of dimensional variables, and instead are treating the function as “a subject for social decision” (Atkinson and Bourguignon 1982 p. 190).

> 24 See also Kolm (1977), Tsui (1995), and Weymark (2006) for traditional derivations of a (relative) normative multidimensional inequality measure from a welfare function.

> 25 On partial indices, see Foster and Sen (1997, p. 168-9).

11
denote the set by I. The object of study is M, the set of two-stage measures with components and . Which axioms are satisfied by these measures? Can we identify a MM: XX→RR subclass of M that is both intuitive and axiomatically sound? h εε H II εε I The properties defining H and I ensure that every measure in M satisfies the axioms of anonymity, scale invariance, and replication invariance.<sup>26</sup> Our first result takes up the weak uniform majorization axiom.

uniform majorization axiom. **Theorem 1** . Let be a two-stage measure with components and . If satisfies weak uniform majorization, then there exists 1 ) ≫0 such that 1 1 MM h εε H II εε I MM for all . nn nn nn cc= (cc , ... , cc h(vv) = cc vv + ⋯+ cc vv **Proof** . See the Appendix. vv εε VV<sup>⋯</sup> The proof draws on Dardanoni (1995) and begins by showing that any convex combination of allocations in having the same value under h also has the same value under h. Applying this to a certain set of allocations yields a simplex over which h is linear, while the remaining VV<sup>⋯</sup> argument expands the characterization to all of . Theorem 1 identifies the two-stage measures that are consistent with weak uniform majorization; the remaining measures in M VV<sup>⋯</sup> violate this basic axiom and hence are not axiomatically sound.<sup>27</sup> violate this basic axiom and hence are not axiomatically sound.<sup>27</sup> Let L be the subclass of M whose aggregation functions are linear with . The next result describes the axioms satisfied by the measures in L. L. . cc≫0 describes the axioms satisfied by the measures in L. L. . **Theorem 2.** Any two-stage measure satisfies the anonymity, scale invariance, replication invariance, limited uniform majorization, and unfair rearrangement axioms. MM εε L

**Proof** . See the Appendix. The proof shows how each invariance property for follows immediately from the analogous property for . For the limited uniform majorization axiom, when a bistochastic matrix is applied MM to array , the new aggregate vector can be found by applying the same bistochastic matrix to the II xx

> 26 See Theorem 2, below.

> 27 This includes two-stage measures using other CES-type aggregation functions suggested by Maasoumi (1986) and used in empirical applications.

12
aggregate vector of , due to the linearity of h. Consequently, both the weak and strict parts of the axiom follow directly from the transfer axiom and anonymity of . As for the unfair ss xx rearrangement axiom, the linear structure of h might lead one to think that the resulting measure II would not be sensitive to the joint distribution and hence that Yet the proof shows that follows immediately from the Lorenz consistency of whenever is MM(xx̿) = MM(xx). not a permutation of . MM(xx̿) > MM(xx) II xx̿ The intuition can be seen in an example with xx , where c = (2,3)= (2,3) are the coefficients in not a permutation of . , where c = (2,3)= (2,3) are the coefficients in h. Suppose, initially, person 1 has 1 = (2,1) while person 2 has 2 = (1,2), so that the initial nn= dd= 2 aggregate distribution is 8<sup>�. As a result of the unfair rearrangement, we obtain</sup> xx xx<sup>xx̿1= (1,1)</sup> and 2 = (2,2) and hence ss= �<sup>7</sup> 10<sup>�. In other words, the unfair rearrangement of xx translates into</sup> a regressive transfer from xx̿ ss̿ = �, and hence, strictly more inequality according to the Lorenz<sup>5</sup> consistency of . ss These results are summarized in the following corollary. II **Corollary.** A two-stage measure satisfies anonymity, scale invariance, population replication, limited uniform majorization, and unfair rearrangement if and only if . MM εε M Each measure in L is determined by a vector of coefficients and a unidimensional measure MM εε L . An approach to selecting is given in Section V below. The choice of can be guided by a cc≫0 large literature on unidimensional inequality measures. The Lorenz curve, which plays a central II cc II role in that literature, also applies directly to the present environment for fixed . First, it provides a useful graphical depiction of the inequality in as given by the _aggregate Lorenz_ cc _curve_ , or the Lorenz curve applied to the aggregate vector associated with . Second, the xx resulting weak and strict Lorenz criteria can be used to rank arrays. For example, ss has strictly LL ss xx more multidimensional inequality than whenever xx′ ′ for all , with strict inequality for some xx . (2) ss ss LL (pp) ≤LL (pp) pp εε [0,1] pp 13
Indeed, when (2) holds, it follows that every with the same would agree that ) > .<sup>28</sup> MM εε L cc MM(xx<sup>′</sup> MM(xx)Given the purpose of the measure, and the desired characteristics initially posited for it, the simple, linear form of the aggregation function can be viewed as an advantage.<sup>29</sup> It is clearly “neutral” in the ALEP sense discussed in Kannai (1980), so that dimensional variables are neither complements nor substitutes. At the same time, the choice of offers substantial scope for incorporating normative and practical considerations, as we shall see below. Linear cc aggregators often appear in empirical applications and theoretical discussions as part of a parametric family.

