Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Wall Street and
Silicon Valley:
A Delicate Interaction
George-Marios Angeletos
Guido Lorenzoni
Alessandro Pavan
Working Paper 07-25
Septembr 23, 2007
RoomE52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research
Network Paper Collection
littp://ssm.coni/abstract=1016870
at
Wall Street and Silicon Valley:
A
Delicate Interaction*
George-Marios Angeletos
Guido Lorenzoni
Alessandro Pavan
MIT and NBER
MIT and NBER
Northwestern University
September
23,
2007
Abstract
Financial markets look at data on aggregate investment for clues about underlying profitability.
At the same time,
firms' investment
two-way feedback between
financial
depends on expected equity
market prices and investment. In
positive
and normative implications of
change,
when information about new investment
high aggregate investment
investment.
Because
is
"good news"
nous complementarity emerges
(shifts in
this
This generates a
paper we study the
during episodes of intense technological
opportunities
is
highly dispersed.
for profitability, asset prices increase
We
in
investment decisions
show that
Because
with aggregate
firms' incentives to invest in turn increase with asset prices,
to informational reasons.
mentals
this interaction
prices.
— a complementarity that
is
an endogedue purely
complementarity dampens the impact of funda-
this
underlying profitability) and amplifies the impact of noise (correlated errors
in individual assessments of profitability).
efficiency: equilibrium
We
investment reacts too
finally discuss policies that
improve
efficiency
next show that these effects are
little
to fundamentals
symptoms
and too much
of in-
to noise.
We
without requiring any informational advantage on
the government's side.
Keywords: heterogeneous information, complementarity,
*We thank
volatility, inefficiency,
beauty contests.
Hyun Song Shin, Rob Townsend, Jaume Ventura, Ivan Werning and seminar
MIT, the Federal Reserve Board, the 2007 lESE Conference on Complementarities and Information
(Barcelona), the 2007 Minnesota Workshop in Macroeconomic Theory, and the 2007 NBER Summer Institute for
useful comments. Angeletos and Pavan thank the NSF for financial support. Email addresses: angelet@mit.edu;
Olivier Blanchard,
participants at
glorenzo@mit.edu; alepavan@northwestern.edu.
Massachusetts Institute of Technology
Department of Econonnics
Working Paper Series
Wall Street and
Silicon Valley:
A Delicate Interaction
George-Marios Angeletos
Guido Lorenzoni
Alessandro Pavan
Working Paper 07-25
Septembr 23, 2007
RoomE52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research
Network Paper Collection
http://ssm.coni/abstract= 1016870
at
1
Introduction
Financial markets follow closely the release of macroeconomic and sectoral data, looking for signals
about underlying economic fundamentals.
In particular, high current levels of activity tend to
At the same time,
forecast high future profitability, leading to an increase in asset prices.
financial
prices affect the real
economy by changing the
up company, higher
asset prices raise the value of a potential
venture capitalists.
For firms already quoted on the stock market, higher asset prices raise the
incentive to invest for individual firms. For a start-
IPO and
facilitate financing
from
value of equity issues and the market valuation of further investments within the firm.
Both
directions of causation
to real investment
literature
—have
—from
been widely explored
real investment goes
Summers
asset prices goes back to
(1989); the hterature
financial prices
and empirical work.
in existing theoretical
on the impact of macroeconomic news on
(1986) and Cutler, Poterba, and
and from
real activity to financial prices
The
Chen, Roll and Ross
on the impact of asset prices on
back to Brainard and Tobin (1968), Tobin (1969), Bosworth (1975), Abel and
Blanchard (1986), Barro (1990), and Morck, Shleifer and Vishny (1990).
In this paper,
we document novel
teraction of these two channels
when
positive
and normative imphcations stemming from the
agents do not shai'e the same information.
that this interaction generates a feedback
mechanism between the
real
and the
We
first
in-
show
financial sector of
the economy: high investment drives up aggi'egate activity; financial markets interpret this as a
positive signal about future profitability; asset prices increase; this adds fuel to the initial increase
in investment.
We
in real investment
next show that this mechanism can exacerbate non-fundamental movements
and
asset prices,
and can
distort allocative efficiency.
particularly relevant in periods of intense technological change,
viability
and
profitability of
Preview.
We conduct our
first-best efficient
if all
of this technology
The
new
and may
"traders"
technologies
is
exercise within a neoclassical
technology.
sell their
who
economy
A
large
in
which allocations would be
number
of "entrepreneurs" gets
They have dispersed information about the
capital in a competitive financial
profitabihty
market before uncertainty
is
participate in the financial market are also imperfectly informed, but
they observe aggregate investment, which provides a
among
when information regarding the
widely dispersed across the economy.
agents had the same information.
the option to invest in a
realized.
new
This mechanism seems
summary
statistic of the
information dispersed
the entrepreneurs. In this environment, movements in real investment and asset prices are
driven by two types of shocks:
profitability of investment,
"fundamental shocks," reflecting actual changes in the long-run
and "expectational shocks,"
reflecting correlated mistakes in individual
assessments of this profitability.
The
positive contribution of the paper
is
to study
how
the interaction between real and financial
activity affects the transmission of these shocks in equilibrium.
Because high aggregate investment
Digitized by the Internet Archive
in
2011 with funding from
MIT
Libraries
http://www.archive.org/details/wallstreetsilico0725ange
is
"good news"
for profitability, asset prices increase
endogenous complementarity emerges
with aggregate investment. As a
in investment decisions.
An entrepreneur
result,
an
anticipates that the
price at which he might sell his capital will be higher the higher the aggregate level of investment.
He
is
when he expects
thus more willing to invest
others to invest more.
complementarity induces entrepreneurs to rely more on
and
profitability,
less
common
In equilibrium, this
sources of information regarding
on idiosyncratic sources of information. This
is
because
common
sources of
information are relatively better predictors of other entrepreneurs' investment choices, and hence
of future financial prices. For the
to the
common
prior,
and hence
same reason, the entrepreneurs'
choices
become more anchored
changes in the underlying fundamentals.
less sensitive to
It follows
that the feedback between the real and the financial sector of the economy amplifies the impact of
common
expectational shocks while also dampening the impact of fundamental shocks.
The normative
different shocks
is
contribution of the paper
is
to study whether the reaction of the
to
optimal from a social perspective. The mere fact that entrepreneurs care about
the financial market's valuation of their investment does not, on
Indeed, as long as
economy
all
its
own, imply any
inefficiency.
agents share the same information, equilibrium asset prices coincide with the
common
expectation of profitability; whether entrepreneurs try to forecast fundamentals or asset
prices
then completely irrelevant
is
This
is
not the case, however,
to aggregate investment induces a
for efficiency.
when information
made by
the market valuation
other entrepreneurs
is
not.
By
to the dispersion of information
positive effects
too
little
We
(i.e.,
is
independent of the
there are no production externalities or spillovers),
is
not warranted from a social perspective.
also
symptoms
and too much
It
then follows that the
of inefficiency: equilibrium investment reacts
to expectational shocks.
conclude by examining policies that improve efficiency without requiring the government
We
the fact," while information remains dispersed. In particular,
or other policies
aimed
at stabilizing asset prices.
aggregate investment, these policies
which improves
efficiency.
raise welfare,
dampen
we
was
but never achieve
inefficiently
consider interventions "during
consider a tax on financial trades
By moderating
the reaction of asset prices to
the equilibrium impact of non-fundamental shocks,
low to start with.
dampen
It
the equilibrium impact
follows that these pohcies can
full efficiency.
next consider interventions "after the fact,"
results
first
In so doing, however, these policies also
of fundamental shocks, which
on
social returns to investment: while
a given entrepreneur
to have any informational advantage vis-a-vis the market.
We
sensitivity of asset prices
implication, the complementarity that emerges in equilibrium due
documented above are
to fundamental shocks
made by
The
dispersed.
wedge between private and
the fundamental valuation of the investment
investments
is
when
uncertainty has been resolved. Building
from Angeletos and Pavan (2007b), we show that
full efficiency
can be achieved by
introducing a tax on capital holdings that
Although
realized profitability.
real
and
contingent on both realized aggregate investment and
is
sunk by the time these taxes are
financial decisions are
collected, the anticipation of these contingencies affects the incentives entrepreneurs
face "during the fact."
By
and traders
appropriately designing these contingencies, the government can induce
agents to respond efficiently to different sources of information, even
can not directly monitor
if it
these sources of information.
Discussion. The
and
asset-price
US
booms
experience in the second half of the 90's has renewed interest in investment
driven by apparent euphoria regarding
and on the optimal policy response
to these episodes.
that entrepreneurs and corporate managers
ai-e
technologies
A common
view
the Internet),
in policy discussions is
Elements of
view are formalized
this
in Shiller
Bernanke and Gertler (2001), and Dupor (2005). Similar concerns
are currently raised for the investment
boom
in China.
The presumption
detect "irrational exuberance" then leads to the result sought
—that
it
that the government can
should intervene.
While we share the view that expectational errors may play an important
we
(e.g.,
driven by noise traders and other irrational forces
in financial markets, or are irrational themselves.
(2000), Cecchetti et al (2000),
new
also recognize that these errors
may
role in these episodes,
originate from noise in information rather than irrationality.
Furthermore, we doubt the government's ability to assess fundamentals better than the market as
a whole.
Our approach
thus different.
is
On
externality that can justify intervention even
At the same time, on the
activity can amplify the
more
the normative side,
identify
interaction between real
impact of noise and that this amplification
is
stronger
dispersed. This helps explain, without any departure from rationality,
technological change, Hke the 90's,
may
is
largely driven
and
financial
when information
why
is
periods of intense
feature significant non-fundamental volatility.
Because the source of both amplification and
investment
an informational
by a policy maker with no superior information.
we show that the
positive side,
we
inefficiency in our
model
rests
on the property that
by expectations about others' choices rather than about fundamentals,
ovu results are reminiscent of Keynes' famous beauty-contest metaphor:
"...professional investment
may be
likened to those newspaper competitions in which
the competitors have to pick out the six prettiest faces from a hundred photographs,
the prize being awarded to the competitor whose choice most nearly corresponds to the
average preferences of the competitors as a whole; so that each competitor has to pick,
not those faces which he himself finds prettiest, but those which he thinks hkeliest to
catch the fancy of the other competitors..." Keynes (1936,
p. 156).
Implicit in Keynes' argument appears to be a normative judgement that something goes
when investment
is
wrong
driven by higher-order expectations. However, Keynes does not explain
why
^
this
might be the
(2005)
case.
More
and Shin
recently, Allen, Morris
and Cespa and Vives (2007) have
(2005), Bacchetta
revisited rational-expectations
and Wincoop
models of asset pricing and
have shovirn that, at least in certain cases, a mechanism similar to the one articulated by Keynes
increases the impact of the
common
and of common noise on equilibrium
prior
these papers abstract from the real sector of the economy.
positive effects they
our paper
is
the
document are symptoms
first
in the interaction
also
of allocative inefficiency.
However,
do not address whether the
To the best
of our knowledge,
to provide a complete micro-foundation for beauty-contest-like inefficiencies
between
Other related
They
prices.
real
and
literature.
financial activity.
By
focusing on the two-way feedback between real and financial
decisions as a potential explanation of "bubbly" episodes, the paper also relates to two other lines of
work.
One
line studies rational
bubbles in economies with financial frictions or asset shortages
Hammour,
Ventura, 2003; Caballero, 2006; Caballero, Farhi and
The mechanisms studied
papers also generate significant non-fundamental movements, but they are unrelated to
in these
information.
The second and more
closely related line studies speculative fluctuations in prices
investment due to heterogeneous priors regarding profitability
prices are largely driven
Scheinkman and Xiong, 2003;
by expectational shocks regarding others' valuations. This
role of higher-order expectations in our paper.
reflect the social value of investment,
The paper
is
similar to the
However, in these papers asset prices continue to
ensuring that no inefficiency emerges.-^ In our paper, instead,
the impact of higher-order expectations
and
(e.g.,
and
Himmelberg, and Huberman, 2005; Panageas, 2005). In these papers, investment and
Gilchrist,
tion
2006).
(e.g.,
is
also the source of inefficiency.
macroeconomic
also relates to the growing
strategic complementarities (e.g.,
Amato and
literatm-e
on heterogeneous informa-
Shin, 2006; Angeletos
and Pavan, 2007a,b;
Baeriswyl and Conrand, 2007; Hellwig, 2005; Hellwig and Veldkamp, 2007; Lorenzoni, 2006, 2007;
Mackowiak and Wiederholt, 2006; Woodford,
2002).
However, unlike the complementarities con-
sidered in these papers, which originate in monopolistic price competition, production or
spillovers, or other payoff externalities, the
mational externality:
activity
is
it
complementarity documented here
emerges only when information
is
is
demand
due to an
infor-
dispersed and only because aggregate
then a signal of the underlying fundamentals. This specific source of complementarity
is
the key to both the positive and the normative results of our paper.
Also related are Subrahmanyam and Titman (2001), Goldstein and Guembel (2003), and Ozde'"In
Gilchrist,
Himmelberg, and Huberman (2005),
inefficiency
can arise due to the monopoly power of the owners
of the firms issuing speculative stocks.
^Morris and Shin (2002) show that more precise public information can have a detrimental effect on equihbrium
game where the complementarity in individual actions reflects a discrepancy between private and social
objectives; but they do not study the origin of this discrepancy and they assume it to be exogenous to the information
welfare in a
structure. In our environment, instead, the discrepancy between private
ity,
originates in the two-way feedback between real
and
and social objectives, like the complementarand crucially depends on the information
financial decisions
structure; this has also interesting imphcations for the social value of information (see Section 5.3).
.
noren and Yuan (2007). These papers study a variety of feedback
and a
real sector that features of
between a financial market
a form of network externality. Like the papers mentioned in the
previous paragraph, the complementarity in these papers
On
effects
is
exogenous to the information structure.
the other hand, a complementarity that originates in an informational externality
model
in the currency-crises
of Goldstein,
Ozdenoren and Yuan
than ours. In their model, the central bank looks at the
fundamentals.
The
(2007); but
it is
size of attack to learn
is
featured
of a different kind
about the underlying
But
larger the attack, the worse the bank's perception of the fundamentals.
then also the higher the bank's willingness to abandon the peg and hence the higher the incentive
for the individual speculator to attack.
