Common knowledge in model formulation and solution

Question

Economics models usually assume that the structure of the economy is common knowledge among agents.

Mathematically, an event is common knowledge if it lies in the meet of all agents' information sets. (The meet of a family of $\sigma$-algebras is the finest common coarsification of all $\sigma$-algebras in the family. See Aumann 1976.)

However, in the literature when a paper makes the common knowledge assumption, one almost never sees the meet of any information sets anywhere. Let me give an example (Grossman and Stiglitz 1980).

Grossman-Stiglitz model

This is a model where two traders with (essentially) mean-variance utility are asymmetrically informed about the mean. By assumption, structure of the economy is common knowledge and traders are rational. Equilibrium price must therefore, first, clear the market and, second, be consistent with the uninformed trader's expectation. Details given as follows.

Let $I$ and $U$ denote the informed and uninformed traders respectively. The traders have same CARA utility, $u_I(W) = u_U(W) = - e^{-\gamma W}$. The risk-free rate is $r$. The price of risky asset tomorrow is $P_2 \sim \mathcal{N}(\bar{P}, \sigma^2) | \bar{P}$. The informed trader $I$ knows $\bar{P}$; the uninformed trader $U$ only knows the prior distribution $\bar{P} \sim \mathcal{N}(P_0, \sigma_0^2)$.

Let both trader have wealth endowments $E$. (CARA utility has no wealth effect, so $E$ plays no role in portfolio choice below.)

Given today's price $P_1$ for the risky asset, trader's wealth tomorrow is $W(\phi) = rW + \phi (P_2 - rP_1)$ if he choose to hold $\phi$ units of risky asset.

Maximizing expected utility, a traders' demand is

$$ \frac{E[P_2|\mathcal{F}] - rP_1}{\gamma Var(P_2|\mathcal{F})} $$

conditional on his information set $\mathcal{F}$. For $I$, $\mathcal{F}_I = \{ \bar{P}, P_1\} $. For $U$, $\mathcal{F}_U = \{ P_1 \}$.

The supply $x$ of risky asset is distributed $\mathcal{N}(\bar{x}, \sigma_x^2)$.

Equilibrium is a price function $P_1(\bar{P}, x)$ such that

$$ \frac{\bar{P} - r P_1(\bar{P}, x)}{\gamma \sigma^2}+ \frac{E[P_2|P_1(\bar{P}, x)] - r P_1(\bar{P}, x)}{\gamma Var(P_2|P_1(\bar{P}, x))} = x. $$

In other words, market clears and the uninformed trader computes his demand using the correct pricing function in equilibrium.

Now in this CARA-normal setting, things are linear and one can solve for equilibrium by guessing a pricing function

$$ P_1(\bar{P}, x) = A\bar{P} + Bx + C $$

and find $A$, $B$ and $C$ by matching coefficients. For example, compute

$$ E[P_2|P_1] = P_0 + \frac{A \sigma_0^2}{A^2 \sigma_0^2 + B^2 \sigma_x^2} [A(\bar{P} - P_0) + Bx] $$

and

$$ Var(P_2|P_1) = \frac{B^2 \sigma_x^2}{A^2 \sigma_0^2 + B^2 \sigma_x^2}\sigma_0^2 + \sigma^2 $$

Substituting into the market clearing equation and some tedious algebra gives endogenous constants $A$, $B$, and $C$.

Instead, Grossman and Stiglitz do something more elegant. They point out that uninformed trader $U$ can condition on the residual demand for any realization of $(\bar{P},x)$

$$ D_{resid} = x - \frac{\bar{P} - r P_1 }{\gamma \sigma^2}. $$

What's even more clever, they note that, since $P_1$ is observed in in equilibrium, they first conjecture that $P_1$ and

$$ \tilde{D}_{resid} = x - \frac{\bar{P} }{\gamma \sigma^2}. $$

are informationally equivalent. Then equilibrium condition becomes

$$ \frac{ r P_1(\bar{P}, x)}{\gamma \sigma^2}+ \frac{E[P_2|\tilde{D}_{resid}] - r P_1(\bar{P}, x)}{\gamma Var(P_2|\tilde{D}_{resid})} = \tilde{D}_{resid}. $$

Now things are much less tedious and there are no endogenous constants $A$ and $B$ to solve for. $E[P_2|\tilde{D}_{resid}]$ is an affine function of $\tilde{D}_{resid}$ and $Var(P_2|\tilde{D}_{resid})$ is an exogenous constant. So that $P_1(\bar{P}, x)$ can be backed out immediately. Since $P_1$ is an affine function of $\tilde{D}_{resid}$, it is verified ex-post that they are informationally equivalent.

Question

To have the uninformed trader $U$ condition on the residual demand, the common knowledge assumption is used. Or, at least, $U$ knows that $I$ knows so that $U$ can put herself in $I$'s shoes and compute $I$'s demand.

However, the meet of the information sets does not appear anywhere---as it should. Result of $U$'s calculations should be measurable with respect to the meet. If one is to formulate this mathematically, where would that appear?

Answer 1

Two points.

1. Common knowledge is defined by Aumann in terms of partitions, not $\sigma$-algebras. There is generally no natural correspondence between these.

2. The grand state space is trivially common knowledge. So however you conceive of the relevant states, something that holds everywhere, such as the structure of the model, is common knowledge even if agents have only the trivial knowledge partition with one element. This is actually useful even when modeling more explicitly epistemic phenomena. For example, one way to guarantee common knowledge of rationality is to assume agents are rational at each state of the world. This is what Aumann did in his 1987 paper on correlated equilibria.