This paper presents a model with rental housing vacancies in equilibrium. Because of the indivisibility and multi-dimensional heterogeneity of housing units, the housing market is thin. As a result, a typical household entering the market is willing to pay a premium for its most-preferred over its second most-preferred available (vacant) unit. This confers monopoly power on landlords, which they exploit by setting rents above costs. Free entry and exit force profits to zero, with vacancies as the equilibrating mechanism. A nice feature of the model is that housing vacancies are socially useful in expanding the choice set of entering households, though there is no presumption that the market vacancy rate is socially optimal. Thin markets are modeled by assuming an idiosyncratic component to households' tastes over housing units.
The positive and normative properties of the basic model, which assumes costless search, are investigated. Then the model is extended to treat costly search. Finally, directions in which the model could usefully be extended are discussed.