Theil Index - Decomposability

Decomposability

One of the advantages of the Theil index is that it is a weighted average of inequality within subgroups, plus inequality among those subgroups. For example, inequality within the United States is the average inequality within each state, weighted by state income, plus the inequality among states.

If for the Theil-T index the population is divided into certain subgroups and is the income share of group, is the Theil-T index for that subgroup, and is the average income in group, then the Theil index is

$T_T = \sum_{i=1}^m s_i T_{T_i} + \sum_{i=1}^m s_i \log{\frac{\overline{x}_i}{\overline{x}}}$

The formula for the Theil-L index is:

$T_L = \frac{1}{m} \sum_{i=1}^m T_{L_i} + \frac{1}{m} \sum_{i=1}^m \log{\frac{\overline{x}_i}{\overline{x}}}$

Note: This image is not the Theil Index in each area of the United States, but of contributions to the US Theil Index by each area (the Theil Index is always positive, individual contributions to the Theil Index may be negative or positive).

The decomposition of the overall Theil index which identifies the share attributable to the between-region component becomes a helpful tool for the positive analysis of regional inequality as it suggests the relative importance of spatial dimension of inequality.

The decomposability is a property of the Theil index which the more popular Gini coefficient does not offer. The Gini coefficient is more intuitive to many people since it is based on the Lorenz curve. However, it is not easily decomposable like the Theil.