Introduction to Rational Expectations
The concept of a rational expectations was introduced by Muth (1961) who was dissatisfied
with the ad hoc modelling of expectations. He argued that when making predictions rational
agents make the best use of all available information, including the structure of the economic
model generating the observables. Thus in stochastic models, if agents are to avoid making
systematic mistakes, their subjective beliefs about the distribution of variables, both exogenous
and endogenous should be the same as the true conditional distributions implied by the model.
Any equilibrium solution to a model in which agents hold rational expectations could be
described as a rational expectations equilibrium. However in the literature on equilibrium asset
prices the term has come to be used mainly for models in which agents recognise the
informational content of the price at which they can trade.
In a competitive market if all traders have the same information about the likely outcome of
holding an asset then its price only affects an agent’s demands through his budget constraint.
Other agents’ demand schedules are relevant only in as far as they determine the market
clearing price. But where agents have differing information about the likely realisation then the
decisions of other agents may reveal or partially reveal, what their information is. If the actions
of other agents are not directly observable then a trader draws inferences about the signals
which the others have received from the price at which he can trade. Suppose agents agree
about the outcomes which are favourable and which are unfavourable. If the market clearing
price of the asset is high then this suggests that other agents have collectively had good news
and, if it is low, then on balance they must have received bad signals. For any market clearing
price a rational agent picks the quantity he demands to maximise his expected utility given his
beliefs about the possible outcomes where these beliefs are conditioned on the price at which he
can trade and on his own private information.
As a simple example consider the case where there are only two possible outcomes from
holding an asset, which is in fixed supply, and that agents have the same ordinal preferences.
Assume that some traders are informed about the outcome. Their demand schedules are a
function of price and of the realisation of the outcome, so that at any given price they will
demand more if the outcome is good than if it is bad and so it is plausible to conjecture that the
market clearing price will be either high or low. If the equilibrium price is high then the
uninformed can infer that this must be because the informed know the outcome of holding the
asset is good and so they too will demand the amount appropriate to the good outcome, whereas
if it is low, they recognise that the outcome is bad. In the REE the market clearing price fully
reveals the signal that the informed have observed and the equilibrium price will be the same as
it would be were all agents perfectly informed.
Rational Expectations with Differential information and Information Aggregation
In the simple case above there were only two possible outcomes and the informed know, the
actual outcome. Differential information relates to individuals observing different pieces of
information. Where there are many agents who have different information sets then it is not
immediately apparent that price can aggregate the individual signals. In a seminal article
Grossman (1976) introduced the information aggregation topic, and considers the case where the
realisations from holding an asset are normally distributed and agents receive signals which are
independent drawings from a normal distribution with a mean equal to the true outcome. All
agents have the same (constant absolute risk aversion) utility function and each agent acts as a
1
price taker, that is, he does not take into account, the effects of his own demands on the
equilibrium price. Grossman is able to solve for the rational expectations equilibrium and
shows that the price is a sufficient statistic for the set of signals observed by all the traders, in
other words any one agent, knowing the equilibrium price gains no extra information from
observing his own signal. This gives rise to the paradox that price can only incorporate
individual’s information if they take this into account in deciding how much to demand, but
knowing the equilibrium price they are rational to disregard their own information in favour of
the superior signal provided by price. Helwig (1990) dubs this phenomenon agents'
schizophrenia.
Grossman (1976) recognised the paradox that if prices are fully revealing no-one gathers
information. No agent will have any incentive to collect costly information since the
equilibrium price perfectly reflects all information collected and price can he observed for free.
However, if no agent has an individual incentive to collect information, price will not be
informative, in which case it would be in the interests of any one agent to purchase a signal. He
concludes that in the model with costs no rational expectations equilibrium exists.
Grossman recognised that it is necessary to add another source of uncertainty to establish
equilibtrium. Diamond & Verrechia (1981) allow agents to observe their own random
endowment. Helwig (1980) explicitly adds liquidity traders. [As do Grossman and Stiglitz (1980)
in an asymmetric information model]. Helwig (1980) allows for differing risk preferences but
shows that if the number of traders is large so that each individually has negligible effect on
price then the noise in any one agent’s signal is filtered out and the equilibrium price reflects
that information which is common to all agents, that is, the distribution of the price depends on
the mean of the signals received by the agents and on the demands of the noise traders. Admati
(1985) extends Helwig's model to the case of many risky assets. Non-zero covariance between
the returns makes it less easy to infer from prices what signals have been received about
individual assets. Helwig (1982) argues that the problem of no-equilibrium is because prices and
quantities are determined at the same time, and develops a dynamic model in which agents
condition on past prices to re-establish equilibrium.
