International Review of Economics and Finance 20 (2011) 624–632

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International Review of Economics and Finance

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Herding with costly information and signal extraction

Wan-Ru Yang⁎Department of Finance, National University of Kaohsiung, 700, Kaohsiung University Rd., Nanzih District, 811, Kaohsiung, Taiwan

a r t i c l e i n f o

⁎ Tel.: +886 7 5919711.E-mail address: wryang@nuk.edu.tw.

1059-0560/$ – see front matter © 2010 Elsevier Inc.doi:10.1016/j.iref.2010.12.004

a b s t r a c t

Article history:Received 23 August 2009Received in revised form 11 October 2010Accepted 9 November 2010Available online 30 December 2010

Costly signal acquisition compels decision-makers to choose between acquiring private signalsand following their predecessors, which can result in problems associated with signalextraction. The results show that the information externality of the second decision-makerinfluences the efficiency of herd behavior among subsequent decision-makers. If the seconddecision-maker acts differently than his predecessor, the followers take a free ride on his signalacquisition and act correctly. However, if the second investor acts in the same manner as hispredecessor, the followers will acquire the costly signals only if the precision of their privatesignals is significant, otherwise herding is inefficient.

© 2010 Elsevier Inc. All rights reserved.

JEL classifications:C7D81D82D83

Keywords:HerdingInformation acquisitionSignal extraction

1. Introduction

Herding occurs when people make homogeneous decisions that rely on available public information rather than their ownprivate signals. For example, a group of investors follow one another to buy over-performing assets or sell underperforming assetsat the same time. However, only a few investors possess essential signals regarding the value of the assets. Bank runs can beconsidered as another herding phenomenon (Chen, 1999). When informed depositors with bank-specific information begin towithdraw from banks, contagious bank runs may be triggered by uninformed depositors who imitate the actions of one another,and do not obtain their own private signals. The above examples imply that taking the opportunity for a free ride on private signalacquisition is a determinant of herd behavior among decision-makers.

In classic herding literature (e.g., Banerjee, 1992; Bikhchandani, Hirshleifer & Welch, 1992), the private signal of an agent hasno cost and the precision of private signals among agents is indifferent. If the type of private signal is a symmetric binary case,people logically follow the predecessors' actions and ignore their own private signal because signals of different types negate eachother. In the real world, however, people have to incur costs such as those related to effort, time, and money to obtain their ownsignals. If obtaining a private signal is not free, the precision of each agent's private signal may vary. When people are heavilyinfluenced by the less precise signals revealed in the actions of predecessors, the action pattern in equilibrium is far from efficient.The question that arises with regard to this problem is as follows: Why do people tend to take a free ride on the signal acquisitionof their predecessors?

This paper studies the manner in which the herd behavior of investors is influenced by the precision of costly private signalsand signal extraction. In this model, the choices of investors concerning the acquisition of private signals are endogenous and

All rights reserved.

625W.-R. Yang / International Review of Economics and Finance 20 (2011) 624–632

unobservable to the public. This creates the problem associated with signal extraction, in which investors are unable to distinguishthose predecessors with their own private signals from those who are free riders on the costly signal acquisition. The resultsindicate that the behavior of the second investor in acquiring a private signal is the determinant of herding efficiency, implying thecorrect path for investment decisions. If the marginal precision of the private signal cost is significant, the second investor will beinterested in purchasing a costly signal. When the second investor acts differently from the first investor that reveals the precisionof the second investor's signal to the followers, the followers will herd behind the investment decision of the second investor. As aresult, the herding triggered by the third investor is based on dependable previous signals, and the decisions of the followinginvestors are more likely to be correct. However, if the second investor acts in an identical manner to the first, the followers areunable to recognizewhether the second investor takes a free ride on the signal acquisition of the first investor. The successors haveno incentive to acquire private signals when the cost of obtaining a more precise private signal is extremely high or when theybelieve that the signals of their predecessors are accurate enough. Therefore, the followersmay herd behind an erroneous decision,as the second investor is actually a free rider like themselves.

This paper relates to several studies regarding herd behavior. Related studies on herding with costly information includeBurguet and Vives (2000), Feltovich (2002), and Kultti and Miettinen (2006, 2007). Burguet and Vives (2000) began thestudy of whether the acquisition of costly information prevented the accumulation of public information in social learningprocesses. Burguet and Vives showed that incentives for agents to acquire costly private information would not be ruined ifand only if the marginal cost of obtaining private information was zero. Conversely, when the marginal cost was positive,agents relied on public information, and comprehensive information regarding the value of a random variable was notrevealed. The model of Burguet and Vives assumes that the cost of private information is a function of private informationprecision. To introduce the possibility of receiving different signals given the true state of future returns, this paper assumesan inverse cost function. In other words, agents decide how much to pay for private signals rather than select the precisionof private signals. Therefore, the problems associated with signal extraction and taking a free ride on the acquisition ofcostly signals arise in a herding model. This paper shares a similar conclusion with Burguet and Vives that the acquisition ofcostly signals degrades people's incentive to obtain their own private signals. Moreover, this paper demonstrates anadditional outcome that is the excessive acquisition of private signals, which can be observed in the experimental work ofKraemer, Noth and Weber (2006).

