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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Idiosyncratic Sentiments
and Coordination
Failures *
George-Marios Angeletos
Working Paper 08-12
April 16,2008
(Originally titled Private Sunspots
and Idiosyncratic Investor Sentiment)
*REVISED:
Dec.
5,
2008
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Idiosyncratic Sentiments and Coordination Failures*
George-Marios Angeletos
MIT
NBER
and
December
5,
.
2008
Abstract
Coordination models have been used
nomena.
fulfilling.
It is
in
macroeconomics to study a variety of
well understood that, in these models, aggregate fluctuations can be purely self-
In this paper
I
highlight that cross-sectional heterogeneity in expectations regarding
the endogenous prospects of the economy can also emerge as a purely
property. This in turn leads to
some
intriguing positive
rationalize idiosyncratic investor sentiment,
in real
and
(ii) It
self-fulfilling
(i)
It
can
can be the source of significant heterogeneity
any heterogeneity
in individual
economic fundamentals, and despite the presence of a
all
strong incentive to coordinate on the same course of action,
(iii) It
can sustain rich fluctuations
aggregate investment and asset prices, including fluctuations that are smoother than those
often associated with multiple-equilibria models of crises, (iv)
vestors learn slowly
(vi) It
JEL
how
to coordinate
It
can capture the idea that
on a certain course of action,
can render apparent coordination
failures
(v) It
in-
can boost welfare.
evidence of improved efficiency.
codes: D82, D84, E32, Gil.
Keywords: Sunspots, animal
spirits,
complementarity, coordination
failure, self-fulfilling
tations, fluctuations, heterogeneity, correlated equilibrium.
'An
equilibrium
and normative implications:
financial investment choices, even in the absence of
characteristics or information about
in
crises phe-
earlier version
was
entitled "Private Sunspots
and Idiosyncratic Investor Sentiment.'
expec-
1
Introduction
Following the contributions of Shell (1977), Azariadis (1981), Cass and Shell (1983), and
and Dybvig
(1983), a voluminous literature has argued that animal spirits, sunspots, or other forms
of extrinsic uncertainty, can be the cause of aggregate fluctuations
downturns or
Aumann's
financial crises with coordination failures.
(1974, 1987) seminal
work on correlated
certainty can be largely idiosyncratic and that
Nash equilibrium. Since
their implications for
is
Diamond
to contribute
and has associated economic
Motivated by a different
equilibria effectively
showed that extrinsic un-
can help rationalize a larger
it
set of issues,
set of
then, correlated equilibria have been widely studied in
outcomes than
game
theory. Yet,
macroeconomic applications have hardly been explored. The goal of
towards
paper
this
gap by studying the positive and normative implications that
filling this
the introduction of idiosyncratic extrinsic uncertainty can have for a class of models that
is
widely
used in macroeconomics to study coordination failures and crises phenomena.
In particular,
I
consider two closely related models.
The second
that abstracts from financial prices.
The common
essential feature of the
investment choices: an individual investor
invest. Following
Diamond and Dybvig
is
game
a simple real investment
first is
a variant that stylizes trading in financial markets.
is
two models
The
that they introduce strategic complementarity in
is
more
willing to invest
when he expects
others also to
(1983) and Obstfeld (1986, 1996), such a complementarity
could capture the role of coordination in bank runs, speculative currency attacks, and other crises
phenomena. Similar coordination problems could
thick-market, or credit-related externalities.
The
also originate in a variety of production,
particular models considered here are close cousins
of those used in the applied global-games literature by Morris
To
all
assumptions ensure that
had been
all
investors
all
different
would choose exactly the same
strategically independent.
when they do not
when they only have
equilibria in
rule out
any exogenous source of
and share the same
other relevant economic fundamentals.
One may expect
the presence of a complementarity in investment choices:
the same choice
I
investors have identical preferences, face identical constraints,
information about exogenous productivity and
choices
and Shin (1998, 2001) and others.
deliver the central result of this paper in its sharpest form,
heterogeneity:
level of
investment
this conclusion not to
if all
investors find
care about one another's choices,
why
it
different investment choices.
These
if
their
be affected by
optimal to make
should they do anything
a desire to align their choices with one another? Yet, there
which identical investors make
demand,
now
exist
The key
to this apparent
paradox
is
that individual investors
uncertainty about the aggregate level of investment. That
any given point, we
will find different investors
is, if
may now
face idiosyncratic extrinsic
we take a snapshot
of the
economy
at
holding different expectations regarding endogenous
economic outcomes, even though they hold identical expectations regarding
all
exogenous economic
fundamentals. This idiosyncratic variation in "sentiment" or "optimism" regarding the endogenous
prospects of the economy requires neither any differences in information regarding fundamentals
nor any deviation from Baycsian rationality; rather,
Formally, this
is
it
emerges as a
self-fulfilling
achieved by the introduction of "private sunspots".
prophecy.
Like the public sunspots
that are familiar from previous work in macroeconomics, the private sunspots considered in this
paper are payoff-irrelevant random variables. But unlike public sunspots, private sunspots are only
imperfectly correlated across agents and are privately observed by them.
The
equilibria that obtain
in
game theory
model are hybrids
of correlated
with private sunspots are thus closely related to the correlated equilibria introduced
by
Aumann
equilibria
(1974, 1987). In particular, the equilibria of the second
and
(noisy) rational-expectations equilibria:
aggregate sentiment
(i.e.,
the equilibrium price reveals partially the
the average private sunspot) in the market.
As an example, one could imagine the agents measuring the brightness
of the
sun or the tem-
perature outside their houses; idiosyncratic measurement error could then be a natural source of
imperfect correlation.
of clues about
Alternatively, one could imagine the agents reading a
what action other agents are
in search
coordinate on; the choice of what newspaper
what any given newspaper
to read, or the interpretation of
cratic.
likely to
newspaper
However, one need not take these examples too
says, could then be
literally.
somewhat
idiosyn-
Rather, one should think of private
sunspots as modeling devices that permit the construction of equilibria in which different investors
have different degrees of optimism regarding the endogenous prospects of the economy.
One
is
interpretation
is
that private sunspots rationalize idiosyncratic investor sentiment; another
that they capture, in a certain sense, idiosyncratic uncertainty regarding which equilibrium action
other agents are trying to coordinate on. Indeed, while the recent work on global games
and Shin, 1998) has addressed
this issue indirectly
preferred interpretation, there
is
a
deliver for the class of coordination
self-fulfilling,
number
its
feature of the equilibrium.
of novel positive
models that
this
Morris
by introducing private information about the
underlying payoffs (fundamentals), private sunspots address this issue at
such uncertainty as an integral, and
(e.g.,
paper
heart by generating
But no matter one's
and normative implications that they
is
concerned with.
On
the positive front,
I
highlight that models with
macroeconomic complementarities can gener-
ate significant heterogeneity in real and financial investment choices. Such heterogeneity can obtain
—
even in the absence
tics,
or after controlling for
— any heterogeneity in exogenous individual characteris-
but only to the extent that individual incentives depend strongly enough on forecasts of others'
Furthermore,
choices.
I
show how introducing idiosyncratic
enrich, not only the cross-sectional, but also the aggregate
models considered in
this paper,
extrinsic uncertainty can significantly
outcomes
of these models.
In the
two
with public sunspots aggregate investment and asset prices can only
take two extreme values ("high" and "low"); with private sunspots, instead, aggregate investment
and asset
prices can follow
smooth stochastic processes spanning the
entire interval
between these
two extreme values. Private sunspots can thus generate much smoother aggregate fluctuations than
public sunspots, indeed fluctuations that are
On
the normative front,
and policy conclusions.
I
Like
more reminiscent
of unique-equilibria models.
show that ignoring private sunspots may lead
many
of the pertinent
models on coordination
to erroneous welfare
failures, the
models
considered in this paper feature two equilibria in the absence of sunspots and these equilibria
are Pareto-ranked:
there
is
a "good" equilibrium in which everybody invests, along with a "bad"
equilibrium in which nobody invests.
extreme
levels of investment, achieving
sunspot-less equilibria.
safely
Adding public sunspots only randomizes among those two
convex combinations of the welfare obtained
in
the two
Therefore, as long as one restricts attention to public sunspots, one can
draw two conclusions: that the occurrence
of an investment crash
is
prima-facia evidence of
coordination failure; and that policy interventions that preclude this outcome (at no or small cost)
are
bound
to .improve welfare.
