Economics & Market Microstructure
Theil T Index GE(1) & Theil L Mean Log Deviation GE(0) Income Inequality Formula
Calculate Henri Theil’s (1967) information-theoretic inequality measures — income-weighted Theil T = (1 / N) * SUM((y_i / mu) * LN(y_i / mu)) and population-weighted Mean Log Deviation Theil L = (1 / N) * SUM(LN(mu / y_i)).
Why economists report BOTH Theil T (GE(1)) and Theil L / Mean Log Deviation (GE(0))!
Unlike the Gini coefficient — whose subgroup decomposition leaves an awkward overlapping residual whenever regional income ranges cross — Henri Theil’s (1967) Generalized Entropy pair GE(1) (Theil T) and GE(0) (Theil L, or Mean Log Deviation MLD) decompose cleanly with zero residual into exact Within-Group plus Between-Group inequality!
The two indices weight opposite tails of the income distribution: Theil T = (1 / N) * SUM_{i=1..N} (y_i / mu) * LN(y_i / mu) weights log-ratios by income shares (making T more sensitive to top-tail billionaire concentration, bounded above by LN(N)), whereas Theil L = (1 / N) * SUM_{i=1..N} LN(mu / y_i) = LN(Arithmetic_Mean / Geometric_Mean) weights every person equally (making L more sensitive to bottom-tail deprivation!).
| Entropy Inequality Metric on Incomes [20, 30, 50, 100] (mu = 50) | Formula Structure | Evaluated Nats |
|---|---|---|
| Theil T Index GE(1) (Top-Tail Sensitive) | (1/N) * SUM((y_i / mu) * LN(y_i / mu)) | 0.1783 |
| Theil L Mean Log Deviation GE(0) (Bottom-Tail Sensitive) | (1/N) * SUM(LN(mu / y_i)) | 0.1835 |
| Maximum Possible Theil T for N = 4 (One Person Holds All) | LN(4) | 1.3863 |
Theil T Inequality Index GE(1) = (1 / N) * SUM((y_i / mu) * LN(y_i / mu))
Computes income-share-weighted log deviations across strictly positive incomes B2:B5.
=IF(MIN(B2:B5) <= 0, "Requires Strictly Positive Incomes", ROUND(((B2 / AVERAGE(B2:B5)) * LN(B2 / AVERAGE(B2:B5)) + (B3 / AVERAGE(B2:B5)) * LN(B3 / AVERAGE(B2:B5)) + (B4 / AVERAGE(B2:B5)) * LN(B4 / AVERAGE(B2:B5)) + (B5 / AVERAGE(B2:B5)) * LN(B5 / AVERAGE(B2:B5))) / 4, 4))Theil L Mean Log Deviation GE(0) = (1 / N) * SUM(LN(mu / y_i))
Computes population-weighted log mean-to-income ratios across B2:B5.
=IF(MIN(B2:B5) <= 0, "Requires Strictly Positive Incomes", ROUND((LN(AVERAGE(B2:B5) / B2) + LN(AVERAGE(B2:B5) / B3) + LN(AVERAGE(B2:B5) / B4) + LN(AVERAGE(B2:B5) / B5)) / 4, 4))Try the example
Use the original values or change them to test your understanding.