A theory of the distribution of housing tenure in a city is developed. Het- erogeneous houses are built by a competitive development industry and either rented competitively or sold to households which dier in their income and sort over housing types through a directed search process. In the absence of either nancial or supply restrictions, higher income households are more likely to own and lower quality housing is more likely to be rented. The composition of the housing stock and the rate of home-ownership depend on the distribution of income, the age of the population and construction costs. When calibrated to match average features of housing markets within U.S. cities, observed dif- ferences in these variables account well for the variation observed across cities in home-ownership and the price-rent ratio. A policy designed to improve housing aordability signicantly raises home-ownership among lower income households while lowering the quality supplied to high income households.