To make the problem simpler, let's consider the case where the allowed values of the share of each person is discrete, e.g., integers. Denoting the the total income as $X$, the $s$-th allowed value as $x_{s}$, the total number of people as $N$, and finally, the number of people with shares of $x_{s}$ as $n_{s}$, the following conditions should be satisfied:
\begin{equation}
C_{1} (\{n_{s}\})\equiv\sum_{s} n_{s} - N = 0,
\end{equation}
and
\begin{equation}
C_{2} (\{n_{s}\})\equiv \sum_{s} n_{s} x_{s} - X = 0.
\end{equation}

The goal is to maximize the number of ways to distribute the shares, i.e., 
\begin{equation}
W(\{n_{s}\}) \equiv \frac{N!}{\prod_{s} n_{s}!},
\end{equation}
under the two constraints given above. The method of Lagrange multipliers is a canonical approach for this, and furthermore, one can choose to maximize $\ln W$ instead of $W$ itself, as $\ln$ is a monotone increasing function. That is,
\begin{equation}
\frac{\partial \ln W}{\partial n_{s}} = \lambda_{1} \frac{\partial C_{1}}{\partial n_{s}} + 
\lambda_{2} \frac{\partial C_{1}}{\partial n_{s}} = \lambda_{1} + \lambda_{2} x_{s},
\end{equation}
where $\lambda_{1,2}$ are Lagrange multipliers. Notice that according to [Stirling's formula][1],
\begin{equation}
\ln n! \approx n\ln n - n,
\end{equation}
leading to
\begin{equation}
\frac{d\ln n!}{dn} \approx \ln n.
\end{equation}
Thus,
\begin{equation}
\frac{\partial \ln W}{\partial n_{s}} \approx -\ln n_{s}.
\end{equation}
It then follows that
\begin{equation}
n_{s} \approx \exp\big(-\lambda_{1} - \lambda_{2} x_{s}\big),
\end{equation}
which is an exponential distribution. That this is really a maximum, rather than a minimum or a saddle point, can be seen by considering the Hessian of $\ln W$:
\begin{equation}
\frac{\partial^{2} \ln W}{\partial n_{s}^{2}} = -\frac{1}{n_{s}},
\end{equation}
and
\begin{equation}
\frac{\partial^{2} \ln W}{\partial n_{s}\partial n_{r}} = 0 \quad (s\neq r).
\end{equation}
Hence the Hessian is concave, and what we have found is indeed a maximum.

**Note:** This is exactly how physicists understand the [Boltzmann distribution][2] in statistical mechanics.


  [1]: http://en.wikipedia.org/wiki/Stirling's_approximation
  [2]: http://en.wikipedia.org/wiki/Boltzmann_distribution