Theil index

Formula

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 index is defined as:

T T = T α = 1 = 1 N ∑ i = 1 N x i μ ln ⁡ ( x i μ )

where μ is the mean income:

μ = 1 N ∑ i = 1 N x i

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 ∑ k = 0 W f k = 1 and the Theil index is:

T T = ∑ k = 0 W f k k μ ln ⁡ ( k μ )

where μ is again the mean income:

μ = ∑ k = 0 W k f 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:

T T = ∫ 0 ∞ f ( k ) k μ ln ⁡ ( k μ ) d k

where the mean is:

μ = ∫ 0 ∞ k f ( k ) d k

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

If everyone has the same income, then T_T equals 0. If one person has all the income, then T_T gives the result ln ⁡ N , which is maximum inequality. Dividing T_T by ln ⁡ N can normalize the equation to range from 0 to 1, but then the independence axiom is violated: T [ x ∪ x ] ≠ 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.

Derivation from entropy

The Theil index is derived from Shannon's measure of information entropy 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

S = k ∑ i = 1 N ( p i log ⁡ 1 p i ) = − k ∑ i = 1 N ( p i log ⁡ p i )

where p i is the probability of finding member i from a random sample of the population. In physics, k is Boltzmann's constant. In information theory, when information is given in binary digits, k = 1 and the log base is 2. In physics and also in computation of Theil index, the natural logarithm is chosen as the logarithmic base. When p i is chosen to be income per person x i , it needs to be normalized by dividing by the total population income, N x ¯ . This gives the observed entropy S Theil of a population to be:

S Theil = ∑ i = 1 N ( x i N x ¯ ln ⁡ N x ¯ x i )

The Theil index is T T = S max − S Theil where S max is the theoretical maximum entropy that is reached when all incomes are equal, i.e. x i = x ¯ for all i . This is substituted into S Theil to give S max = ln ⁡ N , a constant determined solely by the population. So the Theil index gives a value in terms of an entropy that measures how far S Theil is away from the "ideal" S max . The index is a "negative entropy" in the sense that it gets smaller as the disorder gets larger, hence it is a measure of order rather than disorder.

When x is in units of population/species, S 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. 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.

Decomposability

One of the advantages of the Theil index is that it is a weighted average of inequality within subgroups, plus inequality between those subgroups. For example, inequality within the United States is the average inequality within each state, weighted by state income, plus the inequality between states.

If for the Theil index the population is divided into m subgroups and s i is the income share of group i , T T i is the Theil index for that subgroup, and x ¯ i is the average income in group i , then the Theil index is

T T = ∑ i = 1 m s i T T i + ∑ i = 1 m s i ln ⁡ x ¯ i x ¯ Note: This image is not the Theil Index in each area of the United States, but of contributions to the US Theil Index by each area (the Theil Index is always positive, individual contributions to the Theil Index may be negative or positive).

The decomposition of the overall 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.

The decomposability is a property of the Theil index which the more popular Gini coefficient does not offer. 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.

Applications

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