Comparative Statics in an Informational Cascade Model of Investment

Transcription

1 preliminary and incomplete Comparative Statics in an Informational Cascade Model of Investment by Tuvana Pastine National University of Ireland, Maynooth and CEPR Jan, 2005 NUI, Maynooth Economics Department Maynooth, Co. Kildare Ireland fax: Abstract This paper is an adaptation of the Chamley-Gale endogenous-timing information-revelation model of investment (Econometrica, 1994). The paper models a game with pure informational externalities where agents can learn by observing others actions. Social learning about the value of the investment project can result in massive social imitation, possibly leading the society to the incorrect choice, to an inefficient cascade. While Chamley and Gale characterize the equilibrium of such a game, this paper yields an analytic approximation to the probability of inefficient cascades and allows for the derivation of comparative statics results. This is useful for two reasons: i)these results indicate that some of the findings from the exogenous-timing herding literature may not necessarily be generalizable to the endogenous-timing framework. ii)the study may be useful in the analysis of a wide variety of applied issues such as IPO pricing, speculative attacks and adoption of new technology.

2 1. Introduction Individuals with limited information typically observe other peoples s actions before making their own decisions. The predecessors actions often contain information about their private experiences, their private signals concerning the state of the nature. In the social learning process, the individual aggregates her own private signal with information collected from her observations of others actions. This process may lead to imitation or to herd behavior where the agent follows the crowd, even when her private information suggests the opposite. If all agents start herding there is an informational cascade 1. In case early movers signals happen to be incorrect then agents may settle on a common inefficient action, resulting in an inefficient cascade. This paper adapts the endogenous-timing information-revelation investment model of Chamley and Gale (1994) to study the factors that make inefficient cascades more likely. This is the first endogenous-timing herding model with homogenous access to information that allows for the derivation of comparative statics results with respect to all key variables; the signal quality, the observation period length, the prior, and the expected project value 2. Chamley (2004) and Devenow and Welch (1996) give an extensive list of empirical phenomena that herding behavior may explain 3. Examples come both from real markets such as R&D investment decisions and from financial markets; among others, analysts recommendation of a particular stock, bank runs and managers decisions to pay dividends may have elements of herding behavior. It is often argued that conformist behavior in financial and real markets may lead to sudden booms and crashes. This paper studies the factors that influence the likelihood of erroneous mass behavior, either when there is an investment boom even though the true value of the project is low (inefficient positive cascade), or when there is an investment collapse even though the true value is high (inefficient negative cascade). 1 In an informational cascade since all follow the crowd, social learning stops. The actions of the new comers do not allow to differentiate between there private signals. 2 In a framework where agents have preferential access to information, Zhang (1997) provides a endogenous timing framework where the first mover is the agent with the highest precision of information. A cascade starts immediately after the first mover, all agents follow the expert leader. 3 Also see the living document by Bikhchandani, Hirshleifer and Welch (1996) for an overview of the theoretical and empirical literature on herding. -1-

3 In seminal papers by Banerjee (1992) and Bikhchandani and Hirshleifer and Welch (1992) each person observes the behavior of the people who went before him where there is an exogenously determined sequence in the moves. These models show that society may settle in an inefficient outcome because valuable information gets trapped at some stage of social learning. Chamley and Gale (1994) prove the existence of herd behavior even when the timing of moves and information revelation is endogenous. In an endogenous-timing framework, the individual agent has an incentive to wait in order to observe the actions of other players. However if everyone were to wait, the agent would rather move early in order to avoid cost of delay. Hence the timing decision is strategic. While Chamley and Gale characterize the equilibrium this waiting game, this paper yields an analytic approximation to the probability of inefficient cascades and allows for the derivation of comparative statics results. This is useful for two reasons: First, it allows a deeper understanding of the relationship between exogenous and endogenous timing herding models. This paper shows that some of the results on exogenous-timing herding models do not necessarily generalize to models with endogenous timing. Secondly, the derivation of comparative static results in the Chamley-Gale model provides a framework that may be useful in the analysis of a wide variety of applied issues. Some of these will be discussed in the conclusion. 2. Chamley-Gale Endogenous-Timing Information-Revelation Game The higher the value of the project the more people there are with an option to invest. However the number of people with an option is unknown to the agent. Each agent with an investment option faces a tradeoff between investing and waiting. If the agent invests he collects the undiscounted payoff but faces the risk of making a loss in case the true value of the project is low. If the agent waits he collects only discounted payoffs but he can make use of information revealed by others actions. In equilibrium, the agent is just indifferent between investing and waiting. The rate of investment each period is stochastic because in equilibrium players randomize. The agent tries to deduce from the rate of investment each period the number of people with an option and hence the value of the project and updates his beliefs. If the agent s beliefs evolve in such a way that the expected value of the project turns negative, the agents -2-

