Theilindexet

The Theil index, denoted T, is an entropy-based measure of income or wealth inequality. Developed by the Dutch economist Henri Theil in 1967, it belongs to the family of generalized entropy indices and is noted for its additive decomposability, which allows total inequality to be split into within-group and between-group components.

Definition and formula: For a population of N individuals with positive incomes x_i, let μ be the mean income. Theil T is defined as T = (1/N) ∑_{i=1}^N (x_i / μ) ln (x_i / μ). The index is nonnegative and equals zero when all incomes are equal; it grows as inequality increases and has a theoretical maximum of ln N.

Interpretation and properties: The Theil index is scale-invariant and sensitive to the distribution’s tail. It is part of the generalized entropy class and can be interpreted as a relative entropy between the observed income distribution and a reference distribution based on the mean.

Decomposition: If the population is partitioned into G groups with n_g individuals, mean incomes μ_g, and group measures T_g, then T = ∑_{g=1}^G (n_g / N) T_g + ∑_{g=1}^G (n_g / N) (μ_g / μ) ln (μ_g / μ).

Applications and limitations: Theil is used in economics for cross-country and time-series comparisons of inequality and relates to Shannon entropy and GE(1). It requires positive incomes and can be influenced by extreme values; its interpretation depends on the chosen grouping and data quality.