Bread and Social Justice: Measurement of Inequality and Social Welfare Using Anthropometrics

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1 Bread and Social Justice: Measurement of Inequality and Social Welfare Using Anthropometrics Mohammad Abu-Zaineh and Ramses H Abul Naga January 11, 2017 Abstract Although abundant in household surveys, two defining properties of anthropometrics (survival thresholds and non-monotonicity) limit their application in studying social welfare and inequality. We examine the effect of working with welfare decreasing variables on properties of the Lorenz curve, the equally distributional equivalent value and the class of Atkinson-Kolm-Sen (AKS) inequality indices. This analysis leads us to introduce new variants of AKS indices and to apply a concave Lorenz and generalized Lorenz curves that are appropriate for welfare decreasing variables. When applied to a group of five Arab countries (Egypt, Morocco, Jordan, Comoros and Yemen), the findings reveal that the ordering of health distributions below and above the optimum health endowment are very different, highlighting the need to undertake separate distributional comparisons for those who may have insufficient calorie intake from those who may be consuming excess amounts of calories. Keywords: Distributional analysis for health data; Lorenz curve; Anthropometrics; Arab uprisings. Corresponding author: Aix-Marseille University, Aix-Marseille School of Economics (AMSE) & INSERM, IRD, SESSTIM, Centre de la Vieille Charité - 2, rue de la Charité Marseille - France. Business School and Health Economics Research Unit, University of Aberdeen, Scotland. 1

2 1 Introduction In their surveys of the relation between health and economic development, Strauss and Thomas (1998) and Deaton (2003) emphasize the positive correlation between stature, (typically adult height) and level of economic development. There is also a well-documented literature on the relation between stature and participation in developing country labour markets (Behrman, 1999). This literature suggests that it is not only calorie intake that increases with income in developing countries, but also the quality of diet as measured for instance by iron intake, protein consumption and intake (Thomas et al., 1996). One type of data used to inform about stature and aspects of nutritional status (especially in developing country surveys) comes under the heading of anthropometrics (Ravallion, 2016). The type of anthropometrics we are concerned with in this paper have two defining properties: (i) survival places lower and upper bounds on their domain of variation, and (ii) they exhibit a nonmonotonic, inverted U, relation with health (and well-being more generally). Other measures of health status share the same properties, examples include blood pressure, mid-upper arm circumference and torso length. These indicators are inexpensive to collect, may be measured to a high level of accuracy, and abound in developing country households surveys. These two defining characteristics have, however, limited the application of anthropometrics in studying social welfare and inequality in developing countries. Recently however Apablaza et al (2015) have worked out the necessary distributional dominance tools for poverty comparisons of non-monotonic variables such as the body mass index (BMI). In this paper, we complement the seminal contribution of Apablaza et al (2015) in three main directions. These extensions are entirely motivated by data considerations, i.e. the defining characteristics of anthropometrics as bounded variables that do not exhibit a monotonic relation with individual welfare. We first study the effect of bounding below by a and above by m an anthropometric welfare increasing indicator on (i) the Lorenz curve, (ii) the equally distributional equivalent endowment and (iii) the class of Atkinson-Kolm-Sen (AKS) inequality indices 1. The purpose is to clarify systematically why anthropometrics exhibit comparatively low levels of variation in comparison to income and expenditure data. Secondly, we study the effect of working with a welfare decreasing anthropometric indicator over a bounded interval [b; m] on the Lorenz curve, the equally distributed equivalent endowment, and the family of AKS inequality indices. Importantly, we show that routine application of standard AKS income inequality indices in the context of welfare decreasing variables results in these indices taking increasing values in relation to Pigou- Dalton progressive transfers (such transfers are intended to decrease the level of inequality.) We argue in particular that in the context of welfare decreasing variables it 1 In examination of convergence across rich and poor countries, there also arises the issue of working with bounded variables such as mortality rates and their effect on the world distribution of well-being (see for instance Seth and Yalonetzky, 2016). 2

3 is more appropriate to work with a new type of Lorenz curve where cumulating data is undertaken in reverse fashion, starting from the largest values (poor health) and summing to smaller values (better health). The reverse summation (i.e. from high to low values) associated with welfare decreasing variables entails an increasing and concave (rather than convex) Lorenz curve, that we simply call the concave Lorenz curve. Our third point of interest in the paper is to discuss the appropriate generalized curve in the context of welfare decreasing variables. Here, the concept of concave generalized Lorenz curve, GLC, introduced by Shorrocks (2009) for the measurement of unemployment duration will turn out to be a useful device 2. In particular, a situation of non-intersecting concave GL curves informs about a welfare ordering of distributions of a monotonic and decreasing welfare indicator. We note nonetheless this last result has an equivalent representation in terms of poverty deficit curves in Apablaza et al (2015). The tools we develop in this paper are particularly well suited to examine with separate lenses the distribution of malnutrition and overweight and their contributions to inequality and social welfare. In Section 2, we spell out the core elements and notations of our analytical framework. Section 3 deals with the problem of bounded variables. Section 4 focuses on the examination of the measurement of inequality with a welfare decreasing variable and Section 5 examines their use in the measurement of social welfare. The results of the paper are illustrated in Section 6 in the context of their application to anthropometric data pertaining to a group of five Arab countries. Specifically, we use data on body mass from Demographic Health Surveys (DHS) to measure social welfare and inequality in Egypt, Yemen, Jordan, Morocco, and the Comoros. When applied to this group of five Arab countries, the findings provide some support for the hypothesis that lack of bread and social justice may have contributed to the Yemeni revolt of In Egypt on the other hand, it would appear to be low levels of social justice rather than lack of bread that may have contributed to the outbreak of the Arab uprising. Section 7 concludes the paper. 2 Framework and notation Two defining features of anthropometrics are that (i) survival renders these variables bounded from below and above, and furthermore (ii) that these variables have a range of values entailing a positive association with welfare, and another range of values generating a negative association with welfare. We begin with a presentation of our conceptual framework and notation, and then we turn to these two defining properties of anthropometrics and dwell on their practical consequences in relation to the measurement of inequality and social welfare. We assume an observation z i on an anthropometric indicator is defined on one of two intervals: either z i [a, m] or z i [m, b]. These intervals divide the 2 Also in the context of measurement of bipolarization, Konsy and Yalonetzky (2016) introduce a reverse hybrid Lorenz curve, where incomes are accumulated from highest to lowest in a particular domain in the distribution. 3

