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Correlated Equilibria
A Note
and Sunspots:
by
Eric Maskin*
and
Jean Tirole**
W.P.
#398
May 1985
Correlated Equilibria and Sunspots
A Note
by
Eric Maskin
and
Jean Tirole
May 1985
University
Massachusetts Institute of Technology
j.j.Harvard
We are grateful to Robert Aumann and Karl Shell for helpful
comnientB.
We thank the NSF and the Sloan Foundation for research
support
1
.
Introduct ion
this note we study a simple finite horizon economy in
In
which extrinsic uncertainty (e.g., sunspots) natters if and only
if it
is
differentially observed by the agents.
In our model
agents have asymmetric information and thus do not share the same
beliefs (theory) about future prices.
5,
the model,
As we discuss
despite its apparent differences,
is
in section
closely
analogous to recent treatments of symmetric information sunspot
equilibria in infinite horizon, overlapping generations models
(c.f.
Shell
[1977],
Azariadis
[1981],
and Azariadis-Guesnerie
[1983].
We provide two examples based on a two-period,
model developed in section
2.
In
the first period,
two-class
agents
observe imperfectly correlated signals and choose production
(investment)
levels before learning market prices.
They trade
their produced good for others' goods in the second period.
the first example (section 3)
signal.
In
only one class of agents observes
There exist equilibria depending on this signal
("sunspot" equilibria)
that differ from the unique certainty
equilibrium, the equilibrium that arises if agents ignore their
signals.
In
the second example (section 4) both types receive
a
iaperfectly correlated signals.
In
this case,
characterize the set of sunspot equilibria.
there is
a
we coapletely
Section
shows that
5
close connection between our finite-horizon sunspot
equilibria and those treated in the overlapping generations
literature.
The Model
2.
There are two equal-sized classes of consumers.
consumer
is
endowed with
is
2
1.
consumers produce good
1
Similarly,
1.
The marginal product of labor
2.
Consumers of both types consume leisure and the good
produced by the other type.
themselves produce.
where
Type
finite amount of leisure.
a
consumers can sacrifice leisure to produce good
type
Each
1.
is
1
They do not consume the good they
Thus consumer i's utility is U (l.,c.)i
his labor and c.
is
his consumption of good
j.
J
Utility functions are concave and twice continuously
dif f erentiable.
Consumers first decide how much labor to allocate to
production,
there is trade in the produced goods.
and then
Trade is competitive.
i,
the price of good
If x.
the per capita quantity of good
(with good
2
*^
is
-I
/
'^
o
•
1
the numeraire)
is
X
1
.
1
Type
=
.
c.
,
i =
l,2,
1
1
consumers receive
i
and consumption are all equal:
labor,
output,
Id equilibrium,
signal
a
some finite set
in
s.
S.
before deciding how much to produce.
The signals s,
or may not be correlated.
denote the labor supply as
function of the signal received.
a
1 ,( B
1
l.(s.)
Let
.
=1
1
,)/!„( Sp
11
.
(
B
.
)
=
X,
(
s
,
)
/x„ (s„
)
may
consumer chooses
i
where
)
x.(s
(1)
s^,
The equilibrium price will be
A type
.
and
l.(s
=
)
arg max
^
E
B
.)
fu\l.,l.
J.J.
1
1 X.(B.)
Is.].
)
1
1
In equilibrium,
11
11
1.(b.) must equal x.(b.).
Thus,
the first order
condition for the consumer's maximization can be expressed as
X;(B
(2)
E
[uhx. (s.),x
1"^'
B. 1^ 1
+ U^(x. (s. ),x .(b .))-^J
.(s .))'
j'
J
2
1
1
J
•
X^(B^)
JX.
J
=
where
U,
J
J
'
denotes the partial derivative of U
0,
with respect to its
kth argument.
Definition
1
:
A
correlated equilibrium associated with signals
{s,,s„}
is
satisfying
a
pair of functions {x,
(2)
(
s,
)
,
x^ b^)
}
Note that uncertainty is extrinsic because the signals do not
directly affect preferences,
technologies or market organization.
Indeed, we can consider equilibria in which consumers simply
ignore their signals.
Definition
2
:
certainty equilibrium
A
equilibrium
{
x,
(
s
,
)
,
is
x^ ( s^
correlated
a
in which,
) }
for i=l,2,
x.(b.) does not depend on s..
