Extremal properties of the Theil and Gini measures of inequality

Two popular inequality measures used in the study of income and wealth distributions are the Gini (G) and Theil (T) indices. Several bounds on these inequality measures are available when only partial information about the distribution is available. However the correlation between them has been less studied. We derive the allowed region for the joint values of (G, T), for both continuous and discrete distributions. This has the form of a lower bound for T at given G. There is no corresponding upper bound, and T can be made as large as desired for given G by choosing an appropriate form of the Lorenz curve. We illustrate the bound for several parametric models of income distribution and Lorenz curves frequently used in the income distribution literature.