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Department of Economics
Working Paper Series
Policy with Dispersed Information
George-Marios Angeletos
Alessandro Pavan
Working Paper 07-28
October 30, 2007
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Policy with Dispersed Information*
George-Marios Angeletos
Alessandro Pavan
MIT and NBER
Northwestern University
October
26,
2007
Abstract
This paper studies policy
relevant fundamentals
— such
in
a class of economies
in
which information about commonly-
as aggregate productivity and
demand
conditions
—
is
dispersed
and can not be centralized by the government. In these economies, the decentralized use
information can
fail
to be efficient either because of discrepancies between private
and
of
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
in
itself
macroeconomic data,
of economic activity). In either case, 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 Taxashev, Harald Uhlig, Xavier Vives, Ivan Werning, and
seminar participants at Chicago, MIT, Toulouse, the 2007 lESE Conference on Complementcirities 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 hospitaUty during the last
stages of the project.
"The peculiar character of
the
problem of a rational economic order
is
determined pre-
knowledge of the circumstances of which we must m,ake use
cisely by the fact that the
never exists in concentrated or integrated form but solely as the dispersed
plete
and frequently contradictory knowledge which
The economic problem of society
is
not given to anyone in
1
Introduction
The
dispersion of information
is
...
"
its totality.
is
all the
bits of
incom-
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
For example, think of the business cycle.
demand
conditions
and pricing decisions. Yet, such information
is
is
crucial for individual consumption, production,
widely dispersed and
through markets, the media, or other channels of communication
As emphasized by Hayek
(1945), such information can not
government.
Instead, society
the utilization of such information.
One can then be
stitution, such as the
their available information in the
what best
may
most
Information about
must
rely
it is
only imperfectly aggregated
in 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 crowd out valuable private information and can also amplify the impact
of
common
noise, resulting in higher
to internalize
how
their
own
non-fundamental
volatility.
Furthermore, individuals
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
by appropriately
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
is
the objective
achieved by appropriately designing the
activity.
specific appUcation,
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 complementaxities,
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
this gives the
we
identify tax
schemes that
key policy result of the paper.
detect by comparing the equilibrium strategy with the
aforementioned socially optimal strategy depend on whether the inefficiency originates in payoflF
or informational externalities.
fundamental
In the
volatility (overreaction to
much
common
manifests
itself in
noise in
excessive non-
noise) or excessive cross-sectional dispersion (over-
In the second case, inefficiency manifests
reaction to idiosyncratic noise).
social learning (too
first case, inefficiency
macroeconomic
itself in
suboptimal
indicators, financial prices, or other channels of
information aggregation)
Yet, the
same policy prescription works
the two sources of inefficiency.
Our key
for either case, as well as for
result
is
economies that combine
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 reafized net-of-taxes return on their
activity to increase with aggregate activity.
It
follows that a negative
own
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.
Next note that a better alignment of individual decisions obtains when agents
common
sources of information, whereas
idiosyncratic sources of information.
It
more
differentiation obtains
follows that the
when
rely
more on
agents rely more on
government can use the contingency of
marginal taxes on aggregate activity to fashion how agents respond to different sources of information. In particular,
when marginal
taxes decrease with aggregate activity, by inducing strategic
complementarity the policy also induces higher
mon
information. Symmetrically,
relative sensitivity of equilibrium actions to
when marginal taxes
com-
increase with aggregate activity, by inducing
strategic substitutability the policy also induces higher relative sensitivity to idiosyncratic infor-
mation.
It follows that,
control
how
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
Discussion.
It is
cient fluctuations in
(1936),
who argued
to the market.
often argued that financial markets overreact to public news, causing
both asset prices and
itself
real investment.
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
tuations in asset prices and investment.^
broader theme that markets
may
demands
of one another, thus causing inefficient fluc-
In this paper, although
we
are partly motivated by the
react inefficiently to available information,
tion to any specific apphcation, nor do
we examine the deeper
we do not
origins of such inefficiency (which,
unavoidably, will be specific to the application of interest). Rather, our goal
remedies that need not be sensitive to the details of the origin of the
our choice to work with a theoretical framework that
is
limit atten-
is
to identify policy
inefliciency.
This explains
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 foundations for these limits,
institutional design,
offers
some
is
and
their potential implications for policy
and
a challenging topic beyond the scope of this paper. Nevertheless, our analysis
relevant 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
policies, or other institutions, that
for
how
to use available information also
provide the mai'ket with more information.
Furthermore, while our analysis focuses on how the contingency of taxes on aggregate activity
can improve the decentrahzed use of information, in practice, the contingency of monetary pohcy
on aggregate
activity
might also help in the same direction. Indeed, the more general insight that
comes out of our analysis
is
how
the anticipation of such contingencies affects the incentives agents
face in using their available information,
Methodological remarks.
and how
The pohcy
between two dominant paradigms:
the
this in turn affects efficiency.
exercise conducted in this paper strikes a balance
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 pubfic finance" paradigm
We
deviate from the
Ramsey
(e.g.,
Kocherlakota, 2005; Golosov, Tsyvinski and Werning, 2006).
tradition by introducing heterogeneous information
and by avoiding
^This point was epitomized in 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
is
closer to the
new
set of available policy instruments.
In this respect, our policy exercise
At the same time, we deviate from the Mirrlees
public finance literature.
tradition by abstracting from redistributive taxation (or social insurance)
policies that correct inefficiencies in the response of equilibrium
this respect, our policy exercise
is
closer to the
Ramsey
and focusing instead on
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
assumed to be common. In
contrast,
tastes, talent, or labor pro-
in this literature, information regarding these
we study environments
which agents have
in
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
society.
The key
difference then
is
that the
regarding these shocks generates strategic uncertainty: agents face
uncertainty regarding aggregate activity.
It is
precisely this uncertainty that
of taxes on aggregate activity essential for restoring efficiency
makes the contingency
— which also explains why the results
presented here are, to the best of our knowledge, completely novel to the pohcy 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
little
of a certain commodity.
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 to certain sources of information.
The analogue
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
nomic contexts goes back to Phelps
literature that studies dispersed information in
(1970), Lucas (1972),
expectations revolution of the 70's and early 80's.
Woodford
More
Townsend
recently,
Another
line of
in
and the
rational-
Reis (2000) and
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
negative result, which has then been used to
have found the opposite
Cornand
macroeco-
(2001) have raised interest on the business cycle implications of combining information
work by studying policy
^See also
(1983),
Mankiw and
heterogeneity with strategic complementarity in pricing decisions.^
^See also
activity.
result.
specific models.'^
make a
In Angeletos
Some
of these papers have found a
case against central-bank transparency; others
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).
be explained by understanding the underlying
conflicting results can
inefficiencies in the equilib-
rium use of information. For that purpose, we considered an abstract framework that
restricted
information to be exogenous but was flexible enough to nest most of the applications examined
in the hterature;
we then used
framework to study the
this
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,
in
which information
first
generalized version of that framework for a different purpose:
efficiency in the decentralized use of information
environments
the
is
partially endogenous;
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.
and Amador and
In this last respect, our analysis complements that in Vives (1993, 1997)
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
results identify policies that
exogenous information
Layout. Section
is
of past aggregate activity.
Our
can control the speed of social learning and also guarantee that any
socially valuable.
2 introduces the baseline framework.
Section 3 studies inefficiencies in the
decentralized use of information due to payoff externalities.
Section 4 identifies tax systems that
remove such
inefficiencies.
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 outhne our
In this section
details of
any
baseline framework:
specific application
but
is
flexible
game
a
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
e
imposes a tax
Let
ip
r^
[0,1],
G
R
and the dispersion
each choosing an action
on each agent
let
K = J kdip{k)
and a^
of individual actions.
C/
:
R^ x E+ x
6 —
>
E.
For concreteness, in what follows
shock."
k^
G M. In addition, there
is
a government, which
subject to the constraint that the budget
i,
u,
some
populated by a continuum of agents of measure one,
is
balanced.
denote the cumulative distribution function of individual actions in the cross-section of
the population and
for
is
The
The
we
=
=
[/(A;
-
K)^dt/)(fc)]^/^ denote, respectively, the
The (reduced-form)
payoff of agent
i
is
given by
U{k^,K,ak,9^)-Ti,
variable
0;
G
often think of
6 CM
k^
external and strategic effects exhibited in
5
(1)
represents a shock to agent
as "investment"
U may
mean
and
6^
as
a,
i's
payoff.
"productivity
originate, not only fi-om preferences
and technologies, but
and the
social networks,
specification
also
from pecuniary
What
like.
assumed above without missing any channels
the analysis of the latter
we
payoffs in transfers,
postponed to Section
is
effect of dispersion
Uaaf^k for
To guarantee
efficient allocation,
decision problem;
(jfc,
^)).^
we assume the
—UkK/Ukk <
which imposes concavity
will
be public
is less
uji
its
[/ is
own
a concave quadratic poly-
level
(i.e.,
Ua{k, K, ak,d)
0,
which imposes concavity
at the individual's
+ 2(7^/^ + Ukk <
and U^k
how
are three stages. In stage
1,
let
structure
f G
given hy h £
F
is
3,
actions
</>
as follows. Let
G ^,
and
w,;
respectively.
G
aggi'egate productivity are
we
Q
with marginal distributions
/ as the "state of the world."*
refer to
F
the agents. Nature then uses / to draw a pair
i,
with the pairs
drawn independently from
own
/.
Each agent
or the distribution /. Note, though, that
productivity
0,,
First,
according to the probability measure J^ which
common knowledge among
di
for di
distribution / describes the joint distribution
is
but does not observe either
pubhcly
denote the information (also the "type" ) of
for {6i,u)i),
The
set of possible distributions
{Oi,(^i)i£[o^\]
under dispersed
and the game ends.^
of {6,Lo) in the cross-section of the population;
only about the agent's
0,
taxes will be collected in stage 3 as a function of information that
denote a joint distribution
H and
Nature draws / from a
+ U^a <
the government announces a
at that stage. ^ In stage 2, agents simultaneously choose their actions
Next,
=
at the planner's problem.^
that specifies
The information
and
<
than one; and Ukk
revealed, taxes are paid, payoflfs are realized,
i.
that
and uniqueness of both the equilibrium and the
existence
information (described below). Finally, in stage
agent
we assume
which ensures that the slope of the individual's best response
1,
Timing and information. There
T
of endogenous information aggregation;
depends only on
following: Ukk
with respect to aggregate activity
rule
that payoffs can be reduced to the
Finally note that, by assuming hnearity of
6.
tractability,
nomial and that the external
all (A;, A',
is
oligopolistic competition,
are ruhng out any redistributive (or insurance) role for taxation.
Payoff restrictions. To maintain
pohcy
though,
crucial,
is
monopoly power,
externalities,
i
{6i,iOi) for
each agent
then observes his own w^,
uji
encodes information, not
but also about the distribution / of productivities and
information in the population.
Although most of the analysis does not
*The external
recjuire
any restriction on the information structure.
effect of dispersion is relevant for certain applications: in
new-keynesian models,
e.g.,
dispersion in
relative prices has a negative welfare effect.
^Although the restriction -UkK/Ukk
intervention, our
main policy
result
<
1 is
necessary for the equilibrium to be unique in the absence of government
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 equihbria in the absence of
policy intervention.
^Because the government has no private information, the announcement of
about the fundamentals;
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
*This
is
with a shght abuse of terminology because / does not describe the specific
e
{6i,lui) for
we
3.
each single agent.
we
find
it
useful to impose a certain "regularity" condition. Fix an information structure
and consider any two payoff structures, U^ and U~. Let
— {U^\Q,F,T) and /c^ 17
We say that the information
for the economy £^
£^
=
:
{U^;Cl,F,!F).
^
whenever U^q/UI^,
such that k^(u))
^tel^kk
for
k'^(uj)
7^
^kx/^kk
^'^
any
lu
£
economic meaning:
it
^
fi
:
be an equilibrium strategy
structure {fl,F,J^)
The
and only
if
if,
C Q
appreciated (in terms of
fully
However
characterize the equilibrium.
it
has a simple
requires that different sensitivities of individual best responses to either the
fundamentals or others' activity result to different equilibrium actions
types.
"regular"
is
economy
for the
exists a positive-measure set fi
This restriction can be
we
E
an equilibrium strategy
»
^kx/^kk^ there
t^
Cl.
primitives of the environment) only once
— R
A;^
F,J-)
{fl,
role of this condition
which changes
to rule out trivial cases in
is
for a positive
measure of
do not
in incentives
lead to any change in the use of information.^
To
illustrate the type of shocks
Gaussian example. Agent
q
shock while
IJ,0
i's
and information structures that we
productivity
given by
is
=
9i
^+q, where
an idiosyncratic productivity shock. The former
is
and variance
a^; the latter
with zero mean and variance
across agents
is i.i.d.
ct^.
allow, consider the following
6
is
Normally distributed with mean
is
and independent
Each agent's information
an aggregate productivity
of
Normally distributed
9,
consists of a private signal Xi
uii
=
O^
about own productivity and a public signal y
= 9+e
The
across agents, Normally distributed with zero
variable
and variance
variance ay.
example, /
is
S,i
cr^,
noises
and s are independent of one another
^i
9 and variance
</>
cr^
example
is
over
+
iOi
ct^.
