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  Practical Issues and Some Stylized Facts in Measuring Inequality of Opportunity

  The discussion in the preceding section has focused on conceptual issues in the definition and measurement of inequality of opportunities. We now turn to the way these principles and approaches to measurement are handled in the empirical literature, and present stylized facts about some specific dimensions of inequality of opportunity.

  The focus will first be on single dimensions of inequality of opportunities, without necessarily making reference to specific outcomes. More direct applications of the measurement tools discussed above will then be considered with various combinations of outcomes (income, earnings) and sets of opportunities. Special emphasis will be put on the measurement of inter-generational transmission of inequality, which has attracted much attention among social scientists, and which may be considered as a particular case of the measurement principles set out above. Emphasis will also be put on labor market discrimination, which raises some interesting questions when studied from the perspective of inequality of opportunity.

  Direct Measures of Some Particular Dimensions of the Inequality of Opportunity

  The measurement of specific dimensions of inequality of opportunities can be undertaken in an autonomous way, without explicit reference to economic outcomes. This direct approach simply consists of analyzing the distribution of particular circumstances, C. Many individual characteristics could be analyzed in this way, provided they are described by some quantitative index. Given its huge importance in the literature on inequality of opportunity, this section focuses on cognitive ability and then briefly considers the difficulty of handling directly other single dimensions of inequality of opportunity.

  Cognitive Ability as an Opportunity and as an Outcome

  The PISA initiative by the OECD provides firsthand data to measure inequality in one of the most important dimensions of individual circumstances: cognitive ability. It now gathers the scores of samples of 15-year-old students in more than 70 advanced and emerging economies in three tests: one on reading (i.e., answering questions about a short text); another on mathematics; and the third on science. This instrument has been fielded at 3-year time intervals since 2000. In addition to students’ answers to these assessment tests, the database also reports information on their family background and on the characteristics of their schools.

  Considering PISA scores as circumstances implicitly supposes that cognitive ability at age 15 is one of the important determinants of future individual economic outcomes, earnings in particular, and acknowledges that it essentially depends on genetic factors and the family context. People cannot be held responsible for that part of their life, so that inequality of PISA scores among 15-year-olds today will be responsible for some of the inequality of opportunity they will face later in their lifetime. But PISA scores may also be seen as the outcome of the educational process and, as such, as dependent on family circumstances, the efforts of the children, and the educational system itself.11 Hence the debate on how schools may correct for the inequality of opportunity arising from family background. It is however the former perspective—i.e., cognitive scores as a circumstance—that is discussed in what follows.

  Much publicity is given at each new edition of PISA to mean scores by country, to the ranking of countries and how rankings change over time. From the viewpoint of measuring inequality of opportunities, however, what matters most is the statistical distribution of these scores, or their disparities across students.

  Figure 5.2 plots the inequality of PISA scores in mathematics, as measured by the coefficient of variation, against the mean score in the 2012 exercise for OECD countries. Interestingly, there is a clear negative relationship between the inequality and the mean of scores (putting aside the three emerging economies, Chile, Mexico, and Turkey, where the coverage of the PISA survey is much lower than in advanced countries, essentially because a nonnegligible proportion of 15-year-olds have already dropped out of school).12 This is presumably because better mean scores are logically obtained by improving the lower tail of the distribution, more so than making improvements to the upper tail. Yet what may be more important is the substantial difference in the inequality of scores for countries in the same range of average scores. For instance, inequality is 30% higher in Belgium than in Finland or Estonia, in the upper part of the scale of mean scores; the same holds true for France compared with Denmark in the middle.

  Figure 5.2. Mean and Coefficient of Variation of PISA Mathematics Scores in OECD Countries, 2012

  Source: OECD (2014), PISA 2012 Results: What Students Know and Can Do. StatLink 2 http://dx.doi.org/10.1787/888933839582.

  Cognitive ability can be considered as a dimension of economic opportunity only insofar as it is a significant determinant of an outcome like earnings or the standard of living of an individual. In this respect, it is important to stress that test scores in surveys like PISA, the OECD’s Survey of Adult Skills (PIAAC), or its predecessor (the International Adult Literacy Survey, IALS), only explain a limited part of earning differences across individuals. Murnane et al. (2000) and Levin (2012) make this point on the basis of US data. According to the former, a 1% increase in high school test scores entails a 2% increase in earnings when students are 31 years old, which is substantial.13 However, the variance of adults’ (log) earnings explained by high school test scores is small, slightly less than 5% for men (Murnane et al., 2000, p. 556). Family background is a more powerful determinant of earnings, all the more so when considering that test scores are very much dependent on the education and income of parents.

  Instead of cross-country comparisons as in Figure 5.2, it would be interesting to see how inequality, or more exactly the whole distribution of scores, changes over time in a given country. Data reported by the OECD in 2012 include the 90/10 inter-decile and 75/25 inter-quartile ratios for the four exercises since 2003. These measures are remarkably constant except for emerging economies where inequality goes down at the same time as mean scores go up due to the lower tail moving up. France is one of the few advanced countries where the 90/10 inter-decile ratio increased significantly over time. As France’s mean score did not change much, this would suggest that good performers do better and bad ones do worse, possibly a clear sign of an increase in inequality of that specific component of opportunities. A careful country-by-country study of the evolution of the whole distribution of test scores might reveal other interesting features. It is surprising that so much emphasis is being put on the evolution of the means without considering distributional features.

