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Estimation of OpportunityInequality in Brazil usingNonparametric Local LogisticRegressionErik Alencar de Figueiredo a & Flávio AugustoZiegelmann ba Department of Economics , Universidade Federal daParaíba , Brazilb Department of Statistics , Universidade Federal doRio Grande do Sul , BrazilPublished online: 27 Oct 2010.

To cite this article: Erik Alencar de Figueiredo & Flávio Augusto Ziegelmann(2010) Estimation of Opportunity Inequality in Brazil using Nonparametric LocalLogistic Regression, The Journal of Development Studies, 46:9, 1593-1606, DOI:10.1080/00220388.2010.500661

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Estimation of Opportunity Inequalityin Brazil using Nonparametric LocalLogistic Regression

ERIK ALENCAR DE FIGUEIREDO* & FLAVIO AUGUSTOZIEGELMANN***Department of Economics, Universidade Federal da Paraıba, Brazil, **Department of Statistics,

Universidade Federal do Rio Grande do Sul, Brazil

Final version received 13 April 2010

ABSTRACT This article measured opportunity inequality in Brazil by combining a series oftheoretical and empirical tools. The database was built using a two-sample instrumental variable(TSIV), developed by Angrist and Krueger. After that, the axiomatic approach put forward byO’Neill et al. was used, in which the estimation of children’s income distribution function isconditional on their fathers’ wages. The inference process was based on nonparametric locallogistic regression. The results indicate that Brazil has a high level of opportunity inequality. Inother words, in the context of intergenerational mobility, those whose fathers belong to lowerincome strata have to expend greater effort in order to attain a certain income level.

I. Introduction

Income inequality is often regarded as undesirable. Theoretical approaches, such asthe one used by Atkinson (1970), posit that, under some assumptions, a lowerinequality level produces a higher social optimum. Philosophical theories of justicetend to focus on the social benefits of a more egalitarian income distribution, basedupon a max–min strategy for an index of primary goods (see, among others, Rawls,1971). In this regard, outlooks on equality rely on greater political and popularappeal, which call for the adoption of income redistribution policies.

However, the relation between inequality and welfare might not be so strong insome societies. Recently, Alesina et al. (2001) argued about the impact of inequalityon the welfare of European and US citizens. Their conclusions point out that incomeinequality has a negative effect on the happiness of European people, but not on thehappiness of the US population, since Americans associate poverty with inefficiency,whereas Europeans reckon it as ‘bad luck’. Such heterogeneity shows that incomedistribution policies may or may not be socially desirable.

Correspondence Address: Erik Alencar de Figueiredo, Department of Economics, Universidade Federal da

Paraıba, Brazil. Email: eafigueiredo@gmail.com

Journal of Development Studies,Vol. 46, No. 9, 1593–1606, October 2010

ISSN 0022-0388 Print/1743-9140 Online/10/091593-14 ª 2010 Taylor & Francis

DOI: 10.1080/00220388.2010.500661

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The latter argument, if taken to the extreme, raises a question: is income inequality asocial problem? The answer necessarily encompasses a conceptual expansion, leadingto the notion of opportunity inequality. Conventional approaches are usually based onthe inequality of results, which is not totally satisfactory given that income differencesmay originate from different opportunities accessible to people during their lifetime.Based on these facts, this study aims to measure opportunity inequality in Brazil.

The choice of Brazil as subject of study in this case is due to at least two aspects.First, Brazil has one of the worst income distributions in the world.1 Second, theBrazilian society has a high social debt, which has been the basis of incomeredistribution and compensation policies.2 Nevertheless, the formulation ofmeasures to fight opportunity inequality runs into a major difficulty, since theseindicators are not calculated by taking into account earnings inequality only, butalso the inequality conditional on the opportunity set for every individual.3

In order to circumvent this problem, the study uses the axiomatic approach putforward by O’Neill et al. (2001). This framework is based on the concept ofintergenerational mobility, allowing for the construction of the opportunity setbased on an individual’s father’s income. This enables calculating children’s incomeindicators conditional on fathers’ income. To do that, a nonparametric statisticalmethod is used. It is based on a local logistic estimation of the income cumulativedistribution function of individuals conditional on fathers’ income.The remainder of the paper is organised as follows. Section II deals with the

concept of opportunity equality, establishing the axioms for the construction of theopportunity set. Section III describes the database and the statistical methodologyused. The major empirical results are shown in Section IV. To conclude, Section Vpresents the final remarks.

