Introduction

This chapter is concerned with the optimal control problem for a linear stochastic continuous time model where expectations are rational, in the sense that they are consistent with the model, and where the model includes forward-looking ‘jump’ variables. The chapter draws upon work for deterministic models by Calvo (1978), Driffill (1982), Miller and Salmon (1985a) and Buiter (1984b) and extends their results to the stochastic case. For the most part, we confine ourselves to solutions with the familiar saddle-path property, but feedback rules which completely stabilise the system (i.e., have no unstable roots) are considered in a later section.

For stochastic models, the closed-loop or feedback representation of policy is particularly useful in policy design. For the standard control problem (without rational expectations), with a quadratic loss function and an infinite time horizon, the optimal policy may be represented as a linear time-invariant feedback rule on the state vector. This is no longer the case when we examine models with rational expectations and forward-looking jump variables. We show that, in such models, optimal feedback control can be expressed and implemented as a form of integral control. The added complexity of the full optimal closed-loop rule strengthens the case for considering simple sub-optimal rules of the type which have recently been advocated, for example, by Vines et al. (1983), Currie and Levine (1985a) and Taylor (1985).

The plan of the chapter is as follows. Section 2 sets out the results of standard control theory in order to provide a framework for the rest of the chapter and to highlight the differences between optimal control in models with and without rational expectations.