4 strictly prefers to wait and so does every one else and the learning process ends with an investment collapse. If the learning process leads the agent to strictly prefer to invest rather than waiting even though the true value of the project would be revealed with certainty next period, then there is a sudden investment boom. Since the learning process is stochastic the outcome may be inefficient in the sense that even when the true value of the project is high (low), there might be an investment collapse (boom). 3. Framework Each of the identical risk neutral agents with an investment option can exercise the option at any date T=0,1,2,... of his choice. All options are identical and indivisible. The investment decision is irreversible. (0,1) is the common discount factor. Each player with an option chooses either to invest now or delay. If the player never invests the payoff is 0. Whether or not the player has an option is private information. Only if the option is exercised information is revealed. When making their decisions players can observe the history of other players investments. The true value of the investment is denoted by V {V H,V L } where V H >0 and V L <0. V=V H with prior probability q* (0,1). This paper adapts the r-fold replica game of Chamley and Gale 4.There are rn potentially knowledgeable players in the market. Only rn of them have an opportunity to undertake an investment project. The results will hold as r. The higher the true value of the project the more agents will have an investment option. (1)a and. is a measure for signal quality. The further apart n L and n H, the more information the agent has about the true value of the project from the mere fact that he has observed the investment option in the first place. 5 4 This corresponds to Section 6 in Chamley and Gale. 5 In order to compare this signal quality measure to the signal quality measure in Banerjee(1992) and Bikhchandani, Hirshleifer and Welch (1992), suppose that each of the rn potential investors receive a signal about the quality of the project. The signal is correct with -3-

5 So far this is a special case of Chamley and Gale. Chamley and Gale assume that the number of people with an option is stochastic but it is more likely to be high when the true value of the project is high. However, here the value of the project is either high or low and there is a one-to-one mapping between V and n. The restriction to only two possible project values will allow us to summarize agents beliefs about the true state of the nature at time T in a single variable: the probability that the project value is high. This mapping will prove to be very convenient in eventually formulating the learning process in a linear fashion Copy-Cat Traders In this paper, we will also include copy-cat traders. There are = rn H people who are randomly assigned to this market and they simply imitate the probability of investment of the active informed traders in the market. It will be shown that the people add noise to the information to be collected from the market. It is not interesting that these copycats follow the active informed - which is assumed. The copycats are not needed for the model to work either as they are not modeled in Chamley and Gale. However they make it harder for the market participants to deduce the true value of the project from the observed purchases-resulting in higher expected purchases per period. This coupled with the following trading technology will allow us to approximate the learning process with a continuous function Trading Technology In Chamley and Gale both orders and processing of orders happen in discrete time. Whereas in our framework agents place discrete-time state-contingent orders which get processed in continuous time. Players place their orders at the beginning of each period. Orders are processed randomly during the period the exact time that an individual order is processed is distributed uniformly in the period. Since information on others actions will be arriving during the period, players are permitted to make their orders (both invest and wait orders) contingent on the flow of information. Payoffs on all orders processed in a period are received at the end of the period. The benefit of moving to continuous-time order processing is that it will allow us to approximate probability p>1/2. If V=V H then prn=rn H get a positive signal. If V=V L then (1-p)rN=rn L get the positive signal. Then. -4-

6 a transformation of the agent s problem as a Wiener process with absorbing boundaries and derive the probability of inefficient cascades. Each invest order comes with a state-contingent wait order. The investment cannot be reversed in case the invest order is already processed. During the interval [T,T+1), if the statecontingent wait order is triggered, then at most M of the newly triggered wait orders are processed, where M is a large but finite number. The number of newly triggered wait orders W may exceed M. In that case, a randomly selected W-M of these newly triggered wait orders are ignored. These are simply continued to be processed as invest orders. One can interpret M as the maximum capacity of the processing agency to accommodate state-contingent orders. Each wait order comes with a state-contingent invest order. During the interval [T, T+1), if the state of the state-contingent invest order is triggered, then at most M of the newly triggered invest orders are processes, and the payoffs are received at time T+1. If the number of the newly triggered invest orders Z is greater than M, then the remaining Z-M are not processed during the period. It will be shown that this form of contingent order will ensure that in equilibrium the expected payoffs from putting an invest or wait order will be the same as the expected payoffs from putting in an invest or wait order in the Chamley and Gale framework. The benefit of this framework is that it will allow us to get analytical approximations for the inefficient cascade probabilities and to study the comparative statics results. 4. Equilibrium We will start out by conjecturing that the equilibrium of this new game mirrors the equilibrium in the game of Chamley and Gale. Then it will be shown that in this conjectured equilibrium the players expected payoffs from their equilibrium strategies are the same as those resulting to players in Chamley and Gale s game and nobody benefits from possible deviations. The focus is only on the symmetric Perfect Bayesian Equilibria. Before describing the equilibrium strategies, let us first introduce some critical values Critical Values The prior probability that V=V H is q*. Denote q t as the subjective probability at time t that the true value is high. Since orders are processed in continuous time, q t evolves in continuous time. The index of time for discrete decision time nodes will be denoted by T. While t +, the index -5-