These conclusions outline what,
I
believe,
is
the conventional wisdom about the welfare and
policy implications of coordination problems in macroeconomics.
is
warranted once private sunspots are allowed. Assuming that aggregate investment
the "good" equilibrium relative to the
in
Nevertheless, neither conclusion
first
best,
I
is
excessive in
construct an equilibrium with private sunspots
which the economy fluctuates between states during which only a subset of the investors invest
("normal times") and states during which nobody invests ("crashes"). Because the aggregate level
of investment
is
now
closer to the first-best level during
normal times,
higher welfare than the equilibrium where everybody invests.
have an incentive not to invest during normal times,
it
this equilibrium
However,
can achieve
for certain individuals to
must be that these individuals believe that
a crash will take place with sufficiently high probability, while
many
other individuals believe the
But then note
opposite.
in equilibrium only
if
that, as long as agents are rational, such heterogeneity in beliefs
is
possible
a crash does materialize with positive probability.
In conclusion, an occasional cra.sh
—what
looks as apparent coordination failure
—
is
actually
boosting welfare by facilitating idiosyncratic uncertainty and thereby providing the necessary
centive that keeps investment from being excessive during normal times.
It
in-
then also follows that
well-intended policies that aim at preventing apparent coordination failures could actually reduce
welfare by eliminating the aforementioned incentive.
Related literature. The
and sunspots
(1986),
mar
is
literature
on macroeconomic complementarities, coordination
voluminous. Key contributions include Azariadis (1981), Azariadis and Guesnerie
Benhabib and Farmer (1984, 1999), Cass and
(1993),
failures,
Shell (1983), Chatterjee,
Cooper and Raviku-
Cooper and John (1988), Cooper (1999), Farmer (1993), Farmer and Woodford (1997),
Diamond and Dybvig
(1988), Kiyotaki
(1983), Guesnerie
and Woodford (1992), Howitt and McAfee (1992), Kiyotaki
and Moore (1997), Matsuyama (1991), Obstfeld (1986, 1996),
Shell (1977, 1989),
and Woodford (1986, 1987, 1991).
To the best
All the aforementioned papers consider only aggregate extrinsic uncertainty.
my
of
knowledge, the only notable exemption of a macro-finance application that features private
extrinsic signals
is
the paper by Jackson and Peck (1991) on speculative trading.
model
studies an overlapping generations
of rational bubbles, in
by a Vickrey auction. Similarly to the present paper, that
their bids
on private extrinsic
signals,
information about these signals.
It
paper
is
1
it
which asset prices are determined
earlier
work allows traders
to condition
does not allow the asset prices to reveal any
then proves the existence of speculative equilibria in which
traders act on these signals and shows
analysis in asset markets.
although
That paper
how
the resulting equilibria can help rationalize technical
Relative to this previous work, the
main contribution
to introduce idiosyncratic extrinsic uncertainty within a different class of
of the present
macroeconomic
applications and to uncover a novel set of positive and normative predictions for these applications. 2
A
secondary difference
is
that here
I
study a setting where equilibrium asset prices are allowed to
aggregate the underlying extrinsic private information in a rational-expectations fashion.
'See also Jackson (1994) for an extension of the existence results in Jackson and Peck (1991); and Aumann, Peck
Shell (1988) and Peck and Shell (1991) for more abstract analyses of the relation between sunspot equilibria in
and
general-equilibrium market economies and correlated equilibria in games.
2
In this regard, marginally related is Solomon (2003). That paper considers a model in which two countries play a
bank-run game, introduces country-specific sunspots (sunspots publicly observed by the residents of one country but
not the residents of the other country), and shows how the correlation of the two country-specific sunspots generates
a twin crisis. That paper does not consider any of the positive and normative properties that I study here.
Research on this particular class of applications has recently been revived by the global-games
contributions of Morris and Shin (1998, 2001, 2003) and others; see, for example, Angeletos et
Angeletos, Hellwig and Pavan (2006), Dasgupta (2007), Goldstein and Pauzner (2005),
al (2007),
Guimaraes and Morris (2007), Heinemann and
Illing (2002), Hellwig,
Mukherji and Tsyvinski (2006),
Rochet and Vives (2004), Ozdenoren and Yuan (2008), and Tarashev (2007). This literature
in-
troduces heterogeneous information regarding the underlying economic fundamentals (the payoff
structure) within the
same
class of coordination
mation about the fundamentals
models as
also generates heterogeneity in beliefs
heterogeneity in beliefs and actions
is
not a purely
heterogeneous infor-
this paper. Clearly,
self-fulfilling
and
actions.
However, this
property as in this paper. Also, the
strategic uncertainty that originates from heterogeneous information about the fundamentals only
reduces the set of equilibria. Indeed, in the limit case often studied in this literature, namely the
limit as private information gets infinitely precise relative to public information, a
rium
is
selected
and
in this equilibrium all agents take the
same
unique equilib-
action. In contrast, the strategic
uncertainty that results from private sunspots expands the set of equilibria while also accommodat-
The
ing heterogeneity in beliefs.
and normative outcomes
Layout. The
and
contribution of the paper
this heterogeneity
rest of the
paper
revisits the set of equilibria
is
is
then to show what kind of positive
can sustain within the applications of
interest.
organized as follows. Section 2 introduces the baseline model
with public sunspots. Section 3 introduces private sunspots and
studies their positive implications.
Section 4 studies a variant model that captures trading in
financial markets. Section 5 turns to
normative implications. Section
The
2
baseline model: a real investment
The economy
i
£
[0, 1],
are
is
6 concludes.
game
populated by a measure- 1 continuum of agents (investors), who are indexed by
endowed with one unit
of wealth each,
and decide how to allocate
this wealth
between
a safe technology and a risky alternative whose return depends on the aggregate level of investment
Let
R
denote the return to the safe technology,
in the risky technology,
K
the aggregate level of investment, and
in that technology.
technology relative to the safe one.
tt,
= U(ku K) =
The
(1
payoff of
-
ki)R
+
i
the investment of agent
ki
A{K
)
the excess return of this
is
k,{R
+ A{K))
i
=R + A{K)k
t
.
The key assumption needed
A(K) <
such that
K
for all
for. the positive results of this
<
A(K) >
k and
strategic complementarity in investment choices
one where
all
and
let
One can then
minimal
in-
for
K
To simplify the
< k and A{K) =
b
-
c
(0, 1)
This assumption introduces
k.
analysis,
>
e
of
two Nash equilibria,
for
I
K
all
agents
henceforth normalize
>
k,
where
b
>
>
c
0.
think of c as parameterizing the cost of investing in the risky technology, k as the
aggregate investment for which the technology becomes profitable, and b as the
gross benefit enjoyed in that event.
3
To
simplify the exposition
each investor can choose either
("don't invest"), so that
Model
>
that there exists a k
and guarantees the existence
the safe technology.
A(K) = -c <
level of
indivisible:
K
for all
is
agents invest their entire wealth in the risky technology and another where
invest their entire wealth
R=
paper
K
is
also the
mass
ki
—
1
(which
I
I
further
henceforth
assume that investment
simply "invest") or
call
—
4
of agents investing.
interpretation and remarks. The key ingredient of the model
ordination problem. Such a coordination problem
runs, speculative currency attacks,
k%
is
is
is
the presence of a co-
the core element of models of
and other macroeconomics
crises (e.g.,
self-fulfilling
bank
Diamond and Dybvig,
1983; Obstfeld, 1986, 1996). Indeed, as mentioned in the introduction, the particular coordination
game we
are employing here
is
nearly identical to the class of incomplete-information
games
re-
cently used by Morris and Shin (1998) and others in the applied global-games literature to study
crises.
Like that recent work, we abstract from institutional details in order to concentrate on
the role of coordination and to keep the analysis tractable in the presence of incomplete information.
