2.1 Introduction

Use of one-dimensional indices of inequality for looking at inequality of well-being of a population or comparing it with that of another population is highly inappropriate. As we have argued in Chapter 1, well-being of a population is a multidimensional phenomenon. Even then most analyses of inequality have restricted themselves to the analyses of only one dimension of individual well-being, mainly income. Realization of this fact has recently motivated researchers to work on multidimensional economic inequality.

One simple approach to the measurement of multidimensional inequality is to examine dimension-by-dimension inequality levels (see, e.g., Atkinson et al., 2002; Fahey et al., 2005 and World Bank, 2005). For instance, if there are two dimensions of well-being, say income and health, we inspect inequality within each dimension. In consequence, by health inequality, we mean a summary measure of differences between people with respect to their health categories. It does not summarize the differences between income gradients of health, the effects of health on income (see Wagstaff et al., 1991; Wilkinson, 1996; Wagstaff, 2002; Lokshin and Ravallion, 2008; and Decancq and Lugo, 2012). (See also O'Donnell et al., 2015.)

But this extremely simple dimension-by-dimension dashboard approach ignores one noteworthy issue of multivariate analysis of inequality, possible correlation, a measure of association, between dimensions (Atkinson and Bourguignon, 1982). Consequently, such an approach leads to an inadequate picture of multidimensional inequality. An example using intertemporal inequality may justify the situation better. In transitional Russia, over a particular reference period, income inequality started increasing initially, reached the peak in 1998, and then showed a decreasing trend (Gorodnichenko et al., 2008). Over the same reference period, inequality levels of health and education demonstrated a considerable increasing trend (Blam and Kovalev, 2006 and Smolentseva, 2007). Of two multidimensional situations (say, two countries or two intertemporal positions of the same country), if for each dimension, one is characterized by at least as high inequality as the other, with strict inequality for at least one dimension, then it can be reasonably argued that the latter is less unequal than the former. But the diverging trends we have noted for Russia do not lead to an overall conclusion on the movement in multidimensional inequality over the period. In fact, with the changeover to a market economy in Russia, there has been an increase in correlation between income, health, and education (Blam and Kovalev, 2006; Smolentseva, 2007; Lokshin and Ravallion, 2008; Decancq and Lugo, 2012). The dashboard-based indices can certainly be combined by some aggregation function to generate a composite index. But such a composite index also fails to take into account interdimensional correlation.

In contrast to the dashboard and the composite index approaches, at the outset, we can arrive at a well-being index for each individual as a function of the individual's achievements in all the dimensions. At the second stage, these individual indices are aggregated across individuals to arrive at an overall welfare (and hence inequality) evaluation. Kolm (1977) refers to this as the individualistic approach. (See also Dutta et al., 2003.) Chakravarty and Lugo (2016) refer to this procedure as the inclusive measure of well-being approach.¹

Another remarkable characteristic of multidimensional inequality analysis is that inequality indices should be consistent with some dominance criteria in the sense of identification of conditions that capacitate us to unambiguously conclude whether one distribution matrix has higher inequality than another. As a result, it becomes necessary to pay some attention to this matter.

The purpose of this chapter is to confer a review of alternative approaches to the evaluation of multidimensional inequality. The indices we scrutinize in this chapter are axiomatic and mostly normatively based. In view of this, we critically evaluate alternative indices with respect to the intuitively reasonable axioms proposed for an index of multidimensional inequality. This, however, should not convey the message that nonnormative indices are insignificant, and the normative indices are meant to supplant them. (See Chakravarty and Lugo, 2016, for a recent discussion on this subject.)

For the sake of completeness, we begin the chapter with a brief survey of univariate inequality indices. This is done in Section 2.2. Section 2.3 is concerned with alternative views on multidimensional inequality. Indices that belong to two general categories, the direct and the inclusive measure of well-being approaches, are analyzed in details. (Lugo (2007) provided a systematic comparison among some of these indices.) Section 2.4 makes some concluding remarks.

2.2 A Review of One-Dimensional Measurement

As a background material for the following sections, in this section, we briefly present a survey of some one-dimensional indices of inequality.² We will follow the notation adopted in Section 1.2.

2.2.1 Normative One-Dimensional Inequality Indices

The set of income distributions in an person society is denoted by , the nonnegative part of the n-dimensional Euclidean space with the origin deleted. We write for the positive part of . The corresponding sets for all possible population sizes are designated by and , respectively. A typical income distribution in an person society is represented by a vector , where indicates person i's income. Since inequality is undefined for a one-person economy, it is assumed throughout the chapter that . More precisely, we assume that , where .

In the remainder of this section, unless stated explicitly, by the domain of an person income distribution , we will mean either or . As in Chapter 1, the mean is symbolized by (or, simply ), and will be used to indicate the nonincreasingly ordered permutation of .

A normative inequality index is related to a particular social welfare function in a negative monotonic way. For a fixed mean income, an increase in inequality is equivalent to a reduction in social welfare and vice versa. The normative approach to income inequality measurement was initiated in the pioneering contributions of Atkinson (1970) and Kolm (1969) and later on popularized by Sen (1973). Given a social welfare function , the Atkinson–Kolm–Sen “equally distributed equivalent” (ede) income associated , where is arbitrary, is defined as that level of income that, if enjoyed by everybody, will make the existing distribution socially indifferent. More precisely,

2.1

Given that W is continuous and increasing, (2.1) can be solved uniquely for and be expressed as , where , being a particular cardinalization of W, possesses all its properties. Contours of are numbered so that for any , .