Layout. Section
,
2 introduces the baseline model. Section 3 characterizes the equilibrium
and
derives the positive implications of the model. Section 4 characterizes the socially efficient use of
information and contrasts
number
considers a
The
2
We
it
to the equilibrium. Section 5 discusses policy implications. Section 6
of extensions. Section 7 concludes. All proofs are in the
baseline
Appendix.
model
consider an environment in which heterogeneously informed agents choose
in a
"new technology" with uncertain
certainty
is
returns.
how much
After investment has taken place, but before un-
resolved, agents trade financial claims
on the returns of the
installed capital.
of agents: "entrepreneurs," who
who can
t
=
1/2;
0,
variance I/ttq
t
get the option to invest in the
we index entrepreneurs by
t
G {0,1,2,3}, and two types
new
technology,
=
1,
(i.e.,
ng
is
G
[0,
1/2]
and traders by
the precision of the prior). This
new technology and
and
i
G (1/2,
"traders,"
is
unknown
to
random
all
new
investment
kf/2 and
technology.
is
Let
ki
mean
/i
>
and
variable represents the exogenous
agents.
denote the investment of entrepreneur
incurred within the period.
is
1].
the "real sector" of the economy operates: each entrepreneur decides
invest in the
is
i
nature draws a random variable 9 from a Normal distribution with
productivity of the
At
first
are four periods,
subsequently purchase claims on the installed capital of the entrepreneurs. Each type
measure
At
this
population during the investment stage.
Timing, actions, and information. There
of
At
was dispersed
point, the observation of aggregate investment partially reveals the information that
in the
to invest
When
how much
The
i.
to
cost of this
choosing investment, entrepreneurs
have access to various signals (sources of information) that are not directly available to the traders.
Some
of these signals
may have mostly
noise (correlated errors).
first
To
simplify,
idiosyncratic noise, while others
we assume that entrepreneurs observe two
one has only idiosyncratic noise and
7
"
may have mostly common
is
given by Xj
=
d
+
^i,
where
^j
is
signals.
Gaussian
The
noise,
independently and identically distributed across agents, independent of
TTx is
y
=
the precision of the idiosyncratic signal).
+
9
where e
£,
with variance
=
^
2,
bility
A e
errors
Liquidity shocks are
(0, 1).
i.i.d.
The
When
tal.
examined
financial
market
is
=
3,
this cash flow
is
+ Cj2 + Ct3,
t
is
his
is
Ui
=
—kf/2
Next, consider a trader and
consumption stream
is
initial
+
i's
is
9ki. If
sell all their
capital to the traders.
1,
K
=
They can then use
j^ kidi.
this
9^
consumption
he
is
(cii,Ci2,Cj3)
owner,
its
is
let q,
hit
=
zero. Payoffs are thus given
in period
t.
=
(cii,Cj2,Ci3)
is
by the shock, he
First, consider
his payoff
—
(0,
—pQi, Oqi), so that his payoff
essential ingredients of the
model are the
is
Ui
=
following:
=
{—kf/2,0,9ki), so
is
(9
Ui
an entrepreneur.
sells all his capital at
(— ^'f /2,pfcj,0), so that
by
=
u,
the price p
—kf/2+pki.
denote the units of installed capital he purchases in period
is (cji, Cj2, c^s)
Remarks. The two
ment
installed capital to the
consumed.
consumption stream
make the
sell their
publicly revealed, each unit of capital gives a cash fiow of 9 to
where cu denotes agent
that his payoff
and
Supplementary Material.
also the fraction of entrepreneurs
is
not hit by the liquidity shock his consumption stream
is
{^jjielo 1/21'
general case where
not allowed to trade any claims on installed
ai'e
Payoffs. All agents are risk neutral and the discount rate
he
given by
the traders meet the entrepreneurs hit by liquidity shocks in the financial market, they
Finally, at
If
is
(i.e.,
competitive and p denotes the price of one unit of installed capi-
observation to update their beliefs about
Cii
and of
The more
in the
and
I/tt.^^
by a "liquidity shock" with proba-
hit
is
by the shock are forced to
hit
observe the aggregate level of investment from period
and
signal).
across agents, so A
For simplicity, entrepreneurs not hit by the shock
capital.'^
is
we assume that each entrepreneur
by the shock. Entreprenem's
hit
common
common
the "financial market" opens: some entrepreneurs
traders. In particular,
noise
across agents, independent of 6
the precision of the
(i.e., -Ky is
have both idiosyncratic and
all signals
At
l/7r.y
The second has only common
common
Gaussian noise,
is
with variance
6,
2.
His
— p)qi.
(i)
the agents
who
investment decisions have dispersed private information, so that aggregate invest-
a signal of the fundamental;
(ii)
there
is
some common source
aggregate investment from perfectly revealing the fundamental to
all
of "noise" that prevents
agents, so that the dispersion
of information does not completely vanish by the time agents meet in the financial market.
The
specific information structure
we have assumed
properties. In particular, the role of the
common
entrepreneurs' assessments of profitability in stage
that the traders face in stage
^We
2:
relax this assumption in Section
is
signal y
1,
a convenient way to capture these two
is
to introduce correlated errors in the
thereby adding noise to the inference problem
in equilibrium, aggregate investment will
move both with the
6.
''Letting the traders observe the entire cross-sectional distribution of investments does not affect the results. This
is
because, in equilibrium, this distribution
contains as
much
is
Normal with known
variance;
information as the entire cross-sectional distribution.
it
then follows that the
mean investment
fundamental 6 and with the
As mentioned above,
imperfectly.
error e, ensuring that aggregate investment reveals d only
in the
Supplementary Material we dispense with the
and instead consider the case where entrepreneurs observe multiple private
signal y
of
common
which have both idiosyncratic and
common
unobserved
common
errors.
and
2)
remain
intact, highlighting that the
of noise, not the specific form of
A
similar
signals, all
a variant that introduces
also consider
shocks to the entrepreneiurs' cost of investment as an alternative source of
noise in aggregate investment. In both cases, our
1
We
common
main
positive
key for our results
and normative
is
results (Corollaries
common
the existence of a
source
it.
remark applies to other simplifying modeling
we could have
For example,
choices.
allowed the entrepreneurs that are not hit by a liquidity shock to participate in the financial market;
we could
further have allowed
all
entrepreneurs to observe a noisy signal of aggregate investment,
or a noisy price signal, at the time they
our results
is
make
What
their investment decisions.^
essential for
is
only that the dispersion of information remains present both at the investment and
at the trading stage.
Also note the "hquidity shock" need not be taken too
when an agent makes an investment
general idea that
literally.
decision,
Its
presence captures the more
be him a start-up entrepreneur or
the manager of a public company, he cares about the market valuation of his investment at some
point in the
life
of the project.
A
start-up entrepreneur
may worry about
the price at which he will
be able to do a future IPO; a corporate manager may be concerned about the price at which the
company
will
be able to issue new shares. In what
follows,
we
interpret
A broadly
as a
measure of
the sensitivity of the firms' investment decisions to forecasts of future equity prices.^
no production
Finally, note that there are
kind:
both the
initial cost {-k'f/2)
spillovers
and no
and the eventual return on
the investment decisions of other agents.
The
direct payoff externalities of
any
capital (Oki) are independent of
strategic complementarity that will be identified in
Section 3.1 originates purely in an informational externality.
A
benchmark with no informational
what happens when the dispersion
market. That
is,
signals {xi}^^iQ
j
suppose that
^i
and
y)
all
frictions. Before
the information that
becomes commonly known and the
i
useful to
it is
examine
of information vanishes at the time of trading in the financial
becomes commonly known
expected payoff of entreprenem:
we proceed,
financial
market
is
dispersed during period
in period 2.
clears
if
(namely, the
The fundamental
and only
in period 1 reduces to E[ui|a;j,y]
1
=
if
p
=
0.
It
E[9\xi,y]ki
6 then also
follows that the
—
kf/2, which in
^See Section 6 for these extensions.
®See Baker, Stein and Wurgler (2003) for complementary evidence that the sensitivity of corporate investment to
is higher in sectors with tighter financing constraints (which here can be interpreted as higher A).
stock prices
turn implies that equilibrium investment
ki
=
E[e\xi,y]
—+
=
T^e
The key
result here
is
2.
Rather,
it
27 This
period
is
measured by
Xi
T^x+
is
+
TTy
TTg
—^+
+
TT^.
y.
TTj,
driven solely by first-order expectations
This result does not require 6 to be perfectly known in period
A).
as long as the
which we henceforth
case,
'T^e
independent of the intensity of the entrepreneurs' concern about
more generally
applies
— n+ —+
+ T^y
that equilibrium investment
regarding the fundamental and
financial prices (as
'^x
given by
is
asymmetry
refer to as the case
of information about 6 vanishes in
with "no informational
frictions,"
provides a convenient reference point for the rest of the analysis.
3
Equilibrium
Individual investment
made by an
is
described by a function
entrepreneur with information
K{e,y)
(x, y).
=
fc
:
M^
^ E so that
fc(x, y)
Aggregate investment
is
denotes the investment
then a function of
j k{x,y)d^{x\9),
where $(x|^) denotes the cumulative distribution function
.(1)-
of x given
Since traders observe
0.
aggregate investment and are risk neutral, the unique market-clearing price
p
is
We
also a function of {9,y).
Definition 1
A
for
all
(symmetric) equilibrium
for
(x, y)
e arg
maxE
where
determined by
is
an investment strategy k{x,y) and a price function
.
[
-
(1
A) Ok
+ Xp {0, y)
k
-
Ir 12
j
x,
y
]
;
all {e,y),
pie,y)
.
where K{e,y)
=
= E[e\K{e,y)],
fk{x,y)d^ix\e).
an arbitrary information structure. Let X,,t denote the information of agent i
Impose that no agent has private information about
in period 2 so that E[^|Ji,2] = E[6|X2] for all
Prom market clearing we then have that p = E[9\T2]- Prom the law of iterated expectations we then have that
^To
clarify this point, consider
in period
i.
is
¥j[6\K],
{x,y),
k
(ii)
K
=
thus define an equilibrium as follows.
p{6, y) that satisfy the following conditions:
(i)
p
is
the latter denotes the expectation of 6 given the observed level of K.^ Since
{0,y),
(9, y):
E[p|Z,,i]
=
t.
E[E[6»|J2]|J»,i]
*Since the price
is
information. Therefore,
in Section 6.
=
E[6l|I,,i] for all
only a function of
we can omit
K
i.
It
and
follows that every entrepreneur chooses k,
K
is
=
E[6iiJ,,i].
publicly observed, the price itself does not reveal any additional
conditioning on p.
The
case where
p conveys additional information
is
examined
.
Condition
(i)
requires that the entrepreneurs' investment strategy be individually rational,
taking as given the equilibrium price function. Condition
requires that the equilibrium price
(ii)
be consistent with rational expectations and individual rationality on the traders'
side, taking as
given the strategy of the entrepreneurs.
As
it is
libria in
which the price function
Definition 2
A
linear.
is
linear equilibrium
restrict attention to equi-
is
an equilibrium in which p{6,y)
is
linear in {d,y)
Endogenous complementarity
3.1
The
optimality condition for the entrepreneurs' strategy can be written as
kix,y)^E[{l-X)e +
The
linearity oi
strategy
is
Xp{e,y) \x,y].
linear in (x,y); that
is,
there are coefficients {f3o,Pi,P2) such that
implication, aggi'egate investment
Observing
K
is
(2)
p{9,y) in {0,y) and of E[0|x,y] in (x,y) then guarantees that the entrepreneurs'
k{x,y)^0o + 0ix +
By
we
often the case in the literature, tractability requires that
is
K=
given by
02y-
0o
+
(3)
(3i6
+
/32j/
—
(in
+ (A +
(^2)^
+
/32£'-
thus informationally equivalent to observing a Gaussian signal z with precision
-k^,
where
Z
=
K-pQ — ^ —
d +
TT
132
,
-7;
—£
,
and
TTj
_
=
^ /3i+fe
Standard Gaussian updating then gives the expectation of 9 given
prior
and the
signal
K
V
,,.
TTy.
(4)
as & weighted average of the
z:
E
[e\K]
""^
=
TTq
Because market clearing
in period 2 requires
+
p =
fi
""'
+
TTj
7751
+
z.
TTz
E[6\K], we conclude that the equiUbrium price
satisfies
p{e,y)
=
jo
+ liK{e,y),
(5)
where
'''^W^/-.e + n.0.+P2
These
results are
summarized
in the following
""^
lemma.
^'^WT^.KW2-
^'^
.
Lemma
In any linear equilibrium, there are coefficients
1
= PQ + PiX + l32y
k{x,y)
>
Moreover, 71
The key
news"
i/
and only
result here
is
if
Pi
+ /32 >
and
piO,y)
=
jq
70,71) such that
+ jiK
{0,y)
0.
K
the relation between
for profitability (in the sense that
(/3o,/3i,/32,
and
p.
Provided that high investment
a higher realization of
of 6), financial prices increase with aggregate investment (71
>
is
"good
K
raises the traders' expectation
0).
This in turn induces strategic
complementarity in investment decisions. Indeed, when the entrepreneurs are choosing a higher
level of investment,
price at
i
=
2.
they are sending a positive signal to the financial market, thus increasing the
But then each entrepreneur's willingness
to invest at
i
=
1 is
higher
when he
expects
a higher level of investment from other entreprenem's, which means precisely that investment choices
are strategic complements.
from replacing condition
Lemma
We
formalize these intuitions in the next result, which follows directly
(5) into
condition
(2).
2 In any linear equilibrium, the investment strategy satisfies
'
kix,y)^E[il-a)K{9) + aK{d,y} \x,y],
where a
=
A71 and k (9)
Condition
(7)
=
(7)
"'°
^
~{_)t
can be interpreted as the best-response condition in the coordination game that
emerges among the entrepreneurs
for
a given price function:
it
describes the optimal strategy for
each individual entrepreneur as a function of his expectation of aggregate investment (the relevant
summary of the
prices.