4 anthropometric variable s domain of definition into two distinct subsets: [a, m] is the interval ranging from critically low values to the optimum 3, and [m, b] is the range pertaining to the optimum up to critically large values. In the context of sugar level for instance, the lower bound for survival is 40 milligrams of glucose per deciliter of blood, and the corresponding critical upper bound is b = 600 milligrams per deciliter (World Health Organization, 2016). In the context of body mass the lower bound is generally taken to be a = 10 kilograms per squared meter, b is approximately equal to 60, while m can generally be any values chosen in the interval of 18.5 to 24.9 (World Health Organization, 2004; 2016). It is important to note that a and b are reference points from which we measure welfare. To draw a distinction between the reference points a and b and the theoretical bounds on survival, we assume that the minimum possible value x can take for a surviving person is a + ɛ x,where 0 ɛ x < m a, and the maximum value is b ɛ y, where 0 ɛ y < b m, where ɛ x and ɛ y are parameter values chosen by the data analyst along side a, b and m. We partition a distribution Z = (z 1,..., z n ) into two separate vectors X and Y. Let n 1 and n 2 := n n 1 respectively denote the number of observations that belong to the two intervals [a, m] and [m, b]. We may now define two vectors X := (x 1,..., x n1 ) [a, m] n1 and Y = (y 1,..., y n2 ) [m, b] n2 and such that Z = [X Y ]. Consider two individuals i and j with nutritional status x i < m, y j > m and such that x i a = b y j. We take the view that despite the fact that x i a = b y j, the welfare levels of two individuals i and j, one being undernourished, and the other over-nourished are not meaningfully comparable. The approach we take in this paper is to measure a particular aspect of the distribution (i.e. inequality and welfare) separately over the two intervals [a, m] and [m, b], leading us to summarize Z using a two-dimensional vector of inequality indices and welfare indices, that we simply call the vector inequality index and the vector social welfare function. Likewise, when attempting to rank two distributions Z A = [X A Y A ] and Z B = [X B Y B ], say, in terms of inequality, we will only conclude that Z A is the more egalitarian distribution, if both X A is more egalitarian than X B and likewise Y A is more egalitarian that Y B. It will also be useful below to order the elements of the vectors X and Y in particular ways. Specifically, we may order the elements of the vector X in decreasing order such that x [1] x [2]... x [n1] or in ascending order such that x (1) x (2)... x (n1). Accordingly, we denote these ordered vectors using dedicated notation: X = (x [1],..., x [n1]) (1) X = (x (1),..., x (n1)) (2) A similar notation is used to order the n 2 elements of the vector Y. In this section and the next sections we dwell further on the framework constructed 3 For most anthropometrics, m, could be defined as a set or a range of values. As this does not raise unresolved conceptual problems in the context of our paper, we shall simplify our exposition by assuming that m is a single point (i.e., unique optimum health level). 4

5 by Apablaza et al. (2015) to measure poverty in relation to anthropometric data. Following these authors, there are three distinct elementary transformations of the distribution Z that contribute to improving social welfare: (i) a restricted Pigou-Dalton transfer on [a, m] n1 or a restricted Pigou-Dalton transfer on [m, b] n2, (ii) an increment of some x i on [a, m] n1 and finally (iii) a decrement (reduction) of some x i on [m, b] n2. Formally, let δ > 0, and consider two observations x l and x k such that x l δ x k + δ. A new distribution X = (x 1,..., x n 1 ) [a, m] n1 is obtained from X via a restricted Pigou-Dalton transfer if x l = x l δ, x k = x k + δ and x i = x i for all i l, k. Restricted Pigou-Dalton transfers on Y [m, b] n2 are similarly defined. X is obtained from X via a single increment if for some δ m x j, x j = x j + δ and x i = x i for all i j. Y = (y1,..., yn2) [m, b] n2 is obtained from Y via a unique decrement if for some δ y j m, yj = y j δ and yi = y i for all i j. Observe that increments and decrements modify the sum total of a distribution, whereas Pigou-Dalton transfers preserve the mean (and sum total) of the original distribution. In the sections below the focus is initially on inequality, where we deal specifically with Pigou-Dalton transfers. There it will be useful to distinguish between Atkinson-Kolm-Sen (AKS) inequality indices constructed in relation to resource concepts that are welfare increasing and those that decrease welfare. For this specific purpose, it useful to define the set of increasing and concave social valuation functions (in short utility functions) U + as well as the set of decreasing and concave social valuation functions V { } U + = u : R + R; u 0 and u 0 (3) { } V = v : R + R; v 0 and v 0 (4) We begin first with a discussion of the effects of bounding a resource variable on the Lorenz curve. 3 Lorenz curve of bounded health variables To deal with the problem of bounded variables, consider first the usual case where we measure inequality in relation to some variable x that is positively associated with welfare. Let µ denote the mean a distribution X [a, m] n1, and define the set { } 1 n 1 X µ = X [a, m] n1 : x i = µ (5) n 1 i=1 For some q [0, 1], it is possible to express the mean endowment of any distribution X in X µ as a weighted sum of the upper and lower bounds on the distribution so that µ = q(a + ɛ) + (1 q)m q [0, 1] (6) 5

