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DEWEY
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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
Efficiency and Welfare with
Complementarities &
Asymmetric Information
George-Marios Angeletos
Alessandro Pavan
Working Paper 05-29
October 31, 2005
RoomE52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
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MASSACHUSETTS INSTITUTE
OP TECHNOLOGY
1
DEC
1
2 2005
LIBRARIES
and Welfare with
Efficiency
Complementarities and Asymmetric Information*
Alessandro Pavan
George-Marios Angeletos
MIT
and
NBER
Northwestern University
October 2005
Abstract
This paper examines equilibrium and welfare
ities,
in
a tractable class of economies with external-
strategic complementarity or substitutability,
and incomplete information. In equilibrium,
complementarity amplifies aggregate
volatility
by increasing the
sensitivity of actions to public
information; substitutability raises cross-sectional dispersion by increasing the sensitivity to pri-
vate information.
To address whether
these effects are undesirable from a welfare perspective,
We
characterize the socially optimal degree of coordination and the efficient use of information.
show how
depend on the primitives of the environment, how they compare
efficient allocations
to equilibrium, and how they can be understood
and dispersion.
equilibrium
We
in
terms of a
between
complements
emerge as information
volatility
When
welfare necessarily increases with the accuracy of information; and
creases [decreases] with the extent to which information
are strategic
social trade-off
next examine the social value of information in equilibrium.
is efficient,
we
[substitutes].
When
is
common
the equilibrium
the gap between equilibrium and
affects
is
and only
if
if
the
it
in-
agents' actions
inefficient, additional effects
efficient allocations.
We
con-
clude with a few applications, including production externalities, Keynesian frictions, inefficient
fluctuations,
JEL
and
efficient
market competition.
codes: C72, D62, D82.
Keywords: Social value of information, coordination, externalities, transparency.
"Earlier versions of this paper were entitled "Social Value of Coordination
discussions and useful comments,
and Information." For stimulating
we thank Daron Acemoglu, Gadi Barlevy, Robert Barro,
Bassetto, Eddie Dekel, Christian Hellwig, Kiminori
Olivier Blanchard,
Ivan Werning, and seminar participants at Harvard, MIT, Michigan, Northwestern, the Federal Reserve
Chicago, the
Bank
of Italy, the 2005
workshop on coordination games
at the
workshop on beauty contests
Cowles Foundation.
at the Isaac
We are grateful to NSF
research grants SES-0519069 and SES-0518810) and to the Federal Reserve
paper was completed.
Marco
Matsuyama, Stephen Morris, Thomas Sargent, Hyun Song Shin,
Bank
Newton
Institute,
for financial
Bank
of
and the 2005
support (collaborative
of Minneapolis
where part of
this
Introduction
1
In
many economic environments, such
as economies with production externalities, incomplete
fi-
nancial markets, or monopolistic competition, the action an agent wishes to take depends on his
expectations not only about the underlying fundamentals but also about other agents' actions. Fur-
thermore, different agents have different information about the fundamentals and hence different
beliefs
about what other agents are doing. Clearly, private incentives to coordinate and information
asymmetries impact equilibrium behavior. But, are these incentives socially warranted, and
decentralized use of information efficient? Also, does
more
the
is
precise information improve efficiency
and welfare?
In this paper
we examine equilibrium and
that feature rich external
and
strategic effects
welfare in a tractable class of concave economies
—
albeit a unique equilibrium
—and asymmetric
in-
formation.
There
(e.g.,
is
a large number of ex-ante identical small agents each taking a continuous decision
investment). Individual payoffs
may depend,
mean, and possibly the dispersion, of activity
strategic effects in the model.
economic fundamentals
and information
is
—
this
not only on one's
in the population
—
own
action, but also
on the
this is the source of external
and
Agents observe noisy private and public signals about the underlying
is
the source of information asymmetry. Finally, payoffs are quadratic
Gaussian, which makes the analysis tractable.
In equilibrium, strategic complementarity raises the sensitivity of actions to public information; strategic substitutability raises
the sensitivity to private information.
Common
noise in pub-
lic
information generates volatility; idiosyncratic noise in private information generates dispersion.
It
follows that complementarity contributes to higher volatility, substitutability to higher disper-
sion.
1
These are interesting positive properties but alone have no normative content
be no presumption that the impact of strategic
volatility
and dispersion
is
on the use of information and thereby on
undesirable from a welfare perspective. To address this issue, one needs
which
to characterize the efficient use of information,
We
effects
—there should
The
our
first
main
maximize ex-ante
define efficient allocations as the ones that
that information can not be centralized.
is
efficient
result.
utility
under the constraint
allocation for a given
economy can be
represented as the equilibrium of a fictitious economy where individual payoffs are manipulated to
reflect social
motives (that
with respect to the
must align
to internalize payoff interdependencies).
activity in this fictitious
their choices for efficiency to obtain;
of coordination.
The
mean
is,
The analogue for the
actual
it
The
slope of best responses
economy measures the extent
defines
economy
what we
call
to
which agents
the (socially) optimal degree
defines the equilibrium degree of coordination.
amplification effects of various sorts of complementarities are the subject of a vast literature.
See Cooper
(1990) for a review of complete-information applications and Morris and Shin (2002, 2003) for incomplete information.
We
how
first
show how the
depends on the primitives of the environment and
efficient allocation
compares to equilibrium allocation. As with equilibrium, complementarity contributes to
it
a positive optimal degree of coordination, substitutability to a negative. But unlike equilibrium,
the optimal degree of coordination also depends on other external payoff effects that are irrelevant
for private incentives.
coordination
is
In the absence of such non-strategic external effects, the optimal degree of
when
higher than the equilibrium one
agents' actions are strategic
complements
(and lower when they are strategic substitutes).
This result highlights the danger in extrapolating positive properties to normative implications:
economies with complementarities, the high sensitivity to public information and the
in
amplification of volatility featured in equilibrium can be socially desirable.
We
next relate the optimal degree of coordination to the
of payoff concavity, both aggregate volatility
When
comparing allocations that
sensitivity to
volatility
and
differ in their effective
degree of coordination
idiosyncratic noise
The
use of information. Because
cross-sectional dispersion induce welfare losses.
— the planner
common and
dispersion.
and
efficient
resolution of this trade-off
is
— and hence
effectively faces a trade-off
in their
between
reflected in the optimal degree of coor-
dination: the latter increases with social aversion to dispersion
and decreases with
social aversion
to volatility.
Our second main
we
find
it
result
is
a characterization of the social value of information. For this purpose,
useful to parameterize the information structure by the level
noise in the agents' forecasts of the underlying fundamentals.
information with the precision of these forecasts, that
is,
We
and the composition
identify the accuracy of available
the reciprocal of total noise, and
transparency (or commonality) with the correlation of forecast error across agents, that
extent to which noise
is
common. Since
in the
of
is,
its
the
absence of external effects welfare depends only on
the level and not on the composition of noise, this parametrization seems most appropriate from a
theoretical point of view.
When
2
the equilibrium
is efficient,
welfare necessarily increases with the accuracy of information.
Moreover, welfare increases [decreases] with the transparency of information
if
and only
if
agents'
actions are strategic complements [substitutes]. Efficiency thus implies a clear relationship between
the form of strategic interaction and the social value of information.
When the equilibrium is inefficient,
efficient allocations. Its welfare effects
but also on two key aspects of
information
may
also affect the
gap between equilibrium and
then depend, not only on the form of strategic interaction,
this gap: the discrepancy
between optimal and equilibrium degrees of
coordination, and the correlation between first-best and complete-information equilibrium activity.
"This parametrization
is
also appropriate for
contemplating whether to transmit information
about the release of more or
idiosyncratic or
common
less
in
some applied
questions.
Think,
for
example, of a central banker
a transparent or ambiguous way. This need not be simply a choice
information, but rather a choice about the extent to which individuals will adopt
interpretations of the
same piece
of information.
We
conclude the paper by illustrating
how our
results can help
understand the
inefficiencies of
equilibrium and the social value of information in specific applications.
model of production
In a typical
choices, coordination
is
spillovers
inefficiently low.
where complementarities emerge
in
investment
Moreover, welfare unambiguously increases with either
the accuracy or the transparency of information
— a case for timely provision of relevant information
by the government or the media.
The same
coordination
result
is
appears to hold in standard Keynesian monetary economies.
inefficiently
In contrast,
high and transparency can reduce welfare in economies resembling
Keynes' beauty-contest parable for financial markets. Furthermore, in economies where equilibrium
fluctuations are largely inefficient even under complete information, welfare
accuracy and transparency
Finally,
is efficient
may
—from a social perspective, ignorance could be a
we consider an example
of a competitive production
decrease with both
bless.
economy where the equilibrium
even under incomplete information. Since individual actions are strategic substitutes,
welfare increases with accuracy but decreases with transparency
—perhaps a case
for "constructive
ambiguity" in central bank communication.
Related literature. To the
best of our knowledge, this paper
The
welfare analysis for the class of economies considered here.
is
the
first
to conduct a complete
closest ascendants are
Cooper and
John (1988), who examine economies with complementarities but complete information, and Vives
(1988),
of the
who shows
more general
However,
leifer
efficiency of equilibria in a class of competitive
this
paper
value of information.
is
certainly not the
how
More
first
to
examine the
distributional effects can drive a
recently,
and more
strategic complementarity.
wedge between the private and
in
social
and Shin (2002)
an economy that resembles a "beauty contest"
Angeletos and Pavan (2004) and Hellwig (2005),
on the other hand, provide counterexamples where public information
economy with
a special case
social value of information. Hirsh-
closely related to this paper, Morris
show that public information can reduce welfare
complementarity
is
class considered here (see Section 6.4).
(1971) highlights
and that features
economies that
is
socially valuable despite
— a real economy with investment complementarities in the
pricing complementarities in the second.
