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The paper develops a simple two-period model in which homelessness arises endogenously. There is a non-convexity in the housing market, so some agents optimally choose not to consume housing. In the model, homelessness leads to lower labor productivity in the future. Housing is thus an investment good, but borrowing constraints may prevent agents from being able to finance this investment. The borrowing constraints and the productivity loss combine to generate a homelessness trap.