appendix, we describe some alternative approaches to solving the problem of maximizing between-group inequality for the Gini coefficient.<sup>16</sup> The same appendix also shows that without restrictions on the income distribution, no group order can be a priori excluded as a candidate for the BGI maximizing one.

In general, maximum BGI need not increase if the number of groups grows.<sup>17</sup> However, BGI cannot decline if more groups are obtained via proper sub-divisions of the existing groups.<sup>18</sup> In the limit, when every individual constitutes her own group, maximum BGI equals total inequality, and consequently _Rb=Rb'_ .

A possibly more appealing benchmark against which to evaluate between-group inequality can be obtained by introducing one more restriction. In addition to fixing the number of groups and their relative sizes, we can also arrange the groups under examination according to their observed mean incomes, keeping their ‘pecking order’ unchanged.<sup>19</sup> In many cases, there is a well-understood hierarchy of population groups in terms of their mean incomes. Comparing actual between-group inequality to a counterfactual maximum BGI which preserves the actual, observed, rank ordering of groups is conceivably of greater interest than a counterfactual which allows for random

> 16 The choice of Gini is for computational convenience and is not uncommon in the literature that precedes us, such as Davies and Shorrocks (1989).

> 17 In the case of the Gini coefficient, the difference between total inequality and maximum BGI can be crudely bounded and the bounds are functions of the population share of the largest group, and not by the number of groups (see the appendix for a proof).  Maximum BGI will stay bounded away from total inequality unless the maximum group size becomes sufficiently small.  This implies that although the expected value of between group inequality might increase as the number of groups increases (Shorrocks and Wan (2004), proposition 3), this value may be well below total inequality if one of the groups remains very large.  For example, with a lognormal (0,1) distribution and one group occupying 70% of the population while every other individual constitutes a separate group (implying effectively an infinite number of groups), the maximum possible between group inequality (measured by GE(0)) would be 0.373 – well below the inequality level of 0.5 in this distribution. For related results, see Davies and Shorrocks (1989) and Shorrocks and Wan (2004).

> 18 Starting from a BGI-maximizing group order, split-up a group in such a way that mean incomes in the new groups are equal to the old parent group.  BGI among these groups is equal to the old BGI.  Obviously, if the incomes of the two new groups are non-overlapping, then BGI will increase.  Hence a refinement of a particular grouping will generally lead to an increase in BGI.  See also Shorrocks and Wan (2004), proposition 2.

> 19 Ordering population groups by their mean incomes using, say, a household survey would introduce a possible difficulty due to sampling variability.  In other words, our ability to order groups by mean income (or consumption) could be limited by the fact that some of the group means are statistically indistinguishable from each other.  For the time being, we ignore the standard errors associated with the observed group means.

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