=
As
{xi,y) assigns
it
9i is
= ^+e
—
Aiki
-
equivalent to imposing that the variances
\k1, with Ai
=
6^
+ aK,
K
= f kidi,
and
of
mean and
In this
c^.
9i
=
and
is
Normal
in Xj
with mean
imposing the "regularity" condition
clear in Section 3,
a^ and ay are positive and
CTq,
Applications. The following examples are directly nested
• Ui
as well as of ^
mean
Normal with mean 9 and variance a^ and
measure one to y
become
will
in the market.
public noise, Normally distributed with zero
is
a distribution whose marginal h over
whose marginal
in this
i.i.d.
whereas the variable e
The
is
idiosyncratic noise,
about the average productivity
+ ^i
9
in our
+
<;,,
finite.
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
.
TTi
^i is
=
TT*
effect,
while ^
is
a
.
The
scalar a
common productivity
shock
an idiosyncratic productivity shock.
-
(pi
- p*f
,
with p*
=
a9
+
{I
-
a) P,
P =
J Pidi, a G
(0, 1),
and n* G M. This
example captures the incomplete-information new-Keynesian business-cycle models of Woodford (2002),
Mackowiak and Wiederholt
(2006), Baeriswyl
firms suffer a loss whenever their price (pi) deviates from
^As
it
will
become
clear in Sections 3
uniqueness of the optimal policy (not
its
and
4,
the only result that
existence).
is
and Conrand
some target
affected
if
level [p*),
we
and
others:
which
in turn
(2007),
relax this condition
is
the
depends on the aggregate price
level (P).
nominal demand conditions, while
in pricing decisions (a.k.a.
•
TTi
—
{a
—
1
with
K = J kidi, and (a,
Cournot and Bertrand games studied
• Ui
(for
— —
{I
-
r) [ki
This example
is
-
9)"
-
-
r{Li
and Heinemann and Cornand
L), with Li
game
This game
c the degi'ee of strategic substitutabihty
among
— J
or cost shock, ki the quantity (for
{kj
firms' decisions.
- kif dj, L = J Lidi, and
is
More
is
it
concerned
will
become
—any model
following best-response structure:
in
some
in
for
who
framework nests
=
the primary goal of this paper
without centralizing the information that
restrict attention to
and deeper micro-foundations behind
mechanisms
government and then the government
is
is
is
all
in
"details," the lessons
we
these applications.
to study policies that
improve welfare
dispersed in the population, throughout the analysis
tax systems that utilize only information that
rule out direct
in Section 4, this
in the
from one application to another. Nevertheless, by conducting our exercise within
Remarks. Because
we
least as far as
E^[A{e,,e,K)]
provide in the subsequent sections are likely to hold across
so doing,
—at
^
an abstract framework that does not take any particular stand on these
we
second-guess the
will
which the agents' interaction can be summarized
linear function A. Clearly, the institutional details
this structure vary
will
own
a stylized fashion, Keynes' idea that financial
clear in Section 3.2, our
k^
for
his
of others.
generally, as
equihbrium
(0, 1).
higher than the average distance in the population
markets involve a zero-sum race between professional investors
demands
£
r
studied in Morris and Shin (2002), Svensson (2005)
supposed to capture,
is
and
Raith (1996), and various other
an agent faces a cost whenever the distance of
(2006):
action from the actions of others (Li)
(L).
i,
Bertrand)
(for
the beauty-contest
£ R^. This example nests the large
common demand
Bertrand) set by firm
Cournot) or complementarity
6, c)
in Vives (1984, 1998),
this context, 9i represents a
Cournot) or the price
(for
a determines the degree of strategic complementarity
"real rigidities").
+ bOi + cK) ki - \k'^,
10 papers. In
In this context, 6 represents exogenous aggregate
is
in the public
which the agents send reports about
collects taxes
domain. In
their types to the
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 all forms of aggregation
only means that the government
readily reinterpret
some
is
not
itself
and exchange of
a channel of communication.
of the exogenous information as the result of certain
types of information aggregation; for example, some or
8
all
of the agents
may
observe a signal about
the underlying fundamentals that
financial market.
What
the outcome of an opinion poll or the price of an unmodeled
is
crucial for the baseline analysis
is
is
given agent does not depend on other agents' strategies, nor
these alternative cases are considered in Sections 6 and
work on the
Finally, note that, while our prior
2007) and virtually
all
that the information available to any
is it
affected
by government
policies;
7.
social value of information (Angeletos
and Pavan,
the related literature (Morris and Shin, 2002, etc.) have limited attention
to the case of perfectly correlated shocks and to a specific Gaussian information structure, here
allow the shocks to be imperfectly correlated and
we
we
consider more general information structures.
This will permit us to highlight that our main policy results need not be sensitive to the specific
details of the information structure.
Nevertheless,
Gaussian example considered above, while
correlated
=
(ct^
we
will still illustrate certain insights in the
restricting, for simplicity, the shocks to
be perfectly
.
0).
.
Decentralized use of information
3
we show how the equilibrium and the
In this section
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
Before
we proceed
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
knowledge.
exists, is unique,
and
is
We
can then show that the complete-information equilibrium strategy
given by
ki
where 9
=
=
K
{9i,9)
=
Ko
+
KiOi
+
K2O,
f9dh{9) denotes aggregate productivity and where the
coefficients
(ko,ki,K2) are
determined by the payoff structure.^° Note that, by definition, an agent's payoff depends only on
his
own
productivity; that an agent's equihbrium action depends also on average productivity
^"The characterization of the
allocation,
which
is
in turn
in
coefficients (recfci, K2)
the proof of Proposition
1.
These
,
as well as of the coefficients (k3,kJ,k5) for the first-best
coefficients
depend on the reduced-form payoff structure U,
depends on the particular application under consideration. For the purposes of our
understanding the specific values of these coefficients
is
is
not essential.
analysis, however,
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
where the
first-best allocation exists,
is
unique, and
given by
is
determined by the payoff structure. Note that, as with
coefficients {kq, Kj, K2) are again
equilibrium, the first-best action prescribed to an agent depends both on his
own
productivity and
on aggregate productivity; however, the dependence on aggTegate productivity now emerges only
because of external
Now
effects.
suppose that information
is
common
incomplete but
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
1
ki
—
V
denotes the
E[k
Suppose
information
all
maximizes welfare
{9i,6) \V], while the strategy that
common
information
=
economies there can be room
k*
,
there
However, as we
information remains common.
given by
is
k^
=
strategy is given by
E[k* {di,d)
IT'],
where
set.
which k
Clearly, in economies in
common. The unique equilibrium
is
is
will
for policy intervention
no room
show
for policy intervention as long as
few sections, even in these
in the next
once information
dispersed.-'^
is
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(cj)
with K{4>)
=
J^ k
Consider the following
is
dispersed and
when
define an equilibrium in the standard Bayes-Nash fashion.
a (measurable) strategy
= argmax
and
(cj') d(t>{u;')
when information
ak{4>)
coefficient,
=
E[
:
fi
— M
{u;')
is
-
such
>
[/(fc,K((^),o-fc(g!)),6i)
[J^ [k
which
/j
1
a;
that,
for
all a;
G Q,
(2)
],
K{(f))]'^d(t){io')]^/^
for
all
(/>
G $.
the slope of an individual's best response with
respect to aggi'egate activity:
a=——.
(3)
This coefficient meastues the degree of strategic complementarity or substitutability among individual actions.
^^None of our
on economies
in
The
equilibrium use of information can then be characterized as follows.
results requires
re
^
re*.
Indeed, the reader
may
henceforth assume
which inefficiency emerges only under dispersed information.
10
re
=
re*
if
he/she wishes to focus
Proposition 2 The equilibrium
=
€ n, with K{4>)
all a;
unique, and satisfies
is
= E[K{e,e)+a-{K{(P)-K{8,0))
k{ij)
for
strategy exists,
J^ k
(u)') d(l){u)')
for
all
\uj]
e^.
4>
Although Proposition 2 does not provide a closed-form solution
is
an insightful representation of
it.
To
see this, recall that n{9i,6)
taken had information been complete. Next, note that
K{(j))
actions are strategically independent (q
=
— K(d, 6)
actions
ai'e
complements (a >
strategic
while he does the opposite
behavior
0.
the average deviation in the
When
is
simply his expectation of the
instead actions are interdependent
basis of his expectation of the average deviation in the population.
The
actions
ai'e
would have been under complete information,
it
strategic substitutes (a
permit more alignment of actions when a
tilted to
is
if
coefficient
a thus
how much
also captures
In particular,
the agent adjusts his action upwards whenever he
0),
expects aggi-egate activity to be higher than what
a <
would have
the agent adjusts his action away from the aforementioned certainty-equivalence bench-
0),
mark on the
if
is
i
it
a form of certainty equivalence continues to
0),
action he would have taken under complete information.
^
the action agent
is
hold: an agent's equilibrium action under incomplete information
(q
the equilibrium strategy,
for
what they would have done under complete information.
actions other agents are taking relative to
When
(4)
<
0).
>
In other words, ecjuilibrium
and more
differentiation
when
agents value aligning their choices with one
another.
how information
Proposition 2 has direct implications for
common
relying on
is
used
Because
in equilibrium.
sources of information facihtates alignment of individual choices, while relying
on idiosyncratic sources
actions to the former
inhibits
it,
strategic complementarity increases the sensitivity of equilibrium
and reduces the
sensitivity to the latter, while the converse
clearly, consider
the case where the agents' shocks are perfectly correlated and
is
true for strategic
substitutability.
To
see this
information
of agent
is
i
is
more
Gaussian. In this example,
consists of a private signal Xi
idiosyncratic noise
~
and e
where the
^
and where
= 9+
iV (0,0-^).
k{x,y)
=
ire
=
+
TTy
+
{1
(Jg'^, i^y
-
=
a)
t:^'
ay"^,
9 for
£,i
all
i,
where 9
and a public
The unique
^
and n^
N
is
(12,
a^)
— 9 + e,
,
and the information
where
£,t
^
N
[fi,
ay)
then
+ 7yy + 7x2;],
„
,
(5)
by
ne
=
^
signal y
equilibrium
Ko-F (ki -hK2) [7^M
coefficients (7^2,7^,73;) are given
TT0
=
9i
+ TTy +
{1
-
a)
TTx'
""
a~^ denote the precisions
public signal, and the private signal.
,-
11
K0+Try
of,
+
{1
-
a)Trx'
respectively, the prior, the
A higher
a.
reduces 7^ and raises 7^ and 7^: stronger complementarity
and the public
tion towards the prior
Next, note that
activity.
7a;
+
7i/
=
the use of informar
tilts
signal because they are relatively better predictors of others'
—
1
so that the equilibrium action of agent
7/^,
can also be
i
expressed in terms of the underlying fundamentals and noises as follows:
h =
By
Ko
+
+
(ki
K2) [7^//
+
(1
- 7^)^ +
7!/£
+ lxii\
,
strengthening the anchoring effect of the prior, stronger complementarity dampens the overall
sensitivity of individual actions to changes in the underlying fundamentals;
By
increases "inertia."
impact of
Finally,
common
increasing the reliance to noisy public information,
noise
and hence increases the non-fundamental
by reducing the reliance on noisy private information
and hence reduces the non-fundamental
noise
it
in other
words,
it
also amplifies the
it
volatility of aggregate activity.
mitigates the impact of idiosyncratic
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.
E^
=
j"E[9\iu]d4>{u})
shocks and, for any n
>
2, let
Corollary
1
expectation.
Let
denote the average of the agents' expectations of their
E^ = J E[E"''^\iu]d(l){uj)
The equilibrium strategy
is
denote the corresponding n-th order average
given by
k{iu)^E[K{e,e)\Lj\
w/iere
(9
The
under
=
X;^=i
((1
-
own
vw,
(7)
Q^)""~^) ^"•
equilibrium action under incomplete information thus has the same structure as the one
common
information replacing 6 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 centrahzed.
strategy the efficient strategy, or the efficient use of information.