But whereas that work focuses on
fundamentals, here we
In other
macro
intrinsic private signal regarding the underlying
shift focus to extrinsic private signals.
applications, similar coordination problems originate in production,
thick-market, or credit-market externalities.
externalities; Kiyotaki (1988)
and Smith
(1998), Kiyotaki
credit frictions; Chatterjee,
tion;
economic
and Cooper (1999)
for
The deeper foundations
See, for example,
and Woodford (1991)
for
Diamond
aggregate
demand
and Moore (1997), and Matsuyama (2007)
Cooper and Ravikumar (1993)
an excellent review of the
of the coordination
for
role of
problem
for
demand,
(1976) for thick-market
externalities; Azariadis
complementarities due to
complementarities in business forma-
complementarities
in
macroeconomics.
are, of course, specific to
each particular
application. However, modeling these foundations does not appear to be essential for the purposes
3
4
All these parameters are
common knowledge
— there
is
no uncertainty about the economic fundamentals.
assumption is completely inconsequential.
Clearly, as long as agents are risk neutral, the indivisibility
What
of this paper.
may
essential
only that the coordination problem opens the door to extrinsic
is
At the same time, note that the framework introduced so
uncertainty.
prices
is
—a
limit idiosyncratic extrinsic uncertainty
applications of interest.
I
possibility that
with this issue in Section
will deal
is
To
5
and the best response
ki
It
follows that there exist exactly
i)
and another
nobody
We now
making and
random
(cd.f.)
F
level of
1
= BR{K) =
two
one
equilibria:
=
K=
in
if
K>
k
if
K<
k
which everybody invests
for all z).
6
perfectly coordinate on the
same course
:
§ —
->
[0, 1].
s,
whose support
is
§ C
K
all
sustained by
expectation
what choices other
action.
all
investors observe a payoff-
and whose cumulative distribution function
investment can be stochastic. However, because
level of
is
self-fulfilling
Because the investors can now condition their choices on
no uncertainty about the equilibrium
K = 1 for
=
(fcj
The one equilibrium
In either case, investors face no uncertainty about
variable
is
simply
i is
introduce public sunspots. Before making their choices,
irrelevant
is
most
equilibria in the absence of
expectation that everybody will invest; the other by the
will invest.
investors are
of investor
which nobody invests (& t
in
self-fulfilling
that
evidently relevant for
see this, note that, in the absence of sunspots, the aggregate level of investment
deterministic,
the
from how
4.
Public sunspots. As noted above, the model admits exactly two
sunspots.
far abstracts
s is
s,
the aggregate
publicly observed, the investors face
investment and continue to make identical choices.
Equilibria with public sunspots are thus merely lotteries over the two sunspot-less equilibria.
Proposition
1
For any equilibrium with public sunspots, there
with probability p and K(s)
an
equilibrium, in
Beliefs
=
which K{s)
—
with probability
1
and actions vary across
1
—
p.
exists
ap 6
[0, 1]
such that K{s)
Conversely, for any p G
with probability p and K(s)
=
[0, 1],
with probability
1
—
equilibria, or across realizations of the public sunspot,
=
1
there exists
p.
but never in
the cross-section of investors: in any given equilibrium and for any given realization of the sunspot,
investors share the
all
5
same "sentiment"
(i.e.,
the same belief about
all
endogenous outcomes), can
Because there is a continuum of investors, this is true even if investors follow mixed strategies.
A(k) = 0, there also exists a mixed-strategy equilibrium in which each investor invests with probabilit}'
k. I have ruled out this equilibrium by assuming A(k) ^ 0.
This is not essential for any of the results: in the
aforementioned mixed-strategy equilibrium, investors do not face any uncertainty and share the same beliefs.
6
When
perfectly forecast one another's choices, and end
how none
section shows
The next
action.
for private sunspots.
*
introduce private sunspots as follows. First, "Nature" draws a payoff-irrelevant random variable
§
:
—
The support
not observed by any investor.
s that is
F
need to hold once we allow
Private sunspots and idiosyncratic sentiment
3
I
of these properties
up taking exactly the same
Conditional on
m
s,
is i.i.d.
These variables define what
unobserved
is
S
C K and
its c.d.f.
is
Then, each investor privately observes a payoff-irrelevant random variable m.
[0,1].
>
of this variable
common
sunspot
MCR and
across investors, with support
I
call "private
s. I
sunspots":
henceforth
call (§, F,
*
c.d.f.
:
M
x S
—
>
[0, 1].
they are private signals of the underlying
M, $)
the "sunspot structure" and define an
equilibrium as follows.
Definition
a strategy k
An
1
equilibrium with private sunspots consists of a sunspot structure (S, F,
M—
:
>
{0, 1}
\I>)
and
such that
k(m) G arg max
'
M,
U(k, K(s))dP(s\m.)
VmG
fce{o,i}y s
with K(s)
= JM
conditional on
k(m)d'i(m\s) Vs G
m
(as implied by
S,
B ayes'
Note that the sunspot structure
environment. Rather,
it is
and with P(s\m) denoting
rule).
(§, F,
M,
\P)
is
not part of the exogenous primitives of the
a modeling device that permits the construction of equilibria that sustain
endogenous stochastic variation, not only
In the remainder of this section,
I
in the aggregate,
but also
in
Gaussian sunspots. Suppose
s
is
>
noise,
across investors and independent of
i.i.d.
private signal observed by investor
only
if
error.
his private
is.
m, =
with variance a ;
temperature
The next proposition then constructs
measurement
i
4
s,
of s as the "brightness of the sun" or the "average
measurement
to.
drawn form a Normal distribution with mean
variance a^
The
the cross-section of agents.
consider a specific Gaussian sunspot structure that best illustrates
the novel positive properties equilibria with private sunspots can lead
0.
of the posterior about s
the c.d.f.
s
+
>
in a city"
equilibria
el
0.
where
,
/i s
£j is
6R
Normal
One can then
and
Ei
is
think
as idiosyncratic
where an investor invests
of the brightness of the sun or the temperature
and
if
and
sufficiently high.
In these equilibria, an investor's private sunspot captures his idiosyncratic sentiment regarding the
prospects of the economy: the higher m, the higher the investor's expectation of the aggregate level
of investment.
Proposition 2 For any
(i)
An
(ii)
(Hi)
(fi s ,o- s
investor invests
The aggregate
when
level of
m>
m* and
investment
not when
m<
Proof. Let
<E>
denote the
K (s)
> k
if
c.d.f.
if
and only
Normal
of the standard
and only
s
if
>
s*
Because both the prior about
some m* £ R.
for
m>
if
s
and the
s*,
signal
the aggregate level of investment,
distribution.
m*. Aggregate investment
s
—
-
c
=
6$
it is
latter
is
strictly increasing in
thus necessary and sufficient that
7^^ m
*
m*
then given by
(1)
+<t £ $
-1
(k,)
(2)
+C7 e
[i s
—
s*
and variance Var[s|m]
\Ja7 +<ri
cr s
Note that the
is
m
follows that the expected return from investing conditional on signal
s*|m)
Suppose there exists an
m are Gaussian, the posterior about s conditional on m
(7 S
=6Pr(s>
(0, 1).
where
= m*
Normal with mean E[s|m]
E[A(K{s))\m]
on
(0, 1).
such that an investor invests
and therefore
It
,
stochastic, with full support
is
K(s) = Prfm > m*\s) = $
is
m*
The cross-sectional distribution of expectations regarding
E[A'|m], has full support on
m*
,a € ), there exists an equilibrium in which the following are true:
m
+ ^=fe^' -
E[yl|m*]
s*
=
=
2
+ ae
2
l
)
.
T/^5
+cr c
0,
(a s
is
m. For the proposed strategy to be part
satisfies
=
of
an equilibrium,
or equivalently
^n^
1
(I)
(3)
Substituting s* from condition (2) into (3) and rearranging gives
^-/*.-»-{4^*"
1
W + yi+^*"
1
(f)}.
(4)
which completes the proof of part
follows from part
of
Note that
771
(ii)
then follows from condition
s
have
full
In the equilibria constructed above, this
different expectations
about the aggregate
individual investment choices
means that
who have
received
level of investment,
which
different investors also
m>
in
m*. The end result
turn sustain different
and choices can not be traced to any heterogeneity
this heterogeneity in expectations
endowments, technologies, or payoff-relevant information),
can be interpreted as idiosyncratic: variation
"sentiment" or "optimism".
in
with regard to the endogenous prospects of the economy.
in
that
—a sharp difference from the case with public sunspots.
in primitive characteristics (preferences,
it
m and
(iii)
about the distribution of the signals
different investors hold different expectations
in the population.
