GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.137

The Benefit of Mixing Private Noise into PublicInformation in Beauty Contest Games

Hiroki Arato and Tomoya Nakamura

May 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

The Benefit of Mixing Private Noise into PublicInformation in Beauty Contest Games∗
Hiroki Arato†
Tokyo Metropolitan University

Tomoya Nakamura‡
Graduate School of Economics, Kyoto University
Institute of Social and Economic Research, Osaka University

May 27, 2010

Abstract
The authorities often disclose their economic forecasts to the public ambiguously. The more ambiguous the way in which they present the information, themore variously market participants interpret such announcements. Hence, an ambiguous public announcement adds some private noise to the authorities’ economic
forecasts. Using Keynesian beauty contest games, as in Morris and Shin (2002), we
ﬁnd that the addition of private noise to public information is often socially beneﬁcial. We also derive the optimal information dissemination policy and ﬁnd thatthe authorities should acquire information that is as precise as possible, but theyshould make ambiguous announcements to a degree that reﬂects the uncertainty intheir forecast.
∗

We would like to thank Hideshi Itoh, Noritaka Kudoh, Makoto Saito, Tadashi Sekiguchi, and Akihisa
Shibata; the participants of Winter Institute 2010 sponsored by the global COE program of Hitotsubashi University, Young Economist Conference by the global COE program of Osaka University, and theJapanese Economic Association Autumn Meeting 2009 at Senshu University; and the seminar participants at Hitotsubashi University, Senshu University, Tokyo Institute of Technology, Tokyo Metropolitan
University, and Bank of Japan for their helpful comments. Any remaining errors are the sole responsibility of the authors. The authors are also grateful for research grants provided by the research project
entitled: Understanding Inﬂation Dynamics of the Japanese Economy, funded by JSPS Grant-in-Aid for
Creative Scientiﬁc Research (18GS0101), and the global COE programs of Hitotsubashi University andOsaka University.
†
E-mail address: arato@tmu.ac.jp‡
E-mail address: tomoya a 25e@yahoo.co.jp

1Keywords: Beauty contest games; Information structure; Public announcement;
Social welfare
JEL classiﬁcation: C72, D82, D83, and E5

1

Introduction

Central bankers as well as governments are often required to disclose transparent
information about the underlying economic fundamentals. It is widely believed that
if market participants have more accurate information, they can behave more accurately. In some situations, this is true. The provision of such additional informationby the authorities allows the participants to predict the fundamentals with greater
accuracy. Hence, if the participants are concerned only about the fundamentals,
their behavior tends to reﬂect the fundamentals more accurately, and thus, in thissituation, it is socially beneﬁcial for them to possess more information.
However, in many situations, it is observed that market participants behave
strategically, and for this reason, the participants are concerned about not only
the fundamentals but also other participants’ behavior. Morris and Shin (2002)
consider the problem of public information dissemination in such a situation using“beauty contest games.” In their model, each agent has a coordination motive arising from strategic complementarity, but the coordination motive becomes socially
ineﬃcient through the zero-sum structure. Each agent receives two signals about
the underlying fundamentals. One is a public signal from the authorities, which isshared by all agents. In this model, a public signal is a signal that is perfectly correlated among agents. Another is private information, which is perfectly uncorrelated
among agents and cannot be observed by other agents. In this situation, Morrisand Shin (2002) show that the authorities’ public information dissemination could
lower social welfare. The public information serves as a focal point for the belief ofthe other agents. Equivalently, for each agent, the public information is more usefulthan private information for the formation of “higher-order expectations,” that is,an agent’s expectation about others’ expectations of the others’ expectations of · · ·of fundamentals. Hence, when there is strategic complementarity, the agents place
more weight on public information than does the social planner when choosing
his/her action. In other words, the agents overreact to public information. Becauseof this, when the public information is highly noisy, an agent’s action deviates fromthe fundamentals on average as a result of the public information dissemination.
In consideration of this behavior, the authorities should not disclose information
if its precision is suﬃciently low relative to the agents’ private information andshould disclose the information as precisely as possible otherwise. Therefore, in
their model, the optimal announcement policy has a “bang-bang” solution feature.
We may regard the stock market to be a case described by the model studied byMorris and Shin (2002). Certainly, the dividends of stocks reﬂect the fundamentals(the productivity of ﬁrms) to some extent. Therefore, market participants are concerned about the fundamentals. In addition, there exists strategic complementarity

