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Bubbles and CrashesAuthor(s): Dilip Abreu and Markus K. BrunnermeierSource: Econometrica, Vol. 71, No. 1 (Jan., 2003), pp. 173-204Published by: The Econometric SocietyStable URL: http://www.jstor.org/stable/3082044.

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Econometnica,Vol. 71, No. 1 (January, 2003), 173-204

BUBBLES AND CRASHESBY DILIP ABREU AND MARKUS K. BRUNNERMEIER1

We present a model in which an asset bubble can persist despite the presence of ratio-nal arbitrageurs. The resilience of the bubble stems from the inability of arbitrageurs totemporarily coordinate their selling strategies. This synchronizationproblem together withthe individual incentive to time the marketresults in the persistence of bubbles over a sub-stantial period. Since the derived trading equilibrium is unique, our model rationalizes theexistence of bubbles in a strong sense. The model also provides a natural setting in whichnews events, by enabling synchronization, can have a disproportionate impact relative totheir intrinsic informational content.KEYWORDS: Bubbles, crashes, temporal coordination, synchronization, market timing,'overreaction', limits to arbitrage, behavioral finance.

1. INTRODUCTIONNOTORIOUS EXAMPLES OF BUBBLES followed by crashes include the Dutchtulip mania of the 1630's, the South Sea bubble of 1719-1720 and more recentlythe Internet bubble which peaked in early 2000. Standard neoclassical theory pre-cludes the existence of bubbles, by backwards induction arguments in finite hori-zon models and a transversalitycondition in infinite horizon models (cf. Santosand Woodford (1997)). Similar conclusions emerge from 'no trade theorems'in settings with asymmetric information (cf. Milgrom and Stokey (1982), Tirole(1982)).2 All agents in these models are assumed to be rational. The efficientmarketshypothesisalso implies the absence of bubbles. Many proponents of thisview (see, for instance, Fama (1965)) are quite willing to admit that behav-ioral/boundedly rational traders are active in the market place. However, theyargue that the existence of sufficiently many well-informed arbitrageurs guaran-tees that any potential mispricing induced by behavioral traders will be corrected.We develop a model that challenges the efficient markets perspective. In par-ticular, we argue that bubbles can survive despite the presence of rational arbi-trageurswho are collectively both well-informed and well-financed. The backdropof our analysis is a world in which there are some 'behavioral' agents variouslysubject to animal spirits, fads and fashions, overconfidence and related psycho-logical biases that might lead to momentum trading, trend chasing, and the like.

1 We thank seminar participants at various universities and conferences, anonymous referees, andespecially Haluk Ergin, Ming Huang, Sebastian Ludmer, Stephen Morris, Jose Scheinkman, AndreiShleifer, and an editor for helpful comments. The authors acknowledge financial support from theNational Science Foundation and Princeton's CEPS program.2 A more extensive review of the literature on bubbles can be found in Brunnermeier (2001).

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174 D. ABREU AND M. BRUNNERMEIERThere is by now a large literature that documents and models such behavior.3We do not investigate why behavioral biases needing rational corrections arise inthe first place; for this we rely on the body of work very partially documented infootnote 3. We study the impact of rational arbitragein this setting.We suppose that rational arbitrageurs understand that the market will eventu-ally collapse but meanwhile would like to ride the bubble as it continues to growand generate high returns. Ideally, they would like to exit the market just prior tothe crash, to beat the gun in Keynes' colorful phrase. However, market timingis a difficult task. Our arbitrageurs realize that they will, for a variety of reasons,come up with different solutions to this optimal timing problem. This dispersionof exit strategies and the consequent lack of synchronization are precisely whatpermit the bubble to grow, despite the fact that the bubble bursts as soon as asufficient mass of traders sells out. This scenario is the starting point of our work.We present a model that formalizes the synchronization problem describedabove. Our approach emphasizes two elements: dispersion of opinion amongrational arbitrageursand the need for coordination.We assume that the price surpasses the fundamental value at a random pointin time to. Thereafter, rational arbitrageursbecome sequentially aware that theprice has departed from fundamentals. Arbitrageurs do not know whether theyhave learnt this information early or late relativeto other rational arbitrageurs.Inaddition to its literal interpretation, the assumption of sequential awareness canbe viewed as a metaphor for a variety of factors such as asymmetricinformation,differences in viewpoints, etc. that, in our model, find expression in the feature weseek to explore and emphasize, that is, temporal miscoordination. Arbitrageurshave common priors about the underlying structure of the model.The coordination element in our model is that selling pressure only bursts thebubble when a sufficient mass of arbitrageurs have sold out. Arbitrageurs facefinancial constraints that limit their stock holding as well as their maximumshort-position. This limits the price impact of each arbitrageur. Large price movementscan only occur if the accumulated selling pressure exceeds some threshold K.In other words, a permanent shift in price levels requires a coordinated attack.In this respect, our model shares some features with the static second genera-tion models of currency attacks in the international finance literature (Obstfeld(1996)).Thus, our model has both elements of cooperation and of competition. On theone hand, at least a fraction K of arbitrageursneed to sell out in order for thebubble to burst; this is the coordination/cooperative aspect. On the other hand,arbitrageursare competitive since at most a fraction K < 1 of them can leave themarket prior to the crash.In the equilibrium of our model, arbitrageursstay in the market until the sub-jective probability that the bubble will burst in the next trading round is suf-ficiently high. Arbitrageurs who get out of the market just prior to the crash

3 See, for instance, Daniel, Hirshleifer, and Subrahmanyam (1998), Hirshleifer (2001), Odean(1998), Thaler (1991), Shiller (2000), and Shleifer (2000).



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BUBBLES AND CRASHES 175make the highest profit. Arbitrageurswho leave the market very earlymake someprofit, but forgo much of the higher rate of appreciation of the bubble. Arbi-trageurs who stay in the market too long lose all capital gains that result fromthe bubble's appreciation.Our results regarding market timing seem to fit well with popular accountsof the behavior of hedge fund managers during the recent internet bubble.45For example, when Stanley Druckenmiller,who managed George Soros' $8.2 bil-lion Quantum Fund, was asked why he didn't get out of internet stocks earliereven though he knew that technology stocks were overvalued, he replied that hethought the party wasn't going to end so quickly. In his words We thought itwas the eighth inning, and it was the ninth. Faced with mounting losses, Druck-enmiller resigned as Quantum's fund manager in April 2000.6Rational arbitrageursride the bubble even though they know that the bubblewill burst for exogenous reasons by some time to+ T if it has not succumbed toendogenous selling pressure prior to that time. Here to is the unknown time atwhich the price path surpasses the fundamental value and arbitrageursstart get-ting aware of the mispricing. In equilibrium each arbitrageursells out after wait-ing for some interval after she becomes aware of the mispricing. If arbitrageurs'opinions are sufficiently dispersed, there exists an equilibrium in which the bub-ble never bursts prior to to+ -. Even long after the bubble begins and after allagents are aware of the bubble, it is nevertheless the case that endogenous sell-ing pressure is never high enough to burst the bubble. For more moderate levelsof dispersion of opinion (or equivalently for smaller K) endogenous selling pres-sure advances the date at which the bubble eventually collapses. Nevertheless,the bubble grows for a substantial period of time. Moreover, these equilibria areunique (modulo a natural refinement). Thus, there is a striking failure of thebackwards induction argument that would yield immediate collapse in a stan-dard model. The synchronization 'problem' arbitrageurs face is from their pointof view a blessing rather than a curse. After becoming aware of the bubble, theyalways have the option to sell out, but rather optimally choose to ride the bubbleover some interval.Our model provides a natural setting in which news events can have a dispro-portionate impact relative to their intrinsic informational content. This is because

