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Technology
Department of Economics
Working Paper Series
Massachusetts
BEAUTY CONTESTS
Institute of
AND
IRRATIONAL EXUBERANCE:
A NEOCLASSICAL APPROACH
George-Marios Angeletos
Guido Lorenzoni
Alessandro Pavan
Working Paper 10-04
March 5, 2010
Room
E52-251
50 Memorial Drive
Cambridge,
MA 021 42
This paper can be downloaded without charge from the
Social Science Research
Network Paper Collection
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at
Beauty Contests and Irrational Exuberance:
a Neoclassical Approach*
George-Marios Angeletos
MIT
and
Guido Lorenzoni
Alessandro Pavan
MIT and NBER
Northwestern University
NBER
March
5,
2010
Abstract
The
arrival of new, unfamiliar, investment opportunities
movements
financial
in asset prices
markets look
at
and
real
economic
activity.
the real sector for signals about the profitability of the new investment
opportunities, and vice versa. In this paper,
the incentives that agents face
we
is
often associated with "exuberant"
is
During these episodes of high uncertainty,
when making
find that the sensitivity of equilibrium
we study how such information
their real
economic decisions.
symptoms
impact
spillovers
the positive front,
outcomes to noise and to higher-order uncertainty
amplified, exacerbating the disconnect from fundamentals.
that these effects are
On
of constrained inefficiency;
On
the normative front,
we then
we
find
identify policies that can
improve welfare without requiring the government to have any informational advantage vis-a-vis
the market. At the heart of these results
economy
to behave as a
like "irrational
is
a distortion that induces a conventional neoclassical
Keynesian "beauty contest" and
to exhibit fluctuations that
may
exuberance" to an outside observer.
Keywords: mispricing, heterogeneous information, information-driven complementarities,
ity,
look
volatil-
inefficiency, bea,uty contests.
'This paper subsumes and extends an earlier paper that circulated as a 2007 NBER working paper under the title
"Wall Street and Silicon Valley: a Delicate Interaction." We thank Olivier Blanchard, Stephen Morris, Hyun Song
Shin, Rob Townsend, Jaume Ventura, Ivan Werning, and seminar participants at MIT, the Federal Reserve Board,
the Federal Reserve
Bank
of Boston, the 2007
IESE Conference on Complementarities and Information
the 2007 Minnesota Workshop in Macroeconomic Theory, and the 2007
Angeletos and Pavan thank the
alepavan@northwestern.edu.
NSF
for financial support.
(Barcelona),
NBER Summer Institute for useful comments.
Email addresses: angelet@mit.edu; glorenzo@mit.edu;
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium Member Libraries
http://www.archive.org/details/beautycontestsirOOange
I
Introduction
1
The
arrival of
in the late 90s, or
ments
— a novel technology the Internet
emerging economies — often associated with large
move-
new, unfamiliar, investment opportunities
new markets
in asset prices
and
real
in
like
e.g.,
joint
is
economic
activity.
Many
them
oncile with fundamentals. Instead, they interpret
observers find these
as
movements hard
temporary waves of "exuberance" which
appear to get reinforced as market participants look at one another's behavior
the profitability of the
new investments. Furthermore,
to rec-
financial
markets are blamed
destabilizing role, as the agents in charge of real investment decisions
become
regarding
for clues
for playing a
"overly" concerned
about the short-run valuation of their capital instead of paying attention to the fundamentals
—
popular argument that can be traced back to Keynes' famous "beauty contest" metaphor.
Understanding these episodes, while also capturing the aforementioned
away from the
paradigm of
"exuberance," "sentiments," and the
for
and
A
neoclassical
it is
efficient
like;
ideas, requires
markets: within this paradigm, there
is
to
assume that the observed phenomena are
the beliefs and the behavior of irrational agents.
we maintain the axiom
of rationality;
ing dispersed information;
no room
asset prices only reflect the underlying fundamentals;
irrelevant whether real investment decisions are driven by fundamentals or
simple alternative
is
moving
We
2
do not go
we depart from the
prices.
1
driven, to a large extent, by
in that direction here.
neoclassical
and we focus on the information
by asset
Instead,
paradigm only by introduc-
spillovers that
emerge when
financial
markets look at real economic activity as a signal of the underlying fundamentals.
This modeling approach
reflects, in part,
a matter of preferred methodology.
fortable with policy prescriptions that rely heavily
We
are
uncom-
on the presumption that the government has a
superior ability to evaluate the economy's fundamentals, relative to the market mechanism.
importantly,
mation
is
it
helps us capture two important aspects of the
likely to
phenomena
Most
of interest: that infor-
be particularly dispersed because of the absence of previous social learning; and
that market participants seem anxious to look at one another's behavior, and at various indicators
of
economic
activity, for clues
Our contribution
agents face
economy
is
then to study
when making
to noise
is
that these effects are
about the underlying
their real
profitability.
how such information
investment decisions.
spillovers
First,
we
impact the incentives that
find that the response of the
amplified, exacerbating the disconnect from fundamentals.
symptoms
Second, we find
of inefficiency even relative to a constrained planning problem that
respects the diversity of beliefs and the dispersion of information. Combined, these results provide
a theory of rational exuberance
neoclassical
l
2
See, e.g.,
economy
to
behave
and uncover a mechanism that induces an otherwise conventional
like
a Keynesian beauty contest.
Lucas (1978) and Abel and Blanchard (1986).
al. (2000), Shiller (2000), Bernanke and Gertler (2001), Dupor (2005), Akerlof and
Cecchetti et
Shiller (2009).
Preview of model and
"entrepreneurs"
perfect,
The
results.
who each must
real sector of our
how much
decide
to invest in a
and heterogeneous, information about the
The
new technology on
large
number
of
the basis of im-
profitability of this technology. In a subsequent
stage, but before uncertainty is resolved, these entrepreneurs
financial market.
model features a
may
sell their
capital in a competitive
"traders" in this market are also imperfectly informed, but get to observe a
signal of the entrepreneurs' early investment activity.
At the core of
omy.
On
this
model
is
a two-way interaction between financial markets and the
the one hand, entrepreneurs base their
the price at which they
in charge of real
may
sell their capital.
investment decisions
initial
real econ-
investment decisions on their expectations of
This captures more broadly the idea that the agents
—be they the CEO of a public corporation,
—are concerned about the future market valuation
or the
owner of
On
the other hand,
traders look at the entrepreneurs' activity as a signal of the profitability of the
new investment
a start-up
opportunity. This captures
of
more broadly
of their capital.
the idea that financial markets follow closely the release
macroeconomic and sectoral data, and constantly monitor corporate outcomes, looking
about the underlying economic fundamentals. The
that, as in
identifies
On
any Walrasian setting,
is
not by
an information spillover that
the positive side,
we
By
(or,
more broadly,
any source of variation
represents a pecuniary externality
first effect
the source of any distortion.
at the heart of our positive
mechanism that
identify a
noise relative to fundamentals.
opportunity
is
itself
for clues
The second
and normative
effect
results.
amplifies the response of the
economy
to
"fundamentals" we mean the profitability of the new investment
technologies, preferences
outcomes that
in equilibrium
is
and endowments). By "noise" we mean
orthogonal to the fundamentals.
In our
model, such non-fundamental variation can originate from correlated errors in information about
the fundamentals; from higher-order uncertainty; and, in certain cases, from sunspots.
To understand
in the entrepreneurs' information; this
Suppose
for a
where
this result, consider the case
moment
is
"noise" originates
we concentrate on
the case
from correlated errors
for the
bulk of our analysis.
that the entrepreneurs' decisions were driven merely by their opinions about
the fundamentals. In equilibrium, aggregate investment would then depend on the entrepreneurs'
average opinion and would therefore send a signal to the financial market about the underlying
fundamentals. In general, this signal
entrepreneur
—may not be able to
is
tell
going to be noisy: any given agent
whether high aggregate investment
— be he a trader or an
is
caused by a positive
shock to fundamentals, or by a positive correlated error in the entrepreneurs' opinions. Nevertheless,
relative to the typical trader, the typical entrepreneur
the noise in this signal. This
is
because the signal
is
bound
itself is
the collective choice of the entrepreneurs;
the noise in this signal thus originates in the entrepreneurs'
entrepreneurs, as a group, are playing a signaling
game
to have superior information about
own
private information. In effect, the
vis-a-vis the traders.
This observation
cial
is
crucial, for
market are partly predictable by the typical entrepreneur. In particular, whenever there
itive correlated error in
will
implies that the (rational) pricing errors that occur in the finan-
it
more than
his
a pos-
the entrepreneurs' information about the fundamentals, each entrepreneur
expect the average opinion of the other entrepreneurs
increase
is
own
—and hence aggregate investment — to
But then the entrepreneur
opinion.
will also
expect the financial market
to overprice his capital. This in turn creates a speculative incentive for the entrepreneur to invest
more than what warranted from
what may look
like
his expectation of the
fundamentals
As
"exuberant" investment to an outside observer.
same, their collective "exuberance"
may
—and thereby to
engage
in
entrepreneurs do the
all
trigger asset prices to inflate, because the traders will
The anticipation
perceive this exuberance in part as a signal of good fundamentals.
prices can feed back to further exuberance in real
economic
activity,
and so
of inflated
on.
This argument highlights more generally the role of higher-order uncertainty between the agents
participating in the real and in the financial sector of the economy. Within the context of our model,
this
means the
about the traders'
be able to
When
following.
sell his
an entrepreneur decides how much to
about the fundamentals
beliefs
capital in the financial market.
invest,
he must form beliefs
which he
in order to predict the price at
A
trader, in turn,
must form
will
about the
beliefs
entrepreneurs' beliefs in order to interpret the signal conveyed by aggregate investment.
It
follows
that investment and asset prices are driven by the higher-order beliefs of both these two groups.
The key
contribution of our analysis
two sectors of the economy
and asset
is
in highlighting
may heighten the
how
the information spillovers between the
impact of such higher-order uncertainty on investment
prices, thereby exacerbating their disconnect
from fundamentals, while also providing a
micro- foundation to Keynes' "beauty contest."
Interestingly, these effects obtain without
they are
all
and take
infinitesimal
prices
Thus, despite a certain similarity in
any of the agents being
and aggregate outcomes
flavor,
as
strategic, in the sense that
exogenous to their choices.
our results are distinct from those in the financial
microstructure literature, which, in the tradition of Kyle (1985), focuses on
manipulate asset
"invisible
prices.
hand" that
,
3
is,
of the general-equilibrium interaction of multiple small players.
symptoms
side, the
question of interest
of inefficiency. This question
is
is
whether the aforementioned positive
central to our formalization of "exuber-
ance" and "beauty contests," for these ideas typically involve
that something
To address
is
a,
more or
less explicit,
judgement
going "wrong" in the market and that the government should intervene.
this question,
we
consider the problem faced by a planner
who has
agents' incentives but has no informational advantage vis-a-vis the market
3
large players can
Rather, the manipulation effects in our model are the by-product of the
Turning to the normative
results are also
how
See, e.g., Goldstein
and Guembel (2008), which emphasizes how
this
full
—either
power on the
in the
form
of
manipulation could distort real investment.
additional information, or in the form of the power to centralize the information that
in the
economy.
4
We
dispersed
is
then show that this planner would dictate to the entrepreneurs to ignore the
expected mispricing in the financial market and
all
the consequent higher-order uncertainty, and
instead base their investment decisions solely on their expectations of the fundamentals.
This
because any gain that the entrepreneurs can make by exploiting any predictable mispricing
a private rent
—a zero-sum transfer from one group of agents to the other,
wedge between the private and
is
is
only
which only creates a
social return to investment. It follows that our positive results
have
a clear normative counterpart.
We conclude our
is
analysis
by identifying
restricted to use only information that
is
policies that
improve welfare even when the government
policies that stabilize asset prices, like those often
show how simple
advocated in practice, can lead to higher welfare;
We
but we also identify some important limitations of such
policies.
sophisticated policies can do better, possibly restoring
full efficiency.
Related literature.
We first
already in the public domain.
then discuss how certain more
Morris and Shin (2002) recently put forth the idea that models that
combine strategic complementarity with dispersed information can capture the
role of higher-order
uncertainty in Keynes' beauty contest metaphor, spawning a rich literature. However, by lacking
specific micro-foundations, this earlier
origin of strategic complementarity,
ciency of the equilibrium,
Wincoop
of
(2005),
dynamic
if
work did not address the
positive question of
and the normative question of what
any. Subsequent
is
what
the cause of the
is
the
ineffi-
work by Allen, Morris and Shin (2006), Bacchetta and
and Cespa and Vives (2009) has addressed the positive question within the context
asset-pricing models, but has abstracted from real economic activity
the normative question.
Relative to this literature, our contribution
is
and has
left
aside
to address the aforemen-
tioned questions within a micro-founded, neoclassical framework of the interaction between real
and
financial activity.
What opens
the door to non-trivial effects from higher-order uncertainty in our setting
the combination of trading and information spillovers
among
different
groups of agents (the en-
trepreneurs and the traders). In this respect, our formalization of "beauty contests"
to
Townsend
(1984),
who was
the
first
is
is
connected
to highlight the role of higher-order beliefs in settings with
dispersed information and endogenous learning. But while
Townsend studied a framework with no
trade and no other payoff links across agents, bur results rest on the presence of trading opportunities:
if
and the
the entrepreneurs never sold their capital in the financial market, both the amplification
inefficiency
Our paper
Amador and
4
would vanish.
also connects to the
voluminous literature on herding and social learning
Weill, 2008; Banerjee, 1992;
Chamley, 2004; Vives, 2008).
This planning problem builds on the notion of constrained efficiency studied
environments with dispersed information.
for a class of
in
We
share with this
(e.g.,
liter-
Angeletos and Pavan (2007, 2009)
ature the broader idea that the economy
may
feature heightened sensitivity to certain sources of
information, leading to increased non- fundamental volatility; see especially Chari and
for
an application to financial markets and
to investment
Loisel,
Pommeret and
booms. However, the mechanism we identify
in this literature
is
is
Kehoe (2003)
Portier (2009) for an application
distinct.
The key
distortion featured
the failure of individual agents to internalize the impact of their
own
actions
on
the information available to other agents, which in turn affects the efficiency of the decisions taken
by the
latter.