The
decisions:
strategy of other entreprenem^s) taking as given the impact of the latter on financial
,
coefficient
a then measures the degree
the higher a, the higher the slope of the best response of individual investment to
aggregate investment, that
choices.
of strategic complementarity in investment
The
is,
the higher the incentive of entreprenem-s to align their investment
function k{9), on the other hand, captures the impact of the fundamental on the
individual return of investment for given K, normalized by
A
1
—
a.^
similar best-response condition characterizes the class of linear-quadratic
games examined
in
Angeletos and Pavan (2007a), including the special case of Morris and Shin (2002). However, there
are two important differences. First, while in those
is
games the degree
exogenously determined by the payoff structure, here
it is
of strategic complementarity
a
endogenously determined as an integral
^This normalization serves two purposes. First, it identifies k{6) with the complete-information equiUbrium level of
investment in the game among the entrepreneurs, for given price function. Second, it ensures that the unconditional
mean
of investment
is
given by Kk{x,y)
=
Ek,(6).
10
part of the equilibrium. Second, while in those games the degree of complementarity
of the information structure, here
complementarity in our setup
How much
is
it
is
independent
actually originates in the dispersion of information. In fact, the
solely
due to the informational content of aggregate investment.
information aggregate investment conveys about 6 determines the coefficient 71
captures the sensitivity of prices to aggregate investment.
which
,
In turn, the coefficient 71 pins
down
the value of a, which captures the degree of complementarity in the entrepreneurs' investment
decisions. In the absence of informational frictions (the
.section), aggregate
of
benchmark case examined
in the previous
investment provides no information to the traders, prices are thus independent
K, and the complementarity
in investment decisions
absent.
is
When
instead information
is
dispersed, aggregate investment becomes a signal of 9, prices respond to aggregate investment and
a complementarity in investment decisions emerges. The more informative aggregate investment
about
9,
is
the stronger the complementarity.
Because the complementarity depends on the informational content of aggregate investment,
which
in turn
a we need
depends on the entrepreneurs'
to solve a fixed-point problem.
complementarity
is
to determine the equilibrium value of
strategies,
Before doing
so,
we
first
show how
this
endogenous
instrumental in understanding the incentives entrepreneurs face in using their
available sources of information.
Lemma
3 In any linear equilibrium,
02
/?!
Therefore, provided that
common
(3\,(32
>
_
T[y_
TT^ 1
0, the sensitivity
-a
(8)
.
of the entrepreneurs' equilibrium strategy to the
signal relative to the idiosyncratic signal
is
higher the higher the equilibrium degree of
complementarity.
Let us provide some intuition for this result. Consider an entrepreneur's best response to the
strategy that other entrepreneurs follow, holding fixed the price function. Suppose that the other
entrepreneurs' strategy
given by
K
{9, y)
=
0o
is
k
(x, y)
=
+ (3i9 + (32y
(3o
+ I3ix + /?2y,
with
/3i
,
/32
>
and an agent's best predictor
0.^°
Aggregate investment
of aggi-egate investment
is
then
is
'"This condition means that investment responds positively to both signals. Since an entrepreneur's expectation of
it is quite natural to expect this condition to be satisfied in equilibrium. Below we will
show that an equilibrium that satisfies this condition always exists and that this equilibrium is unique for A small
enough. For A high enough, however, it is possible to construct equilibria in which entrepreneurs find it optimal to
9 increases with either signal,
react negatively to a signal because they expect others to
do the same.
11
The
signal y helps predict aggregate investment
on the term P2y- Therefore,
common
the
now
while the
common
both through E[^|x,y] and directly through
its effect
private signal x helps predict aggregate investment only through
signal y
how much
relative to
a means
recall that a higher
is
it
a stronger incentive for an individual entrepreneur to align his
optimal to rely more heavily on the
the former that best helps
them
follows that
It
common
when a
is
higher entrepreneurs
signal y relative to the private signal x, for
align their choice with the choice of others.
it
^^
Equilibrium characterization
3.2
As noted
On
the two signals help predict the fundamental,
a better predictor of aggregate investment than the private signal x. But
is
investment choice with that of other entrepreneurs.
find
E[fl|a;,y],
earher, completing the equilibrium characterization requires solving a fixed-point problem.
how entrepreneurs
the one hand,
use their available information depends on a, the endogenous
complementarity induced by the response of prices to aggregate investment.
how
sensitive asset prices are to aggregate investment,
how
informative aggregate investment
is
On
the other hand,
and hence how strong a
is,
depends on
about the fundamental, which in turn depends on how
entrepreneurs use their available information in the
This fixed-point problem captures
first place.
the essence of the two-way feedback between the real and the financial sector in our model.
solution
Lemma
is
Its
provided in the following lemma.
4 There
exist functions
F M
:
x
x M^.
(0, 1)
^M
and
G R
:
x
(0, 1)
x R3_
^ ]r5
g^^j^ ^j^^^
the following are true:
(i)
In any linear equilibrium,
(32/fSi
solves
•P'
01
while (/3o,/5i,/52,7o,7i)
('^^^
(iii)
unique solution
;A,7r0,7r:,,7ry
(9)
)
one solution at some
there exists a cutoff
X
/32//3i
=
A
>
T^y/TTx',
(7re,7ra;,7ry)
>
such that (9) admits a
X < X;
(iv) There exists
The
at least
{'rte,''^x,''^x),
if
—
\t3i
= G(/32M; A,7r0,7r^,7ry),•
Equation (9) has
For any
1
an open
set
S such
that (9) admits multiple solutions if {X,ne,T^x,''^x)
fixed-point problem that leads to the equilibrium characterization
the variable b
=
(32/ Pi,
the two signals. Given
b,
which represents the
is
set
up
in
G S.
terms of
relative sensitivity of entrepreneurial investment to
we can determine the
sensitivity of the price to aggregate investment 71.
"^^A similar property holds for the more general information structures considered in the Supplementary Material: a
stronger complementarity shifts the use of information towards the signals whose errors are relatively more correlated
across agents.
12
Given
71,
we can then determine the complementarity a and then the
responses to the two signals. These steps describe the mapping
the intuition for part
our economy.
Parts
(i)
of the
lemma: the fixed points of
it
used in
Lemma
4 and provide
identify all the linear equilibria of
then characterize the fixed points of F, establishing than a linear
(ii)-(iv)
equilibrium always exist and, although
The
F
F
sensitivity h of individual best
it is
not always unique,
possibility of multiple equilibria for high values of
illustrates the potential strength of the
A
is
unique
it is
for
A small enough.
interesting for several reasons. First,
two-way feedback between
and financial
real
activity. Sec-
ond, this multiplicity originates solely from an informational externality rather than from the more
familiar payoff effects featured in coordination
and Obstfeld
both
models of
crises
a la
Diamond and Dybvig
(1984)
(1996). Finally, this multiplicity can induce additional non-fundamental volatility in
real investment
and
financial prices.
However, the possibility of multiple equilibria
is
not central to our analysis.
When
unique equilibrium, the key positive and normative predictions documented in Corollaries
below necessarily hold.
When
is
a
and
2
there
1
there are multiple equilibria, these predictions continue to hold for
any equilibrium that
satisfies
the natural property that investment increases with both signals. For
the rest of the paper
we thus
leave aside the possibility of multiplicity
the equilibrium
Proposition
1
is
unique.
The next
proposition then summarizes some key equilibrium properties.
There always exists a linear equilibrium in which the following properties are true:
Individual investment increases with both signals
(i)
price increases with aggregate investment (^i
(Hi) The sensitivity of investment to the
TTy/TTx
>
and
is
(i)
and hence
0^
the equilibrium
satisfies
common
< a<
1
and
is
increasing in A;
signal relative to the private
is
higher than
this equilibrium to be the
unique linear equilibrium.
guarantees that individual investment increases with both signals, which in turn ensures
that high aggregate investment
positive.
>
increasing in A.
Moreover, A small enough suffices for
Part
(/3i,/32
0);
The equilibrium degree of complementarity
(ii)
and focus on the case where
Part
about financial
(ii)
is
necessarily "good news" for profitability
further establishes that
prices.
Combining
this
with
a
and hence that a
is
higher the stronger the entrepreneurs' concern
is
Lemma
3 then gives part
(iii),
which
spells out the
implications for the equilibrium use of information.
3.3
To
Impact of fundamental and expectational shocks
further appreciate the positive implications of informational frictions
thereof
—
it is
useful to rewrite aggregate investment as
13
—and the complementarity
Aggregate investment thus depends on two types of shocks: fundamental shocks, captured by
and expectational
by
shocks, captured
e.
How
entrepreneurs use available information affects
investment respond to these shocks: the sensitivity to fundamentals
while the sensitivity to expectational shocks
When
information
is
>
is
positive,
In contrast,
i^y/i^x-
following
Corollary
is
and hence the
mental shocks
fundamental
when there
signal
are no informational frictions, prices do not react to
is
zero,
and hence
/32//3i
=
T^y/T^x-
is
of expectational shocks relative to funda-
higher in the presence of informational frictions.
the key positive prediction of the paper:
fundamental
volatility relative to
volatility;
regression of aggregate investment on expected profits.
increases with A, this amplification effect
prices.
+/32,
then an immediate implication.
is
This result
/9i
common
relative sensitivity to the
(Main positive prediction) The impact
1
sum
governed by 02-
is
aggregate investment, the equihbrium degree of complementarity
The
governed by the
how
dispersed, prices react positively to aggregate investment, the equilibrium
degree of complementarity
satisfies Ih/I^i
is
6,
is
informational frictions amplify non-
that
is,
they reduce the R-square of a
Importantly, because the equihbrium
a
stronger the more entrepreneurs care about asset
^^
Corollary
1
regards the relative impact of the two shocks.
The next proposition
reinforces this
finding by examining the absolute impact of the two shocks.
Proposition 2 There
exists
A >
and the following comparative
(i)
higher A reduces Pi
(ii)
higher A increases
The key
{6, y)
=
K
+ p2,
(32,
To
that,
for
A G
there
(0, A],
is
a unique linear equilibrium
thus dampening the impact of fundamental shocks;
thus amplifying the impact of expectational shocks.
is
again the role of the complementarity for the equilibrium
see this, suppose for a
{0, y) in all states.
all
statics hold:
intuition for these results
use of information.
p
such
The
fc(x,y)
moment
that 70
=
and 71
—
1,
meaning that
entrepreneurs' best response then reduces to
= E[ {l-\)e + \K{e,y)
\
x,y],
(10)
This result extends to the more general information structures considered in the Supplementary Material. Because
have common errors, there are multiple "expectational shocks." We can then decompose the total variance
of investment in two components: the one that is explained by 9 (which defines "fundamental volatiUty"), and the
one that is explained by the combination of aU common errors (which defines "non-fundamental volatihty"). We then
show that the ratio of the latter to the former is higher with dispersed information (in which case a > 0) than in the
^
all signals
frictionless
benchmark
(in
which case a
=
0).
14
now
so that the degree of complementarity
unique solution to (10)
/3o
=
It is
k (x,y)
TT-
n + TTxil
:
TTe
is
>^)
+ n-y M,
'
=
Po
Pi
=
+ j3\x + P2y,
r,
+ 7r.^(l :
TTg
show that the
easily
with
TT"^
+ TTy
A)
and
'
'
02
TTg
+ 7r3;(l -
common
private signal (captured by
/3i )
signal (captured
.
We
private signal.
common
That
it
is
P2), while
common
also increases the reliance
signal: the prior
by
it
of A. This
is
also
on the prior
is
same reason
for exactly the
profits,
9, for
9, irrespective
otherwise the traders would
which would be a contradiction. But then the average investment
that, because a higher A increases
anchoring
as for
a relative good predictor of others' investment choices.
be equal to the average of
less sensitive to
and
and dampens the reliance on the
signal
because the average price must equal the mean of
make on average non-zero
(3o)
decreases the sensitivity to the
However, note that the average return on investment coincides with the mean of
is
+ TTy
have aheady given the intuition for the result that a stronger
complementarity amplifies the reliance on the
must
'
A)
then immediate that a higher A increases the sensitivity to the prior (captured by
the sensitivity to the
the
One can then
coincides with A.
9,
that
/?o, it
is, /3o
+
+
{Pi
P2)
sum
also reduces the
M
niust equal ^. It
/3i
+ /32-
is
then immediate
In simple words, investment
changes in fundamentals simply because the complementarity strengthens the
effect of the prior.
These intuitions would be exact
were exogenous to
A. In
if
70
=
and
our model, the price
is
71
=
1,
or
more generally
if
these coefficients
an increasing function of aggregate investment, but
the coefficients 70 and 71 depend on A. This explains
why
these intuitions are incomplete
and why
the absolute effects documented in Proposition 2 hold only for a subset of the parameter space.
However, the prediction regarding the amplification of the relative impact of non-fundamental
shocks (Corollary
1)
holds true more generally.
4
Constrained efficiency
The
analysis so far has focused
its
on the positive properties of the equilibrium.
normative properties by examining whether there
is
We now
study
an allocation that, given the underlying
information structure, leads to higher welfare.
The
question of interest here
is
whether society can do better, relative to equihbrium, by having
the agents use their available information in a different
by giving the agents more information.
as in Angeletos
We
way
—not whether society can do better
thus adopt the same constrained efficiency concept
and Pavan (2007a,b): we consider the
allocation that maximizes ex-ante welfare
subject to the sole constraint that the choice of each agent must depend only on the information
available to that agent. In other words,
we
let
the planner dictate
15
how
agents use their available
information, but
doing,
we do not
the planner transfer information from one agent to another. In so
let
we momentarily disregard
implement the
will identify
tax systems that
an equilibrium.
efficient allocation as
Note that the payments
we
incentive constraints; later on
market represent pure transfers between the en-
in the financial
trepreneurs and the traders and therefore do not affect ex-ante
We
utifity.^'^
can thus focus on
the investment strategy and define the efficient allocation as follows.
Definition 3 The
Eu^
=
withK{e,y)
J
maximizes ex-ante
efficient allocation is a strategy k{x,y) that
y'^[{l-X)0k{x,y)-'^kix,yf]d^{x\9)+'^[eXK{0,y)]\d^i9,y)
utility for
an arbitrary strategy. The
term
first
m
the payoff of an entrepreneur with information {x,y); the second term
when aggxegate investment
the payoff of a trader
(11)
'
fk{x,y)d^{x\e).
Condition (11) gives ex-ante
is
utility
is
K{9,y);
finally 4*
in
square brackets
square brackets
is
denotes the cumulative
distribution function of the joint distribution of {0,y). Note that the transfer of capital from the
entrepreneurs that are hit by the liquidity shock to the traders does not affect the return to
investment.
