We consider a random matching model where heterogeneous agents endogenously choose to invest time and real resources in education. Generically in the space of the economies, there is an open interval of possible lengths of schooling such that, at at least one of the associated steady states equilibria, some agents, but not all of them, choose to invest. Regular steady state equilibria are constrained Pareto inefficient in a strong sense. The Hosios (1990) condition is neither necessary, nor sufficient, for constrained Pareto optimality. We also provide restrictions on the fundamentals, which are sufficient to guarantee that equilibria are characterized by overeducation (undereducation), and present some results on their comparative static properties.