II. Opportunity Equality

Roemer (1998) states that income inequality among individuals is originated fromtwo sources: responsibility and non-responsibility factors.4 The former are concernedwith variables that denote the effort made by individuals, such as level of education,decision to migrate, number of hours worked in a year, among others. The latter arebeyond the control of economic agents and are represented by variables ofcircumstance, that is, family background (level of education, fathers’ occupation),and individual attributes such as race, gender, age or place of birth, among others.5

The distinction between these two factors has a considerable implication: onlyinequality arising from the variables of circumstance is socially undesirable. And thatis where ‘offensive’ and ‘inoffensive’ inequalities ensue from. Under this approach, itis possible to decompose the inequality of a given income distribution, I(ya), bydeveloping two counterfactual distributions: one that is free of inequality related tocircumstances, I(yc), and one in which no inequality results from the effort, I(ye). It iseasy to perceive that opportunity inequality will be represented by I(ye).Notwithstanding, the development of counterfactual distributions is not an easy

task. Recently, Ramos and Van de Gaer (2009) have cited a number of difficultiessurrounding this empirical challenge, starting with the definition of effort. Wheneffort is regarded as an observable factor, studies encounter problems withendogeneity and identification of functional forms of earnings equations. On the

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other hand, the approach that considers effort to be an unobservable factor relies onassumptions about the behaviour of agents who belong to the same class ofcircumstances. Quite often, the validity of Roemer’s identification axiom (Roemer,1996) is assumed. Moreover, regardless of how effort variables are defined,approaches are open to fierce criticisms. A major criticism is that the calculationof inequality indicators for the counterfactual distribution, I(ye), does not complywith the axiom of anonymity (Devooght, 2008).

Therefore, the present study used the approach developed by O’Neill et al. (2001).In brief, the method consists in assessing an individual’s social status, taking his setof opportunities (or variables of circumstances) into account. To achieve that, it isnecessary to define the set of opportunities and to establish a relationship betweenthis set and the conditional distribution functions.6

Following O’Neill et al. (2001) we will consider that the opportunity set for anindividual, Sx, is determined by his non-responsibility characteristics, x. The fact isthat the results, or the income level of an individual, will depend on his effort, orresponsibility characteristics, conditional on the opportunity set. These results maybe summarised by z¼ y[e, x], where z is the utility or the income throughout life ande is a variable that represents effort.7

Assume that the distribution function of e is continuous. Some axioms should alsobe postulated.

SINC (Strictly Increasing): y[e,x] is strictly increasing in e.

The interpretation of this axiom is straightforward: a higher effort will give rise toa higher level of utility. By defining F�z zjxð Þ and F�e ejxð Þ as cumulative distributionfunctions of z and e conditional on x, respectively, then the SINC axiom can bewritten as:

F�z y½e; x�jxð Þ ¼ F�e ejxð Þ: ð1Þ

that is, the level of effort of an individual conditional on a class of the opportunityset will be lower than an a-th percentile of the distribution of this effort if and only ifthe result is smaller than an a-th percentile of the distribution of this resultconditional on the same class of the opportunity set.

IND (Independence): F�e ejxð Þ is independent from x.

This means that no differences are assumed in the effort distribution functionsbetween different types of individuals, that is, between individuals with different non-responsibility characteristics. Admitting that such assumption is not valid would bethe same as stating that an individual would exhibit different levels of effortdepending on his (non-responsibility) characteristics.

Finally, Equation (1) and IND provide the following axiom:

RIA (Roemer’s Identification Axiom): F�z y½e0; x1�jx1ð Þ ¼F�z y½e00; x2�jx2ð Þ ) e0 ¼ e00.

This axiom informs that two people with different opportunity levels, but with thesame distribution percentile within their type, have the same level of effort.