2

in the market, because when many agents buy a stock, its price rises, and hence, anagent who has already bought the stock obtains capital gain. Hence, agents are also
concerned about others’ actions. However, such a coordination motive arising fromstrategic complementarity is not always socially beneﬁcial, because such behavior
may be the source of bubbles and ﬁnancial crises. Thus, if this information significantly aﬀects the agents’ behavior, the authorities should handle the information
dissemination carefully.
As an argument against Morris and Shin (2002), it is rare that the authorities do
not disclose any routine information in a given case. Instead, the authorities often
announce their information ambiguously. Alan Greenspan, the former Chairman of
the Federal Reserve, made the following statement in the Wall Street Journal:
Since I have become a central banker I’ve learned to mumble with great
incoherence. If I seem unduly clear to you, you must have misunderstood
what I said. (Alan Greenspan, Wall Street Journal, 1987)
How should we understand this quotation? It is theoretically true that agents
overreact to perfectly correlated information in the economy with strategic complementarities. If a coordination motive arising from strategic complementarity issocially undesirable, a clear announcement by the authorities, which provides purely
public information to the agents, could result in a worse outcome because agents
overreact to public information. For this reason, we conjecture that Greenspan’s
statement is correct. Ambiguous announcements by the authorities could havevaried interpretations among agents. Thus, an ambiguous announcement by theauthorities corresponds to mixing private noise into public information. Hence, anambiguous announcement by the authorities could prevent an overreaction by theagents. However, preventing an overreaction by an ambiguous announcement of
the authorities would not always improve social welfare, because an ambiguous announcement implies a degradation of information about the fundamentals. Hence,
the welfare eﬀects of ambiguous announcements should be investigated formally.
Moreover, when the authorities are allowed to disclose their information vaguely,
it is not clear how they should acquire information; that is, the welfare eﬀects ofcommon noise among agents should be reconsidered. In this paper, we addressthese two questions.
We use the same payoﬀ structure as Morris and Shin (2002) but extend theinformation structure so that both signals are allowed to be imperfectly correlated.1
To do this, we assume that both the signals received by agents contain two kindsof noise, common noise and private noise. This assumption of the information
structure can be interpreted as follows: all agents receive information about thefundamentals from two information sources. For example, One is a private thinktank and another is the authorities (the government or the central bank). Each
1

Myatt and Wallace (2008) use a similar information structure, but study a Lucas-Phelps island
economy. Hence, their results are partially diﬀerent from ours.

3information source researches the fundamentals. If a source has a suﬃciently goodability to conduct such research, it can predict the fundamentals with small error.
Then, the source releases its prediction to market participants. If the source states
the prediction very clearly, market participants would receive common informationfrom the source. Hence, the common noise of the signal results from the imperfectnature of the source’s research. In other words, (the reciprocal of ) the varianceof the common noise reﬂects the source’s research ability. Alternatively, if thesource states its prediction ambiguously, the market participants would interpretthe message in various ways. In other words, the ambiguity of the announcementby the source is equivalent to an additional private noise. Hence, (the reciprocalof ) the variance of the private noise reﬂects the degree of clarity of the source’s
announcement.
Using our model, we consider the welfare eﬀect of reseach ability and clarity of
public predictions. We ﬁnd that the welfare eﬀect of the ability can be U-shaped,
as found in Morris and Shin (2002). Therefore, if authorities can only choose
research ability, the bang-bang solution still exists. However, we also ﬁnd that if
the authorities can choose clarity as well as ability, then the bang-bang solution
disappears. Our model indicates that it is optimal that the authorities always
acquire information with as high a precision as possible but disclose it with the
ambiguity appropriate to its research ability. This means that there is a beneﬁt of
mixing private noise with public information. In our model, the control of clarity
is given to the authorities as a free option. According to our results, even in thesituation where Morris and Shin (2002) conclude that the authorities should hide
their information, the authorities should use the option to disclose information
ambiguously in order to improve welfare.

