# Muth exposition of the rational expectations hypothesis

I am reading in statistical decision theory and stumbled upon the rational expectations literature (rationality with incomplete information->dynamic problem->N.L Stokey->husband). The assumption that subjective expectation approximates the objective probabilities without adaptive learning seems almost ridiculous if one considers that the whole enterprise of statistics is to learn from the past to infer about the future.

Nevertheless, as explained clearly in the answer to another question, Muth (1961) proposed the hypothesis of rational expectations as a purely descriptive model, to facilitate explanation of certain market behavior, however unrealistic it might be to generalize this hypothesis to all behavior.

Please refer to the full text of the paper.

If I understood it correctly, section 3 of the paper is an exposition of how such a rational expectations hypothesis, as the author proposed and shortly justified in section 2, can be applied to analyze several market situations.

I had difficulty understanding the reasoning around equations 3.3-3.4. In particular:

Referring to (3.3) we see that if $\frac{\gamma}{\beta}\neq-1$ the rationality assumption (3.4) implies that $p_t^e=0$, or that the expected price equals the equilibrium price.

What does the last part of the sentence mean? That equation (3.4) holds? How can $\frac{\gamma}{\beta}\neq-1$, $p_t^e\neq0$ and equations (3.3) and (3.4) hold together?

If I understand his exposition as imposing the rational expectations hypothesis (equation 3.4) on the market equilibrium price (equation 3.3), then the solution would be that either $\frac{\gamma}{\beta}=-1$ or that $p_t^e=0$. What does this mean? Or is he trying to show something else?

Muth assumes a model of

"...short-period price variations in an isolated market with a fixed production lag of a commodity which cannot be stored".

It is useful to remember that the model's equations are expressed as deviations from equilibrium values. So in a bit more clear notation than the original (a star denotes long-run equilibrium value)

\begin{align} & D_t-D^* = -\beta (p_t-p^*) & {\rm (Demand)}\\ & S_t-S^* = \gamma (p^e_t-p^*) + u_t & {\rm (Supply)}\\ & D_t = S_t,\;\; D^* = S^* & {\rm (Market \;Equilibirum)} \end{align}

Production is determined one period before, based on expected future price, but final supply is also subject to random shocks, $u_t$, with $E_{t-1}u_t =0$. $p^e_t$ is expected price but we have not yet make any assumption on how it is formed, or to what is equal.

Eliminating quantities through market equilibrium we obtain

$$p_t-p^* = -\frac {\gamma}{\beta} (p^e_t-p^*) - u_t \tag {3.2}$$

Taking expectations conditional on time $t-1$ we obtain

$$E_{t-1}p_t-p^* = -\frac {\gamma}{\beta} (p^e_t-p^*) \tag {3.3}$$

Rearranging and subtracting $p^e_t$ from both sides we see that equation $(3.3)$ leads to

$$p^e_t-E_{t-1}p_t = (1+\gamma/\beta) (p^e_t-p^*) \tag {3.3a}$$

If $\gamma / \beta =-1$ we obtain, without making any assumption on how expectations are formed but as a solution to the model, that $p^e_t=E_{t-1}p_t$. But this is uninteresting, being a very specific configuration of demand and supply responses. Assume then that $\gamma / \beta \neq -1$.

Then this way to write the relation (not in Muth's paper), shows clearly that if $$p^e_t \neq E_{t-1}p_t \implies p^e_t \neq p^*$$ and that $$p^e_t = E_{t-1}p_t \implies p^e_t = p^*$$

Throughout the paper Muth treats $E_{t-1}p_t$ as the theory's prediction, a best prediction (and it is, in the sense of being the minimizer of mean squared error of prediction). Given this Muth argues as follows: if "market expectations" $p^e_t$ (i.e. some concept of "average", "prevailing" expectations) were not equal to the "best" prediction, then recurring pure-profit opportunities would exist, for someone that used $E_{t-1}p_t$ as his own expectation, while all others used some other expectations formation rule. But, is it reasonable to argue that the market as a whole is outperformed by some "wise man"? Is it reasonable to argue that the firms and the businessmen and any other people whose livelihood depends on the workings of this specific market, would not really try hard to be as efficient and as accurate as possible regarding their predictions? It doesn't sound too convincing, especially since we are talking about the collective wisdom of all market participants here.

So making the assumption $p^e_t = E_{t-1}p_t$ (i.e. imposing the RE Hypothesis) appears reasonable, and this leads to

$$p^e_t =p^*$$

(remember the right-hand side is long-run equilibrium price, not next-period's - we are not looking at period-by-period perfect foresight here).

Now use this result on the initial equations describing the market, and eventually obtain the determination of the short-run equilibrium price as

$$p_t = p^* -(1/\beta)u_t$$ This happens because we have imposed REH. In other words the imposition of REH brings about the result that current equilibrium price remains "attracted" and "chained" to long-run equilibrium, fluctuating randomly but not explosively.

Also we have

$$p_t = p^e_t -(1/\beta)u_t$$

which also means than in unconditional expected-value terms

$$E(p_t) = E(p^e_t)$$

"On average" (intertemporally), the price expectation will equal the actual price.

In one move Muth obtained two extremely powerful results:
a) Markets do not explode
b) Market participants on average and "as a whole" predict correctly.

And really, if markets did tend to explode rather than not explode, they would not be around for thousands of years, as they are. And if market participants were predicting consistently poorly, we would have seen much more personal financial ruins than we do.

What REH does not do well, is in helping model and analyze short run and transitional dynamics. It remains a long-run concept, a "long-run view" if you will, and this is why Adaptive Learning emerged, and this is why we are currently researching (in a frenzy), other expectations formation hypotheses.

• Thanks for the very precise answer! Indeed Muth stressed that the model is in deviations, and following your explication, it is clear that what he meant is, imposing his rationality assumption (3.4) on eq. (3.3), and dismissing the case of γ/β=−1, we have deviation p_t^e=0, i.e. the expected price equals the long-run equilibrium price. This is not just an artifact of assuming equilibrium-centered demand and supply, as this only restricts expectation to move proportionally to what is reasonable prediction, which can still explode away from equilibrium, if everyone is dumb. Very interesting! – Xiaoeu Mar 18 '15 at 8:14