7 . So, at discrete time nodes when t=t, q t =q T. Bayes s rule is assumed to describe the agents method of updating the probabilities. At the beginning of the game, the probability that the project has a high value q T=0, conditional on having received an investment opportunity, is given by: (2)b Since, (2) can be rewritten as, (3)c The game is of interest if initially the expected value of the project is positive: (4zs) Otherwise each agent would strictly prefer to wait and the game would end immediately with an investment collapse. It will be useful to introduce two critical values for the subjective probability. Define q_ as the probability where the expected value of the project is zero: q_ V H +(1- q_ ) V L =0 When q T < q_, the expected value of investment is negative. So the agent will strictly prefer to wait. Since everyone who has not yet invested is identical they all prefer to wait and the game effectively ends. Investment stops for good. Define q _ as the probability where the agent is just indifferent between investing now and waiting even though information about the true value of the project is to be fully revealed with certainty next period: _ q VH +(1- _ q ) V L = _ q V H When q T >q _ the agent will strictly prefer to invest now. And so will all identical players, and the game ends with an investment boom where all players with an option invest. The game will be said to be active when q_< q t <q _. (5)d (6)e -6-

8 4.2. Learning The player s actions depend on the publicly observed history of the game which is described by the sequence of the number of people who invested during each period. Following the notation in Chamley and Gale, for any history h, let (h) denote the probability that a player who has not yet invested does so after observing the history h. In the active phase of the game, it must be that 0< (h)<1. Assume for a moment that an agent expects all people with an investment opportunity to invest this period. Then he would strictly prefer to wait to be able to learn the value of the project for sure. But so would everyone else. Hence (h) 1. If he expects nobody else to invest this period, there would be no learning this period, so as long as expected value from investment is positive he would strictly prefer to invest now. 6 But so would everyone else. Hence (h) 0, by contradiction. In equilibrium, 0< (h)<1, such that players are just indifferent between waiting and investing now. Notice that is the endogenous information revelation parameter. If were zero, no information would be revealed. If were equal to one, the number of people who invest would fully reveal information about the value of the project. As r, the number of people putting in invest orders at a decision node is given by the Poisson approximation to the binomial distribution. The parameter of the Poisson distribution is the mean number of invest orders by the investors rn plus the expected number of invest orders by the noise traders rn H. When V=V H, the probability that k players invest at a decision node given is denoted by f H (k, ): (7)f and p.d.f. is denoted by f L (k, ) when V=V L : (8zt) 6 While intuitive, this argument assumes that is not optimal to wait for information that may arrive several periods later. This is in fact the case. The optimal program will have a onestep property where at any period the agent is willing to make a once and for all invest-not invest decision. See Chamley and Gale proposition 3 for the proof. -7-

9 If n L were equal to n H, then =1 and the two probability density functions would collapse together. In such an extreme case the quality of the signal k would be nil and the signal would not reveal any information. The agent tries to deduce from the number of people who invest in each period which distribution the observation comes from. Define k T as the number of invest orders put in at the decision node T-1. Assuming no contingencies are triggered, k T is total number of people who invest in the time interval [T-1,T) and it is common knowledge at decision node T. The history up to time T is h T. The that makes the agents indifferent between investing and waiting at the decision node T is T. Bayesian learning suggests that at time T, when the agent observes k T people investing, the subjective probability will evolve following: (9)g Notice that (1+ )rn is finite. If it were infinite the observation of the rate of investment in one period would already reveal the true value of n, and hence V by the law of large numbers. In such a case all players would strictly prefer to wait, implying =0, which is a contradiction. Chamley and Gale prove that in equilibrium is independent of both r and the total number of people who have already invested 7. The basic intuition is that the individuals learning is equivalent to learning from sequence of samples. Since rn is finite, the rate of investment is very small compared to the size of the economy. Therefore one can think of the sampling simply as sampling with replacement. The equilibrium at the decision node T, will solely depend on history captured by q T-1 and k T Equilibrium Strategies Let us first assume that the institutional setup restricts the agents to only use q_ and _ q as their triggers for the contingency orders. In Appendix C, this assumption is relaxed. The equilibrium of the game with any finite set of possible contingency trigger points with cardinality greater than one and which contains both and is shown to yield the same boundary crossing probabilities as the baseline model. 7 See Proposition 8 and the proof in Chamley and Gale. -8-

