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DEWEY j'
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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Policy with Dispersed Information
George-Marios Angeletos
Alessandro Pavan
Working Paper 07-27
October 30, 2007
RoomE52-251
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Cambridge,
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.M415
at
Policy with Dispersed Information*
Alessandro Pavan
George-Marios Angeletos
MIT
and
NBER
Northwestern University
October 26, 2007
Abstract
This paper studies policy
relevant fundamentals
—such
in
a class of economies
as aggregate productivity
and can not be centralized by the government.
information can
fail
in
which information about commonly-
and demand conditions
—
dispersed
is
In these economies, the decentralized use of
to be efficient either because of discrepancies between private
and
social
payoffs, or because of informational externalities. In the first case, inefficiency manifests itself in
excessive non-fundamental volatility (overreaction to
common
noise) or excessive cross-sectional
dispersion (overreaction to idiosyncratic noise). In the second case, inefficiency manifests
in
suboptimal social learning (low quality of information contained
financial prices,
is
identified:
and other indicators of economic
in
activity). In either case,
itself
macroeconomic data,
a novel role
for policy
the government can improve welfare by manipulating the incentives agents face
when deciding how
to use their available sources of information.
Our key
result
is
that this can
be achieved by appropriately designing the contingency of marginal taxes on aggregate
activity.
This contingency permits the government to control the reaction of equilibrium to different
types of noise, to improve the quality of information in prices and macro data, and, in overall,
to restore efficiency in the decentralized use of information.
JEL
codes: C72, D62, D82.
Keywords: Optimal
policy, private information, complementarities, information externalities,
social learning, efficiency.
'For comments and useful suggestions, we thank Olivier Blanchard, William Fuchs,
Jr.,
Ken Judd, Robert
E. Lucas,
Robert Shimer, Nancy Stokey, Jean Tirole, Nikola Tarashev, Harald Uhlig, Xavier Vives, Ivan Werning, and
seminar participants at Chicago, MIT, Toulouse, the 2007 IESE Conference on Complementarities and Information
and the 3rd CSEF-IGIER Symposium on Economics and
Institutions.
We
are very grateful to
NSF
for financial
support. Pavan also thanks the Department of Economics of the University of Chicago for hospitality during the last
stages of the project.
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium Member Libraries
http://www.archive.org/details/policywithdisperOOange
"The peculiar character of
plete
problem of a rational economic order
and frequently contradictory knowledge which
not given to anyone in
1
Introduction
The
dispersion of information
determined pre-
but solely as the dispersed bits of incom-
form
exists in concentrated or integrated
The economic problem of society
is
is
knowledge of the circumstances of which we must make use
cisely by the fact that the
never
the
is
...
its totality.
is
"
all the
separate individuals possess.
a problem of the utilization of knowledge which
(Friedreich A. Hayek, 1945)
an essential part of the economic problem faced by
society.
This
concerns not only the idiosyncratic needs and means of different households and firms, but also
commonly-relevant fundamentals.
aggregate productivity and
and pricing
For example, think of the business cycle.
demand
conditions
decisions. Yet, such information
is
is
crucial for individual consumption, production,
widely dispersed and
it is
through markets, the media, or other channels of communication in
As emphasized by Hayek
(1945), such information can not
such as the government.
Instead, society
the utilization of such information.
One can then be
stitution,
their available information in the
what best
may
must
rely
only imperfectly aggregated
society.
be centralized within a single
in-
on decentralized mechanisms
for
assured that rational agents will always use
privately-efficient way. This, however,
need not coincide with
serves social interests. For example, complementarities in investment or pricing decisions
induce firms to overreact to public information, because public information helps forecast the
decisions of others; this can
of
most
Information about
common
crowd out valuable private information and can
noise, resulting in higher
to internalize
how
their
own
non-fundamental
volatility.
also amplify the
Furthermore, individuals
impact
may
fail
choices affect the information contained in financial prices or other
indicators of aggregate activity; this can lead to inefficient social learning about the underlying
economic fundamentals.
A
novel role for policy then emerges: even
if
the government cannot centralize the information
dispersed in society and can not otherwise collect and communicate information, there
that
is
may
exist policies that
improve
efficiency in the decentralized use of information
manipulating the incentives faced by individual agents. Identifying such policies
of this paper.
Our key
result
is
to
show how
efficiency
contingency of marginal taxes on realized aggregate
Preview. Rather than focusing on a
policy lesson.
We
is
by appropriately
is
the objective
achieved by appropriately designing the
activity.
specific application,
we seek
to highlight a
more general
thus conduct our analysis within an abstract, but tractable, class of games that
allow for two sources of inefficiency in the decentralized use of information: payoff externalities and
informational externalities.
The former
are short-cuts for a variety of strategic
and other exter-
nal effects featured in applications, such as those originating in production spillovers, investment
.
or pricing complementarities,
monopoly power,
social networking,
potential discrepancies between private and social payoffs.
The
and the
like;
they summarize
latter reflect the (imperfect) aggre-
gation of information obtained through financial prices, the publication of data on macroeconomic
activity, or other
Our
forms of social learning.
analysis then proceeds in two steps.
First,
we compare the equilibrium
in the
absence
of policy intervention with the allocation that maximizes welfare subject to the sole constraint
that information can not be directly transferred from one agent to another; this helps detect the
potential inefficiencies in the decentralized use of information. Next,
implement the
an equilibrium;
efficient strategy as
The symptoms
of inefficiency that
we
detect
this gives the
we
identify tax schemes that
key policy result of the paper.
by comparing the equilibrium strategy with the
aforementioned socially optimal strategy depend on whether the inefficiency originates in payoff or
informational externalities.
fundamental
In the
volatility (overreaction to
reaction to idiosyncratic noise).
social learning (too
much
first case,
common
inefficiency manifests itself in excessive non-
noise) or excessive cross-sectional dispersion (over-
In the second case, inefficiency manifests itself in suboptimal
noise in macroeconomic indicators, financial prices, or other channels of
information aggregation)
Yet, the same,policy prescription works for either case, as well as for economies that combine
the two sources of inefficiency.
Our key
result
is
that the government can control
how
agents
use different sources of information in equilibrium by making the marginal tax rate contingent
on aggregate
An
activity.
appropriate design of the tax system then restores efficiency in the
decentralized use of information, irrespective of the specific source of inefficiency.
The
logic
behind
this result
with realized aggregate
activity,
is
simple.
When
individuals expect marginal taxes to decrease
they also expect the realized net-of-taxes return on their own
activity to increase with aggregate activity.
It
follows that a negative
dependence of marginal
taxes on aggregate activity imputes strategic complementarity in individual choices: agents have
an incentive to align their choices with those of others.
Symmetrically, a positive dependence
imputes strategic substitutability: agents have an incentive to differentiate their choices.
of individual decisions obtains
when
agents rely more on
sources of information, whereas more differentiation obtains
when
agents rely more on
Next note that a better alignment
common
idiosyncratic sources of information.
It follows
that the government can use the contingency of
marginal taxes on aggregate activity to fashion how agents respond to different sources of
mation. In particular,
when marginal
infor-
taxes decrease with aggregate activity, by inducing strategic
com-
complementarity the policy also induces higher
relative sensitivity of equilibrium actions to
mon
taxes increase with aggregate activity, by inducing
information. Symmetrically,
when marginal
strategic substitutability the policy also induces higher relative sensitivity to idiosyncratic infor-
mation.
It follows that,
how
control
by appropriately designing the aforementioned contingency, the government can
agents use their available information. In so doing, the government can control, not
only the impact of noise on equilibrium activity, but also the quality of information contained in
prices
and macroeconomic data. The government can thereby improve welfare even without
centralizing or
communicating information to the market.
Discussion.
It is
cient fluctuations in
who argued
(1936),
often argued that financial markets overreact to public news, causing
both asset prices and
real investment.
tuations in asset prices and investment.
broader theme that markets
any
ineffi-
This idea goes back at least to Keynes
that professional investors, instead of focusing on the long-run fundamental
value of the assets, try to second-guess the
tion to
itself
may
1
demands
of one another, thus causing inefficient fluc-
In this paper, although
we
are partly motivated
react inefficiently to available information,
specific application, nor
do we examine the deeper
we do not
by the
limit atten-
origins of such inefficiency (which,
unavoidably, will be specific to the application of interest). Rather, our goal
is
to identify policy
remedies that need not be sensitive to the details of the origin of the inefficiency. This explains
our choice to work with a theoretical framework that
is
abstract and flexible enough to allow for a
variety of distortions in the decentralized use of information.
Our
analysis also takes as exogenous the limits society faces in aggregating dispersed informa-
tion. Investigating the
institutional design,
offers
some relevant
is
foundations for these limits, and their potential implications for policy and
a challenging topic beyond the scope of this paper. Nevertheless, our analysis
insights. In
our class of economies, equilibrium welfare
may
decrease with addi-
tional information because of possible inefficiencies in the equilibrium use of information. However,
once these inefficiencies have been removed, more information can only improve welfare. Therefore,
policies that provide the
complement
market with the right incentives
for
policies, or other institutions, that provide the
how
to use available information also
market with more information.
Furthermore, while our analysis focuses on how the contingency of taxes on aggregate activity
contingency of monetary policy
can improve the decentralized use of information,
in practice, the
on aggregate activity might
direction. Indeed, the
comes out
of our analysis
also help in the
how
is
same
more general
insight that
the anticipation of such contingencies affects the incentives agents
face in using their available information,
Methodological remarks.
The
between two dominant paradigms:
and how
this in turn affects efficiency.
policy exercise conducted in this paper strikes a balance
the
Ramsey
tradition to optimal policy
(e.g.,
Barro, 1979,
Lucas and Stokey, 1983, Chari, Christiano and Kehoe, 1994); and the Mirrlees tradition, or the
"new public finance" paradigm
We
x
deviate from the
Ramsey
This point was epitomized
in
(e.g.,
Kocherlakota, 2005; Golosov, Tsyvinski and Werning, 2006).
tradition
by introducing heterogeneous information and by avoiding
Keynes' famous beauty-contest metaphor
for financial
markets, which highlighted
the potential role of higher-order expectations. Elements of this role have recently been formalized in Allen, Morris
and Shin (2003), Bacchetta and Wincoop (2005), and Angeletos, Lorenzoni and Pavan (2007).
ad hoc restrictions on the set of available policy instruments. In this respect, our policy exercise
closer to the
is
tradition
new
public finance literature.
At the same time, we deviate from the Mirrlees
by abstracting from redistributive taxation
policies that correct inefficiencies in the response of equilibrium
this respect,
our policy exercise
is
closer to the
and focusing instead on
(or social insurance)
Ramsey
outcomes to aggregate shocks. In
tradition.
Furthermore, the Mirrlees literature studies environments in which agents have private infor-
mation regarding purely idiosyncratic shocks, such as an agent's own
whenever aggregate shocks are featured
ductivity;
shocks
is
tastes, talent, or labor pro-
information regarding these
in this literature,
assumed to be common. In contrast, we study environments
in
which agents have
dis-
persed information regarding aggregate shocks; these shocks could be about either the underlying
fundamentals or the distribution of information in
absence of
common knowledge
The key
difference then
is
that the
regarding these shocks generates strategic uncertainty: agents face
uncertainty regarding aggregate activity.
of taxes
society.
on aggregate activity essential
It is precisely this
uncertainty that makes the contingency
for restoring efficiency
—which also explains why the results
presented here are, to the best of our knowledge, completely novel to the policy literature.
Finally, note that the policies
make agents
we
identify resemble Pigou-like taxes in the sense that they
internalize externalities. However, they are different with regard to
the underlying distortion and the
way they
both the nature of
restore efficiency. In standard Pigou-like contexts, the
market produces/consumes too much or too
of a certain commodity.
little
The Pigou remedy
is
then to impose a tax or subsidy on this commodity. In our context, instead, the market reacts too
much
or too
little
The analogue
to certain sources of information.
of the
Pigou remedy would then
consist in imposing a tax or subsidy directly on the use of these sources of information. However,
this
seems practically impossible. Our contribution
indirectly
is
to
show how the same goal can be achieved
by appropriately designing the contingency of taxes on aggregate
Other related
literature.
The
activity.
literature that studies dispersed information in
macroeco-
nomic contexts goes back to Phelps (1970), Lucas (1972), Townsend (1983), and the rational-
More
expectations revolution of the 70's and early 80's.
Woodford
recently,
Mankiw and
Reis (2000) and
(2001) have raised interest on the business cycle implications of combining information
heterogeneity with strategic complementarity in pricing decisions. 2
work by studying policy
Another
line of
in
We
complement
this line of
environments that share this key combination.
work, following Morris and Shin (2002), has examined whether welfare increases
with more precise public information within specific models. 3 Some of these papers have found a
negative result, which has then been used to
have found the opposite
2
3
See also
See also
Cornand
result.
make a
case against central-bank transparency; others
In Angeletos and Pavan (2007)
we showed how these apparently
Amato and Shin (2006), Hellwig (2005), Lorenzoni (2006), and Mackowiak and Wiederholt (2006).
Amador and Weil (2007), Angeletos and Pavan (2004), Baeriswyl and Conrand (2007), Heinemann and
(2006), Hellwig (2005),
Roca
(2006),
and Svensson
(2005).
conflicting results can
rium use
be explained by understanding the underlying
For that purpose,
of information.
we considered an
inefficiencies in the equilib-
abstract framework that restricted
information to be exogenous but was flexible enough to nest most of the applications examined
in the literature;
we then used
framework
this
to study the social value of information
(i.e.,
comparative statics of equilibrium welfare with respect to the information structure). In the
part of the present paper,
how
to study
we use a
priately designed tax system. In the second part of the paper,
which information
in
first
generalized version of that framework for a different purpose:
efficiency in the decentralized use of information
environments
the
partially endogenous;
is
can be restored through an appro-
we extend
the analysis to
we then show how
dynamic
similar policies also
correct inefficiencies that originate from informational externalities.
In this last respect, our analysis complements that in Vives (1993, 1997) and
Amador and
Weill (2007). These papers study the speed of social learning and the social value of information
in
a dynamic economy where agents learn from noisy observations of past aggregate activity.
results identify policies that can control the speed of social learning
exogenous information
Layout. Section
is
also guarantee that
any
socially valuable.
2 introduces the baseline
framework. Section 3 studies
decentralized use of information due to payoff externalities.
remove such
and
Our
inefficiencies.
inefficiencies in the
Section 4 identifies tax systems that
Section 5 extends the analysis to dynamic settings and Section 6 to
settings with informational externalities.
Section 7 discusses implications for the social value of
information. Section 8 concludes. All proofs are in the Appendix.
2
The
baseline static framework
we
In this section
any
details of
outline our baseline framework:
specific application
but
is
flexible
a
game
that abstracts from the institutional
enough to capture the
role of strategic interactions,
external payoff effects, and dispersed information in a variety of applications.
Actions and payoffs. The economy
indexed by
£
i
imposes a tax
Let
tp
T{
K
on each agent
and the dispersion
£ E. In addition, there
let
K = f hdip(k)
and Ok
of individual actions.
U M2
:
x
M+
x
G —
>
R.
The
=
=
[J (k
The
— K) 2 dtp(k)] 1
The (reduced-form)
a government, which
is
balanced.
