# **Box 2. A Simple and Intuitive Example of HOI**

Consider two countries, A and B, each with a total population of 100 children. Each country has two groups of children, I and II, which consist of the top 50 percent and bottom 50 percent by per capita income, respectively. The coverage rate of school enrollment (or the average enrollment rate) for both countries is 0.6, that is, 60 children attend school in each country. The table below shows the number of children going to school in each group for each country.

Given the total coverage rate, the principle of equality of opportunity will hold true for each country if each of the two groups in each country has the same rate of coverage, that is, if each group has 30 children going to school. But in reality, group II has 20 enrollments in country A and 25 in country B. This suggests that firstly, opportunities are unequally distributed, and secondly, that inequality of opportunities is higher in country A. The D-index is the share of total enrollments that is ―misallocated,‖ namely

||Number of ch<br>10 years enrol<br>|ildren ages 6 to<br>led in school<br>|
|---|---|---|
||Country A|Country B|
|Groups by circumstance,<br>income|(100<br>children)|(100<br>children)|
|Group I|40|35|
|(top 50% by income)|||
|Group II|20|25|
|(bottom 50% by income)|||
|Total|60|60|

10/60 and 5/60 for A and B, respectively. Therefore, HOI _A_ = _C_ 0 (1 – _D_ ) = 0.6 * (1 – 10/60) = 0.50; HOI _B_ = _C_ 0 (1 – _D_ ) = 0.6 * (1 – 5/60) = 0.55.

Thus, even though both countries have equal coverage rates for enrollment, the higher inequality of opportunity in country A leads to the D-index being higher for A than for B, and HOI being higher for B than for A. It is also easy to see that HOI will increase in a country if: (i) the number of enrollments in each group increases equally (in proportionate or absolute terms); (ii) the enrollment for any group increases without decreasing the coverage rates of the other group; and (iii) the enrollment for group II increases, keeping the total number of children enrolled unchanged (implying enrollment in group I reduces by an equivalent amount). These three features relate to the ―scale,‖ ―Pareto improvement,‖ and ―redistribution‖ properties of HOI, respectively—properties that are intuitively appealing.

_Source_ : Authors

**The HOI is an inequality-sensitive coverage rate** in the sense that it improves when inequality decreases with a fixed number of opportunities in a society, or when the number of opportunities increases and inequality stays constant. In more formal terms, the properties of the HOI guarantee that the improvement in the index is sensitive to (i) the overall coverage—when the coverage for all groups increases by factor _k,_ the HOI increases by the same factor; (ii) Pareto improvements—when the coverage for one group increases without decreasing the coverage rates of other groups, the HOI increases; and (iii) redistribution of opportunities—when the coverage rate of a vulnerable group increases for a constant overall coverage rate, there is a decrease in inequality and an increase in the HOI. To compute HOI for a particular opportunity for the children of a country, household survey data are essential. To allow computation of HOI for education and health opportunities, the survey must have a minimum set of information, at the individual (child) or household level, as appropriate. Examples of these would be whether the child is attending school, grade level and last grade completed, and health

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