Instead, in our model, the key distortion
is
the one that rests on
how
the antici-
pation of the signaling role of investment affects the entrepreneurs' incentives. In this regard, the
mechanics are more closely related to those
in the signaling literature (e.g., Spence, 1973)
than to
those in the herding literature. Note, though, that the "senders" in our model (the entrepreneurs)
are non-strategic in the sense that the actions of each one alone do not affect the beliefs of the "receivers" (the traders);
hand", that affects the
Our paper
it is
only the collective behavior of the former, coordinated by an "invisible
beliefs of the latter.
also adds to the growing
contribution in this regard
is
macroeconomic
literature
on dispersed information. 5 Our
to identify novel positive and normative implications of the two-way
interaction between real and financial activity. Closely related in this regard
Goldstein, Ozdenoren and
namely from the
financial
Yuan
(2009). This paper focuses
market to the
is
the recent paper by
on the opposite information
real sector. In so doing,
it
spillover,
complements our paper, but
it
does not consider the amplification and inefficiency effects that are at the core of our contribution.
Tinn (2009) considers an informational
spillover similar to the
one
in
our paper, but within a setting
that does not feature our amplification and inefficiency results. La'O (2010) studies a model where
the two-way interaction between real and financial activity rests on collateral constraints, as in
Kiyotaki and Moore (1997), rather than information spillovers, as in our paper.
Finally,
by touching on the broader themes of heterogeneous
beliefs, speculation,
and mispricing,
our paper connects to Scheinkman and Xiong (2003), Panageas (2005), and Geanakoplos (2009).
This line of work abstracts from asymmetric information and instead models the heterogeneity
of beliefs with heterogeneous priors.
By
abstracting from asymmetric information, this literature
rules out the information spillovers that are at the core of our positive
results,
and normative
results.
Our
on the other hand, are driven by the asymmetry of information, but do not hinge on the
assumption of a
common
prior:
they can be extended to settings with multiple priors, provided
that one allows for endogenous learning stemming from the presence of dispersed information.
Layout. The
rest of the
paper
3 characterizes the equilibrium
is
organized as follows. Section 2 introduces the model. Section
and delivers the key positive
results provide a neoclassical formalization of "exuberance"
5
results.
Section 4 discusses
and "beauty contests"
.
how our
Section 5
See, e.g., Amador and Weill (2009), Angeletos and La'O (2009a), Hellwig (2006), Lorenzoni (2010), Mackowiak
and Wiederholt (2009), Mankiw and Reis (2009), Veldkamp (2009), and the references therein.
characterizes the constrained efficient allocation,
contrasts
it
it
to the equilibrium,
may
Section 6 discusses possible extensions that
policy implications.
Appendix
of the paper. Section 7 concludes.
A
contains
all
and discusses
help reinforce the message
proofs, while
Appendix B proves the
robustness of our results to richer payoff and information structures.
The model
2
Our model
features a single
round of real investment followed by a
with information flowing from the former to the
of agents, "entrepreneurs"
i
E
[0,
1/2]
and traders by
and
"traders."
6
1].
i
(1/2,
Timing and key choices. Time
The economy
latter.
Each type
is
single
financial trading,
populated by two types
is
measure 1/2; we index entrepreneurs by
of
divided in four periods,
is
round of
£
t
{0, 1,2,3}.
At
t
=
a new
0,
investment opportunity, or "technology," becomes available. The profitability of this technology
is
determined by a random variable
model.
drawn from a Normal
It is
common
the
unknown
At
t
to
=
prior (with
all
1,
agents.
ttq
distribution with
being the precision).
new
mean
The
\x
>
and variance l/ng, which
random
realization 9 of this
defines
variable
is
6
the "real sector" of the
to invest in the
This random variable defines the "fundamentals" in our
9.
technology. Let
economy
operates: each entrepreneur gets the opportunity
denote the investment of entrepreneur
ki
investment in terms of the consumption good
is
kf/2.
7
When
i.
The
cost of this
choosing investment, entrepreneurs
have access to various sources of information (signals) that are not directly available to the traders.
The
noise in
some
of these signals
may be
We
be correlated across entrepreneurs.
in
Appendix B. Here,
to simplify,
purely idiosyncratic noise:
it is
we
mostly idiosyncratic, while for other signals the noise
consider a general information structure along these lines
let
the entrepreneurs observe only two signals.
given by X{
=
+ &,
where
&
is
it is
given by y
independent of 9 and of
=
9
{£i}ie[o,i/2]i
+ e,
where e
is
with variance
to introduce a source of non-fundamental
Gaussian noise,
l/ir y
movements
.
The one has
Gaussian noise, independently and
identically distributed across agents, independent of 6, with variance 1/tt x
correlated noise:
may
The key
.
The
common
other has perfectly
across entrepreneurs,
role of the correlated error i
in aggregate investment.
is
8
Throughout, we use "tildes" to denote random variables and drop them when denoting realizations.
One can easily reinterpret this cost as the disutility of effort necessary to produce ki.
8
Such a correlated error, in turn, can have various origins. As discussed in Section 6.3, private signals about the
actions of agents that moved in the past may lead in equilibrium to signals about 6 with correlated errors. More
broadly, network effects, social learning, and information cascades may also explain this correlation. Alternatively,
as emphasized in Hellwig and Veldkamp (2009) and Myatt and Wallace (2009), strategic complementarity
like the
one that, as we will show, emerges endogenously in our economy by itself generates an incentive for the agents to
collect correlated sources of information. See also Dow, Goldstein, and Guembel (2010), Froot, Scharfstein, and Stein
(1992), and Veldkamp (2006) for complementary justifications.
—
—
At
—
t
traders. 9
some entrepreneurs
the "financial market" operates:
2,
transfer their capital to the
These trades may be motivated by a variety of reasons unrelated to private information.
To keep the analysis
tractable,
we model these trades
an idiosyncratic shock with probability A G
consumption
at
—
t
3
by
hit
is
Entrepreneurs hit by this shock do not value
[0, 1].
and have no choice but to
Each entrepreneur
as follows.
sell all their capital at t
—
For simplicity,
2.
the entrepreneurs not hit by the shock are precluded from trading; this last assumption can be
relaxed provided that the equilibrium price does not
From now
in Section 6.3).
on,
we
refer to this
become
perfectly revealing (see the discussion
shock as a "liquidity shock"; but we think of
more
it
broadly as a modeling device that helps us capture the concern that the agents in charge of real
investment have about future equity prices. 10
The
market
financial
is
competitive and the market-clearing price
denoted by
is
When
p.
the
traders meet the entrepreneurs hit by the liquidity shocks in the financial market, they observe the
quantity of capital that these entrepreneurs bring to the market. Since A
K
to observing the aggregate level of investment,
other signals that the real sector
production and sale data.
Any
11
The
may be
—
ui
1/ttu
While
.
9
+
where
r],
u> is
fj
is
This
kidi.
may have about
the fundamentals
as an exogenous signal,
it is
6,
traders have dispersed private signals about
Finally, at i
3,
the fundamental 9
new technology and
the
to capture
is
i and
is
{6}ie[o,i/2]i
in the financial
To
simplify the exposition,
B
Preferences, endowments, and markets.
agent
i's
is
consumption
zero:
t,
while
An
alternative
way
in each period.
a
Si is
m =
random
it
to the ones
is
held
by an
its
ownership.)
endowment
Moreover, they are risk neutral and
en
+
ca
+
s,;Ci3,
where en denotes
variable that takes value
if
the agent
installed capital
to introduce this concern within the context of start-ups rests on the presence of efficiency
gains from transferring capital ownership from the agents
company
as the
we assume that each
9
Throughout, we do not explicitly model the distinction between trading financial claims over the
and trading the capital goods themselves; in our framework, this distinction is irrelevant.
10
it
market when the
All agents receive an exogenous
preferences are given by
in period
with variance
for extensions that allow for richer
technologies and for the marginal product of capital to depend on
good
in a public
publicly revealed and production takes place using
the installed capital.
of the (nonstorable) consumption
8.
8.
entrepreneur or by a trader. (See Section 6.2 and Appendix
their discount rate
more broadly
summarized
unit of capital delivers 9 units of the consumption good, irrespective of whether
e
equivalent
is
straightforward to reinterpret
outcome of the aggregation of information that may take place
—
this
sending to the financial market, including preliminary
Gaussian noise, independent of
modeled here
meant
is
known,
traders then use this observation to update their beliefs about
other information that the traders
signal
= J
is
who have a comparative advantage
in
who have a comparative advantage in starting a new
running it at later stages of development (e.g., Holmes
and Schmitz, 1990).
n In
K
,
is
equilibrium, the cross-sectional distribution of k is Normal with known variance; observing the
thus informationally equivalent to observing the entire cross-sectional distribution of investment.
t
mean
level,
is
an entrepreneur
hit
by a
and value
liquidity shock
Finally, in addition to the
otherwise.
1
aforementioned financial market, we allow the following markets to operate in each period:
market
consumption good (which
for the
and a market
for insurance contracts
is
also the numeraire); a
market
on the entrepreneurs' idiosyncratic
for a riskless
liquidity shocks.
a
bond;
Given
the assumption of risk neutrality, these additional markets will be irrelevant for either investment
decisions or asset prices in equilibrium. Rather, they are introduced so as to clarify that the only
essential
market imperfection we impose
the fundamentals
and the
—our results
will
the one that limits the aggregation of information about
is
not be driven by borrowing constraints, incomplete risk-sharing,
like.
Equilibrium
3
Because of the assumptions of linear preferences and zero discounting, the equilibrium
rate
is
zero in
all
periods
and
risk-free
the consumption allocations and the trades of bonds
all states;
and insurance contracts are indeterminate; and the agents' expected
utility
reduces to the expected
present value of their net income flows. For entrepreneurs hit by the liquidity shock, net income flows
sum up
to
3e+pki — kf/2, while
for entrepreneurs not hit
"surviving entrepreneurs"), they
utility at the
time of investment
Since this objective
as a function of
sum up
is
given,
to 3e + 8ki
up
strictly concave, the
is
x and
by the shock (henceforth
— kf/2.
Therefore, each entrepreneur's expected
to a constant, by E[uj|:rj, y\
— E[(l— \)8+\p— \k\ \x{, y\.
investment choice of an entrepreneur can be expressed
the two signals observed by the entrepreneur.
y,
also referred to as
Aggregate investment
then a function of two aggregate shocks, the fundamental 8 and the correlated error
A
trader's net
income
flow,
on the other hand,
the position he takes in the financial market.
and the aggregate capital
E[iij|A',
lo\
=
(E[£>|A',
u]
—
A',
his
expected
is
given by 3e
and
ui
=
9
+
1],
A
- pq
t
,
where
qi
utility at the
time of trading
is,
up
—
E[9\K,ui}
12
Since
the equilibrium price can be expressed as a function of
we
denotes
w
to a constant,
p)qi. It follows that the market-clearing price in the financial
these observations in mind,
Definition 1
6q%
e.
Since the trader observes the exogenous signal
pinned down by the traders' expectation of the fundamental: p
of {9,e)
+
is
K
is
[8, e, rj).
market
is
a function
With
all
define our equilibrium concept as follows.
(linear rational- expectations) equilibrium,
an aggregate investment function K(9,
e),
is
an individual investment strategy k(x,
and a price function
p(8, e,
rf)
y),
that jointly satisfy the fol-
lowing conditions:
'"Since no trader has private information and the entrepreneurs
who sell their capital have perfectly inelastic
any information, which explains why we omit conditioning on p
when describing the traders' expectations. Also, any value /(el can be observed in equilibrium, which explains
why we do not have to worry about describing out-of-equilibrium beliefs.
supplies, the market-clearing price does not reveal
for
(i)
all
(x,y),
k (x,y) £
(ii)
for
argmaxE
(1
- \)8k
=
/
+
Xp(8,i
fj)k
>
U2
— \k
x,V
all [8,s),
K{6,e)
k(x,y)d<l>(x,y\8,s),
where §(x,y\8,£) denotes the joint cumulative distribution function of x andy, given 8 and
e;
(Hi) for all (8,e,i]),
p(8,e,rt)
where
K = K(8, e)
and
ui
=
8
=E
K,
-\- r/;
(iv) there exist scalars (3q, (3g
and
E
such
for
that,
all (8,e),
K{8,e)=Po + PoO + pE e.
Condition
(i)
requires that the entrepreneurs' investment strategy be individually rational,
Condition
taking as given the equilibrium price function.
Condition
investment.
(iii)
(ii)
just the definition of aggregate
is
requires that the equilibrium price be consistent with market clear-
ing and rational expectations on the traders' side, taking as given the collective behavior of the
entrepreneurs. Finally, condition
ature, this linearity
is
imposes linearity; as usual in the rational-expectations
(iv)
liter-
necessary for maintaining tractability. (In our setting, linearity of the price
function implies linearity of the aggregate investment, and vice versa.)
we henceforth drop the
.
Also,
simplify the language,
and "rational-expectations" and
qualifications "linear"
rium concept simply as "equilibrium"
To
we
refer to 8
shock" and the "noise shock", and to the coefficients
f3g
and
refer to
s as, respectively, the
and @£
our equilib-
"fundamental
as the "responses" of aggregate
investment to these shocks.
3.1
A benchmark
with no information spillovers
For comparison purposes, we
real
and the
vanishes (nu
The
financial sector
—
financial
>
is
now
absent. In particular, suppose that the noise in the traders' signal
oo), so that 8 is
known
market then clears
of an entrepreneur
who
consider a case in which the information spillover between the
if
at the time of trading
and only
receives signals x
entrepreneur's optimal investment
is
if
p
and y
=
is
=
E[8\x,
y]
=
S
his expectation of 8:
+
5x x
role of
K vanishes.
by implication, the expected payoff
simply E[#|x, y]k — k 2 /2.
pinned down by
k (x, y)
8 and,
and the signaling
cj
+ 5y y
It
follows that the
where
5q
=
=
5X
r&fJ.,
?*, 5 y
=
^, and n =
+ n x + n y By implication,
= Sx + Sy and 5E = 5y
Tro
given by K(8, e)
—
Proposition
In the absence of information spillovers, the equilibrium
1
<5n
+
+
5g8
5e e, where Sg
aggregate investment
.
is
.
is
unique and investment
pinned down merely by the entrepreneurs expectations of the fundamentals.
is
'
This result establishes that, in the absence of information spillovers,
rium outcomes whether investment
driven by the entrepreneurs' expectations of the fundamental
is
or by their expectations of the asset price.
ventional neoclassical economy, leaving no
In this respect, our
room
Importantly, this result does not require 8 to be
known by
is
behavior cannot possibly convey any additional information about
this
benchmark
any con-
more generally
applies
for this infor-
which case the entrepreneurs'
#.
13
From now
on,
we
refer to
as the case with no information spillovers.
