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Journal of Financial Economics

Journal of Financial Economics 111 (2014) 137–157

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journal homepage: www.elsevier.com/locate/jfec

Investment busts, reputation, and the temptation to blendin with the crowd$

Steven R. Grenadier a, Andrey Malenko b, Ilya A. Strebulaev a,c,n

a Stanford University, Graduate School of Business, United Statesb MIT Sloan School of Management, United Statesc NBER, United States

a r t i c l e i n f o

Article history:Received 17 August 2012Received in revised form8 May 2013Accepted 14 May 2013Available online 19 September 2013

Keywords:AbandonmentReal optionsSignalingAsymmetric informationReputation

JEL classification:D81D82G31G33

5X/$ - see front matter & 2013 Elsevier B.V.x.doi.org/10.1016/j.jfineco.2013.09.002

thank Cecilia Bustamante (the referee), Barchang Wu (discussants) for many helpful sunts, we also thank Judson Caskey, Robert Gibalenko, Marc Martos-Vila, Erwan Morellec,

ants at Boston University, Higher School ofs School, Massachusetts Institute of TechnoOhio State University, Stanford University, Unity of Notre Dame, University of Oklahoma,iversity of Texas – Austin, University of Visin – Madison, and Washington Universitants at the 2013 AFA Meeting in San DieGroup Meeting at Harvard, and the 2012 AConference.esponding author at: Stanford University,s, 655 Knight Way, Stanford, CA 94305, Unit650 725 8239; fax: þ1 650 725 7979.ail address: istrebulaev@stanford.edu (I.A. St

a b s t r a c t

We provide a real-options model of an industry in which agents time abandonment of theirprojects in an effort to protect their reputations. Agents delay abandonment attempting tosignal quality. When a public common shock forces abandonment of a small fraction ofprojects irrespective of agents' quality, many agents abandon their projects strategically evenif they are unaffected by the shock. Such “blending in with the crowd” effect creates anadditional incentive to delay abandonment ahead of the shock, leading to accumulation of“living dead” projects, which further amplifies the shock. The potential for moderate publiccommon shocks often improves agents' values.

& 2013 Elsevier B.V. All rights reserved.

All rights reserved.

ney Hartman-Glaserggestions. For usefulbons, Deborah Lucas,Kelly Shue, seminarEconomics, London

logy, New Economiciversity of Michigan,University of Roche-rginia, University ofy in St. Louis, andgo, the 6th FinanceSU Sonoran Winter

Graduate School ofed States.

rebulaev).

1. Introduction

None love the messenger who brings bad news.Love can be even harder to find if the bad news speaksill of the messenger's abilities. When decision makers facethe unappealing task of revealing unsuccessful outcomesthat impact their reputations, delay may be their firstinstinct.1 Delay becomes even more enticing if they canwait for an industry-wide common shock to hide indivi-dual failings and instead “blend in with the crowd” byabandoning their projects strategically when some high-quality projects have to be terminated. In this paper, weargue that real investment decisions can be substantiallyaffected by such a blend-in-with-the-crowd mechanism.More importantly, this may have far-reaching repercussions

1 See, e.g., Miller (2002) and Kothari, Shu, and Wysocki (2009).

2 Two famous examples are Solyndra and Evergreen Solar. Solyndramanufactured panels based on copper indium gallium selenide (CIGS)cells. Evergreen Solar's technology was based on polysilicon, but it usedconsiderably less polysilicon in its manufacturing process thancompetitors.

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157138

for the dynamics of whole industries and provide anexplanation for industry-wide investment busts followingsmall and not necessarily negative common shocks.

The economic mechanism works as follows. When anagent operates a risky activity, such as a research anddevelopment (R&D) project or a start-up investment, a keydecision is when to abandon it if it has not yet paid off.If higher-ability agents run better projects, abandonmentis perceived as a negative signal about ability. As aconsequence, the agent has incentives to delay abandon-ment. If, in addition, informativeness of abandonmentvaries over time, the agent has incentives to time aban-donment when its informativeness is the lowest. This isthe case when the industry is hit by a common shock sothat at least some high-ability agents are forced to aban-don their projects. In an attempt to fool outsiders intobelieving that their projects also got hit by the shock,agents managing bad enough projects “blend in with thecrowd” and abandon their projects even if they areunaffected by the shock. In a dynamic environment,expectations of a common shock in the future createincentives for agents of lower quality to further delayabandonment in hope of blending in with forced abandon-ments at the time of the shock. This also leads higher-qualityagents to delay their abandonment to separate themselvesfrom lower types. This delay creates “living dead” projectsoutstanding when the common shock arrives, therebyamplifying its effect even more. As a result, even smallcommon shocks can lead to massive abandonments. Theseshocks need not be negative on aggregate: the key require-ment is that they have a negative effect on a fraction ofprojects run by high-ability agents, forcing their termination.

To aid in the intuition in the model, consider thefollowing three examples off abandonment options inthe context of the temptation to blend in with the crowd.

Example 1. Strategic home mortgage defaults in the pre-sence of systematic shocks to incomes and home prices.

