Stochastic Dominance in Polarization Work in progress. Please do not quote

Transcription

1 Stochastic Dominance in Polarization Work in progress. Please do not quote Andre-Marie TAPTUE 17 juillet 2013 Departement D Économique and CIRPÉE, Université Laval, Canada. andre-marie.taptue.1@ulaval.ca 1

2 Abstract Poor distribution of income leads to a low level of social welfare. Moreover, it can generate social tensions if it leads to the formation of antagonistic groups. This situation arises because poor distribution of income induces the decline of the middle class in favour of two extreme groups of poor and rich. When this arrives, it is said that society is polarized. The two groups may involve in social conflict where the rich want to protect their interest and where the poor want to protest against a disadvantage social order. Hence the importance of comparing the degrees of polarization in two income distributions to choose the one to apply. Sometimes, comparison is affected by an arbitrary parameter in the polarization index. To avoid arbitrariness in choosing parameter that influences the conclusion, we propose the approach of stochastic dominance in polarization for comparing the degrees of polarization in two distributions of income. We apply our methodology to data of five countries that are USA, Canada, Mexico, Italy and Greece drawn from the Luxembourg Income Study databases. Three types of dominance came out : the perfect dominance where one country stochastically dominates another in the dimension of alienation, identification and the two joined together, the limited dominance where there is dominance just in one dimension and no dimension when no dimension exhibits dominance between two countries. Key words : Income, polarization, social conflict, alienation, identification, stochastic dominance 2

3 Table of contents 1 Introduction 4 2 Literature Definitions and issue Different aspects of polarization Existing works on comparing polarization Modeling stochastic dominance in polarization Choice of the polarization index Review of some polarization indices Polarization index for dominance Stochastic dominance at first order and particular cases Effects of identification and alienation on antagonism Stochastic dominance at first order Particular cases of unidimensional dominance Application to income Stochastic dominance in income polarization Measuring alienation and identification Estimating the dominance surfaces Empirical illustration Data Results Concluding remark 38 6 Appendices 42 3

4 1 Introduction Income distribution is a major concern for policy makers. Indeed, poor distribution of income leads to a low level of social welfare. Moreover, it can generate social tensions if it leads to the formation of antagonistic groups of individuals that have difficulties living together.this is called a polarized society. Esteban and Ray (1999) and Esteban and Ray (2009) have shown, for example the nature of the possible links between polarization of society and social unrest. This suggests that it is important to assess the distribution of income both from the point of view of (usually) social well-being and that of polarization. In this line, several studies have been conducted on income distribution especially in two subjects : inequality and polarization. In the domain of inequality, the task has consisted of assessing the difference between individuals incomes. Once the degree of inequality addressed, policy makers should take decisions for transferring income from some individuals to others in order to reduce the level of inequality. Such that, when income is transferred from an individual to another with lower income, inequality and also poverty decrease following the transfer principle of Pigou-Dalton. Thus, some policies induce the movement of population who leaves some groups of income to another. This population movement may create two other problematic situations linked with polarization. For instance, if the policy induces the movement of the population from the middle to the extremes of the distribution, then inequality increases. It also leads to two and large extreme poles of individuals and a small group in the middle, increasing polarization. In other situation where the movement of individuals is such that population below the median income and the one above it cluster each around their local means income, inequality decreases but there is still two poles emerging from that distribution of income giving rise to polarization. So, distribution of income may lead to two or more groups of interests where coexistence can be difficult. One group can see the others as a threat to its economic and social interests and this may lead to social unrest. In the objective to protect the interest of his group, each individual contributes an amount of money to his group for facing the threat of the other (Esteban and Ray, 2008). The intensity of the conflict is inversely linked to the number of groups in society. In case of many groups, their sizes are small and the intensity of conflict is low. But with a small number of groups, their sizes are large giving then the possibility to contribute more to conflict if any and accentuate its intensity. The intensity of conflict is an increasing function of polarization. In fact, a small number of groups in society is associated with high degree of polarization. Hence the relationship between the intensity of conflict and polarization. Because of that relationship between polarization and conflict, study of income polarization 4

5 has attracted much attention from economists in the last twenty years. It began with an attempt for its measurement proposed by Foster and Wolfson (1992) and Esteban and Ray (1994). Since then, many indices of polarization have been proposed in order to assess the degree of polarization in a society. Compared to studies in poverty and inequality, those in polarization have not gone farther especially with regard to robust comparison, partial ordering and statistical tests. We will extend to these aspects the study in polarization in our second and third essay. This chapter focuses on the stochastic dominance in polarization that refers to a set of relationships that may hold between a pair of distributions (Davidson, 2006). It is defined as a degree and nature of separateness of two distributions (Anderson, 2004). We propose inside a methodology about to compare robustly the degrees of polarization in two distributions of income in order to say if one always generates more polarization (and is more likely to generate social unrest) than another whatever the arguments in the polarization index. Robustness is needed to guard against the uncertainty and the lack of agreement regarding the choice of arguments in comparing polarization. We consider the case where the distribution of income is continuous, in the alienationidentification framework. Our main objective is to show how to make robust comparison over classes of polarization indices and ranges of alienation and identification. In the next section on literature, we begin by defining the principal concepts that are used over the paper followed with the presentation of the issue of the paper. It is presented inside different aspects of polarization that have been covered in the field and some works that have been done in the comparison of polarization. In section 3, we review some existing income polarization indices to guide the choice of the one to be used fro comparison. We characterize in this section the class of the polarization indices adaptable to robust comparison and we develop the methodology about. Section 4 is about the empirical illustration of the methodology developed in section 3. It shows how to measure the basic elements of polarization that are alienation and identification as function of income. We provide an application of the approach using data of USA, Canada, Mexico, Italy and Greece drawn from the Luxembourg Income Study Databases. Finally, section 5 concludes the paper. It sums up the idea of the works, the methodology and the main results obtained from the application. 5

6 2 Literature 2.1 Definitions and issue Before pursuing in the review of literature, we need to define some concepts that comprehension is essential to read the paper. The first one is polarization : this concept describes the situation where population is split in two or more groups considering the distribution of income, ethnicity, religion, skin colour, etc. It is usually defined as the declining of the middle class when income is under consideration. The second concept to be mastered is alienation : It is the degree of hostility and envy that an individual can have towards another just because they do not belong to the same group of income or social characteristic. The third concept is identification used to characterize the degree at which an individual feels to belong to a group and to involve in protecting its interest. The fourth and last one is stochastic dominance. Anderson (2004) interprets the stochastic dominance as one way to define the degree and the nature of separateness of two distributions. The problematic of the paper stems from the risk that polarization has to induce social unrest in society. As stated in the introduction, the distribution of income always leads to several groups in society. Individuals of the same group have the same income and similar socioeconomic interests. But, individuals of different groups have dissimilar socio-economic interests. In the case of antagonistic groups, similar members feel identified to each other and alienated towards dissimilar members of the other groups. A society that presents such characteristic is said to be polarized and may generate social unrest and civil wars (Esteban and Ray, 1994). The emergence of social unrest is then related to the degree of polarization in income distribution (Esteban and Ray, 1999) and (Esteban and Ray, 2009). There exists an increasing relation between polarization and social conflict. When the distribution of income induces small number of groups in society, polarization is high especially for two groups and meantime, the intensity of conflict is large (Esteban and Ray, 2008). Even in social area, a link has been established between religious, ethnic polarization and social revolt (Reynal-Querol, 2002) and, civil and ethnic wars (Garcia-Montalvo and Reynal- Querol, 2002). As the distribution of income derives from an economic or income policy, government may pay attention to the one that gives the lowest level of polarization in the process of choosing its policy. This implies to compare the degrees of polarization in the associated distributions of income before making the choice. 6