'
2
denote, respectively, the
payoff of agent
i
is
variable
9{
C R
£
fcj
external and strategic effects exhibited in
(1)
represents a shock to agent
as "investment"
U may
mean
given by
U(ki,K,ak ,8i) -n,
For concreteness, in what follows we often think of
shock."
is
subject to the constraint that the budget
i,
Ui
some
ki
denote the cumulative distribution function of individual actions in the cross-section of
the population and
for
populated by a continuum of agents of measure one,
each choosing an action
[0,1],
£
is
and
64
i's
payoff.
as a "productivity
originate, not only
from preferences
and technologies, but
and the
social networks,
specification
from pecuniary externalities, monopoly power, oligopolistic competition,
also
What
like.
is
we
payoffs in transfers,
postponed to Section
is
nomial and that the external
f° r
an
(k,
K,
4
decision problem;
effect of dispersion
we assume the
— Ukn/Ukk <
which imposes concavity
be public at that stage.
is less
and
Next,
oji
let
how
6
given by h 6
F
0,
and the
which imposes concavity at the individual's
+ 2UkK + Ukk <
are three stages. In stage
2,
and Ukk
+ Uaa <
0,
1,
the government announces a
agents simultaneously choose their actions under dispersed
is
actions and aggregate productivity are publicly
3,
and the game ends. 7
e
as follows. Let W;
(f>
e $,
respectively.
of (0,u>) in the cross-section of the population;
Q
denote the information (also the "type") of
The
we
set of possible distributions
8
/ as the "state of the world."
refer to
F
according to the probability measure
common knowledge among
the agents. Nature then uses / to
i,
with the pairs
drawn independently from
[6i,Ui)i£[o,i]
but does not observe either
own
Q{
for 9i
distribution / describes the joint distribution
is
only about the agent's
Ua {k, K, o^-, 0) —
denote a joint distribution for (#j,w;), with marginal distributions
H and
Nature draws / from a
level (i.e.,
taxes will be collected in stage 3 as a function of information that
In stage
structure
f £
<
than one; and Ukk
revealed, taxes are paid, payoffs are realized,
i.
own
a concave quadratic poly-
existence and uniqueness of both the equilibrium
information (described below). Finally, in stage
agent
is
5
at the planner's problem.
that specifies
The information
its
U
which ensures that the slope of the individual's best response
Timing and information. There
will
we assume that
depends only on
following: Ukk
1>
with respect to aggregate activity
T
Finally note that, by assuming linearity of
6.
tractability,
To guarantee
5)).
crfc,
efficient allocation,
policy rule
that payoffs can be reduced to the
are ruling out any redistributive (or insurance) role for taxation.
Payoff restrictions. To maintain
(Tk-
is
assumed above without missing any channels of endogenous information aggregation;
the analysis of the latter
Uaa
though,
crucial,
/.
draw a
Each agent
First,
T which
pair (9i,Ui) for each agent
i
then observes his
own
u>j,
or the distribution /. Note, though, that w, encodes information, not
productivity
#j,
but also about the distribution / of productivities and
information in the population.
Although most of the analysis does not require any
4
The
external effect of dispersion
is
restriction
on the information structure,
relevant for certain applications: in new-keynesian models,
e.g.,
dispersion in
relative prices has a negative welfare effect.
5
Although the
intervention, our
restriction
—Uki</Ukk <
main policy
1 is
necessary for the equilibrium to be unique in the absence of government
result does not rely
on
this restriction:
the policies identified in Section 4 implement
the efficient allocation as the unique equilibrium even in economies that feature multiple equilibria in the absence of
policy intervention.
6
Because the government has no private information, the announcement of
about the fundamentals;
7
its
only role
is
T
does not convey any information
to affect the agents' incentives.
Throughout, we do not require idiosyncratic productivities to be revealed
at stage
3.
Also, in Section 4.3
consider an extension that adds noise to the observation of actions and aggregate productivity at stage
s
This
is
with a slight abuse of terminology because / does not describe the specific
(8i,uJi) for
we
3.
each single agent.
we
find
it
impose a certain "regularity" condition. Fix an information structure (Cl,F,F)
useful to
U
and consider any two payoff structures,
economy £
for the
S
—
2
(U
2
;Cl,F,!F).
such that k
We
;
S7, i^
JF)
,
and k
2
1
A;
.
UkK /Ukk ^ U K /Ukk
or
w 6
we
be an equilibrium strategy
*
equilibrium strategy for the economy
F,
(Cl,
F)
The
role of this condition
9
is
to rule out trivial cases in
and variance a 2,; the
jjlq
i's
productivity
latter
with zero mean and variance a
terms of
has a simple
it
for a positive
which changes
measure of
do not
in incentives
given by
is
9i
— 9+Q, where 9 is an aggregate productivity
.
Normally distributed with mean
and independent
across agents
is i.i.d.
2
is
Each agent's information
u>{
Normally distributed
of 9,
consists of a private signal Xi
—
#i
about own productivity and a public signal y
= 9+e
The
across agents, Normally distributed with zero
variable
idiosyncratic noise,
is
£j
i.i.d.
and variance a 2 whereas the variable e
,
variance a
2
.
example, /
The
is
<f>
over
and
£;
example
is
cjj
=
+ a 2 As
a2
9 and variance
in this
noises
.
e are independent of one another as well as of 9
will
9i is
Normal with mean
measure one to y
(xi,y) assigns
it
about the average productivity
become
= 9+e
=
Aiki
— ^k 2
,
with Ai
—
9i 4-
mean and
K=
aK,
/
kidi, 9i
=
In this
of q.
and variance ae2 and
Normal
2
a ,, a 2 and a 2
Applications. The following examples are directly nested
• Ui
is
and
mean
in Xi
with mean
imposing the "regularity" condition
clear in Section 3,
equivalent to imposing that the variances
and
9
+ £j
in the market.
public noise, Normally distributed with zero
is
a distribution whose marginal h over
whose marginal
CI
9
an idiosyncratic productivity shock. The former
is
C
CI
type of shocks and information structures that we allow, consider the following
Gaussian example. Agent
shock while
if,
requires that different sensitivities of individual best responses to either the
it
illustrate the
However
characterize the equilibrium.
lead to any change in the use of information.
To
and only
if
fully appreciated (in
fundamentals or others' activity result to different equilibrium actions
types.
"regular"
is
there exists a positive-measure set
,
This restriction can be
fi.
primitives of the environment) only once
— R
Cl
:
>
:
2
k (u) for any
economic meaning:
U 2 Let
Cl — R an
and
say that the information structure
2
7^
(u))
(C/
1
1
^ Uke /Ukk
whenever Ul e /Ukk
l
=
l
x
9
in
+
are positive
and
finite.
our framework:
?;,
<
and
a
<
This example
1.
can be interpreted as a stylized version of models with production or network externalities:
the private return to investment (A) increases with aggregate investment (K)
then parametrizes the strength of the spillover
and q
•
7T-
=
is
7T*
effect,
while 9
is
a
common
.
The
productivity shock
an idiosyncratic productivity shock.
-
^ - p*)
2
,
with p*
=
a9
+
(1
-
a) P,
P =
J prfi,
a G
(0, 1),
and
tt*
G R. This
example captures the incomplete-information new-Keynesian business-cycle models
ford (2002),
Mackowiak and Wiederholt
(2006), Baeriswyl
firms suffer a loss whenever their price (pi) deviates from
9
As
it
will
scalar a
become
clear in Sections 3
uniqueness of the optimal policy (not
its
and
4,
the only result that
existence).
is
and Conrand
some
(2007),
of
and
Woodothers:
target level (p*), which in turn
affected
if
we
relax this condition
is
the
depends on the aggregate price
level (P).
nominal demand conditions, while
in pricing decisions (a.k.a.
•
=
TTi
{a
+
with
K = / kidi, and
Cournot and Bertrand games studied
10 papers. In
•
(for
m =
-
(1
—
r) (ki
This example
is
—
6)
-
—
r{Li
(for
and Heinemann and Cornand
This game
is
and
among
in
Raith (1996), and various other
or cost shock,
More
is
the quantity (for
2
L = J Lidi, and
dj,
ki)
r
G
(0, 1).
Morris and Shin (2002), Svensson (2005)
(2006): an agent faces a cost
whenever the distance
of his
own
higher than the average distance in the population
is
supposed to capture,
in
a stylized fashion, Keynes' idea that financial
who
for
will
second-guess the
of others.
generally, as
equilibrium
fcj.
firms' decisions.
markets involve a zero-sum race between professional investors
demands
This example nests the large
.
c the degree of strategic substitutability
= J (kj —
game studied
action from the actions of others (Li)
(L).
i,
Bertrand)
L), with Li
the beauty-contest
R3
e
common demand
Bertrand) set by firm
Cournot) or complementarity
(a, b, c)
in Vives (1984, 1998),
this context, 9i represents a
Cournot) or the price
(for
a determines the degree of strategic complementarity
"real rigidities").
+ cK) ki - \kf,
bOi
—
1
In this context, 8 represents exogenous aggregate
it
concerned
will
become
—any model
clear in Section 3.2, our
framework nests
—at
least as far as
in
which the agents' interaction can be summarized
ki
=E
in the
following best-response structure:
some
for
i
[A(9i ,e,K)]
linear function A. Clearly, the institutional details
this structure
and deeper micro-foundations behind
vary from one application to another. Nevertheless, by conducting our exercise within
an abstract framework that does not take any particular stand on these
will
provide in the subsequent sections are likely to hold across
Remarks. Because
the primary goal of this paper
without centralizing the information that
we
restrict attention to tax
so doing,
we
rule out direct
systems that
mechanisms
government and then the government
in
Section
4, this is
is
we
these applications.
to study policies that improve welfare
is
dispersed in the population, throughout the analysis
utilize
in
all
"details," the lessons
only information that
is
in the public domain. In
which the agents send reports about their types to the
collects taxes
on the basis
of
such reports. As we
will
show
actually without any loss of optimality within our framework as long as the
government does not use such reports to transfer information from one agent to another before
individual actions have been committed.
Note, however, that this does not
information in society;
Indeed,
we could
it
mean
that
we rule out
only means that the government
readily reinterpret
some
of the
is
forms of aggregation and exchange of
not
itself
a channel of communication.
exogenous information as the result of certain
types of information aggregation; for example, some or
8
all
all
of the agents
may
observe a signal about
the underlying fundamentals that
financial market.
What
is
crucial for the baseline analysis
is
given agent does not depend on other agents' strategies, nor
these alternative cases are considered in Sections 6 and
Finally, note that, while our prior
2007) and virtually
all
work on the
and to a
allow the shocks to be imperfectly correlated and
will
that the information available to any
is it
affected
by government
social value of information (Angeletos
specific
we
policies;
7.
the related literature (Morris and Shin, 2002,
to the case of perfectly correlated shocks
This
unmodeled
the outcome of an opinion poll or the price of an
is
etc.)
and Pavan,
have limited attention
Gaussian information structure, here we
consider more general information structures.
permit us to highlight that our main policy results need not be sensitive to the specific
details of the information structure.
Gaussian example considered above, while
=
correlated (of
we
Nevertheless,
will still illustrate certain insights in the
restricting, for simplicity, the shocks to
be perfectly
0).
Decentralized use of information
3
In this section
we show how
the equilibrium and the efficient use of information depend on the
payoff structure U. This permits us to identify inefficiencies in the equilibrium use of information
that originate from payoff effects.
Common-information benchmarks
3.1
we proceed
Before
to the analysis of equilibrium
useful to review the case of
common
and
efficiency
with dispersed information,
it
is
information; this will help isolate the inefficiencies that emerge
only under dispersed information.
To
start,
suppose that information were complete, so that each agent knows both
his
own
productivity and the cross-sectional distribution h of productivities in the population, and this
fact
is
common
exists,
is
knowledge.
unique, and
is
We
can then show that the complete-information equilibrium strategy
given by
fcj
where 9
= J9dh(8)
own
The
in
=
Ko
+
KiOi
+
K2O,
,
10
Note that, by
definition,
an agent's payoff depends only on
productivity; that an agent's equilibrium action depends also on average productivity
characterization of the coefficients (ko,
allocation,
which
K (0i,6)
denotes aggregate productivity and where the coefficients (kq ki,K2) are
determined by the payoff structure.
his
=
is
in the
proof of Proposition
1.
rei,
K2)
,
as well as of the coefficients (k3>
k !j k 2)
f° r
is
the first-best
These coefficients depend on the reduced-form payoff structure U,
turn depends on the particular application under consideration. For the purposes of our analysis, however,
understanding the specific values of these coefficients
is
not essential.
a
because the latter impacts the average action of others, which in turn
affects the incentives of the
individual agent.
We
can further show that the
first-best allocation exists,
=
ki
where the
K*
(0i,
coefficients (k^, fc*,^) are again
unique, and
is
= «5 + K\6i + K*2 6,
6)
is
given by
.
determined by the payoff structure. Note that, as with
equilibrium, the first-best action prescribed to an agent depends both on his
own
productivity and
on aggregate productivity; however, the dependence on aggregate productivity now emerges only
because of external
Now
effects.
suppose that information
is
incomplete but
common
across
all
agents.
Because of the
quadratic specification of payoffs, a form of certainty equivalence holds: the equilibrium and
cient actions
common
under incomplete but
effi-
information are the best predictors of their complete-
information counterparts.
Proposition
ki
=
V
denotes the
E[k
1
Suppose
information
all
(9i,6) \V], while the strategy that
common
information
information remains common.
economies there can be room
maximizes welfare
is
given by
ki
=
strategy
is
given by
E[rc* (Oi,0) \V],
where
set.
which k
Clearly, in e-conomies in
common. The unique equilibrium
is
=
k*
,
there
However, as we
will
for policy intervention
no room
is
show
for policy intervention as long as
next few sections, even in these
in the
once information
is
dispersed. 11
Equilibrium use of information
3.2
We now
there
is
turn to the analysis of equilibrium allocations
We
no policy intervention.
An
Definition 1
equilibrium
is
fc(w)
when information
is
dispersed and
when
define an equilibrium in the standard Bayes-Nash fashion.
a (measurable) strategy k
= argmax R[U(k,K
(</>),
—?
:
fi
k
{<f>),
such
M.
6)
\
u
that,
for
all
u&
Q,
(2)
},
fc
with K(<t>)
= JQ k
and ak ((t>)
(u') dcj){J)
Consider the following
coefficient,
=
[/n [* (a/)
which
is
- Ki^fd^J)} 1 ! 2
for
all
<f>
e $.
the slope of an individual's best response with
respect to aggregate activity:
a=
^r
(3)
k
This coefficient measures the degree of strategic complementarity or substitutability among individual actions.
'None
The
equilibrium use of information can then be characterized as follows.
of our results requires
on economies
in
k
=fc
re*.
Indeed, the reader
may
henceforth assume
which inefficiency emerges only under dispersed information.
10
k,
=
k*
if
he/she wishes to focus
Proposition 2 The equilibrium
all
wefi,
with K(cf>)
— JQ k (a/) dcj>(uj')
Although Proposition
for
all
G $.
(p
To
it.
see this, recall that K,(9i,9)
taken had information been complete. Next, note that
actions other agents are taking relative to
When
K{</>)
— k(9, 9)
actions are strategically independent (a
=
>
0),
a form of certainty equivalence continues to
When
simply his expectation of the
is
instead actions are interdependent
while he does the opposite
behavior
is
The
if
would have been under complete information,
it
coefficient
a
<
actions are strategic substitutes (a
permit more alignment of actions when a
tilted to
how much
thus also captures
In particular,
the agent adjusts his action upwards whenever he
0),
expects aggregate activity to be higher than what
0.
the average deviation in the
is
basis of his expectation of the average deviation in the population.
actions are strategic complements (a
a <
would have
i
the agent adjusts his action away from the aforementioned certainty-equivalence bench-
0),
mark on the
if
it
what they would have done under complete information.
action he would have taken under complete information.
