Equality of Opportunity in Four Measures of Well-Being

2.1 Well-Being

Three overarching theories of well-being exist in the philosophical literature; objective list theory, preference satisfaction theory, and mental state theory (Parfit, 1984; Griffin, 1986). Preference satisfaction theory is the most commonly assumed in economics. It claims that an individual's welfare depends on the degree to which his or her preferences are satisfied. Often, preference orderings are assumed to be revealed through choice behavior. The underlying tenet behind these revealed preferences is that if an agent chooses bundle A over bundle B, then the agent must prefer A over B, and the agent must be better off with A rather than B. Mental state theory takes its starting point in what goes on inside the mind of individuals rather than their observed choices. According to this theory, well-being is the degree to which individuals are happy or the extent to which they experience pleasure over pain. Objective list theory argues that individuals’ lives go well to the degree that they are in possession of certain items on a list, which could be income, education, health, safety, and so on.

In short, mental state theory cares about what individuals feel, (revealed) preferences about what individuals choose, and objective list theory about external factors, which could be independent of the choices and feelings of individuals. Each theory has its advantages and shortcomings. Preference satisfaction theory, at least in the revealed form, can be criticized when individuals’ decision-making is subject to imperfect knowledge or behavioral biases. If individuals have mistaken beliefs about what is best for themselves or lack willpower to choose what is best for themselves, then there is little reason to believe that their choices are a good manifestation of their well-being. Mental state theory can be criticized for its “physical condition neglect” (Sen, 1985), whereby individuals might feel well only because they have adapted to horrible conditions. Objective list theory can be criticized for being elitist in the sense that a set of indicators and weights are chosen somewhat independent of the preferences of individuals and for not accounting for spillovers in well-being levels. All of these critiques can, of course, be counteracted, but doing so would be outside the scope of this paper.

This three-part division of well-being concepts is still very much in use today in both theoretical and empirical literature about well-being—see, for example, the chapter division in The Oxford Handbook of Well-Being and Public Policy (Adler and Fleurbaey, 2016) and the Stanford Encyclopedia of Philosophy on “Well-Being” (Crisp, 2016). We will operationalize a measure of well-being with roots in each of these theories and see if they lead to different conclusions about equality of opportunity. We are not attempting to argue in favor of one of these welfare concepts. Rather, we will take some of the operationalizations of these concepts at face value and investigate if equality-of-opportunity estimations depend on which measure is used.

2.2 Distributive Justice

Until Rawls published his Theory of Justice (Rawls, 1971), the predominant view of justice was defined in utilitarian terms. Under this view, the just outcome maximizes total welfare or, equivalently, equalizes marginal utilities. This view is welfarist in the sense that if all individuals’ welfare levels are known, then no additional information is needed to decide whether one scenario is more desirable than another. Rawls argued against this welfarist view, emphasizing that we should not seek to equalize marginal utilities but, rather, primary goods, which is a broader notion that also encompasses rights and liberties.

A number of subsequent scholars proposed variations of what the right equalizandum ought to be, building on the work of Rawls. Sen (1980) argued that neither utilities nor primary goods were enough to judge outcomes. He concluded that we need to look at what individuals are capable of achieving with these goods, thus advocating for basic capability equality. Dworkin (1981) contended that resources are the right equalizandum, while Arneson (1989) and Cohen (1990) argued that we ought to pursue equality of opportunity for welfare and equal access to advantage (for a more complete account on the developments in distributive justice since Rawls, see Roemer and Trannoy, 2016).

Although these philosophers differ in their preferred equalizandum, they all adhere to the point of view that knowing all individuals’ welfare levels is not sufficient. We need to know how that welfare came about, whether it came from fortunate backgrounds or from factors for which we can hold individuals accountable. As such, they agree on the need to go beyond welfarism and accept some degree of individual responsibility, and thereby some degree of just inequalities. Notably, none of the philosophers have defined the equalizandum in terms of income. Rather, they have considered broader notions than income, such as welfare, advantage, or functionings. Our approach attempts to get a bit closer to these frameworks. In particular, our approach is closely related to that of Arneson (1989), who precisely argued for equalization of opportunities for welfare.3

