Optimal Growth

Before characterizing the equilibrium further, it is useful to look at the optimal growth problem, defined as the capital and consumption path chosen by a benevolent social planner trying to achieve a Pareto optimal outcome.

and k (0) > 0.1 As noted in Chapter 5, versions of the First and Second Welfare Theorems for economies with a continuum of commodities would imply that the solution to this problem should be the same as the equilibrium growth problem of the previous section. However, we

^[1]In the case where the infinite-horizon problem represents dynastic utilities as discussed in Chapter 5, this specification presumes that the social planner gives the same weight to different generations as does the current dynastic decision-maker.

do not need to appeal to these theorems since in this together case it is straightforward to show the equivalence of the two problems.

To do this, let us once again set up the current-value Hamiltonian, which in this case takes the form

with state variable k, control variable c and current-value costate variable μ. As noted in the previous chapter, in the relevant range for the capital stock, this problem satisfies all the assumptions of Theorem 7.14. Consequently, the necessary conditions for an optimal path are:

Repeatingthe same steps as before, it is straightforward to see that these optimality conditions imply

which is identical to (8.20), and the transversality condition

which is, in turn, identical to (8.11).

This establishes that the competitive equilibrium is a Pareto optimum and that the Pareto allocation can be decentralized as a competitive equilibrium. This result is stated in the next proposition:

PROPOSITION 8.1. In the neoclassical growth model described above, with Assumptions 1, 2, 3 and 4', the equilibrium is Pareto optimal and coincides with the optimal growth path maximizing the utility of the representative household.

8.4.