/
the action agent
is
hold: an agent's equilibrium action under incomplete information
(a
(4)
2 does not provide a closed-form solution for the equilibrium strategy,
an insightful representation of
is
unique, and satisfies
is
= E[K(6,6)+a-(K((f>)-K(6,6)) \u]
fc(w)
for
strategy exists,
In other words, equilibrium
0).
>
and more
differentiation
when
agents value aligning their choices with one
another.
how information
Proposition 2 has direct implications for
relying on
common
Because
used in equilibrium.
is
sources of information facilitates alignment of individual choices, while relying
on idiosyncratic sources inhibits
it,
strategic complementarity increases the sensitivity of equilibrium
actions to the former and reduces the sensitivity to the latter, while the converse
is
true for strategic
substitutability.
To
see this
information
of agent
is
i
is
more
clearly, consider the case
Gaussian. In this example,
consists of a private signal xi
idiosyncratic noise
and e
~
(«i
coefficients (7^,7^,71) are given
by
7
=
and where
irg
=
+
7T
+
y
2
o-g
,
(1
ny
and
+
-
=
a)lT x
cr~
2
,
and
where 9
equilibrium
+7T y
= a~ 2
-r (1
-
is
+ lyV +
"Ov
TTg
tt x
i,
«2) [ltiH
~
E?
7T0
k
~ N (/z, af)
a public signal y — 9 + e,
9 for all
The unique
+
k(x, y)
where the
=
= 9+&
iV (0, aVj.
=
9i
where the agents' shocks are perfectly correlated and
7
11
and the information
where
£,
~
N
(/z,
a2 )
then
Ixx]
(5)
,
=
Ot)lT x
denote the precisions
public signal, and the private signal.
,
C
TTQ
of,
+
1
7T
y
-")^
+
(1
-
,
6)
Q)
7T X
respectively, the prior, the
A
higher
a
reduces 7X and raises 7^ and 7 y
tion towards the prior
and the public
Next, note that 7x
activity.
+ 7y =
stronger complementarity
:
tilts
the use of informa-
signal because they are relatively better predictors of others'
1
—
7^, so that the equilibrium action of agent
i
can also be
expressed in terms of the underlying fundamentals and noises as follows:
h =
By
k
+
(«i
+
k 2 ) [7^M
+
(1
-
7m)#
+ 7y£ +
7x&]
>
strengthening the anchoring effect of the prior, stronger complementarity dampens the overall
sensitivity of individual actions to changes in the underlying fundamentals;
increases "inertia."
impact of
Finally,
common
By
increasing the reliance to noisy public information,
noise
and hence increases the non-fundamental
by reducing the reliance on noisy private information
it
in other words,
it
it
also amplifies the
volatility of aggregate activity.
mitigates the impact of idiosyncratic
noise and hence reduces the non-fundamental cross-sectional dispersion in activity.
Beyond the Gaussian example, a closed-form
solution of the equilibrium strategy can be ob-
tained in terms of the hierarchy of beliefs regarding the underlying fundamentals.
Corollary
shocks and,
E =
1
J E[8\u]d(fi(u)) denote the average of the agents' expectations of their own
n
n~
\(J\d4>{u>) denote the corresponding n-th order average
for any n>2, let E = J E[E
1
expectation.
Let
l
The equilibrium strategy
is
given by
k{tu)=E[K(9,§)\uj}
where 9
The
under
=
£~=]
((1
- aja"" 1 )
En
Vw,
(7)
.
equilibrium action under incomplete information thus has the same structure as the one
common
information replacing 9 with
9.
The
latter is
simply a weighted
sum
of the entire
hierarchy of expectations about the underlying shocks, with the weights depending on the degree of
complementarity: the stronger the complementarity, the higher the relative weight on higher order
expectations.
Since higher order expectations tend to be more anchored to
common
information, this suggests that the intuitions provided by the Gaussian example are
3.3
sources of
more
general.
Efficient use of information
We now
turn to the following question. Suppose the government can not centralize information,
or otherwise transfer information from one agent to another, but can manipulate the
use their available information.
Can
the government then improve
way agents
upon the equilibrium use
of
information?
In this section
that
may permit
we address
this question
by bypassing the
details of specific policy instruments
such manipulation and instead characterizing directly the strategy that maximizes
welfare under the sole restriction that information can not be centralized.
strategy the efficient strategy, or the efficient use of information.
12
We
henceforth
call this
Definition 2
An
efficient strategy is a
Because payoffs are linear
mapping k*
in transfers, the latter
:
Cl
—>
R
that
maximizes ex-ante
utility.
impact welfare only through incentives. Hence,
the combination of the efficient strategy with any transfer scheme that makes this strategy incentive
compatible defines the very best the government can do without centralizing information. That
when implementable, the
mechanism among the ones that
direct
to
efficient strategy is welfare-equivalent to
depend only on
planner recommends to each agent
any arbitrary strategy k
this goal, consider
=
and
Q— M
>
:
o%(h)
The term
The term
k\
e
= (K — K)
— K) —
(k
The
= Var(fc - K\h) =
(k
is,
e
+e+
hi
2
/ [k(6;h) - K{h)] dh{9)
activity.
The
action of any given
V{
common
captures the impact of
— K)
noise in information.
is
common
Finally, the
term
captures the non-fundamental variation in individual activity that
is, u;
Given any strategy k
1
Eu =
W
voL
vol
The
let
captures the impact of idiosyncratic noise.
is
12
following result then shows that a similar decomposition applies to ex-ante welfare.
Lemma
The
let
captures the non-fundamental variation in individual activity that
idiosyncratic to the agent; that
where
characterization.
captures the variation in individual activity that reflects variation in fundamentals.
across agents; that
=
show how the
can be decomposed in three components:
h—
Vi
and
its
will
"explained" by the fundamentals. Similarly,
is
be the fundamental components of the mean and the dispersion of
i
focus on
We
E[k{u>)\9,h}
denote the component of individual activity that
K(h)=E[k\h]= I k{9;h)dh{9)
we
next section; here,
in the
k(9;h)
agent
the best incentive-compatible
and not on the reports made by other agents.
can be implemented
efficient strategy
Towards
his report
restrict the actions the
is,
:
Q.
—
>
M, ex-ante utility (welfare)
E[U(k, K,
= Ukk + 2U2 ki< + UKK <
<7 fc)
9)}
and
W
= Var (e) = Var (K) - Vax(K)
first
term
and
+
dis
^W
vol
vol
+
±W
dis
= Ukk + Uda <
dis
=
Var
in (8) captures the welfare effects of the
(v t )
is
-
0,
given by
dis
(8)
and where
Var
(k
- K) -
Var(fc
-
I<).
fundamental-driven variation in
activity.
other two terms capture the welfare effects of the residual variation in activity: vol measures
non-fundamental aggregate
volatility,
which originates
fundamental cross-sectional dispersion, which originates
Note
that,
by construction,
hi, e
and
Vi
in
common
in idiosyncratic noise.
are orthogonal one to the other.
13
noise, while dis
measures non-
The coefficients
W
vo i
and Wdis then summarize the
two types of
sensitivity of welfare to these
noise:
W
vo \
measures social
aversion to non-fundamental volatility, while Wdis measures social aversion to non-fundamental
dispersion.
Both non-fundamental
How much
because of the concavity of payoffs.
in welfare
losses
and non-fundamental dispersion contribute to a reduction
volatility
depends on the
each of
them
contributes to welfare
under examinations: different primitive preferences,
details of the application
technologies and market structures induce different social preferences over volatility
For our purposes, however,
and
Wiii s
Since
suffices to
it
summarize these
and
dispersion.
social preferences in the coefficients
W
vo i
Their relative contribution can then be measured by the following coefficient:
.
W
vo \ captures social aversion to volatility, while Wdis captures social aversion to dispersion,
higher a* can be interpreted as higher aversion to dispersion relative to volatility.
same fundamental-driven variation
Strategies that share the
non-fundamental
common
actions to
to
volatility
common
and dispersion that they induce.
differ in
the levels of
Intuitively, the higher the sensitivity of
and hence the higher the non-fundamental
noise relative to idiosyncratic noise,
One should thus expect
preferences over volatility and dispersion. This insight
is
Proposition 3 The
13
efficient strategy exists, is unique,
=
k(u)
all
may
sources of information relative to idiosyncratic sources, the higher the exposure
of activity relative to its dispersion.
for almost
in activity
u, with K{4>)
—
E[ k*(6, 6)
Jn k (u/)
In equilibrium, an agent's action
equilibrium action; however,
K, with the weight on the
replace k with k* and
it
was
d(f>(uj')
a*
for
(K{<f>)
all
was anchored to
also adjusted
latter given
a with
+
by
a.
A
volatility
the efficient strategy to depend on social
formalized in the following proposition.
and
-
satisfies
\u]
k*(8, 9))
(10)
<f>.
his expectation of k, the
on the basis
complete-information
of his expectation of aggregate activity,
similar result holds for the efficient strategy once
a*. It follows that, just as
a summarized the
we
private value of aligning
actions across agents, a* summarizes the social value of such alignment.
That the
in
turn
is
efficient strategy
is
anchored to k* the
inversely related to the ratio
,
W
first-best action,
vo i/Wdi s reflects
is
quite intuitive.
That a*
our preceding discussion about volatility
and dispersion: the degree of alignment associated with the
efficient
strategy increases with social
aversion to dispersion and decreases with social aversion to volatility.
13
Hereafter,
when we say unique, we mean up
to a zero-measure subset of Q\ this
one has to make with a continuum of types.
14
is
a standard qualification that
Furthermore, just as
a pinned down
down
sources of information, a* pins
more
see this
is
where the
for almost all (x,j/),
*
ne
+ ny + {1 - a*yTTx
follows that, just as
It
and
'
Inefficiency only
3.4
The
k%) [j*^
+ JyV + J*x x]
ly
by
+ ir y +
n9
-
(1
a*)7r x
'
{l-a*)n x
_
TTy
=
(11)
,
lx
irg
+
ir
y
+
(1
-
'
a*)
irx
the equilibrium levels of inertia, non-fundamental volatility
As
structures, the analogue of Corollary
a with
1
for the case
applies for the efficient strategy
a*.
under dispersed information
further appreciate the inefficiencies that can emerge due to the dispersion of information, con-
sider economies in
which k
whenever information
a
+
dispersion, a* determines the levels that are optimal from a social perspective.
once we replace k with k* and
To
(k*
coefficients (7^,7y,7z) are given
a determines
more general information
of
=k* +
kj_
=
To
then given by
k(x,y)
**
the corresponding relative sensitivity of efficient actions.
again the Gaussian example with perfectly correlated shocks.
clearly, consider
efficient strategy
the relative sensitivity of equilibrium actions to different
y£
The
is
=
k*.
common,
In these economies, the equilibrium
leaving no
room
is
(first-best) efficient
for policy intervention. Nevertheless,
whenever
a* inefficiency emerges under dispersed information, opening the door to policy intervention.
,
following
is
then an immediate implication of the results in the preceding sections.
Corollary 2 Consider an economy
(i)
The equilibrium
(ii)
When a >
is efficient
that
is efficient
under dispersed information
a* and information
is
When a <
a* and information
is
if
and only
if
a—
= k*).
a*;
Gaussian, the equilibrium exhibits overreaction
information and excessive non-fundamental
(iii)
under common information (k
to
public
volatility.
Gaussian, the equilibrium exhibits overreaction to private
information and excessive non-fundamental dispersion.
4
Optimal policy
We now
turn to the core contribution of the paper.
affect the decentralized use of information.
efficient use of
We
We
first
explain
how
different tax
schemes
then identify the tax schemes that implement the
information as an equilibrium.
15
Equilibrium with taxes
4.1
Suppose that the information that
fa)ie[o,i] as
become
wen
publicly available at stage 3 includes
is
as aggregate productivity
T
where the function
and
satisfies
Then, without any
= T (ki,K,ak,8)
:
R2
x
R+
— R
x
+ T^ =
loss of optimality (as it will
by
,
a quadratic polynomial in (k,K,9),
is
>
—
the following properties: Ukk
{K,6), and Tkk
all
u
clear in the next subsection), consider policies defined
Ti
o\,
9.
individual actions
all
Tkk
<
— $ k Z.t
0,
k <
it
T (K,K,0,6) =
^>
We
denote the class of policies that satisfy
these properties by T. Finally, note that this class includes both progressive tax schemes
>
0)
and
for
These properties preserve existence and uniqueness of equilibrium,
0.
while also guaranteeing budget balance state- by-state. 15
Tkk
linear in
is
regressive tax schemes
(i.e.,
linear tax-schedule that does not directly
<
with Tkk
0).
16
One should thus
(i.e.,
with
think of this as a non-
depend on individual productivity
(0;),
but
is
contingent
on aggregate outcomes (K,ak,9)
We
then have the following result
structure by setting Ukk
— ~ 1|
Proposition 4 Given any
the general case
T
scheme
tax
G T,
let
l
_ {l-a)KQ-Tk (0,0,0)
1 - a + Tk k + Tk K
The equilibrium strategy
in the appendix).
is
a - Tk R
a
_
we henceforth normalize the payoff
(to simplify the formulas,
+ Tk k
1
.
~
1
.
_
(1
all u>
G
where k(9,
fi,
9)
=
Rq
a)
fa + k 2 - Tk§
)
.
unique, and satisfies
exists, is
k((j)=E[k(e,e)+a(K((l>)-k(9,e))
for
-
l-a + Tk k + TkK
+ Tkk
+
k\9
+
k 2 9 and K(4>)
=
JQ k
There are three instruments that permit the government
\u>]
(13)
{&') d4>{ui') for all
cf>.
to influence the agents' activity: Tkk,
the progressivity of the tax system; TkK, the contingency of marginal taxes on aggregate activity;
and
Tk §,
matter
the contingency of marginal taxes on aggregate productivity. While
equilibrium outcomes, each of them has a distinctive
for
role.
The
all
these instruments
progressivity Tkk
1S
the
only instrument that permits the government to control k\, the sensitivity of the agents' actions
to their information about their
own productivity
14
We
relax this assumption in Section 4.3.
15
To
see
/ T (*, K
,
ak
the
,
6)
latter
d-4,
(k)
property,
=T
(K, K,
note
0, 6)
that,
+ \{Tkk
for
any
shocks.
For given Tkk, the instrument that
cross-sectional
distribution
ip
of
individual
activity,
+ Taa )al
In our environment, the progressivity of the tax system will turn out to affect the decentralized use of information,
but
it
does not interfere with redistributive concerns; this
is
16
because of the linearity of payoffs
in transfers.
permits the government to control the degree of complementarity (a) and thereby the sensitivity of
common
actions to
noise relative to idiosyncratic noise
the contingency
T^k
the sensitivity of individual actions to variations in aggregate productivity
on
of marginal taxes
4.2
is
the contingency
strategy as an equilibrium.
Whenever
this
is
T that
implements the
the case, by the very definition of the efficient strategy,
even for transfer schemes that violate budget balance and/or anonymity, and even
make
agents to send arbitrary messages to the planner and the planner to
What
the agents before they
commit
T that
T* e
Proposition 5 A
essential
is
T
—
such that k\
that K2
=
function
T
«£,
this result
k*.
one allows the
if
the transfers contingent
,
result establishes existence
and uniqueness
the optimal
efficient strategy
T^k
always
increases with
exists, is
a and
unique, and
decreases with a*.
follows from Proposition 4. First, note that there exists a unique Tkk
is
But then there
an d a unique Tk
are then pinned
true
efficient allocation.
implements the
has the property that, for given (k,k*)
The proof of
The next
their choices.
that
is
only that the planner does not send informative messages to
is
implements the
policy in
efficient
no other transfer scheme that can improve upon T*. This
this also guarantees that there is
of a policy
Tk §
strategy
efficient
turn to the question of whether there exists a policy T* £
on these messages.
on aggregate
9.
