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
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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 conditions below, even
though I always assume (for simplicity) that horizons are
infinite. Detailed treatment of such technical questions can be
found in some of the later references. diffusion) process. Once mainly the province of finance
theorists, Ito^ processes have recently been applied to
interesting and otherwise intractable problems in other areas of
economics, for example, exchange-rate dynamics, the theory of the
firm, and endogenous growth theory. Below, I therefore include a
brief and heuristic introduction to continuous-time stochastic
processes, including the one fundamental tool needed for this
type of analysis, Ito^’s chain rule for stochastic differentials.
Don’t be intimidated: the intuition behind Ito^’s Lemma is not
hard to grasp, and the mileage one gets out of it thereafter
truly is amazing.