Kraemer et al. (2006) conducted an experiment to analyze the acquisition of private information by participants at afixed price. In the experiment, participants acquired considerable private information, compared to a Bayesian rationalparticipant. They demonstrated that the attitude of participants toward risk aversion and error on the part of predecessorscould not explain the excessive acquisition of private information. This paper finds a potential reason for the behaviorregarding the acquisition of private information, which was observed in the experiment of Kraemer et al. In a departurefrom the previous research on herding with costly signal acquisition, this model considers the precision of predecessors'private signals, which cannot be observed by the successors. If agents underestimate the precision of prior signals, theirchoice to acquire a private signal is inappropriate.

Most of the literature pertaining to anti-herd behavior focuses on the concern of the agents' reputation. Effinger and Polborn(2001) and Levy (2004) found that decision-makers tended to act against prior information to prove their ability or talent; this istermed as reputation-based anti-herding. This paper shows that people often choose to contradict their predecessors because theirprivate signals are more precise than those of their predecessors. Therefore, one of the main contributions of this paper is topresent another type of anti-herding: information-based anti-herding.

The remainder of this paper is organized as follows. Section 2 introduces themodel. Section 3 presents the behavior of investorswith respect to the acquisition of private signals and investment herding. Further, Section 3 relates the results of this paper to theobservations found in Kraemer et al. (2006). Finally, the concluding remarks are in Section 4.

2. Model

Supposes three dates, t=0, 1, and 2, and a set of investors i=1, 2, …, n, while the “i” stands for an exogenous order in thedecision-making sequence. At date t=0, each investor invests one unit in a risky asset. At t=1, investor i considers whether hewill maintain the investment until t=2. At date t=2, one unit of the investment yieldsR∈{

�R,�R}, where

�R=2and �R=0. If investor

i discontinues the investment at t=1, he obtains one unit of the return. The investment decision of investor i depends on investori's information including a costly private signal and public information. Public information is the observable investment decisions

made by predecessors. The prior belief of the return being�R is

12. Based solely on the prior belief, the investors act indifferently with

respect to continuing or discontinuing the investment.At t=1, before making the investment decision, investor i pays the cost Ci to acquire a private signal Si concerning the

state of the future return from the set {Sg,Sb}, where 0≤Cib1. Let Sg represent a good private signal, in which the futurereturn is high, and Sb is a poor signal, in which the value of the future return is low. Investor i's signal Si=Sg is denoted bySig and Si=Sb is denoted by Si

b. If investor i chooses Ci=0, he takes a free ride on the signal acquisition of his predecessors.In other words, he obtains a costless signal.

The conditional probability distribution of signal Si given the state of the future return can be expressed as follows:

P Si = Sgi jR =�R

� �= P Si = Sbi jR = �R

� �= qi Cið ÞP Si = Sbi jR =

�R

� �= P Si = Sgi jR = �R

� �= 1q i Cið Þ: ð1Þ

626 W.-R. Yang / International Review of Economics and Finance 20 (2011) 624–632

The precision of investor i's private signal, qi(Ci), is related to the signal cost, Ci.1 To simplify the model, each agent is given ahomogeneous precision function. Throughout this paper, I make the following assumptions concerning the function, qi(Ci).

Assumption 1.

1 Burassociatand We

12bqi Cið Þb1; ∀ Ci N 0; i = 1;2;…;n:

Assumption 2.

q′i N 0 and q″i b 0; ∀ Ci N 0; i = 1;2;…;n:

Assumption 3.

q1 C1ð Þ = 12; as C1 = 0:

qi Cið Þ = qi Ci −1ð Þ; as Ci = 0 and Ci−1 N 0; i = 2;…;n:

Assumption 1 implies that the private signal is informative concerning the state of the future return. Assumption 2 implies thata private signal with a higher cost is more precise. Moreover, there is a decline in the marginal precision that the investors receivefrom paying each additional unit of the signal cost. Finally, in Assumption 3, if the signal from the first investor is costless, thesignal is fully uninformative. This implies that the first investor has an incentive to acquire a costly private signal. Beginning withthe second investor, when investor i takes a free ride on the private signal acquisition of his predecessors, the precision of hiscostless signal is equal to that of the predecessor's costly signal. Suppose that investor i's choice regarding the private signalacquisition cannot be observed by other investors. Each investor relies solely on the history of his predecessors' investmentdecisions to infer the types and precision of the prior investors' private signals.

Investor i's investment choice, Xi(Si, Hi), is a function of his private signal Si and the history of the investment decisions beforeinvestor i, where Hi=(X1, X2, …, Xi−1). Let Xi=1 represent that investor i withdraws the investment at t=1, and Xi=0 denotesthat investor i maintains his investment until t=2.