11
1
Because signals play no role in
a
certainty equilibrium, we
t
t
The simplest
can express such an equilirium as a pair {x,,X2}.
kind of nondegenerate correlated equilibrium is one which simply
randomizes over certainty equilibria:
Definition
3
:
perfectly correlated equilibrium
A
{x, (s
,
exists
X
)
,
x- ( s„
(s
)
.
=x
in which,
for all
a
pair
(b,,E2),
there
certainty equilibrium {x,,x„} such that
a
Ill
.
) }
is
.
for i=l
,
2.
Although signals can affect the allocation of resources in
a
perfectly correlated equilibrium, such an equilibrium is
observationally indistinguishable from one where there are no
signals at all.
We shall say that signals matter in a correlated
equilibrium if and only if it is not perfectly correlated.
equilibrium is then said to be imperfect ly correlated
The
.
We now derive necessary conditions for the existence of an
imperfectly correlated equilibrium.
to the presence of Giffen goods.
Existence is intimately tied
Consumption is
a
Giffen good if
it
increases with the price of the good.
leisure is
of terminology,
by
And,
a
slight abuse
Giffen good if the amount of labor
a
supplied decreases with the price of the good it produces.
easily checked that, when consumption is
It
is
Giffen good, so is
a
leisure.
Propos it ion
:
Consider an imperfectly correlated equilibrium.
Leisure must be
Giffen good for at least one
a
Furthermore,
class of consumers.
s,
and
Br,
are independent,
if the signals
consumption must be
a
Giffen good for at least one class of consumers.
Proof
:
For simplicity,
assume that there is
that can be observed by each class:
s^e {b„
,
.
.
.
,
B^}
for class
a
finite number of signals
1
s, e {s,
ffl,
.
.
.
,
s,
}
for class
1;
Rank the signals so that in the
2.
imperfectly correlated equilibrium:
X,
i
^2
^
X
i
m
and
i
x^
1
<:
X
2
•
with at least one strict inequality, where x. is the production
level of class
j
when it observes signal
generality one can assume that x,/x^
i
s .=s
x,/x_.
..
Without loss of
In this case.
1
'^1
1
''l
..
D
^
i
X2
i
..,.
i
1
B
D
"l
'^l
"l
,
^2
"2
^
••
^
n
.
their signal is
1
..
•
Because,
signal
E^
''l
^
-I
•
X2
the distribution of
Thus,
numeraire) that type
1
consumers face when
whatever the signal received by class
faces better terms of trade if its own signal is
E,.
.
fully dominates that when the signal
s,
This implies that,
1
as
i
^^
X2
^^2
with at least one strict inequality.
prices (with good
1
D
.
by assumption,
than for s,,
(2)
1
s,
s,.
class
2,
rather than
consumers produce more for
leisure must be
Assume next that the signals
the expectation in
type
s,
is
a
Giffen good.
and s^ are not correlated.
Then
does not depend on the own signal s..
The
equation
(3)
E
lij
[U^(x.,x.)
X.
+
21JJ1-'
ui(x. ,x .)x ./x.
]
=
must have at least two distinct solutions
x.
for some class
i.
1
The ratio x./x. is the price of i's consumption (produced by j)
in terms of labor.
class
i,
Thus,
if consumption is not a Giffen good for
we have
^1 " ^'ii^ ^ ^iz'^j
<
'
But this last inequality implies that the bracketed expression in
(3)
is
BODotonic Id
x.
for any x..
Thus,
Q K D
solutions
3
.
Example
x^--l^
cannot have two
(3)
.
1
2
X^
1
Xj
Figure
1
.
8
Consumption and, therefore, leisure are Giffen goods for
There exists a unique certainty
(but not for class 2).
class
1
(and,
hence unique perfectly correlated) equilibrium
To
A.
construct an imperfectly correlated equilibrium, choose x„ such
12
that there are several values of x,
that are optimal for class
when the terms of trade are x,/x„.
Let x,
112
Assume that the signal
values.
and define x,
x,(8,) and
=
observed by class
certainty (i.e.,
1
Xp(s,) or z„
it
=
only;
1
x,
=
class
2
2
x„(s,).
observes no signal.
)
,
class
But because class
2
2
produces some quantity between y^ and z„.
can find a number « in
(0,1)
1
2
b,
is
{s,,b,},
We suppose that
x,(8,).
if it observed s,
denote two such
x,
takes on the values
s,
2
and
Under
would produce y„
By continuity one
1
2
2
offers Xp (i.e.,
the
equilibrium conditions are satisfied).
4
.
Example
2
Consider the following symmetric offer curves depicted in
Figure
2.
