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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Optimal Coordination:
Characterization and Policy Implications
Socially
George-Marios Angeletos
Alessandro Pavan
Working Paper 06-35
September 21, 2006
RoomE52-251
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02 142
Cambridge,
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Social Science Research
1
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.
at
^
SS ACHUSETTS INSTITUTE
OF TECHNOLOGY
.LIBRA RIES
f
\
Socially
Optimal Coordination:
Characterization and Policy Implications*
Alessandro Pavan
George-Marios Angeletos
MIT
and
NBER
Northwestern University
September 2006
Abstract
In recent years there has been a growing interest in
macro models with
het-
erogeneity in information and complementarity in actions. These models deliver
promising positive properties, such as heightened inertia and
also raise important normative questions, such as
and
volatility are socially undesirable,
rect the
way agents use information
But they
whether the heightened inertia
whether there
in equilibrium,
volatility.
is
room
for policies
that cor-
and what are the welfare
We
effects
of the information disseminated by the
media or policy makers.
key to answering
the relation between the equilibrium and
all
these questions
is
the socially optimal degrees of coordination.
argue that a
The former summarizes
the private
value from aligning individual decisions, whereas the latter summarizes the value
that society assigns to such an alignment once
all
externalities are internalized.
J EL codes: C72, D62, D82.
Keywords: Dispersed information, coordination, complementarities,
volatility, in-
ertia, efficiency.
'This paper was prepared for the papers and proceedings of the 2006 meeting of the European
Economic Association
and-proceedings piece.
in
Vienna. This extended version includes proofs that are omitted
We
are grateful to
SES-0519069 and SES-0518810).
NSF
for financial
in
the papers-
support (collaborative research grants
Introduction
1
In recent years, there has been a growing interest in models that share the following
two key
features:
damentals; and
game
heterogeneity in information about the underlying economic fun-
(i)
complementarity
(ii)
Examples include the beauty-contest
in actions.
Morris and Shin (2002); the investment games in Angeletos and Pavan (2004)
in
and Angeletos, Lorenzoni and Pavan
by Morris and Shin
in this issue;
(2006); the
common- interest game
and the business-cycle models
Hellwig (2005), Lorenzoni (2005), and Roca (2005).
in
Woodford
volatility
(i.e.,
in the
paper
(2002),
i
These models deliver interesting positive properties, such as inertia
sponse to changes
in the
(i.e.,
slow re-
underlying fundamentals) and heightened non-fundamental
high sensitivity to
common
noise in information about the underlying
fundamentals). But they also raise important normative questions.
heightened inertia or volatility due to complementarity socially undesirable?
1.
Is the
2.
Are there
policies that could
manipulate the way agents use information, and
thereby correct the sensitivity of the equilibrium to both fundamentals and noise?
If yes,
3.
How
how do
these policies look like?
does the incompleteness of information
affect,
welfare?
What
the social
is
value of the information disseminated by prices, market experts, or the media?
Should central banks disclose the information they
make about
the
economy
in a transparent
collect
and
the forecasts they
and timely fashion, or
is
there
room for
"constructive ambiguity"?
To answer these questions, one needs:
(1) to
compare the equilibrium use
of in-
formation with an appropriate constrained efficiency benchmark, namely the use of
available information that maximizes welfare; (2) to identify policies that
implement
the efficient use of information as an equilibrium; and (3) to understand the comparative statics of equilibrium welfare
with respect to the information structure.
Ongoing work (Angeletos and Pavan, 2006a, b) undertakes these tasks
class of
in a
broad
economies with heterogeneous information, externalities, and strategic comple-
mentarity or substitutability in actions, and discusses a variety of applications. In this
'This class of models differs from global games in that the coordination element
that the equilibrium
also
is
is
moderate enough
unique no matter the precision of private and public information. Related are
models with "inattentive" agents,
as in
Mankiw and
Reis (2006).
we
paper,
restrict attention to
information
is
a (sub)class in which inefficiency emerges only
when
incomplete, thus isolating the inefficiencies that originate in the use of
2
information from other possible distortions.
