Directed Random Market: the equilibrium distribution

We find the explicit expression for the equilibrium wealth distribution of the Directed Random Market process, recently introduced by Mart'inez-Mart'inez and L'opez-Ruiz, which turns out to be a Gamma distribution with shape parameter $\frac{1}{2}$. We also prove the convergence of the discrete-time process describing the evolution of the distribution of wealth to the equilibrium distribution.