Because
Finally, part
QED
supports.
hold different expectations about the mass of investors
is
(1).
along with the fact that both the distribution of s conditional on
(ii)
m conditional on
Part
.(i).
It
This optimism
is
does not require any heterogeneity
expectations regarding the exogenous primitives of the environment, nor any deviation from
Bayesian rationality. Rather,
it is
merely, and purely, a self-fulfilling equilibrium property.
Finally, note that, in the equilibria constructed above, the aggregate level of investment has
full
support on the (0,1) interval. In contrast,
in the equilibria
with no or only public sunspots,
and
the aggregate level of investment could take only the extreme values
sunspots permit, not only endogenous heterogeneity
1.
Therefore, private
in the cross-section of the population,
but also
a richer set of aggregate outcomes.
A
simple dynamic extension.
To
better appreciate the aggregate implications of private
sunspots, consider the following dynamic extension.
each period
each investor choses whether to invest
t,
contemporaneous payoff
and A{Kt)
if
Kt <
k.
is
is
t is
The
t is
m,j
)kt,
t
where Kt
investor's intertemporal payoff
structure
given by
white noise,
period
= A(K
is
(kt
i.i.d.
—
st
st
=
is
is
an
infinite
or not (kt
1)
=
b
—
c
>
simply ]Ct^o^ <7r <'
if
pst-i
+u
t
,
where p £
where
et is
white noise,
=
number
0).
of periods.
He then
Kt > k and A(Kt)
w here
wherein the interesting dynamics enter.
In
receives a
6
t
— —c <
(0, 1).
The unobserved sunspot
in
(0, 1) is
the auto-correlation in the sunspot and ut
The
private sunspot observed by an investor in
across time, with variance a\.
+ et,
=
is
the aggregate level of investment in period
the net return to investment, with A(Kt)
The sunspot
period
nt
There
i.i.d.
10
across agents and time, with variance a\.
Now
note that investors
may
learn over time about past realized sunspots by the observation
of past aggregate investment and/or past payoffs.
the learning through payoffs.
across agents
i.i.d.
equilibria in
in a sufficient statistic
m
(i)
t
t
-\)
ignore
signals of past
&, where
&
is
white
These assumptions guarantee the existence of
and
its
we can
find a sequence
and an equilibrium
{m1,at]^L
the entire sequence of private signals up to period
which
,
an investor invests
statistic rh t
aj.
+
l
I
which the information structure remains Gaussian. 7
which the following hold:
(ii)
= §~ (K
a signal x t
t
and time, with variance
Indeed, as shown in the Appendix,
and
analysis tractable,
assume that investors observe noisy private
also
each investor observes in period
investment:
noise,
I
To maintain the
is
in period
Normal,
t if
across investors, with
i.i.d.
and only
if
fh t
>m*
.
t
Along
in
can be summarized
t
mean
s
and variance a\\
this equilibrium, the sufficient
variance at can be constructed recursively as functions oi\fht-\,at-\\mt,Xt).
Moreover, as the history gets arbitrarily long, (ml, at) converges to some time-invariant (m",a.)
We
thus obtain a stationary equilibrium along which aggregate investment
KAs
x
.
)
s ~
= $ i
t
...
is
given by
rn*
Hence, up to a monotone transformation, aggregate investment follows a smooth AR(1) process. 8
Note then that
fictitious
data generated by the present model would be virtually indistinguishThis would not be
able from fictitious data generated by a canonical unique-equilibrium model.
the case
we had ignored
if
fluctuations (between
private sunspots: aggregate investment could then only feature discrete
and
which would be more
1),
telling of multiple equilibria.
private sunspots can help generate very smooth aggregate fluctuations,
smooth changes
fluctuations driven by sunspots from fluctuations driven by
"Learning to coordinate." As another example
can sustain,
I
now
making
of the rich
it
in
We
conclude that
difficult to identify
fundamentals.
dynamics that private sunspots
consider the following variant. Investors continue to receive the exogenous and
'The assumption that investors do not learn from
as follows. Let the payoff of an investor be nt
=
ztkt,
their past payoffs
where z t =
A(K
t
)
is
merely
for
convenience and can be justified
u>t is white noise, i.i.d. across
+oJt and where
both time and agents, with variance a^. Suppose further that zt is privately observed by the investor, independent^'
of his choice of investment; this kills the value for experimentation that would have emerged if zt was observed only
when ki = 1 Then, the observation of n conveys no more information than z which by itself is a noisy private signal
of Kt. Qualitatively, this is much alike the noisy private signal xt that we have already introduced. The only difference
However, letting
is that the information contained z
is not Gaussian, making the updating of beliefs intractable.
Gu, — oo avoids this problem by rendering the signal Zt uninformative. At the same time, because the expectation of
ujt is zero no matter <t^, investors continue to choose kt so as to maximize their expectation of A(Kt)kt. ft follows
that the error introduced by ignoring the information contained in payoffs vanishes as a^, —> oo.
8
To be precise, $~ (Kt) is a Gaussian AR(1).
t
.
t
t
»
1
11
,
endogenous private signals m. and. x considered above, but now the unobserved sunspot remains
t
=
constant over time: p
As
we can
before,
t
and a u
1
=
by
K
t
1/2,
which gives
= $
(s)
so that
ml =
in
=
St
—
t
oo.
>
some deterministic threshold m*.
for all
It
1/2),
t.
It
now
that
K
=
1
and
whenever
follows that,
K
=
t
if
and only
For simplicity, suppose k
follows that aggregate investment in period
1;
a
signals,
s
>
0,
s
<
t
is
if
=
given
decreasing over time and
is
t
Kt{s)
and whenever
and decreasing over time, asymptotically converging to
Recall
bounded
is
0,
Kt(s)
is
in (1/2, 1)
bounded
and
inside
0.
represent the only two equilibria that are possible in the
absence of private sunspots and that require
We
t.
which an investor invests in period
increasing over time, asymptotically converging to
(0,
s for all
Because of the accumulation of new
[~§l)
converges to zero as
0,
an equilibrium
find
his sufficient statistic rht exceeds
c/b
=
all
investors coordinating on the
same course
of action.
can thus interpret the dynamics that obtain here with private sunspots as situations where
investors slowly learn on which action to coordinate: at any given date,
the "wrong" investment choice
investors
(i.e.,
who makes such a mistake
some
investors are
making
do the opposite of what the majority does), but the fraction of
falls
over time and vanishes in the limit.
Also note that this form of learning can be either exogenous or endogenous:
it
can originate
in
either the signals rat regarding the unobserved sunspot s or the signals xt regarding past aggregate
activity.
We
conclude that private sunspots can capture, not only the idea that agents
perfectly coordinate on the
how
same course
may
fail
to
of action, but also the possibility that agents slowly learn
to do so over time through the observation of one another's actions.
The form
of social learning considered in this
extend the analysis to public signals about either
example
s or
is
past activity.
prices; this bring us to the topic of the
that
is
4
Private sunspots and financial markets
of special interest
The preceding
is
purely private, but one could easily
A
certain kind of public signals
next section.
analysis has been conducted within a simple investment
market interactions.
I
now
that abstracted from
consider a variant model in which investors trade an asset within a com-
petitive financial market. This exercise serves
two purposes.
First,
it
shows how the insights of the
preceding analysis translate in the context of financial markets. Second,
relation can be
game
it
shows how imperfect cor-
accommodated within a rational-expectations-equilibrium framework, where
12
prices
partially reveal the unobserved
common sunspot component
that drives the correlation
among
the
beliefs (the private sunspots) of different investors.
Model
much
set-up. There
is
again a large
An
to trade of a certain financial asset.
and the aggregate investment by K. The
The
later
is
assumed
number
of risk-neutral investors,
who now
individual's investment in the asset
price of the asset
is
denoted by p and
is
its
how
denoted by
ki
dividend by A.
A = A (K).
to increase with aggregate investment in the asset:
decide
Finally, since
investors are risk-neutral, their payoffs arc simply given by
m = Il(k
i
To
rule out infinite positions,
I
assume that
These bounds can be interpreted as the
would be another natural, but
finite positions.)
Without any further
is
denoted by Q,
unobserved supply shock:
et al.
