An annotated reading of rational expectations theory - Part 3

A final comment on Section 3 of Muth's paper

As discussed, this paper now introduces a simple market model (theory?) which is
\begin{align}
D_t &= - \beta P_t\\
S_t &= \gamma P_t^e + U_t\\
D_t &= S_t
\end{align}
where $D_t$ is the demand and $S_t$ is the supply, while $P_t^e$ is the expected market price at time $t$, conditioned on all previous information. Here all $P_t$ is a deviation from the equilibrium, so they are better thought as the amount expected to gain by producing and selling a unit.

The main ambiguity here is  whose expectation that is. It most natural to assume that this is the aggregate expectation of the population, i.e. that the amount produced during the $t$-th period is basically proportional to the price that people expect to get. It's better to make this into a formal assumption about the theory.

Assumption 3. The theory predicts that total production is linear with market expectations, i.e.
\[
P_t^e = \E_\Phi \E_p (P_t \mid x_t),
\]
where $x_t = (P_1, D_1, S_1, \ldots, P_{t-1}, D_{t-1}, S_{t-1})$ is the current information state, which may include some other side-information.

This assumption essentially tells us that the amount of supply is proportional to how much people expect to gain (since the quantities are deviations from the equilibrium) for the goods they produce. At first glance, this appears reasonable, but what decision model does this imply for the producers?

The $U_t$ variable can be taken to be simply zero mean noise in this context, and part of our model $M$. Let us now just equate everything, and write the quantities $U_t, P_t$ in terms of the state $\omega_t$ to obtain:
\begin{equation}
%\gamma P_t^e + U(\omega_t) = - \beta P(\omega_t),
 P(\omega_t) = - \frac{\gamma}{\beta} P_t^e - \frac{1}{\beta} U(\omega_t),
\end{equation}
noting that $P_t^e$ only directly depends on the history of observations and the priors of the population, and not any specific $\omega$.

What is the expected price according to this model? We can take expectations with respect to the model's distribution $M$, to obtain
\begin{align}
\E_M (P_t \mid x_t)
&= \int_{\Omega_t}  P(\omega_t) dM(\omega_t \mid x_t) \\
&=  - \frac{\gamma}{\beta} P_t^e  - \frac{1}{\beta} \int_{\Omega_t}  U(\omega_t) dM(\omega_t \mid x_t) \\
&=  - \frac{\gamma}{\beta} P_t^e  - \frac{1}{\beta}\E_M (U_t \mid x_t)
\end{align}
where $M(\omega_t \mid x_t)$ is the conditional distribution of the model given the information.

Uncorrelated deviations. The paper assumes that we can have $\E_M (U_t \mid x_t) = 0$ for any $x_t$ according to the model $M$. I am not sure how to actually interpret that. This seems to be the case if the actual supply by each provider is proportional to the market price it expects, plus some zero-mean noise due to externalities. This is not necessarily true, but the paper later relaxes this assumption.

Rationality assumption. This boils down to simply
\[
\E_M P_t = P_t^e,
\]
i.e. that the aggregate market prediction agrees with the model's expectation, which actually emplies that the expected price must be the equilibrium price, i.e. $P_t^e = 0$, recalling that quantities are equilibrium deviations. It is unclear whether this is the marginal expectation, or expectations conditioned on something. If the latter, then what?

Finally, can this assumption be relaxed somehow? Though we don't know what it is... precisely. 

Given all of the above assumptions, we can rewrite the  aggregate expectated prediction as
\[
P_t^e = - \frac{1}{\beta + \gamma} \E_M (U_t \mid x_t).
\]
However, I am not sure what this tells us. If we already know $M$, we also must know the beliefs of the agents, and so we already know $P_t^e$. So what is the point?