In recent years, the concept of inequality of opportunity, rather than inequality of achieved states, has received growing attention in the economic literature. The simple advocacy of equal health, for example, fails to hold individuals accountable for their choices. This can be seen as significant limitation.
Equality of opportunity co-opts one of the sharpest ideas in the antiegalitarian arsenal: The notion of responsibility. By compensating for the impact of circumstances beyond individual control, yet holding individuals responsible for the consequences of their choices, equality of opportunity is an appealing compromise between strict equality of health and mere equity of formal rights. It has thus been increasingly advocated by policy makers, as is made clear in World Bank (2005) which focuses on the inequality issue.
This theoretical evolution reflects a number of recent developments in political philosophy, arguably prompted by the seminal work of John Rawls and Amartya Sen. Both Rawls’ equality of social primary goods and Sen’s proposed equality of capabilities move away from the social goal of equalizing subjective welfare. They propose that, once primary goods or capabilities are equally distributed, any residual inequality should be deemed a legitimate consequence of individual choice, hence of individual responsibility. Ronald Dworkin advanced this proposal by arguing that equality of welfare cannot be a valid equity criterion for it fails to make individuals accountable for their preferences , namely, those preferences they are happy to have. The problem thus becomes one of finding the distribution of resources that appropriately compensates individuals for their dissimilar endowments (physical resources, talents, and handicaps), while making them responsible for their preferences . This rationale leads Dworkin to propose the criterion of equality of resources, which attracted important criticisms, such as those raised by Richard Arneson and Gerald Cohen. Cohen shows that Dworkin’s separation between preferences and resources can be intractable in practice: Should one be made responsible for childhood preferences that are chiefly instilled by one’s social environment? This debate has prompted key progresses in social choice theory, which have rendered these new ideas operational within an analytical framework known as the equal-opportunity approach.
The Roemer Model Of Equality Of Opportunity
Equality of opportunity has been given different formal expressions in the social choice literature, such as in early proposals by Marc Fleurbaey and Walter Bossert. A related strand of research focuses on measuring opportunity sets, taking into account the intrinsic value of individual freedom in the ranking of social states. Despite the theoretical appeal of these contributions, they have proved too abstract to prompt related empirical work. Largely for this reason, the workhorse of the applied literature on inequality of opportunity in health has been the model proposed by Roemer (1998, 2002).
The Roemer model sorts all factors influencing individual attainment between a category of effort factors, for which individuals should be held responsible and a category of circumstance factors, which, being beyond individual control, are the source of illegitimate differences in outcomes. It should be noted that, in this framework, effort is not limited to human exertion and comprises all the determinants of health outcomes over which individuals have control. Also, the classification of the determinant of human achievement as either circumstances or effort is partly normative and partly informed by available empirical evidence. In the case of health, we may think of the outcome of interest as health as an adult (H) and define a health production function, H(C,E(C)) where C denotes individual circumstances and E denotes effort. Circumstances affect the health outcomes of individuals and social groups, directly and through their influence on effort factors.
The recent medical and economic evidence on the early determinants of health has emphasized the importance of a number of circumstantial factors. The fetal origin hypothesis stresses the role of parental socioeconomic characteristics as key determinants of in utero fetal growth which, in turn, condition long-term health. The life course models, which emphasize the impact of deprivation in childhood on adult health and longevity, and the pathways models, suggest that health in early life is important mainly because it will condition the socioeconomic position in early adulthood, which explains disease risk later in life. There is also evidence on determinants of health that, although affected by circumstances, are, at least partially, within individual control and therefore constitute effort factors in the context of the Roemer model. Lifestyles such as diet and physical exercise are good examples of such factors.
The Roemer model defines social types consisting of the individuals who share exposure to identical circumstances. Types can thus be defined using the set of observed individual circumstances in the data. In practice, it is up to the researcher to identify circumstances that lead to a meaningful partition of the population of interest. Factors such as parental socioeconomic background and region of birth have often been used by applied economists to partition the population, but other variables such as inborn cognitive ability and childhood health have also been used. It is assumed that the society has a finite number of T types and that, within each type, there is a continuum of individuals. A fundamental aspect in this setting is the fact that the distribution of effort within each type (Ft) is itself a characteristic of that type (t); because this is beyond individual control, it constitutes a circumstance.