Following Maasoumi (1986), h typically has been interpreted as a utility function whose functional form often follows traditional examples from familiar classes. The empirical or normative bases for selecting from among the possibilities has been limited, and in any given application, several different choices for h are usually applied without identifying one, say, as a headline indicator for policy analysis. In contrast, the present paper does not adopt a utility interpretation; instead, analogous to Atkinson (1970) or Atkinson and Bourguignon (1982), it simply views h as a function employed in social evaluation. Its normative content will originate in public policy discussions of the relative importance of specific inequalities rather than from individual preference.

Finally, we should note that the results differ slightly depending on which domain is being assumed. Recall that the domain can either be 1 containing positive arrays or 2 containing nonnegative arrays that have at least one positive entry per column. Domain 1 allows a broader XX XX XX range of components, including those not defined for vectors with zero entries (such as many XX generalized entropy measures and weighted general means), but consequentially yields multidimensional measures that can be used only with positive data. Domain 2 limits consideration to a narrower range of components defined for zero values, but then yields XX measures that apply more broadly.

> 28 This follows directly from the Lorenz consistency of . Arguments entirely analogous to the proof of Theorem 2 show that for any given , the inequality quasiordering associated with the Lorenz criterion satisfies the multidimensional axioms of anonymity, scale invariance, replication invariance, limited uniform majorization and II εε I unfair rearrangement. cc≫0 29 See the discussion in Alkire and Foster (2011, p. 486).

14
# **IV. Multidimensional from Specific Inequalities**

As noted above, a key desirable characteristic of an inequality measure is for it to be easily understood and communicated to others. For multidimensional measures, this could be facilitated by a clear link with specific inequality levels. In this section we show that such a link exists for many two-stage measures in L and for the aggregate Lorenz curve.

The key intuition is found in Shorrocks (1978), who considers the impact of the accounting period on income inequality and measuring _mobility_ as the extent to which inequality falls as the accounting period is extended. Consideration is restricted to measures that are constant-sum convex, which includes the Gini coefficient, the generalized entropy measures, and Atkinson’s II εε I family, among others.<sup>30</sup> Where .1 are (column) vectors listing the incomes of persons over periods, Shorrocks compares the inequality in the total income .nn .1 ) to the yy , ... , yy nn income-share weighted average of the per period income inequalities 1 .1 .nn ), dd II(yy + ⋯+ yy where = ∑ <u>� j�</u> for He notes that the former never exceeds the latter, while the αα II(yy ) + ⋯+ ααnnII(yy.nn μμ yy. two coincide when ααjj kk μμ�yy.kk� are scalar multiples of each other. Mobility can be defined as jj= 1, ... , dd. .jj yy 1 .1 .1 ) ≥0 (3) nn .nn .nn or the extent to which total income inequality falls below the average per period income mm= αα II(yy ) + ⋯+ αα II(yy ) −II(yy + ⋯+ yy inequality due to the “smoothing” effect of aggregation across time periods.

The same logic can be applied in the multidimensional context where the smoothing now occurs across dimensions. Pick any array . Let L<sup>′</sup> denote the set of all measures in L with inequality components that are constant sum convex and select a measure with associated xx εε XX components and . Substituting for in equation (3) converts the final term into MM εε L<sup>′</sup> 1 .1 jj ∙jj, or the multidimensional inequality in array ∙jj , while the cc II cc xx yy first terms become the average specific inequality nn .nn as defined in (1) using weights II(cc xx + ⋯+ cc xx ) = II(ss) = MM(xx) xx = <u>j j</u> for AA(xx) (4) 1 1 cc μμ wwjj cc μμ +⋯+ccddμμdd jj= 1, ... , dd We have the following result.

> 30 His formal results apply to constant-sum _strict_ convex measures; implications of the weaker convexity property are discussed informally (Shorrocks 1978 p. 382).