These works
first
paper, a monetary
illustrate the non-triviality
of the welfare effects of information within the context of specific applications, but do not explain
the general principles underlying the question of interest.
social value of information depends, not only
We
fill
the gap here by showing
on the form of strategic
interaction, but also
how
the
on other
external effects that determine the discrepancy between equilibrium and efficient allocations.
The
literature
on rational expectations has emphasized how the aggregation of disperse private
information in markets can improve allocative efficiency
and Messner and Vives
(2001),
(e.g.,
Grossman, 1981).
on the other hand, highlight how informational
generate inefficiency in the private collection and use of information.
Laffont (1985)
externalities can
Although the information
structure here
is
how
exogenous, the paper provides an input into this line of research by studying
the welfare effects of private and public information depend on payoff externalities.
The paper
about central-bank transparency. While
also contributes to the debate
focused on incentive problems
and Shin (2002, 2005)
and Heinemann and Cornand (2004) argue that central-bank disclosures can lead
markets behave
can improve welfare by reducing price dispersion.
Roca (2005) show that public
information, while in Section 6.4
The
we show
in efficient competitive
rest of the
paper
equilibrium in Section
We
we
In Section 6.3
such disclosures ought to depend on whether the business cycle
made even
to welfare losses
Keynes' "beauty contest"; Svensson (2005) and Woodford (2005) question
like in
the practical relevance of this result; Hellwig (2005) and
effects of
work
Canzoneri, 1985; Atkeson and Kehoe, 2001; Stokey, 2002),
(e.g.,
recent work emphasizes the coordinating role of public information. Morris
if
earlier
highlight that the welfare
under complete
is efficient
that an argument for constructive ambiguity could be
economies.
organized as follows.
is
disclosures
3, efficiency in
turn to applications in Section
Section
We introduce the model in Section 2. We examine
and the
4,
The Appendix
6.
social value of information in Section 5.
includes proofs omitted in the
main
text.
The model
2
Actions and payoffs.
Consider an economy with a measure-one continuum of agents, each
choosing an action k G R. Let
K
section of the population,
\I/
denote the cumulative distribution function for k in the cross-
= J kd^(k)
mean
the
action,
exogenous payoff-relevant variables (the fundamentals), with
U
U(K, K,
M Ar+2 — R
9)
denote utility
(also,
aggregate welfare)
Externality emerges whenever
restrict
— Ukx/Ukk
within
(
— 1,
^
{//<•
+1).
0,
As we
level in the sense that
The
analysis
is
Ukk
simplest
<
when
We
an d
N=
1,
when
Individual utility
1.
Wkk —
but TV
is
given by
— 1,+1)
is
>
Ukk
1
W(K,9) =
whenever UkK
strategic complementarity
(
let
agents choose the same action.
all
next section,
— Uki</Ukk
is
¥" 0-
We
the slope of
necessary and sufficient for the existence
impose concavity
also
a vector of
(1)
will see in the
best responses; restricting this slope within
of a unique stable equilibrium.
R
E
a strictly concave quadratic function. 4,5 Finally, we
is
>
:
N>
(#i, ...,9^)
= U(k,K,6),
u
where
=
and 9
at
both the individual and aggregate
+ %UkK + Ukk <
0.
If
U
were not concave, best
allow us capture the possibility that there are fundamentals
that are relevant for equilibrium but not for efficient allocations, and vice versa.
''That
is,
U(k, K, 9)
'Note that
an external
''In
what
U
effect
=
vXJv' where
depends on
ty
U
is
a (n
only through
+
3)
its first
x (n
+ 3)
and u = (1, k, K, 6).
we extend the model to incorporate
negative-definite matrix
moment (mean
activity);
from the second moment (cross-sectional dispersion) at the end of Section
follows,
we
often refer to
UkK
as the complementarity even
with negative complementarity.
4
if
UkK <
0.
That
is,
we
4.
identify substitutability
responses would not be well-defined; similarly,
well-defined.
Before agents move, nature draws 9 n
=
+
9 -a
en
,
where
and
^n
The common
what
=
we
follows,
and variance a\
= °ZV a n
Sn
If
we
best would not be
{1,...,7V},
from independent
with variances, respectively, a
2
\n )vn and variance a Zn where A n
,
=
realization of 9
e n are, respectively, idiosyncratic
=
2
n
+ £n
9n
and common
and
=
and a n
l
—
9n
let u] n ee
{o~l
+ O^Y
E[9 n \x
a\
l
,y]
= Var
1 '2
E[6i„|x
l
,7/]
=
—
and public
noises,
signals
independent of
2
a'
.
is
Normal with mean
= a~ 2 /a~ 2 and a Zn
(1
not observed
is
5 n )x n
y.
(a~ 2
8
+ a^ 2 )-
1
'2
.
In
Private posteriors,
and variance o 2n where
+ S n zn
l
ee
=
zn
,
-
denote agent
(uj n )
l
(9 n )
l
often identify public information with z rather than with
on the other hand, are Normal with mean
2
The
.
n £
posterior for 9 n given public information alone
+ (1 -
X n yn
9,
fi n
for
,
each n, agents observe private signals x\
for
one another as well as of
E[9 n \y]
mean
distributions with
by the agents. Instead,
yn
first
7
Information.
Normal
W were not concave, the
if
i's
and
Sn
forecast error about 9 n then
,
=
2
Corr (w n ,w{)
i
,
^
j.
Hence, a n measures the total noise in agents' forecasts about the fundamentals and 6 n the extent
to which noise
cr~
2
and
We
its
is
common
9
across agents.
We
accordingly identify the accuracy of information with
transparency (or publicity) with 5 n
.
prefer to parametrize the information structure
reasons.
First, this
mapping between
without any
is
(<J Xn
,&y n ) and
(<5
by
(5n ,an ) rather
all
n ,a n )
only on
o~ n
,
.
If
not 5 n
+ ofn + a~ 2 >
'
it
)
for
two
a one-to-one
and
5n
=
_i_
Vn
-2
CT
_9
_2
e
»
_ 2 £(0,1).
,
(2)
with a change in
strategic interactions, instead, the extent to which information
plays an important role since
point of view,
a Vn
its
there were no externalities and strategic interactions, welfare would depend
With
.
,
:
Second, a change in a Xn or a yri combines a change in the level of noise, a n
composition, 5 n
(cr Xn
loss of generality since, given the prior, there is
-2
=
a~ 2
than
it
affects the structure of higher order beliefs.
thus seems most interesting to separate these two
From
is
public
a theoretical
effects.
For tractability, we have restricted U, and hence W, to be quadratic. For non-quadratic concave environments,
our results represent approximations that are better the lower the noise
hand,
may
introduce effects about which our analysis
is
in
information.
Convexities, on the other
not appropriate.
For example, aggregate convexities can
=
and
generate a social value for lotteries.
L
also
Throughout, we use the convenient vector notation x
drop the superscript
Note that S n
1
is
also
i
whenever
it
=
(y n ),
similarly for all other variables.
We
an increasing transformation of the relative precision of public information.
In the context of specific applications, however,
statics with respect to
(x n ), y
does not create confusion.
(ct x ,ct.)
.
it is
also interesting to translate the results in terms of comparative
See Section 6 for some examples.
Equilibrium allocations
3
Each agent chooses k so
maximize
as to
his expected utility, TEi[U(k,
optimization problem gives the best response for the individual.
Definition
An
1
=
k(x,y)
where K(9,
=
e)
It is useful
E{ k (x, y)
=
K (9) =
is
linear:
The incomplete-information
Proposition
N=
allocation,
{n
>
1
Kn
:
equilibrium
=
1 Let k(9)
+ K\9\ +
kq
kq
^
+
0}
E[ U(k',
all (6, e)
where k
k,
max
The
such
>
K,9)
\
x,
y
K\9\
^
0,
is
+
13
is
solution to this
the equilibrium.
for
that,
all
(x,y),
} ,
n
When
the unique solution to Uk (k, k, 9)
is
...
The
fixed point
R 2N — K
:
9)\x,y\.
the complete-information benchmark.
first
k
is
arg
6,e], for
\
to consider
(unique) equilibrium
quadratic, k
any function k
equilibrium, allocation is
K,
+ kjv#at
9
=
some constants K n G K, n 6
for
is
known, the
Since
0.
{0, 1,
...,
U
is
N}.
12
then characterized as follows.
...
+
Kpj9pf denote the complete-information equilibrium
and
\Ukk\
An
(i)
allocation k
:
— M
R
>
k(x,y)
where K{6,e)
(ii)
=
=
The equilibrium
exists, is
7"
Proof. Part
is
an equilibrium
—
E[ (1
a)K
and only
if
+ aK
x,y
|
=
unique, and
K
=
K
Sn
+
+
^
i
[C
_
1
ln) x n
that
13
«.„
^
U
+ l n zn]
(5)
,
f° r al1
n&N
.
-
(
,
6)
)
:
R 2N
-»
R
and
let
K(9,e)
=
E[k(x,y)\9.
e\.
A
best-
a strategy k'(x, y) that solves the first-order condition
state of the world
Since
(4)
-
n - dF^
- all
n
Take any strategy k
(i).
Kn
a5 n (l - 6 n )
1
is
given by the realizations of
and other aggregates are functions of
12
(x,y)
all
given by
is
E[Uk (k',K,9)\x,y}
"A
for
]
E[k(x,y)\9,e}.
k(x, y)
response
is
is
if
quadratic, k
and only
The assumption
if
N^
= -Uk
Uk e
(6, e)
(0, 0, 0)
8, e,
=
and {£
l
};e[o,i]-
However, since
f
is i.i.d.
across agents,
alone.
/(Ukk
+ UkK
)
and
js n
= -UkS ,J(Ukk + UkK
).
n €
{1,...,JV}. It follows
¥" 0.
avoids the trivial case that the fundamentals are irrelevant for equilibrium.
