GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.112

Coordination Behavior and Optimal Committee Size

Keiichi Morimoto

January 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

Coordination Behavior and Optimal Committee Size∗
Keiichi Morimoto†‡
January 27, 2010

Abstract
How many members should committees consist of? This paper addresses thisquestion in view of imperfect information and coordination behavior among themembers, which is a new approach alternative to introducing information acquisition cost. First, using a simple model, I show that the existence of the coordination
motive dismisses Condorcet’s (1785) suggestion and the ﬁnite optimal size of thecommittee is determined. Second, I provide an application of the mechanism tomonetary policy committees in a basic New Keynesian model. This example will
inspire other applications to policy issues in the dynamic stochastic general equilibrium framework.

Keywords: committee, Condorcet jury theorem, coordination, higher order beliefs
monetary policy
JEL Classiﬁcation: D71; D84; E58∗

I am very grateful to Koichi Futagami for his invaluable advice and encouragement. I also thankYasushi Asako, Junichiro Ishida, Wataru Tamura and the participants of the 108th meeting of Kansai
Macroeconomics Workshop and the seminar at Bank of Japan. Of course, all remaining errors are mine.
I acknowledge ﬁnancial support from the Research Fellowships for Young Scientists of the Japan Society
for the Promotion of Science (JSPS).
†
Japan Society for the Promotion of Science.
‡
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail address: gge014mk@mail2.econ.osaka-u.ac.jp

1

Introduction

How many members should committees consist of? This paper addresses this question inview of imperfect information and coordination behavior among the members, which isa new approach alternative to introducing information acquisition cost. While enlarging
the committee promotes information aggregation eﬀect, it also foments ineﬃcient coordination behavior among the members by reducing the power of the individuals in thewhole of the committee. This ensures the existence of the optimal size of the committee.
The economic theory of committee decision-making has been developing rapidly inrecent years. The background of this stream is not only the progress of game theory
but also that decision-making by committee has played an important role in the actual
economic activities. Many countries traditionally adopt the jury systems and their design
problems have been one of the central subjects of the academic and practical arguments
in committee design. As the latest event, Japan brought the citizen judge system into
eﬀect in 2009 and its design problem was and will be discussed hard. Another outstanding
example is the establishment of the monetary policy committees in many countries. Bankof England and Bank of Japan founded the Monetary Policy Committee in 1997 and the
Policy Board in 1998 respectively and the central banks of the other countries one after
another. Besides, the governments of many countries traditionally call the well-informed
persons to the committees for the important policy issues such as tax reforms and the big
ﬁrms in general hold the meetings to make decisions on the important matters for theirbusiness. Following this trend, the demand for the committee design is growing day byday.
The problem of optimal committee size is one of the important issues in the area ofgroup decision-making. The most fundamental argument is whether or not we can enrich
the performance of the committees (unlimitedly) by increasing the number of the committee members (inﬁnitely). A famous answer for it is Condorcet’s (1785) jury theorem.
The theorem asserts that when the committee members vote honestly by use of their own
information, enlarging the committee always raises the probability that the committee
makes an appropriate decision and it converges to one as the committee size goes toinﬁnity. This implies that ﬁnite optimal sizes of committees do not exist. The result is
intuitive in a sense but there exist several critical arguments against it in recent literature.
They dispute Condorcet’s assertion, mainly focusing on that information acquisition of
the committee members is costly in the real world and this aﬀects the members’ behavior
as follows. 1 When the committee size is large, the committee members are apt to avoid
paying the information acquisition cost and free-ride on the information that the other1

Gerling et al. (2005) provides a brief survey on this topic.

1members provide because their contribution to the decision of the committee is small
relatively to the information acquisition cost. Thus, Condorcet jury theorem can fail tohold under the existence of information acquisition cost.
In this paper, I take an approach diﬀerent from the existing studies to discuss on
the optimal committee size. I focus on the role of coordination behavior among thecommittee members with noisy common information. This is motivated as follows. In
the case of the jury systems, although individual names of jurors are not disclosed, trials by courts are held publicly and their sayings are documented and reported. Whenthe individual jurors face uncertainty, they may avoid distant voting from the generaltendency in the jury. Besides, many organizations have instituted the rules of information disclosure in recent years. In particular, disclosure of public information on thepolicy issues of the government and central bank has usually been a legal mandate inmany developed countries. How does such an institutional trend make a diﬀerence incommittee decision-making? Transparency may generate the incentive of the committee
members to coordinate with the other members because the minutes are often opened tothe public under transparency and hence the individual member faces an accountability
problem. That is, when the information on the decision-making process in the committee
is disclosed to the outsiders, the ineﬃciency of each member’s vote, which is revealed ex
post, is also known to them and it aﬀects the individual valuation or reputation of eachmember. Unless the common information of all the members is perfectly correct, thecoordination behavior can bring an ineﬃcient decision.
There is another reason for taking the approach other than information acquisitioncost. Although the assumption of the existence of information acquisition cost is intuitively plausible, it is questionable whether the assumption ﬁts with design of committees,for example, the jury systems. Indeed, jurors usually question the accused in the courts
but they fundamentally rely on the material and circumstantial evidence provided by theprosecutors and the defense. That is, in general, jurors do not get information drudgingly.
I think that this is one of the limits of introducing information acquisition cost in committee design. Also, the model with costly information does not ﬁt for the committees
consisting of experts. This is because experts are generally well-informed about the issues
concerned in advance and detail information such as data is often provided not by thecommittee members but the staﬀ. Typical examples are the monetary policy committees
and the other policy boards. In fact, Toshiro Muto, who is a deputy governor of Bankof Japan, said that he did not sense the eﬀect of the free-rider problem of information
provision in the meetings according to his experience as an insider of the Policy Board.2
2

See Muto (2007).

2

The setting of the present paper is an attempt to ﬁnd an alternative framework which issuitable to the case where the committee consists of experts.
The main ﬁndings of this paper are as follows. When the committee members have the
incentive to coordinate with the other members, enlarging the committee size foments thecoordination behavior. The reasoning is that each member realizes that her power to theﬁnal decision of the committee is reduced and has to bury herself so as not to stand out
in all the committee members. Thus, while a large committee can absorb idiosyncrasyof noisy decentralized information strongly, it brings a large coordination loss. The
optimal size of the committee is the size which brings the best trade-oﬀ between thepositive and negative eﬀects above and it generally exists in a wide parameter region. The
optimal committee size decreases in the potential coordination motive and the precisionof common information since they increase the dependency of the committee members on
noisy common information. Thus, this paper provides a new explanation for the existence
and properties of the optimal committee size.
Another contribution of this paper to the literature is that the benchmark model has
an explicit application to a concrete economic problem. I extend the benchmark model toa macroeconomic model for monetary policy analysis which starts from decision-making
by the monetary policy committee and investigate its optimal size. In the last decade,many countries established the formal committee for decision-making on monetary policy.
So that, in recent arguments of monetary policy, the institutional design problems ofmonetary policy committee are regarded as important matters and the optimal committee
size is one of them. 3 This paper is the ﬁrst paper which analyzes the optimal size of themonetary policy committee in a formal model of the modern framework for monetary
policy analysis. Since central bank transparency is also a remarkable feature of modern
monetary policy as mentioned above, this paper’s approach is motivated along the trendof central banking.
There is enormous literature on Condorcet jury theorem. Here, I brieﬂy review several
studies relevant to the present paper. It is known that Condorcet jury theorem does nothold under some kinds of strategic voting since sincere voting, which is one of crucial
assumption for the theorem, is inconsistent with equilibrium. Austen-Smith and Banks(1996) show that sincere voting is not attained in equilibrium when the individuals taketheir being pivotal into consideration. Feddersen and Pesendorfer (1998) show that theprobability of false accusation can be increasing in the jury size under unanimity rule.
Several studies suggest that Condorcet jury theorem also fails to hold when informationacquisition of the individuals is costly. 4 Mukhopadhaya (2003) assumes the environment
34Blinder (2007) provides a brief survey on this issue.
It is known that there is a case where the asymptotic eﬃciency as in Condorcet jury theorem can3where the individuals have identical preferences and their information acquisition is costly.
By a numerical method, he shows that welfare can be lower in mixed-strategy equilibria
of large committees than pure-strategy equilibria of small committees. Koriyama and
Szentes (2009) assume the environment similar to Mukhopadhaya (2003) and show thatthe optimal committee size is bounded. They also show that the ineﬃciency of oversizedcommittees is smaller than that of the undersized committees. Thus, the recent literature
mainly focuses on information acquisition cost. The framework of the present paper cantreat the cases of experts committees to which the existence of information acquisition
cost is not suitable.
There are a few studies on optimal size of monetary policy committees. Sibert (2006)
conjectures that the free-rider problem of information acquisition can play a role for the
discussion on the issue. 5 However, it seems disputable in the standpoint of practice ofdecision-making in the monetary policy committees as Muto (2007) suggests. This paper
provides an alternative approach to study optimal size of monetary policy committees andan intuitively plausible answer for the problem in a formal model in modern framework
for monetary policy analysis. Some empirical studies ﬁnd the facts on monetary policy
committees. In particular, Berger and Nitsch (2008) ﬁnd the fact that inﬂation volatility
is U-shaped in the size of the monetary policy committees. This shows that the structureof the monetary policy committees aﬀects the economic outcome in actual. The resultof the present paper explains the fact above by showing the two eﬀects of enlarging
the monetary policy committee: the positive eﬀect of information aggregation and thenegative eﬀect of coordination.
In the rest of this paper is organized as follows. Section 2 provides a simple microeconomic model which describes the substantial mechanism for the determination of the
optimal committee size. Section 3 applies the mechanism in Section 2 to a design of themonetary policy committee in a modern macroeconomic model and provides some policy
implications. Section 4 concludes. All proofs are given in the Appendix.