Noise traders do two things they provide an exogenous motive for trade {otherwise with no gains
from trade we have a Milgrom and Stokey (1982) no-trade problem) and they ensure a non-fully
revealing equilibrium.
Notes on Grossman and Stiglitz (1980)
Efficient Markets Paradox
Grossman (1976) introduces the following paradox: Suppose markets are efficient, such that
prices reflect all available information, and suppose information is costly to acquire; then no
trader will collect information, since it is costly, and the trader can obtain information through
prices for free. But if no trader collects information, then prices cannot reflect all available
information. A paradox!
The Model
2
Grossman and Stiglitz (1980) resolve this paradox. There are two assets in the model: a
safe asset yielding a certain return R, and a risky asset yielding return u,
where u =  + , and E() = 0; var[] = 2. The random variable  is observable at a
cost c, but  is unobservable. There are two types of trader in the market: a proportion 
who are informed (pay a cost c to observe ), and a proportion (1-) who are
uninformed.
All individuals have a utility function which has constant absolute risk aversion (CARA)
Ui = - exp {-aW1i}
(2)
where W1i is the end of period wealth of trader i, and W0i is the trader’s initial wealth,
which is divided between risky and safe assets
W0i = Mi + pXi
(3)
At the end of the period the trader’s wealth will have accumulated to
W1i = Rmi + uXi
(4)
which will be random because the returns on the risky asset are random. Substituting
(4) and (3) into (2) we may write ecxpected utility as
EUi = - E[exp -a {R(W0i - pXi) + uXi}
(5)
The expectations of u of the informed will be conditioned on the information observed
E(u|) =  and var(u|) = var[] = 2. Then expected utility of the informed can be
written1
EUI = - exp {-a[R(W0I - pXI) + E(u|)XI - (a/2)XI2var(u|)]}
ie
(7)
= - exp {-a[RW0I + XI( - Rp) - (a/2)XI22]}
Now maximise expected utility with respect to XI
dEUI/dXI = - exp{.} * [-a( - Rp) + XI a2 2] = 0
So optimal demands of the risky asset are
XI 
  Rp
a 2
(8)
Now consider the demands by the uninformed. They do not observe  directly, but they
do observe equilibrium prices.2 The uninformed conditional distribution for the random
1
Recall if u ~ N(ubar, u2), then E[exp{ut}] = exp{ubar*t + (1/2)t2u2
3
variable u can be written as E[u|p*] and var[u|p*]. As before expected utility can be
written
EUU = - exp {-a[R(W0U - pXU) + E(u|p*)XU - (a/2)XU2var(u|p*)]}
(7’)
Then maximising expected utility and setting the derivative equal to zero yields the
optimal demands of the uninformed
XU 
E[u| p*]  Rp
a var[u| p*]
(8’)
Now consider the market equilibrium for the risky asset. Suppose the per capita supply
of the asset x is also random, where x ~ N(Ex, var(x)). The market clearing condition is
XI + (1- )XU = x
(9)
ie
  Rp *
E[ u| p*]  Rp *

 (1   )
x
2
a 
a var[ u| p*]
We need to solve for p* to obtain the market clearing price.
Conjecture that p* = 1 + 2w ie p* is a linear function of some variable w where
w =  - a2(x - Ex)/
(10a)
If p* is a linear function of w we may write E[u|p*] = E[u|w]and var[u|p*] =
var[u|w]. Then the market equilibrium condition could be solved for p* as

(1   ) E[ u| w ]
w 
 Ex
2
a 
a var[ u| w ]
p* 
 
(1   ) 
 2

 a  a var[ u| w ] 
(A10)
Now provided E[u|w] is a linear function of w and var[u|w] is independent of w
then our conjecture that p* is a linear function of w will be correct. But from the
properties of conditional distributions of normally distributed random variables, see
(A1) and (A2) in the Appendix to the paper
2
E[u|w] = E + 2(w - E)/var(w)
(A11a)
var[u|w] = 2 + 2 - 4/var(w)
(A11b)
var(w) = 2 + [a2/] 2 var(x)
(A11c)
And we assume trade only takes place at equilibrium prices in this single period model
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So it can be seen that E[u|w] is a linear function of w and var[u|w] is independent
of w so that our conjecture that p* is a linear function of w is correct. If we substitute
(A11a) - (A11c) and (10a) back into (A10) we obtain an explicit characterisation of the
rational expectations price function.
