On Bi-Polarization and the Middle Class in Latin America: A Look at the First Decade of the Twenty First Century

1 On Bi-Polarization and the Middle Class in Latin America: A Look at the First Decade of the Twenty First Century. Joseph Deutsch Department of Economics, Bar-Ilan University 52900 Ramat-Gan, Israel Email: [email protected] Jacques Silber Department of Economics, Bar-Ilan University 52900 Ramat-Gan, Israel Email: [email protected] and Gaston Yalonetzky Leeds University Business School (LUBS) University of Leeds, Leeds, United Kingdom. Email: [email protected] October 2012 Preliminary draft. Not to be quoted without the authors' permission. 2 Abstract This paper proposes a new index and graphical representation of the change in bi-polarization and in the relative importance of the middle class that took place in a given country during a given period. These tools extend in fact the concepts of inter-distribution income inequality and Lorenz curves by making a distinction between overall, “pure growth based” and “shape related” distributional changes. The empirical illustration is based on data covering 17 Latin American countries in 2000 and 2009, obtained from the Latinobarómetro surveys for these years. The standard of living of individuals was derived on the basis of correspondence analysis. It appears that these new tools help understanding the changes that took place in the distribution of standards of living during the period analysed. Key Words: bipolarization, generalized Lorenz dominance, interdistributional income inequality, Latin America, Latinobarómetro. J. E. L. Classification: D31 – O15 - O54 3 1. Introduction In a very recent paper C. Kenny (2011) writes that “it is hard to find a set of characteristics or values that are consistently and uniquely middle class across countries and time”. There is indeed in the literature a long list of traits that are supposed to help identifying individuals who belong to the middle class. The literature seems to have stressed the following characteristics of individuals belonging to the middle class. They are supposed to be middle aged people, who invest in their education, have a relatively small number of children (two or three?), spend more on health care, pay taxes, tend to own their own house or apartment as well as one or two cars and, as a consequence, have a significant amount of debt. They are also supposed to have some entrepreneurial spirit, to work in specific occupations, to have stable jobs and the means to avoid poverty even when facing an unexpected shock such as becoming unemployed. They are also expected to hold common values such as a belief in the virtue of democracy (free elections and free speech) and of tolerance (e.g. towards minorities) Finally it is also said that they are optimistic about the future, believe that they are doing better than their parents, belong to the middle spectrum of the distribution of incomes, no matter how such a middle range is defined and are supposed to be one of the main engines of economic growth. Needless to say, including all these characteristics, assuming they are all relevant, to determine who belongs to the middle class is an impossible task, among other reasons because of the scarcity of data sources that would encompass all the potentially relevant variables mentioned previously. In addition there is no agreement, among those who have attempted to define the middle class, about the most important features of the middle class, although the amount of income available is almost always mentioned. 4 But even when the main focus is on the income level, there is no consensus concerning the critical thresholds, those that distinguish the middle class from the poor and from the rich. There is hence a need to be very careful when attempting to assess the size of the middle class in a given country, to find out whether its importance grew over time, to detect its main characteristics or to determine whether the identity of those belonging to the middle class does not change or varies a lot over time. The importance of the middle class is clearly related to the concept of bipolarization. Foster and Wolfson (1992; 2010) recommended making a distinction between four stages when attempting to measure the relative importance of the middle class: - choose a “space” (individual/family/household income, salary, expenditure etc. in an incomeor people-space) - define the "middle" (e.g., the median or the mean income) - fix a range around the middle (identify the middle class by determining a percentage interval above and below the median or the mean) - aggregate the data. Various definitions based on the “income space” have been proposed. Thurow (1984) assumed that the middle class includes the households whose income ranges from 75% to 125% of the median household income. Blackburn and Bloom (1985) recommended using a wider range (60% to 225% of the median). Other ranges have been proposed: 50% to 150% (Davis and Huston, 1992) and two-thirds to four-thirds of men’s median weekly earnings (Lawrence, 1984). Birdsall et al. (2007) suggested including in the middle class those individuals above the equivalent of $10 day in 2005 and at or below the 90th percentile of the income distribution in their own country. In all these cases one computes which share of the total population the middle class includes. 5 Others have preferred to use definitions based on the people space. For Levy (1987), for example, the middle class ranges from the 20th to the 80th percentile. Whatever the definition adopted when using such an approach, one computes here the share in total income of those belonging to the selected population deciles. Graphical representations for representations in both the income and the people space have been proposed by Foster and Wolfson (1992; 2010). The present paper proposes rather a graphical representation of the change in bipolarization which is derived from the concept of inter-distributional change. 