which allows us to normalize the population share and the income share with respect to the sizes of the subgroups h and j.

For both decompositions, by means of pand sjj- the inequality between Gb can be obtained from the differences (рщ - s₍*p as:

When the subgroups overlap, a third residual term is added to the mean- based decompositions, such as the one by Bhattacharya and Mahalanobis, while the measure of G|, is not modified.

From the other side, the distribution-based decompositions, such as the one proposed by Dagum, have a different development: the overlapping component assumes a positive value, which increases by increasing levels of overlapping, and also influences the measurement of G₍₎.

Starting again from the case of к = 2 subgroups, for Bhattacharya and Mahalanobis we have:

while Dagum's decomposition results in:

The third residual term in the decomposition by Bhattacharya and Maha- lanobis is then equal to twice the overlapping component in Dagum's decomposition, with the relevant difference that in the Dagum decomposition the inequality between is also influenced by the presence of overlapping.

Extending the case of к overlapping subgroups leads, for Bhattacharya and Mahalanobis, to:

while, for Dagum's decomposition, we have:

where G, is obtained as in (Equation 4.10).

The comparison between the two approaches is still based on the relation Equation 4.11 and the result

holds also for the case of к overlapping subgroups, thus allowing easy comparison of the two decompositions of the Gini index.

The relation between R and G, can also be exploited in order to derive a relation between the expressions of the inequality between subgroups in the two decompositions. Since

by using R = 2G_t, we obtain

which provides a further element of comparison for both between the two decompositions and between the two approaches.

In the absence of overlapping, the subgroups means, appropriately weighted, are fully informative, while the presence of overlapping could reduce the relevance of the subgroups means as a reference for the inequality between subgroups, depending on how we consider and treat the overlapping units.

TABLE 4.4

Gini Index Decomposition by Bhattacharya-Mahalanobis and by Dagum for the Subgroups A and В in the Four Cases (Zero, Low, High, Maximum Overlapping) Reported in Table 4.1

An Illustrative Example

In this section we follow the example previously introduced, where four different overlapping typologies are analyzed and compared. Our purpose here is to study the Gini index decomposition with respect to different degrees of overlapping.

Table 4.3 reports the population share py, the character share sy, and the Gini index for the subgroups A and В in the four cases analyzed. The pi values are constant, as пл and «g are always the same (60 and 30, respectively) in all four cases. The sy, on the contrary, show the usual dynamic which characterizes the overlapping: initially, when overlapping is absent or low, the Sj are quite different from the pj, while increasing levels of overlapping lead the Sj to converge to the pj, until, when the overlapping is complete, the character share s,- equals the population share p_r A similar behavior interests the Gini indexes, which are different in absence or low overlapping, while being coincident in the fourth case.

The Gini index decomposition for our four cases is illustrated in Table 4.4, for the mean-based evaluations we adopt Bhattacharya and Mahalanobis's proposal, while for the distribution-based evaluations we turn to Dagum's contribution.

Without overlapping, in case I, the Gini index is perfectly decomposable as the sum of G_n, and G/„ with R = G, = 0, and the two decompositions lead to the same results.

The presence of overlapping introduces some differences between the two decompositions, and the results shown in Table 4.4 allow us to highlight three effects related to increasing levels of overlapping between the subgroups.

The first obvious effect refers to R and G_tr which, reflecting the importance on overall inequality related to overlapping units, steadily increase from case I to case IV. Furthermore, by comparing R with G_t, we can observe the relation R = 2 * G/ illustrated in Section 4.3.3.

The second usual effect related to an increasing overlapping is a growing inequality within, as it is clearly shown in Table 4.4. In the first case, G_w is 26.8% of total inequality (G_w/G = 0.0821/0.3067 = 0.2677), a value which increases to 32.8% in the second case, to 43.8% in the third, and up to 55.6% in the last case.

The third effect, the consequence of a greater inequality within and a higher overlapping, is the reduced importance of inequality between on overall inequality. The decrease is not the same for the two decompositions, as it is fully detectable by comparing the third and the sixth column of Table 4.4. From Table 4.4 we can also observe the relation Gi,bm = G;,d - G_tr shown in Section 4.3.3, and therefore obtain the third column as the difference between the sixth and the seventh columns.

The case IV, that is the situation of maximum overlapping, allows to highlight one of the main differences between the two decompositions and the two possible ways to address perfect overlapping highlighted in Section 4.2.1.

Following Bhattacharya and Mahalanobis, in the fourth case the inequality between reaches its minimum, which is equal to zero, and all differences between the two subgroups are attributed to the overlapping component, in their case the residual term R.

In the framework of Dagum's decomposition, the maximum overlapping is evaluated by giving the same importance to the negative (overlapping) and positive (non-overlapping) differences (yg, - улг), which have the same number Пва =з пвл = 788, and also the same intensity, Tba = 2540, as shown in the last column of Table 4.2. It follows that the differences between the two subgroups are equally assigned to the inequality between and to the overlapping component, G_h = G_t = 0.0682, and the minimum of G_h is not zero but Gf.