Conclusion
This chapter introduces the concepts of dynamic stochastic models and rational expectations. Dynamic stochastic models and an appropriate expectations hypothesis (such as rational expectations) are indispensable for modeling conditions in which there is uncertainty about the future and agents exploit all available information.
Unlike the deterministic models with perfect foresight used in previous chapters, in which there was no uncertainty about the future, a more realistic treatment of dynamic macroeconomics requires the modeling of uncertainty and uncertain expectations about the future.We started with a simple expectational model of a competitive market, which illustrates the difference between deterministic and stochastic models and also highlights the role of the treatment of expectations. We distinguished between two expectations hypotheses: adaptive expectations and rational expectations. We determined that the dynamic behavior of the model differs under the two alternative expectational hypotheses.
Finally, we discussed general solution methods for dynamic stochastic models with rational expectations for both exogenous stochastic processes and more general first- and second-order linear models with one endogenous and one exogenous variable. We applied these methods to examples from finance and monetary economics, and also briefly discussed the conditions for unique solutions of multivariate rational expectations models and their alternative solution methods.
The methods illustrated in this chapter are indispensable for understanding and solving dynamic stochastic general equilibrium models and will prove extremely useful in the chapters that follow.
1. The concept of rational expectations is due to Muth [1961] and was introduced to macroeconomics by Lucas [1972]. It has since become the dominant expectations hypothesis in macroeconomics, replacing the concept of adaptive expectations, which was used until then.
The adaptive expectations hypothesis was first used, though not under this name, in the work of Fisher [1911] and later in the work of Koyck [1954]. In received its major impetus as a result of the work of Cagan [1956] on hyperinflation, and subsequently Friedman [1957] on the permanent income hypothesis. As we shall see in chapter 15, before the mid-1970s, it was also used extensively to model the role of expectations in models of the relation between inflation and unemployment (i.e., the Phillips curve).2. See appendix F for an introduction to random variables and stochastic processes, which are the building blocks of dynamic stochastic models.
3. This model has been used to describe agricultural markets because of the lag between planting and harvesting. Kaldor [1934] labeled it the cobweb model, because the pattern of convergence to equilibrium under adaptive expectations resembles a cobweb. Ezekiel [1938], Buchanan [1939], and Nerlove [1958] analyzed the model under various forms of adaptive expectations. The model has since been used extensively, as it has applications in other markets as well. Muth [1961] used this particular model to introduce the concept of rational expectations, arguing that rational producers would not be making systematic mistakes in their formation of expectations, as implied by the adaptive expectations hypothesis.
4. See Ezekiel [1938], Buchanan [1939], and Nerlove [1958] for a more general analysis of this model under various forms of adaptive expectations.
5. See appendix F for an introduction to random variables and stochastic processes.
6. See, for example, the classic text by Box and Jenkins [1976] on forecasting with stationary autoregressive moving average models and nonstationary integrated autoregressive moving average models.
7. A full analysis of this model is given in chapter 12.
8. Recall that, for a small x, ln(1 + x) is approximately equal to x.