Information spillovers
3.2
We now
turn attention to the case of interest, namely when the real sector sends valuable signals
to the financial market.
fixed point
Below, we
how any
equilibrium can be understood as a
and the asset price function (which summarizes the
financial market), focusing
(i.e.,
where
with this property always exists and
Take an arbitrary
coefficients
is
(3g
>
unique
and
j3e
>
for A small
0).
We
(3g
and
/3C
A
.
is
increasing in both the
then prove that an equilibrium
enough.
linear aggregate investment function of the
/?n,
collective behavior
collective behavior of the
on the situations where aggregate investment
fundamentals and the noise
some
describe
first
between the aggregate investment function (which summarizes the
of the real sector)
for
it
a sufficient statistics
in
like
a Keynesian beauty contest.
the traders;
one that the entrepreneurs as a group possess,
for the
economy behaves
for the features of
as long as the information that the traders possess about 9
mation and
irrelevant for equilib-
it is
K {9,e) =
form
central object in our analysis
is
fa
+ Pe® + fie e,
the response of aggregate
investment to noise relative to the fundamental, defined as the ratio
From
ity in
To
at
t
=
the perspective of an outside observer, this ratio determines the fraction of the overall volatil-
aggregate investment that cannot be explained by fundamentals.
clarify this point, consider
2
(i.e.,
Finally, let Xi
[
1|1 / 2 ]2'ii
implies that E[8\l2 ,li]
By
=
is
ip
inversely
an arbitrary information structure. Let 12 be the exogenous information of traders
the information not inferred through A'). Next,
= U i6
Formally,
and assume that Z2
E[0\12 ]. Because
the law of iterated expectations,
K
is
is
let
measurable
in
we then have that EfplZa]
follows that every entrepreneur chooses k t
=
Ji
t
a sufficient statistics
E[8\l,i].
11)
lu
=
be the information of entrepreneur i at t = 1.
for (Z2, 2i ) with respect to 9. This assumption
this also implies that
E[E[5|X2 ,Zi]|Xl(1
]
=
p
=
E[6\l2 K]
E[e|Z,i] for all
,
i
e
=
E[8\l2 }.
[0,1/2]. It
related to the R-square of the regression of the realized
a trader in the model, on the other hand,
tive of
ip
K
on the
determines the informativeness of the signal
that the financial market receives from the real sector. Indeed, as long as
fig
informationally equivalent to observing the following Gaussian signal about
z=«f®>=e +
It
follows that
<p
Var(tpe)/Var(0)
pins
down the
=
-Kg/ir
2
p>
y
.
Now, put aside
for a
^
0,
observing
We
henceforth refer to
tp
K
<pe.
(2)
is
simply
interchangeably as the "relative response to
ratio."
moment
the endogeneity of the aforementioned signal, assume that the
—
traders observe a signal of the form z
6
+
ipe for
some arbitrary
<p
6 R, and consider the
determination of the asset price. Bayesian updating implies that the traders' expectation of 6
weighted average of their prior
where
tt$,
signal
z.
=
nu and n z
,
By
TT
y /<p
mean
2
p,
and
n
/g
=
two signals
their
are the precisions
and
u>
^"r
^ /y
/
,
.
7£
=
—
=
7o
Ty
,
,
,
2 <P,
+
7e6
l
means more
and 7„
=
— —^
(
measure the responses of the
t-t
;
4)
depend on
(p:
because a higher
noise in the signal z but also less sensitivity of the traders' expectation of 6 to this
signal, a higher
its
and the
ui,
+ 7£^ + 7^:
asset price to the underlying shocks. Importantly, these responses
ip
a
z:
the prior, the signal
of, respectively,
is
implication, the equilibrium asset price can be expressed as follows:
P(0>£,V)
where
is
6:
noise-to-signal ratio in aggregate investment: this ratio
and as the "noise-to-signal
noise"
From the perspec-
realized 9.
ip
necessarily reduces the response of the price to the fundamental 6
response to the noise
?/,
while
it
has a non-monotonic
effect
on
its
and increases
response to the noise
e.
Next, consider the incentives faced by the entrepreneurs when they expect the asset price to
satisfy (4).
x and
Optimality requires that individual investment
satisfies
the following condition for
all
y:
k(x,y)
=E
(1
-
\)6
+
\p(6,i,rj)
x,y
(5)
Substituting the asset price from (4) into the entrepreneur's optimality condition
(5),
and noting
that the conditional expectations of 8 and e are linear functions of the signals x and
the conditional expectation of
as a linear function of the
and P'y
.
two
rj
is
zero,
signals:
we
infer that individual
h(x,y)
=
Importantly, these coefficients depend on
11
f3'
+
xx
(3'
p through
+
while
investment can be expressed
p" y, for
y
(4),
y,
some
coefficients
j3'
/3'
,
x
capturing the impact that the
anticipated price behavior has on individual investment incentives. Finally, aggregating across the
entrepreneurs gives
signal sent
<p'
=
/ P'
P'e
.
'
/3'
+ /3'g 9 + p'e e,
real sector
The
latter
is
pinned down by the
which
in
turn depends on
y,
with
can be expressed as
by the
x and
signals
K (9, e) =
fi'
Q
z'
=
=
9
(3'
x
+
(3'
y
+ tp'e,
and
(3'
6
=
(3'
.
y
It
follows that the
with noise-to-signal ratio given by
relative response of individual investment to the
p through
two
(4).
Putting the aforementioned arguments together, we infer that any equilibrium can be under-
stood as a fixed point to a function F that maps each
which
is
K
some
to
= 9 + pe
with noise-to-signal ratio given by some arbitrary
responds by sending a signal
=
z'
+ <p'e
6
£
<p'
when
formally defined in the appendix, has a simple interpretation:
receives a signal z
in
6
<p
with noise-to-signal ratio given by
JR.
This mapping,
the financial market
<p
ip'
G M, the
real sector
=
Of
F(<p).
course,
any equilibrium the signal received by the financial market must coincide with the signal sent by
the real sector, which explains
why
the fixed points of the mapping
F
identify the equilibria of our
economy. Studying the properties of this mapping then permits us to reach the following
which concerns the existence and uniqueness of equilibria
Proposition 2 There always
are all positive.
exists
in
our model.
an equilibrium in which the
>
Furthermore, there exists a cutoff A
0,
result,
coefficients Px, Py, @9, P£,lo,l£,lri
such
that,
for any A
<
A,
this is the
unique equilibrium.
We
conclude that there always exists an equilibrium in which individual investment responds
positively to both the signals
to both the fundamentals 9
x and y and, by implication, aggregate investment responds
and the
price also responds positively to
noise
Whenever
e.
both 9 and
This
e.
is
this
is
positively
the case, the equilibrium asset
because the traders (correctly) perceive
high investment as "good news" about profitability, but cannot distinguish between increases in
investment driven by 9 from those driven by
3.2.1
We now
to noise.
e.
Mispricing, speculation, and amplification
turn to our main positive result, regarding the relative response of equilibrium investment
To
this purpose,
it is
k=E
t
useful to rewrite the entrepreneur's optimal investment as follows:
[0]
where Ej and E t are short-cuts
+ AE
4
\p(6, e, v)
-
&]
=
Ej [§]
t
[E t
[6]
-
9}
(6)
,
for the conditional expectations of, respectively, entrepreneur
the traders. This condition has a simple interpretation.
valuation of a unit of capital.
+ AE
The gap p -
9
=E
t
[9}
that valuation, or the "pricing error" in the market.
-
The
i
and
variable 9 represents the fundamental
9 thus identifies the traders' forecast error of
The component
of investment that
is
driven by
the forecast of this pricing error can then be interpreted as "speculative." For any given expectation
12
of
9,
an entrepreneur
error; this
is
will invest
because he expects to
That entrepreneurs base
tal valuation
in
more
and on
in response to a positive expectation of the traders' forecast
the extra capital in an "overpriced" market.
sell
both on
their investment decisions
fundamen-
their expectation of their
their expectation of the financial price should not surprise. This property holds
any environment where entrepreneurs have the option to
sell their
capital in a financial market.
What
In particular, this property also applies to the benchmark with no information spillovers.
benchmark
distinguishes the present case from that
that permits
them
that the entrepreneurs possess information
is
to forecast the traders' pricing error.
This possibility rests on two properties:
and
signal of the underlying fundamental;
about the sources of variation
in their
that the traders look at aggregate investment as a
(i)
(ii)
that the entrepreneurs possess additional information
investment choices. In particular, note that, for given
positive realization of the noise shock e in the entrepreneurs' information causes a
investment. Since the traders cannot
tell
whether
or by noise, they respond to this investment
this
boom by
boom was
boom
9,
a
aggregate
in
driven by a strong fundamental
bidding the asset price up. However, relative
to the traders, the entrepreneurs have superior information about the origins of the investment
boom. This explains why they can,
and hence
at least in part, forecast the traders' forecast errors
speculate on the market mispricing.
This, in turn, crucially impacts the entrepreneurs' incentives.
Because of the aforementioned
speculative component, each entrepreneur bases his decision on his forecast, not only of
also of
When
e.
their precisions,
it
tt x
comes
and
TT
y
to forecasting
.
When,
contains information on both 9 and
only contains information on
9,
instead,
e,
what distinguishes the two
it
is
simply
which
relatively better predictor than the signal x,
which
e,
the signal
This suggests that an entrepreneur who expects prices to increase
9.
with both the fundamental 9 and the noise e will find
it
optimal to give relatively more weight to
the signal y than what he would have done in the benchmark with no information spillovers
which p does not depend on
e).
noise on aggregate investment
is
Proposition 3 For any of the
Py >
(i)
6y
,
Pg
<
Sg,
and
[3E
>
5E
.
As
all
entrepreneurs find
amplified. This intuition
it
is
optimal to do
(ii)
so,
the impact of the
equilibria identified in Proposition 2, the following is true:
That
is,
relative to the
(in
verified in the following proposition.
benchmark with no information
individual investment responds less to the idiosyncratic signal and
and
but
y,
comes to forecasting the noise
becomes a
x and y
signals
9,
more
f3x
<
5X
,
spillovers,
to the correlated signal,
aggregate investment responds less to fundamental shocks and more to noise shocks.
Proposition 3 illustrates the amplification mechanism generated by the interaction between real
and
financial decisions
mechanism
is
under dispersed information. In Appendix B we show that
this amplification
quite general, in the sense of being present in variants of our model that allow for
richer information
and payoff
structures.
However, we
13
will also see that the
more robust
positive
about the relative response to noise and fundamentals, rather than the absolute
prediction
is
responses.
For this reason,
we
henceforth define the contribution of noise to aggregate volatility
as the fraction of the volatility in aggregate investment that
is
driven by noise, rather than by the
14
fundamentals, and state the main positive prediction of the paper in the following form.
Corollary
1
(Main positive prediction)
tribution of noise to aggregate volatility
Put
it
slightly differently,
of the fundamentals:
will discuss
how
4.
amplified.
the mechanism identified
in the
paper reduces the explanatory power
reduces the R-square of a regression of aggregate investment on
it
this result
contests in Section
is
In the presence of informational spillovers, the con-
6.
We
complements our formalizations of exuberance and Keynesian beauty
Before proceeding to
and the
possibility of multiple equilibria.
3.2.2
Comparative
statics
this,
however, we
first
study certain comparative statics
and multiplicity
In the absence of information spillovers, the strength of the entrepreneurs' concern for asset prices,
as
measured by
crucial.
is
A, is irrelevant for
The next
result
Proposition 4 As long
equilibrium outcomes. With information spillovers, instead,
shows how a higher A reinforces the amplification
get
concern
some
is
equal to Ai for
all
entrepreneurs and
value for the noise-to-signal ratio in aggregate investment.
>
A2
Ai joins the economy. Since this entrepreneur
asset price
remain unchanged. From
(5), it is
this entrepreneur's investment strategy will
is
concern for asset prices.
intuition for this result, consider the following exercise.
for asset prices
effect of Corollary 1.
as the equilibrium remains unique, the contribution of noise to aggregate
volatility increases with \, the strength of the entrepreneurs'
To
it
be
is
let ip\
Now
a
Suppose that the
initial
be the associated equilibrium
new entrepreneur with concern
infinitesimal, aggregate investment
and the
easy to see that, relative to any other entrepreneur,
tilted in favor of the correlated signal y.
The reason
the one discussed before. Relative to the idiosyncratic signal x which contains information only
about
9,
latter
impacts the asset price, a higher concern
the correlated signal y contains information also about the
common
for the latter induces the
error
e.
Because the
entrepreneur to respond
Formally, let K denote the projection of equilibrium A" on 6\ that is, consider the regression of realized investment
— A is orthogonal to the projection A, we have that Var(K) =
on realized fundamentals. Since the residual
Var(K) + Var{K — A). That is, aggregate volatility can the be decomposed in two components: Var(K), which
represents the fundamental component, and Var(K — K), which represents the non-fundamental component. The
contribution of noise to aggregate volatility is then defined as the fraction Var(K - K)/Var(K). In our baseline
- K depends on a single noise shock and the fraction Var(K — K)/Var(K) is simply an
model, the residual
14
K
K
increasing transformation of
residual
K — A',
tp.
model of Appendix B, there are multiple noise shocks driving the
and 2 continue to hold for the fraction Var(K — K)/Var(K).
In the generalized
but the results in Corollaries
1
11
"1
more
to y relative to x. Next, let the concern for asset prices increase to A2 for all entrepreneurs
economy, but continue
in the
noise-to-signal ratio given
y,
by
assume that the signal
to
As
ip\.
all
2 received
by the financial market has a
entrepreneurs start responding more to the correlated signal
aggregate investment starts responding relatively more to the noise
proves that the
mapping F
increasing in A for any given
is
(p.
Formally, this argument
e.
Next, consider what happens as
the traders realize that the entrepreneurs' incentives have changed in the aforementioned manner.
Because aggregate investment has become a noisier signal of the fundamentals, the traders find
optimal to respond
less to
to £ could either increase or
either reinforce or
As a
it.
This
fall.
dampen the
result, the
response of the price to 8
response
its
once acknowledged by the entrepreneurs, can
last effect,
initial effect of
while
falls,
it
the higher
Formally, F(F(tpi)) could be either
A.
higher or lower than F(tpi). However, as long as the equilibrium remains unique, the fixed point of
r
necessarily inherits the comparative statics of F,
Interestingly, however, as
financial sector
may
which proves the
A increases enough, the two-way feedback between the
get sufficiently reinforced that
fixed points are associated
with
self-fulfilling
F may admit multiple
they
make
asset prices
more
is
an open
set
and the
Different
fixed points.
respond more to the correlated signal
sensitive to noise shocks relative to fundamental shocks,
turn justifies their stronger response to the correlated signal
Proposition 5 There
real
prophecies regarding the quality of the signal that the
real sector sends to the financial market: as the entrepreneurs
y,
result.
