The Measurement of Inequality of Opportunity

properly decomposable.13 In fact, for a given {yik }, estimates of between-group inequality can differ for three reasons: (i) the specific inequality index I() used in the decomposition; (ii) the path of the decomposition; and (iii) the decomposition procedure, i.e. whether it is estimated parametrically or non-parametrically.

Point (i) is well-established in the literature. The decomposition of inequality by population subgroup for a given distribution and partition will differ across different members of the Generalized Entropy or Atkinson families. To see point (ii), following Foster and Shneyerov (2000) and Checchi and Peragine (2005), define:

- a smoothed distribution {μik }, corresponding to a particular partition {yik }, as the distribution that arises from replacing yik with the group-specific mean μk .

- a standardized distribution {νik } corresponding to a particular partition {yik } as

the inequality in the original distribution. It summarizes between-group inequality in the partition directly, hence the subscript d.

A standardized distribution, on the other hand, suppresses all between-group inequality, leaving only inequality within-groups. θr =1− I ({νik })I ({yik }), one minus the ratio of inequality in the standardized distribution to the inequality in the original distribution, is therefore another perfectly plausible measure of inequality of opportunity. It computes the between-group inequality in the partition residually, hence the subscript

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13The literature on inequality measurement (e.g. Bourguignon 1979, Shorrocks 1980) has established that total inequality is only additively separable into a between-group component and a within-group component for some indices. The best-known family of additively decomposable measures is the generalized entropy class, which includes the mean log deviation (E(0)) and the Theil entropy index (E(1)). The Gini coefficient is not additively decomposable in the same way. See also Elbers et al. (2008).

14This straightforward, non-parametric decomposition is very similar to Checchi and Peragine (2005), who proposed either to re-weight the distributions of outcome in order to equalize the means of the different circumstances groups (in a “types approach”) or to reweight the means of the individuals who can be considered as having exerted the same efforts (in a “tranches approach”).

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Unfortunately, although there is no obvious reason why either one of these two paths should be preferred to the other, for most inequality indexes they will yield different measures of the share of between-group inequalities (and thus of inequality of opportunity).15 Foster and Shneyerov (2000) characterize the class of inequality measures for which the two methods yield the same results; namely the “path-independent decomposable” class of inequality measures. They show that when the set of inequality indices under consideration is restricted to those that use the arithmetic mean as the reference income, and that satisfy the Pigou-Dalton transfer axiom, this class reduces to a single inequality measure, the mean log deviation, or E(0).

This has a helpful implication for our attempt to measure inequality of opportunity using (4) or (5). It implies that, if we are prepared to adopt the FosterShneyerov path-independence axiom, then the first two of the three previously mentioned reasons for estimates of between-group inequality to vary are eliminated: if we focus on path-independent measures that also satisfy the transfer axiom, we must use the mean log deviation as our inequality index I(), and θd =θr . Differences due both to the use of different aggregation indices and to path-dependence are eliminated simultaneously.16

15 It is easy to see why the two decomposition paths yield different results for other generalized entropy measures. The decomposition of total inequality for these measures can be written as follows:

only eliminate IB. It will also affect IW, by changing the weights associated with the inequality within each sub-group.

16 There is a sense in which measures based on the smoothed distribution correspond to van de Gaer’s concept of “min-of-means”, because they focus exclusively on information about the mean advantage levels of each type, while the measures derived from the standardized distribution are closer to Roemer’s “mean-of-mins” approach, in the sense that it first removes differences between the types, and then compares the (“relative effort”) distributions within each type. See Fleurbaey (2008). In this paper, we deliberately gloss over that philosophical debate, other than to note that we identify a particular measure, axiomatically derived, for which both approaches yield identical results. We see that as an added advantage of our approach.

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sample is sufficiently large relative to the number of cells in the partition {yik }, then we need go no further: the between-group share of inequality for E(0), in a partition defined by a vector of circumstances, is our single scalar estimate of a lower bound for inequality of opportunity. Unfortunately, however, the richer the information set on people’s circumstances, the more cells one would like to include in the partition. As cell numbers increase, cell sizes diminish, leading to the classic problem of data insufficiency for nonparametric estimation. This has led some authors to propose parametric alternatives to the estimation of θd and θr .

In order to construct these alternatives, define a parametrically standardized distribution {~yi }, corresponding to F(y, C), as the distribution that arises from replacing

estimated. Once this has been done, {~yi } is obtained simply by replacing the individual circumstance values in (2) with the sample average for each circumstance variable. A variety of alternative specifications for (2) is possible, of course. Bourguignon et al. (2007) use a log-linear/linear specification of the form:

17 This is (parametrically) analogous to the standardized distribution because, by giving each and every individual the same circumstance variables, it eliminates any inequality between groups that are associated with circumstances.