The Atkinson–Kolm–Sen inequality index associated with is then defined as

2.2

where and are arbitrary. It is bounded between 0 and 1, where the lower bound is attained if incomes are equally distributed. It determines the proportion of total income that could be saved if the society arranged to distribute incomes equally across individuals without any loss of welfare. It can as well be interpreted as the fraction of welfare loss emerging from existence of inequality. The relation shows how we can retrieve welfare from (2.2). It also clearly indicates that the monotonic relationship of welfare with inequality and the mean are decreasing and increasing, respectively.

The index is a relative index, that is, it is invariant under any proportional positive scaling of incomes, if and only if is linear homogeneous (Blackorby and Donaldson, 1978). We will now illustrate (2.2) by providing two examples, under linear homogeneity of . Multidimensional extensions of these indices will be analyzed in the next section. The first illustrative example of (2.2) is the Atkinson index, whose associated welfare function is the symmetric mean of order . For any and , it is defined as

2.3

A progressive transfer of income will reduce by a higher amount, the lower is the value of θ. In the extreme case when , approaches , the relative Rawlsian maximin index of inequality.

The second illustrative example, we choose, is the Donaldson and Weymark (1980) S-Gini index, which corresponds to the S-Gini welfare function. It is formally defined as

2.4

where and are arbitrary. The parametric restriction is necessitated by the Pigou–Dalton transfer principle (see Section 1.3). For , becomes the relative Gini index , which can as well be written as . Thus, is a normalized average of absolute values of all pairwise income differences in a population.³ In the extreme situation when ρ → ∞, approaches the relative maximin index of inequality.

For any and , Kolm's (1976) alternative to , is analytically defined as

2.5

This index achieves its lower bound 0 for the egalitarian income distribution. It is a per capita index in the sense that it determines the per capita income that could be saved if the society redistributed incomes equally without any welfare loss. Since , welfare is related increasingly to the mean and decreasingly to inequality.

The index is an absolute index, that is, it remains invariant under equal absolute changes in all incomes if and only if is unit translatable (Blackorby and Donaldson, 1980). An example of this general formula is the Kolm (1976) index, whose related welfare function is the Kolm–Pollak welfare function (see Chapter 1). For any and , this inequality standard is defined as

2.6

where the positive parameter attaches higher weight to a progressive income transfer as the income of the transfer recipient decreases. In the polar case, as approaches , approaches , absolute maximin index of inequality.

The index given by (2.4) is a compromise relative index – when multiplied by the mean income, it becomes the Donaldson–Weymark absolute S-Gini index . Hence, becomes the absolute Gini index . Evidently, upon dividing the general absolute index by the mean income, we can generate its relative sister (2.4).

2.2.2 Subgroup-Decomposable Indices of Inequality

Sometimes from policy point of view, it may be necessary to move from aggregate inequality to population subgroup-based inequality, where the separation of the population into subgroups is done with respect to a characteristic such as race, religion, sex, ethnic groups, and age. The relevant policy issue here is determination of contribution of inequality within the subgroups to total inequality. The connected assignment can as well be isolation of the subgroups that are more responsible for existing income differences in the country. A related policy matter is investigation of impact of inequality across subgroups on total inequality.

All such policy affairs can be properly addressed by employing a subgroup-decomposable index of inequality. A subgroup decomposable, also popularly known as additively decomposable index of inequality, is one that can be explicitly disintegrated into within-group and between-group terms. The within-group denomination is obtained by aggregating inequality levels of different subgroups. The between-group part is the level of inequality that arises due to variations in mean incomes across subgroups.

An inequality index is said to satisfy subgroup decomposability, more precisely, population subgroup decomposability, if for all , ,

2.7

where , is the population size corresponding to the distribution , , , , , is the positive weight attached to the inequality in , assumed to depend on the vectors and . Equation (2.7) indicates that the population has been segregated into J subgroups, and overall inequality has been expressed as the sum of the within-group and between-group components, given respectively by and , where is arbitrary. The within-group part is the weighted sum of subgroup inequalities. On the other hand, between-group part is the level of inequality that arises if everybody in a subgroup enjoys the subgroup mean income. Clearly, the numbering of subgroups and hence arrangement of x^i,s in are arbitrary. For instance, if we use the notation and to denote, respectively, the income distributions of the males and females in a population, we can alternatively write and for male and female income distributions, respectively. This is a simple matter of notation change.

It may be worthy to mention that the aforementioned policy recommendations that are contingent on subgroup decomposability should be employed under certain restrictions. It is frequently noted that in subgroup decomposition analyses, the between-group term is quite low in comparison with the within-group component. Consequently, policy may be administered toward reduction of within-group inequality. But the impact of the between-group factor may be of high concern for a society. Further, the size of the between-group component is likely to increase with the number of subgroups. The significance of the terms of decomposition may change with the nature of the characteristic partitioning the population. However, for a particular characteristic, if the number of subgroups remains fixed, the analysis of trends of percentage contributions made by the two factors of decomposition may be significant from policy perspective. A high between-group inequality may lead to social conflicts and political unrest.⁴

Shorrocks (1980, 1984) established rigorously that the only parametric family of relative subgroup-decomposable inequality indices is the generalized entropy family given by