Consider a homeowner contemplating a “strategicdefault” on a mortgage, which is the decision to walkaway from a negative-equity mortgage even when one canafford to pay it. As discussed in Guiso, Sapienza, andZingales (2013), whether or not a homeowner strategicallydefaults in the face of significant negative equity shocksdepends on personal factors such as moral and socialconsiderations, which are likely difficult to accuratelydiscern by outsiders. Due to the social stigma attached tostrategic defaults, reputation is harmed by early exercise ofthe default option. Thus, strategic default may be post-poned in hope of blending in with the crowd at timesof market-wide home price or income shocks. As statedin Guiso, Sapienza, and Zingales (2013), “While we doobserve defaults, we cannot observe whether a default isstrategic. Strategic defaulters have all the incentives todisguise themselves as people who cannot afford to pay…”

Example 2. Business closures in the presence of rawmaterial price shocks.

The option to shut down a business represents a classicreal option to abandon. If managerial ability is correlated

with business success and is not perfectly observable,managers will delay shutting down businesses. However,this delay is magnified if common shocks provide anopportunity to pool with higher-ability managers. An exam-ple of such a shock can be drawn from the US solar energyindustry. In 2011, the price of polysilicon, the primary rawmaterial in many solar panels, dropped to less than $40 akilogram from an all-time high of $475 in 2008, following alarge expansion of polysilicon production in China. Thisshock drove out of business many producers (both high-quality and low-quality) that pursued production technolo-gies that depended on non-polysilicon raw materials.2 How-ever, whether or not a firm's demise was due to the rawmaterial price shock was not perfectly observable. For manyprojects, especially early-stage, the propensity to fail follow-ing such as a shock depends on subtle variation in technol-ogy that is difficult for outsiders to decipher, such as theextent to which the technology can be adjusted.

Example 3. Timing of liquidation of toxic assets.

In a spirit similar to Rajan (1994), consider the timingof financial institutions’ liquidations of toxic financial assets.Since reputations matter a great deal, there is an incentive todelay recognition of large declines in asset values. Whileindividual institutions can and do liquidate deteriorated assetsat various points in time, there is a strong incentive to clustersuch actions at moments in which it is difficult for the marketto distinguish between individual characteristics and commonshocks. For example, a negative shock to an asset class as awhole (such as the bursting of a bubble or a shock to marketsentiment) can provide a protective cover to institutionswhose assets were problematic long before the shock. Rajan(1994) discusses how New England banks seemingly under-reported loan loss/earnings announcements during much of1988 and 1989, while clustering their charge-offs at times ofnegative exogenous shocks. By liquidating toxic assets at thesame moment that high-quality firms are forced to do thesame, seemingly small systematic shocks can lead to dispro-portionately large market outcomes.

To explore this mechanism in more detail, we build adynamic signaling model in the real-options framework.The starting point is a cross-section of projects that are upand running. Each project is either successful (with arandom arrival of payoff) or unsuccessful (payoff neverarrives). Neither the agent nor outsiders initially knowwhether the project can succeed. As they observe whetherthe project pays off or not, they update their beliefs aboutthe success of the project in a Bayesian manner. However,each agent has private information about the potentialpayoff from her project, which is correlated with herintrinsic ability. Outsiders are initially aware only aboutthe distribution of ability in the total pool of projects andsubsequently either observe ability at the time of theproject's successful payoff or learn about it by observing

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157 139

abandonment or lack thereof. To continue investment, theagent must spend a flow cost. The decision that the agentfaces is when to abandon the project following a persistentlack of payoff. As higher-ability agents run better projects,they take more time before abandonment. As a conse-quence, when outsiders observe that a project is aban-doned, they lower their estimate of the agent's ability.

We add to this setting a public common shock thatarrives randomly and differentially affects outstandingprojects. When it arrives, a fraction of outstanding projectsis forced to be terminated, irrespective of the agents'abilities. The two requirements on the shock are that itaffects some projects run by high-ability agents and thatthe exposure to the shock is private information of theagent. In this case, because the shock forces abandonmentof some projects run by high-ability agents, abandonmentat the time of the shock is not as informative about theagent's ability as abandonment outside of the shock.Although outsiders observe abandonment, they are unableto distinguish if it occurs because the agent's project isexposed to the shock or because the agent voluntarilyabandons the project that is not exposed to the shock.

In a world with symmetric information about projectquality, agents have no incentives to delay abandonmentin order to signal higher quality nor to pretend to beexposed to a common shock. In the baseline model ofsymmetric information, agents optimally time abandon-ment by trading off the benefits of waiting for a potentialproject payoff with the costs of continuing to incur therunning costs of operation. Given that the agent is indif-ferent between abandoning the project and continuing itfor another instant, arrival of the common shock has noeffect on the agent's payoff.

When we move to the case of asymmetric information,agents now delay abandonment for two reasons. First, sinceoutsiders perceive abandonment as a signal of the poorability of the agent, higher-ability agents delay abandonmentof their projects to the point that lower-ability agents haveno incentives to mimic them. Second, agents with lower-quality projects have an additional incentive to delay aban-donment until the occurrence of the common shock in aneffort to pool with projects that were truly exposed to theshock. Hence, when the shock arrives, agents with badenough projects abandon them even if they are not exposedto the common shock. This effect can be significant. In asimple numerical example, a shock that forces termination ofonly 1% of outstanding projects leads to an abandonment of10% of projects at the time of the shock. A variant of themodel, albeit with different common shocks, can be appliedto explaining an analogous “blending in” phenomenon forwaves of investments.3

3 Consider agents choosing when to invest in irreversible projects,whose payoffs are positively related to ability, although not fullyobservable to outsiders. Suppose a dramatic common shock hits a smallsubset of firms (where the exposure of a particular firm is unobservableto outsiders) causing the project's payoff to jump high enough to ensureimmediate investment. If this exposure is a strong enough signal ofability, then some agents whose projects are not exposed to the shockwill invest to blend-in with the high-ability managers that were exposedto the shock. Thus, a favorable shock that impacts only a few projects canresult in a larger wave of investments.

