Introduction

We have shown in Chapter 4 that the Bellman equation enables us to derive the equations that govern the optimal control policies. However, to have a better understanding of these behavioral functions we often need to know the functional form of the value function. For example, if the indirect utility function is of constant relative risk aversion in Merton's consumption and portfolio model, then we have shown that the share of wealth invested in each risky asset is constant over time. In this chapter we study the methods that determine the functional form of the value function of a stochastic, intertemporal optimization problem.

By definition, the value function depends on the specification of the objective function and the underlying controlled diffusion process. Changing the objective function or the controlled diffusion process would change the functional form of the value function and, hence, the optimal control.

We shall divide economic problems into four different classes of problems. The first class is the one in which the diffusion equation is linear in both state and control variables and the objective function is quadratic. This is the so-called linear–quadratic problem in control theory. The second class is the one in which the controlled diffusion process is linear in both state and control variables and the objective function exhibits hyperbolic absolute risk aversion (HARA). This class of functions contains most of the commonly employed objective functions and therefore deserves a special mention.