This paper sets up and solves a spatial equilibrium model of the housing market. The model is a dynamic version of the canonical compensating-wage differential model of Rosen (1979) and Roback (1982). In contrast with the urban economics tradition of studying house prices in one region given some exogenous outside option of living in the countryside (the “reservation locale”), we model the entire distribution of regions. This has several advantages. First, the model provides predictions for the entire cross-section of income, house prices, and construction.
This facilitates comparison with the data, which are realizations of that joint equilibrium distribution. Second, the outside option of living in the reservation locale is no longer exogenous: instead, its value is determined in equilibrium. This is important when studying the effect of changes in aggregate variables on house prices, because they operate precisely through endogenous changes in the value of the reservation locale. Third, our model has the benefit of numerical tractability. This is useful when we solve for transitional dynamics in order to evaluate the effect of changes in aggregate variables on house prices.
We model metropolitan areas as a collection of geographically separated islands, randomly hit by idiosyncratic and persistent productivity shocks in the non-housing sector. Construction firms can build new houses in any metropolitan area, but new construction is irreversible and is subject to supply regulation, implying that the local housing supply cannot adjust instantly in response to a local productivity shock. We assume households differ in their ability and are fully mobile: they can freely move across metropolitan areas, but are constrained to live in the same area they work. The equilibrium provides the endogenous joint dynamics of cross sectional house prices, construction, labor income, and employment.
In particular, more able households sort into more productive regions. House price differentials between metropolitan areas compensate for the income differential of the marginal, lowest ability household in the location, making him indifferent between staying and moving to the next best metropolitan area. Households also live in smaller and more expensive quarters if they choose to work in higher-income metropolitan areas. Lastly, higher income metropolitan areas have, on average, a larger housing stock and a larger workforce.