4Brunnermeier and Nagel (2002) provide a detailed documentation of hedge funds' stock holdings.5 Kindleberger (1978) notes that even Isaac Newton tried to ride the South Sea Bubble in 1720.He got out of the market at ?7,000 after making a ?3,500 profit, but he decided to re-enter it therebylosing ?20,000 at the end. Frustratedwith his experience, he concluded: I can calculate the motionsof the heavenly bodies, but not the madness of people.6 However, fund managers can also not afford to simply stay away from a rapidly growing bubble.Julian Roberts, manager of the legendary Tiger Hedge Fund, refused to invest in technology stockssince he thought they were overvalued. The Tiger Fund was dissolved in 1999 because its returnscould not keep up with the returns generated by dotcom stocks. A Wall Street analyst who has dealtwith both hedge fund managers vividly summarized the situation: Juliansaid, 'This is irrational and Iwon't play,' and they carried him out feet first. Druckenmiller said, 'This is irrational and I will play,'

and they carried him out feet first. New YorkTimes, April 29, 2000, Another Technology Victim;Top Soros Fund Manager Says He 'Overplayed' Hand.



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176 D. ABREU AND M. BRUNNERMEIERnews events make it possible for agents to synchronize their exit strategies. Ofcourse, large price drops are themselves significant synchronizingevents, and weinvestigate how an initial price drop may lead to a full-fledged collapse. Thusthe model yields a rudimentarytheory of 'overreaction' and 'price cascades' andsuggests a rationale for psychological benchmarks such as 'resistance lines.' Inaddition, our model provides a frameworkfor understanding fads in informationsuch as the (over-)emphasis on trade figures in the eighties and on interest ratesin the nineties.Overall, the idea that the bursting of a bubble requires synchronized action byrational arbitrageurs,who might lack both the incentive and the ability to act in acoordinated way, has important implications. It provides a theoretical argumentfor the existence and persistence of bubbles. It undermines the central presump-tion of the efficient markets perspective that not all agents need to be rational forprices to be efficient, and hence provides further support for behavioral financemodels that do not explicitly model rational arbitrageurs. More generally, ourmodel suggests a new mechanism that limits the effectiveness of arbitrage. Thelatter might be relevant in other contexts; see Abreu and Brunnermeier (2002)for an application of the framework developed in this paper.The remainder of the paper is organized as follows. Section 2 illustrates howthe analysis relates to the literature. In Section 3 we introduce the primitivesof the model and define the equilibrium. Section 4 shows that all arbitrageursemploy trigger strategies in any trading equilibrium. Section 5 demonstrates thatif the dispersion of opinion among arbitrageurs is sufficiently large, they neverburst the finite horizon bubble and it only crashes for exogenous reasons at itsmaximum size. For smaller dispersion of opinion the bubble also persists butarbitrageurs burst it before the end of the horizon. Section 6 investigates therole of synchronizingevents including the special case of price events. Section 7concludes.

2. RELATED LITERATUREThe no-trade theorems provide sufficient conditions for the absence of spec-ulative bubbles. Allen, Morris, and Postlewaite (1993) develop contra-positives

of the no-trade theorems highlighting necessary conditions for the existence ofbubbles. They call a mispricinga bubble if it is mutual knowledge that the price istoo high. Of course, mutual knowledge (that is, all traders know) does not implycommon knowledge (that is, all traders know, that all traders know, and so onad infinitum). They provide illustrative examples that satisfy their conditions andsupport bubbles. In Allen and Gorton (1993) fund managers churnbubbles atthe expense of their less informed client investors (who do not know the skillof the fund manager). They take on an overvalued asset even though they knowthat they might be 'last in line' and hence unable to unload the asset. The orderof trades is random and exogenous in their model. In equilibrium,our sequentialawareness assumption also leads to sequential trading and uncertainty about thetiming of one's trades relative to other arbitrageurs'trades.



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BUBBLES AND CRASHES 177The papers described above assume that all agents are fully rational. In con-trast, our work falls within the class of models in which rational arbitrageurs nter-

act with boundedly rational behavioral traders. In DeLong, Shleifer, Summers,and Waldmann (1990b) rational arbitrageurs push up the price after some ini-tial good news in order to induce behavioral feedback traders to aggressively buystocks in the next period. This delayed reaction by feedback traders allows thearbitrageursto unload their position at a profit. In DeLong, Shleifer, Summers,and Waldmann (1990a) arbitrageurs'risk aversion and short horizons limit theirability to correct the mispricing. In contrast, in our model arbitrageurs initiallydo not even attempt to lean against the mispricing even though they are risk-neutral and infinitely lived. In Shleifer and Vishny (1997), professional fund man-agers forgo profitable long-run arbitrage opportunities because the price mightdepart even further from the fundamental value in the intermediate term. In thatcase, the fund manager would have to report intermediate losses causing clientinvestors to withdraw part of their money which forces her to liquidate at a loss.Knowing this might happen in advance, the fund manager only partially exploitsthe arbitrage opportunity.In these papers rational arbitrageursdo not have the collective ability to cor-rect mispricing either because of their risk aversion or because of exogenouslyassumed capital constraints. In contrast in our paper, the aggregate resources ofall arbitrageurs are sufficient to bring the price back to its fundamental value.Thus, the weakness of arbitrage in our model is particularly striking becausearbitrageurscan jointly correct the mispricing, but it nevertheless persists.Our assumption that a critical mass of speculators is needed to burst a bub-ble has its roots in the currency attack literature. Obstfeld (1996) highlights thenecessity of coordination among speculators to break a currency peg and pointsout the resulting multiplicityof equilibria in a setting with symmetricinformation.Morris and Shin (1998) introduce asymmetric information and derive a uniqueequilibrium by applying the global games approach of Carlsson and van Damme(1993). Both currency attack models are static in the sense that speculators onlydecide whether to attack now or never. In contrast, our model is dynamic. Thedispersion of opinion among arbitrageurs,which we model as sequential aware-ness, can be related to the idea of asynchronous clocks. The latter were firstintroduced in the computer science literature (see Halpern and Moses (1990)).Morris (1995) applies the concept to a dynamic coordination problem in laboreconomics. There are several differences between our papers. His model satisfiesstrategic complementarity and the global games approach applies. Our modelhas elements of both coordination and competition. The latter leads to a pre-emptionmotive that plays a central role in our analysis. It is important for Morris'result that players can only condition on their individual clocks and not on cal-endar time or on their payoffs, whereas in our model they can. In particular,traders learn from the existence of the bubble in our setting. In the richer strat-egy set of our model strategic complementarity is not satisfied and the globalgames approach does not apply. See footnote 15 for a discussion of this point. Inthe typical global game applicationthe symmetric information game has multiple



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178 D. ABREU AND M. BRUNNERMEIERequilibria, and any degree of asymmetric information leads to a unique equi-librium. Thus, there is a discontinuity at the point where asymmetry vanishes.In contrast, in our dynamic model the preemption motive leads to a uniqueequilibrium even under symmetric information. Furthermore, as the extent ofasymmetric information goes to zero, the equilibrium outcome converges to thesymmetric information (no bubble) outcome.

