Existence of Solutions*

The theorems presented so far characterize the properties of a (piecewise continuous) solution to a continuous-time maximization problem. The question of when a solution exists

arises naturally.

Theorem 7.17. (Existence of Solutions) Consider the maximization of (7.)7) subject to (7.)8) and (7.49). Suppose that f and g are continuous in all of their arguments, p > 0, Y and X are compact and F is nonempty. Then a solution to the maximization problem exists.

Proof. The proof follows from the results developed in Appendix Chapter A. In par­ticular, from Part 2 of Theorem A.11, the objective function (7.47) is continuous in the product topology (since the instantaneous payoff function f is the same at each date and is defined over the compact set Ó ?X, thus is uniformly bounded). The constraint set F is also bounded, since Y is compact and is defined by a continuous function, and is therefore, closed and hence also compact in the product topology (see Exercise A.21 in Appendix Chapter A). It is also nonempty by hypothesis. Then by Weierstrass’s Theorem, Theorem A.9, a solution exists. ?

Unfortunately, providing sufficient conditions for the solution to be continuous or piece­wise continuous is much harder. Nevertheless, most economic problems possess enough 290

structure to ensure this. For example, in most of the problems we will encounter Inada- type conditions ensure that optimal controls remain within the interior of the feasible set and consumption-smoothing and no-arbitrage arguments rule out discontinuous controls. In these cases, continuous solutions can be shown to exist. Moreover, using Theorem 7.15, we can often establish that these solutions are unique. Throughout the rest of the book, I will follow the standard practice and assume that it continues solution to this type of maximization problem exists.

7.7.