2.8

For all real values of c, this population replication invariant, symmetric index satisfies the Pigou–Dalton transfer principle. It takes on the minimum value 0 if and only if incomes are equally distributed across individuals. For , it is increasingly related to the Atkinson index. In consequence, for all such values of c, the two indices evaluate inequality in the same way. For , it becomes half the squared coefficient of variation, where the coefficient of variation is defined as the ratio between the standard deviation , the positive square root of the variance , and the ordinary mean . For , becomes Theil's (1967) first index, the entropy index of inequality. On the other hand, for , it coincides with Theil's (1972) second index, also known as the mean logarithmic deviation, the logarithm of the ratio between the ordinary mean and the geometric mean of incomes.⁵

Other notions of population subgroup decomposability have been suggested by many authors, including Bhattacharya and Mahalanobis (1967); Cowell (1980); Blackorby et al. (1981); Foster and Shneyerov (1999); Zheng (2007a); Chakravarty and Tyagarupananda (2009); Ebert (2010); Bosmans and Cowell (2010).⁶

2.3 Multidimensional Inequality Indices

As in Chapter 1, in a society consisting of individuals, achievement of person in dimension is denoted by , where stands for the set of dimensions of well-being. The dimensional matrix , person i's achievement profile, represents a listing of person i's achievements in different dimensions of well-being.

Recall from the notation introduced in Chapter 1 that stands for the ith row of an distribution matrix , whose jth column indicates the distribution of achievements in dimension among individuals, where is arbitrary. Throughout the chapter, we assume that , the mean of , is positive. Since and stand, respectively, for the sets of achievement matrices with nonnegative entries along with the restriction of positive dimensionwise means and positive entries, it is implicit that either or . Consequently, the set of all achievement matrices with dimensions will now be an element of the set of sets (see Chapter 1). Unless specified, all axioms and inequality indices will be stated in terms of the arbitrary set , whose restriction when the population size is is given by .

By a multidimensional inequality index , we mean a nonconstant, nonnegative valued function defined on . Technically, , where for all , the nonnegative real number indicates the level of multidimensional inequality existing in the achievement matrix . Nonconstancy is a vital requirement since it ensures that multidimensional inequality need not be the same across achievement matrices.

It will be clearly established that while some of the indices correspond to the inclusive measure of well-being approach, some others do not. In view of this, following Chakravarty and Lugo (2016), we divide our presentation of the section into two subsections.

2.3.1 The Direct Approach

In the direct approach, the indices are typically axiomatized, where the axioms specify their properties with regard to the individual dimensional achievements. In view of this, at the outset, it is necessary to state the axioms formally and discuss them.

2.3.1.1 Axioms for a Multidimensional Inequality Index

Since each postulate represents a particular notion of value judgment, it is logical to partition the subsection on axioms into several segments.

Invariance Axioms

An invariance axiom depicts a characteristic of the inequality index that keeps its values unchanged under some permissible changes related to inequality.

Following Tsui (1995), our first invariance property can be formally stated as:

1. Strong Ratio-Scale Invariance: For all , where = diag for all .

   The strong ratio-scale invariance condition demands that postmultiplication of the social matrix by a positive diagonal matrix does not change multidimensional inequality. Consequently, inequality remains invariant under any change in the units of measurements of achievements in different dimensions.

   To explain this postulate, as in Chapter 1, we consider three dimensions of well-being, daily adult energy consumption, life expectancy, and income, measured respectively in calories, years, and dollars. Given these dimensions, consider the following social distribution for a three-person society:

   

   The entries in row of shows person i's achievements in the dimensions energy consumption, life expectancy, and income, respectively, where . Now, we can express energy consumption in joules instead of in calories, where 1 cal = 4.18 J. As a result, when expressed in joules, the entries in the first column of , starting from above to below, get transformed into 10 450, 10 868, 10 450, and 11 286, respectively. Next, suppose that we decide to measure life expectancy in months instead of in years, which means that all entries in the second columns of are multiplied by 12. Finally, suppose that incomes earned in dollars are converted into cents, which are obtained by multiplying all entries in the third column of by 100. Evidently, under these scale transformations of the achievements in the dimensions, inequality should remain unaffected. This whole transformation process can be generated by postmultiplying by the diagonal matrix .

   Our new social distribution , which is inequality equivalent to , is given by

   

   More precisely, .

   A weaker form of this postulate, ratio-scale invariance, requires invariance of inequality when the proportionality factors ( values) are the same across dimensions. Under variability of units of measurement of achievements across dimensions, this reasoning claims that independent changes in the units do not affect multidimensional inequality. When units of measurement of some of the dimensional achievements are the same, say, those of “incomes in different states of the world,” then independent variability of the proportionality factors is not appropriate (Weymark, 2006). However, generally dimensional achievements are measured in different units. Consequently, if we require inequality to remain unchanged under changes in the units of measurements of achievements in different dimensions, then the strong form is the appropriate postulate.

2. Strong Translation-Scale Invariance: For all , where is any dimensional matrix with identical rows such that .

   While the strong ratio-scale invariance axiom requires invariability of inequality under proportionate changes in achievements in different dimensions of well-being, the strong translation-scale invariance axiom claims that inequality stays fixed under equal absolute changes in the achievements in the dimensions. The matrix A in the statement of the axiom is a translation matrix. A weaker form of this axiom, translation-scale invariance, demands invariance of inequality when the absolute change is the same for all achievements across dimensions. An inequality index is called relative (respectively, absolute) if it satisfies the ratio-scale invariance axiom (respectively, the translation-scale invariance axiom). These axioms represent two extreme views concerning inequality invariance in the sense that because of nonconstancy assumption, an inequality index cannot satisfy them simultaneously.