2

A Simple Model

This section provides a simple microeconomic model which describes the process ofdecision-making and determination of the ﬁnite optimal size of the committee. It isvery abstract but can grasp clearly the role of coordination behavior in decision-making
hold even if information acquisition is costly. Martinelli (2006) shows that if there is only variable costfor obtaining precision of the signals, then the probability that the committee makes a correct decision
converges to one as the committee size goes to inﬁnity.
5
On the free-rider problem of information as a public good, for example, see Li (2001).

4

by the committee.

2.1

Setup

I set up a benchmark model. The committee consists of N (ex-ante) homogeneous members. It is seated to pursue a target θ ∈ R on behalf of an organization in the background.
6
For example, juries are called to judge criminal suits reasonably in the cause of (social)justice and monetary policy committees are organized to make an appropriate decision
on monetary policy for society’s beneﬁt. The target θ is interpreted as an underlying
state or the committee’s optimal response to it. For instance, θ is the truth of the case in
the trials. 7 In the model of monetary policy by committee which I provide in the next
section, the counterpart of θ is a level of nominal interest rate set in optimal discretionary
policy under perfect information. 8
Each member of the committee has uninformative ﬂat prior about θ over the real line
but receives common and private signals on θ. 9 The common signal is of the form suchasy = θ + η,
where the noise term η is normally distributed with mean zero and variance α−1 . Eachmember knows that the realization of y and the distribution of η are common and known
to everyone. The common signal is interpreted as a content of a staﬀ report or wellbalanced recognition among the committee members especially as experts. The private
signal of an arbitrary member j is also of the standard form such as
xj = θ + ε j ,where the noise term εj is mutually independent and normally distributed with mean
zero and variance β −1 . She knows that the distribution is common to every member
6

I believe that this is a natural motivation for the establishements of the actual committees in manycases according to concrete application (such as the example in section 3) although there are probably
counter examples.
7
Most of literature on problems of juries adopt two state models in which θ = ’innocent’ or θ = ’guilty’.
The model of continuous state admits the case where the jury also participates in the determination of
the appropriate punishment as the citizen judge system in Japan.
8
In a basic New Keynesian model, nominal interest rate set in optimal discretionary policy is linear indemand shock and cost shock. In this example, (innovations of) these shocks are the underlying states.
For detail, see the next section.
9This assumption on the prior about θ is intended for analytical ease. Although this is not so naturalaccording to the application given in the next section, no substantial diﬀerence arises from considering
proper prior. For a detail discussion, see the supplementary note.

5but does not know the realization of the others’ private signals. The private signals
represent the members’ individual views on the target which are generally distinct andnot communicated to the others.
Next, I set the payoﬀ structure of the committee. Although the committee itself
is seated for making a decision near to the true θ under imperfect information, eachcommittee member pursues her own objectives. In this simple model, I assume that eachmember j votes aj ∈ R so as to maximize her own payoﬀ function
uj = −(1 − r)(aj − θ)2 − r(aj − ā)2 ,(1)∑
where r ∈ [0, 1) and ā = N1 N
k=1 ak . The meaning of the payoﬀ function above is asfollows. Each member j has two goals. One is to hit the true target and the other is not
to remove her vote from the average of all. That is, while she honestly tries to contributeto an appropriate decision, she also seeks coordination with the other members eventhough it makes the performance of the committee worse. Thus, I interpret parameterr as the degree of coordination motive of the members. Note that I assume that theobjective of establishing the committee is to grasp the true target and make the decision
as correct as possible. 10 Since the coordination motive distorts the members’ use of
information, it generates only a loss for the performance of the committee.
However, there are some reasons for considering such coordination motive. Usual
committees are established for better decision-making by choosing the delegations fromthe large organizations. For example, ﬁrms hold committee meetings to make decisions
on big bargains or selections of recruits, governments summon well-informed persons
committees for various policy issues and the central banks have the formal policy boardsfor decision-making on monetary policy. In the cases of policy issues, since each member’s
saying or voting in the meeting is often released to the public, she will be at least partiallymotivated to coordinate with the other members. In the cases of the ﬁrms, although therecords are rarely opened formally, what the members said in the meeting usually spreads
from nowhere or can be speculated by the outsiders. Therefore, I interpret r also as ameasure of the indirect eﬀect of transparency on decision-making in the committee.
All the votes are aggregated by a speciﬁc voting rule and it becomes the ﬁnal decisionof the committee. For analytical ease, I assume that the voting rule of the benchmark
case is the arithmetic mean:
1 ∑
â =
aj .
N j=1
N

10

I set the performance measure of the committee along this line soon later.

6(2)This rule is quite simple but enough for grasping the basic mechanism this paper suggests.
Of course, the approach I will take is applicable to the case of more realistic voting rule.
In section 2.4, I analyze the case of the median-voting rule:â = med {aj }.
1≤j≤N

I will show that the basic properties of the model do not change in that case.
I ﬁnally set a performance measure of the committee. Since in this paper I assume
that the committee is seated to make accurate decisions for the beneﬁt of the organizationin the background, a natural measure of the committee’s performance is
W = −(â − θ)2 .

2.2

(3)

Equilibrium

I derive the Bayesian Nash equilibrium of the model. The ﬁrst order condition of eachmember j’s problem isaj =(1 − N1 )r
1−r
E
(θ)
+
Ej (ā),j
1 − Nr
1 − Nrwhere the symbol Ej represents a mathematical expectation conditioned on information
available to member j. Let δ =

1
)r
(1− N
. Then, the FOC above is rewritten asr
1− N

aj = (1 − δ)Ej (θ) + δEj (ā).

(4)

Note that δ is increasing in N . That is, each member places a higher weight on the
average of all the votes relative to the target as the committee becomes larger. What
does cause it? Since the number of the committee members is ﬁnite, each member canpartially control the average of all the votes: see (1). When the committee size is small,she does not have to care about the others’ behavior extremely because her power in thecommittee is relatively large. As the committee becomes large, the power fades away andshe must seek harder for coordination with the other members. In short, the massivenessof the group drives the individuals to bury themselves in it.
Equation (4) provides the conjecture that the equilibrium strategy is linear in common
and private signals. 11 In fact, the following proposition supports for it together withthe uniqueness property.
For a heuristic derivation of equilibrium, put aj = (1 − γ)y + γxj . Here γ is an undetermined
coeﬃcient. Substituting this into (4) and comparing the coeﬃcients of both sides, the response coeﬃcientγ given in Proposition 1 is obtained.
11

7

Proposition 1 There exists a unique equilibrium strategy such that for all y, xj ∈ R,
aj = (1 − γ)y + γxj ,(1−r)β.
where γ = (1− r )(α+β)−r(1−1)( α +β)N

N

N

Proof.
See Appendix A.
This equilibrium strategy brings the essential mechanism for the main result of thepresent paper. The next assertion shows it.
Corollary 1 The response coeﬃcient (γ to private signal
is decreasing in N . Given α, β](1−r)ββand r, it lies in the half-open interval α+(1−r)β, α+β.
Proof.
See Appendix B.
Corollary 1 asserts that each member’s dependency of the common signal increases
as the size of the committee becomes larger. It results from the relationship between
each member’s control of decision-making in the committee and coordination behavior
which I explained above. When the committee becomes larger, each member becomes
more sensitive to common signal to adjust her voting to the others’ more precisely. If
the committee consists of only one person, the coordination motive vanishes and hisβbehavior accords with that of the basic statistical decision-making: γ|N =1 = α+β. As the(1−r)βcommittee size goes to inﬁnity, the response coeﬃcient γ converges to α+(1−r)β. That is,
the equilibrium strategy corresponds to that of the ﬁnite-players version of Morris andShin’s (2002) beauty contest game. In the ﬁnite-players version of Morris and Shin’s(2002) beauty contest game, each player j has the same informational structure as themodel of the present paper and her utility function is 12(−(1 − r)(aj − θ)2 − r aj −121 ∑ )2
ak .
N − 1 k6=j

As an alternative setting, we may assume that her utility function isr ∑−(1 − r)(aj − θ)2 −(aj − ak )2 .
N −1
k6=j

The solution under this utility function is the same as the case of the function in the text although thesocial welfare function changes.