Although benefits from “blending in” are concentratedamong low-ability agents, the possibility of a commonshock delays abandonment of all agents. Because outsidersexpect low types to delay abandonment, early abandon-ment becomes an even worse signal of ability than inthe model with no common shocks. Consequently, higher-ability agents delay abandonment too as a means ofseparating from lower-ability agents. Such suboptimalinvestment in loss-making situations is consistent withsubstantial psychological research, a trait called escalationof commitment or sunk cost fallacy (Arkes and Blumer,1985; Staw and Hoang, 1995). Here, escalation of commit-ment occurs rationally as agents find it optimal to over-invest hoping that a common shock will arrive. Thisimplies that by the time the shock hits, outstandingnegative net present value (NPV) projects can be the normrather than an exception, especially if the shock followed along period of stable times. For a common shock to triggeran investment bust, it need not be negative on aggregate:As long as it is bad enough news for some projects ofhigh-ability agents, it can trigger strategic abandonmentseven if other projects are affected positively by the shock.This can rationalize why industry-specific investmentbusts sometimes do not seem to follow negative funda-mental news.

Although expectations of a common shock in the futuredelay abandonment, once the shock arrives, some agentshave incentives to speed up abandonment, occasionallyterminating sound projects that would be continued ifthere were no information frictions. Intuitively, “blendingin” can be so attractive to the agent that she prefersreceiving a moderate reputation payoff at the time ofthe shock to continuing the project and facing the risk ofabandoning it in a normal time. Both this effect and thedelay outside of the shock amplify the observed effect ofthe shock.

Interestingly, although the presence of common shocksleads to a delay and clustering of abandonments, they arenot necessarily detrimental to an industry. Without acommon shock, abandonments are excessively delayed.The effect of a common shock is thus twofold. On onehand, a common shock's arrival in the future provides anadditional incentive for the agent to delay abandonmenttoday. On the other hand, at the time of the shock someagents, who would otherwise excessively continue invest-ment, abandon their projects. The former effect decreaseseconomic efficiency and the latter can increase it.Although the combined effect is ambiguous, for a reason-able range of parameters, negative common shocks may bevalue-increasing as they play a “cleansing” role.

1.1. Related literature

This paper extends the literature that uses real-optionsmodels to analyze the timing of abandonment of invest-ment projects. Most models do not account for the signal-ing role of abandonment (e.g., Brennan and Schwartz,1985; Lambrecht, 2001; Lambrecht and Myers, 2007).Grenadier and Malenko (2011), Morellec and Schürhoff(2011), and Bustamante (2012) study the signaling roleof exercise timing, as we do in this paper. Our focus on

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157140

separating equilibria and the solution approach are bor-rowed from Grenadier and Malenko (2011). However, in allthese models there are no common shocks, which is ourfocus here.4 Clustering can also arise in pooling equilibriaeven without common shocks, as in Bustamante (2012).However, the economic reasoning behind clustering in ourmodel is different. In pooling equilibria, clustering occursbecause high types find it too costly to separate from lowtypes by pursuing a different strategy. In contrast, hereclustering occurs because low types strategically chooseto abandon projects when a common shock hits. It isthe active role of low types and the relation of clusteringto public common shocks (not necessarily negative) thatdistinguishes clustering in our model from the prior real-options literature.

Several papers study inefficient delay and/or clusteringof decisions due to various information effects. Amongthem, the most closely related models are Rajan (1994)and Acharya, DeMarzo, and Kremer (2011). Rajan (1994)studies coordination between two banks in the recogni-tion of bad loans when managers care about their reputa-tions. As reputation is less important in the adverse stateof the economy, in equilibrium bank managers recognizebad loans if the state is adverse, but not if the state isnormal.5 While both Rajan's and our model have clusteringdue to reputation concerns, there are two importantdifferences. First, Rajan's model is effectively a static modelin that bank managers can make a decision at one pointin time. In contrast, decision-making is dynamic in ourmodel, as abandonment can be undertaken at any time,including before, during, and after any common shockappears. This leads to dynamic effects: For example,expectations of the arrival of the common shock in thefuture feed back into the decision today, delaying aban-donment and adding up to the accumulation of the “livingdead” projects. Second, a common shock in our model hasa differential effect on projects. In contrast, the adversestate in Rajan's model is the same for all bank managers.Thus, models have different implications for what shockstrigger abandonment. For example, our model impliesthat industry-wide investment busts can follow commonshocks that are neutral or even positive on aggregate.

Strategic pooling of low-ability managers unaffected bythe shock with high-ability managers affected by the shockhas similarities with pooling in models of voluntarydisclosure (Dye, 1985; Jung and Kwon, 1988), in whichmanagers with bad news pool with managers that have nonews by not disclosing information. In a dynamic disclo-sure model, Acharya, DeMarzo, and Kremer (2011) showthat bad (but not good) market news triggers immediatedisclosure by firms. While their model also featuresdelay and clustering, the economic mechanism is different,

4 See also Kanodia, Bushman, and Dickhaut (1989) and Boot (1992)for models in which reputation concerns may cause managers to not stopexisting projects. This effect is similar to the delay of abandonment innormal times in our paper. However, common shocks are not studied byKanodia, Bushman, and Dickhaut (1989) and Boot (1992).