6 The distributions that belong to the set X µ have a common defining feature. Firstly, they are (weakly) less equally distributed than the vector ˆX X µ given by ˆX = (µ,..., µ). Since ˆX is the least unequally distributed distribution in the set X µ, we adopt the notation ˆX = (X µ ) where is the mapping that associates with any distribution X X µ the most egalitarian redistribution (the least element) (X µ ) of the set X µ. Conversely, we define a map that associates with any X X µ its most inegalitarian redistribution (its greatest element) (X), and we define X = (X µ ). As such, when working with income data, the least egalitarian distribution has one person appropriating the entire cake, that is, x [1] = n 1 µ, while x [j] = 0 for all j = 2,..., n 1. In the context of a bounded anthropometric variable, survival dictates a limit to inequality: (X µ ) now has, n 1 q individuals with endowment (a + ɛ) and the remaining n 1 (1 q) individuals will have endowment m. To dwell further on the consequences of bounding a resource variable, consider the following hypothetical example associated with Table 1. Example 1 Let there be n 1 = 10 individuals, let a+ɛ = 10, m = and assume the mean level µ, associated with the set of distributions X µ, equals The least egalitarian distribution associated with any distribution X X µ is given by the distribution X 0 with five persons (q = 0.5) having resources a + ɛ = 10, and another five persons having resources m = Each of the distributions X 1 to X 5 are obtained by taking two extreme values (10, 24.9) and replacing them by the mean of the two values, namely (17.45, 17.45). The final distribution X 5 is equal to (X µ ), the least inegalitarian distribution of the set X µ. The Lorenz curves pertaining to this sequence of five distributions are sketched in Figure 1. This simple example illustrates the first defining result pertaining to the Lorenz curves of anthropometrics: the Lorenz curve of a bounded variable such as nutritional status lies within an envelope with upper limit the perfect equality line, and with lower limit the Lorenz curve of the least egalitarian distribution (X µ ), associated with the set of distributions X µ. Continuing with the above example, we report calculations of inequality using an Atkinson-Kolm-Sen family of inequality indices. Assume, as in Apablaza et al (2015), for individuals with endowments x [a, m] individual level nutritional attainment is measured using a family of increasing and concave utility functions, (x a) 1 α u α (x; a, m) =, α 0, α 1 1 α (7) ln(x a), α = 1 and let social welfare be measured by the average level of attainment in X : U α (X) = 1 n 1 u α (x i a) (8) n 1 i=1 Let x E [a, m] denote the equally distributed endowment in the vector X such that welfare is identical to the level of attainment in X. We have that 6

7 u α (x E ; a, m) = U α (X). We note that the social welfare in the distribution X is a simple increasing transformation of the equally distributed equivalent endowment. Accordingly, we define x E = f(x; α, a, m) as an anthropometric welfare index 4. In the context of the family of social welfare functions (8) f(x; α, a, m) is defined as follows: a + 1 n 1 (x i a) 1/(1 α), α 0 n f(x; α, a, m) = 1 i=1 a + exp 1 n 1 (9) ln(x i a), α = 1 n 1 It may be noted of course that f() is homogenous of degree 1 in X, a and m, and is increasing in Pigou-Dalton transfers. Furthermore, for any distribution X X µ, the equally distributed endowment satisfies the inequality a x EDE µ. However, given that the distribution of X cannot achieve a higher level of inequality than that associated with the least egalitarian distribution (X µ ), the greatest lower bound on x E is not a but instead the equally distributed equivalent endowment associated with the distribution (X µ ). In Table 1 for instance, we see that for α = 0.5, the least upper bound on x E, is given by f[ (X µ ); 1 2, a, m] = 14.49, rather than a = 9.8, while of course the upper bound for x E is given by the value µ = Similarly, for α = 2, the lower bound on x E is given by f[ (X µ ); 2, a, m] = 10.19, etc. We summarize the general results illustrated in the above example with the following Proposition. Proposition 1 Let x [a, m] denote a welfare increasing anthropometric variable. Let X µ denote the set of distributions in [a, m] n1 with mean endowment µ, as defined in (5), with least egalitarian distribution X = (X µ ), with most egalitarian distribution ˆX = (X µ ), and where Xand ˆX are distinct distributions. Let ɛ x [0, m a)denote a parameter, and let q [0, 1]such that µ = q(a+ɛ x )+(1 q)m. Then: (i) For any distribution X X µ, the Lorenz curve of X is located within an envelope of curves, with upper bound the perfect equality line, and with lower bound the Lorenz curve L[p; (X µ )] associated with the least egalitarian distribution of the set X µ, where 0 p q L(p; (X µ )) = i=1 (a + ɛ x ) p µ mp [m (a + ɛ x )]q µ q p 1 (10) 4 Note an interesting parallel with the definition of the achievement index in the literature on the measurement of socio-economic inequalities in health: Wagstaff (2002) defines the mean of the health distribution multiplied by one minus the concentration index (the latter is a measure of inequality) as the achievement index. Here, the equally distributed value is one minus the inequality index, multiplied by the mean, that defines the anthropometric welfare index. 7

