On Relative Bi-Polarization and the Middle Class in Latin America. A ...

On Relative Bi-Polarization and the Middle Class

in Latin America.

A Look at the First Decade of the 21st Century.

Joseph Deutsch and Jacques Silber

Bar-Ilan University.

Outline of the talk

1. Introduction: On the middle class and bi-polarization

2. Measuring the change in Bi-polarization

- On the concepts of inter-distributional inequality

and of distributional change curves

- Overall distributional change

- “Pure Growth related” distributional change

- “Shape related” distributional change

- Measuring the change in Bi-polarization

3. An Empirical Illustration: On Changes in Bi-

Polarization in Latin America between 2000 and

- data sources

- estimating individual standards of living via

correspondence analysis

- deriving the three types of distributional change

curves for the various Latin American countries

- measuring the change in bi-polarization

between 2000 and 2009 in these countries

- focusing the analysis of distributional change on

the middle class only

4. The determinants of the standard of living

5. Implementing a Shapley decomposition of the various

types of distributional change and of the change in

bi-polarization

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 consistentlyand uniquely middle class across countries and time”.

There is indeed in the literature a long list of traits that aresupposed to help identifying individuals who belong to themiddle class. It is said that they are

- often 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 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)They are- optimistic about the future- and believe that they are doing better than their parents.They clearly belong to the middle spectrum of the distribution of incomes, nomatter how such a middle range is defined.Finally they are supposed to be one of the main engines of economic growth.

Needless to say, including all these characteristics, assumingthey are all relevant, to determine who belongs to the middleclass is an impossible task, among other reasons because ofthe scarcity of data sources that would encompass all thepotentially relevant variables mentioned previously.

In addition there is no agreement, among those who haveattempted to define the middle class, about the mostimportant features of the middle class, although the amountof income available is almost always mentioned.

But even when the main focus is on the income level, there isno consensus concerning the critical thresholds, those thatdistinguish the middle class from the poor and from therich.

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

income- or 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.

Definitions based on the “income space”

- Thurow (1984) assumed that the middle class includes thehouseholds whose income ranges from 75% to 125% of themedian 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 andHuston, 1992) and two-thirds to four-thirds of men’s medianweekly earnings (Lawrence, 1984).

- Birdsall et al. (2007) suggested including in the middle classthose individuals above the equivalent of $10 day in 2005 and ator below the 90th percentile of the income distribution in theirown country.

In all these cases one computes which share of the totalpopulation the middle class includes.

Definitions based on the people space

- For Levy (1987) the middle class ranges from the 20th

to the 80th percentile.

- Whatever the definition adopted, here one computesthe share in total income of those belonging to theselected population deciles.

Graphical representations: see Foster and Wolfson (1992;2010) for representations in both the income and thepeople space.

2. Measuring Changes in Bi-polarization

The Concepts of Inter-distributional Inequality andLorenz curves

- introduced in the literature by Butler and McDonald(1987)

- Assume two different density functions f(x) and h(x)describing the distribution of income x in a givencountry at two different periods 0 and 1.

- Let F(x) and H(x) be the two distributions functionscorresponding to the two density functions f(x) andh(x).

Plot these two distribution functions F(x) and H(x)respectively on the horizontal and vertical axis of a 1 by1 square, that is, for each income x we plot thepercentage of individuals with an income lower thanor equal to x observed in the distributions F(x) andH(x).

Clearly if the curve obtained is completely below thediagonal, we can conclude that the distribution H(x)first order stochastically dominates the distributionF(x).

More generally if most of the curve lies below thediagonal we can conclude that the population withincome distribution H(x) has an economic advantageover the population with an income distribution F(x)(see, Bishop et al., 2011).

Conversely if most of the curve lies above the diagonal,then we can state that the population with incomedistribution F(x) has an economic advantage over thepopulation with income distribution H(x).

Assume now that F(x) corresponds to the distribution ofincome at time 0 and H(x) to that at time 1. By plottingagain H(x) on the vertical axis versus F(x) on the horizontalaxis we obtain a curve that will represent what could becalled the “distributional change” between time 0 and time1.

We can then compute a summary index of such adistributional change equal to twice the sum of the areaslying between the distributional change curve and thediagonal, making sure any area lying below the diagonal isgiven a positive sign and any area above a negative sign.

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CDF of Latent variable 2000 (pi)

Overall Distributional Change CurveBrazil

- It is also possible to compute directly the value of such asummary index which is in fact related to the Gini index.

- Borrowing ideas from Handcock, M.S. and M. Morris (1999) wenow make a distinction between “pure growth related” and“shape related” distributional change.

Distributional Change in the Case of Pure Growth

- Call and the median incomes corresponding to thedistributions F(x) and H(x) and assume that .

- Let now k(x) be the density function obtained when the densityfunction f(x) is horizontally translated by an amount .Finally let K(x) be the distribution functions corresponding tothe density function k(x). A plot of K(x) on the vertical axisversus that of F(x) on the horizontal axis would then give us a“distributional change curve” that would only be affected bygrowth since K(x) was derived from F(x) by a translation.

Assuming there was positive growth (since wepostulated that ), the distributional change curveobtained will then start at some point A on thehorizontal axis (the segment OA would then representthe proportion of individuals who at time t(corresponding to distribution F(x) ) had an incomelower than the lowest income at time t+1(corresponding to distribution K(x) ).

Such a distributional curve would also end at point Band the segment BC would represent the share of thepopulation who at time t+1 had an income higher thanthe highest income at time t (see Figure 1).

Conversely if there was negative growth so that themedian of the distribution K(x) is smaller than that ofthe distribution F(x), we would get a curve that wouldgenerally start at some point E on the vertical axis. OEwould then represent the proportion of individualswho at time 1 had an income smaller than the smallestincome at time 0. The curve would end on the upperhorizontal axis at some point D and DC wouldrepresent the proportion of individuals who at time 0had an income higher than the highest income at time1.