Using Uk(n,K,0)
for all (x,y).
function must satisfy ¥i[Ukk(k'
— for all 9, and the fact that U is quadratic,
— k) + Uki<{K - k)\x, y\ = 0, or equivalently,
k'(x,y)
for all (x, y). In equilibrium, k'(x,y)
Part
x and
in
Since E[k|x,j/]
(ii).
=
=
Thus suppose the equilibrium
z.
some
coefficients a £
1R,
RN
£
b
x,y
|
}
and
c
it is
natural to look for a fixed point that
is
linear
is
k(x, y)
for
aK
k(x,y), which gives (4).
linear in (x, z),
is
- a)n +
E[ (1
the best-response
=
a
+ b-x + c-z
K
£ R^. 14 Then
(7)
(9, e)
=
a
+b
9
+c
z
and therefore
(4)
reduces to
k(x,y)
=
where k
A
is
=
(1
NxN
the
an equilibrium
a
(1
— a)«o + aa +
((1
— a)n +
=
(I
(«!,...,«„). Substituting E[6|x,j/]
matrix and
is
=
if
aa,
=
bn
Equivalently a
k
,
if a,
b
=
A).x
E[0|x,
+ ac
j/]
+ Az, where
=
b
(I
Kn (l
and
— A)
-
z
the
I is
diagonal matrix with n-th element equal to 5 n
and only
— a)no +
-
ab)
,
N
N
x
identity
we conclude that
(7)
c solve
[(1
a)(l
-
— a)n + a 6]
5„)/[l
and
,
- a (I -
iV}. Note that 6 n + c n = K n always; b n = Cn =
n £ {1,
c,j £ (0, Kn) otherwise. Letting 7 n = Cn/'n n £ (0, 1) for any
...,
c
5 n )\,
=
A [(1
and
c„
whenever k„
n £
N gives
— q)k +
+ ac.
c*6]
= K n 5 n /[1 - q(1 - 6 n
= 0; and 6 n £ (0, k„) and
)],
(5)-(6). Clearly, this is the
unique linear equilibrium. Furthermore, since best responses are linear in E[0|x,y] and
there do not exist equilibria other than this one. (This follows from the
and Shin (2002); our payoffs are more general but the structure of
same argument
beliefs
E[/CT|x, y],
as in Morris
and best responses
is
essentiaUy the same.)
Condition
(4)
has a simple interpretation: an agent's best response
of his expectation of
some given
"target"
and
simply the complete-information equilibrium.
activity, a,
The
is
what we
is
an
afflne
his expectation of aggregate activity.
The
combination
The
target
is
slope of best responses with respect to aggregate
identify with the equilibrium degree of coordination.
sensitivity of the equilibrium allocation to private
and public information depends on both
the degree of coordination and the transparency of information. When
x n and z n are simply the Bayesian weights and hence
-y
n
=
5n
.
a
The term
= 0,
the weights on signals
[a5 n (l
— 5 n )]/[l — a(l — S n
)]
thus measures the excess sensitivity of equilibrium allocations to public information as compared
to the case
where there are no complementarities. Note that
this
term
is
increasing in a. Stronger
complementarities thus lead to a higher relative sensitivity to public information. This
A
dot between two vectors denotes inner product.
is
a direct
implication of the fact that, in equilibrium, public information
of aggregate behavior than private information.
coordinating
If
=
a relatively better predictor
In other words, public information has also a
role.
information were complete
choose k
is
=
(<r„
for all n, or at least for all
n £ N),
K = k. Incomplete information affects equilibrium behavior in two ways.
noise generates (non-fundamental) volatility, that
complete-information level
variation in aggregate activity
is,
The
first is
and only
if
a <
-*
do n da
(ii)
over, the
Var{K — k),
(i) Volatility,
d 2 V-r(K-,)
e -'
( ie
^•
is,
around the
variation in
is
characterized
a and a n and
increases with
10
Proposition 2
if
K
common
measured by Var(K — k), the second by Var(k-K).
Their dependence on the degree of coordination and the information structure
Sn
agents would
First,
Second, idiosyncratic noise generates dispersion, that
k.
the cross-section of the population.
below.
all
<
or 5 n
necessarily increases with
,
^~-- Moreover, the impact of noise on volatility increases with
a
U
Q)>-
Dispersion,
Var(k~ K),
necessarily decreases with
impact of noise on dispersion decreases with a
a and
(i.e.,
—
Sn
and increases with a n More.
—
g„ gQ
<
0).
Higher complementarity thus mitigates the impact of noise on dispersion, and obtains a better
alignment of individual choices, but amplifies aggregate
Higher transparency also reduces
volatility.
dispersion possibly at the expense of higher volatility. Higher accuracy, on the other hand, reduces
both
later.
volatility
and dispersion.
In the next section,
this relates to the
4
We
we turn
will
in
more
detail the welfare effects of information
to the characterization of the efficient allocation
and show how
optimal degree of coordination.
Efficient allocations
The property
that complementarity generates high sensitivity to
plifies volatility, is interesting
on
its
normative question of whether these
is
examine
the efficient use of information.
welfare (expected utility)
among
own.
But
common
this is only a positive property.
effects are socially undesirable,
We
and thereby am-
noise,
To address the
one needs to understand what
define efficient allocations as those that
maximize ex-ante
the ones that are measurable in the agents' decentralized infor-
mation.
Definition 2
An
efficient allocation is a function k
Eu=
f
J(6,e)
In the following,
whenever we say
information counterpart.
:
W 2N — R
>
that
maximizes ex-ante
utility
i U{k(x, y), K{6, e), 6)dP(x\6, e)dP(6, e)
Jx
"volatility"
we mean
volatility of
aggregate activity around
its
complete-
subject to
K(9,e)
where P(9,e) stands for the
distribution of x given 9
We
c.d.f.
and
= f k(x,y)dP(x\9,e),
cooperatively and
e.
is
to
it.
It
thus answers exactly the question of interest for this paper,
namely how allocations and welfare would change
agents were to internalize their payoff interde-
if
pendences and appropriately adjust their use of available information. 16
see in Section
5, it is
The
appropriate for the purposes of this paper.
the solution to the "team problem" where agents choose a strategy
is
commit
all (9, e).
of the joint distribution of(9,e) and P(x\9,e) for the conditional
believe that this notion of efficiency
allocation defined above
for
What
is
more, as we will
precisely this notion of efficiency that helps understand the social value of
information in equilibrium.
We
by deriving a necessary and
start
Lemma
An
1
allocation k
:
E[
WL
2N
—
>
sufficient condition for efficient allocations.
M. is efficient if
and only
if,
for almost
Uk (k(x, y),K, 6) + UK {K, K, 9)\x,y] =
all (x, y)
,
0,
(8)
where K(9,e) =E[k(x,y)\9,e].
Proof. The Lagrangian of the problem in Definition 2 can be written as
A=
U(k(x,y),K(9,e),9)dP(x\9,e)dP(9,s)+
/
l{9,e)
Jx
+ f
The
first
Jx
l
order conditions for K(9,e) and k(x,y) are therefore given by
UK (k(x,y),K(9,e), 9)dP(x\9, e) + \(9,e) =
J
Jx
Noting that
Uk
is
linear in its
can be rewritten as —X(9,s)
strictly
for almost all (9,e)
[Uk (k(x,y),K(9,£),9)-X(9,e)}dP(9,e\x,y)=0
/
is
K(9,e)- f k(x,y) dP(x\9,e) dP{6,
X(9,e)
J(e,s)
=
arguments and that K(9,e)
for
= J
Uk(K{9,s),K(9,£),9). Replacing
concave and the constraint
is
linear, (8) is
almost ah
(9)
(x, y)
(10)
k(x,y) dP(x\9,e), condition
(9)
this into (10) gives (8). Since
both necessary and
sufficient,
U
which completes
the proof.
113
Our
efficiency
that information
concept
is
is
the same as in Radner (1962) or Vives (1988) and shares with Hayek (1945) the idea
disperse and can not be
communicated
to a "center".
Clearly, this
concepts that assume costless communication and focus on incentive constraints
Myerson, 1983).
(e.g.,
is
different
Mirrlees, 1971;
from efficiency
Holmstrom and
The
This result has a simple interpretation.
commonly known and
case where 9
is
U(K, K,
thus solves the the first-order condition
9). It
Uk(K, K,
We
=
9)
0.
17
which corresponds to the
first-best allocation,
henceforth denoted by k*(0), maximizes
is
Wk{K, 9) =
The incomplete-information counterpart
0,
or equivalently
of this condition
can then expand this condition to characterize the
W(K,9) =
Uk{K, K,
9)
+
is (8).
efficient allocation
under incomplete
information in a similar fashion as with equilibrium.
Proposition 3 Let k*
*
Kn
^
^
0}
(9)
=
Kq
+
k\9\
+
+ k*n 9n
...
0, and
,
\ukk
An
allocation k
:
R 2N —
=
k(x,y)
>
E[
(1
\
and only
is efficient if
TSL
{n
W
U
2UkK + UKK = n „
2a4 KK
„ * _
=
a
(i)
N* =
denote the first-best allocation,
- a*)K*
+ a*K
>
1
:
nn
(11)
if
\x,y]
for almost
all (x,y),
(12)
where K{9,e) =¥,[k(x,y)\9,e}.
The
(ii)
efficient allocation exists, is essentially unique,
k(x, y)
=
Kq
+ Y]
„
r,»
Kn K 1 -
}~-y
< = 5n + f- 5n{
a*
5n
{I
1
and
l*n) x n
is
given by
+ 7n zn]
(13)
,
forallneN*.