=
does not observe s,,
such that when (b,,s,) have
respective probab i lit ieB (es,l-a), class
1
offer curvi
1
x^=l^
'
8
^1
Figure
2
There are three certainty equilibria, B,C,D.
(x,,^^)-
a)
offer cu rve
Choose
h
X,
{x,
k
<
X,
}
Let B
such that
for k
>
h
and
1.
Ab
illustrated in Figure 2, k can assume only three
3.
More generally, however, k aight take on any
values, 1,2, and
number of values.
10
(x,,x„) belongs to I's offer curve.
b)
Choose {^2} analogously
Assume that:
receives
signal
i)
each class
ii)
these signals are not perfectly correlated;
i
the probability of
iii)
s
.
is
a
s.
e
{s.}.
in
particular,
positive whatever the signal received by
the agents are infinitely risk averse.
k
k
The offers {x.(s.)} form an inperfectly correlated
Because of the extreme risk aversion, each class
equilibrium.
behaves as if the other class
X..
J
always offered the lowest value
Are there other imperfectly correlated equilibria?
that there are not,
{Xr,}}.
consider
a
i
set of equilibrium offers
Because of infinite risk aversion,
To see
{{x.},
(x,,Xp) must form a
certainty equilibrium if s. has positive conditional probability
given any
s
.
But C and
.
D
cannot be candidates for this
equilibrium.
In particular,
Giffen good.
Thus,
2,
there exists
a
at C,
leisure is locally not
a
given the corresponding labor supply by class
unique point on the offer curve for class
Similarly, we can rule out
B.
Hence,
all the imperfectly
correlated equilibria are as described above.
1.
11
5
.
D
iscuBS ion
This note emphasizes the role of imperfect ly correlated
signals in generating equilibria that could not occur without
extrinsic uncertainty.
nature are important.
were independent,
Both the correlation and its imperfect
On the one hand,
if all
only certainty equilibria would be possible (as
long as agents were small relative to the economy).
hand,
if,
signals
agents'
in our model,
On the other
all agents observed the same signal,
all
equilibria would be randomizations among certainty equilibria.
This last feature distinguishes our approach from that of
Cass-Shell [1983] and Peck-Shell [1985].
latter two papers,
beforehand,
In
the models of the
However,
all agents observe the same signal.
they trade securities that are contingent on the
realization of this signal.
Thus the signal induces wealth
effects that can give rise to new equilibria.
This is true
regardless of the set of securities that are available.
Cass-
Shell avoid the implications of the First Fundamental Welfare
Theorem by supposing that some agents are born only after the
securities market closes, whereas Peck-Shell, posit non-Walrasian
trade in the post-sunspot market.
In our model,
by contrast,
there are no securities and hence no wealth effects.
Thus a
signal observed by everyone can create no equilibrium outside the
set of certainty equilibria.
12
The reader may be disturbed by our assunption that a large
class of consumers observe exactly the same signal and yet the
the signal is unobservable to others.
Why,
for example,
could
a
member of the informed class not be persuaded (or bribed) to
divulge his information?
There are at least two ways around this
difficulty.
First,
one could construct
a
model in which the members of
a
given class do not observe the same thing, but rather private
signals that are only imperfectly correlated with each other.
long as there is some correlation,
equilibrium in such
As
model
a
with large numbers would be much like that in the present
formulation.
the acquisition of information by
However,
outsiders would be more difficult, since learning the value of
any single agent's signal would be of little benefit.
Alternatively, one could consider
a
instead
model in which,
of the large number of traders we have assumed in each class,
there are only a few,
In such a model,
i.e.,
a
model of monopolistic competition.
bribery might be difficult since large agents
would be likely to take into account the effect on equilibrium of
selling information.
We can suppose that,
after observing their
signals, agents choose production levels in Cournot fashion.
Then,
since,
formally speaking, the agents are playing
a
game,
the Cournot outcome is merely a correlated equilibirum in the
sense of Aumann [1974].
Moreover, by increasing the number of
13
agents in each class,
[1978],
one can show,
a
la
Novshek-Sonnenschein
that the Cournot equilibrium converges to the Walrasian
equilibrium that we considered
in
sections 2-4.
As we Mentioned in the introduction,
there
is
a
close analogy
between our finite-horizon nodel and those of the literature on
sunspots in overlapping-generations economies.
over lapping-generat ions literature,
at any given tine,
and,
It
is
the
agents live for two periods,
an old and young geneneration coexist.
assumed that sunspot activity (the signal)
is
observed by
sense in which information is
everybody, but there is
a
nonetheless asymmetric.