8The second objective of each player is the average of the others’ actions, which does
not include her own action. In this case, each player can not control the average andhence has to care about the others’ behavior more greatly than the case where she cando it. Therefore, the basic beauty contest game can be regarded as a limiting case of thecoordination game in this paper.

2.3

Optimal Size of the CommitteeBy (3) and Proposition 1, the decision of the committee is∑Nâ = (1 − γ)y + γ

k=1 xkN.

(5)Thus, substituting (5) into (3), the expected performance of the committee is
∑N
[(
)2 ]
k=1 (θ + εk )E(W |θ) = −E (1 − γ)(θ + η) + γ
−θ
N
∑N
[(
)2 ]
k=1 εk= −E (1 − γ)η + γN−1β= −(1 − γ)2 α−1 − γ 2.
N
I investigate the relationship between the size of the committee and its performance.
Considering the continuation of E(W |θ) with respect to N , I obtain∂E(W |θ)∂γ
β −1 ∂γ= 2(1 − γ)α−1− 2γ·+ γ 2 β −1 N −2 .
∂N
∂N
N ∂N

(6)

The meaning of (6) is clear in view of Corollary 1. Enlarging the committee has three∂γeﬀects on its performance. Note that, since ∂N< 0 by Corollary 1, the ﬁrst term isnegative and the second and third terms are positive. The ﬁrst term represents theindirect negative eﬀect due to the stronger coordination behavior. The second term is
the indirect positive eﬀect owing to the decrease of the members’ dependency on the
noisy private signals. The third term is the direct positive eﬀect from the decrease of thevolatility due to the noisy private signals by averaging larger samples.
The next proposition provides a necessary and suﬃcient condition for the existenceof the optimal committee size under the average-voting rule.
Proposition 2 Under the average-voting rule, there exists a ﬁnite optimal size of the
5r−1
committee if and only if the parameter set satisﬁes r > 15 and αβ < (1−r)
2.
9Proof.
See Appendix C.
The parameter condition above means that the degree of coordination motive, r, is
not so weak and the precision of common (private, resp.) information relative to private(common) information is large (small).
This result is intuitively plausible. When the coordination motive is strong, each member depends highly on common information to approximate her own voting to the others’.
Besides, when the relative precision of common information is larger, each member also
places a higher weight on common information to hit the true target more accurately. 13
The negative eﬀect of enlarging the committee is ampliﬁed in such a case. Then, thisdominates the positive eﬀects in the limit since the marginal contribution of averaging
reduces as the committee becomes larger.
In fact, the expected performance E(W |θ) is single-peaked in the committee size N
under the necessary and suﬃcient condition for the existence of the optimal size. 14[Figure 1 about here.]
Figure 1 gives a numerical example of this relationship. 15 When the committee size issmall, the direct positive eﬀect of averaging is very large and it (and the indirect positiveeﬀect) dominates the negative eﬀect. However, as the committee size becomes larger, thedirect positive eﬀect becomes smaller and the negative eﬀect becomes relatively larger.
Thus, the expected performance has the single peak in the committee size. According
to the discussion above, I obtain the following characterization of the optimal committee
size.
Corollary 2 Suppose that the parameter set satisﬁes the necessary and suﬃcient condition for the existence of the optimal committee size under the average-voting rule: r > 15
5r−1∗
and αβ < (1−r)
is given by the following equation.
2 . Then, the optimal size NN∗ =argmaxN ∈{N− (Ñ ),N+ (Ñ )}E(W |θ),where Ñ ∈ R is the solution of the equationγ
1γ 2 ( ∂γ 2 )−1 β
=
·
−
N
1−γ N
2(1 − γ) ∂N
α13This mechanism necessarily works since I assume r < 1.
A formal proof of this fact is given in the necessity part of the proof of Proposition 2.
15
To depict Figure 1, I set r = 0.4, α = 1.25 and β = 1.
14

10

with respect to N ∈ R, N− (Ñ ) is the maximal integer which is not larger than Ñ and
N+ (Ñ ) is the minimal integer which is not smaller than Ñ .
Proof.
It is obtained immediately from the proof of Proposition 2. See Appendix C.
∂γ
Note that Ñ depends only on r and the precision ratio αβ because γ and ∂N
does
16
so.
Since Ñ is the single peak of the continuation of E(W |θ) with respect to N , N ∗is mainly determined by them. However, rigorously speaking, N ∗ depends on not the
precision ratio but the pair (α, β) in general. This is because E(W |θ) can not be written
as the function of r and αβ and it is impossible to decide which of N− (Ñ ) and N+ (Ñ ) the
optimal size N ∗ is equal to only with the values of r and αβ .
Next, I investigate two relationships between the parameters and the optimal size.
Although it is diﬃcult to obtain them in analytical ways due to discreteness of thecommittee size, I can ﬁnd the robust qualitative results below.
First, I ﬁx α = 1.25 and β = 1 as in Figure 1 and calculate the optimal sizes forvarious values of r in [0.3, 0.99]. 17[Figure 2 about here.]
Figure 2 illustrates the result. The optimal committee size is non-increasing in the
degree of coordination motive, r. When r is small and hence the coordination motive
is weak, the negative eﬀect of enlarging the committee is small. The importance ofenhancing the positive eﬀect of information aggregation is then relatively large. Therefore, the optimal size of the committee is very large for small r. As r becomes larger,
the optimal size decreases rapidly since the negative eﬀect of coordination acceleratingly
swells. Figure 2 provides another interesting fact. It illustrates that the optimal size ofthe committee is one for suﬃciently large r. That is, if the coordination motive is verystrong and adding a committee member is too costly, then a single decision-maker can be
optimal to choose a correct alternative. When the members bury themselves and follow
the incorrect common sense with little criticism, appointing a sincere individual may be
a good measure to reach a right conclusion.
Second, I set r = 0.4 as in Figure 1 and calculate the optimal sizes for various valuesof α (β, resp.) in [1, 5] with β ﬁxed to 2 (α ﬁxed to 3).
[Figure 3 about here.]16
17

See the necessity part of the proof of Proposition 2.
When α = 1.25 and β = 1, the existence condition is met for all 0.3 ≤ r ≤ 0.99.

11Figure 3 illustrates the result. Given the precision of private information (common
information, resp.), the optimal committee size is non-increasing (non-decreasing) in the
precision of common information (private information). Given the degree of coordination
motive, when the precision of common information is large relative to that of private
information, each member places a high weight on common information. Then, enlarging
the committee strongly foments the coordination behavior among the members. The optimal size is hence non-increasing in α. Contrary, since such a negative eﬀect of enlarging
the committee is small when the precision of private information is large, the gain of
information aggregation is relatively large. Thus, the optimal size is non-decreasing in
β. The result above suggests that we should establish a small committee to reduce thecoordination loss when the common sense (or the staﬀ report) on the issue concerned isprecise to some extent.

2.4Median-Voting RuleIn the last of this section, I consider the case where the voting rule is the median-voting
rule. The basic properties of the model do not change even under this voting rule. The
discussion below ensures the robustness of the results in section 2.3 in a wide class ofvoting rules.
Since each committee member’s behavior is invariant in voting rules, the decision of
the committee isâ =med {(1 − γ)y + γxj }1≤j≤N= θ + (1 − γ)η + γ med {εj }.
1≤j≤N

Thus, the expected performance of the committee is
[(
)2 ]
2 −1
2
E(W |θ) = −(1 − γ) α − γ E
med {εj }.
1≤j≤N

The expectation in the second term of the right-hand side has no analytical expression forﬁnite N . However, since the distribution of med1≤j≤N {εj } is approximately the normal−1
distribution with mean 0 and variance πβ2N for suﬃciently large N (med1≤j≤N {εj } ∝−1
N (0, πβ2N )), I obtain
E(W |θ) ≈ −(1 − γ)2 α−1 − γ 2

12

πβ −1
2N

(7)

for suﬃciently large N . 18 Thus, I obtain a suﬃcient condition for the existence of the
ﬁnite optimal committee size under the median-voting rule by use of this asymptotic
property.
Corollary 3 Under the median-voting rule, there exists a ﬁnite optimal size of the comπ
mittee if the parameter set satisﬁes r > π+8.
and αβ < (π+8)r−π
π(1−r)2
Proof.
See Appendix D.
The intuition for this suﬃcient condition is similar to Proposition 2 but there is aquantitative diﬀerence between the median-voting rule and the average-voting rule. One
can show that the parameter region which satisﬁes the suﬃcient condition under the
median-voting rule is smaller than that of the average-voting rule. Figure 4 illustrates it.
[Figure 4 about here.]
The cause of this result is the diﬀerence between statistical properties of mean and−1
median. That is, as seen in med1≤j≤N {εj } ∝ N (0, πβ2N ), sample median converges inprobability more slowly than sample mean. 19 Therefore, under the median-voting rule,
it is more important to promote the positive eﬀect of information aggregation than under
the average-voting rule. This makes the existence condition under the median-voting rulestricter.
A necessary condition along the line of Proposition 2 can not be obtained under the
median-voting rule because there is no analytical expression of the distribution of sample
median for small sample: see the necessity part of the proof of Proposition 2. However,the basic properties under the average-voting rule will be robust since the distribution ofsample median approaches that of sample mean very quickly. 20 Moreover, the proofs of
Proposition 2 and Corollary 3 imply the following. In general, we can obtain a similar(suﬃcient) condition for the existence of the optimal committee size when we adopt as
18

It is known that when n random variables Xi (i = 1, ..., n) identically and independently follow a
distribution with median M and density function f , the distribution of Yn = med1≤i≤n {Xi } with large
1sample is approximately the normal distribution with mean M and variance 4n(f (M))2 . As a special case,2
when each Xi follow the normal distribution with mean µ and variance σ identically and independently,
2
the distribution of the sample median Yn is approximately N (µ, πσ2n ) for large n. For a detail explanation,
see Kenney and Keeping (1962).
∑N−1
19
Since N1 j=1 εj ∼ N (0, βN ), the variance of sample mean is π2 times as small as that of sample
median.
20
For a detail explanation on this fact, see Maritz and Jarret (1978).