3. THE MODELHistorically, bubbles have often emerged in periods of productivityenhancingstructuralchange. Examples include the railwayboom, the electricity boom, andthe recent internet and telecommunication boom. In the latter case, and in many

of the historical examples, sophisticated marketparticipantsgraduallyunderstoodthat the immediate economic impact of these structuralchanges was limited andthat their full implementation would take a long time. They also realized thatonly a few of the companies engaged in the new technology would survive inthe long run. On the other hand, less sophisticated traders over-optimisticallybelieved that a 'paradigm shift' or a 'new economy' would lead to permanentlyhigher growth rates.We assume the price process depicted in Figure 1. This price process reflectsthe scenario outlined above, and may be interpreted as follows. Prior to t = 0

Pt/Pt =egt

/ ~~~~~~~113(t- to))Pt

to to +rpK to + 7 to +random K traders all traders bubbleburstsstartingare awareof are awareof forexogenouspoint the bubble the bubble reasonsmaximum life-span of the bubble r

FIGURE 1.-Illustration of price paths.



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BUBBLES AND CRASHES 179the stock price index coincides with its fundamental value, which grows at therisk-free interest rate r, and rational arbitrageursare fully invested in the stockmarket. Without loss of generality, we normalize the starting point to t = 0 andthe stock market price at t = 0 to po = 1. From t = 0 onwards the stock pricePt grows at a rate of g > r, that is Pt = egt.7 This higher growth rate may beviewed as emerging from a series of unusual positive shocks that graduallymakeinvestors more and more optimistic about future prospects. Until some randomtime to, the higher price increase is justified by the fundamental development.8We assume that to is exponentially distributed on [0, oo) with the cumulative dis-tribution function P(to)= 1-e-Ato. Nevertheless, the price continues to increaseat the faster rate g even after to. Hence, from to onwards, only some fraction(1 -,3(.)) of the price is justified by fundamentals, while the fraction ,3(.) reflectsthe bubblecomponent. We assume that ,3(): [0, T]~-* [0, ,3] is a strictlyincreas-ing and continuous function of t - to, the time elapsed since the price departedfrom fundamentals. For the special case where the fundamental value egto+r(t-to)coincides with the price path at to and always grows at a rate r thereafter,,3(t - to) = 1 - e(gr)(t-to)The price Pt = egt is kept above its fundamental value by irrationallyexuber-ant behavioral traders. They believe in a new economy paradigm and thinkthat the price will grow at a rate of g in perpetuity.9 As soon as the cumula-tive selling pressure by rational arbitrageurs exceeds K, the absorption capacityof behavioral traders, the price drops by a fraction ,3(.) to its post-crash price.From this point onwards, the price grows at a rate of r.10Even if the selling pres-sure never exceeds K we assume that the bubble bursts for exogenous reasons assoon as it reaches its maximum size ,3. This translates to a final date, since ,3(.)is strictly increasing. Let us denote this date by to+ r. Note that this assumptionof a final date is arguablythe least conducive to the persistence of bubbles. In aclassical model it would lead to an immediate collapse for the usual backwardsprogramming reasons.Another important element of our analysis is that rational arbitrageursbecomesequentially informed that the fundamental value has not kept up with the growthof the stock price index. More specifically, a new cohort of rational arbitrageursof mass 1/ij becomes 'aware' of the mispricing in each instant t from to untilto + ,q. That is, [t0, to+ ,1] forms the 'awareness window.' Since to is random, an

7 The analysis can be directly extended to a setting that incorporates a stochastic price process witha finite number of possible price paths and where the price grows in expectations at a rate of g.8 Kindleberger (1978) refers to the phase of fundamentally good news as 'displacement.'9The price process we assume is a modeling simplification that facilitates a clean and simpleanalysis. One, admittedly simplistic, setting that rationalizes this process is a world with behavioraltraders who are risk-neutral, but wealth constrained, and who require a rate of return of g in order toinvest in the stock market. If they are overconfident (in the sense that they are mistakenly convincedthat they will receive early warning of a change in fundamentals), they will purchase any quantity ofshares up to their aggregate absorption capacity K whenever the price falls below eg'. See also theremark at the end of this section.10We refer to fundamental value as the price that will emerge after the bubble bursts. Notice thatfor 18(0)= 0, there is no drop in fundamentals at to.



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180 D. ABREU AND M. BRUNNERMEIERV(tolti) V(toltj) C(tOltk)

ti-r7 tJ-77 ti tj tk tFIGURE2.-Sequential awareness.

individual arbitrageurdoes not know how many other arbitrageurshave receivedthe signal before or after her. An agent who becomes aware of the bubble attj has a posterior distribution for to with support [ti -q, ti]. Each agent viewsthe market from the relative perspective of her own ti. We will refer to thearbitrageurwho learns of the mispricing at time ti as arbitrageur ti. From theperspective of arbitrageur ti the truncated distribution of to is

-Atn - eA(ti-to)~~OIiJ -

Figure 2 depicts the distributions of to for arbitrageurs ti, tj, and tk, wheretk = ti + 7q,so that arbitrageurtk is the last possible type to become aware of thebubble given arbitrageurti's information. Viewed more abstractly,the set of pos-sible types of arbitrageursis [0, oo). Nature's choice of to determines the 'active'types ti E [to, to+ 71]in the economy. As noted earlier, we view this specificationas a modeling device that captures temporal miscoordination arising from differ-ences of opinion and information. We assume that

A g-r1-e-AqK


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BUBBLES AND CRASHES 181Let o-(t, ti) denote the selling pressure of arbitrageurti at t and the functionu: [0, oo)x [0, oo)~-* [0, 1] a strategy profile. The strategy of a traderwho became

aware of the bubble at time ti is given by the mapping o{((, ti): [0, ti + f] + [0, 1].Note that [1 - o-(t, ti)] is trader ti's stock holding at time t. For arbitrary u-, thefunction o-(t, ) need not be measurable in ti. We confine attention to strategyprofiles that guarantee a measurable function o(t, .). Notice that no individualdeviation from such a profile has any impact on the measurabilityproperty. Theaggregate selling pressure of all arbitrageurs at time t > to is given by s(t, to) =min{t,t?+, o(t, ti) dti. Let

T*(to)= inf{tls(t, to) > K or t = to+ 7}denote the bursting time of the bubble for a given realization of to. Recall P(. I i)denotes arbitrageur ti's beliefs about to given that to E [ti-71, tij. Hence, herbeliefs about the bursting date are given by

H7(tIti)= f dk(toIti).T*(to) ti.DEFINITION 2: The function T(ti) = inf{tIo-(t, ti) > 0} denotes the first instantat which arbitrageurti sells any of her shares.Arbitrageur to+ qK reduces her holdings for the first time at T(to+ 'qK). Since,by Corollary 1, all arbitrageurs who became aware of the mispricing prior to

to+ 'qK are also completely out of the market at T(to + 'qK), the bubble burstswhen trader to+ qK sells out her shares, provided that it did not already burstearlier for exogenous reasons.COROLLARY 2: The bubble bursts at T*(to) = min{T(to + rqK), to+ r}.We will show that in any equilibrium, each arbitrageur ti can rule out certain

realizations of to at the time when she first sells out her shares. For this purpose,we define t`(ti).



