Introduction to Modern Economic Growth This implies that equation (6.48) has a unique solution It can be easily verified that a higher β, by making the entrepreneur more patient, increases the cutoff threshold R 6.9 Taking Stock This chapter has been concerned with basic dynamic programming techniques for discrete time 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 economics more generally A good understanding of these techniques is essential for an appreciation of the mechanics of economic growth, i.e., 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 relegated to the Mathematical Appendix This chapter also presented a number of applications of dynamic programming, including a preliminary but detailed 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 These parallels will be developed further in Chapter We have also briefly discussed the decentralization of the optimal growth path and the problem of utility maximization in a dynamic competitive equilibrium Finally, we presented a model of searching for ideas or for better techniques While this is not a topic typically covered in growth or introductory macro textbooks, it provides a tractable application of dynamic programming techniques and is also useful as an introduction to models in which ideas and technologies are endogenous objects 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 call for modeling using undiscounted objective functions (i.e., β = rather than β ∈ (0, 1)) More important, throughout we 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 models, where the state vector grows over time Almost all of the results presented 305