2.3 Equality of Opportunity with the Norm-Based Approach

The philosophical theories of distributive justice have been operationalized in economics through the works of Roemer (1993), Van de gaer (1993), and Fleurbaey (1994), amongst others. The starting point in many of these operationalizations is to consider a population, , and a distribution of an outcome variable for this population, . Often, y is considered to be income, but here we will take welfare/well-being as the outcome, such that is the welfare of individual . An individual's outcome is assumed to be a product of two sets of variables: circumstances, , and effort, . Circumstances are the factors outside the realm of control for the individual, the factors for which one ought not to hold an individual responsible. These are often taken to be gender, region of birth, parental education, parental income, and so on. Effort variables are the factors for which one ought to hold an individual responsible. The well-being of individual i is thus assumed to be given by . We consider the well-being levels to be cardinal and interpersonally comparable, but our setting also works in an ordinal framework, where we convert the welfare levels into welfare ranks.

Based on this setup, the literature proceeds by measuring the extent to which the outcome variable is driven by circumstance or effort variables. To do so, we will rely on the norm-based approach (Ramos and Van de gaer, 2016), also called the fairness gap in Fleurbaey and Schokkaert (2009). This approach uses fair allocation rules to assign each individual a fair outcome, which only depends on his or her effort level, and calculates inequality of opportunity as the divergence between the actual outcomes and the fair outcomes. The more the two distributions diverge, the less individuals’ effort is associated with their well-being and the more circumstances are shaping individuals’ outcomes.

The axiomatic literature on fair allocations has put forward two criteria that fair allocation rules ideally ought to satisfy, these being the compensation principle and the reward principle. The compensation principle states that differences in well-being due to differences in circumstances should be eliminated. The reward principle is concerned with the proper reward of effort for individuals with the same circumstances. Unfortunately, these two criteria are mutually incompatible (Bossert, 1995; Fleurbaey, 1995). The literature has proposed allocation rules that weaken these two principles in order to make them compatible. Two such rules are the Egalitarian Equivalent principle (Bossert and Fleurbaey, 1996) and the Generalized Proportionality principle (Cappelen and Tungodden, 2017).

Both principles start by defining a norm vector of circumstance variables, , and calculate the well-being that individuals would have had, if they had this norm level of circumstances, . This neutralizes the effect of circumstances on well-being and the differences in can therefore be thought of as expressing fair advantage. As the mean of may differ from the mean of the actual outcomes, and since resources should not be wasted and cannot come out of nothing, some adjustment is necessary. The two fair allocation rules deal with this in different ways. To estimate the distribution of fair outcomes, which we call z, the Egalitarian Equivalent (EE) principle distributes the difference between the size of the two distributions equally across individuals:

(1)

where . This principle has the advantage that if the well-being function is linearly separable in circumstances and effort, such that , then the fair well-being levels do not depend on which norm vector of circumstances is chosen and the expression boils down to

(2)

The Generalized Proportionality (GP) principle, instead, respects the well-being shares of the fair well-being allocation by scaling them by the same factor:

(3)

An individual's fair outcome only depends on his or her effort variables and reflects how much the individual ideally is entitled to under the given allocation principles.4 The more aligned the fair outcomes and the actual outcomes are, the lower is the inequality of opportunity. Conversely, the more they diverge, the greater is the inequality of opportunity. We can measure the extent to which the two distributions diverge from each other by employing a divergence measure, D(y‖z), which evaluates the divergence between the two distributions, y and z.

Magdalou and Nock (2011) (MN) provide a framework for axiomatically grounded divergence measures.5 They put forth a class of divergence measures between an outcome distribution, y, and a reference distribution, z:

(4)

where, for all , the function ϕ(c) is given by the following:

(5)

s is a sensitivity parameter, similar to the one we find in standard inequality measurement. This class boils down to the generalized entropy class of standard inequality measures if z is assumed to be the mean of the actual distribution. The class satisfies partial symmetry (i.e. it is invariant to permutations of pairs) along with other relevant properties. By using the function ϕ from equation (5) in equation (4), one obtains

(6)

Cowell (1985) suggests a different class of divergence measures, which he calls measures of distributional change, that satisfies different properties.6 The measure equivalent to (4) is as follows:

(7)

By using the function ϕ from equation (5) in equation (7), can be written as follows:

(8)

where “−1” is added when s ≠ 0, 1 to obtain a proper generalization of the generalized class of inequality indices. Devooght (2008) provides an empirical application of this class to equality of opportunity in Belgium.