(2006),
assumed
bank
(in
,
to
where
its sole role is
less tractable,
its
in [k,k], for
some
finite
k and
k.
way
=
to ensure that investors take
and k
=
1.
Finally, the
supply of
be an increasing function of the price and of some
uGllCR.
The. shock u can also be interpreted as
to introduce noise in the price.
Ozdenoren and Yuan (2007), and Tarashev (2006)
crises.
The
to study
bank
runs, speculative
price can then be interpreted as the stock price
the context of bank runs), the domestic interest rate or the peso forward (in the
context of currency attacks, or the stock price of a
among
.
Close cousins of this model have been used by Angeletos and Werning (2006), Hellwig
currency attacks, and other financial
of a
bounded
loss of generality, let k
Q — Q (p,u)
the impact of "noise traders";
Remarks.
is
ki is
i
borrowing and short-selling constraints. (Allowing
result of
for risk aversion
the asset, which
= [A(K)-p]k
,K,p)
institutional stock holders (in
financial markets).
As
in
company that
faces a coordination
Ozdenoren and Yuan's application on feedback
the baseline model, the positive dependence of
A
on
K
is
problem
effects in
the source of the
coordination problem. At the same time, the endogeneity of the price, along with an upward-sloping
supply
for the asset, will
strategic substitutability
be the source of a negative pecuniary externality
— among the traders:
—and
of a certain
form of
the more other traders invest in the asset, the higher
the price an individual trader has to pay in order to invest in the asset.
Note that such an adverse price
effect
emerges quite naturally within the context of speculative
currency attacks: as long as the speculative attack
(i.e.,
as long as
K
<
k), the
is
not sufficiently strong to trigger a collapse
more the others speculate, the higher the individual
13
cost of speculation,
simply because the speculation typically leads to an increase
in
domestic interests rates and thereby
on the cost of borrowing and short-selling the domestic bond. More generally, such an adverse price
any asset market
effect is generic to
another set of agents
Alternatively, the
(e.g.,
set of agents (e.g., speculators) trades
whose net supply
noise traders)
model
In this case
technology.
which one
in
interpreted as one of real
is
complementarity
A(K)
in
externalities, while the adverse price effect could
of the asset
—rather than
is
upward
financial
against
sloping.
—investment in a new
could emerge from production or thick-market
emerge from investors competing
for
a scarce input
(capital, labor, oil, etc.).
Rational-expectations equilibria with private sunspots.
same
are introduced in the
common
signal
sunspot variable
rrii
G M, which
is
s
fashion as in the baseline model:
We
nature
introduce private sunspots
first
draws an unobserved
G S from some distribution F; nature then sends each agent
drawn
across agents from a conditional distribution
i.i.d.
i
a private
$ These
variables
.
are payoff-irrelevant and are independent of the supply shock u; they are once again devices that
introduce aggregate and idiosyncratic extrinsic uncertainty.
What
asset
is
novel here relative to the model of the previous section
market may
*publicly reveal information
is
that the price that clears the
about these sunspot variables.
This motivates the
following equilibrium definition, which introduces private sunspots within an otherwise-standard
rational-expectations equilibrium concept.
Definition 2 A rational- expectations equilibrium with private sunspots consists of a sunspot structure (S, F,
and a
M,
,
3/),
a price function
belief (c.d.f.)
/j,
:
S x
R
(i) Beliefs are consistent
(ii)
Given the
beliefs
and
(Hi) Given the
:
x
§xM
K—
>
M.,
[0, 1],
an individual demand function k
:
MxE—
»
[k,k],
such that the following hold:
the price function, the
U.(k,
/
demand function
satisfies individual rationality:
K (s,P (s,u)) ,P (s,u))'dfj,( s,u\m,p)
V(m,p)
7§xu
k(m;p)dty(m\s) Vs G
demand
>
—
with Bayes rule given the equilibrium price function.
fce[o,i]
= fM
M
x
k(m,p) G arg max
where K(s,p)
P
§.
function, the price function
satisfi.es
K{s,P(s,u)) = Q(s,u)
14
market- clearing:
V(s,u).
As
in
most rational-expectations models, the analysis
and functional forms.
of distributional assumptions
u
N
~
(0, <t„)
~
s
,
independent of both
if
K
>
some
k and A{K)
scalar A
>
s
and
s
=
,
af)
u.
I
and
,
rrii
=
+
s
£i,
otherwise, for some scalar k €
of the standardized
c.d.f.
equilibria in which investors'
fact that
and
supply
As a
is
demand
result, the
e.i
N (0, of
~
)
uncertainty
and
(0, 1);
is
Gaussian:
and
across agents
is i.i.d.
A and Q A(K) =
:
Q (p, u) = $
+ \$~
(u
l
(p))
,
Normal
for
noise in prices), the price
is
$
distribution.
establishes the existence of rational-expectations
demand
for the asset is increasing in
increasing in u, this ensures that the equilibrium price
is
1
functions are decreasing in the price and increasing in their
aggregate
Because the supply shock u
u.
where
all
further impose the following functional forms for
Equilibrium analysis. The next proposition
private sunspots.
thus assume that
I
This scalar parameterizes the price elasticity of the supply of the asset, while
0.
denotes again the.
N (p
intractable without an "artful" choice
is
unobserved
(recall, this
is
s.
Along with the
increasing in both s
shock captures more generally any
only a noise indicator of the underlying
common
sunspot component
s.
This ensures that, although investors do learn something about one another's' investment choices
from the observed
price,
they continue to face some residual idiosyncratic uncertainty regarding one
another's investment choices, and hence about the eventual dividend of the asset. As a result, these
equilibria feature different investors finding
even though they
component
all
it
optimal to make different portfolio choices,
strictly
share the same preferences, constraints, and beliefs regarding any exogenous
of asset returns
—heterogeneity
in portfolio choices originates
merely
in self-fulfilling
heterogeneity in beliefs regarding the endogenous component of asset returns.
Proposition 3 For any (c u ,A),
there exists a rational- expectations with private sunspots in which
the following are true:
(i)
An
investor's equilibrium
demand
for the asset
k(m,p) =
,
1
if
is
m
given by
> m*
(p)
otherwise
where
m*
asset,
K(s,p),
(ii)
of s
and
(p) is
a continuous increasing function of p.
is
continuously increasing in
The equilibrium price
is
given by p
s
By
implication, the aggregate
and continuously decreasing in
= P (s, u)
a continuously decreasing function of u.
15
,
where
P
is
demand for
the
p.
a continuously increasing function
.
Proof. Consider a sunspot structure such that \a E (o~ 2
m*
suppose there exists an
such that an investor invests
(p)
proposed strategy, aggregate demand
for all (s,u). Since the function
the observation of p
is
m*
signal
= —o e u
m, and the public
also Gaussian, with
= m*(p) +
CT E
A<E>"
=
signal z are
E[s|m,p]
=
<Jp
0St
dividend conditional on signal
E[A|m,p]
if
and only
if
m
is
,
m
> m*
1.
Next,
Given the
(p).
m* (p)+a E A$ -1
(p)
(and so are a e
in equilibrium
=
1
(p)
2
cr
+
s
,
= s — ae u,
A,
and $),
n,
(5)
a 2 Because the prior about
.
s,
conditional on
s
the private
m and p
+ ^m+^z(p)
^V^
a
a
a
=
is
m
*
(
(a~ 2
+ a~ 2 + o~ 2 )~
=
^[^{E[s\m,p}--m''(p)-a^-
,
(p)
p)
=
m.
It
where m**
(^
(p)
^+a
£
.
$
Pr
[s
> m*
1
(p)
l2
.
