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.A/1h45
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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Private Sunspots and Idiosyncratic
Investor Sentiment
George-Marios Angeletos
Working Paper 08-12
April 16,2008
RoomE52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research
Network Paper Collection
29348
http://ssrn. com/abstract^
1
1
at
Private Sunspots and Idiosyncratic Investor Sentiment
George-Marios Angeletos
MIT
and
NBER
April 16, 2008
Abstract
This paper shows how rational investors can have different degrees of optimism regarding the
prospects of the economy, even
fundamentals.
The key
can emerge as a purely
complements. This
class of
in
is
if
they share exactly the same information regarding
(ii)
It
economic
that heterogeneity in expectations regarding endogenous outcomes
self-fulfilling
equilibrium property
when investment
choices are strategic
turn has interesting novel positive and normative implications for a wide
models that feature such complementarities:
sentiment,
all
(i) It
can rationalize idiosyncratic investor
can be the source of significant heterogeneity
choices, even in the absence of
any heterogeneity
in real
and
financial investment
in individual characteristics
presence of a strong incentive to coordinate on the same course of action,
(iii) It
and despite the
can sustain rich
fluctuations in aggregate investment and asset prices, including fluctuations that are smoother
than those often associated with multiple-equilibria models,
investors learn slowly
(vi) It
JEL
how
(iv) It
can capture the idea that
to coordinate on a certain course of action,
can render apparent coordination failures evidence of improved
codes: D82,
(v) It
can boost welfare,
efficiency.
D84, E32, Gil.
Keywords: Sunspots, animal
spirits,
complementarity, coordination
tations, fluctuations, heterogeneity, correlated equilibrium.
failure, self-fulfilling
expec-
Introduction
1
Going back
to Kejaies,
many have argued
that animal spirits, market sentiments, or other forms of
extrinsic uncertainty can be the cause of aggregate fluctuations.^ In this
paper
argue that extrinsic
I
uncertainty can be largely idiosyncratic and can therefore also be the source of heterogeneity in real
and
financial in\'estment choices.
sentiment.
I
game
In so doing,
this exercise within
two closely related models. The
that abstracts from financial prices.
The common
mentarity
propose a rational theory of idiosyncratic investor
then explore some novel positive and normative implications.
I
conduct
markets.
I
in
The second
essential feature of the
is
two models
investment choices: an individual investor
first is
a simple real investment
a variant that stylizes trading in financial
is
that they allow for strategic comple-
is
more
willing to invest
when he expects
others also to invest. Such a complementarity could originate in a variety' of production, demand,
thick-market, or credit-related externalities analyzed in prior work.
To
deliver the central result of this
heterogeneity:
all
paper
in its sharpest form,
assumptions ensure that
all
investors
all
One may expect
the presence of a complementarity in investment choices:
when they only have a
The key
why
which identical investors make
is
uncertainty about the aggregate level of investment. That
may now
is,
if
optimal to make
now
exist
face idiosyncratic extrinsic
we take a snapshot
will find different investors holding different
in
of the
economy
at
expectations regarding endogenous
all
exogenous economic
"optimism" regarding the endogenous prospects of the
requires neither any differences in information regarding fundamentals nor any deviation
from Bayesian rationality; rather,
'The
it
should they do anything
economic outcomes, even though they hold identical expectations regarding
economy
their
different investment choices.
that individual investors
fundamentals. This idiosyncratic variation
These
if
not to be affected by
investors find
choices,
investment
desire to align their choices with one another? Yet, there
to this apparent paradox
any given point, we
level of
this conclusion
if all
same choice when they do not care about one another's
equilibria in
and share the same
other relevant economic fundamentals.
would choose exactly the same
choices had been strategically independent.
different
any exogenous source of
rule out
investors have identical preferences, face identical constraints,
information about exogenous productivity and
the
I
role of extrinsic uncertainty has
generations economies
(e.g.,
it
emerges as a
self-fulfilling
prophecy.
been formahzed within two related but distinct classes of models: overlapping
Azariadis, 1981, Azariadis and Guesnerie, 1986) and models with complementarities
(e.g., Benhabib and Farmer, 1984, Obstfeld, 1986; Chatterjee, Cooper and Ravikumar, 1993; Cooper and John, 1988,
Kiyotaki and Moore, 1997; Matsuyama, 1991; Weill, 1989).
Formally, this
used
is
achieved
b}'
the introduction of "private simspots".
Like the public sunspots
previous work, the private sunspots considered in this paper are payoff-irrelevant
in
variables.
random
But unlike public sunspots, private sunspots are only imperfectly correlated across agents
and are privately observed by them. The equilibria that obtain with private sunspots are thus closely
related to the correlated equilibria introduced in
game theory by Aumann
(1974, 1987):
when
there
are endogenous prices, the equilibria considered in this paper are hybrids of correlated equilibria
and rational-expectations
equilibria.
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
to read, or the interpretation of
cratic.
likely to
newspaper
in
search
coordinate on; the choice of what newspaper
what any given 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 ecjuilibrium
played. Indeed, while the pertinent work has been criticized for assuming
each individual agent
may
possibility that
be uncertain what action other agents are trying to coordinate, private
sunspots address this issue at
eciuilibrium.
away the
is
its
heart by generating such uncertainty as an integral feature of the
But no matter what interpretation one
gives to private sunspots, there
is
a
number
of
novel positive and normative implications that they deliver.
On
the positive front,
I
highlight that models with
ate significant heterogeneity in real
—or
and
macroeconomic complementarities can gener-
financial investment choices.
— any heterogeneity
Such heterogeneity can obtain
exogenous individual characteris-
even
in the
tics,
but only to the extent that individual incentives depend strongly enough on forecasts of others'
choices.
absence
It is
after controlling for
in
thus symptomatic of the "beauty-contest" character of financial and real investment
emphasized by Keynes.
Furthermore,
I
show how introducing idiosyncratic
extrinsic uncertainty can significantly enrich,
not only the cross-sectional, but also the aggregate outcomes of these models. In the two models
considered in this paper, with public sunspots aggregate investment and asset prices can only take
two extreme values
("high''
asset prices can follow
On
and
"low");
with private sunspots, instead, aggregate investment and
Private sunspots can thus generate
extreme values.
pubhc
and
smooth stochastic processes sparming the
sunspots, indeed fluctuations that are
the normative front,
I
much smoother aggregate
show that ignoring private sunspots may lead
in this
absence of sunspots: a "good" (Pareto-dominant) one in
which nobody
invests.
fluctuations than
more reminiscent of unique-equilibria models.
The models considered
policj' conclusions.
between these two
entire interval
to erroneous welfare
paper feature exactly two equilibria
which everybody
invests;
in
the
and a "bad" one
Adding public sunspots only randomizes among those two extreme
in
levels of
investment, achieving convex combinations of the welfare obtained in the two sunspot-less equilibria.
Therefore, as long as one restricts attention to public sunspots, one can safely draw two conclusions:
that the occurrence of an investment crash
policy interventions that preclude this
Neither conclusion
is
is
prima-facia evidence of coordination failure; and that
outcome
(at
no or small cost) are bound
warranted once one allows
for private sunspots.
that the aggregate level of investment in the "good" equilibrium
best.
Then one can construct an equilibrium with
is
to
improve welfare.