5 Agarwal and Kolev (2012) present a model similar to Rajan's (1994)to study clustering of corporate layoffs. They also provide empiricalevidence of such clustering.

because disclosure is assumed to be truthful, as in othermodels of voluntary disclosure. Clustering is also presentin the equity issuance model of Korajczyk, Lucas, andMcDonald (1992), which is based on time-varying firm-level adverse selection, and in models of herding due tolimited information of each agent (e.g., Banerjee, 1992).6

The decision when to abandon a project under privateinformation has similarities to the decision when to sell anasset in the presence of adverse selection. Thus, the paperis also related to a recent literature on repeated lemonsmarkets (e.g., Daley and Green, 2012; Fuchs and Skrzypacz,2013; Kurlat, 2013).

The remainder of the paper is organized as follows.Section 2 describes the setup of the model and solvesthe benchmark case of symmetric information. Section 3provides the solution for the main case of asymmetricinformation. Section 4 discusses implications of the model.Section 5 examines pooling equilibria, in which a set ofagents abandons their projects simultaneously even out-side of the common shock, and shows that many of theinsights are not specific to the separating equilibrium.Finally, Section 6 concludes.

2. Model setup

A risk-neutral agent operates an investment projectand decides when to abandon it. We model investment as anexperimentation problem, which is a common approach tomodel entrepreneurial and R&D activity.7 Specifically, theproject can be successful or unsuccessful. If the project issuccessful, it provides a single payoff of θ40 to the agent ata Poisson event that arrives with intensity λ40. If theproject is unsuccessful, it never pays off. At any point intime t, prior to the payoff arrival, both the agent andoutsiders assess the likelihood that the project is successfulby pðtÞ, which is updated in a Bayesian fashion as shownbelow. The initial prior is pð0Þ ¼ p0Að0;1Þ. We define thepotential project payoff θ to be the quality of the project andassume that it is the private information of the agent.Outsiders do not know θ except for its prior distribution,which is common knowledge. It is given by the probabilitydensity function (p.d.f.) f ðθÞ and the cumulative distributionfunction (c.d.f.) FðθÞ with full support on ½θ ; θ �; θ4θ40. Inpractice, the quality of the project is likely to be positivelybut imperfectly correlated with the ability of the agent. Forsimplicity, we assume it coincides with θ. To keep the projectafloat, the agent must pay cost c per unit time. Thus, theproject ends either when it pays off or when it is abandonedby the agent. The discount rate is r40.8

6 Models with delay due to information reasons also includeGennotte and Trueman (1996), in which management delays disclosureof negative news until the end of trading hours, and Milbradt (2012), inwhich an institution delays realizing losses due to fair value accountingand the balance sheet constraint.

7 E.g., see Bergemann and Hege (1998, 2005), Miao and Wang (2007),Manso (2011), and Bouvard (forthcoming).

8 An alternative way of modeling private information is to assumethat the quality of the project positively affects the probability of theproject being successful. This alternative model has similar economicsand results but features a more involved solution. The critical driver is

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157 141

The agent cares about her reputation in the eyes ofoutsiders. When a successful project pays off, all partiesobserve θ. For simplicity, we assume that the reputationpayoff upon the project payoff is γθ, where γ40 measuresthe importance of reputation. However, in the eventof abandonment outsiders do not observe θ, because theproject payoff is not realized. Instead, at any time t beforethe project either pays off or is abandoned, outsiders holdthe conditional expectation E½θjIt �, where It is the pasthistory that outsiders know at time t. The reputationpayoff upon abandonment at any time τ is thereforeγE½θjIτ�.

The decision to abandon is publicly observed andirreversible. The project may be abandoned at anytime for either of two reasons. First, it may be abandonedbecause the agent chooses to do so. Abandoning theproject allows the agent to stop paying out the flow of cat the cost of forgoing the potential payoff θ. Second, theproject may be abandoned exogenously because it getsexposed to a common shock that impacts many projects atonce. In such an event, the agent has no choice but toabandon. The common shock arrives as a Poisson eventwith intensity μ40. For tractability, we assume that therecan be only one shock. A project is exposed to the commonshock with probability q. We assume that the arrival of thecommon shock is publicly observable, but whether theproject is exposed to it is learned privately by the agentupon the arrival of the shock. For simplicity, we assumethat if the project is not exposed to the shock, its potentialpayoff is unaffected. However, this assumption is notnecessary for our results: They hold even if the shockpositively affects the potential payoff of some projects,implying that sector-specific investment busts need notfollow bad news. The key requirement is that the shockforces abandonment of some projects indiscriminately ofthe agents' quality.

We assume that it is optimal to begin operating allprojects. Specifically, even the worst project should beoperated for at least an instant. A necessary and sufficientcondition for this is

λ41p0

cθþrγ

� �: ð1Þ

2.1. Learning about success of the project

As prospects of the project are uncertain, an importantstate variable is the belief that the project is successful. LetpðtÞ denote the posterior probability that the project issuccessful given that the project has not paid off for time tsince inception. Over a short period between t and tþdt,the project pays off with probability λ dt, if it is successful,and zero, otherwise. Hence, the posterior probability pðtþdtÞconditional on not getting a payoff during ½t; tþdt� is equal to

p tþdtð Þ ¼ ð1�λ dtÞpðtÞð1�λ dtÞpðtÞþ1�pðtÞ: ð2Þ

(footnote continued)that abandonment reveals negative information about the ability of theagent to outsiders.