2. Measuring Changes in Bi-polarization 2.1. The Concept of Overall Distributional Change and Inter-distributional Inequality and Lorenz curves The concepts of Inter-distributional inequality and Lorenz Curves were introduced in the literature by Butler and McDonald (1987) and may be summarized as follows. Assume two different density functions )(xf and )(xh describing the distribution of income x in a given country at two different periods 0 and 1. Let )(xF and )(xH be the two distributions functions corresponding to the two density functions )(xf and )(xh . These two distribution functions )(xF and )(xH will now be plotted respectively on the horizontal and vertical axis of a 1 by 1 square. In other words for each income x we plot the percentage of individuals with an income lower than or equal to x observed in the distributions )(xF and )(xH . If the “distributional change curve” obtained happens to be completely below the diagonal, we can certainly conclude that the distribution first order stochastically dominates the distribution More generally if most of the curve lies below the diagonal we can conclude that the population with the income distribution has an economic advantage over the population with an income distribution (see, Bishop et al., 2011). If however 6 most of the curve lies above the diagonal we would conclude that the population with the income distribution has an economic advantage over the population with an income distribution . 2.2. Distributional Change in the Case of Pure Growth Let us now call f x m and h x m the median incomes corresponding to the distributions and and let us, for example, assume that f x h xmm  . Let now )(xk be the density function obtained when the density function )(xf is horizontally translated by an amount )( f x h xmm  . Finally let )(xK be the distribution functions corresponding to the density function )(xk . A plot of on the vertical axis versus that of on the horizontal axis would then give us a “distributional change curve” that would only be affected by growth (assuming that refers to time and to time ) since was derived from by a translation. Assuming there was positive growth (since we postulated that f x h xmm  ), the distribution change curve obtained will then start at some point A on the horizontal axis (see, Figure 1). The segment OA would then represent the proportion of individuals who at time t (corresponding to distribution ) had an income lower than the lowest income at time t+1 (corresponding to distribution ). Such a distributional curve would also end at point B and the segment BC would represent the share of the population who at time t+1 had an income higher than the highest income at time t (see Figure 1). In the particular and exceptional case where the lowest income at time t+1 would be higher than the highest income at time t, the distributional change curve would become identical to the broken curve OFC. In such a case we know that there would be no overlap between the distributions )(xf and )(xk and the index of distributional change , defined as being equal to twice the area between the distributional curve and the diagonal, would evidently be equal to 1. The 7 complement to one of the distributional change index may hence be considered as a measure of the degree of overlap between the distributions and in the case of positive growth over time. Figure 1: Distributional Change Curve in the Case of Pure Positive Growth O O A O B O O O O O C O O O O O F 8 Conversely if there was negative growth so that the median of the distribution K(x) is smaller than that of the distribution F(x), we would get a curve that would generally start at some point E on the vertical axis. OE would then represent the proportion of individuals who at time 1 had an income smaller than the smallest income at time 0. The curve would end on the upper horizontal axis at some point D and DC would represent the proportion of individuals who at time 0 had an income higher than the highest income at time 1. In the particular and exceptional case where the highest income at time t+1 would be lower than the lowest income at time t, the distributional change curve would become identical to the broken curve OGC. In such a case we know that there would be no overlap between the distributions f(x) and k(x) and the index of distributional change defined previously would evidently tend towards -1. The complement to -1 of such an index would hence be a measure of the degree of overlap in the case of negative growth. 9 Figure 2: Distributional Change Curve in the Case of Pure Negative Growth O O C E D G 16 Figure 3: Overall distributional change, by country. 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Argentina 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Bolivia 17 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Brazil 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Colombia 18 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Costa Rica 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Chile 19 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Ecuador 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve El Salvador 20 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Guatemala 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Honduras 21 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Mexico 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Nicaragua 22 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Panama 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Paraguay 23 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Peru 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Uruguay 24 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Venezuela 25 “Pure growth related” distributional change The results concerning the “pure growth related” distributional change are also given in Table 1. It appears that as a whole growth was highest in Chile, Argentina, El Salvador, Uruguay, Costa Rica and Brazil. On the other side the countries where as a whole growth was negative were Guatemala, Honduras, Nicaragua, Venezuela and Paraguay. Table 2 gives information on the length of the “non-overlapping” segments while Table 3a gives the variation between 2000 and 2009 in the median standard of living of the different countries. Table 3b compares the ranking of the countries according to the average growth rate during the period 2001-2010 and to the value of the “pure growth related” index of distributional change. The index of rank correlation between these two measures turns out to be equal to 0.47. We then show in Figure 4 the “pure growth related” distributional change curves for the various countries. 