Suppose,
in particular,
excessive relative to the
private sunspots in which the
economy
first
fluctuates
between states during which only a subset of the investors invest ("normal times") and states during
which nobody invests ("crashes").
first-best level
level of
investment
is
now
closer to the
during normal times, this equilibrium can achieve higher welfare than the equilibrium
where everybody
normal times,
Because the aggregate
it
invests.
However,
for certain individuals to
must be that these individuals
high probability, while
many
have an incentive not to invest during
believe that a crash will take place with sufficiently
other individuals believe the opposite. But then note that, as long as
agents are rational, such heterogeneity in beliefs
is
possible in equilibrium only
if
crashes do happen
with positive probability.
Therefore, an occasional crash
— what looks as apparent coordination
failure
—
is
actually boost-
ing welfare by facilitating idiosjarcratic uncertainty and thereby providing the necessary incentive
that keeps investment from being excessive dining normal times.
It
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),
is
voluminous.
literature
Key
•
on macroeconomic complementarities, coordination
failures,
contributions include Azariadis (1981), Azariadis and Guesnerie
Benhabib and Farmer (1984), Cass and
Shell (198.3), Chatterjee,
Cooper and Ravikumar
Cooper and John
(1993),
Diamond and Dybvig
(1988),
(1983), Guesnerie
and Woodford (1992),
Howitt and McAfee (1992), Kiyotaki (1988), Kiyotaki and Moore (1997), Matsuyama (1991), Obstfeld
(1986,
1996),
and Woodford (1986, 1987, 1991).-
idiosyncratic extrinsic uncertainty. Although
of this class of models,
it
clearly illustrates
this
how
Aumann's
In so doing, the paper builds on
Although the main contribution
of these earher
as well as their welfare implications.
(1974, 1987) seminal
work on correlated
here
is
equilibria.
to identif)' a set of positive and normative implications that
is
have not been considered by prior applied work, a secondary contribution
correlation can be
works considers
the introduction of such uncertainty can enrich the
and aggregate outcomes of these models,
cross-sectional
None
paper uses only a highly stylized representative
accommodated within rational-expectations
is
equilibria.
that equilibrium prices convey information about the underlying
to
show how imperfect
The conceptual
common components
issue
of the
imperfect correlation devices that different agents observe, so that each agent's beliefs about these
sunspots are endogenous to the strategies of other agents.
between
beliefs
and strategies that
Layout. The
and
rest of the
paper
revisits the set of equilibria
This introduces a fixed-point element
is
absent in standard correlated equilibria.
is
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 6 concludes.
The
2
baseline model: a real investment
The economy
i
e
[0, 1],
are
is
game
populated by a measure- 1 continuimr of agents (investors), who are indexed by
endowed with one unit
of wealth each,
a safe technology and a risky alternative.
The
and decide how
to allocate this wealth
safe technology delivers a return
what, while the return of the risky technology depends on the aggregate
techirology.
The
payoff of investor'
TTi
where
ki
=
n{ki,K)
denotes investor
investment, and
A{K)
i's
i
=
is
(1
level of
R
between
no matter
investment in that
given by
- k,)R + k{R + A{K)) = R + A{K)k„
investment
in the risky technologj',
K
denotes the aggregate level of
the excess return of this technology relative to the safe one.
See Benhabib and Farmer (1999) and Cooper (1999)
for excellent reviews.
4
The key assumption needed
such that -4(A')
<
for all
K
for the positive results of this
<
k and A{I\) >
strategic complementarity in investment choices
one where
and
A{K) = -c <
let
One can then
minimal
think of
level of
c as
and guarantees the existence of two Nash
To
also the
Model
ki
mass
—
mentarity
(which
1
"^
ity
is
6
>
c
for A'
henceforth
I
work.
in prior applied
assume that investment
call
simply "invest") or
is
=
fc,
The
core element
is
At,
where
6
>
c
>
0.
indivisible:
each investor
("don't invest"), so that
the presence of strategic comple-
production, investment, or portfolio choices.
demand,
Such complementarity could
or thick-market externalities, as well as in credit
What
is
essentia.l
is
only that such complementarity opens the door to extrinsic
Note, however, that the framework introduced so far abstracts from
other signals of aggregate activity)
may
I
will deal
with this issue
Public sunspots. As noted above, the model admits exactly two
To
how
prices (or
limit idiosyncratic extrinsic uncertainty, a possibility that
evidently relevant for most applications of interest.
in Section 4.
equilibria in the absence of
see this, note that, in the absence of sunspots, the aggregate level of investment
deterministic,*^
and the best response of investor
k,
= BR (K)
i
is
parameters are
is
simply
1
if
A'
>
if
A'
<
—
common knowledge there is no uncertainty about the economic fundamentals.
no bite here because agents are risk neutral.
example, Diamond (1976) for thick-market externalities; Kiyotaki (1988) and Woodford (1991)
All these
*
henceforth normalize
For the purposes of this paper, modeling the deeper foundations of such complementar-
not essential.
sunspots.
agents
of agents investing.'*
in individual
uncertainty.
is
and
further
I
originate in a plethora of production,
frictions.^
k.
I
>
all
interpretation. This model can be interpreted as a highly stylized version of a variety
models considered
of
<
-
eciuilibria,
aggregate investment for which the technology becomes profitable, and b as the
can choose either
is
K
for
simplify the analysis,
A{K) =
(0, 1)
This assumption introduces
k.
parameterizing the cost of investing in the risky technology, k as the
gross benefit enjoyed in that event.
K
>
for all A'
that there exists a k G
is
agents invest their entire wealth in the risky technology and another where
all
invest their entire wealth in the safe technology.
R =
paper
Indivisibility has
^See, for
demand
for ag-
and Smith (1998), Kiyotaki and Moore (1997), and Matsuyama (2007) for
complementarities due to credit frictions; Chatterjee, Cooper and Ravikumar (1993) for complementarities in business
formation; Diamond and Dybvig (1983) and Obstfeld (1986, 1996) for coordinated bank runs and currency attacks;
and Cooper (1999) for an excellent review of the role of complementarities in macroeconomics.
^Because there is a continuum of investors, this is true even if investors follow mixed strategies.
gregate
externalities; Azariadis
follows that
It
one
in
=
[ki
investors, necessarily
all
which ever^'body invests
=
A'
everybody
make
—
the
K =
same choice and there
for
1
The one equilibrium
?').''
for all
{ki
all
i)
sustained by the
is
two
equilibria:
which nobody invests
in
self-fulfflling
the ether by the self-fulfilling expectation that
will invest;
exist exactly
and another
nobody
expectation that
will invest.
In either
no uncertainty about what choices other investors are making and perfectly
case, investors face
coordinate on the same course action.
Now
let
us introduce public sunspots. Before investors
a payoff-irrelevant random variable
function
on
s,
(c.d.f.)
F
is
§
:
—
whose support
is
make
their choices, they publicly observe
M
and whose cumulative distribution
S C
Because the investors can now follow strategies that
[0, 1].
>
s,
the aggregate level of investment can be stochastic. However, because
contingent
ai'e
publicly observed,
s is
the investors continue to face no uncertainty about the ecjuilibrium level of investment and continue
to
make
identical choices.
As a
result, equilibria
with public sunspots are merely lotteries over the
two sunspot-less
equilibria.