It
follows that the expected
(p)
is
- m*
Along with
(«)
<t e
$
-1
(k.)
|
m,p]
it
optimal to invest
if
and only
the unique solution to
(p) ,p]
]
+
(K)))
follows that an investor finds
(E[s|m**
= m*
l
is
=
<D
m**
=
where a pos t
poat
post
Pr [K(s,p) > K\m,p]
(p)
(6)
s
=
increasing in
> m**
In any equilibrium,
2
2
oJ a~ >
mean
and variance Var[s|m,p]
the latter
satisfy
Gaussian, the posterior about
all
post
(6),
p must
Equivalently,
.
noise with variance a 2
'
By
m
if
2
informationally equivalent to the observation of the signal
Normal
is
and only
if
l'
given by
common knowledge
is
z(p)
where n
is
K (s) = Q {p,u)
Market clearing imposes
+ a~ 2 +<j~ 2 a~ 2 )~
+
(p)
- aE
(6), this gives
[\a E a post a-
16
^
2
-
l]
l
(K))j
=
p
a unique solution for m*(p):
a s2 cr po\ t $
'
(p)
(7)
We
conclude that there exists a unique equilibrium demand function, which
k{m,p)
=
j
1
if
m*
By assumption,
(p) as in (7).
otherwise
A(T £ (jp OS tO'~
uously increasing in p and hence the equilibrium
Along with the
in p.
m > m* (p)
(
!
with
2
>
which guarantees that
1,
demand
supply of the asset
fact that the
is
for the asset
m*
from
(p)
p
(7) into (5)
and solving
(
[\a e a post
which
is
continuously increasing in
s
(cr„
2
and therefore
smooth fluctuations
demand-supply
(2006), Barlevy
once again,
fail
+
-aE $0-2)
demand
intersects only once with supply.
This
in asset prices.
intersections,
-
is
l
a contin-
continuously decreasing
is
— P (s
,
u)
The
.
latter
is
found
{K)
\
-1
l]
and continuously decreasing
In the equilibria constructed above, the aggregate
in its price
p, s
(p) is
Doing so gives
for p.
s-aE u-
r>
— P
(s.u)\-k
— <P
m*
continuously increasing in p, this also
guarantees that there exists a unique equilibrium price function, p
by substituting
given by
is
a~a'
'post
p
in u.
,
QED
for the asset
globally decreasing
is
Moreover, these equilibria feature only
unlike the backward-bending
demand
and discrete price changes (crashes) featured
in
functions, multiple
Angeletos and Werning
and Veronesi (2003), or Ozdenoren and Yuan (2008). Therefore, an outsider could,
to distinguish empirically this
model from a smoother, unique-equilibrium model
of
the financial market.
5
Private sunspots and efficiency
In the baseline
model
of Sections 2
and
3,
the best sunspot-less equilibrium (the one in which
K—
1)
coincides with the first-best allocation. This, however, need not be the case in general. Investment
booms may sometimes be
activities, or
having adverse price
with high investment
among
all
excessive, leading to inefficient, bubbles, crowding out of other productive
(K —
equilibria with
1)
no
In a certain sense, this
is
effects.
may
For any of these reasons, the sunspot-less equilibrium
feature inefficiently high investment, even
if it is it
is
the best
(or only public) sunspots.
precisely the case in the financial-market
model, a proper welfare analysis
is
complicated by the fact that
17
I
model
of Section
4.
In that
have assumed an exogenous supply
of the asset:
have not modeled the "noise traders" that
I
behind
lie
this supply.
We
can nevertheless
bypass this complication by focusing on the welfare of the investors that have been modeled
of the latter as domestic agents
and the ones behind the supply
as "unloved" foreigners.
—think
Note then
that higher aggregate investment implies a higher price at which the asset can be acquired.
result,
in
although domestic investors are better
which
K£
K—
0,
the equilibrium in which
off in
they would have been even better
off if
K
—
As a
than the one
1
they could somehow coordinate on some
they would have then guaranteed the same rate of return at a lower price.
(k, 1), for
Whenever
there are such inefficiencies,
and implement the
correct these inefficiencies
is
it
natural to think about Pigou-like policies that
first-best allocation as
an equilibrium (although not
necessarily the unique one). Suppose, though, that such policies are unavailable, too costly, or far
from perfect,
for reasons that are
beyond the scope of
this paper.
I
will
now show how
private
sunspots, unlike public sunspots, can then improve welfare.
Towards
investment
this goal, consider the following variant of the baseline
is
now
1-c-hK
j
I
,
(0, ^]
a congestion
in Section 4.
The
net return to
given by
A(K) =
where k 6
model.
and h >
effect:
0.
9
The
baseline
-c -
hK
model
is
if
K
if
K<k
>
k
nested with h
(8)
=
0.
Allowing h
>
introduces
a negative externality and a source of local substitutability like those featured
As noted
already, such a congestion effect
emerges naturally within the context of
speculative currency attacks and other financial crises, in the form of an adverse price effect; one
can thus interpret h also as a measure of the strength of this price
In fact,
what the model
was featured
Without
in the
model
this abstraction,
private sunspots; instead,
of this section does
is
precisely to allow for the adverse price effect that
of Section 4 while also abstracting
it
effect.
from the informational
would be impossible to characterize the
we would have
role of prices.
set of equilibria with arbitrary
to limit attention within the class of Gaussian sunspots.
For the purposes of this section, however, we prefer to pay the cost of this abstraction for the benefit
of identifj'ing the best equilibrium
Before doing that,
level of
9
I
among
all
equilibria with arbitrary private sunspots.
revisit the set of equilibria
with public sunspots and also characterize the
investment that maximizes the investors' welfare.
Letting b
=
1
is
merely a normalization, while k < ^ simplifies a step
18
in
the proof of Proposition
5.
Proposition 4 Suppose
(i)
G (^5^1
ft
1
—
c)
There exist only two sunspot-less
The equilibrium
(ii)
in which
K — 0,
which
in
K=
.
equilibria,
A(K) >
for all
agents investing
is
[re,
Now
1).
w(K) denote
let
(ii),
strictly
—
note that w{\)
assumption h <
concave
—
1
<
k;
>
—
re,
(iii),
-
(1
— h and
c
has an upward
it
continuous); and thereafter
K
1
Finally, for part
c.
K
for
In particular, for
—
w' (K)
1
—
c
A')II(0,
=
tu(0)
note that
jump
— 2hK,
^r
2h
v
'
The key
1
if
and only
result here
is
if ft
>
is
G
<
and, as long as h
[0, re)
when
1
-
c,
the fraction of
at
not too low
(ft
0,
A) = KA{K).
so that the result follows again
w (K)
K=k
is
(at
continuous, strictly decreasing, and
which point
it is
but not
right-
so that the first-best level of investment
G
(re,
1)
if 1
-
if 1
-c-2/i<
if 1
-
c
left-
c
given by
is
- 2hn <
-
that, as long as the congestion effect
> ^^),
from the
2ft
<
1
-
c
-
(9)
2/ire
>
^. QED
continue to exist exactly two equilibria
effect
K
again continuous and strictly concave, but possibly non-monotonic.
it is
K* = argmax w(K)
K
A" <
for all
welfare (ex-ante utility)
re
Therefore,
1).
[re,
:
w{K) = KU(l,K) +
For part
K* G
is
A(K) <
follows from the fact that
KG
K
K = 0.
achieves higher welfare (ex-ante utility) than the equilibrium
1
(Hi) The first-best level of aggregate investment
(i)
and another with
1
any equilibrium with public sunspots.
as well as than
Proof. Part
K=
one with
in
is
not too high
(ft
<
1
-
c),
there
the absence of sunspots; but, as long as the congestion
neither equilibrium
not improve upon those two equilibria
is
is
clear: public
first-best efficient.
That public sunspots can
sunspots only attain convex combinations of
the welfare levels attained by the two sunspot-less equilibria and they are thus dominated by the
equilibrium in which
As noted
K=
1.
This, however,
in the Introduction, this
is
is
not true once we allow for private sunspots.
not completely surprising given
correlated equilibria in general sustain a large set of outcomes than
needs to verify that this
is
Nash
Aumann's
equilibria.
result that
However, one
indeed the case for the particular model considered here, as well as to
19
appreciate this possibility within the particular model.
To accomplish both
same time,
tasks at the
I
opt to characterize the best possible equilibrium that can obtain with private sunspots. This permits
me
model
to identify, for the
one allows
for private
at hand,
sunspots
what equilibrium properties
when
are necessary for efficiency
—and then to contrast them with those that obtain
if
one ignores
private sunspots.
Proposition 5 Suppose h £
maximize
equilibria that
p*
<
such that
1,
(i)
K{s)
—
all
—
(1
—
\fc, 1
c)
,
allow for private sunspots, and consider the set of
K* <
There exists a unique pair (q",p*), with
welfare.