8 (ii) For any distribution X X µ, the equally distributed equivalent endowment x E = f(x; α, a, m) satisfies the inequality a + ɛ f[ (X µ ); α, a, m] x E µ (11) Recall that in relation to the AKS family of inequality indices, we measure inequality, following Atkinson (1970), by the index ι 1 (X; α) = µ x E µ (12) In the light of the above results, it clearly follows that for any vector X [a, m] n1 the lowest bound on inequality is 0. For the upper bound on the AKS index, we may rearrange (11), to obtain the following rule of thumb: ι 1 (X; α) 1 a + ɛ m This inequality shows in a simple manner how the maximum value the AKS index takes relates to the bounds a and m defining the variable of interest. While the above upper bound on the level of inequality is easily calculated, it is also possible to derive alternative upper bounds from (11), that lend themselves to easier interpretations. An alternative upper bound on the AKS index may clearly be given by the value of the inequality index at the least egalitarian distribution (X µ ), a value that may well fall short of 1. We observe for instance in Example 1, that inequality is maximal at X 0 = (X µ ). The upper bound on the AKS index, (inequality evaluated at X 0 ) is equal to 0.17 when α = 0.5, and is equal to 0.42 when α = 2. In our applications section we also suggest to normalize the inequality index in the light of (ii) of Proposition 1, to work with an additional variant of the AKS index. Let x E denote the equivalent endowment associated with the least egalitarian distribution (X µ ): x E satisfies the identity x E = f[ (X µ ); α, a, m], and let x E = f(x; α, a, m). One possible normalization of the AKS index for the X distribution takes the form (13) ι N 1 (X; α) = µ x E µ x E (14) with the feature that for any vector X X µ the normalized index ι 1(X; α) takes values ranging between 0 (at the distribution (X µ ) and 1 (at the distribution (X µ )). Returning to Example 1, ι N 1 (X; α) now ranges between 0 at X 5 = (X µ ) and 1 at X 0 = (X µ ). At X 3 for instance, the AKS index takes the value 0.07, indicating that same social welfare level of X 3 could be reached with 93% of the mean endowment, if the distribution of resources were purely egalitarian. Additionally, we compute ι N 1 (X 3 ; 0.5) = 0.43, indicating that inequality reaches 43% of its maximum value. Computing the normalized index ι N 1 (X; α) alongside the AKS index provides useful additional information about the relative magnitude of inequality in a given distribution. 8

9 4 Measurement of inequality with a welfare decreasing variable We now turn our focus to the examination of the measurement of inequality in relation to the class of social valuation functions V. That is we examine the context of observations falling in the welfare decreasing range, where larger values of y are associated with lower values of welfare. It is important to reflect on the consequences of working with the class V in relation to the measurement of inequality and social welfare. The focus in this section is on the measurement of inequality: clearly the conventional Lorenz curve is still informative about inequality orderings in relation to welfare increasing or decreasing social valuation functions, as long as they remain Schur concave. In particular, the Lorenz curve may be used to investigate inequality orderings for any social welfare function constructed as average well-being in relation to valuation functions in the classes V or U +. Nonetheless, Unlike in the class U + where the equally distributed equivalent value is increasing in Pigou-Dalton transfers, we show below that the equally distributed endowment associated with the class V is decreasing in Pigou- Dalton transfers. For this reason, conventional AKS inequality indices (constructed as one minus the ratio of the equally distributed equivalent value to the mean) will be increasing in Pigou-Dalton transfers in relation to the class V (instead of being decreasing). Consider two ordered vectors associated with a distribution Y : the vector Y = (y [1],... y [n2]) is the decreasing rearrangement of Y while Y = (y (1),... y (n2)) is the increasing rearrangement of Y. The purpose of working with the decreasing rearrangement of Y is the following: if we want to maintain the well established and meaningful practice of summing resources starting from the worst off individuals, the analogue to summing incomes in increasing order is to sum y values in decreasing order. This point will become clearer in the next section when we investigate welfare orderings in relation to the class V. Associated with the distribution Y, we can construct two types of Lorenz curves: the conventional Lorenz curve associated with the vector Y, evaluated n 2p n 2 at some ordinate p [0, 1] may be defined as LC(p; Y ) = y (j) / y j, and it is also the case that LC(p; Y ) = LC(p; Y ). However, we may also construct a Lorenz curve in relation to the decreasing rearrangement Y, that we shall call the concave Lorenz curve: LC(p; Y ) = n 2p y [j] j=1 n 2 j=1 j=1 (15) y j j=1 9

10 Clearly, Y and Y have the same mean, and will belong to the set of distributions { } 1 n 2 Y γ = Y [m, b] n2 : y i = γ (16) n 2 i=1 Just as in the previous section, for some q [0, 1], it is possible to express the mean endowment of any distribution Y in Y γ as a weighted sum of the upper and lower bounds on the distribution so that γ = q(b ɛ y ) + (1 q)m q [0, 1] (17) The distributions that belong to the set Y γ have a common defining feature. Firstly, they are (weakly) less equally distributed than the distribution Ŷ Y γ given by Ŷ = (γ,..., γ). Since Ŷ is the most egalitarian distribution in the set Y γ, we write Ŷ = (Y γ) where (.) is the mapping that associates with a given distribution Y the most egalitarian redistribution (its least element) (Y γ ). To illustrate further the consequences of a welfare decreasing resource variable, we consider another hypothetical example, associated with Table 2. Example 2 Consider n 2 = 10 individuals, b = 60.2, ɛ y = 0.2, and m = 24.9 as in Example 1. Assume the mean level γ, associated with the set of distributions Y γ, equals The least egalitarian distribution associated with any distribution Y Y γ is given by the distribution Y 0 with five persons (q = 0.5) having resources b ɛ y = 60, and another five persons having resources m = Each of the distributions Y 1 to Y 5 are obtained by taking two extreme values (60, 24.9) and replacing them by their mean i.e. (42.45, 42.45). The final distribution Y 5 is equal to (Y γ ), the least inegalitarian distribution of the set Y γ. The Lorenz curves pertaining to this sequence of five distributions are sketched in Figure 2. There are three main points that the above simple example serves to illustrate: The concave Lorenz curve associated with the least egalitarian distribution lies above the equality line, and has a concave increasing shape as a result of summing values in decreasing order. Successive Pigou Dalton transfers displace the concave Lorenz curve downwards in the direction of the equality line. In Table 2, we calculate the equally distributed equivalent endowment using the family of decreasing and concave utility functions of Apablaza et al. (2015) (b y) 1 β, β 0, β 1 v β (y; b, m) = 1 β (18) ln(b y), β = 1 and we let social welfare be measured by the average level of attainment in Y : V β (Y ) = 1 n 2 v β (y i b) (19) n 2 i=1 10

