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  • A social welfare approach to the measurement of inter-generational mobility has been proposed by Atkinson (1981) and Atkinson and Bourguignon (1982), which differs somewhat from the inequality of opportunity analytical framework presented here. It consists of defining social welfare on father-son pairs, so that each cell of the transition matrix is given a utility U(Yi, Yj) and the social welfare of society is defined by the mean value of this utility, weighted by the transition probabilities, pij. The simplest case is when U( ) is additive in the earnings of parents and children. Atkinson and Bourguignon (1982) derived dominance criteria to compare transition matrices on that basis, depending on the properties of the function U( ).

  • A similar line of thought has been pursued by Kanbur and Stiglitz (2015), who extend the preceding approach by considering a steady state of an economy consisting of infinitely lived dynasties under the assumption of constant transition matrix and earnings distribution across generations. Within that framework, they identify a social welfare based dominance criterion of one matrix over another or, in other words, of one stationary state of an economy with some inter-generational mobility feature over another stationary state with a different mobility matrix.

  In the perspective of inequality of opportunity, there are problems with the last two approaches. On the one hand, the assumption of a fully stationary economy and a social welfare dominance comparison based on dynasties with an infinite number of generations seems extreme, even though the stationarity assumption is often implicit in statements about inter-generational earnings mobility. On the other hand, it is a problem that the very nature of circumstances is used to make comparisons of outcomes across groups of individuals with identical circumstances. In other words, the mobility of children with rich parents may matter less than that of poor parents. What should matter from the point of view of inequality of opportunity is how different the distribution of earnings is across parents’ earnings levels, with no particular importance being given to those levels.

  In summary, there is some ambiguity about the way in which inequality of opportunity corresponding to nonparametric specifications of the inter-generational mobility of earnings can be measured. There are various ways mobility can be evaluated or transition matrices compared based on social welfare criteria. But the link with measures of inequality of opportunity of the kind that can be derived from simple parametric models of type (9)—at least under the assumption of homoscedasticity of the residual term, v—is unclear. For this reason, the rest of this section looks at the parametric case.

  Data Requirements

  A priori, the data requirement to estimate the IGE or the parents-children earnings transition matrix seems extremely demanding. One should observe the earnings of parents, generally the father, and that of the children, generally the sons, at more or less the same age or during the same period of their lifecycle. Long panel databases extending over 20 years and more would allow this to be done. For instance, the PSID in the United States has been collecting data on the same families and their descendants for almost 50 years. The British and the German household panels also extend over 25 years and more. In some countries, register data, most often tax data, allow researchers to follow people throughout their lifetime and from one generation to the next, but only a few countries have open and anonymized register data at this stage.

  However, panel data are not really necessary to estimate model (9). The availability of repeated cross-sections over long periods is sufficient. Moreover, they permit the IGE to be estimated in a consistent—i.e., asymptotically unbiased—manner, something which is not certain with panel data.

  To see this, it must be noted that the observation of Log y−1 is most likely to include measurement errors or, at least, transitory components of fathers’ earnings that are unlikely to have had any effect on their sons’ earnings. Estimating γ in (9) with OLS and without precaution for measurement error will thus lead to the so-called attenuation bias, a bias that has been shown to be quite substantial in inter-generational mobility studies.21 The solution is to “instrument” Log y−1 by regressing that variable on some fathers’ or parents’ characteristics at that time, Z, at the same date, say t–1 and to use the predicted rather than the observed value when estimating (9) with OLS. Thus, if the parents’ characteristics Z are observed at time t in the same database as their children’s earnings, and if an earlier cross-section is available at time t–1 this allows us to estimate the log earnings of adults with characteristics Z. Running OLS on (9) using the predicted earnings of the parents at time t–1 will yield an asymptotically unbiased estimator of the IGE. Through this so-called two-sample instrumental variable (TSIV) estimation strategy (Björklund and Jäntti, 1997), repeated cross-sections with information on respondents’ parents and covering a long-enough period are sufficient to estimate the IGE.22

  There are two important caveats to the preceding method. First, the instrumental variable approach just sketched is valid only to the extent that the instrument Z may be assumed to be orthogonal to the income of the children. It must be recognized, however, that this is unlikely to be the case, as most observable parents’ characteristics, like education, occupation, wealth, etc., may be thought of as influencing the economic achievements of children. Second, even if the TSIV strategy did allow the IGE to be estimated consistently with repeated cross-sections rather than with panel data, it would not permit the corresponding inequality of opportunity to be estimated as defined by (10). This is because the variance of the instrumented earnings of parents is not the same as the variance of their true earnings.

  Measurement errors are also likely to affect the estimation of mobility measures, social welfare dominance tests, and inequality of opportunity through the transition probability matrix methods mentioned above. In that case, both the error on fathers and sons matter. The former may be responsible for misclassifying fathers in the income scale, whereas the latter introduces noise in the transition probabilities.