13

a voting rule any statistics which has the consistency as an estimator of the true statewith respect to private signals and converges at an order not lower than sample mean. 21

3

Monetary Policy Committee

In this section, I provide a simple application of the model in Section 1. Following
Morimoto (2009) basically, I set up the model for monetary policy analysis which starts
from decision-making by the monetary policy committee. For a detail explanation of themodel setting, see Morimoto (2009).

3.1

Macroeconomic Model

As the underlying macroeconomic model, I adopt a basic New Keynesian model. The
model consists of the two stochastic diﬀerence equations
1(it − Et πt+1 ) + ut ,σ
= βEt πt+1 + κxt + et ,xt = Et xt+1 −

(8)

πt

(9)

together with a monetary policy rule. Here, xt , πt , it , ut , et are output gap, inﬂation rate,
nominal interest rate, demand shock and cost shock in period t respectively and β, σ, κ arepositive parameters. Parameters β, σ, κ represent the discount rate, constant elasticity ofintertemporal substitution and impact of one unit output gap on inﬂation, respectively.
I assume that ut and et follow AR(1) processes in such a way thatut = ρu ut−1 + ϕt ,et = ρe et−1 + ψt ,where ρu , ρe ∈ [0, 1) and the innovations ϕt and ψt are normally distributed with mean
zero and variances σϕ2 and σψ2 respectively. Once a setting rule of nominal interest rateis speciﬁed, macro dynamics of the model economy is determined as sequences of output
gap and inﬂation rate under the policy rule.
As in most works in optimal monetary policy, I adopt the following social loss function
as the welfare measure.
L ≡ V (π) + λV (x),
21

(10)

An estimator for a parameter is said to satisfy consistency if it converges in probability to the true
value of the parameter.

14where V (π) and V (x) are asymptotic variances of inﬂation rate and output gap and λ is
the weight that society places on output gap relative to inﬂation. 22

3.2

Interest Rate Setting by Committee

Next, I set up the process of decision-making on interest rate setting in the monetary
policy committee. The committee consists of N (ex-ante) homogeneous members. The
informational structure of the committee is as follows. Each committee member faces
information imperfectness about innovations of demand shock and cost shock. For simplicity, I assume that each committee member has improper ﬂat prior about them over
the real line. 23 In the end of period t − 1, each member receives two kinds of signals on
innovations of demand shock and cost shock in period t. One is common signal and theother is private signal. Each member j 0 s common signal is of the standard form such as
ϕct = ϕt + µt ,
ψtc = ψt + νt ,where the noise terms of µt and νt are independently and normally distributed with mean
zero and variance σµ2 and σν2 respectively. There are some sources of common signalsof the committee members. The committee members probably have a kind of wellbalanced recognition on economic states as macroeconomists. Besides, in most actual
central banks, the ﬁrst step of or one of preparations for the meeting of monetary policy
committees is the staﬀ report on the present conditions and future developments of theeconomies.
Each member j’s private signal is of the standard form such as
ϕjt = ϕt + εjt ,22

In the usual analysis in the New Keynesian literature, the welfare measure is the second orderapproximation of the household’s utility function. In the basic model, it is proportional to
E0

∞
∑

(
)
β t πt2 + λx2t .

t=0

In this paper, however, I focus only on the average performance of monetary policy and reset it tolim (1 − β)E0

β→1

∞
∑

(
)
β t πt2 + λx2t = L.

t=0

23

Although it is more natural to assume that each member knows the distribution of innovations,ϕt ∼ N (0, σϕ2 ) and ψt ∼ N (0, σψ2 ), this does not change the basic properties below and only complicatesmanipulations. For details, see the supplementary note.

15ψtj = ψt + ηtj ,where the noise terms of εjt and ηtj are independently and normally distributed with mean
zero and variance σε2 and ση2 respectively.
In the meeting of the monetary policy committee, based on her own information, eachmember j votes the level of nominal interest rate in period t, ijt , to maximize her own
expected payoﬀ. The function form of the payoﬀ function is assumed to be−r(ijt − i∗t )2 − (1 − r)(ijt − īt )2 ,

(11)

∑
j
∗
where r ∈ [0, 1), īt = N1 N
j=1 it and it is the nominal interest rate in period t set in
optimal discretionary policy under perfect information by a single policy maker. That is,i∗t is the solution of the following linear-quadratic problem. 24
min πt2 + λx2t ,1(it − Et πt+1 ) + ut ,σπt = βEt πt+1 + κxt + et .

s.t. xt = Et xt+1 −Solving this problem, I obtain the analytical expression of i∗t such thati∗t = σut + Φet ,ce +(1−ρe )σκwhere Φ = λλcρ(1−βρ
2 .
e )+κ
Nominal interest rate in period t, it , is determined by aggregating the voting rates witha speciﬁc rule. As the benchmark case, I assume that the voting rule is the arithmetic
mean:

1 ∑ j
i.
N j=1 t
N

it =

(12)

This is quite simple but suﬃcient for the purpose of this paper. In the actual institutions,however, the majority rule is often adopted and hence it is natural to consider the caseof the median-voting rule:it = med {ijt }.
1≤j≤N

(13)

I will show that as in the simple model of the previous section, the basic result does notchange under the median-voting rule.
24

For a detail explanation of this issue, see Walsh (2003).

16

3.3

Equilibrium Dynamics and Macroeconomic Volatility

Now let us see the equilibrium strategy of the subgame in the committee. The ﬁrst ordercondition of member j’s problem isjj
ijt = (1 − δ)Et−1(i∗t ) + δEt−1(īt ),jwhere Et−1
is a mathematical expectation conditioned on information available to member
j in the end of period t − 1. Since
]
[
]
[
ση−2
σµ−2
σε−2
σν−2
j
j
j
c
c
∗
ϕ +
ϕ + Φ ρe et−1 + −2
ψ +
ψ ,Et−1 (it ) = σ ρu ut−1 + −2
σε + σµ−2 t σε−2 + σµ−2 t
ση + σν−2 t ση−2 + σν−2 t

I obtain the following proposition immediately from Proposition 1.
Proposition 3 There exists a unique equilibrium strategy of the form such that
ijt = σ[ρu ut−1 + γu ϕjt + (1 − γu )ϕct ] + Φ[ρe et−1 + γe ψtj + (1 − γe )ψtc ],where γu =(1−r)σε−2

−2
σµ
−2
r
1
(1− N
)(σµ
+σε−2 )−r(1− N)( N+σε−2 )and γe =(1−r)ση−2
−2
σν
−2
−2
r
1
(1− N )(σν +ση )−r(1− N)( N+ση−2 )

(14).

Substituting (14) into (12), I obtain the following equilibrium nominal interest rate.
it = σ[ut + γu ε̃t + (1 − γu )µt ] + Φ[et + γe η̃t + (1 − γe )νt ]= i∗t + σ[γu ε̃t + (1 − γu )µt ] + Φ[γe η̃t + (1 − γe )νt ],

(15)

∑
∑N j
j
1
where ε̃t = N1 N
ε
and
η̃
=
t
t
j=1
j=1 ηt . The second and third terms of (15) representN
the ineﬃciency of interest rate setting due to imperfect information and coordination
behavior among the committee members.
Macro dynamics of the model economy is given by (8), (9) and (15). Note that since
the relevant state variables in period t are et , ε̃t , η̃t , µt and νt , equilibrium output gap
and inﬂation rate is linear in them. Thus, by the method of undetermined coeﬃcients, I
obtain the following equilibrium output gap and inﬂation rate. 25[]
γe Φ(1 − γe )Φ
κet − γu ε̃t +η̃t + (1 − γu )µt +νt , (16)xt = − cλ (1 − βρe ) + κ2
σ
σ
[
]
λc
γe Φ
(1 − γe )Φ
πt =et − κ γu ε̃t +η̃t + (1 − γu )µt +νt . (17)λc (1 − βρe ) + κ2σσ25

For the detail of the calculation, see Appendix E.