(14)
)
In equilibrium, each agent's action was an affine combination of his expectation of k, the
complete-information equilibrium action, and of his expectation of aggregate activity.
true here for the efficient allocation
this sense, condition (12) is the
idea
if
we
replace k with
«;*,
the first-best action, and
The same
a with
is
a*. In
analogue for efficiency of the best response for equilibrium. This
formalized by the following.
is
Proposition 4 Given an economy e
=
(U; a,
S,
ji,
erg)
G S,
U (e)
let
be the set of functions U'
such
agents perceived their payoffs to be U' rather than U, the equilibrium would coincide with
that, if
the efficient allocation for e.
For every
(i)
(ii)
For every
Part
(i)
U
In
some
may be
U'
non-empty.
eW (e)
only
if a'
= —U'kK /U'kk
equals a*
be represented
as the equilibrium of a fictitious
16
Part
individual incentives are manipulated so as to coincide with social incentives.
and hence
-WkbJWkk,
18
e,
is
states that the efficient allocation can
game where
''Since
U (e)
e,
n e
{1,
cases, this
W
...,
is
quadratic, k* (9)
N}.
may
=
It follows that «£
also suggest a
way
kJ
^
to
+
if
k\9i
+
...
and only
+
if
implement the
n 9 n where Kq
k.*
WK B„
,
efficient allocation.
able to use taxes and subsidies to fashion individual best-responses.
10
= —Wk
= Uk e„ + UK e„ #
(0,0)
JW'kk and K~n
=
0.
For example, the government
(ii),
on the other hand, explains why we identify a* with the optimal degree of coordination: a*
describes the level of complementarity that agents should perceive
to obtain as
an equilibrium outcome, that
The counterpart
is,
of optimal coordination
mal degree of coordination, the higher the
Corollary
The
1
equilibrium one, which
externalities were to be internalized.
the efficient use of information: the higher the opti-
is
sensitivity of efficient allocations to public information.
and only
true if
is turn, is
second-order non-strategic
it
the optimal degree of coordination
if the
complementarity
is
higher than the
high enough relative to
is
effects:
Proposition 3 and Corollary
environment and how
if
and only
>jn \/neNDN*
In
is
the efficient allocation were
relative sensitivity of the efficient allocation to public information is higher than
that of the equilibrium allocation if
of coordination
if all
if
1
>a
a*
<^=>
show how the
4=>
efficient allocation
UkK > -Ukk-
(15)
depends on the primitives of the
compares to the equilibrium one. As with equilibrium, the optimal degree
But unlike equilibrium, the optimal
increasing in the complementarity, UkK,-
degree of coordination depends also on
Ukk, a second-order
external effect that does not affect
private incentives. In the absence of such an effect, the optimal degree of coordination
absolute value) than the equilibrium one (a*
=
is
higher (in
2a) reflecting the internalization of the externality
generated by the complementarity.
To understand
better the forces behind the determination of the optimal degree of coordination,
an alternative representation
expressed as
Eu = EW(k*,
9)
is
\EKK\ Var{K _
Kl +
volatility
and dispersion generate welfare
on
volatility is
respect to individual activity. Note that
and second-order external
losses follows directly
Wkk =
(Wkk =
equally to welfare losses. Complementarity
level,
Wkk,
given by
Ukk
effects (in the sense that
the curvature of individual utility
volatility.
(
i 6)
£* captures the welfare
namely those due to aggregate
to aggregate activity, while the weight on dispersion
the individual
_ A").
volatility
and
cross-
19
ences. Naturally, the weight
its
\IM Var{k
9) is ex-ante utility in the first-best allocation, while
sectional dispersion.
can be
,
losses associated with incomplete information,
That
utility) at the efficient allocation
- L* where
r=
Note that EW(re*,
Welfare (ex-ante
useful.
is
the curvature of welfare with respect
given by Ukk, the curvature of utility with
+ ^UkK + Ukk- When there are no strategic
UkK = Ukk — 0), aggregate welfare inher-
Ukk), so that volatility and dispersion contribute
(UkK >
0) helps offset the diminishing returns faced at
thus reducing concavity in the aggregate
The converse
is
from concavity of prefer-
true for substitutability
(UkK <
Condition (16) follows from a Taylor expansion around k
11
=
K=
(Wkk) an d
0) or external
therefore the weight on
concavity
k*(5); see the Appendix.
(Ukk <
0).
Volatility
is
generated by
common
tive sensitivity of allocations to public information
dampens
by idiosyncratic
noise, dispersion
—equivalently,
The
dispersion at the expense of higher volatility.
noise. Increasing the rela-
raising the degree of coordination
efficient
use of information reflects the
resolution of this trade-off.
Corollary 2 The optimal degree of coordination equals one minus
the weight that welfare assigns
to volatility relative to dispersion:
Extension. In some applications of
effect
in
on individual
We
on the other hand, dispersion has positive external
can easily accommodate such an
=
f(k —
K)
d^(k).
Uk k + 2£/
dispersion with
2
CT
£*
The optimal degree
Then
20
all
effect
effect (see Section 6.2).
— and we do so for the rest of the paper—provided
9,
aV) with
Ua
2
£
K
being a constant
our results go through once we replace the welfare weight on
In particular, welfare
is
IWWI
= ±^K±Var(K-K*)+-
now
\Ukk
of coordination
beauty contest of Morris and Shin
(see Hellwig, 2005). In the
that dispersion enters linearly in the utility function: U(k, K,
and o\
has a direct external
For example, price dispersion has a negative effect on individual utility
utility.
New-Keynesian monetary models
(2002),
interest, cross-sectional dispersion
given by
Eu = EW(k*, 8) —
£*, where
+ 2Ua 2\
—^-Var(k-K).
(18)
is
WKK
a
Ukk + 2Ua
2
Finally condition (15) becomes
> In Vn
In
Note that a*
A
higher
Ukk
of dispersion.
is
e
N n N*
increasing in
<=>
Ukk
(
a*
^> UkK
>a
Both these
Ua
higher Ua
or v) an(i decreasing in
decreases the social cost of volatility, while a
effects are external
and non-strategic
coordination without affecting private incentives.
> -Ukk +
The former
(or u).
2
i
—they
2L^2.
This
is
intuitive.
decreases the social cost
affect the social value of
contributes to a higher optimal
degree of coordination, the latter to a lower.
Efficient economies.
We
conclude this section with necessary and sufficient conditions for
the equilibrium to be efficient under incomplete information.
20
In analogy to
Wkk
<
0,
we impose Ukk
+ 2Ua 2 <
0,
negatively on dispersion.
12
which
is
necessary and sufficient for welfare to depend
Proposition 5 The equilibrium
is efficient if
and only
if
=
and
a
k(-)
or,
UkK +UKK -2Ua2 =
equivalent^,
0,
k
for
k*(-)
=
a*,
UK (0,0,0) = Uk (0,0,0) UkK /Ukk andUK e„ = Uk g UkK /Ukk
n
,
all n.
The
But
condition k(-)
«;*()
means that the equilibrium
under complete information.
is efficient
under complete information alone does not guarantee
efficiency
What
information.
=
also
is
needed
efficiency
efficiency in the use of information
is
addition the equilibrium and the optimal degrees of coordination coincide.
under incomplete
which obtains when in
21
Social value of information
5
We now
examine the impact of information on equilibrium welfare (allowing
for U„2
k
with economies where the equilibrium
because efficiency
is
is
^
0), starting
This provides a useful benchmark, not only
efficient.
always an excellent starting point, but also because in our class of economies
efficiency implies a clear relation
between the form of strategic interaction and the
social value of
information.
Proposition 6 Suppose
with
<r
n
the equilibrium is efficient.
and increases [decreases] with
,
5 n if
For any n £
and only
if
TV,
welfare necessarily decreases
agents' actions are strategic complements
[substitutes].
As highlighted
allocation
is
An increase
welfare.
in the previous section, the
summarized
in
on
in the
impact of noise on
for given 5 n raises
An increase in
5 n for given
both
Note
that,
That accuracy
the
when
is
is
volatility
with either o Xn or a Vn
Corollary 3 Suppose
implies that,
,
for
is
(see condition (18)).
Such a substitution
equivalent to a reduction in dispersion,
is
higher than that of volatility, that
beneficial can then
and dispersion
efficient
and dispersion and therefore necessarily reduces
,
the equilibrium aUocation
same observation
volatility
a n on the other hand,
possibly at the expenses of volatility.
social cost of dispersion
impact of information on welfare at the
is efficient, it
welfare-improving
is, if
and only
if
a*
if
>
and only
if
the
0.
maximizes ex-ante expected
utility.
be obtained also an implication of Blackwell's theorem. Indeed,
when
the equilibrium
is efficient,
welfare necessarily decreases
any n £ N.
the equilibrium is efficient.
Welfare necessarily increases with the precision
of either private or public information.
Note that a and a* depend on
U
but not on (a,
S)
.
This explains
the payoff structure alone, as shown in Proposition 5 above.
13
why
efficiency
can be checked on the basis of
In economies where the equilibrium
complicated for two reasons.
the optimal one {a
and dispersion
is
^
is
inefficient,
the equilibrium degree of coordination need not coincide with
First,
resolved.
^
in addition to those associated with volatility
and
the role
ofa^a*,
re*),
(18)
differ
from the socially
thus introducing first-order welfare losses
dispersion.
moment
maintaining for a
=
re
sum
with incomplete information continue to be the weighted
22
trade-off between volatility
may
Second, the equilibrium level of activity
(re
first
way the
a*), thus introducing inefficiency in the
optimal one even under complete information
Consider
the welfare effects of information are more
re*.
The
welfare losses associated
and
of volatility
dispersion, as in
For given a, a higher a* means a lower relative weight on volatility and hence a lower
.
cost associated with an increase in
(Proposition
<5
n
It
follows that, relatively to the efficiency
low coordination (a
6), inefficiently
while inefficiently high coordination (a
a ^ a* does not
23
.
>
<
benchmark
a*) increases the social value of transparency,
a*) reduces
affect the value of accuracy: a lower
it.
