Revisiting the Lucas model

We revisit the influential economic growth model by Lucas (1988) ["On the mechanics of economic development." Journal of Monetary Economics, 22(1):3-42], assuming that households optimally allocate consumption and education over the life-cycle given an exogenous interest rate and exogenous wages. We show that in such a partial equilibrium setting, the original two-state (physical capital and human capital) optimization problem can be decomposed into two single-state optimal control models. This transformation allows us to rigorously prove the existence of a singular control describing the allocation of education time along a balanced growth path. We derive a constructive condition for a singular control to exist and show that under this condition infinitely many singular controls are optimal in the individual household problem. In contrast to the original general equilibrium framework in which an agent always chooses part-time education and part-time work, in our framework such an agent might find it optimal to allocate her whole available time to education at the beginning of her life and to focus on labor supply only when she is older.