   Suppose that in , while calorie consumption decreases by 10, income and life expectancy increase respectively by 50 dollars and 5 years. Then the dimensional matrix with identical rows is given by

   

   Then according to this second invariance axiom, , where

   

   Next, suppose that with these three dimensions of well-being, of two achievement matrices and , the former is regarded as less unequal than the latter. Now, assume that the units of energy consumption, income, and life expectancy are converted, respectively, from calorie to joules, dollars to cents, and years to months. Then it is natural that inequality ranking between the two matrices remains unchanged. The third invariance axiom, which was suggested by Zheng (2007a,b) in the univariate context and extended by Diez et al. (2008) and Chakravarty and D'Ambrosio (2013) to the multidimensional context, demands this.

3. Strong Unit Consistency (UCO): For any implies that for all for all .

   A ratio-scale invariant index is unit consistent, but the converse is not true (see Zheng, 2007a).

   The next two invariance axioms, formally stated next without elaborations, are inequality counterparts of the correspondingproperties of a social welfare function (see Chapter 1).

4. Symmetry: For all , where is any permutation matrix of order .
5. Population Replication Invariance: For all , where is the l-fold replication , that is, the achievement matrix is obtained by placing sequentially from top to below times, being any integer.

Distributional Axioms

A welfare function is a summary statistic of the levels of satisfaction enjoyed by the individuals in a society from their achievements in different dimensions. In contrast, an inequality index quantifies interpersonal differences existing in the distribution of dimensional achievements. Taking cue from one-dimensional case, we can say that, under ceteris paribus assumptions, multidimensional inequality should have a negative monotonic association with multidimensional welfare. Consequently, the following two distributional axioms, whose welfare twins have been stated in the earlier chapter, seem quite sensible.

1. Multidimensional Transfer: For all , if is obtained from by Pigou–Dalton bundle of progressive transfers, then .

   Since progressive transfers make the achievement distributions more equitable, inequality should go down under such transfers. The multidimensional transfer axiom states this analytically. Similarly, if the value of an inequality index reduces under a uniform Pigou–Dalton majorization operation, we say that it fulfills the uniform majorization principle.

2. Increasing Inequality under Correlation-Increasing Switch: For all , if is obtained from by a correlation-increasing switch, then if the dimensions underlying the switch are substitutes.

If the two dimensions involving the switch are substitutes, then one counterbalances the deficiency of the other. Now, one of the two persons affected by the switch has at least high achievements as the other in all dimensions, with strict inequality for at least one dimension. Consequently, interdimensional correlation goes up, since before the switch, the former person had lower achievement in one dimension. As a result, inequality should go up if the dimensions underlying the switch are substitutes. In contrast, inequality decreases or remains unchanged, if the two dimensions that are primitive to the switch are complements or independents (see Bourguignon and Chakravarty, 2003).

Technical Axioms

There is no inequality in the distribution of dimensional achievements across individuals if everybody possesses the average level of achievements in each dimension. Formally, for any , let denote the achievement matrix, showing equal distribution of achievements in different dimensions across persons. In other words, each row of is given by the coordinated vector . Since there is no inequality in , should assign the value 0 to . We state this property formally as follows:

1. Normalization: For all .

Since an inequality index is unambiguously nonnegative, this is the situation of minimum inequality, showing the lower bound of the index. It is a cardinality principle because we can assign a different numerical value to the inequality index in this situation by taking an affine transformation of the index. Note that the original and the transformed indices will rank any two distribution matrices in the same way. Consequently, no information is lost if the inequality index is subjected to an affine transformation. However, because of the affine transformation taken, nonnegativity of the index value may not be ensured.

Finally, the following supposition is self-explanatory.

1. Continuity: For all varies continuously with respect to dimensional achievements.

2.3.1.2 Examples of Indices

The first example we analyze here is the one that corresponds to the multidimensional generalized entropy family characterized by Tsui (1998), using the multidimensional, aggregative principle, formally stated as follows:

Multidimensional Aggregative Principle

For all , , , , where the aggregative function is continuous and increasing in first two arguments, and are respectively the vectors of means of dimensions corresponding to the achievement matrices and , and , that is, the dimensional distribution matrix is obtained by placing the matrices and from above to below.

Since and , it follows that . If there are female workers and male workers in a society, then this postulate shows how we can calculate overall inequality in terms of multidimensional inequality levels of the two sexes and the corresponding vectors of dimensionwise means. As argued in the earlier section, such decomposition becomes useful for judging the impact of subgroup inequalities on overall inequality.

Tsui (1998) characterized the following aggregative multidimensional generalized entropy family on :

2.9a

where and are arbitrary; , and for all are constants. The parameters , are required to satisfy some restrictions for inequality to decrease under a uniform majorization operation and to increase under a correlation-increasing switch. For instance, if , then , , and . Evidently, this symmetric, population replication invariant index is continuous and normalized.

If sensitivity to a correlation-increasing swap is not incorporated, then the resulting index will be (2.9a), or

2.9b

or,

2.9c

where and are arbitrary; and all the parameters in (2.9b) are required to obey some restrictions. In (2.9c), for all . For , the necessary parametric constraints in (2.9b) are , , and . In (2.9a), for , we require only and . The inequality is no longer necessary now.