7 However, comparison is sometime affected by the presence of some arbitrary parameter in the polarization indices. The numerical comparison of two indices gives a total order that can change depending on the parameter, leaving out the robustness. Thus, order obtained may be reversed between two indices following the changing of any admissible parameter in the index. For instance, when ordering many distributions, Duclos et al. (2004) find that the order between two polarization indices changes according to the value of the polarization sensitivity. Other empirical studies on polarization confirm that view on polarization sensitivity. It is the case in Esteban et al. (2007). In general, the reversing of the ordering is not specific to the polarization sensitivity. It may also occur for any argument in the polarization index formula. If polarization index is set as function of alienation and identification as arguments, the order between two indices may change from one value of the arguments to another. To avoid this instability, the approach of stochastic dominance is proposed like in poverty for instance, to establish a steady and robust order between two distributions of income when polarization is under consideration, regardless of the arguments in the index, especially alienation and identification. This concept appeared primary in the domain of finance in order to choose the best portfolio and in the domain of social welfare to maximize utility (Anderson, 2004). The basic works on stochastic dominance are attributed to Fishburn (1980), Shorrocks (1983) and Foster and Shorrocks (1988). In the domain of inequality and poverty it consists of saying if one distribution of income always generates more poverty than another whatever the poverty line. If so, the distribution that always has the higher poverty index is said to be stochastically dominated by the other. In practice, the analysis leads to stochastic dominance curves. At first order, its setting remains largely on the cumulative distribution function to compare the degrees of poverty of two distributions. Over a set of poverty lines, the cumulative distribution curve is draw and indicates the sense of dominance if any. We adapt this concept to polarization comparisons. Then it will be possible to say if one distribution is always more polarized than another or not over the ranges of the latter arguments (alienation and identification). If the rank between two polarization indices is maintained throughout all indices, the distribution that always has the higher degree of polarization is said to be stochastically dominated by the other. Two main advantages are usually cited as reasons to use stochastic dominance. The first one already mentioned is to avoid arbitrariness for example in choosing the parameter in the polarization indices or the poverty line in the case of inequality and poverty. The second advantage concerns the irrelevance of the index to be used. The same ranking is obtained, in the situation where dominance exists, with all indices consistent with the ordering. A third advantage may be added : if one distribution has induced social unrest, the comparison using stochastic 7

8 dominance to another distribution helps indicate if the second is close to generate social unrest too. So, if income polarization is proved to be among the causes of social unrest in Syria, Yemen, Tunisia or Egypt, it can be assumed that other distributions of income close to one of these countries will also generate social unrest. Hence, the importance of making comparison by stochastic dominance in order to assess income distributions. 2.2 Different aspects of polarization The importance attributed to the phenomenon of polarization and its consequence lead many economists to investigate polarization in a large aspect. A part the aspect of income, several works have been done to prove that society can be polarized even in the social and political dimensions. In each of these areas, study is focused on the formation of two or more antagonistic groups. The economic polarization is connected with situations where groups are formed based on the distribution of income. Following that distribution, population clusters around many groups. Individuals with almost the same income are grouped together. This aspect of polarization was first modelled by Esteban and Ray (1994) where they propose a synthetic polarization index when the distribution of income is discrete. They use the alienation-identification framework with the contention that polarization under these two basic elements is linked with social unrest. The polarization index proposed is a mean of all antagonisms in society. It is a typical case where polarization measurement is axiomatically characterized. Polarization is also measured when the distribution of income is continuous Duclos et al. (2004). A class of polarization measures was axiomatically derived by the latter authors remaining in the alienation and identification framework. Their index is still a sum all effective antagonisms obtained from alienation and identification within groups. This last index solves two problems encountered in the first one about the discontinuity of the distribution and the assumption of people already clustered in the existing groups. The number of groups is not really known in the process of polarization. It can be considered endogenous in the measurement of polarization. This idea is tested in Esteban et al. (2007) as an extension of the polarization measure of Esteban and Ray (1994). The problem of considering pre-existing groups is corrected by identifying the efficient cut-offs of the population clusters. In another aspect, the phenomenon of polarization was identified by the declining of the middle class that may arrive in two situations : when the poor get poorer and the rich richer, and when the poor and the rich cluster each one around their local means. The first situation is qualified as increased spread and the second one as increased bipolarity (Foster and Wolfson, 8

9 1992). The two considerations induce bi-polarization, situation in which the middle class tends to disappear. Based on the measurement of the middle, the latter authors and Wolfson (1994) propose an index of polarization that is also function the Gini index of inequality. In many approaches, polarization is function of the deviation of individual incomes from the median. In the line of Wolfson (1994), polarization is measured as function of relative deviation of incomes from the median by Wang and Tsui (2000). But it may also be characterized as intermediate polarization between relative and absolute median deviation (Chakravarty and D Ambrosio, 2010). Still in the dimension of income, inequality indices have been used to derive general ethical index in polarization measurement by Chakravarty and Majumder (2001) giving indication about close links between the two phenomena. When the attribute under consideration is a social characteristic of the population such as religion, ethnicity, skin colour, professional categories or region, we talk of social polarization where population can rather be clustered in social groups. The members from a given group present the same interests and those from different groups have distinct interests. Given that distinct groups have opposite interests, each group views another one as a threat for its existence. Then tensions are always present between groups and may escalate into civil wars. This is especially the case of religious and ethnic polarization modeled by Reynal-Querol (2002) where it was possible to develop an index of social polarization. She contents that religiously divided societies are more prone to intense conflicts because religious identity is fixed and nonnegotiable. Social polarization index as the one of Reynal-Querol (2002) is linked to the one of Esteban and Ray (1994) when one substitutes the Euclidean distance in the latter index by the discrete distance. Even if the variable is categorical and ordinal with each individual having a possibility to be in only one category, society can be polarized and the polarization index can be function of frequency or the percentage of the population in each category (Apouey, 2007). The measurement of polarization should also take into account the alienation within group and that between group (Permanyer, 2010). Animosity may persist inside group composed of the same ethnic or religious people where radical members may alienate the more moderate members. So, setting alienation within group to zero as Reynal-Querol (2002) can be inconsistent. In other points of view, polarization is overestimated if it does not account the inside group division (D Ambrosio and Permanyer, 2011). For instance a society with two social groups where all the members of the first group are black and those of the second group are white is more polarized than a society with the two social groups where half of the members of the first group 9

10 is black and half of the members of the second group are also black. Difference should be made between social polarization and fractionalization. In fractionalization, groups are not necessary antagonistic. So, the fractionalization index doest not contain any sensitivity parameter. It is interpreted as the probability that two randomly drawn individuals do not belong to the same ethnic group. Some works have focused on the fact that polarization can also be a consequence of the combined effect of the economic and social dimensions. Hence, the two dimensions must be combined for the measurement of this phenomenon. When population cluster into many groups, one can consider on the one hand group polarization, computing income polarization in each group and on the other hand explained polarization as a portion of polarization not explained by social characteristic (Gradín, 2000). In the same line, the ratio of between-group inequality to within-group inequality can be regarded as a scalar polarization index (Zhang and Kanbur, 2001). Polarization measured like this last one captures the average distance between the groups in relation to income differentials within groups. In this perspective, within-group inequality represents the spread of the distributions in the subgroups, and between-group inequality is a measure of the distance between the group means. Another aspect of the measurement of multidimensional polarization using the decomposition of the indices into intra-group and inter-group components of inequality is attributed to Gigliarano and Mosler (2009). In a multidimensional polarization measurement, continuous variables can be mixed with discrete variables. For instance, one can used household adult equivalent income, adult life expectancy as continuous variables and education level as discrete one (Anderson, 2011). Finally, polarization may come from political area. In political science, polarization concerns political parties. For instance, the U.S.A is said to be politically (bi) polarized across its congress. The congress is divided into two groups of republicans and democrats. Each subject to be discussed yields two tendencies that come out as the views of each party. This aspect of the polarization of the U.S.A was analyzed by McCarty et al. (2006). They define polarization as a separation of politic parties into liberal and conservative camps. There is no middle positions on political decisions meaning the disappearance of the moderates in congress. The two parties have pulled apart and the point of view of any member of a group is commonly shared by his peers. The economic inequality also conducts to political polarization. Then rich people will support congressmen or parliamentarians who are opposed to redistribution and poor people will support those who have opposite position. In voting a law, a position of parliamentarians may be to protect the interests of the economic group supporting them. So the interests of the middle voters is no longer represented. Two senators from the same state and party tend to be very similar in their positions (Poole and 10

11 Rosenthal, 1984). In contrast, senators from the same state but from different parties are highly dissimilar, suggesting that each party represents an extreme support coalition in the state. The two extreme groups may sometimes be opposed such that the economic groups supporting them fall in social revolt. 2.3 Existing works on comparing polarization Two essential works mentioned at the beginning of the paper have dealt with polarization comparison between many countries. In fact, Duclos et al. (2004) compared the degree of polarization between 21 countries for five different values of polarization sensitivity. From one value of the sensitivity to the other, the order between countries changes. Another comparison is done in Esteban et al. (2007) between five countries for three distinct values of polarization sensitivity and gives the same result. Then the static comparison although it is complete, is not robust. Conclusion largely depends on the parameter in the index. Some other authors such as Foster and Wolfson (1992), Anderson (2004) and Duclos and Échevin (2005) have also attempted to extend their works, of measuring polarization, to the domain of polarization comparison. When polarization is seen as the declining of the middle class, dominance between two distributions can be function of the middle class measurement in each of them. The approach was implemented by Foster and Wolfson (1992). Their methodology focuses on two aspects of polarization : the increased spread that concerns the population movement away from the median income and the increased bi-polarity relative to the median of income which arrives when people below and above the median cluster each around the mean of their incomes. The two population movements lead to two polarization curves indicating when one distribution has unambiguously more polarization than another. The first polarization curve is based on the definition of the middle class and uses the cumulative distribution function. For a distribution F, it is defined as : S F (q) = ỹ(q) ỹ(0.5) = 1 m F F 1 (q) F 1 (0.5) (1) for the q-th population percentile. For each q, S F (q) is the distance that separates the median income (m F ) from the income of the person situated at the q-th percentile. The second degree polarization curve is obtained by integrating the first one : 0.5 B F (q) = S F (p)dp. (2) q 11