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the error term of the model is suppressed. This counterfactual distribution is also obtained by estimating a specific parametric model for (2), and suppressing within-group inequality by replacing yi with its prediction, given the vector of circumstances C. In a reduced-form framework, and under the functional form assumptions above, the

Unlike θdN and θrN , θdP and θrP rely on specific functional form assumptions. In addition to the possible sample-size insufficiency for non-parametric estimation, there is another reason why the costs of such a parametric approximation may be worth bearing: the parametric approach permits the estimation of the partial effects of one (or a subset) of the circumstance variables, controlling for the others, by constructing alternative counterfactual distributions, such as:

in the case of a parametrically standardized decomposition. This allows us to compute circumstance J-specific inequality shares:

The existence of this trade-off between parametric and non-parametric methods - with non-parametric decompositions being more flexible (with no functional form assumptions) but more data intensive, while the parametric approach is less data intensive but relies on (potentially restrictive) functional form assumptions – points to two things: the need to investigate their comparative performance on the same data set (as we do below), and the possibility that the methods may usefully complement each other.

A final methodological consideration refers to the interpretation of these measures, given the realistic possibility that not all relevant circumstance variables may

18 The measure computed for Brazil by Bourguignon et al. (2007) was an example of a parametric residual measure: θrP .

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be observed. A “true” measure of inequality of opportunity, as conceptually defined in equations (3) – (5), would require that all relevant circumstance variables be included in the vector C. This is unlikely to be the case in practice for almost any conceivable data set, and certainly for the six countries we study below. The empirical estimates defined in this section – whether direct or residual, and parametric or not – should be interpreted as lower-bound estimates of inequality of opportunity. To see why they are lower bounds, note that including an additional element in vector C causes each and every cell in the partition {yik }to be further subdivided (into at least another two cells). This can not lower the between-group inequality share and, unless the additional element is orthogonal to the measure of advantage, will raise it.

An additional reason why the measure is a lower-bound is that the partitioning of the population into categories within each circumstance variable in C is very coarse in this paper. An example is the classification of parental occupations into only two cells: “agricultural worker” or “other”. For most circumstance variables, international comparability required aiming for “common denominator”, relatively aggregated classifications. Like adding other circumstance variables, further subdivision of these categories within each circumstance might also increase (but could not reduce) the share of inequality attributed to opportunities.

Similarly, in the parametric case, notice that equation (7) is a reduced-form specification, intended to capture both the direct and indirect effects of circumstances on advantage. Adding another element of the vector C to this specification (or further refining the set of dummy variables for each circumstance) must reduce the variance of the residual and increase the variance (or any other inequality measure) accounted for by the set of observed circumstances.19

In what follows, we calculate three of the four members of the relative-Θ class – namely θdN , θrN and θrP - for the distributions of earnings, income and consumption in

19 A possible misunderstanding would be to argue that, because certain omitted circumstances might be negatively correlated with the observed vector C, the parametric measure need not be a lower bound. It is of course possible that the share of inequality attributed to a specific set of (observed) circumstances is overestimated. This might happen if omitted circumstance variables are negatively correlated with the observed ones. But the R2 of regression (7) can not fall by including these other circumstance variables, so that the estimate is a lower-bound for the share of inequality attributed to all circumstances (rather than to the observed subset), analogously to the non-parametric case.

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recent household surveys from six Latin American countries. We show that the two nonparametric indices differ for decomposable inequality measures other than E(0), but focus our discussion on that index, for which θdN = θrN . We then examine the differences between the non-parametric and parametric estimates. Although, as expected, these differences are larger for smaller sample sizes, they are generally quite small, suggesting that our path-independent estimates of inequality of opportunity are also methodologically robust to the choice of estimation procedure. Before presenting the results for each concept in Sections 5 and 6, the next section briefly describes the data sets.

4.The Data and the Partition by Circumstances

We use data from six nationally representative household surveys in Latin America, namely the Brazilian Pesquisa Nacional por Amostra de Domicílios (PNAD) 1996; the Colombian Encuesta de Calidad de Vida (ECV) 2003; the Ecuadorian Encuesta Condiciones de Vida (ECV) 2006; the Guatemalan Encuesta Nacional sobre Condiciones de Vida (ENCOVI) 2000; the Panamanian Encuesta de Niveles de Vida (ENV) 2003; and the Peruvian Encuesta Nacional de Hogares (ENAHO) 2001. The ENAHO and the PNAD are original national surveys, while the others are LSMS-type surveys. This particular group of surveys were selected for containing information on family background and, more specifically, on parents’ education, father’s occupation, or both.20

In all countries, we restrict the sample to individuals aged 30 to 49, which are the cohorts with the highest proportion of employed persons.21 Sample sizes for each survey, both before and after excluding observations with missing data, are reported in Table 1. Sample sizes with complete information range from about 6,000 (for Panama) to 72,000 observations (for Brazil), for the analysis of income and consumption, and from about

20The Encuesta Nacional sobre Niveles de Vida de los Hogares (MxFLS) 2002 contains a similar set of variables. Unfortunately its simple size proved to be too small relative to the partition to generate reliable results.