32 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (vi) CDF of Latent variable 2000 (pi) Distributional Change Curve - Pure Growth Ecuador 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (vi) CDF of Latent variable 2000 (pi) Distributional Change Curve - Pure Growth El Salvador 33 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (vi) CDF of Latent variable 2000 (pi) Distributional Change Curve - Pure Growth Guatemala 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (vi) CDF of Latent variable 2000 (pi) Distributional Change Curve - Pure Growth Honduras 34 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (vi) CDF of Latent variable 2000 (pi) Distributional Change Curve - Pure Growth Mexico 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (vi) CDF of Latent variable 2000 (pi) Distributional Change Curve - Pure Growth Nicaragua 35 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (vi) CDF of Latent variable 2000 (pi) Distributional Change Curve - Pure Growth Panama 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (vi) CDF of Latent variable 2000 (pi) Distributional Change Curve - Pure Growth Paraguay 36 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (vi) CDF of Latent variable 2000 (pi) Distributional Change Curve - Pure Growth Peru 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (vi) CDF of Latent variable 2000 (pi) Distributional Change Curve - Pure Growth Uruguay 37 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (vi) CDF of Latent variable 2000 (pi) Distributional Change Curve - Pure Growth Venezuela 38 “Shape related” distributional change We now turn to the analysis of “shape related” distributional change. Table 1 indicates that this type of distributional change was highest (and positive) for Guatemala, Uruguay, Nicaragua and Venezuela and lowest (and negative) for Argentina, Chile, Ecuador, Panama, El Salvador and Costa Rica. Figure 5 presents these “shape related” distributional change curves for the various countries and allows one to make a more detailed analysis of the observed change. Let us take a look for example at the case of Guatemala. One observes that from a pure change in shape point of view almost everyone in 2009 would have had a higher standard of living than in 2000. There was thus between 2000 and 2009 among the “poor” a shift of the observations towards the median. We also observe that among the “rich”there was a rightward shift of the observations toward higher standards of living. Another interesting case is Ecuador where the “shape related” distributional change was almost nil for those having a standard of living above the median whereas for those with in 2000 a standard of living below the median, there was a downard shift towards the lowest levels of standard of living. 39 Figure 5: “Shape Effect” Distributional Change Curves 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (vi) Distributional Change Curve - Shape Effect Argentina 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (vi) Distributional Change Curve - Shape Effect Bolivia 40 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (vi) Distributional Change Curve - Shape Effect Brazil 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (vi) Distributional Change Curve - Shape Effect Colombia 41 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (vi) Distributional Change Curve - Shape Effect Costa Rica 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (vi) Distributional Change Curve - Shape Effect Chile 48 distributional curve corresponding to the rich is mostly below the diagonal, since this implies that the rich have become richer. Table 4: Value of the “Change in Bi-Polarization” Index for the Various Countries. “Change in BiPolarization” Index Rank Argentina 0.0142 4 Bolivia 0.0584 14 Brazil 0.0370 10 Colombia 0.0276 7 Costa_Rica 0.0155 5 Chile 0.0371 11 Ecuador 0.0806 17 El_Salvador 0.0801 16 Guatemala 0.0204 6 Honduras 0.0080 3 Mexico 0.0288 8 Nicaragua -0.0042 2 Panama 0.0746 15 Paraguay 0.0541 13 Peru 0.0509 12 Uruguay 0.0332 9 Venezuela -0.0473 1 Table 4 then indicates that bipolarization increased the most in Ecuador and El Salvador whereas it decreased only in Venezuela and Nicaragua. Figure 4 indicates that in the case of Venezuela what happened was essentially an improvement in the standards of living of the poor, the same being true for Nicaragua though the changes were less important there. In Ecuador and El Salvador, on the contrary, Figure 4 shows that there was a clear deterioration in the standards of living of the poor. 49 The Case of the Middle Class To have a better view of what happened to the middle class we will now truncate the distributional change curve and ignore the lowest 20% and highest 10% of the distributions of standard of living in 2000 and 2009. The corresponding curves are given in Figure C-1 in Appendix C. We will limit our analysis to some countries for which the overall distributional change curve covering the whole population was partly above and partly below the diagonal. Let us take the case of Brazil, for example. The corresponding curve shows now clearly that the lower middle class (more or less the lower 30% of what we defined as middle class) were in a worse situation in 2009 than in 2000. This was also true of the very upper middle class (the upper 25% of what was defined as middle class). But for those located in the middle of this truncated distribution the standard of living was clearly higher in 2009 than in 2000. Similar conclusions may be drawn when looking at the graph for Costa Rica in Figure C-1. For Chile the picture is less clear-cut. The lower middle class (lower 15% of the middle class) and the upper middle class (upper 30% of the middle class) were worse off in 2009 so that as a whole one cannot say that those we defined as belonging to the middle class were better off in 2009. Finally in Argentina we observe that everyone in the middle class, but the upper 22-23%, was better off in 2009. 