12 This method ranks two distributions the same as the method of measuring the middle class where the distribution having smaller middle class has unambiguously larger polarization. In the second aspect, based on the average distance from the median, one distribution has more polarization and is ranked above another one when this average distance is larger. It is considered as the stochastic dominance in the case of polarization. The two polarization curves signal unambiguous ranking in polarization like Lorenz curves in inequality. The methodology was empirically applied to income and earning data of U.S.A and Canada and found that the U.S.A is more polarized than Canada. In another approach implemented by Wolfson (1994), the polarization curve is drawn using the transformation of the cumulative distribution function. The idea is borrowed from inequality where, if the axes of the cumulative distribution function are exchanged such that the cumulative population is on the horizontal axis and the income on the vertical axis normalized by the mean income, then the result is a curve called Jan Pen s (1973) parade. After dividing this curve through the mean income, the integration moving to right from the origin results on the traditional Lorenz curve. Following the same technic, the polarization dominance curve can be drawn using the transformation of the cumulative distribution function. The transformation gives the first polarization curve showing for any population percentile how far its income is from the median. It also indicates for a whole population how spread out is the income distribution from the middle. The integration of this curve from the middle point where by construction the height of the curve is zero, to both sides along the horizontal axis gives the second polarization curve. These two curves induce a partial ordering in polarization. Anderson (2004) analyses the link between the stochastic dominance in polarization and various notions of polarization. For two distributions of income, he introduces the concept of right and left separation in analyzing the polarization ordering. In the situation of the right separation, distribution F is always at least at the right of distribution G and, it is always at least at its left if the distributions exhibit left separation. The idea was illustrated with four empirical examples. The first one compares the income distributions of single females and single parent mothers. The second example is the convergence of growth rates seen as the depolarization of the world economy. The third example measures the effect of the unionization on the female and male wage distributions. The fourth and last example studies the polarization of the differences of the spouse educational levels. Polarization dominance may also use a function that the argument is the relative deviation of income from the median. In this spirit, Duclos and Échevin (2005) construct first order bipolarization dominance and first symmetric polarization dominance where for each value of 12

13 income, polarization is a function of its relative deviation from the median. They also constructed the second polarization curves and applied their methodology to some countries of Luxembourg Income Study databases. Chakravarty and D Ambrosio (2010) study the polarization ordering of income distributions using a concept of intermediate polarization and introducing the concept of intermediate ordering by constructing intermediate polarization curve. It is defined as the generalization of the absolute and the relative indices of polarization. An empirical illustration was done using the European Community Household Panel (ECHP) with 8 waves in four countries. It permits to test on the one hand the dominance between the countries for each wave and on the other hand the dominance over the waves within a country. Chakravarty and Maharaj (2009) apply the notion of polarization ordering to social polarization index, especially the one of Reynal-Querol (2002). From this setting a population distribution generates more polarization if the polarization index is far from zero, the minimum of the Reynal-Querol index and close to one, its maximum. With this consideration, they introduce the proximity of two distributions in term of euclidean distance. The partial ordering was also derived in the case of ordinal data by Apouey (2007). Applied to the self assessment health (SAH) data, the polarization measures she proposes satisfy the movement away from the median and the increased bi-polarity properties. All these works are limited because in some aspect, the comparison is static and not robust. The order obtained may reverse following the value of the parameter in the index. In another aspect, the framework of alienation and identification is not considered in the comparison. The absence of these two elements constitutes in our point of view a principal limit of the literature in polarization ordering. The degree of animosity and the intensity of belonging to a particular group are the causes of social unrest that polarization induces in society. So, any polarization measure that can be used for polarization dominance should consider the variation these elements. In general, polarization index would still suffer the lack of robustness when used for comparison if one decides to compute polarization for a fixed values of the arguments. Two distinct values may lead to two distinct orders between polarization degrees of two distributions. A robust conclusion can not be drawn from the entire range of the arguments. Then the necessity to firstly propose a function linked with polarization index and that takes alienation and identification as arguments and, secondly develop stochastic dominance in polarization based on that function. 13

14 3 Modeling stochastic dominance in polarization 3.1 Choice of the polarization index Review of some polarization indices The first model of polarization based on the identification-alienation framework is attributed to Esteban and Ray (1994). For the discrete income distribution, the polarization index is the sum of all effective antagonisms between groups in society. n n P (π, y) = π i π j T (I(π i ), a(δ(y i, y j ))) (3) i=1 j=1 where n is the number of the groups in the society, π = (π 1, π 2,, π n ) is the vector of the population sizes of each group, y = (y 1, y 2,, y n ) the vector of the logarithm of each group income ; T is the antagonism function in the population, I the identification function, a the alienation function between two groups and δ(y i, y j ) stands for the absolute distance function y i y j. Duclos et al. (2004) consider rather the situation where the distribution of income is continuous while remaining on the identification-alienation framework and proposed the following index. P = T (f(x), x y )f(x)f(y)dxdy (4) where f(x) is the density of identification (intensity of the population) whose income is x and x y measures the alienation of the individuals with income x towards those with income y and vice versa. It is considered that f is in the class of continuous densities defined in R +. Another aspect of population grouping around local means of their respective incomes was examined by Wolfson (1994) and characterized as bi-polarization. It arrives in two situations : when population move away from the middle, both sides of the income median, to the extremes and, when below and above the income median population clusters around their local income means. He proposes a bi-polarization index deduced from the index proposed by Foster and Wolfson (1992). Recall that the polarization index elaborated by the latter is : 1 P F W = 2 B F (q)dq (5) 0 14

15 where B F (q) is defined by equation (2). It is twice the area beneath the second degree polarization curve. Wolfson (1994) proposes the following scalar index of bi-polarization : where P W = 2 µ (2D G) (6) m D = (µ U µ L ) 1 µ (7) whith µ U the mean income of the share above the median, µ L the mean income of the share below the median and µ the mean of all income. D is the relative median deviation, G the Gini coefficient and, m the median income. This way to measure polarization has many extensions that are found in Rodríguez and Salas (2003), Chakravarty and Majumder (2001) and Wang and Tsui (2000). The above works focus solely on income as an argument on which people can form groups and feel identified with similar members and alienated towards dissimilar individuals. However it is admitted that social tensions may arise between members of different social groups formed from social characteristics such as religious, ethnicity and race (Horowitz, 1985). Following this reasoning, social polarization was modelled by Reynal-Querol (2002) who constructed the social polarization index and applied it to religious and ethnic groups. The index of polarization proposed is : n RQ = 1 (0.5 π i ) 2 π i /0.25 (8) i=1 where n is the number of social groups and i the proportion of population that belongs to group i= 1,, n. Reynal-Querol normalizes alienation to a constant in her model, advancing the difficulty to calculate distances between groups. But Permanyer (2010) contests this argument saying that fixing the size of groups, the higher the distance between them, the higher the level of social tension. He links social polarization to distances between groups based on radicalism degree each individual has in his group. He proposes two axiomatically characterized indices of social polarization which integrate alienation between social groups. 15

16 Apouey (2007) develops a model of social polarization in a case of categorical data. She proposes to analyze polarization when population is split in disjoint hierarchical categories. Taking an example of self assessment health, she constructs two social polarization indices one with the number of individuals in each category and another one with the cumulative distribution function of population split among categories. D Ambrosio and Permanyer (2011) define social polarization in the context of nominal and categorical data. They assume that polarization changes when the members of group are intrinsically different according to another social characteristic. They use the overlapping notion between social groups to define alienation and express the polarization index as proportional to the sum of all effective antagonism in society. In some situations, polarization may be viewed as depending on economic and social variables that may interact and generate social unrest. Thus it seems important to introduce the multidimensional polarization. In addressing the question, Anderson (2011) conceptualizes an index of polarization with continuous variables in a dimension and discrete variables in another dimension. His works were preceded by that of Gigliarano and Mosler (2009) who introduce the multivariate polarization indices to measure effects of non-income attributes like wealth and education. They use the decomposition by sub-groups of certain indices of multivariate inequality to develop an index of multidimensional polarization. Makdissi et al. (2008) also mix income and social characteristics to assess polarization. They developed a polarization index in two dimensions, ethnicity and income where measuring ethnic polarization consists of using Esteban and Ray s (1994) index while assigning to each individual the average income of the population. Inside each ethnic group, polarization is measured using the approach of Duclos et al. (2004). The last polarization indices that we did not give the explicit form are not in the scope of the paper for any possible utilization. We wish to mention their existence to clarify the context in which we are to choose the index for the dominance Polarization index for dominance In modelling stochastic dominance, we first need an index of polarization. We rely in this paper on the existing indices that the explicit forms are presented above the choice. Given a large number of existing indices there is an embarrassment for making the choice. But in our contention, stochastic dominance in polarization must take into account the fundamental elements of its definition. These are the identification and the alienation concepts. 16

