On a multidimensional diversity ranking with ordinal variables

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1 On a multidimensional diversity ranking with ordinal variables Mariateresa Ciommi, Casilda Lasso de la Vega, Ernesto Savaglio VERY PRELIMINAR VERSION - JUNE 2013 Abstract In the last ten years, the measurement of diversity has become an increasingly important issue in the literature. On measuring diversity, two different approaches have been proposed. The first one assumes that the individuals in the population are exogenously classified into groups. Then diversity relies on the percentage of individuals belonging to each group. The second one defines diversity as the aggregation of diversity between any two individuals by introducing a distance between pairs. This paper assumes this viewpoint. An axiomatic characterization of a family of diversity ordering based on the counting approach is carried out. These orderings are founded on two very straightforward axioms: monotonicity and α separability. Finally, an empirical application based on EU-SILC data for 15 European countries for 2004 and 2009 completes the analysis of diversity. JEL Classification: I30, I32, D63 Key Words: Diversity measurement, counting approach, ordinal and categorical data, orderings. 1 Introduction In the last years, the definition and the measurement of diversity has became an increasingly important issue in the literature. Several indices have been proposed in different contexts: biology, sociology, cultural, economics, and so on. But is diversity good or bad? The answer is not unambiguous and depends on the context. In general, in a society diversity would be m.ciommi@univpm.it Department of Economic and Social Sciences, Università Politecnica delle Marche IT) casilda.lassodelavega@ehu.es - University of the Basque Country U.P.V./E.H.U. and BRIDGE Research Group ES). ernesto@unich.it Department of Economics University G.d Annunzio Chieti-Pescara IT). 1

2 desirable. In fact people as well a groups would obtain benefit from having different skills, different talents, different options or different points of view. For instance, as suggested by Bossert, Pattanaik and Xu [9], in the case of political options for the voters in a country, having diversity means have more options to choose and then, more freedom. In this case, divesity is something to preserve. However, in a context of integration we are interesting in eliminating diversity. In addition, depending on the specific context, the term diversity has different interpretations. For instance, Biologists use this word to denote the number of variety of species present in a region and their spatial distribution.whereas Sociologists define the diversity of a society as the probability that two randomly selected group members belong to the same category. Measuring diversity has always been a challenging task for both economic theorists and policy makers. In particular, the role of ethnic, cultural or social diversity with respect to economic performance has received increasing attention from economists in recent years See, Alesina and La Ferrara [2] for a review of the literature). For us, diversity focuses on the differences of endowments among individuals. We affi rm that two individuals are similar if they have the same bundle of attributes otherwise there exists a degree of diversity among them. Then, the purpose of this work is to investigate some of the theoretical approaches proposed in the study of diversity and introduce a new role for ranking multivariate distributions in terms of their diversity. In fact, the measurement of diversity over time enables us to evaluate the impact of the implementation of a certain policy and helps policy makers to address the best strategy to contribute to the creation of a society more cohesive. This paper is related to several strands of the literature. Firstly, it is naturally related to the theoretical economic literature on the measurement of diversity. In this framework many scholars have made important contributions: Weitzman [21] derives the diversity of a set from pairwise dissimilarity between its elements. His work is based on the primitive notion of a cardinal numeric measure of distance between living creatures. Nehering and Puppe [13] generalize this measure; in their work, they propose deriving the Weitzman s distance functions from an a prior grouping of the objects into collections of weighted attributes. Afterwards, Bossert, Pattanaik and Xu [9] propose an axiomatic characterization of Weitzman s work. After this pioneer work, many researchers have been interested in measuring diversity based on an ordinal distance. Pattanaik and Xu [14] and Bossert, Pattanaik and Xu [9] provide and improve a distance function that establishes if objects are similar or dissimilar, and measures the diversity of different sets. Bervoets and Gravel [6] provide an axiomatic characterization of two different ways of ranking sets on the basis of the diversity that they offer. Finally, Pattanaik and Xu 2

