  We discuss various limits of a simple random exchange model that can be used
for the distribution of wealth. We start from a discrete state space - discrete
time version of this model and, under suitable scaling, we show its functional
convergence to a continuous space - discrete time model. Then, we show a
thermodynamic limit of the empirical distribution to the solution of a kinetic
equation of Boltzmann type. We solve this equation and we show that the
solutions coincide with the appropriate limits of the invariant measure for the
Markov chain. In this way we complete Boltzmann's program of deriving kinetic
equations from random dynamics for this simple model. Three families of
invariant measures for the mean field limit are discovered and we show that
only two of those families can be obtained as limits of the discrete system and
the third is extraneous. Finally, we cast our results in the framework of
integer partitions and strengthen some results already available in the
literature.