17 Any measure of dominance on polarization that does not consider these two phenomena seems to have left something essential for the robust comparison and the core of polarization. More specifically, any measure of polarization dominance must evaluate the effect of identification and alienation variations on the degree of polarization. Thus comparison of income distributions with respect to polarization depends on the variation of these two variables. This consideration restricts the set of indices in which we can choose our index. For instance, the polarization indices built in the domain of social polarization are excluded. The latter indices normalize alienation between groups to a constant. Then, leaving the class of social indices, we remain with the indices of pure income polarization without any social aspect. The reference is made here to the polarization index of Esteban and Ray (1994) in the discrete case and the one Duclos et al. (2004) when the income distribution is continuous. However the polarization measure of Esteban and Ray (1994) generates two major problems. The first one is about discontinuity of the distribution and the second one attributed to the assumption that individuals are supposed already bunched in the income groups. To avoid facing this two problems we concentrate on polarization index of Duclos et al. (2004) where the income distribution is supposed continuous. Given the income distribution of a population A, the general form of the polarization index is proportional to the sum of all effective antagonisms between different points of the distribution : P A = p(f(x), x y )df (x, y) (9) where p(.,.) is the antagonism function taking alienation and identification as argument, f(x) is the density of the identification and x y the alienation between two groups with income x and y respectively. With a change of variables, this polarization index can be written as P A = p(a, i)df A (a, i) (10) representing the general aggregation function of alienation noted a and identification noted i across population. F A (a, i) is the joint distribution function of a and i, both supposed nonnegative. 17

18 When income is the only variable under consideration, alienation and identification are function of income and income alone. In the foregoing of the paper we concentrate only on that special case where the distribution of income is represented by its cumulative distribution function F A (y). As the consequence of that hypothesis the polarization index is PA = p(a A (y), i A (y))df A (y) (11) where p(.,.) is the antagonism function, a A (y) a measure of alienation from any norm in A, and i A (y) a measure of identification of the individuals whose income is y. The alienation function can be the distance from almost any norm and this norm can also be y specific. 3.2 Stochastic dominance at first order and particular cases Effects of identification and alienation on antagonism and alienation are the two basic elements on which depends antagonism. According to Esteban and Ray (1994) and following Duclos et al. (2004), the identificationalienation framework is adaptable for studying polarization. For polarization to stand in a society there must be a high degree of homogeneity within each group and a high degree of heterogeneity across groups. The homogeneity within group relates to identification while the heterogeneity across groups is linked to alienation. Since antagonism is function of alienation and identification and polarization an aggregation of antagonism generated by groups, it is worth analyzing their impact on antagonism and especially looking at how antagonism in an society changes when these elements increase. Proposition 3.1. Increasing identification increases antagonism. Keeping alienation constant we first want to see how antagonism is modified when identification increases. In the modelling of income polarization, Duclos et al. (2004) assume that : Intra-group homogeneity accentuates polarization. Which means that, given an income distribution that generates many groups in society, if the degree of homogeneity increases in a group, leaving alienation in all groups constant, then polarization in a society increases too, in a group depends on the number of similar individuals and the common characteristic they possess. 18

19 Given any group g, the last assumption leads to the implicit form of the identification function : i = i(n g, y) where n g is the population size of the group and "y" the common income level. Fixing the total size of population, if an individual changes the group then antagonism changes. Suppose for instance that the value of income of an individual increases. Then he changes the group moving from the one with lower income to the another one with higher income. First, identification in the older group decreases due to loss of a member and identification in the second group increases due to the arriving of a new member. This is supported by the conclusion of Esteban and Ray (1994) that the within-group identification is an increasing and continuous function of the group s size. Because the income of the individual has increased, his contribution to identification in the new group is greater than what it was in the old group. In the end, the overall identification increases. Since the number of the members in the second group increases with the new member, they may feel more confident to organize themselves in case of social unrest. They also have more power, given their number and the increasing of their income, to protest against some decisions in a society. It is the case of a new worker who begins to own salary or any other individual who, for any reason, begins to own more income than before. Such a person moves to a new group which s identification increases. As consequence of increasing identification, the overall antagonism in the society increases because its component attributed to the second group has increased absolutely more than the decrease of its component attributed to the first group. Then antagonism is an increasing and continuous function of identification. Figure 1 illustrates this case. p 2 = p(a, i) i 0. (12) Proposition 3.2. Antagonism is convex in identification. We first imagine a situation where identification increases. Following proposition 3.1, antagonism is accentuated in a society. So, if we consider the additional part of antagonism induced by the increasing of identification, we state that if identification grows for the second time then this additional antagonism also increases. Indeed the second increasing of identification leads to the surplus of antagonism. This surplus increases the overall antagonism by augmenting the initial level of antagonism and the additional part of antagonism induced by the first identification increasing. This is so because the arriving of a new member in a group affects all the 19

20 Figure 1 Increasing identification increases antagonism identification alienation members, including the one who entered the group just before, reinforcing their power and their confidence. As consequence, the part of the antagonism of each member increases. Hence the marginal antagonism grows. It is illustrated by figure 2. Proposition 3.3. Increasing distances increases antagonism. p 22 = 2 p(a, i) ( i) 2 0. (13) Suppose a situation where the distribution of income leads to many antagonistic groups. Taking one group, if the intra-variance remains constant and the intra-mean increases then antagonism increases. The increasing of the mean income corresponds to the jump or the movement of the group to the right. This movement happens when the income of the group has increased. This increasing of income of the group reinforces the capability of its members to protest against any decision that threats their interests. The jump of the group increases the distance between that group and the neighbouring ones. Since alienation is an increasing function of distance, it increases too. Thus antagonism must increase following the property that it is continuous and increasing on alienation (Duclos et al., 2004) : p 1 = p(a, i) a 0. It is illustrated by figure 3. (14) 20

21 Figure 2 Antagonism is convex in identification identification identification Figure 3 Increasing distances increases antagonism alienation alienation Proposition 3.4. Increasing identification impacts antagonism more as greater distances increase. Consider a population distributed in many (income) groups where some have greater distances (higher income) from the rest of the groups. Suppose that while greater distances between groups (the incomes of the groups situated above the mean or the mean income) increase, identification also increases. However, greater distances also mean greater alienation such that, if they increase, alienation increases too and therefore antagonism must also increase. If simulta- 21

22 neously identification increases then antagonism increase (propositions 3.1 and 3.3) more due to the combined effect. This increasing of antagonism is explained by : p 12 = 2 p(a, i) i a 0. (15) Figure 4 Increasing identification with greater distances increases antagonism identification alienation Stochastic dominance at first order We now ask how we can order polarization indices between two income distributions from two populations A and B, over some classes of polarization measures. Let Π 1 be a class of polarization indices defined by : Π 1 = {P p 1, p 2 and p 12 0} (16) where P is a polarization index as defined in (10). This class of polarization verifies propositions 3.1 to 3.4. Given two distributions from A and B, we are looking at the conditions under which it can be said that distribution A always has more polarization than distribution B whatever the values of a and i, considering the polarization indices member of Π 1 ; i.e the conditions under which, for P Π 1 we have P = P A P B > 0 and that must be fulfilled whatever the values of alienation and identification. 22

23 The polarization index P gives the finite value representing the level of polarization in a society. It is integrating on alienation and identification and the result does not depend on these variables. Then, the necessity to define a comparison framework in which identification and alienation vary in the result in a way that two distinct values give two distinct results. Contrary to studies of stochastic dominance in poverty where only one factor, the poverty threshold, varies, two variables are under consideration in the case of polarization. The stochastic dominance as defined in poverty using the cumulative distribution function is not applicable in the case of polarization. The condition for comparing two distributions in terms of polarization must consider the space of two dimensions. We introduce a new concept of polarization surface (H(.,.)) that can be calculated for any two values of alienation and identification in a given distribution of income. For instance, suppose alienation and identification are independent ; For a pair of alienation and identification values (z a, z i ) 0, let H A (z a, z i ) the polarization surface from (z a, z i ) defined by : H A (z a, z i ) = 1 (F A (z a, ) + F A (, z i ) F A (z a, z i )). (17) By construction, it is located in the nonnegative orthan of R 2. Figure 5 Dominance surface when alienation and identification vary identification H A (z a, z i ) z i z a alienation Figure 5 indicates that H A (z a, z i ) = df A (a, i) (18) z a z i H A (z a, z i ) = I(a z a )I(i z i )df A (a, i) (19) 23