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That being said, define pi ¼ F�z zjxð Þ as the cumulative distribution of result (incomelevel) z conditional on non-responsibility characteristics, x. Assume that this func-tion is strictly increasing in z. Thus, F��1z pjxð Þ expresses the income level obtained byan individual of type x who belongs to percentile 100* p. On account of RIA,observing the value of F��1z pjxð Þ will be the same as observing the value of y[e,x].Therefore, F��1z pjxð Þ will provide information on responsibility characteristics and

non-responsibility characteristics of individuals. O’Neill et al. (2001) assert that thisallows us to draw income as a function of p 2 [0,1] for different values of x. Theopportunity set for a particular type x is given by the outcomes someone of type xcan obtain by varying his responsibility characteristic e or p. This way, theopportunity set of individuals of type x will be

Sx ¼ ðz; pÞ 2 ðRþ � ½0; 1�Þjz ¼ F��1z pjxð Þ� �

; ð2Þ

whereRþ is the set of non-negative real numbers.The visual inspectionof set inEquation(2) will show the level of opportunity inequality of different types of individuals and alsotowhat extent different options or levels of effort yield different results (income). Finally,one should determine the elements of x. A possibility is to consider the context ofintergenerational mobility, as dealt with in Van de Gaer et al. (1998) and O’Neill et al.(2001). Thus, z will represent the children’s income and x the fathers’ income.The literature mentions some methods for estimating children’s income,

conditional on their fathers’. Bourguignon et al. (2007) propose a multivariateparametric method in which the endogeneity problem is approached usinginstrumental variables. As previously highlighted, this strategy is open to criticismsabout the specification of functional forms, in addition to other limitations. Eventhough Lefranc et al. (2008) do not use fathers’ income as variable of circumstance,they establish a relationship between responsibility and non-responsibility variablesusing stochastic dominance tests. Pistolesi (2009), on the other hand, facesspecification problems while using a nonparametric model.Given these possibilities, the present study utilises a bivariate model where the

estimation of children’s income conditional on fathers’ income will be made using anonparametric method, namely, the nonparametric local logistic regression. Therationale for this choice lies on the statistical properties of the resulting estimatedcumulative distribution function, as discussed in the next section. All in all, thestatistical approach adopted here seems more adequate than the one used, forinstance, by O’Neill et al. (2001).

III. Data and Statistical Methodology

This section presents the main elements of the empirical strategy and statisticalmethodology used in this study. First, there is a brief outline of the database used,and thereafter the statistical approach is described.

Data

The data used in this study were obtained from the Brazilian National HouseholdSurvey (PNAD). This survey has been conducted by the Brazilian Institute of

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Geography and Statistics (IBGE) since the late 1960s and consists of a basicquestionnaire that includes questions about household and personal characteristics,such as family size, household income, educational background, number of workinghours, personal income, among others. In some years, some special characteristics areinvestigated and then summarised in supplemental issues. These special character-istics include, for instance, health, food safety, child labour and social mobility.

The latest supplement contains data on individuals and their fathers, and wasapplied in years 1973, 1976, 1982, 1988 and 1996. More specifically, it givesinformation on education, schooling and parent’s and children’s occupation.However, the major problem with this survey is the lack of a ‘father’s wage’variable. That is, in a study of intergenerational mobility, it is not possible to regresschildren’s income on fathers’ income.

Due to the lack of this information, it is necessary to use a method that allowsbuilding an approximation for fathers’ income. This is provided by the two-sampleinstrumental variable (TSIV), designed by Angrist and Krueger (1992). In summary,one seeks to estimate fathers’ income (predicted wage) based on characteristics suchas educational and professional background.

To do that we considered two samples, called respectively fathers’ sample andchildren’s sample. Both contain information on male household heads aged 25 to 65years, with a workload of 40 hours or more per week in all jobs, who live in theurban zone.