The third functional form (2.9c) is a straightforward multidimensional extension of Theil's second index. In (2.9c), individual dimensional indices constituting a dashboard are aggregated to arrive at a composite index. This dashboard consisting of Theil's second indices for different dimensional indices may be named as the mean logarithmic deviation dashboard. This family consisting of (2.9a)–(2.9c) may be regarded as a multidimensional extension of the Shorrocks (1980) univariate generalized entropy family.

Lasso de la Vega et al. (2010) characterized the multidimensional aggregative, unit-consistent inequality index that decreases under a uniform majorization transformation and increases under a correlation-increasing switch. For arbitrary , , it is defined as

2.10a

where , , and for all . I_DVU becomes a relative index ( in (2.9)) if and only if . A nonzero value of makes I_DVU a unit-consistent index, which is not relative. For any value of , is not an absolute index. An equiproportionate increase in the quantities of an attribute for different individuals will increase or decrease inequality unambiguously according as or . Evidently, (2.10a) is the unit-consistent variant of (2.9).

As Diez et al. (2008) demonstrated, if the issue of interdimensional correlation is given up, then the resulting inequality index will be (2.10a), where , , and have to be chosen such that is strictly convex, or

2.10b

where , , and have to be chosen such that is strictly convex, or

2.10c

where and for all . The functional forms given by (2.10b) or (2.10c) may be treated as unit-consistent twins of (2.9b) and (2.9c), respectively. For , all these indices are multidimensional generalizations of Zheng's (2007a) one-dimensional unit-consistent indices.

In another highly interesting contribution, Lasso de la Vega et al. (2010) established that a multidimensional aggregative relative inequality index satisfies the multidimensional transfer principle if and only if it will be either

2.11a

where all , , are arbitrary; either for all or for all or

2.11b

where all , , are arbitrary and for all . In (2.9a), if we choose and , then satisfies the multidimensional transfer axiom for . Hence, in this special case, the functional form in (2.9a) is formally equivalent to that in (2.11a). In addition, when the number of dimensions is two or more, (2.11b) coincides with the composite index (2.9c).

The general procedure of first aggregating across all individuals for each dimension, and then across dimensions at the second step, is also followed in the multidimensional generalized Gini index characterized by Gajdos and Weymark (2005). However, in the alternative approaches we have analyzed so far in this part of the subsection, no concept of social welfare has been utilized. In contrast, Gajdos and Weymark's (2005) derivation has an explicit normative basis.

To define the index formally, we assume that . Recall that for all and , stands for the achievement matrix, each of whose jth column entries is , where . Now, define tacitly by the equation . The strong Pareto principle and continuity ensure that defined via the ethical indifference condition is well defined. In words, is a positive scalar, which, when multiplied by the ideal social matrix , makes the matrix , showing the current distribution of totals of dimensional achievements across persons, ethically indifferent. It is a multidimensional translation of the Atkinson–Kolm–Sen ede income. The positive scalar attains its upper bounded 1 when each attribute is equally distributed among the individuals, that is, when (Weymark, 2006).

Kolm's (1977) multidimensional inequality index is defined as

2.12

where and are arbitrary. Suppose that following some policy recommendation, the society has decided to move its current achievement distributions, as registered by , to . Analytically, such an operation can be performed by premultiplying by the bistochastic matrix, each of whose entry is . This process can as well be executed sequentially. Suppose that is reformed into by some equitable transformation, say, some uniform majorization operation. In consequence, the welfare position of the society improves. A second equitable reformation at the next step improves it further. Continuing this way, finally, we arrive at . Consequently, the standard determines the proportion of welfare improvement enjoyed by the society if it decides to move the actual matrix of achievement profiles to the ideal matrix . Equivalently, determines the fraction of welfare lost through unequal distribution of dimensional achievement totals across persons. We can also say that it ascertains the proportion of total achievements in each dimension that could be saved if the society distributed these totals for different dimensions equally among persons without any loss of welfare. In the polar case where there is only one dimension, reduces to the one-dimensional Atkinson–Kolm–Sen inequality index. If the welfare function obeys continuity, symmetry, the strong Pareto principle and decreases under a Pigou–Dalton bundle of progressive transfers, then the continuous, symmetric index satisfies multidimensional transfer axiom. It is bounded between 0 and 1, where the lower bound is achieved if the distribution of achievements for each dimension is equal among persons. It is a relative index if is linear homogeneous. In view of equation (2.12), we can interpret as the Kolm multidimensional equality index and under linear homogeneity, .

To illustrate the construction of , we note that for the distribution matrix considered earlier, the associated matrix is given by

Then is defined implicitly by the equation

We will analyze the multidimensional Kolm relative indices associated with the Gajdos–Weymark homothetic welfare functions given by Eqs (1.10) and (1.11). Let us first illustrate the calculation of for the generalized Gini welfare standard given by (1.10). We replace in the equation of by and equate the resulting expression for welfare function with the actual form of the function to get

from which it follows that

2.13a

where . By a similar calculation,

2.13b

where .

Consequently, the corresponding family of multidimensional Kolm (1977) relative inequality indices turns out to be

2.14a

where . By a similar calculation,

2.14b

where ; and are arbitrary, and . The coefficients and in (2.13a)–(2.14b) fulfill the following restrictions: for all with ; for all pairs with increasing and for all ; and as before, for any , . The functional forms, specified in (2.14a) and (2.14b), are the Gajdos–Weymark multidimensional generalized relative Gini inequality indices.