In the particular and exceptional case where the lowestincome at time t+1 would be higher than the highestincome at time t, the distributional change curvewould become identical to the broken curve OFC. Insuch a case we know that there would be no overlapbetween the distributions f(x) and k(x) and the indexof distributional change defined previously wouldevidently tend towards 1.

Conversely if in the case of negative growth thehighest income at time 1 is smaller than the smallestincome at time 0, we would get the curve OGC.

Distributional Change: The “Pure Shape Effect”

- Assume now that we compare the cumulative distributionsH(x) on the vertical axis and K(x) on the horizontal axis. Bydefinition these two distributions have the same medianincomes.

- If for each income x we now plot the percentage ofindividuals with an income lower than or equal to xobserved in the distributions H(x) and K(x), we will findout that, as expected, the distributional change curveobtained will pass through the point (0.5,0.5) since thesetwo distributions have the same median.

- We also know that, by definition, the slope of this curve ispositive. As a consequence up to the point (0.5,0.5) thecurve will be located in the lower left square of size 0.5 by0.5 while beyond the point (0.5, 0.5) the curve will belocated in the upper right square of size 0.5 by 0.5.

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CDF of Latent variable 2000 (vi)

Distributional Change Curve - Shape EffectPeru

By computing again (twice) the sum of the signedareas between this distributional change curve and thediagonal we would obtain a measure of the amount ofdistributional change that occurs when it is assumedthat only the shape (and not its location) of theincome distribution varied between times t and t+1.

Measuring the change in bi-polarization

We can also derive from the plot of H(x) on the vertical axis versusK(x) on the horizontal axis a measure of change in bi-polarization.

More precisely if for any income below the median we know that theproportion of individuals with an income lower than or equal to x ishigher for the distribution H(x) than for the distribution K(x), then, upto the median, the curve obtained will not only be in the lower leftsquare of size 0.5 by 0.5 but it will also lie above the diagonal.

Conversely if for any income above the median the proportion ofindividuals with an income higher than or equal to x is lower for thedistribution H(x) than for the distribution K(x), then, beyond themedian, the curve obtained will not only be in the upper right square ofsize 0.5 by 0.5 but it will also lie below the diagonal. In fact the moredistant the curve obtained is from the diagonal in these two 0.5 by 0.5squares, the more bi-polarization there is in the distribution H(x) incomparison to the distribution K(x).

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We therefore measure the change in bi-polarization by the sumof the areas lying between the curve and the diagonal, in the two0.5 by 0.5 squares previously defined.

Since the curve may cross once or more the diagonal, we defineas follows the sign given to the areas lying between the diagonaland the curve:

In the lower left square of size 0.5 by 0.5, any area lying betweenthe curve and the diagonal which is located below the diagonalwill be given a negative sign while any area lying between thediagonal and the curve which is located above the diagonal willbe given a positive sign.

Conversely in the upper right square of size 0.5 by 0.5, any arealying between the curve and the diagonal which is located belowthe diagonal will be given a positive sign while any area lyingbetween the curve and the diagonal which is located above thediagonal will be given a negative sign.

The sum of all these signed areas will then be considered as ameasure of the change in bi-polarization.

Since the higher the change in bipolarization, thelower the relative importance of the middle class attime 1 when compared to time 0, we have here ameasure of the change in the relative importance ofthe middle class that took place between two timeperiods (ignoring the impact of economic growth).

3. An Empirical Illustration: On Changes in Bi-Polarization in Latin America between 2000 and 2009

The Data sources

- We used the 2000 and 2009 Latinobarómetro surveys

- These surveys do not provide any data on the actual income of individuals.

- But they provide information on the durable goods which individuals haveas well as on their access to a certain number of services.

- Eleven durables goods or types of access to basic services were taken intoaccount: television, refrigerator, home, personal computer, washingmachine, phone, car, second home, access to drinking water, access to hotwater and sewage facilities.

- To estimate individual standards of living we used correspondenceanalysis.

Correspondence analysis

Correspondence Analysis (CA) was first developed by Benzécri, (1973). It aims atanalyzing simple two-way tables where some measure of correspondence isassumed to exist between the rows and columns. It allows one to obtain a graphicaldisplay of row and column points in biplots, which helps discovering somestructural relationships that may exist between the variables and the observations.

Correspondence analysis (CA) is in a certain way similar to principal componentsanalysis (PCA) but principal components analysis should be used when the variablesare continuous, whereas correspondence analysis is mainly applied to the case ofcontingency tables, that is, when the variables take only the values of 0 or 1.

If we asume a contingency table with, for example, I rows and J columns,correspondence analysis will give us a plot of (I+J) points, I points corresponding tothe rows and J points to the columns. It can be proven that if two row points areclose on the bi-plot, then their conditional distributions across the columns aresimilar whereas if two column points are close this implies that their conditionaldistributions across the rows are similar.

The main outputs of correspondence analysis are principal components which are orthogonaland each component turns out to be a linear combination of the variables on one hand, theobservations on the other.

Like PCA, CA is a data exploration technique that uncovers correlation patterns across setsof variables described by single components named principal components.

The main difference between such approaches and standard econometric approaches is thatthe dependent variable is unobserved. We therefore assume that welfare is amultidimensional latent variable.

There are at least two advantages in using CA in addition to its suitability forcategorical data.

The first is that MCA gives more weight to indicators with a smaller number of“hits”. In other words if, for example, for a dimension of the standard of living likehaving a refrigerator, we observe that only a few individuals have a refrigerator, thenthese individuals will be given a higher weight.

The second property is reciprocal bi-additivity. This property, which we alreadymentioned, states that

- the composite “standard of living” score of an individual is the simple average ofthe factorial weights of the “standard of living” categories for this individual.

- the weight of a given dimension of “standard of living” is the simple average of thecomposite “standard of living” scores of the population units that belong to the givendimension.

