Abstract
The empirical analysis of inequality of opportunity centres on disparities between social types, defined by the exposure to circumstances beyond individual control. Despite this, its main theoretical foundation—the Roemer model—does not indicate how to carry out, in practice, the required partition of the population into such types. This paper operationalises this definition of social types using a latent classes approach. Our specification is embedded in a probabilistic extension of the canonical Roemer model, which assumes that the relevant population consists of a finite number of latent types, from which each individual can be treated as a random draw. This makes possible the use of the full set of circumstances in the data, allows for unobserved individual heterogeneity and does not require an ex-ante specification of the number of types by the researcher. Our approach is illustrated by an empirical application featuring a large UK cohort study that was used in earlier literature to examine inequalities of opportunity in a wide array of social outcomes.
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Notes
This problem is clearly alleviated in cases where the researcher is solely interested in a particular type of inequality, such as gender or racial disparities. Yet, even in these special cases, empirical analyses often suggest that more complex stratifications are needed, involving a wider set of circumstances than gender and ethnicity (see, for example, Johnson 2010 on the characterization of racial segregation in the US).
Ad hoc solutions are particularly hard to defend in the context of ex-ante analyses of inequality of opportunity, since these centre on the measurement of inequality between social types (the term ex-ante refers to the fact that this approach can be used in cases where circumstances are known, but effort has not been exerted by the individuals—see Fleurbaey 2008 for details).
Although our approach does not entirely eliminate arbitrariness in the definition of types that characterizes earlier work in this field (this is inherent to the operationalization of the responsibility cut, distinguishing between circumstances and effort), it reduces it in three major ways. First, by allowing the researcher to use a much richer set of information, eliminating the need to, more or less arbitrarily, omit certain circumstances. Second, by taking into account unobserved heterogeneity in circumstances. Third, by providing applied researchers with a well-established statistical method for treating the information circumstances. This is far less arbitrary than the ad hoc selection of a small number of circumstances in the data.
Note that the level of effort depends on the whole policy (see Roemer 2003).
The use of rank \(\pi \) as an interpersonally comparable measure of effort is precisely justified in Roemer (2003).
It should be noted that the van de Gaer and Roemer approaches are equivalent in cases I which there is a type dominated by all other types for each degree of effort.
TIP curve originated in the poverty literature (see Jenkins and Lambert 1997) and has more recently been applied to the analysis of economic inequality, including inequality of opportunity.
He also proposed to minimize the maximum inequality throughout the different levels of relative effort and the inequality between the average outcome of each type of individuals.
In general, these complete orderings allow decomposing total inequality into inequality of opportunity and inequality of effort components, as shown by Ruiz-Castillo (2003), Checchi and Peragine (2010) and Ferreira and Gignoux (2011). Using an ex-ante criterion, the population is partitioned according to individuals’ circumstances and inequality of opportunity is evaluated in terms of differences between individuals endowed with the same circumstances (the between-group component of overall inequality). Adopting an ex-post approach, the population is firstly partitioned into types, according to individuals’ circumstances, and then each type is further subdivided according to personal effort. Correspondingly, inequality of opportunity is measured betwen individuals who have exerted the same effort (the within-group component of overall inequality).
If \(C=1\) all individuals share exposure to the same set of observed circumstances; If \(C=\hbox {N}\), it is not possible to find two individuals in society with the same set of observed circumstances.
Note that the canonical model implicitly assumes that the sets \(t\) and \(c\) coincide, and therefore, all probabilities collapse to either zero or one and \(v_i^t (\phi )=v_i^c (\phi )\) for \(t=c = 1,\ldots ,T\).
It should however be noted that the latent classes approach is an effective method for defining social types irrespective of the particular characteristics of this probabilistic extension of the Roemer model. It is fully compatible with it, but the use of latent class models would be entirely justified on the basis of its practical expediency, as made clear in Sect. 1.