If for all and , then (2.14a) reduces to

2.14c

where is the relative Gini index of the distribution of achievements in dimension . If we denote by , then we can rewrite (2.14c) as . Consequently, we can say that in this particular case, the individual dimensional indices framing the relative Gini inequality dashboard are aggregated to arrive at the Gajdos–Weymark multidimensional generalized relative Gini index. Following the discussion in Section 1.5, we can say that this is a frequency-based weighting scheme.

Clearly, it satisfies the following factor decomposability postulate:

Factor Decomposability

For all and , , where for all , with at least one being positive, and .

This postulate is quite alluring from policy standpoint. It becomes helpful in judging the contributions of individual dimensions to overall inequality and hence to locate the dimensions that cause higher inequality (see Shorrocks, 1982; Chantreuil and Trannoy 2013; Chakravarty and Lugo, 2016). The decomposition (2.14c) of the Gajdos–Weymark multidimensional generalized relative Gini index is different from the arithmetic average of dimensionwise relative Gini indices (see Koshevoy and Mosler, 1997).

Similarly, when for for all , (2.14b) becomes

2.14d

for arbitrary and . In this case, we subtract the geometric mean of relative Gini equality indices across dimensions from 1. This variant of factor decomposability states that the geometric mean of individual dimensional relative Gini equality indices produces the multidimensional generalized relative Gini equality index .

It will now be worthwhile to discuss the absolute sister of suggested by Tsui (1995), which may be treated as a multidimensional translation of Kolm's (1976), one-dimensional inequality index. For and , let be the scalar that solves the equation

2.15

where is the dimensional achievement matrix, each of whose entries equals 1. Satisfaction of continuity and the strong Pareto Principle by make sure that is well defined.

Then Tsui's multidimensional inequality index is defined as

2.16

where and are arbitrary. measures inequality by the amount that can be taken away from each individual from the achievement in each dimension of the ideal achievement matrix such that the resulting matrix becomes ethically indifferent to the original distribution matrix. For , corresponds exactly to . It is bounded from below by 0. This lower bound is achieved if . It becomes an absolute index if is translatable. Translatability of establishes that .

To illustrate the calculation of , we consider again the distribution matrix and the associated ideal matrix . Then is implicitly defined by

We will now calculate for the Gajdos–Weymark translatable welfare function given by (1.13). For this purpose, we need to replace in the equation of by and equate the resulting expression for the welfare function with the actual form of the function to get

where ; and are arbitrary, ; so that

2.17a

where ; and are arbitrary, . By a similar calculation, for the welfare function in (1.13),

2.17b

where ; and are arbitrary, . The coefficients and in (2.17a) and (2.17b) fulfill the following restrictions: the sequence is positive, for all pairs with increasing and for all ; and as before, for any , . The functional forms (2.17a) and (2.17b) are the Gajdos–Weymark multidimensional generalized absolute Gini inequality indices.

If for all , then (2.17b) becomes

2.17c

where and are arbitrary. In (2.17c), dimensionwise indices constituting the absolute Gini dashboard are averaged across dimensions to get the multidimensional generalized absolute index. The satisfaction of the factor decomposability property by the aggregated index in (2.17c) is evident. Implicit under the choice of this weighting sequence is the normative assessment that claims that all the metrics are equally important.

Several attempts to extend the relative Gini index to the multidimensional framework have generally been adopted by two broad procedures that rely on alternative formulations of the index.⁷ Koshevoy and Mosler (1996) considered Lorenz zonoid as a multidimensional generalization of the one-dimensional Lorenz curve. A multidimensional Gini index can be calculated using the volume of the Lorenz zonoid. Arnold (1987), Koshevoy and Mosler (1997) and Anderson (2004) extended the Gini definition based on pairwise distances by proposing a multidimensional distance measure with the objective of measuring distances across the vectors of outcomes. (See also Koshevoy (1997) and Banerjee (2010).) List (1999) defined the compensation matrix associated with by replacing , the entry of , by for all pairs . The multidimensional Gini index introduced by List is defined as a normalized value of the one-dimensional Gini index applied to the vector of nonnegative real numbers, with a positive mean, generated by taking some nonnegative-valued, continuous, increasing, and strictly concave transformation of the rows of the compensation matrix. However, it is not clear if all such indices can be interpreted in terms of loss of welfare since they have not been related to any notion of social welfare.

2.3.2 The Inclusive Measure of Well-being Approach

In this subsection, we study several multidimensional inequality indices that are explicitly dependent on the inclusive measure of well-being approach. According to this approach, which Kolm (1977) refers to as the individualistic approach, a real number summarizing the well-being of a person in all dimensions of well-being is specified. These summary metrics of well-beings across persons are then combined using some aggregation rule to arrive at a level of well-being for the society as a whole. The society-level aggregated well-being is then employed to determine the extent of multidimensional inequality in the society.

As in Chapter 1, for any , represents the extent of well-being derived by person from possession of . For any , , we refer to as the portfolio of well-being standards of the population.

The job of a social welfare function here is to rank achievement matrices in terms of underlying well-being portfolios. More precisely, for all , , is regarded as at least as good as if and only if arrays the well-being portfolio associated as at least as good as that corresponding to (see Chakravarty and Lugo, 2016). This approach is, in fact, the welfarist approach, which involves aggregation of individual well-being indices (see Chapter 1).