The determination of investor i's decisions is performed in two stages. First, investor i decides whether to acquire a privatesignal, and then he chooseswhether tomaintain his investment until t=2. The expected payoff to investor iwith a private signal isD̂i.

D̂i = P Sgi jS1; S2; :::Si−1� �

EUi Xi;Ci jSgi ;X1;X2;…; Xi−1

� �+ P Sbi jS1; S2; :::Si−1

� �EUi Xi; Ci jSbi ;X1;X2;…;Xi−1

� �:

ð2Þ

In Expression (2), P(Sig|S1,S2,...Si−1) is the probability of obtaining a good private signal given the types of the first (i−1)predecessors' signals that are inferred from the observable investment decisions of the predecessors, (X1, X2, …, Xi−1). Let EUi(Xi,Ci|Sig, X1, X2,…., Xi−1) be the expected payoff from maintaining or discontinuing the investment to investor i with a good privatesignal, which is shown in Expression (3).

EUi Xi;Ci jSgi ;X1; X2;…:; Xi−1� �

=1−Ci;Xi = 1

2P�R jHi; S

gi

� �−Ci;Xi = 0

:

(ð3Þ

Investor i adopts Bayes' rule to update his belief concerning the state of the future return, P(�R |Hi, Si

g), depending on his good

private signal and the history of the predecessors' investment actions. The representation of EUi(Xi, Ci|Sib, X1, X2,…., Xi−1) is similarto Expression (3), which is the expected payoff to investor i with a bad signal. Denote ˜Di to be the expected payoff to investor iwithout acquiring a private signal.

D̃i =1;Xi = 1EUi Xi jX1;X2; :::;Xi−1ð Þ = 2P

�R jHi

� �;Xi = 0 :

�ð4Þ

If investor i does not acquire a private signal, he updates his belief concerning the state of the future return, P(�R|Hi), based only

on the public information, Hi.

guet and Vives(2000) assumed that the cost of private signal acquisition is determined by the precision of the signal. In their model, precision ised with the variance of the private signal. This study follows the definition of precision of private signals presented by Bikhchandani et al. (1992), Küblerizsäcker (2003). They defined the precision as the probability of obtaining a good signal at a good state and a bad signal at a bad state.

627W.-R. Yang / International Review of Economics and Finance 20 (2011) 624–632

This paper analyzes each investor's choice concerning the investment and acquisition of private signal by using backwardinduction. First, this paper finds investor i's optimal investment decision with and without a private signal. Then, given theinvestment decision, this paper examineswhether investor i chooses to acquire a private signal. The investors' strategies regardingthe investment and acquisition of private signals are studied in the next section.

3. Information acquisition and decision rule

This section studies the manner in which investor i chooses to acquire a private signal. Further, the herd behavior of investorswith regard to making the investment decisions is also investigated.

3.1. First investor

The first investor is the first agent to make an investment decision. Therefore, his action is based solely on his private signal, S1.Now, this paper considers how the first investor makes the investment decision after acquiring a private signal. When the firstinvestor acquires a good signal, S1

g, and maintains his investment until t=2, his expected payoff is

EU1 X1 = 0;C1 jSg1� �

= 2P�R jSg1� �

−C1 = 2q1 C1ð Þ−C1 ð5Þ

where P(�R|S1

g)=P

�R� �

P Sg1 j�R

� �P

�R� �

P Sg1 j�R

� �+ P �R

� �P Sg1 j�R� �. Based on Assumption 1, 2q1(C1) is greater than 1 as C1N0. If the first investor

withdraws the investment at t=1, his payoff is EU1(X1=1, C1|S1g)=1−C1. Hence, the first investor with a good private signalprefers to maintain the investment because the expected payoff from maintaining the investment is greater than that fromdiscontinuing the investment. Conversely, if the first investor has a poor signal, S1b, his expected payoff from maintaining theinvestment is

EU1 X1 = 0;C1 jSb1� �

= 2P�R jSb1

� �–C1 = 2 1:−q1 C1ð Þð Þ−C1: ð6Þ

Discontinuing the investment is the optimal choice to the first investor with a poor signal because it yields 1−C1N2 (1.−q1(C1))−C1.

Next, given the optimal investment decision, we can calculate the expected payoff of the first investor after acquiring a privatesignal. Assume that, to the first investor, the probability of obtaining a good signal is equal to that of obtaining a poor signal, which

means P(S1g)=P(S1b)=12. When the first investor chooses to acquire a costly private signal, his expected payoff is

D̂1 = P Sg1ÞEU1 X1 = 0;C1 jSg1� �

+ P Sb1� �

EU1 X1 = 1;C1 jSb1� �

=12

2q1 C1ð Þ−C1ð Þ + 12

1−C1ð Þ = 12

+ q1 C1ð Þ−C1:

�ð7Þ

If the first investor does not acquire a private signal, he acts only based on his prior belief concerning the future return. In

Expression (4), P(�R|H1) becomes P(

�R)=

12. Therefore, the payoff to the first investor without a private signal is equal to 1, which is

independent of the first investor's investment action.Consider the time at which the first investor has the incentive to acquire a costly private signal. The condition regarding the

acquisition of private signals is characterized in Lemma 1.

