Theil Inequality Decomposition
Also known as: Theil Index, Theil's T and L, Generalized Entropy Decomposition, Within-Between Inequality Decomposition
The Theil index, introduced by Henri Theil in 1967 by importing Shannon's information theory into economics, measures income inequality as the divergence between each unit's income share and its population share. Its defining advantage is exact additive decomposability: total inequality splits cleanly into a within-group component (inequality inside each subgroup) and a between-group component (inequality between subgroup means). Theil's T and its companion L (mean log deviation) are the two best-known members of the generalized-entropy class, which Anthony Shorrocks showed in 1980 to be the only inequality measures that are additively decomposable in this way.
Key highlights
- Exactly additively decomposable into within-group and between-group inequality with no residual, the defining advantage over the Gini coefficient.
- Grounded in information theory and characterized axiomatically (Shorrocks 1980) as essentially the unique decomposable inequality class.
- A tunable sensitivity parameter beta lets the analyst emphasize the bottom (low beta) or top (high beta) of the distribution.
- Satisfies the standard inequality axioms — scale invariance, the transfer principle, population replication, and symmetry.
Intuition
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How it works
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When to use it
Use Theil decomposition when you have individual or household income (or consumption) data, a meaningful partition of the population into groups, and you want to attribute total inequality exactly to within-group and between-group sources. It is the standard tool for questions like 'how much of national inequality is between regions versus within them?' or 'how much is explained by education?' Theil's T is appropriate when top-end sensitivity matters; the mean log deviation L is preferred when you want population-share weights, a clean residual-free decomposition, and consistency with Shapley factor decompositions. Report which GE member you use, since the between/within split and its interpretation depend on beta. For non-nested factor attribution (multiple overlapping characteristics) combine GE decomposition with regression-based or Shapley methods.
Strengths & limitations
- Exactly additively decomposable into within-group and between-group inequality with no residual, the defining advantage over the Gini coefficient.
- Grounded in information theory and characterized axiomatically (Shorrocks 1980) as essentially the unique decomposable inequality class.
- A tunable sensitivity parameter beta lets the analyst emphasize the bottom (low beta) or top (high beta) of the distribution.
- Satisfies the standard inequality axioms — scale invariance, the transfer principle, population replication, and symmetry.
- Undefined for non-positive incomes (T and L use logarithms), requiring censoring or use of consumption data.
- Less intuitive to communicate than the Gini coefficient because it is in entropy units with no direct geometric reading.
- The within-group weights for general beta are neither pure population nor pure income shares, complicating interpretation away from beta = 0.
- Sensitive to the top of the distribution (for T) where survey data are least reliable and top incomes are often under-captured.
Common pitfalls
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Applications
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Frequently asked
What is the difference between Theil's T and Theil's L?
Both are generalized-entropy measures but with different sensitivity. Theil's T (GE(1)) weights log income ratios by income shares, making it more responsive to changes at the top of the distribution. The mean log deviation L (GE(0)) averages log inverse income ratios and is more responsive to the bottom. In decompositions, L is often preferred because its within-group weights are the pure population shares (summing to one), giving a cleaner, path-independent split, whereas T uses income-share weights.
Why is Theil decomposable when the Gini coefficient is not?
The Gini is based on pairwise differences across the whole distribution, so when you split the population into groups the between- and within-group parts do not add up to the total — there is a residual that depends on how much the group income ranges overlap. Theil and the broader generalized-entropy class are built from sums of individual entropy-like terms, which are additive by construction, so the within and between components sum exactly to the total with no overlap residual. Shorrocks (1980) showed the GE class is essentially the only one with this property.
How should I choose the sensitivity parameter beta?
Choose beta to match where in the distribution you care most about inequality. Low or negative beta (including GE(0), the mean log deviation) emphasizes the bottom of the distribution; beta = 1 (Theil's T) is balanced toward the top; beta = 2 (half the squared coefficient of variation) is very top-sensitive and is the only member defined for zero incomes. Because the within/between split depends on beta, report results for more than one value to show robustness, and prefer GE(0) when you want population-share weights.
Sources
- 1.Theil, H. (1967). Economics and Information Theory. Amsterdam: North-Holland.ISBN 9780444814630
- 2.Shorrocks, A. F. (1980). The class of additively decomposable inequality measures. Econometrica, 48(3), 613–625.
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ScholarGate. (2026, June 22). Theil Inequality Decomposition. ScholarGate. https://scholargate.app/economics/theil-inequality-decomposition