17

The bracketed parts of the equilibrium output gap and inﬂation rate are the ineﬃcienteconomic ﬂuctuations due to the ineﬃcient interest rate setting by the committee. Notethat the second term of (17) is κ times as large as that of (16). This means that theineﬃcient interest rate setting brings the ineﬃcient output gap ﬂuctuation, −[γu ε̃t +γe Φη̃ + (1 − γu )µt + (1−γσe )Φ νt ], and it hits on inﬂation through the aggregate supply
σ trelation, the New Keynesian Phillips curve.
To ﬁnd equilibrium social loss, let us calculate the asymptotic variances of the output
gap and inﬂation rate. After some calculations, I obtain
[
]2 2
[
]
σψ
κ
1 2 2 ( γe Φ )2 2
V (x) =+γ σ +σηλc (1 − βρe ) + κ2 1 − ρ2e N u ε
σ
[
(
)2 ]
(1 − γe )Φ
2 2(18)+ (1 − γu ) σµ +
σν2 ,
σ
[
]2 2
[
]
σψ
λc
κ2 2 2 ( γe Φ )2 2
V (π) =+γ σ +σηλc (1 − βρe ) + κ2 1 − ρ2e
N u ε
σ
)2 ]
[
(
(1 − γe )Φ
2
2 2
σν2 .
(19)
+κ (1 − γu ) σµ +σThe ﬁrst terms of (18) and (19) are due to cost shock, one of the economic fundamentals.
The second (third, resp.) terms are due to noisiness of private (common) information of
the committee members, which is one of the non-fundamentals. By (10), (18) and (19),the social loss in equilibrium is[()2)2 ] σ 2
(
λc
κ
ψ
L =
+λ c
λc (1 − βρe ) + κ2
λ (1 − βρe ) + κ2
1 − ρ2e()2 )][ (( γ Φ )2 ) ((1 − γe )Φ
1
e
2 2
2
2 2
2
σν2 .
+(κ + λ)γ σ +ση + (1 − γu ) σµ +
N u ε
σ
σ
The social loss L seems somewhat complicated but its meaning is clear. The ﬁrst term
is equal to the social loss under optimal discretionary policy under perfect information.
Since I adopt optimal policy under discretion as the optimality concept, it is not relevantto the performance of the monetary policy committee. The second term is the social
loss generated by the ineﬃcient interest rate setting due to imperfect information andcoordination behavior among the members. Therefore, the second term of L should be
regarded as the performance measure of the monetary policy committee in this model.
Note that it corresponds just to E(W ) in the previous section.

18

3.4

Optimal Size of MPC

In the rest of this section, I investigate an existence condition of the optimal size of themonetary policy committee and its property. As mentioned above, the optimal size is
the size N ∈ N which maximizes the performance measure of the committee:
] [
(
[
)2 ]
(1 − γe )Φ
1 2 2 ( γe Φ )2 2
2 2
γ σ +
ση − (1 − γu ) σµ +
P ≡−
σν2 .
N u ε
σ
σ
Note that when the response coeﬃcients γu and γe are independent of the committee
size N , P is increasing in N and the ﬁrst term of P converges to zero as N goes toinﬁnity. This is a variant of Condorcet’s assertion. Although the ineﬃciency of interest
rate setting does not disappear because of noisy common information, the idiosyncratic
noise of private information is perfectly absorbed by averaging the large sample in thelimit. 26
In general cases, γu and γe depend on N and hence P is not necessarily increasingin N . So that, the optimal size of the monetary policy committee can exist. The next
proposition provides both of necessary and suﬃcient conditions separately.
Proposition 4 The following statements on the existence of the optimal size of the monetary policy committee hold under the average-voting rule.
1. There exists a ﬁnite optimal size of the monetary policy committee if the parameter
−2
ση−2
5r−1
5r−1
set satisﬁes that r > 15 , σσε−2 < (1−r)< (1−r)
2 and
2.
σ −2
µ

ν

2. There exists a ﬁnite optimal size of the monetary policy committee only if the pa−2
ση−2
5r−1
5r−1
rameter set satisﬁes that r > 15 and σσε−2 < (1−r)< (1−r)
2 or
2.
σ −2
µ

ν

Proof.
See Appendix F.
Because this model includes two economic shocks, demand and cost shock, it is diﬃcult
to ﬁnd exactly a necessary and suﬃcient condition along the line of Proposition 2. 27
However, the necessary condition above is close to the suﬃcient condition. This is because
the mechanism for the existence of the optimal size is similar to that of the simple model−2
in the previous section. Note that according to Figure 4, if r > 51 , the condition that σσε−2 <5r−1(1−r)2

µ

σ −2

5r−1
and ση−2 < (1−r)
2 is normally satisﬁed because common information is interpretedν
as well-balanced recognition among the economist or the staﬀ report and hence it does
26
27

See Corollary 1 and 7 of Morimoto (2009).
See the proofs of Proposition 4 and Corollary 4.

19not seem far more imprecise than the members’ private information. Therefore, it canbe said that the monetary policy committees should not be so large under central banktransparency and a reputation eﬀect if we consider a source of the coordination motive
as such.
The properties of the optimal size of the monetary policy committee are basicallysimilar to those of the simple model. So that, I do not report numerical examples here.
First, the optimal size N ∗ is non-increasing in r. That is, the stronger what foments thecoordination behavior among the committee members is, the smaller the monetary policy committees should be. In the standpoint mentioned above, it is optimal to promote
eﬃcient use of the members’ information by holding the monetary policy committees
to small groups when central bank transparency and a reputation eﬀect is large. Second, given the precision of private information, the optimal size is non-increasing in the
precision of common information. In the context of the monetary policy committee, itsuggests that the committee size should be small when the staﬀ report or common understanding of general economists on the present and future economic states is reliable
to some extent.
Next, I provide a suﬃcient condition for the existence of the optimal size of themonetary policy committee under the median-voting rule, which is similar to the case of
the average-voting rule as in the simple model.
−2

σ −2

µ

ν

π
Corollary 4 If the parameter set satisﬁes that r > π+8, σσε−2 < (π+8)r−π
and ση−2 <π(1−r)2(π+8)r−π, then there exists a ﬁnite optimal size of the monetary policy committee.
π(1−r)2

Proof.
See Appendix G.
As in the simple model in the previous section, the parameter condition for the existence of the optimal committee size under the median voting rule is stricter than thatof the average-voting rule. It suggests that the the eﬀect of information aggregation ismore valuable for the monetary policy committees which adopt the majority rule.

3.5

A Positive Implication

Finally, I provide a positive implication of the model for the actual monetary policy
committees. Using a data set on the characteristics of the monetary policy committees
in more than 30 countries from 1960 through 2000, Berger and Nitsch (2008) report thatinﬂation volatility is U-shaped in the size of the monetary policy committee. In thepresent paper, the inﬂation volatility exhibits a similar behavior as in the case of the20

simple model since the the part of variace of inﬂation which depends on the committee
size is
[
)2 ]
]
[
(
κ2 2 2 ( γe Φ )2 2
(1 − γe )Φ
2
2 2
γ σ +
ση + κ (1 − γu ) σµ +
σν2 .
N u ε
σ
σ
Since Berger and Nitsch (2008) also report that similar results are obtained about output
growth, one of the real terms, I conjecture that the property results from a feature of thefunctions of committee decision-making for economic stabilization. The present paper
gives an explanation for the property in terms of the positive eﬀect of information aggretion and the negative eﬀect of fomenting the ineﬃcient coordination which an increaseof the committee size brings to the performance of the monetary policy committee.

4Conclusion

How many members should the committees consist of? The present paper provides ananswer for this question in committee design, focusing on coordination behavior among
the members based on higher order beliefs. The approach in this paper is an alternative
to incorporating information acquisition costs into the payoﬀ structures, which is usuallyadopted in literature. This paper also analyzes in a formal model the optimal size ofmonetary policy committees, which is one of the most important issues of monetary
policy design.
There are a few remaining problems. The ﬁrst one is to construct a theoreticalfoundation for the members’ coordination motive. It seems very signiﬁcant in the area ofcommittee design. Especially, I think transparency and reputation can play an importantrole for it. The second one is to ﬁnd applications of the mechanism given in this paper toother economic problems. Considering the structure of this paper’s model, the mechanism
will be applicable to the models in which the optimal action of the committee under
perfect information is a linear function of states. I believe that such situations are notrare in the economic phenomena of our interest.

21

Appendices
Appendix A: Proof of Proposition 1
By the iterated substituion of F.O.C. (4), I obtain 28
aj = (1 − δ)Ej (θ) + (1 − δ)δEj (Ē(θ)) + (1 − δ)δ 2 Ej (Ē 2 (θ)) + · · ·∞∑= (1 − δ)δ s Ej (Ē s (θ)).

