Taking Stock
This chapter has been concerned with basic dynamic programming techniques for discretetime infinite-dimensional problems. These techniques are not only essential for the study of economic growth, but are widely used in many diverse areas of macroeconomics and economic dynamics more generally.
A good understanding of these techniques is essential for an appreciation of the mechanics of economic growth. In particular, they will shed light on how different models of economic growth work, how they can be improved and how they can be taken to the data. For this reason, this chapter is part of the main body of the text, rather than being relegated to the Appendices at the end.This chapter also presented a number of applications of dynamic programming, including a preliminary analysis of the one-sector optimal growth problem. The reader will have already noted the parallels between this model and the basic Solow model discussed in Chapter 2. These parallels will be developed further in Chapter 8. I also briefly discussed the decentralization of the optimal growth path and the problem of utility maximization in a dynamic competitive equilibrium.
It is important to emphasize that the treatment in this chapter has assumed away a number of difficult technical issues. First, the focus has been on discounted problems, which are simpler than undiscounted problems. In economics, very few situations involve undiscounted objective functions (β = 1 rather than β ∈ (0,1)). More important, throughout I have assumed that payoffs are bounded and the state vector x belongs to a compact subset of the Euclidean space, X. This rules out many interesting problems, such as endogenous growth, where the state vector grows over time. Almost all of the results presented here have equivalents for these cases, but these require somewhat more advanced treatments.
6.12.