Optimal Growth

Before characterizing the competitive equilibrium further, it is useful to look at the op­timal growth problem. Recall that this involves characterizing the capital-labor ratio and consumption path that maximizes the utility of the representative household.

subject to

and k (0) > 0.^[14] 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, in the present context there is no need to appeal to these theorems since it is straightforward to characterize both allocations and show their equivalence.

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. With the same argument as in Section 7.7 in the previous chapter, it can be shown that Theorems 7.13 and 7.14 can be applied to this problem and characterize the unique optimal growth path (in particular, recall Exercise 7.25). Consequently, the necessary and sufficient conditions for

this optimal path are:

Repeating the same steps as before, we can combine the first two optimality conditions in

(8.33) and obtain (8.27) for the path of consumption of the representative household. In addition, once again integrating the second first-order condition,

Combining this with the first condition in (8.33) evaluated at t = 0 implies that μ (0) = u' (c (0)) > 0. Substituting this expression into the transversality condition (the third condi­tion in (8.33)), simplifying and canceling out μ (0) > 0, we obtain (8.28).

These steps establish that the competitive equilibrium is a Pareto optimum and that the optimal growth path 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.