15
**Theorem 3** . Select any and define using (1) and (4). Then for any we have , with if the normalized vectors of are identical MM εε L<sup>′</sup> AA(xx) xx εε XX for . ∙jj jj mm(xx) = AA(xx) −MM(xx) ≥0 mm(xx) = 0 xx /μμ xx **Proof** jj= 1, ... , dd. See the Appendix. The theorem shows that the average specific inequality level is generally larger, and certainly no smaller, than the multidimensional inequality level , with their difference being AA(xx) the Shorrocks mobility measure , assessed here across dimensions rather than through time. MM(xx) In the special case where the column vectors of are multiples of one another (hence ordered in mm(xx) the same way and with identical shapes), the aggregate vector will also be a multiple, so that xx and hence . ss MM(xx) = AA(xx)Given any in L′ the mobility term can be usefully broken down into two independent terms mm(xx) = 0 that respectively reflect the association among dimensional distributions and their relative MM shapes. The _rearrangement_ term measures the pure effect of positive association on multidimensional inequality as one moves from to the completely aligned RR(xx) = MM(xx̿) −MM(xx) version . By Theorem 2 and the unfair rearrangement axiom we know that . The xx _structural_ term measures the mobility associated with the completely xx̿ RR(xx) ≥0 aligned vector . It is always nonnegative by Theorem 3 given weights 1 from (4). In SS(xx) = AA(xx̿) −MM(xx̿) addition, the anonymity of ensures that . With these observations, the next result nn xx̿ ww , ... , ww follows immediately from Theorem 3. II AA(xx) = AA(xx̿)

follows immediately from Theorem 3. **Theorem 4** . Select any L′ and define 1 using (3). Then for any we have nn 1 ∙1MM εε ww , ... , ww xx εε XX (5) nn ∙nn where RMM(xx) = ww II(xx ) + ⋯+ ww II(xx ) −RR(xx) −SS(xx) Equation (5) provides a general expression linking multidimensional to specific inequalities, with (xx), SS(xx) ≥0 the nonnegative terms and accounting for the extent to which falls below . If is not a permutation of , we know that by the unfair rearrangement axiom, in RR(xx) SS(xx) MM(xx) AA(xx) which case the rearrangement term is impacting measured inequality; if the columns of are not xx̿ xx RR(xx) > 0 scalar multiples of each other, and inequality measure is sensitive to their different shapes, then xx̿ II 16
by Theorem 3 and so the structural term is impacting measured inequality.<sup>31</sup> As an array evolves over time, changes in multidimensional inequality can be viewed in terms of four SS(xx) > 0 factors: (i) changes in the specific inequality levels, (ii) changes in the weights on the specific xx inequalities through dimensional means, (iii) changes in positive association across dimensions, and (iv) changes in the shapes of the dimensional distributions. The Gini coefficient is the most common Lorenz-consistent measures, due in part to its intuitive interpretations and its clear link to the Lorenz curve. When applied to ordered vectors, GG the Gini becomes a linear function, which simplifies its associated two-stage multidimensional measure and expression (5) above. Let denote the two-stage measure based on the Gini coefficient , and let its associated rearrangement term. We have the following GG MM εε L<sup>′</sup> expression for . GG GG RR (xx) GG **Theorem 5** . Consider MM (xx) and define using (4). Then for any , we have expression for . **Theorem 5** and define 1 using (4). Then for any , we have GG nn 1 MM∙1 εε L′ ww , ... , ww xx εε XX (6) GG nn ∙nn GG where MM (xx) = ww. GG(xx ) + ⋯+ ww GG(xx ) −RR (xx) GG **Proof** . See the Appendix. RR (xx) ≥0

**Proof** The key step of the proof uses the linear structure of over ordered vectors to show that , eliminating the structural term from (5).<sup>32</sup> Theorem 5 offers a GG remarkably straightforward expression for multidimensional inequality when BB GG GG is used: is SS (xx) = AA (xx̿) −MM (xx̿) = 0 the average specific Gini minus a term that is positive for any GG that is GG MM (xx) not a permutation of , but falls to zero as a positive association in GG GG GG rises towards its maximum RR (xx) = MM (xx̿) −MM (xx) xx level in . xx̿ xx Suppose that a linear xx̿ with has been selected in the first stage. Rather than choosing a particular in the second stage, the Lorenz curve can be used to depict multidimensional h εε H cc≫0 inequality and make comparisons. Given with aggregate vector , the aggregate Lorenz II εε I 1 curve for can be defined as for , where is the mean xx εε XX0 ss pp ss ss ss LL xx LL (pp) = μμ(ss) ∫QQ (rr)ddrr pp εε [0,1] μμ(ss)

> 31 Measures with the needed sensitivity include those that are constant-sum _strictly_ convex.

