An annotated reading of rational expectations theory - Part 2

Whereby, having previously guessed a general model from the assertions of Muth's original paper, I now proceed to look at the paper itself.

To summarise, my own take on those assertions would be that they imply:

1. Each agent $i$ has his own prior belief $p_i(\omega)$ about the state of the world.
2. This prior belief is generated independently of those of other agents from some distribution $\Psi$.
3. This belief is conditioned on some information $x$, which is common to all; so each agent has a posterior belief $p_i(\omega \mid x)$.
4. The agents are interested in predicting some quantity $y$, which depends deterministically on the state of the world. Let us call this $f(\omega)$. 
5. Given the posterior $p_i(\omega \mid x)$, each agent can calculate a corresponding posterior distribution for the quantity itself. This can be done by firstly sampling $\omega$ from the posterior and then calculating $f(\omega)$. 
6. A more limited prediction is to just calculate the expectation of $f$ itself. For simplicity, let's call $f_i(x) = \E_{p_i} (f \mid x)$ the expected value of $f$ for the $i$-th agent, given information $x$.
7. There is some "theory" $\theta$ which predicts a particular value for $f$, let us call this $\theta(f, x)$
8. The prior distribution $\Psi$ is such that, for any information $x$, we have that $\theta(f, x) = \E_{\Psi} \E_{p_i} (f \mid x)$.

Section 3 of the paper discusses the problem of price fluctuations. Here we are specifically talking about some kind of time series, although that is not strictly speaking following from the discussion previously.

Here I immediately run into a problem. The paper presents a set of demand supply equations, where it assumes that the demand $D_t$ is equal to the supply $S_t$, i.e. $D_t = S_t$. There is also a market price $P_t$ for a single good, as well as a fluctuation term $U_t$. This is all fine. However, the supply is defined as
\[
S_t = \gamma P_t^e + U_t,
\]
where $P_t^e$ is defined to be "the market price expected to prevail during the t-th period on the basis of information available through the (t-1)'st period". I can't help but feel that this is quite imprecise, so let us turn this back into the basic framework I outlined above.

At time $t$, the state of the world is $\omega_t$. Agents are allowed to make inferences about the complete state of the world $\omega = (\omega_1, \ldots, \omega_t, \omega_{t+1}, \ldots)$.
I assume that the model that we have (again here I am not sure if that describes the actual system or our model of the system) is with respect to the aggregate expectation, so that if $P_t = f(\omega_t)$ then the first interpretation is this.

Interpretation 1. The "price expected to prevail" is the aggregate population expectation
\[
P_t^e = \E_{\Psi} \E_{p_i} (f \mid x).
\]
where $x = (S_1,ldots, S_{t-1}, D_1, \ldots, D_{t-1}, P_1, \ldots, P_{t-1})$ I am not quite sure that's what is intended, though. It is equally likely that the equations represent a "true" model of some sort.

Interpretation 2. The "price expected to prevail" is that of the "true" model
Here there is some other true model which places probability $M$ upon world states, and then
\[
P_t^e = \E_{M} (f \mid x).
\]

Not clear at all what is meant.