Deriving the distributional change curves:By comparing the standards of living at times 0 and 1, we can now derivethe various distributional change curves.

The Overall Distributional ChangeThe results of the computation of the measures of the overall, “pure

growth related” and “shape related” distributional change aresummarized in Table 1.

It appears that the overall distributional change was highest (positivevalues) in Uruguay, Venezuela, Mexico, Argentina, Chile and Peru.

The highest negative values were observed in Honduras, Panama,Paraguay, Bolivia, Colombia and Nicaragua, these negative valuesmeaning that in these countries, as a whole, the standard of living waslower in 2009 than in 2000, although this was not necessarily true forall the segments of the population.

In Colombia, for example, the situation was worse in 2009 than in 2000for everyone. In Bolivia this was true for everyone but the richest 5 to8%, in Nicaragua for everyone but the poorest 3% and richest 2% andin Paraguay for everyone but the richest 25%.

We also give graphical illustrations.

Table 1: Value of the overall, “pure growth related” and “shape related” indices of

distributional change for the period 2000-2009 in various Latin American countries.

Country Overall Growth Shape Effect

Argentina 0.0427 0.1161 -0.0663

Bolivia -0.0900 -0.1315 0.0443

Brazil -0.0050 0.0692 -0.0678

Colombia -0.0815 -0.0771 -0.0232

Costa_Rica 0.0188 0.0932 -0.0715

Chile 0.0377 0.1195 -0.0647

Ecuador -0.0226 -0.0316 0.0123

El_Salvador 0.0098 0.0280 -0.0069

Guatemala -0.2137 -0.3493 0.1568

Honduras -0.1252 -0.1479 0.0275

Mexico 0.0498 0.0520 0.0004

Nicaragua -0.0788 -0.1667 0.1106

Panama -0.0984 -0.1827 0.0874

Paraguay -0.0961 -0.1315 0.0200

Peru 0.0364 0.0820 -0.0257

Uruguay 0.0789 0.1445 -0.0631

Venezuela 0.0521 0.0804 -0.0276

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Overall Distributional Change CurveArgentina

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Overall Distributional Change CurveBolivia

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Overall Distributional Change CurveColombia

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Overall Distributional Change CurveCosta Rica

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Overall Distributional Change CurveChile

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Overall Distributional Change CurveEcuador

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Overall Distributional Change CurveEl Salvador

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Overall Distributional Change CurveGuatemala

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Overall Distributional Change CurveHonduras

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Overall Distributional Change CurveMexico

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Overall Distributional Change CurveNicaragua

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Overall Distributional Change CurvePanama

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Overall Distributional Change CurveParaguay

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Overall Distributional Change CurvePeru

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Overall Distributional Change CurveUruguay

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Overall Distributional Change CurveVenezuela

“Pure growth related” distributional change

The results concerning the “pure growth related” distributionalchange are also given in Table 1.

It appears that as a whole growth was highest in Uruguay, Chile,Argentina, Costa Rica and Peru.

On the other side the countries where as a whole growth was themost negative were Guatemala, Panama, Nicaragua, Honduras,Paraguay and Bolivia

Table 2 gives information on the length of the “non overlapping”segments while Table 3a gives the variation between 2000 and2009 in the median standard of living of the different countries.

Table 3b compares the ranking of the countries according to theaverage growth rate during the period 2001-2010 and to the valueof the “pure growth related” index of distributional change. Theindex of rank correlation turns out to be equal to 0.68.

We then show some of the “pure growth related” distributionalchange curves.

Argentina 0.0427 0.1161 -0.0663

Bolivia -0.0900 -0.1315 0.0443

Brazil -0.0050 0.0692 -0.0678

Colombia -0.0815 -0.0771 -0.0232

Costa_Rica 0.0188 0.0932 -0.0715

Chile 0.0377 0.1195 -0.0647

Ecuador -0.0226 -0.0316 0.0123

El_Salvador 0.0098 0.0280 -0.0069

Guatemala -0.2137 -0.3493 0.1568

Honduras -0.1252 -0.1479 0.0275

Mexico 0.0498 0.0520 0.0004

Nicaragua -0.0788 -0.1667 0.1106

Panama -0.0984 -0.1827 0.0874

Paraguay -0.0961 -0.1315 0.0200

Peru 0.0364 0.0820 -0.0257

Uruguay 0.0789 0.1445 -0.0631

Venezuela 0.0521 0.0804 -0.0276

Table 2: Non Overlapping Segments

of the “growth related” distributional change curves.

OverallPositiveGrowth.HorizontalSegment(OA)

OverallPositiveGrowth.VerticalSegment(CD)

OverallNegativeGrowth.VerticalSegment(OE)

OverallNegativeGrowth.HorizontalSegment(FD)

Argentina 1.3 0.5

Bolivia 18.6 0.3

Brazil 0.7 0.1

Colombia 0.4 0.1

Costa_Rica 0.7 0.7

Chile 4.3 0.9

Ecuador 0.5 0.1

El_Salvador 2.8 0.1

Guatemala 31.8 21.4

Honduras 11.4 0.2

Mexico 0.2 0.1

Nicaragua 41.1 0.3

Panama 11.5 0.9

Paraguay 2.3 3.4

Peru 0.6 0.1

Uruguay 1.0 1.0

Venezuela 3.1 0.5

Table 3a: Variation in the Median Standard of Living in Various Latin AmericanCountries between 2000 and 2009.