In general, it is not possible to compare directly the levels of effort expended by individuals from different types because circumstances partly determine outcomes. For example, the number of times per week one does physical exercise is partly determined by individual choice (effort) and also influenced by parental background, social milieu, and peer pressure (circumstances). Thus, two individuals who exercise exactly the same number of times per week, may be interpreted as exerting very different levels of effort, depending on their circumstances. To make the degree of effort expended by individuals of different types comparable, Roemer proposes the definition of quantiles of the within-type effort distribution (e.g., the distribution of weekly frequency of physical exercise within each type): Two individuals from different types are deemed to have exerted the same degree of effort if they sit at the same quantile (π) of their type’s distribution of effort. When effort is observed, this definition is directly applicable. However, if effort is unobservable, an additional assumption is required: By assuming that the average outcome, health in this case, is monotonically increasing in effort, i.e., that healthy lifestyles are a positive contribution to the health stock, effort becomes the residual determinant of health once types are fixed; therefore, those who sit at the πth quantile of the outcome distribution also sit, on average, at the πth quantile of the distribution of effort within this type.
How is the equality-of-opportunity policy characterized in this framework? Ideally, this policy should ensure identical health across types at identical levels of effort. Let us assume that, given our health production function, the highest health level attainable by type (t) given quantile level of effort (π) and policy (φ) is given by the indirect outcome function vt(π, φ). In this setting, the equality-of-opportunity policy pth equalizes the highest attainable health level across types for identical values of p, i.e., vt (π,φE-opp)=vt`(π,φE-opp).
In addition, because the resources available for policy interventions are generally finite, one also needs to ensure that φE-opp is feasible. However, this poses a problem: As shown in Roemer (2002), it will not be possible, in general, to find an equality-of-opportunity policy that simultaneously satisfies the feasibility requirement. Thus, in practice, instead of literally equalizing v between types at each π, one maximizes the minimum value of v across types at each π.
But we are not finished yet. In general one is not interested in finding the equality of opportunity for a sole particular value of π: Healthcare policy does not usually apply only to those at say, the qth quantile of weekly frequency of physical exercise. The problem is that there are different optimal policies for different values of π, even if interest in the subset of efficient policies is restricted. So how is the equality-of-opportunity policy found? A number of compromise solutions have been suggested in the literature. The most widely used in practice (proposed by Roemer (2002)), consists of aggregating over all policies (each defined for a particular value of π) and giving each of them the same weight.
Ex Ante And Ex Post Inequality
So far, this account of inequality of opportunity has focused on inequalities between groups of individuals, called types, who share exposure to identical circumstances. This approach is the most prevalent in applied work and is known as the ex-ante approach. The term ex ante refers to the fact that this approach can be used in cases where circumstances are known, but effort has not (yet) been exerted by the individuals.
There is, however, an alternative approach to the concept of equality of opportunity. Assume that effort is observed. The population of interest can thus be split into groups, known as tranches, which correspond to levels of exerted effort (e.g., number of times per week one does physical exercise). In this setting, inequality of opportunity corresponds to differences in outcomes within each tranche, i.e., amongst individual who have exerted the same level of effort. The source of unjust inequalities is still the variation across individual circumstances, but this line of research is known as the ex post approach, because it requires information on the level of effort already exerted by individuals.
An important question is the extent to which the ex-ante and the ex post approaches are similar. Although they share points in common, they are fundamentally different. As mentioned earlier, equality of opportunity requires the elimination of differences in outcomes that are due to circumstances but not to effort. This is known as the principle of compensation. It should however be noticed that this principle of compensation leads to different compensatory policies according to whether one takes the ex ante or the ex post approach. In the ex ante case, compensation requires transferring resources from individuals in the most advantaged types to people in least advantaged ones. But in the ex post approach, the required transfers are within-tranche transfers, amongst individuals who exert the same level of effort. Thus, ex ante and ex post compensation are generally incompatible.
Another important aspect concerns the definition of fair rewards to effort. Individuals with the same circumstances are considered. The theory of equality of opportunity described so far is silent regarding the fair way of rewarding different levels of effort amongst such individuals. There is at present an intense debate on the way to combine the compensation principle with a suitable reward principle, but a definite solution has not yet been reached. Fleurbaey and Schokkaert (2012) describe a number of possible avenues for achieving this within the framework of equality of oportunity, although J. Roemer has recently argued that the definition of what constitutes fair rewards to effort should instead come from an ancillary theory, which limits the degree of inequality that is acceptable. In addition, Fleurbaey and Peragine have shown that the available options for combining the principles of compensation and reward depend vitally on whether the researcher adopts the ex ante or the ex post approach. Although, in general, the principles of compensation and reward are theoretically incompatible, this conflict can be avoided when one adopts the ex ante approach (but not the ex post one).