Lemma 1. The first investor acquires a private signal if and only if the marginal precision of the private signal cost,Δq1 C1ð ÞΔC1

, is greater

than 1, where Δq1(C1)=q1(C1)−q1(0)=q1(C1)−12and ΔC1=C1−0.

Proof. The payoff to the first investor with a private signal isD̂1=12+q1(C1)−C1. The payoff to the first investor without a private

signal isD̃1=1. The expected payoff with a private signal,D̂1, exceeds the expected payoff without a signal,D̃1, if and only ifΔq1 C1ð ÞΔC1is greater than 1. Q.E.D.

In Lemma 1, let Δq1(C1) represent the difference between the precision of the first investor's private signal, S1, and that of thecostless signal. Denote ΔC1 to be the change in the amount of the signal cost. If the first investor does not acquire a costly private

signal, the precision of the costless signal,q1(0), is12. To the first investor, acquiring a costly private signal is preferable to owning a

costless signal when the marginal precision of costs,Δq1 C1ð ÞΔC1

, is greater than 1. Thus, the first investor will be interested in

acquiring a costly private signal if paying the cost notably increases the precision of the first investor's signal.

628 W.-R. Yang / International Review of Economics and Finance 20 (2011) 624–632

3.2. Second investor

Based on the investment action of the first investor, the second investor infers the type of private signal owned by the firstinvestor and considers whether or not to take a free ride on the signal acquisition of his predecessor.2

When the first investor maintains the investment at t=1, which is denoted by X1=0, the second investor anticipates that hispredecessor has received a good signal, S1g.3 Without acquiring a private signal, the second investor's belief in receiving a high

future return, P(�R|X1=0)=P(

�R|S1g), is greater than

12. Therefore, the second investor acts in the same manner as the first investor

did, and his expected payoff D̃2 can be expressed as

2 Whfunctionwhere 0

3 The4 In t

the con5 Ass

6 If q1this pap

model.7 P(S

D̃2 = EU2 X2 = 0 jX1ð Þ = 2P�R jX1� �

= 2P�R jSg1� �

= 2q1 C1ð Þ:4 ð8Þ

Next, consider the investment behavior of the second investor with a private signal. Suppose that the first investor maintainsthe investment at t=1 and the second investor acquires a good signal. Then, the second investor updates his belief of obtaining ahigh return, P(

�R|X1,S2g), represented in Expression (9).

P�R jX1; S

g2

� �=

P�R� �

P Sg1; Sg2 j�R

� �P

�R

� �P Sg1; S

g2 j�R

� �+ P �R

� �P Sg1; S

g2 j�R

� �

=q1 C1ð Þq2 C2ð Þ

1q 1 C1ð Þ−q2 C2ð Þ + 2q1 C1ð Þq2 C2ð Þ :5

ð9Þ

The belief, P(�R|X1, S2

g), is greater than12; therefore, the second investor also maintains the investment if he obtains a good

private signal.If the first investor maintains the investment at t=1 but the second investor obtains a poor signal, the second investor's

posterior belief of receiving a high return is

P�R jX1; S

b2

� �=

P�R

� �P Sg1; S

b2 j�R

� �P

�R

� �P Sg1; S

b2 j�R

� �+ P �R

� �P Sg1; S

b2 j�R

� �

=q1 C1ð Þ 1−q2 C2ð Þ½ �

q1 C1ð Þ + q2 C2ð Þ−2q1 C1ð Þq2 C2ð Þ :

ð10Þ

In Expression (10), the precision of the second investor's signal, q2(C2), influences the second investor's investment decision.Lemma 2 shows how the second investor with a poor private signal makes the investment decision.

Lemma 2. If the second investor receives a poor signal, S2b, he maintains the investment until t=2 as q1(C1)Nq2(C2), but discontinues

the investment at t=1 as q1(C1)bq2(C2).6

Proof. See the Appendix.Although the second investor obtains a poor signal, he acts in the same manner as the first investor did because his private

signal is less accurate than his predecessor's. On the other hand, if the second investor's poor signal is more precise than the firstinvestor's good one, his investment action is different from that of the first investor.

For the second investor, the probability of acquiring a specific signal when the first investor obtains a good signal is P(S2g|S1

g) andP(S2b|S1g).7 Given the optimal investment choice for each type of private signal, the second investor's expected payoff from acquiringa private signal is

D̂2 = P Sg2 jSg1� �

EU2 X2 = 0;C2 jX1; Sg2

� �+ P Sb2 jSg1

� �EU2 X2 = 1;C2 jX1; S

b2

� �= q1 C1ð Þ + q2 C2ð Þ−C2:

ð11Þ

en the first investor does not acquire a private signal, the marginal precision of the cost,Δq1 C1ð ÞΔC1

, is less than 1 for any C1 and 0bC1b1. The precisionof each investor is homogeneous for 0bCib1. Therefore, investor i cannot find a cost Ci that satisfies the condition of private signal acquisition,

Δq1 Cið ÞΔCi

bCib1, i=2, 3,…, n. It is trivial that the followers do not acquire their private signals if they infer that the first investor tends to own a costless signalsecond investor infers that the first investor has received a poor signal if the first investor discontinues the investment.his paper, to the followers, the signal costs of predecessors are anticipated values. The followers anticipate the lowest value of the signal cost that satisfiesditions at which the predecessors are willing to acquire their private signals.ume P S1; S2; :::Si j

�R

� �= P S1 j

�R

� �P S2 j

�R

� �:::P Si j

�R

� �, i=1, 2,.., n.

