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Housing affordability is the main policy challenge for many large cities in the world. Zoning changes, rent control, housing vouchers, and tax credits are the main levers employed by policy makers. But how effective are they at combatting the affordability crisis? We build a new framework to evaluate the effect of these policies on the well-being of its citizens. It endogenizes house prices, rents, construction, labor supply, output, income and wealth inequality, as well as the location decisions of households. Its main novel features are risk, risk aversion, and incomplete risk-sharing. We calibrate the model to the New York MSA, incorporating current zoning and affordable housing policies. Housing affordability policies carry substantial insurance value but cause misallocation in labor and housing markets. Housing affordability policies that enhance access to this insurance especially for the neediest households create large net welfare gains.