inequalities are deemed fair and which unfair is guided by the two key principles of compensation and reward. When an ex-ante view is taken of the compensation principle, it is possible to construct counterfactual distributions that do eliminate all fair inequality, and contain all unfair inequality. Examples include the between-types and direct unfairness distributions in Tables 2 and 3. When an ex-post view of compensation is adopted instead, it is not possible to construct counterfactual distributions that completely eliminate all fair inequality, and contain all unfair inequality. Two examples of distributions that are consistent with ex-post compensation, but not with the reward principle, are the within-tranches and fairness gap matrices in Tables 4 and 5.

# **5. Empirical applications**<sup>**20**</sup>

As noted in Section 3, the informational requirements for applying the direct approach to the measurement of inequality of opportunity – involving the observation of full choice sets – are very demanding and, to the best of our knowledge, that approach has never been applied empirically. Indirect approaches, on the other hand, rely on the observation of actual circumstances, outcomes and (possibly) efforts, and have been taken to real data. Indeed, empirical applications of all four approaches reviewed in Section 4 exist.

Almas et al. (2011) and Devooght (2008) use norm-based measures, based on Direct Unfairness and Fairness Gap. Checchi and Peragine (2010) and Aaberge et al. (2011) apply the within-tranches measure (as well as the between-types inequality) to estimate inequality of opportunity in Italy and in Norway respectively. But neither ex-post compensation nor norm-based measures have been computed systematically, or comparably, across many countries.<sup>21</sup> Most measures of inequality of opportunity computed in practice have followed ex-ante approaches and, in particular, the between-types approach.

The version of the ex-ante approach with vi = μi – that is, the between-types measure _I(_<sup>~</sup> _xBT )_ – has been applied to at least forty countries, by a number of authors. The specific inequality index _I(.)_ does vary across some of the papers but most use the mean logarithmic deviation, following Checchi and Peragine (2010) and Ferreira and Gignoux (2011). In a few cases, as noted below, the Gini index, the Theil (T) index and even the variance have been used.

Despite these differences, as well as a variety of caveats on data comparability across – or even within – studies, there is a set of eight papers that report the between-types measure and which constitutes, to our knowledge, the most closely comparable sources on actual I. Op. measures across countries. These eight papers - Checchi et al. (2010); Ferreira and Gignoux (2011); Ferreira et al. (2011); Pistolesi (2009); Singh (2011); Belhaj-Hassine (2012), Cogneau and Mesple-Somps (2008) and Piraino (2012) - cover forty-one countries, ranging from Guinea and

> 20 This section draws heavily on Brunori, Ferreira and Peragine (2013).

> 21 Though see Björklund et al. (2012) for an interesting application of ex post inequality measures to long-run incomes in Sweden.

**_22_**