One way of implementing the inclusive measure approach in the current context, is to specify a social welfare function , which is then employed to design the underlying inequality index. The first illustration we provide for this approach is the multidimensional Atkinson index, characterized by Tsui (1995).

Let us calculate the values for the homothetic welfare functions , characterized by Tsui (1995), where are of the forms (1.7) and (1.8). For the utility function in (1.7) we have, , from which it follows that . Similarly, for (1.8), we derive . In these specifications of , it is assumed that and are arbitrary.

By plugging these forms of into (2.12), we get the following functional forms of the Kolm inequality quantifier associated respectively with the welfare functions (1.7) and (1.8):

2.18a

and

2.18b

where and are arbitrary, for all , and is appropriately restricted so that it increases with respect to a correlation-increasing swap and decreases under a uniform majorization change. For instance, if , and should hold. This symmetric, normalized, population replication invariant relative index is the multidimensional Atkinson index of inequality. For , the formula coincides with the one-dimensional Atkinson (1970) index.

Lasso de la Vega et al. (2010) established that if we invoke the multidimensional transfer principle instead of the uniform majorization principle as the redistributive criterion, then the parametric restrictions in (2.18a) and (2.18b) become and for all . The restriction here is much simpler than that required in (2.18a).

Our subsequent illustration of the Kolm index is based on the double-class constant elasticity of substitution (CES) social welfare function proposed by Bosmans et al. (2015). Since some of the notation used in the current subsection are slightly different from that employed in Section 1.6, we explicitly mention the individual well-being metrics chosen by these authors. This also makes our discussion here self-contained.

The identical individual well-being standard considered is

2.19a

or,

2.19b

where , , and are arbitrary, for all , and . The parameter is related to the elasticity of substation between dimensions. The CES between any two dimensions is given by . The private welfare standards given by (2.19a) and (2.19b) are continuous, increasing, linear homogeneous, and strictly concave.

In each of (2.19a) and (2.19b), a CES aggregation is employed to arrive at the social well-being standard. The corresponding social welfare function is defined as

2.20a

or,

2.20b

At the second stage, another CES aggregator is invoked on the individual well-being levels for determining the social welfare function. These welfare standards are continuous, linear homogeneous, and strictly S-concave. They obey the strong Pareto principle as well.

For the welfare function defined by (2.20a), the appropriate form of satisfies the equation

which leads to

2.21

The resulting inequality quantifier turns out to be

2.22

and . (For reasons stated in the discussion after Eq. (1.17), we do not pursue the cases and further.)

The relative, symmetric, replication invariant Bosmans–Decancq–Ooghe multidimensional inequality index, given by (2.22), changes under a uniform majorization transformation to the correct direction and increases under a correlation-increasing switch if and only if .

Following Bosmans et al. (2015), we will now provide Graaff's (1977) decomposition of equality into efficiency and equity components for the general equality index and then apply it to (2.21). We follow the inclusive measure of well-being approach. For this, assume that both and are continuous, increasing in individual achievements, and linear homogeneous. Further, strict concavity and strict S-concavity are assumed respectively for the individual and social well-being standards. For any and , denote the dimensional vector of means of achievements in different dimensions by .

The level of equity associated with can be measured by the smallest fraction of the dimensional achievement totals in necessary to uphold the level of social welfare indicated by . Formally,

Observe that if we denote the distribution associated with by , then it is immediate that and . It, thus, follows that equity in the social matrix is obtained by applying the definition of to the matrix . Observe also that “ must coincide with ” in Definition 2.2 (Bosmans et al., 2015, p. 97). Combining and , we get . Since, in view of definition of , , it follows that .

The efficiency and equity components associated with (2.21) are given respectively by and .

The next illustration of this approach, we provide, is based on the Decancq and Lugo (2012) multidimensional S-Gini welfare function , defined by Eq. (1.19). For this, let us first substitute for in (1.19) and equate the resulting expression with the actual form of to obtain

from which we deduce the following form of the equality index:

2.23

Consequently, the Decancq–Lugo multidimensional S-Gini inequality index becomes

2.24

where and are arbitrary; is the rank of individual in the distribution of well-being levels , which are assumed to be nonincreasingly ordered; is the positive weight attached to his transformed achievement in dimension , ; and are parameters.

Decancq and Lugo (2012) proposed an alternative multidimensional S-Gini welfare function , defined by Eq. (1.18). The related Kolm multidimensional inequality measure, defined next, is not a member of the family that uses the direct approach since the underlying welfare evaluation is done in two stages. It is not a representative of the inclusive measure of well-being class as well. Instead, it relies on dimension-by-dimension dashboard formulation. However, we provide a discussion on this for the sake of completeness.

For the welfare standard specified in (1.18), it follows that

which produces

2.25

where and are arbitrary; is the rank of individual in nonincreasingly ordered permutation of the distribution ; , , and are the same as in (2.24). For , the two indices reduce the Donaldson and Weymark (1980) S-Gini inequality index. In general, one cannot claim that the two indices will make a distribution matrix equally unequal (see Decancq and Lugo, 2012, p. 733).

The next element of the inclusive measure of well-being category we scrutinize is the multidimensional Kolm–Pollak absolute index characterized by Tsui (1995). It corresponds to the utilitarian social welfare function , defined by Eq. (1.9). Recall that the underlying utility function is given by

where and are arbitrary, and is an arbitrary constant. The parameters and , , are chosen such that is increasing and strictly concave (see Chapter 1).