10 PROPOSITION 1: Let T be described by Equation (?), the following equilibrium strategy supports a symmetric Perfect Bayesian Equilibrium: a) If subjective probability is sufficiently low q T q_, put in a wait order with a state-contingent _ invest order. If in the time interval [T,T+1), q t q, the state-contingent invest order is triggered. b) If subjective probability is sufficiently high, q T q _, put in an invest order with a statecontingent wait order. If in the time interval [T,T+1), q t q_, the state-contingent wait order is triggered. c)if subjective probability is q_<q T <q _, with probability T, put in an invest order with a statecontingent wait order. If in the time interval [T,T+1), q t q_, the state-contingent wait order is triggered. With probability (1- T ) put in a wait order with a state-contingent invest order. If in the time interval [T,T+1), q t q _ the state-contingent invest order is triggered. PROOF: a) By equation (5), when the subjective probability is q_, the expected value of the project is just equal to zero. Hence the agent strictly prefers to wait when q T <q_. If in the time interval [T,T+1), q t q _ the expected value of the project would be so high that the agent would prefer to invest. b) By equation (6), when the subjective probability is _ q, the expected value of investing now is just equal to waiting one more period assuming that information about the true value of the project were to be reveal for sure next period. Hence, when q t q _, the agent prefers to invest right away. If in the time interval [T,T+1) new information were to arrive such that q t q_ the agent would prefer to wait. c) If subjective probability is q_<q T <q _, the expected value of investing is positive but the agent will also consider waiting in order to learn about the true value of the project. In equilibrium the agents is just indifferent between investing now and waiting. See the beginning of section 3.2 for the discussion of the non-existence of pure-strategy equilibrium. i) The agent with an investment option who has not yet exercised his option will put an invest order at time T with probability T. If however in the time interval [T,T+1), q t falls below q_, the agent would prefer to wait. ii) The agent with an investment option who has not yet exercised her option will put a wait order at time T with probability (1- T ). If however in the time interval [T,T+1), q t rises above _ q the agent would prefer to invest. -9-

11 The equilibrium strategies of this game yield the same payoffs as in Chamley and Gale even though in Chamley and Gale orders and processing of orders happen in discrete time and hence orders are not state-contingent. The possible deviations from the equilibrium strategies do not improve the payoffs. When q T q_, while the wait order would come with a state-contingent invest order, the contingency order will never be triggered in equilibrium. Once q T q_ all identical agents with an investment opportunity will prefer to wait. This becomes an absorbing state and the investment ends for good. No new information can be received in the time interval [T,T+1) to increase q t above _ q. Hence in this case the payoffs would be the same in this game and in Chamley and Gale. When q T q_, notice that this would be an absorbing state as well. All agents with an investment option would prefer to invest. Since r, the rate of information flow would be a continuous variable and the true value of rn and hence V, would be revealed at once. If V=V H, agents subjective probability would remain above _ q. If V=V L, the subjective probability would immediately drop down below q_. All agents state contingent orders would be triggered at once but only M of them would be able to stop the investment. The game would end with all investing except for those lucky M people. However M is small compared to rn. Hence the probability of being one of those M people being able to stop investment would be zero as r. Therefore also in this case the payoffs would be the same in this game and in Chamley and Gale. In the active phase of the game, when q_<q T <q _, with probability T, the agent puts in an invest order with a state-contingent wait order. If in the time interval [T,T+1), q t q_, the statecontingent wait order is triggered. Once the contingency is triggered all unprocessed invest orders would convert into wait orders. Since M is a large number, investment would stop for good.???? When q_<q T <q _, with probability (1- T ) the agent puts in a wait order with a statecontingent invest order. If in the time interval [T,T+1), q t q _ the state-contingent invest order is triggered. If q t rises above _ q, then in fact all agents would now prefer to invest all at once. M is very large but finite, whereas r. Hence M newly arrived invest orders would be processed this period. All the rest would be processed next period. At time T, the agent realizes that the is an infinitely small probability that his invest order would be processed if the state is triggered. Hence equation (H) continues to define T. 5. Information Cascades -10-

12 The subjective probability evolves as a result of observational learning from the rate of investment each period, which is a stochastic variable. Chamley and Gale prove that eventually the game will end with an information cascade 8. If the subjective probability hits q_ before _ q, the game ends with an investment collapse. If the subjective probability hits _ q before q_, the game ends with an investment boom. We are particularly interested in the probability of inefficient cascades. The measures of interest are then the probability that the process hits q_ before _ q when V= V H, and the probability that the process hits _ q before q_ when V= V L. The first is an inefficient negative cascade and the latter is an inefficient positive cascade Transformation In order to obtain the boundary crossing probabilities, we will need to transform the problem into an equivalent problem that is tractable. Subjective probabilities evolve following (9), substitute f H (k T ; T-1 ) and f L (k T ; T-1 ) into (9). Cancel out k T factorial from the numerator and denominator. Take the inverse of both the left and right hand side of the equation and subtract one from each side. Now plugging in for yields, (10)k Taking the natural logarithm of both sides and multiplying both sides by minus one yields: (11)m where k T is distributed Poisson with the parameter T-1 (1+ )rn H when the true value of the project is high and it is distributed Poisson with the parameter T-1 ( + )rn L when the true value of the project is low. For large rn, Poisson distribution can be approximated by the normal 8 Proposition 8 in Chamley and Gale. 9 While the paper will discuss both inefficient outcomes, notice that only inefficient negative cascade would be categorized as inefficient herding. Here agents that receive a signal would invest if learning were not permitted (see Eq 4). Since herding by definition is acting against one s own signal, only when the crowd chooses not to invest can we talk about inefficient herding. -11-

