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 297
material covered here may have been less familiar than the discrete time optimization methods presented in the previous chapter.
Part of the difficulty arises from the fact that optimization here is with respect to functions, even when the horizon is finite (rather than with respect to vectors or infinite sequences as in the discrete time case). This introduces a range of complications and some technical difficulties, which are not of great interest in the context of economic applications. As a result, this chapter has provided an overview of the main results, with an emphasis on those that are most useful in economic applications, together with some of the proofs. These proofs are included to provide the readers with a sense of where the results come from and to enable them to develop a better feel for their intuition.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. Irrespective 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 our 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.Finally, to avoid having the current chapter just on techniques, we also introduced a number of economically substantive applications of optimal control. These include the intertemporal problem of a consumer, the problem of finding the optimal consumption path of a non-renewable resource and the q-theory of investment. We also used the q-theory of investment to illustrate how transitional dynamics can be analyzed in economic problems involving dynamic optimization (and corresponding boundary conditions at infinity). A detailed analysis of optimal and equilibrium growth is left for the next chapter.
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. We next turn to economically more substantive issues.
7.10.