Related Literature
In this literature, most studies assume a restrictive information structure, which
consists of purely private and purely public information. They investigate the welfare eﬀect under various payoﬀ structures and ﬁnd implications against Morris andShin (2002). Angeletos and Pavan (2004) assume payoﬀs that have investment
externalities, and Hellwig (2005) assumes a monopolistic competition payoﬀ structure. They conclude that purely public information always improves social welfare.
Angeletos and Pavan (2007) consider a more general market environment. They
parameterize the cases when purely public information improves the welfare.
The exceptions are Cornand and Heinemann (2008) and Myatt and Wallace(2008). They focus on the welfare eﬀects under a diﬀerent information structurefrom Morris and Shin (2002).
Cornand and Heinemann (2008) study the optimal dissemination range of public announcement. They conclude that the authorities should always choose as higha precision of public information as possible and public information should be disseminated to only some of the agents. Hence, from Cornand and Heinemann (2008)4and our study, we ﬁnd that disseminating public information to some of the agentsand mixing private noise into public information have a similar welfare implication.
Myatt and Wallace (2008) extend a standard information structure. Their extension is similar to ours. However, their study mainly focuses on a Lucas-Phelps
island economy. They emphasize that in the Lucas island economy, the output gap
is an appropriate measure of macroeconomic performance. On the other hand, as
in Morris and Shin (2002), we deﬁne the welfare measure as the heterogeneity ofequilibrium action. We agree that the output gap is a more correct welfare measureof the macroeconomy. However, we focus on the stock market, and believe that thewelfare measure of heterogeneity is appropriate for three reasons. First, in stock
markets, the authorities would care about the realized price as well as the stabilityof the price. We can divide the welfare measure, as in Morris and Shin (2002), into
two parts, the loss from volatility (the average deviation of average action from thefundamentals) and dispersion (the average deviation of each agents’ action fromaverage action). The volatility part corresponds to the gap between the realized
price and fundamentals, and the dispersion of the actions represents the dispersionof the bid prices. It can be interpreted as the market stability measure. Hence,
we think that heterogeneity of actions is appropriate for the eﬃciency measure ofstock markets. Second, in reality, the central bank communicates to stock market
participants rather than the production sectors. Hence, when central bank communication is considered, the stock market eﬃciency measure of heterogeneity is more
appropriate for the objective function. Third, stock market conditions strongly
aﬀect macroeconomic performance through the ﬁrms’ and households’ ﬁnancial positions and transmission mechanism of the monetary policy. Hence, even if we think
about macroeconomic performance, stock market conditions are still important forthe central bank and the government.

2

The Model

2.1

Payoﬀ Structure

In the economy, there is one measure of agents indexed by i ∈ [0, 1]. Each agent
chooses an action ai ∈ R. We write a for the action proﬁle over all agents. As in
Morris and Shin (2002), we assume the following payoﬀ structure that describes
Keynesian beauty contest games:ui (a, θ) = −(1 − r)(ai − θ)2 − r(Li − L),(1)where θ is the state of the economy (fundamentals), r ∈ [0, 1) is a constant, and
∫ 1
∫ 1
2
Li ≡
(aj − ai ) dj,
L≡
Lj dj.
00

The ﬁrst term of i’s payoﬀ represents that agent i suﬀers a loss from the distance
between his action ai and economic fundamentals θ. Li is referred to as beauty

5contest term. The term implies that agent i incurs a loss from the distance between
his action and the others’ action. Hence, agents have a coordination motive. Then,
we have agent i’s best response function:ai = (1 − r)Ei (θ) + rEi (ā),(2)

where Ei represents the∫ 1 agent i’s expectation operator conditional on his available
information and ā ≡ 0 aj dj represents the average action over all agents. From(2), parameter r can be understood as the degree of agent i’s coordination motivein this economy.
We assume that social welfare is deﬁned as the normalized aggregation of payoﬀ
over the whole population:
∫ 1
∫ 1
1
ui di = −(ai − θ)di.
(3)
W (a|θ) ≡1−r 00
Note that the utility loss generated by the beauty contest term disappears in thissocial welfare function; hence, the socially optimal action isai,opt = Ei (θ).

