Following up on this post by Noah, I revisited the rational expectations literature. This is an attempt to do some annotated reading.
Muth's paper from 1961 opens by saying that "the expectations of firms ... tend to be distributed, for the same information set, about the prediction of the theory."
How can we formalise this more precisely? Adopting a point of view where each economic agent $i$ has some prior belief $p_i$ over the set of possible states of the world $\Omega$, we need to define what we mean by expectations. Let $F$ be a function space of functions $f : \Omega \to Y$. For example, one particular $f$ could be the dollar price of gold per ounce for different world states $\omega$., To take expectations, we need to define a prior belief $p$ over the possible states of the world, with a posterior belief $p(\omega \mid x)$ given some information $x$. Combining those, the expected value of some $f \in F$, given information $x$ is:
\[
\E_p(f \mid x) = \int_\Omega f(\omega) d p(\omega \mid x).
\]
The assertion that economic agents' expectations are distributed about the prediction of the theory, can also be quantified. First we need to identify what we mean by prediction of the theory.
Definition 1. A theory $\theta$ is a function $\theta : F \times X \to Y$ where $F$ is the set of economic quantities we wish to measure, $X$ is the set of information states and $Y$ is the set of possible predictions.
Assumption 1. (a) Each economic agent $i$ has a prior belief $p_i$ drawn independently from a prior distribution $\Psi$. (b) For any information state $x \in X$, the corresponding set of posterior distributions $p_i(\omega \mid x)$ satisfies
\[
\E_{\Psi} \E_p(f \mid x) = \theta(f, x).
\]
This is a quite strong assumption. Note that the outer expectation is with respect to the population distribution of subjective beliefs while the inner one is with respect to the actual beliefs of each agent.
When the number of agents is large, then
\[
\frac{1}{N} \sum_{i=1}^N \E_{p_i}(f \mid x) \approx \theta(f, x).
\]
I am not sure yet if that's what the paper is really about (I just scanned it once and now going through it) but I am tempted to make guesses. The actual paper never explicitly talks about the above, but that's what I am guessing it assumes behind the scenes. The remaining sections seem to be about a specific market problem, and then about very specific expectations/belief models for the agents.
(to be continued)
Muth's paper from 1961 opens by saying that "the expectations of firms ... tend to be distributed, for the same information set, about the prediction of the theory."
How can we formalise this more precisely? Adopting a point of view where each economic agent $i$ has some prior belief $p_i$ over the set of possible states of the world $\Omega$, we need to define what we mean by expectations. Let $F$ be a function space of functions $f : \Omega \to Y$. For example, one particular $f$ could be the dollar price of gold per ounce for different world states $\omega$., To take expectations, we need to define a prior belief $p$ over the possible states of the world, with a posterior belief $p(\omega \mid x)$ given some information $x$. Combining those, the expected value of some $f \in F$, given information $x$ is:
\[
\E_p(f \mid x) = \int_\Omega f(\omega) d p(\omega \mid x).
\]
The assertion that economic agents' expectations are distributed about the prediction of the theory, can also be quantified. First we need to identify what we mean by prediction of the theory.
Definition 1. A theory $\theta$ is a function $\theta : F \times X \to Y$ where $F$ is the set of economic quantities we wish to measure, $X$ is the set of information states and $Y$ is the set of possible predictions.
Assumption 1. (a) Each economic agent $i$ has a prior belief $p_i$ drawn independently from a prior distribution $\Psi$. (b) For any information state $x \in X$, the corresponding set of posterior distributions $p_i(\omega \mid x)$ satisfies
\[
\E_{\Psi} \E_p(f \mid x) = \theta(f, x).
\]
This is a quite strong assumption. Note that the outer expectation is with respect to the population distribution of subjective beliefs while the inner one is with respect to the actual beliefs of each agent.
When the number of agents is large, then
\[
\frac{1}{N} \sum_{i=1}^N \E_{p_i}(f \mid x) \approx \theta(f, x).
\]
I am not sure yet if that's what the paper is really about (I just scanned it once and now going through it) but I am tempted to make guesses. The actual paper never explicitly talks about the above, but that's what I am guessing it assumes behind the scenes. The remaining sections seem to be about a specific market problem, and then about very specific expectations/belief models for the agents.
Assumptions in section 2 of the paper
The paper concludes section 2 by making assumptions about:
1. normal random disturbances. I have no idea what those disturbances are supposed to be. Of what?
2. Certainly equivalence. This probably means that the optimal investment choice depends only on the posterior expectation of $f$ and not on the complete posterior belief.
3. The system equations are linear. I have no idea how to interpret this.
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