The concavity of the value function of the extended Barro-Becker model

In this paper, the model of endogenous fertility proposed by Becker and Barro (1988), and extended by Benhabib and Nishimura (1989) is considered. However, in their model, the uniqueness of the optimal path may fail to hold. Furthermore, the value function may not be concave, because of the variable discount factor with respect to the choice made about fertility. In this paper, we show that under a set of mild conditions, based on the assumption that the cost of raising a child is non-constant, there exists a unique optimal path and the value function is concave and continuously differentiable. We also show the existence of the unique steady state and a monotonically optimal path, and confirm that the steady state is saddle point stable.