We consider a random matching model where heterogeneous agents choose optimally to invest time and real resources in education. Generically, there is a steady state equilibrium, where some agents, but not all of them, invest. Regular steady state equilibria are constrained inefficient in a strong sense. The Hosios (1990) condition is neither necessary, nor sufficient, for constrained efficiency. We also provide restrictions on the fundamentals sufficient to guarantee that equilibria are characterized by overeducation (or undereducation), present some results on their comparative statics properties, and discuss the nature of welfare improving policies.