<
and
<
economy
fluc-
q*
1
these equilibria are characterized by the followinq properties:
q* with probability p*
and K(s)
=
with probability
1
— p*
thai
;
tuates between "normal times", events during which aggregate investment
is
is,
the
positive,
and
"crashes",
events during which investment collapses to zero.
(ii) q*
and p* decrease with
c or h; that
is,
the probability of a crash increases,
and
the level of
investment in normal times decreases, as fundamentals get worse.
Proof. By the revelation
by a
c.d.f.
F
:
[0, 1]
—
»
[0, 1]
principle,
any equilibrium with private sunspots can be represented
such as the following hold:
first,
"Nature" draws q from F; next, a
"mediator" sends private messages that say "invest" to a fraction q of the population, while
private messages that say "don't invest" to the remaining fraction
individually rational to follow the action
identify the best equilibria
recommended
1
—
by studying the distributions
F
sends
finally, investors find it
q;
in their respective
it
messages. 10
We
can thus
that maximize welfare (ex-ante utility)
subject to the relevant incentive-compatibility constraints.
Take any F. Let
p,\
(resp., /iq)
be the
c.d.f.
the message "invest" (resp., "don't invest").
By
of the posterior
for'
an investor
who
receives
Bayes' rule,
f*(l-qi)dF(q>)
^q'dF(q')
Jo
about q
q'dF{q')
J Q (1
~q')dF{q')
For the recommended actions to be incentive-compatible, the expected net return from investing
must be
positive conditional
on the message "invest" and negative conditional on the message "don't
Restricting attention to pure strategies
is
immaterial because of the continuum of agents.
20
invest":
/
A (q) dfi\
W{F)=
where
w (q) = qA
(ex-ante utility)
>
(g)
=
r (q)
q)
A
0.
R (F) = f
Jo
and
(q)
best equilibria are thus identified by maximizing
F
F
For any
.
(10), these constraints
r
dF (q) <
(<?)
that satisfy these constraints
is
W(F)
(q)
dF
(q)
=
I
first
W(F) >
subject to
show that any solution
F
a contradiction, take any
F (q) = lim^^- F (q)
F
assigns to q G
panel of Figure
has an upward
it's
maximum
for all q
G
F
> R(F).
=
q
[k,
1,
the mass that
=
at q
= K*
By
K*) and
[0,
F (q) = F (q)
F
G
1). It
[re,
is
it is
decreasing in q for q G
-
[w (0)
w
(<?)]
F
11
1]. It
-R(F)=
4-
is
increasing for
h(l
— 2k) >
all
R(F) <
k)
F
is
0.
F
Towards
by letting
thus constructed from
and to q
= K*
> K*. As
all
F
the mass that
illustrated in the left
<
k; it
again continuous and strictly concave, reaching
for all q
G
[0, re)
and
w {K*) > w (q)
dF
(q)
+ [
[w [K*)
- w
{q)\
dF
(g)
>
0.
Jk
it
[0, re);
[re,
that maximize
improves welfare:
1,
has an upward
r (q)
jump
r
[r
(0)
[0,
k)
-
r (q)}
dF
(q)
q £
is
=
at q
follows that the variation
J0
r(q)
[0,
w (0) > w (q)
follows that
implication, the variation
increasing in q for q G
n That
> K*\
does not
G (0,K*).
to q
Clearly,
continuous and strictly decreasing in q for q
Next, as illustrated in the right panel of Figure
as long as c
for q
assigns to q £
and thereafter
re;
W(F) - W(F) = f
Jo
R(F)
that satisfy
[0, 1]
>
problem assigns zero measure
w (q) = qA (q)
the function
at q
—
K*}, while not affecting the mass assigned to q
jump
[«,!].
all
[0. 1]
F
that violates this property and construct a variation
€
for q
to this
:
.
W(F) >
non-empty and the constraint
the set of non-decreasing functions
by reassigning to
0,
W (F)
bind at the optimum. The remainder of the proof thus characterizes the functions
W{F) among
reduce to
that satisfies these constraints, welfare
+ (l-q)U(0,q)}dF(q)= f w
Jo
[qU(l,q)
f
Jo
the set of
Using
given as follows:
is
En=
The
-
(1
<
(q)
w(q)dF{q)>0
(
Jo
and
(q)
A (g) d(j,Q
and fQ
+ f
Jk
F
(1
=
— q)A(q)
re;
and
it
is
continuous and strictly
is
continuous and strictly
relaxes incentive compatibility:
[r
(K*) -
r (q)]
dF (q) <
0.
guaranteed by the assumption that K < 1/2 and holds more generally
0.
21
Figure
F
Since the variation
q
6
(0,
and a
q
The
1:
both
is
G
:
> K*. That
[A"*, 1]
is,
1
—p
->
feasible
We
A*) can be optimal.
c.d.f.
functions w(q)
[0, 1]
and welfare-improving, no
such that
F (q) =
the mass assigned to q
is
> A*.
It
Consider now the subproblem of choosing
b (p)
= —
—
(1
multiplier A p
that, for
R (G) <
any A p >
0,
the function
G
[0, 1]
G
x [K*
,
w
assigns
1]
all
q)A(q)
that assigns positive measure to
G
G
is
=
/^..
=
{q)
(1
-
p)
>
the mass assigned to q
p e
(1
-
+p
p) r (0)
This
(0, 1].
+pG (q)
A'*,
for
G
and
is
r(q)dG(q).
/
the same as maximizing
is
w (g) dG (q) R{G) =
,
maxg JK
f£. r
[w {q)
= argmax 9 gf^-.
q
W{p,
if it
.
- X p r (q)]dG
(q)
continuous and strictly concave
(q) is
but then (11) would be violated, since r (K*)
(11).
< K* and F
R (F) =
—
q)
qp
=
.
We
(1
= {l-p)r{0)+pr
must bind:
Lagrange multiplier associated with
p
for given
solves
that maximizes
(11)
0,
for q
[0, 1]
Jk"
measure to
R(p, q)
Note that the constraint
—
(q)
dG {q)
,
and
a convex optimization problem, there exists a Lagrange
— \p r
(q)
=
unique qp such that qp
exists a
implies that the optimal
(p, q)
is
such that the optimal
and therefore there
a pair
this
-p
1
and
where 1^ (G)
,
Because
p) r (0) /p.
>
6 (p)
F
(1
then also follows that
W{F) = {l-p)w(0)+p f w{q)dG(q)
Jk"
subject to
=
qA{q) and r(q)
conclude that, for any optimal F, there exists a scalar p G
the distribution of q conditional on q
W (G)
=
—
(1
Using the
22
i['u;
p)
(g)
w (0) + pw
<
Xp r
(q)],
over [A'*,
(q)
,
F
with
subject to
[IX)
optimum would be
w (0) =
1]
which in turn
0.
- K*)A(K*) >
fact that
—
(q)
in q
can thus identify any optimal
did not, the
=
1
But now note
.
0.
Thus,
and
r (0)
let
=
(p,q)
A*
=
>
.4 (0)
(1, A""),
be the
=
-c, the
Figure
Comparative
2:
first-order conditions for q
w'(q*)-Xr'{q*){
Recall that
<
r' (q)
q*
=
0,
or equivalently A*
K*.
If
=
p*
1,
or equivalently A*
-
(1
-
c
-
instead p*
2ft) / (1
<
if
q*
= K*
=
if
q*
€ (K*, 1)
>
if
q*
=
for all q
6
> c
-
-
(1
(1
-
c
<
ft)
c
—
(1
then (11) gives r
.1,
W (q*) -
+ c]
\[r (q*)
fo)
But then the
1.
(q*)
(I
—
ft)
/c;
=
left
part of (12) gives
—
c
c (1
h)
.
=
Hence, p*
1
=
-
p*) /p*
>
0,
which guarantees
ft
>
1
-
yfc\ if
this solution satisfies
instead
ft
<
1
—
A*
ft-(l-
ft
Note then that
ft^c"
=
p
%fc,
<j*
the
w (1) -
<
1
optimum
is
1
=
(1
1
A*
[r (1)
+
c]
- A*r'(l) >
if
c
if
that- q*
q*
<
<
(1
1
—
1
-
ft)
>
0,
and
2
.