> 32 The proof also applies to any that is linear over ordered vectors, such as the generalized Gini measures (Weymark 1981). II εε I

17
of and is its quantile function.<sup>33</sup> For , let and respectively denote the quantile function and Lorenz curve of the ss th variable in jj . Multidimensional jj ss QQ : [0,1] →RR jj= 1, ... , dd QQ (rr) LL (pp) inequality in is evaluated using its aggregate Lorenz curve while specific inequalities are ∙jj jj xx xx evaluated using the specific Lorenz curves ; in each case greater inequality is indicated by a ss xx LL lower Lorenz curve. jj LL Multidimensional and specific inequalities are linked for this case as well. Let denote the weighted average of the specific Lorenz curves for using the weights from (4) so that AA LL 1 1 for xx . (7) AA nn nn It can be shown that LL (pp) = ww LL (pp) + ⋯+ ww is itself a Lorenz curve. Indeed, let LL (pp) pp εε [0,1] be the completely aligned version of and denote its aggregate vector by AA , the associated quantile function by , and the LL (pp) xx̿ aggregate Lorenz curve by . Then ss̿ xx ss̿ QQ 1 ss̿ LL 0 1 1 pp LLAA(pp) = ∑cc1jj jjμμjj ∫[cc QQ (rr) + ⋯+ ccnnQQnn(rr)]ddrr = 0 for (8) pp μμss� ∫QQss̿(rr)ddrr= LLss̿(pp) pp εε [0,1] so that the average Lorenz curve is identical to , the aggregate Lorenz curve of . This is analogous to what was found above for AA and relies on the linearity of ss̿ in the ordered incomes LL LL xx̿ of . Now define the _rearrangement function_ GG by for MM LL , and note that it graphically depicts the inequality-reducing impact of dampened LL LL ss ss̿ QQ RR : [0,1] →RR RR (pp) = LL (pp) −LL (pp) association in moving from to .<sup>34</sup> We have the following result. pp εε [0,1] **Theorem 6** . Consider any linear xx̿ xx with and define 1 using (3). Then for any , we have nn h εε H cc≫0 ww , ... , ww xx εε XX 1 1 for (9) ss nn nn LL where LL (pp) = ww. LL (pp) + ⋯+ ww LL (pp) + RR (pp) pp εε [0,1] LL RR (pp) ≥0

> 33 A quantile function is a generalized inverse of the cdf of a distribution; it lists the income of each person against the percentile of the person, ranging from lowest to highest.

> 34 Integrating measures the average distance or the area between the two Lorenz curves, and hence is half of the rearrangement term for the Gini coefficient. LL RR (pp)

18
# **Proof** . See the Appendix.

A Lorenz curve can be used in place of a numerical inequality measure in the two-stage approach, with representing multidimensional inequality in . Expression (9) shows how multidimensional inequality ss can be additively decomposed into a weighted average of LL xx specific Lorenz curves plus a nonnegative rearrangement function ss reflecting the extent to AA LL LL which dampens positive association in . The final term vanishes for all LL when is a RR permutation of , but otherwise is strictly positive for some . xx xx̿ pp xx̿ **V. Calibration** xx pp The above results have established the properties of these multidimensional inequality measures and explored their links with specific inequalities. We now turn to the task of implementing the measures with data. As with other measures using the joint distribution, data need to be drawn from a single source to construct the vector of achievements for each person. Dimensional variables must be viewed as cardinal variables that are comparable across people so that meaningful specific inequalities and Lorenz curves can be constructed.<sup>35</sup> In keeping with the stated purpose of the measure, data should be available in a base period and over time. We assume the existence of an initial array and several subsequent arrays for some . xx= xx<sup>1</sup> xx<sup>2</sup> , ... , xx<sup>TT</sup> TT≥2Taking positive multiples of variables leaves specific inequalities unchanged but can directly Taking positive multiples of variables leaves specific inequalities unchanged but can directly impact multidimensional inequality. In empirical applications, attention is often paid to rescaling variables to make them “comparable” in some sense. For example, variables might be rescaled to a common range such as [0,1]; or normalized by dividing by the mean or another indicator of size. The resulting variables are then posited to be comparable even though no measuring rod related to inequality has been invoked. In addition, the very rescaling process that asserts comparability across dimensions can, if reapplied to each round of data, reduce comparability across time.

> 35 More precisely, dimensional variables must be ratio scales; note that this fixes a natural zero value for each variable, which in the present context also has implications for comparability across variables. Related issues are discussed in Alkire and Foster (2010).