13 distribution. Notice that k T 0. However the normal distribution assigns positive probability to events with k T <0. Hence this approximation is less than perfect for small rn. Define w T as: (12)p Notice that w T is an increasing monotonic transformation of q T. Plugging (12) into (11), we get a transformed problem: (13)zh where k T * is distributed normal with mean and variance 2 : (14zu) (15zy) H >0 by Lemma A1. And L <0 by Lemma A3 in Appendix A. Notice the effect of the noise traders: keeping the rate of information flow constant, an increase in, leads to a weaker drift. That is, the positive drift H declines and the negative drift L goes up. The higher the less informative an observation is about the true value of the project. Both ( 2 ) H and ( 2 ) L go down with an increase in keeping constant. Each observation has less informational content hence the beliefs don t get updated a lot. Therefore keeping constant, an increase in would make he agents strictly prefer to wait. Since in equilibrium agents are just indifferent between investing -12-

14 and waiting, a higher must be associated with a higher. This will be of significance when we discuss an approximation for the closed form solutions of inefficient cascades. The transformation (12) of the lower bound given by (5), of the upper bound given by (6) and of the starting point given by (3) yield : The lower bound: q_ w_ w_ = (16)q The upper bound: _ q w _ w _ = (17)r The starting point: q 0 w 0 w 0 = (18)s Since initially the expected value of the project is positive (4), w_ <w 0. And w 0 <w _ examining (6) and (3) together. Individual learning is a stochastic process with independent increments. This process is a well known description of individual learning in cognitive psychology. In much of that literature individuals modeled as learning through random sampling. This characterization of the learning process is then used to explain laboratory evidence on individual response times and error rates. The present paper shows that even with fully rational agents group behavior will resemble individual behavior with boundedly rational agents of the type used in cognitive psychology Boundary Crossing Probabilities with constant The individual learning process follows the equation (13) where the error term is distributed approximately normal with mean and variance 2. Both the mean and the variance of the process depend T-1 and hence they depend on the history of the game. They are not constant. Now we are going to examine a different process. In this modified problem, we will examine the processes described by equations (13) and (14) and (15) yet with a constant (0,1), implying a constant drift and variance. In section 5.3, we will prove that the process with the endogenously determined T will yield identical boundary crossing probabilities as in the modified problem with fixed. 10 See Luce (1986) for an introduction to this literature. One curious evidence from laboratory experiments shows that subjects tend to behave as if boundaries drift towards each other. In our framework this would be the case if the value of the project decays during the decision process. -13-

15 Note that orders are processed in continuous time and the processing time of each order is distributed uniformly over the period [T,T+1). So we can define w t as a continuous variable which coincides with w T when t=t. Denote as the stochastic term which is distributed normal where and 2 (given by Equations (14) or (15) depending on the true value of the project ) are respectively the drift velocity and the power of the noise of the process. Assuming T-1 = T, w t can be approximated by a Wiener process 11 : (19)zz Equations (16), (17) and(18) give the starting point and the bounds. We can easily compute the boundary crossing probabilities. i) Probability of hitting w_ before w _ when V=V H : In this case, the drift is positive, H >0. The probability of hitting w_ before w _ is given by 12 : (20)t (14)defines H. and ( 2 ) H. Plug in Equations (16), (17) and(18) for w_, w _, w 0 and multiply the numerator and the denominator of (20) by : (21)u where by Lemma A1 in Appendix A. Notice that H and hence this probability is independent of. While an increase in leads to an increase in, it also leads to an increase in 2. And these two effects counterbalance each other in the determination of the boundary crossing probabilities. ii) Probability of hitting w_ before w _ when V=V L : In this case the drift is negative, L <0. 11 FOOTNOTE 12 See Karlin and Taylor (1975). -14-

16 (22)v Plug in (15), (16), (17), (18), then: (23)w where independent of. by Lemma A3 in Appendix A. Once again L and hence this probability is 5.3. Inefficient Cascade Probabilities for the Original Problem PROPOSITION 2: The boundary crossing probabilities of the original problem are equal to the boundary crossing probabilities found using a Wiener process, (21) and (23) of the modified problem. Proof: In the actual learning process the parameter is updated at each The boundary crossing probabilities for this process can be reconstructed iteratively using Lemma C1 in Appendix C. Starting with the Wiener process with absorbing boundaries defined in (16), (17) and (18), create a process where the parameter changes to (which is stochastic) at t=1 and stays constant thereafter. From the lemma this new process has the same transition probabilities as the original process. Since after t=1 the process is a Wiener process we can do this again after one more period and the new process will also have the same transition probabilities. Iterating this argument yields the result. Let us denote probability of an inefficient negative cascade by Prob(INC) which is equal to the probability that the process hits q_ before _ q when V= V H. By Propostion 2, Prob(INC) is given by equation (21). Prob(IPC) denotes the probability of an inefficient positive cascade and it is the probability that the process hits _ q before q_ when V= V L. By Proposition 2, Prob(INC) is given by equation (23). -15-