It
follows that (11) can
be rewritten compactly as^^
Eu=^E[Vik{x,y),e)]^'^E[E[V{k{x,y),e)\x,y]],
where
V {k,6) =
9k
—
^k^.
Prom
trepreneurs' payoffs are V{k,9).
for
almost
all
x and
Proposition 3 The
y,
the society's viewpoint, A
It is
irrelevant
then obvious that a strategy k{x,y)
k{x,y) maximizes E[V{k,9)\x,y].
efficient
is
investment strategy
k{x,y) =E[9\x,y]
it
is
as
efficient if
following result
is
if
the en-
and only
if,
then immediate.
given by
is
=
The
is
and
6on
+ Six + 5iy,
where
T^'e
+'n'x
+%
TTq
-I- TTj;
-t- TTj,
TTe
'^By ex-ante utility, we mean before the realization of any random variable, including those that determine whether
an agent will be an entrepreneur or a trader. However, note that, because utility is transferable, any strategy k{x,y)
that maximizes ex-ante utility also maximizes any weighted average of the expected utility of an entrepreneur and of
a trader. By the same token, any strategy k(x, y) that improves upon the equilibrium in terms of ex-ante utihty can
yield a Pareto improvement. It suffices, for example, to let the entrepreneurs and the traders continue to trade at a
price p{6, y) = K{9\K{e, y)), where K(d, y) = J k {x, y) d<^{x\9).
'^^It
suffices to substitute the expression for
K (9, y)
in (11).
16
Note that the
efficient strategy
not for informational frictions.
It
would have coincided with the equilibrium
strategy*
were
if it
follows that our earlier positive results admit a normative inter-
pretation.-'^
Corollary 2 (Main normative prediction) In
the presence of informational frictions, the im-
pact of expectational shocks relative to fundamental shocks
is
inefficiently high.
Policy implications
5
Having
First,
-
identified a potential source of inefficiency,
we
we now analyze the
consider interventions "during the fact," by which
market at
i
=
2,
"after the fact,"
when
about 9 has been resolved. In both
At the end
we
of this section, however,
we
consider policies
on information that becomes public
policies contingent
cases,
informational advantage vis-a-vis the private sector, which
analysis.
interventions in the financial
uncertainty about 6 has not been resolved yet. Next,
by which we mean
after uncertainty
we mean
effect of different policies.
at
i
=
3,
we impose that the government has no
is
our preferred benchmark for policy
also consider situations
where the government can
directly affect the information available to the agents.
Interventions "during the fact": price stabilization
5.1
We
start
by considering
policies
aimed
at reducing asset-price volatility. In particular,
government imposes a proportional tax r on financial trades (purchases of
tax can depend on the price, which
is
public information.
suppose the
capital) at date 2. This
For simplicity,
takes the following
it
linear form:
r(p)
where
{to,ti) are scalars.
The
Tax revenues
=
ro
+ rip,
are rebated as a
equilibrium price in the financial market
is
now
(12)
.
lump sum.
given hy p
=
M[9\K]
—
{tq
+
Tip)
;
equiva-
lently,
P
=
7^
1
where 70 and
effect is to
-t-
(E [9\K]
71 are given, as before,
dampen
-
to)
=
-^
+
1
Ti
by
(6).
(70
+ liK -
tq)
(13)
,
Ti
When
the tax
is
pro-cyclical
the response of asset prices to the traders' expectation of
9,
(i.e.,
ri
>
0),
and thereby
its
their
response to the news contained in aggregate investment. In equilibrium, this tends to reduce the
degree of complementarity in investment decisions. To see this more clearly, note that the degree
^^Corollary 2 presumes that the equiUbrium
equilibrium in which /3i,02
>
0.
is
unique.
When
there are multiple equilibria, the result holds for any
Since the efficient allocation satisfies
is efficient.
17
/3i,/32
>
0,
this also ensures that
no equilibrium
Figure
of complementarity
is
now
1:
The impact
of price-stabilization policies.
given by
A71
a
1
If 7i
were exogenous,
Lemma
in
3 that
/32//3i,
it
+
ri
.
would be immediate that a decreases with
the relative weight on
common
t\.
But
if
equihbrium
is
t\
on
a.
we know from
falls,
sources of information, must also
turn means that aggregate investment becomes more informative about
counteracting the direct effect of
a
0,
fall.
This
so that 71 increases,
However, one can show that, at least as long as the
unique, the direct effect dominates, guaranteeing that the equilibrium degree of
complementarity decreases with ti}^
We
conclude that a higher
ti,
by reducing the degree
of complementarity, necessarily reduces
the relative impact of expectational shocks. However, by reducing the overall sensitivity of prices
to
all
sources of variation in investment, a higher ri also reduces the impact of fundamental shocks.
As argued
in the previous section, in the absence of
sensitive to expectational shocks
and
poUcy intervention, investment
insufficiently sensitive to
fundamental shocks.
is
It
excessively
follows that
the welfare consequences of the tax are ambiguous: while reducing the impact of expectational
shocks improves
efficiency,
These intuitions are
reducing the impact of fundamental shocks has the opposite
illustrated in Figure 1
where
for
each value of
^This follows from an argument similar to the one that establishes that
18
a
is
ti
,
effect.
the value of tq
monotonic
in A.
is
chosen
optimally to maximize welfare.
tion policy considered here
The top panel
and under the constrained
the sensitivity to expectational shocks
drawn
depicts the difference in welfare under the stabiliza-
for a baseline set of parameters:
j32
and to fundamental shocks
=
-Kg
n^
—
iiy
—
=
and A
1
(/?i
(A,7r5i,7r.i;,7ry)
and
from
(0,1)
x M^.
induces a lower
and a lower
is
equihbrium without
policy, reflecting the trade off discussed above.
While these numerical
results,
it
/?2
able to establish this result formally. However,
-^ oo)
is
its
/3i
+ (32
as
qualitative
compared
it is
(i.e.,
easy to show that
ti
full
>
0),
to the
we have not been
price stabilization
(i.e.,
never optimal. In this limit, prices cease to react to aggregate investment, the strategic
=
(1
—
X)E[9\x,y].
implication, the relative sensitivity of investment to expectational shocks /32/(/3i
efficient level,
At
is
we have randomly drawn
complementarity disappears, and equilibrium investment reduces to k{x,y)
level.
figure
which span the entire parameter space, reveal that the optimal
policy always involves a strictly positive degree of price stabilization
T\
The
/?2)-^^
For each such vector, we have found
that the optimal t\
positive
+
However,
0.5.
features are robust across a wide set of parametrizations. In particular,
10,000 parameter vectors
the bottom panels depict
efficient allocation;
but
fundamental
overall sensitivity to the
its
a marginal increase
this point,
is
+
is
(32)
A times lower than at the
in the relative sensitivity implies only
By
at its
efficient
a second-order
welfare loss, while a marginal increase in the overall sensitivity implies a first-order welfare gain.
It
follows that
never optimal to fully stabilize the price.
it is
Proposition 4 A
tax that stabilizes prices can increase welfare; however, a tax that completely
eliminates price volatility
is
never optimal.
"
.
,.
'
Interventions "after the fact": corrective taxation
5.2
Suppose now that the government imposes a proportional tax t on
tax
is
now paid by
The advantage
be made contingent on
We
all
of introducing a tax in period 3
information which
is
where (to,ti,T2) are
The
acquired
that the tax rate r can
now
K
and
publicly available in that period, including
scalars;
=
To
+ T^0 +
T2K,
(14)
tax revenues are again rebated in a lump-sum fashion.
shows that these simple tax schemes can implement the constrained
^'^Note that to affects the unconditional average of k (x, y), but has
signals
is
who
3.
focus on linear tax schemes of the form
T{e,K)
result
asset holdings in period
the entrepreneurs not hit by the liquidity shock and by the traders
capital in period 2.
9.
,
x and
y, i.e.,
on
/3i
and
/32.
We
henceforth concentrate on
19
ri.
no
effect
on the
The
following
efficient allocation.
sensitivity of investment to the
Proposition 5 There
exists a
unique linear tax scheme that implements the efficient allocation as
an equilibrium. The optimal tax
The
satisfies tq
intuition behind this result
is
<
<
0, ti
and
T2
>
0.
that the government can control the degTee of strategic
complementarity perceived by the agents by appropriately designing the contingency of the marginal
tax rate r on aggregate investment: the higher the elasticity to of the marginal tax rate with respect
to
K, the lower the degree
of complementai-ity in investment choices
of equilibrium investment to
common
Although
same time, adjust
ri to
is
analogous
However, the government now has an extra
the elasticity rj of the tax to the realized fundamental.
government can thus induce the optimal
while, at the
the lower the sensitivity
noise relative to idiosyncratic noise. This effect
to that of the stabilization policies discussed above.
instrument available:
a and
Through
T2 the
obtain the optimal absolute sensitivities to each shock.
this result does not require
+
relative sensitivity to expectational shocks /32/(/3i
any informational advantage on the government's
02)
^*
side,
it
assumes that the government observes perfectly the fundamental 6 and the agents' capital holdings
at the time taxes are collected. However, the result easily extends to situations
tities are
observed with measurement error. In particular, suppose that in stage 3 the government
only observes 9
while
e
where these quan-
and
r/j
=
are
6
+
e
and
Si
measurement
=
+
Si
rji
for
each
i,
where
errors, possibly correlated
6 and of the agents' information in period
2.
Then
let
K
Si
is
the capital holding of agent
with one another, but independent of
be the cross-sectional mean of
the government's measure of aggregate investment) and consider a proportional tax on
following form: t{9,K)
unique
=
tq
+
t^O
+
T2K.
set of coefficients (ro,ri,r2) that
that these coefficients continue to satisfy
To
recap, the key insight here
public signals of 6 and
is
It is
<
Q
s,
Si
(i.e.,
of the
then easy to check that there continues to exist a
implement the
t\
<
efficient allocation as
an equilibrium and
T2-
that the government can use the contingency of the tax rate on
K that will be revealed at stage 3 to achieve efficiency in the decentralized use
of information during stage
1.
Although
this information
becomes
available only after all investment
decisions are sunk, by promising a specific policy response to this information the government
able to manipulate
i,
how entrepreneurs
investment decisions, even
if it
use their available sources of information
when making
is
their
cannot directly monitor these sources of information.^^
^®The optimal tq is then chosen to induce the optimal level of unconditional average investment. Similar tax
schemes implement the efficient investment strategy in all the extensions considered in Section 6.
"'^These intuitions, and the implementation result in Proposition 5, build on Angeletos and Pavan (2007b). This
paper considers optimal policy within a rich, but abstract, class of economies with dispersed information on correlated
values.
See also Lorenzoni (2007) for monetary policy in a business-cycle model with dispersed information on
underlying productivity.
20
Optimal release of information
5.3
We now
turn to policies that affect the information available to the agents. This seems relevant
given the role of the government in collecting (and releasing) macroeconomic data.
To capture
noise, that
is,
this role,
suppose that
in stage 2 traders
they observe
K=K+
where
ry
is
can only observe average investment with
aggregate measurement error, which
with mean zero and variance
is
r],
a random variable, independent of
Suppose now that the government can
l/7r,,."°
the macroeconomic data available to financial traders.
the weight that traders assign to
by which the government
is
K
when estimating
By changing
other shocks,
all
affect the precision
nn of
the government determines
tt,,
future profitability. This
is
another channel
able to affect the degree of strategic complementarity in investment
decisions.
Indeed, the choice of n^
is
formally equivalent to the choice of ri in the setup with a tax on
financial transactions (Section 5.1). For each value
activity, there is
versa.
is
To
a value
rj of
tt^
of the precision of the signal about aggregate
the tax elasticity that induces the same equilibrium strategy, and vice
see this, note that in
any hnear equilibrium, the observation oi
K = Po+{P\+ Pi)9+ (52s:+'n
informationally equivalent to the observation of a signal
K-00
+
/3i
,
A+
/32
^
/?2
,
/92
1
/5i
+ /32
with precision
The
equilibrium price
and hence the degree
model.
By changing
is
then given by p{0,y,r])
=
70
+ -/i[K{9,y) +
77],
a
of strategic complementarity remains equal to
the value of
tt^,
with 70 and 71 given by
=
(6),
A71, as in the baseline
the government can then directly manipulate 71 and thus the
degree of strategic complementarity perceived by the entrepreneurs.^^
We
conclude that the choice of
of Ti: decreasing
reduces
its
tt^
tt^ is
subject to the
response to fundamental shocks.
is
trade-offs
emphasized
for the choice
reduces the relative response of investment to expectational shocks, but
The
mediate degree of release of macroeconomic data
such data
same
results of Section 5.1 then imply that
may be
an
it
also
inter-
optimal even when the cost of collecting
zero.^^
^°The equilibrium characterization for this case is a straightforward extension of the baseline
^'See the Supplementary Material for the proof of this claim.
^^Note, however, that this holds only as long as the equilibrium
is inefficient.
in Section 5.2 are in place, thus guaranteeing that the equilibrium
21
is efficient,
If,
case.
instead, the policies considered
then a higher
tt,,
is
always welfare
which
Finally, one could consider policies
fundamental
affect directly the agents'
In particular, the government can collect
d.
information regarding the
some information about 9
in period
and
1
decide whether to disclose this information to the entrepreneurs, or to both the entrepreneurs and
the traders. In the
first case,
the policy corresponds to an increase in the precision of the signal y in
the baseline model. Although entrepreneurs have a more precise estimate of the fundamental, this
information
is
not shared with the traders. Therefore, this policy could exacerbate the asymmetry
of information
and could magnify the feed-back
effects
between investment and asset
prices,
with
possible negative consequences on social welfare.
In the second case, instead, the policy corresponds to an increase in the precision of the
prior in the baseline model. This policy
is
socially beneficial for
two reasons:
quality of the information available to the entrepreneurs and hence
their decisions to the fundamental.
endogenous signal
K
Second,
it
it
first, it
common
improves the
permits them to better align
reduces the reliance of financial markets on the
in their estimate of the fundamental.
This second
effect
tends to reduce
the degree of strategic complementarity in investment decisions, and hence also the discrepancy
between equihbrium and
efficient allocations.