3 [15] suggest an ordinal distance function to develop a notion of dominance between sets, and to characterize a specific ranking rule, belonging to a class of rules, that satisfies the property of dominance, for use in ranking sets in terms of diversity. All these contributions take into account a prior notion of proximity or dissimilarity. This work also relates to the literature on ethnic diversity. In this field two papers play, in our opinion, a relevant role: on the one hand, the works by Alesina, Devleeschauwer, Easterly and Kurlat [1] and by Bossert, D Ambrosio and La Ferrara [7] on the other. The first one proposes three measures of fractionalizations: one linguistic, one related to religion and the last one that combines skin color and language. The importance of this paper is that combines qualitative data. The second paper characterizes a more general version of the ethno-linguistic fractionalization index. In particular, the index proposed in our work presents some strong connections with the general formulation of the Bossert, D Ambrosio and La Ferrara index. The aim of the paper is to define and axiomatically characterize a family of counting diversity ordering induced by a multidimensional index of diversity. The family of diversity indices that induces the orderings are based on the number of attributes in which individuals differ. So, we proposed to apply in a diversity context the well-known counting approach introduced by Atkinson [5] in a context of deprivation, and applied by Chakravarty and D Ambrosio [10] and Bossert, D Ambrosio and Peragine [8] in a context of social exclusion, and by Alkire and Foster [3] and Lasso de la Vega [12] in a multidimensional poverty context. The counting approach is a suitable method when the variables are ordinal and categorical. We observe that most of the data available on social conditions, for instance of individuals in a region or country, involve categorical or ordinal variables. On the other hand, to our knowledge, the diversity indices proposed in the literature work well only with quantitative variables. So doing, we have showed that the counting approach also fits well in the diversity context. The paper is organized as follows: Section 2 introduces basic notions and definitions. In particular it shows that the proposed index reduces to the Simpon Diversity index when the number of attributes is equal to one. In addition a discussion of the two axioms characterizing the ordering is provided. Section 3 presents the main results. Section 4 provides an empirical application by comparing several European Countries in two different years. Finally, section 5 concludes the paper. 2 Notation and definitions We consider individuals who differ in gender, political choice, religion, level of education, mother tongue, race, etc. and we compute how much they are dissimilar. Analytically, this means to 3

4 compare discrete multidimensional distributions of ordinal or categorical variables in terms of diversity. A discrete multidimensional distribution of k 1 personal attributes on a finite population of n 2 individuals can be represented as a n k matrix X = [x ij ] whose generic entry x i,j denotes the quantity of the j-th attribute allocated to the i-th individual. We denote with M n, k) Z n,k + the set of all n k matrices, we say that the j-th column of a matrix X M n, k) is the marginal) distribution of the j-th attribute among n people and that its i-th row represents the set of personal attributes of individual i, namely her endowment. We proceed with our analysis of diversity by first introducing a measure of the individual similarity which, for any X M n, k) and for any pair of individuals i, j {1, 2,..., n}, computes the number c ij of attributes in which two individuals i and j do not differ, namely: c ij = {h x ih = x jh for h = 1, 2,..., k}. We observe that c ij = 1, 2,..., k, c ii = k and c ij = c ji for any i and j. The computation of c ij for all i, j allows us to construct the so-called coincidence matrix C X = [c i,j ] i,,...,n consisting in a n n symmetric matrix, with all k on the main diagonal and whose generic element is a measure of the similarity of any pair of individuals in the population under scrutiny. denote with C = X M n,k) C X the set of all the feasible coincidence matrices. We collect all the information provided by a coincidence matrix in a function S n αx, n, k) : C [0, 1] that associates to each C X a real number in the range [0, 1] according to the following rule: S n αx, n, k) = 1 1 n 2 k α n i=1 We n c α i,j with α N {0, 1}. 1) Expression 1) identifies an aggregate measure of diversity for a society X of n individuals endowed with k attributes. We refer to 1) as a class of Multidimensional Counting Diversity Indices hereafter MCDI). Remark 1. It is worth noticing that for k = 1 expression 1) entails S n X, n, k) = 1 n c ij for any α. Since n n i=1 c ij = 2 N n in i 1) i=1 2 + n = N i=1 [n in i 1)] + 1 n n 2 i=1 N i=1 n i = N i=1 n2 i, we obtain: S n X, n, 1) = 1 1 n 2 that is the celebrated Simpson Diversity Index, 1 N i=1 n 2 i 2) with N that denotes the number of groups n 1, n 2,..., n N ) in which a population of n individuals is partitioned according to the the k-th 1 The diversity index of Simpson [19]) was originally introduced as a measure of concentration and it is now largely used as a diversity measure. In biology, it is a dominance index because it gives more weight to the common or dominant species. It takes into account both species richness it measures the number of different kinds of organisms) and evenness of abundance it compares the similarity of the population size for all species). 4