The fathers’ sample was constructed using the 1976 PNAD data. These dataprovide information about (hourly) wage,8 years of schooling and occupation. Thefirst step was to create dummy variables for occupation and schooling. The 927occupations described in the survey were classified into six categories according toPastore and Silva (1999).9 Education was divided into seven categories: no education(less than one year of schooling); incomplete lower elementary education (from oneto three years of schooling); complete lower elementary education (four years ofschooling); incomplete upper elementary education (from five to seven years ofschooling); complete upper elementary education (eight years of schooling);incomplete or complete high school education (nine to 11 years of schooling) andincomplete or complete college education (over 11 years of schooling).

The ultimate goal of this stage is to perform the regression of the logarithm ofwage on dummy variables for education and occupation. Given that some studies,including those by Menezes Filho et al. (2003) and Ferreira and Veloso (2006),highlight the change in the school premium in several cohorts throughout time inBrazil, it is important to include dummies for year of birth and for the relationshipbetween occupation and schooling. Finally, the synthetic profile of fathers was builtusing a wage equation.10

The second stage consisted in collecting information on male household headsaged 25 to 65 years, who worked 40 hours or more a week in all jobs, and lived in theurban zone, by using the 1996 PNAD supplement. This database is referred to as‘children’s sample’. This sample includes information on children’s income and onfathers’ education and occupation. The sample comprises 25,927 individuals withfive years of schooling and an average hourly wage of 8.8 Reais (minimum wagevalues realised in 1996). Fathers’ low educational level is a major feature. Around 52per cent have less than three years of schooling. Because of that, most (60%) hold

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jobs in sectors where human capital stock is quite low (categories 1 to 3). Finally, thewages for the ‘synthetic’ fathers were based on the coefficients estimated in the firststage (in the fathers’ sample). Thus, we obtained the two variables of interest of thisstudy: children’s wage and fathers’ predicted wage.

Estimation of the Conditional Distribution Function using Local Logistic Regression

The correct specification of a parametric model is an important challenge forestimation and inference processes. Certainly, the perfect specification of thefunctional form is the ‘first best’ for the empirical strategy; nonetheless, theory doesnot always provide the necessary tools to attain that (Li and Racine, 2007). Thus,such a hindrance opens the path for nonparametric models as an importantalternative. Concisely, the assumption that the functional form is known and can bedescribed from a finite number of parameters is replaced by a model in which theunknown underlying function to be estimated (not specified in the model) is onlysupposed to possess a few derivatives at the points of interest. In other words, for thespecific case of this study, the function F��1z pjxð Þ described in section II will beestimated using a method that does not impose a priori knowledge about itsfunctional form. Nonparametric methods have been widely used in economicsstudies, especially in microeconometrics.11 Although these techniques are based onsomehow complex mathematical derivations, their results are easily understood,even by those who show no interest in modelling details. In short, the proposedmethod in this section is used to estimate the cumulative distribution of anindividual, conditional on his father’s income. As previously mentioned, the use ofthis tool can be justified by the fact that the variables are continuous and thefunctional form that establishes a relationship between them is unknown.In statistical terms, nonparametric methods are quite flexible for the estimation of

unknown curves. In this context, kernel smoothing tools provide estimators that are bothflexible and intuitive, besides having good asymptotic properties. Silverman (1986) andFan andGijbels (1996) comprehensively describe these techniques, while Simonoff (1996)and Bowman and Azzalini (1997) introduce a more intuitive and applied approach.Our approach here is based on local regression via kernel smoothing. Informally

speaking, the idea is to run local regressions for each point on a grid constructed fora covariate instead of running a global parametric regression. In this sense, theunknown underlying function is locally replaced by its Taylor’s series approximationto a certain order. On the top of that, to estimate a function value at a point x, thosesample points, let’s say (Xi,Yi), whose X variable assumes values near to x will havemore weight to estimate the function value at x. Therefore sample points with firstcoordinate far from the x point at which the function is to be estimated will have asmall influence on the estimated value.In this article, as well as in a wide variety of statistical problems, the objective

curve to be estimated is the cumulative conditional distribution function. Consider,for instance, the estimation of the quantile function of Z given X, using a randomsample with pairs fðX1;Z1Þ; . . . ; ðXn;ZnÞg. Yu and Jones (1998) suggest using the‘double kernel’ approach for the local linear regression. O’Neill et al. (2001) use amultiple-step procedure, whereby they first estimate the joint and marginal densitiesand then the joint, marginal and conditional distributions.