The two different classes of divergence measures, and , coincide when s = 1 (the last two terms in when s = 1 cancel each other out in our case). For this reason, we are going to use s=1 as our main specification. Parameters s = 0 and s = 2 with will be used as robustness checks. One of the features of is that a progressive transfer in the actual distribution, y, reduces the divergence between y and z as long as the individuals involved in the transfer have the same reference, z. It is a kind of priority given to the worse-off individuals when the individuals involved in the transfer share the same z. Moreover, if (and only if) s < 2, the further down the distribution of y such a transfer takes place, the more the divergence between z and y is reduced. This property resembles the principle of diminishing transfers in the context of inequality measurement, which holds for the class of entropy indices when s < 2. When s = 2, the measure is ordinally equivalent to the Euclidian distance, and it is thus insensitive to the position on the distribution where the progressive transfer (among individuals with the same reference) takes place. Thus, the parameter value s = 2 can be seen as a threshold. Contrary to this, the parameter value s = 0 yields a measure that is more sensitive to transfers lower down the distribution than our baseline measure with s = 1.

As another robustness check, we will use a divergence measure that is a generalization of the standard Gini coefficient developed by Almås et al. (2011), called the Unfairness Gini, :

(9)

As a final robustness check, we will use a very simple measure of measure of the divergence; the rank correlation between welfare levels and . A large correlation suggests a high degree of inequality of opportunity, since individuals with the best circumstances also have the highest welfare levels. A correlation of 0, in turn, reflects that the quality of individuals’ circumstances is not correlated with their welfare. If we use income as the outcome variable and consider parental income as the only circumstance variable, then this measure boils down to a frequently used measure of immobility; Spearman's correlation between parents’ and children's income level (see, e.g., Chetty et al., 2014).

3.1 Constructing Welfare Variables

We use four different welfare variables in the analysis. First, we use log incomes. This is the most frequently used outcome variable in equality-of-opportunity studies. We use it as a baseline for comparison to the other well-being measures. We use annual net household income expressed in 2010 constant EUR adjusted for family size using the OECD equivalence scales. The other three welfare variables are rooted in the three concepts of well-being that Parfit (1984) and Griffin (1986) put forward.

The second welfare measure we use is life satisfaction, which has roots in mental state theory. Life satisfaction is the answer to the question “How satisfied are you with your life, all things considered?” The answer categories range from 0 (completely dissatisfied) to 10 (completely satisfied). For the purpose of this study, we consider the answers to be interpersonally comparable. This is not meant as an endorsement of this particular account of well-being but, rather, as an inquiry into how inequality-of-opportunity estimates would look if one accepted these assumptions.

The third welfare measure we use is a multidimensional welfare measure, which has roots in objective list theory. To construct the measure of multidimensional welfare, we partly follow Decancq and Neumann (2016). We consider four dimensions; income, health, leisure, and employment.7 Income is measured the same way as above. Employment is a binary variable, taking the value of 1 if the respondent had a job at the time of the survey. Leisure is measured as the number of daily hours spent on leisure (capped at 6 hours). Health is itself a composite index, composed of (1) an indicator for whether the individual is disabled, (2) the number of doctor's appointments the respondent had last year, and (3) the number of inpatient nights in hospitals that the respondent had last year. To aggregate these sub-dimensions into one health dimension, we regress a health satisfaction question on the three variables and use the coefficients as weights. The health satisfaction variable is composed of answers to how satisfied individuals are with their health on a scale from 0 (not at all satisfied) to 10 (completely satisfied). For the income, leisure, and health dimensions, we standardize the values such that the highest observed level is 1 and the lowest observed level is 0. Now we have four dimensions each bounded between 0 and 1. To arrive at the final multidimensional index, we simply add these four together.