This
facilitates
tions above
is
coordination.
the main message of this paper: the key to answering
identified
down
pins
the ques-
the relation between the equilibrium and the socially optimal degrees of
The former
identified with the slope of
is
respect to others' activity, and pins
is
all
down
an agent's best response with
the equilibrium use of information; the latter
with the slope that would make agents internalize
The former summarizes
the efficient use of information.
from aligning individual decisions; the
externalities,
all
latter
and
the private value
summarizes the value that society assigns
to such an alignment.
A
2
There
simple model
is
a continuum of agents, indexed by
choosing an action
£
fcj
which makes transfers
Payoffs. Agent
R
tj
i's
(
G
K
is
f
(kj
—
ki)
L= J
Lidi
and uniformly distributed over
e -g-i think of k as investment).
to the agents, subject to
payoff
Ui
9
i
is
m + U,
= ~{h -
2
9)
is
is
also a
is
budget balance, J
tidi
each
government,
=
0.
where
-
r (Li
an exogenous random fundamental
dj
There
[0, 1],
the mean-square distance of
- L) (e.g.,
i's
r*L.
aggregate productivity), Li
=
action from other agents' actions,
the average of these distances, and r and r* are non-negative scalars.
This payoff structure has a simple interpretation. The term
value of taking an action that
is
—
(ki
9)
aligned with the fundamentals, whereas the term Li
introduces a private value to aligning one's action to those of others
strategic complementarity.
The term
which controls the discrepancy,
if
captures the
any,
— the
source of
Z, on the other hand, introduces an externality
between the private and the
social value of such
alignment.
2
Actually, the results presented here extend to economies where the distance between the complete-
information equilibrium and the first-best
is
non-zero, as long as this distance
is
invariant with 6 (see
Angeletos and Pavan, 2006a,b.) Thus, one can think of the class considered here as capturing businesscycle models in which, under complete information, the "output gap"
the business cycle.
is
non-zero but constant over
Indeed, since
payoffs were
is
as
is
given by
if
L
does not depend on agent
u^
"1
e
11
w=
= — (fcj -
9)
— rL
—
I
[—(kj
/
Uidi
Hence, from a social perspective
9)
—
2
it
2
as
is
action,
z's
t
from a strategic viewpoint
Aggregate welfare, on the other hand,
.
2
—
—
9)
r*Li]
di.
payoffs were given by
if
it
uf ctal = — (fcj
-
r*Li. In this sense, r parametrizes the private value of aligning choices, while r*
parametrizes the social value of such alignment.
Remark. While
the particular payoffs assumed here admit a convenient inter-
pretation, the key assumption
only that inefficiency vanishes once information
is
is
complete (Angeletos and Pavan, 2006a, b). Here, the complete-information equilibrium
and the
both given by
first-best allocation are
Information.
mean
3
is
it
important to allow agents
for concreteness,
9;
we adopt a Gaussian
Before agents move, nature draws 9 from a Normal distribution with
and variance
/i
6,\/i.
For the purposes of this paper
to have heterogeneous information about
specification.
=
fcj
The
c|.
agents observe private signals
are, respectively, idiosyncratic
realization of 9
a?;
= 9+&
is
not observed by the agents. Instead,
and a public
and common Normal
as well as of 9, with variances a\
and a
signal y
noises,
= 9 + s,
where £ l and e
independent of one another
2
,.
Equilibrium degree of coordination
3
We
start
by characterizing equilibrium without government intervention
we reintroduce the government
i)\
The best response
of agent
i
ki
where a
=
r/(l
+
r)
K =
and
in Section
solves
J
kjdj.