6
Extensions
Our
analysis has identified a
Both
effects
then contribute to higher welfare.
mechanism through which the dispersion
of information induces
com-
plementarity in real investment choices, amphfication of non-fundamental disturbances, and
market outcomes,
efficiency of
assumptions to
sumed that the
illustrate this
traders'
all at
once.
mechanism
demand
In the baseline model,
we have made a number
in the simplest possible way.
for installed capital
is
we have
In particular,
perfectly elastic, that entrepreneurs
in-
of
as-
who
are not hit by the liquidity shock do not trade in the financial market, and that the traders' valu-
ation of the asset coincides with that of the entrepreneurs. In this section,
we
relax each of these
assumptions.
We
first
extend the model to allow
This extension
is
interesting because
for the traders'
and lowers the ex-ante incentive
In a second extension,
the financial market.
we
market
for capital to
be downward sloping.
introduces a potential source of strategic substitutability
it
in the entrepreneurs' investment decisions:
installed capital in the financial
demand
when aggregate investment
also higher, putting a
is
is
higher, the supply of
downward pressure on
asset prices
to invest.
allow entrepreneurs not hit by the liquidity shock to participate in
This extension
is
interesting for
two reasons:
first,
it
allows for
the entrepreneurs' information to be aggregated in the financial market; second,
non-trivial allocative role for prices.
improving.
22
it
some
of
introduces a
Although some interesting
paper (Corollaries
1
and
2)
and normative predictions of the
differences arise, the key positive
remain valid in both extensions: as long as the dispersion of information
does not completely vanish in the financial market, the signaling
effect of
aggregate investment
continues to be the source of amplification and inefficiency in the response of the equilibrium to
common
sources of noise.
we
Finally,
consider a variant that introduces shocks to the financial-market valuation of the
This variant brings the paper closer to the recent literature on "mispricing"
installed capital.
and "bubbly"
assumed
asset prices.
in the baseline
also helps clarify that the details of the information structure
It
model are not
any source of common noise
essential:
we
in the information that
aggregate investment conveys about the underlying fundamentals opens the door to amplification
and
'
inefficiency.
The
6.1
supply-side effect of capital: a source of strategic substitutability
We modify the benchmark
at the price p,
is
now
model
as follows.
term
(/>
given by
a positive scalar.
is
in (15),
q(p,
K) —
is
(f)
4>(E [0\K]
-
p).
As
E[^|i<']
in the
=
70
= XK,
= {e-p)qi- —q},
On
the presence of the last
finite price elasticity for
for
some
in
so the equilibrium price
=
captures the price elasticity of this
any
where the demand
coefficients 70
and
71.
Ai^c
=
now
demand
given
function
infinitely elastic,
E[^|i<r].
However, unlike in the benchmark
Market clearing now requires that
,
-
is
is
A
linear equilibrium the traders' expectation of 9
is
E[9\K]
units of capital.
the traders' demand, which
to the special case
benchmark model,
,
(f)
g,
70
+
(71
- ^)
,r
.
^
A-
(16)
that aggregate investment has two opposing effects on the price of installed capital, p.
the one hand,
other hand,
two
is
(15)
.
-
benchmark model
difference with the
The parameter
p
It follows
units of capital
-
The
+ 7iiv
qi
"
model, the equilibrium price does not coincide with
q{p,K)
who buys
which represents a transaction cost associated to the purchase of
-^ oo.^^
given by
i,
'
and our benchmark model corresponds
i.e.
net payoff of trader
:
convex transaction cost ensures a
by
The
'
Ui
where
,.
;
effects
it
it
raises the traders' expectation of 9, thereby
raises the
pushing the price up.
On
the
supply of capital, thereby pulling the price down. The strength of these
determines whether investment choices are strategic complements or substitutes.
^^A more familiar way of introducing a finitely elastic demand is to assume risk aversion. The alternative we use
here captures the same key positive and normative properties namely, demands are finitely elastic and individual
payoffs are concave in own portfolio positions but has the advantage of keeping the analysis tractable by making
—
—
the elasticity of demands invariant to the level of uncertainty.
23
Proposition 6
(i)
hi any linear equilibrium, the investment strategy satisfies
k{x,y)=E[{l-a)K{e) + aK{0,y) \x,y],
with
a =
A71
-
and k
A"/(/>
A small enough
(ii)
and for 71
(6»)
-
suffices for the equilibrium to be unique, for
of complementarity
by now famihar, informational
model.
The second term,
demand
for
the asset
sufficiently
choices
[iTx^l^+xy^,
investment to increase with
9,
to be positive.
The degree
is
=
(17)
is
a
is
now
sum
—X~/cj), captiues the
first
term A71 captures
the,
simple supply-side effect that emerges once the
information contained in aggregate investment
finitely elastic. If either the
strategic substitutes (a
The
of two terms.
investment on asset prices documented in the benchmark
effect of
poor (low 71) or the price
become
the
elasticity of
<
0).
demand
is
sufficiently
low (low 0), investment
However, the question of interest here
is
not whether
investment choices are strategic complements or substitutes, but how the positive and normative
properties of the equilibrium are affected by the dispersion of information.
implications that emerge in this extension are essentially the
satisfies k{x, y)
Pl
Provided that investment increases with both
good news
when
that
for 9
(i.e.,
71
0), in
f^o
a —
—X^/(j).
It
a happens
benchmark model.
3 immediately extends to the
+ (3ix + (32y
with
-a
signals,
then aggi^egate investment
which case Proposition 6 implies that a
>
is
—X'^/cf).
=
necessarily
In contrast,
—
{X~/4>)K, so
follows that the dispersion of information increases the value of
a and hence
there are no informational frictions, the equilibrium price
amplifies the impact of
if
>
TTx I
=
as in the
Lemma
First, consider the positive properties of the equilibrium.
modified model: equihbrium investment
same
In this respect, the
common
to be negative.
is
simply p
9
expectational shocks relative to that of fundamental shocks, even
We
conclude that Corollary
1,
which summarizes the key positive
predictions of the model, continues to hold.
Next, consider the normative properties. Because of the convexity of the transaction costs,
is
necessary for efficiency that
for
Eu
alH e
=
(1/2,
1].
/ {5/
Ex-ante
[(1
-
all
utility
A)efc(x,y)
traders take the
same position
in the financial
market:
g,
it
— XK
then takes the form
- \k{x,yf]
d^{x\9)
- H-'h''-h^'^'^'~
24
+
\
[9XK{9,y)
- ^[XK{9,y)]
d^{mi)
and the
investment strategy
efficient
Proposition 7 The
= -\^/4><0,
To understand
X{9 —
it
—
\K/(j))
reflects
6
the function k{x,y) that maximizes (19).
investment strategy
efficient
k {x,y)
where a*
is
=
k* (0)
= E[{16/
{l
+
a*) K* (6)
X^/cl))
If
the unique linear solution to
+
a* K{d,y)\x,y],
this result, note that the social return to investment
-
X^K/4>.
The new term,
is
given by
benchmark model,
relative to the
XK
(20)
^ J k{x,y)d^{x\9).
and K{0,y)
,
the cost associated with transferring
the traders.
is
is
— A)^
(1
4-
-\'^K/<p and
units of the asset from the entrepreneurs to
information were complete, efficiency would require that each agent equates his
marginal cost of investing to the social return to investment, which would give k
The analogue under incomplete information
is
=
6
—
}?K/4>?'^
that each agent equates the marginal cost to the
expected social return:
k{x,y)=¥.[6-{\^/<i>)K{e,y)
Reairanging this condition gives
The key
effect
finding here
on the private and
x,y].
(21)
(20).
that the introduction of
is
\
downward
social returns to investment. This
is
sloping
demands has a symmetric
simply because the negative pecuniary
externahty caused by the higher supply of capital perfectly reflects the social cost associated with
having traders absorb this additional capital. As a
result,
it is
only the informational effect that
generates a discrepancy between the equilibrium and the efficient allocation.
As
in the
benchmark model,
this discrepancy manifests itself in the response of equilibrium to
expectational and fundamental shocks. Indeed, while equihbrium investment satisfies (18), efficient
investment
satisfies
k{x,y)
—
Pq
+ 0lx + 02y
with
Because
in
any equilibrium
in
which
/3i,/32
>
the complementarity satisfies a*
relative sensitivity of the equilibrium strategy to
that Corollary
2,
1
TTy
/?2
common
noise
is
inefficiently high.
< q <
We
1,
the
conclude
which summarizes the key normative predictions of the model, continues to hold.^^
^*Note that under full information the optimal level of investment would be equal to re* (6).
^^As common in competitive environments, there are other forms of pecuniary externalities that could induce
strategic substitutabihty in the entrepreneurs' investment decisions, even with a perfectly elastic demand for capital.
For example, suppose that, in order to complete their investment, entrepreneurs need to purchase certain inputs
whose aggregate supply is imperfectly elastic (e.g., labor or land). Higher aggregate investment then implies higher
aggregate demand for these inputs, and hence higher input prices and lower entrepreneurial returns, once again
inducing strategic substitutabihty in the entrepreneurs' investment choices. However, such pecuniary externalities do
not, on their own, cause discrepancies between private and social returns. Indeed, it is easy to construct variants of
25
6.2
Information aggregation through prices
The
analysis so fax has imposed that the entrepreneurs
who
are not hit by the hquidity shock
can not access the financial market. Apart from being unrealistic,
possibility that the price in the financial
is
dispersed
among
the entrepreneurs.
by allowing entrepreneurs not
analysis
market aggregates, at
To address
by the
hit
assumption rules out the
this
least partly, the information that
this possibility, in this section
we extend the
liquidity shock to participate in the financial
market.
To guarantee downward
we assume
sloping demands,
that traders and entrepreneurs alike incur
a transaction cost for trading in the financial market of the same type as in the previous section. ^^
Thus, the payoff of an entrepreneur
the
first
period, and trades
qi
who
i
not hit by a liquidity shock, has invested
is
units in the second period,
is
units in
fc;
given by
'
1
while the payoff of a trader
their
demand
traders,
expectation of 6 based on
the aggregate
K
in the
for the asset in the
on the other hand,
demand
is
2
given by (15), as in the previous section.
i is
Because the observation of
1
2
second period perfectly reveals 9 to every entrepreneur,'^^
second period reduces to qE
given by
q-r
=
(j){E[d\K,p]
—
p).
=
4>
{9
—
p).
The demand
Note that traders now form
K and on the information revealed by the equilibrium price p?'^
for the asset
is,
\ {\
—
\)
qe
+
of the
\qT and the aggregate supply
is
their
Because
\\K, market
clearing implies
It
follows that the joint observation of
asymmetry
of information thus vanishes
the previous section, this
is
K
and p perfectly
reveals 9 to the traders as well.
and the equilibrium price
satisfies
by the
liquidity
shock {XK/2)
is
the traders and the entrepreneurs not hit by the liquidity shock.
coincides with the social return to investment
is
no
= 9—
j,/2-\)
^^- ^^
^'^
just the social return to investment, adjusted for the fact that the total
capital of the entrepreneurs hit
This result
p
The
difi:erent
it
now
equally distributed
Because the equilibrium price
follows that the equilibrium
from what we established
among
for the frictionless
is efficient.
benchmark
at the
end
the model that capture such sources of strategic substitutability while retaining the property that the informational
effect of
aggregate investment
is
the sole source of amplification and inefficiency, as in the example analyzed here.
hit by the liquidity shock do not pay the transaction cost for the units of the
^^We assume that the entrepreneurs
asset that they have to sell in the second period; this simplification has
^^This presumes that entrepreneurs use their private information
which
is
no impact on the
results.
when deciding how much
to invest
(i.e.
/3i
^
0),
indeed true in equilibrium.
^*In the
benchmark model,
as well as in the extension
examined
the information revealed by the equilibrium price simply because
had symmetric information.
26
all
in the previous section,
we did not condition on
agents trading voluntary in the financial market
of Section
2:
if
the dispersion of information vanishes at the time of trade in the financial market,
equihbrium investment
this result hinges
is
driven merely by first-order expectations of 6 and
on the equilibrium price perfectly revealing
To make
0.
However,
is efficient.
this clear, in the subse-
quent analysis we introduce an additional source of noise, which prevents prices from being perfectly
revealing.
Assume
to
them
that the cost of trading for the entrepreneurs
at the time they trade but
of an entrepreneur not hit
by the
which
is
to is
assumed
what
liquidity shock
that
is
revealed
is
now
given by
1
to be independent of
we look
cu,
not observed by the traders. In particular, the payoff
1
where
subject to a shock
is
all
other
random
variables, with E[a;]
=0
and
I/ar[a;]
=
we continue
to denote the
investment strategy by k{x,y) and we denote by p{9,y,Lo) the equihbrium price.
Because the
In
follows,
at linear rational expectations equilibria;
observation of aggregate investment in the second period continues to reveal 9 to the entrepreneurs
(but not to the traders), asset
and qr
—
4>iE [6\K,p]
—
demands can be written
p) for the traders.
as
q^^;
=
</>
(0
-
tj
—
p) for the entrepreneurs
Market clearing then implies that the equilibrium price
is
p
Once
again, the price
is
(^
_
6,
However, because the shock
^)
_ -^^XK.
u
is
not
known
ensuring that the informational effect of
of the fundamental reemerges.
(22).
i=A
(22)
a weighted average of the traders' and of the entrepreneurs' valuation of the
asset, net of trading costs.
longer perfectly reveals
= 5^E [0\K,p] +
This
effect is
At the same time, the supply-side
captured by the
effect of
K
is
first
also present
K
to the traders, the price no
on the traders' expectation
term
and
is
in the right-hand-side of
captured by the
last
term
in (22).
While the supply-side
effect
induces strategic substitutability, the informational effect induces
complementarity. In any rational expectations equilibrium in which the price
E [0|/C,p]
is
is
linear in {9,y,oj),
the projection of 6 on {K,p) and, by (22), the price p can be expressed as a linear combi-
nation of {K,6,uj).
It follows that, for
any
linear equilibrium, there exist coefficients (70,71,72,73)
such that^^
K[e\K,p]^jo + liK + j20 +
K
is
-/3io.
(23)
^^Note that 71, which captures the effect of
on the traders' expectation of 6, now combines the information that
directly revealed to the traders by the observation of
with the information that is revealed to them through the
K
observation of the equilibrium price.