  Stylized Facts

  The best illustration of this literature on the measurement of inter-generational mobility is the well-known “Great Gatsby” curve, building on the work of Miles Corak and popularized by Alan B. Krueger. It plots estimates of IGE against the level of inequality—in the contemporaneous generation—for a set of developed and developing countries. This curve is shown in Figure 5.3 below.

  Along the vertical scale of the chart, one observes a rather wide dispersion of the estimated IGE, from 0.2 for Nordic countries—Sweden being a little above that level—to 0.5 in the United States and 0.6 in Latin American countries. If it is assumed that the inequality of earnings is similar among parents and children, then (10) suggests that a consistent measure of inequality of opportunity is the square of the IGE, or the R2 of the regression of the log of sons’ earnings over the log of fathers’. Then it can be seen that inequality of opportunity corresponding to the fathers’ earnings alone is extremely low in Nordic countries, amounting to less than 4% of the variance of log earnings, while it is more substantial in the United States, amounting to 25% of sons’ earnings inequality. Yet this would still leave considerable room for mobility if it were the case that no other circumstance, orthogonal to parents’ earnings, constrained children’s earnings, which seems unlikely.

  The plot also shows a strong correlation between earnings immobility (i.e., IGE) or inequality of opportunity, and the degree of inequality, as measured by the Gini coefficient of household disposable income, at a point in time. Several explanations have been given for the negative correlation shown in Figure 5.3. The most frequent one relies on some convexity in the relationship between parents’ income and their investment in the human capital of their children, or possibly some unequal access to quality schooling depending on parents’ income. If rich parents invest a higher proportion of their own income in the education of their children, or if only the children of parents above some level of income have access to good quality schools, then more income ineq
uality among parents should generate less inter-generational mobility.

  The preceding argument is equivalent to assuming some nonlinearity in the basic model (9) or, more exactly, that the IGE may depend on the level of income. If the IGE increases with income, as just suggested, then the linear approximation (9) would indeed yield an OLS estimate of the IGE that increases with the degree of income inequality.23

  Figure 5.3. The Great Gatsby Curve

  Source: Corak, M. (2012), “Here is the source for the Great Gatsby Curve,” in the Alan B. Krueger speech at the Center for American Progress on January 12, https://milescorak.com/2012/01/12/here-is-the-source-for-the-great-gatsby-curve-in-the-alan-krueger-speech-at-the-center-for-american-progress/.

  That the IGE may vary with the level of income is shown in the case of the United States by Landersø and Heckman (2016), p. 22, as can be seen in Figure 5.4 below.24

  Another, more mechanical, explanation of the negative slope of the Great Gatsby curve is based on (10) above. If one compares two countries where income inequality was the same in the older generation, then it is the case that, with the same correlation coefficient (R2) between the earnings of the two generations, the IGE will be higher in the country with the highest inequality today. For instance, Landersø and Heckman (2016, p. 17) show that the IGE in Denmark would be much bigger than what it is if the distribution of children’s earnings was identical to the US distribution. This again illustrates the difference between the concept of immobility as described by IGE and that of inequality of opportunity related to parents’ earnings. Yet it is unlikely that Figure 5.3 would be fundamentally different if the IGE were replaced by the inequality of opportunity as defined in (10).25

  Figure 5.4. Nonlinear Inter-generational Elasticity in the United States

  Source: Landersø, R. and J.J. Heckman (2016), “The Scandinavian fantasy: The sources of intergenerational mobility in Denmark and the US,” Center for the Economics of Human Development, University of Chicago.

  Other available explanations of the upward sloping Gatsby curve go from mobility to inequality rather than the opposite. For instance, Berman (2016) stresses that if the distribution of the residual term, v, is constant, model (9) leads to a steady-state distribution of earnings whose inequality is given by: . Thus, inequality of income increases with IGE. As a matter of fact, it should be noted that this property does not hold only at a steady state. Among two societies with the same distribution of earnings in one generation, inequality will be higher, all things being equal, in the society where parents transmit more of their earning capacity to their children. Formally, (9) implies that:

  which is increasing in γ.

  One might ask whether this positive relationship between inequality and inter-generational immobility may also hold inter-temporally. Interestingly enough, Aaronsson and Mazumder (2008) show that trends in wage inequality between 1940 and 2000 in the United States coincide with trends in IGE, with a compression in the first part of the period and increasing disparities in the later part.26 However, not enough data are available to test that hypothesis on a cross-country basis.