Rewriting (2) and taking the limit dt-0, we obtain thefollowing differential equation:

dpðtÞ ¼ �λpðtÞð1�pðtÞÞ dt; ð3Þwith the boundary condition

pð0Þ ¼ p0: ð4ÞThe solution to (3)–(4) is

p tð Þ ¼ p0ð1�p0Þeλtþp0

: ð5Þ

The dynamics of the belief process are intuitive. As long asthe project does not pay off, the belief that it is successfulcontinuously decreases over time. The speed of learning isproportional to λ, the intensity with which the successfulproject pays off. Once the project pays off, the belief that it issuccessful jumps up to one. Note that because the agent'sprivate information does not enter the belief evolutionprocess (5), the agent and outsiders always share the samebelief about the likelihood of success.

Eq. (5) implies that there is a one-to-one mappingbetween pðtÞ and the time passed since inception of theproject t. We can invert (5) to get the time since inceptionof the project given posterior p:

t pð Þ ¼ 1λln

p01�p0

1�pp

� �: ð6Þ

Thus, either the belief about the success probability of theproject p or the time passed since inception of the project tcan be used as a state variable. In what follows, we use t asa more intuitive state variable, but all valuation equationscan be equivalently rewritten in terms of p.

2.2. Benchmark case

As a benchmark, consider the case of symmetric infor-mation in which outsiders also know θ. We break up thevalue function into two regions depending on whether ornot the common shock has yet occurred. Let Vn

bðt; θÞ andVn

aðt; θÞ denote the value to the agent before and after thearrival of the common shock, respectively, conditional onthe project not paying off for time t since inception.Working backwards in time, consider first the problemafter the common shock has occurred. By Itô's lemma forjump processes (see, e.g., Shreve, 2004), in the range priorto abandonment, the evolution of Vn

aðtÞ is given by

dVn

a t; θð Þ ¼ ∂Vn

aðt; θÞ∂t

dtþ θþγθ�Vn

a t; θð Þ� �dN tð Þ; ð7Þ

where NðtÞ is a point process describing the arrival of theproject payoff. In (7), the first term reflects the changein the expected value due to learning about the success ofthe project, and the last term reflects the change in theexpected value if the project pays off (the agent receivesthe project payoff θ, the reputation payoff γθ, and forgoesVn

aðt; θÞ). Because the agent believes that the project issuccessful with probability pðtÞ, Et ½dNðtÞ� ¼ pðtÞλ dt. Divid-ing (7) by dt and taking the expectation:

EtdVn

aðt; θÞdt

� �¼ ∂Vn

aðt; θÞ∂t

þ θþγθ�Vn

a t; θð Þ� �λp tð Þ: ð8Þ

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157142

As keeping the project afloat requires a cash outflow of cper unit time, Et ½dVn

aðt; θÞ��c dt must equal rVn

aðt; θÞ dt. Thisyields the following differential equation:

rþλp tð Þð ÞVn

a t; θð Þ ¼ �cþ∂Vn

aðt; θÞ∂t

þλ 1þγð Þθp tð Þ: ð9Þ

Let τnaðθÞ denote the time since inception of the project atwhich the agent optimally abandons her project, if it hasnot paid off yet. Eq. (9) is solved subject to the followingboundary conditions:

Vn

aðτnaðθÞ; θÞ ¼ γθ; ð10Þ

∂Vn

aðτnaðθÞ; θÞ∂t

¼ 0: ð11Þ

The first equation is the value-matching condition. Itreflects the fact that upon exercise of the abandonmentoption, the value to the agent equals her reputation payoff.The second equation is the smooth-pasting condition. Itensures that the exercise trigger maximizes the agent'svalue (see, e.g., Dumas, 1991).

Evaluating (9) at τnaðθÞ yields

rγθþc¼ λθp0ð1�p0ÞeλτnaðθÞ þp0

: ð12Þ

The left-hand side of (12) denotes the costs of delayingabandonment for an additional instant: the cost of keepingthe project afloat and the delay in getting the reputationpayoff. The right-hand side of (12) denotes the benefitof delaying abandonment for an additional instant: thepotential payoff from the project. Continuation of invest-ment is optimal as long as the latter value is higher thanthe former, and condition (1) ensures that it is always thecase at the onset of the project. The agent abandons theproject when the costs and the benefit of delaying aban-donment for an additional instant coincide. Eq. (12)implies the following abandonment threshold:

τna θð Þ ¼ 1λln

p01�p0

λθ

cþrγθ�1

� �� �: ð13Þ

The comparative statics of the abandonment threshold areintuitive: abandonment occurs earlier if the waiting cost cis higher, the success intensity λ of the project is lower,the discount rate r is higher, and reputation γ is moreimportant.

