A model of frictional assignment is developed to study the composition of housing units and households across city-level ownership and rental markets. Heterogeneous houses are built by a competitive development industry and either rented competitively or sold through directed search to households which differ in wealth and sort over housing types. Even in the absence of financial restrictions and constraints on house characteristics, higher income households are more likely to own and lower quality housing is more likely to be rented. When calibrated to match average features of housing markets within U.S. cities, the model is qualitatively consistent with U.S. data on the relationships between observed differences in median income, inequality, median household age, and construction/land costs across cities and both home-ownership and the average cost of owning vs. renting. Policies designed to improve housing affordability raise both housing quality and ownership for lower income households while lowering housing quality (but not ownership) for high income ones.
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