  The same type of analysis may be undertaken with other economic outcomes. However, to keep with the spirit of model (9) it is important to make sure that the same variable can be observed for both parents and children. For instance, having parents’ earnings on the right-hand side of the equation and income per capita (or per adult equivalent) on the left-hand side is interesting, but the interpretation is not anymore in terms of inter-generational transmission of earning potential, as income per capita also depends on family size, marriage, and labor supply. There is also an issue with the period of observation of the right-hand variable. Presumably, one would expect parents’ income to influence the life-time earnings of children. This may not be reflected when observing children during a short period at some stage of their life.

  Chetty et al. (2014a) address this issue when analyzing the spatial heterogeneity of inter-generational income mobility in the United States, as they indeed use lifetime pre-tax family income as income variables in both generations as drawn from administrative tax data. They find considerable spatial variation of income mobility across “commuting zones”: “The probability that a child reaches the top quintile of the national income distribution starting from a family in the bottom quintile is 4.4% in Charlotte (North Carolina) but 12.9% in San Jose (California).”

  Taking advantage of the length of register data, Chetty et al. (2014b) also study the time evolution of inter-generational mobility, in effect the rank correlation between fathers’ and sons’ earnings. They find no significant change across birth cohorts born between 1971 and 1982. This is in line with the results found earlier by Lee and Solon (2009) for the US using the PSID panel data set for cohorts born between 1952 and 1975. Both results diverge somewhat from Aaronsson and Mazumder (2008). Nonconsensual results are also found in other countries, as shown by the critiques by Goldthrope (2012) to the finding by Blanden et al. (2011) that mobility would have fallen in the United Kingdom.

  Another interesting concept, closer to the sociological view on mobility, has recently been studied by Chetty et al. (2017). “Absolute mobility” is defined as the proportion of 30-year-old children whose real income is higher than their parents’ when they were 30. Combining register data since 1970 with assumptions on rank correlation together with cross-sectional data for the period before, absolute mobility has declined continuously from the 1940 to the 1965 birth cohort—i.e., the baby boomers. It then stabilized but fell again soon because of the financial crisis—i.e., for cohorts born in the late 1970s.

  Somewhat surprisingly, much less work has been done on the inter-generational transmission of wealth inequality and on the key role of inheritance in the inequality of opportunity. This is in part due to the availability of data. Typical household surveys generally do not include data on wealth. When they do, they do not necessarily include data on parents, or they are not repeated over a period long enough to apply the TSIV methodology. As for panel data, some waves of PSID do include wealth questionnaires. They have been used by Charles and Hurst (2003) to estimate an IGE for wealth. Unfortunately, no information is available that would allow to correct for measurement error bias. The British and German household panels do include data on wealth but the number of observations is too small to estimate IGE for wealth at mid-life, an age at which the wealth concept becomes relevant for both parents and children. One could also think of using estate statistics, but these actually lack relevance as their link to inequality of opportunity is through the heirs, whose wealth is not observed.

  In Nordic countries, several recent studies of inter-generational wealth dynamics have relied on administrative data. Boserup, Kopczuk, and Kreiner (2014) provide estimates of the wealth IGE at mid-life in Denmark, and Adermon, Lindahl, and Waldenström (2015) do the same for Sweden. Both studies cover more than two generations. In both countries, the wealth IGE estimates are comparable and of limited size (around 0.3), a value comparable to the earnings IGE in Sweden but twice as large in Denmark.

  Generalized Inter-generational Mobility Analysis and Inequality of Observed Opportunities

  Parental income is only one of the circumstances affecting the economic outcome of an individual, even though it may be correlated with other circumstances. To provide a more complete picture, fathers’ earnings, y–1 in equation (9), can be replaced or complemented by a vector of variables referring to the parental characteristics of an individual in the current generation. Labor force or household surveys often give information on the parents of respondents (education, occupation, residence, age when respondent was 10). Rather than using the TSIV approach to estimate parental earnings or income based on these characteristics, one may simply measure the related inequality of opportunity by the share of the inequality of income or earnings in the current generation that is accounted for by parents’ characteristics, including the determinants of their own earnings.


  Formally, model (9) is replaced by:

  where yi is a particular economic outcome; Zi a vector of variables that include all observed parental characteristics and some other characteristics beyond individual control, like gender; and vi all the unobserved determinants of the economic outcome that are orthogonal to Zi. In agreement with the definition (7), the R2 statistics of that regression may be interpreted as the inequality of opportunities associated with individual characteristics in Zi when measuring the inequality of outcomes with the variance of logarithms. Some authors prefer using other measures of inequality.27

  In comparison with model (9), model (11) may be considered as a model of “generalized” mobility in the sense that more parental characteristics are taken into account that do not necessarily include the outcome being explained in the current generation. It can also be noted that this model is identical to the model used in the inter-generational mobility analysis of earnings when instrumenting the earnings of the parent by a set of characteristics available in the database—i.e., the TSIV approach. Model (11) would then correspond to the “reduced” form of the earnings mobility model, necessarily less restrictive than the structural form (9).