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184 D. ABREU AND M. BRUNNERMEIERDEFINITION 3: The function touPP(ti) denotes the lower bound of the supportof trader ti's posterior beliefs about to, at T(ti).Lemma 2 derives a lower bound for t'uPP(ti).LEMMA 2 (Preemption): In equilibrium,arbitrageuri believesat time T(ti) thatat most a mass K of arbitrageursbecame aware of the bubbleprior to her. That is,

tupp (ti) >ti - IK.

To see this, suppose to the contrary that arbitrageurti believes that tocan alsobe strictly smaller than ti - IqK. For these to's,the selling pressure at T(ti) strictlyexceeds K. Consequently, if the bubble bursts at T(ti), arbitrageur ti does notreceive the pre-crash price for her holding of the asset. Furthermore, arbitrageurti has to believe that the bubble bursts with strictly positive probability at T(ti).Hence, she has a strict incentive to preempt and sell out slightly prior to T(ti). Incontrast to the proof in the Appendix, this intuitive argument implicitly assumesT(ti) > ti.The Preemption Lemma allows us to derive further properties of the burstingtime T*(.). Lemmas 3 and 4 in the Appendix show that T*(.) is strictlyincreasingand continuous. It follows that T*-1(.): [T*(O),oo) ~- [0, oo), the inverse of T*(.),is well defined. Lemma 5 in the Appendix establishes that T(.) is also continuous.Let BC(t) denote the event that the bubble has not burst until time t.

LEMMA 6 (Zero Probability): For all ti > 0, arbitrageur i believes that the bub-ble bursts with probabilityzero at the instant T(ti). That is, Pr[T*-1(T(ti))Iti,Bc(T(ti))] = Ofor all ti > 0.The following proposition establishes that in any equilibrium arbitrageursnec-essarily use trigger-strategies. That is, each arbitrageur ti sells out at T(ti) andnever re-enters the market.PROPOSITION 1 (Trigger-strategy):In equilibrium,arbitrageur i maintains the

maximumshortposition for all t > T(ti), until the bubble bursts.PROOF: Suppose not, and that there exists an equilibriumin which arbitrageurti sells out at T(ti) and re-enters the market later. Given transaction costs c > 0and the preceding lemma, in equilibrium arbitrageur ti must stay out of themarket for a strictlypositive time interval. She will stay out of the market at leastuntil type ti + ? exits the market, for some ? > 0 (independent of ti).12 By thecut-off property arbitrageurti cannot re-enter the market until after arbitrageur12Exiting the market and paying the transaction cost c can only be justified if the bubble burstswith a certain probability. Since the bubble bursts with zero probabilityexactly when trader ti enters,

she has to stay out of the market until at least a certain mass of (younger) traders e also exit themarket. Note s is independent of ti.



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BUBBLES AND CRASHES 185t1+ s first re-enters. The same reasoning applies to arbitrageurti+ s with respectto arbitrageur ti + 2?. Proceeding this way we conclude that arbitrageur ti staysout of the market until the bubble bursts at to+ z or arbitrageurti+ 71K exits andthen re-enters the market. Of course, the latest possible date at which the bubblebursts from arbitrageurti's viewpoint is when ti+ 7K exits the market. Q.E.D.

We have proved that T*-1(.) exists and is strictlyincreasing and continuous. Weconfine attention to equilibria for which the latter function is absolutelycontinu-ous such that H(t) = c(T*l(t)) is also absolutely continuous. Let 7r(t) denoteits associated density. Recall that P(to) = 1 - e-Ato and H(tItJ) is arbitrageurti'sconditional cumulative distribution function of the bursting date at time ti. Sim-ilarly,7r(tlti)denotes the associatedconditionaldensity.The trigger-strategycharacterization greatly simplifies the analysis of equilib-rium and the specification of payoffs. As noted earlier the payoff to selling outat time t ist

e-rs(l -3(s - T*-l(s)))p(s)I7T(sltJ)ds + e-rtp(t)(1 - H(t1ti)) - C.Differentiating the payoff function with respect to t yields the sell-out condi-tion stated in Lemma 7. Note that

h(tlti) = (t Iti)htt)-1 - H(tItj)is the hazard rate that the bubble will burst at t.

LEMMA7 (Sell-out Condition): If arbitrageur ti's subjective hazard rate issmaller than the 'cost-benefitratio,' i.e.h(tl|ti) < g(- -(t)r

traderti will choose to hold the maximum long position at t. Conversely, fh(tl|ti) > g(- -(t)r

she will trade to the maximum shortposition.1313One could easily extend the analysis to capture the reputationalpenalty, due to relative perfor-mance evaluation, that institutional investors face for staying out of the market while the bubblegrows at the expected rate g. In a setting with a 'reputational penalty' equal to a fraction k of theprice level, the term flti+T[ftSe-rukp(u) du]ir(sIti) ds needs to be added to the payoff and the sell-outcondition generalizes to

h(tlti) < (>) g r t)The reputational term does not modify the structure of the results, but would be relevant in obtainingrealistic calibrations of the model.



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186 D. ABREU AND M. BRUNNERMEIERRecall that if the bubble bursts at t its size is precisely /3(t- T*-1 t))p(t).To understand the sell-out condition intuitively, consider the first-order benefits

and costs of attack at t versus t + A, respectively. The benefits are given byAh(tItJ)[p(t)3(t - T*-1(t))], the size of the bubble times the probabilitythat thebubble will burst over the small interval A. In the case that the bubble does notburst, the costs of being out of the market for a short interval A are(1 - Ah(tIti)) P(t + P())-rp (t)) .

Note thatp(t + A)- p(t) - rp(t) > 0'A

since the bubble appreciates faster than the riskfree rate. Dividing by Ap(t) andletting A -+ 0 yields the attack condition. Lemma 7 also conforms with our earlierresult that trader ti either wholeheartedly attacks or holds the maximum longposition.5. PERSISTENCE OF BUBBLES

The impossibilityof bubbles emerging in standard asset pricing models is mosttransparentin a finite horizon setting in which they are ruled out by a straightfor-ward backward induction argument. Interestingly, a similar argument involvingthe iterative removal of non-best-response symmetric trigger-strategies pro-vides some intuition for the main results of this section.Even if no arbitrageursells her shares, the bubble ultimately bursts at to+ z byassumption. Since each arbitrageurbecomes aware of the mispricing only afterto, she knows for sure that the bubble will never last beyond ti + r. But it mightburst even before this time since from arbitrageurti's point of view, the ultimateburstingdate to+ - is distributedbetween ti+r - 'qand ti+ r. If the bubble burstsat to+ I, arbitrageurti's best response is to exit the market before ti + -. Morespecifically, let ti+ rl be arbitrageurti's best response exit time if she conjecturesthat other arbitrageursnever attack, and consequently that the bubble bursts onlyat to+ r. The conjecture that all arbitrageursride the bubble for rl periods leadsto a new bursting date and a new best response exit time ti + r2. Proceeding inthis way, we inductivelyobtain r3, r and so on. In the standard model this yieldslim2 7n = 0, which precludes the emergence of bubbles. We show in Section 5.1that this backward induction procedure has no bite if

A g-r1 - e-A1

In particular, limn,2 7n = r and furthermore, the bubble only bursts at to+ f.Conversely, forA g-r1 - e-A7K 1



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BUBBLES AND CRASHES 187we show in Section 5.2 that lim 3 n< IT' but lim,3O rn > 0. Hence, this back-ward procedure does bite, but not as much as in the classical case. Note thatthis induction argument is restricted to symmetric strategies. The formal analysisbelow employs a different reasoning that also addresses nonsymmetric strategies.Sequential awareness blunts the competition amongst arbitrageursupon whichthe efficient markets view is premised and leads to sufficient ambiguity amongstarbitrageursfor a strategy of market timing to be profitably pursued. Hence, fromthe arbitrageurs' point of view, the lack of synchronization is not a problembut rather a blessing.

