To this end, we start with the Lagrangean approach in Section 2, deriving the maximum principle for discrete time problems. We aim to take advantage of the foundation already developed, utilizing as much as possible the optimization theory of Chapter 5. The goals of this supplementary chapter are therefore more modest. Although many of the necessary prerequisites are contained in earlier chapters, some essential elements (such as integration) are missing. A rigorous treatment of dynamic optimization (especially optimal control theory) is quite difficult. Consequently, we call this the Lagrangean approach. The third approach to dynamic optimization extends the Lagrangean technique of static optimization to dynamic problems. Dynamic programming has already been explored in some detail to illustrate the material of Chapter 2 (Example 2.32). The second principle approach, dynamic programming, was developed at the same time, primarily to deal with optimization in discrete time. This was generalized under the stimulus of the space race in the late 1950s to develop optimal control theory, the most common technique for dealing with models in continuous time. The classic approach is based on the calculus of variations, a centuries-old extension of calculus to infinite-dimensional space. Another factor complicating the study of dynamic optimization is the existence of three distinct approaches, all of which are used in practice. This need not be seen as an unrewarding chore-the additional complexity of dynamic models adds to their interest, and many interesting examples can be given. Dynamic models are increasingly employed in economic theory and practice, and the student of economics needs to be familiar with their analysis. This requires a more sophisticated theory and additional solution techniques. On the other hand, many dynamic models have no finite time horizon or are couched in continuous time, so that the underlying space is infinite-dimensional. On the one hand, the repetitive nature of dynamic models adds additional structure to the model which can be exploited in analyzing the solution. While the same principles of optimization apply to dynamic models, new considerations arise. Many economic models involve optimization over time. Chapter 5 deals essentially with static optimization, that is optimal choice at a single point of time.
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