The Theil index, derived by econometrician Henri Theil, is a statistic used to measure economic inequality.

Mathematics

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The formula is

$T=\backslash sum\_\{i=1\}^N\; \backslash left(\; \backslash frac\{x\_i\}\{\backslash sum\_\{j=1\}^N\; x\_j\}\; \backslash cdot\; \backslash ln\{\backslash frac\{x\_i\}\{\backslash overline\{x\}\}\}\; \backslash right)$

where $x\_i$ is the income of the $i$th person, $\backslash overline\{x\}$ is the mean income, and $N$ is the number of people. The first term inside the sum can be considered the individual's share of aggregate income, and the second term is that person's income relative to the mean. If everyone has the same (i.e., mean) income, then the index = 0. If one person has all the income, then the index = $\backslash ln\{N\}$.

The Theil index is derived from Shannon's measure of information entropy. Letting $T$ be the Theil Index and $S$ be Shannon's measure,

$T=\backslash ln(N)-S$

Shannon derived his entropy measure in terms of the probability of an event occurring. This can be interpreted in the Theil as the probability a dollar drawn at random from the population came from a specific individual. This is the same as the first term, the individual's share of aggregate income.

Decomposability

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One of the advantages of the Theil index is that it is the weighted sum of inequality within subgroups. For example, inequality within the United States is the sum of each state's inequality weighted by the state's income relative to the entire country.

If the population is divided into $m$ certain subgroups and $s^k$ is the income share of group $k$, $T\_k$ is the Theil index for that subgroup, and $\backslash overline\{x\}^k$ is the average income in group $k$, then the Theil index is

$T\; =\; \backslash sum\_\{k=1\}^m\; s^k\; T\_k\; +\; \backslash sum\_\{k=1\}^m\; s^k\; \backslash ln\{\backslash frac\{\backslash overline\{x\}^k\}\{\backslash overline\{x\}\}\}$

Therefore, one can say that a certain group "contributes" a certain amount of inequality to the whole.

Another, more popular, measure of inequality is the Gini coefficient. The Gini coefficient is more intuitive to many people since it is based on the Lorenz curve. However, it is not easily decomposable like the Theil.

Application of the Theil Index

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Theil's index takes an equal distribution for reference which is similar to distributions in statistical physics. An index for an actual system is an actual redundancy, that is, the difference between maximum entropy and actual entropy of that system.

Theil's measure can be converted into one of the indexes of Anthony Barnes Atkinson. James E. Foster used such a measure to replace the Gini coefficient in Amartya Sen's welfare function W=f(income,inequality). The income e.g. is the average income for individuals in a group of income earners. Thus, Foster's welfare function can be computed directly from the Theil index T, if the conversion is included into the computation of the average per capita welfare function:

$W\; =\; \backslash overline\{income\}\; \backslash times\; \{e^\{-T\}\}$

References

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Introduction to the Theil index from the University of Texas

Econometrics | Information theory | Economic indicators | Welfare economics | 戴尔指数