Implementation of the
We now
of taxes
and T^k, the instrument that permits the government to control
Finally, for given T^k
activity.
is
also exists a
(0, 0, 0)
unique T^k such that a
such that Rq
down by budget
—
Kq.
The
—
a*, a unique
Tk g
such
rest of the parameters of the policy
balance, establishing that there exists a unique policy
that implements the efficient strategy as an equilibrium. 17
The optimal
policy has the property that, keeping k and n* constant, the optimal
with a and decreases with a* This
.
tarity perceived
by the agents,
in information.
If
we thus look
it
is
reduces the sensitivity of individual decisions to
across economies that share the
(i.e.,
properties under incomplete information
TkK
is
(i.e.,
17
The uniqueness
common
same equilibrium and
they feature the same
k,
and k*) but
noise
efficiency
differ in these
they feature different a and a*), we then find that
higher in economies that exhibit a larger discrepancy between the private and
the social value of aligning choices; equivalently, the optimal
non-fundamental
increases
because a higher TkK, by reducing the degree of complemen-
properties under complete information
the optimal
T^k
volatility of the equilibrium relative to its
T^k
is
higher the more excessive the
non-fundamental dispersion.
result holds for "regular" information structures. For "non-regular" information structures, the
degree of complementarity
is
irrelevant, leaving
one degree of indeterminacy. See Section 4.5
17
for
such a case.
Implementation with measurement error
4.3
The
policies considered so far
do not require that the government have superior information than
the agents at any time: the taxes are conditioned on information that
are levied. However, the preceding analysis has
many
In contrast, for
time.
applications
it
is
public at the time taxes
assumed that actions are perfectly revealed
may be more
at that
appropriate to assume that actions, as
well as aggregate fundamentals, are only imperfectly observed.
To accommodate
we now consider a
this possibility,
productivity to be observed with noise in stage
in stage 3 the
noise while
government
— and
an idiosyncratic
Vi is
the true aggregate productivity
9
=
9
+
errors
<;,
where
? is
2.
We
—observes
ki
if
=
the government
a
2
.
— as well
as
let
K = K+
77
+
fcj
a'i
77
i
chooses
+
in stage 2,
fcj
where
v{,
and a 2
,.
77
and a
2
=
o~\
+a
2
is
a
then
common
Similarly, instead of
any other agent
—observes a signal
All these noises could be interpreted as
to be independent of the fundamentals
then
agent
with respective variances
noise,
9,
In particular,
3.
other agents
noise with variance
and are assumed
have in stage
all
variant that allows actions and average
measurement
and of the information that agents
denote, respectively, the cross-sectional
average and dispersion of k, and consider tax schedules of the form
tz
where the function
T
T
is
assumed
E T). The tax an agent expects
E[T(fc 2 ,A-,a fc ,0)M
= T(ki,K,a k ,9),
to satisfy the
pay
to
is
same properties
as in the previous section
(i.e.
then given by:
= mF{ki,K,akt 9)\u)i)
+ 2(Tkk + 2TkK + TKK )a 2 + ±(Tkk + Taa )av2 +
1
ri
The
last three
terms
in (14)
Tgecjl
capture the impact of measurement errors on the expected tax. Because
these terms are independent of the agents' actions, they have no impact on individual incentives
and hence they do not
interfere
with the incentives provided by the tax system.
It follows that,
not only the noise does not interfere with the ability to implement the efficient strategy, but also
it
does not affect the properties of the optimal tax system.
Of
course, the last property relies on the noise being additive
actions).
When,
instead, the noise
is
multiplicative,
it
measurement error once again does not
efficient strategy,
although
Proposition 6 The
it
now may
efficient strategy
this result, for the
without any significant
undo the
incentive effects of the noise.
interfere with the ability to
affect the details of the
It
implementable the
optimal tax system.
can be implemented regardless of whether activity and fun-
damentals are observed with measurement
Given
separable from the agents'
does impact incentives. However, by appro-
priately adjusting the policy, the government can fully
follows that
(i.e.,
error.
remainder of the analysis we can abstract from measurement error
loss of generality.
18
—
.
Inefficiency only under dispersed information
4.4
As mentioned
in Section 1, financial-market observers
how information
are often concerned about
many
work
believe that financial markets
—
at
both the academic and the policy front
processed in financial markets. In particular, while
is
many
well on average,
also feel that financial markets
often overreact to noisy public news, causing excessive non-fundamental volatility in both asset
and
prices
At some
real investment.
the restriction that k
=
k*
and a >
level, this possibility
a*.
More
generally,
can be captured in our framework by
economies
which k
in
—
k* but
a ^ a*
offer
an interesting benchmark because in these economies policy intervention becomes desirable
only
when information
dispersed.
is
We now
thus examine the properties of optimal policy for this
special class economies.
Because a
^
a* means that the equilibrium
is
inefficient in its relative
sources of information and hence to different types of noise,
co- varies with the
=
letting e
components of
K—K
it is
is
necessary that the marginal tax
by the fundamentals. In
activity that are not explained
denote the component of activity that
response to different
due to common
particular,
we have the
noise,
following result.
Corollary 3 Consider economies
When a >
converse
is
a*
,
the optimal policy has
true
This result
under dispersed information.
in which inefficiency emerges only
T^k >
and the marginal tax co-varies
positively with
e.
The
when a < a*
is
intuitive. In situations in which,
if it
were not
for policy intervention, the equilib-
rium would exhibit overreaction to common sources of information and excessive non-fundamental
volatility,
the optimal marginal tax co- varies positively with the
fundamental
volatility.
from overreacting to
It is
common
noise that drives this non-
then precisely this property of the tax system that discourages agents
common
sources of information and
dampens non-fundamental
Note, however, that this appealing property of the tax system
is
volatility.
achieved only in an indirect
way, without any need to monitor and quantify the various sources of information available to the
agents.
This
particular,
is
done by making marginal taxes contingent on observable aggregate outcomes. In
when a > a*
,
the optimal policy reduces the complementarity in individual actions,
and thereby dampens the reaction
of the equilibrium to
common
of marginal taxes on realized aggregate activity positive (T^k
noise,
>
0).
by making the contingency
The optimal
guarantees that the overall response of the equilibrium to aggregate productivity
by making the marginal tax
4.5
not distorted
also a decreasing function of realized aggregate productivity
(Tk g
<
0).
Idiosyncratic vs aggregate shocks
Another special case of
the
is
policy then
new
interest
is
public finance literature.
the case of independent private values typically considered in
Studying
this particular case helps appreciate
19
how our
policy
depend on the dispersion
results
of information regarding aggregate shocks, as opposed to purely
idiosyncratic shocks.
Within our framework,
that
9i is
this case
can be captured as follows:
the private information of agent
distribution h
is
common
i
and
<f>
knowledge, so that there
—
is
that h, and hence
8, is
commonly known and
that the unique equilibrium
by k
=
(uji)
efficient allocations.
to condition the tax
What
renders
environment
is
is
As a
k* (Qi,Q)
We
is
no uncertainty about the cross-sectional
given by k (wj)
result, neither
also
common
the fact that
=
K
(0i,6~)
a matters
a and a* and hence
,
common
is
&i
is
known
to agent
i,
this implies
Similarly, the efficient allocation
.
is
it is
not necessary
18
also the contingency
in the
TkK, irrelevant
aforementioned
not per se the fact that private values are independent but rather that there
no strategic uncertainty: no agent faces uncertainty regarding the distribution of actions
population. Indeed,
that h
if
we
relax either the assumption that
common knowledge
is
distribution of actions
is
also
#; is
known
but maintain the assumption that
common knowledge in equilibrium.
is
to agent
common
behavior nor a* matters for the
We
efficient allocation
i
in the
or the assumption
knowledge, then the
a matters
— and therefore nor T^k
for equilibrium
essential for achieving
is
conclude that the key distinctive property of the correlated-value environments
consider in this paper
is
is
This in turn eliminates the problem
of forecasting aggregate activity, once again guaranteeing that neither
efficiency.
given
nor a* matters for
for equilibrium behavior,
activity.
dis-
knowledge. Together with the fact
conclude that in the case of independent private values
system on realized aggregate
so
6{,
second, suppose that the cross-sectional
h;
tribution of types in society. Because 0, the cross-sectional distribution of information,
knowledge, in equilibrium aggregate investment
=
suppose that Ui
first,
we
the strategic uncertainty created by the dispersion of information regarding
aggregate shocks.
Corollary 4 When
edge,
the cross-sectional distribution of information in society
the contingency of taxes
on aggregate
((f))
activity (TkK) is i}°t essential for
is
common
knowl-
implementing the
efficient strategy.
5
A
The
analysis so far has been confined to a static game.
dynamic economy
be embedded
in a
dynamic setting with a more macro
helps further appreciate
1
and
how our
results can
be relevant
We now
flavor.
show how
This serves two goals.
for applications.
Second,
it
9.
Moreover, should the tax be
k.2
made
First,
it
accommodates
In particular, the efficient allocation can be implemented with a tax schedule that depends only on
through
game can
this static
own
activity
contingent on K, this contingency would matter for equilibrium behavior only
(the sensitivity of the complete-information equilibrium to 9), not through
the contingency on 9 should then be adjusted to perfectly offset the effect of
20
TkK on
a
k.2-
(the degree of alignment);
the possibility that interesting dynamics in actions originate in the dynamics of information.
we
this section,
start
by taking the dynamics
analysis of endogenous information in Section
There are
N+
>
periods, with TV
1
chooses an action
Investing ki
t
G (kij)
mean and
G
and
F
6j jt
and
,
=
t
1,
...,
his level of
N, each agent
consumption,
£ R, which we henceforth interpret as capital invested
ki >t
costs
t
at are the
shock, and
6.
In each period
2.
of investment in a riskless discount bond,
and
to the
up
Set
5.1
we turn
of information as entirely exogenous;
In
in period
and
t
F (hij, Kt,
delivers
the dispersion of activities in period
To
are real-valued functions.
o>t,
t,
is
A,i+i
The agent
.
also
in a risky technology.
t+1, where Kt
an exogenous productivity
is
simplify the exposition,
that 9 t denotes the
it
Aij+i) in period
that the productivity shocks are perfectly correlated across agents, so that
(The timing convention we adopt here
Q
chooses his level
i
common
we henceforth impose
—
./U.t+i
shock that
&t f° r
is
au *»*
relevant for
period-^ decisions.)
The
agent's period-i budget
Ci,t
+ G{k
i>t )
given by
is
+
=
qt b iit
F(ki tt -i,Kt-i,at -i,9t-i)
+b
iit
where qt denotes the period-i price of discount bonds (the reciprocal
and r^ denotes the period-t taxes the agent pays
rate)
19
government.
-n
-\
it
,
period— t
of the
risk-free
to (or the transfers he receives from) the
Finally, the agent's intertemporal preferences are given
by
N+l
where Ut
is
a real-valued function.
This framework
is
quite flexible.
can be nested by setting U(c,k)
assumed
a firm)
costs
X >
.
is
in those statics
G (k) =
and then
c
(e.g.,
nested by letting
U (c, k) =
c,
F (k, K,
t
to Cj,n+i
agent's
ct,
9)
=
benchmark
—G(k) + (3F(k,K,a,6) equal the
9k,
and
effort, in
payoffs
as the profit of
model with convex investment
G (k) =
which case
k
+
it
x/c
2
,
for
some constant
would be natural to
let
U (c, k) = c — H(k), with H{k) = k representing the disutility of effort. Finally, one
allow U and G to depend on (K,a,9), as to capture externalities in leisure, pecuniary
2
externalities in the costs of investment,
For
in the static
—G{k) + @F(k, K,a,8) can be interpreted
could also interpret k as individual
and
could further
19
letting
Alternatively, a stylized version of the neoclassical growth
One
0.
examples
=
we considered
All the applications
= N+
= F (ki
1,
t
N,
endowment
we impose that
ki,t
Kn, on, An+i) —
=
&i,t
G (0)
and so on.
=
4- bi,/v
for all
—
to zero so that F(ki,o,Ko,(Jo,Ai)
i,
in
which case the last-period budget constraint reduces
tj,n+i- Furthermore, for
=
bj,o
21
=
for all
i.
t
=
1,
without
loss,
we normalize each
If
agents had
common
information about the shocks in
periods, then the analysis could
all
G
proceed essentially without any further restrictions on the functions F,
where agents have heterogeneous information. To keep the analysis
are interested in cases
we impose two
we assume
restrictions. First,
some function H. Second, we
for
and U. Here, however, we
that
U
—c—
H(k),
the function
U
in the
bond market
clears
consumption: U(c,k)
linear in
is
tractable,
let
V {k, K, a, 0) = -[G (fc) + H{k)} + 0F {k, K, a, 9)
and assume that
static
V satisfies
model. The
and only
= /3
if q%
the same properties with respect to
first restriction
ensures that, in
which case the demand
(in
life-time utility of agent
(in
i
(k,
K,
a, 9) as
periods and states, the
all
bond
for the risk-free
is
if
indeterminate) and that the
the absence of taxes) reduces to
N
Ui^Y^p^V {kiiU K ,a ,6
t
t
t)
t=\
The second
restriction then permits to extend our previous static analysis to the
The key
knows
this,
in
period
t
is
to rule out informational externalities
about
{&t,Wi,t}tLi>
in period
with marginal distributions
the cross-sectional distribution of
6
LOi tt
{6t,u)i t}tL\ in
F
by
4>t
6
i,
with
t
for all
t,
then irrelevant. 20
It is
i
then without
efficiency, to restrict attention to strategies that
Given that information
is
exogenous
in all dates
efficient allocations parallels that in the static
Vjt (k,
k, 0, 6)
—
and
let
equilibrium action in period
20
An
distribution / also describes
drawn independently from
and
t.
/.
This ensures that there
depend only on
is
Finally,
we
nothing to
—whether such actions
loss of generality, for either
equilibrium or
w;^.
t
and
states, the analysis of
benchmark. Let k
= argmax K V (k, k, 0,9)
k* (6)
would be k
(9 t )
,
;
if
both the equilibrium and
denote the (unique) solution
(9)
information were complete, the
while the first-best action would be k* (9 t )
alternative that would also guarantee that agents do not learn anything about (9 S
of past actions
21
for
Equilibrium, efficiency and policy
5.2
to
To ensure
denote a joint distribution
learn about (8 s ,4> s )g =t from the observation of other agents' (past) actions
is
t.
the population. First, Nature draws / from a set
{9t,(^i,t}tLi
(a>i,t-i,0t-i,0t-i) belongs to Ui
are observable
<
The (exogenous) information
The
$t-
in periods s
what an agent
according to the probability measure T. Nature then uses / to draw a
sequence {Oti^i^tLi f° r each agent
assume that
F
Let / G
fit-
for u> lit given
t
of possible distributions
possibility that
of the information structure as follows.
represented by
is
t
—the
depends on the actions other agents took
Ot
we model the dynamics
an agent
of
here
dynamic model.
is
to
assume that
Both k and k" are
for all
t
linear functions of
>
6.