On
the other hand, the possibility that
a n reduces both
volatility
and dispersion and
therefore necessarily increases welfare.
Consider next the role of re
^
re*
,
information. In equilibrium, welfare
£ = -Cov {K -
re,
Wk
(k, 9))
in
which case the equilibrium
is
given by
+
\^d
Eu = EW(re,
Var(K -
are the welfare losses due to incomplete information.
24
re)
+
The
9)
—
is inefficient
C,
where
—^——^-
Var(k - K)
•
two terms
last
efficient
(re
=
re*
and hence Wx(re,
of the fact that small deviations
9)
K—
re,
=
0),
When the
the covariance term
around a
complete-information equilibrium
term contributes to a welfare
first-order effect.
maximum
is inefficient
loss or gain; this
are the familiar
The
covariance term,
complete-information equilibrium
is
zero; this
have zero first-order
is
merely an implication
is
effects.
due to externalities (Wk(k,6)
is
(19)
£
in
welfare losses associated with volatility and dispersion (second-order effects).
on the other hand, captures a novel
even under complete
7^ 0),
But when the
the covariance
because a positive [negative] correlation between
the "error" in aggregate activity due to incomplete information, and Wjk(k,6), the social
return to activity, mitigates [exacerbates] the first-order losses associated with externalities.
As shown
in the
Appendix (Proof
Cov (K -
re,
Wk
(k, 9))
of Proposition 7), this covariance
= \WKK \Cov (K -
re, re*
-
re)
term can be expressed as
= \WKK Y,
\
0»«n
(
20 )
neJV
"As
k
= k*
a / a'
as long as
a) /a, which
we assume
obvious from the derivation of (16) in the Appendix, (16) and similarly (IS) extend to
when
This can also be seen from (19) below noting that Wk(k,9) =
23
Recall from Proposition 2 that volatility increases with <5„ if and only if a <
.
k.
=
k*.
or 6
<
(1
-
here in order to simplify the discussion. In the alternative case, welfare necessarily increases with S n (when K
2,1
Condition (19) follows from a Taylor expansion around
14
K = k{9)\
see Appendix.
=
k").
where, for
n £ N,
all
—k
k*
— k, k #_„
Var(K|0_ n
Cov
71
~^T
4,71
2
=
n <y\
<j>
'
K - K, K
Gov (
The
.
coefficients
7j
I
9- n
)
.
,
n capture the covariation between
aggregate "error" due to incomplete information, and
while the coefficients
)
n
1
with 0_ n standing for (Oj)j^ n
|
)
— - a + aoT^
=
vn
k*
(
K
—
the
k,
the complete-information equilibrium,
k,
—
n capture the covariation between the latter and k*
the efficiency gap
k,
under complete information.
A
6.
lower a n always implies a v n closer to zero, for less noise brings
But how
and
4>
n
0.
positive
and
(j)
n
As a
result,
when
is
K
Intuitively, less noise brings
(f>
n
the welfare contribution of a lower
but negative when
>
<j>
n
<
<j
any given
for
means getting
4>
n
K
but further away when
>
n through the covariance
Combining
0.
the correlation between equilibrium and
this
with the
effect of
necessarily increases welfare
best
first
is
term in
<r
(19)
is
n on volatility
when
<p n
>
positive) but can reduce welfare
(i.e.,
when
sufficiently negative.
The impact
of S n
on v n on the other hand, depends on the sign
,
K
transparency increases the covariance between
How
when
closer to k*
we conclude that higher accuracy
dispersion,
when
also
k
which in turn depends on the correlation between complete- information equilibrium
first best.
<
closer to
depends on whether getting A' closer to k
this affects welfare
closer to k*,
K
this in turn affects welfare
and k when a
depends again on the sign of
of the complementarity: higher
>
<f>
n
but decreases
.
it
when a <
0.
Hence, as evident from (20),
the sign of the effect of 8 n on first-order welfare losses depends on the sign of the product of a
and
4> n
-
Combining
with the
this
covariance term dominates for
[low] suffices for
<p
effects of S n
on
volatility
n sufficiently away from zero,
the welfare effect of 5 to have the
same
and dispersion, and noting that the
we conclude
that
<p
n sufficiently high
[opposite] sign as a.
These insights are verified in the following complete characterization of the welfare
effects of
information.
Proposition 7 There
4>
<
0,
exist functions
<f>,
<f>',4>,
<j>
:
(— 1,
1)
x (-co,
1)
—
>
M, with
4>
<
4>
and
<f>'
<
such that the following are true for any n 6 TV:
[Strategic Independece]
and only
[Strategic
if
a*
>
[a*
When a =
<
welfare increases [decreases] with 5 n for all
0,
if
and only
if
q>
n
[Strategic Substitutability]
((?n,5n) if
and only
if
<p
n
if
0]
Complementarity] When a
(o„,5 n )
(cr n ,5 n )
>
</>(a,
a*)
S
When a G
<
<f)(a,a*)
n
[<f>
(
welfare increases [decreases] with 5 n for all
(0,1),
<
</>(a,a*)].
— 1,0),
[4> n
>
welfare increases [decreases]
4>(a a*)].
}
15
with 5 n for
all
Welfare decreases [increases] with a n for
[Accuracy]
[<t><£
= —\
while
4>
<
<
if
By
(ft,
<p'
if
=
a*
and only
and only
if
Proposition
5,
;
(ii)
a*
if
< -a
a*
and
are invariant with £
<j>, <j)
,
whenever a
(f)
2
a £
for
> a
2
y£
/(I
-
and
;
0(= K n )
N
n £
and only
a € — 1,0),
(Hi) for
as long as
efficient if
is
N, then
a = a* and
cj>
The
—
is
if
and only
if
If
6.
and only
n
for all
=
=
k*
a* and k
the only inefficiency
this inefficiency
—
a
if
does not
is,
affect the
a*
following case
is
/(l
—
2
< a
a*
= =
tf>
2a),
while
,
still
,
in
which case
^
Kq or
comparative statics of
Proposition 6 continues to hold for
n £ N. Away from
is
benchmark, Proposition
this
be understood as a function of a, a*, and
—which
is
what we did
—
k,
the complete-
either constant or positively correlated with k.
0'
*
n which case Cov (k* — k,k) >
creases with the accuracy of information, whereas
it if
2
also instrumental for understanding the social value of information.
Corollary 4 Suppose (4>n)n£N —
and decreases with
=
either that «o
is
following sufficient conditions are then immediate for the case where k*
information efficiency gap,
The
<
We- conclude that understanding the efficient use of information
in the previous section
(j>
a > 1/2 or a* > —
if
4>
(
5 implies that the social value of information can
(<A n ) n6 jv-
(a, a*)
<j>
2a).
the equilibrium
some n
for
if
equilibrium welfare with respect to (S n ,a n ) for n G TV; that
all
>
d>
satisfy the following properties: (i)
<
(0, 1),
the welfare effects of information are given by
that k*
if
(a, a*)]
The functions
4>
(a n ,8 n ) if and only
all
< a<
it
increases with
0.
its
Then, welfare always in-
transparency
if
a*
> a >
0,
0.
also interesting, as
it
contrasts with the Blackwell-like result
we encoun-
tered earlier for efficient economies.
Corollary 5 Suppose
<fi
n
< —1/2 and a =
private or public information about 6 n
6
we show how our
efficient allocations
simplicity, in
6.1
=
0.
Welfare decreases with the precision of either
.
Applications
In this section,
and
a*
results
and the welfare
most cases we assume a
may
help understand the relation between equilibrium
effects of
single
information in specific contexts of interest. For
fundamental variable (TV
=
1)
and drop the index
n.
Investment complementarities
The canonical model
of production externalities can be nested by interpreting k as investment
and
defining individual payoffs as follows:
U(k, K, 9)
= A(K,
16
9)k
-
c(k),
(21)
where A(K,
9)
=
and c(k)
6 £ R,
—
(1
=
a)9
+ aK
represents the private return to investment, with a 6
2
k /2 the private cost of investment.
20
(0,
1/2) and
Variants of this specification appear in
Bryant (1983), Romer (1986), Matsuyama (1992), Acemoglu (1993), and Benhabib and Farmer
(1994), as
weU
as
models of network externalities and
important ingredient
investment
=
k*
}~2 9,
level of investment
and hence
the more so the higher
there
is
9.
= tSt >
<j>
under complete information
That
0-
investment
is,
= — 1, UkK =
Furthermore, Ukk
is
a >
this class of
is
k
= 6,
inefficiently
0,
and
equilibrium degree of coordination
that the equilibrium one:
a
=
a
is
positive
>
and a*
and the optimal one
=
2a > a >
0.
is
Using
models.
whereas the
low
for all 9
Ukk = Ua 2 =
a positive complementarity but no other second-order external effect.
we have the
The
that the private return to investment increases with the aggregate level of
is
—the source of both complementarity and externality in
The equilibrium
is
spillovers in technology adoption.
It
0.
first
>
0,
best
and
That
is,
follows that the
also positive
and indeed higher
this together
with Corollary
4,
following result.
Corollary 6 In
the investment example described above, coordination
fare unambiguously increases with both the accuracy
and
inefficiently low
and
wel-
the transparency of information.
In this example the agents' private desire to coordinate
actually not strong enough.
is
is,
not only socially warranted, but
then intuitive that higher transparency, or more precise public
It is
26
information, necessarily increases welfare by facilitating better coordination.
Economies with
—where complementarities emerge through
or other types of market incompleteness — are often related to
frictions in financial
lateral constraints, missing assets,
markets
economies with investment complementarities
priate for
many
positive questions,
it
col-
the one considered here. Although this
like
need not be so
for
is
appro-
normative purposes. As the examples we
study in the next two sections highlight, the result here depends on the absence of certain secondorder external effects and on a sufficiently strong correlation between equilibrium and first-best
Whether these properties
activity.
are shared by
mainstream incomplete-market models
is
an open
question.