17 6. Comparative Statics We will start with the comparative statics that behave same as in exogenous-timing models and then move on to the new comparative statics results we learn from this endogenous-timing framework The Prior Expected Value: In an exogenous-timing herding framework Welch (1992) shows that as the prior expected value from investment goes up, early movers are more likely to invest. Hence there is a higher change that the society ends up in a positive cascade. In our endogenous-timing framework, we get the same direction of results. The reason is discussed below. PROPOSITION 3: As the prior expected value increases (q* or V H or V L ), it becomes more likely that all agents with an option undertake the project. It becomes less likely that there is an investment collapse when the true value is high, Prob(INC). It becomes more likely that there is an investment boom even when the true value is low, Prob(IPC). Proof: See Appendix B. The prior probability q* gives the initial beliefs of the agent before learning starts. It can take the interpretation of reputation. The comparative statics results indicate that the better the initial reputation of the investment project, the higher chances it will have to be undertaken by masses even when the true value of the project is low. As the prior expected value of the project goes up either due to an increase in V H or V L, the upper bound _ q and the lower bound q_ both decrease see equations (6) and (5), while the starting point q 0 is unchanged. Therefore the probability of hitting the upper bound before hitting the lower bound increases. When there is potentially a lot to gain and little to loose, much would be lost due to discounting while waiting, so agents would be more prone to investing before they are certain it is a good project (q _ ). And their beliefs about the odds of the project being a high value project does not need to be as high for the agents to strictly prefer to wait. Hence the -16-

18 likelihood of an inefficient positive cascade goes up and the likelihood of an inefficient negative cascade goes down Discounting: Discounting doesn t play a role in exogenous timing models. Examination of this issue requires an endogenous timing model. This is the first endogenous-timing paper with comparative statics results on discounting. The agent makes a choice between investing now or later. If the agent waits, he can learn by observing other people s actions, however the payoff gets discounted. All else constant, as people get more patient, goes up, they will be more willing to wait. Since waiting induces learning, one might be tempted to conclude that higher would be associated with a smaller probability of an inefficient negative cascade. However this is not the case. PROPOSITION 4: As agents become more patient ( ), when the true value is high, it become more likely that there is an investment collapse, Prob(INC). When the true value is low, it becomes less likely that there is an investment boom, Prob(IPC). Proof: The probability of an inefficient negative cascade goes up as goes up, (24)ab On the other hand, the probability of an inefficient positive cascade goes down as goes up. (25)ac An increase in has a direct and an indirect effect. The indirect effect is through the rate of information flow. As agents get more patient, at the ongoing information flow, they would strictly prefer to wait. However as argued earlier, =0 cannot be sustained in equilibrium. So the rate of information flow goes down such that people are just indifferent between waiting and investing. When the true value of the project is high V=V H, a weaker information flow implies -17-

19 a weaker drift velocity H which simply increases the likelihood of a negative cascade. However at the same time the weaker information flow decreases the noise ( 2 ) H in the learning process. Each observation will have a small influence on the updating process. This reduces the likelihood of a negative cascade. And these two opposing effects exactly counterbalance each other. cancels out from the probability of inefficient cascade (see equations (21) and (23)). The indirect effect through the information flow is therefore nullified. The spirit of the story is the same for the case when V=V L. The direct effect of an increase in is through the upper bound. A higher yields a higher upper bound _ q, leaving the starting point and the lower bound unchanged. Since investors are more patient, they are willing to wait until they are almost certain about the project before they buy. This makes the inefficient negative cascade more likely, and inefficient positive cascade less likely Quality of Information: To understand the role of signal quality, first notice that agents who receive the investment option would all undertake the investment if there were no social learning (equation x). It is through social learning that the possibility of an investment collapse arises. The increase in the signal quality affects the outcome of the game through two channels; i) it increases the confidence of the agent in his own signal. Keeping the level of signal quality of the rest of the people constant, as the signal quality of the agent goes up, the agent is more likely to undertake the project, hence the Prob(INC) and Prob(IPC). ii) it increases the confidence of the agent in the observational learning since others have high quality signals. Now keeping the signal quality of the agent constant, as the signal quality of the rest of the players increases the agent becomes more likely not to undertake the project, hence Prob(INC) and Prob(IPC). These two channels with opposing forces can be examined below. Let us first examine the probability of an inefficient negative cascade. Equation (20) can be rewritten as: (26)ad Notice that w_ and w _ are independent of. So, -18-

