${1 \over T} e^{-i/T}$ for Boltzmann-Gibbs distribution There is a book from Tom Carter on entropy. In the Economics I application (page 111), he ingeniously computes that the distribution of fixed amount of M money over N individual tends to 
$$p_i = {1 \over T}\, e^{-i/T}$$
The temperature, $T = M/N$, is the average amount of money per individual. 
The author calls this a Boltzmann-Gibbs distribution. I wanted to find a plot and compare it with Pareto distribution and Planck black body radiation. The problem is however, and this is my question, Wikipedia does not have it! Actually, there is an article on Boltzmann distribution, which is also called Gibbs distribution, but I see nothing similar to $p_i = 1/T\, e^{-i/T}$ there. At least I cannot see how ${N_i \over N} = {g_i e^{-E_i/(k_BT)} \over Z(T)}$ with $N=\sum_i N_i,$ and partition function, $Z(T)=\sum_i g_i e^{-E_i/(k_BT)}$ can be reduced to it. I do not see a plot $p_i(T)$ either. Are they related?
 A: Check the derivation of the Boltzmann distribution from the microcanonical ensemble on the Wikipedia page "Maxwell-Boltzmann Statistics".
We suppose that the "wealth classes" of individuals are discretised, so that, for example, we find the number of individuals with $m_1 = \$500$, the number with $m_2 = \$10000$  and so forth: as an approximation we restrict the wealth classes to discrete values. We don't even have to have the wealth classes equispaced.
Now, look at the derivation on the Wiki page, and there will be a precise analogy between that derivation of the Boltzmann distribution from the microcanonical ensemble, and the derivation of Tom Carter's formula. "Amount of Money Held" $m_i$ forms a precise analogy with "Energy" and "Number of Particles in each state" with "Number of Individuals in each Wealth Class", call it $n_i$. What you'll get is that the number of arrangements, helped by Stirling's formula, is:
$$\log\Omega \approx N\log(N)-N - \sum_j \left(n_j\,\log n_j-n_j\right) = N\log(N) - \sum_j n_j\,\log n_j$$
and it is constrained by the same two constraints: the total number of individuals is constant:
$$\sum_j n_j = N = const$$
and the total amount of money is constant:
$$\sum_j n_j m_j = M = const$$
so that you'll get, in analogy with the BD:
$$p_i = \mathcal{Z}^{-1} e^{-\beta\,m_j}$$
where the partition function $\mathcal{Z}$ and $\beta$ come from the Lagrange multipliers for the two constraints.
The underlying assumption is that all the arrangements of allocating shares of money to indivuals are equally likely, something I highly doubt as wealthier people tend to rig the rules and the conditions to make themselves wealthier! However, let's assume it's true. This formula is general for general, unevenly spread wealth classes. We now need to assume what these wealth classes are. Let's now assume they are evenly spaced: $m_j = j\,\delta$. Then we must calculate $\mathcal{Z}$ to make all the $p_j$ sum up to unity. The result is:
$$p_j = (1-e^{-\delta\,\beta}) e^{-j\,\beta\,\delta}\qquad(1)$$
The mean amount of money $M/N$ held by each individual is 
$$\frac{M}{N}=\sum_{j=0}^\infty j \,\delta\,p_j = \frac{1}{e^{\beta\,\delta}-1}\qquad(2)$$
and if we make $\delta$ very small to give us many different money classes, then the (2) approximates to $N/M = \beta \delta$, and (1) approximates to $p_j = \delta\,\beta\,e^{-j\,\beta\,\delta}$, thus we get:
$$p_j \approx \frac{N}{M} e^{-j\frac{N}{M}}$$
which is your formula.