S C R5
such
that,
for
which
in
y.
all (A, TTg,ir x
,
Tr
y
,
w ) e S, there exist
7r
multiple equilibria.
This multiplicity originates merely from the information spillover between the real and the
financial sector of the
economy.
It is
thus distinct from the one that emerges in coordination
models of crises such as Diamond and Dybvig (1983) and Obstfeld (1996). Rather,
one
in
it is
Gennotte and Leland (1990) and Barlevy and Veronesi (2003). These papers
closer to the
also
document
multiplicity results that originate in information spillovers. However, these papers abstract
economic activity and focus on
spillovers that
and uninformed traders. In our
between the
both
real
real sector
and the
emerge within the
financial market,
setting, instead, the multiplicity rests
financial
market and can manifest
from
real
between informed
on the two-way feedback
itself as
sunspot volatility in
investment and asset prices. Clearly, this possibility only reinforces the message of our
paper: the
mechanism we have
identified can contribute to significant
non-fundamental
volatility,
not only by amplifying the impact of correlated errors in information, but also by opening the door
to additional volatility driven
by sunspots.
15
4
Beauty contests and exuberance
In the preceding analysis,
equilibria.
We now
we studied the economy from
the perspective of rational-expectations
look at the problem from a different angle, one that permits us to uncover the
can play in our
role that higher-order uncertainty
setting.
This in turn helps explain the way in
which our framework provides a formalization of the notions of "beauty contests" and "exuberance."
We
do so
in three steps.
game among the
coordination
differences from, previous
class of coordination
clarify that
"noise"
4.1
work that has attempted
real
but also
metaphor with a certain
and the
is
best understood as a signaling-cum-trade
financial sector of the economy. Finally,
in
our
we
game
discuss the various forms that
our economy.
game among
coordination
in
of the information spillover with the option to trade; this helps
and "exuberance" may take
A
to capture Keynes'
to,
games. Second, we explain that the role of higher-order uncertainty
our formalization of "beauty contests"
between the
represented as a
entrepreneurs; this helps us highlight certain similarities
on the combination
setting rests
we show how our Walrasian economy can be
First,
Substituting condition (2) into condition
the entrepreneurs
(3),
we can
express the traders' expectation of the funda-
mentals, and therefore the equilibrium price, as a linear function of aggregate investment. Replacing
the resulting expression into the entrepreneurs' optimality condition
Proposition 6 In any
equilibrium,
there exist scalars
acq,
the following result.
(5) leads to
Kg and
a such
that the equilibrium
investment choices solve the following fixed-point problem:
k(x,y)
Furthermore,
a >
if
and only
if
=E
facilitates
+
K, g
6
high investment
a positive signal about 9), which in turn
This result
KQ
is
+
aK{6,i)
is
x,y
(7)
,
"good news" about profitability
(i.e.,
necessarily the case whenever the equilibrium
a certain game-theoretic representation of our economy:
is
we
if
conveys
unique.
fix
the
equilibrium response of the financial market, the equilibrium investment decisions can be repre-
sented as the Perfect Bayesian Equilibrium of a coordination
best responses given by condition (7) and with the coefficient
complementarity
in this
game.
game among the
entrepreneurs, with
a measuring the degree
Importantly, the origin of the coordination motive
of strategic
is
the infor-
mational spillover between the real and the financial sector. Although each entrepreneur alone
is
too small to have any impact on market prices, the entrepreneurs as a group can influence the
beliefs of the traders
and hence the equilibrium
price.
16
This naturally leads to a complementarity
investment decisions: the higher the aggregate investment, the higher the traders' expec-
in their
tation about the profitability of the
which each entrepreneur
will
this in turn puts constraints
new investment
be able to
opportunity, and hence the higher the price at
sell his capital.
Of
on the extent that the entrepreneurs can
collectively
"manipulate" the
traders' beliefs. Nevertheless, as the above proposition establishes, a coordination motive
among
and
course, the traders recognize this,
is
present
the entrepreneurs as a long as the traders look at investment as a signal of the underlying
fundamentals.
This game-theoretic representation, in turn, establishes a useful parallel between our model and
a class of games with incomplete information and linear best responses analyzed,
As
by Morris and Shin (2002) and Angeletos and Pavan (2007, 2009).
work, strategic complementarity (a
earlier
>
0) tilts
it
is
among
others,
known from
this
the equilibrium use of information towards
correlated sources of information, which in turn offers an alternative angle on
what
drives the
result in Proposition 3 regarding the response of individual investment to the different signals.
Furthermore, Morris and Shin (2002), and a growing body of research thereafter, have associated
the aforementioned class of linear-quadratic games with Keynes' beauty contest metaphor.
This
approach puts aside the micro-foundations of what these games represent and instead focuses on
the fact that these games help capture the dependence of equilibrium outcomes on higher-order
From
beliefs.
this perspective,
and
if
one treats the equilibrium response of the financial market as
exogenous, our economy can be interpreted as a beauty contest
However,
tarity in our
it is
important to note that, in contrast to this
economy
is
among
the entrepreneurs.
earlier work, the strategic
complemen-
endogenous: as already mentioned, the complementarity originates in the
information spillover between the real and the financial sector, not in any direct payoff externality,
production spillover, and the
15
The beauty contest game between the
4.2
An
like.
even more appealing
—
at least in our
view
real
and the
financial sector
— formalization of "beauty contests"
obtains
if
one
does not abstract from the micro- foundations and instead focuses on the endogeneity of the response
of the financial market.
15
With
and
For example,
labor market.
"beauty contest" then emerges as a game between the real and the
market interactions, the degree of strategic complementarity a can be either positive
Section 6.2 we consider a variant of our baseline model that introduces a competitive
In this variant, higher aggregate investment raises the demand for labor, which in turn increases
richer payoffs
or negative.
A
richer
in
equilibrium wages and reduces the expected return on capital.
As
a result, a source of strategic substitutability
emerges. Alternatively, Angeletos and La'O (2009a) consider a Walrasian economy
in which specialization
commodities introduce strategic complementarity (a > 0) even in the absence of an
information spillover. However, as we show in Appendix B, the following is true within a rich class of environments
that can accomodate the aforementioned effects: relative to a situation without information spillovers, the signaling
(a
<
0)
and trade
of differentiated
role of aggregate investment necessarily tilts the entrepreneurs' best responses in the direction of
more
strategic
complementarity (or less strategic substitutability), exactly as in the baseline model considered here. In this sense,
our result that the information spillover is a source of strategic complementarity is quite robust.
17
economy, that
financial sector of the
rather than as a
To
game among the
more
see this
clearly,
between the entrepreneurs, as a group, and the
entrepreneurs.
if
they hold on their capital, and
the price at which they
p,
own
beliefs
about
their first-order beliefs
If
9,
may
capital in the financial market. It follows that the entrepreneurs only need to forecast the
in turn,
is
pinned down by
This implies that the behavior of the entrepreneurs
is
pinned down by
fundamentals and the behavior of the traders. The traders' behavior,
their
traders,
note that in our model the entrepreneurs' payoffs depend only on
the exogenous return they receive
sell their
is,
9.
about 9 and by their second-order
beliefs
about the traders'
beliefs
about
9.
the traders' beliefs about the fundamentals had been exogenous, this would have been the
end of the story
do matter
—
beliefs of
in our setting
sector of the
higher order would have not mattered. In contrast, higher-order beliefs
because the informational
economy makes the
signal conveyed
spillover
traders' beliefs endogenous.
between the real and the
Indeed, in order to interpret the
by aggregate investment, the traders must form
behind the entrepreneurs' actions.
actions are pinned
down by
their
By
beliefs
own
about the entrepreneurs' first-order
first-
beliefs
about the entrepreneurs' second-order
about the driving forces
the argument in the preceding paragraph, the entrepreneurs'
and second-order
beliefs.
about the fundamentals, and hence their behavior, depend on
beliefs
financial
It
beliefs
their second-order beliefs
(i)
about the fundamentals and
(ii)
about their own first-order
to forecast the behavior of the traders, the entrepreneurs
must form
follows that the traders'
their third-order beliefs
beliefs.
beliefs
But
then, in order
about the traders'
higher-order beliefs, and so on.
These observations make clear that higher-order uncertainty plays a
role in our
economy only
because of the information spillover from the entrepreneurs to the traders. Furthermore, the presence of strategic uncertainty within the group of the entrepreneurs, while empirically appealing,
is
not strictly needed for our mechanism. In the variant with heterogeneous priors studied in Section
6.1,
we can
allow the entrepreneurs to share the
ization of "beauty contests"
is
same information. In these
respects, our formal-
perhaps more closely related to Townsend (1984), who emphasizes
the role of higher-order uncertainty in settings with endogenous learning, than to Morris and Shin
(2002),
4.3
who emphasize
the role of strategic uncertainty in coordination games.
Noise and exuberance
The preceding
equilibrium,
discussion helps recognize the following.
we assumed a particular information
To
facilitate the characterization of the
structure that permitted us to guess and verify
the fixed point directly instead of working with the infinite regression of higher-order
role of higher-order uncertainty
then manifested
itself
The
only in the amplification of the correlated
errors in the entrepreneurs' information about the fundamentals.
18
beliefs.
However, with more general
information structures, "noise" or "exuberance" could also originate from shocks to higher-order
beliefs.
Indeed, following Angeletos and La'O (2009b), one could introduce shocks that
move
higher-
order beliefs without affecting either the fundamentals or any agent's exogenous information about
the fundamentals. These shocks would then cause fluctuations in investment and asset prices that
would look a
sunspot fluctuations, even though they would not be triggered by correlation
lot like
devices and they would not rest on equilibrium multiplicity.
Combining these observations with the
of "exuberance"
we propose
and
(iii)
the formalization
paper can take any of the following three forms:
in this
of correlated errors in information about the fundamentals;
to higher-order beliefs;
we conclude that
results of Section 3,
(ii)
(i)
amplification
fluctuations originating from shocks
sunspot fluctuations. Either of these forms captures variation
in
equilibrium outcomes that likely would seem "hard to reconcile with fundamentals" in the eyes of
an outside observer; they are possible
spillover
in our setting because,
between the two sectors of the economy. In Section
form of "exuberance" that can obtain
in
and only because, of the information
6.1,
we
discuss a fourth
complementary
a variant of our model with heterogeneous priors
—
this
variant,
permits us to reinterpret the correlated error e of our baseline model as a form of "bias" in
beliefs,
without, however, abandoning the axiom of rationality.
Finally,
whereas the entire preceding discussion focuses on positive aspects, we believe that a
proper formalization of "exuberance," "sentiments," "beauty contests," and the
like
should also
capture the normative aspects of these ideas. Indeed, references to these notions typically come
together with an argument
tioning of the
—more or
less explicit
—that there
economy and that the government should
is
something "wrong"
intervene.
in the func-
Keynes himself brought up
his
famous beauty-contest metaphor, and talked more generally about "animal
an
explicit
attempt to make a case that the market mechanism can be
spirits," as
inefficient,
part of
not merely to
describe market behavior. In this regard, the normative results of the next section are an integral
part of our formalizations of "exuberance" and "beauty contests." This
work where the normative aspect
and Wincoop, 2005) or imposed
is
in
either ignored (e.g., Allen, Morris
an ad hoc manner
(e.g.,
is
in contrast to previous
and Shin, 2006; Bacchetta
Morris and Shin, 2002).
and Policy
5
Efficiency
The
analysis so far focused
on the positive properties of the equilibrium.
normative properties and to policy implications.
1!)
We now
turn to
its
Constrained efficiency
5.1
In the environment considered in this paper, the government could obviously improve upon the
competitive equilibrium
it
public
—
this
could collect
all
the information dispersed in the economy and
would remove any asymmetry of information and would achieve the
cation. In practice,
question
if it
it
we address
seems implausible that the government be able to perform
here
is
make
first-best allo-
this task.
16
The
whether the government can improve upon the equilibrium by manip-
ulating the agents' incentives through taxes, regulation, and other policy interventions.
consider a notion of constrained efficiency that
is
We
thus
designed to address this question, without getting
into the details of specific policy instruments. Namely,
we
consider a planner
who
can dictate to the
how
to use their available information but that cannot transfer information from one agent
to another.
Angeletos and Pavan (2007, 2009) propose and study such a notion of constrained
agents
efficiency within a class of
games with dispersed information;
here,
we adapt
this notion to the
Walrasian economy under consideration by embedding the aforementioned information constraint
into otherwise standard definitions of feasible
Definition 2
A
and Pareto-optimal
allocations.
feasible allocation is a collection of investment choices k t
,
one for each entrepreneur,
together with a collection of consumption choices c#, one for each entrepreneur and for each trader
in each period, that jointly satisfy the following constraints:
(i)
resource feasibility:
Ci\di
<
—
e
};ki di,
I
JiZ[0,l/2]
i€[0,l]
ci2di
<
Ctfdi
<
Jig [0,1]
/
e
+
Jie[o,i]
with
Cj3
=
all
i
such that
Si
=
t
(i.e.,
for
all
entrepreneurs hit by the shock).
informational feasibility: for each entrepreneur
(ii)
(xi,y), Cj2
trader
for
dk di
/
7ie[o,i/2]
i
G
is
contingent on {xi,
(1/2,1], en
is
y, Si,
K, uj), and
non- contingent, c^
is
Cjs
is
i
G
[0,1/2],
en and
are contingent on
ki
contingent on (xi,y,s z K,u>,0); for each
,
contingent on (K,ui), and C&
is
contingent on
(K,u,6).
Definition 3
An
efficient allocation is a feasible allocation that is not Pare.to
dominated by any
other feasible allocation.
16
Why the government may not be able to centralize the information that is dispersed in the economy is an
important and difficult question that, as emphasized by Hayek, rests at the heart of the market mechanism. Clearly,
this question is beyond the scope of this paper.
'JO
Because of the linearity of preferences
consumption, efficiency leaves the distribution of
in
consumption across periods indeterminate. Moreover, the distribution of consumption across agents
will
depend
strategy
in general
on the point chosen on the Pareto
uniquely determined and
is
is
Q=
J
Ok-idi
measures the
efficient
investment
the one that maximizes the following welfare objective:
W=E
where
However, the
frontier.
level of
Q-
kfdi
\
aggregate output at
t
=
i
3 and where |
J.
kfdi represents
the social cost of producing this level of aggregate output. Equivalently,
W = E[ek(x,y)-±k(x,y)
which leads to the following characterization of the
Proposition 7 The
efficient
is
given by
=E
=
6
all
The
(x, y)
\<i\x,
(x,y), where the coefficients 5o, S x
efficient
,
y]
and
+
(8)
],
investment strategy.
investment strategy
k
almost
efficient
2
5x x
5 y are the
+
S x y,
same
(9)
as in Proposition
investment strategy thus coincides with the equilibrium strategy
with no information
spillovers.