=
for all
5.
d^Ui/dki
= {l-a)E
(£*
l
9
—
0,
which reduces to
+ aE,K
That
is,
the best response of an agent
(1)
is
a
weighted average of his expectation of 9 and his expectation of aggregate activity K. 4
The
equilibrium
is
then given by the fixed point to
Also note that the payoff structure assumed here
is
nested in the more general one considered in
= U(ki,K,o\,K) = —(1 + r)kf + 2k 9 +
2
mean, and al = J{k-j — K) dj the dispersion,
Angeletos and Pavan (2006a,b) by rewriting payoffs as Ui
2rk
x
K—
(2r*
— r)a k — rK —
9
,
where
K
= J kjdj
is
the
of activity in the population.
4
For any random variable
z,
we
let E,-2
=
Et[z|xi,j/].
this best-response condition.
t
The
K
slope of best responses with respect to aggregate activity
a simple increasing function of
tions with one another;
equilibrium behavior.
accordingly
call
a
0,
condition (1) reduces to
5
are the familiar Bayesian weights.
simply the best predictor of
kj
the equilibrium degree of coordination.
E,#
That
When
9.
=
is
to coordination impacts
This coefficient plays a key role on how agents use information
When a —
which here
agents care to align their ac-
summarizes how the private value
it
We
how much
captures
r,
',
is,
instead
—
in equilibrium.
+ X y y + X x x where (A^, A y A x
when a = 0, the equilibrium action is
A^/ii
a >
i,
.
the equilibrium
0,
is
)
given by the
linear strategy
h=l i^ + l
l
where the
That
is,
+ lxX
i
(2)
,
by
coefficients (7^,7j/,7a;) are given
1
yy
> Ay
— a\ x
7X
,
—<A
=
1
- aX x
x
(3)
.
a positive degree of coordination increases the reliance of equilibrium actions to
the prior and to public information, and decreases the reliance to private information. 6
The
logic for this result
is
simple.
The
prior
and the public signal are
better predictors of others' activity than the private signal.
The higher
relatively
more
a, the
agents care to align their choices, and hence the more they find
it
optimal to rely on
and
"yx
decreases with a.
y.
It
follows that
both 7^ and jy increase with a, whereas
The equilibrium use
of information in turn determines
sponds to both fundamentals and
write aggregate activity as
follows that a higher a,
activity to the
K=
noise.
7^/i
+
Using 7^
(1
—
7^) 9
by increasing 7^ and
fundamental and increases
its
-y
y
,
how aggregate
[i
activity re-
+ jx + jy = 1 and y — 9 + e, we can
+ jy e, where e is common noise. It
reduces the sensitivity of aggregate
sensitivity to
higher degree of coordination increases both inertia and
common
volatility.
noise.
That
is,
a
At the same time,
because a higher a reduces the reliance on private information, and hence the sensitivity to idiosyncratic noise, it also
reduces the dispersion of activity in the cross-section
of the population.
5
6
a e 2 /cr 2 A y = a y 2 /a 2 and \ x
See the Appendix for the derivation of (2) and
That
is,
AM
=
,
,
s
a x 2 /a
(3).
2
,
where a
2
=
aB
2
+ ay 2 +
2
cr x
.
Socially optimal degree of coordination
4
We
next turn to the characterization of a particular constrained
This allocation
behind the
lation. It
the strategy that maximizes ex-ante welfare
is
veil of
efficient allocation.
expected
(i.e.,
utility
ignorance) taking as given the dispersion of information in the popu-
can be represented as the solution to a planner's problem, where the planner
can perfectly control how each agent uses
his available information,
but can not transfer
information from one agent to another.
As
it
turns out, the efficient allocation
hi
=
where a*
2r* / (1
+
2r*)
.
=
(1
the strategy that satisfies
is
- a*) Erf + a*EiK,
Condition
is
(4)
the analog of (1) with a* replacing a.
This suggests a simple interpretation of condition
a* then summarizes
how much
we
The
efficient allocation
can be
of coordination perceived
by the
(4).
implemented by manipulating the equilibrium degree
agents (for example, through taxes, as
(4)
will see in the
The
next section).
coefficient
We
the planner wants the agents to align their actions.
accordingly identify a* with the (socially) optimal degree of coordination. 7
Just as
a
pins
down
the equilibrium use of information, a* pins
use of information: the efficient allocation
where the
if
between the equilibrium and the
any,
determined merely by the discrepancy,
if
a > a*
.