19
Our approach leaves variables in their original forms and accounts for the relative importance of variables, and their comparability, through the calibration of from the aggregation function. The calibration assumes that policymakers can express the relative importance of specific cc inequalities using positive normative weights 1 summing to 1. The weights are then applied to data in the base period to obtain a _normative_ nn average Lorenz curve, namely νν , ... , νν applied to data in the base period to obtain a _normative_ average Lorenz curve, namely 1 1 (10) NN nn nn where LL for (pp) = νν LL (pp) + ⋯+ νν are the specific Lorenz curves from period 1. For example, if weights LL (pp) are equal, jj will be the simple average of the specific Lorenz curves; while if a weight of 0.5 is LL jj= 1, ... , dd placed on income and the remaining 0.5 is equally split among the rest, the resulting NN will LL more closely resemble the income Lorenz curve. The normative Lorenz curve provides a NN LL snapshot of average specific inequality in the base year using weights that reflect policy priorities.

Expression (9) linking multidimensional and specific inequalities contains a second average Lorenz curve which depends on the choice of coefficients 1 through its weights. The calibration approach selects these coefficients to ensure that the latter curve is the same as the AA nn LL cc , ... , cc former. Given the associated quantile functions and means for we can rewrite expression (10) as jj jj QQ (rr) μμ jj= 1, ... , dd, <u>1 d</u> 0 1 0 (11) pp νν pp νν NN μμ1 μμdd nn which through (8) suggests the use of coefficients LL (pp) = ∫ QQ (rr)ddrr+ ⋯+ ∫ QQ= (rr)ddrrj for . Indeed, applying these νν jj μμjj coefficients to equation (7) yields for the base period. In this way, the baseline cc jj= 1, ... , dd for evaluating the evolution of inequality can reflect the policy priorities embodied in the AA NN LL (pp) = LL (pp) normative weights 1 . An analogous relationship holds for the measure associated with the Gini coefficient. Substituting nn = <u>j</u> into the weights = <u>1μ1</u> used in GG νν , ... , νν jj μμjjνν jj ∑ccjjμμjjccjj MM GG immediately yields 1 .1 cc ). So, for example, if the measure had two ww AA (xx) dimensions (say health and income) with equal normative weights, each Gini point in health GG nn .nn AA (xx) = νν II(xx ) + ⋯+ νν II(xx would have the same value in as a Gini point in income. Through the calibration, which accounts for both the weights and the base year means, the effective measuring rod becomes GG AA (xx) units of specific inequalities.

20
Once the coefficient vector has been fixed, the measure L′ (and ) can be applied to distributions for for the purpose of evaluating multidimensional inequality ss cc MM εε LL through time. The analysis it provides is consistent with the core axioms of multidimensional xx<sup>tt</sup> tt= 1, ... , TT inequality, while its intuitive aggregation structure helps interpret results. The link with specific inequalities likewise helps in understanding the evolution of multidimensional inequality via changes in specific inequalities, dimensional means, and joint distribution. Depending on the unidimensional inequality measure used, further analysis is possible. For example, can help evaluate the robustness of trends to the choice of unidimensional measure; while the use of a ss LL decomposable allows multidimensional inequality to be decomposed into traditional withingroup and between-group terms. II

# **VI. Illustrations.**

We begin with a series of simulated results to illustrate the properties of the approach. The data generation process for our simulation is based on multivariate log-normal distributions, a class of distributions that approximates well many policy-relevant variables. We generate data for outcome variables over time periods by populating samples for each period with dd= 2 100,000 random draws of outcome pairs from the bivariate log-normal distribution with TT= 2 given means and covariance matrix.<sup>36</sup> In the resulting arrays for , the distribution of nn= the first outcome is more unequal than the distribution of the second outcome. Outcomes have xx<sup>tt</sup> tt= 1, 2 equal normative weights so that = 1 for where is the mean of 1. The resulting is applied to for ccjj 2μμ to obtain aggregate vector jj jj= 1, 2, μμjj , while corresponds to the xx.jj = xx.jj completely aligned version of . cc xx<sup>tt</sup> tt= 1, 2 ss<sup>tt</sup> ss̿<sup>tt</sup> In Figure 1, the solid line depicts the “actual” aggregate Lorenz curve xx̿<sup>tt</sup> xx<sup>tt</sup> , while the dashed tt line depicts the completely aligned aggregate (or average) Lorenz curve ss for each time LL (pp) tt period _t_ = 1, 2. Figure 1 shows the simulated effect of the proportional increase (growth) in the ss̿ LL (pp) mean of the first outcome in period 2, namely, both 2 and 2 shift downwards, indicating an increase in multidimensional (and average) inequality between the two periods. ss ss̿ LL (pp) LL (pp)

> 36 We use Stata command drawnorm to generate a bivariate normal distribution of two variables with the given parameters. We then exponentiate these variables to obtain the bivariate log-normal distribution (Stata 2023).