5 attribute. As long as MCDI reduces to 2) for k = 1, the class of multidimensional diversity indices proposed here could be interpreted as an extension of the Simpson diversity index to the case in which individuals differ for more than one characteristic. In what follows, we axiomatically characterize the diversity ordering d that, for analogy with the measure 1), we call Multidimensional Counting Diversity Ordering MCDO). The MCDO is defined on multidimensional discrete distributions and it is induced by the computation of the MC DI, namely: 2 Definition 1 For any X M n, k) and Y M n, k ) X d Y if and only if S n αx, n, k) S n αy, n, k ) 3) with α N \ {0, 1}. In words, we say that a society X is at least as much multidimensionally) diverse as a society Y if the sum of the number of attributes in which any pair of individuals i, j {1, 2,..., n} coincide in the former is not greater than that in the latter. Let us take the following order-preserving transformation: and let us define with d ij = k α c α ij i and j, 3 S α X, n, k) = n 2 k α S n αx, n, k) = n 2 k α n i=1 n c α ij 4) the measure of the level of diversity between individual and with D X = [d ij ] Z n n + the corresponding diversity matrix. We denote with s X n = s X 1, s X 2,..., s X n ) T the n-th dimensional column vector or diversity profile whose components s X i, the so-called diversity scores, represent the diversity of an individual with respect to the society in which she lives, namely: s X i n n = nk α c α ij = k α c α ij) n = d ij. 5) So, expression 1) can be restated as: n n n n S α X, n, k) = n 2 k α c α ij = k α c α ij) n n = d ij, 6) and the MCDO defined in 3) reduces to: i=1 i=1 i=1 X d Y if and only if S α X, n, k) S α Y, n, k ). 7) 2 The symmetric and asymmetric component of d are donoted with d and d respectively. 3 Since the value of d ij depends on α, we refer to d ij as a α-dependent measure of diversity or simply a measure of the level of diversity between individuals. 5

6 Thus, according to condition 7, in a society X there is at least as much multidimensional) diversity as in the society Y if and only if the sum of the individual diversity d ij for any pair i, j {1, 2,..., n} in the former is not smaller than the corresponding sum in the latter. The axiomatic characterization of the diversity criterion underlying 7) involves only two very compelling properties. We denote with M n, k), ) a binary relational system such that, for any X, Y M n, k), X Y means that distribution X shows at least as much diversity as distribution Y. 4 first plausible axiom for a diversity ranking claims that two identical societies are equivalent in terms of diversity, however if we add one individual to a society, then the diversity should not decrease. Analitically: Axiom 1 M - Monotonicity) For all X, Y M n, k) if x i,j = y i,j for any i = 1,..., n and j = 1,..., k, then X Y. For all X,Y such that Y = X\ [x i ] with x i Z k +, X Y. We consider now a society Z M n, k) to which we add a group of identical individuals g in order to obtain a new distribution matrix Z = Z {g} with a populazion size of n + n g. If we compute the diversity matrix for Z, we obtain the following block matrix: G Z The D Z = D Z 8) G T Z 0 where the block 0 is the null matrix representing the diversity among individuals in the group g, G Z is the matrix whose elements express the diversity between individuals in Z and in g and G T Z is the transpose of G Z. 5 Computing the diversity scores s Z i in which individual i belongs to society Z: s Z i = n+n g k α c α ) i,j for i [1, 2,..., n] from the case in which she belongs to the new group g: s Z i = n+n g Since all the individuals in g are identical then equation 9) becomes: s Z i = n n+n k α c α ) g i,j + j=n+1 for Z, we distinguish the case k α c α ) i,j for i [n + 1, n + 2,..., n + ng ] 9) n+n g j=n+1 k α c α i,j) = n ) k α c α i,j = 0 for i [n + 1, n + 2,..., n + ng ], k α c α ) i,j for i [n + 1, n + 2,..., n + ng ]. 4 The symmetric ) and asymmetric ) part of have the usual meaning. 5 Notice that all the columns of G Z and of course all the rows of G T Z ) are equal. 6