The answer
1.
If
a > a*
,
would
mentarity in their actions.
implication,
efficient
use of information
between a and a*: the
to public information
a and
is
sensitivity of the
inefficiently
high
But
if
a <
were
to perceive a
lower comple-
a*, then the heightened levels of inertia
but
it
and
anything but excessive.
for the derivation of condition (4).
A
similar condition characterizes the efficient
use of information in the richer class of economies considered in Angeletos and Pavan (2006a).
general, the
is
a*
be higher if agents
volatility featured in equilibrium are
Appendix
By
a*.
then the inertia and the volatility featured in equilibrium are
inefficiently high; welfare
'See the
a with
to the first question raised in the introduction thus
reduces to a simple comparison between
Result
any,
mean and
equilibrium allocation to the prior
and only
if
the efficient
given by
coefficients (7„,7y,7x) are as in (3) replacing
the discrepancy,
if
is
down
mapping from the underlying payoff structure
to the coefficient a*
is
remains true that a* summarizes the social value of aligning choices across agents.
6
In
not a simple as here,
Optimal policy
5
When
the equilibrium use of information
inefficient
is
/
(a
a'), a novel role for policy
emerges: welfare can be enhanced with policies that manipulate the agents' incentives
In our framework, this can be achieved with a relatively
to align their decisions.
simple linear tax system, which
system
— the key
is
to
make
is
the incomplete-information analogue of a Pigou tax
the tax rate contingent on ex-post aggregate activity.
Consider the following tax scheme. Transfers take place at the end of the game,
made
once agents have
directly, or
it
infers
it
government either observes 9
their choices; at that point, the
by observing
K
and
y.
The
made
transfer
to agent
is
i
then
given by
ti
where r (K,
is
9)
is
= -r(K,9)k + T(K,9),
l
T (K, 9)
the marginal tax rate and
for
some tk,t$ G K; the
=
coefficients
rate to aggregate activity
(2
+
2r)(TK
tk and
tq
The
transfer.
tax rate
Agent
i
K + Tg9),
parametrize the sensitivity of the tax
and to the fundamental, while the term
normalization. Finally, budget balance imposes
T (K, 9) =
anticipates that the tax rate he will
aggregate activity K. His best response
h-
if
lump-sum
given by
T(K,9)
It
a
(1
-a-
is
re )
r
(A', 9)
pay per unit of
2
+
2r
is
just a
K.
fc;
depend on
will
thus given by
E
t
9
+
(a
- rK ) E
t
A'.
follows that the proposed policy implements the efficient allocation as an equilibrium
and only
by the
if
tk —
ce
—
a*
difference between
Result
2.
Any
= — tq.
a and
Hence, the optimal contingency of r on
K
is
dictated
a*.
inefficiency in the inertia or volatility of the equilibrium allocation
can be corrected by appropriately designing the contingency of the marginal tax rate on
aggregate activity.
The optimal tax
rate
must increase with
K
if
and only
if
a > a*
Social value of information
6
We now
show how the
relation
statics of equilibrium welfare
We
find
with respect to information. 8
decompose any change
useful to
it
between a and a* helps understand the comparative
accuracy and a commonality
We
effect.
in the information structure into
an
identify the accuracy of available information
with the reciprocal of the total noise in the agents' forecasts of the fundamental, and
9
That
commonality with the correlation of noise across agents.
its
—
Ejf?
denote agent
commonality
Welfare
as 5
—
forecast error,
i's
Corr(uii,u>j), for
i
we
^
a~ 2
define accuracy as
letting
is,
— 1/Var (tJi)
if
they knew
common noise
that
j.
is,
(i.e.,
These errors manifest themselves
6.
variation in aggregate activity
sectional dispersion, that
is,
in
what they
two dimen-
noise in public information) generates non-fundamental
Second, idiosyncratic noise
level 6.
and
lower under incomplete than under complete information because the
is
would have done
volatility,
=
9
noise in the agents' information induces "errors" in their actions relative to
sions. First,
u){
(i.e.,
K relative to the complete-information
noise in private information) generates cross-
variation in individual activity k across agents.