Partial Orderings And Inequality Measures
How can inequality of opportunity be identified in practice? A number of different approaches have been proposed, based on partial equality-of-opportunity orderings. The most widely used in applied work defines equality of opportunity based on stochastic dominance conditions. The rationale is the following. Denoting by F(.) the cumulative distribution function of health (CFD), a literal translation of the idea of equality of opportunity would correspond to the situation in which the distribution of health outcomes does not depend on social types, i.e., F(·|t)=F(·|t`). This condition is, however, unlikely to hold in any society and hence is too stringent to be applied to real data. Instead one could assume that the data be deemed consistent with the existence of inequality of opportunity when the social advantage provided by different circumstances can be unequivocally ranked by stochastic dominance criteria, i.e., F(·|t)>SDF(·|t`). First-order stochastic dominance (FSD) holds for the whole class of increasing utility functions; thus if the distribution of health outcomes of type t FSD dominates that of type t’, this means that all individuals with an utility function that is increasing in health (i.e., who prefer better health to worse health) would prefer the outcomes of type t to those of type t’. Although one may extend this partial ordering to second- and even third-order stochastic dominance criteria, most of the applied literature has been focused on first-order comparisons. These are better suited for the ordinal outcomes that are often used in health economics, such as self-assessed health. Moreover, in addition to their clear meaning in terms of welfare and preferences , these conditions have an important attractive feature: They are statistically testable in practice.
Partial orderings are useful but often inconclusive, hence complete orderings have been proposed to measure inequality of opportunity. In this literature, an analysis of inequality of opportunity in Brazil carried out by Bourguignon et al. (2007) prompted a number of methods collectively known as the parametric approach to the measurement of inequality of opporunity. The idea is intuitive. Earlier, the definition of the health production function of individual health outcomes, H = f(C,E(C)) was given. The same specification applies to the health outcomes of social groups. Thus, a parametric regression model can be used to estimate the counterfactual distribution of outcomes that would be brought about by assigning the same circumstances to all the individuals, i.e., H=f(C,E(C)). Inequality of opportunity can then be measured by an index Г= 1- H/H.
A different approach, known as nonparametric supposes that one replaces each individual outcome in H by one’s type-specific mean (μt), obtaining the smoothed distribution of outcomes HC. This eliminates, by construction, all within-type inequality, hence a relative inequality index I(HC) measures exclusively between-types disparities, which constitute inequality of opportunity. Alternatively, one may replace the outcome of each individual i outcome (hi) by μ/μ` hj, where μ is the mean in the population of interest, obtaining the standardized distribution HE. In this case, all the between-types inequality is eliminated, leaving solely within-type inequality. As a result, inequality of opportunity corresponds to the difference between the total inequality in health outcomes, I(H), and the inequality measured by I(HE).
Two important practical issues arise in this context. The first concerns the choice of an appropriate inequality index I, given that, in general, the smoothing and standardizing approaches lead to different results. There is a class of measures (known as path-independent decomposable measures, for which these two approaches lead to the same result. Amongst this family of measures, the mean-log deviation has been very widely used in applied work. The second issue of interest is that of choosing between the parametric and the nonparametric approaches. Nonparametric methods are, in general, more robust in the sense that they do not depend on parametric assumptions. They are, however, more data-hungry: When the information on the circumstances set is rich the number of types increases, leading to data insufficiency. This is less of a problem for the parametric approach, which, in addition, allows the estimation of partial effects, namely, circumstance-specific inequality shares. Nonetheless, this comes at the expense of an increased reliance on structural assumptions.
Another index that has been increasingly used in applied work is the Gini-opportunity index proposed by Lefranc et al. (2009). This is a modified Gini coefficient that quantifies the inequality between the different types’ opportunity sets. The area underneath a type’s generalized Lorenz curve, and hence the value of its Sen evaluation function Aj = μj (1-Gj) constitutes a cardinal measure of this type’s opportunity set (Gj denotes the Gini coefficient and mj the average outcome within that social type). Thus, in the context of inequality of opportunity, one may rank types (not individuals) according to their respective values of the Sen evaluation. For any pair of types, denoted i and j, and starting from the one with the smallest value of the Sen evaluation function, the Gini-Opportunity index across types i to k is defined as: G -Opp =1/μ∑ki ∑i<j[pipj(Aj-Ai)]. This index gives the weighted average of the differences between the types’ opportunity sets in which the weights (p) are the sample weights of the different types. The value of all these indices is highly sensitive to the number of types; this can be a problem because, as seen before, the number of types is, in practice, defined subjectively by the researcher.