21For Brazil, we further restrict the sample to household heads and their spouses, as the family background information was collected only for these individuals.

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4,000 to 50,000 for the analysis of earnings (when only employed individuals are retained).22

The surveys contain information on a common set of circumstances: (a) three variables related to family background: father’s and mother’s education and father’s occupation during the person’s childhood; (b) ethnicity (or race); and (c) region of birth (or type of area of birth). The only exceptions are that the father’s occupation variable is not available for Colombia and Peru, and results must be interpreted with this caveat in mind. We also use gender as a circumstance variable in the analysis of earnings. Parental education variables are coded into three categories: no education (or unknown), primary (incomplete or complete, depending on the country), and complete primary or secondary and more.23 Father’s occupation is recoded into two categories: agricultural workers and others. Ethnicity (coded in two categories) is captured either by self-reported ethnicity or by the ability to speak an indigenous language. Region of birth is coded into three broad regions (one being generally the capital area) but is captured by the type of area (urban/rural) for Panama. Table 2 describes the specific definitions of the circumstance variables in each survey in greater detail. Table 3 (Panel A) presents the corresponding descriptive statistics.

The number of categories for each circumstance variable was restricted to three or fewer, so as to reduce the number of “circumstances group” cells with zero or very few observations. As discussed in Section 3, this is important for the non-parametric analysis, which relies on the quality of the estimates for conditional means in these cells and their sampling variation may be very high for cells containing few observations. This greater sampling variance might artificially inflate the estimated inequality between groups, thereby inducing an over-estimation of inequality of opportunity. Table 4 shows the maximum number of cells in each survey, the number of cells actually observed (i.e. the complement of the number of empty cells), the mean cell size and the proportion of cells

22Employment rates are defined as the ratio of all employed individuals to all persons in the 30-49 agegroup. These rates for men and women (respectively) are: 0.90 and 0.55 in Brazil; 0.91 and 0.62 in Colombia; 0.97 and 0.72 in Ecuador; 0.96 and 0.51 in Guatemala; 0.91 and 0.53 in Panama; and 0.94 and 0.72 in Peru.

23Whether complete primary attainment was included as part of the middle or upper grouping for parental education depends on relative group sizes. An effort was made to prevent the top grouping from being too small relative to other countries, to enhance comparability. None of the results is particularly sensitive to these decisions.

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with fewer than five observations. Despite observing only six circumstance variables and exercising considerable parsimony in the partitioning of the population, we still have two surveys – from Guatemala and Panama – for which over 20% (40%) of cells have fewer than five observations in the income/consumption (earnings) analysis. By contrast, Brazil’s national PNAD survey, with a sample size one order of magnitude larger, has 6%-8% of cells with fewer than five observations. This reflects the limited sample sizes of LSMS surveys, and underscores the importance of the parametric estimates in validating (or refuting) the non-parametric results presented below.

Turning to the advantage variables, labor earnings are measured on an individual basis as monthly earnings from all occupations, including the monetary value of various in-kind payments. Family incomes and consumption are measured as per capita household income (from all sources) and per capita aggregate household consumption. Aggregates for family incomes are computed as the sum of all household members’ individual incomes, and include all jobs earnings plus other incomes such as those from assets, pensions and transfers.24

Consumption expenditure data is not available for Brazil. Elsewhere, the reference period is the year, but some expenditures are captured on a weekly or monthly basis. Consumption aggregates do differ across surveys in some respects. In particular, income and consumption are adjusted for differences in the local cost of living in most LSMS datasets (Ecuador, Guatemala, and Panama but not Colombia) and in the Peruvian ENAHO dataset. LSMS surveys (Colombia, Ecuador, Guatemala and Panama) and the ENAHO survey also include imputed rents for owner-occupied housing in both consumption and income aggregates, whereas the PNAD does not. Table 3 (Panel B) reports means and standard deviations for these three economic advantage variables.

5.Inequality in Earnings Opportunity

In most societies, remuneration for one’s work in the labor market is an important component of overall income, and thus a key determinant of a person’s command over

24 The reference period for the earnings of self-employed workers, which is the month in Brazil, Colombia, and Peru; depends on the frequency of payments in Panama; and is the year elsewhere. For wage earnings the reference period is the month in all surveys.