11 Let y E [m, b] denote the equally distributed endowment in the distribution Y such that welfare is identical to the level of attainment in Y. We have that v β (y E ; m, b) = V β (Y ), where y E = g(y ; β, m, b), and the latter function is defined as follows: b 1 n 2 (b y j ) 1/(1 β), β 0, β 1 n 2 j=1 g(y ; β, m, b) = b exp 1 n 1 (20) ln(b y j ), β = 1 n 2 It may be noted of course that g() is homogenous of degree 1 in Y, m and b, meaning that if we rescale the Y values together with m and b, then the y E is rescaled accordingly. More importantly, it is to be noted that g() is decreasing in Pigou-Dalton transfers. Returning to Example 2, and taking for instance β = 0.5, the equally distributed equivalent endowment ranges from 50.0 for the least egalitarian distribution Y 0, and decreases as we undertake Pigou-Dalton transfers, until it equals the mean at Y 5, the most egalitarian distribution. Of course, as a result, Pigou-Dalton transfers will increase social welfare, as the function v β () is decreasing in y. When β = 0.5, social welfare stands at 0.50 at the distribution Y 0 and rises to reach 0.68 at the egalitarian distribution Y 5. Similarly, for β = 2, the equally distributed endowment y E ranges from = g(y 0, 2, m, b); to = g(y 5, 2, m, b), the mean value. The third point the example illustrates is the following: The equally distributed equivalent endowment y E is generally larger than the mean γ of the distribution, and is decreasing in Pigou-Dalton transfers. j=1 We summarize these results with the help of Proposition 2. Proposition 2 Let y [m, b] denote a welfare decreasing anthropometric variable. Let Y γ denote the set of distributions in [m, b] n2 with mean endowment γ, as defined in (16), with least egalitarian distribution Ỹ = (Y γ), with most egalitarian distribution Ŷ = (Y γ), and where Ỹ and Ŷ are distinct distributions. Let ɛ y [0, b m) denote a parameter, and q [0, 1] such that γ = q(b ɛ y ) + (1 q)m. Then, (i) for any distribution Y Y γ, the Lorenz curve of Y is an increasing and concave function located above the equality line. (ii) The Lorenz curve of Y Y γ lies within an envelope of curves, with upper bound the Lorenz curve L[p; (Y γ )] associated with the least egalitarian distribution of the set Y γ, where (b ɛ y ) p p q γ L(p; (Y γ )) = q(b ɛ y ) + (p q)m (21) p q γ and with lower bound the equality line. 11

12 (iii) For any distribution Y Y γ, the equally distributed equivalent endowment y E = g(y ; β, m, b) is decreasing in Pigou Dalton transfers and satisfies the inequality γ y E g[ (Y γ ); β, m, b] b ɛ y (22) In the light of (iii) it follows clearly that the standard AKS index of inequality (14) defined in relation to welfare increasing variables such as income will be increasing in Pigou-Dalton transfers in our current context of the Y distribution. For this reason, in the context of welfare decreasing variables we suggest instead to construct the AKS inequality index in the reciprocal form ι 2 (Y ; β) = y E γ y E (23) In this form, the index is now decreasing in Pigou-Dalton transfers on [m, b] as is required from an equality preferring inequality index. Because the Lorenz curve of Y is bounded, as indicated in (ii) of the above Proposition, we also suggest, as we previously did so in the context of the X distribution, to normalize the inequality index and to work with an additional variant of the AKS index (23). Let ỹ E denote the equally distributed equivalent endowment associated with the least egalitarian distribution: ỹ E satisfies the identity ỹ E = g[ (Y γ ); β, m, b], and let y E = g(y ; β, m, b). The normalized version of the AKS index for the Y distribution that we suggest to work with takes the form ι N 2 (Y ; β) = ( ye γ ỹ E γ ).ỹe y E (24) with the feature that for any vector Y Y γ the normalized index ι N 2 (Y ; β) takes values ranging between 0 (at the distribution (Y γ ) and 1 (at the distribution (Y γ )). The index ι N 2 (Y ; β) informs about the level of inequality the distribution Y attains, as a percentage of the maximum level of inequality associated with a distribution defined over [m, b] and with mean γ. 5 Measurement of social welfare and the concave generalized Lorenz curve Having discussed the implications of anthropometric variables for the measurement of inequality, we examine their use in the measurement of social welfare. In this section it is necessary as above to distinguish between social welfare functions constructed in relation to the class U + from those constructed in relation to the class V, and we refer the reader to Apablaza et al (2015) for a complete discussion in the context of poverty of measurement. The focus here is on the measurement of social welfare in relation to welfare decreasing variables, where it was shown by Shorrocks (2009, 1983) that : 12