It should finally be noted that a good measure of inequality of opportunity in health should be able to bring together multiple circumstances and, given that health is an inherently multidimensional concept, multiple dimensions of health outcomes. This also applies to the case of inequality of opportunity in healthcare, which incorporates a number of different dimensions, such as general practitioner visits and specialist visits. Rosa Dias and Yalonetzky (2013) have recently addressed this issue by drawing on the segregation literature and proposing new measures that are applicable when health (or healthcare) is proxied by a finite number of ordinal indicators.
Inequality Of Opportunity In Health Economics: Theoretical Contributions And Empirical Evidence
Theoretical Contributions In Health Economics
It is possible to argue that inequality of opportunity is already the implicit equity concept in some earlier contributions in health economics, such as Alan Williams’ fair innings argument and the Rawlsian approach to the measurement of health inequalities proposed in Bommier and Stecklov (2002). Yet, although the volume of applied research on inequality of opportunity in health has grown rapidly over the past few years, the amount of theoretical work, has been comparatively smaller.
Fleurbaey and Schokkaert (2009) make an important contribution toward incorporating the analysis of inequality of opportunity in health in the broader framework of responsibility-sensitive egalitarianism. They propose analyzing inequality of opportunity in health within the framework of a complex structural model that encompasses simultaneously the demand for health, lifestyle and healthcare, labor supply, and income distribution. In this model, the health stock depends on a range of factors, encompassing the consumption of healthcare and other goods, job characteristics, socioeconomic background, genes, and unanticipated health shocks. Labor income is endogenous and depends on various factors including individual ability. The demand for healthcare also depends on multiple factors, including supply-side variables and individual demand for supplementary health insurance.
This model can be solved in two stages. First, individuals decide on their desired level of supplementary health insurance. Second, for that level of insurance coverage, they maximize utility subject to income constraints, time constraints, and to the supply of healthcare constraints. This allows for the joint determination of the demand for health care, consumption goods, and individual labor supply. Finally, armed with the optimal values for these, the optimal levels of health, income, and utility are endogenously determined by the model.
This complex structural model is the most encompassing framework proposed for the analysis of unfair health inequalities (including inequality of opportunity). However, the multiple and reciprocal causal relationships that it embodies poses serious operational challenges to the empirical identification of the model.
Another aspect that has received attention in the health economics literature relates to the fact that, in practice, it is often not possible to observe the full set of relevant circumstances influencing health outcomes. Fleurbaey has shown that this issue, known as the partial-circumstance problem, may bias the measurement of inequality of opportunity in health. At present, there has not yet been found a reliable way to derive theoretical bounds for this bias. Rosa Dias (2010) examines the practical relevance of this matter by proposing a simple behavioral model of inequality of opportunity in health that integrates Roemer’s framework of inequality of opportunity with the Grossman model of health capital and demand for health. The model generates a recursive system of equations for the health stock and each of a series of effort factors such as the weekly consumption of calorific food, alcohol, and the weekly frequency of physical exercise. To take into account the role of unobserved heterogeneity, the system is then jointly estimated by full information maximum likelihood with freely correlated errors. The results suggest that, when unobserved heterogeneity in the set of circumstances is taken into account, the estimates of the recursive relationship between circumstances, effort, and health outcomes change considerably, thereby corroborating the empirical relevance of the partial-circumstance problem. Garc´ıa-Gomez et al. (2012) use an analogous estimation strategy to implement the framework of Fleurbaey and Schokkaert (2009), thereby modeling the channels through which circumstances affect health outcomes in adulthood. Armed with this behavioral model, Garc´ıa-Gomez et al. (2012) showed that distinguishing between these different channels is useful not only as a means of avoiding the partial-circumstances problem, but also in order to perform a sensitivity analysis of the results with respect to different normative positions regarding the factors that should be considered, i.e., circumstances and effort.