If
and, along
:
" v^)^
(13)
2'
ft-(l-Vc)
y/c)'
and p* <
and only
if
(12)
this rules out
0,
possible only
is
the latter in turn holds
=
p*
and A* >
with (12), gives the following unique non-negative solution (q*,p*, A*)
9
if
while the right part of (12) gives w' (1)
/c,
-
=
Together with w' (A")
=
p*
ifp*e(o,i)
o
>
2h) /
c
if
=
<
1
=
-
following:
<
[AT*, 1].
-
0.65
0.6
statics of best private-sunspot equilibrium.
<
<
0.55
0.5
and p reduce to the
(11) implies q*
-
0.45
0.4
35
if
and only
attained with q*
if
=
c
1
>
(1
—
and p*
ft)
=
2
,
1.
or equivalently
QED
This result establishes that the best equilibrium with private sunspots has the economy alternating between "normal times",
"crashes",
i.e.,
states during
i.e.,
states during which a large fraction of the population invests,
which nobody
invests.
23
From
and
the perspective of the pertinent macroeco-
nomics
literature, this
seems quite, paradoxical: the occurrence of a crash
is
considered prima- facia
evidence of a coordinate failure, for the best equilibrium with no or only public sunspots would
never feature a crash.
The key
to this apparent
paradox
is
the incentive effect that the possibility of
a crash has during normal times. In particular, the fact that that
many but
not
all
investors invest
during normal times contributes towards higher welfare than in the best sunspot-less equilibrium:
the level of investment during normal times
is
now
However,
closer to the first-best level.
individuals to have an incentive not to invest during normal times,
it
must be that these individuals
many
believe that a crash will take place with sufficiently high probability, while
believe the opposite. In turn, such heterogeneity in beliefs
is
for certain
other individuals
possible in equilibrium only
if
crashes
do happen with positive probability, which explains the result.
To
further illustrate the economics behind the determination of the best equilibrium, Figure 2
considers
similar.
its
comparative statics with respect to
The dashed
line gives p*
A'*, the first-best level of
as
1
—
y/c
<
h,
,
c;
while the solid line gives q* For comparison, the dotted lined gives
.
investment
.
Note that
and that thereafter both
q*
and the probability
also falls,
sunspot-less equilibrium
always, that p*
falls,
c.
<
1
and
q*
<
1
as soon
In words, as the fundamentals
the equilibrium level of investment during
of a crash increases. This
is
true even though the best
invariant with the fundamentals.
is
Conclusion
6
The
pertinent macroeconomics literature on coordination failures and crises has focused on aggregate
extrinsic uncertainty.
In this paper, building
showed how the introduction
by
> K*
p* and q* decrease with
worsen, so that the first-best level of investment
normal times
the comparative statics with respect to h are
this literature uncovers
upon Aumann's concept
of correlated equilibria,
of idiosyncratic extrinsic uncertainty within the class of
some
intriguing positive
and normative properties
I
models used
that, albeit being
generic to this class of models, have not been appreciated within macroeconomics.
In one sense, the private sunspots considered in this paper capture the idea that agents face
uncertainty about which equilibrium
upon which other agents
is
played: each individual does not
are trying to coordinate.
This
is
know what
is
the action
similar to the strategic uncertainty
featured in the recent work on global games, but does not inhibit equilibrium multiplicity.
another sense, they capture idiosyncratic variation
24
in
investor sentiment:
In
different agents hold
endogenous prospects of the economy.
different expectations regarding the
Private sunspots can
thereby sustain significant heterogeneity in choices even in the absence of any heterogeneity in
primitive characteristics or hard information. These possibilities were absent from previous work:
in equilibria with public sunspots, all agents share the
face
same
about endogenous outcomes,
beliefs
no uncertainty about what other investors are doing, and play the same action.
Another intriguing
possibility
is
that, with private sunspots, social learning
can take place
with regard to endogenous coordination rather that exogenous fundamentals. In particular, asset
may
prices or other indicators of aggregate activity
among
prospects of the economy and better coordination
Finally, private sunspots
if
may
unrest the conventional
may be
is
nothing to be
situations, occasional
necessary for facilitating idiosyncratic uncertainty and thereby improv-
ing efficiency during normal times.
is
there
wisdom regarding the normative proper-
macroeconomic applications with coordination problems. In certain
investment crashes
failure
the agents, even
endogenous
them regarding the exogenous economic fundamentals.
learned from
ties of
facilitate better predictability of the
When
this
the case, what looks ex post as a coordination
is
actually contributing towards higher ex-ante welfare; and policies aimed at preventing
such apparent coordination failures
efficiency during
may
backfire by eliminating a social
mechanism that improves
normal times.
Appendix: dynamics and learning
Consider the dynamic extension of Section
with ut
and x
t
~
A/"(0,
= $~
}
s\
<7 U ),
(Kt)
+ £s,
Proposition 6 There
~
Af(0,cr s ), and a s
with e
exists a
information of an investor at
Normal with mean
Proof. The proof
st
is
t
t
~ N{0,cr £
)
3.
The sunspot
= —-. The
and
£
sequence {ml,at}'^L
with respect to
and variance a 2 and
,
(ii)
by induction. The result
St
private signals are given by
~ AA(0, c^).
(
is
Let
cr~
,
a^
trivially holds at
with the static benchmark. Thus suppose the result holds at
t
t
—
>
= oj
m< =
2
,
+
ps t
St
ut,
+ £t
a e = a~ 2
.
that (i) the private
in a sufficient statistic rht that is
an investor invests
25
au =
2
and an equilibrium such
summarized
—
an AR(1): st+\
s t follows
1.
at
1,
t
if
and only
since the
first
Then, Kt(s l)
if rht
>
m
t-
period coincides
=$
St
f
~m
ancj
t
J
x t+ i
=
-j^{st
— ml) +£t+ii where
with precision
0tct^. It follows
sufficient statistic
m +\
that
t
m *+i =
Normal with mean
is
a„(l+Q £ )/3
f
(a„+(i+« t )ft) (
(
But then, by a similar argument
+
f
which proves that the
Now
<
and only
-<>•
=
if
result holds at
the sequence
(ii)
/
m
[
»
t
-
*
au
+
summarized
(1
+
it is
t
+
1
= /^j,
2
<7
t
+1
_\1
a
W, Xt ) ) + ^T m
1
+ ae
where
„ii
14
,
,
*
+1
(
15
.
a4)ft
indeed a continuation equilibrium that an
1
(i
)}
(
•
is
nested with p
—
1
and a u
=
>
{/3t}£?.
oo, in
0.
finite
It
(a u n^,
,
follows that
a e ), the function T(0)
(i)
converges to this fixed point for any
infinitely long,
both m* and
which case
(15) reduces to 0t+i
k
=
=
c/b
1/2 into (16) gives mj
=
for all
t,
as claimed in the
>
initial
0o
such
>
0.
at converge.
= r(A) —
main
strictly
is
there exists a unique
then immediate that at decreases monotonically over time and asymptotes to
letting
i6 )
and completes the induction argument.
Finally, consider the variant with learning over a constant underlying sunspot (st
is
)
,
This proves the claim that, as the history becomes
This
a
in
> m*+1 where
rht+i
note that, for any p e (0,1) and any
T(0) and
Gaussian signal about s t
effectively a
{%Sr^v) + \l i+ ^~
increasing and strictly concave, with T(0)
that
is
and variance
ttt
i
as in Proposition 2,
1 if
m*n =
st+i
I+^ m + TT^
= r(A) =
ft+i
t
so that xt+\
,
t
that the private information regarding st+i can be
2
p
ft +1
investor invests in period
— a
0t
as
=
s for all t).
(!+&£) fit+&et
—
>
It
oo. Finally,
text.
References
|1]
Angeletos, George-Marios, Christian Hellwig, and Alessandro Pavan (2007), ''Dynamic Global
Games
of
Regime Change: Learning,
Mutliplicity and
Timing
of Attacks," Econometrica 75,
711-756.
(2]
Angeletos, George-Marios, and Ivan Werning (2006), "Crises and Prices: Information Aggregation, Multiplicity
and
Volatility,"
American Economic Review
26
96.
Economic Theory 25 380-96.
[3]
Azariadis, Costas (1981), "Self-fulfilling Prophecies," Journal of
[4]
Azariadis, Costas, and Roger Guesnerie (1986), "Sunspots and Cycles," Review of
Economic
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