In any period,
a
In
paper asset against consumption.
the old generation trades
The price of the paper asset
depends on the labor supply by the young generation, which in
turn is influenced by current sunspot activity.
assumption
is
that,
when they buy the asset,
The crucial
members of old
generation do not know what the sunspot activity will be.
Thus
they choose their labor supply decision without knowing the
sunspot conditions on which the young generation's labor supply
is based.
In this
sense,
the young generation has private
information.
We now develop this point of view more formally.
the simplest overlapping generations model,
in Azariadis - Guesnerie
[1983]
studied for example
and Grandmont
single consumption good and paper asset.
A
Consider
[1985].
There is
typical consumer
a
14
He supplies labor (1) when young,
lives for two periods.
His utility fuction is U(l,c).
consunes (c) only when old.
produces one unit of output with one unit of labor.
competitive wage
is
one.
buys the paper asset at
at price P^.-i-
If M
is
constraints are
budget
*
(4)
c
If he
1
born at time
is
and c =p.
p.M,
=
t
the consumer
.-.M.
t+1
t
,
1
.
s
e
{s
,
,
.
.
,8
)
e,
1
s^
=
Definition
4
I
:
^t
A
~
^^^
interpreted as the
,
The evolution of activity is governed
by a Markov process with transition matrix
T
=
{t.
.}
where
is
a
price
stationary sunspot equilibrium
labor supply function x(s)
a
such that
x(s)
=
arg aax
[Ud.l^lil-
E
^
P(b)
|
s]
and
p(s)M
Equation
=
(6)
t.
'
function p(s) and
(6)
his
(P,^i/P,)l.
=
level of sunspot activity.
(5)
the
the quantity of asset he purchases,
Now consider a signal
Pr{s^_^^
t,
Thus,
He
and sells it in period t+1
at price p
t
and
x(s)
defines equilibrium in the paper asset market,
.
=
15
Substituting
whereas (5) deterBines the optiBun labor supply.
into (6), we can identify a stationary sunspot equilibrium
(5)
with a labor supply function x(») satisfying
x(b) maximizes
(7)
[U(1,I^4^
X s
E
i
Comparing
equilibrium
(7)
is
)
s]
I
and (1) we see that
nothing but
a
.
^
stationary sunspot
a
symmetric correlated equilibrium in
which both classes receive signals in the same signal space.
particular,
deterministic cycle (as in Grandmont [1985])
a
corresponds to
Thus,
In
a
perfectly correlated equilibrium.
despite the difference in interpretation,
there is
a
strong connection between sunspot and correlated equilibria.
To
develop this connection further, consider a signal space
{s
,...,B
i,j,
c.
and an nxn symmetric matrix C
}
.6[0,1]
and EZc
.
.
=
1.
C
is
a
=
{c. .),
where for all
correlation matrix;
it
gives the joint probability of signals received by two classes of
traders in
a
correlated equilibrium.
there is a corresponding unique matrix
Because
T
=
is
C
{t. .}
ij
symmetric,
of conditional
probabilities of one class's signal given that of the other.
Thus,
one can construct a stationary sunspot equilibrium from a
correlated equilibrium associated with
matrix
C.
a
symmetric correlation
Conversely, by deriving the matrix
transition matrix
T,
C
from the Markov
one can a construct correlated equilibrium
from a stationary sunspot equilibrium.
16
References
R.J. [1974], "Subjectivity and Correlation in Randomized
Strategies," Journal of Mathematical Economics Vol. 1, pp.
AumanD,
.
67-96.
Azariadis, C. [1981], "Self-f ullf il 1 ing Prophecies," Journal of
Economic Theory 25, pp. 380-396.
.
Azariadis,
Mimeo
C.
Guesnerie [1983], "Sunspots and Cycles,"
and R.
Shell [1983], "Do Sunspots Matter?," Journal of
Political Economy pp. 193-227.
Cass,
D.
and K.
,
Grandmont, J.M. [1985], "On Endogenous Competitive Business
Cycles," Econometr ica pp. 995-1046.
.
J. and K. Shell [1985], "Market Uncertainty: Sunspot
Equilibria in Imperfectly Competitive Economies," CARESS W.P.
Peck,
85-21
Novshek, W. and E. Sonnenschein [1978], "Cournot and Walras
Equilibrium," Journal of Economic Theory 19, pp. 223-266.
,
Shell,
K.
[1977],
"Monnaie et allocation intertemporelle
memorandum.
^355
u
I
3
,
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