"Beauty contests"
6.2
Keynes contended that
vs.
financial
other Keynesian frictions
markets often behave
like
"beauty contests" in the sense that
traders try to forecast and outbid one another's forecasts, but this motive
"This
on
8,
"
is
the example
we examined
thus omitting the effect of
Cov
in
(k,
.
(presumably) not
Angeletos and Pavan (2004), although there we computed welfare conditional
K
—
k.)
on welfare
Translating these results in terms of a x and a y
reduction in either a x or a y
is
,
it
is
losses.
easy to show that welfare unambiguously increases with a
Hence, both public and private information are beneficial in this example. However, a
higher a, by increasing the value of transparency, increases the welfare gain of public information and decreases that
of private information.
17
warranted from a social perspective because
Capturing
microfoundations
this idea with proper
following Morris and Shin (2002),
=
a >
=
a* and «()
«*(•).
due to some (unspecified) market imperfection.
it is
The
is
is
an open question, but one possible shortcut,
economy" as an economy
to define a "beauty-contest
means that the
condition
first
which
in
private motive to coordinate
not
is
warranted from a social perspective; the second means that the inefficiency of equilibrium vanishes
becomes complete. By Proposition 7 we then have the
as information
Corollary 7 In beauty-contest economies,
welfare
is
following.
increasing in accuracy but non-monotonic in
transparency.
The
Ui
where
6
£
K
and
r
G
i's
(0, l).
follows that k*
Note how
-(1 -
r)
this
=
- Of -
(fe
L = L(k
x
action from other agents' actions,
2
This example
'
U(k, K,
It
=
the underlying fundamental,
is
distance of agent
of Li,
assumed by Morris and Shin (2002)
specific payoff structure
=
k
6,
9,
a\)
Ukk
=
=
is
(Li
-
-
L)
= J (k' — k )~ d^(k') is the mean squareL = J L(k)d' i/(k) is the cross-sectional mean
% )
t
i
nested in our framework with
-(1 -
-2,
r
given by
is
r)
UkK —
•
(k
2
0)
-
r
•
(k
Ukk = — 2r,
2r,
example features two external
-
- K) 2 +
£/_2
=
r,
r
a\.
effects that tilt the trade-off
between
dispersion in the opposite direction than the complementarity. In particular,
the social cost of
volatility,
while
\Ja i
>
=
and hence a
r
= a*.
>
k
volatility
Ukk <
are non-strategic, in the sense that they do not affect private incentives,
increases
Both
decreases the social cost of dispersion.
and
effects
and both contribute to
reducing the social value of coordination. In the specific example considered by Morris and Shin
(2002), these effects perfectly offset the impact of the complementarity, so that the optimal level
of coordination
is
zero
— which explains why transparency, and thereby public information, can be
welfare- reducing.
Keyensian
competition or incomplete markets are in the heart of
frictions such as monopolistic
various macroeconomic complementarities (a.k.a. "multipliers" or "accelerators"). These frictions
share with beauty contests the idea that complementarity originates in some market imperfection.
However, the normative properties of beauty contests need not be shared by other Keynesian
frictions.
"'The
first
term
in u,
captures the value of taking an action close to a fundamental "target"
introduces a private value for taking an action close to others' actions, whereas the
no
social value in doing so.
social perspective,
it
is
as
if
Indeed, aggregating across agents gives
utility
were simply u
=
—(k —
8)~, in
w = —(1 —
r)
The
Li term
L term ensures that
f(k —
which case there
8.
is
0)"<i\t'(fc),
there
of course no social value to
coordination.
28
Note that Li
= / ((*:' - K) - {h - K)f d<H
(k')
=
(fc
-
18
A')
2
+
a\,
L=
2
<j\,
is
so that, from a
and L, - I
=
{K -
A')
2
-
o*.
Consider, for example, new-Keynesian monetary models where complementarity emerges in
pricing decisions
(e.g.,
Woodford, 2003; Hell wig, 2005; Lorenzoni, 2005; Roca, 2005).
goods results in a negative externality from cross-
class of models, imperfect substitutability across
sectional dispersion in prices
(U
i
<
which
0),
In this
in turn contributes to a higher optimal degree of
k
coordination
—the opposite of what happens
an excellent analysis of
information
is
beauty contest above. Hellwig (2005) provides
in the
He shows
this class of models.
that the optimal sensitivity to public
Moreover, the business cycle
higher than the equilibrium one.
is efficient
complete information. Translating these properties in our framework gives a* > a
in
which
case,
by Corollary
4,
>
and
under
<j)
=
0,
and transparency. This helps
welfare increases with both accuracy
understand why, unlike in Morris and Shin (2002), public information
welfare improving in
is
Hellwig (2005) and Roca (2005).
Inefficient fluctuations
6.3
The
focus in the previous section was on
trade-off between volatility
To
isolate the
gap k* — k
tilt
the
turn focus to first-order effects. In particular,
co- varies negatively
with k
—that
is,
economies
inefficiently deep.
impact of
external effects (JJkK
efficiency
the complementarity and second-order effects
We now
and dispersion.
we consider economies where the
where recessions are
how
first-order effects
— Ukk — Ua 2 =
0), so
^
(<j>
0),
that a*
we abstract from
=a=
0.
strategic
and second-order
From Proposition 7 then
<p
< —1/2
is
necessary and sufficient for welfare to decrease with accuracy and be independent of transparency.
Corollary 8 Suppose
in the sense that
Cov
that a*
=a=
(k, k*)
<
and
^Var(K,).
that equilibrium fluctuations are sufficiently inefficient
Then
welfare decreases with either private or public
information.
As an example,
consider an
economy where
8
—
(61,62) 6
and where agents engage
ffi"
investment activity without complementarity but for which private and social returns
U(k,
some A £
for
(0, 1).
follows that k
—
8\,
The
A', 9,
a%)
=
61k
2
-
k /2
private return to investment
while k*
=
(1
—
A) 8\
+
+
is
X{8 2
the discrepancy between private and social returns
is
9\, while
and
social returns
is
o~e 2
If
=
0,
so that k*
we maintain that the
is
=
—A.
But
if
close to zero, then
decreases with either private or public information about 8\
and
the social return
If
small enough, then
increases with either private or public information about 8\.
correlation between private
.
A
4>
A
and k*
is
19
low enough but
cf>
a*
89 3= 9\. It
<
> —1/2 and
l
1/2,
=a>
is
0,
welfare
meaning that the
= — 1 < —1/2 and
special case of this
let
is
A < 1/2, meaning that
constant and the entire fluctuation in investment
correlation between k
differ:
- h)K,
and hence <^
X82,
in an
is
welfare
when A
=
1
inefficient.
then welfare continues to
The
on the
recent debate on the merits of transparency in central bank communication has focused
role of complementarities in
new-Keynesian models
The
2005; Woodford, 2005; Hellwig, 2005; Roca, 2005).
suggest that this debate might be
somewhat misfocused
(e.g.,
Morris and Shin, 2002; Svensson,
results of this
—a
critical role is
and the previous section
played by the inefficiency
of equilibrium fluctuations.
For example, we conjecture that the result in Hellwig (2005) and Roca (2005) that public
information has a positive effect on welfare
in these models. In
up introduces an
shocks
—which
relies
on the property that the business cycle
standard new-Keynesian models
efficiency gap.
As long
(e.g.,
business cycle. But
if
fluctuations are driven by productivity, taste, or
the business cycle
efficient
Woodford, 2003) the monopolistic mark-
the case in Hellwig (2005) and Roca (2005)
is
is
—
this
monetary
gap remains constant over the
driven by shocks in mark-ups or the "labor wedge,"
is
it
seems possible that providing markets with information that helps predict these shocks can reduce
welfare. This
an interesting question that we leave open
is
Efficient competitive
6.4
The examples considered
We now
economies
so far feature either positive complementarity or
There
is
is
of inefficiency.
under both complete and incomplete information. 30
efficient
a continuum of households, each consisting of a consumer and a producer, and two
commodities. Let qn and
q^i
sumer
i).
living in household
denote the respective quantities purchased by consumer
v(q, 6)
=
6q
-
2
bq /2, 6 e R,
and
b
=
l
>
the price of good
are the profits of producer
terms of good
2.
i
1
relative to
v(q ll ,9)
0,
pqu
is
i
(the con-
His preferences are given by
u
where p
some form
turn to competitive economies where agents' choices are strategic substitutes and where
the equilibrium
where
for future research.
+
q 2l
(22)
,
while his budget
+ q2i = e +
good
2, e is
is
(23)
TTi,
an exogenous endowment of good
(the producer living in household
i),
2,
and
7r,-
which are also denominated in
Profits in turn are given by
n=pki-c(ki)
decrease with accuracy but
now
it
also decreases with transparency
(24)
— which strengthens particularly the case against
public information.
30
We
constructed this class of quadratic competitive economies independently but then found out that Vives (19SS)
had proved
efficiency of equilibria for exactly this class long before us.
only the welfare effects of information are novel here.
20
Hence, with regard to this particular
class,
where
good
ki
denotes the quantity of good
with
2,
c (k)
The random
=
2
k /2
produced by household
1
i
and
c(k) the cost in terms of
31
demand
variable 9 represents a shock in the relative
for the
two goods. Exchange
and consumption take place once 9 has become common knowledge. On the contrary, production
when information
takes place at an earlier stage,
Consumer
=
qn
K
consume the same quantity
for all
and p
i
=
— bK, where
9
-pK + e +
= bK
restated as Ui
=
v(K,
This example
is
thus nested in our model with
9)
U(k, K,
=
which case k*
in
and a*
=
a
= -b <
k,
=
9/
(1
+
TTi
9,
b)
,
a\)
2
/2
K=
l
= {9- bK)k -
Ukk
t
=
-1,
UkK
—
1
where
bqu.
