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
where pi is the probability of finding member i from a random sample of the population. In physics, k is Boltzmann's constant. In information theory k=1 if it is in terms of bits and the log base is 2. Physics and the Theil index have chosen the natural logarithm as the logarithmic base. When pi is chosen to be income per person (xi), it needs to be normalized by dividing by the total population income, N*avg(x). This gives the observed entropy of a Theil population to be:
The Theil index is TT = Smax - STheil where the theoretical maximum entropy Smax is when all incomes are equal, i.e. each xi = average xi = a constant. This is substituted into STheil to give Smax = ln(N) for TT, a constant determined solely by the population. So the Theil index gives a value in terms of an entropy that measures how far STheil is away from the "ideal" Smax. The index is a "negative entropy" in the sense that it gets smaller as the disorder gets larger, so it is a measure of order rather than disorder.
When x is in units of population/species, 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.
| Notation | Information Theory | Theil Index TT |
|---|---|---|
| N | number of unique characters | number of individuals |
| i | a particular character | a particular individual |
| xi | characteri count | income of individuali |
| N*avg(x) | total characters in document | total income in population |
| TT | unused information space | unused potential in price mechanism |
| data compression | progressive tax |
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Famous quotes containing the word entropy:
“Just as the constant increase of entropy is the basic law of the universe, so it is the basic law of life to be ever more highly structured and to struggle against entropy.”
—Václav Havel (b. 1936)