Proposition
For any equilibrium with public sunspots. there
1
with probability p and K{s)
an
—
which K{s)
equilibrium^ in
with probability
=
1
and actions vary across
Beliefs
I
—
p.
exists
ap ^
such that K{s)
[0, 1]
Conversely, for any p G
=
with probability p and K{s)
—
equilibria, or across realizations of the public sunspot,
1
there exists
[0, 1],
with probability I
—
p.
but never in
the cross-section of investors: in any given equilibrium and for any given realization of the sunspot,
all
investors share the
same "sentiment"
(i.e.,
the
same
perfectly forecast one another's choices, and end
section shows
about
all
endogenous outcomes), can
up taking exactly the same
of these properties need to hold once
we allow
action.
The next
for private sunspots.
Private sunspots and idiosyncratic sentiment
3
I
how none
belief
introduce private sunspots as follows. First, ''Nature" draws a payoff-irrelevant random variable
not observed by any investor.
s that
is
F
—
:
S
Conditional on
^When
.4(/y)
=
s,
0,
aggregate investment
A(k)
=
results.
0.
I
of this variable
is
S C
R
and
its
c.d.f.
Then, each investor privately observes a payoff-irrelevant random variable
[0, 1].
>
The support
m
is
i.i.d.
across investors, with support
M
C M and
c.d.f.
'I'
:
M
x S
—
rn..
[0, 1].
there also exists a mixed-strategy equilibrium in each investor invests with probability
is
is
k\
then k and investors are indeed indifferent between investing an.d not investing so long as
have ruled out this equilibrium by assuming A{k)
j^ 0.
This, however,
is
not essential for any of the
These variables define what
I
unobserved conmion sunspot
s.
call
I
they are private signals of the underlying
"private sunspots":
henceforth
call (§, F,
M, ^)
the "sunspot structure" and define an
equilibrium as follows.
Definition
An
1
equiUbrium with private sunspots consists of a sunspot structure (S,F,
a measurable strategy
fc
:
M—
{0, 1}
>
A:(m) G arg
M, $) and
such that
max
Vm
]l(k,K(s])dP(s\m.)
G M,
fc6{0,l}7§
with K{s)
—
f'^k{m)d^(m\s) Vs G
S,
and with P(sjm) denoting the
of the posterior about s
c.d.f.
conditimial on rn [as implied by Bayes' rule).
Note that the sunspot structure (§,F, M,\E')
environment. Rather,
it is
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
consider a specific Gaussian sunspot structure that best illustrates
the no^'el positive properties equilibria with private sunspots can lead
Gaussian sunspots. Suppose
variance a~
>
noise,
a.cross investors
i.i.d.
The
0.
the cross-section of agents.
s is
to.
drawn form a Normal distribution with mean
vii
=
with variance
a'^
private signal observed by investor
and independent of
s,
i
is
s
+
>
of s as the "brightness of the sun" or the "average temperature in a city"
measurement
only
if
error.
his private
The next
St,
0.
where
/ia
£i
is
G
Ci as
of the brightness of the sun or the temperature
is
think
idiosyncratic
proposition then constructs equilibria where an investor invests
measurement
and
Normal
One can then
and
R
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
(i)
An
(li)
(Hi)
a.ny (f.is.as.a^). there exists
investor invests
The aggregate
when
level of
m-
> m* and
investment
is
an
equilibrium, in
not when
m
< m*
,
(0, 1).
for som.e
stochastic, with full support
The cross-sectional distribution of expectations regarding
¥.[K\m], has full support on
which the following are true:
on
m* G
M.
(0,1).
the aggregate level of investment,
Proof. Let
m*
$
denote the
of the standard
c.d.f.
such that an investor invests
and only
if
=
A'(s)
and therefore
K (s)
> k
if
and only
s
>
s*
Because both the prior about
s
=
m > m*
if
Pr(77i> m*ls)
if
Normal
.
Aggregate investment
= $
and the signal
m
(2)
are Gaussian, the posterior about s conditional on
-z^—z^m + —z^—r^Us - s*
follows that the expected return from investing conditional
Note that
tlie
latter
6Pr
is
(s
>
s*\m)
-
c
strictly increasing in
6$
(
^JuT'^
E
;jr?^m* + ;;^?^M. -
s*
Substituting s* from condition
m*
(2) into (3)
follows from part
of
777.
conditional on
Note that
m
(ii)
s
(i).
[Ajm*]
Part
(ii)
=
0,
=
(a~'^
full
supports.
(3)
gives
(4)
,
then follows from condition
(1).
Finally, part
about the aggregate
individual investment choices
m
(iii)
and that
QED
about the distribution of the signals
In the equilibria constructed above, this
different expectations
.s*l
•
(f )
along with the fact that both the distribution of s conditional on
have
+ ar'^)"^.
is
or equivalently
+ Jl + ^$-i(f)
hold different expectations about the mass of investors
is
m
signal
\-^^^m + -rf^/i, -
different investors hold different expectations
in the population.
on
= y=f=ff *"'
and rearranging
(K)
which completes the proof of part
+ a7^
and variance Varfslml
m
m. For the proposed strategy to be part of an equilibrium,
satisfies
thus necessary and sufficient that
it is
=
(1)
= m* + ae^~\K).
Normal with mean Efslml
=
then given by
where
s*,
It
[A{K{s))\m]
is
an
exists
j,
I
is
E
Suppose there
distribution.
level of
means that
who have
received
different investors also
m>
777*.
The end
result
investment, which in turn sustain different
— a sharp difference from the case with public sunspots.
it
and choices can not be traced to any heterogeneity
this heterogeneity in expectations
Because
in primitive characteristics (preferences,
endowments, technologies, or payoff-relevant information),
This optimism
can be interpreted as idiosyncratic variation in "sentiment" or "optimism".
with regard to the endogenous prospects of the economy.
in expectations regarding the
Bayesian rationality. Rather,
is
does not require any heterogeneity
It
exogenous primitives of the environment, nor any deviation from
merely, and purely, a self-fulfilling equilibrium property.
it is
Finally, note that, in the equilibria constructed above, the aggregate level of investment has
full
In contrast, in the equilibria with no or only public sunspots,
support on the (0,1) interval.
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
sunspots, consider the following dynamic extension,
each period
each investor choses whether to invest
t,
contemporaneous payoff
and A{Kt)
if
<
/\t
K.
The
is
t is
t
Now
is
structure
rnt
i.i.d.
—
st
sj
=
+
c(,
where
is
where p G
Ui,
et is
may
the learning through payoffs.
i.i.d.
+
across time, with variance
of past aggregate investment
noise,
where Kt
an
is
1) or
infinite
not
=
b
—
c
>
li
0).
of private
of periods.
He then
In
receives a
/3
G
t
= —c <
k and A{Kt)
(0, 1).
The unobserved sunspot
in
(0, 1) is
the auto-correlation in the sunspot and Uf
The
private sunspot observed by an investor in
across agents
i.i.d.
and time, with variance
a^.
learn over time about past realized sunspots by the observation
and/or past
I
=
Kt >
simply ^fZo^^'^t, where
ct^.
white noise,
[kt
number
the a.ggregate level of investment in period
is
wherein the interesting dynamics enter.
is
pSi-\
note that investors
investment:
-4(/\;)/c<,
investor's intertemporal payoff
given by
white noise,
period
=
TCt
—
{kt
the net return to investment, with A{Kt)
is
The sunspot
period
There
also
payoffs.