21
<!-- Start of picture text -->
1.0 == _L,(q) - Lorenz for observed distribution (t:)<br>= =— =  £,(q)- Lorenz for alligned distribution (t:)<br>——_L.(q) - Lorenz for observed distribution (tz)<br>==- = Liq) - Lorenz for alligned distribution (tz)<br>08 U<br>2— ij<br>g Ui<br>= !y<br>z= 4<br>< t;,<br>a 06 1,<br>J 4<br>fo} F<br><xOo 7yf<br>ZS ‘,<br>o—] oA vMy4<br>&u<br>SE Cf<br>=] “7<br>oO it<br>0.2 Sls47<br>Z a<br>4 Sig2 4<br>Sid - (t:) Average distance between L.(q) and L4(q): 0.045<br>i (te) Average distance between L.(q) and L..(q): 0.037<br>0.0<br>0.0 0.2 0.4 0.6 0.8 1.0<br>Proportion of population (q)<br><!-- End of picture text -->
<!-- Start of picture text -->
1.0 == _L,(q) - Lorenz for observed distribution (t:)<br>= =— =  £,(q)- Lorenz for alligned distribution (t:)<br>——_L.(q) - Lorenz for observed distribution (tz)<br>==- = Liq) - Lorenz for alligned distribution (tz) ly<br>g— 08 44<br>2oij4<br>c yf<br>_ ‘7<br>fo} /<br>@ fi<br>£ 4<br>Oo yfVd<br>ZS4oA Vs//<br>&4“4<br>3s 4, 7<br>0.2 Soa<br>44<br>“4<br>ad<br>“ee“<br>Siva ~ (ti) Average distance between L.(q) and £.(q):" 0.045<br>a (te) Average distance between L.(q) and L.x(q): 0.050<br>0.0<br>0.0 0.2 0.4 0.6 0.8 1.0<br>Proportion of population (q)<br><!-- End of picture text -->
<!-- Start of picture text -->
1.0 == _L,(q) - Lorenz for observed distribution (t:)<br>= =— =  £,(q)- Lorenz for alligned distribution (t:)<br>——_L.(q) - Lorenz for observed distribution (tz)<br>0.8 ==- = Liq) - Lorenz for alligned distribution (tz) Ht/<br>fe)L Li<br>@1<br>2<br>z 77<br>ao 06 /<br>J 4<br>fo} f<br>o4<br>gv4<br>Oo i; ¢<br>a’<br>E=] oo<br>oO 7 7<br>0.2 Fis<br>7 A<br>a<br>A a<br>gor (ti) Average distance between L,(q) and Li(q): 0.045<br>‘ (te) Average distance between L.(q) and L.(q): 0.027<br>0.0<br>0.0 0.2 0.4 0.6 0.8 1.0<br>Proportion of population (q)<br><!-- End of picture text -->
<u>Table 1: Changes in multidimensional inequality and components for three scenarios.</u><sup>*</sup>

||**M**|**LD**|**T**|**heil**|**G**|**ini**|
|---|---|---|---|---|---|---|
||Period 1|Period 2|Period 1|Period 2|Period 1|Period 2|
|_Scenario 1_<br><br><br>|_Mean in_|_come of the first_|_outcome increas_|_es from 100 in Pe_|_riod 1 to 200 in_|_Period 2_|
|M(x�)<br><br>|0.270|0.323|0.281|0.335|0.400|0.434|
|M(x)<br><br>|0.158|0.212|0.170|0.235|0.311|0.360|
|A(x)<br><br>|0.302|0.353|0.302|0.354|0.400|0.434|
|R(x)<br>|0.112|0.111|0.111|0.100|0.089|0.074|
|S(x)<br>|0.032|0.030|0.021|0.019|0.000|0.000|
|_Scenario 2_<br><br>|_Inequality of th_|_e second outcom_|_e increases from_|_Gini 0.3 in Perio_|_d 1 to Gini 0.4 i_|_n Period 2_|
|M(x�)<br><br>|0.270|0.356|0.281|0.360|0.400|0.450|
|M(x)<br><br>|0.158|0.204|0.170|0.211|0.311|0.349|
|A(x)<br><br>|0.302|0.365|0.302|0.366|0.400|0.450|
|R(x)<br><br>|0.112|0.152|0.111|0.149|0.089|0.101|
|S(x)<br>|0.032|0.009|0.021|0.006|0.000|0.000|
|_Scenario 3_<br><br><br>|_Corre_|_lation between o_|_utcomes increase_|_s from 0 in Perio_|_d 1 to_≈_0.3 in P_|_eriod 2_|
|M(x�)<br><br>|0.270|0.270|0.281|0.282|0.400|0.400|
|M(x)<br><br>|0.158|0.200|0.170|0.211|0.311|0.347|
|A(x)<br><br>|0.302|0.302|0.302|0.303|0.400|0.400|
|R(x)<br>|0.112|0.070|0.111|0.071|0.089|0.053|
|S(x)<br>*) The simulate<br>mean 30, Gini|0.032<br>d distributions ha<br>0.3; Correlation: p|0.032<br>ve the following<br>eriod 1: 0; perio|0.021<br>parameters: First<br>d 2: ≈ 0.3.|0.021<br>outcome: mean 1|0.000<br>00, Gini 0.5; S|0.000<br>econd outcome:|