(4)

This implies that, in this model, the action that reﬂects only the fundamentals issocially optimal.
If the information is complete, ai = ai,opt = θ. Hence, under complete information, the equilibrium allocation in this economy is eﬃcient and can achieve theﬁrst-best allocation. We consider the case of incomplete information below.

2.2

Information Structure

For simplicity, we assume that the agents have an improper prior distribution of
the fundamentals θ. In Morris and Shin (2002), each agent potentially receives two
kinds of signals regarding θ. One is a purely private signal that is independent
2
among agents, xi = θ + ϵxi with ϵxi ∼ N (0, σx,an). The other is a purely public
2 ).
signal that is perfectly correlated among agents, y = θ + η y with η y ∼ N (0, σy,ac
We extend this information structure. First, consider public signal y. Morrisand Shin (2002) interpret this as the information released by the authorities. This
assumption implies that the authorities acquire information and disclose it as itis. However, in reality, information acquisition and disclosure are diﬀerent policy
tools. Statistical staﬀs or researchers obtain the information and report to top
oﬃcials. The top oﬃcials decide not only whether to release it but also with howmuch clarity they should disclose it to the public. To separate these two policies,
we assume the signal received by agent i from the authorities is such thatyi ≡ y + ϵyi = θ + η y + ϵyi ,62with ϵyi ∼ N (0, σy,an).

As explained in Section 1, we can interpret η y as an information acquisition error,2and σy,ac
as the authorities’ research ability.2 We assume the authorities can controly2
2 , ∞) and σ̄ 2
σy,ac ∈ [σ̄y,ac
y,ac > 0. The second error term ϵi can be regarded as2announcement error and the authorities can choose the value of σy,an∈ [0, ∞).
This means that the authorities can freely choose the expression of disseminating
information to the public and control the variance of the interpretations of her
2
2
message. We refer to σy.an
as announcement clarity.3 Note that when σy,an= 0,
yi = y becomes purely public information. Hence, this extension nests Morris andShin (2002).
To make our analysis general, we assume that the signal from the other information source (for example, a private think tank), xi , is such thatxi ≡ θ + η x + ϵxi ,2with η x ∼ N (0, σx,ac),22
and assume that σx,acand σx,anare given for the authorities.

3 Eﬃcient and Equilibrium Information Allocation
In this section, we derive an equilibrium strategy and socially optimal information
allocation under the payoﬀ and information structure given in the previous section.
2
2
Deﬁne ψk ≡ (σk,ac+ σk,an)−1 , k ∈ {x, y}, as signal k’s precision. By using ψk ,agent i’s expectation of the value of θ can be derived as:
Ei (θ) = E(θ|xi , yi ) =ψx xi + ψy yi.
ψx + ψy(5)

Hence, from (4), agent i’s socially optimal information allocation isai,opt (xi , yi ) =ψx xi + ψy yi.
ψx + ψy(6)

In equilibrium, however, agents do not behave according to (6) because they areconcerned about the average action as well as the fundamentals. Then, the signals’
correlations among agents play an important role to estimate average action andhence, an equilibrium strategy for the following reason. If the signal is uncorrelated
among agents, agent i cannot know the realized value of the signal that other agentsreceive. On the other hand, if the signals are correlated among agents, then thesignal that agent i receives has some information about the realized values of thesignal that other agents receive. Because all agents know that the others move byusing such correlated signals, the correlated signal can use to predict the average22 /(σ 2
action. Deﬁne ρk ≡ σk,ack,ac + σk,an ) as signal k’s correlation among agents andwe then have the following proposition about an equilibrium strategy.
232Note that high σy,acmeans low ability.
2
Note that high σy,anmeans low clarity.

