# 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 H_{t} and H_{t}, 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, K_{t}, H_{t}, 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

I

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 s_{t}, H_{t}+_{1}, and M_{t}, given *H _{t+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 s_{t}, H_{t}+_{1}, and M_{t} are respectively

s cannot be zero; otherwise, c^{2} 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.