Next, consider the problem of optimal abandonmentbefore the arrival of the common shock. The evolution ofVn

bðt; θÞ is given by

dVn

b t; θð Þ ¼ ∂Vn

bðt; θÞ∂t

dtþ θþγθ�Vn

b t; θð Þ� �dN tð Þþ q γθð½

�Vn

bðt; θÞÞþð1�qÞðVn

aðt; θÞ�Vn

bðt; θÞÞ� dMðtÞ;ð14Þ

where M(t) is a point process describing the arrival of thecommon shock. Compared to (7), Eq. (14) includes anadditional term reflecting the effect of a potential arrivalof the common shock. Upon its arrival, one of the twoscenarios takes place. With probability q, the project isexposed to the shock, in which case forced abandonmenttakes place and the agent gets the reputation payoff γθ.With probability 1�q, the project is not exposed to theshock, in which case the agent gets the after-shock

expected value Vn

aðt; θÞ. Dividing (14) by dt, taking theexpectation, and equating it with cþrVn

bðt; θÞ , we obtainthe following differential equation:

rþλp tð Þþμð ÞVn

b t; θð Þ ¼ �cþ∂Vn

bðt; θÞ∂t

þλ 1þγð Þθp tð Þ

þμðqγθþð1�qÞVn

aðt; θÞÞ: ð15Þ

Analogous to τnaðθÞ, let τnbðθÞ denote the time since inceptionat which the agent optimally abandons her project, if it hasnot paid off yet. By analogy with (10) and (11), Eq. (15)is solved subject to boundary conditions Vn

bðτnbðθÞ; θÞ ¼ γθand ∂Vn

bðτnbðθÞ; θÞ=∂t ¼ 0. Evaluating (15) at τnbðθÞ and rear-ranging the terms gives us

rγθþc¼ λθp0ð1�p0Þeλτ

n

bðθÞ þp0

þμ 1�qð Þ Vn

a τnb θð Þ; θ� ��γθ� �

: ð16Þ

It is easy to see that τnbðθÞ ¼ τnaðθÞ satisfies (16). It is alsothe unique solution, because the right-hand side of (16) isdecreasing in τnbðθÞ. Thus, the abandonment policy of theagent is the same both before and after the arrival of thecommon shock. This may seem surprising: casual intuitionmay suggest that a possibility of the common shock in thefuture adds to the cost of delaying abandonment and leadsto earlier abandonment prior to the shock. However, thisintuition is incorrect. The reason is that the abandonmentthreshold is determined by the indifference condition ofthe agent. When the agent is indifferent between aban-doning the project and continuing it for another instant,arrival of the common shock has no effect on the agent'spayoff: it simply forces her to accept one option of the twothat she is indifferent between. Hence, the optimal aban-donment policy of the agent is unaffected by the potentialarrival of the common shock. We denote this full-information trigger by τnðθÞ ¼ τnaðθÞ ¼ τnbðθÞ. In the absenceof information frictions between the agent and outsiders,the only consequence of the common shock is to introducenoise in abandonment by forcing the agent to abandon theproject prematurely if it gets exposed to the shock.Although the abandonment thresholds are the samebefore and after the arrival of the shock, the value func-tions are not. Because the common shock may triggerabandonment at a suboptimal time, Vn

bðt; θÞoVn

aðt; θÞ forany toτnðθÞ.

The optimal abandonment strategy in the benchmarkcase is summarized in Proposition 1:

Proposition 1. Suppose that θ is known to both the agent andoutsiders. Then, the optimal strategy of the agent is toabandon the project if it has not paid out for time τnðθÞ.When the common shock arrives, the project is abandoned ifand only if it is exposed to the shock.

Let us see how τnðθÞ depends on θ:

ddθ

τn θð Þ ¼ 1λθ

rγθþ c�1� c

ðrγθþcÞ240; ð17Þ

because τnðθÞ40. Abandonment in the benchmark casesatisfies three properties. First, the equilibrium abandon-ment strategy is optimal in that there is no other abandon-ment strategy that yields a higher expected payoff tothe agent. Second, better projects are abandoned later.

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157 143

Intuitively, if the potential payoff from the project ishigher, then the agent optimally waits longer beforescrapping it. Finally, the common shock affects abandon-ment if and only if the project is exposed to the shock.

3. Private information case

In this section, we provide the solution for the maincase of asymmetric information between the agent andoutsiders. Even though there is no revelation of θwhen theproject is abandoned, outsiders try to infer θ from obser-ving the timing of abandonment. In turn, the agentattempts to manipulate her abandonment decision toshape the belief of outsiders. If there were no commonshock, this would lead to a separating equilibrium, as inGrenadier and Malenko (2011): each type delays abandon-ment in the hope of fooling outsiders into believing thather type is higher, but in equilibrium outsiders learn θwith certainty. However, with a common shock, theequilibrium is more interesting: the agent may choose totime abandonment to the arrival of the common shock. Bydoing this, she blends in with agents of higher types thatare forced to abandon due to their exposure to the shock.

Because we assume only one common shock, we solvethe model by backward induction. First, we consider thetiming of abandonment after the shock has already arrivedat time s in the past. In this case, no future shock arrival ispossible, so the solution is standard. Second, we considerthe problem of abandoning the project at the time of theshock. Finally, we consider the problem of abandoning theproject prior to the arrival of the shock.