(20)

s=0

To calculate this inﬁnite series, I use the following lemma.
Lemma 1 For all j and s,
Ej (Ē s (θ)) = (1 − µs )y + µs xj ,(21)β( α +β)sN
where µs = (α+β)
s+1 .

Proof of Lemma 1.
I prove it by induction. Choose arbitrary member j. The assertion obviously holds
when s = 0. Suppose that it holds for an arbitrary s. Then, by (21), I obtain
1 ∑
(θ) =
Ek (Ē s (θ))
N k=1
∑N
xk
= (1 − µs )y + µs k=1 .
N
N

Ē

s+1

Therefore,
Ej (Ē

s+1( ∑N x )k=1 k(θ)) = (1 − µs )y + µs Ej
N(1
N − 1 αy + βxj )= (1 − µs )y + µs
xj +
·
N
N
α+β
]
[
)
(1
(
α
N −1
β )
N −1
·
µs y + µs+·
xj .
= 1− 1−
N
α+β
N
N
α+β
α

+β

β
α
By 1 − NN−1 · α+β= N1 + NN−1 · α+β= Nα+β , I obtain
α
α
+β )
+β
1− N
µs y + µs N
xj
α+β
α+β
= (1 − µs+1 )y + µs+1 xj .

Ej (Ē s+1 (θ)) =(The symbol Ē s (θ) denotes an s-th order average expectation. That is, for an arbitrary s ∈ {1, 2, ...},)(∑NĒ (θ) = N1 k=1 Ek Ē s−1 (θ) .
28

s

22

This completes the proof of Lemma 1.
Substituting (21) into (20), I obtain
aj = (1 − δ)

∞
∑

]
[
δ s (1 − µs )y + µs xj

s=0

[
(∑
)
]
∞
∞
∑
s
s
= 1 − (1 − δ)δ µs xj .
δ µs y + (1 − δ)
s=0

s=0

After some algebraic manipulations, I obtain 29(1 − δ)∞∑δ s µs = γ,

s=0where γ is deﬁned in Proposition 1.

Q.E.D.

Appendix B: Proof of Corollary 1
The denominator of γ can be reduced to
[
]
(2
1 )
(1 − r)α + 1 − r− 2 β.
N
N
Considering its continuation with respect to N , I obtain
1 )∂ (2− 2 = −2N −3 (N − 1) ≤ 0.
∂N N
NThis shows that the denominator of γ is increasing in N and positive. 30 Therefore, γ isdecreasing in N .
(1−r)ββIt is immediately obtained that γ = α+βwhen N = 1 and limN →∞ γ = α+(1−r)β.
Q.E.D.

Appendix C: Proof of Proposition 2(Suﬃciency)
Since E(W |θ) converges as N goes to inﬁnity, a ﬁnite optimal size of the committee)< 0 for suﬃciently large N . I derive an explicit expression of this suﬃcient
exists if ∂E(W∂N
condition.
∑∞

29

s

Note that the inﬁnite series

30

It is larger than or equal to (1 − r)(α + β) > 0.

s=0 δ µs converges since

23

∑∞

β ∑∞
s=0 δ µs = α+β
s=0
s

(

)s

1
)r
(1− N
α
·( N
+β)
r
1− N

.

Equation (6) can be reduced to
[
(
)]
γ
1
γ 2 ( ∂γ 2 )−1∂E(W |θ)−1 ∂γ β= 2(1 − γ)β
−
·
−
N
.
∂N
∂N α
1−γ N
2(1 − γ) ∂N

(22)

∂γ
Note that ∂N< 0 for all N by Corollary 1 and that the second term of the bracketedpart of (22) converges to zero as N goes inﬁnity. Thus, ifβγ 2 ( ∂γ 2 )−1,> − limN
N →∞ 2(1 − γ) ∂N
α

(23)

|θ)
then ∂E(W< 0 for suﬃciently large N , which ensures the existence of a ﬁnite optimal∂Nsize of the committee.
Let us ﬁnd the parameter condition equivalent to (23). After some calculations, I
obtain∂γ 22r(1 − r)αβN =−.
N →∞ ∂N[α + (1 − r)β]2
lim(1−r)β, I obtain
By limN →∞ γ = α+(1−r)βγ 2 ( ∂γ 2 )−1β]
β 1−r [limN
· 1 + (1 − r) .
=− ·N →∞ 2(1 − γ) ∂Nα4rαThus, (23) if and only if
1>

1−r [
β]
· 1 + (1 − r) ,
4r
α

(24)

5r−1
which is equivalent to that r > 15 and αβ < (1−r)
2.

(Necessity)
I will show that E(W |θ) is monotonically increasing in N unless the parameter set
5r−1
satisﬁes that r > 15 and αβ < (1−r)
2 . Deﬁne the function f : (1, ∞) → R byf (N ) =γ
1γ 2 ( ∂γ 2 )−1
·
−
N
,
1−γ N
2(1 − γ) ∂N

for all N ∈ (1, ∞).

Note that f (N ) is the second term of the bracketed part of (22). Since γ is decreasing in
N by Corollary 1, the ﬁrst term of f (N ) is decreasing in N . Since
2r(1 − r)(1 − N1 )αβ∂γ 2
N = −[
]2 ,
∂N
r
1
α
(1 − N )(α + β) − r(1 − N )( N + β)24

by use of the deﬁnition of γ, I obtain−

γ 2 ( ∂γ 2 )−1 (1 − r)2 β 2
1
N
·
.
=
2(1 − γ) ∂N
1−γ2r(1 − r)(1 − N1 )αβ

Since γ is decreasing in N by Corollary 1, the denominator of the right-hand side above
is increasing in N . This implies that the second term of f (N ) is also decreasing in N .
Therefore, f is a decreasing function.
)
Note that limN →1 ∂E(W
> 0 by limN →1 f (N ) = +∞. Thus, since f is monotonically∂N∂E(W )
decreasing, ∂N > 0 for all N unlessβ> lim f (N ).
α N →∞
Immediately from the calculation in the necessity part, this condition is equivalent to
5r−1
that r > 51 and αβ < (1−r)
Q.E.D.
2.

Appendix D: Proof of Corollary 3
Since equation (7) holds for suﬃciently large N , according to the proof of Proposition 2,
the next inequality is a suﬃcient condition for the existence of a ﬁnite optimal size.
βπγ 2 ( ∂γ 2 )−1
>
lim
N
.
α
2 N →∞ 2(1 − γ) ∂N(25)Hence, similar to (24), inequality (25) is equivalent to1>β]
π 1 − r[·1 + (1 − r) ,
2
4r
α

π
which is equivalent to that r > π+8and αβ < (π+8)r−π.
π(1−r)2(26)
Q.E.D.

Appendix E: Derivation of (16), (17), (18) and (19)Macroeconomic dynamics of the artiﬁcial economy is given by the following system ofstochastic diﬀerence equations.
1(it − Et πt+1 ) + ut ,σ
= βEt πt+1 + κxt + et ,xt = Et xt+1 −

(27)

πt

(28)

it = σ[ut + γu ε̃t + (1 − γu )µt ] + Φ[et + γe η̃t + (1 − γe )νt ].

25

(29)

Since the relevant state variables in period t are et , ε̃t , η̃t , µt and νt , the solution will beof the form
xt = Ax et + Bx ε̃t + Cx η̃t + Dx µt + Ex νt ,(30)

πt = Aπ et + Bπ ε̃t + Cπ η̃t + Dπ µt + Eπ νt ,(31)

where Ak , Bk , Ck , Dk and Ek (k = x, π) are undetermined coeﬃcients. Substituting(30),(31) into (27),(29) and then (29) into (27) and comparing the coeﬃcients of bothsides, I obtainΦ − ρe Aπγe Φ, Bx = −γu , Cx = −,σσ(1 − γe )Φ
= −(1 − γu ), Ex = −.
σ

Ax = ρe Ax −
Dx

(32)

Substituting (30),(31) into (28) and comparing the coeﬃcients of both sides,Aπ = βρe Aπ + κAx + 1, Bπ = κBx , Cπ = κCx , Dπ = κDx , Eπ = κEx .

(33)

Solving the ﬁrst equations of (32) and (33), Ax and Aπ turn out to beκ,Ax = − cλ (1 − βρe ) + κ2

Aπ =

λc
.
λc (1 − βρe ) + κ2

Thus, I obtain equilibrium output gap and inﬂation rate in period t in the text.
Finally, I calculate asymptotic variances of output gap and inﬂation rate. By et =σ2ρe et−1 + ψt , asymptotic variance of et is 1−ρψ 2 . Besides, since et , εjt , ηtj , µt and νt areemutually independent, each covariance of them is zero. Noting the two facts above, I
ﬁnd asymptotic variance of output gap in the text. Similarly, I can calculate asymptotic
variance of inﬂation rate in the text.