Table 2 presents results for the three inequality measures when the outcome variables in period 2 are based on a simple transformation of the period 1 data: namely, the second variable is unchanged from period 1 while the first is obtained from its period 1 value by an additive or a proportional increase. A uniform increment of 20 units lowers multidimensional inequality, as is seen in the top panel of Table 2: declines from _0.160_ to _0.130_ , declines from _0.173_ to _0.144_ , and from _0.312_ to _0.284_ MMLLMM . That increment also results in a drop in the TTheeiiee MM MM rearrangement component for all three indexes, and the structural component declines for MLD GGiinnii MM and Theil, and remains at 0 for Gini. In contrast, a proportional increase in the first outcome by 20% increases multidimensional inequality as seen in the bottom panel of Table 2. grows from _0.158_ to _0.171_ , grows from _0.170_ to _0.185_ , and grows from _0.311_ to _0.323_ MMLLMM , MM while both the rearrangement and structural mobility components change only slightly. TTheeiiee GG MM MM 25
Table 2: Changes in multidimensional inequality from additive or proportional changes in the first outcome.<sup>*</sup>

||**M**<br>|**LD**<br>|**Th**<br>|**eil**<br>|**G**<br>|**ini**<br>|
|---|---|---|---|---|---|---|
||Period 1|Period 2|Period 1|Period 2|Period 1|Period 2|
|_Scenario 1_<br><br><br>|_The first o_|_utcome variable_|_additively incre_|_ases from 100 in_|_Period 1 to 120 i_|_n Period 2_|
|M(x�)<br><br>|0.270|0.216|0.281|0.234|0.400|0.364|
|M(x)<br><br><br>|0.158|0.129|0.170|0.142|0.311|0.283|
|A(x)<br><br>|0.302|0.222|0.302|0.242|0.400|0.364|
|R(x)<br><br>|0.112|0.087|0.111|0.092|0.089|0.081|
|S(x)<br><br>|0.032|0.006|0.021|0.008|0.000|0.000|
|_Scenario 2_<br><br><br>|_The first outco_|_me variable prop_|_ortionally increa_|_ses from 100 in P_|_eriod 1 to 120 i_|_n Period 2_|
|M(x�)<br><br>|0.270|0.284|0.281|0.296|0.400|0.409|
|M(x)<br><br>|0.158|0.171|0.170|0.185|0.311|0.323|
|A(x)<br><br>|0.302|0.316|0.302|0.316|0.400|0.409|
|R(x)<br>|0.112|0.113|0.111|0.111|0.089|0.086|
|S(x)<br>*) The simulate<br>mean 30, Gini|0.032<br>d distributions ha<br>0.3; Correlation: p|0.032<br>ve the following<br>eriod 1: 0; period|0.021<br>parameters: First<br>2: ≈ 0.3.|0.020<br>outcome: mean|0.000<br>100, Gini 0.5; Se|0.000<br>cond outcome:|

We now apply the methods to analyze changes in multidimensional inequality in Azerbaijan from 2016 to 2018. We use data from the second (2016) and the fourth (2023) rounds of the Life in Transition Survey (LITS). The LITS is a survey run by the European Bank of Reconstruction and Development and the World Bank covering the so-called “transition countries” of Europe and Central Asia and several comparator countries of Western Europe, the Middle East, and North Africa (EBRD 2023). The survey included a nationally representative sample of around 1,000 households in Azerbaijan in the second and fourth rounds. Azerbaijan experienced rapid growth in per capita GDP between 2016 and 2023, which makes the example of this country helpful in illustrating the properties of the methods. We construct the multidimensional inequality index for three dimensions captured by the monthly per capita income (in 2017 PPP terms), years of education, and respondent’s health GGiinnii MM assessment. The measure is calibrated for 2016 using normative weights of 1⁄2 for the income dimension and 1⁄4 for the education and health dimensions. Table 3 presents the specific and multidimensional inequality levels for Azerbaijan in 2016 and 2023. The mean monthly per capita income increased by almost 59 percent from about 852 PPP dollars in 2016 to 1350 PPP dollars in 2023. The average years of education and health self-assessment remained relatively stable. Income growth was accompanied by an increase in income inequality, from a Gini of 0.253 in 2016 to 0.339 in 2023. The inequality in years of education grew while the inequality in health assessment slightly declined.