Median Standard of Living2000 2009 Difference

Argentina 0.2042 0.2322 0.0280Bolivia 0.1436 0.0974 -0.0462Brazil 0.1896 0.2087 0.0191Colombia 0.1436 0.1263 -0.0173Costa_Rica 0.1710 0.1971 0.0261Chile 0.1896 0.2204 0.0308Ecuador 0.1087 0.1034 -0.0053El_Salvador 0.0974 0.1012 0.0038Guatemala 0.2042 0.0541 -0.1502Honduras 0.1477 0.0974 -0.0503Mexico 0.1710 0.1846 0.0136Nicaragua 0.0974 0.0459 -0.0515Panama 0.1971 0.1354 -0.0617Paraguay 0.1738 0.1355 -0.0384Peru 0.0974 0.1117 0.0143Uruguay 0.1842 0.2219 0.0377Venezuela 0.1710 0.1971 0.0261

Table 3b: Ranking of per capita GDP growth rates versus ranking

of “pure growth related” index of distributional change.

Meangrowth rate2001-2010

Rank ofmean

growth rate

“Pure growthrelated”

distributionalchange index

Rank of “Puregrowth related”

distributionalchange index

Rankdifference

Argentina 3.61% 3 0.1161 3 0

Bolivia 2.10% 11 -0.1315 14 -3

Brazil 2.46% 8 0.0692 7 1

Colombia 2.53% 7 -0.0771 11 -4

Costa_Rica 2.38% 9 0.0932 5 4

Chile 2.66% 6 0.1195 2 4

Ecuador 2.98% 5 -0.0316 10 -5

El_Salvador 1.57% 14 0.0280 9 5

Guatemala 0.80% 16 -0.3493 17 -1

Honduras 2.01% 12 0.0520 8 4

Mexico 0.58% 17 -0.1667 15 2

Nicaragua 1.59% 13 -0.1827 16 -3

Panama 4.47% 2 -0.1315 12 -10

Paraguay 2.16% 10 0.0820 6 4

Peru 4.48% 1 0.1445 1 0

Uruguay 3.09% 4 0.1161 3 1

Venezuela 1.55% 15 -0.1315 12 3

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Distributional Change Curve - Pure GrowthArgentina

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Distributional Change Curve - Pure GrowthBolivia

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Distributional Change Curve - Pure GrowthBrazil

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Distributional Change Curve - Pure GrowthColombia

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Distributional Change Curve - Pure GrowthCosta Rica

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Distributional Change Curve - Pure GrowthChile

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Distributional Change Curve - Pure GrowthEcuador

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Distributional Change Curve - Pure GrowthEl Salvador

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Distributional Change Curve - Pure GrowthGuatemala

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Distributional Change Curve - Pure GrowthHonduras

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Distributional Change Curve - Pure GrowthMexico

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Distributional Change Curve - Pure GrowthNicaragua

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Distributional Change Curve - Pure GrowthPanama

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Distributional Change Curve - Pure GrowthParaguay

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Distributional Change Curve - Pure GrowthPeru

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Distributional Change Curve - Pure GrowthUruguay

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Distributional Change Curve - Pure GrowthVenezuela

“Shape related” distributional change

We now turn to the analysis of “shape related” distributionalchange. Table 1 indicates that this type of distributional changewas highest (and positive) for Guatemala, Nicaragua andPanama and lowest (and negative) for Costa Rica, Brazil,Argentina, Chile and Uruguay.

Let us take a look for example at the case of Guatemala. Oneobserves that from a pure change in shape point of view morethan 35% of the individuals would have had in 2000 a standardof living smaller than the smallest standard of living in 2009while more than 20% of the individuals would have had in 2009a standard of living higher than the highest standard of living.

There was thus between 2000 and 2009 among the “poor” a shift ofthe observations towards the median.

We also observe that among the “rich”there was a rightward shift ofthe observations toward higher standards of living.

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Distributional Change Curve - Shape EffectGuatemala

In the case of Nicaragua we observe that more than40% of the individuals would have had in 2000 astandard of living smaller than the smallest standardof living in 2009, a shift which again indicates a movetowards the median standard of living among the“poor”.

The shifts are less drastic for the Panama, anothercountry with an important rightward shift in standardsof living from a “shape effect” point of view.

Argentina 0.0427 0.1161 -0.0663

Bolivia -0.0900 -0.1315 0.0443

Brazil -0.0050 0.0692 -0.0678

Colombia -0.0815 -0.0771 -0.0232

Costa_Rica 0.0188 0.0932 -0.0715

Chile 0.0377 0.1195 -0.0647

Ecuador -0.0226 -0.0316 0.0123

El_Salvador 0.0098 0.0280 -0.0069

Guatemala -0.2137 -0.3493 0.1568

Honduras -0.1252 -0.1479 0.0275

Mexico 0.0498 0.0520 0.0004

Nicaragua -0.0788 -0.1667 0.1106

Panama -0.0984 -0.1827 0.0874

Paraguay -0.0961 -0.1315 0.0200

Peru 0.0364 0.0820 -0.0257

Uruguay 0.0789 0.1445 -0.0631

Venezuela 0.0521 0.0804 -0.0276

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Distributional Change Curve - Shape EffectArgentina

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Distributional Change Curve - Shape EffectBolivia

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Distributional Change Curve - Shape EffectBrazil

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Distributional Change Curve - Shape EffectColombia

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Distributional Change Curve - Shape EffectCosta Rica

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Distributional Change Curve - Shape EffectChile

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Distributional Change Curve - Shape EffectEcuador

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Distributional Change Curve - Shape EffectEl Salvador

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Distributional Change Curve - Shape EffectGuatemala

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Distributional Change Curve - Shape EffectHonduras

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Distributional Change Curve - Shape EffectMexico

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Distributional Change Curve - Shape EffectNicaragua

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Distributional Change Curve - Shape EffectPanama

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Distributional Change Curve - Shape EffectParaguay

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Distributional Change Curve - Shape EffectUruguay

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Distributional Change Curve - Shape EffectVenezuela

Variations in Bi-PolarizationThe observations that were just made concerning the “shape related distributionalchange” have evidently implications concerning the variation between 2000 and2009 in the degree of bi-polarization of the distribution of standards of living.A new measure of change in bi-polarization was previously proposed.Table 4 gives the value of this index for the various Latin American countries.We recall that a distinction has to be made between the “poor”, those whosestandard of living is below the median, and the “rich”, those with a standard ofliving higher than the median.If the curve for the poor is mostly above the diagonal, then the poor have become“poorer” (assuming the two distributions compared have the same standard ofliving) so that bipolarization increases.On the other hand bipolarization will increase if the “shape related” distributionalcurve corresponding to the rich is mostly below the diagonal, since this impliesthat the rich have become richer.Table 4 then indicates that bipolarization increased the most in Peru whereas itdecreased the most in Nicaragua. In the case of Peru we observe that whathappened was essentially a decrease in the standards of living of the poor whilefor Nicaragua there was a striking increase in the standard of living of the poor,assuming evidently in both cases no change in the overall median standard ofliving.

