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The Journal of Finance
excess return on the other asset. This occurs even though there is no market-
wide risk: There will be three units of the good tomorrow for sure. The trader
with better beliefs holds a better portfolio; he overweights his portfolio in
the asset with a positive excess return.
This simple example shows that if traders have differing beliefs they per-
ceive differing risk-return trade-offs and they may choose to hold idiosyn-
cratic risk. Each trader believes that he is being compensated for the risk he
holds. In fact, at least one of these traders has incorrect beliefs. So his per-
ceptions about the risk-return trade-off are also incorrect. Next, we intro-
duce private information into the economy so that neither trader has incorrect
4
beliefs—they have a common prior and more or less information.
1
_
1
_
The common prior on states is ~ , !. Trader one, the informed trader,
receives a private signal y ʦ $1,2% with probability on each signal. If y ϭ 1,
then the conditional probability of state one is , and if y ϭ 2, the conditional
probability of state one is . Trader two is uninformed, but he knows this
2
2
1
2
_
3
4
_
1
_
4
structure and he uses this knowledge, along with equilibrium prices, to make
any inferences that he can about the informed trader’s information. Unless
we introduce further randomness into the economy, prices will reveal the
informed trader’s signal and, in equilibrium, traders will have common be-
liefs. To prevent this uninteresting case, we use the standard device of noisy
5
supply. The aggregate supply of assets one and two is given by the random
1
_
variable x ʦ $~305,1!,~1,305!% with probability on each supply vector. This
2
random supply is equally divided across the traders to form their initial
6
endowments of assets. We assume that x and y are uncorrelated. So
there are four states of the world at time t, z ʦ Z ϭ $z , z , z , z % ϭ
1
2
3
4
$
~1,~305,1!!,~1,~1,305!!,~2,~305,1!!,~2,~1,305!!%, each of which is equally likely.
Calculation shows that rational expectations equilibrium prices and shadow
risk-free rates are as shown in Table I. Equilibrium prices in states z and
2
z3 are equal so the date t equilibrium is nonrevealing in these date t states.
Prices in each of states z and z differ from all others, so if the date t state
1
4
is one of these, the equilibrium is revealing. Thus, equilibrium beliefs of
3
4
_
1
4
_
_ _
1 1
the uninformed trader are ~ , ! in state z , ~ , ! in states z and z , and
1
2
2
2
3
4
We do not believe that traders’ differing beliefs necessarily come from this type of common
prior structure. Disagreement about probabilities seems far more natural than does a common
prior. When traders disagree, market prices provide information about others beliefs, but with-
out some further structure it is not clear how or if traders should use this to change their own
beliefs. We use the standard common beliefs and information structure to analyze the effects of
private versus public information. The analysis can be done without common priors.
5
An alternative that works equally well is to introduce noisy traders. In our analysis, this is
easily done by having some traders whose beliefs are random and who do not learn from prices.
6
We assume that traders do not make an inference about the state of the world from their
endowment. Alternatively, we could assume that the uninformed trader has a constant endow-
ment and that only the informed trader’s endowment varies. We do not do this only because it
complicates the calculations. Another standard alternative is to allow a random exogenous sup-
ply of the assets. We do not do this only because then we would have a partial equilibrium
model, which is more difficult to compare to the usual consumption-based asset-pricing structure.