5.1. Exogenous CrashesRecall that arbitrageursbecome aware of the bubble sequentially in a random

order and furthermore have a nondegenerate posterior distribution over to. Allarbitrageursbecome 'aware' of the bubble during the interval [to, to+ 71],wherewe have interpreted q to be a measure of differences in opinion and other het-erogeneities across players. From to+ qK onwards, more than K arbitrageursareaware of the bubble and have collectively the ability to burst it.We show that ifA g-r1 - e-A1

then in the unique trading equilibrium the bubble only bursts for exogenousreasons when it reaches its maximum size ,B relative to the price. In this casethe endogenous selling pressure of the rational arbitrageurs has absolutely noinfluence on the time at which the bubble bursts.DEFINITION 4: The function r(ti) := T(ti) - ti denotes the lengthof time arbi-trageur ti chooses to ride the bubble subsequent to becoming aware of the mis-pricing.It is worth noting that our result holds despite the fact that it is possible withinour model for traders to coordinate selling out on particular dates, say Friday,13th of April, 2001 by adopting (asymmetric) strategies that entail nonconstant

r(ti).PROPOSITION 2: Suppose

A g-r1- e-,1K- 3

Then thereexistsa unique tradingequilibrium.In this equilibriumall traderssell outI =- 1l g - rg- r-A,B

periods after theybecome aware of the bubbleand stay out of the marketthereafter.Nevertheless, for all to, the bubble_burstsor exogenous reasons precisely when itreaches its maximumpossible size ,B.



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188 D. ABREU AND M. BRUNNERMEIERPROOF: Step 1: r' defines a symmetricequilibrium. Suppose arbitrageur tibelieves that the bubble bursts at to+ ;. Conditioning on to E [ti - -q,ti] her

posterior distribution over to iseAn - eA(t-to)

and her posterior distribution over bursting dates t = ti + r isH'(ti It') -=eAn - eA(-T)17+Tjj+ -Iti) eAq _ 1

at ti. The corresponding hazard rate is

h(ti + rlti) 1-HI(ti+It)- ItSuppose that the bubble bursts for exogenous reasons. Then = z in the expres-sion for the hazard rate. The latter is depicted in Figure 3 as an increasing func-tion of r. Since the exogenous bursting date to+ f is independent of -, so is thecost-benefit ratio of the sell-out condition (Lemma 7). This ratio equals (g - r)//3and is represented by the horizontal line in Figure 3. The point of intersectionI1 solves h(ti+ r1ti) = (g- r)//3. Clearly,

7' = r- Inr g )A-r-ArBy the sell-out lemma arbitrageurti is optimally out of the market for r > r' andconversely for r < r' she holds on to her shares. Under our assumed condition

AL~~~~~~~1-e-A(r-)||

11_ _ _ _ _ _Exogenous crash.I /3

T1 ~~~~TFIGURE 3.-Exogenous crash.



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190 D. ABREU AND M. BRUNNERMEIERearlier, 72 < T1 periods after they became aware of the bubble. Proceeding inthis way leads to a decreasing sequence r', r2, T3,. .. which converges to somer* that in fact defines the unique symmetric trigger-strategy Perfect BayesianNash equilibrium. The iteration of this argument does not eliminate bubbles.The reason is that the iterative procedure loses bite gradually. As the burstingdate advances, the size of the bubble also diminishes, which in turn increases theincentive to ride the bubble by reducing the cost of not exiting prior to the crash.Though the bubble bursts for endogenous reasons it may survive for a substantialperiod; at the time of the crash the bubble component is

1 - e-AnK (g-r)ABy an appropriate choice of parameter values, the latter term can be made arbi-trarilyclose to ,3, which in turn can be chosen to be close to 1. Note that for thespecial case where the fundamental value is egto at to and grows at a rate of rthereafter, 3(t - to) = 1e--(g-r)(t-t') and

g-r A(A-(g-r)(1 e-A71K))-K.Proposition 3 derives a symmetricequilibrium,where each arbitrageur sells hershares r* periods after she becomes aware of the bubble. It also demonstratesthat this trading equilibrium is unique.PROPOSITION 3: Suppose

A g-r1-e-A?7K

Then there existsa unique trading equilibrium, n which arbitrageurs i with ti > 'qKleave the market7* :-W( bgA?K -)-K

1 _e-A71Kperiods after theybecome aware of the bubble.All arbitrageurs i such that ti < 17Ksell out at 7)K+ r*. Hence, the bubble bursts when it is a fraction

1 - e -,?1K -gr)Aof thepre-crashprice.

PROOF: Step 1: r* defines a symmetricequilibrium. Suppose that all arbi-trageurswith ti > 7)K sell out their shares at ti+ r and arbitrageurswith ti


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BUBBLES AND CRASHES 191The cost-benefit ratio (g - r)/,3('qK + r) of the sell-out condition (Lemma 7)is declining in r, since 8(qK + r) is (by assumption) an increasing function. See

Figure 4. In equilibriumA _ g-r1-e-Ak?K -(qK +r*)-

Hence,

1_e-A77K

Note that the assumption thatA g-r1-e-AnK 0, while the conditionA g-r

1-e-A7E 18implies that r* < - 7K. Hence, r* indeed defines a symmetric equilibrium. Thenext steps establish that this equilibrium is unique.Step 2: The bubblealways bursts or endogenousreasons when

A g-r1-e-AnK f3

g-r

h*= A~~1-e-A77K

FIGURE 4EndogenouscrashTFIGURE 4.-Endogenous crash.



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192 D. ABREU AND M. BRUNNERMEIERLet r1 solve

h(ti +'r' I i) A g-r-1e-Ak(~-'Each trader would exit the market at ti+ r1 if she believed that the bubble wouldburst for exogenous reasons when it reached its maximumpossible size ,B.Underour assumptions to+ r' + 7K < to+ i. Hence, the bubble does not alwaysburst forendogenous reasons only if r(ti) > r' for at least some ti. Consider arbitrageurtj, where tj = argmax,.{r(ti)}. We establish Step 2 by showing that r(tj) > 'leads to a contradiction. Observe that:(i) By the Preemption Lemma t'upp(tj) > tj -rK.(ii) If tsuPP(tj) > tj - rlK, then arbitrageur tj's hazard rate at T(tj) > tj + T'that the bubble will burst for exogenous reasons strictly exceeds (g - r)//3. Sincethis is also the case at T(tj) - E for sufficiently small E > 0, she has an incentiveto sell out strictly prior to T(tj).(iii) Finally, suppose tsuPP(tj) = tj -7K. Since r(tj) > r(ti) for all ti, the hazardrate that the bubble will burst at T(tj) is at least A/(1 -e-A?7K). Since the bubblebursts at to +7K + T(tj) > to+K +r 1, the size of the bubble 3(7K ++r(tj)) exceeds/3(r1K + r'). It follows that the sell-out condition for arbitrageur tj is violated atT(tj).Step 3: Minimum and maximum of r(ti) coincidefor ti > 7/K. By Step 2, T(to+7/K) < to+ z v to> 0. That is, the bubble only bursts for endogenous reasons. Bythe Preemption Lemma, tsupp(ti) > ti - 7/K. Furthermore, tsupp(ti) ti - 7/K canbe ruled out since arbitrageur ti would be strictly better off selling out at someE > 0 time after her ostensible equilibrium sell out date T(ti). (Recall that byLemma 5, T(.) is continuous.) Hence, given tsuPP(ti) = ti -7K, arbitrageur ti'sconditional density of to at T(ti) is