It
1.
in the
benchmark
follows that our key positive result has a normative counterpart.
Corollary 2 (Main normative prediction) In
the presence of information spillovers, the con-
tribution of noise to aggregate volatility is inefficiently high.
As anticipated
in the previous section, this result provides the
tion of "exuberance"
and "beauty
contests."
The
intuition for
this'
normative basis of our formalizaresult
is
quite simple.
The agents
investment decisions possess information that permits them to forecast not only
in charge of real
the long-run profitability of their investments but also the mispricing of this profitability by other
agents at subsequent stages of financial trades.
turn gives
from a
rise to
a "speculative return," which
social viewpoint.
of information
away from
The market
markets.
By
aggregation
formation
is
efficiency,
is
possibility of forecasting such a mispricing in
however purely private and hence not warranted
tilts
the
we mean the
way entrepreneurs respond
to their sources
with negative implications for welfare.
friction that sustains this inefficiency
this
is
Such a private benefit
The
is
only the absence of perfectly revealing
following. In Walrasian settings, the absence of perfect information
tightly connected to missing markets:
perfectly revealed through prices,
and a
21
when markets
are complete,
first-best allocation
is
all
relevant in-
obtained (Grossman,
1981). In this sense, our normative result necessarily rests on
some missing market. However,
has nothing to do with borrowing constraints, incomplete risk-sharing and the
only essential market friction
is
like.
the one that limits the aggregation of information:
emerges robustly as long as the financial market looks at the
this
Rather, the
inefficiency
real sector's activity as a signal of the
underlying fundamentals. 17
Finally, note that Corollary 2
have focused on.
When
presumes that the equilibrium
is
the case
we
means
that,
Since this property
9.
when
clearly satisfied
is
there are multiple equilibria,
of
all
them
by the
efficient
are inefficient.
Policy implications
5.2
While the preceding analysis suggests that there
exist policies that
improve upon equilibrium welfare
without requiring the government to centralize the information that
is
does not spell out the details of the specific policies that permit to do
aimed
may
at reducing asset price volatility
Our
achieve this goal.
motivated by two considerations. First, there
is
unique, which
there are multiple equilibria, the result holds for any equilibrium in which
aggregate investment increases with
allocation, this also
is
is
dispersed in the economy,
so.
We now show how
it
policies
focus on this class of policies
a vivid debate on whether central bankers, or
governments more generally, should try to tame "exuberant" movements
in asset prices.
Second,
such policies look a priori plausible in our setting, since the inefficiency in our model rests on how
financial
markets respond to the signals sent by the real
sector.
Consider a proportional tax r on financial trades at
more broadly a
variety of policies that
may
may
we
let
For simplicity, the tax
to
is
meant
to capture
is
assumed to be paid by the buyers (here
may be
contingent on the level of asset
the tax rate r be contingent on p:
T
where
This tax
2,
include not only taxes on capital gains but
the traders). To capture the idea that policy intervention
prices,
—
introduce a "wedge" between the asset price and the
underlying private valuations of the asset; this
also regulatory interventions.
t
and tp are
The equilibrium
scalars chosen
=
T
(p)
=
+ Tp p,
T
(10)
by the government. 18
price in the financial
market now
P=—+—
1
T„
satisfies
E[0|A>]-TO
V
p
=
E[0\K,
.
u>]
-
r
(p),
which yields
(11)
/
For example, we could allow the entrepreneurs to trade at t = 1 securities ("futures") whose returns are correlated
9; as long the price of these securities is not perfectly revealing of 9, the financial market at t = 2 would continue
with
to look at the equilibrium
1
The revenues
distribution of this
K
collected
as a signal of 9,
by
lump-sum
this
and our
results
would go through.
tax are rebated as a lump-sum transfer.
transfer
is
irrelevant.
22
Because of linear preferences, the
hi
If
the tax
is
procyclical, in the sense that
to the traders' expectation of 8,
rp >
and thereby
0, its effect is
e,
investment
to
K
because entrepreneurs assign relative
K becomes a more
precise signal of the fundamentals 9,
and thereby also to the noise
were not the case, the second
on
We
infer that policies
real
economic
effect
aimed
However, the
e.
first effect
would not emerge
As
level, this
some
less
making
weight to
prices
y,
namely
it is
aggregate
more responsive
—
must always dominate
for, if
that
in the first place.
at stabilizing asset prices can
extent,
tends to
this happens, a second,
dampen
the relative impact of noise
Furthermore, a wide range of numerical results suggest that
activity.
desirable to do so to
At the aggregate
(5).
responsive to noise relatively to fundamentals.
countervailing effect emerges:
the response of asset prices
thereby also dampening the relative bias towards
the correlated signal in the entrepreneurs' best responses
less
dampen
to the information contained in aggregate investment.
This dampens the price response to the noise
make investment
to
optimal to
set
rp £
(0, co).
19
reduce the impact of noise only by reducing the response of asset prices to
it is
always
However, these policies
all
sources of variation
in the traders' expectations of their valuation of capital. In so doing, they also reduce the response
of asset prices to the fundamentals themselves.
As the
real sector anticipates this, the absolute
response of real economic activity to fundamentals also goes down, which entails a welfare
since that response
was already
inefficiently low. It follows that this
improve welfare, but cannot possibly restore
Proposition 8 Consider
loss,
kind of policy intervention can
efficiency.
the policies defined by (10).
As
long as the equilibrium, remains unique,
a higher tp necessarily reduces the contribution of noise to aggregate volatility and, in so doing, can
improve welfare. However, no policy in
The
this class
can implement the constrained- efficient allocation.
analysis above thus provides a rationale for policies
without invoking either the presence of irrational forces
wisdom on the
side of the government.
such policies: they
economy
this
it
participants or any superior
highlights an important limitation of
only by also dampening the response of the
to fundamentals.
The government, however, may do
By
at reducing asset price volatility,
among market
At the same time,
may tame unwanted exuberance
aimed
we mean
better by considering
policies that are contingent
on a wider
more sophisticated
policy interventions.
set of publicly-available signals
about both
the exogenous fundamentals and the endogenous level of economic activity. In particular, consider
a tax on capital trades whose rate
19
is
contingent, not only on the asset price p, but also on aggregate
is positive, we have found this to be
we have randomly drawn 10,000 values of the parameter
vector (A, -Re, tt x 7r u 7^, A) from (0, 1) x M.\. For each such vector, we have numerically computed the value of tp that
maximizes welfare and we have found this to be strictly positive. At the same time, we could show that a policy of
While we have not been able to prove a formal
result that the optimal tp
the case for an extensive search of the parameter space:
,
full
20
,
price stabilization
The equilibrium
is
(i.e.,
tp
—
>
oo)
is
never optimal.
necessarily unique
if
A and rp are small enough.
23
investment K:
r
where To,rp ,Tfc are
By
scalars.
of investment on asset prices
trepreneurs' noise
prices respond
— r(p, K) =
choosing tk
>
+ Tp p + tk K,
t
0,
and thereby ensure that
more strongly to
all
asset prices
0,
the signaling effect
no longer respond to the en-
the government can ensure that asset
other sources of information that the traders have about the
fundamentals (here summarized in the signal
In fact, conditioning the tax on the asset price
u>).
accomplishes the same as conditioning the tax on the signal
8.
dampen
the government can
At the same time, by choosing tp <
e.
(12)
ui,
In terms of the game-theoretic representation of Proposition
and thereby on the fundamentals
6, this
means that an appropriate
combination of tk and tv permits the government to control separately a, the degree of strategic
complementarity in investment decisions, and
of) the fundamentals. It
relative
Kg, the sensitivity of best responses to (expectations
then follows that these contingencies permit the government to reduce the
impact of the noise while at the same time raising the absolute impact of the fundamentals,
therefore restoring
full efficiency.
Proposition 9 Consider
the policies defined by (12).
These policies can control separately the
response of aggregate investment to noise and fundamentals. Furthermore, there exists a policy in
this class that
implements the constrained
efficient allocation as the
unique equilibrium.
This result highlights the distinct role that state-contingent policies can play
economy
the decentralized use of information, and thereby the response of the
fundamental and noise shocks, when information
focusing on taxes on financial trades,
its
is
dispersed.
applicability
eventual capital returns (or firms' profits).
If this
is
tax
it
can
affect
is
non-contingent, then
(e.g.,
the price p and aggregate investment
6K
as of
t
different states of nature; this
is
way
can
it
across
affect the
all
states of
the average level of investment and the average level of the price,
certain public signals
—
this insight
broader. For example, consider a tax on
but cannot affect their response to the underlying shocks. In contrast,
aggregate output
to the underlying
While we illustrated
incentives faced by the entrepreneurs and/or the traders only in a uniform
nature. In so doing,
in controlling
3),
K
if
this tax
as of
t
—
2,
is
contingent on
or the realized
then this tax can impact incentives in a different way across
because different states of nature, and different information
are associated with different expectations at
t
—
1
regarding these contingencies.
It
sets,
follows that
these contingencies can help control the response of the economy to the underlying fundamental
and noise shocks, much
well
alike the taxes
on
financial trades studied above.
21
Whether such state-contingent policies are time-consistent or politically feasible is an important question, but
beyond the scope of this paper. Also, the ability of such state-contingent policies to restore full efficiency may
well rest on special features of our model, such as the absence of risk aversion
and the
ability of the
government to
perfectly observe the signals that the real sector sends to financial markets. However, the (weaker) result that these
24
We
conclude this section by considering policies that directly affect the information available
In particular, towards capturing the role of the government in collecting various
to the market.
data on economic activity at either the sectorial or the macroeconomic
of our
model where the financial market observes only a noisy
statistic of
K.
that the government can control the quality, or precision, of this statistic.
the precision of this signal, the
fundamental.
It
more the weight that the
the response of asset prices to aggregate investment
it
—
is
subject to essentially the
price-stabilization policies:
same
22
We
trade-offs as those
it
Suppose further
Clearly, the higher
when estimating the
this statistic to
essentially in the
with the price-stabilization policies considered above.
precision
traders assign to
government can use the precision of
follows that the
consider a variant
level,
same way
as
manipulate
it
could do
optimal
infer that the choice of the
emphasized
for the
aforementioned
increasing the precision of this statistic increases the response of real
economic activity to the fundamentals, but also exacerbates the relative impact of
intermediate quality of macroeconomic statistics
the cost of improving this quality
is
may
noise.
thus be optimal in our context, even
An
when
negligible.
Discussion and extensions
6
In this section
we
discuss various extensions that help reinforce the message of the paper.
by providing a possible reinterpretation of our
priors.
We
We
start
results within a variant that introduces heterogeneous
then continue by enriching the "real" and the "financial" side of the economy.
Heterogeneous priors
6.1
Our
analysis has
imposed that
this
assumption
may be hard
unfamiliarity of the
all
agents share a
common prior. While
new investment
opportunities, different agents
of rationality
accommodate
macroeconomics,
have different priors about
about the informativeness of available signals
priors about the joint distribution of the fundamentals
analysis can
may
in
Rather, because of the
to justify during the episodes of interest.
their likely profitability, as well as
how our
standard
and the
this possibility, while at the
and a non-paternalistic approach
available signals).
(i.e.,
different
We now
discuss
same time maintaining the axiom
to policy.
how agents respond to their different sources of information, and in so doing control the
impact of noise and fundamental shocks, is not sensitive to the details of our model. See Angeletos and Pavan (2009)
for the broader applicability of this insight, and Angeletos and La'O (2009a) and Lorenzoni (2010) for applications
contingencies can control
in canonical business-cycle
22
models.
is a formal equivalence between the two policies. Let the statistic of aggregate
Gaussian noise with variance 1/ttv For each 7r c there is a price elasticity rp
of the tax in (10) that induces the same response of the price to aggregate investment in the economy with perfect
observability of
and vice versa.
In fact,
it is
easy to show that there
investment be A"
K
=
K+
<;,
where
?
is
,
,
25
It is
easy to show that our results extend to a variant of our model where the entrepreneurs
and the traders have
is
different priors
about
and where these priors are common knowledge. This
because both the amplification result of Corollary
1
and the
inefficiency result of Corollary 2 are
driven merely by the presence of an information spillover between the two groups, which remains
present irrespective of whether or not these groups share the same prior on
Perhaps more
if
we
interestingly, a
complementary form
may emerge
of "exuberance"
in our setting
allow for heterogeneous priors about the informativeness of available signals. In particular,
consider a variant where the signal y
informative of the fundamentals only in the minds of
is
the entrepreneurs; the traders, instead, believe that y
prior
6.
=
that y
is
are mutually
+
while the traders' prior
e,
known and
the rest of the model
variation in the entrepreneurs' beliefs that
is
is
is
pure noise. Formally, the entrepreneurs'
that y
—
These differences
e.
unchanged.
Under
in prior beliefs
this specification, y causes
considered "unjustified" from the traders' perspective.
However, from the entrepreneurs' perspective,
information about 6 which
is
is
the traders' refusal to believe that y contains
it is
"unjustified." Furthermore, these differences in opinions are mutually
accepted: the agents have agreed to disagree. Finally, these differences in opinions do not involve
any form of naivete or
irrationality: given his prior
and
his information,
each agent forms rational
expectations about the fundamentals, the prices, and the other agents' actions; and this fact
common
knowledge.
Because the entrepreneurs believe that y
react to
it
even
if
y.
is
informative about
is
they would find
it.
On
it if
about
6.
when the
In contrast,
far
is
uncorrelated
y impacts the entrepreneurs' own
traders do not directly observe y
aggregate investment as a signal of the fundamentals, they are not able to
in
optimal to
they could directly
follows that, in the absence of an information spillover, the price
It
it
the other hand, because
pure noise, they would themselves not react to
with y while aggregate investment responds to y only in so
beliefs
#,
they believe that the price did not correlate with
the traders believe that y
observe
is
and instead look
tell
at
apart movements
investment that are driven by signals that the traders themselves consider informative about 9
(here captured
by the
signals x)
be an unjustified "bias"
in the
from the movements that are driven by what the traders believe to
in the entrepreneurs' beliefs (here
presence of an information spillover, the price
the entrepreneurs' incentive to respond to
that the interpretation of "noise"
correlated error in their
own
eyes of the traders, "noise"
is
is
now
y.
is
captured by the signal
correlated with y, which in turn reinforces
In other words, Corollary
different.
y). It follows that,
From
1
continues to hold, except
the eyes of the entrepreneurs, "noise"
is
information, exactly as in the benchmark model, whereas from the
now synonymous
to a "bias" in the entrepreneurs' beliefs.