13 While the conventional generalized Lorenz curve is still informative about welfare orderings in relation to the class U +, it is no longer informative about welfare orderings in relation to the class of social valuation functions V. For the purpose of investigating welfare orderings in the class of welfare decreasing variables V, we adopt the concave generalized Lorenz curve (the concave Lorenz curve multiplied by the mean of the distribution). Specifically, we shall show that: In comparing two vectors Y A and Y B (with possibly different mean endowment levels) social welfare is higher in Y B than Y A for all decreasing and equality preferring social welfare functions if and only if the concave generalized Lorenz curve of Y B lies below the concave generalized Lorenz curve of Y A. Multiplying the concave Lorenz curve by the mean γ of a distribution Y, we obtain the concave generalized Lorenz curve of Y as follows: n 2p Ψ(p; Y ) = y [j)] (25) The function Ψ(p; Y ) is similar to the generalized Lorenz curve, with the important difference that the summation of the elements is undertaken in reverse fashion, from the largest to the smallest value of the distribution Y. Clearly, the conventional generalized Lorenz curve associated with the vector Y is the function Ψ(p; Y ), with the summation of the elements undertaken from the smallest to the largest y value. Returning to Example 2, the concave generalized Lorenz curves pertaining to the five distributions Y 0 to Y 5 will be ordered in the same manner as their concave Lorenz curves, as all five distributions have the same mean. It follows therefore that the distribution Y 5 with the bottom concave Lorenz curve is associated with the highest welfare level. The welfare ordering of distributions in going from Y 0 to Y 5 is achieved by undertaking successive Pigou-Dalton transfers. More generally, when a pair of distributions Y A and Y B have different mean endowment levels and the concave generalized curve of Y B lies below that of Y A, Y B may be obtained from Y A using a sequence of Pigou-Dalton transfers and or decrements (as defined in Section 2). The following result is a straightforward extension of Shorrocks (2009) Proposition 3 Let Y A, Y B [m, b] n2 denote two distributions. Then the following three statements are equivalent: j=1 (i) Ψ(p; Y A ) Ψ(p, Y B ) for all p [0, 1], (ii) In the family of social welfare functions (19), V β (Y A ) V β (Y B ) for all β 0, (iii) Y B is obtained from Y A via a finite sequence of Pigou-Dalton transfers, and or decrements. 13

14 6 From the lower and upper tails to the distributional comparisons A remark is due before we turn to our empirical examination of inequality and social welfare in a group of Arab countries. We have thus far discussed separately the question of distributional comparisons in the X and Y domains that are below and above the optimum value, m. As argued earlier we do not assume that well-being is meaningfully comparable below and above the optimum value, m, in the context of anthropometrics. In turn, this entails that when we cardinalize social preferences with a given pair of (α, β) values, any given social welfare or inequality vector does not entail a complete ordering of distributions Z. To observe that the resulting order is incomplete, consider two hypothetical distributions Z A and Z B such that the welfare of those below the optimum is higher in distribution Z A, but the welfare of those above the optimum is higher in Z B. Such patterns do not allow us to rank the distributions Z A and Z B in terms of welfare. These patterns will be encountered in the section below when we compare the distribution of body mass pertaining to Egypt and Yemen with those of other countries. For this reason, if it is deemed necessary to compare the entire distribution of well-being across two countries, we are led to construct vectors of social welfare and inequality indices. Formally, this would lead us to conclude that welfare is higher in country A than in country B if and only if the generalized Lorenz curve X A lies above the GLC of Y A and the concave GLC of Y A lies below that of GLC of Y B. 7 Illustrative application: Inequality and social welfare in five Arab countries. Our illustrative application makes use of anthropometric data on adult (nonpregnant) women of reproductive age (15 to 49). The analysis is performed using data from the latest available Demographic and Health Surveys (DHS) conducted in five Arab countries: Egypt (2015), Yemen (2013), Jordan (2012), Comoros (2012) and Morocco (2004). The BMI is calculated by the authors as the weight in kilograms divided by the square of height in meters. The proposed methodology demands that we set (survival thresholds) values for the lower and upper bounds of the BMI as well as a cut-off point (or an optimal value) that separates the lower- and upper-tail distributions. Given that there is no consensus on these values in the literature, with the proposed range being often between 10.0 and We set the value of the lower bound to be equal to a = 9.80 and upper bounds b = (with ɛ x = 0.2 = ɛ y ) while the cut-off (or optimal) value, m, is fixed at This is in line with the recommendation of the World Health Organization (2004; 2016) suggesting that a BMI above indicates overweight. As such, X, the lower tail distribution, is used to measure 14