12
We
henceforth call this
Definition 2
An
mapping
efficient strategy is a
Because payoffs are
A;*
linear in transfers, the latter
:
n— R
>
maximizes ex-ante
that
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 centraUzing information. That
when implementable,
direct
to
the efficient strategy
mechanism among the ones that
depend only on
his report
is
welfare-equivalent to the best incentive-compatible
restrict the actions the
Towards
planner recommends to each agent
and not on the reports made by other
strategy can be implemented in the next section; here,
eflficient
this goal, consider
is,
any arbitrary strategy
A:
:
il
we
— M
focus on
and
»
We
agents.
its
will
show how the
characterization.
let
k{e;h)^E[k{Lj)\e,h]
denote the component of individual activity that
=
k{h)
= f k {9; h) dh{e)
E[k\h]
and
"explained" by the fundamentals. Similarly,
is
a|(/!,)
=
Var(fc
- k\h) = /
be the fundamental components of the mean and the dispersion of
agent
i
The term
The term
The
action of any given
e
{k
= {K — K)
+ e + Vi
~ki
— K) —
(k
is,
captures the non- fundamental variation in individual activity that
e
captures the impact of
— K) captmes
idiosyncratic to the agent; that
The
=
captures the variation in individual activity that reflects variation in fundamentals.
fcj
across agents; that
=
activity.
- k{h)fdh{e)
h)
can be decomposed in three components:
ki
Vi
[k {0;
is,
common
noise in information.
is
common
Finally, the
the non-fundamental variation in individual activity that
captures the impact of idiosyncratic noise.
i),
is
^^
1
Given any strategy
fc
:
H
^ M,
ex-ante utility (welfare)
is
given by
Eu^E[uCk,k,ak,9)] + ^Wyorvol + ^Wd^,-dis
where W^oi
vol
The
term
following result then shows that a similar decomposition applies to ex-ante welfare.
Lemma
The
let
=
=
Ukk
Var
first
+ 2C/2fcA' + Ukk <
(e)
term
=
Var
(A')
and Wdts
- Var(X)
and
=
Ukk
dis
=
+
Var
U^a <
{vi)
=
0,
(8)
and where
Var
(fc
- K) -
Var(fc
- K).
in (8) captinres the welfare effects of the fundamental-driven variation in activity.
other two terms capture the welfare effects of the residual variation in activity: vol measures
non-fundamental aggregate
fundamental
volatility,
which originates
cross- sectional dispersion,
^Note that, by construction,
ki, e
and
Vi
which originates
in
common
in idiosyncratic noise.
are orthogonal one to the other.
13
noise, while dis
measures non-
The coefficients Wyoi
and Wdis then summarize the
two types of
sensitivity of welfare to these
noise: Wyoi
measures social
aversion to non-fundamental volatility, while Wdis measures social aversion to non-fundamental
dispersion.
Both non-fundamental
in welfare
losses
volatility
and non-fundamental dispersion contribute to a reduction
How much
because of the concavity of payoffs.
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 and dispersion.
For our purposes, however,
and Wdis- Their
it
suffices to
relative contribution
summarize these
—
social preferences in the coefficients Wyoi
can then be measured by the following
coefficient:
'^>
I;-
Since Wyoi 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.
Strategies that share the
non-fundamental
actions to
to
volatility
common
same fundamental-driven variation
and dispersion that they induce.
of activity relative to
its
Proposition 3 The
dispersion.
One should thus expect
is
efficient strategy exists, is unique,^^
= E[K*{eJ) +
••
'
-
all u},
differ in
the levels of
Intuitively, the higher the sensitivity of
and hence the higher the non-fundamental
to idiosyncratic noise,
preferences over volatihty and dispersion. This insight
for almost
may
sources of information relative to idiosyncratic sources, the higher the exposure
common noise relative
'
in activity
k(uj)
= Jq^ (a;') d4>{uj')
with K{(j))
the efficient strategy to depend on social
formalized in the following proposition.
and
satisfies
a* {K{^)-K*i9,e))
for
all
volatility
\iv]
(10)
(p.
In equilibrium, an agent's action was anchored to his expectation of k, the complete-information
equihbrium action; however,
K, with
it
was
also adjusted
the weight on the latter given by a.
replace k with k*
and a with
A
on the basis of his expectation of aggregate
activity,
similar result holds for the efficient strategy once
a*. It follows that, just as
a summarized
we
the 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
first-best action,
is
quite intuitive.
That a*
inversely related to the ratio Wyoi /Wdis reflects 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.
''Hereafter,
when we
say unique,
we mean up
to a zero-measure subset of
one has to make with a continuum of types.
14
fi;
this
is
a standard quaUfication that
i
I
:
'.
.
Furthermore, just as a pinned down the relative sensitivity of equihbrium actions to different
down
sources of information, a* pins
more
see this
clearly, consider
efficient strategy is
again the Gaussian example with perfectly correlated shocks.
almost
'''
all
ng
{x,y),
+ TTy +
where the
{1
-
follows that, just as
It
-
a*)
'y
7T^'
3.4
To
k,
K*2)
b;/^
+ YyV +
+
7ry
+ (l-a*)7rx'
(11)
,
by
'^
TTg
+
tt^
+
(1
-
a*)
tt^'
^
'
the equilibrium levels of inertia, non-fundamental volatility
levels that are
a with
7x2;]
optimal from a social perspective. As for the case
1
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
Q
+
[l^l
structures, the analogue of Corollary
with k* and
Inefficiency only
+
7re
a determines
more general information
once we replace
kS
coefficients (7^,7^,7^) are given
and dispersion, a* determines the
of
The
then given by
fc(x, y)
for
To
the corresponding relative sensitivity of efficient actions.
7^
The
is
=
k*
common,
In these economies, the equilibrium
.
leaving no
room
is
(first-best)
for policy intervention. Nevertheless,
efficient
whenever
Q*, 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)
is efficient
When a > a* and
that
is
When a <
under common information (k
under dispersed information
information
is
a* and information
is
if
and only
if
a—
=
k*).
a*-,
Gaussian, the equilibrium exhibits overreaction to public
information and excessive non-fundamental
(iii)
efficient
volatility.
Gaussian, the equilibrium exhibits overreaction to private
information and excessive non-fundamental dispersion.
4
Optimal policy
We now
tvirn to the core contribution of the paper.
affect the decentralized use of information.
efficient
We
We
first
explain
how
different tax
schemes
then identify the tax schemes that implement the
use of information as an equilibrium.
15
Equilibrium with taxes
4.1
Suppose that the information that
all
and
{K,
6}"^
Then, without any
clear in the next subsection), consider policies defined
where the function
erf,
pubhcly available at stage 3 includes
as well as aggregate productivity
(^i)i€[o,i]
become
is
satisfies
:
6
R" x M+ x
^
M
+
T^a
=
individual actions
loss of optimality (as it will
by
a quadratic polynomial in {k,K,9),
is
-
the following properties: Ukk
and T^k
9),
T
all
Tkk
<
-%^;E^ <
0,
it
is
T{K,K,0,e) =
1,
We
denote the class of policies that satisfy
these properties by T. Finally, note that this class includes both progressive tax schemes
>
and regressive tax schemes
0)
for
These properties preserve existence and uniqueness of equilibrium,
0.
while also guaranteeing budget balance state- by-state. ^^
Tkk
hnear in
(i.e.,
linear tax-schedule that does not directly
<
with Tkk
0).^^
One should thus
(i.e.,
with
think of this as a non-
depend on individual productivity
but
(^,),
is
contingent
on aggregate outcomes {K,ak,0)
We
then have the following result (to simplify the formulas, we henceforth normalize the payoff
structiure
by setting Ukk
= — 1;
Proposition 4 Given any
tax
the general case
T
scheme
£ T,
let
a - TkK
,
a
1
_ (l-a)Ko-n- (0,0,0)
l-a + Tkk + TkK
.
The equilibrium strategy
exists, is
in the appendix).
is
+ Tkk
1
,
l
+
-
_
(1
all a;
e Q, where k{6, 9)
=
ko
g)
(m +
K2)
-
T^e
.
l-a + Tkk + TkK
Tkk
unique, and satisfies
k{io)^E[k{9,9)+a{K{^)-R{e,9))
for
-
+
k\9
-f
k29 and
K{<j>)
=
f^ k
\
to
(13)
]
{lj') dcpico')
for
all
(f).
There are three instruments that permit the government to influence the agents'
activity: Tkk,
the progressivity of the tax system; TkK, the contingency of marginal taxes on aggi'egate activity;
and
r^.g,
matter
the contingency of marginal taxes on aggregate productivity. While
for
equilibrium outcomes, each of them has a distinctive
role.
The
all
these instruments
progressivity Tkk
is
the
only instrument that permits the government to control ki, the sensitivity of the agents' actions
to their information about their
own productivity
^*We
relax this assumption in Section 4.3.
^^To
see
the
latter
property,
note
that,
for
any
shocks.
For given Tkk, the instrument that
cross-sectional
distribution
tp
of
individual
activity,
jT{k,K,ak,e)d^{k) = T{K,K,0,e)+^{Tkk + T^oW.
^®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
and T^k, the instrument that permits the government to control
Finally, for given T^k
activity.
the contingency Ti-k of taxes on aggregate
is
the sensitivity of individual actions to variations in aggi'egate productivity
of marginal taxes
4.2
on
the contingency
efficient strategy
turn to the question of whether there exists a policy T* G
strategy as an equilibrium.
Whenever
this also guarantees that there
this
is
T
that implements the efhcient
the case, by the very definition of the efficient strategy,
no other transfer scheme that can improve upon T*. This
is
even for transfer schemes that violate budget balance and/or anonymity, and even
agents to send arbitrary messages to the planner and the planner to
on these messages. What
is
essential
is
T
T* G
Proposition 5 A
has the property
such that Ki
that K2
=
function
K,2,
T
=
policy in
T
that
The next
implements the
for given {k,k*)
this result
Kj.
true
one allows the
transfers contingent
result establishes existence
and uniqueness
that implements the efficient allocation.
that,
The proof of
make the
if
is
only that the planner does not send informative messages to
the agents before they commit their choices.
of a policy
Tj^g
0.
Implementation of the
We now
is
is
follows
But then there
and a unique
are then pinned
,
the optimal
efficient strategy
T^x
from Proposition
always
increases with
4. First,
such that kq
down by budget
=
Kq.
The
a and
unique, and
decreases with a*.
note that there exists a unique Tkk
also exists a unique Tf^n such that
T/^ (0, 0, 0)
exists, is
a = a*
rest of the
a.
,
unique
Tf^g
such
parameters of the policy
balance, establishing that there exists a unique policy
that implements the efficient strategy as an equihbrium.^^
The optimal
policy has the property that, keeping k
with Q and decreases with a*. This
tarity perceived
by the
in information.
If
agents,
we thus look
it
is
because a higher T^k, by reducing the degree of complemen-
reduces the sensitivity of individual decisions to
across economies that share the
properties under complete information
(i.e.,
properties under incomplete information
the optimal T^fc
is
and k* constant, the optimal Tkx increases
they feature different a and a*),
noise
efficiency
differ in these
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 T/.^
non- fundamental volatility of the equilibrium relative to
^^The uniqueness
same equilibrium and
they feature the same k and k*) but
(i.e.,
common
its
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
many
In contrast, for
applications
it
pubhc
at the time taxes
assumed that actions are perfectly revealed
are levied. However, the preceding analysis has
time.
is
may be more
at that
appropriate to assume that actions, as
well as aggregate fundamentals, are only imperfectly observed.
To accommodate
we now
this possibility,
productivity to be observed with noise in stage
in stage 3 the
noise while
government
—and
an idiosyncratic
Ui is
the true aggregate productivity
9
=
6
errors
+
where
<;,
<;
2.
We
ki
=
agent
+
ki
i
chooses
+
rj
Vi,
ki in
where
77
stage
is
a
2,
then
common
the government
cr^.
— as well as any other agent —observes a signal
measurement
All these noises could be interpreted as
be independent of the fundamentals and of the information that agents
to
then
—observes
if
with respective variances a^ and a^. Similarly, instead of
noise,
(9,
In particular,
3.
other agents
noise with variance
is
and are assumed
have in stage
all
consider a variant that allows actions and average
let
K=K+
rj
=
and a^
a^
+ cr^
denote, respectively, the cross-sectional
average and dispersion of k, and consider tax schedules of the form
where the function
T &T). The
T
is
assumed to
satisfy the
tax an agent expects to pay
E[T{ki,K,ak,e)\io^]
=
is
same properties
as in the previous section
(i.e.
then given by:
'
E[T{ki,K,ak,e)\uJi]
^^
+i
The last
{Tkk
+ 2TkK + Tkk) ^l +
\ {Tkk
+ TaaWv +
Tee<y-,
three terms in (14) 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 abihty to implement the efRcient strategy, but also
it
does not affect the properties of the optimal tax system.
Of
course, the last property rehes
actions).
When,
instead, the noise
is
on the noise being additive
multiplicative,
it
measurement
efficient strategy,
it
now may
efficient strategy
affect the details of the
this result, for the
without any significant
effects of the noise.
It
optimal tax system.
can he implemented regardless of whether activity and fun-
damentals are observed with measurement
Given
undo the incentive
error once again does not interfere with the ability to implementable the
although
Proposition 6 The
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
—
.
under dispersed information
Inefficiency only
4.4
As mentioi^ed
in Section
1,
financial- market observers
are often concerned about
many
how information
work
believe that financial markets
— at both the academic and the policy front
processed in financial markets. In particular, while
is
well
on average, many also
feel
that financial markets
often overreact to noisy public news, causing excessive non-fundamental volatility in both asset
prices
and
real investment.