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A drawback of such methods is that they do not guarantee that the estimatedfunctions are non-decreasing and restricted to the interval [0,1]. In this regard, Hallet al. (1999) propose an estimator that satisfies these conditions. This estimator isknown as local logistic estimator of the conditional distribution function. The idea isto create an indicator response variable which assumes only values 0 or 1, and locallyregressing it via a logistic function against the covariate of interest. Note that thistype of logistic estimator is a particular case of a broader class, which we maydesignate as functional class. Ziegelmann (2002) uses this class of estimators toguarantee a non-negative estimator for conditional variance.

Let fðXi;ZiÞg be a random two-dimensional sample of size n. Our aim is toestimate the conditional distribution function pðzjxÞ � PðZi � zjXi ¼ xÞ. Note that,if we write Wi ¼ IðZi � zÞ where I(.) is the indicator function, then

EðWijXi ¼ xÞ ¼ pðzjxÞ:

Now assume that, for a fixed z, p(zjx) has r71 continuous derivatives. Therefore,we will choose the following logistic form to approximate p(zjx) locally (that is, inthe neighbourhood of x)

Lðx; yÞ ¼ exp fpðu� x; yÞg1þ exp fpðu� x; yÞg ; ð3Þ

where pðu� x; yÞ ¼ y0 þ y1ðu� xÞ þ :::þ yr�1ðu� xÞr�1 for u in the neighbourhoodof x.

This way, adjusting this model locally to the data of indicator functionWi will giveus the estimator bpðzjxÞ � exp fpð0;byÞg=f1þ exp fpð0;byÞgg ¼ exp fby0g=f1=expfby0gg, where by0 denotes the value of y0 which minimises

Xn1

Wi �exp fpðXi � x; yÞg

1þ exp fpðXi � x; yÞg

� �2

KhðXi � xÞ; ð4Þ

where KhðxÞ ¼ ð1=hÞKðx=hÞ, in which K(x) is a kernel function (traditionally asymmetric probability density function around 0) and h is the smoothing parameter,or bandwidth, which controls the level of complexity of the estimated curve. Hallet al. (1999) derive the asymptotic properties of this estimator in the broader contextof time series, where the case of an independant and identically distributed sample isa particular case.

In this paper K(.) will be the probability density function of a standard normaldistribution, while the smoothing parameter h will be chosen by cross-validation,that is so as to minimise something like an out-of-sample ‘forecast’ error (see Fanand Gijbels, 1996, for details).

IV. Results

This section presents the major results of this study. First, the cumulativenonparametric conditional distribution functions described in section III areestimated. After that, the results are briefly discussed.

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Inference

As previously seen, the opportunity set was estimated using the income levelestimated for the fathers. That is, the context of intergenerational mobility wasconsidered. In a recent study, Ferreira and Veloso (2006) showed that Brazil has lowintergenerational income mobility. In other words, fathers’ income is a determiningfactor for children’s income. In brief, elasticities ranged between 0.58 and 0.73,depending on the tools used in the regressions.Actually, these results indicate high opportunity inequality in Brazil. However, to

confirm this assumption, it is necessary to take into account the ‘children’s effort’variable. Table 1 shows the first sign of opportunity inequality for Brazil. This tablesummarises the results for the cumulative distribution conditional on children’sincome vis-a-vis the fathers’ income. The values were calculated using the toolsdescribed in section III. The analysis is simple; consider the first column of results,and in this case, fathers belong to the fifth percentile, that is, the poorest 5 per cent.Therefore, the estimated probability for children belonging to the poorest 25 percent, for example, amounts to approximately 0.43. In the same column, note that theestimated probability for the children finding themselves below the 95th percentileamounts to nearly 0.99. The differences between these two probabilities becomemore evident when we compare them with the results shown in the last column,where fathers belong to the 95th percentile. Note that, under this circumstance, theestimated probability for children being among the poorest 5 per cent is virtuallyequal to zero. In turn, the estimated probability for them to be on the top of thedistribution (the richest 5%) is 0.242 (1–0.758).In brief, there is some strong evidence that confirms the major conclusion drawn