7 Thus, the aggregate measure of multidimensional diversity for the individuals in the group g with respect to the individuals in the society Z is: n n g s Z i = n g k α c α i,j) for i [n + 1, n + 2,..., n + n g ]. Finally, if X and Y are two distribution matrices obtained by adding two groups of identical individuals g and f to X M n, k) and Y M n, k) respectively and if n g s X i = n f s Y for i g and j f, i.e. the value of the multidimensional) diversity for the two groups of individuals added to X and Y is the same, then we require that the multidimensional diversity ranking of X and Y be consistent with that of X and Y, namely: Axiom 2 S - Separability) For any X M n, k) and Y M n, k) and for any two groups g and f of n g and n f identical individuals, if n g s X i = n f s Y j, then X Y if and only if X {g} = X Y = Y {f}. j 3 Results We show that the axioms M and S are necessary and suffi cient conditions to characterize the Multidimensional Counting Diversity Ordering in 3). Theorem 1 A diversity total ordering satisfies M and S if and only if = d. Moreover, our result is tight. In order to prove this result, we consider a society X with n individuals, all equal but one, who differ in only one attribute and a society X composed of two individuals who differ in k > 1 attributes. A straighforward computation of S α X, n, k) for X entails that S α X, n, 1) = 2n 1). We prove that to any society X can be associated a X that shows the same level of diversity, namely: Lemma 1 For all X M n, k) there exists a X M n, 1) such that S α X, n, k) = S α X, n, 1). We observe that, for any X, S α X, 2, k) = 2k α. Moreover, if the number of individuals in a simple society X is equal to k α + 1, where k is the number of attributes of the two-people society X, then S α X, n, 1) = S α X, 2, k), i.e.: Lemma 2 For any X M 2, k), let X M SαX,2,k) 2 + 1, 1) be the simple society associated to X according to Lemma 1. Then, for any multidimensional diversity total ordering satisfying axiom S, X X 7

8 Now, we prove Theorem 1: Proof of Theorem 1 To prove that d is a total ordering satisfying Monotonicity and Separability is straightforward. Suppose that is a total order that satisfies M and S, we need to show that for all X M n + 1, k) and its associated matrix X : S α X, n + 1, k) = S α X, n + 1, 1) X X 10) By induction, if n = 1, X M 2, k) then 10) follows from the Lemma 2. Suppose 10) holds for n 1, then 10) holds for all society X M n, k). Indeed, n n n 1 S α X, n, k) = k α c α ) n 1 ij = k α c α ij) n + 2 k α c α nj) i=1 i=1 11) We denote by X\ {x n } the society X in which we remove the n-th individual. The individual diversity score s X n is then s X n = n k α c α nj). By Lemma 1, Sα X\ {x n }, n 1, k) = S α X\ {x n }), n, 1) and by induction we have X\ {x n } X\ {x n }). Suppose now to add to X\ {x n } an individual and in X\ {x n }) to add a group g of identical individuals, i.e. X\ {x n }) {x n } = X and X\ {x n }) {g} = X. Then, the diversity of the new individual in X is 1 s X n while, in the simple society, we have exactly s X n individuals with the same diversity value that is 1). Thus, the level of diversity is s X n 1. Therefore, by Separability, we have that X X. Now, let X M n, k) and Y M m, k ). By Lemma 1, it is possible to find two societies X M n, 1) with n = SαX,n,k) and Y M m, 1) with m = SαY,m,k ) such that S α X, n, k) = S α X, n, 1) and S α Y, m, k ) = S α Y, m, 1). Thus, by the previous result, we obtain: X X and Y Y. 12) Using Lemma 1, we have: then, by Monotonicity, S α X, n, 1) = 2 n 1) S α Y, m, 1) = 2 m 1) 13) S α X, n, 1) S α Y, m, 1) n m and X Y S α X, n, 1) S α Y, m, 1). 8