In consequence, must satisfy the equation

2.26

from which it follows that

2.27

Since the index in (2.27) coincides with univariate Kolm index if , it may be treated as the multidimensional analog of the Kolm (1976) index.

A variant of inclusive measure of well-being approach was suggested by Maasoumi (1986). He directly employed a two-stage aggregation procedure to suggest a multidimensional inequality index on the domain M₃. His approach relies on the theory of information. The index is constructed first by aggregating dimension-by-dimension achievements for each person and then applying an aggregation across persons. The dimensional achievements of individual i are aggregated first using a utility function , where the positive weights add up to 1 across dimensions, the parameter is related to substitutability between dimensions. For , we get the Cobb–Douglas utility function .

At the second stage Maasoumi (1986) employed Shorrocks (1980) generalized entropy-type aggregation on σ_is. That is, the multidimensional inequality index proposed by Maasoumi (1986) is given by

2.28

Here denotes the mean of the utility vector , and the parameter c reflects different perceptions of inequality. Dardanoni (1996) demonstrated that the Maasoumi multidimensional index given by (2.28) may not respond properly to a change desired under a uniform majorization operation. However, I_MM is symmetric, population replication invariant, and ratio-scale invariant.

Bourguignon (1999) suggested an inequality index that can be interpreted in terms of fraction of welfare lost through unequal distribution of achievements. In order to explain the role of interdimensional association, for simplicity of exposition, we assume at the outset that . The utility function associated with the symmetric utilitarian social welfare function is given by

2.29

where and are arbitrary, is an inequality sensitivity parameter, reflect the degree of substitutability between the two dimensions, and is the positive weight assigned to the achievement in the dimension , where .

The Bourguignon (1999) inequality index is defined as

2.30

When achievements are equally distributed across persons, welfare is maximized, and this value is given by . Consequently, in this case, inequality is 0. Inequality is positive for any distribution matrix, which is different from . Hence, determines the proportion of welfare loss arising from unequal distribution of achievements. This symmetric, population replication invariant, and uniformly majorized statistic satisfies the ratio-scale invariance axiom. For a correlation-increasing switch, condition to reduce inequality is . By strict quasiconcavity of U, we have and . Values of these parameters can be chosen appropriately to ensure increasing or decreasing inequality under a correlation-increasing switch. This is an advantage of this index. The flexibility of choice of parameters enables us to look at interdimensional association from different perspectives. One common feature of the functions (2.28) and (2.30) is that although they specify individual well-being levels at the first stage, their final aggregations do not permit us to interpret them in the inclusive measure of well-being framework directly.

2.4 Concluding Remarks

Several functional forms representing multivariate inequality have been suggested in the literature. This chapter may be regarded as an endeavor toward the presentation of an analytical scrutiny on them.

A great deal remains to be explored. There have been some efforts to study ordinal inequality in the univariate case.⁸ This area remains to be investigated in the multivariate context.

A second aspect of worth investigation is ordering of achievement matrices for a fixed number of dimensions with respect to multidimensional inequality. Since several multivariate indices have been suggested in the literature, ranking of two different matrices by two different indices may be of opposite directional (see Decancq and Lugo, 2012). Muller and Trannoy (2012) addressed the problem of multidimensional inequality comparisons from a compensation perspective. Under some well-defined notion of compensation, they developed ordering conditions in terms of second order stochastic dominance. A similar line of investigation here is to order achievement matrices in terms of inequality by taking into account some notion of redistributive criterion, interdimensional association and variability of the population size, and totals of achievements across dimensions.

Analysis of univariate inequality in an uncertain environment has been performed in the literature (see, e.g., Ben-Porath et al., 1997). Weymark (2006) argued that it is possible to develop similar analysis in the multivariate structure. In Chapter 6, we consider an uncertain framework and study expected poverty that results from the risk of an individual's dimensional achievements falling below corresponding threshold limits that are exogenously given.

One very important issue that we have not discussed in this chapter is inequality in the distribution of opportunities. An opportunity is assumed to be desirable in the sense that with greater opportunity, quality of a person's living condition does not go down. Reduction of disparities in the distribution of opportunities is an unquestioned doctrine of distributive justice.

Opportunities can be specified in various forms. For instance, an expansion in a person's income/wealth may increase his opportunities in many ways, say, greater access to higher education, better health care, higher chances for holiday trips, and so on. Evidently, an individual's opportunities are described by a set rather than by numbers, as specified in the person's multidimensional achievement profile.

Opportunities or advantages, which are also referred to as outcomes, can be circumstantial and noncircumstantial or effort-based (see Roemer, 1998; Weymark, 2003; Yalonetzky, 2012). Opportunities identified by the former category are given exogenously. These are the outcomes over which individuals cannot exert control. Examples include ethnicity, gender, religion, parental education. They all are circumstantial opportunities. Combinations of such outcomes determine types of individuals. Opportunities spotted by the latter category do not mean that their attainability by some individuals of a specific type makes them enjoyable by all persons of the type. Their collection constitutes the class of noncircumstantial opportunities. Examples are health status, education, and earning. Consequently, as a source of opportunity inequality, circumstantial opportunities should not be held responsible; they are ethically irrelevant to the measurement of opportunity inequality (see Hild and Voorhoeve, 2004 and Yalonetzky, 2012).⁹

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