1, uJi
it
is
a sufficient statistic for
In particular, k.(9)
=
ko
+ (m +
:
,
<t>s)^=t
1
Next,
from the observation
(^i,s,<^s,^s)s=i) with respect to
= kJ + (kJ +
U with V.
and k'(8)
K.2)
the coefficients (ko,ki,K2, «o iK*,^) determined as in the proof of Proposition
22
(u>i t,
21
.
replacing
k,1)6 with
let
a
with
W
vo i
=
Vki<
=
+ 2VkK + Vkk
Vkk
W
a=1 ~w^'
*
,
and
'v^
=
and Wdis
+ Vaa
Vjtfc
\
voi
,
once again, a summarizes the private value
of aligning choices (the equilibrium degree of complementarity), while a*
summarizes the
The equilibrium
value of such alignment (the relative social aversion to dispersion and volatility).
and
under incomplete information are then characterized
efficient allocations
propositions, which are direct extensions of Propositions 2 and
Proposition 7 The equilibrium
w* €
two
3.
unique, and satisfies, for
is
in the following
all
periods
t
and
all
fit,
=
fct(wt)
withK
t
{4> t )
E[ K (e t )
+ a-(K
(<j> t
t
)-
\ut
Kt (6 t ))
],
=/fct (u/)#t(w').
Proposition 8 The
all
strategy exists,
social
efficient strategy exists, is unique,
and
for
satisfies,
periods
all
t
and almost
wt £ Of,
kt(6H)
with
K
t
((p t )
The
efficient
efficiency
about
= fQ
K
t
kt
=
E[ k* (6
8t
)
+ a* (K
t
{&) - k*
(6t ))
\
u
t
],
(to') d(f> t (u>').
strategy can be implemented in a similar fashion as in Section
can be induced in period
and
t
t
by making taxes
that becomes publicly available at
t
in
period
+ 1. As
t
+
1
when they make
In particular,
contingent on information
in the static
model, the optimal policy
does not require any informational advantage on the side of the government.
the agents anticipating
4.
It
merely depends on
their decision that the marginal tax they will
pay in the
future will be contingent on public information about aggregate economic conditions.
6
Informational externalities
A
key functioning of modern economies that
is
missed in our preceding analysis
is
the aggregation
of dispersed information in various indicators of aggregate activity, such as financial prices, trade
volume, aggregate employment, output and investment data.
What
that the informational content of these indicators depends on the
information in the
first place:
informative aggregate activity
mation aggregation and
taken into account
We
study
the
is
more individuals
on
crucial for our purposes
way agents use
The
its
more
various channels of infor-
social learning thus introduce informational externalities that
and
is
their available
their private information, the
of the underlying fundamentals.
when determining
this issue,
rely
is
have to be
the socially optimal use of information.
policy implications, within a variant of the
dynamic framework
introduced in the previous section. Past shocks are no longer directly revealed to the agents. Rather,
23
the agents learn about these shocks from the observation of a noisy signal of past aggregate activity.
This signal
a proxy for the informational role of macroeconomic data, financial prices, and other
is
channels of information aggregation and social learning.
up
6.1
Set
To be
able to analyze the endogenous dynamics of information in a tractable way,
some
rifice
we have permitted
of the generality
exogenous information to be Gaussian.
all
and variance
\x
The
ag.
In particular,
and we assume that
realization of 8
— 9 + et
public signal y t
9,
and private
is
it is
we assume that the component
t
t
and
of the
we
constant over time,
drawn from a Normal distribution with mean
= # + &,*,
where e t
~ Af(0, cr^
t )
agents observe a
t,
and
~ Af{0, cr^
£j jt
and independent of
are noises, independent of one another, independent across time,
also independently
is
never revealed. Instead, at any date
signals Xi
sac-
preceding analysis: we henceforth restrict
in the
fundamentals about which the agents have heterogeneous information
denote this component by
we must
any date
identically distributed across agents. In addition, at
t
>
2,
)
^
with
9,
t
agents
observe the following three random variables, which affect payoffs and convey information about
K _i
9:
t
and
at
~
=
K _i
t
+
T] t
M(0, a^ t )
a
,
t
=
-i
a
are shocks,
t
-i
+v
t
A =
and
,
common
9
t
+
at
,
where
Cij
The
variable
mean
A
+ G{k
i>t )
+
qt b iit
= F(k
That the
it
t-i,Kt-i,a -i,A
t
variables Kt-i
F) coincide with the signals about past
t
),
)
activity
rest of the
model
ft^U (ci t,ki,t)
,
is
where
t
as in Section 5.
~
J\f{Q,al
t ),
and independent
given by
is
t
vt
+b
it
t-i
-r
iit
.
is
the underlying
and Ot-l that enter period— t income (through
not essential.
is
observation of income does not perfectly reveal either 9 or
J2t=i
Af(Q,tf
can be interpreted as the period— t productivity shock, while 9
t
(trend) productivity.
The
~
across agents, independent across time,
any other random variable. The period-i budget of the agent
of
rj t
K -i-
essential
is
that the
payoff of an agent
is
given by
is
t
The intertemporal
U (c^t, fc^t) = c^— h{ki t)-
What
22
That preferences
are linear in consumption
t
ensures once again that q t
—
(3,
that the trades of riskless bonds and the timing of consumption are
indeterminate, and that the intertemporal payoff of an agent (in the absence of taxes) reduces to
N+l
Y,0
t
~1
V-(kiitl Kt i *i,Ai+
i
t=i
V (k, K, a, A) =
where
the
22
same
+ /3F
(k,K,a,A)
.
The
function
V
is
quadratic and satisfies
restrictions as in the previous sections.
Also, the results that follow do not depend on whether the signals about past actions are public or private.
we could
In particular,
Ai,t
—[G(k) + H(k)}
=
6
+
a,i lt
,
allow the agents to receive private signals Ki,t-\
in addition to, or in substitution for, the
=
Kt-i
+
aforementioned public signals.
24
7?i,t,
ot-\
=
ot-\
+
1^,0
an d
,
Equilibrium
6.2
The
is
between the economy of
essential difference
the endogeneity of information:
information about 8
the strategy agents follow in period
(Kt ,at) and hence
contained in
is
t
in Section 5
determines
how much
behavior in periods
affects the agents'
In the absence of policy, this informational externality
on.
and the one examined
this section
t+ 1
not internalized by the agents: the
is
fact that the use of information in the present affects the information available in the future does
—
solution to Vk (k,k,0,6)
—
Thus, letting once again k(0),
not alter private incentives.
and a
Proposition 9 The equilibrium
=
kq
+
+
(«i
K2)9 denote the unique
—VkKJVkk-, we have the following result.
strategy exists
and
unique. Let {^j,^t} i=1 denote the (unique)
is
Then, for
information structure generated by the equilibrium strategy.
all
t,
the strategy
and
the
information structure jointly satisfy
= E[K(6) + a-(Kt
kt(ut )
for
all u) t
e Hi, with
K
t
= JQ k
(4> t )
1
t
(to
)
d(p t {uj')
for
(ct>t
all
)-K(9)) \ut
$
<E
<j) t
t
(15)
]
-
This result does not require the information structure to be Gaussian. However, once we
8
and the exogenous noises
signals of past activity
available in
is
to be Gaussian, this result ensures that the information contained in the
also Gaussian. All the information
any given period can then be summarized
and the other
for
restrict
two
in
— exogenous and endogenous—that
sufficient statistics,
one
for
is
the private
the public signals; the dynamics of these two statistics admit a simple recursive
structure and the equilibrium strategy reduces to an affine combination of the two.
Proposition 10 The equilibrium
strategy
h,t (ui,t)
=
is
given by
k
{ltXi,t
(1
-
Tt
y
+
It) Yt)
with
•*
The
variables
Xij and
= (l-a)TTf
n
^
+7lf
and public information about 8
Yt are sufficient statistics for all the private
that is available to agent
i
in period
while nf
t,
and
tt\
are their respective precisions.
The
sufficient
statistics are given recursively by
Xi,t
=
(J-j
—^-Ai.t-i + —r x
7rf_!
7Tf
i,t
Y=
and
t
iff
a'}
a~ 2
7
y-Yt-i + -%- yt + -By-A
t +
y
-k\_
—
vrf
iri
where
Vt
=
K _i
t
-
k
-.
—+
{ki
+
(«1
25
k2 )
v
(l
K2 )7i_l
ir
t
- lt-\)Y -i
t
2
t
-i («i
+ ^2? g£?
y
_
—Vt
(17)
IT?
,
/1Q
(18)
and
Similarly, the precisions irf
a linear transformation of the signal of past activity.
is
are
ir\
given recursively by
TT?=Kti+°x
2
t
^=
and
Finally, the initial conditions are Xifi
The
logic
behind condition (16)
is
—
0,
the
7rf_ 1
l^o
same
+a- 2 + a- 2 +
=
t
Mi 7o
t
—
as the one
0, tTq
(
+AC 2 )
Kl
=
and
2
2
7t
=
7Tq
we encountered
-i^
o~
2
4
(19)
-
.
e
in the static
benchmark.
For given degree of complementarity a, the relative sensitivity 7* of the equilibrium strategy to
private information increases with the precision of private information and decreases with the
a
precision of public information. At the same, for given precisions, a higher
tilts
the equilibrium
strategy away from private information and towards public information, as agents find
it
optimal
to better align their choices.
Conditions (17)-(19), on the other hand, describe the endogenous evolution of the information,
which can be understood as
noise, aggregate activity in period t
K -i =k
t
Y ~\
Because
t
K -\ +r]
t
t
now note
is
publicly
in period
t is
First note that, because
follows.
—
1 is
+
(«i
known (and
is
given by
+ K2
)
= 0+
7
—
simply a Gaussian signal with precision
be combined
in the sufficient statistics
averages of
all
+
(lt-10
(1
-
Y -\)
7t-i)
t
k-i
.
and
7t-i), observing
t
7f t
in the sufficient statistic
Y
t
.
=
2
7 _j
Xi
t
But
Vt,
r
t,
(k\
+
while
K2)
all
o~~ t
.
It follows
that
all
private
public signals can be combined
Condition (17) then states that these
statistics are
simply weighted
the available signals, with the weights dictated by the respective precisions of these
signals, while condition (19) states that the precisions of these statistics are
precisions of the
component
The key property
increasing in 7t_i.
23
to notice
This
is
simply the sums of the
signals.
pends on the strategy followed
is
K -\ =
informationally-equivalent to observing the variable y t defined in (18).
that
signals can
it
so are the coefficients kq, k\,
Vt
which
Xi -\ equals 6 plus idiosyncratic
is
that the precision of information available in one period de-
in previous periods.
In particular, for
all t,
7Tj
and thereby nf
is
because the informative content of the signals of aggregate activity
higher the more sensitive the strategies of the agents to their private information. 24 This
important informational externality that the equilibrium
fails
is
an
to internalize in the absence of policy
intervention.
23
When
9 changes over time, the period-t precision of information regarding the period-t fundamental need be
monotonic over time; but
24
A
it
remains an increasing function of the sensitivities of past strategies to private information.
similar property typically holds in rational-expectation-equilibria models:
price increases with the sensitivity of individual asset
demands
26
the information contained in the
to private information.
,
and policy
Efficiency
6.3
We now
turn to the policy implications of the aforementioned informational externality by charac-
terizing the strategy that
maximizes ex-ante
taking into account this externality. However,
utility
we now
unlike the cases of exogenous information examined in the previous section,
restrict at-
tention to strategies that are linear in the available private signals. Without this restriction, the
endogenous signals are no longer Gaussian and the analysis becomes intractable.
Suppose,
for
follow in period
t
a moment, that the government
affects the
information available in subsequent periods. Suppose further that the
summarized
period-i private and public information were
and
respective precisions nf
7rf
,
so that
^N =
E
and
let
2Vfc/<-
k* (6)
= argmax K V(k, k, 0,6) =
+ VKKJKYkk +
Vera)-
to recognize that the strategy the agents
fails
in sufficient statistics
Xi^ and
Vj with
^ ^,
+
+
k*q
(k*
+
k%)9 and a*
=
—
1
W
vo i/Wdi s
=
Proposition 8 would then imply that the efficient strategy
1
is
—
{V^k
+
given by
with
(l-«*)7Tf
,
1
Now
+ 7^
(1-0*) 71?
suppose the government takes into account the endogeneity of the information. As long
as welfare in the subsequent periods
is
increasing in the precision of available information,
be desirable to adjust the current use of information so as to induce more learning
periods.
Because more learning
is
it
should
in subsequent-
achieved only by the aggregation of private information, this
suggests that the informational externality raises the sensitivity of efficient strategies to private
information.
The
following result verifies this intuition.
Proposition 11 The
linear strategy that maximizes ex-ante utility
Kt Kt) =
k*
(
7£jfM
+
is
given by
- tT) Yt )
(1
where
(l-a*)7i?
It
(1
for
all
all
t
< N,
the private
-
a*)
while 7^*
7T?
=
(1
+ n? - P (1 —
a*)
it
x
n/
[(1
- 7£ x ) 2
a*) (l
—
a*)
ir
x
N+
Trfyrf
7r^]
and public information about 6 available
to
.
Xi
{^y
t
t
agent
and
i
2
(«J
Y
t
+
k*)
2
2
a~ t+1
are sufficient statistics for
in period
t,
while
ir
x
and
their respective precisions; they are obtained recursively using (17)-(19), replacing (7t,Ko>
with (7t**, Kq, Kj, K2)
27
irf
are
K i> K 2)
The key
result here
that, holding constant the current precisions of private
is
information, the optimal weight on private information
higher than
is
what
it
and public
would have been had
information in the subsequent periods been exogenous:
**
7i
As anticipated,
on private information
mation and hence higher welfare
The
in
<**)*?
(l-a*)n? + nl
from the internalization of the informational externality: by
this follows directly
raising the reliance
>
in
one period, society achieves higher precision of infor-
subsequent periods. 25
following alternative representation of the optimal strategy helps translate the result here
terms of the degree of complementarity that the policy must induce in equilibrium.
in
Proposition 12 Consider
mation structure.
=
a*^
a*
efficient strategy
fc?(wt)
for almost
As
u €
t
in the case
how much
own
all
and
let {fi t
There exists a unique sequence {a£*} t=1
such that the
,
the efficient linear strategy
fit,
with
=
Kt(<f>t)
and
E[ k* {9)
=
Jq k%
,$ t } i=1
<
with a^*
,
be the associated infor-
a* for
all t
<
N
the information structure jointly satisfy, for all
+ a? (K
(&) - k*
(a/)
for
t
d(f> t {ui')
all
(9t ))
</> t
e $t
|
u
t
and
t,
(20)
]
.
without informational externalities, the weight a*t * in condition (20) summarizes
society would like the agents to factor their expectations of other agents' choices in their
choices.
Unlike the case without informational externalities, this weight
now depends on
the
information structure. Nevertheless, condition (20) remains a valid and insightful representation
of the optimal strategy: the result that a*[*
informational externality
is
<
a* highlights that having the agents internalize the
isomorphic to having them perceive a lower complementarity in their
actions than the one they should have perceived
had information been exogenous. This
in turn
guides policy analysis: the optimal linear strategy can be implemented with similar tax schemes
as in the
benchmark model, but now the
must be higher than what
it
Corollary 5 Informational
sensitivity
T^k
of the marginal tax to aggregate activity
would have been with exogenous information.
externalities
unambiguously contribute
to a
higher optimal sensitivity
of the marginal tax to aggregate activity.
This result
is
true irrespective of the specific payoff interdependencies and irrespective of
whether the equilibrium would have been
it
easily extends to richer
25
Of
hold at
=
had information been exogenous.
