  We study the Immediate Exchange model, recently introduced by Heinsalu and
Patriarca [Eur. Phys. J. B 87: 170 (2014)], who showed by simulations that the
wealth distribution in this model converges to a Gamma distribution with shape
parameter $2$. Here we justify this conclusion analytically, in the
infinite-population limit. An infinite-population version of the model is
derived, describing the evolution of the wealth distribution in terms of
iterations of a nonlinear operator on the space of probability densities. It is
proved that the Gamma distributions with shape parameter $2$ are fixed points
of this operator, and that, starting with an arbitrary wealth distribution, the
process converges to one of these fixed points. We also discuss the mixed model
introduced in the same paper, in which exchanges are either bidirectional or
unidirectional with fixed probability. We prove that, although, as found by
Heinsalu and Patriarca, the equilibrium distribution can be closely fit by
Gamma distributions, the equilibrium distribution for this model is {\it{not}}
a Gamma distribution.