=
the restriction that k
k*
At some
level, this possibility
and a > a*. More
offer
an interesting benchmark because
only
when information
is
dispersed.
our framework by
in
which k
generally, economies in
in these
We now
can be captured
=
k* but
a ^
a*
economies policy intervention becomes desirable
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 €
=
K
necessary that the marginal tax
components of activity that are not explained by the fundamentals. In
K
—
it is
response to different
denote the component of activity that
is
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
Ti^j^
>
and
the marginal tax co-varies positively with
is
intuitive. In situations in which,
to
common
if it
were not
volatility. It is
from overreacting to
for policy intervention, the equilib-
sources of information and excessive non-fundamental
the optimal marginal tax co- varies positively with the
fundamental
The
e.
when a < a*
rium would exhibit overreaction
volatility,
under dispersed information.
in which inefficiency emerges only
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
and thereby dampens the reaction
pohcy reduces the complementarity
of the equihbrium to
common
of marginal taxes on realized aggregate activity positive [Tj^k
noise,
>
in individual actions,
by making the contingency
0).
The optimal
guarantees that the overall response of the equilibrium to aggi-egate productivity
by making the marginal tax
4.5
not distorted
also a decreasing function of reaUzed aggi'egate productivity {Tj^g
<
0).
Idiosyncratic vs aggregate shocks
Another special case
the
is
policy then
new
of interest
is
public finance literature.
the case of independent private values typically considered in
Studying this particular case helps appreciate how our policy
19
results
depend on the dispersion of information regarding aggregate shocks, as opposed
idiosyncratic shocks.
,
Within our framework,
that
this case
can be captured as follows:
the private information of agent
6i is
distribution h
is
common
and
i
cj),
and hence
that the unique equilibrium
by
=
k{u}i)
K* {8i,d)
.
given hy k{u)i)
is
As a
We
efficient allocations.
is
is
result, neither
=
k
common
renders
{0i,9)
environment
is
that h
distribution of actions
we
is
known
to agent
z,
this implies
is
given
it is
not necessary
^^
ai'e
also
independent but rather that there
is
faces uncertainty regarding the distribution of actions in the
relax either the assumption that
0, is
known
but maintain the assumption that
common knowledge
behavior nor a* matters for the
We
common
equilibrium behavior, nor a* matters for
for
in equilibrium.
is
(j)
to agent
common
efficient allocation
i
or the assumption
knowledge, then the
This in turn eliminates the problem
of forecasting aggregate activity, once again guaranteeing that neither
efficiency.
dis-
,
common knowledge
is
so
a* and hence also the contingency T^k, irrelevant in the aforementioned
no strategic uncertainty: no agent
if
is
Similarly, the efficient allocation
.
not per se the fact that private values
population. Indeed,
is
conclude that in the case of independent private values
a and
$i,
knowledge. Together with the fact
fact that 6i
a matters
—
oji
second, suppose that the cross-sectional
h\
to condition the tax system on realized aggregate activity.
What
suppose that
no uncertainty about the cross-sectional
also
commonly known and the
6, is
—
first,
the cross-sectional distribution of information,
knowledge, in equilibrium aggregate investment
/i,
cj)
knowledge, so that there
tribution of types in society. Because
that
to purely
a matters
—and therefore nor T^pc
for
equilibrium
essential for achieving
is
conclude that the key distinctive property of the correlated-value environments we
consider in this paper
is
the strategic uncertainty created by the dispersion of information regarding
aggregate shocks.
Corollary 4 When
the cross-sectional distribution of information in society
edge, the contingency of taxes
on aggregate activity {T^k)
is
(</>)
is
common
knowl-
not essential for implementing the
efficient strategy.
'
•
\
5
A
The
analysis so far has been confined to a static game.
dynamic economy
be embedded
in a
dynamic
helps fiu-ther appreciate
setting with a
how om-
results
^*In particular, the efficient allocation can be
and
9.
Moreover, should the tax be
through
k;2
more macro
can be relevant
We now
flavor.
show how
this static
This serves two goals.
for applications.
Second,
it
game can
First,
it
accommodates
implemented with a tax schedule that depends only on own activity
made contingent on K,
this contingency
would matter
(the sensitivity of the complete-information equihbrium to 9), not through
the contingency on 6 should then be adjusted to perfectly offset the effect of
20
TkK on
for
a
ti2-
equihbrium behavior only
(the degree of alignment);
the possibility that interesting dynamics in actions originate in the dynamics of information.
we
this section,
by taking the dynamics of information
start
analysis of endogenous information in Section
There are
+
A'^
1
periods, with
>
A''
chooses an action
and
6.
In each period
2.
of investment in a riskless discount bond,
Investing
to the
up
Set
5.1
we turn
as entirely exogenous;
In
shock, and
G (ki^t)
mean and
at are the
G
and
F
and
—
1,
each agent
...,N,
his level of
consumption,
£ ^, which we henceforth interpret as capital invested
k^^t
costs
ki^t
b^^t,
t
in period
t
and
F {ki^t, K^, at, A^^t+i)
delivers
the dispersion of activities in period
To
are real-valued functions.
Ai^t+i
t,
chooses his level
i
in
is
that
Of
in period
i
+
1,
where Kt
an exogenous productivity
is
we henceforth impose
simplify the exposition,
common
denotes the
also
a risky technology.
that the productivity shocks are perfectly correlated across agents, so that Ai^t+i
(The timing convention we adopt here
The agent
Cj^j.
—
shock that
^t for all i,t.
is
relevant for
period-i decisions.)
The
agent's period-i budget
Ci,t
where
rate)
+
G{ki^t)
+
is
given by
qtbi,t
= F {ki^t-i,Kt-i,at-i,6t-i) + 6i,t-i -
n^t,
denotes the period-t price of discount bonds (the reciprocal of the period— i risk-free
qt
and
denotes the period-i
Tj^j
tax;es
the agent pays to (or the transfers he receives from) the
goverimient.^^ Finally, the agent's intertemporal preferences are given by
Ui^Y.f3'-'U{c^,t,k,^t).
where Ut
is
a real-valued function.
This framework
is
can be nested by setting U{c, k)
assumed
we considered
quite flexible. All the applications
in those statics
examples
=
c
and then
(e.g.,
+
letting -~G{k)
(3F{k,
—G{k) + (3F{k,K,a,6) can be
in the static
K,
a, 6)
benchmark
equal the payoffs
interpreted as the profit of
a firm). Alternatively, a stylized version of the neoclassical growth model with convex investment
costs
X >
nested by letting
is
One
0.
G (k) =
c,
F [k, K, a, 6) —
could also interpret k as individual
and
could further
U (c, k) =
and
effort, in
and so
=
k
+
it
xfc^, for
some constant
would be natural to
let
on.
-I-
/ci,t
i,
to Ci,N+i
agent's
(fc)
which case
= A/^ 1, we impose that
= bi,t = for all in which case
= F (fci,Af,i^Af,(7Ar, An+i) — G (0) +bi,N — Ti,jv+i. Furthermore,
endowment to zero so that F{ki,o, KQ,ao, Ai) = 6,,o = for ah
t
G
U {c,k) = c- H{k), with H{k) = k"^ representing the disutility of effort. Finally, one
allow U and G to depend on (A', a, 9), as to capture externalities in leisure, pecuniary
externalities in the costs of investment,
'^For
Ok,
i.
21
the last-period budget constraint reduces
for
i
=
1,
without
loss,
we normalize each
If
common
agents had
information about the shocks in
periods, then the analysis could
all
proceed essentially without any further restrictions on the functions F,
for
we assume
restrictions. First,
H
some function
.
Second, we
V
static
model. The
and only
\i qt
—
satisfies
(in
c
— H{k),
ensures that, in
all
the function
periods and states, the
bond market
the risk-free
for
K,
a, 6) as
{k,
bond
is
U
in the
clears
if
indeterminate) and that the
the absence of taxes) reduces to
(in
i
—
= -[G (k) + H{k)] + (3F (k, K, a,0)
which case the demand
life-time utility of agent
analysis tractable,
consumption: U{c,k)
linear in
is
the same properties with respect to
first restriction
(5
U
we
let
V{k,K,cj,9)
and assume that
that
U. Here, however,
To keep the
are interested in cases where agents have heterogeneous information.
we impose two
G and
N
-
W,
The second
restriction then permits to extend our previous static analysis to the
The key
knows
this,
in
J]/3*-iF(/c,,i,i^4,ai,et).
here
period
is
to rule out informational externalities
about
t
{6t,uJi^t}tLi^
is
t
possibility that
depends on the actions other agents took
9t
we model the dynamics
of an agent in period
—the
with marginal distributions
for
G
Wj^j
given by
uji^t
F
Let f G
f2(.
(f)t
in periods s
what an agent
<
i.
To ensure
The (exogenous) information
of the information structure as follows.
represented by
dynamic model.
denote a joint distribution for
G $f. The distribution / also describes
the cross-sectional distribution of {6t,iOi^t}tLi in the population. First, Nature draws / from a set
of possible distributions
sequence
F
{6tjUJi,t}tLi for
assume that
according to the probability measure
each agent
{iOi^t-iift^t-i^St-i)
i,
with
belongs to
{6t,L0i^t}tLi
uji^t:
for all
i
Nature then uses / to draw a
J-.
drawn independently from
and
t.
This ensures that there
learn about {9s,(t>s)^=t from the observation of other agents' (past) actions
are observable
irrelevant. ^°
then
is
It is
then without loss of generality,
efficiency, to restrict attention to strategies that
depend only on w^
is
Finally,
we
nothing to
—whether such actions
for either ecjuilibrium or
f.
Equilibrium, efficiency cind policy
5.2
Given that information
is
exogenous in
to Vk{K,K,0,9)
=
equihbrium action
An
and
in
let
period
t
all
dates and states, the analysis of both the equilibrium and
the static benchmark. Let k (9) denote the (unique) solution
efficient allocations parallels that in
^°
/.
= axgmax^V {k,k, 0,9)
k* (9)
would be k
{9t)
,
;
if
information were complete, the
while the first-best action would be k*
{9t)
."^
Next,
alternative that would also guarantee that agents do not learn anything about {Os,4>s)s=t from the observation
of past actions
is
to
assume that
for all
i
>
1, uit^t
is
a sufficient statistic for
(a;,,!, (aJi,s,'/>s,^s)s=i)
with respect to
i6s,<t>s)s=f
^^Both K and k' are Unear functions of
the coefficients
(«;o,«;i,fi;2,«;o,«;i,
6.
In particular, k(S)
=
ko
+ (ki + K2)
^2) determined as in the proof of Proposition
22
= kq + [ki + K2)0
U with V.
and n*{6)
1
replacing
with
let
a=
with Wyol
^
Vkk
+ 2VkK +
-—-
and Wdis
V/tA'
=
+ Va^;
V^k
=1--^,
a
and
a summarizes
once again,
of aligning choices (the equilibrium degree of complementarity), while a*
the private value
summarizes the
The equilibrium
value of such alignment (the relative social aversion to dispersion and volatihty).
and
under incomplete information are then characterized in the following two
efficient allocations
propositions, which
^
Kt{(t>t)
= J kt
unique, and satisfies, for
is
periods
all
t
and
all
= E[k {9t) + a
all ujt
G
efficient strategy exists, is unique,
ktiujt)
The
-
\ujt],
Kt (Ot))
and
satisfies,
for
periods
all
t
and almost
-
fit,
Kt{(l)t)
{Kt {^t)
{uj') dcptiuj').
Proposition 8 The
with
strategy exists,
3.
fit,
ktiujt)
with
and
direct extensions of Propositions 2
aj-e
Proposition 7 The equilibrium
U!t
social
=
=
n'^*{Ot)
a*-{Kt{<Pt)-K*{et))
J^kt{uj')d(pt{Lo').
efficient strategy
,
can be implemented
efHciency can be induced in period
about Kt and
+
9t
t
in
,
a similar fashion as in Section
by making taxes
that becomes pubhcly available at
\uJt],
f
in
period
+ 1. As
t
+
1
when they make
In particular,
contingent on information
in the static model, the
does not require any informational advantage on the side of the government.
the agents anticipating
4.
It
optimal policy
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 this
crucial for om- purposes
way agents use
their available
is
of the underlying fundamentals.
The
various channels of infor-
social learning thus introduce informational externalities that
and
is
the more individuals rely on their private information, the more
when determining
issue,
is
its
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
is
a proxy for the informational role of macroeconomic data, financial prices, and other
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
we assume
/i
and variance
public signal yt
The
u'l.