9 Hence, equation 12) and Lemma 2 entail: X Y S α X, n, k) S α Y, m, k ) that completes the proof. 4 Empirical findings In what follows, we apply MCDO to rank some) European Countries in terms of multidimensional diversity. In order to do that, we implement a replacement subsampling bootstrap see e.g. Efron and Tibshirani [11] and Swanepoel [20]) to the cross-sectional micro data collected at household level by the European Union Statistics on Income and Living Conditions EU-SILC) sample survey. 6 The dataset covers the period with all the households residing in the territory of the Member States at the time of data collection as reference population. We consider 15 countries 13 EU member states, Norway and Island) and the first i.e. 2004) and the last i.e. 2009) year of the data collection. The multidimensional diversity among States is computed with respect to eight attributes related to the social exclusion physical and social environment; housing and non-housing related arrears; non-monetary household deprivation indicators) and housing dwelling type, tenure status and housing conditions). 7 For each country and for both the years, we construct a matrix data set) whose columns represent the variables distributed across the households rows). 6 For a detailed description of the EU-SILC Project, see 7 The list of the eight considered attributes is the following: 1) Crime violence or vandalism in the area: the objective of this variable is to assess whether the respondent feels crime, violence or vandalism in the area to be a problem for her household. Area refers to the place situated close to the place of residence. Answers are dichotomous. 2) Arrears on utility bills: whether the household has been in arrears on utility bills electricity, water, gas) in last 12 months. Answers are dichotomous. This is the variable with the most no answered question missing value) since represents the most private question. 3) Capacity to aff ord paying for one week annual holiday away from home: this question is about ability to pay, regardless of whether the household actually wants the item. Answers are dichotomous. 4) Capacity to face unexpected financial expenses: answers are dichotomous. 5) Ability to make ends meet: the household respondent s assessment of the level of diffi culty experienced by the household in making ends meet. Answers ranges from 1 with great diffi culty) to 6 very easily). 6) Dwelling type: answers take the following values: 1=detached house, 2=semi-detached or terraced house, 3=apartment or flat in a building with less than 10 dwellings, 4=apartment or flat in a building with 10 or more dwellings. 7) Tenure status: possible answers are 1 Owner), 2 Tenant or subtenant paying rent at prevailing or market rate), 3 Accommodation is rented at a reduced rate -lower price that the market price-) and 4 accommodation is provided for free). 8) Having a computer : the household is considered to possess it if any member possesses it. It includes a 9

10 In order to reduce the effect of random sampling errors in our bootstrap procedure, we perform s = times the following experiment: a) for all the analyzed countries, we randomly select re-samples with replacement) a population of m = 100 individuals from the matrix dataset; b) for each of these samples, we compute S α X, n, k), 8 that is our statistics of interest and we analyze the empirical distribution of the results; c) finally, we calculate the mean value of each of the s replicates of the experiment and the obtained value is chosen to represent the value of the multidimensional diversity for a given countries in a given year. Ranking of 15 EU countries according to MCDO Table 1 with the ranking of the EU member States for 2004 and 2009 illustrates our results. We observe that Greece is the most diverse country in both years, while the Northern countries i.e. Iceland, Sweden, Norway, Denmark) achieve the best performance. In particular, after five years, some countries reverse their relative position: Portugal is ranked at the 6th place in 2004 and at the 3rd in 2009; Austria moves from the 7th place in 2004 to the 4th place in 2009; France occupies the 2nd place in 2004 but it decreases its multidimensional diversity up to the 6th place in 2009; Belgium positively switches from the 3th place in 2004 to the 8th place in Other countries like Greece that is the most diverse country) and Estonia that occupies the 9th position) have the same ranking in both years:. Finally, some countries permute their positions: Finland and Luxembourg are respectively 10 and 11 in 2004 and 11 and 10 in 2009; portable computer or a desktop computer, and if it is provided only for work purpose, this does not count as possession of the item. Answers value are: 1 yes), 2 no - cannot afford) and 3 no - other reason). 8 The MCDI gives different values depending on α. However, any increasing tranformation preserves the ordering so, in order to simplify calculations, we compute S 2 X, n, k). 10

11 Figure 1: Boxplots Iceland, Sweden, Norway and Denmark, although they permute their positions, they remain the countries with a low level of multidimensional diversity. The comparison of the empirical distribution of the s multidimensional diversity) values for each country see the boxplots in Figure 2) in terms of median, dispersion and shape of the data does not show a clear-cut ordering of the 15 States under scrutiny. Thus, we check the robustness of the ranking in Figure 1 by using a parametric and a non-parametric test. We verify using several different techniques) that all the distributions of the s-data are normal, then we apply a pair-wise t-test to the difference between the means of the s-data of any pair of consecutively-ranked countries. 9 The non-parametric test consists in constructing a new vector whose entries are the differences between the ordered components of each consecutive pair of the s-data vectors. For each of these new vector distributions, we compute the percentiles 2.5 and Table 3 shows the results obtained by the two tests We observe that when tests are not statistically significant, two or more) countries have the same ranking. 9 The results of these controls can be provided by the authors upon request. 11