The fourth welfare measure, equivalent incomes, is based on preference satisfaction theory. In short, equivalent incomes are the incomes that individuals need together with a reference bundle, to make them indifferent to their actual bundle. Although preferences are often estimated from choice behavior, this is difficult when the arguments cover a wide array of dimensions of well-being. An alternative method to recover preferences, used by Decancq et al. (2015), is to regress life satisfaction on the dimensions of well-being and interpret the weights as marginal rates of substitution. The resulting utility functions seem to be highly correlated with the utility functions one would recover from choice behavior (Akay et al., 2017).8 This approach easily accommodates preference heterogeneity by allowing for interactions between sociodemographic characteristics, w, and the various dimensions, dim. We follow this approach and use the following subset of the circumstance and effort variables as preference heterogeneity parameters, w: birth location, sex, age, work hours, self-employed, and public sector worker. As the non-income dimensions, dim, we consider health, employment, and leisure, as in the multidimensional index. Our life satisfaction regression looks as follows:

(10)

Output from this regression is shown in Table A.2 in the Appendix. In order to calculate the equivalent incomes, we first select a reference vector, , of all dimensions other than income. Here, we choose the mean outcome (the mode for categorical variables), since this avoids favoring any extreme marginal rates of substitution.9 Then, we calculate the income needed together with for this joint bundle to make individuals indifferent to their actual bundle. That is, we isolate below:

(11)

The result is an interpersonally comparable measure of individual welfare that takes differences in preferences into account.

By employing equivalent incomes as a welfare measure in our analysis, we are implicitly taking sides in a rich philosophical debate about whether individuals should be held responsible for their preferences. Our empirical approach implicitly deems differences in well-being arising from preference heterogeneity unfair if they stem from circumstance variables, but fair if they stem from effort variables. This is in contrast to most applications of equivalent incomes, which hold that individuals should be responsible for all their preferences. We can amend our approach such that individuals are held responsible for all their preferences by decomposing the equivalent income measure into a part that is due to preference heterogeneity and a part that does not incorporate preference heterogeneity. We will do so as a robustness check. A detailed discussion of this method along with an illustration is given in Appendix A.3. Still, our usage of equivalent incomes as a welfare measure that is regressed on income and circumstance variables is in contrast to the typical use of equivalent incomes, which incorporate issues of unfairness directly into the measure. We simply interpret equivalent incomes as another measure of well-being to be used in our analysis.10

Histograms of the four final welfare measures are presented in Figure 1.

3.2 Estimating Equality of Opportunity

For our empirical specification, we consider well-being to be a linearly separable function of effort and circumstance variables:

(12)

Two important issues remain unsettled. The first is the issue of how to interpret the error term, . The error contains omitted effort variables, omitted circumstance variables, measurement error, and general uncertainty. It is unclear whether this should be considered to be within individual control. This is an important decision, as it accounts for most of the variation in the welfare levels. In our baseline specification, we follow the inequality-of-opportunity literature and consider it an effort variable, but as a robustness check we shift it to the other side of the responsibility cut.

The other unsettled issue is what to do with the correlation between the effort and circumstance variables. Individuals’ effort levels are partly determined by their own choices and partly by their circumstances. Years of education, for example, is partly influenced by individuals’ social background and partly by an individual's own choices. We follow Roemer's approach and consider this correlation to be outside the realm of individual responsibility (Roemer, 1998).11 In practice, this means that prior to estimating the impact of circumstances and efforts on well-being, we perform an auxiliary regression of the following form:

(13)

We perform such a regression for each effort variable and use the residuals from these regressions as our effort variables in our main regression, which then becomes

(14)

Due to the Frisch–Waugh–Lowell theorem, the coefficients on the effort variables will be the same in (12) and (14). The coefficients on the circumstance variables will be different, as they in (14) also incorporate the indirect effect of circumstances on effort.12 We will later report specifications where we omit this auxiliary regression.