This extension permits us to capture the idea that many market players often appear to believe that
they "know better" than the rest of the market, while at same time recognizing that
26
other market players
may
also think in a symmetric way.
model speculative movements
recent literature that uses heterogeneous priors to
(Scheinkman and Xiong, 2003) and
differentiates
23
this literature allows for
how information
the equilibrium impact of certain "biases" as some agents cannot
investment or asset price
spillover that
boom
is
spillovers
is
at the
may
amplify
apart whether the observed
tell
driven by "hard information" or by the "biases" of other agents.
Furthermore, our approach has distinct normative implications. To appreciate
the concept of Pareto optimality allows for subjective probabilities.
and the speculative forces studied
open the door
heterogeneous
In contrast, by combining heterogeneous priors with dispersed informa-
our approach helps uncover a novel positive result:
priors,
in asset prices
Nevertheless, our analysis
symmetric information, thus ruling out the information
core of our analysis.
tion,
real investment (Panageas, 2005).
from this literature in one crucial respect:
priors but imposes
brings our paper closer to the
It also
that
follows that heterogeneous
aforementioned literature, do not by themselves
Rather, one has also to take a paternalistic stand that
to policy intervention.
the priors of some agents are "wrong"
in the
It
this, recall
—a stand that we have sought to
avoid.
In contrast, once
heterogeneous priors are combined with the information spillovers we have highlighted, the consequent amplification of the perceived "biases"
non-paternalistic planner
who
undesirable even under the perspective of a
is
evaluates each agent's ex-ante utility using the agent's
own
prior,
not some other prior that he judges more appropriate. In other words, the property that agents
respond to their own "biases"
symptom
not per se a
is
of inefficiency under a non-paternalistic
perspective; but the property that information spillovers amplifies this response
Richer specification of the real sector
6.2
We now
consider a certain variant of the real sector of our model, one that introduces a competitive
labor market.
The production technology
uses as inputs not only capital as in the baseline model,
but also labor. For tractability, this technology
requiring n units of labor at
n >
0.
labor
is
The
t
=
net return to capital
3 in order to
is
thus r
supplied only by the traders.
24
labor. Their intertemporal budget,
where the
last
m—
{r—
p)qi
assumed to be
H
is
where
w
for
some
denotes the wage rate. For simplicity,
now
given by u %
=
cji+Cj2+Cj3 — H{ti),
a strictly convex function representing the disutility of
on the other hand,
+ wli -
Leontief, with one unit of capital
produce 8 units of the consumption good,
— 9 — um,
term represents labor income.
to a constant, as
is
Their preferences are
where 4 denotes labor supply and where
up
is.
It
is
given by
follows that
H(£i).
23
The
c,;i
we can
+Cj2 + cj3
=
2>e+{r — p)qi+wli,
express the payoff of a trader,
payoff of a surviving entrepreneur, on the
In particular, the aforementioned papers allow different agents to disagree on the informativeness of different
exogenous signals about the fundamentals, but assume that all the signals are commonly observed. It follows that
there is nothing to be learned from observing the behavior of other agents.
24
Allowing the entrepreneurs to also supply labor, or introducing a third class of agents ("workers") whose only
role in the economy is to supply labor would not change the results in any significant way.
27
other hand,
We
is
=
in
rki — hkf, while that of
(at
[1/2,1], while labor
=
t
for labor
w = H'(nK), and r —
- pqh
as u = V {9 K, q
i
t
{ )
)
By
nK).
is
[1/2, 1],
expressed
6
e
(9,K,k)
side,
=
[6
- H'{nK)n]k.
= \K
l
vestment
= V^(9,K,XK)
(at
t
—
1).
and where
m = pki—^kf.
is
First, consider
wages
= w
i
for all
G
Leontief assumption,
—
ti
£
— nK
for all
follows that the payoff of a trader can
be
- H'(nK)n]q + H'(nK)nK - H(nK),
can be expressed as U{ = V e (6,K,ki) — ^ fy where
where V\6, K,
q)
=
[9
,
=
t
2).
Optimality on the traders'
in equilibrium, implies that
=E
p
where v
It
Next, consider asset prices (at
along with the fact that q
^e
= n ^ Ow
implication, in equilibrium,
- H'(nK)n.
while the payoff of a surviving entrepreneur
V
by a liquidity shock
Optimality of labor supply requires that H'(li)
3).
market clearing requires that ^gn/2 11^
demand
the aggregate
6
hit
characterize the equilibrium of this variant by backward induction.
and employment
i
an entrepreneur
E
t [-]
t
t
[v
(13)
],
Finally, consider in-
denotes the traders' expectation.
Substituting the price into the entrepreneurs' optimality condition gives the
following equilibrium condition:
ki^ilwil+XEilEtlv^-v*],
where W{
=
(1
- X)V^(9, K,
ki)
+
XV* (9, K, XK) and where
E,-[-]
(14)
denotes the entrepreneur's expec-
tation.
These results have a simple interpretation.
ities,
or the reduced-form payoffs,
solved out.
The dependence
First,
the functions
V
e
V
and
f
represent indirect
util-
which emerge once equilibrium employment and wages have been
of these payoffs on
K
reflects the
pecuniary externalities associated
with the labor market: higher aggregate investment raises the demand
for labor,
wages and depressing the return to investment.
l
Second, the variable v
thereby raising
represents the marginal
valuation of capital for a trader. Third, the variable Wi represents the marginal valuation for an
entrepreneur
who
expects to get a price equal to v l
when he
sells his capital.
In finance jargon,
these variables represent the "fundamental valuation" of capital, or the "fundamental return" to
investment. Finally, the expression
the error that the traders
that
is,
term
in condition (14)
Ejfi)']
make
—
v\ represents the pricing error in
in estimating their
own
the financial market,
valuation of the asset.
The
last
then represents the entrepreneurs' forecast of such a pricing error; this term
captures a "speculative" return component akin to the one in the baseline model.
As
in the baseline model, the pricing error
is
forecastable in the eyes of the entrepreneurs only
because they have private information about the sources of variation behind the signal (aggregate
investment) that they collectively send to the financial market.
28
Furthermore, this speculative
component
once again a zero-sum transfer between the traders and the entrepreneurs. These
is
observations suggest that the present variant continues to feature essentially the same kind of
and
amplification
Appendix B, where we show that
also to
more
We
inefficiency highlighted in the baseline model.
Corollaries
flexible specifications of the
and 2 extend not only
1
reduced-form payoffs
verify these intuitions in
to the present variant, but
V and V
e
1
that
may summarize
the
'Inner workings" of the real sector of the economy.
Apart from illustrating the robustness of our insights
to richer specifications of the real side of
the economy, these results help clarify the notions of "fundamental" and "speculative" returns to
investment. While these notions were tightly connected to the
same exogenous random
the baseline model, here they are allowed to have deeper micro- foundations.
boom.
A
Furthermore, these
how an episode
of "exuberance" can be associated not only with high investment
asset prices, but also
with high employment, high wages, and an all-around economic
results indicate
and high
variable in
similar reasoning then implies that the signals that the real sector sends to the financial
market need not be limited to aggregate investment; they should be interpreted more broadly as
the information, in part public and in part private, that the real sector conveys to financial-market
participants about real economic activity.
6.3
Richer specification of the financial market
Next,
we
discuss extensions that
may
help capture the role of the financial market as a provider of
information, as opposed to a simple receiver, as in the baseline model. This possibility introduces
additional feedbacks between real and financial activity, which
may
actually reinforce the message
of our paper.
To
9,
suppose that traders are
start,
they observe private signals of the form
model of the
financial
Grossman and
market
Stiglitz (1976)
u>i
—
for the asset
is
9
+
closer to the literature
and Hellwig
rfr,
where
all
rn
is
noise.
This would make the
on rational expectations
(1980). Specifically,
display constant absolute risk aversion and that
demand
an exogenous public signal about
risk averse and, instead of
random
assume that the
in the tradition of
traders' preferences
variables are Gaussian.
Each
trader's
then given by
ft
=
*&^,
(15)
rvan[6]
where
E
z
\9]
=
risk aversion.
E[9\wi,p, K], Vari{9}
=
Var[9\LUi,p,
K] and where F
Note that the market clearing price now serves as a
the information dispersed
among
sources of variation in the
demand
is
the coefficient of absolute
signal about
9, for it
aggregates
the traders. However, as long as there are additional unobserved
or supply of the asset, the equilibrium price will not be perfectly
L"J
revealing. For example, suppose the noises
u>i
which
is
a noisy signal of
shock are allowed to trade
common shock
that
is
in
are correlated: the price will then reveal the average
suppose that the entrepreneurs not
Finally,
6.
rji
by the
hit
liquidity
the financial market, but their valuations are subject to an additional
Once
not observable by the traders.
again, this shock guarantees that
the price does not perfectly reveal the fundamental to the traders.
K as a signal of
perfectly revealing, the traders will continue to use
our information-driven complementarity would
still
be present
25
As long
as the price
is
not
Therefore, the key source of
9.
in this extension.
Furthermore, because our mechanism reduces the informativeness of aggregate investment,
also implies that the traders
end up
faced by each trader, Var^d],
higher risk
premium
have about
9.
for
it
The
higher.
When
this
is
less
information: the conditional uncertainty
the case, each trader will not only require a
holding the asset, but also react
less to
any private information she may
equilibrium price will then also do a worse job in aggregating this information,
be relatively more sensitive to other sources of aggregate
will
mechanism may
side, this
for
is
with
in equilibrium
noise.
It
follows that our
also reduce the informational efficiency of the financial market.
means that our mechanism may
non-fundamental
it
On
volatility.
premia
raise risk
in financial
On
the positive
markets and amplify their
the normative side, the increased uncertainty
may
also exacerbate
the misallocation of the asset, which in turn would reinforce our welfare implications: the planner
would
like
entrepreneurs to react more to the fundamentals and
emphasized
in the baseline model,
to the financial
but also because
this
less to noise,
would transmit more precise information
market and thereby improve informational and allocative
Next, suppose that
we introduce
efficiency.
a second round of real investment decisions, which takes
place after the financial market closes.
round of investment
not only for the reasons
Now
information would travel not only from the
to the financial market, but also from the latter to the second
first
round of
investment. This would capture the role of asset prices in guiding investment decisions by revealing
valuable information that
managers
(e.g.,
Dow
is
dispersed in the marketplace and not directly available to corporate
and Gorton, 1997, Subrahmanyam and Titman, 1999, Chen, Goldstein and
Jiang, 2007). Importantly, our
financial markets.
This
is
mechanism would then imply a deterioration
a direct implication of the argument
made
in this functioning of
in the previous
paragraph
round of
real invest-
regarding the informational efficiency of asset prices.
Alternatively, suppose that
we introduce a
financial
ment. This market includes some informed traders,
market before the
who may
or
may
rounds of trading, as well as some uninformed liquidity traders, whose
first
not be present at subsequent
role
is
to preclude perfect in-
formation aggregation. Suppose further that the entrepreneurs observe some private signals about
~5
Whether
or not the price reveals 6 to the entrepreneurs
reveal 8 to the traders, so that the traders continue to use
is
K
not crucial. The key
as a signal of
8.
For a
is
that the price does not perfectly
analysis of an extension along
full
these lines, see Section 6.2 of the earlier version of this paper, Angeletos, Lorenzoni and Pavan (2007).
30
the trading positions of the informed traders; think of this assumption as a parable for the fact
that investors and firm
managers are often anxious
by some key informed big players
in financial
to collect information about the positions taken
markets. 26 Then, we could re-interpret some of the
exogenous signals that the entrepreneurs receive in our model as imperfect private learning about
the actions of these big players.
signals
—and thereby the
In this case, the origin of the correlation in the entrepreneurs'
initial source of "noise" or
errors of these early traders.
"exuberance" in our model
—could well be the
27
Furthermore, because these earlier traders
may
themselves have some private information about
the sources of variation behind the signals they send to entrepreneurs and later traders, they
may
also
be able to forecast the errors made by these subsequent agents.
An
information-driven
complementarity similar to the one that emerges in the entrepreneurs' investment decisions may
then emerge also in the traders' positions. This complementarity,
in turn, is likely to
be stronger the
higher the degree of short-termism of the traders: the more the early traders' portfolio choices are
driven by forecasts of future pricing errors, as opposed to forecasts of the fundamentals, the stronger
the complementarity in their choices,
our model.
An
much
what happens
alike
in the case of the entrepreneurs in
extension along these lines could thus not only reinforce our results, but also bring
our analysis closer to Froot, Scharfstein, and Stein (1992), Allen, Morris and Shin (2006), and other
papers that study the implications of short-termism in financial markets.
Conclusion
7
This paper examined the interaction between the real and the financial sector of a neoclassical
economy with dispersed information about the
profitability of a
new investment
conveying a positive signal about profitability, higher aggregate investment
higher real activity
—increases asset
way feedback between
real
and
prices,
which
financial activity
—
opportunity.
or,
more
By
broadly,
in turn raises the incentives to invest. This two-
makes
real
economic decisions sensitive to higher-
order expectations and amplifies the impact of noise on equilibrium outcomes. As a result, economic
agents
may behave
as if they
exhibit fluctuations that
were engaged
may appear
in
a Keynesian "beauty contest" and the economy
in the eyes of
26
an external observer as
if
may
they were the product
For the sake of this discussion, ignore the additional effects that may obtain when these big players attempt to
manipulate asset prices and/or real economic activity.
2
Indeed, suppose that the entrepreneurs, in addition to the market-clearing price in the first round of trading,
observe two purely idiosyncratic signals, one about the fundamentals 8 and one about the position Q of the early
informed traders. Let these signals be i, = 8 + f, and y, = Q + ?, and impose that the noises f, and c, are
independent across i and of any other random variable. Next, suppose that the equilibrium value of Q is a linear
function of the early traders' average forecast of 8, which in turn is a linear function of 8 itself and some noise e:
Q = j/)q + i/i 1 5 + i/j2£. The observation of the signal y, is then informational equivalent to the observation of the signal
x l2 = —{yi - i/>o) = 8 + ^-e + -j-Si- Clearly, this is a private signal about 8, whose error is correlated across the
entrepreneurs. A mechanism similar to the one in the baseline model is therefore once again at play.