Table 4: Change in the Bi-Polarization Index.

Country Value of the Index Ranking of the countriesArgentina 0.0118 9Bolivia -0.0012 7Brazil 0.0312 14Colombia 0.0226 12Costa_Rica 0.0095 8Chile 0.0300 13Ecuador -0.0078 5El_Salvador 0.0131 11Guatemala -0.0292 2Honduras -0.0089 4Mexico 0.0367 15Nicaragua -0.0863 1Panama -0.0065 6Paraguay 0.0398 16Peru 0.0847 17Uruguay 0.0127 10Venezuela -0.0226 3

The Case of the Middle Class

To have a better view of what happened to the middleclass we will now truncate the distributional changecurve and ignore the lowest 20% and highest 10% ofthe distributions of standard of living in 2000 and2009.

We will limit our analysis to some countries for whichthe overall distributional change curve covering thewhole population was partly above and partly belowthe diagonal.

Let us take the case of Brazil, for example. The corresponding curveshows now clearly that the lower middle class (more or less the lower30% of what we defined as middle class) were in a worse situation in2009 than in 2000.

This was also true of the very upper middle class (the upper 5% of whatwas defined as middle class).

The bulk of the middle class was however in a better situation in 2009than in 2000 and this appears clearly in Table 5 since the measure ofoverall distributional change for the middle class was positive andequal to 0.0239.

A similarly look at the graph for Costa Rica shows that in this countrypractically everyone in the middle class was better off in 2009.

For Chile the picture is less clear-cut. The lower middle class (lower25% of the middle class) and the upper middle class (upper 25% of themiddle class) were worse off in 2009 so that as a whole one cannot saythat those we defined as belonging to the middle class were better offin 2009.

Finally in Mexico we observe that everyone in the middle class wasbetter off in 2009 but the lower 15%.

Overall Distributional Change Curves specific to the middle class, that is, ignoring theindividuals with the lowest (bottom 20%) and highest (top 10%) standard of living.

0 20 40 60 80 100x1

Overall Distributional Change CurveCosta Rica

0 20 40 60 80 100x1

Overall Distributional Change CurveChile

0 20 40 60 80 100x1

Overall Distributional Change CurveMexico

“Growth related” distributional changeHere evidently the picture is the same as that obtained when looking atthe whole population (the “growth related” distributional change curve iseither completely above or completely below the diagonal).

“Shape related” distributional changeHere again we may want to take a look at some interesting specific cases.Let us start with Argentina. It appears that the lower middle class and theupper middle class (at the exception of the upper 5% of the middle class)would have been better off whereas the “middle middle class” as a wholewould have been somehow worse off in 2009, had there been only a“shape related” distributional change.For Brazil and Chile the picture is quite similar: the upper and lowermiddle class would have been better off in 2009 but for the “middlemiddle class” no clear conclusions may be drawn.The situation was different in Mexico because there the lower half of themiddle class would have been better off while as a whole the upper halfwould have been worse off in 2009, had there been only a “shape related”distributional change.

“Shape Related” Distributional Change Curves specific to the middle class, that is, ignoring theindividuals with the lowest (bottom 20%) and highest (top 10%) standard of living.

0 20 40 60 80 100x1

Distributional Change Curve - Shape EffectArgentina

0 20 40 60 80 100x1

Distributional Change Curve - Shape EffectBrazil

0 20 40 60 80 100x1

Distributional Change Curve - Shape EffectChile

0 20 40 60 80 100x1

Distributional Change Curve - Shape EffectMexico

Table 5: Value of the overall, “growth related” and “shape related” indices of distributional

change for the middle class between 2000 and 2009 (various Latin American countries).

Argentina 0.1045 0.1305 -0.0143

Bolivia -0.1736 -0.1548 -0.0366

Brazil 0.0239 0.0883 -0.0518

Colombia -0.0949 -0.0918 -0.0395

Costa_Rica 0.0585 0.1290 -0.0760

Chile 0.0028 0.0730 -0.0536

Ecuador -0.0256 -0.0352 0.0114

El_Salvador 0.0200 0.0937 -0.0616

Guatemala -0.3307 -0.4284 0.1295

Honduras -0.1735 -0.2492 0.1166

Mexico 0.0729 0.0853 -0.0024

Nicaragua -0.1990 -0.1927 0.0576

Panama -0.1762 -0.2333 0.0881

Paraguay -0.1438 -0.2102 0.0236

Peru 0.0532 0.1231 -0.0244

Uruguay 0.1414 0.2253 -0.0843

Venezuela 0.0762 0.0856 -0.0116

Table 6: Change in Bi-Polarization Index

for the Middle Class

Value of the Index Ranking of thecountries

Argentina 0.0103 (by increasingvalues)

Bolivia 0.0532 9

Brazil 0.0180 15

Colombia 0.0256 11

Costa_Rica 0.0086 12

Chile 0.0018 8

Ecuador -0.0054 7

El_Salvador 0.0306 4

Guatemala -0.0022 13

Honduras -0.0296 5

Mexico 0.0489 3

Nicaragua -0.0630 14

Panama 0.0005 1

Paraguay 0.0593 6

Peru 0.0857 16

Uruguay 0.0129 17

Venezuela -0.0361 10

4. The determinants of the standard of living

Table 7 gives the results of a regression where thedependent variable is the individual standard of living,as derived from correspondence analysis, while theexplanatory variables are respectively