(C1) is equal to q2(C2), then P(�R|X1,S2b)=

12and the second investorwith a poor signal is indifferent as towhether hemaintains orwithdraws the investment. In

er, the investors prefer to obtainmore precise signal or onlydependon the actions of their predecessors. Therefore, the case of q1(C1)=q2(C2) is neglected in the

2g|S1

g)=

P Sg1 ;Sg2 j�Rð ÞP �

Rð ÞP Sg1ð Þ +

P Sg1 ;Sg2 j�Rð ÞP �

Rð ÞP Sg1ð Þ =1−q1(C1)−q2(C2) 2q1(C1)q2(C2). P(S2b|S1g)=

P Sg1 ;Sb2 j

�Rð ÞP �

Rð ÞP Sg1ð Þ +

P Sg1 ;Sb2 j

�Rð ÞP �

Rð ÞP Sg1ð Þ =q1(C1)+q2(C2)−2q1(C1)q2(C2).

,.

629W.-R. Yang / International Review of Economics and Finance 20 (2011) 624–632

Lemma 3 summarizes the second investor's behavior with respect to the acquisition of private signal.

Lemma 3. Assume that the first investor maintains the investment at t=1. The second investor acquires a private signal if and only if

the marginal precision of the private signal cost,Δq2 C2ð ÞΔC2

, is greater than 1, where Δq2(C2)=q2(C2)−q2(0)=q2(C2)−q1(C1) andΔC2=C2−0.

Proof. See the Appendix.

The condition for acquiring a private signal in Lemma 3 is similar to that of Lemma 1. For the second investor, the precision ofhis costless signal is equivalent to that of the first investor's private signal. The difference between the precision of the secondinvestor's private signal, S2, and that of the first investor's signal, S1, is represented by Δq2(C2). If the cost of the private signal, C2,makes the second investor's private signal more reliable than the first investors', the second investor prefers to acquire a privatesignal than to be a free rider. Further, the second investor withdraws the investment depending on the signal S2b andmaintains theinvestment depending on the signal S2

g.

3.3. Third investor

The third investor does not know the precision of the previous investors' signals but can observe their investment decisions.The third investor considers whether acquiring a private signal is an optimal choice, depending on all available information, whichincludes the investment actions of the first two investors.When the first investormaintains his investment but the second investordiscontinues the investment at t=1, the third investor receives the signals (S1g, S2b) based on the decisions of his predeces-sors. According to the decisions of the first two investors, the third investor infers that the probability of obtaining a high futurereturn, P(

�R|X1,X2)=P(

�R|S1g, S2b), is less than

12because he realizes that the signal of the second investor is more precise than that of

the first investor, by observing a conflict among prior investment decisions. In other words, the third investor expects that thefuture return is more likely to be low. To the third investor, the probability of receiving a poor signal conditional on a low futurereturn, P(S3=S3

b|R=R )=q3(C3), is greater than12. Therefore, the third investor prefers to follow the investment action of the

second investor, rather than acquire a costly signal that has the same type as the signal of the second investor. Similarly, thesuccessors to the third investor make the same inference regarding the signals precision the first two predecessors as the thirdinvestor did; hence, they also take a free ride on the private signal acquisition of the second investor. Consequently, a herding ofdiscontinuing investment begins with the third investor.

If both the first and second investors maintain their investments, the third investor infers that the combination of hispredecessors' private signals is (S1

g, S2g) or (S1

g, ϕ2).8 Let ϕ2 denote that the second investor has no costly private signal. The problemassociated with signal extraction arises when the predecessors make identical investment decisions. In this case, the third investoris unable to distinguish whether the second investor is a free rider of the private signal acquisition, based on the investmentactions of the first two investors.

Consider the behavior of the third investor with regard to acquiring a costly signal, as the choice of the second investor to be afree rider is ambiguous. When the third investor chooses to follow the first investor's investment action, his expected payoff ofbeing a free rider is D̃3=EU3(X3=0|X1)=2q1(C1).

