Financial & Economic Analysis
Theil T & Theil L (Mean Log Deviation) Entropy Inequality Indices
Quantify revenue, wage, or regional resource concentration using Theil’s GE(1) T index, GE(0) Mean Log Deviation L, and normalized [0, 1] Shannon inequality share.
Why Theil’s entropy indices succeed where Gini cannot decompose additively
Unlike the Gini coefficient—whose subgroup overlap term prevents clean within-group vs between-group additive decomposition—Henri Theil’s (1967) information-theoretic inequality measures belong to the Generalized Entropy family and decompose exactly into Within-Region + Between-Region inequality.
For N strictly positive observations x_i with mean mu: Theil’s T index (GE(1), upper-tail sensitive) averages (x_i / mu) * LN(x_i / mu) across all observations, bounded between 0 (perfect equality) and LN(N) (one unit holds 100%). Theil’s L index (GE(0), Mean Log Deviation, lower-tail sensitive) averages LN(mu / x_i) = LN(arithmetic_mean / geometric_mean).
| Inequality Metric | Formula | Values [20, 40, 60, 120] | Equal [60, 60, 60, 60] |
|---|---|---|---|
| Mean (mu) | AVERAGE(B2:B5) | 60.00 | 60.00 |
| Theil T Index GE(1) | ROUND(AVERAGE(C2:C5), 4) | 0.1874 | 0.0000 |
| Theil L Index GE(0) | ROUND(AVERAGE(D2:D5), 4) | 0.2027 | 0.0000 |
| Normalized T / LN(N) | Theil_T / LN(N) | 0.1352 (13.52% of max) | 0.0000 |
Row-Level Theil T & Theil L Entropy Terms
Compute each unit’s relative entropy contribution (B2 / $G$2) * LN(B2 / $G$2) for GE(1) and LN($G$2 / B2) for GE(0), then take AVERAGE across the column.
=(B2 / $G$2) * LN(B2 / $G$2)Normalized [0, 1] Upper-Bound Entropy Share
Divide Theil T by its theoretical maximum LN(COUNT(B2:B5)) to express concentration on a bounded 0-to-1 scale across panels of different sizes.
=ROUND(G3 / LN(COUNT(B2:B5)), 4)Try the example
Use the original values or change them to test your understanding.