Theil index

For a population of N "agents" each with characteristic x, the situation may be represented by the list x_i (i = 1,...,N) where x_i is the characteristic of agent i. For example, if the characteristic is income, then x_i is the income of agent i.

The Theil T index is defined as[4]

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

and the Theil L index is defined as[4]

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

where ${\displaystyle \mu }$ is the mean income:

${\displaystyle \mu ={\frac {1}{N}}\sum _{i=1}^{N}x_{i}}$

The Theil L formula represents the logarithm of the geometric-mean of the ratio: (mean income)/(income i), over all the incomes included in the summation. ...obviously a relevant fact for any range of incomes on the same side of the mean income. .

...showing that this form of the Theil has obvious, intuitive, plausible and natural justification, rather than just being justified in terms of entropy. .

Because a transfer from a larger income to a smaller one will change the smaller income's ratio more than it changes the larger income's ratio, the transfer-principle is satisfied by this index.

Of course, if desired, a weighting factor, such as (mean income)/(income i) could be included in the terms of the summation (as in the Theil-T formula above, with the income-ratios inverted), to weight the index in favor of more strongly counting changes to income-ratios in which income i differs from the mean income by a larger factor.

In Theil T, each income-ratio's logarithm is weighted by a factor equal to that income-ratio's own value. And so, if the income-ratio is 2, then the index's value is affected as if there were two of that person. ...a reasonable weighting if each income-ratio's importance is judged to be proportional to its own value...the factor by which a particular income differs from the mean income.

Equivalently, if the situation is characterized by a discrete distribution function f_k (k = 0,...,W) where f_k is the fraction of the population with income k and W = Nμ is the total income, then ${\displaystyle \sum _{k=0}^{W}f_{k}=1}$ and the Theil index is:

${\displaystyle T_{T}=\sum _{k=0}^{W}\,f_{k}\,{\frac {k}{\mu }}\ln \left({\frac {k}{\mu }}\right)}$

where ${\displaystyle \mu }$ is again the mean income:

${\displaystyle \mu =\sum _{k=0}^{W}kf_{k}}$

Note that in this case income k is an integer and k=1 represents the smallest increment of income possible (e.g., cents).

if the situation is characterized by a continuous distribution function f(k) (supported from 0 to infinity) where f(k) dk is the fraction of the population with income k to k + dk, then the Theil index is:

${\displaystyle T_{T}=\int _{0}^{\infty }f(k){\frac {k}{\mu }}\ln \left({\frac {k}{\mu }}\right)dk}$

where the mean is:

${\displaystyle \mu =\int _{0}^{\infty }kf(k)\,dk}$

Theil indices for some common continuous probability distributions are given in the table below:

Income distribution function	PDF(x) (x ≥ 0)	Theil coefficient (nats)
Dirac delta function	${\displaystyle \delta (x-x_{0}),\,x_{0}>0}$	0
Uniform distribution	${\displaystyle {\begin{cases}{\frac {1}{b-a}}&a\leq x\leq b\\0&{\text{otherwise}}\end{cases}}}$	${\displaystyle \ln \left({\frac {2a}{(a+b){\sqrt {e}}}}\right)+{\frac {b^{2}}{b^{2}-a^{2}}}\ln(b/a)}$
Exponential distribution	${\displaystyle \lambda e^{-x\lambda },\,\,x>0}$	${\displaystyle 1-}$ ${\displaystyle \gamma }$
Log-normal distribution	${\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}e^{(-(\ln(x)-\mu )^{2})/\sigma ^{2}}}$	${\displaystyle {\frac {\sigma ^{2}}{2}}}$
Pareto distribution	${\displaystyle {\begin{cases}{\frac {\alpha k^{\alpha }}{x^{\alpha +1}}}&x\geq k\\0&x<k\end{cases}}}$	${\displaystyle \ln(1\!-\!1/\alpha )+{\frac {1}{\alpha -1}}}$    (α>1)
Chi-squared distribution	${\displaystyle {\frac {2^{-k/2}e^{-x/2}x^{k/2-1}}{\Gamma (k/2)}}}$	${\displaystyle \ln(2/k)+}$ ${\displaystyle \psi ^{(0)}}$${\displaystyle (1\!+\!k/2)}$
Gamma distribution	${\displaystyle {\frac {e^{-x/\theta }x^{k-1}\theta ^{-k}}{\Gamma (k)}}}$	${\displaystyle \psi ^{(0)}}$${\displaystyle (1+k)-\ln(k)}$
Weibull distribution	${\displaystyle {\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}}}$	${\displaystyle {\frac {1}{k}}}$ ${\displaystyle \psi ^{(0)}}$${\displaystyle (1+1/k)-\ln \left(\Gamma (1+1/k)\right)}$

