Appendix B: Measures of Regional Inequality

The measurement of regional disparities is an arduous task; no single statistical measure is able to capture its myriad dimensions. This paper has applied some widely used measures to highlight various dimensions of regional income inequality. The selected measures are briefly described in the following paragraphs.

1) Maximum to minimum ratio (MMR)

A comparison of the per capita GRDP (gross regional domestic product, or other index such as per capita consumption or per capita income) of the region with the highest income to that of the region with the lowest income (minimum per capita GRDP) provides a measure of the range of the disparity between them. Maximum to minimum ratio (MMR) provides a quick, easy to comprehend, and politically powerful measure of regional income inequality.

2) The Gini index is widely used in the inequality literature. Following Shankar and Shah (2003), we compute the unweighted Gini index as follows:

where y_i and y_j are the GRDPs per capita of regions i and j, respectively. n is the number of regions, and y_u is the unweighted mean of the per capita GRDPs. Gu varies from 0 for perfect equality to 1 for perfect inequality.

The unweighted Gini index takes every region in the same magnitude, i.e., every region was taken as one equal unit regardless of its population size. The weighted Gini index, which weights the regions’ per capita GRDPs based on their respective population proportions, is calculated as shown below:

where y = GDP / P is the national mean per capita GDP. P_i and P_j are the populations of regions i and j, respectively. P is the national population, and n the number of regions. Gu varies from 0 for perfect equality to 1 - (P_* / P) for perfect inequality, where P_* represents the population of the region which produced the total GDP. If P_* is small compared to P, i.e., if the region with a small proportion of the population produced all the GDP, then the value for perfect inequality would approach 1.

3) Another commonly used measure of inequality is the Theil index. Following Theil (1967), it is computed as follows:

where γ_i is the GDP share of region i and p_i is the population share of region i. For equal per capita GRDPs, i.e., with GRDPs proportional to regional populations, this index takes a value of 0. For a case where region i produces the entire GDP, Theil becomes log (P/P_i) , where P is the total population of the country, and P_i is the population of region i. Note here that as the population share of region i decreases, Theil increases if region i produces the entire GDP.

Similar to the ^Theil_T index, we can compute the ^Theil_L index, which uses the population share as a weight, i.e.,

Compared to other measures of inequality such as Gini, CV (coefficient of variation), and Rw (relative mean deviation), Theil indexes satisfy several desirable properties, i.e., they are additively decomposable, and satisfy mean independence (or income-zero-homogeneity), the principle of population replication (or population-size independence), and the Pigou- Dalton principle of transfers (Bourguignon, 1979; Shorrocks, 1980, Akita et al., 1999). An inequality index is said to be additively decomposable if total inequality can be written as the sum of between-group and within-group inequality. Mean independence implies that the index remains unchanged if everyone’s expenditure is changed by the same proportion, while population-size independence means that the index remains unchanged if the number of households at each expenditure level is changed by the same proportion, i.e., the index depends only on the relative population frequencies at each expenditure level, not the absolute population frequencies. Finally, the Pigou-Dalton principle of transfers implies that any expenditure transfer from a richer to a poorer household that does not reverse their relative ranks in expenditures reduces the value of the index.

Suppose that the regions are grouped into mutually exclusive and collectively exhaustive groups and each group can be divided into several small sub-regions. The Theil index can be decomposed into within-group and between-group components as follows:

where the meanings of pi and ?i are the same as above, I is the number of groups, pij is population share of sub-region j in group i, ?i is the GDP share of sub-region j in group i, LB is the between-group component of the Theil index L and measures the extent of inequality due solely to differences in the group mean per capita GDP. Lw is the within-group component of the Theil index L and is defined by a weighted average of within-group Theil indexes with the weights being the population shares of the groups P_ij.

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