To maintain the
analysis tractable,
assume that investors observe noisy private
each investor observes in period
f
a signal
=
3'(
^~^{Kt-i) +
^t,
I
ignore
signals of past
where
^j
is
white
across agents and time, with variance a?. These assumptions guarantee the existence of
equilibria in which the information structure remains Gaussian.^
^The assumption that
investors do not learn from their past payoffs
both time and agents, with variance
a^,.
=
is
merely for convenience and can be justified
zt
=
-^(/^i) +oJt
Suppose further that
Zt
is
as follows. Let the payoff of an investor be nt
Ztkt,
where
and where
tOt
is
white noise,
i.i.d.
across
privately observed by the investor, independently
would have emerged if zt was observed only
zt, which by itself is a noisy private signal
of h't. 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 cj is not Gaussian, making the updating of beliefs intractable.
of his choice of investment; this kills the value for experimentation that
when
kt
=
1.
Then, the observation of
ttj
conveys no more information than
Indeed, as
shown
which the following hold:
in
(i)
an investor invests
(ii)
statistic
mt and
find
a sequence {ml,at]'^Q and an equilibrium
the entire sequence of private signals
a sufficient statistic m^, which
and
we can
the Appendix,
in
iir
is
Normal,
period
t if
and
i.i.d.
onl}' if
up
to period
fh-t
> m^. Along
can be summarized
,s
and variance af;
this equilibrium, the sufficient
variance at can be constructed recursively as functions of (m^-i,
its
Moreover, as the history gets arbitrarily long, {rn^.at) converges to
We
t
mean
across inA'estors, with
in
o-(_i;
some time-invariant
thus obtain a stationary equilibrium along which aggregate investment
is
m^,
xt).
(m*,cr.)
given by
Hence, up to a monotone transformation, aggregate investment follows a smooth AR(1) process.^
Note then that
fictitious
data generated by the present model would be virtually indistinguish-
able from fictitious data generated by a canonical unique-equilibrium model.
the case
if
we
and
discrete fluctuations (between
1),
This would not be
with public simspots, aggregate investment features
ha,d ignored private sunspots:
which would be more
telling of multiple equilibria.
We
conclude that private sunspots can help generate very smooth aggregate fluctuations, making
difficult to identif}' fluctuations
it
driven by sunspots from fluctuations driven by smooth changes in
the underlying fundamentals.
"Learning to coordinate." As another example
can sustain,
I
now
=
constant over time: p
before,
we can
777,(
and
and du
1
find
=
1/2,
by Kt{s)
which gives
— ^
i-^]
.
converges to zero as
ttIj
=
xt considered above,
0,
so that St
an ecjuilibrium
his sufficient statistic int exceeds
c/b
=
in
=
but no the unobserved sunspot remains
s for all
t
It
t.
It
1/2),
^ oo avoids this problem
iUt
is
cr^,
follows that,
by rendering the signal
whenever
1;
s
signals, at
precise,
>
0,
zt
h'tis)
and whenever
is
if
simplicity,
and only
suppose k
s
<
t
is
if
=
given
is
bounded
0, I<^t{s) is
in
(1/2,1) and
bounded
inside
0.
uninformative. At the same time, because the expectation of
investors continue to choose kt so as to
^~^(Kt)
t
decreasing over time and
is
maximize
their expectation of A{Kt)kt-
that the error introduced by ignoring the information contained in payoffs vanishes as
^To be
period
follows that aggregate investment in period
and decreasing over time, asymptotically converging to
(0,
a^
in
some deterministic threshold m^. For
for all
Because of the accumulation of new
-^ cc.
t.
which an investor invests
increasing over time, asymptotically converging to
zero no matter
dynamics that private sunspots
consider the following variant. Investors continue to receive the exogenous and
endogenous private signals
As
of the rich
a Gaussian AR(1).
10
cr^
—^ oo.
It
follows
now
Recall
that
K
—
1
and
K
—
represent the only two equilibria that are possible in the
absence of private sunspots and that require
We
investors coordinating on the
all
same course
of action.
can thus interpret the dj'namics that obtain here with private sunspots as situations where
some
investors slowly learn on which action to coordinate: at any given date,
the "wrong" investment choice
who makes such a mistake
investors
investors are
making
do the opposite of what the majority does), but the fraction of
(i.e.,
falls
over time and vanishes in the limit.
Also note that this form of learning can be either exogenous or endogenous:
can originate
it
in
either the signals nit 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 ma,y
same course
perfectly coordinate on the
to do so over time through the observation of one another's actions.-"^
4
Private sunspots and financial markets
market
analysis has been conducted within a simple investment
interactions.
I
now
game
that abstracted from
consider a variant model in which investors trade an asset within a com-
petitive financial market. This exercise serves
preceding analysis translate
relation can be
to
of action, but also the possibility that agents slowly learn
how
The preceding
fail
two purposes.
First,
it
shows how the insights of the
in the context of financial markets. Second,
it
shows how imperfect
accommodated within a rational-expectations-equilibrium framework, where
partially revea.l the unobserved
common
sunspot component that drives the correlation
cor-
prices
among
the
beliefs (the private sunspots) of diiTerent investors.
Model
much
set-up. There
is
again a large
to trade of a certain financial asset.
number
An
individuaFs investment
and the aggregate investment by K. The price
The
later
this
is
is
meant
The form
signals
assumed
of risk-neutral investors,
of the asset
is
in
who now
the asset
denoted by p and
to increase with aggregate investment in the asset:
A=A
is
its
decide
how
denoted by
ki
dividend by A.
(A').
Once
again,
to capture, in a crude way, a variety of feedback effects identified in prior work.-'^
of social learning considered here
about either
s or
past activity,
A
is
purely private, but one could easily extend the analysis to public
certain kind of public signals that
is
of special interest
is
prices; this bring
us to the topic of the next section.
'
'
For example, Ozdenoren and Yuan (2008) argue that the higher the position of institutional investors
in
the stock
company, the better the monitoring of the management of that company, and hence the higher return
and the higher the demand for that stock; Subrahmanyam and Titman (2001) stress the role of complementarities
among the customers, suppliers, or employees of a company; and many others emphasize feedback effects from stock
prices to capital availability, and therefrom to firm profitability and ba,ck to stock prices.
of a particular
11
Since investors are risk-neutral, their payoffs are simply given by
= [A{K)-p]k,.
TT,=U{k,J<,p)
To
rule out infinite positions,
bounds can be interpreted
assume that
Without any further
which
is
denoted by Q,
unobserved supply shock;
Q=
the impact of "noise traders";
fc,
is
bounded
assumed
Q{p,u)
,
which
is
drawn
some
UC
—
Finally, the
1.
The shock u can
M.
(Allowing for
to ensure that investors take finite
and k
first
supply of the
and of some
also be interpreted as
draws an unobserved
distribution F; nature then sends each agent
aggregate and idiosyncratic extrinsic uncertainty.
the previous section
=
fc
These
k.
i
common
a private signal
sunspot
m, € M,
across agents from a conditional distribution $. These variables are payoff-
i.i.d.
and are independent of the supply shock
irrelevant
k and
finite
to introduce noise in the price.