= 9 + et
and we assume that
6,
realization of 9
and private
it is
signals
Xi^f
= + ^j,f,
where
et
~ A/'(0, ay^)
t,
agents observe a
and
~ A^(0, a^i)
^j,t
are noises, independent of one another, independent across time, and independent of
and
also independently
9:
kt-i
and
Ot
~
=
Kt-i
+
A/'(0, o"^
^
Tjt,
)
=
variables,
+
at-i
Ut,
common
The
variable.
Ci,t
The
at-i
random
are shocks,
random
of any other
any date
identically distributed across agents. In addition, at
observe the following three
which
^
and At
9
+
at,
where
77;
~
>2,
^i^t
agents
X{0,(jl^t),
and independent
+ b^^t-l-T^,t
is
the underlying
That the variables Kt-i and at-i that enter period— i income (through
(trend) productivity.
F) coincide with the
with
given by
is
variable At can be interpreted as the period— i productivity shock, while 9
mean
~
^f(0,o'^t)^ ^t
across agents, independent across time,
F{ki^t^l,Kt-l,at-l,At)
t
9,
and convey information about
affect payoffs
period-^ budget of the agent
+ G{k^^t) + qtbi,t^
we
drawn from a Normal distribution with mean
never revealed. Instead, at any date
is
of the
constant over time,
is
sac-
restrict
component
that the
fundamentals about which the agents have heterogeneous information
denote this component by
we henceforth
in the preceding analysis:
In particular,
we must
signals
about past activity
is
not essential.
What
essential
is
that the
payoff of an agent
is
given by
is
observation of income does not perfectly reveal either 9 or i\i_i.^^
The
rest of the
Xltll^ /?*"' t/
model
(cj^t, ki^t)
1
ensures once again that
is
where
qt
—
as in Section
U (cj^j,
/?,
fcj^t)
=
5.
The intertemporal
Ci^t^h{ki,t)-
That preferences are
linear in
consumption
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
J2P'''v{k^,t.Kt,at,At+l),
t=i
where
V
(fc,
K,
a,
A)
= -[G (fc) + H{k)] + (3F
(fc,
K, a, A)
.
The
function
V
is
quadratic and satisfies
the same 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.
In particular,
Aj,t
= 6 + at^t,
we could
allow the agents to receive private signals K,,t-i
in addition to, or in substitution for, the
=
Kt-i
+
aforementioned public signads.
24
7?i,t,
fft-i
=
at-i
+
Vi,t,
and
,
Equilibrium
6.2
The
is
essential difference
between the economy of
the endogeneity of information:
information about
is
the strategy agents follow in period
t
in Section 5
determines how
much
contained in (Kt^^t) a^nd hence affects the agents' behavior in periods
In the absence of policy, this informational externality
on.
and the one examined
this section
fact that the use of information in the present affects the information available in the future
solution to T4 {k,k,0,6)
=
and a
Proposition 9 The equilibrium
=
—
k,q
+
and
is
+
{kj
—Vkx/Vkk, we have the following
strategy exists
+
1
not internalized by the agents: the
is
not alter private incentives. Thus, letting once again K{d)
t
does
K2)0 denote the unique
result.
unique. Let {^t,^t}tLi denote the (unique)
information structure generated by the equilibrium strategy.
Then, for
all t, the strategy
and
the
information structure jointly satisfy
^ E[Kie) + a-{Kt{(Pt)-Km
ktiojt)
for
all ujt
^ ^t, with
Kt{4>t)
=
Jq
kt {lo')
d(j)t{ijj')
for
all
4>t
&
,-
\u>t]
(15)
^t-
This result does not require the information structure to be Gaussian. However, once we
6
and the exogenous noises to be Gaussian,
signals of past activity
available in
this result ensures that the information contained in the
also Gaussian. All the information
is
any given period can then be summarized
and the other
for
restrict
two
in
—exogenous and endogenous—that
sufficient statistics,
one
is
for the private
the public signals; the dynamics of these two statistics admit a simple recursive
structm'e and the equilibrium strategy reduces to an affine combination of the two.
Proposition 10 The equilibrium
-
with
''
strategy
^i,t (Wi,i) ==
is
given by
K
(7t^i,i
(1
-
'
7i) ^t)
>
.
-
(1
The variables
that
+
is
q)
TTf
+ Trf
Xi^t o-nd Yt are sufficient statistics for all the private
available to agent
i
in period
while
t,
irf
and
and public information about 9
n^ are their respective precisions.
The
sufficient
statistics are given recursively by
^.,
= ^AV-i
^x,,
^Aj,i_i + —
^Xi,i
and
Y,
y^
_,„-,-
^'"^
=—
%F,-,
^.
+ ^y, + ^A, +
where
~
Vt
_
=
Kt-i
- KQ-,
+
—+—
K2){l-'Yt-i)yt-i
^
{ki
[Kl
,
25
K2) 7t-i
^"'
+
"^^
"^"^
y,
(17)
/^a^
U«J
is
a linear transformation of the signal of past activity.
Similarly, the precisions
and
-n'f
are
-k"
given recursively by
Trf
=
7rf_i
+
and
cT-2
^rf
Finally, the initial conditions are Xifi
The
behind condition
logic
is
(16)
=
=
+ ^-f + a^; +
7rf_i
0, Yifl
=
/u,
70
=
the same as the one
0, ttq
(ki
=
+ K2)Nti<T-?.
and
ttq
we encountered
=
(19)
a-g'^.
in the static
benchmark.
For given degree of complementarity a, the relative sensitivity 74 of the equilibrium strategy to
private information increases with the precision of private information and decreases with the
precision of
pubhc information. At the same,
for given precisions,
a higher
a
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
Kt-l
Because Yt-i
Kt-i
+
rif
now note
is
publicly
in period
t is
First note that, because Xj_4_i equals 9 plus idiosyncratic
follows.
=
-
t
Ko
1 is
given by
+ {Ki+K2)ht-lO+il--ft-l)Yt-i).
known (and
so are the coefRcients kq, ki, K2 and 7f-i), observing Kt-i
informationally-equivalent to observing the variable yt defined in (18).
is
(K1
simply a Gaussian signal with precision
signals can
be combined
averages of
all
+K2)7t-1
ttj
=
7^_j {k\
+
K2)
cr~f
.
It
follows that all private
in the sufficient statistic Xi^t, while all public signals
in the siifRcient statistics Yf.
Condition (17) then states that these
can be combined
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
to notice
increasing in jt-i-^^ This
is
simply the sums of the
signals.
pends on the strategy followed
is
But
that
'
which
=
is
that the precision of information available in one period de-
in previous periods.
In particular, for
all t, nt
and thereby
yrf
is
because the informative content of the signals of aggregate activity
higher the more sensitive the strategies of the agents to their private information.^'* This
important informational externality that the equilibrium
fails
is
an
to internalize in the absence of policy
intervention.
^^When
9 changes over time, the period-i precision of information regarding the period-i fundamental need be
monotonic over time; but
it
remains an increasing function of the sensitivities of past strategies to private information.
^*A 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.
M
;
,
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
unlike the cases of exogenous information examined in the previous section,
we now
restrict at-
tention to strategies that are linear in the available private signals. Without this restriction, the
endogenous signals
Suppose,
a moment, that the government
for
follow in period
no longer Gaussian and the analysis becomes
ai-e
fails
to recognize that the strategy the agents
subsequent periods. Suppose further that the
affects the information available in
t
intractable.
period-i private and public information were summarized in sufficient statistics Xi^t
respective precisions
and
k* {6)
let
2Vfc/f
-I-
=
7rf
and
7rf,
+
Ktct)-
Yt
with
so that
=
argmax,^V{K,,K,0,6)
VKK)/{Vkk
and
Kq
+
(k|
+
^2)^ and a*
=
1
- Wyoi/Wdis =
Proposition 8 would then imply that the efficient strategy
=
ki,t (a;^,t)
K* (7;X,.i
+
-
(1
-
1
is
{Vkk
+
given by
7*) Yt)
with
il-a*)7rf
7i
Now
(l-a*)7rf
+ 7rr
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
periods.
Because more learning
is
more learning
as to induce
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
The
information.
following result verifies this intuition.
Proposition 11 The
'
=
K* (7r*Xi,i
+
(1
-
7?*) Yt)
given by
...
,
'
*
all t
all the
itOi,t)
utility is
'
where
for
ht
^
'
maximizes ex-ante
linear strategy that
< N,
(1
-
while
a*)
TTf
7^ =
+ ^f _
(1
—
/3
(1
a*) tt^/
_
[(1
a*) (1
—
-
7;;i)'
a*) nf^
TTf ttI'
+ tt^]
.
«i)-' (^ + K\f a'^^,
Xi^t
private and public information about 6 available to agent
and
i
Yt are sufficient statistics for
in period
t,
while nf
their respective precisions; they are obtained recursively using (17)-(19), replacing
with {-ft*,K.Q,Kl,K2)
27
-
and
{-yt,
no,
vrf
are
1^1, 1^2)
The key
result here
that, holding constant the cuii'ent precisions of private
is
information, the optimal weight on private information
is
higher than what
it
and public
would have been had
information in the subsequent periods been exogenous:
As
anticipated, this follows directly from the internaUzation of the informational externality:
raising the reliance
on private information
mation and hence higher welfare
The
in
in
in
one period, society achieves higher precision of
subsequent
infor-
periods.'^^
following alternative representation of the optimal strategy helps translate the result here
terms of the degree of complementarity that the policy must induce
Proposition 12 Consider
mation structure.
a^ —
the efficient linear strategy
and
let
There exists a unique sequence {o^rii^i
'
in equilibrium.
{^t,^t}'^=i be the associated infor'"'^^'^
^t*
<
<^*
f°''"
clU t
<
a*, such that the efficient strategy and the information structure jointly satisfy, for
N
for almost
ut &
society
choices.
fit,
with Kt{(pt)
=
K
Jq
{uj') d(j)t{u>')
for
all
4>t
without informational externalities, the weight
in the case
how much
own
all
would
and
all t,
k*ii^t)=n'^*iO) + ar-{Kt{<f>t)-K*{9t))\oJt]
As
by
(20)
^ ^t-
aj"* in
condition (20) summarizes
the agents to factor their expectations of other agents' choices in their
like
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 externaUty
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 hnear strategy can be implemented with similai' tax schemes
£is
in the
benchmark model, but now the
must be higher than what
it
sensitivity
T^^
of the marginal tax to aggregate activity
would have been with exogenous information.
Corollciry 5 Informational 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
Of
hold at
t
had information been exogenous.
Moreover,
Gaussian information structures, with multiple private and public signals
course, this informational externality
=
efficient
is
absent in the very
A^.
28
last period,
which explains why the result does not
of aggregate activity.
It relies
merely on two properties:
(i)
by increasing the sensitivity of actions to private information; and
learning
We
is
TkK induces more
that a higher
learning
that the social value of such
(ii)
positive because the optimal policy removes any inefficiencies in the use of information.
further discuss the importance of this last point in the next section, where
we study how
the policies we have identified affect the social value of information. Before turning to this issue,
however, we study the properties of optimal policy for a special case of interest: economies in which
inefficiency
emerges only because of informational externalities.
This special case
is
motivated by the following observations. In certain settings
economies with no externalities), one
may
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
still
same time, when information
represents an important benchmark. At the
expect that individual market participants
fail
to internalize
how
is
dispersed, one
may
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
inefficiency
The
In our set up, these settings correspond to economies in which
(k,
a)
=
(«;*,a*), so that private
and
social payoffs coincide
emerges only because of informational externalities.
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, the
optimal policy
This result
may be
is
in which inefficiency emerges only because of informational externali-
such that TkK
>
0.
relevant for understanding optimal policy over the business cycle. Consider
standard real-business-cycle models in which
all
firms and households share the
The assumption
regarding aggregate productivity and taste shocks.
same information
of frictionless competitive
markets along with the absence of direct payoff externahties 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.
by having marginal taxes increase with
Our
fail
to internalize
results then suggest the following
realized aggregate
macroeconomic
how
let
This
their
poUcy remedy:
activity
and decrease
with realized productivity, the government can improve the information contained in prices and
macro data, and can thereby reduce the non-fundamental component
of the business cycle
fluctuations that are driven by noise in information regarding aggregate
conditions).
•'
29
demand and
(i.e.,
the
productivity
J
•
7
Implications for the social value of information
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 as the Bureau
Bureau, or the Federal Reserve Banks), then
desirable to
communicate such information
it
is
Labor
macroeco-
Statistics, the
US Census
important to understand whether
to the market.
Indeed, as long the equilibrium use of information
available information can have a detrimental effect
of
(e.g.
is
A
positive answer
inefficient,
is
an increase
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
volatihty.
is
However, this can not be the case
if
the equilibrium use of information
is efficient.
This
because the equihbrium strategy then coincides with the solution to a planning problem where
the planner directly controls
how
agents use their available information.
An argument
analogous
to Blackwell's theorem then guarantees that additional information can not reduce welfare.^®
following result
is
Corollary 7 In
then a direct implication of these observations.
general,
more precise information can reduce
The
,.
welfare.