It
=
7T;
follows that
=
pk{-c(ki)
i's
(6
utility
can be
-bK)k{-k 2 /2.
+ bK 2 /2 + e,
2
k /2
Ukk =
-b,
—
9
which together with market clearing
,
f kd^(k).
,
=
b,
Ua =
2
=
and therefore
0,
0.
der complete information, the economy
in
good
of
+e+n
That the complete-information equilibrium
economy
incomplete.
chooses (qu,q2i) so as to maximize (22) subject to (23), which gives p
i
Clearly, all households
gives
is still
which the
first
is
is
efficient (k
=
Un-
k*) should not be a surprise.
merely an example of a complete-markets competitive
What
welfare theorem applies.
is
interesting
is
that the equilibrium
under incomplete information, despite the absence of ex-ante com-
remains (constrained)
efficient
plete markets. This
because the strategic substitutability perceived by the agents coincides with
is
=
the one that the planner would have liked them to perceive (a*
a)
32
The
.
following
is
then a
direct implication of Proposition 6.
Corollary 9 In the competitive economy described above,
unambiguously decreases with both 6 and
may be
This result
monetary
a single
policy. If
common
we
relevant for the debate on transparency vs.
interpret "transparent" central
even
"
0).
It is
for efficient
bank disclosures
makes a case
be reminiscent of Morris and Shin (2002), but
= a <
constructive ambiguity in
as information that admits
=0
<
is
different.
a), here
perhaps more surprising that a case
it
This
for constructive ambiguity.
is
Whereas the
due to
for constructive
result there
was
efficient substitutability
ambiguity can be
made
competitive economies.
Implicit behind this cost function
is
a quadratic production frontier.
The
resource constraints are therefore given
by
= Ji fc an d Jz <?2i = e — | f, k^ for good 1 and 2, respectively.
Ji Qu
"The equilibrium would be inefficient if we had defined welfare as producer surplus
for
open economies that are net exporters of good
is,
and welfare
interpretation and "ambiguous" disclosures as information that admits multiple
driven by inefficiently high coordination (a*
(a*
is efficient
a.
idiosyncratic interpretations, then the result above
may
the equilibrium
2.
In this case, u;
=
7r,
alone,
and therefore a"
which
may be
relevant
= —2b < a = — b <
0;
that
the cooperative solution between the producers would involve stronger substitutability (and hence less sensitivity
to public information) than equilibrium.
21
Finally, this last result
welfare
damaging even
in
competitive economies where the use of information
example, suppose that an exogenous increase in the informativeness of prices
by a reduction
in the
may be
opens up the possibility that the informative role of prices
impact of noisy traders
Due
costly collection of private information.
markets
in financial
is
efficient.
—caused
for
For
example
—leads agents to reduce their
to strategic substitutability, such a substitution of
private information for public information could reduce welfare even
if
agents' overall uncertainty
reduces.
model
Clearly, the
in this
paper does not allow for information aggregation through prices, but
extending the results in this direction seems a promising hne for future research.
Appendix
Proof of Proposition
From
2.
condition
k[x,y)
K(6,
S)
(5),
=
kq
+ Yl
Kn
[(1
- 7 Js„ + jn zn
=
K
+ J2
*n
[(1
- 7n)0n +
]
In^n]
n€N
Hence k -
Var(x n -
K =
6n )
=
T, n€ ^ K n
[(l
a\ i0 Var(zn -
- j n )(x n 9n )
=
2
a Zn
=
{
K
and
9 n )}
a y^
- k = EnGW K n7n(~n -
+ a g„)
=
and 5 n
2
a~ /a~
Using
n )-
together with
,
we have
(6),
Var(k -
A
)
^
=
f
Kn
[(1
- 7 „)
<J
Var(K-K) =
which gives the
neN
9
J2 K nri*i =
£>'
n£N
n£N
9
Kn
(l-a) 2 (l-<5 n
{l _ a + a6n)2
Uk (K*, k*,9) + Ukk
°n,
<5n
_(l-a + a5 n
:al
)
Part
3.
(k(x,
(i).
y)-K*) +
Since
U
is
UkK (K -
quadratic, (8) can be rewritten as
K *)+
+ Uk {k\ k\9) + (UkK + Ukk) (K Of
2
)
result.
Proof of Proposition
E[
^
=
neN
33
2
course, for this to be true
it
must be that there
is
some
inefficiency in the collection
otherwise Blackwell's theorem would again imply that any exogenous information
22
«*)
is
beneficial.
\x,y]
=
=
0.
of information, or
Using
Wk (k*, 9) = Uk(K*, k*,6) + Uk(k*, k*,
E[
follows from the
Part
(ii)
with
«*()
Proof of Proposition
the equilibrium
is
+ ^Uk K + Ukk
Ukk
same steps
the unique function k
...
+
K^tf/v
,
=
1,
=
—
>
x,y
|
}
= 0,
re',
9)
=
a with a* and
U\
agents perceive payoffs to be
that solves
E.
(l-a)K +a'K
Uk (n',
KQ
2jV
When
(ii).
From
0.
\x,y],
(25)
= -U'kK /U'kk
E[k(x,y)\e,e], a'
the unique solution to (25)
k(x, y)
«*)
gives (12).
part
first
M
:
E[
=
where tf(0,e)
the unique solution to
is
Proposition
K 27V
G
the above reduces to
as in the proof of Proposition (1) replacing
Consider
4.
k(x,y)
for all (x,y)
0,
y)-K*) + (2UkK + UKK )(K -
(k(x,
Wkk =
which together with
re(-)
Ukk
=
9)
and
k'(9)
=
k'
+
k'^i
+
the same arguments as in the proof of
given by
is
+ Xl ne ^,
K»
[(
X
~
"f'n)
+ l'n z n)
Xn
,
where
-
1
For this to coincide with the
that
=
re'(-)
k*(-)
For part
diate that
(i) it
=
re'(-)
and that
=
K
is
re*(-),
dn
efficient allocation for all (x,y)
=
a'
and
-
a*,
which proves part
=
a'
(16).
a*,
K,
9)
=
G
M.
,
it is
necessary and sufficient
(ii).
+ Uk(K, K, 6)k,
U(k, K, 9)
in
which case
it is
imme-
u
Since
U
is
quadratic, a second-order Taylor expansion around
exact:
U(k, K,
It
1
suffices to let U'(k,
Proof of Condition
k
a'
9)
- U(K, K, 9) + Uk (K, K, 9)
follows that ex-ante utility
is
=
U(K, K,
9).
k(x,y) and
A
K =
-K) +
^-
{k
-
A')
2
.
given by
Eu = E[W(K,
where k
(k
Ukk^u^ - T
K
+ -y-E[{k
.,
9)}
K(9,e) are shortcuts
quadratic expansion of
N
2i
and W[K,9)
for the efficient allocation
W(K, 9) around
re*,
which
is
exact since
U
quadratic, gives
W(K,6) = W(k*,9) +
Wk {k*,B)
23
(A -
re*)
+
^~
(K -
re*)
2
.
and thus
=
W are
By
definition of k*,
Eu
At the
Efc
follows that
0. It
W
Ukk
KK wurj
^2
= EW(k*,8) + —~^-E[(K-k*^]
-
k
efficient allocation,
= EA =
=
Wk(k*,Q)
=
k*
,
^2 n eN' K *n
Ek* and therefore E[(A - k*)
—
[(1
ln)( x n
= Var{K - k*)
2
\
~
-E[(k-K) 2
+ 7n(~« —
E[(k - A') 2 =
#n)
and
]
#n)]
implying that
Var(k - A), which
gives the result.
Proof of Proposition
with the definitions of
The
5.
a and a*
«;*(•),
«(•),
Proof of Proposition
.
Suppose
6.
from the proof of Proposition 4 together
result follows directly
=
«;(•)
=
and a
«*(•)
a* and consider the set
K. of
allocations that satisfy
for
some
a'
<
1,
or equivalently
71
k(x,y)
=
E[(l
fc(z, y)
=
k
+
-a')« +
_ in) x n
Y^neN K « K 1
— Ek =
+ Yn ~n]
where
i
n (l-5 n
°"."
^+ a'5
—
Sn)
=
)
"
or
1
(
Clearly, the equilibrium (and efficient) allocation
allocation in K, EA'
a'if|x,y]
is
nested with a'
Ek, ex-ante welfare can be written as
IWifK-l
C = i!i££lyar (#-«)
+
= a(=
a*). Since for any
Eu = B1F
21/^1
1^ + —
^-Var(Jfc-A')
=
-
(k, 5)
—
£,
any
where
2U
—+ —
^fi,
\Ukk
a 2\
with
=
fi
Using
T/ar(A-«)
=
- A)
=
Var(fc
we have
(1
- Q*)Var (A -
^
^<
^
=
K
^
_
[(1
^ar(fc
=
(a
£V
_
2
- A).
g
/(
_
,2 [(J
_ yj2
2
{(J
/{1
_
^
^
that
Note that Eu depends on
allocation
~
+
£^ ^
y^^j
^n&N "|
l'n
«)
is
nested with a'
In solves dQ/d"f'n
=
0;
=
and
a'
a*,
that
it
l-5n
Sn
for
(<J,i,cr„),
must be that
n G
a'
TV,
=
n
f
only through Q.
Since the efficient
a* maximizes Eu, or equivalently that
is,
(1
_V)2$ =
On
24
iz<
1
-
0„
(26)
Q
Next note that
Sn
.
By
and hence Eu
increases,
is
d5 n
we thus have
(26),
the case
if
and only
if
U
Using the
K,
(k,
9, a'l)
fact that
a*
>
U
Since
and only
if
if
7*/(l - 7*)
>
[<]
<5 7l
/(l
- 6n ), which
a* (by efficiency) then gives the result.
A,
[k,
— K)'\8, e] and
E[(/c
9,
a2 )
A,
8,
is
quadratic in k and linear in a 2
0)(k
hence Eaj.