Private sunspots are introduced as before; nature
variable s £ § from
some
to be an increasing function of the price
where u £
its sole role is
way
less tractable,
loss of generality, let
is
in [k,k], for
borrowing and short-selling constraints.
as the result of
would be another natural, but
risk aversion
positions.)
asset,
I
u\
they are once again devices that introduce
What
is
novel here relative to the model of
that the price that clears the asset market
is
may
publicly reveal information
about these sunspot variables. This motivates the following equilibrium definition, which introduces
private sunspots within an otherwise-standard rational-expectations equilibrium concept.
Definition 2 A rational- expectations equilibrium with private sunspots consists of a sunspot structure (S, F,
and
M,
a price function
v[/),
a belief (c.d.f.)
/z
:
S x
M
(i) Beliefs are consistent
(ii)
Given the
/c
(??),
beliefs
p) G arg
and
max
x
P
Mx
;
M ^ M, an individual demand function
—
M
[0, 1], such that the following hold:
Sx
fc
(iii)
=
Jj^
M
x
K—
*
[k,k],
>
with Bayes rule given the equilibrium, price function.
the price function, the
Il{k,
/
demand function
satisfies individual rationality:
K {s,P {s,u)) P {s.u)) dfi{s,u\m,p)
,
\/{rn,p),
fce{o.i}ysxU
where K{s,p)
;
k[m.,p)d^{m\s) Vs £
S.
Given the demand function, the price function
satisfies
K{s,P{s,u)) = Q{s,u)
12
market- clearing:
V(s,u).
As
in
most rational-expectations models, the
and functional forms.
of distributional assumptions
u
^ N
(0,(t2)
,
~
s
independent of both
if
K
>
some
K and -4(A')
scalar A
>
(/i.sjCTg)
and
s
=
u.
I
and m,
,
=
+
s
~
A''
otherwise, for
some
£
scalar k
(0, 1):
fact that supply
(OjCXg)
and
c.d.f.
of the standardized
Normal
all
uncertainty
is
Q (p,u)
is
Gaussian:
and
across agents
i.i.d.
A and Q A{K) =
:
—$
(u
+
/\<I>"-^
(p))
,
As a
is
result, the
aggregate
for
noise in prices), the price
is
is
<E>
establishes the existence of rational-expectations
demand
in the price
for the asset
is
and increasing
increasing in
increasing in u, this ensures that the equilibrium price
Becciuse the supply shock u
1
distribution.
which investors' demand functions are decreasing
private sunspots.
u.
£,,
further impose the following functional forms for
Equilibrium analysis. The next proposition
equilibria in
and
thus assume that
I
where
e^,
intractable without an "artful" choice
is
This scalar parameterizes the price elasticity of the supply of the asset, while
0.
denotes again the
N
anal3'sis
unobserved
(recall, this
is
s.
in their
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.
equilibria feature different investors finding
even though they
component
all
As a
result, these
optimal to make different portfolio choices,
share the same preferences, constraints, and beliefs regarding any exogenous
of asset returns
heterogeneity
strictly
it
in beliefs
— heterogeneity
in portfolio choices originates
merely
in self-fulfilling
regarding the endogenous component of asset returns.
Proposition 3 For any
((7,j,/\),
there exists a rational- expectations with private sunspots
m
which
the following are true:
An
(i)
investor's equilibrium
demand
for the asset
k {m,p)
is
given by
,lifm> m* [p)
otherwise
where
m*
(p) is
a continuous increasing function of p.
asset, A'(s,p), is continuously increasing in s
(ii)
The equilibrium price
is
given by p
By
implication, the aggregate
and continuously decreasing
= P
{s,u)
of s and a continuously decreasing function of u.
13
,
where
P
is
demand for
the
in p.
a continuously increasing function
Proof. Consider a sunspot structure such that
suppose there exists an m*
demand
is
K (s) = Q
for all {s,u). Since the function
the observation of p
is
signal
= —a^u
771,
Normal
is
+ a^A^"^
(p)
noise with variance
=
o"~
=
and variance Far[s|m.,p]
m
(6),
=
[.4|77?,p]
the latter
if
m
>
is
satisfy
m*
(p)+(JeA<J>~'' (p)
in equilibrium
(p)
(jrcr„.
""post
$
771**
m*
,
=
s
+
(and so are
=
CTejA,
s—a^u,
and $),
n,
(5)
Because the prior about
s,
the private
is
(p)
(p)
=
=
M5
It
777,**
=
-
771*
(p)
.
Pr
(p)
[s
-
> m*
(p)
(6)
^^
follows that the expected
is
+
u.^^'^^k)
\
m,p\
a,-J>-i(K,)))
follows that an investor finds
(p)
{E[s\m**
771*
-
(p)
'^post
+ '^r^ + '^n")"
(""J'^
k|7T7,p]
(E[s|7n,p]
777.
where
(^
^
Cpost
Pr [K{s,p) >
increasing in
77i** (p)
In any equilibrium,
Given the
rn* {p).
is
= $ (-L-
and only
must
^Ms + ^m + ^^
=
where
cTp^^f,
dividend conditional on signal
By
>
Gaussian, the posterior about s conditional on rn and p
all
""post
if
m
mean
E[s|m,p]
E
if
5>
Equivalently, p
.
= m*
and the public signal z are
also Gaussian, with
=
common knowledge
is
and only
Next,
informationally equivalent to the observation of the signal
z (p)
where n
u)
(p,
m*
if
1.
given by
K (s,p)
Market clearing imposes
+ cr~^ + iT~-f7,7")~-'/-cr~^a^~ >
such that an investor invests
(p)
proposed strategy, aggregate
Acr£(cr~'^
it
optimal to invest
if
and only
the unique solution to
(p) ,p]
-
Along with
+ (r,^-\K) +
777,*
(p)
- as^-\h'))) = p
(6), this gives
[Xa,apost(r-^
14;
-
l]
a unique solution for m*{p):
a^s^'^lt^'^ (p)
(")
We
conclude that there exists a unique equilibrium
k {m,p)
1
=
if
demand
m
>
function and
777*
is
given by
(p)
otherwise
with
777*
By assumption,
(p) as in (7).
uously increasing
in
>
1'
which guarantees that
p and hence the equilibrium demand
Along with the
in p.
Xa^apostf^n'
supply of the asset
fact that the
for the asset
777*
from
(p)
p
(7) into (5)
= PT-,f(s.u) = ^
^
;f.
and solving
s
for p.
- a,u -
I
'
[Xa.apost (o-„^
which
is
continuously increasing in
s
+
p = P{s,u) The
.
this also
p,
latter
is
found
Doing so gives
-
CTe^'H^^)
0-2)
-
1] cr~cr.p^^^^
and continuousl}' decreasing
In the equilibria constructed above, the aggregate
in its price
ps
a contin-
is
(p)
continuously decreasing
continuously increasing in
is
guarantees that there exists a unique equilibrium price function,
by substitutiirg
is
777*
demand
in u.
QED
for the asset is globally
decreasing
and therefore intersects only once with supply, Moreover, these equilibria feature only
smooth fluctuations
demand-supply
This
in asset prices.
intersections,
is
unlike the backward-bending
demand
and discrete price changes (crashes) featured
in
functions, multiple
Angeletos and Werning
(2006), Barlevy and Veronesi (2003), or Ozdenoren and Yuan (2008). Therefore, an outsider could,
once again,
fail
to detect any ob\'ious
symptoms
of multiplicity and could
fail
to identify 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
no
In a certain sense, this
is
1)
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
15
I
model
of Section
4.