Both types
of errors contribute to lower welfare.
An
increase in accuracy for given commonality
—reduces
composition of noise
welfare.
An
in total noise for given
both types of errors and hence necessarily increases
increase in commonality for given accuracy, on the other hand, substitutes
one type of error
for another:
Whether
volatility.
—a reduction
in equilibrium
this increases welfare
it
decreases dispersion but can increase
depends again on the relation between the
equilibrium and the socially optimal degree of coordination.
Result
mation,
welfare
3.
(ii)
is
(i)
If
Equilibrium welfare necessarily increases with the accuracy of infor-
a < a*
welfare also increases with commonality;
,
non-monotonic
To understand part
if
instead
a > a*
,
in commonality.
(ii),
note that,
when
the planner chooses the optimal degree
of coordination, he effectively faces a trade off
between dispersion and
volatility:
the
higher the degree of coordination perceived by the agents, the lower the sensitivity
8
We
focus on equilibrium without government. Since the optimal policy restores any efficiency in
the equilibrium use of information, the welfare effects of information in an economy with optimal policy
coincide with those in an
It is
economy where a = a*.
2
o~ 2 = a^ + Oy 2 + u^T 2 while
easy to check that
,
5
=
2
(<Jg
+ o"^ 2
)
/cr~
2
.
and the higher the
to idiosyncratic noise
sensitivity to
lower the dispersion and the higher the volatility.
common
When
by substituting dispersion
is
ways
its effect
on
a*, higher
welfare.
may
When
^
inefficient (a
is
this inefficiency.
(a
=
a*), higher commonality,
When,
an increase
0,
so
instead, the equi-
a*), the welfare effect of
Intuitively,
>
(provided a*
commonality
commonality
in
al-
a closer alignment of decisions, but whether this improves efficiency
depends on whether there
a <
efficient
is
a strictly positive value to aligning choices).
facilitates
a higher a*
volatility:
for volatility, necessarily raises welfare
librium use of information
depends on
and
willingness to substitute dispersion for volatility.
the equilibrium use of information
that there
and hence the
follows that the optimal degree
It
of coordination reflects social preferences over dispersion
means a higher
noise,
is
too
little
commonality mitigates the
instead
a >
a*, higher
much alignment
or too
When
to start with.
and therefore necessarily
inefficiency
commonality exacerbates the
raises
inefficiency
and
thereby reduce welfare.
Having understood the
understand the welfare
social value of accuracy
effects of
any change
and commonality,
it is
now easy
to
example, suppose that
in information. For
a prompt release of news by the media, more transparency in central-bank communications, or
more timely publication
of
macroeconomic
by the administration,
statistics
result in an increase in the precision of available public information, keeping constant
This induces an increase
the precision of private information.
and the commonality
of information.
By
Result
3,
both the accuracy
in
the increase in accuracy necessarily
boosts welfare; but the increase in commonality can decrease welfare
rium degree of coordination
is
inefficiently high.
The
following
is
if
the equilib-
then an immediate
implication.
Result 3b. More precise public information
but can decrease welfare if
As another example
of
a
is
how
the relation between
maker
communicating to the market: very
but
—precisely because they are
fine
too fine
ways; and simpler messages that convey
are simpler
trade-off
a <
a*,
sufficiently higher than a*.
of information, suppose that a policy
of
necessarily increases welfare if
a and a*
faces the choice
affects the social value
between two possible ways
messages that convey a
— are
less
— admit a common interpretation.
between accuracy and commonality;
likely to
lot of
be interpreted
information but
a*
>
in idiosyncratic
—precisely because they
Then, the policy maker
if
information
a, so that
effectively faces a
commonality
is
socially
may
desirable, he
well opt for the coarser messages. (See Morris
and Shin (2006)
for
a
motivation and further implications.)