  The same kind of analysis, based on different tests, is being performed for students at younger ages by a number of organizations, for instance, the Progress in International Reading Literacy Study, led by the International Association for the Evaluation of Educational Achievement. However, the results have not received as much publicity, even though they are equally, if not more, relevant in the analysis of inequality of opportunity, as numerous studies have shown that differences in individual cognitive abilities appear very early in life. Pre-school tests already show high variability among children depending on their family background, and several studies have shown that these differences might have long-lasting effects, as school systems would at best compensate for only part of them. Experiments with preschool programs aimed at leveling the playing field—like the Perry program or the Abecedarian program in the United States—have provided evidence of this (Kautz et al., 2014). As shown in recent work by Heckman,14 these preschool inequalities are due in large part both to “parenting”—i.e., the care the parents devote to their young children—and to health factors.

  Other Single Dimensions of the Inequality of Opportunity

  Initial inequality in noncognitive skills is also important throughout people’s lifetime, and may be considered as another dimension of inequality of opportunity, even though no synthetic measure is actually available, which makes comparing societies over space or over time difficult
.

  Health status is another dimension of childhood circumstances related to family background, and another component of human capital. In the same way that cognitive skills at age 15 influence future earnings and are heterogeneous across young people, health status at earlier ages is known to potentially influence the whole career of people and to be heterogeneous too.15 The difficulty here is to monitor inequality in health status. There is an important literature, for instance, on birth weights as a predictor of health status, future education achievements, and adult earning levels (e.g., Currie, 2009). The same may be true of anthropometric indices at early ages, although most indices are strongly influenced by weight at birth. It is somewhat surprising that more attention is not given to the evolution of inequality in these indices and, as for educational test scores, their dependency on parental characteristics.

  Another single dimension of the inequality of opportunity, different from human capital, is inherited financial capital. It is certainly possible to measure inequality of inheritance flows during a given period. Wolff (2015) does so for the United States using data from the University of Michigan’s Panel Study of Income Dynamics (PSID) and provides Gini coefficients of inheritance flows both for the whole population and among recipients. However, this is not very informative because of the heterogeneity in the age of inheritors. The extent to which inheriting the wealth of one’s parents at the age of 55, something frequent these days, may be considered as a component of inequality of opportunity is not totally clear, except if, somehow, one has been able to borrow, effectively or virtually, much earlier in life against this future wealth flow. As the credit market is highly imperfect, and the inheritance date or the amount to be inherited highly uncertain, it is not even certain it would make much sense to try to estimate something like the inequality in discounted expected inheritance flows of all 25-year-old individuals. Further, donations as well as inheritances would have to be taken into account.

  Inheritance is a dimension of the inequality of opportunity whose inequality is difficult to evaluate as such, even though, when considering cohorts beyond a certain age, it is a key factor shaping inequality in economic outcomes, like income or standard of living, that are seldom reported in surveys.

  Outcome-Based Measures of Inequality of Opportunities

  Rather than considering the inequality of various dimensions of opportunities in an isolated way, it is possible to measure it indirectly through their overall effect on the inequality of the outcomes under study, using a relationship of type (2) above. It was seen that, in various ways, this relationship provides an indirect scalar metric of inequality of opportunity. Various illustrations of this approach are shown below, while also reporting stylized facts on some key components of inequality of opportunities.

  Inter-generational Mobility of Earnings

  Much work has been devoted to the estimation of models of type (2) where the outcome yi is the (full-time) earnings of an observed individual, i, and Ci is the (log) (full-time) earnings of their parents, most often their father, observed roughly at the same age. Denoting the latter by y−1,i, the basic specification of the model is thus:

  where a is a constant and vi is a zero mean random term, standing for all unobserved earnings determinants independent of fathers’ earnings. The coefficient E1 summarizes all the channels through which fathers’ earnings, and their own determinants like education, may affect sons’ earnings.

  This model, reminiscent of the famous Galton (1886) analysis of the correlation of height across generations, is generally presented as belonging to the literature on inter-generational mobility, with the least square estimate being interpreted as the inter-generational elasticity (IGE), or the degree of immobility across generations. Equivalently, in a Galtonian spirit, the coefficient 1 − is interpreted as the speed of “regression toward the mean.”

  This approach to inter-generational mobility is based on a parametric specification and the estimation of a specific parameter. Nonparametric specifications come under the form of mobility matrices showing the probabilities, pij, for the earnings of sons whose fathers’ earnings are in income bracket i to be in bracket j. We consider these two approaches in turn.