24 where I(.) is an indicator function equals one if the argument is true and zero if not. After defining this dominance area, it remains the problem of establishing a link between the difference of two polarization indices and that of the two corresponding dominance surfaces ( P and H). We first show that P = H for all P Π 1 and the stochastic dominance is now defined with the polarization surface. As consequence, P 0 P Π 1 H(z a, z i ) 0 (z a, z i ) 0. (20) Proposition 3.5. ( Π 1 dominance) Consider two income distributions from two populations A and B. H(z a, z i ) = H A (z a, z i ) H B (z a, z i ) 0 z a, z i 0 Then, P > 0, P Π 1 iff and H(z a, z i ) = H A (z a, z i ) H B (z a, z i ) > 0 for some z a, z i 0.. The proof is given in appendix. When this condition is met, it is said that distribution B stochastically dominates distribution A at first-order. It is equivalent to say that, for any parameter of the function p, distribution A has more polarization than distribution B. It is a partial ordering since it is possible to have two distributions where H(z a, z i ) 0 nor H(z a, z i ) 0 for all (z a, z i ). There may exist (z 1a, z 1i ) and (z 2a, z 2i ) such that H(z 1a, z 1i ) > 0 and H(z 2a, z 2i ) < Particular cases of unidimensional dominance The comparison can be done separately in each dimension, the first one where identification is supposed to be constant and the second one where it is alienation that is supposed to be constant. Case 1 : (Inequality) dominance When there is no identification or if identification does not vary in the distribution, antagonism depends only on the vector of alienations. Therefore the polarization index set as the summation of antagonism components is the inequality index since it is equivalent to the Gini index. We suppose that identification is constant in the polarization index such that p 2 = 0 and p 12 = 0. 24

25 Let I 1 be a class of alienation or inequality measures obtained by setting p 2 = 0 and p 12 = 0 in (16) : I 1 = {P p 1 0, p 2 = 0 and p 12 = 0}. (21) For illustration, we suppose identification fixed at z i. It follows that H A (z a, z i ) = C Since a and i are supposed independent, H A (z a, z i ) = = z a df a (a) = CH A (z a, 0). (22) z a z a = (1 F i (z i )) = CH A (z a, 0) z i df A (a, i) (23) f a (a)da f i (i)di z } i {{} = 1 F i (z i ) z a f a (a)da = C z a df a (a) where C = 1 F i (z i ), f a and f i are the marginal distributions of a and i with F a respectively F i their cumulative distribution functions. It is proportional to the area from z a to the right under the curve of the density function of alienation. Notice that, if identification is set to zero, then H A (z a, 0) = z a df a (a) (24) The Dominance surface for the stochastic dominance is illustrated in figure 6. Note that (22) can be written as H A (z a, z i ) = C I(a z a )df a (a), z a 0, (25) expression that will be used for estimation. Proposition 3.6. Consider two income distributions from A and B. H(z a, z i ) = H A (z a, z i ) H B (z a, z i ) 0 z a 0 Then, P > 0, P I 1 iff and H(z a, z i ) = H A (z a, z i ) H B (z a, z i ) > 0 for some z a 0. 25

26 Figure 6 Dominance surface when alienation varies and identification is set to zero Frequency density curve : f a (z a ) H A (z a, 0) z a - Case 2 : dominance We focus now on the situation where alienation between groups is constant. The animosity between members of different groups does not vary. Therefore antagonism is just function of a vector of identification. In consequence p 1 = 0 and p 12 = 0. Let M 1 be a class of polarization measures obtained by setting p 1 = 0 and p 12 = 0 in (16) : M 1 = {P p 1 = 0, p 2 0 and p 12 = 0}. (26) For illustration we suppose alienation set to a constant. H A (z a, z i ) = K Indeed, from the independence of a and i, H A (z a, z i ) = = z i df i (i) = KH A (0, z i ). (27) z a z i df A (a, i) (28) f a (a)da z } a {{} = 1 F a (z a ) = (1 F a (z a )) = KH A (0, z i ) z i z i f i (i)di f i (i)di = K where K = 1 F a (z a ). It is proportional to the area from z i to the right under the curve of the density function of identification. z i df i (i) 26

27 Notice that, if alienation is set to zero then H A (0, z i ) = z i df i (i), z i 0. (29) 7. For that case, the Dominance surface for the stochastic dominance is illustrated in figure Figure 7 Dominance surface when identification varies and alienation is set to zero Frequency Density curve : f i (z i ) H A (0, z i ) z i - Note also that (27) can be written as H A (z a, z i ) = K I(i z i )df i (i), z i 0, (30) expression that will be used for estimation. Proposition 3.7. Consider two distributions A and B. H(z a, z i ) = H A (z a, z i ) H B (z a, z i ) 0 z i 0 Then, P > 0, P M 1 iff and H(z a, z i ) = H A (z a, z i ) H B (z a, z i ) > 0 for some z i 0. 4 Application to income 4.1 Stochastic dominance in income polarization Now, we are going to focus on a specific case of income distribution. and alienation depend exclusively on the distribution of income characterized by its cumulative 27

28 distribution function F A (y). Then we consider rather the polarization index defined in equation (11) : PA = p(a A (y), i A (y))df A (y). For an income distribution from population A, equation (19) defining the dominance surface can be rewritten as : H A (z a, z i ) = I(a A (y) z a )I(i A (y) z i )df A (a A (y), i A (y)) (31) which leads to HA(z a, z i ) = I(a A (y) z a )I(i A (y) z i )df A (y). (32) The dominance condition is then given by : P > 0, P Π 1 iff H (z a, z i ) 0 z a and z i 0. (33) Corrolary 1 : In the case of income distribution (see equation (11) for the polarization index), the condition for the stochastic dominance in polarization is defined by : H (z a, z i ) = HA (z a, z i ) HB (z a, z i ) 0 z a and z i 0 P > 0, P Π 1 iff and H (z a, z i ) = HA (z a, z i ) HB (z a, z i ) > 0 for some z a, z i Measuring alienation and identification We propose some functional forms of the alienation and identification functions. The one of alienation are suggested by the work of Foster and Wolfson (1992) who considered the distance from the median. We consider in addition the replication principle which stipulate that polarization does not change under any k-fold replication of the population maintaining the same income distribution. Due to this principle the natural requirement may be that alienation and identification functions are homogeneous of degree zero in income. First approach : Stochastic dominance using Foster-Wolfson indices or median income dominance If we use the income normalized by the median, we obtain a class of indices that are in line with the class of Foster-Wolfson indices. These indices are obtained by setting p 1 > 0 and p 1,1 < 0. It recaps the situation where the antagonism function is strictly concave in alienation. a A (y) = y Q A (0.5) 1 (34) i A (y) = µ A f A (yµ A ) (35) 28

29 where Q A (0.5) is the median income. In that case the polarization index is given by : PA W = p( y/q A (0.5) 1, µ A f A (yµ A ))df A (y). (36) The Dominance surface H W A (z a, z i ) associated is : HA W (z a, z i ) = ( ) y I Q A (0.5) 1 z a I (µ A f A (yµ A ) z i ) df A (y). (37) Corrolary 3 : In the case of income distribution and when alienation function is median normalized and identification function mean scaled, the condition for the stochastic dominance is : H W (z a, z i )=HA W (z a, z i ) HB W (z a, z i ) 0 z a and z i 0 P W > 0, P W Π 1 iff and H W (z a, z i )=HA W (z a, z i ) HB W (z a, z i ) > 0 for some z a, z i 0. Second approach : Mean income polarization dominance Here, income is normalized by its mean. a A (y) = y 1 µ A (38) i A (y) = µ A f A (yµ A ) (39) where µ A is the mean income and f A (y) its density. The corresponding polarization index is given by : PA N = ( y p 1 µ A ), µ Af A (yµ A ) df A (y). (40) Given this form of the polarization index, the dominance surface H N A (z a, z i ) is defined by : HA N (z a, z i ) = ( y I 1 µ z a A ) I (µ A f A (yµ A ) z i ) df A (y). (41) Corrolary 2 : In the case of income distribution and when alienation function is mean normalized and identification function mean scaled, the condition for the stochastic dominance is : 29