Note that if \(\left\{ {C_1 ,\ldots ,C_K } \right\} \) are all binary, then system consisting of Eqs. (12) and (13) constitutes a well-known special case known in the literature as a Rasch model (see Rasch 1961). Also it should be noted that, in our model, type membership does not depend on the distribution of the outcome of interest, although this feature could be easily incorporated in a latent class specification, as shown in Cameron and Trivedi (2005, pp. 622–25). That kind of specification, in which class membership depends on the outcome of interest, has been widely used in the health economics literature, for example to model healthcare utilisation in the presence of unobserved heterogeneity. Bago d’Uva (2006) uses latent class models to estimate, simultaneously, healthcare utilisation (the outcome of interest) and class membership probabilities. For simplicity we do this in our empirical illustration and thus refer the reader to Cameron and Trivedi (2005) and references therein.
Alternative classification rules have been suggested for cases in which, for a substantial share of the sample, the highest and the second highest posterior probabilities of type membership are particularly close; an in-depth discussion of these can be found in Vermunt and Magidson (2004).
These small area data are available under a special licence, which imposes restrictions on the handling and usage of the data. Details can be found at http://www.cls.ioe.ac.uk/studies.asp?section=0001000200030015.
The childhood morbidity index is the sum of points, where one point is attributed to the occurrence of each of the following medical conditions: infectious diseases; ear and throat problems; recurrent acute illnesses; acute illnesses (other); asthma, bronchitis and wheezing; allergies; chronic diseases (medical); chronic physical or mental handicaps; chronic sensory illnesses; injuries; psychosocial problems; psychosomatic problems; other childhood morbidity (unspecified).
Most variables in the local area data used to characterize the socioeconomic milieu of the cohort-members (e.g., percentage of unemployed, social housing tenants, and skilled–unskilled workers) are continuous, negative skewed and feature a large number of zeroes. In practice this leads to a very large number of empty cells, causing numerical problems in the computation of the variance-covariance matrix, thereby making estimates inefficient and reducing the power of statistical significance and goodness of fit tests. We dichotomize these variables as shown in the table; as a robustness check we also estimated the model using a series of different partitions (such as terciles and quartiles) and the results are not affected. This robustness analysis is shown in the working paper version of this article, available at http://www.jgabriel.net/page3.htm.
\(BIC\left( \hat{\psi } \right) =-2L\left( \hat{\psi } \right) +\log \left( n \right) \upsilon \).
The details of these analyses are available in the working paper version of this article, downloadable from http://www.jgabriel.net/page3.htm.
These are shown on the last row of Table 8. Armed with these probabilities, and denoting the required shares by \(X_t \), these are obtained from the system of equations: \(\left\{ {{\begin{array}{l} {{\begin{array}{l} {\log \left( {X_2 /X_1 } \right) =-0.974} \\ {\log \left( {X_3 /X_2 } \right) =0.485} \\ {\log \left( {X_4 /X_3 } \right) =-0.0531} \\ \end{array} }} \\ {\log \left( {X_5 /X_4 } \right) =-0.462} \\ {X_1 +X_2 +X_3 +X_4 +X_5 =1} \\ \end{array} }} \right. \).
For clarity, we have restricted the number and categories of circumstance variables shown in Table 3. The full table of estimates of posterior probabilities is available from the authors.
In addition, Stevenson and Wolfers (2008) suggest that the measurement of inequality in life satisfaction is particularly sensitive to how narrowly social groups are defined in practice. This provides an additional (empirical) motivation for examining the definition of social types in context of life satisfaction.
It should be acknowledged, however, that the issue of reporting heterogeneity in life satisfaction remains controversial. Nonetheless, as mentioned above, our goal is to illustrate the latent classes approach, not to develop a full-fledged empirical analysis to clarify this controversy. We thus assume, for simplicity, that there is no reporting bias in life satisfaction, thereby treating this variable as a measurable quantity. This simplifying assumption is also convenient to make our example consistent with most of the applied work on the Roemer model, which focuses on more objectively measurable outcomes, such as education, health or income.
The results of these tests are available upon request from the authors.
The advantages of LCMs are methodological and explained above; they do not depend on the identification of stochastic dominance relationships in any particular example, such as this empirical illustration.
This example covers the case where the two approaches lead to a different number of types (three in the ad hoc definition and five in the latent classes one), since this is the most frequent in empirical applications. Although an artificial example comparing the same number of types under the two approaches could be constructed, this would be incongruous: one of the main advantages of the LCM approach is to provide the researcher with the number of types (rather than this having to be defined ex-ante); using a number of types that is different from our best model would thus be incoherent.