The rational expectations price depends (linearly) on the information  and the random
supply x. Importantly the uninformed when they observe price, are not able to infer the
value of , because of the noise in the supply. Note that if there was no random supply
(x=Ex) then from (10a) w=, and observing prices would be equivalent to observing 
- prices would be fully revealing. So that if information was costly no trader would pay
to observe , but then (A10) could not depend upon : the paradox! The resolution to
the paradox is to introduce some additional noise into the model (random supply) so
that the price is not fully revealing. When the uninformed observe a high price they
cannot be sure whether it is because the informed have observed a high  and have high
demands, or whether the supply
Informativeness of Prices
From (10) we may write
var[w|] = [a2/] 2var(x)
(11)
and how well informed uninformed traders can become from observing p* is measured
by var[w|]. When var[w|] is zero, w and  are perfectly correlated, and the price
system is perfectly informative. On the otherhand when var[w|] is very large there
are many realisations of w that are associated with a given , and the price system is
relatively uninformative. From (11) a high var(x) leads to an imprecise price system.
Similarly the term [a2/]will also determine the precision of the price system. When
a is small (not very risk averse) or 2 is small (so that the information  is very precise)
or  is large (lots of informed traders) then [a2/] will be small. In which case the
aggregate demand of informed traders is very responsive to realisations of . So the
movement in price is likely to be due to movements in aggregate demand of informed
traders via the realisation of . So the uninformed can infer that changes in price are
due to changes in , and the price system is precise.
The actual informativeness of the price system is measured by the correlation
coefficient between p* and . This correlation coefficient is defined as
p*, = Cov(p*, )/p*
and since p* = 1 + 2w then Cov(p*,) = 22, p* = 22var(w), so
p*,2 = 1/(1+m) where m = [a2/] 2[var(x)/ 2]
Information Quality
The quality of information is measured by the correlation coefficient between  and u.
u,2 = n/(1+n) where n = 2 / 2
5
Equilibrium in the Information Acquisition Market
We can now examine equilibrium in the market when  is an endogenous variable. The
realised wealth levels of the informed and uninformed are
W1I = R(W0I -c)+ XI(u - Rp*)
W1U = RW0U + XU(u - Rp*)
These wealth levels can be substituted back into the respective utility functions to
obtain the ratio of expected utilities as
EV (W1I )
var(u| )
 eac
  ( )
EV (W1I )
var(u| w )
(14)
For equilibrium in the information acquisition market we require that the expected
utilities of the informed and uninformed are equal ie (*) = 1. When  is low we know
that the price system is relatively uninformative, so there is an incentive to become
informed, when  is high the market price is very informative so tyere is no incentive to
be informed. Then * will be the equilibrium proportion of the market who will choose
to purchase information.
EV(WI)/EV(W
U)
()
1
*

() is a strictly monotonically increasing function of . Since we have defined
expected utility as negative, when  is small then the expected utility of the informed is
just a little bit negative. Whereas the expected utility of the uninformed is a lot negative
so the ratio EV(WI)/EV(WU) is small in absolute terms. As  increases the expected
utility of the uninformed gets progressively less negative and hence smaller.
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Comparative Statics Properties
1) An increase in the quality of information n increases the informativeness of the price
system
2) A decrease in c increases the informativeness of the price system
3) A decrease in a increases the informativeness of the price system
Conclusions
The Grossman-Stiglitz paper is truly seminal. It characterises a rational expectations
equilibrium, and demonstrates that the equilibrium price is not fully revealing - markets
cannot be strong form efficient, since this would destroy the incentive to collect
information. Price is made not fully revealing because of the addition of some “noise”.
In G/S this is through a random supply, but the insight is that there has to be some
source of noise. They also determine the equilibrium number of traders who will choose
to become informed. They also examine the quality of information, and the
informativeness of prices.
The G/S model is a model of asymmetric information rather than differential
information. In G/S there is only a single piece of information , as opposed to a traders
receiving individual signals, for example trader i might observe si =  + I. In which
case there would be a concern as to whether price aggregates all of the information in
the individual signals. An advantage of the G/S set-up is that asymmetric models do not
have to worry about Helwig’s (1980) 'schizophrenia' problem, that a trader who
observes an individually specific piece of information might trade strategically. If
traders are all observing the same signal, then with competitive traders, each trader has
to act immediately since otherwise the other traders will act to make the signal
redundant.
