In this chapter we will review some applications of dynamic optimization to economics. In Section 1 we develop two models of search to illustrate the use of dynamic programming in a stochastic setting. Section 2 analyzes the decision problem faced by a social planner who maximizes the utility of an infinitely-lived representative agent in a one-good neoclassical economy. In Section 3 we study the optimal investment policy of a competitive firm when the installation of capital is costly. Finally, in Section 4 we develop the Cass-Koopmans model of a dynamic competitive economy and use it to analyze the welfare cost of factor taxes. Section 5 concludes with a series of problems.
Search Models
Search theory provides a simple and yet interesting application of dynamic programming to economics. In the basic search model, wage offers drawn from a given distribution arrive at fixed or random intervals, and an agent simply decides whether to accept one of them and become employed or reject them and continue searching for a better opportunity. We have, then, a very simple problem in stochastic dynamic programming: The control is simply a take-it-or-leave-it decision, and the distribution of the state variables (the offers) is time-invariant and does not depend on either the state or the control.
The first part of this section introduces the basic “microeconomic” model of job search. In addition to its interest as an application of dynamic programming, this model provides a useful counterpoint to the neoclassical model of a competitive labor market.
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