I am jealous of the nice working paper written by Ravi Kanbur entitled, [Shannon-Theil-Rawls: Information Theory, and the Veil of Ignorance](https://www.ecineq.org/milano/WP/ECINEQ2024-669.pdf)[^kanbur] Suppose there are $N$ individuals in a population and the income of the $i$th individual is denoted $x_i$. Let $\bar x$ denote the average income, and let $q_i = x_i /(N \bar x)$ be the $i$th individual's share of the population's total income. The *Shannon entropy* of the income distribution is defined as follows: $ \mathcal{S} \coloneqq \sum_{i=1}^N q_i \ln(1/q_i). $ Notice that if income is evenly split among all $N$ individuals, then $q_i = 1/N$ and the Shannon entropy attains its maximum value $ \mathcal{S}_{\mathrm{max}} = \frac{1}{N} \sum_{i = 1}^N \ln(N) = \ln(N). $ The difference between this maximum entropy value to the Shannon entropy of a discrete distribution is the distribution's Theil index: $ \begin{aligned} \mathcal{T} &\coloneqq \mathcal{S}_{\mathrm{max}} - \mathcal{S} \\ &= \ln(N) - \sum_{i=1}^N \frac{x_i}{N \bar x} \ln \left(\frac{N \bar x}{x_i}\right) \\ &= \ln(N) - \sum_{i=1}^N \frac{x_i}{N \bar x} \ln \big(N\big) - \sum_{i=1}^N \frac{x_i}{N \bar x} \ln \left(\frac{x}{x_i}\right) \\ &= \frac{1}{N} \sum_{i=1}^N \frac{x_i}{\bar x} \, \ln\Big(\frac{x_i}{\bar x}\Big). \end{aligned} $ This final form of the finite-population Theil can be thought of as an expected value. Let $X$ denote an income drawn uniformly at random from among the $N$ individuals. Then the Theil can be expressed as $ \mathcal{T} = \mathbb{E}\left(\frac{X}{\mathbb{E}(X)} \ln\left(\frac{X}{\mathbb{E}(X)}\right)\right). $ This formulation can be found in Theil's original book on the subject[^theil-book]. ###### Extension of the definition to continuous income distributions. The random variable representation for the Theil index readily translates to a situation in which we want to define the Theil for a continuous income distribution with pdf $f(x)$. Namely, $ \mathcal{T} = \mathbb{E}\left(\frac{X}{\mu} \ln\left(\frac{X}{\mu}\right)\right) = \int_0^\infty \frac{x}{\mu} \ln \left(\frac{x}{\mu}\right) f(x) \mathrm{d} x, $ where $\mu = \mathbb{E}(X)$. Using the notation $\mathcal{H} = \mathbb{E}(X \ln(X))$, where $X \sim f$, we can write the Theil in its simplest form: $ \mathcal{T} = \frac{\mathcal{H}}{\mu} - \ln(\mu). $ --- [^kanbur]: Kanbur, Ravi. _[Shannon-Theil-Rawls: Information Theory, Inequality and the Veil of Ignorance](https://www.ecineq.org/milano/WP/ECINEQ2024-669.pdf)_, ECINE Working Paper series (2024). [^theil-book]: Theil, Henry. *Economics and Information Theory*. North Holland. (1967)