In that
have assumed an exogenous supply
of the asset:
modeled the "noise traders" that he behind
ha.ve not
I
We
this supply.
can nevertheless
bypass this complication by focusing on the welfare of the investors that have been modeled
and the ones behind the supply
of the latter as domestic agents
as "unloved" foreigners.
— think
Note then
that higher aggregate investment implies a higher price at which the asset can be acquired. As a
although domestic investors are better
result,
in
which
K
£
K
=
they would have been even better off
0,
(k, 1), for
off in the
equilibrium in which
if
they could
K
=
than the one
1
somehow coordinate on some
they would ha.ve then guaranteed the same rate of return at a lower price.
Whenever there
are such inefficiencies,
it
natural to think about Pigou-like policies that
is
correct these inefficiencies and implement the 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.
will
I
now show how
private
sunspots, unlike public sunspots, can then improve welfare.
Towards
investment
now
is
1
f
{
G
(0, ^]
a congestion
or,
more
and h >
effect:
0.-^^
The
baseline
-
hK
hK
model
is
The equilibrium
K = 0,
in which
if
1
—
c)
in
which
K
—
K
<
K.
(8)
=
0.
Allowing h
>
introduces
1
equilibria,
one with
(i)
for all
agents investing
I
-
c
"Letting 6
=
follows from the fact that ^(A')
K
is
€
[«,!].
K: w{K)
— h and
Finally, for part
1
is
1
and another with
K=
0.
as well as than any equilibrium, with public sunspots.
<
>
K=
achieves higher welfare (ex-ante utility) tlian the equilibrium
Proof. Part
=
>
-
.
is
w{l)
K
K
nested with h
(Hi) The first-best level of aggregate investm-ent
.4(A')
if
a negative externality similar to the pecuniary externality featured in Section 4
There exist only two sunspot-less
(ii)
c
generally, a source of inefficiency in the best sunspot-less equilibrium.
Proposition 4 Suppose h G (^^,
(i)
-
-c -
1
k,
net return to
given by
A{K) =
where
The
model.
this goal, consider the following variant of the baseline
(iii),
w{0)
~
Now
=
0,
note that
merely
a
let
K* G
[k,1).
for all
KG
[0,
uj(K) denote welfare (ex-ante
A'TI(1, A') 4- (1
-
A')n(0, A')
=
k) and, as long as
utility)
when
A'^(A'). For part
/i
(K)
is
1
—
c,
the fraction of
(ii),
note that
so that the result follows a.gain from the assumption h
w
<
<
1
—
c.
continuous, strictly decreasing, and strictly concave for
normalization, while k
<
I,
simplifies a step in the prool' of Proposition
16
6.
K
<
it
k:,
thereafter
for
K
>
jump
has an upward
K,
it
K
a.t
—
k
which point
(at
A'*
=
=
—
1
arg
- 2hK,
c
so that the first-best level of investment
m^x w{K) =
i^zs
I
g (^ j)
1
The key
right- but not left-continuous);
again continuous and strictly concave, but possibly non-monotonic.
is
w' {K)
Therefore, A'*
it is
<
1
if
and only
result here
if 1
-
c
-
2/iK
if 1
-
c
-
2/?
<
if 1
-
c
-
2/i
>
In particular,
is
given by
-
c
<
<
1
- 2hK
(9)
h> ^. QED
if
that, as long as the congestion effect
is
and
is
<
not too high {h
1
-
c),
there
continue to exist exactly two equilibria in the absence of sunspots; but, as long as the congestion
effect
is
not too low {h
>
^y^), neither equilibrium
not improve upon those two equilibria
is
is
clear: public
That public sunspots can
first-best efficient.
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 A'
=
Proposition 5 Whenever
1.
/i
This, however,
is
—
c)
£
—
(1
tain strictly higher welfare than
\/c, 1
any of the
not true once we allow for pri\'ate sunspots.
,
there exist equilibria with private sunspots thai sus-
equilibria with
no or only public sunspots.
This result can be established with a specific example.
Here,
acterizing the best possible equilibrium with private simspots.
equilibrium properties are necessary for efficiency
to contrast
them with those that one
Proposition 6 Suppose h G
[1
equilibria that m.aximize welfare.
p*
<
1,
(i)
such that
K{s)
=
all
—
identifies
\/c,l
—
c)
,
if
when one
one
I
go one step further by char-
This permits
me
to identif}^
allows for private sunspots
what
— and then
restricts attention to public sunspots.
allow for private sunspots, and consider the set of
There exists a unique pair {q*,p*), with
K* <
q*
<
1
and
<
these equilibria are characterized by the following properties:
q* with probability p*
and K[s)
=
with probability
1
-p*
;
tuates between "normal tim.es", events during which aggregate investment
that
is
is,
the
positive,
economy flucand
"crashes",
events during which investment collapses to zero.
(ii) q* a.nd p*
decrease with c or h; that
is.
the probability of a crash increases,
investm.ent in norma.l times decreases, as fundamentals get worse.
and the
level of
Proof. By the revelation
by a
F
c.d.f.
:
-^
[0, 1]
any equilibrium with private sunspots can be represented
principle,
[0, 1]
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
recommended
I
—
identify the best equilibria by studying the distributions
F
sends
investors find
finally,
q;
in their respective
it
messages.
^'^
We
it
can thus
that maximize welfare (ex-ante utility)
subject to the relevant incentive-compatibility constraints.
Take any F. Let
(resp., ido)
fj.i
be the
c.d.f.
By
the message "invest" (resp., "don't invest").
and
—1
who
receives
Bayes' rule,
J^q'dFjq')
=
(q)
/^i
of the posterior about q for an investor
uq (q)
^
f^{l-q')dFiq')
—,
10
J^il-g')dF{q')
f^q'dF{q')
For the recommended actions to be incentive-compatible, the expected net return from investing
must be
positive conditional
invest": Jq
A
{q)
d/j,i
(q)
W {F)~
>
on the message
and
w
i
A (g)
Jq
dF
(q)
{q)
>
—
q)
"invest"
dfio (?)
<
and negative conditional on the message "don't
0.
Using
R (F) = f
Jo
and
Jo
where
w
(q)
= qA
(ex-ante utility)
(q)
and
(I
A {q)
.
For any
I
[qU{l,q)
Jo
F
dF (g) <
that satisfy these constraints
is
W{F)
(q)
dF
subject to
^
{q)
W {F)
W{F) >
non-empty and the constraint
I
first
the set of non-decreasing functions
show that any solution
a contradiction, take any
F [q) =
lim^_^-
F (q)
by reassigning to g
=
F
for q
all
G
to
tliis
F
:
[0,1] -^ [0, 1]
W{F) >
that satisfy
problem assigns zero measure to
q
.
> R{F).
bind at the optimum. The remainder of the proof thus characterizes the functions
W{F) among
0,
that satisfies these constraints, welfare
+ {l-q)n{0,q)]dF{q)^ f w
Jo
best equilibria are thus identified by maximizing
the set of
F
r (q)
reduce to
given as follows:
is
Ett=
The
=
r {q)
(10), these constraints
F
[0.