7

Proposition 1. The equilibrium exists, is unique, and is linear in xi and yi . Suchan equilibrium strategy can be written asai (xi , yi ) ≡βx ψx xi + βy ψy yi,
βx ψx + βy ψy(7)where βk ≡ (1 − rρk )−1 .
Proof. See Appendix.
Since r > 0, βk is increasing with ρk . Hence, we ﬁnd that the agents react tothe relatively correlated signal more strongly than the Bayesian (socially optimal)
weight.
From (6) and (7), we ﬁnd that if βx ̸= βy , agents allocate the two signals
2 , σ2
ineﬃciently. Recall that (σy,acy,an ) is the authorities’ information policy. Hence,
the authorities can control βy and set the equilibrium weight near the socially
optimal weight. However, changing βy involves a change of ψy . If the authorities
have to lower ψy in order to set βy near βx , the welfare eﬀect is ambiguous becausesuch a policy makes it diﬃcult for agents to predict the fundamentals. Hence, what
we have to do next is to check the marginal welfare eﬀects of ability and clarity andﬁnd their optimal pairs.

4

Welfare and the Optimal Policy

Substituting (7) into (3), the expected welfare is given as[∫ 1](ai − θ)di θ
E[W (a|θ)] = −E0=−ψx βx2 + ψy βy2(ψx βx + ψy βy )222≡ −Ω(σy,ac, σy,an).

(8)

Examining (8), we can know the welfare eﬀect of authorities’ ability and announcement clarity. First, we consider the eﬀect of ability given clarity:∂Ω<02∂σy,ac2⇔ σy,an<2222+ (2r − 1)σx,an]2−(1 − r)σx,ac+ σx,an[(1 − r)σx,ac
2
.
σ
−
y,ac
2
2
2
2
(1 + 2r)σx,ac+ (1 − r)−1 σx,an(1 + r)[(2r + 1)σx,ac+ (1 − r)−1 σx,an](9)2 /σ 2
We can verify that when (2r − 1)/(1 − r) > σx,acx,an , the welfare eﬀect of ability
is U-shape. Figure 1 shows the welfare eﬀect of ability graphically.

8

[Insert Figure 1 here.]2Note that we can translate the case of σy,ac= ∞ as no announcement. Because
the signal that has zero precision means that such an information is junk, no agent
uses that signal to determine his action. Hence, the authorities determine her
2announcement policy as to whether the welfare surpasses EW (σy,ac= ∞). We can2222easily obtain the critical value σ̃y,ac deﬁned as EW (σy,ac = σ̃y,ac ) = EW (σy,ac
=
∞):
σ2

2
σ̃y,ac≡2 (1 − r + x,an )2
σx,ac
σ2

2
2
(1 + r)σx,ac+ σx,an
2
+
σy,an
.
2
2
2
2
σ
x,an
−(1
−
r)
σ
+
(3r
−
1)σ
x,ac
x,an
−(1 − r)2 + (3r − 1)
x,ac

2
σx,ac

The next proposition sums up the comparative statics about the ability.
2 /σ 2
Proposition 2. Assume that (2r − 1)/(1 − r) > σx,acx,an and ﬁx the clarity. If22σ̄y,ac ≥ σ̃y,ac , the welfare improves by her announcement. On the other hand, if
2
2 , the welfare is degraded relative to no announcement.
σ̄y,ac< σ̃y,ac2Note that Morris and Shin (2002) is the case of perfect clarity, σy,ac= 0. Thenthe “bang-bang solution” is optimal: disclose their information if the ability is
suﬃciently high, or hold it back otherwise.
Next, we consider the welfare eﬀect of clarity:∂Ω<02∂σy,an2⇔ σy,an<2222−(1 − r)2 σx,ac+ (3r − 1)σx,an[(1 − r)σx,ac+ σx,an
]2
2
σ
−
.
y,ac
2
2
2
2
(1 + r)σx,ac(1 + r)σx,ac+ σx,an+ σx,an(10)

2 /σ 2
When (3r − 1)/(1 − r)2 > σx,acx,an and
2
2
˜y,ac
σy,ac
> σ̃
≡

2
2
[(1 − r)σx,ac+ σx,an]2,22−(1 − r)2 σx,ac+ (3r − 1)σx,an2. We depict
the welfare eﬀect of clarity is an inverted U-shape with respect to σy,anthe fact in Figure 2.