However, policies that
store efficiency in the decentralized use of information also guarantee a positive social value for
re-
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
is
where
not warranted from a social perspective,
causing overreaction to public news.^*^ This example has lead to a renewed debate on the merits
of transparency in central
in isolation
bank communications.^® 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.
^^
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
^^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).
^^Their example
inefficiency
is
nested in our baseline static framework with k
emerges only under dispersed information and manifests
^*See, for example,
Amato and
Heinemann and Conrand
Woodford (2005).
^^An exception
is
=
k*
itself in
and q > a*;
it
is
an economy where
excessive non-fundamental volatility.
Shin (2006), Angeletos and Pavan (2004, 2007), Baeriswyl and Conrand (2007),
(2006), Hellwig (2005), Morris
and Shin (2002, 2005), Roca (2006), Svensson (2005),
Baeriswyl and Conrand (2007), which focuses on the signaling effects of monetary policy.
30
activity
and can thereby slow down
Amato and Shin
Shin (2005) and
A
social learning.'^°
(2006).
Our
theme
similar
is
explored in Morris and
imply that the government can improve the
results
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
conditions or the profitability of available technologies —
highly dispersed
Information about commonly-relevant fundamentals
in society,
is
perfectly aggregated through markets,
institution.
ized
As
first
necessarily
mean
for
an
this
means that
effective utilization of
society has to rely
such information.
and
on decentral-
However, this does not
social incentives in the use of such information
the equilibrium's response to certain sources of information
The key
only im-
that the government should not interfere with the decentralized use of informa-
tion: to the extent that private
non-fundamental
is
and can not be centralized by the government or any other
emphasized by Hayek (1945),
market mechanisms
and demand
volatility, excessive dispersion, or
may be
do not coincide,
leading to excessive
inefficient,
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
disseminate information or create
We
if it
new channels through which information
cannot
is
itself collect
and
aggregated in society.
established this result within an abstract but fiexible 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
If
volatility, 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 of marginal taxes contingent
for
any economy featuring dispersed information
regarding commonly-relevant fundamentals.
^"Amador and Weill
models are nested
(2007) extend Vives (1993, 1997) to situations with both private and pubhc learning.
in our analysis of Section 6 as special cases that rule out payoff externalities.
31
Both
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.
contingent taxes and subsequently
activity. Alternatively, the
make
For example, the government could
first collect
non-
rebates whose magnitude depends on realized aggregate
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
their pricing
and produc-
on how they anticipate monetary policy to 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
stabihzation purposes, but also to improve the information that
data. Further exploring
how
design of monetaiy policy
is
the policy objectives
a
fi'uitful
Bank can use the
we have studied
is
contained in prices and macro
in this paper filter in the optimal
direction for future research.
32
for the familiar
9
Appendix
Proof of Proposition
We
1.
prove the result in three steps.
equihbrium and the
respectively, the complete-information
common
results to the case of incomplete but
Step
1.
any given
0.
U
first-best allocation.
Step 3 extends the
information.
i is
Aggregating across agents
(i.e.,
(0, 0, 0)
-I-
.
and
(21) with (22),
_
+
[Ukk
6i),
is
UkeO^
=
equivalent to
=
0.
(21)
we thus have that
+ UkK]K +
Uk96
=
Q.
(22)
letting
_ Uke
^'-^Ihk
(0,0,0)
t/fc
Ukkh + UkKK +
integrating over
Uk{Q,Q,Q)
'
k, for
pinned down by the first-order condition Uk [h, K,6i)
quadratic in {k,K,9), the first-order condition
is
Uk
Combining
and 2 characterize,
1
Consider the complete-information equilibrium. Because payoffs are concave in
K the optimal action for agent
Because
Steps
^'- -{Uk,
+ UkKy
Uke
,„„,
^^^^
^'=-{Uuu + UkK)~^''
,
gives the result.
Step
2.
A
Next, consider the first-best allocation.
of a strategy k
:
Q
H
x
R
^>
K{h)
=
Jk{6\
and dispersion
h)
dh
(6)
combination
and a system of budget-balanced transfers across agents. For any
given cross-sectional distribution of productivities h £
let
feasible allocation consists of a
and
ct/,(/i)
=
(/
h)
[k [6]
H
and any given strategy
- K{h)f dh
fc
x //
:
{6)^^^ be the corresponding
behind the
utility
strategy
k.
An
allocation
w{k;h). Because
U
is
is
thus
M,
mean
for simplicity
L=
efficient
^{Ua,
if
+ UkkH +
f k{0)dh
{0)
Uk {K, K, 0)
\
—
if
the strategy k
Uca'^kj
\uee<yl
of
fc,
K
+
Uek
-\-2ti
L with
[k{0)
{U,,
+
f/fcfc)
33
U
-
h.
+M=
x
+
dh
Then
H
—
>
M.
maximizes
A
=
(0)
- K0
let
Kf dh {0) -
respect to
[7^^^
Q
[k (6) 0]
-K]^0
+ Uk {K, K, 0) -
:
we then have that
and ak on
~k\-ix(J [k{0)
denote the Lagrangian for this problem. Optimizing
Uke0
depends only on the
in transfers, ex-ante utihty
and only
we dropped the dependence
w-x(
(24)
ignorance (equivalently, welfare under an utilitarian aggre-
quadratic, and Ua{k,K,ak,0)
w = U{K, K, 0, §) +
where
veil of
Because of the quasi-linearity of payoffs
gator).
>
of activity in the cross section of the population. Next, let
w{k;h)= j U{k{e\h),K{h),(7k{h),e)dh{e)
denote ex-ante
—
al
K, a^ and k{0) we have that
;'
,'i.i
Combining, we have
UkeO
tliat
+ Uk{K, K, 9) + Uk{K, K, 6) -
UkeO
+
{U^^
+
Ukk)
[k(.9)
-K] =
Q
(25)
or equivalently
UkeO + Uk{0,
0, 0)
Integrating over 9
K + UKeO + {U^a + Ukk) [Kd) - K{h)]
+ Uk (0, 0, 0) + [U^k + 2UkK + Ukk]
(26)
we then have that
^^
^
UkjO,
+
0, 0)
Uk{0,
(Ukb
0, 0)
-[UkK + 2UkK + UKK]
+
Uke)
-[UkK + 2UkK + UKK]
,
.
^
'
'
'
Substituting (27) into (26), and letting
+ ^K (0,0,0)
+ 2UkK + Ukk)
Uke
t/fc(0,0,0)
°
~
-{Ukk
"'
^
- -
{Ukk
+
Uke
U..)
"'
~
'
+ UKe
*
+ 2UkK + Ukk)
-{Ukk
"''
.„„.
^
'
gives the result.
Step
Now
3.
suppose that information
common, the aggregate
for agent
i is
K
activity
also
is
commonly known
+ Ukkh + UkKK +
V denotes the commonly- available information set
agents,
and noting that the
+
(0, 0, 0)
[Ukk
Note that the above two conditions are
9i
in equilibrium.
+ UkK] K +
i is
=
Since
all
information
The first-order
is
condition
^,
(whatever this
is
Uk0nOi\'P]
£[^17^]. It is
Aggregating this across
is).
simply E[^|P], we get
=
identical to conditions (21)
and 9 have been replaced by E[0i|P] and
action for agent
UkenOi[P]
cross-sectional average of E[^,|'P]
Uk
that
incomplete but common.
thus given by
Uk{^.Q.^)
where
is
0.
and
(22),
except for the fact
thus immediate that the equilibrium
given by
h^¥.[K{9M'P]-
A
similar
argument implies that the
efficient action is
given by
ki^¥.[K*{9,,9)\'P],
which completes the proof of the
Proof of Proposition
result.
We
2.
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
1.
A;'
:
Q— M
»
such that,
fc
:
Q — R and
>
for all
a;,
let K{(j))
k' (w) solves
= j k [uj)
.
A
best-response
is
a
the first-order condition
nUk{k',K{(j,),9)\Lo]=Q.
34
dcf) [uj)
(29)
Using the fact that Uk{k', K,
where k stands
=
O for
E[Ukk
—
or equivalently k'{u})
which gives
Step
2.
+
Uk{K{e, e),K{e, 9), 6)
Ukk
-
{k'
+ UkK {K -
k{0, d))
all
(0,0)
(k'
-
E[k{0, 0)
(29) reduces to
,
+ UkK
k{0, 0))
+ a {K
- k{0,0))
{(f>)
-k{0,0))\u] =
{!< (^)
to].
\
0,
What
remains to prove
is
that the equilibrium exists and
is
the class of payoff functions
U
that satisfy the properties specified in Section
=
{U,Q,Q,,F)
equiUbrium strategies
an economy
e'
efficient strategy for
measure-zero set of w.
Proof of Corollary
K{4>)
=
fact that ki
/ E[
=
R-om
1.
k{0, 0)
—
k,2{1
k*
U
-
a,s
exists
an equilibrium
(4),
U eU.
and
for the
is
As shown
aK{0, 0)
u) ]d(^{uj)
\
+ a f E[
K
denote
An economy
2.
e'
\
set of
coincide with the
in Proposition 3, the
exists
{(P)
is
efficient strategies
uniquely determined for
economy e
U
immediate that the
we have that aggregate investment
and
is
all
but a
unique.
satisfies
iojd^iu).
a) /a,
= {l-a)Ko +
Ki0,0)-aKi0,0)
Ki0.
that
K{(t))^{l-a)Ko + KiE^ + a fElKicp)
\oj]d<l){uj)
-
and hence that
E[K
Iterating
it is
and a* corresponding to
as long
with U' &
e'
follows that
It
U
Moreover, U' e
any economy
use of information. Let
9, Q, F) coincides with the set of
{?7,
{U',Q,Q,F) such that the
e.
efficient
conditions (4) and (10),
=
the economy e
for
—
By comparing
.
K and a corresponding to
It follows
uj,
unique; this can be done
which characterizes the
Using the
k{uj) for all
(4).
3,
for
=
In equilibrium, k'(u;)
with the help of Proposition
given by e
Kid, 9)),
complete-information equiUbrium allocation and the fact that k solves
for the
UkiK{e,e),K{e,e),e)
=
6)
{(f))
\u]^{l-
and then substituting
a)Ko
+
+ aE[
KiE[E^\uj]
f E[
K
(4>)
\
J
]d(p(uj')
\
u
].
into (4) gives
oo
;,.
'
':
k{uj)
.
= E[Ko +
Ki0
+
KiY^a''E'' \u],
(30)
n=l
which together with ki
=
Lemma
1.
Proof of
Eu =
E[ U{k,K,ak,0)
K2(1
A
+
— a)/a and
the definition of
gives the result.
Taylor expansion oi U(k, K, 0-^,0) around {k,K,ak, 0) gives
Uk{k,K,ak,0){k
+\Ukk{k - kf + \ukk{K -
-
k)
kf +
+ UK{k,k,ak,0){K - K) +
\u„„{ak - dkf
m
Ua{k,k,Uk,0){<Jk
+ UkKik - k){K - K)
].
-
-
..
<Jfc)
-
f'
..
By
the law of iterated expectations,
E[K\h]^E[E[K\9,h]\h]=E[E[E[k{uj)\4>]\e,h]\h]
=
It follows
=
that E[K\
E[kie;h)\h]
=
k{h).
E[k] and that
E[Uk{kk,ak,e){k Furthermore, because
=
E[E[k{Lj)\e,h]\h]
U
is
linear in
-
=
Ua{k,k,ak,0)
(t|,
Ucr{k,k,ak,0){ak
= E[UK{k,k,dk,6){K -
k)]
+
6-fc)
-Ucraiak
U^a^kio'k
=
k)\
-
0.
^k)- It follows that
- a^f = -U^^
[f^l
-
^l]
Next, note that
- kf =
{k
=
Because E[k]
E[K]
[{k
=
- K) -
=
E[k]
- k)f + (K -
{k
kf + 2{K - k)[{k -K)-
E[k] and because {K - k)
is
orthogonal to
[{k
{k
-
k)].
- K) -
{k
-
K)],
we
then have that
E[{k
-
=
icf]
Vax[{k
- K) -
{k
- k)] +
Ys.r[K
-
K].
Finally, note that
E[{k
because
(fc
— K)
Combining
-k){K- k)] = E[{K - kf~] + E[(fc - K)(K - k)] = E[{K - kf]
is
{K — k).
orthogonal to
all
the above results,
Eu =
E[U{k, k, ak,e)]
+^[/fcfcVaj:[A'
Using the
we thus have that
+ ^Uaa Wl -
- k] +
^l]
it/A'A-Var(A'
+
lUkky&v[{k - K)
-
A')
-
+ UkK^MK -
{k
-
k)]
A).
fact that
Var[(Jt
and rearranging, then
-
A')
-
(fc
- A)] =
Var[fe
- A] -
Var[fc
- A] =
al
-
af
gives the expression in (8).
Proof of Proposition
3.
