Taking Stock

This chapter has reviewed the basic tools of dynamic optimization in continuous time. By its nature, this has been a technical (and unfortunately somewhat dry) chapter. The ma­terial covered here may have been less familiar than the discrete-time optimization methods presented in the previous chapter.

It is useful to recap the main approach developed in this chapter. Most of the problems in economic growth and macroeconomics require the use of discounted infinite-horizon optimal control methods. Theorem 7.13 provides necessary conditions for an interior continuous solu­tion to such problems. Theorem 7.14 provides sufficient conditions, related to the concavity of the maximized Hamiltonian, for such a solution to be a global or unique global maximum (these conditions require the existence of a candidate solution, since they use information on the costate variable of this solution). Importantly, the conditions in Theorem 7.14 are more straightforward to verify than those in Theorem 7.13 (in particular, than Assumption 7.1). Therefore, the following strategy will be used in all cases in this book.

(1) We start with the necessary conditions in Theorem 7.13 to construct a candidate solution.

(2) Once a candidate path has been located, we then verify that the concavity conditions in Theorem 7.14 are satisfied. If they are, then we have located a path that is

optimal. If, in addition, the maximized Hamiltonian is strictly concave, then this is the unique optimal solution.

It is also worth noting that while the basic ideas of optimal control may be a little less familiar than those of discrete-time dynamic programming, these methods are used in much of growth theory and in other areas of macroeconomics. Moreover, while some problems natu­rally lend themselves to analysis in discrete time, other problems become easier in continuous time. Some argue that this is indeed the case for growth theory. Regardless of whether one agrees with this assessment, it is important to have a good command of both discrete-time and continuous-time models in macroeconomics, since it should be the context and economic questions that dictate which type of model one should write down, not the force of habit. This motivated my choice of giving roughly equal weight to the two sets of techniques.

There is another reason for studying optimal control. The most powerful theorem in optimal control, Pontryagin’s Maximum Principle, is as much an economic result as a mathe­matical result. As discussed above, the Maximum Principle has a very natural interpretation both in terms of maximizing flow returns plus the value of the stock, and also in terms of an asset value equation for the value of the maximization problem. These economic intu­itions are not only useful in illustrating the essence of this mathematical technique, but they also provide a useful perspective on a large set of questions that involve the use of dynamic optimization techniques in macroeconomics, labor economics, finance and other fields.

This chapter also concludes our exposition of the “foundations” of growth theory, which comprised general equilibrium foundations of aggregative models as well as an introduction to mathematical tools necessary for dynamic economic analysis. I next turn to economically more substantive issues.

7.10.