Moreover,
Gaussian information structures, with multiple private and public signals
course, this informational externality
t
efficient
is
absent in the very
N.
28
last period,
which explains why the
result does not
on two properties:
of aggregate activity. It relies merely
by increasing the
learning
We
is
that a higher
(i)
sensitivity of actions to private information;
positive because the optimal policy removes
any
and
(ii)
T^k induces more
that the social value of such
inefficiencies in the use of information.
further discuss the importance of this last point in the next section, where
the policies
however,
we have
inefficiency
we study how
Before turning to this issue,
identified affect the social value of information.
we study the
learning
properties of optimal policy for a special case of interest: economies in which
emerges only because of informational externalities.
This special case
is
motivated by the following observations. In certain settings
may
economies with no externalities), one
or near perfect, coincidence of private
and
(e.g.,
Walrasian
expect that competitive market forces achieve a perfect,
social payoffs.
Although such a coincidence may
fail
to
obtain in the absence of complete markets or in the presence of other market distortions, this case
same
represents an important benchmark. At the
still
expect that individual market participants
fail
time,
when information
to internalize
how
is
may
dispersed, one
their choices affect the quality of
information contained in financial prices, macroeconomic indicators, and other endogenous signals
of the underlying fundamentals.
information
and
is
endogenous but where
(k,ch)
=
and
(k*,cy*), so that private
social payoffs coincide
emerges only because of informational externalities.
inefficiency
The
In our set up, these settings correspond to economies in which
following
is
then an immediate implication of Corollary 5 along with the fact that, for
these economies, the optimal tax would be zero had information been exogenous.
Corollary 6 In economies
ties,,
in which inefficiency emerges only because of informational externali-
the optimal policy is such that
This result
may be
T^k >
0.
relevant for understanding optimal policy over the business cycle. Consider
standard real-business-cycle models
in
which
all
firms
regarding aggregate productivity and taste shocks.
and households share the same information
The assumption
of frictionless competitive
markets along with the absence of direct payoff externalities then guarantees that the equilibrium
business cycle
is
efficient.
Now
consider a small, yet realistic, modification of these models:
information be dispersed and only imperfectly aggregated through prices and macro data.
modification
is
likely to
render the business cycle inefficient as agents
choices affect the information of others.
Our
fail
to internalize
how
results then suggest the following policy
let
This
their
remedy:
by having marginal taxes increase with realized aggregate macroeconomic activity and decrease
with realized productivity, the government can improve the information contained
macro data, and can thereby reduce the non-fundamental component
fluctuations that are driven
by noise
of the business cycle
in information regarding aggregate
conditions).
29
in prices
demand and
(i.e.,
and
the
productivity
Implications for the social value of information
7
Throughout our
if
analysis,
we have
ruled out policies that convey information to the agents. However,
the government possess information that
is
not directly available to the market
nomic data collected by government agencies such
Bureau, or the Federal Reserve Banks), then
desirable to
communicate such information
it
as the
is
Bureau
of
Labor
(e.g.
macroeco-
Statistics, the
US Census
important to understand whether
to the market.
Indeed, as long the equilibrium use of information
is
A
positive answer
is
an increase
inefficient,
available information can have a detrimental effect on welfare. For example,
if
it
is
socially
not obvious.
in the precision of
a >
may
a*, agents
overreact to any additional public information, exacerbating the already excessive non-fundamental
volatility.
is
However, this can not be the case
if
the equilibrium use of information
is efficient.
This
because the equilibrium strategy then coincides with the solution to a planning problem where
the planner directly controls
to Blackwell's
how
agents use their available information.
An argument
analogous
theorem then guarantees that additional information can not reduce welfare. 26 The
following result
is
Corollary 7 In
store efficiency
then a direct implication of these observations.
general,
iji
more
precise information can reduce welfare.
However,
policies that re-
the decentralized use of information also guarantee a positive social value for
any
information disseminated by policy makers or other institutions.
In an influential paper, Morris and Shin (2002) used an elegant example to illustrate the
possibility that
more
precise public information can reduce welfare: a "beauty-contest" game,
the strategic complementarity perceived by the agents
causing overreaction to public news.
of transparency in central
in isolation
27
is
where
not warranted from a social perspective,
This example has lead to a renewed debate on the merits
bank communications. 28 Whereas
this question has
been studied largely
from other aspects of policy making, our results indicate that a central bank's optimal
communication policy
is
far
from orthogonal to the corrective
In a related but different line of reasoning,
Amador and
roles of
monetary and
fiscal policies.
29
Weill (2007) argue that, by crowding
out private information, an increase in the precision of exogenous public information can reduce
the precision of the endogenous information contained in prices and other indicators of economic
26
For further details on how the welfare effect of additional information depends on the inefficiencies,
if
any, of the
equilibrium use of information, see Angeletos and Pavan (2007).
27
Their example
inefficiency
28
is
nested in our baseline static framework with k
emerges only under dispersed information and manifests
See, for example,
Amato and Shin
Heinemann and Conrand
Woodford
29
An
(2006), Angeletos
=
k* and
itself in
a >
a";
it
is
an economy where
excessive non-fundamental volatility.
and Pavan (2004, 2007), Baeriswyl and Conrand (2007),
(2006), Hellwig (2005), Morris
and Shin (2002, 2005), Roca (2006), Svensson (2005),
(2005).
exception
is
Baeriswyl and Conrand (2007), which focuses on the signaling effects of monetary policy.
30
activity
and can thereby slow down
Shin (2005) and
Amato and
social learning.
30
A
theme
similar
is
explored in Morris and
Shin (2006). Our results imply that the government can improve the
informational content of prices, can raise the speed of social learning, and can guarantee that any
public information
it
disseminates
is
welfare improving, once
it
sets in place policies that correct
the underlying inefficiency in the decentralized use of information.
Concluding remarks
8
—such aggregate productivity
highly dispersed in
conditions or the profitability of available technologies —
Information about commonly-relevant fundamentals
and demand
society,
is
is
only im-
perfectly aggregated through markets, and can not be centralized by the government or any other
As
institution.
ized
first
emphasized by Hayek (1945),
market mechanisms
necessarily
mean
for
an
this
means that
effective utilization of
such information.
and
social incentives in the use of
the equilibrium's response to certain sources of information
The key
on decentral-
However, this does not
that the government should not interfere with the decentralized use of informa-
tion: to the extent that private
non-fundamental
society has to rely
volatility, excessive dispersion, or
such information do not coincide,
may be
inefficient,
leading to excessive
suboptimal social learning.
contribution of the paper was to identify policies that correct such inefficiencies: by
appropriately designing the contingency of marginal taxes on aggregate activity, along with other
properties of the tax system, the government can manipulate the incentives the agents face in using
different sources of information
and can thereby improve welfare even
new channels through which information
disseminate information or create
We
if it
cannot
is
itself collect
and
aggregated in society.
established this result within an abstract but flexible framework in order to highlight the
potential generality of the insight.
Of
course, the details of the optimal contingency will
the details of the application under examination.
news and excessive non-fundamental
volatility,
If
as
the key inefficiencies are overreaction to public
it
is
often argued to be the case for financial
markets, then marginal taxes must increase with aggregate activity.
inefficiency
is
depend on
The same
is
true
if
the key
the failure of markets to internalize the endogeneity of the information contained
in financial prices
and macroeconomic data.
so that agents rely less on
common
In both cases,
it
is
desirable to provide incentives
sources of information; this can be achieved by introducing
a positive contingency of marginal taxes on signals of aggregate activity.
The opposite
policy
prescription applies to markets exhibiting overreaction to private information and excessive crosssectional dispersion. Nevertheless, the key principle
on aggregate economic conditions
—remains valid
—the optimality
for
of marginal taxes contingent
any economy featuring dispersed information
regarding commonly-relevant fundamentals.
30
Amador and
Weill (2007) extend Vives (1993, 1997) to situations with both private and public learning. Both
models are nested
in
our analysis of Section 6 as special cases that rule out payoff externalities.
31
Examining the
practical, political-economy, considerations that
tion of explicit aggregate contingencies in the tax code
However,
it
is
is
clearly
might complicate the introduc-
beyond the scope
of this paper.
important to note that there are various direct and indirect ways through which
such contingencies can obtain in practice.
For example, the government could
first collect
non-
contingent taxes and subsequently make rebates whose magnitude depends on realized aggregate
activity. Alternatively, the
tax code could be revised over time on the basis of past macroeconomic
performance. To the extent that such rebates or revisions are systematic, they can have similar
incentive effects as the type of contingencies envisioned in this paper.
Moreover, the contingency of monetary policy to macroeconomic performance could serve a
similar role:
how
firms use different sources of information
tion choices depends
when making
on how they anticipate monetary policy
to
their pricing
and produc-
respond to the information about
aggregate employment, output and prices that arrives over time. For example, to the extent that
higher interest rates have similar incentive effects as higher taxes, the Central
contingency of
its
interest-rate policy
on aggregate economic conditions, not only
stabilization purposes, but also to improve the information that
data. Further exploring
how
design of monetary policy
is
Bank can use the
the policy objectives
we have studied
is
32
contained in prices and macro
in this
a fruitful direction for future research.
for the familiar
paper
filter in
the optimal
Appendix
9
Proof of Proposition
We
1.
respectively, the complete-information equilibrium
results to the case of incomplete
Step
1.
any given
0.
but
common
U
first-best allocation.
Step 3 extends the
information.
i
is
pinned down by the first-order condition
quadratic in (k,K,9), the first-order condition
is
Uk
Aggregating across agents
(0, 0, 0)
integrating over
(i.e.,
with
(21)
K
(22),
and
is
[Ukk
Uk
(ki,
k, for
K,9{)
=
equivalent to
+ Ukk ki + UkK K + Uke 9i =
Uk (0, 0, 0) +
Combining
and the
and 2 characterize,
1
Consider the complete-information equilibrium. Because payoffs are concave in
K the optimal action for agent
Because
Steps
prove the result in three steps.
0.
(21)
we thus have that
9i),
+ UkK
K + Uke 9 =
]
0.
(22)
letting
C4 (0,0,0)
Uke
°--(ukk + ukK y M -=!w
Uk e
K2
~-(ukk + ukK
,,„*
(23)
Ku
)
gives the result.
Step
2.
A
Next, consider the first-best allocation.
of a strategy k
x
:
H
—* R and
feasible allocation consists of
a system of budget-balanced transfers across agents. For any
given cross-sectional distribution of productivities h G
let
K{h)
= J k {6; h) dh (9)
and dispersion of activity
and ak (h)
=
(J
[k {9;
w(k;h)=
strategy
k.
An
allocation
w(k;h). Because
U
is
2
- K{h)} dh
veil of
is
thus efficient
quadratic, and
(6>))
if
and only
if
Ua (k,K,a k ,9) = Uaa a k
,
denote the Lagrangian
the dependence of k,
( I k{9)dh
for this
(24)
(6)-k\-ix[I
problem. Optimizing
B
\k[6)
L with
[k
h.
- (Uaa
respect to
-U9k 9 + X =
+ Ukk )+fi =
33
(9)9}
[9)
H
—
>
K
maximizes
that
dh
Then
- K} 2 dh
Uke 9 -X-2fi \k(9) -K\ =
Uk (K, K, 9) + UK [K, K, 9)
x
:
we then have
and a k on
=w-A
R,
let
the strategy k
K
L
*
ex-ante utility depends only on the
in transfers,
+U kU
we dropped
H—
be the corresponding mean
Next,
w = U(K,K,0,0) + \{Uau + Ukk )4 + -Uoeo e2
for simplicity
1/2
x
:
ignorance (equivalently, welfare under an utilitarian aggre-
l
where
and any given strategy k
U{k{9;h),K{h),a k (h),9)dh(9)
Because of the quasi-linearity of payoffs
gator).
h)
H
in the cross section of the population.
denote ex-ante utility behind the
a combination
(9)
- K9
let
-
a\
K, a\ and k(9) we have that
Combining, we have that
Uk6 9 + Uk (K, K, §) + UK (K, K, §) - Ukg 6 +
{Uaa
+ Ukk
[k{6)
)
- K] =
(25)
or equivalently
Uke 6 + Uk (Q, 0, 0) + UK (0, 0, 0) + [UkK + 2UkK + UKK ]K + UKB + {Uaa + Ukk
Integrating over 6
[HO)
)
- K(h)}
(26)
we then have that
= Uk (0,0,0) + UK (0,0,0)
{Urn
+
^
Uke) g
Substituting (27) into (26), and letting
.
"°
_ Uk (0,0,0) + t/fc (0,0,0)
" -(C/ + 2Cfc* + C/k/c)
Kl
~ ~
'
fcfc
^
(Ukk
+ Uaa)
~Uk e + UK6
K2
^ m
*
~ -{Ukk + 2UkK + Ukk)
'
,
[
'
gives the result.
Step
Now
3.
suppose that information
common,
the aggregate activity
for agent
i
is
commonly known
also
is
in equilibrium.
+ Ukkh + UkK K + Ukg E[0i\P] =
(0, 0, 0)
V denotes the commonly-available
agents,
incomplete but common.
+ pkk +
(o, o, o)
UkK)
E^T
7
]
is
and 6 have been replaced by
action for agent
i
is
E[5j|7 ?
similar
is
first-order condition
]
and Ef^lT3 ].
It is
is).
Aggregating this across
(21)
0.
and
(22),
except for the fact
thus immediate that the equilibrium
given by
ki
A
information
simply E[#|P], we get
k + u^ne^v] =
Note that the above two conditions are identical to conditions
0i
The
all
0,
information set (whatever this
and noting that the cross-sectional average of
uk
that
Since
thus given by
Uk
where
K
is
argument implies that the
efficient action is
=E[K*(0
ki
which completes the proof of the
Proof of Proposition
2.
= E[K(6i,0)\P].
i
given by
,§)\P],
result.
We
prove the result in two steps. Step
1
proves that condition
(4)
characterizes any equilibrium. Step 2 proves existence and uniqueness.
Step
strategy
Take any strategy k
1.
k'
:
Q— K
>
such that,
:
Q.
for all
— R
u>,
>
and
let K((j>)
k' (u>) solves
(u) dip
(to)
.
A
best-response
is
a
the first-order condition
E[Uk(k',K(<l>),e)\u]
34
= Jk
= 0.
(29)
Using the
Uk (k',K,9) = Uk (K,(9
fact that
where k stands
for all (9,9)
E[Ukk
or equivalently k'(u>)
Step
2.
,
,
,
+ Ukk
-
(k'
+ UkK
k{9,9))
—
-
(k'
(29) reduces to
,
k(0, §))
+ a (K
E[k(9, 9)
+ UkK (K
(<f>)
—
{$)
k(9,9))
- k
u>].
|
(8,
= 0,
$))\u]
What remains
to prove
is
that the equilibrium exists and
the class of payoff functions
U
that satisfy the properties specified in Section
=
(U,
Q,Q,F)
.
By comparing
equilibrium strategies for the economy e
an economy
e'
=
efficient strategy for
measure-zero set of
=
conditions (4) and (10),
(U, 0,
fi,
U
use of information. Let
efficient
denote
An economy
2.
immediate that the
it is
F) coincides with the
is
set of
set of efficient strategies
(f/',0,fi,F) such that the k* and a* corresponding to e' coincide with the
k and a corresponding
any economy
uj.