'
31
of "irrational exuberance." Importantly, these effects are
by the dispersion of information, and obtain
While both
irrationality
tion policies that are contingent only
of inefficiency, are driven purely
an otherwise conventional, neoclassical,
and our mechanism can
not rest on any presumption of superior
tals
in
symptoms
wisdom on
justify policy intervention, our
the government's side.
on publicly-available
signals
We showed how stabiliza-
about the exogenous fundamen-
Through a proper design
of such contingencies, the gov-
ernment can dampen the impact of noise on equilibrium outcomes, improve
cases,
approach does
indirectly tax/subsidize the response of economic
and the endogenous economic activity can
agents to different sources of information.
setting.
welfare, and, in certain
even attain a certain constrained-efficiency target.
The
analyzed in this paper are likely to be stronger during periods of intense techno-
effects
logical or institutional change,
opportunities
is
likely to
when the information about
be highly dispersed. At some
the profitability of
level, this
new investment
seems consistent with the recent
experiences surrounding the internet revolution or the explosion of investment opportunities in
emerging economies. Our mechanism may, however, also be relevant
tions. Indeed,
information regarding aggregate supply and
dispersed, as indicated by surveys of forecasts
release of key
to the ones
macroeconomic
documented
statistics.
in this paper
ordinary cyclical fluctua-
conditions seems to be widely
financial markets' anxiety preceding the
This opens the door to the possibility that
may
effects similar
operate over the business cycle.
Finally, our analysis has left aside credit
finance relevant for investment decisions.
and by the
demand
for
market
We
spillover as the only source of amplification
frictions that
have done so
and
make the
availability of outside
in order to focus
inefficiency.
on the informational
However, the episodes of interest
appear to involve important interactions between credit markets, asset
prices,
and investment. An
extension of our model that introduces collateral constraints as in Kiyotaki and Moore (1997)
reveal additional amplification effects
multiplier.
asset prices,
Episodes of "exuberance"
but also
in
coming from the
may
interaction of our
then manifest, not only
exuberant credit booms.
32
in
mechanism with a
may
credit
exuberant investment and
Appendix A: proofs
Proof of Proposition
The proof proceeds
2.
the equilibrium characterization in the main text.
proves existence of a equilibrium with
[3X) (3y
,
/?#,
pE
Step
in three steps.
the details of
1 fills in
Step 2 analyzes the fixed point problem and
jg,
,
7£ 7^ >
Step 3 proves uniqueness of the
0.
,
equilibrium for A small enough.
Step
A'
First note that there exist
1.
no equilibria
which
in
0.
Indeed, in any such equilibrium,
would convey no information to the financial market. The equilibrium price would then simply
be equal to
E[#|uj].
Because this
is
an increasing function of
would then impose that they react positively to both
that
fig
—
be equal to
K
would perfectly reveal 9 to the traders
that responds positively to both x and
equilibrium must satisfy
fig
analysis in the
=
j3 x
+ (3y ^
main
text,
the facts that
=
0,
nz
=
ny /^p
2
,
E[6\x,
y]
=
and
+
coefficients
(/?o,
Ac,
(3y )
A,
=
f3£
(3y
^
are given
=
?y
-
0.
-
fi)
+
0o
Indeed, in any such
it
optimal to follow a linear strategy
j3 £
=
0.
Hence, any
0.
given by
is
5 X (x
=
any equilibrium the
in
price
Substituting (4) into
(5).
5 y (y
-
E[i\x,y]
/j,),
is
=
y
is
(5),
—
given by
and using
E[9\x,
y],
and
given by
+ px x + (5y y,
by the following:
(l-/?x -/3 y M
(16)
)
ITy
V
Any
find
we have that the entrepreneurs' investment strategy
k (x, y)
where the
=
assumption
which case the equilibrium price would
in
we then have that
jj,
j5E
contradicting the assumption that
y,
and the entrepreneurs' investment strategy
E[ijj|x,y]
signals, thus contradicting the
But then again each entrepreneur would
0.
From the
the entrepreneurs' best responses
8,
Next, note that there exists no equilibrium in which
0.
equilibrium,
(4)
=
f3g
+
(fi"
V
[TTg
+
7T
W) J
(nv % + ','f^
+
J
V
(
<P
{Kg
(18)
»
TToj)
equilibrium must thus satisfy (16)-(18), along with
*
Step
2.
To
Px
+ Py
establish existence of a equilibrium in which
33
f3x
,
(3 y
>
0, let b
= Py /P.v
-
Dividing (18)
by
and noting that
(17)
—
<p
+ b)
6/(1
yields
2
=
6
i
i
S^
*»+(&)
Using the definitions of Sx and
5y
,
of (19)
equivalent to the following function of
is
_5y
mapping
It is
—
=
f2
b
1
=
*"
.
—
>
limj-^+oo
A
F
F (6)
follows that
finite. It
tp'
well defined
is
one can then show that the right-hand side
(1
+
6) b
F
monotone transformation
a
b is
F(b) identifies the mapping
easy to see that
*
[(l-\)(l-6x )+n]b 2 +(2-\)6y b + 5y j
Noting that
0.
1
(19)
6:
\
~^\ 1+
F(6)
where
+ *«)-*&(% + lV«)
(*«
as in Proposition
,
^--^.
—
mentioned
T(tp)
of
main
in the
(p,
we have
text.
and continuous over R+, with F(Sy /5x
has at least one fixed point
b
>
5y /S x
that the
Given
.
>
)
S y /5 X
this value of
existence of a equilibrium can be established by construction. First, the equilibrium value of
obtained substituting
of
(16).
7tj
given by
(3X is
Finally,
in the
Step
@y /b and
from the
main
=
p
text,
+ b)
fact that 0g
it is
immediate
To prove uniqueness,
3.
into (18)
also positive.
is
can be seen directly from
Px an d
b/ {1
(17).
Py must correspond
=
(3 X
and
Given
+ (3y
and
first
=
e
,
the equilibrium value of
and from the formulas
(3 y ,
is
Next, the equilibrium value
notice that there exist no equilibria in which
This in turn implies that
A'
>
the right-hand side of (20)
is
strictly positive, for
is
j3y
f3y
6,
/?o is
given by
7e and
for 70,
to see that all these coefficients are also positive.
Next, note there exists
F
and
(3X
all
such that,
for
any A G
any
[0, A']
-
A) (1
-
5X )
+
fi]6
2
+
This
the denominator in the fraction in
(2
-
A) 6 y b
+
when A G
that,
defined and continuously differentiable over the entire real
5X {[(1
0.
defined in (20).
G R. This implies
b
@x —
equilibria, irrespective of the sign of
F
to a fixed point of the function
,
the function
clearly positive.
is
and
[0, A'],
with
line,
5y }
Moreover,
lim F{b)
b—»— 00
=
lim F(b)
b—>+oo
=FOQ =
case,
F
is
suppose 5y
=
(1
-
A) (1
-
8X )
+
f2.
^
(1
-
A) (1
-
Sx )
+
Q.
Then F'
<
Sy
+
(1
A'.
-
a) (1
We now
-
sx )
+n
>
IT-.
need to consider two cases.
First,
— -1/2. In this
respectively, by F_ = F( — 1/2) and F = F^,. Second,
(6) has two zeros, respectively at b = 61 and at b = 62,
The function
bounded from below and above,
1
X
Thus, from now one, restrict attention to A
suppose 6y
-^-
F
34
then has a global
minimum
at b
where
-sy - v/ [(i-\)(i-5x -5y )+m>
and
s
b
-
iy
-
(1
-
A) (1
iy) - n
-
a
<y
-jy
+ v[(i-A)(i-^-^)+nF;
5y-(l-\)(l-5x -6y)-n
s
62
"
"
^ (1 — A) 5q + O, the function F then has a global minimum at F = F (62) and a global
maximum at F = F (b{). It is easy to check that in all the cases considered both F and F converge
to 5y /6 x as A — 0. But then F converges uniformly to 5 y /S x as A — 0. It follows that for any
When
5y
>
>
>
e
[S
there exists a A
0,
y /5X
~e,5y /5x +
Now, with a
Fon
slight
and A >
1.
It
>
£
— e,
follows that,
A
if
<
A,
Proof of Proposition
<p
fix
£
<
From
(0,1).
5X
and
>
fiy
has no fixed point outside the interval
continuous in b at
F
(5
with
=
A)
(6;
F (6; A),
{5y /5x ,0)
such that dF{b; X)/db
(0, A]
to highlight the dependence of
<
F is
+e],
.
-i, 5y /6 x +i]
that there exist e
b
$
continuous and differentiable in
[d
6,
—
y /S x
£, 5y /6 x
>
+ e],
with dF(b; X)/db
<
result.
In any of the equilibria identified in Proposition
conditions (17) and (18) in the proof of that proposition,
6y
0. It
we have
[5y /6x
has at most one fixed point, which establishes the
3.
=
£
the following are true: for any
[0, A],
y /5 x
and 8F(Sy /5x ;0)/db
for all b
1
results with the continuity of F(-; A),
A £
all
[(^/tfa:
is
and A £
such that, for
7^ b; for 6
F (b)
abuse of notation, replace
Combining these
[0, A],
F
A,
e].
follows that there exist i
and A £
<
so that, whenever A
A'
Notice that 8F(b; X)/db
A.
F (6; A)
<
2,
we have
that
then follows that
it
Moreover, the two inequalities imply
LP
1
-
_
fiy
Sy
^
7Ty
5X
fi x
<p
=
>
Finally,
=
=
fix
+
ox
+ o yv +
t>
x
fiy
J
+ dy +
7T0
'
2.
A',
For any A £
A" £
=
+
TT
last inequality follows
Proof of Proposition
[0,
[0,
=
>
TTy
4.
X
A',
^y
(1
+
TT
y
from
TT
y
ip)n e
W+
Lp)ir
y
-
7T-,
+
and
ip
2
if7T x
+
r
;
(tT0
7T
<
dx
+
+
d
^x
+
^y ^y
y
—,
W
r
^u)
y
W)
and
is
b"
F (b; A)
introduced in the proof of Proposition
Take any pair
continuously differentiable over R.
be the unique solutions to
F (6; A) =
b,
respectively
A" (existence and uniqueness of such solutions follows directly from Proposition
for
A
2).
Furthermore, as shown in the proof of Proposition
A'
^8
^ui)
+ tpne
+
+ <f~
TT
tpirx
— A
(21).
F(;A)
let b'
r
7T}
</>
-
-
(1
Consider the function
A), the function
A) with A"
and A
+
Tfx
<pir e
A
TT0
where the
+
+
+
nx
ipiTy
A
35
2,
F {b, A') - b >
for all b
£
[0, b').
Simple
algebra then shows that
>
that b"
b'
The
.
dF (6; A) /dX >
any
for
result in the proposition
b
>
Proposition
n +^"+ 7r
2;
for
Consider the function
5.
we
convenience
Take the parameters
•
is
Sx 5 y
,
,
=
ft)
proof of Proposition
Moreover, at the point
2.
properties, together with the properties that
>
in
(
—00, ensure that,
— 00,62) and
F [b) -
that
b
one
such that
F
6y
62,
we have
Sy
,
,
61
>
=
Because
(.75, .2, .1, .1)
F
b\
and
62 are as defined in the
(A,
irg, ix x
,
7r
y
Proof of Proposition
,
u) S
ir
E
,
ft.)
<
62
(b\
is "strict"
0.
,
These
5 y ft)
,
>
in the sense
A, 5x ,5y ,ft) in
admits at least three fixed points whenever (X,5x ,5y ,ft) £ B.
(A, 5 x ,S
y
<
F (b; A, Sx
there necessarily exists an open set
,
=
admits at least one fixed point
continuous in
is
ft
these parameters the function
and lim(,__ 00
F
and the
parameters, with
all
that F(b2; X,5 x ,5 y ,ft)
ft)
b)
introduced in the proof of
Furthermore, each of these three fixed point
in (62,0).
,
ft)
where
,
,
(A, 5X , 5
ft)
y
,
With
(.75, .2, .1, .1).
F (0; A, 6X
proposition then follows by noting that for any
parameters
,
addition to a fixed point in (S y /5X +00),
in
changes sign around them.
neighborhood of
92.
if
follows
It
0.
an increasing function of
F (b; A, 5X
F is defined and continuous over the entire real line and bo <
=
>
+
are highlighting here the dependence on
(A,
if b
6/(1
then follows from the fact that
fact that the contribution of noise to aggregate volatility
Proof of Proposition
with strict inequality
0,
BC
The
an open
(0, l)
3
x
R
result in the
G B, there corresponds a unique
set of
5
.
Substituting condition (2) into condition (3) gives the equilibrium
6.
and the
price as a linear function of aggregate investment
p(6,
s, 77)
=
70
traders'
exogenous
signal:
+ jK K(e, e) + 7w w,
(22)
with
Clearly,
9.
The
and
jk >
if
and only
result then follows
letting kq
=
A70, Kg
=
Proof of Proposition
7.
Proof of Proposition
8.
contribution of noise.
using
(3),
if (3g
>
0,
which means that high investment
from substituting
1
-
A
+ X-y^
is
a positive signal about
(22) into the entrepreneurs' optimality condition (5)
and a
=
Xjk-
This follows directly from the main text.
Consider the
first
claim,
namely that a high rp reduces the
relative
Substituting the price (11) into the entrepreneurs' best response (5) and
one obtains a system of equations for
/3o, (3X
and
Following similar steps as in the proof of that proposition,
36
(3y
it is
,
as in the
possible to
proof of Proposition
2.
show that a equilibrium
is
=
characterized by a ratio b
((1
* =
that satisfies b
//3x
= F (b;
+
b (1
F(b;9)
with
f3y
+ tp
1/ (1
easily see that
-
A)
(<J
+ 5y +
n)
+
Afi*) b 2
A*
6)
(2 (1
-
F (b; *) =
b
with
from such a solution, one can construct a equilibrium in which
b
>
F (b;
0,
•)
is
increasing in *.
equilibrium value of
We
increasing in *. Equivalently, the ratio
b is
proof of Proposition
one can
2,
>
b
6 y /d x
f3X) j3 y
>
0.
and
is
— „^„
tp
,
is
A*)
6y
one can then
(ii)
that, starting
Furthermore, for any
is
unique, the
and hence the
decreasing in tp Lastly, following steps
also verify that the equilibrium
+
X
2,
thus conclude that, as long as the equilibrium
contribution of noise to aggregate activity,
in the
+ A*) 5y b + {l-
A)
Following steps similar to those in the proof of Proposition
).
there always exists a solution to
(i)
+
*) where
relative
similar' to
indeed unique
those
when A and
tp are small enough.
may
Next, the claim that a positive tp
increase welfare can be established by numerical example;
namely that there
see footnote 19. Finally, consider the last claim,
implement the constrained
efficient allocation as
suppose that the opposite
with the efficiency condition
(9),
no policy as
in (10) that
can
an equilibrium. Towards delivering a contradiction,
Comparing the entrepreneurs' equilibrium best responses
true.
is
is
(6)
one can immediately see that efficiency requires that the following
condition holds:
>-
E
=
x,y
for all x, y.