30 H N (z a, z i ) = HA N(z a, z i ) HB N(z a, z i ) 0 z a and z i 0 P N > 0, P N Π 1 iff and H N (z a, z i ) = HA N(z a, z i ) HB N(z a, z i ) > 0 for some z a, z i 0. Third approach : Stochastic dominance using Pigou-Dalton inequlity indices If the alienation and identification functions are instead given by : the polarization index is defined by a A (y) = y µ A (42) i A (y) = µ A f A (yµ A ), (43) PA I = p( y µ A, µ A f A (yµ A ))df A (y). (44) This class of indices is obtained for p 1,1 > 0 ; it is qualified as a subclass of the entire class of classical Pigou-Dalton inequality indices. The Dominance surface H I A (z a, z i ) associated is : HA(z I a, z i ) = I ( ) y z a I (µ A f A (yµ A ) z i ) df A (y). (45) µ A Corrolary 4 : In the case where the polarization index comes from a subclass of the class of classical Pigou-Dalton inequality indices, the condition for the stochastic dominance is defined by : H I (z a, z i ) = HA I (z a, z i ) HB I (z a, z i ) 0 z a and z i 0 P I > 0, P I Π 1 iff and H I (z a, z i ) = HA I (z a, z i ) HB I (z a, z i ) > 0 for some z a, z i Estimating the dominance surfaces We want to apply our methodology in the case of income dominance. The first step is to estimate the dominance surfaces in three situations : when alienation and identification vary, when only alienation changes while identification is fixed and when only identification varies while alienation is fixed. 30

31 First case : Estimation of the dominance surface when alienation and identification vary Suppose we have a random sample of N independent and identically distributed observations of income (y k, k = 1,, N) drawn from population A. When alienation and identification vary recall that the polarization surface is given by (32) : H A(z a, z i ) = I(a A (y) z a )I(i A (y) z i )df A (y). For a nonstochastic point (z a, z i ), an estimator of H A (z a, z i ) is Ĥ A(z a, z i ) = I(â A (y) z a )I(î A (y) z i )d ˆF A (y) = N N 1/N I(â A (y k ) z a )I(î A (y t ) z i ). (46) k=1 t=1 and alienation are not observed, but depend both on income that is random. Thus they need to be estimated. depends on mean or median income such that the estimation of the latter is necessary for that of the former. N ˆµ A = N 1 y k (47) k=1 ˆm A = ˆF 1 A (1 2 ) = inf{y ˆF A (y) 1 }. (48) 2 depends on the density function of income. i A (y k ) = µ A f A (y k µ A ). An estimator of identification is then given by : î A (y k ) = ˆµ A ˆfA (y k ˆµ A ). (49) The income density f is estimated non parametrically using the normal kernel density : 31

32 ˆf A (y k ˆµ A ) = 1 N A h A N A j=1 ( 1 exp 0.5 yj ) y k ˆµ 2 A h (50) 2π where h is the bandwith of Silverman estimated by : h A = 1.06ˆσ A N 1 5 A (51) with ˆσ A the estimator of the standard deviation of the income distribution. Second case : Estimation of the dominance surface when only alienation or identification varies Suppose we have a random sample of N independent and identically distributed observations of income (y k, k = 1,, N) drawn from population A. We consider first that only alienation varies and identification is fixed to be constant. The dominance surface is defined by (25) and its estimator is : ĤA(z a, z i ) = Ĉ I(â A (y) z a )d ˆF a (a A (y)) (52) = N 1 (1 ˆF N i (z i )) I(â A (y k ) z a ) (53) k=1 ) N = N (1 1 N I(î A (y k ) z i ) I(â A (y k ) z a ) (54) k=1 k=1 where î(y k ) is given by equation (49) and â(y k ) used equation (47) or (48). If we consider now that only identification changes when alienation is constant, the dominance surface is given by (30) and an estimator is : ĤA(z a, z i ) = ˆK I(î A (y) z i )d ˆF i (i A (y)) (55) = N 1 (1 ˆF N a (z a )) I(î A (y k ) z i ) (56) k=1 ) N = N (1 1 N I(â A (y k ) z a ) I(î A (y k ) z i ) (57) k=1 k=1 where î(y k ) is still given by equation (49) and â(y k ) still used equation (47) of (48). 32

33 4.2 Empirical illustration Data We apply the methodology using data drawn from the Luxembourg Income Study (LIS) data sets for several countries among which Canada and the U.S.A for the year The disposable household income is taken as a disposable income defined as a post tax and transfer income. We should have applied purchasing power parities to the data in order to convert the national currencies into the same unity. But the currency unity does not matter here as we work with the relative income. Also for this reason, we did not divide data by the inflation rate when comparing two different years. Following the idea of Davidson and Duclos (2000), household income is divided by an adult equivalent scale h 0.5 where h is the household size. This operation allows to have a comparable incomes for individuals living in households of different sizes. We also weighted all household observations by the household weight "HPOPWGT inflated". In the program, tis is equivalent of using the household weight "hweight normalized" multiplied by the number of the persons in the household unit that is done automatically. Sample sizes for each country are clustered in the following table. Table 1 Country codes and sample sizes Countries abbreviation year Sample sizes USA us Canada ca Mexico mx Italy it Greece gr We assume that during the survey, observations were drawn using simple random sampling that can avoid selection bias. We learnt from Davidson and Duclos (2000) that the LIS data are drawn from a complex sampling structure with stratification, clustering and nondeterministic rates. If we know this design and access data, our methodology can be adapted to deal with statistical issues. A few percentage (less than 0.5%) of income is negative in each country. We should have set these values to zero but we did not, considering that negative income means that the individual has to pay money to the Government at the end of the period. Such an individual does not earn enough and, be indebted can reinforce his level of alienation and identification. In the same line of Davidson and Duclos (2000), we did not account for the measurement errors due to contaminated data. 33

34 On the one hand, we consider two measurements of alienation : the median alienation where alienation is measured as a relative deviation of the income from the median and the mean alienation where its is measured as the relative deviation from the mean income. On the other hand, we begin with the alienation dominance in polarization, followed with the identification dominance in polarization and we finish with a combination of the two variables. We begin with the comparison between USA and Canada where we apply the three aspects of polarization. We realize that using the median-alienation or the mean-alienation does not change the overall result. What changes is just the interval of dominance if any. We use the case of USA and Canada to justify this statement. Thus, we only apply the median-alienation for the comparison. One more justification of that choice is also that the median is robust in the sense that any change in the income values does not modify the value of the median but modifies the value of the mean income. In the last consideration, we choose USA as a benchmark country to which we compare other countries Results The estimation allows to estimate the minimum and the maximum of identification and alienation in each country. In the particular case of median-alienation, among the five countries that are USA, Canada, Mexico, Italy and Greece, the lowest level of identification is exhibited by Mexico and stands at and the highest level is found in Greece and stands at The lowest level of alienation is zero for all the five countries and the highest level is in Mexico. The maximum are the values from which polarization does not change and converges to its limits. Instead of computing bootstrap for testing the convergence, these estimates of the intervals of variation of alienation and identification are used as a clear indication that the estimators of the polarization surfaces converge. Table 2 Interval of variation of and Countries Min identification Max identification Min alienation Max alienation USA Canada Mexico Italy Greece Canada stochastically dominates USA in a restricted interval of alienation in the case of 34

35 median-alienation (see the corresponding figures at the appendix). The same pattern is shown for the identification dominance. The curve of Canada is always below the one of USA until a certain value of identification. The model gives another information about large values of identification and alienation. When these two variables take large values, the two countries have the same degrees of polarization that are zero. Hence the median-alienation and the identification alienation are respectively nil when the income distribution leads respectively to high levels of alienation and identification. This is an awaiting results since the dominance surface is the portion above alienation and/or identification. If alienation or identification is very high in society, just a few values of alienation computed as median-alienation or identification would be above. Accounting the two dimensions reveals an ambiguous conclusion when we reduce the USA polarization surface from the Canada one. The sense of the dominance is not clear. The difference is positive for some values of the both variables and negative for other such that one can not say if one country always presents larger polarization based on the first graphic. But when we reverse the difference, considering rather Canada minus USA, the difference is negative for almost all the values of the variables. This last difference shows clearly the dominance of Canada over USA where the polarization degree is always larger. Consider rather the mean-alienation. The patterns of dominance does not change. In the sense of mean-alienation, of identification and the joined variables, Canada still dominates USA but in a small interval of identification and alienation. Indeed, the dominance curves move to the left in the mean-alienation dominance and identification dominance. The result does not change for the dominance in the two dimensions. The next step is to take USA as a benchmark and compare to the other countries using the median-alienation and identification. In general, the dominance in each of the two dimensions separately is an indication of the dominance in the two dimensions when they are combined. But, if there is non-dominance in one of the dimension of alienation or identification, one can expect the non-dominance when the two dimensions are joined together. When comparing polarization in USA and Mexico, there is no dominance in the alienation dimension, nor in the identification dimension. For a small values of alienation, Mexico stochastically dominates USA, but the median-alienation curves is confounded at many points. The identification dominance curves cross for the two countries. Before a certain value of identification, USA stochastically dominates Mexico and above this value, it is Mexico that dominates. In consequence, no country stochastically dominates the other in the two dimensions when they are combined. The sign of the difference between the polarization surfaces change significantly when we compute USA minus Mexico and also for Mexico minus USA. USA stochastically dominates Italy in the dimension of alienation. Its curves is always 35