27
Using
Xp,
(23), (22),
we reach the
Proposition 8
and the
fact that the private return to investment
a=
and for
As
"
kix,y)^E[il-a)Kie) + aK{9,y)
^
A small enough
(ii)
and .(9)
=
'T^XT^/f
6*
+
-
investment to increase with
6,
71 to be positive.
in the previous section,
a combines an
informational effect (captured by
—
supply-side effect always contributes to strategic
,,2-a)
'^'^^
)-
substitutability, while the informational effect contributes to strategic
high investment
is
good news
for 6
(i.e.
71
>
Once
0).
We now
turn to the characterization of the
concept we use
is
again, the overall effect
and only
ambiguous, but
economy. The efficiency
now we need
mimic the information aggregation that the market achieves through
is
if
^^^^ ^
continues to hold.
1
efficient allocation for this
the same as in the preceding sections; however,
j^a'^'i)
complementarity
the role of informational frictions remains the same as before: Corollary
to
A)
\x,y],
suffices for the equilibrium to be unique, for
supply-side effect (captured by
if
—
(1
In any linear equilibrium, the investment strategy satisfies
(i)
^7i
the expectation of
following characterization result.
.
where
is
to allow the planner
prices.
We
thus proceed
as follows.
First,
we
define an allocation as a collection of strategies k{x,y), qe {x,y,K,p,u!)
and qr {K,p),
along with a shadow-price function p{0,y,u)) with the following interpretation: in the
an entrepreneur with signals (x,y) invests k{x,y);
realizations of aggregate investment
of capital held
t
=
1) is
K = K [6, y)
by an entrepreneur not
hit
in the
second period,
and the shadow
by a liquidity shock
price
(in
p
all
first
period,
agents observe the
— p{9, y, uj)
;
the
amount
addition to the one chosen at
then given by qs {x,y,K,p,ijj), while the amount of capital held by a trader
is
given by
qT{K,p).
Next,
we say that the
XK {9,y) =
{1
-
X)
allocation
I
is
feasible
if
if,
for all {9,y,uj)
qEix,y,K{9,y),uj,p{9,y,uj))d^ {x\9)
As with equilibrium,
this constraint plays
two
resource constraint
not violated; second,
it
is
and only
roles:
first,
it
+
•
,
,
qT{K{9,y),p{9,y,uj)).
(24)
guarantees that the second-period
defines the technology that
is
used to generate the
endogenous public signal (equivalently, the extent to which information can be aggregated through
the shadow price).
Finally, for
any given k{x,y), qs {x,y,K,p,u!) and qT{K,p), ex ante
28
utility
can be computed
as
Eu =
W
(fc,
qs, qr), where
=
W{k,qE,qT)
^J J{-^k{x,yf
+
Xp{e,y,Lj)k{x,y)
+ il-X)ek{x,y) +
+ {1-X)R{9- oj,q^{x,y,K{e,y),u,p{0,y,uj))}d^
{x\e)
d^
{0,y,uj)
+ ljR{e,q^{K{e,y),p{6,y,uj)))}d^ie,y,uj),
R {v, q) =
where
vq
—
and where K{6, y)
(f / {24>)
= f k{x, y)d^ {x\0) We
.
then define an
efficient
allocation as follows.
An
Definition 4
efficient allocation is a collection of strategies k{x,y),
along with a shadow price function p{9 y
,
,
uj)
,
that jointly
qs (x.y, K,p,oj) andqxiK.p),
maximize ex-ante
utility,
Eu =
W {k,qE,qT),
subject to the feasibility constraint [24)-
Because
is
utility is transferable, the
to provide
an endogenous public
shadow price does not
upon which the
signal
allocation of the asset in the period 2
can be conditioned. The next lemma then characterizes the
Lemma
affect payoffs directly; its sole function
efficient allocation of
the asset.
5 The efficient allocation in the second period satisfies
XK
To understand
this result,
period. For any given
K,
(Plo
suppose
^
for
efficiency in the
a
XK
,
moment
{l-X)(Puj
,^^,
that information were complete in the second
second period would require that
all
entrepreneurs hold
the same qs and that {qT^qs) maximize
Oqr
subject to (1
information
price
is
^
+
(1
-
ei^s
A)
-
LoqE
-
—q%
I
Aiv. Clearly, the solution to this
problem
is
(25). In
our environment,
incomplete but the same allocation can be induced through the following shadow-
and demand functions: p{9,y,u})
\K _
2-A
— X)qE + qT =
- ^^T
——
^Ix
^
(lE{x,y,K,p,uj)
— ^^ — ^x'
^^'^ Q.t{K,p)
=
„ 30
f-
We now
characterize the efficient investment decisions. Using
Lemma
5,
ex ante utility reduces
to
E„ = lE{-l.= +
«-^^(Ai.)^} + il-^*.J.
(26)
^"Note that the proposed shadow price is also the unique market-clearing price given the proposed demand functions.
efficient trades can thus be implemented by inducing these demand functions through an appropriately designed
tax system and then letting the agents trade in the market.
The
29
Except
two minor differences
for
—the smaller weight on (AAT)^
with absorbing the fixed supply AA'
split across
of
u)
in the
a larger pool of agents, and the
second period
last
term
now
for the fact that
in (26),
affects the allocation of capital across entrepreneurs
which adjusts the cost associated
,
this quantity
which captures how the
and traders
ante utility has the same structure as in (19) in the previous section.
in the
The
is
volatility
second period
following result
is
—exthen
immediate.
Proposition 9 The
efficient
investment strategy
is
the unique linear solution to
k{x,y)^E[{l-a*)K*{d) + a*K{e,y)
where a*
= --^(^y
Comparing the
i^*
= j^0,
{0)
and K{e,y)
(27)
= j k{x,y)d^{x\6).
with the equilibrium one, we have that, once again, as long as
efficient strategy
investment increases with both signals, so that high investment
a remains
\x,y],
is
good news
for profitability,
then
higher than a*, in which case the key normative prediction of the paper, as summarized
by Corollary
2,
continues to hold.
Financial-market shocks
6.3
In the specifications considered so
We now
the installed capital.
have different valuations.
fai',
entrepreneurs and traders share the same valuation for
develop a variant of the model in which entrepreneurs and traders
In this variant, additional non- fundamental volatility originates from
correlated errors in the entrepreneurs' expectations about the traders' valuations; once again, our
mechanism
recent
We
t
=
3
amplifies the impact of these errors. This variant thus helps connect our
work on speculative trading a
Harrison and Kreps
is
given by {6
+ u))ki,
This random variable
is
where
in the
a;
is
The
traders' utility in period
a random variable, independent of
a private-value component in the traders' valuation.
different discount factor, or
valuations a la Harrison and Kreps (1978). For our purposes, what matters
in the traders' utility
is
taken as given by the social planner; that
preference orderings revealed by the agents' trading decisions.
u!
and simply
call
it
and of any other
economy. Normally distributed with mean zero and variance
from the hedging motive of the traders, from a
uj
to the
(1978).'^^
consider the following modification of the baseline model.
exogenous random variable
(j^.
la
model
We
is,
is
It
can originate
from heterogeneous
that the presence of
the planner respects the
thus choose a neutral label for
a "financial market shock."
We also modify the entrepreneurs'
information
set, to
allow for information regarding
investment decisions. In particular, the entrepreneurs observe a
^See Scheinkman and Xiong (2003), Gilchrist, Himmelberg, and
30
common
Huberman
signal
(2005),
uj
to affect
w — w + C; where Q is
and Panageas
(2005).
.
common
The
cr3.
this
is
noise,
independent of any other exogenous random variable
w
signal
Xi
—
economy, with variance
observed by the entrepreneurs but not by the traders; as in the baseline model,
is
a shortcut for introducing correlated errors in the entrepreneurs' expectations regarding the
financial-market shock.
about
in the
6,
we remove the common
+ ^i,
where
on common expectational shocks about
Finally, to focus
signal y
:
oj
rather than
the entrepreneurs observe only private signals about
6,
idiosyncratic noise as in the baseline model.
^j is
In this environment, the asset price in period two
is
given by
'
p^E[0\K,uj]+oj.
'
.
,
It follows
that equilibrium investment choices depend not only on the entrepreneurs' expectations
on
of 6, but also
investment
by
K {9, w)
is
=
their expectations of
given by k {x,w)
i3o
+ (3i6 + f32W.
=
f3o
t^;
:
there exist coefficients {/3o,Pi,P2) such that individual
+ Pix + p2W
and, by implication, aggregate investment
is
given
Following similar steps as in the basehne model, leads to the following
result.
Proposition 10
(i)
In any equilibrium, there exist a scalar
a >
and a function
k{9,u>) such
that
k{x,w)
(ii)
both 9
A small enough
=E[
(1
-a)K,{e,oj)+aK{e,w)
suffices for the equilibrium to be
\
x,w].
unique and for investment to increase with
and w.
(in) The efficient investm.ent satisfies
k
{x,
w)
—E
9
[
+
x,w]
Xuj
I
(iv) In
to 9
any equilibrium in which investment increases with both 9 and w, investment underreacts
and overreacts
to
w.
In this economy, entrepreneurs pay too
financial market.
The reason
interpret high investment as
Because the noise
is
much
essentially the
good news
attention to their signals regarding shocks in the
same
as in the
for 9, financial prices increase
in the entrepreneurs' signals
When
benchmark model.
traders
with aggregate investment.
about the financial market shock
ui is
correlated,
these signals are relatively better predictors of aggregate investment than the signals about
9.
By
implication, entrepreneurs' investment decisions are oversensitive to information about financial
market shocks
an increase
is
relative to information
in investment that
about their fundamental valuation
6.
was purely driven by expectations regarding
amplified.
31
Through
financial
this channel,
market shock
Absent informational
of investment to 6
traders'
and
a;
frictions
would be
(i.e., if
efficient.
known
Since
at the time of financial trade)), the response
can be interpreted as the difference between the
u)
and the entrepreneurs' fundamental valuations of the
efficiency results obtained in richer
asset, this case
is
reminiscent of the
models of "bubbles" based on heterogeneous
Panageas (2006) derives a similar
ulai',
^ were
efl&ciency result for a
valuations in a 9- theory model of investment.
The
priors; in partic-
model that introduces heterogeneous
interesting novelty here
is
that inefficiency arises
once we introduce dispersed information. Traders are then uncertain whether high investment
is
driven by good fundamentals or by the entrepreneurs' expectations of speculative valuations. This
uncertainty opens the door to our feedback effect between financial prices and investment, creating
inefficiency in the response of investment to different sources of information.
Other extensions
6.4
An important
function of stock prices
information that
(e.g.,
Dow and
This
effect is
made
by
is
is
to guide corporate investment choices
by revealing valuable
dispersed in the marketplace and not directly available to corporate managers
Gorton, 1997; Subrahmanyam and Titman, 1999; Chen, Goldstein and Jiang, 2007).
absent in the preceding analysis, because the entrepreneius' investment choices are
before the opening of the financial market. However,
letting the entrepreneurs
in the financial market."*^
make an
we can
easily incorporate
such an
effect
additional investment in stage two, after observing the price
Provided that the dispersion of information does not vanish, the source of
complementarity and inefficiency we have documented remains. Interestingly, though, an additional
information externality emerges:
if all
agents were to increase their reliance on idiosyncratic sources
of information, then the information contained in prices
improve the
would be more
precise,
which
in
turn would
efficiency of the investment decisions that follow the observation of these prices. Clearly,
this informational externality only reinforces the conclusion that agents rely too
sources of information, and hence that non- fundamental volatility
is
much
to
common
inefficiently high.
Throughout the preceding extensions, we have maintained the assumption that traders cannot
directly invest in the
on
this
assumption.
new technology during
the
first
period.
Cleaiiy, our results
do not hinge
For example, consider the benchmark model and suppose that each trader
j chooses first-period real investment kj at cost fc|/2
and then trades an additional
qj units in
the second-period financial market. Neither the equilibrium price in the financial market nor the
entrepreneurs' choices in the
now
first
period are affected;
includes the investment of the traders, which
is
we could introduce a
financial
market
that happens
simply given by
More
the information structure in the second period.
^^Alternatively,
all
in stage
instead of a noisy price signal.
32
generally,
1
or
let
is
fcr
that aggregate investment
—
JE^,
which does not
affect
one could drop the distinction
entrepreneurs observe a noisy signal of
K
between entrepreneurs and traders altogether and simply
who
make
first
and then trade
real investment decisions
about
talk
differentially
financial claims
Next, consider the assumption that a fraction A of the entrepreneurs
and
is
forced to
capital in the financial market; this
sell their
informed agents
on the installed
is
hit
by a
capital.
liquidity
shock
was a modeling device that ensured that
the private return to first-period investment depends on (anticipated) second-period financial prices
while ensuring tractabihty.
If
one were to drop the assumption of risk neutrality, or assume that
the second-period transaction costs depend on gToss positions, or introduce short-sale constraints
in the financial market, then the profits an agent could do in the financial market
on how much capital he enters the market with;
first-period investment
this in turn
depend on expectations of future
would depend
would ensure that private returns to
financial prices, even in the absence of
liquidity shocks.^^
Finally, consider the
Clearly,
what
is
assumption that
essential
is
only that there
dispersed information. For example,
trepreneur ihe
we could then
tively,
6i
= 9 + Vi,
where 9
is
the
a
is
we could
let
1.
is
perfectly correlated across entrepreneurs.
common component about which
the productivity of the
common component and Vi
also let the entreprenem's' signals
we could introduce common and
during period
profitability
be
is
new technology
for en-
an idiosyncratic component;
plus noise instead of 9 plus noise. Alterna-
di
idiosyncratic shocks to the entrepreneurs' cost of investment
In this case, unobservable
common
shocks to the cost of investment would also
act as a source of noise in the information that aggregate investment conveys about
playing the same role as the correlated errors in the entrepreneurs' signals about
7
agents have
9, essentially
9.^'^
Conclusion
This paper examined the interaction between real and financial decisions in an economy
information about underlying profitabiUty
itability,
is
dispersed.
By conveying a
in
which
positive signal about prof-
higher aggregate investment stimulates higher asset prices, which in turn raise the incen-
tives to invest.
This creates an endogenous complementarity, making investment decisions sensitive
In turn, this can
to higher-order expectations.
amplify the impact of
common
dampen
the impact of fundamental shocks and
expectational shocks. Importantly,
all
these effects are
symptoms
of inefficiency.