50 4. The link with traditional bi-polarization curves Let and be two distributions of a continuous variable and we define [ ] for distribution and, likewise, for . If stands for the median then we can standardize the distributions by dividing each value by the median. Such division yields the variable and we will have, e.g. , and . Define now the spread from the median: | |. Following Foster and Wolfson (2009) let us compare the following two (first-order) change-in-polarization indices. The one based on the first-order polarization curve appears in the middle graph in Figure 6 and is defined as: ∫[ ] ∫ [ ] { } ∫ [ ] { } The one based on distributional change curves is given in the right graph in Figure 6 and is defined as: ∫[ ] ∫[ ] 51 Net positive values mean that exhibits relatively less polarization than (although there may be compensation effects operating at different percentiles of the distributions). Net negative values mean that exhibits relatively more polarization. We may note the proportionality relationship between and . When the spread difference [ ] increases, the right-hand side of has to increase as well and that can only be accomplished by the widening of some of the percentile gaps, below the median and/or above the median. Hence , which is a function of both sets of gaps, also increases when a spread difference increases. Both and can be expressed as functions of the percentile gaps, but while the first index is a weighted sum of these gaps in which the weights are , the second index is a weighted sum of percentile gaps in which the weights are . Note also that a Pigou-Dalton transfer across the median should reduce polarization and increase and if represents the pre-transfer distribution and represents the post-transfer distribution. 52 Figure 6: First-order polarization curves and distributional change curves 𝑧 𝑞 𝑞𝐵 𝑞𝐴 𝐵 𝐴 𝐴 𝐵 𝑆 𝐴 𝐵 𝑙 𝑙 𝐹 𝐴 𝑙 𝐹𝐵 𝑙 𝐹 𝐴 𝑙 𝐹 𝐴 𝑙 𝐹𝐵 𝑙 𝑞 53 Foster and Wolfson (2009) discuss also what happens if a Pigou-Dalton transfer takes place on one side of the median. If the transfer preserves ranks, then it is easy to show that is insensitive to it, as it is only measuring polarization with respect to the median. However such transfer should increase bi-polarization as it concentrates the distribution on the side of the median where the transfer took place. An index that is sensitive to these transfers and thus measures changes in bipolarization can be constructed using second-order polarization curves. Let us define the cumulative spread from the median as |∫ | |∫| | |. Let us now compare the following two (first-order) change-in-polarization indices. The first one is based on the second-order polarization curve (middle graph in Figure 2)and is defined as: ∫[ ] ∫ [ ] { } ∫ [ ] { } The second one is based on the concept of cumulative relative distribution (right graph in Figure 1) and may be defined as: ∫[ ] ∫[ ] 54 Here also we may note the proportionality relationship between and . It should also be stressed that the indices attach more weight to spreads closer to the median. Therefore if a PigouDalton transfer occurs on one side of the median, representing the pre-transfer distribution and the post-transfer distribution, then the indices will take a negative value thereby showing an increase in bipolarization. The empirical illustration given in Section 3 dealt only with a “change in first order bipolarization”. We plan in the near future to complete this illustration by looking also at “second order changes in bi-polarization”. 55 Figure 2: Second-order polarization curves and cumulative relative distributions 𝑆 𝑞 𝑞𝐵 𝑞𝐴 𝐵 𝐴 𝐴 𝐵 𝐶 𝐴 𝐵 𝑘 𝐹 𝐴 𝑘 𝐹 𝐴 𝑘 𝐹𝐵 𝑘 𝑞 𝐴 𝐵 56 5. Concluding Comments This paper proposed a new index and graphical representation of the change in bi-polarization and in the relative importance of the middle class that took place in a given country during a given period. These tools extend in fact the concepts of inter-distribution income inequality and Lorenz curves by making a distinction between overall, “pure growth based” and “shape related” distributional changes. The empirical illustration was based on data covering 17 Latin American countries in 2000 and 2009, obtained from the Latinobarómetro surveys for these years. The standard of living of individuals was derived on the basis of correspondence analysis. It seems that the new tools proposed in this paper help understanding the changes that took place in the distribution of standards of living in Latin America during the period analysed. They also suggest a new way of determining what happened there to the middle class between 2000 and 2009. This empirical analysis was limited to the case of a “first order change in bi-polarization”. In future work we plan to extend this analysis to the case of a “second order change in bipolarization”. 57 Bibliography Benzécri, J.-P. and F. 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(2007) “The Decline of the Middle Class: An International Perspective,” Journal of Economic Issues 41: 181-201. 64 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Costa Rica 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Chile 65 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Ecuador 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve El Salvador 66 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Guatemala 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Honduras 67 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Mexico 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Nicaragua 68 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Panama 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Paraguay 69 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Peru 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Uruguay 70 020 40 60 80 100 020 40 60 80 100 x1 CDF of Latent variable 2009 (wi) CDF of Latent variable 2000 (pi) Overall Distributional Change Curve Venezuela