Notes on Continuous Auctions and Insider Trading by Kyle (1985)
Risk neutral insider maximises expected profits, where profits are
 = (v - p) x
An insider observes v the terminal value of an asset, v~N(p0, S0) and along with noise
trades u~N(0, su2), submits demands x. The market maker observes the total order flow y
= x + u, and sets a price p(y).
Equilibrium is defined by two conditions:
1)
Insider max expected profits
2)
Market price is efficient, i.e. p = E[v|y]
Market maker competition is not modelled but Bertrand competition is assumed to take
place between market makers when they announce the price at which they are willing to
trade.
Suppose
p(y) =  + y
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x(v) =  + v
Given this linear price function, the insider maximises
E[  | v] = E([v -  - (x + u)]x | v)
Differentiating with respect to the insider's demands
dE
= v -  - 2x = 0
dx
i.e.
x(v) = -

1
+
v
2
2
so that  = -/2 and  = 1/2.
The market efficiency condition can be written
p =  + y = E[v | y =  + v + u]
which can be thought of as a regression equation
v =  + y + error
 = Cov(y,v)/var y
 = E(v) -  E(y)
 = p0 - ( + p0)
where
and
so
 =
 0
 - p0 = - (  +  p0 )
 0 +  2u
2
We have four equations and four unknowns; solving yields
 =
 2u
0
8
 =
1 0
2  2u
2
 = - p0  u
0
 = p0
1/λ is a measure of the depth of the market, i.e. the order flow necessary to induce prices
to rise or fall by one dollar. We can now identify the equilibrium trade for an insider
x = (v - p0 )
 2u
0
and expected profits conditional upon v are
(v - p0 )2  2u
E[  | v] =
2
0
and the unconditional profits are
E =
1
2
0  u
2
Market liquidity is "a slippery and elusive concept, in part because it encompasses a
number of transactional properties of markets"
Tightness is the cost of turning around a position in a short period of time
Depth is the size of an order flow innovation required to change prices by a given amount.
Resiliency is the speed at which prices recover from a random shock.
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Rational Expectations Equilibria with hedging demands
There are two investors in the model: a speculator and a hedger indexed by i. Both investors have
CAR utility functions which depends on terminal wealth U= -exp{-bW1i}, where terminal wealth is
generated by
W1i = zi + mi(v - p)
In the case of the speculator zi is zero, but for the hedger zi is interpreted as a random endowment.
There is a risk-free asset which pays a zero rate of return and the risky asset pays a random dividend
v. Each agent will choose a porfolio mi to maximise
EV = E(W1) - (b/2)Var(W1)
The speculator observes a signal s, about the dividend v, where v=s+
In equilibrium price will be set by a competitive market maker equal to the expected value of the
dividend, given the total order flow X, and we conjecture that E(v|X) = s + z
Given the conjecture then in equilibrium the uninformed hedger will have the same information as
the speculator, so the optimal portfolio for each type of investor is
mi = {E(v|s) - p - bCov(,z)}/ b Var(v|s)
The only difference in the portfolio for the two types of investors is that the hedger has a covariance
term, because the hedger wishes to offset his endowment shock.
The aggregate order flow X=m1 + m2, so
X = {2s(1-)-2z-bCov(,z)}/ bVar()
then the price is given as
p = E(v|X) = {Cov(v,X)/Var(X)}. X
The covariance of X and v, is given by
Cov(s+,X) = 2{(1-)Var(s) - Cov(z.)}/ b Var()
and the variance of X is
Var(X) = {4(1-)2Var(s) + 42Var(z)} / {bVar()}2
Line up coefficients and
 = 1 and  = Cov(z,) / Var(z)
so the price function is p = s + {Cov(z,) / Var(z)}. z
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Review of Economic Studies, Vol. 67, no. 1, 79-90.
Dow, J. and R. Rahi (2003) “Informed Trading, Investment, and Welfare”, Journal of
Business, 76 (2003), 439-454.
Fama, E.F. (1991) "Efficient Capital Markets", Journal of Finance, vol. 46, 1575-1617.
Grossman S.J. (1976) "On the Efficiency of Competitive Stock Markets where Trades
have Diverse Information" Journal of Finance.
Grossman S.J. and J.E. Stiglitz (1980) "On the Impossibility of Informationally Efficient
Markets" American Economic Review.
Helwig M. (1980) "On the Aggregation of Information in Competitive Markets" Journal
of Economic Theory.
Kyle, A. (1989) "Informed Speculation with Imperfect Competition", Review of
Economic Studies, vol. 56.
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Economy" Econometrica.
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