A3 Economic Growth and Efficiency
A3.1 The First Best Solution
We first analyze the growth path that would be chosen by a central planner who maximizes an intertemporal social welfare function. The objective of the planner at time t is the same as that of the altruistic individual, the “head of the family,” living at time t. Since the planner does not discriminate between Ht and Ht, the maximization problem faced by the planner is
Solving c2 in Eq. (5.A6) and substituting the objective function, we obtain the following first-order conditions for the planner’s optimization problem by calculating the derivatives with respect to C, Kt, Ht, respectively.
together with the transversality condition,
Equations (5.A11.1, 5.A11.2, 5.A11.3, and 5.A11.4) imply that the economy moves right from the first period on a path of balanced growth. The optimal growth rate, у*, is given by
where r* is given by
A3.2 Optimizing Behavior in the Market Economy
An individual born at time t solves the following problem of maximization. She or he chooses st, Ht+1, and Mt, given Ht+1 in Eq. (5.A2). Substituting Eqs. (5.A2), (5. A4.1'), and (5.A4.2) into (5.A10), we have
The optimality conditions with respect to st, Ht+1, and Mt are respectively
s cannot be zero; otherwise, c2 would be zero, which is inconsistent with optimizing behavior. H cannot be zero either; otherwise, Y would be zero, which is inconsistent with optimizing behavior. However, M could become zero. If the private marginal return of educational investment is higher than the private marginal return of bequests at M = 0, intergenerational transfer is operated only in the form of human capital investment.