Abstract This chapter is devoted to the analysis of solutions of linear rational-expectations models. Successively introducing various types of expectations (perfect, naive, adaptive, and rational) in the Muth model, the final reduced forms and the linear stationary solutions are compared. The main solution techniques for rational-expectations models are reviewed on the Cagan model. The “non-uniqueness problem” is also discussed. The reduced form of a very general linear model is given and the linear stationary solutions are parametrically described. The parameters have a simple interpretation and allow for statistical applications. Finally, a generalization to multivariate rational-expectations models is given. If no invertibility conditions are imposed on the structural coefficient matrices, the solution techniques used in the univariate case become insufficient. A method is suggested to obtain the general solution of a multivariate model and to characterize the dimension of the solutions space.

Introduction

The problem of modeling the mechanism by which economic agents form their expectations is fundamental in macroeconomic theory. In many models, it is essential to include expectations of future variables. However, such expectations are often unobservable. Therefore, assumptions on their formation are needed to complete the specification.

The rational-expectation hypothesis was introduced in a seminal paper by Muth (1961). Several years later the assumption was incorporated in many macromodels (e.g., Sargent and Wallace 1975, 1976; Lucas 1976; Taylor 1979). In these models, expectations are optimal predictions given all the available information. Rational expectations are thus based on an information set that may be chosen by the model builder.