- the area of residence (capital, urban center, other)

- the marital status (married, single, other)

- the age

- the gender

- the education

- the father’s education

- the country of residence

Table 7: The determinants of individual standard of living

Variable Mean S. Deviation Coefficients t values

Constant 0.16561 0.09630 0.05002 14.60Urban center 0.21901 0.41358 0.01386 10.35Capital 0.09190 0.28888 0.02255 12.49Married 0.57736 0.49398 0.01333 8.35Single 0.31483 0.46445 0.01584 8.54Age 38.83331 15.93828 0.00073 20.63Male 0.48095 0.49964 0.00469 5.04Education 10.20867 4.71762 0.00594 48.19Father’s

education6.81147 5.06393 0.00417 36.02

Bolivia 0.06129 0.23985 -0.04526 -17.33Brazil 0.06404 0.24483 0.01352 5.21Colombia 0.06313 0.24320 -0.03205 -12.37Costa Rica 0.04681 0.21123 0.00789 2.82Chile 0.06317 0.24326 -0.00376 -1.46Ecuador 0.06649 0.24913 -0.05353 -20.97El Salvador 0.05204 0.22211 -0.04368 -15.95Guatemala 0.04957 0.21705 -0.01135 -4.07Honduras 0.05549 0.22893 -0.03103 -11.50Mexico 0.06943 0.25419 -0.00584 -2.29Nicaragua 0.05079 0.21957 -0.07153 -25.91Panama 0.05267 0.22337 -0.02209 -8.17Paraguay 0.05176 0.22154 -0.02025 -7.41Peru 0.06031 0.23807 -0.08108 -30.99Uruguay 0.06407 0.24489 0.00576 2.22Venezuela 0.06480 0.24616 -0.00780 -2.94y2009 0.51648 0.49973 0.00265 2.42

27706.02 R 0.276502 R F value= 489 N=31,916

As expected the standard of living of an individual,other things constant, is higher in the capital, formarried individuals and males, increases with one’sage and education and one’s father’s education.

Ceteris paribus it is also highest in Brazil, Argentina,Costa Rica and Uruguay and lowest in Peru,Nicaragua, Ecuador, Bolivia and El Salvador.

6. A Shapley Decomposition of the Determinants of Distributional Changeand Change in Bipolarization

On the Shapley decomposition.

The case of four variables:Let us give a simple illustration where the index I depends on four determinantsa,b,c,d. Table 1 below gives all the possible ways of ordering these four elements.4

Table 1: The 24 ways of ordering four elementsa appears in the first positiona appears in the second positiona appears in the third positiona appears in the fourth positionabcd bacd bcad bcdaabdc badc bdac bdcaacbd cabd cbad cbdaacdb cadb cdab cdbaadbc dabc dbac dbcaadcb dacb dcab dcbaThe so-called Shapley decomposition looks at all possible elimination sequences. AsTable 1 indicates, there are 6 cases where a appears in the first position, 6 where itappears in the second, etc…If we look at the various ways of eliminating a, we can say that if a is eliminated firstits contribution to the indicator will be equal to the difference between the value ofthe indicator when all four determinants are different from zero and its value when ais equal to zero. In that case (a eliminated first) the contribution of a will be written as

C(a/a eliminated first) = [I(a0, b0, c0, d0) – I(a=0, b0, c0, d0)]Clearly, looking at the first column of the table above, there are six such cases out of24 possible orderings.5

The case where a is eliminated second is based on the orderings given in the secondcolumn of this table. There are three possibilities:- b is eliminated first and a second, so that the contribution of a in this case willbe written as

C(a/a eliminated second and b first)=[I(a0,b=0,c0,d0)–I(a=0,b=0,c0,d0)]and this case appears twice in the second column of the table.- c is eliminated first and a second so that the contribution of a in this case will bewritten as

C(a/a eliminated second and c first)=[I(a0,b0,c=0,d0)–I(a=0,b0,c=0,d0)]and this case appears also twice in the second column of the table.- d is eliminated first and a second so that the contribution of a in this case will bewritten as

C(a/a eliminated second and d first)=[I(a0,b0,c0,d=0)–I(a=0,b0,c0,d=0)]and this case appears also twice in the second column of the table.6

The case where a is eliminated third is based on the orderings given in the thirdcolumn of this table. There are again three possibilities:- b is eliminated first, c second and a third or c is eliminated first, b second anda third. In both cases the contribution of a will be written asC(a/a eliminated third and b and c first or second)

= [I(a0,b=0,c=0,d0)–I(a=0,b=0,c=0,d0)]- b is eliminated first, d second and a third or d is eliminated first, b second and athird. In both cases the contribution of a will be written as

C(a/a eliminated third and b and d first or second)

= [I(a0,b=0,c0,d=0)–I(a=0,b=0,c0,d=0)]- c is eliminated first, d second and a third or d is eliminated first, c second and athird. In both cases the contribution of a will be written asC(a/a eliminated third and c and d first or second)

= [I(a0,b0,c=0,d=0)–I(a=0,b0,c=0,d=0)]Finally the cases where a is eliminated last (fourth) appear in the fourth column of thetable above. There are 6 such cases but in each of these cases the contribution of amay be expressed as7

C(a/a eliminated fourth while b,c and d eliminated first, second or third)

= [I(a0,b=0,c=0,d=0)–I(a=0,b=0,c=0,d=0)]Taking all the 24 cases into account we therefore conclude that the overallcontribution of a to the indicator I may be expressed as

C(a) ={ (6/24) [I(a0, b0, c0, d0) – I(a=0, b0, c0, d0)]