=
^(k - A)
- A) +
— A')-], we
E[(k
2
,
+ Ua *o\.
have that ex-ante
utility
given
Eu = EW(K,
A
—
Using a
[<]0.
(19).
[<]0
= U(K, K, 9, 0) + Uk (K,
=
o\
dEu/d5 n >
that
Proof of Condition
is
Finally, consider the effect of
.
the envelope theorem,
dd n
Using
any o n
decreases, with
Taylor expansion of
W( A, 8)
Ukk + 2Ua
+
9)
2
k
-
E[(k
-
A')
2
].
K = k then gives
around
Wk
W{K, 8) = W(k, 9) +
9)(K -
{k,
k)
—^(A - k)
+
2
and hence
Eu = EW(k,
In equilibrium,
E[(A -
k)}
9)
+ E[Wk (k, 9) (A -
EK =
Ek =
= Var(K -
2
Step
(f>,
</>',
E[(A -
E[Wk(k,
= Var(k -
2
}
9)
k)
2
g*
+
]
(K -
Wkk <
E[(fc
- A) 2
].
= Cov\Wk{k, 9), (K -
k)}
A), which gives the
prove the result in three steps. Step
k)],
result.
computes the welfare
1
statics.
Step 3 characterizes
(j>
The property
1.
and
<j),
that
W
is
quadratic, along with \\\-(k* 9)
,
=
(by definition of the
first
imply that
0,
Wk
It
•
due to incomplete information. Step 2 derives the comparative
the bounds
best),
We
7.
^p
therefore,
and E[(k - K)
k)
Proof of Proposition
losses
and
Eac
+
k,)}
(k, 9)
=
WK K
(
*,
9)
+
WKK -{k-k*) = \WKk\
(«*
-
«)
•
follows that
Cov(K Since A"
-
k
=
£ Kn
-y
n (z n
-
k,
Wk
9 n ), z n
mutually orthogonal whenever n
COV (K - K,K* -
K)
7^ j,
= \WKk\ Cov (K -
(k, 9))
-
9n
=
[X n (e n )
(1
- A„)(^ t
_
-
k)
9 n )]
(27)
.
,
and
(e n ,£j, 9 n 9 3 ) are
,
we have
= Cov
(
£ K n Jn (zn - 9 n
= H( K n~=
+
k, k*
)
,
£
K n) K n f n C0V (9 n Z n
,
(
Kn ~ K n)
-
8n )
8n
=
Z (^T^yhnl-(l-K)Var(9 n
25
)
)}
=
Using
=
4> n
«-
Kn )/Kn
=
7„
,
- a + ad n ), and
5 n /(l
- A n Var
(1
=
(8 n )
)
(cr
g
^/a z ^)aj n = a\ n =
a n/5 n we have that
,
l
C0V(K-K,K*-K,)= Y.^nl--,
-^-T K l a l]
1 - a + a °n
I
(28)
j
ir,
while
Cov
(
K - k, k
0_„
|
=
)
K^7 n Cou
(*„
-
n
6>
,
= --
n)
n
— a + aoT-^
n
1
Next, as in the proof of Proposition
crn
.
2,
5
Var (A'"^
tt
E n - nZnXv
+
=
^—i
z
=
Substituting (27)-(30) into (19), using v
(1
(29)
ad n J-
(l-a)i(l-6 n
A
(1 — a + ao n
—i
—
a
(1
n€N
2
)
2
)
a*)\Ukk
+ 2 f/
2
,
1
and rearranging, we get
k
+ 2Uol
- 2^ A (a, a
\Uk k
L=
k
\
<5
0,,,
,
n)
<<
n6Af
where
=
(1
Note that KW(k,9)
is
A
Step
2.
(a,
a
,
0, d)
welfare with respect to (5n
,
o~ n
)
dC
a*) [20(1
independent of
\Uk k
+ 2Ual
(5 7l ,a n )
Kn A{a,a
+ 2Ua2\
2
thus need to understand the sign of
By
2
(l
-
*)
•
and hence the comparative
\
2
85 n
We
q)
coincide with the opposite of those of £. Note that
\Ukk
dC
- a + a5) + 5] + (l9
(1 — a + ao)~
,
=
q~2
-
,4>n ,a n )
dA(a,a*,<f>n ,5n )
Kn °n2
85 n
A and 8A/8d n
.
(31),
dA
q 2 [(l ~
=
J n )(l
- a) -
gn]
-
(l-a +
S5„
When a =
0, this
-a- aS n
a*(l
a6 n )
-
)
2a<t>(l
-
3
reduces to
dA
_
_
,
<9<5„
and hence,
When
for
any n £ N, 8C/85 n > [<]0
instead
if
and only
if
a*
<
[>]0.
a^O,
dA
85 n
2(1
[1
-q*)
- q + a5 n
2q[/(q,a*,d" n )
2
)
26
-
<f>
n },
q*)(l
- a + aJ n )
(31)
statics of
/(a a
<
Since a*
su7n[<9£/(9<5 n
1,
=
]
=
J)
'
'
or
(p
n G (0,
a <
0.
a >
and
- 0j.
sign\f(a,a*,5 n )
and
/(a, a*, 5)
alternates sign as S n varies between
is
necessary and sufficient for
V£ n when a
<
0,
whereas
n
d£/d5 n >
for
V(5„
dC/da"^
Finally, note that
/(a, a*,
<5).
<5e[o,i]
cp
<f>
Let
= max
<^(q,q*)
<
Hence,
dL/d5 n <
dC/dS n
then
</>),
2a(l- a + a5)(l-a*)
sign[a]
4>(a,a*) = min
<5e[o,i]
If
ai{(l-5)(l-a)-5}-a*(l-a-a6)
-
n
>
when a <
0.
>
[<]
*
«=
(p
(p is
dC/d5 n >
if
>
ra
—
-
<5
—
+ J(l- Q *)
<
- a + ad)
—
2(1
'
V5„ when a
[<] g(a, a*,
(l-Q.) 2 (l-^)
,
o(a,a
,o);
yv
no matter whether a >
1,
dC/d5 n <
necessary and sufficient for
and only
if
and
a*)(l
>
and
for
V£ n when
where
n ),
0.
Letting
a*)= min
6'(a,
and
<?(a, a*, 5)
</>
= max
(a, a*)
d"£[0,l]
we
get that dC/do'^
as £ n varies
Step
3.
if 4> n
G
>
[<
((/>',
(ft
0]
a*
=
instead a*
and when a* <
Consider
a*
<
1)
G
[0, 1] if 4> n
a,
first
>
a,
/
is
f
>
4>
[<
</>'],
whereas dC/do^ alternates sign
is
=
a*)
a*)
<j)(a,
in 5, with
(l-a)
^g =
=
both / and 5 are independent of
4>'(a,
When
for all 5„
Note that both / and g are monotonic
a,
5),
).
g/
When
g(a, a*,
<5e[0,l]
=
and
5,
=
a*)
<p(a,
,_
<p
(a, a*)
=
f(a,a*,l),
(a, a*)
=
g(a,a*,0),
=
f(a,a*,0)
=
g(a, a*,
=
f(a,a*,0)<j>(a,a*)
4/ (a, a*)
=
g(a,a*,l)
strictly decreasing in 5
<
4>
=
f(a,a*,l)<j>(a a*)
4>'(a,a*)
=
g(a, a* ,0)
>
(0, 1). If
a*
>
5,
<
<p
(a, a*)
a, then
a2 +
(1
1).
- 2a)a* >
(using the fact that
and therefore
#a,a*) <
<Ha, a*)
=
so that
and
£(a,a*)
a G
0.
(and g strictly decreasing) in
strictly increasing
±(a,a*)
the case
— —— <
/(a, a*,
27
1)
=
Zl
2a(l —
J_ <
cv*)
0.
If
<
instead a*
a,
*\
At
#«,a
=
)
Since
—a
/(l
ft
/(a,a
<
and therefore
2
then
—
a > 1/2 or a* >
if
-a
2
-n
,
if
/(l
-
a*
<
<
a, then a*
and only
<
<
<
we conclude
if
If
and only
>
a*
a = _
2;
^
< a2
while
if
a*
)
,
-
)
=
/(«,«
> -a 2 /(l - 2a),
or a*
1/2,
a G ( — 1,0).
/<„, „-.<»
if
<
<f>
(
and hence
(1
a > 1/2
and
2a).
^o
- 2a)a*
<
_
2a(1
a. }
+
-
whenever a
Next, consider the case
Ha, a') =
Q2
=
1)
and only
<
2a)
*
if
while
a*
> a
- a2
q*
-
2a(1
<
if
(0, 1),
_ a«)
and only
<p
<
if
if
a'
> a2
and only
.
if
2
.
a, then
< *., C) =
cfi
<
if
=
/(a, a', 1)
and only
a*
if
f^fff
< — a 2 /(l —
2a). If instead
a~ and hence
*
0(a,a*)
—
We
<ji
a G
that, for
=
,0)
conclude that, for a 6
<
j>(a,a*)
(0, 1),
<
tfi
=
if
and only
if
2
° ~
"*
(— 2a)(l
—
=
f(a,a*,0)
a*
<
a",
<
0.
a*)
and © <
if
and only
if
a*
<
2
-a /(l-2a).
Finally, note that
g( a
Q *.°)
.
=-
(
Together with the monotonicity of
whenever a
<
a*,
and
lI^ <°
2(
0'
and
ff(a,a*,l)
= -|<0.
)
5, this
implies that
> —1/2 whenever a >
a*
.
$ <
<j>
<
for all (a^a*),
<p
< — 1/2
u
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'
2 3
!'
u
o
30
Bank
of Kansas City at Jacksons Hole,
I
4 2006
Date Due
MIT LIBRARIES
""|ll||l|||ll
III|IPI|II|I
3 9080 02617 9595
:
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•