We use a quantitative spatial equilibrium model to evaluate the distributional and welfare impacts of a recent temporary rent control policy in Berlin, Germany. We calibrate the model to key features of Berlin's housing market, in particular the recent gentrification of inner city locations. As expected, gentrification benefits rich homeowners, while poor renter households lose. Our counterfactual analysis mimicks the rent control policy. We find that this policy reduces welfare for rich and poor households and in fact, the percentage change in welfare is largest for the poorest households. We also study alternative affordable housing policies such as subsidies and re-zoning policies, which are better suited to address the adverse consequences of gentrification.