These
effects are likely to
be stronger during periods of intense technological change, when the
dispersion of information about the potential of the
ysis therefore predicts that such periods
new
technologies
is
particularly high.
come hand-in-hand with episodes
33
anal-
of high non-fundamental
may feature additional deviations from the first best
which may introduce novel effects in addition to the ones we have considered.
34
Such an extension is studied in the Supplementary Material.
^^Note, however, that these extensions
straints),
Our
(e.g.,
short-sale con-
volatility
and comovement
in investment
and
asset prices.
At some
level, this
seems consistent with
the recent experiences surrounding the internet revolution or the explosion of investment opportunities in
China.
What
looks like irrational exuberance
actually be the amplified, but rational,
While both explanations open the door to policy intervention,
response to noise in information.
the one suggested by our theory
may
is
not based on any presumption of "intelligence superiority" on
the government's side.
Our mechanism
also represents a likely source of
non-fundamental
volatility
the business cycle. Indeed, information regarding aggregate supply and
be widely dispersed
release of key
is
in the population,
macroeconomic
an important direction
statistics.
and
demand
inefficiency over
conditions seems to
which explains the financial markets' anxiety preceding the
Extending the analysis to richer business-cycle frameworks
for future research.
34
Appendix: Proofs omitted
Proof of Lemma
The
3.
main text
in the
derivations of
and
/?i
are in the proof of the
(52
Lemma
4.
Rearranging
(30), gives
-
1
Using
(29),
Proof of
a—
=
A71 and 52/5\
Lemma
TTy/iTx,
A71
then gives the
The proof proceeds
4.
result.
the dependence of
Part
(i).
F
and
G
on
and
(iv).
(A, iTg,iTx, TTy)
and
(ii),
Substituting K{9, y)
(iii)
=
Po
(x, y)
=
(1
-
=
(1
- A+
=
[(1
+
in equilibrium the
conditions must hold
A)
E [e\x, y] +
let
it
=
ttq
into (7)
[(1
A7i/?i
-A+
A70
+
A71 (Po
E [^la:, y] +
A7i/3i)
- A+
)
5ofi
A7i/3i)
+
<5i]
A70
A70
+
are used in the last steps.
+ tTx + tt^,
and using
=
6q
Trg/n, 61
E [6\x, y] —
=
JqM
+
ttx/t^,
^i^
and
+ ^2y
+ PiE [e\x, y] + /J-^y)
+
+ A7i/32y
A7i/3o
A7i/3o]
x+[{l-X +
above must coincide with Pq
+
A7i/3i) 62
+ pix + P2y
+
A71/32]
for all
y
x and
y,
the following
.
.
/3o
= (l-A +
A7i/3i)<5o/i
+
A7o
+ A7i/3o,
(28)
Pi^{l-\ + \-fiPi)5u
^2
It is
We
We
(i).
-
k
Because
by proving part
Throughout, to simpUfy notation, we suppress
+ Pi^ + P2y
gives
start
F which
continue with some auxiliary results regarding the function
conclude by estabhshing parts
We
in several steps.
(29)
-
= (l-A + A7iA)52 + A7i/32.
immediate that any equilibrium must
71/31
satisfy Pi
^
0.
Then
82 (1 + b)
= M^) = TT,^^^^^^^^
(5o62 + 52 (1 + bf
let b
=
(30)
P2I Pi- From
(4)
and
(6),
(31)
while from (29) and (30),
h=
—+
<5i
^71
(l-A +
35
A^
A7i/3i)5i-
C32-)
^
_
-
)
)
=
Substituting (31) into (32) gives b
F(6), where
^^r
^
~
^
F
B=
follows that, in
any
the set of
is
(1
+ 6)6
A) {So
+
62) 62
e
M
such that
all 6
+
(2
1
^^^^
"
-
A),526
1
- A+
+
^2
J
A7i/3i j^ 0.
Using
(31), the
M
{6 G
-
(1
:
+ (52)6^ +
X)(6o
linear equilibrium, 6
(2
-
+
X)S2b
627^ 0}.
necessarily a fixed point of F, while the coefficients
is
given by the following conditions:
aJ^e
A)/52)7o!7i)
(/3o,
-
(1
by
latter is given
It
^
I
(5i
Note that the domain of
A
= [l-A + A/i(6)]<5i
/3i
=
/32
(34)
'
=
6/3i
- A + Xh
6[1
^^
[1
/3i
- A+
{b)]5i
(35)
^
A/i (6)](5i
Po^{l-X + Xh{b))6o^i +
^0
+ ^2(1 +
'
(37)
^^^^-TT2
5)^
Conditions (34)-(38) uniquely define the function G.
=
Auxiliary results. Let g{b)
M
:
g{b)
A=
^
0} and
{S2Xf -
(1
complement
its
4(5o<52(l
-
A).
If
—
=
Be
is
A<
X){do
0,
+
S2)b'^
(2
-
X)S2b
+
62;
the domain of
F
is IB
=
G
{6
M g{b) = 0}. Note that the discriminant of g{b) is
= 0; if A = 0, then B^ = {- 2(i-a)?£^.) }' Anally,
G
{6
+
:
then B^
A > 0, then B^ = { 2fi -'a K^t+fe ~ ifi - AK^+fe } because there are values for (<5o,^2, A) that
make A negative, zero, or positive, all three cases are possible in general. However, because A is
continuous in A and A = -48062 <
when A = 0, A small enough suffices for Mc — 0. Moreover,
if
'
because g{b)
The
>
function
where 0i
(6)
=
>
60
F
[^2
for
is
-
any
6
>
(1
-
A)
So]b'^
+
hm F(6)= hm
F i-l) =
F{0)
=
R+ C B
always.
continuously differentiable over
6—>-oo
and
0,
-
2^26
=
+
Moreover,
F(6)
=
62-
Foo
entire
domain, with
aJ^
F'(6)
6^+00
62/61
its
Ml + - — ^
=
"i
<F {62/51).
[
(1
A) [do
+
Ji^
j >
02) )
di
,
36
,,
,
/u'-'.:
(;.
=
Consider the case 62
A <
Because
the function
0,
and increasing
{I
>
for b
—
F
is
Part
>
=
62/ Si
Part
then
let
-f"
=
<
=
A)<5o^2
62,
,
at bi
it
=
6
< -1/2
decreasing for b
is
-
^2 7^ (1
A) ^o-
Then
—1/2.
^i
=
(6)
where
_-52+
,
"""^ ^^
=
^/il^^JjS^
62-il-X)6o
and a
local
minimum
at 62-
F is continuous over R+, with F {52/Si) > 52/5i
equation F (b) = b admits at least one solution at
we have that
follows that the
It
cxd.
at 6
maximum
the preceding results
(^)
and
61
52-{l-X)So
then reaches a local
By
(iij.
and hmf,_oo
b
F
admits a unique solution at
defined over the entire real hne,
-52 - v/(l -
_
=
,
^'
function
=
(pi (6)
—1/2. Next, consider the alternative case,
admits exactly two solutions, at 6
The
Then
X)So.
TTy/TTx.
(Hi).
Fix any (61,62) G
F =
F( — 1/2) and
(0, 1)^.
F =
If
Foo-
If,
A
such that ^2
is
instead, A
=
-
(1
A) ^o,
such that 62
is
^
where
—
(l
—
60
-
1
+ 62),
let F =
(61
X)6o, then
F = max{Foo,F (61)}. It is easy to check that both F and F converge to
A — 0. Since F is continuous over its entire domain, B, and A small enough suffices for
we have that A small enough also suffices for F to be bounded in [F,F]. But then F
min{Foo,F(62)} and
62/61 as
B =
>
M,
converges uniformly to 62/61 as A -+
that,
whenever A <
Now
<
B = R and F
any
G
77
1 for all b
A
(e)
has no fixed point outside the interval [62/61
—
e,
62/61
=
0. It
G
there exist i
(0, 1),
-
[^2/^1
62/61
i,
+ i]
Combining the aforementioned
and A >
F (b) ^
=
note that the function F'(6;A)
that, for
77
A,
there exists A
such that,
b;
G
for b
foUows that,
for
[62/61
A <
any A G
for
A,
is
=
{rj)
>
and
all
A G
[0, A],
{62/61, +00),
F (0) >
such that
we have
7?
—
—
i,
F
has at most one fixed point. Together with the fact that
62/61
+ f], F
is
(ii)),
and lim;,^-oo
continuous and differentiable in
this proves part
F (6) >
< F'
such
+ e].
follows
(5;
<
A)
that there exist e
the following are true: for any b ^ [^2/^1
<
properties that
(r/)
results with the continuity of F,
continuous over the entire real hne) and F(52)
It is
A
—1 < —
>
[0, A].
=
(iv).
=
and A
easy to check that {61,62, X)
Part
>
continuous at {62/61,0) with F'{62/6i;0)
s
has at least one fixed point (from part
is
follows that for any e
0. It
>
<
0.
implies that
+
e],
<
1.
It
with F'
F
(6)
necessarily
B = M
(so that
F
These properties, together with the
-00, ensure that,
F admits at least one fixed point in (—00, 62)
£,62/61
(iii).
(.2, .1,.75)
62
b,
>
and one
in addition to
a fixed point in
in (62, 0). Indeed, in this
example
F admits exactly three fixed point, which are "strict" in the sense that F (6) — 6 changes sign around
them. Because F is continuous in {b,6i,62,X) in an open neighborhood of {61,62, X) = (.2,.1,.75)
there necessarily exists an open set S C (0,1)"^ such that F admits three fixed points whenever
,
{6i,62,X)GS.
m
Proof of Proposition
1.
R-om Lemma
4,
there always equilibrium in which b
37
>
i^y/'Kx-
Pick
—
the equilibrium that corresponds to the highest solution to F(b; X)
denote this solution.
a <
that
Lemma
3
>
h{h)
suffices for
from
and pait
>
in 6 over
+00) and
the highest solution to F(6; A)
6
>
6(A). Part
Proof of Proposition
(A)
/3o
a >
>
0i (A)
,
(A)
/32
,
(6;
all
from 71
follows
and
+
26)
+
((5o
+ ^2)
6^)
A)
-
6}
=
-
6
is
continuous
-00. Since 6(A) G {7ry/Trx,+oo)
F (6; A) — 6 <
that F increases
for
is
any
with
A.
=
6
71 (A) the corresponding equilibrium coefficients,
these functions are continuous.
Using conditions (34)-(38), the sensitivity of investment to the realization of 6
(i).
0;
from
Let 6 (A) denote the unique fixed point to F(6; A)
A.
70 (A)
A
increases with
then necessarily the case that
0, it is
,
follows
note that
{62 (1
{F
+00)
{i^y/T^x,
>
Next, note that the function F(6; A)
0.
Take any A <
2.
as given by (34)-(38). Note that
Part
=
6
a
that
€
let 6(A)
(36) observing that 6
then follows from this property together with the fact
(iii),
and denote by
+ h)
note that
(ii),
0; finally,
First,
satisfies limfe^+co
—
>
/32//3i
6(1
dF (6; A) /d\ >
suffices for
(TTy/TTa,-,
(52
For part
0.
which we prove next.
(iii),
dF [h- A) _
so that 6
>
for /3i,/32,7i
along with
(8)
and
follows from conditions (34), (35)
(i)
and hence
follows
1
Pai't
and
b
is
given
by
/3i(A)
where VF
We
(31).
=
(A)
can compute
can then use
upon
=
that 6(0)
A e
this to
request),
(0, A),
we
(A)
6'
and
(A)
W
W
<
ly
=
(0)
W
1
5i
[-1
that
(ii).
+
I3[
=
From
/i(6(A))
(0)
=
81
+
and
+
h{82/8i)]
0.
Proof of Proposition
4.
(0)
gives
(3'2
(0)
Thus consider the second
The
6) (1
- A + Aft(6))j^ and
—
Theorem
to
F (6, A) — 6. We
1,
W
.
this ensures that there exists
(A)
<
0;
that
we have that
is, (3i
<
0,
first
(A)
/3i
=
+
P2
=
82/81
[1
is
A G
(0, A]
lower than
- A + Xh
claim
is
(6 (A))](5i
and h{82/8i)
which together with the
result then follows
The
h{b) defined as in
W" (A) After some tedious algebra (which is available
and W" (0) = -tttt^t^^^^ < 0. Together with the fact
A/i'(6(A)6'(A))]. Since 6(0)
[-1
+
such that, for aU
81+82,
its
value in
decreasing in A.
is
condition (34),
>
(3'2
(1
and
=
{0)
the frictionless benchmark, and
Part
(A)
(0)
and hence
=
W(A)(<5i+<52),
b" (A) applying the Implicit Function
compute
find that
82/81
PF
A), with u;(6,A)
(6 (A),
it;
+ ^2(A) =
result
=
and hence
^^^^j^^g^'^^
from part
from the local continuity of
I3'2
(i)
<
1,
that
(3[
(A)
=
we have
(3[
(0)
+
(A) in A.
proved by the numerical example in the main text.
claim. Given any hnear strategy k{x, y)
38
=
/3o
+ PiX + /92y>
ex-ante utihty
is
given by
1
= E
2Eu
=
--k{x,y)
-\(3i
suppose prices are
+ek{x,y)
(39)
+ f3o{l-(3i-(32)i^-
^;92Vi
Now
2
+
+
(/3i+/32)a^
fully stabilized a.tp
— p.
^/3f TT-i
-
+ /32)'2_-l
^ (A
TT,-
l-2(A+/32)
(/9i+/32)
Substituting p{9, y)
= p into
the entrepreneurs'
best response (2) gives the following coefficients for the equilibrium investment strategy:
/3o
Note that p
=
(1
-
+
A) So^i
A
\p,
two terms
affects only the first
welfare that can be achieved with
=
(1
-
in (40)
A)
through
price stabilization
full
02^(1-
and
<5i,
on
its effect
A) 82.
/3o.
(40)
Hence, the maximal
obtained by choosing p so that
is
/3o
=
l-(l-A)(<5i+<52).
Next, note that for any a G
(0, 1)
and any
€ R, there
6
induces an equilibrium in which the investment strategy
Po
=
=
/3i
1
To
and
J— ao\
/?2
exists a
given
is
—
poUcy t{p)
tq
+
Tip that
by'^^
=
I
—
(41)
.
a
see this, suppose that, given (tq, ti), the entrepreneurs follow the linear strategy defined in (41).
