Extremal properties of the Theil and Gini measures of inequality

Abstract

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.

Appendix: Proof of Proposition 1

The starting point of the proof is the following representation of the Theil index, which was given in Rohde (2007)

$$\begin{aligned} T = \int _0^1 L'(r) \log L'(r) dr \,. \end{aligned}$$

(26)

The Lorenz curve is defined in terms of the probability density function p(y) as

$$\begin{aligned} L(r) = \frac{\int _0^x y p(y) dy }{ \int _0^\infty y p(y) dy} \end{aligned}$$

(27)

where x is the solution of $\int _0^x p(y) dy = r$.

We would like to find the extremal values of the Theil index at constant Gini coefficient

$$\begin{aligned} \frac{1}{2}(1-G) = \int _0^1 L(r) dr \end{aligned}$$

(28)

This is a constrained variational problem over the space of the Lorenz curves L(r) with fixed points $L(0)=0, L(1)=1$, satisfying the general properties of the Lorenz curve: i) increasing $L'(r)\ge 0$, and ii) convex $L''(r)\ge 0$.

The Gini coefficient constraint (28) is taken into account by introducing a Lagrange multiplier $\alpha$ and studying the extremal points of the functional

$$\begin{aligned} \Lambda [L] = \int _0^1 L'(r) \log L'(r) dr + \alpha \int _0^1 L(r) dr \end{aligned}$$

(29)

Requiring the vanishing of the functional derivative $\frac{\delta \Lambda }{\delta L} =0$ gives the Euler–Lagrange equation

$$\begin{aligned} \frac{d}{dr} \left( \log L'(r) + 1 \right) + \alpha = 0 \end{aligned}$$

(30)

The solution of this equation is

$$\begin{aligned} L'(r) = c e^{\alpha r} \end{aligned}$$

(31)

Imposing the boundary conditions $L(0)=0,L(1)=1$ this gives the extremal Lorenz curve

$$\begin{aligned} L(r) = \frac{e^{\alpha r} - 1}{e^\alpha - 1} \end{aligned}$$

(32)

This reproduces (5). The optimal function satisfies also the increasing and convexity properties so it is in the space of admissible Lorenz functions.

Substituting into the expressions of the Gini and Theil measures this gives also the relations (3) and (4). This completes the proof of Proposition 1.