Ae A7-(ti- 7KIti, Bc(T(ti))) - eAk7Kwhich is independent of ti. Let ti E argminr(ti) and t-iE argmaxr(ti) and supposethat maxr(ti) > minr(ti). By the continuity of T(.) shown in Lemma 5 and thedefinitions of ti and ti, it follows that

HI(T(ti) + AIti,Bc(T(t ))) < H(T(t-1)+ Ti, BC(T(t1i))) for all A > 0,and conversely for A < 0. Consequently,

h(T(tj)jtj, Bc(T(ti))) < h(T(t-i)jt-i, Bc(T(tij))).However,

/3(7/K + r(t )) < /3(7/K + r(ij)).Thus, the sell-out condition cannot be satisfied for both arbitrageursti and ti, acontradiction.



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BUBBLES AND CRASHES 193Step 4: For ti < 7K, T(ti) = T(QrK).Clearly, no ti should sell out prior to T(VqK)and by the cut-off property will sell out at T(QrK). Q.E.D.Notice that the maximum 'relative' bubble size /3* increases as the disper-sion of opinions among arbitrageurs7q ncreases. The comparative statics for theabsorption capacity of the behavioral traders K are the same as for q. A larger

K requires more coordination among arbitrageurs and thus extends the bubblesize. Finally, /3* s also increasing in the 'excess growth rate' of the bubble (g - r).The faster the bubble appreciates, the more tempting it is to ride it.REMARKS:

* The usual backwardinduction argument ruling out bubbles begins at a ter-minal date that is common knowledge. In our model, prior to the actual burstingof the bubble it is never common knowledge even among a mass K < 1 of thearbitrageurs that a bubble exists. To see this, note that at to+ 11K, (a mass of)K traders know of the bubble. That is, at to+ 77K the existence of the bubble ismutual knowledge among K traders. But they do not know that K traders knowof the bubble. This is the case only at to+ 2qjK.More generally, nth level mutualknowledge among K arbitrageurs is achieved precisely at to+ nfK. See Rubin-stein (1989) for an early exploration of the crucial distinction between commonknowledge and high levels of mutual knowledge.* Note that in contrast to Tirole (1982) and Milgrom and Stokey (1982), ourtrading game is not a zero-sum game. The profits of rational arbitrageurscomeat the expense of the behavioral traders.

6. SYNCHRONIZING EVENTSA central theme of our analysis is that dispersion of opinion makes it difficultfor arbitrageurs to synchronize selling out of the bubble asset. This difficultyprovides crucial cover for profitable 'ridingof the bubble.' A natural conjecture isthat the presence of synchronizingevents punctures this cover. Perhaps the mostobvious such events are dates like Friday, April 13th, 2001. In fact, the analysisofthe preceding section allows arbitrageurs to condition on absolute time. Never-theless, equilibrium was found to be unique and symmetric: all arbitrageurs waitthe same number of periods r* after becoming aware of the mispricing beforeattacking. Hence, perfectly anticipated events have no impact on the analysis.In Section 6.1 we consider unanticipated synchronizing events. The importantimplications of this analysis are that:(i) news may have an impact disproportionate to any intrinsic (fundamental)informational content;(ii) fads and fashions may arise in the use of different kinds of information;(iii) while synchronizing events facilitate coordinated attacks, when suchattacks fail, they lead to a temporary strengthening of the bubble.Finally in Section 6.2 we consider price events. These may also serve as syn-

chronization devices and lead to full correction (a price cascade) or marketrebounds.



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194 D. ABREU AND M. BRUNNERMEIER6.1. UninformativeEvents

Since the primaryemphasis of this section is to understand the role of 'news'in causing price changes beyond its informational content, we restrict our formalanalysis to nonfundamental events that serve as pure coordination devices. Inparticular, we extend the earlier model to allow for the arrivalof synchronizingsignals at a Poisson arrival rate 0.It is natural to assume that only arbitrageurs who are aware of the bub-ble observe synchronizing events. Indeed, we assume that such events are onlyobserved by traders who became aware of the bubble more than Te > 0 periodsago. The stronger assumption that Te > 0 captures in a simple way the idea thatarbitrageurs become more wary the longer they have been aware of the bubble,and, hence, look out more vigilantly for signals that might precipitate a burstingof the bubble. An alternative justification for some arbitrageurs not recognizingsynchronizing events is suggested by the second remark at the end of this sub-section.Synchronizingevents both allow arbitrageursto synchronize sell outs and alsoconvey valuable information in the event that attacks are unsuccessful. The bub-ble will survive a synchronized sell out if an insufficient number of arbitrageurshave observed the synchronizing event, that is, if to is sufficiently high. Let tebe the date of such a synchronizing event. By the cut-off property, there existsTe> re such that all arbitrageursti who observed the sunspot at least Te periodsafter becoming aware of the mispricing, sell out after observing the synchro-nizing event at te. If the bubble does not burst then all arbitrageurs learn thatto > te- e- 7K. It follows that all arbitrageurs ti with ti < te - Te will re-enterthe market until type te - Te first exits in equilibrium, or until the next synchro-nizing event occurs. Even traders who left the market prior to the arrivalof thesynchronizingevent buy back their shares.In contrast to the previous sections, there is no hope of finding a unique equi-librium in this generalized setting. For example, the equilibrium behavior of theprevious section would be exactly replicated if all arbitrageurswere to simplyignore all synchronizing events, and there are potentially numerous interme-diate levels of responsiveness to synchronizing signals. We focus on responsiveequilibria.

DEFINITION 6: A 'responsiveequilibrium'is a trading equilibrium in whicheach arbitrageurbelieves that all other traders will synchronize (sell out) at eachsynchronizing event.This raises the bar for our analysis since bubbles are harder to sustain in aresponsive equilibrium.This is the polar opposite of the equilibrium in which allarbitrageursignore all synchronizing events.We establish that there exists a unique responsive equilibrium. It is symmetric

in the sense that as long as an arbitrageur does not observe a synchronizingevent, each arbitrageurstays in the market for a fixed number r**periods after



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BUBBLES AND CRASHES 195becoming aware of the bubble. We assume that 0 < (g - r)/f, to ensure that thepossible occurrence of a synchronizingevent does not by itself justify selling out.As in Subsection 5.2 (endogenous crashes), we focus on the parameter valuesthat satisfy

g-r A g-r,B < 1-e-AnK ti + Te. Furthermore, he staysout of the market or all t > ti+ r**exceptin theevent that the last synchronizedattackfailed in which case she re-entersthe marketfor the interval t E (te, te + -**-re), unless a new synchronizingevent occurs in theinterim.

Absent the arrival of a synchronizingevent the bubble will burst at to+ 77K+ T**.The arrival of synchronizing events might accelerate the bursting date. Aftera failed sell out at the latest synchronizing event te, even traders who startedselling out prior to te, that is, traders with ti < te - r**, buy back shares. Sellouts only resume at te + 7** - 'e. 'Market sentiment' bounces back after a failedsynchronized attack. However, even in the event of failed attacks and re-entry byarbitrageursinto the market, the bubble bursts no later than to+ 77K+ T**.The proof of Proposition 4 is summarized in Appendix A.2. Notice that giventhe structure of our setup, the equilibrium is fully characterized by r**: In anytrading equilibrium, all arbitrageurs who sold out after the last synchronizingevent will re-enter the market after a failed attack and sell out again exactlywhen the 'first' trader, who did not participate in the synchronized sell out, exitsthe market.

