We analyze equilibrium housing prices in an overlapping generations model with perfect housing and rental markets. The economy exhibits a two-stage phase transition: as the income of home buyers rises, the equilibrium regime changes from fundamental to bubble possibility, where fundamental and bubbly equilibria can coexist. With even higher incomes, fundamental equilibria disappear and housing bubbles become a necessity. Even with low current incomes, housing bubbles may emerge if home buyers have access to credit or have high future income expectations. Contrary to widely-held beliefs, fundamental equilibria in the possibility regime are inefficient despite housing being a productive non-reproducible asset.