[Insert Figure 2 here.]
The intuition is as follows. A more ambiguous announcement by the authorities
lowers its correlation and precision. When the clarity of the authorities is high, yi ismore correlated than xi (so that βy > βx ). Then, a more ambiguous announcement

9moves its correlation toward another signal’s correlation. Hence, the ineﬃciency of
information use is reduced. This positive welfare eﬀect exceeds the negative one by
the information degradation when yi is much more correlated than xi . However, as
the clarity decreases, the positive welfare eﬀect by the lower correlation becomes
weaker, and becomes even more negative when yi is less correlated than xi . Hence,
the total marginal welfare eﬀect by the more ambiguous announcement is positive
when the clarity is high but negative when the clarity is low.
2 /σ 22
˜2
Proposition 3. Assume that (3r − 1)/(1 − r)2 > σx,acx,an and σy,ac > σ̃y,ac , andﬁx the ability of the authorities. Then, the welfare eﬀect of disseminating ambiguous
2
information to the public is an inverted U-shape. Hence, the optimal clarity σ̃y,an
is strictly positive.

Note that the clarity in our model is regarded as the option that is granted tothe authorities in the model of Morris and Shin (2002) at no cost. Therefore, if
welfare is decreased by using this option, there is no need to exercise it. However,
Proposition 3 states that the optimal clarities exist in the interval [0, ∞). Thismeans that this option has the ability to improve social welfare.
[Insert Figure 3 here.]Here, we bundle the two eﬀects of the two policies in one ﬁgure. Figure 3
2 , σ2
illustrates the social welfare contours in (σy,acy,an )-space. Note that from (9)2
and (10), there is the line that represents ∂EW/∂σy,an= 0 upper than the line2
represents ∂EW/∂σy,ac = 0. The curves represent iso-welfare sets. The arrows
show the direction of welfare improvement. Recall that this beauty contest game iseﬃcient and achieve the ﬁrst best in the case of complete information. Therefore,
the origin is the point of ﬁrst best. Recall also that the authorities can freely choose
22
2 , ∞).
clarity σy,anin [0, ∞), but the choice of ability σy,ac
is restricted in [σ̄y,ac
Finally, we have our main proposition.
2
2 , ∞) and σ 2∈ [σ̄y,ac
Proposition 4. Assume σy,acy,an ∈ [0, ∞). Then, (i) the2
2 /σ 2
2
optimal ability is equal to σ̄y,ac ; (ii) if (3r − 1)/(1 − r)2 > σx,acx,an and σ̄y,ac >222
˜y,ac , then the optimal clarity σ̃y,an is strictly positive; and (iii) σ̃y,an is (weakly)σ̃
2 .
increasing in σ̄y,ac(i) says that the authorities should acquire as a precise information as possible.
This means that the “bang-bang solution” of Morris and Shin (2002) disappears.
(ii) implies that it is desirable that the authorities announce her information ambiguously to some extent when the ability is low. (iii) means that the lower theauthorities’ ability, the more ambiguous she should be in her announcement. We
think that proposition 4 supports Greenspan’s comments.

