Theil index

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The Theil index belongs to the class of inequality measures and was developed by the econometrician Henri Theil . It is used for the statistical description of income and wealth distributions .

definition

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For income earners , the average income and Theil indices are defined under the convention as follows: ${\displaystyle n}$ ${\displaystyle y_{1},\dots ,y_{n}}$ ${\displaystyle \mu ={\frac {1}{n}}\sum _{i=1}^{n}y_{i}}$ ${\displaystyle T_{L},T_{T},T_{S}}$ ${\displaystyle 0\cdot \ln {0}=0}$

${\displaystyle T_{T}=T_{\alpha =1}={\frac {1}{n}}\sum _{i=1}^{n}\left({\frac {y_{i}}{\mu }}\cdot \ln {\frac {y_{i}}{\mu }}\right)}$

${\displaystyle T_{L}=T_{\alpha =0}=MLD={\frac {1}{n}}\sum _{i=1}^{n}\left(\ln {\frac {\mu }{y_{i}}}\right)}$

${\displaystyle T_{S}={\frac {1}{n}}\sum _{i=1}^{n}\left[{\frac {1}{2}}\left({\frac {y_{i}}{\mu }}-1\right)\ln(y_{i})\right]}$

MLD stands for mean log deviation. The relationships apply

${\displaystyle T_{L}(y)=T_{T}\left({\frac {1}{y}}\right)}$

${\displaystyle T_{S}={\frac {T_{T}+T_{L}}{2}}}$

${\displaystyle 0\leq T_{L}}$

${\displaystyle T_{L}=0\quad \Leftrightarrow \quad T_{T}=0\quad \Leftrightarrow \quad T_{S}=0\quad \Leftrightarrow \quad y_{i}=\mu \quad {\text{für alle }}i}$

${\displaystyle T_{T}=\ln(n)\quad \Leftrightarrow \quad y_{i}=n\mu \quad {\text{für ein }}i}$

${\displaystyle T_{S}(y)=T_{S}\left({\frac {1}{y}}\right)}$

Relationships / derivatives

Claude Shannon developed his entropy measure from the probability of an event occurring. Theil derived his index from it. The Theil index can be understood as the probability that a euro withdrawn from a population comes from a particular individual. This is the same as the first expression: an individual's share of total income.

If this is Shannon's measure, then applies ${\displaystyle S}$

${\displaystyle T=\ln(N)-S}$.

${\displaystyle e^{T}}$is a measure of uniform distribution, with an associated measure of inequality . ${\displaystyle \left(1-e^{T}\right)}$

Dismantling

The Theil index aggregates the weighted sum of the inequalities of subgroups. For example, the unequal distribution in Germany can be calculated from the unequal distributions in the countries.

If the population can be divided into subgroups and the income share of a subgroup is the total income, then describes the inequality in the subgroup and is the average income of the subgroup . The part index is then ${\displaystyle m}$ ${\displaystyle s_{k}}$ ${\displaystyle k}$ ${\displaystyle T_{k}}$ ${\displaystyle {\overline {x}}_{k}}$ ${\displaystyle k}$ ${\displaystyle T_{k}}$

${\displaystyle T=\sum _{k=1}^{m}s_{k}T_{k}+\sum _{k=1}^{m}s_{k}\ln {\frac {{\overline {x}}_{k}}{\overline {x}}}}$.

Described in this way, the Theil index is the “contribution” of the subgroup to the inequality in the entire group. ${\displaystyle T_{k}}$

A more popular measure is the Gini coefficient , but the Theil index is also used in addition to the Gini coefficient in federal wealth and poverty reports. The Gini coefficient is not as decomposable as the Theil coefficient.

literature

• Henri Theil: The Information Approach to Demand Analysis . In: Econometrica , Vol. 33, No. 1, January 1965, ISSN  0012-9682 , pp. 67-87 ( JSTOR 1911889 )

Web links

• Introduction to the Theil index from the University of Texas (PDF; 1.4 MB)

Individual evidence