These models have been widely used also in the happiness literature.
References
Aaberge R, Mogstad M, Peragine V (2011) Measuring long-term inequality of opportunity. J Public Econ 95:193–204
Arneson R (1989) Equality and equal opportunity for welfare. Philos Stud 56:77–93
Aitken M, Rubin DB (1985) Estimation and hypothesis testing in finite mixture models. J R Statist Soc B 47:67–75
Bago d’Uva T (2006) Latent class models for utilisation of health care. Health Econ 15(4):329–343
Baker M, Melino A (2000) Duration dependence and nonparametric heterogeneity: a monte carlo study. J Econom 96(2):357–393
Becchetti L, Massari R, Naticchioni P (2011) The drivers of happiness inequality: suggestions for promoting social cohesion. IZA discussion paper 7153
Beegle K, Himelein K, Ravallion M (2009) Frame-of-reference bias in subjective welfare regressions. World Bank policy research working paper 4904
Betts J, Roemer JE (2007) Equalizing opportunity for racial and socioeconomic groups in the United States through educational finance reform. In: Peterson P (ed) Schools and the equal opportunity problem. The MIT Press, Cambridge, pp 209–237
Björklund A, Jäntti M, Roemer JE (2012) Equality of Opportunity and the distribution of Long-Run Income in Sweden. Soc Choice Welf 39(2–3):675–695
Bourguignon F, Ferreira F, Walton M (2007) Equity, efficiency and inequality traps: a research agenda. J Econ Inequal 5:235–256
Brereton F, Clinch JP, Ferreira S (2008) Happiness, geography and the environment. Ecol Econ 65(2):386–396
Clark A, Fleche S, Senik C (2012) The great happiness moderation. IZA discussion paper 6761
Cameron C, Trivedi P (2005) Microeconometrics: methods and applications. Cambridge University Press, New York
Carneiro P, Crawford C, Goodman A (2007) The impact of cognitive and noncognitive skills on later outcomes, CEE discussion papers
Chantreuil F, Trannoy A (2013) Inequality decomposition values: the trade-off between marginality and consistency. J Econ Inequal 11(1):83–98
Checchi D, Peragine V (2010) Inequality of opportunity in Italy. J Econ Inequal 8:429–450
Clark A, Flèche S, Senik C (2012) The Great Happiness Moderation. SOEP papers on multidisciplinary panel data research 468, DIW Berlin
Cohen GA (1989) On the currency of egalitarian justice. Ethics 99:906–944
Cunha F, Heckman J (2007) The technology of skill formation. Am Econ Rev 97(2):31–47
Dardanoni V, Li Donni P (2012) Incentive and selection effects of Medigap insurance on inpatient care. J Health Econ 31(3):457–470
Dearden L, Ferri J, Meghir C (2002) The effect of school quality on educational attainment and wages. Rev Econ Statist 84:1–20
Deaton A (2012) What does the empirical evidence tell us about the injustice of health inequalities? In Nordheim OF et al. (eds) Inequalities in health: ethics and measurement. Oxford University Press, Oxford
Dolan P, Peasgood T, White M (2008) Do we really know what makes us happy? A review of the economic literature on the factors associated with subjective well-being. J Econ Psychol 29:94–122
Dworkin R (1981a) What is equality? Part 1: equality of welfare. Philos Public Affairs 10:185–246
Dworkin R (1981b) What is equality? Part 2: equality of resources. Philos Public Affairs 10:283–345
Ferreira F, Gignoux J (2011) The measurement of inequality of opportunity: theory and an application to Latin America. Revi Income Wealth 57:622–657
Ferrer-i-Carbonell A, Frijters P (2004) How important is methodology for the estimates of the determinants of happiness? Econ J 114:641–659
Fleurbaey M (2008) Fairness, responsibility, and welfare. Oxford University Press, Oxford
Fleurbaey M, Schokkaert E (2009) Unfair inequalities in health and health care. J Health Econ 28:73–90
Fleurbaey M, Peragine V (2013) Ex ante versus ex post equality of opportunity. Economica 80(317):118–130
Forcina A (2008) Identifiability of extended latent class models with individual covariates. Comput Statist Data Anal 52(12):5263–5268