Appendix F: Proof of Proposition 4(1. The Suﬃcient Condition)
I prove it in the same way as the proof of Proposition 3. By the deﬁnition of theperformance measure of the monetary policy committee, the counterpart to (22) is
{
[ −2
( ∂γ
)−1 ]
γu
1
γu2
∂P
u
2 ∂γu σε2= 2 (1 − γu )σε
−
·
+
N
∂N
∂N σµ−2 1 − γu N2(1 − γu ) ∂N)−1 ]}( )2 [ σ −2
( ∂γ
γe
1
γe2
e
η
2 ∂γe Φ2+(1 − γe )ση−·+N. (34)∂N σσν−2 1 − γe N
2(1 − γe ) ∂N
26

∂L
Hence, a suﬃcient condition for that ∂N> 0 for suﬃciently large N is
( ∂γ
)−1
σε−2
γu2
u
2
>
lim
N
σµ−2 N →∞ 2(1 − γu ) ∂N

and

( ∂γ
)−1
ση−2
γe2
e
2
> lim
N
.
σν−2 N →∞ 2(1 − γe ) ∂N−2

5r−1
Similar to the proof of Proposition 3, this is equivalent to that r > 15 , σσε−2 < (1−r)
2 and
µ

ση−2
5r−1
< (1−r)2.
σν−2(2. The Necessary Condition)
According to the proof of Proposition 2, if both of the ﬁrst and the second terms of(35) converge to some non-negative numbers, then the optimal size does not exists. Thus,
for the existence of the optimal size, it is necessary for at least one of them to converge toa negative number. Similar to the proof of Proposition 3, this is equivalent to r > 51 and

ση−2
σε−2
σε−2
5r−1
5r−1
5r−1
1
1
−2 < (1−r)2 or r > 5 and−2 < (1−r)2 , which can be reduced to r > 5 and−2 < (1−r)2
σµ
σν
σµ
σ −2
5r−1
or ση−2 < (1−r)
Q.E.D.
2.
νAppendix G: Proof of Corollary 4
Note that med1≤j≤N {ϕjt } = ϕt +med1≤j≤N {εjt } and med1≤j≤N {ψtj } = ψt +med1≤j≤N {ηtj }.
Thus, by (13) and (14), nominal interest rate under median-voting rule isit = i∗t + zt + σ(1 − γu )µt + Φ(1 − γe )νt ,where zt = med1≤j≤N {σγu εjt + Φγe ηtj }.
By the same way as in Appendix E, I obtain the following equilibrium output gap
and inﬂation rate.
[]κ1(1 − γe )Φ
xt = − cet −zt + (1 − γu )µt +νt ,λ (1 − βρe ) + κ2
σ
σ
[
]
λc
1
(1 − γe )Φ
πt =et − κ zt + (1 − γu )µt +νt .
λc (1 − βρe ) + κ2σσ
The asymptotic variances of the equilibrium output gap and inﬂation rate are
(
[
]2 2
[
)2 ]
σψ
(1 − γe )Φ
κ
1
2 2
V (x) =+ V (zt ) + (1 − γu ) σµ +σν2 ,λc (1 − βρe ) + κ2 1 − ρ2e σ 2
σ
(
[
]2 2
[
)2 ]
σψ
(1 − γe )Φ
λc
κ2
2
2 2
V (π) =+ V (zt ) + κ (1 − γu ) σµ +σν2 ,λc (1 − βρe ) + κ2 1 − ρ2e σ 2σ27which give the social loss in equilibrium by (11). Since()π(σ 2 γu2 σε2 + Φ2 γe2 ση2 )zt ∝ N 0,,
2N
I obtain that
]
[
]2 2
σψ
π 1 2 2 ( γe Φ )2 2
κ
γ σ +
ση
V (x) ≈+ ·λc (1 − βρe ) + κ2 1 − ρ2e
2 N u εσ[()2 ](1 − γe )Φ
2 2+ (1 − γu ) σµ +
σν2 ,
σ
[
]2 2
[
]
σψ
λc
π κ2 2 2 ( γe Φ )2 2
V (π) ≈
γ σ +
ση
+ ·
λc (1 − βρe ) + κ2 1 − ρ2e
2 N u εσ)2 ][((1 − γe )Φ
2
2 2
σν2
+κ (1 − γu ) σµ +σ[

for suﬃciently large N .
Therefore, according to the proof of Corollary 2 and Proposition 4, I obtain the resultof Corollary 4. 31
Q.E.D.

31

See Appendix D and F.

28

References
Austen-Smith, D. and J.S. Banks (1996), ”Information aggregation, rationality, and theCondorcet jury theorem”, American Political Science Review, 90, 34-45.
Berger, H. and V. Nitsch (2008), ”Too many cooks? Committees in monetary policy”,
mimeo.
Blinder, A. (2007), ”Monetary policy by committee: Why and how?”, European Journalof Political Economy, 23, 106-123.
Condorcet, M. (1785), ”Essai sur l’application de l’analyse á la probabilité des decisions
rendues a al pluralité de voix ”, Paris: L’Imprimerie Royale.
Feddersen, T. and W. Pesendorfer (1998), ”Convicting the innocent: The inferiority of
unanimous jury verdicts under strategic voting”, American Political Science Review,
92, 23-35.
Gerling, K., Grüner, H.P., Kiel, A. and E. Schulte (2005), ”Information acquisition anddecision making in committees: A survey”, European Journal of Polical Economy, 21,563-597.
Kenney, J.F., and E.S. Keeping (1962), ”Mathematics of statistics”, Part 2, Van Nostrand, Princeton.
Koriyama, Y. and B. Szentes (2009), ”A resurrection of the Condorcet jury theorem”,
Theoretical Economics, 4, 227-252.
Li, H. (2001), ”A theory of conservatism”, Journal of Political Economy, 109, 617-636.
Maritz, J.S., and R.G. Jarrett (1978), ”A note on estimating the variance of the sample
median”, Journal of the American Statistical Association, 73, 194-196.
Martinelli, C. (2006), ”Would rational voters acquire costly information?”, Journal of
Economic Theory, 129, 225-251.
Morimoto, K. (2009), ”Optimal structure of monetary policy committees”, mimeo.
Morris, S. and H.S. Shin (2002), ”Social value of public information”, American Economic Review, 92, 1521-1534.
Mukhopadhaya, K. (2003), ”Jury size and the free rider problem”, Journal of Law, Economics, and Organization, 19, 24-44.
29

Muto, T. (2007), ”How do central banks make decisions?: Monetary policy bycommittee”, Summary of a speech at the Spring Meeting of the Japan
Society of Monetary Economics in Chiba on May 12, 2007, readable at
http://www.boj.or.jp/en/type/press/koen07/ko0709b.htm.
Sibert, A. (2006), ”Central banking by committee”, International Finance, 9, 145-168.
Walsh, C.E. (2003), ”Monetary theory and policy”, 2nd ed., MIT Press, Cambridge, MA.

30

A Supplementary Note for ”Coordination Behavior
and Optimal Committee Size”∗
Keiichi Morimoto†‡
January 27, 2010

Abstract
This is a supplementary note for Morimoto (2010) ”Coordination Behavior andOptimal Committee Size”. Here, I treats the case where each committee member
has proper prior on underlying state θ.

Keywords: beauty contest, proper prior, Condorcet jury theorem
JEL Classiﬁcation: D71; D84; E58

1

Incorporating Proper Prior into the Simple Model

Along the line of Morris and Shin (2002), Morimoto (2010) assumes each member’s improper ﬂat prior about the underlying state over the real line. However, it is more natural
to adopt some informative prior about the underlying state according to applications tocommittees of experts. I analyze such a case here and show that the results of Morimoto(2010) do not change. For a detail of the notations and model setting, see Morimoto(2010).
Suppose that the underlying state θ is normally distributed with mean zero andvariance ω −1 > 0 and each committee member knows this.

∗I am very grateful to Wataru Tamura for his helpful comment. I acknowledge ﬁnancial support fromthe Research Fellowships for Young Scientists of the Japan Society for the Promotion of Science (JSPS).
†
Japan Society for the Promotion of Science.
‡
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan. E-mail address: gge014mk@mail2.econ.osaka-u.ac.jp

1

1.1

A Derivation of the Unique Equilibrium Strategy

Recall that the ﬁrst order condition for each member problem is
aj = (1 − δ)Ej (θ) + δEj (ā).