7
We
Conclusion
argued that the relation between the private and the social value to coordination
the key to answering
all
the normative questions raised in the introduction
the inertia and volatility featured in equilibrium are inefficient,
in correcting
We
how
agents use information, and what
is
what
is
is
— whether
the role of policy
the social value of information.
illustrated this point within the context of a specific example,
which admitted
a simple parametrization of the private and social values of coordination. In general,
the mapping from the payoff structure to the equilibrium and optimal degrees of coordination need not be as simple as in the example considered here.
insight extends:
Yet, the
main
the private value of aligning choices can be read from the slope of
best responses, while the social value of such alignment can be read from the slope of
best responses that would
make
agents internalize
all
externalities. (See Angeletos
and
Pavan, 2006a.)
For example, in the beauty-contest models of Morris and Shin (2002) and Angeletos,
Lorenzoni and Pavan (2006), the complementarity perceived by the agents
warranted from a social perspective (a >
but a*
=
0).
This explains
why
is
not
in these
models welfare may decrease with higher commonality, and hence may also decrease
with more precise public information. In contrast,
wig (2005) and Roca (2005), the
the private one (a*
dispersion in prices
firms.
As a
>
a).
This
in the business-cycle
social value of coordination turns out to
is
because individual
utility falls
models
of Hell-
be higher than
with cross-sectional
— an externality that raises the social value of aligning prices across
result, the
heightened inertia and volatility featured in these models due
to the complementarity in pricing decisions are anything but excessive;
necessarily increases with
more
precise public information.
10
and welfare
Appendix
Proof of Conditions
(2), for
some
K = 7M
/i
+
(2)- (3). Suppose that
coefficients (7M ,7 y ,7 x ).
"fy
y
+ jxO
Given
all
this strategy, aggregate activity
and the best-response condition
h = (l-a)E [9} + aE ll ^ + ly y + lx 9}
= (l-a + a~fx )E [9] + a'y ,n + a'y
= [(1 - a + a-yx A M + cry,,] + [(1 - a -f ajx
l
l
IJ
f
For the proposed strategy to be an equilibrium,
)
it
Xy
+ ajy
y
i
reduces to
=
(1
- a+
Q7.T ) A^
+
7y
=
(1
- a+
a-fx ) Xy
+ ajy
jx
=
(1
- a + ajx
,
)
]
+
[(1
-a+
Xx x{
a-yx )
.
]
must be that
7^
7y 7X )
Xx
OOV,
,
.
gives (3). (For a discussion of
when
this linear strategy
the unique equilibrium, see Angeletos and Pavan, 2006a.)
Proof of Condition
welfare
is
f
where F(x\9,y)
c.d.f.
k (xi,y)
,
and
is
the
-
I
{
J(9,y)
the
Consider an arbitrary strategy k
(4).
Uxi
c.d.f.
R2 —
>
R. Ex-ante
(hi
-
2
- r*Lt] dF{x,\9.y)\ dG{0,y)
9)
L
J
J
of the conditional distribution of x given (9,y)
= f
of the joint distribution of (9,y), Li
kj
—
k {x,j,y)
.
(fcj
—
kj)
f
I
{k
2
+
k
2
~2k
l
two functions, k
Ew
(x, y)
dF
and
:
(fc
(x,y)
(xj\6,y)
,
fc,
=
1
2
J k
G(9,y)
k J )dF(x J \9,y)dF(x \9,y)
k(x,y) dF(x\9,y)~2K(9,y)
K (9, y) =
dF
,
Note that
I LidF(xi\e,y)
where
:
given by
Ew=
is
agent
j,y
/J,
)
Solving the above for (7^,
(1) for
given by
is
[
i
is
agents follow a linear strategy as in
(x\9, y)
.
2
Hence, we can think of the planner as choosing
K
- Of - 2r*k
so as to
(x,y)
z
dF
maximize
(x\9,y)
+ 2r*K
(9,
yf
\
dG (6,y)
(o,y)
(5)
11
= f
subject to the constraint K{9,y)
k (x, y) dF(x\9,y)
problem, taking the first-order conditions
for this
.