Frey B, Stutzer A (2002) What can economists learn from happiness research? J Econ Lit 40(2):402–435
Frijters P, Johnston D, Shields M (2011) Life satisfaction dynamics with quarterly life event data. Scandinavian J Econ 113(1):190–211
Galindo-Rueda F, Vignoles A (2005) The declining relative importance of ability in predicting educational attainment. J Human Resour 40:335–353
Goodman LA (1974) Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika 61:215–231
Hagenaars JA, McCutcheon AL (2002) Applied latent class analysis models. Cambridge University Press, Cambridge
Heckman J (2007) The economics, technology and neuroscience of human capability formation. Proc Nat Acad Sci 14:13250–13255
Helliwell J, Layard R, Sachs J (2013) World happiness report. Sustainable Development Solutions Network
Jenkins SP, Lambert PJ (1997) Three ‘I’s of poverty curves, with an analysis of UK poverty trends. Oxford Econ Papers 49:317–327
Johnson R (2010) The health returns to education policies: from preschool to high school and beyond. American Economic Review, papers and proceedings, pp 188–194
Jones A, Roemer JE, Rosa Dias P (2014) Equalising opportunity in health through educational policy. Soc Choice Welf 43(3):521–545
Lefranc A, Pistolesi N, Trannoy A (2008) Inequality of opportunities vs. inequality of outcomes: are western societies all alike? Rev Income Wealth 54:513–546
Lefranc A, Pistolesi N, Trannoy A (2009) Equality of opportunity and luck: definitions and testable conditions, with an application to income in France. J Public Econ 93:1189–1207
Li Donni P, Peragine V, Pignataro G (2014) Ex-ante and ex-post measurement of equality of opportunity in health: a normative decomposition. Health Econ 23:182–198
Luongo P (2011) The implication of partial observability of circumstances on the measurement of IOp. Res Econ Inequal 19:23–49
Marrero AG, Rodríguez JG (2011) Inequality of opportunity in the U.S.: trends and decomposition. Res Econ Inequal 19:217–246
Marrero AG, Rodríguez JG (2012) Inequality of opportunity in Europe. Rev Income Wealth 58(4):597–621
Marrero AG, Rodríguez JG (2013) Inequality of opportunity and growth. J Dev Econ 104:107–122
MacCulloch R, di Tella R (2005) Partisan social happiness. R Econ Stud 72:367–393
Moreno-Ternero D (2007) On the design of equal-opportunity policies. Investigaciones Económicas 31:351–374
Niehues N, Peichl A (2014) Upper bounds of inequality of opportunity: theory and evidence for Germany and the US. Soc Choice Welf 43(1):73–99
Oswald A, Wu S (2009) Objective confirmation of subjective measures of human well-being: evidence from the USA. Science 317:576–579
Oswald AJ, De Neve J-E (2012) Estimating the influence of life satisfaction and positive affect on later income using sibling fixed-effects. Proc Nat Acad Sci (PNAS) 49:19953–19958
Peragine V (2002) Opportunity egalitarianism and income inequality. Math Soc Sci 44:45–60
Peragine V (2004) Ranking of income distributions according to equality of opportunity. J Income Inequal 2:11–30
Peragine V, Savaglio E, Vannucci S (2009) Poverty rankings of opportunity profiles. Working papers ECINEQ, 115
Peragine V, Palmisano F, Brunori P (2011) Economic growth and equality of opportunity. Working papers 232, ECINEQ, Society for the Study of Economic Inequality
Power C, Peckham C (1987) Childhood morbidity and adult ill-health. J Epidemiol Commun Health 44:69–74
Ramos X, van de Gaer D (2012) Empirical approaches to inequality of opportunity: principles, measures, and evidence. IZA discussion papers 6672, Institute for the Study of Labor (IZA)
Rayo L, Becker G (2007a) Habits, peers, and happiness: an evolutionary perspective. Am Econ Rev 97(2):487–491
Rayo L, Becker GS (2007b) Evolutionary efficiency and happiness. J Polit Econ 115(2):302–337
Rasch G (1961) On general laws and the meaning of measurement in psychology, pp 321–334. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, IV. University of California Press, Berkeley