If everyone has the same income, then T_T equals 0. If one person has all the income, then T_T gives the result ${\displaystyle \ln N}$, which is maximum inequality. Dividing T_T by ${\displaystyle \ln N}$ can normalize the equation to range from 0 to 1, but then the independence axiom is violated: ${\displaystyle T[x\cup x]\neq T[x]}$ and does not qualify as a measure of inequality.

The Theil index measures an entropic "distance" the population is away from the egalitarian state of everyone having the same income. The numerical result is in terms of negative entropy so that a higher number indicates more order that is further away from the complete equality. Formulating the index to represent negative entropy instead of entropy allows it to be a measure of inequality rather than equality.

The Theil index is derived from Shannon's measure of information entropy ${\displaystyle S}$, where entropy is a measure of randomness in a given set of information. In information theory, physics, and the Theil index, the general form of entropy is

${\displaystyle S=k\sum _{i=1}^{N}\left(p_{i}\log _{a}\left({\frac {1}{p_{i}}}\right)\right)=-k\sum _{i=1}^{N}\left(p_{i}\log _{a}\left({p_{i}}\right)\right)}$

where

• ${\displaystyle i}$ is an individual item from the set (such as an individual member from a population, or an individual byte from a computer file).
• ${\displaystyle p_{i}}$ is the probability of finding ${\displaystyle i}$ from a random sample from the set.
• ${\displaystyle k}$ is a constant.[note 1]
• ${\displaystyle \log _{a}\left({x}\right)}$ is a logarithm with a base equal to ${\displaystyle a}$.[note 2]

When looking at the distribution of income in a population, ${\displaystyle p_{i}}$ is equal to the ratio of a particular individual's income to the total income of the entire population. This gives the observed entropy ${\displaystyle S_{\text{Theil}}}$ of a population to be:

${\displaystyle S_{\text{Theil}}=\sum _{i=1}^{N}\left({\frac {x_{i}}{N{\bar {x}}}}\ln \left({\frac {N{\bar {x}}}{x_{i}}}\right)\right)}$

where

• ${\displaystyle x_{i}}$ is the income of a particular individual.
• ${\displaystyle \left(N{\bar {x}}\right)}$ is the total income of the entire population, with

• ${\displaystyle N}$ being the number of individuals in the population.
• ${\displaystyle {\bar {x}}}$ ("x bar") being the average income of the population.

• ${\displaystyle \ln \left(x\right)}$ is the natural logarithm of ${\displaystyle x}$: ${\displaystyle \left(\log _{e}\left(x\right)\right)}$.

The Theil index ${\displaystyle T_{T}}$ measures how far the observed entropy (${\displaystyle S_{\text{Theil}}}$, which represents how randomly income is distributed) is from the highest possible entropy (${\displaystyle S_{\text{max}}=\ln \left({N}\right)}$,[note 3] which represents income being maximally distributed amongst individuals in the population– a distribution analogous to the [most likely] outcome of an infinite number of random coin tosses: an equal distribution of heads and tails). Therefore, the Theil index is the difference between the theoretical maximum entropy (which would be reached if the incomes of every individual were equal) minus the observed entropy:

${\displaystyle T_{T}=S_{\text{max}}-S_{\text{Theil}}=\ln \left({N}\right)-S_{\text{Theil}}}$

When ${\displaystyle x}$ is in units of population/species, ${\displaystyle S_{\text{Theil}}}$ is a measure of biodiversity and is called the Shannon index. If the Theil index is used with x=population/species, it is a measure of inequality of population among a set of species, or "bio-isolation" as opposed to "wealth isolation".