20 (27)ae The first term is positive. It relates to the first channel. (28)af since while and. But the second term is negative. It relates to the second channel. since by Lemma A2 and by Lemma B1 in the Appendix A and B. Therefore the are two forces working in opposite directions. as, Let us now examine the probability of inefficient positive cascade. Rewrite Equation (22) (29)ag So, (30)ah The first term is negative. It relates to the first channel. <0 (31)ai -19-

21 since and since L <0 and. But the second term is positive. It relates to the second channel. since by Lemma A4 and by Lemma B2 in the Appendix A and B. Hence there are two opposing effects. PROPOSITION 5: As the signal quality improves might go up or down depending on the parameter values., the likelihood of an inefficient cascade Proof: Proposition 5 will be proven in two steps. Step 1: >0, when w 0 w_. when w 0 is not to close to either of the bounds, w_<<w 0 << w _. Proof: See Appendix B Step 2: <0, when w 0 w _. when w 0 is not to close to either of the bounds, w_<<w 0 << w _. Proof: See Appendix B. Depending on the initial beliefs of the agent the own signal quality effect can dominate the signal quality of the society effect. Initially before learning starts if the agent only weakly -20-

22 believes that the project has a high value (a low q*), then the overall effect of an increase in the signal quality ( ) is to decrease the probability of not undertaking the project when the true value is high (Prob(INC) ). Initially before learning starts if the agent strongly believes that the project is a high value project (a high q*), then an increase in the signal quality ( ) leads to an increase in the probability of undertaking the project when the true value is low (Prob(IPC) ). If however the agent does not have a strong prior such that w 0 is not too close to either of the bounds the social learning channel becomes more important in determining the direction of the effect of signal quality on the outcome of the game. Hence depending on the initial beliefs, an increase in the signal quality might increase or decrease the probability of an inefficient cascade. 7. Discussion The comparative statics results from this endogenous timing herding model may be able to shed some light on a variety of questions from different fields of economics. The parameters of the model, the discount factor, the prior beliefs, the signal quality and the expected value of the project can take different interpretations depending on the market under consideration Initial Public Offerings The IPO market is a fixed-price common-value good market where later potential investors can observe the investment decisions of early investors. One of the puzzles in this market is the strongly documented underpricing. 13 And casual observation of the IPO market shows that offerings occasionally fail because there is too low of a demand. Both these features are consistent with our herding model of investment. The IPO price may be set low in order to decrease the probability of a negative cascade by increasing the expected value to potential investors. However even then the probability of a negative cascade where the offering fails remains positive Financial versus Real Markets 13 See Beatty and Ritter (1986). 14 Welch (1992) examines the price setting by an informed seller of an IPO where buyers cascade. When there is inside information, Welch (1992) can explain why an optimally priced IPO might fail. -21-

23 This paper suggests that financial markets might be more prone to inefficient collapses than the real markets. While simply represents the discount factor, it may also be regarded as capturing the time required to process investment decisions. Keeping the rate of time preference constant, as the time to process investment decisions increases so does the distance between the time periods in the model, leading to a lower. In financial markets, the administrative and technological systems may be faster to react to agents investment decisions than in real markets. Hence in financial markets the relevant would be larger than in real markets. The paper suggests that as goes up the likelihood of an inefficient collapse would be higher. In a more fully developed model for the purposes, one could analyze the effect of liquidity on the probability of inefficient collapses. The more liquid market might imply a higher since expected time to trade would be shorter. Hence a financial market that is open to the world markets and hence with higher liquidity might be more prone to inefficient collapse. This possibility is often suggested in the discussion of hot money and exchange rate/debt crises and the model presented here may be adaptable to give some meat to that discussion Speculative Attacks The model may help to gain further understanding of the importance of the reputation of a government pursuing a fixed exchange rate regime. Each agent has a one unit of domestic currency for possible investment in the foreign exchange market. The higher the potential speculative gains the more players are aware of the opportunity. Investment in foreign currency will have a low expected payoff if in reality the fundamentals of the economy are good. It will be high if in reality the fundamentals of the economy are bad. Agents can observe the amount of speculative purchases from the monetary authority each period. The model would suggest that it is possible that a speculative attack is staged even when economic fundamentals are good. The possibility of such an inefficient cascade would decline however with the good reputation of the government. Give REFERENCE 7.4. Advertising, Warrantees and Buy-Back Options This paper suggests that firms producing goods of identical qualities will face different chances of success depending on their reputation q*. Hence in markets with social learning and herding behavior, firms would have an additional incentive to invest in reputation building activities such as advertising. Another key variable in the analysis is the expected value of the project. It is well -22-