by Ferreira and Veloso (2006): fathers’ income level seems to be a crucial factor forthe determination of children’s income. Nevertheless, how much of this behaviour isdue to non-responsibility characteristics?12 The answer to this question is provided inFigure 1, where we have the estimates for the Brazilian opportunity set. This figureshows the results for the estimated probabilities of children having the same incomeas or a lower income than a given relative income,13 since their fathers belong to the5th, 50th and 95th percentiles. In this case, we set the parent’s percentile, say, at 5 percent, and observe the cumulative probabilities of their children in relation to theirrelative income. One of the results from section II, RIA, indicates that two people

Table 1. Cumulative conditional probabilities – fathers and children

Fathers’ percentiles

Children’s percentiles 5th 25th 50th 75th 95th

5th 0.076 0.108 0.054 0.005 0.00025th 0.433 0.375 0.237 0.107 0.09350th 0.683 0.679 0.506 0.321 0.18575th 0.909 0.889 0.775 0.632 0.51195th 0.993 0.987 0.977 0.947 0.758

Source: Estimates made by the authors based on PNAD microdata (IBGE, 1976 and 1996).

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with different levels of opportunity, but belonging to the same distribution percentilewithin their type, exhibit the same effort level. This axiom allows us to establish anincome level where individuals with different opportunity sets can be assessed. Therationale behind it is simple: what is the level of average effort each individual shouldhave in order to get to this distribution percentile?

Figure 1 indicates that the higher the fathers’ income level, the lower the effortlevel children have to make to obtain a relative income equal to 1. Therefore, if aparent belongs to the 95th percentile, his child must have an effort level ofapproximately 0.19 to reach the average income. On the other hand, if the parent ispoor (5th percentile), his child should have an effort level of approximately 0.70 toobtain a relative income equal to one. In other words, between the extremes of theopportunity set, effort has to be more than three times greater in order to obtain thesame income level.

In terms of the significance of the differences between cumulative conditionaldistribution functions, we used the Kolmogorov-Smirnov test for two independentsamples. In this regard, we then provide a justification for its use. Although thesamples that originated the estimates were actually the same for all curves (a priorinot characterising independence), only one part of the sample is used for theestimation of each one of the curves (local estimate). This means that, depending onthe fathers’ percentile, only fathers’ income values close to the percentile at issue areused for the estimation conditional on that percentile. Considering the smoothingparameters obtained by cross-validation, there are no overlapping observationstaken into account in the three curves of Figure 1. This way, we have the equivalentto independent samples, with approximately 400 observations used for eachestimated curve. Thus, using the traditional Kolmogorov-Smirnov test for the twopairs of curves (5th percentile against the 50th percentile and the 50th percentile

Figure 1. Estimate for the opportunity set in Brazil.Note: The horizontal axis corresponds to the relative income of children (z), whereas thevertical axis represents the cumulative probability, P(Z5¼ zjX¼ x), for x in different

percentiles for the fathers.Source: Estimates made by the authors based on PNAD microdata (IBGE, 1976 and 1996).

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against the 95th percentile), we may conclude that the curves should be different at a5 per cent significance level.The magnitude of Brazilian opportunity inequality is evident when compared with

the results obtained for the USA. According to O’Neill et al. (2001), for the USA, thedifference in effort between individuals belonging to the 25th and 75th percentilesamounts to approximately 56 per cent. Brazilian data, shown in Table 1, reveal a 112per cent difference ([0.679/0.321]-1).In summary, results strongly suggest that, besides having a striking income

inequality, Brazil also has a high level of opportunity inequality. In other words, asignificant share of this inequality is associated with non-responsibility factors, morespecifically with fathers’ income in this study. This heats the debate aboutcompensatory policies. The subsequent section presents some results available fromthe literature and raises some questions about the compensatory policies adopted inBrazil.