36 under or at the same level as the one of Italy. This is a typical case where the dominance needs to be tested. The distance that separates the two curves at any point of alienation is very tiny. In consequence a statistical test will be necessary to draw conclusion. In the dimension of identification, there is no dominance as the two curves cross. With the two dimensions, we are not expected the dominance. But the graph of the USA polarization surface minus the one of Italy shows that if identification is above a threshold, USA stochastically dominates Italy. The comparison of USA and Greece gives the same figures as the comparison of USA and Italy. The USA median-dominance curve is situated under or at the same level of the Greece median-dominance curve. The conclusion would also be drawn after a statistical test. The two curves cross in the identification dominance and it is possible to have USA dominates Greece in two dimensions if identification is set above certain threshold. We can conclude about the dominance of Mexico by Canada in the alienation dimension only when alienation is strictly above some limit since the their median-alienation curves cross. The dominance is clear in the identification dominance where the Canada curve lies always under the Mexico curve with a large distance between the two curves. If alienation is set above certain limit, then we would have dominance of Mexico by Canada in the two dimensions of alienation of identification. This result is clear in the graph where Canada polarization values are subtracted from Mexico polarization values throughout the range of alienation and identification. The difference is negative for all the range of identification whenever alienation is above a threshold. Canada stochastically dominates Italy in the three dimensions. In the dimension of alienation, Canada curve clearly lies under Italy curve. The same result appears in the dimension of identification. In the joined two dimensions, the difference in any sense indicates dominance. The difference of Italy polarization degrees and Canada ones is always positive where as it is always negative when taking Canada minus Italy. The dominance can be qualified as strict dominance a least for specific values of alienation and identification. If the maximum values of these variables are well defined, the two curves do not touch each other in any dimension and the difference of the polarization surfaces in any sense is strictly negative of strictly positive. Canada also stochastically dominates Greece in the same way as it does for Italy. We can still talk of strict stochastic dominance when alienation and identification intervals are well defined. The alienation and identification curves of Canada always lie under those of Greece in the two dimensions. In the joined two dimensions, the difference of the polarization degrees between Canada and Greece is almost everywhere negative. That is, Greece always exhibits higher degree of polarization than Canada. Mexico stochastically dominates Italy but only in the alienation dominance where its curve 36

37 lies below the one of the latter. The two curves cross in the dimension of identification and the sign of the difference between polarization surfaces changes depending of the values of the two variables. Mexico also stochastically dominates Greece only in the alienation dominance. The two curves touch each other but the Mexico curve is still below the one of Greece. In the dimension of identification, their curves cross and the difference between polarization surfaces in any sense changes the sign depending on the identification and alienation values. Finally, given the resemblance of the above results on the comparisons between Italy, Greece and other countries, one may expect no dominance between Italy and Greece in the three aspects. Their curves are confounded in the alienation dominance. Nevertheless, Italy stochastically dominates Greece in the dimension of identification where its curve is below the one of Greece. Table 3 Median- and dominance for five countries Countries Median- Median- and polarization polarization polarization USA & Canada Canada USA Canada USA Canada USA USA & Mexico Mexico USA Mexico USA Mexico USA USA Mexico USA Mexico USA & Italy USA Italy Italy USA Italy USA USA Italy USA Italy USA & Greece USA Greece Greece USA Greece USA USA Greece USA Greece Canada & Mexico Canada Mexico Canada Mexico Canada Mexico Mexico Canda Mexico Canada Canada & Italy Canada Italy Canada Italy Canada Italy Canada & Greece Canada Greece Canada Greece Canada Greece Mexico & Italy Mexico Italy Mexico Italy Mexico Italy Italy Mexico Italy Mexico Mexico & Greece Mexico Greece Mexico Greece Mexico Greece Greece Mexico Greece Mexico Italy & Greece Italy Greece Italy Greece Italy Greece Greece Italy Country A country B country A stochastically dominates country B Greece Italy Country A country B country A does not stochastically dominate country B 37

38 5 Concluding remark This paper addresses the problem of comparing robustly the polarization levels in two income distributions. We begin with some fundamental definition and the issue of the paper. In the literature, we have first sum up different notions of polarization and the works that have been conducted on polarization comparison. We propose to apply the stochastic dominance principle in comparing polarization to avoid instability of that the order may suffer from the arbitrariness of some parameter in the index. In the process of setting the stochastic dominance in polarization, we make an overview of the existing polarization indices especially for income distribution. Our development reach to the dominance surface, function of alienation and identification that can be compared rather than the polarization index. In addition to the dominance in the two dimensions, we show how dominance can be done in each of the dimension alone. We apply our methodology to five countries that are USA, Canada, Mexico, Italy and Greece. Using median or mean alienation does not change the sense of the dominance. So, we use just the median alienation because of the robustness of the median income. We found that dominance is frequent in the alienation dominance and, rare in the dimension of identification and both of them. Canada stochastically dominates USA in the dimension of alienation, identification as well as in both and Mexico only in the identification dominance. Mexico stochastically dominates USA but only in alienation dimension. No country dominates the other in the two other aspects. But it is the USA that dominates stochastically Italy and Greece still only in the alienation dominance without dominance in any sense in the two other dimensions. Canada dominates Italy and Greece in the three dimensions Mexico dominates this two countries only in the alienation dominance. Finally, Italy stochastically dominates Greece in the dimension of identification and the two countries have almost the same level of polarization based on median-alienation. The result of the dominance raises the question of testing for the dominance. At some points of alienation and identification, the curves and surfaces touch and sometimes, the distance between curves or surfaces is very small limiting the conclusion about real dominance. Hence, the dominance between two curves or two surfaces must be confirmed by a statistical test. We intend to realize the statistical inference and statistical tests in polarization dominance in another paper. 38

39 References Anderson, G. (2004). Toward an empirical analysis of polarization. Journal of Econometrics, 122 :1 26. Anderson, G. (june 27, 2011). Polarization measurement and inference in many dimensions when subgroups cannot be identified. Economics, The Open-Access, Open-Assessment E-Journal, N Apouey, B. (2007). Measuring health polarization with self-assessed health data. Health Economics, 16 : Chakravarty, S. R. and D Ambrosio, C. (mars, 2010). Polarization ordering of income distributions. The Review of Income and Wealth, 56, N1. Chakravarty, S. R. and Maharaj, B. (September, 2009). polarization. ECINEQ Working Papers, 134. A study on the rq index of ethnic Chakravarty, S. R. and Majumder, A. (2001). Inequality, polarization and welfare : Theory and applications. Australian Economic Papers. D Ambrosio, C. and Permanyer, I. (2011). Measuring social polarization with ordinal and cardinal data. Journal of Public Economi Theory. Davidson, R. (2006). Stochastic dominance. Davidson, R. and Duclos, J.-Y. (2000). Statistical inference for stochastic dominance and for the measurement of poverty and inequality. Econometrica, 68 N 6 : Duclos, J. Y., Esteban, J., and Ray, D. (2004). Polarization : Concepts, measurement, estimation. Econometrica, 72, N 6 : Duclos, J.-Y. and Échevin, D. (2005). Bi-polarization comparisona. Economic Letters, 87 : Esteban, J., Gradín, C., and Ray, D. (2007). An extension of a measure of polarization, with an application to the income distribution of five OECD countries. The Journal of Economic Inequality, 5 :1 19. Esteban, J.-M. and Ray, D. (1999). Conflict and distribution. Journal of Economic Theory, 87 :

40 Esteban, J.-M. and Ray, D. (2008). Polarization, fractionalization and conflict. Journal of Peace Research, 45 N 2 : Esteban, J.-M. and Ray, D. (2009). Linking conflict to inequality and polarization. Journal of Economic Literature Classification Numbers : D74, D31, D74, D31. Esteban, J.-M. and Ray, D. (Jul. 1994). On the measurement of polarization. Econometrica, 62 : Fishburn, P. C. (Feb., 1980). Stochastic dominance and moments of distributions. Mathematics of Operations Research, 5, N 1 : Foster, J. E. and Shorrocks, A. F. (Jan. 1988). Poverty ordering. Econometrica, 56, Issue 1 : Foster, J. E. and Wolfson, M. C. (1992). Polarization and the decline of the middle class : Canada and the u.s. OPHI working paper N. 31, Journal of Economic Inequality (2010), 8 : Garcia-Montalvo, J. and Reynal-Querol, M. (2002). Why ethnic fractionalization? polarization, ethnic conflict and growth. Gigliarano, C. and Mosler, K. (2009). Constructing indices of multivariate polarization. Journal of Econmic Inequality. Gradín, C. (February, 2000). Polarization by subpopulation in spain : Review of Income and Wealth, 46. Horowitz, D. L. (1985). Ethnic group in conflict. University of California Press. Makdissi, P., Roy, T., and Savard, L. (2008). An ethnic polarization measure with an application to ivory coast data. Groupe de Recherche en Économie et Développement International : GREDI, Working Paper, McCarty, N., Poole, K. T., and Rosenthal, H. (2006). Polarized America : The Dance of Ideology and Unequal Riches. MIT Press. Permanyer, I. (2010). The conceptualization and measurement of social polarization. Journal of Economic Inequality. Poole, K. T. and Rosenthal, H. (1984). The polarization of american politics. The journal of politics, 46 :