Substituting the equilibrium price (11), this condition can be rewritten as
1
K
1
By
Proposition
K—
5q
+
E[d\K.
=
z,y
TO
TD
for all x, y.
(23)
the fact that the policy implements the efficient allocation in turn implies that
7,
+ 6q6 + 5£ £.
This implies that E[0
are as in Section 4 with
/?o
=
<5q,
j3x
—
1
5X
A', w]
,
and
=
70
(3y
=
+ jkK + 7^ lj
5y
.
7o
+ IK
(
60
+
Se 6
+
5e i
I
coefficients (7o,7A',7w)
Therefore, (23) can be rewritten as
1
E
where the
+
-y
u uj - r
\x,y
0.
1
Taking unconditional expectations, one can then see that
t—
1
+ Tp
[70
+ IK
(<5o
to
+ ^M) + 7w/-i -
and tp must
T
]
-
/i
=
satisfy
0.
Subtracting side by side the last two equations, after some manipulation, one obtains that
1+T,
{iK&e
+ lw) -
1
E
fj.\x,y
-
1
37
+ Tp IK
S E E[i\x,y}=0.
Substituting for the terms in expectation yields
1
1
+
{.ixbe
1
1
Given that
+ t,
this condition
•7/*-^ [y
1
Substituting the
first
-m-
must hold
1
.1
+ Tp
+
lu)
-
<5x(2-
for all
+T,
and given that
n)
+
sy(y
-
m)]
-
5 y (y
fi)]
=
o for an
i
1
+ T,
7if(5 e
=
0,
Sy)
=
0.
1
1
+ 7——1k5s
^y
y-
2-%
must be that
y, it
(1
(24)
1 T" 7 Tj
7K^ =
0.
cannot be true given that, when investment
We
a contradiction.
-
condition into the second gives
>
S£
-
a*)
+ Iw) -
1
last condition
(x
[5 x
x and
hh'dd
+ lw) -
(li<5e
-
1+Tp
This
1
7V
and tp must be
finite to
is efficient,
ensure that (24)
is
7/f
is
satisfied.
necessarily positive,
Therefore, there
is
conclude that a simple stabilization policy as then one given in (10) cannot
implement the constrained
Proof of Proposition
9.
efficient allocation as a
To
competitive equilibrium.
establish the result,
suffices to
it
show that there
exists a policy of
the type given in (12) such that, under this policy, there exists a competitive equilibrium in which
E\p-6\x,y}
To
see that this
must
satisfy
p
is
P
if
for allx,y.
(25)
indeed the case, note that, under any policy as in (12), the equilibrium price
= K[9\K,uj} — tq —
Next note that
=0
rp p
=
1
- tkK.
+ T„
Equivalently,
K,u>]
-
T - TK
K
(26)
the policy (to,tp ,tk-) implements the constrained efficient allocation, then
(27)
with coefficients (7o,7k,7w) as in Section 4 with
(3q
=
6$,
Px =
5X
into (26), one can then easily see that the policy with coefficients
T
=
70,
tk = IK,
38
rp
=
-yul
-
1.
,
and
j3y
=
5y
,
Replacing (27)
is
such that p
time
(25)
1,
—
—
6
lu
—
6
we then have that
=
Since the entrepreneurs possess no information on the shock
1].
E[7J|x,y]
Finally, note that
is satisfied.
7W which implies that 7^ €
,
=
which
tk >
verifies that,
and rp <
under the identified
0; this last
result follows
r\
at
policy, condition
from the definition of
(0, 1).
Appendix B: a generalized model
This appendix considers a generalization of our model that helps illustrate the robustness of our
key positive and normative results (as stated in Corollaries
some
made
of the related claims
main
in the
1
and
signal observed
s-th signal.
entrepreneur
...,
5} and write them
by entrepreneur
i.
=
Let p s
as x, 5
=
+ £«,
Corr(^i s ,^j s ), for
i.e.,
=
given the signals Xi
S >
where
^
i
j,
1 signals
&a
about
9.
Index
the error in the s-th
is
denote the correlation in the
on the information available to
Finally, let E;[-] denote the expectation conditional
i,
also forms the basis for
text.
Information structure. Suppose that the entrepreneurs observe
these signals by s £ {1,
and
2)
(xn,...,Xig).
Payoff structure. For a surviving entrepreneur, the value of holding k units of capital in period 3
is
V e (9, K, k). An entrepreneur's payoff is then V e (9, A, k)-k 2 /2
and pk —
if
he
is
not hit by a liquidity shock,
k~/2 otherwise. For a trader, on the other hand, the value of holding q units of capital
V'(0, K, q), so that his payoff is
V
t
A,
(9,
reduced-form payoffs that the agents
the focus of our analysis.
may
For example,
summarize the interaction of the agents
To maintain
tractability,
we impose
—pq. The functions
q)
V
e
and
V
f
are
meant
in the variant considered in Section 6.2, these payoffs
in a
competitive labor market during the production stage.
that these functions a linear-quadratic.
Vkk
<
Vkg >
interpreted as better fundamentals.
0,
so that higher 9
is
further impose that
To guarantee that
and V*q <
we
VL >
to capture the
obtain through a variety of market interactions outside
individual decision problems are concave,
and
is
0.
Finally,
We
also let
we assume
that
(1
where
Q =
A A'.
Section 6.2.
inefficiency:
idea,
X)Vjl-(9,
This assumption
effects featured in the
externalities that
-
is
K, K)
in certain
14(0, K, Q)
=
(28)
motivated by the following considerations.
reduced-form payoffs
emerge
+
V
e
and
V
t
are
market interactions,
The
external
meant to capture only the pecuniary
like in
the case of the labor market in
In Walrasian settings, such pecuniary externalities need not be the source of any
they often wash out at the aggregate.
and thereby to
We
impose (28) only
in
order to capture this
isolate the information spillover as the only source of inefficiency.
Equilibrium and efficiency.
The
following proposition provides a characterization of the equi-
librium and efficient allocations for the
more general model described above.
39
Proposition 10
(i)
For any equilibrium, there
the payoff structure, the information structure,
exist scalars kq, Kg,
and
and a (these scalars depend on
the particular equilibrium) such that investment
satisfies the following:
= Ei[Ko + K e 8 + aK].
ki
(ii)
There exists scalars Kq, Kq, and a* (these scalars depend only on the payoff structure) such
that, in the
unique constrained efficient allocation, investment
The
(Hi)
There exists a constant
if
and only
ip
>
if
such
«
=
a >
that,
«-
+
a > a*
if
and only
The equilibrium
is
constrained, efficient if
implication,
(v)
Parts
(i)
(30)
relative contribution of noise to aggregate volatility is higher in equilibrium than in the
constrained- efficient allocation
(iv)
satisfies the following:
= Ei[K*o + K*e + a*K}.
ki
By
(29)
and
pendix and the
(ii)
if
a*.
for any equilibrium,
*A«
(3D
high investment serves as a positive signal about
and only
if
there are
no information
0.
spillovers.
provide a certain isomorphism between the generalized economy of this ap-
games studied
class of
in
Angeletos and Pavan (2007, 2009).
The
scalar
a can
then interpreted as the degree of complementarity featured in any given equilibrium, while a* can
be interpreted as the degree of complementarity featured in the efficiency allocation.
Part
(hi)
points out that comparing the relative impact of noise in any equilibrium with that in the efficient
allocation
is
synonymous
is
efficient
comparing a with a*. Part
a with a*
of discrepancy between
equilibrium
to
is
high investment
is
1
when and only when the information
and
2,
is
spillovers (amplification)
Proof of Proposition
and normative
t
The combination
results of the baseline model, as
continue to hold in the more general model consider here: as long as
both higher than 'what
and excessive from a
10.
Part
(i).
it
would have been
=E
t
in the
absence of information
social perspective (inefficiency).
Clearly, the equilibrium price
p
E
spillover vanishes.
"good news" about the underlying fundamentals, the relative impact of noise on
equilibrium investment
where
then establishes that the only source
the information spillover, while part (v) points out that the
of these results then imply that the key positive
stated in Corollaries
(iv)
must
satisfy
[Vt(8,K,\K)}
denotes the traders' expectation, given their available information.
40
By
implication, the
equilibrium level of investment must satisfy
h=
Ei\w(e,
k,
h)]
+
AEi Et[vf(d, k, xk)\ -
where Ej denotes the expectation of entrepreneur
w{9, K, k)
Note that
=
must hold
this condition
(1
vq\e, k, xk)
(32)
and where
i
- X)Vke {6, K,
irrespective of
k)
+ W*{d, K,
XK).
whether there are information
spillovers
from
the real sector to the financial market; in fact, this condition holds for any information structure.
Because the functions
Ve
and
V
1
w
are both linear-quadratic, the function
w(9, K, k)
=
and Wk
linear,
= Vk k <
0.
Next, because
we have that the
K
is
itself linear:
+ wgd + wkK + w k k,
ioo
(33)
where wo, wg, Wk, Wk are scalars pinned down by the payoff structure, with wg
e
is
known
=
{l
— X)V£g + XVL >
to the traders, and because the function V*
traders' error in forecasting their valuations
is
is
proportional to their error
in forecasting 6:
E
t
[V*[0,
K, XK)} - V*{B, K, XK)
Substituting (33) and (34) into (32),
we have
= V£
(e, [§]
-
d)
(34)
that, in equilibrium, the investment strategy
must
satisfy
Ei
1
wo
- wk
1
w°
.e + JOX-K + xitivM-e
- Wk
(35)
where
1
- Wk
Next, note that, in any equilibrium, the entrepreneurs' investment strategy
h=
Po
+ ^2 @sXi
is
given by
'-
s=l
for
some
scalars po, Pi,
,@s- By
implication, aggregate investment
is
given by
K = Po + Pe{9 + e)
where
e
=
entrepreneurs' signals. It
—
errors in the
J2s=i J~ £ s is a weighted average of the correlated
is a Gaussian signal of 9, which
follows that, in the eyes of the traders,
J2s=i Pa &n ^ wnere e
K
41
can be written as
in turn implies that their forecast of 9
E
where u
and only
is
t
[0]
=
K[6\lj,
=
K]
+ 1lj lo + 1K K
70
the exogenous information of the traders and where 70,
>
if /3$
0.
u>,
their forecast of
Substituting (36) into (35) and using
=
Ej[a>]
Q
=
;
1
Xipjo,
\-
-w k
k6
=
l-Wk
Consider the constrained
(ii).
k,q
and a defined as
£
(1/2,
This
1].
is
is
if
1
at
t
= E;[0].
=
- wk
+
Mhk
(37)
First, note that, irrespective of the
t
=
the planner always finds
1,
2 uniformly across the traders: q %
17.
= XK
for
It follows
given by
W=E
is
XK
9: E,[(D]
a direct implication of the concavity of payoffs with respect to
that the welfare objective
Clearly, this
>
follows:
a =
and
A^(7u
efficient allocation.
optimal to allocate the supply of capital
all i
are scalars, with 7^-
7^
we conclude that the investment strategy must
information structure and irrespective of the investment strategy at
it
,
coincides with their forecast of
Ej[#],
indeed satisfy condition (29), with the scalars kq,
K
W
(36)
Next, note that, since the entrepreneurs have no information about the error in
the traders' exogenous signal
Part
follows:
2
!W
k +
(1
- X)V e {6, K,
k)
+V
l
(9,
K, XK)
the same as welfare in a variant economy where there
is
only one class of agents,
say the entrepreneurs, whose payoffs are given by
U=
This variant economy
-ifc 2
+ (l- X)V e {9,K,k) + V (9,K,XK)
t
nested in the class of economies studied
is
in
Angeletos and Pavan (2007,
2009). Following similar steps as in the proof of Proposition 2 in Angeletos
easy to check that the constrained
efficient allocation is
e, \uk (~e,
Using the definition of
E,
-ki
+
(1
U
in
k, h)
(1
+ uK (9, k, A')] =
follows:
- x)v%(9, k, k) + vk(9, k, xk) + xv e (d, k, xk)
q
or, equivalently,
/Cj
—
i&Li
(2007),
w(§, k, h)
+
(1
- \)v%{6, k, k) +
42
it is
pinned down by the following condition:
our setting, the above can be rewritten as
- x)vke (e, k, h) +
and Pavan
vK {l a\ xk)
l
=
Under the assumption introduced
=E [w{e,K,ki)].
ki
Using
(33),
we conclude
above reduces to
in (28), the
(38)
t
that the efficient investment
must indeed
satisfy (30),
with the scalars Kq,
Kg and a* defined as follows:
k
=
1
Finally, existence
same arguments
Part
(iv).
=
- Wk
1
>
Part
(v).
formation
u>
if f3g
>
(39)
.
1
1
of Angeletos and
as the equilibrium
(iii)
and
- Wk
Pavan
(2009).
efficient allocations of the class of
games studied
then follows from Proposition 3 of that paper.
This part follows directly from the definition of a
and only
if
a =
and
fixed point to condition (30) follows from essentially the
the observation that, in any given equilibrium,
7A'
,
parts imply that the equilibrium and efficient allocations of our
Angeletos and Pavan (2009). Part
Part
k8
as in the proof of Proposition
economy can be understood
in
,
and uniqueness of the
The preceding
(Hi).
- Wk
jk
in (37)
and a*
in (39), along
gives the slope of Kt[8] with respect to
with
K
and
0.
Consider the case with no information
spillovers,
which
is
nested by letting the
in-
that the traders possess be a sufficient statistics for the entire information that the
entrepreneurs collectively possess.
In this case,
spective of the investment strategy.
From
(35),
HL t
[0\uj,K}
= E
t
[9\u]
and
Ej[Et[0]
—
9}
=
0,
irre-
one can then immediately see that the equilibrium
allocation coincides with the constrained efficient allocation and, by implication,
is
also unique.
Conversely, consider the case with information spillovers. That the efficient allocation cannot be
an equilibrium follows directly from part
(v)
along with the fact that investment
is
a positive signal
of 6 along the efficient allocation.
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