《Dynamic Optimization in Continuous-Time Economic Models》.pdf

Dynamic Optimization in Continuous-Time Economic Models (A Guide for the Perplexed) Maurice Obstfeld* University of California at Berkeley First Draft: April 1992 *I thank the National Science Foundation for research support. I. Introduction The assumption that economic activity takes place continuously is a convenient abstraction in many applications. In others, such as the study of financial-market equilibrium, the assumption of continuous trading corresponds closely to reality. Regardless of motivation, continuous-time modeling allows application of a powerful mathematical tool, the theory of optimal dynamic control. The basic idea of optimal control theory is easy to grasp-- indeed it follows from elementary principles similar to those that underlie standard static optimization problems. The purpose of these notes is twofold. First, I present intuitive derivations of the first-order necessary conditions that characterize the solutions of basic continuous-time optimization problems. Second, I show why very similar conditions apply in deterministic and stochastic environments alike. 1 A simple unified treatment of continuous-time deterministic and stochastic optimization requires some restrictions on the form that economic uncertainty takes. The stochastic models I discuss below will assume that uncertainty evolves continuously ^ according to a type of process known as an Ito (or Gaussian 1When the optimization is done over a finite time horizon, the usual second-order sufficient conditions generalize immediately. (These second-order conditions will be valid in all problems examined here.) When the horizon is infinite, however, some additional terminal conditions are needed to ensure optimality. I make only passing reference to these con