+ (2/24) [I(a0,b=0,c0,d0)–I(a=0,b=0,c0,d0)]

+ (2/24) [I(a0,b0,c=0,d0)–I(a=0,b0,c=0,d0)]

+ (2/24) [I(a0,b0,c0,d=0)–I(a=0,b0,c0,d=0)]

+ (2/24) [I(a0,b=0,c=0,d0)–I(a=0,b=0,c=0,d0)]

+ (2/24) [I(a0,b=0,c0,d=0)–I(a=0,b=0,c0,d=0)]

+ (2/24) [I(a0,b0,c=0,d=0)–I(a=0,b0,c=0,d=0)]

+ (6/24) [I(a0,b=0,c=0,d=0)–I(a=0,b=0,c=0,d=0)]}One can naturally derive in a similar way the overall contributions C(b) of b, C(c) of cand C(d) of d to the value of the indicator I. Moreover it is also easy to verify that

C(a) + C(b) + C(c) + C(d) = I(a0, b0, c0, d0)

where I(a0, b0, c0, d0) is in fact the original value of the index I.

The results of such decompositions are given in tables8 to 11. It seems however that more interesting resultscould have been obtained, had we implemented thewhole analysis (correspondence analysis, regression,Shapley decomposition) separately for each country.

Another possibility is to implement a moresophisticated Shapley decomposition, one that makesa distinction on one hand between individual valuesand mean values at the country level, on the otherhand between mean values at the country level andoverall mean values (at the level of the wholeLatinóbarometro sample).

Table 8: Shapley Decomposition of the Index of Overall Distributional Change.

Circumstances Efforts OtherExplanatoryVariables

Residual Total

Argentina -0.00372 -0.00377 0.13561 -0.07903 0.04909

Bolivia -0.06565 -0.10392 0.14435 -0.06307 -0.08828

Brazil -0.05679 -0.06245 0.12408 -0.00773 -0.00289

Colombia -0.04834 -0.10212 0.12161 -0.05005 -0.07890

Costa_Rica -0.00185 -0.07488 0.09264 0.00534 0.02126

Chile -0.03086 -0.02665 0.14931 -0.04770 0.04410

Ecuador -0.06557 -0.10702 0.11625 0.03479 -0.02155

El_Salvador -0.10720 -0.09362 0.09621 0.11670 0.01208

Guatemala -0.10641 -0.16574 0.05916 0.00008 -0.21290

Honduras -0.04806 -0.14449 0.08147 -0.01248 -0.12356

Mexico -0.00418 -0.03755 0.08551 0.00959 0.05337

Nicaragua -0.06334 -0.18182 0.12245 0.04488 -0.07783

Panama -0.07113 -0.13585 0.11029 -0.00026 -0.09694

Paraguay -0.02702 -0.07836 0.07432 -0.06356 -0.09461

Peru -0.07919 -0.11310 0.16252 0.06724 0.03748

Uruguay 0.02607 -0.03246 0.11906 -0.03122 0.08145

Venezuela -0.03687 0.03003 0.01009 0.05276 0.05601

Residual Total

Argentina -7.6 -7.7 276.2 -161.0 100

Bolivia 74.4 117.7 -163.5 71.4 100

Brazil 1967.3 2163.5 -4298.5 267.7 100

Colombia 61.3 129.4 -154.1 63.4 100

Costa_Rica -8.7 -352.3 435.8 25.1 100

Chile -70.0 -60.4 338.5 -108.2 100

Ecuador 304.3 496.6 -539.5 -161.4 100

El_Salvador -887.1 -774.7 796.1 965.7 100

Guatemala 50.0 77.8 -27.8 0.0 100

Honduras 38.9 116.9 -65.9 10.1 100

Mexico -7.8 -70.4 160.2 18.0 100

Nicaragua 81.4 233.6 -157.3 -57.7 100

Panama 73.4 140.1 -113.8 0.3 100

Paraguay 28.6 82.8 -78.6 67.2 100

Peru -211.3 -301.8 433.7 179.4 100

Uruguay 32.0 -39.9 146.2 -38.3 100

Venezuela -65.8 53.6 18.0 94.2 100

Table 9: Shapley Decomposition of the Index of “Pure Growth Related”

Distributional Change.

Residual Total

Argentina -0.00725 0.03014 0.14557 -0.05240 0.11607

Bolivia -0.08211 -0.14622 0.16868 -0.07137 -0.13102

Brazil -0.05132 -0.04943 0.15423 0.01573 0.06921

Colombia -0.04616 -0.13852 0.14187 -0.03426 -0.07707

Costa_Rica 0.02377 -0.08809 0.11799 0.03840 0.09207

Chile -0.01753 -0.00915 0.16367 -0.01742 0.11957

Ecuador -0.06221 -0.17416 0.14558 0.05918 -0.03161

El_Salvador -0.12162 -0.11953 0.12541 0.14426 0.02851

Guatemala -0.11791 -0.20276 0.07160 -0.10011 -0.34918

Honduras -0.07060 -0.15006 0.11725 -0.04396 -0.14737

Mexico 0.00030 -0.03681 0.08306 0.00541 0.05196

Nicaragua -0.09452 -0.22485 0.12981 0.02277 -0.16679

Panama -0.08519 -0.16863 0.10668 -0.03180 -0.17894

Paraguay -0.02457 -0.10351 0.08939 -0.09286 -0.13154

Peru -0.14590 -0.10285 0.21481 0.11564 0.08171

Uruguay 0.04207 -0.02512 0.14894 -0.02281 0.14308

Venezuela -0.02984 0.04039 0.01242 0.05733 0.08029

Residual Total

Argentina -6.2 26.0 125.4 -45.1 100

Bolivia 62.7 111.6 -128.7 54.5 100

Brazil -74.2 -71.4 222.8 22.7 100

Colombia 59.9 179.7 -184.1 44.5 100

Costa_Rica 25.8 -95.7 128.2 41.7 100

Chile -14.7 -7.7 136.9 -14.6 100

Ecuador 196.8 550.9 -460.5 -187.2 100

El_Salvador -426.5 -419.2 439.8 505.9 100

Guatemala 33.8 58.1 -20.5 28.7 100

Honduras 47.9 101.8 -79.6 29.8 100

Mexico 0.6 -70.9 159.9 10.4 100

Nicaragua 56.7 134.8 -77.8 -13.7 100

Panama 47.6 94.2 -59.6 17.8 100

Paraguay 18.7 78.7 -68.0 70.6 100

Peru -178.6 -125.9 262.9 141.5 100

Uruguay 29.4 -17.6 104.1 -15.9 100

Venezuela -37.2 50.3 15.5 71.4 100

Table 10: Shapley Decomposition of the Index of “Shape Related”

Distributional Change.