15 inequality and social welfare of the undernourished sub-sample (including those who are classified severely; moderately, and mildly thin) up to the normal weight (i.e., normal nutritional status). The upper tail distribution Y then is used to measure inequality and social welfare in relation to individuals ranging from optimum body mass to excess body mass. After cleaning the data from missing and miscoded values on the variable of interest, the respective sample sizes are as follows: Egypt (n = 7187), Jordan (n = 11080), Morocco (n = 16916), Yemen (n = 22947) and Comoros (n = 5083). 7.1 The lower tail distribution We begin by examining social welfare in the five countries. The convex generalized Lorenz curves pertaining to the lower tail distributions of the five countries are plotted in Figure 3. To read the findings, we can observe that the generalized Lorenz curve of the X distribution with highest social welfare would take the form of a straight line starting at zero with a slope equals m. In turn, this would entail the mean of the distribution µ approaching m = 24.90, the anthropometric social welfare index, x E, approaching µ and the inequality index, ι 1 (X; α), approaching zero. We can view the process of improvement in the distribution of undernutrition as one where the generalized Lorenz curve approaches from below the generalized Lorenz curve of the optimum distribution X = (m,..., m). We report in Table 3 results pertaining to inequality and social welfare indices in relation to two values for the inequality aversion parameter: α = 1 and 2. The anthropometric social welfare index x E, ranks social welfare as highest in Egypt (21.829), followed by Morocco (21.233), Comoros (21.186), Jordan (21.184), with social welfare being lowest in Yemen (19.627). Note that the mean µ ranks the five countries in the same fashion. An inspection of generalized Lorenz curves is useful to investigate the robustness of these findings to alternative parametrizations of the social welfare functions. As the curves of the five distributions exhibit intersections in the bottom three deciles, it is theoretically plausible to find alternative rankings of the five countries in relation for instance to large values of alpha. We turn to inequality in the lower tail distribution. Consider first the classic version of the AKS index. A computation of AKS indices, ι 1 (X; 1), reveals that inequality is less than 2.0%, and is highest in Yemen and lowest in Egypt. Recall however from Proposition 1 that measuring the variable of interest on the domain [a, m] restricts the maximum value the AKS index can take. In this sense, computing the normalized variant of the AKS indices, ι N 1 (X; α), provides additional insights. For α = 1, inequality in Yemen as measured by the normalized index, is the highest at approximately 5.3%. This is followed by 3.8% in the Comoros and Jordan, while it is the lowest in Morocco and Egypt (4.7% and 3.3%, respectively). Seen from this perspective, the results suggest that inequality in the lower tail of the distribution of body mass is far from being insignificant. We discuss more briefly the findings related to α = 3 (see the second panel 15

16 of Table 3) where similar trends can generally be observed. Inequality remains highest in Yemen, with the normalized variant of the AKS index, ι N 1 (X; 3) standing at 14.4% of its maximum value, and lowest in Egypt (5.4%). The welfare ranking of the countries, as measured by the anthropometric social welfare index, x E, remains the same, with social welfare being highest in Egypt (x E = 21.37) and lowest in Yemen (x E = 18.57). We summarize this discussion by observing that the findings set Yemen apart from the other four countries, in that Yemen has a comparatively low level of social welfare and higher inequality as measured by the two variant of the AKS index. 7.2 The upper tail distribution We now turn to the upper tail [m, b] of the distribution. To interpret the results reported in Table 4, it may conceptually be useful to think of economic development in this context as the process involving the concave generalized Lorenz curve approaching (from above) a straight line starting at zero, with a slope equal to m, and inequality going to zero. This straight line is the concave generalized Lorenz curve of the upper tail distribution Y = (m,..., m). To begin with, it is worth noting that the mean of the distribution is highest in Egypt (32.4) and lowest in Morocco (28.8). Consider first the results pertaining to β = 1. Recalling here that welfare is decreasing in y, we find that the anthropometric welfare index y E ranks social welfare as lowest in Egypt (y E = 33.17) and highest in Morocco (y E =29.06). For β = 1, we also find that y E equals 31.6 in Jordan, 29.7 in the Comoros, 29.6 in Yemen. Turning now to inequality, we find that the AKS index for welfare decreasing variables,ι 2 (Y ; β = 1), takes the highest value in the context of Egypt: 2.3%. In contrast, this same figure stands at 0.8% in the context of Morocco, and inequality is between the two values in the context of the other three countries. Once again, the magnitudes of inequalities in the upper tail of the distribution of body mass appear to be higher when we consider the normalized variant of the AKS indices, ι N 2 (Y ; β). For β = 1, inequality in Egypt, is the highest at approximately 7.1%. This is followed by 5.4% in Jordan and 4.1% in the Comoros, while it is the lowest in Morocco (about 3.0%). In accordance with the above results, the concave generalized Lorenz curve of Egypt lies above the other four curves (see Figure 4). The corresponding curve for Jordan lies above that of Morocco, Yemen and the Comoros, with the remaining three concave generalized Lorenz curves crossing at the bottom deciles of the Y distribution. The findings for social welfare and inequality pertaining to β = 3 are qualitatively similar to those discussed above (see the second panel of Table 4). 7.3 A tentative overview It is tempting to draw together the summary statistics of the lower and upper tail distributions (social welfare and inequality) in synthetic diagrams. In Figure 5 pertaining to the lower tail distribution, we plot on the horizontal axis the level 16

17 of equality (i.e. one minus the normalized AKS index) and on the vertical axis the ratio µ/m of the mean to the optimum value. The latter ratio is intended to capture average health attainment level. In the context of Figure 6, pertaining to the upper tail distribution, the attainment ratio takes the reciprocal form m/µ and the equality is measured by one minus the normalized AKS inequality index for welfare decreasing variables. Figure 5 exhibits a tentative ranking of the five countries under consideration in terms of the level of equality (calculated in relation to the aversion parameter set at α = 3), versus the corresponding attainment ratio µ/m. The diagram reveals clearly that, Morocco, Jordan and the Comoros display very similar distributional patterns with the attainment ratio, being in the range of 0.86 and the level of equality being in the range of 92 to 93%. It is interesting to note that Egypt and Yemen stand well apart from these three countries. Specifically, while Egypt has a comparable attainment ratio (88%) it achieves substantially more equality (97%) than the group of three countries. On the other hand, Yemen has a considerably lower health attainment ratio (80%), and furthermore less equality (80.6%) than the other three countries. Turning to the upper tail distribution, Figure 6 exhibits the relevant ranking of the five countries under consideration in terms of the level of equality (calculated in relation to the aversion parameter set at β = 3), versus the corresponding attainment ratio m/µ. The diagram reveals more salient differences between the countries, with Egypt now moving from top to bottom and Morocco ranking top in terms of both the equality and attainment dimensions. Jordan has a more favorable distribution than Egypt, and both Yemen and the Comoros fair better than Jordan in terms of equality and attainment. Overall, we have not found any pairwise comparisons of countries where country A had a higher level of inequality and attainment in the upper and lower tails of the distributions. Nonetheless, we note that the cardinal welfare rankings of the five countries in figures 5 and 6 are in broad agreement with the ordinal the rankings of the countries in terms of convex and concave generalized Lorenz curves. 8 Conclusion Although abundant in household surveys, the use of anthropometrics in the study of social welfare and inequality has so far been limited. The type of anthropometrics we examined in this paper had two defining properties: survival places lower and upper bounds on their domain of variation, and they exhibit a non-monotonic, inverted U, relation with well-being. We have studied the effect of bounding an anthropometric welfare increasing indicator on the Lorenz curve, the equally distributional equivalent endowment and the class of Atkinson-Kolm-Sen (AKS) inequality indices. The purpose of this analysis was to clarify systematically why anthropometrics exhibit comparatively low levels of variation in comparison to income and expenditure data. We then have turned our attention to studying the effect of working with a 17