K{4>)=
fact that Ki
U
to e. Moreover, U' £
e'
as long as
U
with U' £
exists
U
I"e[
=
From
1.
k{9 ,9)
—
k,2(1
(4),
G U. As shown
and
follows that an equilibrium for the
It
Proof of Corollary
<i>
\
]d<p(u))
+a
in Proposition 3, the
uniquely determined for
is
economy e
exists
we have that aggregate investment
- ok(9 ,8)
K
f E[
(<j>)
|
and
is
all
but a
unique.
satisfies
u]d4>{u).
a)/a,
k(9,9)
It follows
all ui,
unique; this can be done
is
which characterizes the
Using the
k(u) for
(4).
3,
for
=
In equilibrium, k'(u>)
with the help of Proposition
given by e
(K-k(O,0)),
the complete-information equilibrium allocation and the fact that k solves
for
Uk (K{9,6),K{6,6),6) =
which gives
9) k{9 9) 9)
,
- olk%6) =
(1
-
a)K
+
ki6.
that
K{(f>)
=
-^Ko + KiE +a
1
(1
[e[K(<P)
u]d<j){u)
I
and hence that
E[K((j>)
Iterating
\u]
= {l-a)K + KiE[E
and then substituting
1
\(j]
+
aE[ fE[K{^>)
|
J
]d(f>(oj')
\
u
].
into (4) gives
oo
k(ij)
= E{K +
K1 9
n
+ K J2 an E
1
w],
(30)
I
n=l
which together with K\
—
Lemma
1.
Proof of
K2(l
A
—
a) /a and the definition of 9 gives the result.
Taylor expansion of U(k, K, a k ,9) around
(k,
K, a k ,9)
gives
En - E[UCk,K,ak ,9) + Uk Ck,K,a k ,9)(k-k) + UK (k,k,a k ,9)(K-K) + Ua (k,K,a k ,9)(a k
+\ukk {k - kf + ~UKK (K -
2
l
K) + -Uaa {a k - b k f +
35
UkK (k - K)(K - K)
].
-a k
)
a
By
a
the law of iterated expectations,
=
E[K\h] =E[E[K\e,h]\h]
=
It
follows that
=
E[K]
,
E[E[k(u)\6,h]\h] =E[k(6\h)\h]
=
K(h).
E[K] and that
E[Uk (k, K, a k ,9)(k Furthermore, because
E[E[E[k(u)\<f>]\e h]\h]
U
linear in o\,
is
Ua {k,K,a k ,9)(a k -
= E[UK (k, K,
k)]
k ,6)(K
-
Ua (k,K,a k ,8) = Uaa a k (a k ~
-Uaa (a k -
ak ) +
ak ) 2
=
K)}
0.
<?>) It follows that
= -Uaa
[a k
- ak
]
Next, note that
-
{k
Because E[k]
k)
=
2
=
E[K]
[(k
=
-K)=
E[k]
(k
-
+ (K- K) 2 + 2{K -
K)} 2
E[K] and because {K
- K)
is
K)\{k
-K)-(k- K)}.
orthogonal to
\{k
-K)-(k- K)},
we
then have that
-
E[(fc
k)
=
2
}
-K)-
Var[(fc
- K)} + Vai[K -
(k
K).
Finally, note that,
E[(fc
because (k
— K)
Combining
- K){K is
all
Eu
=
= E[(K - k) 2 +
K)\
}
the above results,
we thus have that
l
k ,6)}
2
+ -Uaa
l
[ak
- ak2 + ^Ukk Vac[{k - K) ]
(k
-
K)]
1
+ -Ukk Vzx[K -K}+ -Ukk Vm{K -K) + UkK V&r(K Using the
}
[K — k).
orthogonal to
E[U(k, K,
- K)(K - K)\ = E[[K - K) 2
E[(k
K).
fact that
Var[(fc
-K)-
(k
-
K)]
and rearranging, then gives the expression
Proof of Proposition
3.
A
=
- K] -
Var[&
- K] =
a\
-
b\
in (8).
strategy
Eu=
Var[fc
is
efficient
if
and only
if it
maximizes
U(k(u),K(cj>),a k (cP),6)df(cj,6)dnf),
Jf Jn,e
with
K(<f>)
=
J^
distribution over
k(u)d(f> (lo)
D,
and
generated by
<y k ((j))
/.
=
The
[Jn [k(uj)
strict
—
2
K((f>)]
d(fi
(to)]
with
(f>
denoting the marginal
concavity and the quadratic specification of
36
U
ensures
that a solution to this problem exists and
and
distribution of
distribution
J-
Z
for
almost
all u>.
G
Let
denote the distribution of 9 conditional on
(6\(j>)
The Lagrangian
'.
unique
is
(<f>)
denote the marginal
by their joint
as implied
<j),
problem can be written as
for this
A = UIq In U(k(u), K(cj>), uk {<l>),e)d<t>{u)dZ{d\(j>)dG{<j>)
+ /4 \(<f>) [K(4>) - Jn k(u>)d<j> (u)] dG(<f>)
~ fniH") ~ K(<t>)M (w)] dG(cf>)
+ U M)
HM
Therefore, the
and
first
order conditions with respect to K(4>}, a k ((j>), and
sufficient for optimality, are given
by the
following:
(31)
for
almost
Ua (k(u), K(4>),a k (cP),6)d4> (_j) dZ{9\4>) +
all
X(4>)
- 2r,mk
(w)
-
almost
Using the facts that Ujc(k,K,6)
almost
Ua (k,K,a k ,9) = Ua acr k
(j>)
UK (K(<j>),K(<f ),8)dZ(e\4>) = -UK (K
)
/
= -\Uaa
Substituting the above into (33),
it
satisfies
where,
_____
ment
note that,
is
when
given by
Proposition
1,
K
all
=
all us
£
fi
strategy k
Jq k
(u>) d<f> (u>)
,
and
(<f>),
K(4>), 6)
:
Q— K
>
is
and only
efficient if
-
fc
on
w and
of
(34)
K
on
<f>.
Because both
can be rewritten as
+ UkK (K -
+ Uke (9 + (UkK + UKK )-(K-K*(9,9)) + Ucra (k-K)\u>} =
(k
k*(9, §))
if
:
the dependence of
linear, condition (34)
Uk {K*{6, e),K*(9, 9), 9) + Ukk
+UK (K*(9,6),K*(9,9),9)
Now
=
ui.
Uk (k, K, 6) + UK (K, K, 9) + Uaa [k -K]\u] =
Uk{k,K,8) and UK(k,K,9) are
E[
we conclude that the
we have dropped
for simplicity,
u
all
.
the following condition for almost
E[
=
conditional on
Je
7]{(p)
<f)
conditions (31) and (32) reduce to
,
= -
(9,
linear in its arguments, that K{<j>)
is
all
(33)
denotes the cumulative distribution function of
H4>)
=
K(</>))]dP(0,<l>\Lj)
for
that
{4>)
(32)
/9x , [Uk (k(u>),K(<l>),0) -
cf>\ui)
<f>
2r1 {4>)a k
for
where P(9,
which are necessary
UK {k{u), K(<j>), 8)d$ (w) dZ{6\<t>) + A(0) =
Je In
IQ Jn
k(u>),
k*(S, 9))
9)
(35)
0.
agents follow the first-best allocation, then in each state aggregate invest-
k*(8,8). Replacing k*(9,6~) and k*(9,8) into condition (25) in the proof of
we thus have
that the
first
best strategy solves
Uk (K^9j),K*(9,9),9) + UK (K%9j),K%l9),9) + Uke (9-9) + (Uaa + Ukk )[K*(9,9)-K*(9,8)} =0
(36)
37
a
,
.
Substituting (36) into (35) and rearranging gives (10).
Proof of Corollary
Part
2.
degenerate" information structure T, given n
unique solution to (10)
is
if
and only
a=
=
k*
(2)
and
(3):
for
any "non-
the unique solution to (4) coincides with the
,
a*. Next, consider parts
(ii)„
and
(hi).
When
information
Gaussian and shocks are perfectly correlated
=
n
+
(ki
+ k2
K =
K
+
(rti
+
k
K=
=
k
It
if
from Propositions
follows
(i)
+
k
p
) [if
fj,
K 2 ) bVM
(«i
+
k2 )
+ 7j,y + 7iz]
+
+ 7x)0 +
(ly
[7M /i
+
(7y
7y £]
+ 7X
,
0]
)
follows that
vol
The
=
from
results then follow directly
Proof of Proposition
U
+ k2
[(kj
7y
(6)
]
0^ and dis
and
K, ak
,
9, 6)
=
[(ki
+
2
k, 2 ) -y
x}
<j\
(12).
Given any policy
4.
(k,
2
)
T
6 T,
= U (k, K,
k ,8)
let
-
T(k, K, a k 6)
,
denote an agent's payoff, net of taxes.
Now
offs
let
k
x
:
H
are given by U.
—
>
R
Because
Uk (k(9;h),K(h),6,6) =
0,
denote the complete-information equilibrium strategy when pay-
U
is
concave in
=
with K{h)
k,
k(9;h) must solve the first-order condition
Jk{9';h)dh{9'). Because
U
is
quadratic in
(it,
K,
9,9),
the first-order condition can be rewritten as
Uk
Integrating over
9,
(0, 0, 0, 0)
we then have
+ Ukk k (9; h) + UkK K (h) + Uke 8 + UkS 8 =
(37)
+ Ukk + Uk K K(h) + Uke + Uk e-
with (38) then gives k
K
=
(9;
Uk {0,0,0,0)
Ukk
(37)
that
154(0,0,0,0)
Combining
0.
+
^fetf
h)
—
k(9, 9)
=
Rq
+
Uk (0,0,0) -Tk
~Uk k - UkK
Uk e
-Ukk + Tkk
-Ukk
Uke + UkS
Ki =
- Ukk + UkK
R\9
+
=
0.
(38)
k 2 9 with
(0,0,0)
+ Tfcfc + TkK
Uke
«i
K-2
38
-Ukk
Uk e - T,ke
~ UkK + Tk k + 7)kK
K\
Using (23) the coefficients (ko,Ri,K2) can then be conveniently rewritten as follows:
-Q-
1
=
«i
:
~
1
Tkk
w TkK
~
k
Jk
(ui) d<i> (u>)
4 °)
m
and
let
Tkk
^-r g
fc
m
~ a ~ urk Tkk ~
2 kK
then gives the formulas in the proposition.
Next, consider the game under incomplete information. Take any strategy k
K{4>) —.
(
T-^-^i
)
1
Ukk — — 1
k
(l-q)( Kl+ « 2 +
_
Normalizing
w
.
A
best-response
is
a strategy
k'
— R
fi
:
>
such that, for
:
— R
fl
>
all to, k' (uj)
solves
the first-order condition
E[Uk (k', K(cf>), 6, 6)
Using the fact that
and the
fact that
Uk {k'
,
=
0.
(42)
= Uk {k{9, 9),k{9, 9), 9, 9) + Ukk (k' - k{9, §)) + UkK (K-k(9, §))
Uk (k(9,9),k{9,9),9,9) = for all (6,6) (42) reduces to
K,
h solves
\uj}
9, 9)
,
k'{to)
=
E[k(e, 6)
+ a(K
-k(9,9))
{<f>)
\
u]
with
a
TkK
= UkK ~ TkK = +tt
-Ukk + Tkk
l-jfcTkk
UkK_
=
-
-Ukk
In equilibrium, k'(u>)
follows
=
k(uj) for all u,
from the same arguments as
Proof of Proposition
(13).
in step 2 in the
efficient strategy,
a =
only
is
easily
a*,
Ko
=
Kq,
it is
ACi
shown from conditions
=
is
unique
2.
J-.
For the equilibrium
K*,
K2
T that
and
=
«2-
(
satisfies (44)
(43).
and that
First, note that k\
44 )
this policy
—
re|
if
is
and
if
Because
a*
Ukk + Uaa <
if
That
this also guarantees that
l
/K\)
= -Uaa
Ukk — Tkk <
0.
(45)
.
Next, note that, given
Tkk = —Uaa
,
and only
TkK = -Ukk
0.
=
and
necessary and sufficient that
(39)-(41)
Tkk = Ukk {\-n
a
to (13) exists
proof of Proposition
thus suffices to prove that there exists a policy T* £
unique. This
That a solution
Take any generic information structure
5.
with policy to coincide with the
It
which gives
m
'
Tkk and Tk j<
(a
satisfy
-
a*)
- Tkk a* = -Ukk a + (Ukk + Uaa
— $ K ^^ K <
1
)
a*
= Uau - UkK - UKK
then follows from the fact that
With (Tkk ,Tk x) thus determined, and with both
39
Tk §
and
W
Tk (0,0,0)
vo i
.
(46)
= Ukk +2Uk ^+UxK <
being unconstrained,
it
is
a
a
then immediate that there exist a unique
^0
=
T
fc
-
(1
(0, 0, 0)
and that
T to balance
Tkk + Taa — 0.
that this
is
Finally, for
such that £2
—
Tk (0, 0, 0)
^2 an<^ a unique
such that
by
Kq\ these are given
Tk g = Ukk
Tk §
a)
[« + k*2 )
- Ukk
(1
-
a) («S
(«!
+
-
«o)
«a)]
- (T
fe
*
+ TfcK
- {Tkk + TkK )
the budget (state by state)
Along with the other properties
«3)
4 = -UK
= -UKe
(0, 0, 0)
identified above,
it is
(47)
,
.
T (K, K,0,6) =0
must be that
it
+
(kJ
)
for all
(K, 6)
then easy to verify
equivalent to imposing the following:
=
T(0, 0,0,0)
= Tm =
7X0,0,0)
TK (0,0,0) = -T (0,0,0) = UK (0,0,0)
Tkk = -%TkK - Tkk = -Uaa + 2Uk K + ZUkk
fc
TK§ = ~ Tk§ - UK8
=
Taa
This also implies that the policy
-Tkk
T is
unique. Finally, that
Ukk <
a* follows directly from (46) along with the facts that
Proof of Proposition
6.
TkK
the case of multiplicative measurement error,
—
6(1
E[T(ku K,
+
c).
A
and Wdi s
let k\
=
k\{\
do not
=
E[T(ku K,
k
,
6)+\Tkk
(a
2
+ av2
a
Proposition 4 thus continues to hold with k and
K
K = K(l +
+ + Vi),
77
)
k
2
+ Uaa <
0.
<J k
,6)
77),
a\
=
a\(l
For
+ rj) 2
then gives
^TkKG%hK^\TK KolK2 ^\taa c^Gl\
Wi
redefined as follows:
Uk (0,0,0) -Tk (0,0,0)
2
+ Tkk (1--+
-Ukk - UkK + TkK (l + a
Uk6
-Ukk + Tkk (l + a 2 + a 2
UkB ~ Tk g
-Ukk -UkK + TkK (l + a 2 )+Tkk (l +
UkK -TkK (l+a 2
-Ukk +Tkk (l+a 2 + a 2
)
«i
Ukk
decreases with
affect individual decisions.
Taylor expansion of T(ki,K, a k ,6) around T(ki, K,
k ,6)\u)i}
=
a and
For the case of additive measurement error, the result follows
directly from (14) noting that the last three terms in (14)
and 6
increases with
cr
2
+
a2 )
)
-
=
2
=
a
a2 + a2 )
'
)
)
By
implication, Proposition 5 also continues to hold, although
2
depend on a and a
2
,
m
40
now
the optimal tax contingencies
]
Proof of Corollary
Prom
3.
and
(45), (46)
we have that
(47),
the restriction k
=
implies
«;*
that
Tkk =
The tax system
is
TkK = -Ukk {a-a*),
0,
thus linear in
k,
K=
and K.
