John Fraser Muth (1930-2005) was an American economist known as the father of Rational Expectations revolution in Economics primarily due to his 1961 publication: “Rational Expectations and the Theory of Price Movements.”

He believed that the kind of information used and how it is put together to frame an estimate of future conditions is important to understand because the character of dynamic processes is typically very sensitive to the way expectations are influenced by the actual course of events. Furthermore, he stressed that it is often necessary to make sensible predictions about the way expectations would change when either the amount of available information or the structure of the system is changed.

Clearly, in the Theory of Rational Expectations, expectations reflect the optimal forecast (the best guess of the future) using all available information. This implied that expectations will be identical to optimal forecasts. This theory can be applied to the Financial Markets.

The Hypotheses

The Theory of Rational Expectations puts forth two hypotheses as concluded from studies of expectation data:

I. Average expectations in an industry are more accurate than naive models and as accurate as elaborate equation systems.

II. Reported expectations generally underestimate the extent of changes that actually take place.

Muth maintained that expectations are informed predictions of future events similar to predictions of a relevant economic theory. More precisely, the hypothesis can be rephrased as: expectations of firms tend to be distributed for the same information set about the prediction of the theory.

The assertions about the hypothesis are as follows:

I. Information is scarce and the economic system does not waste it;

II. The way expectations are formed depends specifically on the structure of the relevant system describing the economy.

III. A ‘public prediction’ as noted by Grunberg and Modigliani will have no substantial effects on the operations of the economic system (unless such prediction is based on inside information).

Because the type of expectational behavior postulated is purposeful, and the absence of avoidable errors is necessary for optimality on the part of agents in the modeled economy, Muth chose the term Rational Expectations to describe his hypothesis. As it happens, Muth’s ideas were not immediately embraced in economics profession. The appreciation of the Theory of Rational Expectations was greatly enhanced in the early 1970s by a number of path-breaking papers by Robert E. Lucas Jr., in which the rational expectations notion was extended and applied to important issues in macroeconomics and the financial markets. Further contributions to rational expectations theory was done by Lucas (1981), Thomas J. sergeant (1973) and Robert J. Barro (1977) and they helped spread the concept of rational economics to mainstream economics (McCallum 1989).

I give an algebraic representation to hypothesis 1: That the agents are accurate and avoid all systematic errors. I further consider an agent forming his expectation at time of (the price in the next period caused by inflation) and denote the expectation by .

The possible expectational error in time becomes:

With the condition that this error will not be systematically related to any information possessed by the agent in period when the expectation was formed.

Achieving this condition would result in the assumption that expectations believed by agents are equal to the mean of probability distribution of the variable being forecast, given available information.

Theoretical Framework

Thus, the expectational hypothesis we seek can be adopted by assuming that, for any variable and any period :

In words, the condition requires that the subjective expectation (forecast) of held by agents in be objective to the (mathematical) expectation of conditional on .

If the reader chooses to ask where the actual probability distribution of came from; it is of course unknown. But the economist wishing to use this expectational hypothesis in the content of the model, has in the course of constructing that model adopted his own view of how is generated.

Proof of Condition (1) Being True

We find the average expectational error over a large number of periods if condition 1 is utilized by first finding the mean of distribution of values. That is:

Where

The law reflects the common sense idea that the expectation of an expectation that will be based on more information than is currently available will simply be the expectation given the lesser available information.

Let denote any variable whose value is known to agents at . Thus is an element of . Covariance which is the mean of distribution of the product is expressed as:

Since is an element of , it is true that .

Applying the law of iterated expectations, we say that the final term in (3) equals . This shows, that the covariance is zero:

Hence the expectational error is, in a statistical sense, unrelated to any element of . Thus it was shown that the adoption of assumption (1) will, in fact imply that the systematic expectational errors will be absent.

Application of The Theory of Rational Expectations: A case of The Cagan Model of Hyperinflation

Here I illustrate the case of inflation in a dynamic environment by studying a familiar example, the Cagan model of inflation under the assumption that expectations are formed rationally. It turns out that the strongest evidence in favor of The Quantity Theory of Money comes from the episodes of hyperinflation. Hence, we restrict the analysis on the relationship between the money stock and the price level.

The Cagan model is expressed as the demand for money also called real cash balances.

where

= money stock

= average price level

= parameters of the equation

, the stochastic disturbance term is purely random and possesses the following properties of zero mean and constant variance:

In literature, an error term with these properties is often called white noise error term.

= the inflation rate between period t and period t+1

= the subjective expectation formed in the period t of period t+1

= the change in the subjective expectation formed in the period t of period t+1

To explain the rationality assumption we make modifications to equation (1).

We adopt a simplified notation where replaces for any variable as the expectation (the mean) of the variable’s distribution conditional upon information available in period and use it to simplify equations (5) and (6) together as:

With the agents fully informed of the workings of the economy, we note that since is known to agents at , the expectation (or mean) equals itself. The expected inflation rate is equal to :

After expanding I have:

Showing that equals by definition:

Rewriting equation (7) using the rule gives:

Now making the subject by rearranging thus:

With being determined exogenously (12) is not a finite solution because it contains the expectational variable for average price level: . We attempt to reduce the algebraic expansion by comparing it to , adjusting for the next period’s forecast taking the mean and collecting the like terms:

Substituting expression (17) for in expression (12) then rearranging to give the following:

Expression (12) could be rewritten for applied, and the result substituted for . This would bring in . The process could then be further repeated. By repeating the process indefinitely, all future terms could be eliminated, giving a solution for in terms of the disturbance variable , the money stock and the expectation of all future values and will look exactly like this:

This emphasizes that the level of the price in period depends on the values of the money stock that are expected for each period in the entire infinite future.

Theoretical Justification

Accurate expectations are desirable, and there are strong incentives for people to try to make them equal to optimal forecasts by using all available information. The incentives for equating expectations with optimal forecasts are especially strong in financial markets. In these markets, people with better forecasts of the future get richer. In the 1970s and 1980s, economists, including Robert Lucas of the University of Chicago and Thomas Sargent, now at New York University, used the rational expectations theory discussed to examine why activist policies appear to have performed so poorly. Their analysis cast doubt on whether macroeconomic models can be used to evaluate the potential effects of policy and on whether policy can be effective when the public expects that it will be implemented. Because the analysis of Lucas and Sargent had such strong implications for the way policy should be conducted, it has been labeled the rational expectations revolution. The application of The Theory of Rational Expectations to financial markets (where it is called the Efficient Market Hypothesis or Theory of Efficient Capital Markets) is thus particularly useful.

Summary Notes

Rational expectations theories were developed in response to perceived flaws in theories based on adaptive expectations. Under Adaptive Expectations Theories, expectations of the future value of an economic variable are based on past values. For example, people would be assumed to predict inflation by looking at inflation last year and in previous years. So, if the economy suffers from constantly rising inflation rates (perhaps due to government policies), people would be assumed to always underestimate inflation. This may be regarded as unrealistic – surely rational individuals would sooner or later realize the trend and take it into account in forming their expectations. Hence, macroeconomics research papers ought to be written in line with expectations regarding future outcomes.

FURTHER READINGS

McCallum, B. T. (1989): Monetary Economics Theory and Policy, Macmillan Publishing Company, Newyork.

Mishkin, S. F. (2004): The Economics of Money, Banking and Financial Markets, 7th ed., Addison-Wesley Series in Economics.

Muth, J. F. (1961): Rational Expectations and the Theory of Price Movements

See Notes on The Cagan Model (PDF), Money Demand, The Cagan Model, Testing Rational Expectations in Turkey (PDF)


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