The Theil index measures what is called redundancy in information theory.[4] It is the left over "information space" that was not utilized to convey information, which reduces the effectiveness of the price signal. The Theil index is a measure of the redundancy of income (or other measure of wealth) in some individuals. Redundancy in some individuals implies scarcity in others. A high Theil index indicates the total income is not distributed evenly among individuals in the same way an uncompressed text file does not have a similar number of byte locations assigned to the available unique byte characters.

Notation	Information theory	Theil index TT
${\displaystyle N}$	number of unique characters	number of individuals
${\displaystyle i}$	a particular character	a particular individual
${\displaystyle x_{i}}$	count of ith character	income of ith individual
${\displaystyle N{\bar {x}}}$	total characters in document	total income in population
${\displaystyle T_{T}}$	unused information space	unused potential in price mechanism
	data compression	progressive tax

According to the World Bank,

"The best-known entropy measures are Theil’s T (${\displaystyle T_{T}}$) and Theil’s L (${\displaystyle T_{L}}$), both of which allow one to decompose inequality into the part that is due to inequality within areas (e.g. urban, rural) and the part that is due to differences between areas (e.g. the rural-urban income gap). Typically at least three-quarters of inequality in a country is due to within-group inequality, and the remaining quarter to between-group differences."[5]

If the population is divided into ${\displaystyle m}$ subgroups and

• ${\displaystyle s_{i}}$ is the income share of group ${\displaystyle i}$,
• ${\displaystyle N}$ is the total population and ${\displaystyle N_{i}}$ is the population of group ${\displaystyle i}$,
• ${\displaystyle T_{i}}$ is the Theil index for that subgroup,
• ${\displaystyle {\overline {x}}_{i}}$ is the average income in group ${\displaystyle i}$, and
• ${\displaystyle \mu }$ is the average income of the population,

then Theil's T index is

${\displaystyle T_{T}=\sum _{i=1}^{m}s_{i}T_{i}+\sum _{i=1}^{m}s_{i}\ln {\frac {{\overline {x}}_{i}}{\mu }}}$ for ${\displaystyle s_{i}={\frac {N_{i}}{N}}{\frac {{\overline {x}}_{i}}{\mu }}}$

For example, inequality within the United States is the average inequality within each state, weighted by state income, plus the inequality between states.

Map of economic inequality in the United States using the Theil Index. A high positive theil index indicates more income than population while a negative value shows more population than income. A value of zero shows equality between population and income.

Note: This image is not the Theil Index in each area of the United States, but of contributions to the Theil Index for the U.S. by each area. The Theil Index is always positive, although individual contributions to the Theil Index may be negative or positive.

The decomposition of the Theil index which identifies the share attributable to the between-region component becomes a helpful tool for the positive analysis of regional inequality as it suggests the relative importance of spatial dimension of inequality.[6]

In addition to multitude of economic applications, the Theil index has been applied to assess performance of irrigation systems[8] and distribution of software metrics.[9]

• Generalized entropy index
• Atkinson index
• Gini coefficient
• Hoover index
• Income inequality metrics
• Suits index
• Wealth condensation
• Diversity index

• Software:
  • Free Online Calculator computes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset
  • Free Calculator: Online and downloadable scripts (Python and Lua) for Atkinson, Gini, and Hoover inequalities
  • Users of the R data analysis software can install the "ineq" package which allows for computation of a variety of inequality indices including Gini, Atkinson, Theil.
  • A MATLAB Inequality Package, including code for computing Gini, Atkinson, Theil indexes and for plotting the Lorenz Curve. Many examples are available.