In the first and third stages of the problem, our focus ison the separating Bayesian Nash equilibrium. There arealso pooling or semi-pooling equilibria. In these equilibria,several types abandon projects at the same time evenwithout the common shock due to self-fulfilling beliefs.There are two reasons for our focus on the separatingequilibrium. First, because we are interested in howcommon shocks trigger waves of strategic abandonments,it is natural to abstract from possible waves of abandon-ments in other periods. Second, pooling and semi-poolingequilibria typically do not survive the D1 equilibriumrefinement, which intuitively requires outsiders to placezero posterior belief on type θ for any off-equilibriumaction whenever there is another type θ′ with a strongerincentive to deviate.9 In the Appendix we provide intuitiveoff-the-equilibrium-path beliefs that satisfy the D1 refine-ment. Such beliefs satisfy an intuitive property that theoutsiders' best guess of the agent's type evolves continu-ously.10 In contrast, a pooling equilibrium requires out-siders' belief to change discontinuously over time if

9 See Cho and Sobel (1990) and Ramey (1996) for the proof of thisresult for signaling models with discrete types and a continuum of types,respectively. In the Appendix, we also provide the off-the-equilibrium-path beliefs that support the equilibrium and satisfy the D1 refinement.

10 In addition, the beliefs are such that outsiders identify the mostextreme actions with the most extreme types. Any agent that abandonseven earlier then the lowest type does in equilibrium is perceived to bethe worst type, and any agent that delays abandonment beyond thehighest type is perceived to be the highest type.

no common shock arrives. Because of this, we find theseparating equilibrium in the first and third stages of theproblem as more empirically plausible, so we relate ourimplications to it. However, we extend the analysis byconsidering pooling in the first and third stages of theproblem in Section 5. It shows that a common shocktriggers strategic abandonments by some unaffected typesin this case too. We provide a heuristic analysis in thissection and offer formal proofs in the Appendix.

3.1. Abandonment after the arrival of the shock

Consider the decision of type θ to abandon the projectafter the common shock has already occurred at time s inthe past. Conjecture that the distribution of types rightafter the common shock is truncated at some type θ frombelow. This conjecture is verified later. Then, right after thecommon shock at time s, outsiders believe that θ isdistributed over ½θ ; θ � with p.d.f. f ðθÞ=ð1�FðθÞÞ. In thespecial case of s¼0 and θ ¼ θ , this case corresponds tothe overall game if there were no common shocks.

We first solve for the agent's abandonment strategyfor a given belief function of outsiders, and then applythe rational expectations condition that the abandonmentstrategy and the belief function of outsiders must beconsistent with each other. Specifically, suppose that out-siders believe that type θ abandons the project once it hasnot paid off for time τaðθÞ, which is a monotonic anddifferentiable function of θ. Thus, if the agent abandonsthe project at time τ, outsiders infer that the type is~θðτÞ ¼ τ�1

a ðτÞ. Let Vaðt; ~θ ; θ; τÞ denote the value to the agent,conditional on the project not paying off for time t, for afixed belief of outsiders upon abandonment ~θ , wheretA ½s; τ� is the time since inception of the project, θ is thetrue type, and τ is the abandonment time. In the regionprior to abandonment, Vaðt; ~θ ; θ; τÞ satisfies a differentialequation identical to (9):

rþλp tð Þð ÞVa ¼ �cþ∂Va

∂tþλ 1þγð Þθp tð Þ: ð18Þ

Upon abandonment of the project, the agent gets thereputation payoff of γ ~θ . It implies the following boundarycondition:

Vaðτ; ~θ; θ; τÞ ¼ γ ~θ : ð19ÞSolving (18) subject to boundary (19) yields the value tothe agent for a given abandonment threshold and thebelief of outsiders:

Va t; ~θ ; θ; τ� �¼ eðrþ λÞtp tð ÞU ~θ ; θ; τ

� ��crþ λ

rþλ

crþθþγθ

� p tð Þ;

ð20Þwhere:

U ~θ ; θ; τ� �¼ 1�p0

p0e� rτ c

rþγ ~θ

� þe�ðrþ λÞτ c�λð1þγÞθ

rþλþγ ~θ

� �:

ð21ÞGiven the hypothesized inference function τaðθÞ, we cansubstitute ~θðτÞ ¼ τ�1

a ðτÞ for ~θ in (20)–(21) and take the first-order condition with respect to τ:

�c�rγ ~θðτÞþλpðτÞðθþγðθ� ~θðτÞÞÞþγ ~θ′ðτÞ ¼ 0: ð22Þ

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157144

Eq. (22) reflects the costs and benefits of postponingthe abandonment. The first two terms reflect the costs ofwaiting an additional instant dt: paying a cost c dt anddelaying the reputation payoff. The third term reflects thebenefits of waiting an additional instant: the possibilitythat the project will pay off. The probability of the projectpaying off over the next instant is λpðτÞ dt, and the payoffexceeds the abandonment payoff by θþγðθ� ~θðτÞÞ. Finally,the last term reflects the effect of waiting an instant on theoutsiders' inference of the agent's type: abandoning theproject an instant later changes the reputation payoff byγ ~θ′ðτÞ dt.

Applying the rational expectations condition that inequilibrium outsiders' beliefs must be consistent with theagent's abandonment strategy ð ~θðτaðθÞÞ ¼ θÞ, we get theequilibrium differential equation:

dτadθ

¼ γ

cþrγθ�λθpðτaðθÞÞ; θA θ ; θ

h i: ð23Þ

This equation is solved subject to the standard initialvalue condition that the lowest type in the post-shockhistory, θ , abandons at the symmetric information threshold:

τaðθÞ ¼maxfτnðθÞ; sg; ð24Þ

where τnð�Þ is given by (13). Intuitively, if the lowest type didnot abandon at the symmetric information threshold, shewould find it optimal to deviate to it. This deviation not onlywould improve the direct payoff from exercise, but alsocould improve the reputation payoff, since the current beliefis already as bad as possible. If τnðθÞZs, then abandonmentat τnðθÞ is feasible. If τnðθÞos, then abandonment at τnðθÞ isnot feasible. In this case, the most preferred abandonmenttiming of type θ is immediately after the common shock.