A'*)
and
Restricting attention to pure strategies
is
F (q) = F [q)
F
assigns to g
for q
G
[0,
> K*; F
k)
is
F
Towards
by letting
thus constructed from
and to g
= K*
immaterial because of the continuum of agents.
18'
0.
(0,/\'*).
that violates this property and construct a variation
the mass that
does not
that maximize
R{F) <
G
Clearly,
all
F
the mass that
.
Figure
F
G
assigns to q
panel of Figure
it's
maximum
G
for all q
at q
—
&t q
=
By
[k, 1].
w
the function
jump
has an upward
=
functions w{q)
A'*
e
{q)
= qA
[q) is
and thereafter
k;
[k, 1). It
[w (0)
/
continuous and
it
follows that
implication, the variation
W{F) - W{F) =
-
u'
((?)]
is
F
dF
{q)
1,
has an upward
+
=
r (q)
jump
it
follows that the variation
(1
at q
F
q
G
(0, A'*)
and a
q
c.d.f.
G
:
> K* That
-^
[A'*, 1]
is,
.
I
—p
is
We
[0, 1]
conclude that,
such that
the distribution of g conditional on g
''That r(q)
as long as c
+
is
increasing for
h(l
-
2k)
>
for
all
>
A'*. It
f w{q)dG{q)
JK'
q
<
£
for all q
k)
[0,
w
and
(A'*)
k; it
> w
-w
— q)A{q)
~
and
k;
it
(q)]
dF
>
(q)
(q)
0.
is
continuous and strictly
is
continuous and strictly
[r
{!<*)-
r
{q)]dF
<
{q)
0.
£
[0, ac) is
=
F
that assigns positive measure to
any optimal F, there exists a scalar p G
F (q) = 1 - p
the mass assigned to q
WiF)^{l-p)w{0)+p
decreasing in q for q
strictl_y
both feasible and welfare-improving, no
can be optimal.
illustrated in the left
Jk
7o
is
.
relaxes incentive compatibility:
R{F)-R[F)= r[r{0)-riq)]dF(q)+ [^
F
> K* As
[w {!<*)
/
decreasing in g for g S
It
q)A{q)
improves welfare:
increasing in q for q G [0,k.);^^
[/i;,!].
—
(1
again continuous and strictly concave, reaching
w (0) > w {q)
Next, as illustrated in the right panel of Figure
Since the variation
=
qA{q) and r{q)
while not affecting the mass assigned to q
[k, A'*],
1,
The
1:
0,
for q
p
is
< K* and F
{q)
=
{1
-
the mass assigned to g
p)
>
[0,1]
+ pG (g)
A'*,
and
for
G
is
then also follows that
and
R{F) =
{I
-
p) r {0)
+p
I
r {q)
dG
[q]
JK'
guaranteed by the assumption that k < 1/2 and holds more generally
0.
19
G
Consider now the subproblem of choosing
W (G)
b{p)
- p)r
(1
>
multipHer Xp
that, for
and
<b (p)
RiG)
subject to
= —
Because
(0) /p.
W (G)
where
,
this
w
the function
0,
a pair
G
{p, q)
x
[0, 1]
G
[A'*,
assigns
—
Note that the constraint
must
(11)
bind:
but then (11) would be violated, since
Lagrange multiplier associated with
first-order conditions for q
<
if
<
r' (q)
=
0,
or equivalently A*
A'*. If
=
p*
(1
c
-
instead p*
2h) /
<
g*
for all q
G
=
(1
> —
-
c
-
(1
(1
/7.)
XpV {q)]dG
Note then that
ft
>
1
—
^/c]
if
=
[q)
But now note
.
(1
—
+ pr
-
,
turn
F
with
— p)w
{1
[A'*, 1]
in
(q)
<
+ pw
(0)
XpV
(g)],
subject to
(q)
0.
(11)
optimum would be
>
A'*). 4 (A'*)
w
[0]
=
+
c]<
Thus,
0.
and r
w{q*)-
— c—
—
c
<
—
(1
Together with
But then the
1.
=
(p, g)
(1,K*),
>
A*
let
be the
= A (0) = — c,
(0)
the
left
u.-'
=
(A'*)
c
-
—
/i)
=
—
c
—
c (1
li)
.
Hence, p*
p") /p*
>
0,
ifp*G(0,l)
>
if
p*
=
1
X*
h
the
<
I
(1
- ^cY
w (1)
and p*
optimum
is
20
<
1 if
and only
if
/i
-
and only
attained with q*
A*
-
(1)
g*
<
<
q*
[r
+ c] >
>
X*t' {!)
if
c
if
(12)
1
this rules out
0,
—
possible only
is
(1
=
=
p*
and A* >
which guarantees that
h^/d
h
i/c,
if
part of (12) gives
the latter in turn holds
/c;
1
<1 —
=
<
=
/i)/c, while the right part of (12) gives w' {!)
2/i) / (1
-
X[r{q*)
1
this solution satisfies q*
instead h
and
,
which
with (12), gives the following unique non-negative solution {q*.p*,X*)
q
(q)
can thus identify any optimal
A'*
=
then (11) gives r [q*)
1,
We
Using the fact that
[A'*,l].
1, (11) implies q*
or equivalently A*
-
=
g*
if
<
=
(0)
=
if(?*G(A'M)
>
q*
dG
/^,. r {q)
and p reduce to the following:
w'{qn-Xr'{q*){ =
Recall that
qp-
did not, the
if it
(K*)
r
(11).
—
(q)
argmaXgg[/^'.j]['U) (q)
=
q
= {l-p)r
R{p., q)
the same as maximizing
continuous and strictly concave in q over
(q) is
that maximizes W{p, q)
1]
is
,
ma.XG Jj^,[w
solves
measure to
all
This
(0, 1].
w (q) dG (q) RiG) =
/^..
therefore thei-e exists a unique qp such that qp
implies that the optimal
-
G
- Apr
(g)
=
p G
a convex optimization problem, there exists a Lagrange
is
such that the optimal
any Xp >
for given
(1
I
=
0,
and
1
- hf
.
If
and, along
:
- V~cf
- v^)-
(13)
(1
if
=
c
1
>
(1
—
a,nd p*
h)"
=
or equivalently
,
1
.
QED
Figure
Comparative
2:
statics of best private-sunspot equilibrium.
This result establishes that the best equilibrium with private sunspots has the economy
nating between "normal times",
i.e.,
and
which nobody
"crashes",
ature, this
i.e.,
which a large fraction of the population
states during
failure, for
The key
is
considered prima-facia evidence of
the best equilibrium with no or only public sunspots would never feature
to this apparent
paradox
is
the incentive effect that the possibility of a crash
has during normal times. In particular, the fact that that
normal times contributes towards higher welfare than
of investment during
invests,
invests. R'onr the perspective of the pertinent liter-
seems quite paradoxical: the occurrence of a crash
a coordinate
a crash.
states during
alter-
normal times
now
is
many but
not
all
investors invest during
in the best sunspot-less equilibrium: the level
closer to the first-best level.
viduals to have an incentive not to invest during normal times,
it
However,
must be that these individuals
believe that a crash will take place with sufficiently high probability, while
believe the opposite. In turn, such heterogeneity in beliefs
is
for certain indi-
many
other individuals
possible in equilibrium only
if
crashes
do happen with positive probability, which explains the result.