Residual Total

Argentina -0.00017 -0.03539 -0.00465 -0.02607 -0.06629

Bolivia 0.01654 0.03216 -0.02024 0.01568 0.04414

Brazil -0.00870 -0.01662 -0.01617 -0.02627 -0.06777

Colombia -0.00857 0.02041 -0.00452 -0.03053 -0.02322

Costa_Rica -0.02742 0.01529 -0.02432 -0.03638 -0.07283

Chile -0.01336 -0.01716 -0.00387 -0.03039 -0.06478

Ecuador -0.00285 0.05346 -0.02104 -0.01740 0.01217

El_Salvador 0.02721 0.02154 -0.02811 -0.02833 -0.00769

Guatemala 0.03285 0.04473 -0.01691 0.09616 0.15683

Honduras 0.02821 0.00855 -0.04315 0.03388 0.02749

Mexico -0.00612 -0.00250 0.00351 0.00535 0.00024

Nicaragua 0.03785 0.04472 -0.00562 0.03352 0.11047

Panama 0.01555 0.01735 0.00820 0.04607 0.08717

Paraguay -0.00028 0.02450 -0.01244 0.00824 0.02002

Peru 0.05472 -0.02056 -0.02525 -0.03564 -0.02674

Uruguay -0.01390 -0.01396 -0.02110 -0.01442 -0.06337

Venezuela -0.00716 -0.01295 -0.00157 -0.00593 -0.02761

Residual Total

Argentina 0.3 53.4 7.0 39.3 100

Bolivia 37.5 72.9 -45.9 35.5 100

Brazil 12.8 24.5 23.9 38.8 100

Colombia 36.9 -87.9 19.5 131.5 100

Costa_Rica 37.6 -21.0 33.4 49.9 100

Chile 20.6 26.5 6.0 46.9 100

Ecuador -23.4 439.2 -172.9 -142.9 100

El_Salvador -353.8 -280.2 365.5 368.4 100

Guatemala 20.9 28.5 -10.8 61.3 100

Honduras 102.6 31.1 -157.0 123.3 100

Mexico -2528.2 -1032.1 1449.1 2211.2 100

Nicaragua 34.3 40.5 -5.1 30.3 100

Panama 17.8 19.9 9.4 52.9 100

Paraguay -1.4 122.4 -62.1 41.2 100

Peru -204.7 76.9 94.5 133.3 100

Uruguay 21.9 22.0 33.3 22.8 100

Venezuela 25.9 46.9 5.7 21.5 100

Table 11: Shapley Decomposition of the Index of Change in Bipolarization.

Residual Total

Argentina -0.00584 -0.02109 0.01131 0.02748 0.01187

Bolivia -0.03234 0.03182 0.02269 -0.02340 -0.00123

Brazil -0.01421 0.00133 0.01981 0.02428 0.03121

Colombia -0.01972 -0.00128 0.02478 0.01888 0.02266

Costa_Rica -0.00651 -0.00089 0.01161 0.00957 0.01379

Chile 0.01035 -0.00395 0.01546 0.00751 0.02937

Ecuador -0.01768 0.02064 0.02118 -0.03180 -0.00767

El_Salvador -0.07791 -0.00181 0.02018 0.07424 0.01470

Guatemala -0.03537 -0.00223 0.01413 -0.00598 -0.02945

Honduras -0.04669 -0.00477 0.02951 0.01300 -0.00895

Mexico -0.00016 0.00263 0.00036 0.03414 0.03698

Nicaragua -0.05374 -0.00600 0.02555 -0.05122 -0.08540

Panama -0.00688 0.01340 0.01802 -0.03317 -0.00862

Paraguay -0.01027 -0.00652 0.00880 0.04772 0.03973

Peru 0.01844 0.01827 0.03962 0.01066 0.08699

Uruguay -0.00729 -0.01985 0.02447 0.01562 0.01295

Venezuela -0.04880 -0.02461 0.02031 0.03088 -0.02222

Residual Total

Argentina -49.2 -177.8 95.3 231.6 100

Bolivia 2632.2 -2589.8 -1847.4 1905.0 100

Brazil -45.5 4.3 63.5 77.8 100

Colombia -87.0 -5.7 109.3 83.3 100

Costa_Rica -47.2 -6.5 84.2 69.5 100

Chile 35.2 -13.5 52.6 25.6 100

Ecuador 230.6 -269.1 -276.1 414.7 100

El_Salvador -530.1 -12.3 137.3 505.1 100

Guatemala 120.1 7.6 -48.0 20.3 100

Honduras 521.7 53.3 -329.7 -145.3 100

Mexico -0.4 7.1 1.0 92.3 100

Nicaragua 62.9 7.0 -29.9 60.0 100

Panama 79.8 -155.4 -209.0 384.6 100

Paraguay -25.8 -16.4 22.2 120.1 100

Peru 21.2 21.0 45.5 12.3 100

Uruguay -56.3 -153.3 189.0 120.6 100

Venezuela 219.7 110.8 -91.4 -139.0 100

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