  Parametric Representation of Inter-generational Mobility

  To see more clearly the relationship between IGE and inequality of opportunity, one may consider applying directly the alternative definition of inequality of opportunities provided in the preceding section of this chapter. Applying (4)–(7) to model (9) and assuming that the inequality measure M{ } is scale invariant, it can be shown that:

  where the notation ^ refers to least square estimates. In the particular case where M{ } is the variance of logarithms, VL, it turns out that the two measures in absolute terms are identical because of the additivity property of the variance:

  while in relative terms:

  where R2 is the measure of the explanatory power of the independent variables in regression (9) or, in the present case, the square of the correlation coefficient between the (log) earnings of parents and children.

  It can be seen that there is a difference between the inter-generational mobility of earnings (IGE) and inequality of opportunity linked to fathers’ earnings. The former is proportional to the latter with a coefficient equal to the ratio of the inequality of children’s earnings to that of their fathers.17 In other words, it is only in a world where the inequality of earnings does not change across generations that both the inequality of opportunity, based on the variance of logarithm, and the IGE coincide.

  In their study of geographical differences in inter-generational mobility in the United States, Chetty et al. (2014a) use the ranks of parents and children in the earnings distribution as a relative measure of mobility. Based on the “copula” of the joint distribution of the (log) earnings of the two generations—i.e., the joint distribution of father/children ranks in their respective earnings distribution—this measure is independent of the marginal distributions of log earnings. It turns out that the rank-rank correlation is not very different from the log earnings correlation for reasonable small values of the latter. Through (10), it is thus possible to recover the IGE from the rank-rank correlation.

  Nonparametric Representation: Mobility Matrices

  Another way of representing the inter-generational mobility of earnings is through a transition matrix representing the way a two-generation dynasty transitions from a given earnings level for fathers to another (or the same earnings) level for sons. Let N earning brackets be denoted by Yk, and denote Pij the probability that the sons of fathers in bracket Yi find themselves in bracket Yj. The distribution of earnings is given by the total rows and columns of the matrix P = {pij}, but it is also possible to break free from these distributions by defining the income brackets as the quantiles (deciles, vintiles … of the distribution of the earnings for fathers on the one hand, and for sons on the other.18

  Whether the transition or “mobility” matrix is defined in terms of income brackets or quantiles is not without implications for the interpretation that may be given to the comparison of two matrices. When referring to income brackets, the matrix shows “absolute” mobility, i.e., the probability that children’s earnings could be higher, or lower, than their parents’, a concern of many parents today. Conversely, defining the matrix in terms of quantiles permits to analyze “relative” mobility, irrespectively of earnings levels. The difference between the two approaches lies essentially in the fact that the latter does not take into account the change in the distribution of earnings across generations.19

  Table 5.1. Inter-generational Transition Matrix for Earnings

  StatLink 2 http://dx.doi.org/10.1787/888933839601.

  There is a huge literature on how to draw mobility indicators from such a representation of the influence of parents’ earnings on children’s earnings—see Fields and Ok (1999) or the survey by Jäntti and Jenkins (2015). For instance, mobility is often measured by one minus the trace of the mobility
matrix. Shorrocks (1978) suggested a “Normalised Trace” measure given by , where A is the matrix P with rows normalized to 1—i.e., the N probabilities pij are divided by the row sum pi. Other measures are based on the expected number of jumps from one bracket, or decile, to another.

  Rather than comparing transition matrices on the basis of mobility indices, some dominance criteria have also been developed, which may lead to incomplete ordering and, thus, to cases of noncomparability between two matrices. For instance, the diagonal criterion says there is less mobility in a transition matrix than in another if all diagonal elements, rather than their sum, are smaller in the former than in the latter. Shorrocks (1978) proposed a stronger criterion, the “strong diagonal view,” according to which there is more mobility in matrix A than in B if .

  Although related, this kind of measure based on the transition probabilities has only an indirect link with measures of inequality of opportunity in the sense that it is not expressed in terms of the distribution of outcomes, which logically should be here the distribution of children’s earnings. There are various ways in which such a link may be established:

  • The Roemer inequality of opportunity formula (8) would be one way, although apparently seldom used, mostly because the transition matrix consistent with it would be conceptually different from P above. Indeed, the children’s earnings brackets should be row-dependent so as to correspond to the deciles—or other quantiles—of the distribution of earnings among children from parents in a given earnings bracket or quantile.

  • An alternative would consist of associating to each row of the matrix a scalar depending on the mean earnings and its distribution within the row. In a more general context, Van de Gaer (1993) suggested measuring the inequality of opportunity by the inequality of the mean earnings across “types,” i.e., fathers’ earnings here, which is actually a measure of type Idu as defined in (4). Lefranc, Pistolesi, and Trannoy (2009) argued in favor of combining the mean with some inequality measure within type. More generally, one could consider the observed distribution of children’s earnings with the same father’s earnings as being that of the ex ante random earnings of the typical child in that type. Then one would associate to each row of the transition matrix the certainty equivalent of the distribution of earnings in that row for a given level of risk aversion. This would be equivalent to associating to each row of the matrix the equivalently distributed earnings (EDE) for that row, in the sense of Atkinson (1970), and then defining inequality of opportunity as the inequality of these EDEs across rows.20