9
positive,
It follows
which
that
turn
in
is
Proof of Corollary
E
n of 9 (which
because
<f)
is
true
and only
if
Because
4.
in
<p) is
common knowledge and
E =
=
l
Combining these
E\E[6\u]\<l>]
results,
e
Cov(Tk (K,9),e)
measurable
is
and hence
E[A"|/i]
(
+ k2 )
Kl
.
with a marginal tax rate given by
Tk (K, 9) = Tk
Next, recall that
Tk§ = -TkK
and
<j>
also
= K — K is orthogonal to any function of h,
= Tk xVar(e), which is positive if and only
if
is
+ TkK K + Tk §9.
(0, 0)
a >
a*
= E [9\h] — E [9\h,
=E
E\E[9\uj,<t>]\<f>]
is
knowledge, the n-th order average expectation
common knowledge and
because 9
Tk x
if
m
.
common
including
[9\<t>]
equal to
<f>]
,
E
l
,
for all n.
Moreover,
we have that
= E [E [6\h,<f>]
\<f>]
=E[S\4>]
,
we have that
oo
§
= Y,
((1
- a)"""
En = E
1
)
[6\<£\
=E
[S\u]
.
n=\
From
Corollary
1 it
then follows that the unique equilibrium
=E
k (w)
in
which case a
is
irrelevant. Similar
is
also irrelevant.
= E[ K *
Along with the
given by
,
arguments imply that the
k(uj)
so that a*
[k (9, 9) \u]
is
(0,5) \u]
efficient allocation is
,
result in Proposition 4,
strategy can always be implemented with a tax scheme for which
Proof of Propositions 7 and
Proof of Proposition
first
8.
period, that the equilibrium strategy at
Proposition
2.
Now
consider
t
=
2.
t
t
=
—
1 is
for
t
=
2 establishes the result.
41
efficient
0.
is
3.
exogenous in the
unique and solves (15) follows directly from
structure
1.
is
now endogenous but uniquely
That the equilibrium strategy
unique and solves (15) thus follows again from Proposition
>
Tk K =
Because information
1.
The information
determined by the unique equilibrium strategy
t
we then have that the
These are direct extensions of Propositions 2 and
Start with
9.
given by
2.
at
t
Repeating the same argument
=
2
is
for all
,.
Proof of Proposition
=
with Wij
(xi
t
Start with
10.
Xu =
—
k(8)
Ko
+
x iA
(ki
=
1.
In the
period, information
first
exogenous
is
i,yi,Ai). Standard Gaussian updating then gives
E[e\u iA
where
t
,
,
+
=
wf
cr"
2
we then have
«2)^,
i,1
^
=
Yi
,
^X
=
]
+
^
Y
1
(48)
,
!
^i
Trf
=
that the unique solution to (15)
is
h (wi,i) = ko +
+
(asi
+
^fz/i
+
ac 2 )
+
(7i*i,i
and
(1
-
a^"
2
+ a" 2 +
a"?. Using
given by
(49)
7i ) *i)
with
^[(l-aK]/[(l-aK + <].
7l
To
by guessing that the equilibrium strategy
see this, start
with
Finally, use the latter along
Next, consider
signal
is
t
=
(15)
and
Ko
+
(k\
+
some
K2) (71 #
coefficient 71.
+
(1
—
71)
ii)'-
(48) to derive the equilibrium 71.
=
In the second period, 0^2
2.
=
K\
Next, use this guess to compute aggregate activity as
satisfies (49) for
U
u>i i
(xi
t
2,y2,A2,Ki,a\). The endogenous
<
given by
K =
The information about
+ «a)
+
(«i
# contained in
K\
x
k
is
+
(71 ^
-
(1
7i) *i)
+ V2
thus the same as that contained in
*
y -«o-(^+^)(i- 71 )y1 =
+
.
J
1
(«i
where
772
= t/2/[(ki + K2)7i]
is
+
K2J71
Gaussian noise with variance
on the other hand, conveys no information about
which
is
common
knowledge.
It
8,
cr?
2
—
°"n
2/ ( K i
=
(k\
The
+
follows that the period-2 public information about 8 can
signal o\,
K2)
7
2
2
o"
^
be summa-
on (yi,Ai,Ki,ai,y2,A2)
Gaussian with mean
V
^2 =
Cx^
-\y
4
-4*i +
^l
and precision
summarized
^ = n\ +
o-"
2
+ &~ \ + 7 2
,
in the sufficient statistic
7
A +
+ -ir^2
^2
2
*2
is
^2) 7i
because (49) implies that o\
rized in a sufficient statistic Yi such that the posterior about 8 conditional
is
+
(«i
+
K2)
2
2
2
2 .
(*1+«2)X—yi
y
*2
u~\- Similarly, the private information can be
Xj^ such that the
posterior about 8 conditional on
Gaussian with mean
—
X-i.2
—
^i,l H
5" x i,2
IT,
and precision
Trf == 7rf
+
-2
o~
x 2
.
The unique
h (wi,2) = «o +
with 72
=
[(1
-
a)
Trf]
/ [(1
-
a)
Trf
+
tt^]
solution to (15)
(«i
+
«2) (72^i,2
.
42
is
+
then given by
(1
-
72)
Y2
)
(2:;, 1,2^,2)
It is
t
>
immediate that the construction
of the equilibrium strategy for
t
—
We
3 with the statistics X^t an Yt defined recursively as in the proposition.
unique equilibrium strategy
=
7i
[(1
-
a)
J [(1 -
a)
Trf]
Proof of Proposition
conclude that the
is
ki,t (ui,t)
with
2 applies also to any
11.
=
Ko
+
ir
itf
+
y
«2) (ltXitt
+
(1
-
It)
Y
t)
,
m
.
t
+
(«i
]
We prove the result in two steps.
Part
(i)
characterizes the efficient
linear strategy in the absence of payoff externalities; this helps isolate the role of informational
externalities. Part
Part
(i).
(ii)
then extends the result to general payoff structures.
Suppose
V (k,K,a,6)
does not depend on (K,a) and, without any further loss of
generality, let
V (k, K, a,0) = Let ht
—
(k
-
2
6)
.
{yi,A\,Ki,...,yt-\,At-i,Kt-i,yt,At} denote the public history in period
agents follow a strategy
k
=
and suppose
t
{k t }^=1 such that
t
kt {Ui,t)
=
Pi
+ Y^ Qt,rXi,r,
ih)
T=l
where
P
t
(ht) is
a deterministic function of h t and
Q tiT
are deterministic coefficients. It follows that
t
=
ki,t
and hence Kt
=
Pt
+ 7t# + Vt+i,
Pt
where Pt
+ ltO +
^2, Qt,r€i,T,
a shortcut for Pt (ht) and 74
is
is
defined as
t
= ^2Qt, T
It
-
T=l
Next consider welfare. Given any
w =E[v
t
linear strategy, ex-ante utility
(k hU
A t+1
=
))
E[v (h,u
0)}
-
<
is
Eu =
*N
J2t=i
w
t<
where
i+1
and where
E[v(k ht ,9)}
= E E
= E
Now
consider a strategy
{(p +
(P +
t
7t
t,
k = {kt}^ that
pick an arbitrary function
P
t
+
Y!T=1 QtMr)~ey
and any
is
t
a variation of the
and
let ki iS (u>i tS )
coefficients Qt, T
-
Pt (ht)
43
+
—
initial
strategy k
ki tS (u>i iS ) for all s
such that 5Z r =i Qt,r
t
h(ui,t)
,h t
e-ef-J2 ,QlAr
constructed as follows. First, pick an arbitrary
period
itO
t
*}2Qt, T Xi,T-
<
=
=
t.
{kt}t=i
Next, in
7*' an<^
^
P
.
Finally, for all s
>
let
t,
s
k s [Ui
t
=
s)
+ /
Ps [hs)
Qs,r x i,r,
y
r=l
A
where the functions'
are such that
P
By
- ps
(...,/?*,...)
s
^
construction, at any period s
-
[...,kt
implication the same per-period welfare level wt, as the
t
(h t
strategy k.
initial
that, for all
is
+p
k induces the same outcomes, and by
the strategy
t,
condition for the strategy k to be efficient
(h t )
t
and
t
It
follows that a necessary
all ht,
2
(Pt ,(Qt,r)r=i)
argmax=E -(P + lt e-e) -Y^t=1 QIA,t
Pt.Qt.r
e
YX=\
s.t.
This in turn
is
the case
and only
if
Next note
Kt+rjt
that, because
= Pt+ JtO +
rj t
for all
if,
= (l- lt )E[6\h
Pt(h t )
ht
t
P
t
is
t
and
t
it
all ht
Q t<T =
and
}
=
Q't.r
€ T
'
7t
_ 2 Vr.
(50)
public information, the observation in period
t
+
1
of
K
t
=
informationally equivalent to the observation of a signal
is
—
=
yt+i
=
-
9
+
fj
(51)
t+ i
It
where
r)t+\
=
tt\,
where
Y
t
and
posterior about 6 in period
7if
71?
initial
conditions Yi
O
[<
n
= <_! +
=
It^nt+v
^
Gaussian with mean
t is
iiq
and
ttj'
a„ it
+
= c^ 2
a
E
7Tt
=
71"^
-f 71?,
[6\uiit ]
=
ttit
2
with
initial
conditions X^i
+ 7j-i^,i
]
Yt
and
>
Similarly, the private posteriors are
.
-^-^x
i>t
+
-^n,
-2
= -^-Xht -i + -^-x
=
t
2
where
x
A"j, t
E [0\h =
71?
71?
2
mean
and precision
given any
follo"ws that,
are defined recursively by
7Tf
Ty
with
~
Gaussian noise with precision &I t +i
common
linear strategy, the
precision
iS
TJt+i/lt
i^i and
7rf
= ov
iit
j
44
and
nf
=
*?_!
+
a-"
2
,
Gaussian with
Now
note that
a" 2
ax,j
r=l 2-fj=l
which together with
We
(50) gives
conclude that a linear strategy k maximizes ex-ante
Kt) =
ki, t
for
some
coefficient
7's,
,
t>
Eu =
9).
WFB -
when
note that,
=
«l)]
and
t
all
w^,
(52)
Tfe-Xi,t
agents follow a strategy as in (52),
-{^(7r?)-i
^
and hence
is
for all
(
E[w(fc i
Wfb
+
7;) it
if,
7 6R.
To determine the optimal
where
-
(1
only
utility
1
/J*"
+
2
(l-r>i)
2
1
(7r?)-
}
1
+
{ 7i (tt?)-
-
(1
2
7i )
1
(vrf)-
}
,
the first-best level of welfare (the one obtained under complete information about
Because the evolution of nf does not depend on
the choice of the optimal linear strategy
74,
reduces to the following problem:
'"^li/l^wr'+^-^wr
s.t.
with
=
condition 7^
initial
<t7
,
=^ + A
7rf+1
where A4
=
+ a-
i
+ a~
cr~t
2
7
is
t
2
1
}
V<
the exogenous change in the precision of
public information.
Consider the value functions L4
L
t
(n?,n?)
R+ — R+
mm
=
{7.}."=,
s.t.
For
all t
< N, L
.
t
(7rf,7rf)
must
S=t
y
n s+1
74 is
2
7s
«)- +
1
(l-7,)
2
(
rD-
=
tt«
1
7
l
}
J
+ A s + a- 2 7 2 Vs>i
satisfy
s.t.
and hence the optimal
^ ^{
mm {7? (vrf)-
L, (7rf.Tr?)
defined by
>
:
7rf+1
1
=
+
(1
v
7r
t
-
lt
+A +
t
1
f
(Trf)"
^7
the solution to the following
+ /?L t+1
K
2
FOC:
1^5^4 + 1^+!
,K)--(.-,)(.r+^^=».
;
1
.
-1
,
45
,
+1)7r?+1 )}
^From the envelope theorem,
dL t+i
_
K
2
,,
-(l-
7f+1 )
+1
't+
from the low of motion
Finally,
for irf
,
0*1+1
It
follows that, for
-v**
N—
<
all i
=
t
2
+ vrf - fi (1 - T^) Trfvrf (^+1
nff/ (nfj
no more an informational
Part
(ii).
(«;*,,«*,«*,)
as
m
(9)
74,
information.
with
Ex
Eu =
where
Waa
<
WFB
must
=
and
^^
<
0j
'
"?
£
simply because this
,
a- 2+1
is
+
*?
the last period and hence there
V
U
(re, re,
with V.
=
0, 6)
A
similar
re*,)
(ytXit
re*,
let
+
(kj
+
argument
re*.)#
as in part
(i)
ensures that the
=
{u it )
+
k*q
(kJ
+
+
(i
-
7t)
y*;»
4
is
and public
then given by
^
{& +
2
«5)
/?
V7
an d
WKK/Waa =
(re*
(0)
,
0,0)
1
1
(vrf)"
{^f (7,)
+
^
(1
-
is
the first-best level of welfare.
—
a*,
we conclude
2
7,)
W)-
1
}
Using the
mi
£
£?
s.t.
^K
_1
2
,
vrf+1
=
Trf
("f )
+ A, +
+
(re*
(i
+
-
<**) (1
2
re*.)
(tt? .tt?)
= min{ 7 2
s.t.
The
FOC
1
(Trf)"
7rf+1
=
+
(1
ttJVj
-
=
a*)
Trf
(1
-
2
7i
+A +
t
)
- it?
KT
1
t,
we have
KT* + ^+1 (*?+i,*l+i)}
(re*
+
2
re*;)
a" 2 ^2
for 74 gives
7,
«,- -
(1
-Od- W)- +
7,)
46
}
a~ 2 7i2 V*
Letting Lj (Trf,^) denote the associated value function in period
t
fact that
that the optimal 7's must solve the
following problem:
L
is
satisfy
£f=1
t_1
max V
arg
and
Y* are the relevant sufficient statistics of available private
ante utility
Wfb +
Wkk
0,
.Xji
=
(^S) replacing
h
some
ix
Consider now the more general payoffs
efficient linear strategy
for
v
+ N
)
>
"2
)
externality.
re*
with
satisfies
t
11
^=
N,
j
7T
=
rrf
Finally, for
2
.,..
the optimal
1,
)-
^^=0.
The envelope
condition for nf+1 gives
8L
d7T t+i
while the law of motion for
= -(l-^)(l-7t + ifW+1 )"
+1 gives
7r^
=
It follows
2 (kJ
+
k*2 )
2
°J+1 7t-
that the optimal 7's satisfy
**
7
(l-<**)*f
="
tt?
+
(1
-
a*)
Trf
- 0(1 -
a*) (1
- t^) 2 Trf tt?
2
(tiJ^)"
(/cf
+
k*2 )
2
a~ 2t+l
'
which completes the proof.
Proof of Proposition
12.
Let {ft*}u=i be the coefficients that characterize the efficient
linear strategy as in Proposition 11
and
let {7ff,7rJ'} _
f 1
and public information generated by the
letting a£*
The
efficient linear strategy.
result then follows
from
be the unique solution to
(l-ar>f
(l-arw+Tr?
In fact,
be the corresponding precisions of private
it is
7i
then and only then the unique solution to (20) coincides with the strategy obtained
in
Proposition 11.
Proof of Corollary
Gaussian signals studied
12,
where
it
is
shown
7.
Consider the environments with both exogenous and endogenous
in Section 6.
The
that, for all periods
from the proof of Proposition
result follows directly
t,
the present-value welfare losses
the efficient linear strategy are decreasing functions of
public information available in the beginning of period
t.
irf
L
t
and n^ the precisions
,
obtained along
of private
and
Putting aside informational externalities,
the result can also be established for non-Gaussian signals using a Blackwell-like argument for the
planner's problem that characterizes the efficient strategy.
47
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