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private goods. Some argue that it is also directly related to self-esteem and social status. Furthermore, it is affected both by one’s own choices and efforts, and by exogenous circumstances. Labor earnings would therefore certainly qualify as an “advantage” concept in Roemer’s terms.

As noted by Lefranc et al. (2006), Roemer’s concept of equality of opportunity for earnings would require that the distribution of earnings conditional on any circumstance variable be identical to the marginal distribution – i.e. that there should be no difference across the earnings distributions estimated for particular population subgroups defined according to their circumstances.25 Figure 1 shows the distributions of earnings conditional on mother’s education (Panel A) and on ethnicity (Panel B), for our six countries. At least in the case of mother’s education, first order dominance should be satisfied in most countries as conditional distribution functions never cross.26 The distances between circumstance groups appear particularly high in Brazil and Panama. There are also pronounced visual differences in the ethnicity panel, with the distance between ethnic groups being higher in Guatemala and Brazil.

The scalar inequality measures developed in Section 3 can help us quantify these inter-country differences, as well as allowing us to consider the combined (and partial) effects of all six circumstance variables. Table 5 presents our main results for earnings opportunity. The first row contains the estimates of overall earnings inequality in each of the six countries, using three indices from the generalized entropy class of measures: the mean log deviation, E(0); Theil index, E(1); and half the square of the coefficient of variation, E(2).27 Below each point estimate, we report bootstrapped standard errors, computed using 100 replicates of the sample, taking into account weights, stratification and clustering for each re-sampling. Since they refer only to a selected sub-sample of the working population, as described in Table 1 (which largely corresponds to 30-49 yearolds), these are not national measures of earnings inequality. Nevertheless, the indices are

25This is the strongest concept of equality of opportunity. Lefranc et al. (2006) allow for weaker concepts, such as there being no firstor second-order stochastic dominance between the conditional distributions.

26These figures are included as an illustration only, and are not the focus of our analysis. Statistical tests for stochastic dominance (Davidson and Duclos, 2000) are therefore not reported in this paper.

27As noted earlier, all generalized entropy measures are additively decomposable by population subgroup.

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generally high: mean log deviations are 0.572 in Panama, 0.608 in Colombia, 0.616 in Brazil, 0.638 in Ecuador, 0.675 in Peru and 0.786 in Guatemala.28

The next two rows of Table 5 present our non-parametric estimates of the opportunity share of inequality in earnings, given by θdN and θrN , on the basis of the partitions described in Section 4 and Table 2. We report the measures for all three inequality indices (E(0), E(1), and E(2)). As noted in Section 3, we focus our discussion on E(0), the only measure for which θdN = θrN . By this measure, differences in observed opportunity account for 20% of total earnings inequality in Colombia, 21% in Peru, 25% in Panama, 26% in Ecuador, 29% in Guatemala and 35% in Brazil. The differences between Brazil and any other country, with the exception of Guatemala, are statistically significant at 5% (as are the differences between Guatemala and Peru or Colombia).29 In other words: a fairly coarse partition of the population by six circumstance variables accounts for between one-fifth and one-third of total earnings inequality.

It is interesting to note that this ranking is different from the ranking of overall earnings inequality. In particular, Brazil, which has only the fourth highest earnings inequality in the sample, has by far the largest opportunity share of that inequality. Similarly, Panama, which has the lowest earnings inequality, ranks above Colombia and Peru in opportunity share. Such re-ranking suggests that inequality of opportunity and inequality of outcomes are not simply different ways of measuring the same thing: they capture different features of distribution in a society.

The remainder of Table 5 presents the results of the parametric decompositions, namely θrP and θrJ , for J set to equal each individual circumstance variable in our partition: gender, ethnicity, father’s occupation, father’s education, mother’s education and birthplace.30 Still focusing on the path-independent measure E(0), the parametric

28The differences between Ecuador and Peru, as well as between Brazil, Colombia and Ecuador, or Brazil, Colombia and Panama, are insignificant at the 5% level, on the basis of the bootstrapped standard errors.

29The opportunity shares tend to be somewhat higher for the Theil index (E(1)); and much lower for E(2). In the case of the latter, this is due in part to its greater sensitivity to income gaps at the top of the distribution. Notice also that the path-dependence is most pronounces for E(2), given the squaring of relative means built into its weighting scheme for the within-group component. See footnote 13.

30As discussed in Section 3, these indices are based on the OLS estimates of equation (7). The reducedform estimates for all six countries are presented in Table 6. All coefficients have the expected signs and most are quite strong. Since this is a reduced-form equation, these coefficients can not be interpreted causally, and capture both the partial direct effects of C on y, and indirect effects through E. As noted

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