105Conclusion

We extend the information structure as in Morris and Shin (2002) and show thatthe authorities’ ambiguous announcement could be beneﬁcial to social welfare. Our
main conclusions are presented in proposition 4. An ambiguous announcementmeans adding private noise into the purely public information. If the agents overreact to the public information because of strategic complementarities, then adding
private noise to public information prevents the agents from overreacting. Suchan announcement policy could be desirable when the authorities have an imperfect
information acquisition ability. Moreover, our result suggests that the lower theability, the greater the beneﬁt from the more ambiguous announcement.
There are some open questions in this paper. First, we assume a full-commitment
announcement policy. Allowing that the authorities act strategically, she may tella lie to improve ex post social welfare after the value of signal is realized. Second,the complete information and the common knowledge about ability and clarity arestrong assumptions. In reality, it is natural that market participants cannot know
these precise values. Finally, our result may depend on the speciﬁcation of thepayoﬀ structure, as in Morris and Shin (2002). In this literature, it is known thatthe welfare implication of information is highly dependent on the payoﬀ structure.4
However, we think that this research oﬀers some kind of contribution. First, ourmodel has simple and clear implications and some potential for extension. Second,
our conclusion that an ambiguous announcement can be beneﬁcial suggests that weneed to consider in more detail not only whether the government should disclose
but also how the government should disclose.

References
Angeletos, G.-M. and A. Pavan (2004) “Transparency of information and coordination in economies with investment complementarities,” American Economic
Review 94(2), pp. 91–98.
Angeletos, G.-M. and A. Pavan (2007) “Eﬃcient use of information and social valueof information,” Econometrica 75(4), pp. 1103–1142.
Cornand, C. and F. Heinemann (2008) “Optimal degree of public information dissemination,” Economic Journal 118, pp. 718–742.
Hellwig, C. (2005) “Heterogeneous information and the welfare eﬀects of public
information disclosures.” UCLA working paper.
Morris, S. and H. S. Shin (2002) “Social Value of Public Information,” American
Economic Review 92(5), pp. 1521–34.
4

See Angeletos and Pavan (2007).

11

Myatt, D. P. and C. Wallace (2008) “On the sources and value of information:
public announcements and macroeconomic performance.” Working paper, Oxford University.

Appendix: Proof of Proposition 1
The existence, uniqueness, and linearity are the same reasons as in this standard
literature.5 Here,∫ 1 we show the derivation of (7).
Deﬁne x ≡ 0 xi di = θ + η x . We can translate yi into
yi = θ + η x − η x + η y + ϵyi = x − η x + η y + ϵyi .
This means that yi has information about x. Similarly,
xi = θ + η y − η y + η x + ϵxi = y − η y + η x + ϵxi .
222 /(σ 22
For each k ∈ {x, y}, deﬁne ψk ≡ (σk,ac+ σk,an)−1 , and ρk ≡ σk,ack,ac + σk,an ).
We have
ψx xi + ψy yi,
Ei (θ) =ψx + ψy(ψx + ψy ρx )xi + ψy (1 − ρx )yi
Ei (x) =, andψx + ψyψx (1 − ρy )xi + (ψy + ψx ρy )yi
Ei (y) =.
ψx + ψy

Because of the assumptions of the normal distribution for all error terms andthe quadratic payoﬀ function, we can use the method of undetermined coeﬃcients,
as in Morris and Shin (2002). We assume
ai = λxi + (1 − λ)yi .

(11)Then, the average action isā = λx + (1 − λ)y.
Hence, from (2),ai = (1 − r)Ei [θ] + rEi [ā]
= (1 − r)Ei (θ) + rλEi (x) + rλEi (y).

(12)

Substituting each expected value into (12) and comparing (12) with (11), we have
λ=

βx ψx
,
βx ψx + βy ψyand 1 − λ =βy ψy,βx ψx + βy ψywhere βk ≡ (1 − rρk )−1 .
5

See Morris and Shin (2002) and footnote 5 in Angeletos and Pavan (2007).

12

EW

2
W (σy,ac= ∞)2
σ̃y,achigh

2
σ̄y,ac2
σy,acability

Figure 1: welfare eﬀect of information acquisition

EW

clear

2
σ̃y,an2
σy,anannouncement

ambiguous

Figure 2: welfare eﬀect of ambiguous announcement

13

low

∂EW
=0
2
∂σy,an2
σy,anambiguous

Optimal∂EW=02∂σy,ac

clear
2
˜y,ac
σ̃

abilityhigh

Figure 3: optimal announcement policy

14

2
σy,ac

2
σ̄y,aclow