We consider a simple stochastic model of a urban rental housing market, in which the interaction of tenants and landlords induces rent fluctuations. We simulate the model numerically and measure the equilibrium rent distribution, which is found to be close to a lognormal law. We also study the influence of the density of agents (or equivalently, the vacancy rate) on the rent distribution. A simplified version of the model, amenable to analytical treatment, is studied and leads to a lognormal distribution of rents. The predicted equilibrium value agrees quantitatively with numerical simulations, while a qualitative agreement is obtained for the standard deviation. The connection with non-equilibrium statistical physics models like ratchets is also emphasized.