To compare who the opportunity-deprived are across the four well-being measures, we rank individuals according to their opportunity profile; that is, according to the quality of their circumstances. We calculate each person's yearly opportunity rank as follows:

(15)

where is the yearly cumulative distribution of individuals’ unfair advantage; that is, the part of their well-being due to circumstances. This allows us to compare the opportunity-deprived across the four well-being measures.

To estimate equality of opportunity over time using the norm-based approach, we first compute each individual's fair outcome, using equation (2):

(16)

That is, we compute each individual's fair outcome as the contribution to well-being from the individual's effort and the average contribution coming from circumstances for all individuals in the sample for the year in question. Second, we utilize our battery of divergence measures to compute the divergence between the fair outcomes and the actual outcomes.

4.1 Who Are the Opportunity-Deprived?

Table A.3 in the Appendix shows the results of the regressions from equation (14). Based on this output and equation (15), we calculate each person's yearly opportunity rank. Table 1 shows the correlations between these opportunity ranks for the four welfare measures. The correlations reveal the extent to which the same people are opportunity-deprived across the four measures. In the table—and throughout the paper—we bootstrap confidence intervals in order to take all derived uncertainty into account, including the uncertainty when constructing the welfare measures. We bootstrap 500 resamples at individual-level clusters.

The opportunity ranks display rather high correlations, suggesting that the same people are opportunity-deprived regardless of how we measure welfare. The welfare measures we have constructed are, of course, partly contained within each other; the income variable is included in both the multidimensional index and the equivalent income measure, and the latter two use the same four dimensions but aggregate them differently. We can analyze the extent to which this is driving the high rank correlations by comparing them with the rank correlation between the welfare levels, which are shown in Table 2. In all cases, the correlations are higher when we look at . This indicates that the high-opportunity rank correlations are not driven solely by the measures’ interrelatedness. This also suggests that the way in which welfare is measured matters more if we target the welfare-deprived than if we target the opportunity-deprived.

Since individuals’ opportunities are unobservable, policymakers may have to assist the opportunity-deprived indirectly. One way of doing so is by targeting individuals with circumstance profiles that are highly correlated with having low opportunities. We can use the opportunity ranks to test if the characteristics of the opportunity-deprived are similar across the four welfare measures. To do so, we calculate the average opportunity rank for individuals with a given circumstance. The results are shown in Figure 2. The lower the average opportunity rank, the fewer opportunities individuals with the given circumstance have, and the more this circumstance is a potential factor that policymakers can use to target the opportunity-deprived. If the average rank is less than 50, then the particular group has less than average opportunities.

There are many similarities across the welfare measures. Individuals with low-educated parents and individuals whose father was a blue-collar worker or not employed have low opportunities. The same applies to individuals who grew up in the countryside, individuals born in East Germany, short individuals, females, and individuals with many siblings.

Meaningful differences emerge only in two places, for people born abroad and for different age groups. People born abroad are more opportunity-deprived in all measures but life satisfaction. A possible explanation for this is that people born abroad understand the life satisfaction scale more optimistically than Germans. Alternatively, it may be because people born abroad tend to have other circumstances, which are particularly good for life satisfaction. We can indirectly check which effect is driving the result by excluding place of birth from the regression, calculating new opportunity ranks, and recomputing the average opportunity rank of individuals born abroad. With this approach, the direct channel from being born abroad on life satisfaction is omitted. Now the average opportunity rank of people born abroad falls below 50 (not shown in the figure). This suggests that the previous higher rank was solely driven by the direct positive effect of being born abroad on life satisfaction, and that individuals born abroad fare worse than people born in Germany with respect to the remaining circumstances. Hence, it may be that the high-opportunity rank with respect to life satisfaction for people born abroad is solely due to scaling effects.

With respect to age, young people are opportunity-deprived in income but not in the multidimensional index. This is hardly surprising, as the multidimensional index includes health, and young people on average are healthier. Since young people have a lower preference for health (as shown in Table A.2 in the Appendix), young people are not faring better with equivalent incomes, despite the fact that they are more healthy. With respect to life satisfaction, the middle-aged are opportunity deprived while the young and people above 55 are doing better. This mimics the well-known U-curved relationship between subjective well-being and age (Blanchflower and Oswald, 2008; Clark et al., 1996).