On the contrary, if the third investor tends toacquire aprivate signal, theprobabilityof receivingagoodsignal, S3g, isλP(S3

g|S1g)+(1−λ)

P(S3g|S1

g, S2g).Moreover, theprobability of obtaining apoor signal, S3b, isλP(S3b|S1

g)+(1−λ) P(S3b|S1g, S2

g). Letλ represent the third investor andhis successors' estimations concerning the probability that the second investor is a free rider, where 0bλb1. Based on Section 3.2, if thesecond investor has no incentive to acquire a more precise signal, he will choose to be a free rider, which implies λ=P(C2bC1).9

Given the possible information sets, the posterior belief of the third investor with a good signal, S3g, is P(�R|X1, S3

g)=P(�R|S1

g, S3g) or

P(�R|X1, X2, S3

g)=P(�R|S1

g, S2g, S3

g). Both P(�R|S1g, S3g) and P( R|S1

g, S2g, S3

g) are greater than12; therefore, the third investor with a good

signal also maintains his investment until t=2.On the other hand, the third investor considers the precision of his private signal if he receives a poor signal while his two

predecessors receive good ones. When the third investor obtains a poor signal, S3b, his posterior belief of receiving a high futurereturn is

8 If th9 In S

conside

P�R jSg1; Sg2; Sb3

� �=

q1 C1ð Þq2 C2ð Þ 1−q3 C3ð Þ½ �q3 C3ð Þ + q1 C1ð Þq2 C2ð Þ−q2 C2ð Þq3 C3ð Þ−q1 C1ð Þq3 C3ð Þ : ð12Þ

Ifq3 C3ð Þ

1−q3 C3ð Þ≤q1 C1ð Þq2 C2ð Þ

1−q1 C1ð Þð Þ 1−q2 C2ð Þ2� �, the third investor's belief, P(

�R|S1g, S2g, S3b), is greater than

12; thus, the third investor

makes the same investment decision as his predecessors did. Conversely, the third investor with a poor signal prefers to withdraw

e first and second investors discontinue their investments, the third investor infers that his predecessors' private signals are (S1b, S2b) or (S1b, ϕ2).ection 3.2, we find that the precision of the first investor's private signal influences the second investor's incentive of being a free rider. Hence, this paperrs λ=P(C2bC1) rather than λ=P(C2=0).

630 W.-R. Yang / International Review of Economics and Finance 20 (2011) 624–632

the investment at t=1 asq3 C3ð Þ

1−q3 C3ð ÞNq1 C1ð Þq2 C2ð Þ

1−q1 C1ð Þð Þ 1−q2 C2ð Þ2� �. The third investor is willing to acquire a more precise signal as he

expects to act differently than his predecessors did after receiving a poor signal. Therefore, the third investor is interested in payingthe private signal cost if his optimal choice based on a poor signal and the decisions of predecessors is to discontinue theinvestment.10

Denote D̂3 as the third investor's expected payoff from acquiring a private signal.

10 If thguesses

11 P(S

12 P(S

D̂3 = λ P Sg3 jSg1� �

EU3 X3 = 0;C3 jX1; Sg3

� �+ P Sb3 jSg1

� �EU3 X3 = 1;C3 jX1; S

b3

� �h i11;12

+ 1�λð Þ P Sg3 jSg1; Sg2� �

EU3 X3 = 0;C3 jX1;X2; Sg3

� �+ P Sb3 jSg1; Sg2

� �EU3 X3 = 1;C3 jX1;X2; S

b3

� �h i= 2 λP Sg3 jSg1

� �P

�R jSg1; Sg3� �

+ 1−λð ÞP Sg3 jSg1; Sg2� �

P�R jSg1; Sg2; Sg3� ��

+ λP Sb3 jSg1� �

+ 1−λð ÞP Sb3 jSg1; Sg2� �

−C3:

ð13Þ

Proposition 1 characterizes the behavior of the third investor with respect to the acquisition of a private signal and investment.

Proposition 1. Assume that the first and second investors maintain their investments at t=1. If the signal cost, C3, satisfies the

conditions,q3 C3ð Þ

1−q3 C3ð ÞNq1 C1ð Þq2 C2ð Þ

1−q1 C1ð Þð Þ 1−q2 C2ð Þð Þ and C3bC3⁎=[α(C1,C2,C3)−2q1(C1)]−λ[α(C1,C2,C3)−q1(C1)−q3(C3)], where α(C1,

C2,C3)=2q1 C1ð Þq2 C2ð Þq3 C3ð Þ + q3 C3ð Þ + q1 C1ð Þq2 C2ð Þ−q1 C1ð Þq3 C3ð Þ−q2 C2ð Þq3 C3ð Þ

1−q1 C1ð Þ−q2 C2ð Þ + 2q1 C1ð Þq2 C2ð Þ , the third investor acquires a private signal.

Then, the third investor withdraws the investment based on signal S3b and maintains the investment based on S3

g.

Proof. If the signal cost, C3, satisfiesq3 C3ð Þ

1−q3 C3ð ÞNq1 C1ð Þq2 C2ð Þ

1−q1 C1ð Þð Þ 1−q2 C2ð Þð Þ, the probability, P(�R|S1

g, S2g, S3b), is less than

12. This implies

that the private signal of the third investor is so precise such that the private signal dominates the investment decision of the thirdinvestor. Next, we calculate whether paying the signal cost, C3, is preferable to being a free rider of private signal acquisition. If C3 isless than the threshold value C3⁎, acquiring a private signal is beneficial because the expected payoff after acquiring a private signal,D̂3, is greater than the expected payoff without a private signal, D̃3. Q.E.D.