24 known that warranty and buy back options signal higher product quality. Hence these options increase the expected value from investing in the product both directly and indirectly through signaling. By offering warranty and buy back options, firms can increase the chances of positive cascades where purchases of the product booms. All else equal, firms that do not offer these options will have a relatively high probability of facing a collapse of purchases. This model suggests that in markets where there is social learning these marketing tools will have even a bigger significance. 8. Conclusion This paper allows us to analyze factors that make inefficient cascades more likely. The results derived from the exogenous-timing herding models are shown not to necessarily carry over to a setting with endogenous timing. In Banerjee (1992) and Bikhchandani and Hirshleifer and Welch (1992), the agent has to make a choice between two alternatives. The agent does not have the chance to postpone his decision. Using the terminology of this paper, these exogenous-timing models intrinsically have one critical bound. The alternative with the higher expected payoff is picked. In this endogenous-timing paper, there are two alternatives as well: invest and don t invest. But the agent has a chance to postpone his decision in order to learn more. Hence there are two critical bounds. It is the existence of the upper bound in the endogenous-timing model that leads to richer results. The agent might strictly prefer one alternative over the other if forced to make a decision, yet the agent might still choose to wait in order to learn more about the true value of the project. When the sequence of the moves is predetermined, an improvement in the signal quality decreases the probability of an inefficient cascade. Both higher own signal quality and higher signal quality of the society leads to a decline in the likelihood of an inefficient outcome, since early movers will be more likely to have taken the correct action. However when agents can postpone their decision, there is a third force that is not present in the previous discussion. As the signal quality of the society improves, the agent will be more willing to wait because he knows information to arrive is of good quality, making inefficient negative cascade more likely. Depending on the initial values, either of the effect can overwhelm the other. The effect of the discount factor cannot be analyzed in an exogenous-timing model, since the agent is not allowed to postpone his decision. While waiting the agent learns from the society, -23-

25 so one might be tempted to conclude that the more patient agents get the smaller would be the probability of an inefficient negative cascade. The paper shows the opposite to be the case. An increase in patience leads to a higher upper bound. Agents are simply more willing to wait until they are almost certain about the project before they buy. This makes inefficient negative cascades more likely. REFERENCES Banerjee (1992), A Simple Model of Herd Behavior, Quarterly Journal of Economics, 107:3, Beatty and Ritter (1986): Investment Banking, Reputation and the Underpricing of Initial Public Offerings, Journal of Financial Economics 15, Bikhchandani, Hirshleifer and Welch (1992): A Theory of Fads, Fashion, Custom, and Cultural Change as Informational Cascades, Journal of Political Economy 100:5, Bikhchandani, Hirshleifer and Welch (1996): Informational Cascades and Rational Herding: An Annotated Bibliography, Working Paper:UCLA/Anderson and Ohio State University and Yale/SOM (Note: The World-Wide Web: Chamley and Gale (1994): Information Revelation and Strategic Delay in a Model of Investment, Econometrica, 62, No.5, Devenow and Welch (1996): Rational Herding in Financial Economics, European Economic Review, 40, Karin and Taylor (1975): A First Course in Stochastic Processes, New York: Academic Press. Luce (1986): Response Times: Their Role in Inferring Elementary Mental Organization, New York: Oxford University Press. Nelson (2002): Persistence and Reversal in Herd Behavior: Theory and Application to the Decision to Go Public, The Review of Financial Studies, 15:1,

26 Welch (1992): Sequential Sales, Learning and Cascades, The Journal of Finance, 47:2, Zhang (1997): Strategic Delay and the Onset of Investment Cascades, RAND Journal of Economics, 28, No.1, APPENDIX Appendix A: Lemma A1: Proof: Define. Note that f(1)=0. Since f( )>0 for 0< <1. So,. Lemma A2: Proof: Define. Note that f(1)=0 and. evaluated at =1,.. Hence for 0< <1 and so f( )<0, so. -25-

27 Lemma A3: Proof: Define. Note that f(1)=0. Since f( ) <0 for 0< <1. So. Lemma A4: Proof: have the same numerator. In Claim A2 it is shown that the numerator is negative. Since the denominator is negative as well,. Appendix B: Proof of Proposition 3: Since H >0 and, and -26-

28 since L <0 and. In line with exogenous-timing herding literature, and and and Lemma B1: Proof: The term inside the brackets is positive. Examine the properties of the following function: Let where x>w_. Then notice that the term inside the brackets is equal to. If, then and. Let where z<0 for x {w _, w 0 }. The term in the parenthesis is then equal f(z=0)=0. And for all z<0. Hence. Therefore. -27-

29 Lemma B2: Proof: The term inside the brackets is positive. Examine the properties of the following function: Let where x>w_ then notice that the term inside the brackets is equal to. If, then since w _ >w 0. Let where z>0 for x {w _, w 0 }. The term in the parenthesis is then equal f(z=0)=0. And for all z>0. Hence. Therefore Proof of Proposition 5: Step 1: >0, when w 0 w_. when w 0 is not to close to either of the bounds, w_<<w 0 << w _. Proof: w 0 is a function of q*. All other terms in Eq 1 are invariant of q*. w 0 w_ is equivalent to When w 0 w_ the second term (Eq 3) of Eq 1, goes to zero. The first term (Eq 2) remains positive. Hence when w 0 w_. -28-