Discussion

The results shown in the previous section are in agreement with earlier evidence.Bourguignon et al. (2007), for instance, conclude that non-responsibility factorsaccount for over one quarter of income inequality in Brazil. To obtain that result,the authors decomposed income by using counterfactual distributions andconsidering effort as an observable variable.Using a multivariate model, Ferreira and Gignoux (2008) investigate Brazil and

another five Latin American countries, observing that opportunity inequalityrepresents, on average, nearly 30 per cent of income inequality in those countries. Ina recent study, Figueiredo and Netto Junior (2010) adopt the method based onstochastic dominance and corroborate the statistical significance of the Brazilianopportunity inequality. Thereafter, the authors calculate the level of Brazilianopportunity inequality. Based on that, it was possible to make an ordinalcomparison with international inequality levels, described in Lefranc et al. (2008).In short, the Brazilian index is more than twice as high as the highest level obtainedin developed countries.It is common knowledge that the methods used in these studies, including those

suggested here, are open to harsh criticism. As pointed out, the calculation ofindicators for counterfactual distributions does not follow the axiom of anonymity(Devooght, 2008). On the other hand, bivariate models, as the one used above, canreveal biases especially from the omission of variables of circumstance.14 In otherwords, obtaining opportunity inequality indicators is open to strong criticism.Furthermore, the calculation of these indicators constitutes a smaller share of the

debate on opportunity inequality. The big challenge lies on the selection of anoptimal compensatory policy. Put differently, what political strategy should beadopted after detecting a high level of ‘offensive’ inequality?In the Brazilian case, income inequality has been redressed by redistribution

policies such as ‘Bolsa Famılia.’15 This strategy has been argued against as far as itscoverage (see Soares, 2009) and its conditionalities (see Medeiros et al., 2008) areconcerned. However, its impacts on the reduction of inequality have proved to beremarkable.16 According to Barros et al. (2007), Bolsa Famılia accounted for about

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20 per cent of the decrease in inequality in Brazil in the first half of the 2000s. Itseffect on poverty is nonetheless subtler. In the same period, the programme managedto reduce the percentage of poor people by only 1.7 percentage points, from 21.7 percent to 20 per cent of the population (Soares and Satyro, 2009).

Nevertheless, some authors, including Soares and Satyro (2009), warn against theprogramme’s lack of creation of new opportunities. This means that thecompensation strategy should offer qualification, training, family counselling,microcredit, local development and other programmes whose goal is to enablefamilies to escape poverty by their own means. As a matter of fact, despite theimportance of the issues raised by the authors, the adoption of an efficientcompensatory policy for promoting opportunity equality needs to go a step further.

First of all, the compensation of an individual for occasional welfare losses due toresponsibility factors is not acceptable. Therefore, a compensatory policy will onlybe efficient if it is possible to make a distinction between the factors that determineinequality. In this sense, cash transfers will act as neutralising factors for non-responsibility variables. According to Bossert et al. (1999), compensation theoriesbasically follow two paths. One approach attempts to integrate some idea ofindividual responsibility into the formulation of general social welfare functions. Thesecond approach is based on the definition of allocation rules that are more or lessspecific to the various economic models that are being investigated.

Amongst the large number of research studies about this issue, special attentionshould be paid to the approaches proposed by Roemer (1993) and Van de Gaer(1993), in which the framework of the social welfare function and the alternativesuggested by Bossert et al. (1999), heavily focused on specific compensatory rules,17

are contemplated. Roughly speaking, the model proposed by Roemer-Van de Gaeris based on tax systems with distinct values for each group of circumstances. Bossertet al. (1999) show that this approach has important limitations and suggest analternative based on individual talents and on responsibility variables. In brief, thetheoretical discussion on compensatory policies is comprehensive and includes thedifficulties in the distinction between responsibility and non-responsibility variablesand the limitations related to the tools necessary for detecting opportunityinequality.

From an empirical standpoint, Roemer et al. (2003) assess the tax systems of 10developed countries. Their variables of circumstance include fathers’ occupation andlevel of education. In summary, the tax systems is inefficient in promotingopportunity equality in most of the investigated countries. Betts and Roemer(2005) estimate the expenditure with education that would be able to equaliseopportunities among the various groups of circumstances. Their conclusions revealthat obtaining small gains with opportunity equality has a significant impact ongovernment budgets.