Proposition 1 provides the requirements for the precision regarding the third investor's private signal and the threshold valueof the signal cost, C3⁎. If one of the conditions in Proposition 1 does not hold true, the third investor does not acquire a private signal,but follows the actions of his predecessors in maintaining the investment. With regard to the condition of signal precision, the

critical level,q1 C1ð Þq2 C2ð Þ

1−q1 C1ð Þð Þ 1−q2 C2ð Þð Þ, increases with the precision of the first two investors' signals. Therefore, the probability that

signal cost, C3, satisfies the condition of signal precision diminishes as the signals of the first two predecessors increase in precision.In other words, when the third investor believes that the private signals of predecessors are highly valuable, he has less incentiveto acquire his own private signal.

3.4. Fourth investor

Suppose that the first two investors maintain their investments, but the third investor withdraws his investment at t=1. Byobserving the actions of his predecessors, the fourth investor infers that the third investor has acquired a more precise signal thanthe first and second predecessors did. Therefore, information aggregation stops at the third investor, and a herding ofdiscontinuing investment begins with the fourth investor.

However, if each of the first three investors choose to maintain their investments, the fourth investor anticipates that thecombination of his predecessors' private signals is (S1

g, S2g, S3g), (S1

g, ϕ2, S3g), (S1

g, S2g, ϕ3), or (S1

g, ϕ2, ϕ3). This is equivalent tosaying that the fourth investor is unable to distinguish which predecessor other than the first investor is a free rider. Thefourth investor's incentive to acquire a private signal is the same as that of the third investor. The fourth investor is willingto purchase a private signal if he acts differently from his predecessors after receiving a poor signal. The conditionconcerning the signal precision, which leads the fourth investor to a different belief in the possibility of obtaining a high

future return, isq4 C4ð Þ

1−q4 C4ð ÞNq1 C1ð Þq2 C2ð Þq3 C3ð Þ

1−q1 C1ð Þð Þ 1−q2 C2ð Þð Þ 1−q3 C3ð Þð Þ. Nevertheless, if the fourth investor's signal cost, C4,

contributing to a more precise private signal exceeds the threshold value, C4⁎, the fourth investor prefers to take a freeride on the signal acquisition of his predecessors.

e signal cost, C3, satisfies the conditionq3 C3ð Þ

1−q3 C3ð ÞNq1 C1ð Þq2 C2ð Þ

1−q1 C1ð Þð Þ 1−q2 C2ð Þð Þ, the third investor also has the incentive to acquire a private signal when hethat his predecessors' signals are S1

g and ϕ2.

3g|S1

g, S2g)=

P Sg1 ;Sg2 ;S

g3 j

�Rð ÞP �

Rð ÞP Sg1 ;S

g2ð Þ +

P Sg1 ;Sg2 ;S

g3 j�Rð ÞP �Rð Þ

P Sg1 ;Sg2ð Þ =1−q1 C1ð Þ−q2 C2ð Þ−q3 C3ð Þ + q1 C1ð Þq2 C2ð Þ + q1 C1ð Þq3 C3ð Þ + q2 C2ð Þq3 C3ð Þ

1−q1 C1ð Þ−q2 C2ð Þ + 2q1 C1ð Þq2 C2ð Þ :.

3b|S1

g, S2g)=

P Sg1 ;Sg2 ;S

b3 j

�Rð ÞP �

Rð ÞP Sg1 ;S

g2ð Þ +

P Sg1 ;Sg2 ;S

b3 j�Rð ÞP �Rð Þ

P Sg1 ;Sg2ð Þ =

q3 C3ð Þ + q1 C1ð Þq2 C2ð Þ−q1 C1ð Þq3 C3ð Þ−q2 C2ð Þq3 C3ð Þ1−q1 C1ð Þ−q2 C2ð Þ + 2q1 C1ð Þq2 C2ð Þ .

631W.-R. Yang / International Review of Economics and Finance 20 (2011) 624–632

Based on the characteristics of the fourth investor's behavior with respect to acquiring a costly private signal, Proposition 2illustrates investor i's actions regarding the investment and acquisition of private signal.

Proposition 2. Assume that the first and second investors maintain their investments at t=1. To investor i, i=4, 5, ….

(i) As Xi−1=Xi−2, if investor i's signal cost Ci satisfiesq Cið Þ

1−q Cið ÞNq1 C1ð Þq2 C2ð Þq3 C3ð Þ⋅⋅⋅qi Ci−1ð Þ

1−q1 C1ð Þð Þ 1−q2 C2ð Þð Þ 1−q3 C3ð Þð Þ⋅⋅⋅ 1−qi−1 Ci−1ð Þð Þ and CibCi*,

where Ci* is the threshold level of the signal cost at whichD̂i ND̃i, investor i will acquire a private signal, and the investment decision

will depend on the type of his signal .(ii) If Xi−1≠Xi−2, investor i and his followers do not acquire any private signal, and Xi−1=Xi=Xi+1=Xi+2=…. A herding

begins with investor i.