26
<u>Table 3: Specific and multidimensional inequalities in Azerbaijan, 2016-2023.</u>

||**2016**|**2023**|
|---|---|---|
|**Specific Inequalities**<br>|||
|**Income**|||
|Mean|852.46|1384.44|
|Gini|0.253|0.339|
|**Education (years)**|||
|Mean|10.304|11.091|
|Gini|0.094|0.115|
|**Health**|||
|Mean<br>|3.448|3.511|
|Gini<br>|0.166|0.159|
|**Multidimensional Inequality**<br> <br><br>|||
|A(x) = M(x)<br><br>|0.181|0.260|
|<br> M(x)<br>|0.144|0.230|
|<br><br> m(x) = R(x)<br>|0.037<br>|0.029<br>|

Multidimensional inequality as measured by increased significantly between 2016 and 2023, from 0.144 to 0.230. This increase is due to (i) changes in the specific inequalities, (ii) GGiinnii MM changes in the effective weights as dimensional means change, and (iii) changes in the rearrangement term. Holding (ii) and (iii) fixed and altering specific inequalities from the 2016 to the 2023 levels raises the average specific inequality from 0.181 to 0.238. Incorporating the 2023 means alters the effective weights from 0.50, 0.25, and 0.25, respectively, to 0.61, 0.20, and 0.19, thus increasing the average further to 0.260. Finally, the rearrangement term fell slightly from 0.037 to 0.029, reflecting greater alignment of dimensions and ensuring that the increase in multidimensional inequality (namely, from 0.144 in 2016 to 0.230 in 2023) is more pronounced than the increase in average specific inequality.

Figure 4 depicts the associated Lorenz curves for 2016 and 2023. The solid curves represent , the Lorenz curve of the aggregate distribution from the actual data array. The dashed tt curve is ss associated with the aligned aggregate distribution or, equivalently, the LL (pp) ss<sup>tt</sup> tt weighted average of the specific Lorenz curves. The vertical difference between the solid and ss̿ LL (pp) ss̿<sup>tt</sup> dashed curve is the Lorenz rearrangement function for the year, while the average vertical difference is clearly linked to the Gini rearrangement term. Each 2023 curve is below its respective 2016 curve, indicating that multidimensional inequality is unambiguously higher in 2023 than in 2016. Replacing the Gini with any Lorenz-consistent inequality index would preserve the conclusion that multidimensional inequality rose from 2016 to 2023. II 27
<!-- Start of picture text -->
1.0 —————_ L,(q) - Lorenz for observed distribution (t:)<br>— = = £,(q)- Lorenz for alligned distribution (t:) /<br>———__L.(q) - Lorenz for observed distribution (tz) /,<br>=== Lxq@) - Lorenz for alligned distribution (tz)<br>0.8<br>—<br>2 ;<br>8 /<br>ne} //<br>&7<br>raelo}&06 . 7 /;<br>@4<br>“<br>s<br>Oo yt<br>g 0.4 1;<br>& 7<br>fj ha<br>E ‘7<br>=] 7<br>oO Ja<br>0.2 fp<br>‘7<br>Ga<br>(7 S 7 (t1) Average distance between L.(q) and L,(q):a 0.018<br>—< (te) Average distance between L.(q) and L.(q): 0.018<br>0.0<br>0.0 0.2 0.4 0.6 0.8 1.0<br>Proportion of population (q)<br><!-- End of picture text -->
term together shape multidimensional inequality, while an empirical example from Azerbaijan illustrates how multidimensional inequality evolved during a period of rapid economic growth.

There are several concepts related to multidimensional inequality that the paper does not address. We have not relied on a prior notion of welfare in our approach, and the functional form is not reliant on traditional representations of welfare, utility functions, or even preferences. While there is significant ongoing work in these areas, our paper is not intended to contribute to the associated lines of research. The capability approach provides a natural guide for expanding consideration to other evaluation spaces. As there are multiple capabilities, our multidimensional approach to inequality might be seen as moving toward the same general goal. But our approach does not account for individual variations in conversion factors between resources and outcomes, nor the value of having many choices, both of which are central to the capability approach (Foster and Sen 1997, Basu and Lopez-Calva, 2011). In addition, there is a burgeoning literature on inequality with ordinal variables.<sup>37</sup> In contrast, the methods presented in this paper require cardinal variables, while the link with specific inequalities also requires inequality values to have cardinal significance. Consequently, they are not applicable to the many cases where cardinality assumptions are unable to be maintained.

> 37 See Allison and Foster (2004), for example.

29