Based on this body of evidence, the conclusion is that the development of a policyfor promotion of opportunity equality requires vigorous debate. In the Braziliancase, the first step is to assess the efficiency of income redistribution policies in forcesince the second half of the 1990s. Then, it is necessary to investigate how the fiscalregime contributes to reducing opportunity inequality. Finally, the effects ofalternative compensatory policies have to be simulated based on the studiesdeveloped by Roemer et al. (2003) and Betts and Roemer (2005).

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In sum, the debate about opportunity inequality in Brazil has just begun. Theresearch agenda is extensive and requires not only improvement of the techniquesused to measure inequality, but also the assessment of current income redistributionpolicies and the suggestion of possible alternatives.

V. Final Remarks

This article measured opportunity inequality in Brazil. An approach based onintergenerational mobility was used as the main factor for the construction of theopportunity set. The database could only be built due to the use of the two-sampleinstrumental variable (TSIV), developed by Angrist and Krueger (1992). After theconstruction of the vectors using the children’s and fathers’ wages, a nonparametriclocal logistic regression was used to estimate the children’s income distributionconditional on fathers’ wage.This information, along with the axiomatic approach described in O’Neill et al.

(2001), allowed inferring on the level of effort put in by individuals conditional onthe level of opportunity. Then it was detected that Brazil seems to have a high levelof opportunity inequality. In other words, those individuals whose fathers belong tolower income strata have to expend greater effort in order to attain a certain incomelevel. However, the detection of a high level of opportunity inequality does not putan end to the debate. Therefore, the next step, left as a suggestion for furtherresearch, would be to discuss the necessity to adopt compensatory policies.

Acknowledgements

The authors thank professors Fernando A. Veloso and Sergio Ferreira. We take fullresponsibility for any errors or omissions. The first author wishes to thank thefinancial support of CNPq, by the Project 475225/2009-0.

Notes

1. See United Nations Development Program (2006).

2. From the second half of the 1990s, federal governments have adopted a series of income redistribution

policies, classified as ‘affirmative policies’. Among them, one may cite the system for preferred

admission for racial minorities (blacks and Indians) at public universities, food grants and school

attendance programmes.

3. Bossert et al. (1999) demonstrate that this procedure is not an easy task.

4. Alternative approaches define opportunity equality using the set of opportunities to which individuals

have access. For further details, see Pattanaik and Xu (1990) and Kranich (1996).

5. For a list of variables used in empirical studies, see Ramos and Van de Gaer (2009).

6. Recently, Lefranc et al. (2008) developed a model that assesses opportunity equality based on

conditional distributions and on stochastic dominance tests.

7. It should be underscored that notation z¼ y[e,x] excludes random factors (for example luck). For

further details, see Lefranc et al. (2009).

8. Earnings from all jobs divided by the number of working hours.

9. The classification follows the level of intensity of human capital required by the occupation. For

example, category 1 (lower) is made up of agricultural labourers, fishermen, wood cutters, and so

forth. Category 6 includes top-level managers, magistrates, those with higher education, and so forth.

10. The regression has 83 covariates and is robustly estimated under autocorrelation and hetero-

skedasticity. The results were omitted due to space restrictions.

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11. A list of applications is provided in Cameron and Trivedi (2005).

12. Note that the study conducted by Ferreira and Veloso (2006) only investigates intergenerational

mobility, overlooking opportunity inequality issues.

13. That is, the income normalised by the mean.

14. Fleurbaey (1998) discusses the limitations of bivariate models.

15. Bolsa Famılia was created in October 2003. It consists of direct cash transfer to poor families (poverty

is defined according to per capita income levels), conditional on some requirements (prenatal care for

pregnant women, maintenance of children in school, among others).

16. For a more in-depth discussion on the effects of this type of policies on the levels of poverty and

inequality, see Bourguignon et al. (2002).

17. For a survey of the literature, see Fleurbaey and Maniquet (2011).

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