In Proposition 2(i), the critical level,q1 C1ð Þq2 C2ð Þq3 C3ð Þ⋅⋅⋅qi Ci−1ð Þ

1−q1 C1ð Þð Þ 1−q2 C2ð Þð Þ 1−q3 C3ð Þð Þ⋅⋅⋅ 1−qi−1 Ci−1ð Þð Þ, increases with investor i's position in

the sequence of making decisions. This implies that the probability of the signal cost Ci satisfying the condition of signal precisiondiminishes. Therefore, the investors in the later stages of the decisions sequence are less likely to acquire private signals. Theintuition behind this result is that when all predecessors act in the same manner, the possibility that successors will obtain moreprecise private signals decreases; hence, the investors intend to rely only on the decisions of predecessors.

Kraemer et al. (2006) found that, in their experiment with regard to information aggregation, half of the participantsoverestimated the value of their private signals, thereby contributing to the excessive acquisition of private information. In theirexperiment, the prediction and acquisition behavior of the predecessors was based on public information; however, the type andstrength of the predecessors' signals were unobservable. Thus, according to the conditions concerning the acquisition of privatesignals derived in themodel, this paper concludes that the anticipation errors on the part of participants can result in the excessiveacquisition of private signals observed in Kraemer et al.'s experiments. In this paper, because the signal costs of predecessors donot constitute public information, the successors may underestimate the precision of the predecessors' private signals. Accordingto Proposition 2(i), we can examine how investor i's anticipation concerning the precision of the previous signals influences the

critical level,q1 C1ð Þq2 C2ð Þq3 C3ð Þ⋅⋅⋅qi Ci−1ð Þ

1−q1 C1ð Þð Þ 1−q2 C2ð Þð Þ 1−q3 C3ð Þð Þ⋅⋅⋅ 1−qi−1 Ci−1ð Þð Þ. If investor i underestimates the precision of his predecessors'

private signals, the critical level of the signal precision declines. Then, investor i has a greater incentive to acquire his own privatesignal. Hence, when investors make errors in their expectation regarding the precision of the prior signals, they overestimate thevalues of their private signals and instance of unnecessary signal acquisition increases, despite the fact that their decisions arerational.

4. Conclusions

This paper demonstrates that the unobservable precision of the predecessors' private signals plays a critical role in theacquisition of costly private signals. First, by making this assumption, we find that the problem associated with signal extractionexists in the process of acquiring private signals. As a result, herding begins with the first few investors, and successors followincorrect previous decisions. Thus, this paper provides strong evidence to support the conjecture of Banerjee (1992) thatendogenous information acquisition exacerbates herding inefficiency. Second, under this assumption, one of the results in thismodel is consistent with the findings of Kraemer et al. (2006). This paper provides the underlying reason that the precision of priorsignals is undervalued to explain why the excessive acquisition of private signals arises in the experiment of Kraemer et al.

Further, in this paper, the actions of predecessors do not create payoff externality to followers, and the order in which investorsmake decisions is exogenous. Therefore, the present model is able to be extended in the following manner. First, the behavior ofdecision-makers regarding the acquisition of costly signals can be studied from the perspective of payoff externality. For example,when banks serve their depositors on a “first-come, first-served” basis, creating negative payoff externality for depositors who arelate to withdraw, the depositors may have less incentive to collect private signals and less information is accumulated in themarket. Second, future studies can include endogenous order in the sequence of acquiring private signals and making investmentdecisions. Because acquiring private signals and making investment decisions may not occur at the same time, it is interesting toexamine whether the number of free riders regarding the acquisition of private signals increases. Finally, the previous empiricalliterature related herd behavior with information asymmetries among market participants (Dvorak, 2005; Grinblatt & Keloharju,2000; Jeon &Moffett, 2010). The herding model in this paper can be extended to study the relationship between the acquisition ofcostly information and private information advantage. By modifying the present model, we are able to investigate whether theinformation asymmetry is more severe and the herd behavior of investors with information advantage is different from that ofother investors.

Acknowledgements

This research is supported by the Taiwan National Science Council (NSC 96-2415-H-390-005-MY2). The author would like tothank the seminar participants at the 2009 WEAI conference in Vancouver and the anonymous referee for the invaluablesuggestions. Any remaining errors are the responsibility of the author.

632 W.-R. Yang / International Review of Economics and Finance 20 (2011) 624–632

Appendix

Proof of Lemma 2. If q1(C1)Nq2(C2), the second investor's belief, P(�R|X1,S2b), is greater than

12; therefore, the second investor

maintains the investment. If q1(C1)bq2(C2), P(�R|X1,S2b)b

12; hence, the second investor discontinues the investment. Q.E.D.

Proof of Lemma 3. The expected payoff with private signal, D̂2, exceeds the expected payoff of being a free rider,D̃2, if and only ifΔq2 C2ð ÞΔC2

N1. Q.E.D.

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