41 Reynal-Querol, M. (February, 2002). Ethnicity, political systems and civil wars. Journal of Conflict Resolution, 46 N. 1 : Rodríguez, J. G. and Salas, R. (2003). Extended bi-polarization and inequality measures. Inequality, Welfare and Poverty : Theory and Measurement, 9 : Shorrocks, A. F. (1983). Ranking income distributions. Economica, 50, N 197 :3 17. Wang, Y.-Q. and Tsui, K.-Y. (2000). Polarization ordering and new classes of polarization indices. Journal of Public Economic Theory, 2 (3) : Wolfson, M. C. (1994). When inequalities diverge. The American Economic Review, 84, N 2 : Zhang, X. and Kanbur, R. (2001). What difference do polarization measures make? an application to china. Journal of Development Studies, 37 :3 :

42 6 Appendices Proposition 3.5 : We want to prove that P > 0 P Π 1 iff H(z a, z i ) > 0 z a 0, z i 0 where Π 1 = {P p 1, p 2, p 12 0}. Let f(a,i) be a joint density function for the random vector (a,i) and F the associated cumulative distributive function. P = = 0 0 ( 0 0 p(a, i)df (a, i) (58) ) p(a, i)f(a, i)da di (59) We first proceed to a change of variables, defining α = a and ι = i such that the polarization measure in (59) can be reexpressed as P = = = = ( 0 0 p( α, ι)df ( α, ι) (60) p( α, ι)dh( α, ι) (61) ) p( α, ι)f( α, ι)dα dι (62) g(ι)dι (63) We now need to compute g(ι). g(ι) = 0 p( α, ι)f( α, ι)dα. (64) We integrate (64) by parts with respect to α which gives : 42

43 α g(ι) = p( α, ι) + where Φ 1 (α, ι) = 0 ( p 1 ( α, ι) f( a, ι)da α=0 α= α = p( α, ι)φ 1 (α, ι) α=0 α Hence, g(ι) = p(0, ι)φ 1 (0, ι) + noting that Φ 1 (, ι) = 0. ) f( a, ι)da dα (65) 0 α= + f( a, ι)da and p 1 ( α, ι) = 0 p 1 ( α, ι)φ 1 (α, ι)dα (66) p( α, ι) ( α) = p(a, i). a p 1 ( α, ι)φ 1 (α, ι)dα (67) We replace this expression of g(ι) (67) in equation (63) and obtain 0 [ ] 0 [ 0 ] P = p(0, ι)φ 1 (0, ι) dι + p 1 ( α, ι)φ 1 (α, ι)dα dι (68) = P 1 + P 2 (69) The term P 1 is explained as follow : P 1 = = 0 0 [ ] p(0, ι)φ 1 (0, ι) dι (70) [ 0 ] p(0, ι) f( a, ι)da dι (71) The integration of (71) by parts with respect to ι gives : ι 0 0 P 1 = p(0, ι) f( a, i)dadi ι=0 ι= + = p(0, ι)h(0, ι) ι=0 0 ι= + [ ι 0 ] p 2 (0, ι) f( a, i)dadi dι (72) p 2 (0, ι)h(0, ι)dι (73) 0 = p(0, 0) + p 2 (0, ι)h(0, ι)dι (74) noting that H(0, 0) = 1 and H(0, ) = 0 ; with p 2 (0, ι) = p(0, ι) ( ι) The term P 2 is developed as follow : = p(0,i) i. 43

44 P 2 = = = = 0 [ 0 0 [ 0 0 [ 0 0 ] p 1 ( α, ι)φ 1 (α, ι)dα dι (75) α p 1 ( α, ι) p 1 ( α, ι) α ] f( a, ι)dadα dι (76) ] f( a, ι)dadι dα (77) h(α)dα (78) We need to explained h(α). h(α) = 0 We then integrate (79) by parts with respects to ι. ( α ) p 1 ( α, ι) f( a, ι)da dι (79) ι h(α) = p 1 ( α, ι) + 0 α [ p 12 ( α, ι) ι f( a, i)dadi ι=0 ι= α = p 1 ( α, ι)h( α, ι) ι=0 ι= + ] f( a, i)dadi dι (80) 0 [ ] p 12 ( α, ι)h( α, ι) dι (81) 0 [ ] = p 1 ( α, 0)H( α, 0) + p 12 ( α, ι)h( α, ι) dι (82) Noting that H( α, ) = 0 ; with p 12 ( α, ι) = 2 p( α, ι) ( ι) ( α) = 2 p(a,i) i a. Putting this expression (82) in (78) we obtain P 2 = 0 0 [ 0 [ ] ] p 1 ( α, 0)H( α, 0)dα + p 12 ( α, ι)h( α, ι) dι dα (83) The addition of (74) et (83) gives an explicit form of the polarization index. Thus, for a given distribution named A, we have 0 P A = p(0, 0) p 1 ( α, 0)H A ( α, 0)dα + p 2 (0, ι)h A (0, ι)dι p 12 ( α, ι)h A ( α, ι)dιdα (84) 44

45 In the set Π 1, at least one of the tree term p 1, p 2 or p 12 is strictly positive. If not the index is constant from one distribution to another. Consider a polarization index P Π 1 and two distributions A and B. Then P = P A P B = + = p 1 ( α, 0)(H A ( α, 0) H B ( α, 0))dα p 2 (0, ι)(h A (0, ι) H B (0, ι))dι p 12 ( α, ι)(h A ( α, ι) H B ( α, ι))dιdα (85) p 1 ( α, 0) H( α, 0)dα + 0 p 2 (0, ι) H(0, ι)dι p 12 ( α, ι) H( α, ι)dιdα (86) Sufficient condition : It is equivalent to say that H( z a, z i ) > 0 z a, z i 0 and to say that H(z a, z i ) > 0 z a, z i 0 using a change of variables. Thus, if H 0 z a, z i 0, then P is the sum of tree positive terms where at least one is strictly positive since in Π 1, we have p 1 0, p 2 0 and p 12 0 with at least one strict inequality and therefore P 0. Necessary condition : Suppose P 0 P Π 1 ; we want to prove that H(z a, z i ) 0 z a, z i 0. Suppose rather that z a et z i 0 H(z a, z i ) 0. In that condition there may exist a polarization index P Π 1 such that P 0 and this must contradict the hypothesis. Indeed, consider a polarization index P Π 1 where p 1 (z a, 0) = p 2 (0, z i ) = 0 and p 12 (z a, z i ) > 0. Since H(z a, z i ) 0 we must have P 0 which contradicts the hypothesis. Thus the necessary condition is verified. Proposition 3.6 : dominance Define the class of indices I 1 as : I 1 = { P/p 1 0, p 2 = 0 } If indices are taken in this class then P = 0 p 1 ( α, 0) H( α, 0)dα (87) since p 2 = 0 = p 12 = 0 and the last two terms of P cancel. 45

46 0 z a 0. With the hypothesis p 1 > 0, we conclude that P > 0 P I 1 iff H( z a, 0) > It is equivalent to say that P > 0 P I 1 iff H(z a, 0) > 0 z a 0 Proposition 3.7 : dominance Define the class of indices M 1 as : M 1 = { P/p 1 = 0, p 2 0 } If the index is taken in this class then P = 0 p 2 (0, ι) H(0, ι)dι (88) since p 1 = 0 = p 12 = 0 and the first and the last terms of P cancel. With the hypothesis p 2 > 0, we conclude that P > 0 P M 1 iff H(0, z i ) > 0 z i 0. It is equivalent to say that P > 0 P M 1 iff H(0, z i ) > 0 z i 0 46

47 Figure 8 Stochastic Dominance USA and Canada Value of polarization Value of polarization Median alienation dominance: USA and Canada threshold hus32_ca hca32_us dominance: USA and Canada threshold threshold 0 Difference of polarization: USA CANADA threshold hus33_ca hca33_us 47

48 Figure 9 Stochastic Dominance USA and Canada Canada U.S.A Canada U.S.A 0 Difference USA-Canada 48