18 welfare decreasing anthropometric indicator over a bounded interval, on these same tools of distributional analysis. Importantly, we have shown that routine application of classic AKS income inequality measures, in the context of welfare decreasing variables, results in these indices taking increasing values in relation to Pigou-Dalton progressive transfers. This analysis has therefore lead us to introduce AKS inequality indices that are appropriate for welfare decreasing variables. We have also argued that, in the context of welfare decreasing variables, it is more appropriate to work with a new type of Lorenz curve where cumulating data is undertaken in reverse fashion, starting from the largest values (poor health) and summing to smaller values (better health). The reverse summation has entailed an increasing and concave (rather than convex) Lorenz curve, that we have called the concave Lorenz curve. Our third point of interest in the paper was to highlight the relevance of the results of Shorrocks (2009) and Apablaza et al. (2015) in the context of welfare decreasing anthropometric variables. Accordingly, we have shown that a situation of non-intersecting concave generalized Lorenz curves informs about a welfare ordering of distributions of a monotonic and decreasing welfare indicator. When applied to a group of five Arab countries (Egypt, Morocco, Jordan, Comoros and Yemen), the findings of the paper provided some support for the hypothesis that lack of bread and social justice may have contributed to the Yemeni revolt of 2011 (Zurayk and Gough, 2014). In Egypt on the other hand, it would appear to have been low levels of social justice rather than lack of bread that may have contributed to the outbreak of the Arab uprising (Kadri, 2014). 9 Appendix This appendix contains proofs of statement (iii) of Proposition 2 and Proposition 3. We begin with a definition of a differentiable Schur-convex function. A real-valued function φ(y 1,..., y n ) defined on a set S R n and differentiable on the interior of S is Schur-convex if for all y i and y j the inequality ( φ (y i y j ) φ ) 0 (26) y i y j is satisfied (see Theorem A.3. of Marshall et al. 2011, p.83). Let Y A = (y A 1,..., y A n ) and Y B = (y B 1,..., y B n ) denote two vectors in S. We say that Y B is weakly majorized by Y A, written Y B w Y A if the following n inequalities are satisfied: k y[i] B i=1 k y[i] A k = 1,..., n (27) i=1 Observe then that the concave generalized Lorenz curve (25) is a graphical device for representing the weak majorization ordering. 18

19 Proof of Proposition 2 Statements (i) and (ii) are easily established by noting that we undertake summations from larger to smaller values. For (iii) the proof consists in showing that (20) is a Schur-convex function. On the basis of-remark 3.A.5 of Marshall et al (2010 p. 85), in showing that (20) is Schurconvex, without loss of generality we may readily consider the case of n 2 = 2 individuals with endowments b > y 1 > y 2 > m. Consider first the case β = 1, and define the function [ ] ln(b y1 ) + ln(b y 2 ) φ β (y 1, y 2 ) = b exp (28) 2 Differentiating φ β with respect to y i, we obtain: [ ] φ β ln(b y1 ) + ln(b y 2 ) 1 = exp y i 2 2(b y i ) (29) Accordingly, it follows that φ β / y 1 φ β / y 2 = b y 2 b y 1 > 1 (30) Hence, φ β (y 1, y 2 ) satisfies the inequality (26) and is a Schur-convex function. Equivalently, φ β (y 1, y 2 ) decreases in Pigou-Dalton transfers. For the case β 1 we follow the above steps to establish that φ β (y 1, y 2 ) is Schur-convex. We thus conclude that the equally distributed equivalent endowment y E of (20) decreases in Pigou-Dalton transfers. Proof of Proposition 3 The equivalence between statements (ii) and (iii) is established in Apablaza et al. (2015). We thus show that (i) (ii). Let Y A and Y B denote two vectors in S. From Theorem 3A8 of Marshall et al. (2010, p. 87), the function φ : S R satisfies the property Y B w Y A = φ(y B ) φ(y A ) (31) if and only if the function φ is increasing and Schur-convex on S. Equivalently, φ satisfies the property Y B w Y A = φ(y A ) φ(y B ) if and only if φ is a decreasing and Schur-concave function on S. Any social welfare function constructed as a weighted sum of functions in the class (4) being decreasing and Schur-concave, we also have that in (19) V β satisfies the property Y B w Y A = V β (Y A ) V β (Y B ) if and only if β References Apablaza M., Bresson, F., Yalonetzky, G., 2016, When more does not necessarily mean better: Health-related illfare comparisons with nonmonotone welbeing relationship. Review of Income and Wealth. Version of Record online: 18 NOV DOI: /roiw