A different, although related, issue concerns the correct way to treat the partial correlations between circumstances and effort. Jusot et al. (2013) examine the practical relevance of this issue for the measurement of inequality of opportunity in health by applying a reduced-form approach to data from a large French survey. Interestingly, their results suggest that adopting fundamentally different normative approaches to this matter makes little difference, in practice, for the measurement of health inequalities.
In recent years the number of applications of the inequalityof-opportunity framework to health has grown rapidly. Rosa Dias (2009) and Trannoy et al. (2010) examine the existence and magnitude of inequality of opportunity in health using, respectively, data from the UK and France. Employing the stochastic dominance testable conditions proposed by Lefranc et al. (2009), they find that, in both countries, there is clear inequality of opportunity in self-reported health between individuals of different parental background (defined according to the father or male head of household’s occupation). Furthermore, these empirical applications show that shifting the focus from inequality in health to inequality of opportunity changes the results significantly: For example, in the case of the UK, Rosa Dias (2009) shows that an unusually rich set of circumstances that include parental background, childhood health, ability, and social development account for just approximately one-fourth of the total inequality in health.
These articles also show that inequality of opportunity in health is substantial in the countries studied: Trannoy et al. (2010) show that a hypothetical complete nullification of the influence of observed circumstances on health would, in the case of France, leads to a 57% points reduction in the selfreported Gini coefficient. Jusot et al. (2010) pursue this line of research further by using data from the 2004 Survey on Health Ageing and Retirement in Europe to compare the extent of inequality of opportunity in health across 10 European countries. Their results suggest that the magnitude of this type of inequality is markedly different between blocks of countries: Inequality of opportunity in self-assessed health is systematically higher in Southern Europe than in Northern European countries. In addition, this article makes clear that there are also differences regarding the most important circumstances in each of the countries.
Another important aspect concerns the evolution of inequality of opportunity in health over the lifecycle: Do circumstances affect health outcomes more heavily in the early years of life, young adulthood, or in old age? Rosa Dias (2009) provides some empirical evidence on this issue, using data from a UK cohort study; results from this study show that the influence of circumstances on self-reported health at 23, 33, 42, and 46 years of age is remarkably constant. This issue has been reexamined in greater depth by Bricard et al. (2012). This article proposes two alternative strategies for quantifying inequality of opportunity in health over the lifecycle. From an ex ante perspective, an aggregate measure of the lifetime health stock is estimated for each individual; inequality of opportunity in this aggregate health is then measured between individuals or groups. Alternatively, from an ex post perspective, health inequalities are measured across individuals at each stage of their lifecycle, before aggregating inequalities over the lifetime. Bricard et al. (2012) show that these two perspectives are grounded on different normative principles, and that they lead to different results when applied to real data.
Finally, an area that is, at present, receiving growing attention is the application of the inequality-of-opportunity framework to the normative evaluation of concrete policy interventions. Figheroa et al. (2012) propose a methodology to evaluate social projects from an equality-of-opportunity perspective by looking at their effect on the distribution of outcomes conditional on observable covariates. They apply this approach to the evaluation of the short-term effects of Mexico’s well-known Oportunidades program on children’s health outcomes. Jones et al. (2012) also proposes a normative framework, but designed for the evaluation of complementary policy interventions such as the health effects of educational interventions. This article grounds this proposal on Roemer’s (2002) model of inequality of opportunity, and applies it to data from a large-scale UK educational reform. Although Figheroa et al. (2012) focus on the evaluation of short-run policy effects, Jones et al. (2012) center on their long-run impact on health and lifestyle.
Although considerable evidence on inequality of opportunity in health has been amassed, there are still important unanswered questions in this field. First, virtually all the available evidence relates to developed countries. It would be interesting to know more about the magnitude, causes, and the channels of influence of inequality of opportunity also in developing countries. Second, further research is needed on the impact of health policy on inequality of opportunity in health. Although over the past years much has been learnt about the size and evolution of this type of inequality, little is still known about the ways to tackle it effectively.
- Bommier, A. and Stecklov, G. (2002). Defining health inequality: Why Rawls succeeds where social welfare theory fails. Journal of Health Economics 21, 497–513.
- Bourguignon, F., Ferreira, F. and Mene´ndez, M. (2007). Inequality of opportunity in Brazil. Review of Income and Wealth 53(4), 585–618.
- Bricard, D., Jusot, F., Tubeuf, S. and Trannoy, A. (2012). Inequality of opportunities in health over the life-cycle: An application to ordered response health variables. Health Economics 21, 129–150.