Then
E[^|if]
The market
=
70
+
fiK, where (70,71) are obtained
clearing price
is
then equal to
P
Replacing (42) and K{9,y)
=
00
i
+n
+ Pi9 + f32y
+ I1K ~
(70
into (2),
entrepreneur consists in following the strategy k {x,y)
- To)
+
+n
A(7o
1
where a
—
A7i/(1
+ ri).
^^Equivalently, for any a e
ft'om (41) using the formulas given in (6).
(1
-
we then have
=
l3o
A) 5oi^
02
= {l-X)52 +
It is
then immediate that there exists a
(0, 1)
X) 61
+ af3i6,
a[f3iS2
and any ko € K, there
the investment strategy satisfies k{x,y)
=
E[(l
—
given by
+ a[f3o + 0iSofA
=
-
that the best response for each
+ (3ix + 02y
/3i
[l
(42)
To)
a)k{d)
+ (32]
(tq, ti)
such that
exists a policy (ro, ti) that sustains
+
aK{6,y)
39
|
x,y] with k.{9)
=
(
-^5^
j3o
—
/3o,
an equiUbrium
j
9
+
ko-
0i
in
—
0i
which
and 02
= P2
Now
(it
=
let 60
choose
suffices to
1
-
-
(1
X){6i
that
rj so
+ 62)
and
ce
for
—
=
a and then adjust tq so that 0q
any a G
b).
[0, 1) let
_
= (l-A + Q/3i)J2
-
P2{a)
^
1
—a
,
and
=
W{a)
-^-b'o
+ bo
1
-/3i(a) -/32(a)]
^(^2(a))'7r;i
+
^
(/3i
(a))' tt^i
W
(/3i(a)
+ /32(a))' tt,"!
+ /32(a))7r,-i
l-i(/3i(a)+/32(a)^
given by W{0), whereas welfare under any pohcy
full price stabilization is
continuously differentiable over
is
^
that implements a hneax strategy as in (41) with a G
note that
it
(,/3i(a)
Pi{a)+p2{a))
Note that welfare under
(ro, Tj)
+
M-
[0, 1).
(0, 1)
and
=
6
5o
is
given by W{a). Next
To prove the second claim
in the proposition
thus suffices to show that
at a
=
0.
dWdPi
dWd0^_
da
dPi da
d/32
da
First note that
dW
dW
W2
Using
dW
/3i(0)
=
(1
-
-bofi
- Att-i -
(/3i
+ ^2)
T^e'
+ ^e' +
-ban
- hT^y^ -
(/3i
+ h)
T^t
+ T't +
=
-
60
=
1
-bofi
+
[1
-
(1
-
are both increasing in a,
it
follows that
(5i,
/32(0)
(1
A) 82
dW
dW
/3i=/3i(0)
Because ^1 and
/32
-
and
A)
002
-
(1
A)
((5i
[l
-
(/^i
+ 02)
/^
[1
-
(/5i
+ /32
\i
A)((5i
+
^2),
we thus have that
+
62)] /i^
+
ATTg-i
=
Att-i
>
0.
/32=/32(0)
dW{0)/da
is
positive,
which establishes
the result.
Proof of Proposition
5.
Rewrite the tax rule as
Ti9,K)
where
(j)o
^ — tq,
0i
= 1 — n,
and
(j)2
=
(01
-
1)^
-T2. Suppose that
40
- ^2K,
all
other entrepreneurs follow the efficient
!
,
= E [0|a:,y]. The
strategy k(x,y)
p{e,
= E [^ -
2/)
(/3o,/3i,/32)
=
K{e,
+ <f>i-yQ +
^(po
E
where we have used
t(^,
equilibrium price
[^
|
K{e,
v))
=
+
70
The
{5o^',5\,52) into (6).
00
+ <^iE [d
K{e,
+ cl,2K{e, y)
y)]
I
+ 4>2)K{0,y)
{4>i^i
K{6,y)]
=
y)]
I
then given by
is
^\K{d,y), with (70-71) determined by substituting
best response for each individual entrepreneur
then to
is
follow the strategy
A;(x,2/)=E[(l-A)(0-r) + Ap|x,y]
= E [(1 =
(1
-
A)
A) 00
For the tax t(9,
K)
+ (t>iO +
(<^o
+
+
A (00
ct>2K)
(43)
+
+
0i7o)
+
A (00
(1
-
A)
0170
0iE
+
[e\x, y]
to implement the efficient allocation,
strategy in (43) coincides with k{x,y)
(1-A)02 + A(0i7i +
02)
=
— E [^|2:,y]
(1-
O,
A) 01
which
,
=
+
(0i7i
+
[(1
it is
is
-
|
+
A) 00
A (0i7i
+
02)]
E
[K\x,
y]
thus necessary and sufficient that the
possible
and
1,
x,y]
^2) A')
(1
-
if
and only
A) 0o
+
A (0o
if
+
0i7o)
_
7i-
=
0.
Equivalently,
To
The
= -00 =
_
.
70:
from the
result then follows
ri
= l-0i = - ^
<
fact that 70
<
71
and
when
T2
= -02 =
the entrepreneurs follow the efficient
allocation.
Proof of Proposition
For part
6.
entrepreneurs' best response
In the limit, as A —*
By
+ T^zt3i+02
0,
we have that
i.e.
6,
71
>
0, i.e.
A >
Substituting (4) into (6) gives
^
+
^o, /3i
—
_
+ /32f TTy
{pi
>
such that,
(/3l+/32)52
0^60
^i, /Jo
—*
for all
A G
^2,
+
+ (32r 52
{(3i
and hence
(0, A), (/3i
71 -^
+ (32) >
the traders' expectation of ^ increases with K, and
,,
/
\^
0, i.e.
r%i >
0.
investment
a = A(7i — A/0) >
0,
entrepreneurs perceive a complementarity in their investment decisions.
Proof of Proposition
The
P^ne
/3o
(ii).
{Pl+P2)7ry
_
7^e
continuity, then, there exists
increases with
Thus consider part
1
TT,
^
(2).
suffices to substitute the price as in (16) into the
it
(i),
result then follows
7.
Let
V{k,K,e) = -^k^ + 0k- ^K"^. From
from Proposition 3
in
(19),
Eu = ^EV{k,K,9).
Angeletos and Pavan (2007a), noting that k*
41
{9)
=
«
argmaocK
V {K,K,e) = ^^^6
Proof of Proposition
are coefficients
price
is
{(3o,
Prom
8.
= Vkk =
^t^^+'^X^K+v^
any equilibrium
(2), in
01,02) such that k{x,y)
—
in
which p
+ (3ix + I32y- Prom
/Jq
->?I4>.
is
(22)
linear in (0,j/,a;), there
and
(23), the equilibrium
then
-
p{9, y, u)
for
= l-
and a*
P{Kid, y),e, uj)
=
r,o
+ m^id, y) + ^e +
(44)
rfycu.
some (??o,^i,%,%)-
Now
consider the optimality of the traders' strategies.
mation that K{8,
y) reveals
about 9
is
same
the
K{9,
y)
is
the
same
=
vr,
''"yiWhile
)
to
(
/3l+/?2
the information that p{9,y,u)) reveals about 9 given
1
—\p{9,y,u:)-'no-r)iK{9,y)\
= 9A
is TTg
E[9
\
= ^
(
1
n^.
7^3
uj
12
V2
distributed with
infor-
as that of a signal
s^
whose precision
benchmark model, the
02
^
01+02
is
in the
as that of a signal
J^(g,y)-/?0
whose precision
As
A
trader
who
observes
K
and p thus believes that 9
is
normally
mean
K{9,y),p{9,y,uj)]
/\
=
=
^le
+
/"'-^ z^
/l\^
^
+ 71 -^(^, y) + 726* + IzuJ
70
where
70
—
=
,
71
— 0,0
Me
=
^
^0
+
'n-z+
TTe
+
TT,
+ ^2
TTs /9l
^^^>
(46)
K
72
73=
(47)
+ TTs
/\
^.
42
(48)
tS''^'
(
.;,,
,
Combining
we then have that
(22) with (44)
'70
-
Vi
=
(49)
2-A
(50)
- A V'
2
m =
x\
/
1
4>)
2!,(72 + l-A)
(51)
m = ;7^(73-l + A).
Lastly, consider the
the strategy k {x,y)
(2),
Po+Pix+P2y =
(1
optimahty of the entrepreneurs' investment
—
Po
+
The
Pix
02
equilibrium
=
[1
-A+
A7?i/3i
+
Xv2]SolJ-e
^1
=
(1
- A+
A77i/3i
+
Ar?2)
(1- A +
A77i/3i
+
A%)^"2
is
0,
and even
+
if
strategies.
and only
(/?o,/3i,/32)
AtJo
+
must
From condition
satisfy
if (/3o,/3i,/52)
satisfy the following:
Aryi^o
(53)
(54)
<5i
+
A7yi/?2
(55)
'
a thus a solution to (45)-(55).
existence of a linear equilibrium
>
individually rational
is
/3o
following steps similar to those in the
for 7i
+
- X)E[e\x,y]+XE\p(e,y,uj)\x,y].ThaXis,
/32 ==
A linear
(52)
uniqueness for A small enough can be established
its
benchmark model. Here we prove that A small enough
a > 0.
2
^ii^ j TTy and
(\
for
and
/
2
\
= ^
^s
TT;^
)
(
=
TTe
+
TT,+
TTs
Pi
+ (^2
+ /?2) ^y
+ P2? T^y +
(/3l
Phe +
into (46) gives
TTu;
1
TT,
71
=
suffices
{Pl
PW
(/9l+/32)^2
+
^|5o
In the limit, as A
^
0,
we have
that
/3o
^
continuity, then, there exists A
2^
(71
-
^) >
^ ^ii
/?2
,,J^'
~*
and hence
'^2:
(^L±M^^-^^515q
« =
+ /32)'52 +
<^0i /^i
7x^
By
(/3i
>
+
{5i
+ 8if 62
such that,
0.
43
>
0.
TT^+TTy+Tra:
for all
A G
(0, A),
(/3i
+
/32)
>
0,
71
>
and
,'-!
Proof of Proposition
The
Let
9.
result then follows for the
Proof of Proposition
10.
same argument
Part
In any equilibrium in which the price p(9,uj,w)
(i).
{6,iij,w), there are coefficients il3o,(3i,/32)
know
but do not know either
oj
signal z with precision
with
TT(^
= a7
.
It
tt,,
C,
is
hnear in
such that the investment strategy can be written as
k {x, w)
implying that aggregate investment
as in the proof of Proposition 7.
satisfies
=
l3o
+ Pix + (32W,
K{6, w)
oy 6, observing
=
K
/3o
is
+ /^i^ + /?2'^ + (^iQ-
For the traders,
who
then equivalent to observing a Gaussian
where
follows that the equilibrium price satisfies
p{e,uj,w)
=
E[e\K,u;]+u!
=
+ jiK
70
{9,w)
+
-
(1
7i/?2)w,
(56)
where
=
7o
7^ and 7i
A*
=
——,
r
=
7
^.
(57)
Substituting (56) into the entrepreneurs' best response gives
k{x,w)
= {1-
\)E[9\x,w]
+
XE\p{e,u},w)\x,w]
= {l-X)E[e\x,w] + X-ro +
which can be rewritten as
a =
Finally, that
Part
(ii).
k{x,w)
a >
is
A71
shown
Substituting
=
A
(70
in part
and
(i)
k
XiiE[K{0,w)\x,w]
+
X{l--fi(32)E[u;\x,w]
of the proposition by letting
{0,ui)
= (1-
A)6'
+
A7o
+ A(l-7i/32)w
I-A71
in the next part.
K {9,w) =
+ 7i/3o) +
(1
/3o
-A+
+ l3i9 + P2W
into (58) gives
X-iil3i)E[9\x,w]
44
+
X-iil32W
+
A
(1
-
71/32)
E
[u\x,w]
(58)
.'; "(;
.
Using the facts that
CTfl
^/(<7g ^
+
cTj:
^)Mi
E[^|x,ti>]
<^'i
k(x,w)
^
o'x-
=
=
+ ^x ^))
^
"/('''e
+
A(7o
+A
[77
=
E[^|x]
a-iid
(1-7/)
For this strategy to coincide with k
(x,
Six and E[a;|x,u;]
=
77
<T,
+ (l-A +
7i/9o)
+
+
Sq
+ cr, ^),
l{(jj-
A7iA)(5o
E[a;|w;]
=
r]w,
where
60
=
the above reduces to
+ (l-A +
A7i/3i)5ix
71/32] u;
—
w)
'^
=
(3o
+ PiX + l32W,
it is
necessary and sufficient that the
coefficients {Po,(3i,f32) solve the following system:
By
00
=
01
= (l-A +
A7i/3i)<5i,
(60)
P2
=
-
(61)
+ 7i/3o) + (l-A +
A(7o
A
+
[77
(1
7?)
A7i/3i)<5o
(59)
7i/?2]
(57),
7i/3i
2^''"
=
.
-e
(I)
which together with
(60) guarantees that
/3i
6(0,1),
.
e
(62)
+ n,
R-orn (60) and (61)
(0, (5i).
we then
get
,2
^'
^l-A)((|)'7r, +
^-
7rc)+A7rc)
or equivalently
VA/3i'
A/3i
where
+
y-HQh-
F(6;A)
It is
then easy to show that,
in a
neighborhood of
for
tl-V
7rij6
^\'^"(l-A)A27re62 + ^J-
A small enough,
F
has a unique fixed point and this fixed point
0^^ri
is
..
"
A/3i
Along with the
Part
(Hi).
fact that
The
/3i
>
always, this guarantees that 02
social planner's
optimality condition stated in part
Part
(iv).
From part
(iii)
5i
problem can be
(iii)
up
(x,
w)
=
is
given by
I3q+ Plx
45
^ for
A small enough.
as in the baseline model, giving the
of the proposition.
the efficient strategy
A:
set
>
+ 02W
}'S
:,.'
:./
with
Po
We
=
So,
PI
have aheady shown, in the proof of part
underreacts to
9.
Next, note that Pi
equilibrium in which P2
>
>
-
and
Si
(ii),
p;
that Pi
imphes
71
>
<
=
Si
A77.
—
Prom
0.
P^
,
which means that investment
(61)
it
then follows that, in any
also the case that
0, it is
p2
=
Xr]
+ X{l-r])^ip2>Xv^p2,
which means that investment overreacts to w.
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