REMARKS:* The news events above are special in that they convey no informationabout the value of the asset. Nevertheless, they facilitate synchronization. Animmediate consequence is that, in general, news events may have an impact quiteout of proportion to their fundamental content.* The issue of what events are considered by market participants to be syn-chronizing events is a subtle one. There is obviously an unlimited supply ofpotential synchronizing events (such as the weather, the levels of various eco-nomic indicators, etc.). But which do participants view as relevant? Their veryabundance emphasizes the difficulty and suggests a higher level synchronizationproblem, and the possibility of multiple equilibria.* The above issue is closely related to fads and fashion in the acquisitionand use of information. For example, trade figures drove the market duringthe 1980's. In contrast, in the late 1990's Alan Greenspan's statements moved



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196 D. ABREU AND M. BRUNNERMEIERstock prices, while trade figures were ignored. The model in this section may beextended to allow for multiple types of signals, A, B, C, etc. In some equilibria,arbitrageursmay condition on A and C, in others only on B, etc. More interest-ingly, some equilibria may initially condition on A and then after some historiesswitch to B and subsequently back to A, etc. Such a model would yield a rudi-mentary theory of fads and fashion. These themes are reminiscent of Keynes'(1936) comparison of the stock market with a beauty contest:

professional investment may be likened to those newspaper competitions in which the com-petitors have to pick out the six prettiest faces from a hundred photographs, the prize beingawarded to the competitor whose choice most nearly corresponds to the average preferencesof the competitors as a whole; so that each competitor has to pick, not those faces whichhe himself finds prettiest, but those which he thinks likeliest to catch the fancy of the othercompetitors, all of whom are looking at the problem from the same point of view.

6.2. Price Events-Price Cascades and Market ReboundsArguably, the most visible synchronizing events on Wall Street are large pastprice movements or breaks through psychological resistance lines. In this section,we allow random temporary price drops to occur, which are possibly due to moodchanges by behavioral traders. This enables us to illustrate how a large pricedecline might either lead to a full blown crash or to a rebound. In the latter eventthe bubble is strengthened in the sense that all arbitrageursare 'in the market'for some interval, including those who had previously exited prior to the priceshock.In our simple and highly stylized model, exogenous price drops occur with aPoisson density Opat the end of a random trading round t. We suppose thatbehavioral traders' mood changes temporarily and that they are only willing tohold on to shares if the price is less than or equal to (1 - yp)p,. If the bubbledoes not burst in the 'subsequent' trading round, behavioral traders regain theirconfidence and are willing to sell and buy at a price of Pt until their absorptioncapacity K iS reached or another price drop occurs. Arbitrageurs who exit themarket immediately after a price drop receive (1 - yp)p, per share. Should thesynchronized attack after a price drop fail, arbitrageurscan only buy back theirshares at a price of Pt, In other words, leaving the market even only for an instantis very costly despite the absence of transactions costs, which we set equal tozero. Hence, only traders who are sufficiently certain that the bubble will burstafter the price drop will leave the stock market. Let H:= (tl,... , tp) denotea historyof past temporary price drops. Proposition 5 states that only traders whobecame aware of the mispricing more than rp(Hpn)periods earlier will choose toleave the market and attack the bubble after a price drop. Notice that, although

rp(Hpn)s derived endogenously for the subgame after a price drop, it serves thesame role as the exogenous re in subsection 6.1. The structure of the equilibriumis similar to the unique responsive equilibrium of the preceding subsection, withthe main difference being that all arbitrageursknow H , and HpaffectsAnalogous to the previous subsection, r***denotes how long each arbitrageurrides the bubble after ti if there has been no price drop so far.



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BUBBLES AND CRASHES 197PROPOSITION 5: There existsan equilibrium(rp ), r***) n which arbitrageur iexitsthe marketaftera price dropat t if (n+1) > ti+ rp(Hp). Furthermore, he is

out of the marketat all t > ti+ r***exceptin the event that the last attackfailed, inwhich case she re-enters hemarket or the intervalt E (tn+), tn+) + * -p(HProposition 5 shows that a price drop that is not followed by a crash leads toa rebound and temporarily strengthens the bubble. In this case all arbitrageurscan rule out that the bubble will burst within (4(n+1) (n+1)+ r*** rp(Hp)) forendogenous reasons. Within this time interval, the price grows at a rate of g withcertainty, except if another exogenous price drop occurs. Consequently, all arbi-trageurs re-enter the market after a failed attack and buy back shares at a priceof Pt, even if they have sold them an instant earlier for (1 - yP)p,. Furthermore,

the failed attack at t(n)increases the endogenous threshold rp(Hp). More specifi-cally, a failed attack at t') makes arbitrageursmore cautious about the prospectsof mounting a successful attack after a price drop at (n+1) > t~().In particular, if4,n+1) is close to a failed synchronized sell out at t arbitrageurswill not find itoptimal to exit again at t( .16

7. CONCLUSIONThis paper argues that bubbles can persist even though all rational arbitrageursknow that the price is too high and they jointly have the ability to correct themispricing. Though the bubble will ultimately burst, in the intermediate term,there can be a large and long-lasting departure from fundamental values. A cen-tral, and we believe, realistic, assumption of our model is that there is disper-sion of opinion among rational arbitrageursconcerning the timing of the bubble.This assumption serves both as a general metaphor for differences of opinion,information and beliefs among traders, and, more literally, as a reduced-formmodeling of the temporal expression of heterogeneities amongst traders. Whileit is well understood that appropriate departures from common knowledge willpermit bubbles to persist, we believe that our particularformulation is both nat-ural and parsimonious. The model provides a setting in which 'overreaction' and

self-feeding price drops leading to full-fledged crashes will naturallyarise. It alsoprovides a framework that allows one to rationalize phenomena such as 'resis-tance lines' and fads in information gathering.16The equilibrium described in Proposition 5 is also maximally responsive in that price drops areresponded to 'maximally' in the chronological order in which they appear. However, the bubble ismore likely to burst at t(n+,) in an equilibrium in which the earlier event at t(n) was not respondedto than in the equilibrium where arbitrageurs sold out at t(n) and the attack failed. We do nothave a uniformly most responsive equilibrium in this section, but rather one that is responsive inchronological order to the maximum extent possible. In short, the above equilibrium is not necessarilythe one in which the bubble bursts earliest for any possible sequence of price drops. Note that the

same issue also arises in a setting with unanticipated public events and strictly positive transactionscosts c > 0. We abstracted from these effects in Section 6.1 by assuming c = 0.



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BUBBLES AND CRASHES 199Replacing uperscript by k+ 1 andsuperscript withsuperscript givesus a generalrecursivedefinitionof the payofffunction.Arbitrageuri's expected payofffrom a strategy nvolvingK - 1changes n herportfolio s thengiven bythe function

VK((XK, tK). (X tl))where (xK = 1, tK = ti) denotes her initial position.

