We provide an explanation for the stylized fact that poor households are concentrated in the inner city of most U.S. metropolitan areas. We consider a metropolitan area with an inner city surrounded by a suburb and two income classes. Using numerical simulations, we show that two equilibria typically exist: one in which the inner city has a majority of poor households and the other in which it has a majority of rich households. We argue that the growth path selects the former equilibrium because rich households "jump" to the suburb before poor households "spill" into the suburb. In addition, the model provides an explanation for gentrification: at large metropolitan populations, population growth causes rich households in the city to live in areas previously inhabited by poor households.