Unlike the other categories shown in the figure, individuals will in most cases experience all age categories during their life. It is questionable whether resources should be allocated such that individuals have equal opportunities in every part of their life. For this reason, later on we will place age on the other side of the responsibility cut. This may seem counterintuitive, but it amounts to saying that individuals should have equal opportunities on expectation over their life cycle rather than in every point of their life (for a similar approach, see Almås et al., 2011).

In sum, there is relatively large agreement about who the opportunity-deprived are across the four measures. Hence, if a policymaker strives to target individuals with low opportunities, it matters relatively little how welfare is measured. Although this suggests that equalizing opportunities does not depend on how welfare is measured, it may very well be the case that the public policies needed to prevent inequality of opportunity from arising in the first place depend on which measure of welfare is used. To study this properly, a causal framework is needed, which goes beyond the scope of this paper.

4.2 Equality of Opportunity over Time

Before analyzing how equality of opportunity has evolved over time, and whether this depends on the well-being measure used, we start by analyzing how the level of well-being has evolved over time and how inequality in well-being has developed over time. Panel (a) of Figure 3 shows the level of well-being in Germany from 1992 to 2016. In the figure, and in the time-series figures to follow, we use 3-year moving averages to smoothen out erratic trends. The level of well-being is normalized to 100 in 1992 to foster comparisons between the different measures.

The figure shows that from around 2005, well-being has increased with all four measures, but prior to that, life satisfaction and the multidimensional index followed different patterns than the remaining two measures. Panel (b) of Figure 3 shows the development in inequality in the four welfare measures using the Theil index. Inequality in log incomes has increased substantially over most of the period and inequality in life satisfaction has decreased over the past 10 years, while inequality in the other two measures has followed intermediate trends.11

Thus, the measure of welfare matters both when we look at the development of welfare and the inequality of welfare over the past 25 years in Germany. How do things look for the development of inequality of opportunity over time? This is displayed in Figure 4.

Although the confidence bands are quite wide, the measures follow broadly the same pattern. From 1992 to 2005, inequality of opportunity increased, and from then and until 2016, it gradually fell. Some of the welfare measures, particularly equivalent incomes, are noisier, which can be explained by the added uncertainty from the fairly complicated process of constructing this welfare measure. Our setup does not allow us to speak to the causes of the changes in inequality of opportunity, but we note that the fall from around 2005 coincides with the implementation of the Hartz plan, which could explain the pattern.

4.3 Altering the Responsibility Cut

The analysis above was based on important normative assumptions regarding what individuals were held responsible for. We assumed that individuals were responsible for four variables (4var): their education, work hours, and whether they are (i) self-employed or (ii) work in the public sector. We also assumed that individuals should not be held responsible for the part of these variables that could be accounted for by circumstance variables. That is, the correlation (cor) between circumstances and effort was itself considered outside the control of individuals. We further assumed that the part of individual well-being that was unaccounted for by circumstance or effort variables (residual) was within individual control. Next, we implicitly considered well-being differences across different age groups (age) as unfair. Finally, for the equivalent income measure, we assumed that individuals were not responsible for well-being differences due to preference heterogeneity arising from circumstances (pref).

Our baseline effort set contained 4var and residual. In this section, we try to shift the responsibility cut by altering what goes into the effort set. First, we look at whether the characteristics of the opportunity-deprived change as we change the effort set. This is analyzed in Figures 5 and 6. Figure 5 uses our most narrow effort set, only 4var, while Figure 6 uses the widest possible effort set, 4var, residual, cor, age, pref.

The figures show the same overall pattern as our main results. For the most part, the opportunity-deprived share the same characteristics across all four measures. The only disputes are, once again, for individuals born abroad and for different age groups. The responsibility cut does matter for quantifying the degree to which a particular group is opportunity-deprived. For example, individuals born in East Germany have an average opportunity rank of about 40 with the smallest effort set and 15 with the largest effort set. In other words, East Germans are not doing too poorly if we only hold individuals responsible for four variables (which almost mimics just looking at outcomes), but the more we hold individuals responsible for, the more disadvantaged they are. This suggests that East Germans have relatively high effort levels.