A
strategy
Eu=
is
efHcient
if
and only
if it
maximizes
U{k{Lj),K{<l>),ak{(p),0)dfiu;,0)dnf),
JFJn,e
with K{4>)
=
Jq
distribution over
k{ui)d(t> {u))
fi
and
generated by
aki^P)
/.
=
[/f;['>^(<^)
The strict
-
K{4))\^d4)
(a;)]
with
(p
denoting the marginal
concavity and the quadratic specification of
36
U ensures
that a solution to this problem exists and
distribution of
distribution
Z {9\(j))
and
(f)
^=
unique
almost
for
and
first
imphed by
as
their joint
^/e/j,f/(^-M,A-(,^),afc(</>),e)d(^(u;)dZ(e|0)dG(0)
order conditions with respect to K{4>),
sufficient for optimality, are given
by the
and
(Tk{(t>),
+
(cj) dZ{9\<l,)
X{4>)
k(u}),
which are necessary
=
(31)
almost
for
^ UAHuj),
dG{<P)
following:
/e /n UK{k{Lo)J<{ci)),9)dcj)
Jq
cj),
denote the marginal
problem can be written as
for this
+ /^ \{cj>) [K{4>) - f^ k{uj)dcf> (a;)] dG(,^)
+ UV{<P) Hi^) - QH^) - A'(<^)]2#(a;)]
Therefore, the
G {4>)
Let
all u).
denote the distribution of 9 conditional on
The Lagrangian
J^.
is
K{(l>),ak{c^),e)dcj) {u) dZ{9\cf>)
all
(f)
+ 2'n{cP)ak{4>) =
(32)
for
/ex* [Uk{k{^),K{<P),
9)
-
-
Xi<P)
2r?((/))(fc (u;)
-
almost
iC(<^))]dP(^,
(\>\uj)
denotes the cumulative distribution function of
Using the facts that UK{k,K,9)
that Ua{k,K,ak,9)
=
is
= - [
<\>\uj)
{9,
almost
=
and
all u;
on
0) conditional
linear in its arguments, that K{(j))
Uaa^^k^ conditions (31)
A(<^)
(f)
(33)
for
where P[9,
all
=
J^ k
lo.
{oj) dcp {co)
,
and
(32) reduce to
UK{Kicl>),K{cP),9)dZ{9\cP)
=
-UKiKicf>),K{(l>),e)
Je
= -hUa
V{4>)
O'-'iTcr-
Substituting the above into (33),
it
satisfies
we conclude
the following condition for almost
E[
C/fc(fc,
K, 9)
that the strategy
all u;
£
f2
E[ UkiK*i9,9),K*{9,9),e)
+Uk{i^*{9,9),k*{9,9),9)
Now
ment
note that,
is
when
given by
Proposition
1,
iv
all
=
:
Q
^M
is
efficient if
and only
+ Uk{K, K,e) + Uaa[k-K]\u] =
linear, condition (34)
if
:
where, for simplicity, we have dropped the dependence of
Uk{k,K,9) and Uxik, K,9) are
A;
/c
on
u;
and
of
(34)
K
on
</>.
Because both
can be rewritten as
+ Ukk (k - k*{9,9)) + U^k [K - K*{9j)) +
+ {UkK + Ukk) {K - ti*{9,9)) + U,^{k - K)
Uke{9
|
a;
]
-
9)
=
0.
^^^^
agents follow the first-best allocation, then in each state aggregate invest-
k*{9,9). Replacing k*{9,9)
we thus have that the
first
and k*{9,9)
into condition (25) in the proof of
best strategy solves
Uk{^^l9),K^9,9),9) + UK{K*i9,9),>i*{9,9),9)
+ Uke{9'9) + {U,, + Ukk)[K%9,9)~K*{9,9)] =0
(36)
37
,
Substituting (36) into (35) and rearranging gives (10).
Proof of Corollary
Part
2.
degenerate" information structure
unique solution to (10)
is
if
and only
,
if
a=
=
given «
«;*,
and
(2)
(3):
for
any "non-
the unique solution to (4) coincides with the
a*. Next, consider parts
and
(ii)
When
(iii).
information
Gaussian and shocks are perfectly correlated
=
Ko
+
{ki
+
K2)
i'Jij.H
+ -jyV + JxX]
K =
Kq
+
{ki
+
K2)
triilJ'
+ hy + Ix)
+
+
+ -/x) d]
k
k = KO + (ki + K2)
=
k
It
T
from Propositions
follows
(i)
[yf,fj-
{-^y
,
-yys]
follows that
vol
The
=
results then follow directly
Proof of Proposition
+ K2) jyf al
[(ki
from
(6)
and
[(ki
+
^2)
Ixf
o-|
'
and
-
(12).
T
Given any policy
4.
=
dis
£ T,
let
U [k, K, ak,e,e)=U {k, K, ak,e) -
T{k, K, ak, 0)
denote an agent's payoff, net of taxes.
Now
offs
Ukik
let
k
:
Q
U
are given by
{6; h)
,k{h),
—
H
X
f
M
Because
.
9, 9)
=
0,
denote the complete-information equilibrium strategy
U
is
concave in
fc,
k
^ Jk (9'\ h) dh [9')
with k{h)
must
{6; h)
.
when pay-
solve the first-order condition
Because
U
quadratic in
is
(fc,
K,
9, 9),
the first-order condition can be rewritten as
Uk
Integrating over
9,
(0, 0, 0, 0)
we then have
Uk
Combining
(37)
+
+ UkKK
Ukk~k {0; h)
+
[Ukk
with (38) then gives k
+
{9; h)
Ukx]
=
k (h) +
k{9, 9)
-Ukk
+ UkK
-Ukk
-Ukk
=
Ukk
[Uk0
ko
- UkK +
=
0.
+
+
ki9
Uks] ^
+
=
0.
k29 with
Tkk
+ TkK
+
+ Tkk
Uke
Ki
—
UkfO
Uke
Uke
Kl
Uke
+
[/fc(0,0,0)-T^- (0,0,0)
f/fc(0,0,0,0)
Ko
K2
Uke9
(37)
that
(0, 0, 0, 0)
Ukk
+
{h)
-Ukk
+ UkK
38
Uko-n^ke
-Ki
- UkK + Tkk + TkK
(38)
('
•
'
^.
(
— —
Using
can then be conveniently rewritten as foUows:
(23) the coefRcients (kqi'^Ii^o)
'^o
=
Ki
=
'^2
=
(l-a)ACo+tr77fc
^—
rV^
(0,0,0)
r^=r-
(39)
1
ri^«i
-—
= —1
Normalizing Ukk
r^-^i
rrf^
(41)
then gives the formulas in the proposition.
Next, consider the
K{(t))
+ K2) +^?;-e-
a) (Ki
(1
(40)
game under incomplete
— J k{uj) d(j) {uj) A
.
best-response
information. Take any strategy
a strategy
is
fc'
^ R such that,
Q
:
/c
Q — R and
>
:
for all
uj,
let
k' {cu) solves
the first-order condition
K (0) ,ej)\uj] =
E[Ukik',
Using the
and the
fact that Uk{k',
fact that
K,
=
6, 9)
Uk{k{e, e),k{0,
k solves Ukik{9,0),k{9,e),9,9)
=
k'{Lu)
E[ki9,9)
(42)
+ Ukk
6), 6, 6)
=
0.
{k'
for all (9,9)
,
+ d{K{(P)-k{9,9))
- k{e, 6)) + UkK {K - k{e, 9))
(42) reduces to
\uj]
with
^ UkK ^ UkK - TkK ^ ^ + it-Tf^K
-Ukk + Tkk
-Ukk
i-wrJkk'
-
In equilibrium,
fc'(a;)
follows from the
=
same arguments
Proof of Proposition
a*
ko
,
=
Kq,
it is
ki
=
thus suffices to prove that there exists a policy T* €
unique. This
only
is
easily
shown from conditions
to (13) exists
and
is
unique
2.
J-.
For the equilibrium
necessary and sufficient that
k2
Kj,
T that
and
(39)-(41)
=
h^.
(44)
satisfies (44)
(43).
and that
First, note that ki
this policy
—
kJ
if
is
and
if
Tu-
Because Ukk
a=
That a solution
as in step 2 in the proof of Proposition
efficient strategy,
a=
a*
if
+ Ucra <
C/fcfc(l-Kl/Kl)
this also guarantees that
= -Ukk {a -
That Tkk and TkK
With
=
Ukk
=
(45)
-C/aa.
— Tkk <
0.
Next, note that, given T^k
= —Uaa,
and only
TkK
0.
(13).
Take any generic information structure
5.
with policy to coincide with the
It
which gives
k{u}) for all tu,
{Tkk,
satisfy
a*)
- Tkka* =
- ^^^I^^^' <
1
-UkkOc
+
{Ukk
+
U,a) a*
=
U,, - UkK - Ukk-
then follows firom the fact that Wyoi
= Ukk+'^UkK+UKK <
TkK) thus determined, and with both T^g and Tk (0,0,0) being unconstrained,
39
(46)
it is
then immediate that there exist a unique
'^0
=
^0' these are given
T^g
=
Ukk
-
Tf, (0, 0, 0)
and that
T to balance
Tkk + T^a — 0.
that this
is
=
R,2
+ kD -
a) [{k\
-
Ukk
(1
-
(Ki
+
-
acq)
a) (kS
-
K2)]
-
{Tkk
it
=
re-(o,o,o)
Tj^(0,0,0)
=
-Tfc(0,0,0)
=
-'^k0
era
—
-'kk
6.
0(1
+
T is unique.
%
=
=
=
A
¥.[T{h,k,akM^i]
,
(0, 0, 0)
i\
,
0, ^)
(47)
.
=0 for all
[K, 6)
then easy to verify
it is
o
TkK
<
increases with
and W^js
=
a and
+
Ukk
decreases with
U^a <
0.
terms in (14) do not
let
fcj
=
ki{l
+ + Ui),
'rj
around
affect individual decisions.
K = K{1 +
77), a-|
K, (Jk,6) then
=
a-|(l
For
+ r?)^
Taylor expansion of T{ki,K,
o-fc,
= E[Tiki,K,ak,e)+^Tkk
+ a^) k1+TkK'jlk,K+\TKKCjlK''+\T,,ayk\
(a^
a
6)
gives
^i\
redefined as follows:
Kl
=
"'
_
~
a
- UkK + TkKil + ^P +
+
a2
+ a2)
-Ukk-UkK + TkKa + '^^^)+Tkk{l +
<7^
+
-Ukk
Tkk (1
Uk0
-Ukk + Tkk{l + a^ +
al)
Uke
UkK-TkKJl +
a^^)
+ Tkk{l +
+
-Ukk
<T^,
-
Tk§
~
"'
<^^)
C7i)
implication. Proposition 5 also continues to hold, although
a^.
T(fc,:,
t/fc(0,0,G)-Tfc (0,0,0)
Kq
depend on a^ and
T (i\
= -Uko,
For the case of additive measurement error, the result follows
Proposition 4 thus continues to hold with k and
By
K^)
t/x (0,0,0)
Finally, that
(14) noting that the last three
?).
+
= Uk9
the case of multiplicative measurement error,
=
such that
= -2TkK-Tkk = -U,, + 2UkK + '2Ukk
TkS
Proof of Proposition
(Kt
identified above,
a* follows directly from (46) along with the facts that Ukk
and 9
(0, 0, 0)
folloviring:
r(o, 0,0,0)
This also implies that the policy
+ Uk)
must be that
Along with the other properties
equivalent to imposing the
from
^ unique T^
+ TkK) ^5 = -Uk
{Tkk
the budget (state by state)
Tkk
directly
''''^^'^
1^2
by
(1
Finally, for
such that
Tf,g
u
now
the optimal tax contingencies
'
-
40
Proof of Corollary
Prom
3.
(45), (46)
and
we have that the
(47),
restriction
k
—
k* implies
that
Tkk
The
tax system
=
TkK = -Ukk {a-Oi*), and T^e = -TkK
0,
thus linear in
is
,
Next, recall that
6
and
JK. It
positive,
,
K = E[K\h]
which in turn
is
true
Tk{K,
J?" of 9 (which
n
=
e
is
and only
Because
4.
measurable in 0)
is
common knowledge and
(p is
9)
and hence
if
^2)
with a marginal tax rate given by
foUows that Cov{Tk{K,d),e)
Proof of Corollary
because
fc,
+
(«i
also
= K — K is orthogonal to any function
= TkKVar{e), which is positive if and
if
(p is
+ UkK + T^gO.
(0, 0)
q > a*
only
if
T^k
is
.
common
knowledge, the n-th order average expectation
common knowledge and
equal to E^, for aU n. Moreover,
= E [9\h] = E [9\h,
because 9
of h, including
cf)]
,
we have that
E^ =E[E[9\Lo]\<p]=E[E[9\uj,(P]\(P]=E[9\4>]^E[E[9\h,(t)]\(t>]^E[9\<i)],
Combining these
results,
,
we have
that
00
-.
