The purpose of this paper is to develop and estimate a new equilibrium model of public housing that acknowledges the fact that the demand for public housing may exceed the available supply. We show that ignoring these supply side restrictions leads to an inconsistent estimator of household preferences. We estimate the parameters of the model based on a unique panel data set of low-income households in Pittsburgh. We find that public housing is an attractive option for seniors and exceedingly poor households headed by single mothers. We also find that for each family that leaves public housing there are, on average, 3.8 families that would like to move into the vacated unit. Simple logit demand models that ignore supply side restrictions cannot generate reasonable wait times and wait lists. Demolitions of existing units increase the degree of rationing and potentially result in welfare losses. An unintended consequence of demolitions is that they increase racial segregation in low-income housing communities.