To further
considers
similar.
its
illustrate the
comparative statics with respect to
The dashed
line gives p*
A'*, the first-best level of
as
1
—
^/c
<
economics behind the determination of the best equilibrium, Figure 2
h, aird
,
c;
while the solid line gives q* For comparison, the dotted lined gives
investment
.
.
Note that
q*
> K*
always, that p*
that thereafter both p* and q" decrease with
worsen, so that the first-best level of investment
normal times also
the comparative statics with respect to h are
falls,
sunspot-less equilibriimi
falls,
and the probability of a crash
is
1
and
g*
<
1
as soon
In words, as the fundamentals
the equilibrium level of investment during
increases. This
invariant with the fundamentals.
21
c.
<
is
true even though the best
6
Conclusion
The
pertinent literature on macroeconomic complementarities and endogenous fluctuations has
cused on aggregate extrinsic uncertainty. In this paper
tainty
and showed how
this
fo-
introduced idiosyncratic extrinsic uncer-
can lead to novel positive and normative implications.
In one sense, the private sunspots considered
uncertainty about which equilibrium
upon which other agents
I
is
in this
paper capture the idea that agents face
played; each individual does not
are trying to coordinate.
^^
know what
is
the action
In another sense, they capture idiosyncratic
variation in investor sentiment: different agents hold different expectations regarding the endogenous
prospects of the economy.
public sunspots,
all
These
possibilities
were absent from previous work:
in equilibria
with
agents share the same beliefs about endogenous outcomes, face no uncertainty
about what other investors are doing, and play the same action.
The heterogeneity
of beliefs regarding endogenous economic
outcomes obtained
in this
paper
does not not require any heterogeneity in information about exogenous economic fundamentals, nor
any deviation from Bayesian
Furthermore,
it
rationality.
Rather,
it
obtains as a self-fulfilling eciuilibrium property,
sustains significant heterogeneity in choices even in the absence of any heterogeneity
in primitive characteristics. It
can thus show up as significant "residual" variation
in
any econometric
exercise that attempts to explain the observed heterogeneity in investment or portfolio choices on
the basis of heterogeneity in individual characteristics such as wealth, risk aversion, or information
regarding economic fundamentals. At the same time,
fluctuations, possibly
for
making
it
can sustain richer and smoother aggregate
it
easier for this class of
models to match aggregate data and harder
an econometrician to detect the underlying multiplicity of equilibria.
Another intriguing possibility
is
that, with private sunspots, social learning can regard endoge-
nous coordination rather that exogenous fundamentals.
In particular, asset prices or data about
aggregate activity ma,y facilitated better predictability of the endogenous prospects of the economy
and better coordination among agents, even
if
there
is
nothing to be learned from them regarding
the exogenous economic fundamentals.^^
In this respect, the paper also relates to the recent
and Werning, 2006). This
how
work on global games
(e.g.,
Morris and Shin, 2001; Angeletos
literature introduces heterogeneous information regarding the fundamentals
and studies
the resulting strategic uncertainty affects equilibrium outcomes, in certain cases selecting a unique equilibrium.
In contrast, private sunspots
interesting question
in global
'^The
is
how
accommodate
strategic uncertainty without obstructing equilibrium multiplicity.
the two sources of strategic uncertainty
games with multiple
is
interact
if
An
one introduces private sunspots
equilibria.
role of prices in facilitating coordination has also
context, however, this
may
been emphasized
in
Angeletos and Werning (2006);
in their
only because prices serve as a public signal regarding underlying imobserved fundamentals.
22
wisdom regarding the normative properties
Finally, private sunspots unrest the conventional
environments with coordination problems. In certain cases, occasional investment crashes
of
may be
necessary for facilitating idiosyncratic uncertainty and thereby improving efHciency during normal
times.
When
this
the case, what looks ex post as a coordination failure
is
is
actually contributing
towards higher ex-ante welfare; and policies aimed at preventing such apparent coordination failures
may
backfire
by eliminating a
social
mecha,nism that improves efficiency during normal times.
wide class of models
All these insights are evidently relevant for a
in
macroeconomics that
However, the present paper delivered these insights only within two
feature complementarities.
highly stylized representatives of this class of models.
founded models remains an open direction
Embedding the
analysis within richer micro-
for future research.
Appendix: dynamics and learning
Consider the dynamic extension of Section
~
with Ut
and
xt
~
A/'(0,o"„)! 5i
= ^~^
(K't)
+ ^j,
Proposition 7 There
with
c(
exists a
information of an investor at
Normal
with
mean
Proof. The proof
is
t
=
~h=ist
—
i^t)
summarized
tvith respect to St is
a^
and
,
(li)
by induction. The result
with precision pta^.
+
It
6-1-1
1
where
Pt
=
o'^
an investor invests
trivially holds at
,
m
so that x^+i
is
^
i
=
>
at
1,
1.
is
Normal with mean
Pt+\
=r
/5t)
=
—
st+i
au+
23
+
\
t
if
effectively a
c^)Pt
+
=
and only
cj
ar^.
the private
(i)
if nit
>
is
'm-t-
since the first period coincides
Then,
and variance
,
[l
.Sf
a~-, a^
ut,
-|-
a sufficient statistic m-t that
follows that the private information regarding
sufficient statistic irit^i that
=
+
psf
mf =
are given by
a~~, ac
=
st+i
sequence {m^l,at]'^Q and an equilibrium such that
with the static benchmark. Thus suppose the result holds at
Xt+i
follows an AR(1):
St
and a^ = f^- The private signals
~ .V(0, a^) and ^t ~ A/'(0, ac). Let a-„ =
and variarux
st
The sunspot
3.
^""(0,0"^),
.S(+i
af^-^
A'f(s')
— $
^^'~"^'
)
(
and
Gaussian signal about
can be summarized
=
in
St
a
Pt+i, where
Q'£.
(15)
."
•
,
But then, by a
argument as
similar
investor invests in period
i
+
1
if
in
Proposition
and only
if
2,
it is
>
fht+i
(K.)
indeed a continuation equilibrium that an
where
mj'^j,
+
./l
+ -^$-^(f)
+
^1
which proves that the
Now
result holds at
note that, for any p £
+
t
(0,1)
that
P =
r(/3)
and
and completes the induction argument.
1
and any
increasing and strictly concave, with r(0)
>
0.
It
finite
(ai,,,Q'^,
follows that
av), the function T{p)
(i)
there exists a unique
the sequence {Pt}tLo converges to this fixed point for any
(ii)
This proves the claim that, as the history becomes
infinitely long,
both
77z^
Finally, consider the variant with learning over a constant underlying
This
is
is
nested with p
—
then immediate that
letting
K
=
c/b
—
(16)
,
1
I
at
and a„
=
oo, in
which case
(15) reduces to Pt+i
—
and
sunspot
^{Pt)
decreases monotonically over time and asymptotes to
1/2 into (16) gives
ml =
for all
t,
as
ctj
—
as
strictly
is
>
/?
initial
po
such
>
0.
converge.
(sj
=
s for all t).
{'^+ce^)Pt+aei
^ oo.
It
Finally,
claimed in the main text.
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Expectations and Fluctuations
eds..
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77, 93-98.
New Keynesian
Economics.
in
Aggregate De-
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