=
^
^
((1
-
Q)a"-i)
^" = E
[^|(^]
= E [%]
.
n=l
Prom
Corollary
1 it
;,:-,.
in
which case a
is
then follows that the unique equilibrium
,
.(
.
,
k{u) =E[k(6',^)
1,
irrelevant. Similar
is
also irrelevant.
=
Along with the
given by
,
arguments imply that the
kiuj)
so that a*
\uj]
is
efficient allocation is
E[K*{9,9)\uj],
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
1.
=
0.
for
i
Because information
=
structure
1.
2 establishes the result.
41
is
3.
exogenous in the
is
now endogenous but uniquely
That the equihbrium strategy
unique and solves (15) thus follows again from Proposition
>
T^k
eflBcient
unique and solves (15) follows directly from
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
=
uJi^i
Start with
10.
=
t
1.
In the
first
period, information
exogenous
is
Ai). Standard Gaussian updating then gives
(xj,!, j/i,
IE[%u]-^^U + ^n,
=
k{9)
=
Xi,i
ko
+
x^j,
{ki
+
=
Trf
K2)0,
=
cr-j, Fi
-2
^-2
-2
where
^^ + ^2/1 + -^Ai
we then have
(48)
tt^
=
that the unique solution to (15)
is
and
a^^
_^
^-2
^-2_ ^gj^^g
_^
given by
+ {Kl+K2)hlX^,l + {l-Jl)Yl),
kl{u!^,l)^Ko
(49)
with
7i^[(l-a)7rf]/[(l-a)^f +
To
see this, start
by guessing that the equilibrium strategy
Next, use this guess to compute aggregate activity as
and
Finally, use the latter along with (15)
Next, consider
signal
is
t
=
2.
=
+
(ki
+
6 contained in
Ki
is
Ko
Ki- Ko-
a),:^2
=
^^1,1
K2) (71^
+
(1
+
U
(xi,2,y2,-^2,-^i.
=
^2/[('^i
+ K2)7i]
is
common
knowledge.
-
{I
-
^i)Yi).
^^i).
The endogenous
^'l)
+%
+ K2){l-'yi)Yi
+ K2)Ji
Gaussian noise with variance
It
7l)
6,
cr? 2
=
'^^,2/ ('^i
+
because (49) imphes that
'^2)
cti
—
7i
•
(ki
The signal
+
follows that the period-2 public information about 9 can
K2)
ai,
7iC^,ii
be summa-
on {yi,Ai,Ki,
rized in a sufficient statistic Y2 such that the posterior about 6 conditional
is
+
K2) [jiO
{ki
on the other hand, conveys no information about
is
{ki
thus the same as that contained in
(Ki
which
+
given by
The information about
772
kq
coefficient 71.
(48) to derive the equilibrium 71.
In the second period,
Ki
where
some
satisfies (49) for
=
A'l
7rf].
i5-i,j/2,
^2)
—
Gaussian with mean
V2 + 7i(ki+K2)^T^.y^
^IV ^^^2
^
^^^
^^ = ^^^ +
+
ni
ii^'
.
,
,
,
and precision
summarized
is
tt^
=
tt^
+
cf~1
+
a~^
+
jf (ki
in the sufficient statistic Xi^2
+
K2)^
cr,7J-
Similarly, the private information
conditional on
such that the posterior about
Gaussian with mean
Trf
and precision
Trf
=
ti"!
+ cr^ 2- The
^2 (Wi,2)
With 72
=
[(1
-
a) ^f /
]
[(1
-
a)
unique solution to (15)
=
Trf
Trf
Ko
+
+
(ki
+
K2) (72^i,2
Tr|]
42
is
+
then given by
(1
-
72)
^"2)
,
':
can be
(x^^i, a;j,2)
,
It is
t
>
immediate that the construction of the eciuihbrium strategy
for i
=
3 with the statistics Xi^t an Yj defined recursively as in the proposition.
unique equilibrium strategy
7f
=
-
[(1
a) nf] I
[(1
-
Proof of Proposition
We
any
conclude that the
is
=
ki,t ii^i,t)
with
2 applies also to
q)
+ Trf
vrf
11.
Ko
+
]
+
{ki
K2) (7t^i,t
+
(1
-
It) Yt)
.
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
(ii)
then extends the result to general payoff structures.
Suppose
(i).
V {k,K,a,9)
does not depend on {K,cr) and, without any further
loss of
generality, let
V{k,K,a,e)^~{k-0f
Let hf
=
.
{yi,Ai,Ki,...,yt-i,At-i,Kt-i:yt:^t} denote the public history in period
agents follow a strategy
k =
and suppose
t
{fcf}^i such that
t
h
=
{^l,t)
Pt {ht)
+
^
Qt,TXz,T,
T=l
where Pt
(ht) is
a deterministic function of ht and Qt^r are deterministic coefficients.
h,t
=
-Pi
+
7t^
It
follows that
+ X] QtAhT,
T=l
and hence Kt
=
Pt
+ 7t^ + 7?t+i,
where Pt
is
a shortcut
7t
for Pt {ht)
and
74 is defined as
Y^Qi.r.
r=l
Next consider
welfare.
'
Given any
Wt
= E[v
linear strategy, ex-ante utility
(fci,t,
= E [v {h,t,0)] -
At+i)]
is
Eu —
J2t=i ^t, where
alt+,
and where
E[v{k^^t,e)]
= E E
= E
Now
consider a strategy
k
-{{Pt +
{Pt
=
t,
+ Y!,^,
QtA^,r) - 0}'
,ht
+ itO-e)'-J2[^,QlAr_
{kt}^=i that
constructed as follows. First, pick an arbitrary
period
ite
is
t
and
let ki^s{'^i,s)
=
k
ki^s{^i,s) for all s
<
t.
=
It,
initial
pick an arbitrary function Pt and any coefficients Qt^r such that ^^.^i Qt,T
t
ki(^l,t)
=
Pt (ht)
+
^
T=l
43
Qt,TXi,T-
=
strategy
a variation of the
{fct}f=i
Next, in
and
let
—
Finally, for all s
>
let
t,
s
T=l
where the functions Pg are such that
By
^
construction, at any period s
implication the
same per-period
k
condition for the strategy
welfare level wt, as the initial strategy k.
that, for all
is
argmax^E
s.t.
is
the case
and only
if
=
Pt{ht)
Next note
Kt
+
T]t
=
Pt
+ 'Jt.O +
rjt
for ail
t
and
and
follows that a necessary
all ht,
ht
It
all ht
and
{l-^tW'[0\ht\
that, because Pt
^
y1t=i Qt,T
if,
t
It
+ jtO -9) - Y!^^^ QlAr
[Pt
Pt,Qt.r
This in turn
k induces the same outcomes, and by
the strategy
t,
to be efficient
G
{PtAQt,T)r=i)
+ Pt{ht)
^Ps[...,kt-Pt{.ht)
P,(^...,Kt,..)
Qt,,
=
-2 ^^-
^f
7t
public information, the observation in period
is
(50)
•
t
+
\
oi
Kt
=
informationally equivalent to the observation of a signal
is
yt+i
—
=
=0 + m+i
(51)
It
where
r]t+i
—
rit+i/jt
linear strategy, the
is
common
precision n^, where Yt
and
conditions Yi
—
y-j^t-i
—
v-yt~^
-I
-2
= <-l + (^y,t
+
V
1/
—
t is
lt'^^,t+\- ^^ follows that, given
Gaussian with mean
hq and
,
I
tt^
=
—2
^a,t
y'^t'^
2
TTf
—
irf
+ nf
Xi,t
with
initial
conditions Xj j
,
cr^^. Similarly,
—
and
yt
'
the private posteriors are Gaussian with
where
= -^Xi^t-i +
=
Yt
-2
+ lt-\^ri,t
,
y
mean
and precision
E [d\ht] =
any
are defined recursively by
—
n
initial
posterior about 6 in period
7rf
yt
with
=
Gaussian noise with precision o-^f+i
x'j^i
and nf
-~Xi,t
=
a^-y.
44
and
nf
=
Trf.j
+ ajf
.
!(-
Now
note that
^^2
r=l 2^j=l ^x,j
which together with (50) gives
s
r=l
We
k maximizes
conclude that a Unear strategy
h^t
for
some
coefficient
To determine
"ft
G
=
(wm)
ex-ante utihty only
-
(1
+
It) Yt
if,
for all
t
and
all ui^^t,
(52)
'itX^^t
^
when
the optimal 7's, note that,
E
[v
)-i
=-
{h,,e)]
agents follow a strategy as in (52),
(TTf
{7?
+
(1
-
7t)' i-^!)-'}
and hence
Eu ^ Wfb Ylti
where
6).
WpB
^*"'
b'
+
(^*'^)"'
(^
-
^^)'
^""tV'},
the first-best level of welfare (the one obtained under complete information about
is
Because the evolution of nf does not depend on
the choice of the optimal linear strategy
7t,
reduces to the following problem:
mm
i?E;
(<)"'+ (1-^*)'
./5*-^{^'
(-')"'}
{7,
s-t.
with
initial
condition
=
iv[
a^"^
,
=
Trf^i
where Af
Trf
=
+ Ai + C7-272
cr^j^
-I-
a~J
is
Vi
the exogenous change in the precision of
public information.
Consider the value functions Lt
L,(7rf,^f)
=
:
M+ —
< N,
Lt
i,
(rrf ,7rf)
(Trf, Trf)
must
= min (7?
7f is
4+, =
4 + A, + (T-h^
(Trf)-i
7rf+i
=
+
(1-7^)2
(Trf )-^+/3Li+i(Trf+i,Tr,V)}
+ At + ^tlt
nf
the solution to the following
7,(7r,)
vs>t
satisfy
s.t.
and hence the optimal
(l-7«)'(7rf)-'}
/s=t
s.t.
all t
defined by
ffi+
^""z^-' {7,^(0-^ +
min
1.7s
For
>
-(l-7.)(-/)
45
FOC:
+2^a^^^=0-
,
^From the envelope theorem,
Finally,
from the low of motion
for rrf
-2
It
alH <
follows that, for
= N,
t
(ii).
the optimal 7j satisfies
1,
'y'^
7if
=
nfj/ (tt^
no more an informational
Part
—
_
*,
Finally, for
A^
=
must
information.
Ex
Xn and
where
Wacr
<
Wfb =
0,
Wkk
Et^i
<
U
Vr
is
(k, k, 0, ^)
with V.
A
and
=
similar
kJ
let
+
+ ^2)6
{k.\
argument as
in part
(i)
ensures that the
=
+
k5
{k\
+
K*^) {-itXit
+
(1
-
It) Yt)
,
and public
(^'-'
('^*
iA +
-2) I
(^) '0' ^)
W^K/^aa =
1
'"'
"%
2
(7.)^ (-f
'^')-^
+
"%(!Itf «)-^
2
is
the first-best level of welfare.
—
Q*,
we conclude
Using the
s.t.
(rrf ,7rj')
Lt(7rf,7rf)
FOC
=
^f
+ A, +
(k*
+
K*2f a-jl^
denote the associated value function in period
=
niin{7,'(7rf)-^
s.t.
The
nf^^
Trf+i
=
Vf
t,
we have
+ (l-a*)(l-7*)'K')"'+/3imK+i,^f+i)}
Trf^i
=
7rf
+ Aj +
(k^
+
^2)^ <^nh'i
for 74 gives
^'W)--(I-a-)(.-.,)W)- +
46
fact that
that the optimal 7's must solve the
following problem:
Letting Lt
is
then given by
.
/3*~^^
and
0,
V
Yt are the relevant sufficient statistics of available private
ante utility
E. = W,s +
the last period and hence there
is
satisfy
kt {uJit)
with
max V
arg
in (28) replacing
efficient linear strategy
74,
simply because this
,
Consider now the more general payoffs
with {kq,i<[,K2) as
some
tt^)
externality.
K* [6)
for
+
7rf
i^f^^=0,
The envelope
condition for
gives
7r^^.j
while the law of motion for
gives
7rj'^_j
'^"t+i
It
o
/
*
-2
*\2
,
follows that the optimal 7's satisfy
il-a*)nf
=
7t
nf
+
(1
-
a*)
TTf
-(3
{I
-a*) {I-
j^;,)' nfnf
«i)
(k*
+
ac*)^
a'J^,
which completes the proof.
Proof of Proposition
12.
linear strategy as in Proposition 11
{7"}^i be
Let
and
a"
The
efficient linear strategy.
be the unique solution to
(l-ar)^f
it is
result then follows
from
.
(1
In fact,
be the corresponding precisions of private
let {Trf ,7rj'}f^^
and public information generated by the
letting
the coefficients that characterize the efficient
- ar) nf +
^.,..
^*
TTf
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 Lf obtained along
the efficient linear strategy are decreasing functions of nf and
public information available in the beginning of period
t.
7rf
,
the precisions 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 chai'acterizes the efficient strategy.
47
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