APPENDIXB: DETAILSOF SECTIONLEMMA1 (No partialpurchases r sell-outs):0r(t, ti) E {0, 1} Vt, ti.PROOF: Consideran equilibrium trategyoc(., i) that involvesa changein positionat tk fromthe precedingpositionadoptedat tk+I < tk. (The initialposition s represented y K.) Suppose hat

xk(tk) g {0, 1} and o-(., ti) is optimal. The expected payoff from tk+1 onwards is given by Vk+1(_) plusthe valueof bond holdings. Notice that Vk+l(.), excludinghe transactionostsce rk+l, is linear in xkandfurthermoremustbe strictlypositive.Hence,forxk - xk+l > ( ti - qK.PROOF: Let tosup(ti) = ti -K - a. Forarbitrageuri to believe that a > 0, it is necessarily the casethat for all t E [ti - a, ti), arbitrageurs t1 do not sell out prior to T(ti), since otherwise by Lemma 1andCorollary the bubblewouldhave burstalready.Furthermore,he aggregate ellingpressureatt= T(ti) is sto, > K withstrictlypositiveprobability,ince(by Lemma1 andCorollary ) all traderstk < ti willalso be fullyout of the marketat T(ti). Hence,withstrictlypositiveprobability ll traderstj who firstsell out at T(ti) will only receive the post-crashprice for their sales.Consequently, lltraders t1 E [ti - a, ti) have an incentive to preempt (attack slightly earlier) the possible crash at T(ti),a contradiction. Q.E.D.LEMMA3 (Bursting Time): Thefunction T*(.) is strictly ncreasing.PROOF: Recall T*(to) = min{T(to + qK), to+ -}. Since T(-) is weakly ncreasingby the cut-offproperty, so is T*(.). Now suppose that there exists t- > to such that T*(to)= T*(10).Clearly, to+ X

T*(to). It followsthatfor all to E (to t-o],T(to + 7)K) = T*(to) = T*(to). Let ti = to+ .K By Lemma2t_upp(ti)> to. However, tsupp(ti) t ti -qK, a contradiction. QE.D.

LEMMA4 (Continuity of T*): The function T*: [0, oo) -- [0, oo) is continuous.PROOF: (i) Since T* is increasing,we only need to rule out upward umps.Supposethat thereis an upward jump at to.Then for smallenough E > 0 and s E (to - E, to), T*(s) = T(s + qK). LetT*= limS,f,oT*(s) and T*= limS,/o T*(s), and suppose E < T- T*. Recall that transactions cost certis incurred or any changeof position at t. This precludesan arbitrageurromchangingpositionat t and t' if the probability f a crashbetweent and t' is small relative to c. Hence, for smallenoughE > 0, type ti = to+ K E is strictlybetter off sellingout at T* E > T*thanat T(ti) < T*,a contradiction. QE.D.LEMMA5 (Continuity of T): The function T: [0, oo) -- [0, oo) is continuous.PROOF: This argument is almost identical in structure to the proof of Lemma 4 and is omitted.LEMMA6 (Zero Probability): For all ti > 0, arbitrageur i believes that the bubble bursts with prob-abilityzero at the instant T(ti). That is, Pr[T* l (T(ti))Iti, BC(T(ti))] = O or all ti > 0.



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200 D. ABREU AND M. BRUNNERMEIERPROOF: Observe that the conditional c.d.f.

P[toti, BC(T(ti))] = P(to)-

lp(touPP(ti))gP(ti) - gP(tsupp(ti))Clearly, the latter is continuous. Q.E.D.

APPENDIX C: DETAILS OF SEcTION 6C. 1. Restrictionto Interim-Trigger-Strategies

Let an interim-trigger-strategye one for which an agent follows a trigger-strategy between twosuccessive synchronizing events. We argue that, in equilibrium, agents use interim-trigger-strategies.Let H(tlti) := (t), . . , t()) denote a history of past events at times t(l) < te2) < < t(n)strictly prior to t and observed by arbitrageur ti. Let ten E H(tlti) be the time of the most recent

synchronizing event observed by arbitrageur ti. Let ta equal ti if arbitrageur ti has not observed anysynchronizing event, and equal t(n)otherwise. Denote arbitrageurti's posterior distribution over to att = ta by P( Iti,BC(ta), H(talti)). In other words, trader ti conditions upon her awareness date, thefact that the bubble did not burst prior to ta and the history of observed past synchronizing events if- t(n). The event that the bubble did not burst prior to ta, BC(ta), depends on the realization of toand also on the sequence of synchronizingevents. To simplify notation we denote {ti, Bc (t), H(ta Iti)}by Jt. (ta).DEFINITION : (i) The function Te(ti) = inf{tIlu(t, ti) > 0, H(tIti) = 0} denotes the first instantat whicharbitrageuri sells out any of her shares conditional upon not having observed a synchronizingevent thus far.(ii) The function Te*(to) = min{Te(to+? K), to+ ir} specifies the date at which the bubble bursts,

absent the observation of a synchronizing event by trader to+ qK.The analysis for T(.) and T* .) of Section 4 applies to Te(.) and Te*) in this section. In particular,

Te*.) is strictly increasing and continuous. The arguments are analogous and are not repeated here.Since each arbitrageur incurs positive transactions cost c when selling her shares, and the bubblebursts with zero probability at each instant, she has to leave the market at least for an interval.Arbitrageur ti stays out of the market at least until arbitrageurti+ E exits the market. Recall that theoccurrence of synchronization events alone does not justify the loss in appreciation from exiting overa certain interval, since 063< g - r. Hence, a generalized version of Proposition 1 applies betweensynchronization events, leading to what we have termed interim-trigger-strategies.Furthermore, thedistribution function that the bubble will burst in the absence of further synchronizing events isgiven byH(tIJi(t )) = 'p(Te 1(t)Ii(ta))

Since in Section 6 arbitrageursre-enter the market in equilibrium,transactions costs are potentiallyincurred repeatedly. Keeping track of all transactions costs complicates the analysis in uninterest-ing ways. In what follows we exploit the strategic restrictions obtained above (i.e. interim-trigger-strategies), while setting transactions costs to zero.C.2. Generalized Sell-out Condition

LEMMA 8 (Generalized Sell-out Condition): Suppose arbitrageur i has not observed a failed sellout. Then she sells out at t satisfyingh(tIJJ),3(t Te l(t)) + 0


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204 D. ABREU AND M. BRUNNERMEIEROBSTFELD,M. (1996): Models of Currency Crises with Self-Fulfilling Features, European EconomicReview, 40, 1037-1047.ODEAN, T. (1998): Volume, Volatility, Price and Profit When All Traders Are Above Average,Journal of Finance, 53, 1887-1934.RUBINSTEIN,A. (1989): The Electronic Mail Game: Strategic Behavior Under Almost CommonKnowledge, American Economic Review, 73, 225-244.SANTOS,M. S., AND M. WOODFORD 1997): Rational Asset Pricing Bubbles, Econometrica, 65,19-57.SHILLER,R. J. (2000): IrrationalExuberance. Princeton, New Jersey: Princeton University Press.SHLEIFER,A. (2000): Inefficient Markets-An Introduction to Behavioral Finance. Oxford: OxfordUniversity Press.SHLEIFER,A., AND R. W. VISHNY 1997): The Limits of Arbitrage, Journal of Finance, 52, 35-55.THALER,R. H. (1991): Quasi Rational Economics. New York, NY: Russell Sage Foundation.TROLE, J. (1982): On the Possibility of Speculation under Rational Expectations, Econometrica,50, 1163-1182.

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