- Figheroa, J. L., Van de Gaer, D. and Vandenbossche, J. (2012). Children’s health opportunities and project evaluation: Mexico’s Oportunidades program. CORE Discussion Papers 2012015. Belgium: Universite´ catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Fleurbaey, M. and Schokkaert, E. (2009). Unfair inequalities in health and health care. Journal of Health Economics 28(1), 73–90.
- Fleurbaey, M. and Schokkaert, E. (2012). Equity in health and health care. In Pauly, M., McGuire, T. and Barros, P. P. (eds.) Handbook of Health Economics, vol. 2, pp 1004–1092. North-Holland: Elsevier.
- Garc´ıa-Gomez, P., Schokkaert, E., Van Ourti, T. and Bago D’Uva, T. (2012). Inequality in the face of death. CORE Working Paper 2012/24. (in press).
- Jones, A. M., Roemer, J. E. and Rosa Dias, P. (2012). Equalising opportunity in health through educational policy. Health, Econometrics and Data Group (HEDG) Working Paper. Chicago, USA: The University of Chicago Press.
- Jusot, F., Tubeuf, S. and Trannoy, A. (2010). Inequality of opportunities in health in Europe: Why so much difference across countries? Health, Econometrics and Data Group (HEDG) Working Paper 10/26. (in press).
- Jusot, F., Tubeuf, S. and Trannoy, A. (2013). Circumstances and effort: How important is their correlation for the measurement of inequality of opportunity in health? Health Economics. doi: 10.1002/hec.2896.
- Lefranc, A., Pistolesi, N. and Trannoy, A. (2009). Equality of opportunity and luck: Definitions and testable conditions, with an application to income in France. Journal of Public Economics 93(11–12), 1189–1207.
- Roemer, J. E. (1998). Equality of opportunity. Cambridge, MA: Harvard University Press.
- Roemer, J. E. (2002). Equality of opportunity: A progress report. Social Choice and Welfare 19, 455–471.
- Rosa Dias, P. (2009). Inequality of opportunity in health: Evidence from a UK cohort study. Health Economics 18(9), 1057–1074.
- Rosa Dias, P. (2010). Modelling opportunity in health under partial observability of circumstances. Health Economics 19(3), 252–264.
- Rosa Dias, P. and Yalonetzky, G. (2013). Measuring Inequality of Opportunity in Health When the Health Variable is Discrete and Multidimensional. Oxford: Oxford University Press.
- Trannoy, A., Tubeuf, S., Jusot, F. and Devaux (2010). Inequality of opportunities in health in France: A first pass. Health Economics 19, 921–938.
- Van de Gaer, D., Vandenbossche, J. and Figueroa, J. L. (2012). Children’s health opportunities and project evaluation: Mexico’s Oportunidades program. World Bank Economic Review (forthcoming).
- World Bank (2005). World Development Report. Equity and Development. Washington, DC: The World Bank.
- Arneson, R. (1989). Equality and equal opportunity for welfare. Philosophical Studies 56, 77–93.
- Bossert, W. (1995). Redistribution mechanisms based on individual characteristics. Mathematical Social Sciences 29, 1–17.
- Checchi, D. and Peragine, V. (2010). Inequality of opportunity in Italy. Journal of Economic Inequality 8, 429–450.
- Cohen, G. A. (1989). On the currency of egalitarian justice. Ethics 99, 906–944.
- Dworkin, R. (1981). What is equality? Part 2: Equality of resources. Philosophy & Public Affairs 10, 283–345.
- Ferreira, F. and Gignoux, J. (2011). The measurement of inequality of opportunity: Theory and an application to Latin America. Review of Income and Wealth 57(4), 622–657.
- Fleurbaey, M. (2008). Fairness, responsibility and welfare. Oxford: Oxford University Press.
- Fleurbaey, M. and Peragine, V. (2013). Ex-ante versus ex-post equality of opportunity. Economica 80(317), 118–130.
- Foster, J. and Shneyerov, A. (2000). Path independent inequality measures. Journal of Economic Theory 91(2), 199–222.
- Rawls, J. (1971). A theory of justice. Cambridge, MA: Harvard University Press.
- Sen, A. (1980). Equality of what? In McMurrin, S. (ed.) The tanner lectures on human values 1. Salt Lake City: University of Utah Press.