In this chapter, we consider uncertainty about the future, the concept of dynamic stochastic models, and the hypothesis of rational expectations.
Dynamic stochastic models (and an appropriate expectations hypothesis) are indispensable if one is to model conditions in which there is uncertainty about the future. Unlike the deterministic models with perfect foresight that we have used so far, 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.
Thus, models need not only be dynamic but also stochastic, in the sense that they encompass uncertainty. This chapter introduces the basic concepts and presents the properties and alternative solution methods for such dynamic stochastic models under rational expectations.1Dynamic stochastic models can be contrasted with dynamic deterministic models, which do not allow for uncertainty. A dynamic deterministic model consists of a set of differential or difference equations that describe exactly how the system will evolve over time. Thus, there is no uncertainty about the evolution of such a model economy, unless the model is characterized by multiple equilibria. The two-period models of chapter 2, and the infinite-horizon growth models of chapters 3–8 are examples of such deterministic models. A static stochastic model allows for uncertainty and describes the interactions of random variables. A dynamic stochastic model describes the dynamic interactions among stochastic processes, which constitute the dynamic equivalents of random variables. If a dynamic stochastic model is solved several times, it will not produce identical results because of differences in its stochastic elements. Different runs of a dynamic stochastic model are different realizations of a stochastic process and will in general result in different outcomes. Stochastic models embody both randomness and uncertainty.2
Thus, instead of describing a process that can only evolve in one way (as in the case of solutions of deterministic systems of ordinary differential or difference equations), a dynamic stochastic model exhibits indeterminacy due to randomness.
Even if the initial conditions (or starting points) are known, there are several, often infinitely many, directions in which the process can evolve over time.Like random variables and stochastic processes, dynamic stochastic models can be defined and analyzed in both continuous and discrete time. In the case of discrete time, a dynamic stochastic model determines a sequence of endogenous random variables as functions of exogenous random variables, stochastic processes, and a set of parameters. The approach we take is to model these random variables as random functions of the time index. The random variables of a dynamic stochastic model exhibit dynamic stochastic interdependence.
We start with a simple expectational model of a competitive market, to illustrate the differences between deterministic and stochastic models and also highlight the role of the treatment of expectations. This is a model in which demand is a function of the current price, but supply has to be determined one period in advance, based on the prior expectations of firms about the current price. In addition, supply is subject to a random disturbance, that is not known when supply decisions are made. Hence, prior expectations about the current price partly determine the supply of the commodity in question. The market clears at a price that equates this partly predetermined stochastic supply with current market demand. This simple expectational model has played a very important role in the study of expectations in economics, which is the reason we will examine it here at some length.
We move on to distinguish between two expectations hypotheses. Adaptive expectations, which was the main expectations hypothesis used in macroeconomics before the 1970s, and rational expectations, which is the key expectations hypothesis used since then. We then demonstrate how the dynamic properties of the simple expectational stochastic model differ under the two alternative expectations hypotheses.
We then proceed to illustrate general solution methods for dynamic stochastic models with rational expectations, applying them to models of exogenous univariate stochastic processes and first- and second-order linear stochastic models with one endogenous and one exogenous variable. We apply these methods to both the expectational competitive market model and two additional simple economic models, one from finance and one from monetary economics.
The final part of the chapter introduces multivariate rational expectations models and briefly discusses the conditions for unique solutions and the alternative solution methods used.
In subsequent chapters, the solution methods introduced in this chapter are applied extensively to dynamic stochastic models of aggregate consumption, investment, money, and aggregate fluctuations.
9.1