(1)

First, I ﬁnd the equilibrium strategy by a heuristic method. Since each member’s estimation of the underlying state is partially anchored by the prior θ ∼ N (0, ω −1 ), it
becomesαβω × 0 + αy + βxj
=
y+
xj .
(2)
Ej (θ) =
ω+α+β
ω+α+β
ω+α+β
β
α
Note that ω+α+β+ ω+α+β< 1. Thus, I can conjecture that the equilibrium strategy is of
the form such thataj = γc y + γp xj ,(3)

where γc and γp are undetermined coeﬃcients such that γc + γp < 1. Substituting (2)and (3) into (1), after some calculations, the left-hand size of (1) turns out to be[]((1 − δ)β
1)
β
1−
γp y
+ δγc + δ
ω+α+β
ω+α+β
N
[
]
(
(1 − δ)α1)δα1−
γp xj .
+
+ γp + δ
ω+α+β N
ω+α+βN
Thus, comparing the coeﬃcients of both the sides of (1) and using the deﬁnition of δ, I
obtainγc(1 − 2rN −1 + rN −2 )α
)
(
)[
],
= (
1 − Nr [β + ω + α] − r 1 − N1 β + ω+αN(4)(1 − r)β()[].
[β + ω + α] − r 1 − N1 β + ω+αN(5)γp = (

1 − Nr)Second, I formally show that the strategy above is the unique equilibrium strategy.
Substituting (1) recursively, I obtain
aj = (1 − δ)∞∑

δ s Ej (Ē s (θ)).

(6)

s=0

Thus, I next calculate the s-th order belief Ej (Ē s (θ)). For given s = 0, 1, ..., put the s-thorder belief of arbitrary member j as follows.
Ej (Ē s (θ)) = µs y + νs xj .
2

(7)

Then, since Ē s+1 (θ) = µs y + νs N1
Ej (Ē

s+1∑N

k=1 xk , I obtain(1

N − 1 αy + βxj )(θ)) = µs y + νs
xj +
·
N
N
ω+α+β
[
]
ω+α
(
)
β
α
1
N
= µs +1−
νs y +
νs xj .
ω+α+β
N
ω+α+β

Therefore, the coeﬃcients of higher order beliefs satisfy the simultaneous diﬀerence equation
(
1)
α
1−
νs ,
µs+1 = µs +
ω+α+β
N
ω+α
β
N
νs+1 =
νs ,
ω+α+β
β
α
, ω+α+β). The solution of this equation iswith the initial value (µ0 , ν0 ) = ( ω+α+β
[
]
(
α
1)
β
1 − τsµs =1+ 1−·, for s = 0, 1, ...,
ω+α+β
N ω+α+β 1−τβµs =τ s , for s = 0, 1, ...,
ω+α+β
ω+α

(8)
(9)

+β

N
where τ = ω+α+β. Substituting (7), (8) and (9) into (6) and using the deﬁnition of δ, I
obtain the unique equilibrium strategy of the form given by (3), (4) and (5).
I ﬁnally check that the relationship between the committee size and the equilibrium
strategy.

∂γc
= [
∂N

2r(1 − r)N −2 (1 − N −1 )αβ(1 − 2rN −1 − N −2 )(ω + α) + (1 − r)β

∂γp
= −[
∂N
lim γcN →∞lim γpN →∞1.2]2 > 0,2r(1 − r)N −2 (1 − N −1 )(ω + α)β(1 − 2rN −1 − N −2 )(ω + α) + (1 − r)βαα, γc |N =1 =,=
ω + α + (1 − r)βω+α+β(1 − r)ββ=, γp |N =1 =
ω + α + (1 − r)βω+α+β]2 < 0.

(10)

(11)
(12)
(13)

Proper Prior and the Existence of Optimal Size

Next, I investigate the existence condition for the optimal size of the committee. Since
the distribution of θ is given in this case, it is natural to consider the unconditional3performance of the committee:
E(W ) = −E
= −E

[(
[(

)2
1 ∑
(θ + εk ) − θγc (θ + η) + γp
N k=1
N

]

)2
1 ∑
γc η + γpεk − (1 − (γc + γp ))θ
N k=1= −γc2 α−1 − γp2N

β −1− [1 − (γc + γp )]2 ω −1 .
N

]

(14)

The third term of (14) is generated by the weak response to common and private signals
due to the prior of mean zero. Diﬀerentiating E(W ) with respect to N , I obtain∂E(W )∂γc∂(γc + γp )β −1 ∂γp= −2γc α−1− 2γp·+ γp2 β −1 N −2 + 2[1 − (γc + γp )]ω −1
.
∂N
∂N
N ∂N
∂N
The ﬁrst, second and third terms are the same eﬀects as those of the benchmark model.
The forth term is the negative eﬀect from that each member is more strongly anchoredby the prior of mean zero. 1 Since[(
γp2 ( ∂γp 2 )−1∂E(W )γp 1
β ∂γc /∂N
−1 ∂γp= 2γc β
− ·
−
·
−
N
∂N
∂N
α ∂γp /∂Nγc N
2γc ∂N)]1 β ∂(γc + γp )/∂N,−[1 − (γc + γp )] · ·γc ω∂γp /∂N
along the line of the proof of Proposition 2, the following inequality is a suﬃcient condition
for the existence of the optimal committee size. 2{}{}
γp2 ( ∂γp 2 )−1β ∂γc /∂N
1 β ∂(γc + γp )/∂N
lim − ·> lim −
N.
+ [1 − (γc + γp )] · ·
N →∞
N →∞
α ∂γp /∂N
2γc ∂Nγc ω∂γp /∂N
By (10), (11), (12) and (13), this is equivalent to
5r − 1β + (1 − r)−1 ω<.
α(1 − r)2

1.3

(15)

The Case of Median-voting Rule

Incorporating the proper prior, the suﬃcient condition for the existence of the optimal
size under median-voting rule in Corollary 3 should be modiﬁed in the same way. Similar−2−1∂(γc +γp )= − [( 2r(1−r)N )(1−N )ωβ ]2 < 0.
∂N
1−2rN −1 −N −2 (ω+α)+(1−r)β1

By (10) and (11), I obtain

2

Similar to the proof of Proposition 2, it can be easily checked that this is also a necessary condition.

4

to section 2.4 in the main text, since
π βE(W ) ≈ −γc2 α−1 − γp2−1− [1 − (γc + γp )]2 ω −1

2N
for suﬃciently large N under the median-voting rule, a suﬃcient condition for the existence of the optimal size is}{}{
π γp2 ( ∂γp 2 )−1
1 β ∂(γc + γp )/∂Nβ ∂γc /∂N> lim − ·+ [1 − (γc + γp )] · ·.
lim − ·NN →∞N →∞α ∂γp /∂N
2 2γc ∂Nγc ω∂γp /∂N
This is equivalent to
β + (1 − r)−1 ω(π + 8)r − π.
<απ(1 − r)2

2

(16)

The Model of Monetary Policy Committee

Next, I modify the model of monetary policy committee in section 3 of the main text
along the line of the modiﬁcation of the simple model. In this model, each member ofthe monetary policy committee has prior such that ϕt ∼ N (0, σϕ2 ) and ψt ∼ N (0, σψ2 ).

2.1

The Case of Average-voting Rule

Similar to section 1.2, the suﬃcient condition and necessary condition for the existence−2
σ −2 +(1−r)−1 σϕ
5r−1< (1−r)of the optimal size in Proposition 4 are modiﬁed to ’r > 15 , ε
2 and
σ −2
−2
ση−2 +(1−r)−1 σψ

σν−2

µ

−2
−2
ση−2 +(1−r)−1 σψσε−2 +(1−r)−1 σϕ
5r−1
1
5r−1
5r−1
< (1−r)<or< (1−r)−2−2
2 ’ and ’r > 5 and
2
2’
(1−r)
σµ
σν

respectively.

2.2

The Case of Median-voting Rule

Similar to section 1.3, the suﬃcient condition for the existence of the optimal size inπ,Corollary 4 is modiﬁed to ’r > π+8−2
σε−2 +(1−r)−1 σϕ
−2
σµ

and
< (π+8)r−π
π(1−r)2−2
ση−2 +(1−r)−1 σψ

σν−2

<

(π+8)r−π’.
π(1−r)2

References
Morimoto, K. (2010), ”Coordination Behavior and Optimal Committee Size”, mimeo.
Morris, S. and H.S. Shin (2002), ”Social value of public information”, American Economic Review, 92, 1521-1534.
5

-0.34

-0.36

E(W)

-0.38

-0.4

-0.42

-0.44

expected performance
expected performance with in the limit
-0.46

2

4

6

8

10

12

14

16

18

20

N

Figure 1: The relationship between committee size and expected performance in thesimple model

22
21
20
19
18
17
16
15
14

N∗

13
12
11
10
9
8
7
6
5
4
3
2
1
0
0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

Figure 2: The relationship between degree of coordination motive and optimal size in thesimple model

11

with β = 2
with α = 3

10
9
8

N∗

7
6
5
4
3
2
1

1

1.5

2

2.5

3

3.5

4

4.5

5

α, β

Figure 3: The relationship between precisons of common and private signals and optimal
size in the simple model

100

average-voting rule
median-voting rule

90
80
70

β/ α

60

No finite optimal size

50
40
30
20

r = π/( π+8)10Optimal size exists.
0
0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

r

Figure 4: The regions below the dashed and solid curves are the parameter regions which
satisfy the suﬃcient conditions for the exsistence of optimal committee size in the simple
model under the average and median voting rules respectively. The former contains thelatter.