Next, we look at whether the developments over time depend on where we place the responsibility cut. We try six different specifications, which gradually expand the effort set. The results are displayed in Figure 7. Our primary interest is not whether the trends change as we change the responsibility cut but, rather, whether the four well-being measures follow similar trends regardless of where we place the cut.

Our baseline results are displayed in panel (c). If we no longer hold individuals responsible for the four variables we had deemed effort (panel (b)), then our main results become more clear; inequality of opportunity increased until about 2005 and then decreased for all four measures. If we only hold individuals responsible for the four variables and shift the residual to the other side of the responsibility cut (panel (a)), then the picture looks very different, particularly for log income. Since the four effort variables are able to explain very little of the variance, the measure almost boils down to the development in inequality over time, which, as we have already established, follows different trends. Other studies that have switched the residual to the other side of the responsibility cut likewise found this to have a great impact on the results (see, e.g., Devooght, 2008; Almås et al., 2011).

If we also hold individuals responsible for the correlation between effort and circumstances (panel (d)), then we still see an increase in inequality of opportunity in all measures from 1992 to 2005, but from then on the multidimensional index diverges from the rest. In the other cases where we add more to the responsibility cut (panels (e) and (f)), the overall trend changes, but stays broadly similar across the four well-being measures. In these cases, inequality of opportunity has remained rather flat or decreased a little from 1992 to 2016.

In sum, we find that when characterizing the opportunity-deprived, neither the measure of welfare nor the precise location of the responsibility cut is of great importance. When analyzing developments in equality of opportunity over time, where we place the residual matters quite a bit, while enlarging the effort set further has few implications. Although the trend may change depending on what we hold individuals responsible for, for a given responsibility cut, the well-being measures follow broadly the same trend.

4.4 Robustness Checks

In this section, we test whether our findings are sensitive to using other divergence measures and fair allocation rules. Our primary interest is not whether the trends change as we change the method but, rather, whether the four well-being measures follow similar trends regardless of what method we use. First, we try to use the Generalized Proportional allocation rule rather than the Egalitarian Equivalent allocation rule. Since the results now will depend on which norm vector we use, we show the results using the mode/mean circumstance (for categorical and continuous variables, respectively), as well as the worst and best circumstances, as defined by their bivariate relationship with welfare. The results are shown in Figure A.2 in the Appendix. Although large confidence bands prevent us from nailing down the trends with precision, particularly when the worst circumstances are used as the norm vector, in the other two cases, point estimates suggest that the measures follow broadly the same trend.

Next, we try to use the Magdalou–Nock divergence measures with s = 0 and s = 2 rather than s = 1. The results are shown in Figure A.3 in the Appendix. Again, we face wide confidence intervals. When s = 0, point estimates suggest that inequality of opportunity has decreased over the past 15 years with all measures. Things look a bit different with s = 2, where no trend is visible over the past 10 years with life satisfaction and the multidimensional index, while inequality of opportunity for income decreased. Since s = 2 gives less weight to the bottom of the distribution, this suggests that the improvements in equality of opportunity found in our baseline scenario have mostly arisen since the most opportunity-deprived were given greater chances. At the top of the distribution, the relationship between opportunities and outcomes remains equally strong.

We also try using different divergence measures; the Fairness Gini and the correlation between opportunity ranks and welfare ranks. The latter comes with a number of distinct advantages: (1) the measure is constrained to be between 0 and 1, so there is little need to index the numbers to 1992 = 100; (2) the confidence bands are narrower; (3) it minimizes our reliance on cardinality, which is problematic in the case of life satisfaction; (4) it ensures that we are comparing well-being measures that follow the same distributions; (5) it is more intuitive to understand; (6) it allows that zero and negative well-being levels can easily be included in the analysis; and (7) it generalizes a frequently used measure of intergenerational mobility, and as such the results can be compared with certain intergenerational mobility studies. The results are displayed in Figure A.4 in the Appendix. Both with the Fairness Gini and the Spearman correlation, the results suggest that inequality of opportunity has improved for all four measures over the past 10–15 years.