Rawls J (1971) A theory of justice. Harvard University Press, Cambridge
Rodríguez JG (2008) Partial equality-of-opportunity orderings. Soc Choice Welf 31:435–456
Roemer JE (1993) A pragmatic approach to responsibility for the egalitarian planner. Philos Public Affairs 10:146–166
Roemer JE (1998) Equality of opportunity. Harvard University Press, Cambridge
Roemer JE (2002) Equality of opportunity: a progress report. Soc Choice Welf 19:455–471
Roemer JE (2003) Defending equality of opportunity. The Monist 86(2):261–282
Roemer JE, Aaberge R, Colombino U, Fritzell J, Jenkins S, Marx I, Page M, Pommer E, Ruiz-Castillo J, Tranaes T, Wagner G, Zubiri I (2003) To what extent do fiscal regimes equalize opportunities for income acquisition among citizens? J Public Econ 87:539–565
Rosa Dias P (2009) Inequality of opportunity in health: evidence from a UK cohort study. Health Econ 18:1057–1074
Rosa Dias P (2010) Modelling opportunity in health under partial observability of circumstances. Health Econ 19:252–264
Ruiz-Castillo J (2003) The measurement of the inequality of opportunities. Res Econ Inequal 9:1–34
Sen A (1980) Equality of what? In: McMurrin S (ed) Tanner lectures on human values, vol 1. Cambridge University Press, Cambridge, UK
Sen A (1985) Commodities and capabilities. North Holland, Amsterdam
Stiglitz JE, Sen A, Fitoussi J-P (2009) Report by the commission on the measurement of economic performance and social progress
Tubeuf S, Jusot F, Bricard D (2012) Mediating role of education and lifestyles in the relationship between early-life conditions and health: evidence from the 1958 British cohort. Health Econ 21:129–150
Stevenson B, Wolfers J (2008) Happiness inequality in the United States. J Leg Stud 37:533–579
Stevenson B, Wolfers J (2013) Subjective well-being and income: is there any evidence of satiation? Am Econ Rev 103(3):598–604
van den Berg B, Powdthavee N (2011) Putting different price tags on the same health condition: re-evaluating the well-being valuation approach. J Health Econ 30(5):1032–1043
van Praag BM, Ferrer-i-Carbonell A (2006) An almost integration-free approach to ordered response models. Tinbergen Institute discussion paper 06–047/3
van Praag BM (2007) Perspectives from the happiness literature and the role of new instruments for policy analysis. CESifo Econ Stud 53:42–68
Trannoy A, Tubeuf S, Jusot F, Devaux M (2010) Inequality of opportunities in health in France: a first pass. Health Econ 19(8):921–938
Tubeuf S, Jusot F, Bricard D (2012) Mediating role of education and lifestyles in the relationship between early-life conditions and health: evidence from the 1958 British cohort. Health Econ 21(S1):129–150
van de Gaer D (1993) Equality of opportunity and investment in human capital. Catholic University of Leuven (faculty of economics, no. 92), Leuven
Vermunt JK, Magidson J (2004) Latent class analysis. In: Lewis-Beck MS, Bryman A, Liao TF (eds) The sage encyclopedia of social sciences research methods. Sage, Thousand Oaks, pp 549–553
World Bank (2006) World development report 2006: equity and development. The World Bank and Oxford University Press, Washington, DC
Acknowledgments
We are grateful for comments on earlier versions of this work from John Roemer, Dirk van de gaer, Valentino Dardanoni, Juan Prieto-Rodríguez, Rafael Salas and seminar participants at the University of Oxford. The NCDS database was supplied by the Economic and Social Research Council (ESRC) Data Archive.
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Li Donni, P., Rodríguez, J.G. & Rosa Dias, P. Empirical definition of social types in the analysis of inequality of opportunity: a latent classes approach. Soc Choice Welf 44, 673–701 (2015). https://doi.org/10.1007/s00355-014-0851-6
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DOI: https://doi.org/10.1007/s00355-014-0851-6