12 Ranking according to P-test 5 Conclusions There is a growing interest in defining and measuring diversity. The primary goal of this work was define and characterize a family of counting diversity orderings based on a diversity index that takes in account the number of attributes in which individuals in a society differ. Feature works are addressed in several directions: firstly we would to found dominance conditions in order to guarantee unanimous diversity counting ranking in a counting framework if profiles of diversity of different societies do not intersect. Finally we would to investigate the relationship among diversity, poverty, deprivation and social exclusion: our purpose is to tie the analysis of poverty, in a multidimensional point of view, to the diversity context. 6 Appendix Proof of Lemma 1 For any X M n, k), S α X, n, k) = n n i=1 k α c α ) ij {0, 2, 4,..., nn 1)k α }. We suppose wlog S α X, n, k) = d, then we consider a X M d/2 ) + 1, 1 ), i.e. a simple society with d/2) + 1 individuals who differ in only one attribute and such that d/2 of the individuals are equal and only one is diverse. Hence, the fact that S α X, d 2 + 1, 1) = 2 d 2 ) = d completes the proof. Proof of Lemma 2 For any X M 2, k), we consider the diversity matrices of X and X, 12

13 namely: D X = 0 kα c α D X k α c α = where c = c 12 = c 21. If we take Y M 2, k) and Z M 2, 1) such that individuals in both societies are the same, i.e. c = k in Y and c = 1 in Z, then D Y = D Z. If we add one individual y 3 to Y such that c 31 = c 32 = c, then, by S, the diversity matrix associated to Y = Y {y 3 } is: D Y = 0 0 k α c α 0 0 k α c α k α c α k α c α 0 If we add a group g of identical individuals to Z who are different from the individuals already in Z and whose number n g is equal to k α c α, then the diversity matrix for the new society Z = Z {g} is: D Z = Thus, by S, Y Z. If we remove the second individual from both societies in order to obtain Y = Y \{y 2 } and Z = Z \{z 2 } such that Y M 2, k) and Z M 1 + n g, 1) and we compute the diversity matrices for the two new societies, we have: D Y = 0 kα c α D Z k α c α = Since for {y 2 } and {z 2 } s Y 2 = k α c α = n g = s Z 2 the Separability axiom applies and we obtain the desired result. 13

14 References [1] Alesina A., Easterly W., Kurlat S., Fractionalization, Journal of Economic Growth, 8, 2003), [2] Alesina A., La Ferrara E., Ethnic Diversity and Economic Performance, Journal of Economic Literature, 43,3, 2005), [3] Alkire S., Foster J. Counting and Multidimensional Poverty Measurement, Journal of Public Economics, 95,7-8, 2011), [4] Atkinson A.B., On the measurement of economic inequality, Journal of Economic Theory 2,3 1970), [5] Atkinson A.B., Multidimensional deprivation: Contrasting social welfare and counting approach, Journal of Economic Inequality ), [6] Bervoets S., Gravel N., Appraising diversity with an ordinal notion of Similarity: an Axiomatic Approach, Mathematical Social Sciences, 53,3, 2007), [7] Bossert W., D Ambrosio C., La Ferrara E., A Generalized Index of Fractionalization,Economica, 78, 2011), [8] Bossert W., D Ambrosio C., Peragine V., Deprivation and Social Exclusion, Economica, 74, 2007), [9] Bossert W., Pattanaik P.K., Xu Y., Similarity of options and the measurement of diversity, Journal of Theoretical Politics,2003), [10] Chakravarty S.R., D Ambrosio C, The measurement of social exclusion, Review of Income and Wealth, 52,3, 2006), [11] Efron B, Tibshirani R. 1993), An Introduction to the Bootstrap, Chapman and Hall, New York. [12] Lasso de la Vega M.C., Counting poverty orderings and deprivation curves, Studies in Applied Welfare Analysis, Research on Economic Inequality, 18, 2010), [13] Nehring K., Puppe C., A Theory of Diversity, Econometrica, 70,3, 2002), [14] Pattanaik P.K., Xu Y., On diversity as freedom of choice, Mathematical Social Sciences, 40,2 2000),

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