Setting up the Lagrangian
and
for k (x,y)
K {G,y),
combining
the two (so as to get rid of the lagrange multiplier), and using the law of iterated
expectations,
we
get that the following
must hold
[{-2{k(x,y)-e]-4:r*k(x,y)
almost
for
all (x,
y)
:
+ 4r*K(e,y)}dP(9\x,y) =
0,
Je
where P(8\x,y)
a*
=
is
the
c.d.f.
2r*/(l + 2r*) and rearranging, the above reduces to
problem
is
concave, this condition
Proof of Result
:
M.
—
>
1R,
Using
3.
ex-ante welfare
Ew =
is
both necessary and
it
(5),
\
is
where
E[-]
+ 2r*) E
(Jfe
2
f [k(x,y)-K(8,y)} dF(x\9,y)\dG(6,y)
J
{K(9,y)-e 2 }dG(d,y)
- K) 2 - E {K - 8) 2
=
E[K]
—
E[9], the
Ew = -c {Var
c
=
1
+
2r*
>
0.
,
above reduces to
[k
Moreover, by
Var{k-K)= ]~
{
(1
It
a more detailed
denotes the expectation over (9,y,x). Using the fact that, at the equilibrium
strategy, E[k]
where
sufficient. (For
easy to see that, for any given strategy
-/
(1
Using
Because the maximization
Ux
J(0,y)
= -
y.
given by
+ 2r*)/
_(i
is
(4).
x and
and Pavan, 2006a.)
derivation, see Angeletos
k
of the conditional distribution of 9 given
- K) +
(1
(2)-(3),
a)2{l
a
-J)
2
- a + aS)
2
and
- a) Var [K -
Var(K -
from which the result
9)
=
—
(1
(1
— a + ady
follows.
12
.
we have
follows that
(_
9)}
3
-a+
^ a~
a8) 2
References
[1]
Angeletos, George-Marios, and Alessandro Pavan (2004), "Transparency of In-
formation and Coordination in Economies with Investment Complementarities,"
American Economic Review
[2]
94, 91-98.
Angeletos, George-Marios, and Alessandro Pavan (2006a), "Efficient Use of Infor-
mation, and Social Value of Information,"
[3]
NBER
Working Paper
??.
Angeletos, George-Marios, and Alessandro Pavan (2006b), "Optimal Policy for
Economies with Dispersed Information," MIT/Northwestern University mimeo.
[4]
George-Marios,
Angeletos,
"Beauty
(2006),
Guido
Lorenzoni,
Micro-foundations
Contests:
and
and
Alessandro
Policy
Pavan
Implications,"
MIT/Northwestern University mimeo.
[5]
Hellwig, Christian (2005), "Heterogeneous Information and the Benefits of Public
Information Disclosures,"
[6]
mimeo.
Lorenzoni, Guido (2005), "Imperfect Information, Consumers' Expectations and
Business Cycles,"
[7]
UCLA
MIT
mimeo.
Mankiw, N.Gregory, and Ricardo Reis
Equilibrium"
,
prepared
for the
(2006),
"Sticky Information in General
2006 meeting of the European Economic Associa-
tion.
[8]
Morris, Stephen, and
Hyun Song Shin
"The Social Value
(2002),
of Public Infor-
mation", American Economic Review 92, 1521-1534.
[9]
Morris, Stephen, and
Hyun Song Shin
(2006),
"Optimal Communication", pre-
pared for the 2006 meeting of the European Economic Association.
[10]
Roca, Mauro (2005), "Transparency and Monetary Policy with Imperfect
Common
Knowledge," Columbia University mimeo.
[11]
Woodford, Michael (2002), "Imperfect
Monetary
Policy," in P.
Common Knowledge
Aghion, R. Frydman,
J. Stiglitz,
and the Effects of
and M. Woodford,
eds.,
Knowledge, Information, and Expectations in Modern Macroeconomics, Princeton
University Press.
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