By verifying that Uð ~θ ; θ; τÞ satisfies the regularity con-ditions in Mailath (1987), Proposition 2 shows that thisargument indeed yields a unique (up to the off-equilibrium beliefs) separating equilibrium threshold τaðθÞ.

Proposition 2. Let τaðθÞ be the increasing function that solvesEq. (23) subject to the initial value condition (24). Then, τaðθÞis the abandonment threshold of type θ in the unique (up tothe off-equilibrium beliefs) separating equilibrium.

It immediately follows that the equilibrium abandon-ment threshold after the arrival of the shock depends onthe lowest type in this region, θ , and the arrival time ofthe shock s. Thus, we denote the after-shock equilibriumabandonment threshold by τaðθ; θ ; sÞ. We denote the equi-librium value function, Vaðt; θ; θ; τaðθ; θ ; sÞÞ, by V aðt; θ; θ ; sÞ.

Note that the special case of this problem, s¼0 andθ ¼ θ , corresponds to the whole abandonment problemunder asymmetric information, if there were no commonshocks. Thus, the equilibrium threshold in the modelwithout common shocks is τaðθ; θ ;0Þ. By Proposition 2, itsolves (23) subject to τaðθ; θ ;0Þ ¼ τnðθÞ. Compared to thecase of symmetric information, the agent excessivelydelays abandonment: τaðθ; θ ;0Þ4τnðθÞ for any θ4θ . Werefer to this special case in Section 4.2, in which wedecompose the delay into two effects: delay caused bysignaling incentives of the agent and delay caused by“blending in” with others at the common shock.

3.2. Abandonment upon the arrival of the shock

Consider the decision of type θ to abandon the projectupon the arrival of the common shock. Analogously tothe previous section, conjecture that the agent's timing ofabandonment before the arrival of the common shock isgiven by threshold τbðθÞ, which is increasing in θ. Con-jecture also that τbðθÞZτnðθÞ for any θ, i.e., the agent delaysabandonment of her project compared to the benchmarkcase. Both conjectures are verified in Section 3.3. Giventhese conjectures, the posterior belief of outsiders uponthe arrival of the common shock at time s is that θ isdistributed over ½ �θ ; θ� with p.d.f. f ðθÞ=ð1�Fð �θÞÞ, where�θ ¼ τ�1

b ðsÞ.When the common shock arrives, the project may be

exposed to it (with probability q) or not exposed to it (withprobability 1�q). In the former case, the agent has nochoice but to abandon the project. In the latter case, theagent may abandon the project strategically together withthe exposed projects or postpone the abandonment. If theagent abandons the project immediately, she gets a repu-tation payoff that is independent of her true type. If theagent postpones the abandonment, her expected payoff isincreasing in her true type. Hence, there exists typeθA ½ �θ ; θ � such that the non-exposed project is abandonedif θo θ and not abandoned if θ4 θ . Provided that θoθ ,there can be two scenarios depending on the abandon-ment strategy of the type θþɛ that is marginally higherthan θ:

Scenario 1: Type θþɛ abandons the project immedi-ately after the common shock. This happens if and only if

sZτnðθÞ; ð25Þ

γ

R θ�θ θf ðθÞ dθþq

R θθ θf ðθÞ dθ

FðθÞ�Fð �θÞþqð1�FðθÞÞ¼ γθ: ð26Þ

Condition (25) states that the optimal abandonment timeof type θ is upon the arrival of the common shock orimmediately after. Condition (26) states that type θ isexactly indifferent between blending in and abandoning atthe time of the common shock (the payoff of this strategyis the left-hand side of (26)) and separating and abandon-ing a second after the common shock.

Scenario 2: Type θþɛ abandons at the symmetricinformation threshold τnðθþɛÞ. This happens if and only if

soτnðθÞ; ð27Þ

γ

R θ�θ θf ðθÞ dθþq

R θθ θf ðθÞ dθ

FðθÞ�Fð �θÞþqð1�FðθÞÞ¼ Vn

aðs; θÞ; ð28Þ

where Vn

aðs; θÞ ¼ V aðs; θ ; θ ; sÞ is the value to type θ in thesymmetric information framework, when time s has passedsince the inception of the project. Note that Vn

aðs; θÞ is thevalue in the world of full information, and thus its solution issimple to calculate. Condition (27) states that the optimalabandonment time of type θ is given by threshold τnðθÞ.Condition (28) states that type θ is indifferent betweenabandoning at the time of the common shock and postpon-ing abandonment until her optimal separating threshold.

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157 145

Note that because Vn

aðt; θÞ ¼ γθ for all tZτnðθÞ, condition(26) is a special case of condition (28).11

Scenarios 1 and 2 imply that the equilibrium can have oneof two forms. In the first scenario, abandonment happenscontinuously with positive probability, while in the secondscenario a wave of abandonments triggered by the commonshock is followed by a period of no abandonment. The nextproposition shows that there exists a unique indifferent typeθ in equilibrium, and it is always strictly between �θ and θ:

Proposition 3. At the moment the common shock hits, typesθo θ that are not impacted by the shock voluntarily abandonto pool with those that are impacted by the shock. θAð �θ; θÞ isunique, and computed in the following way. Let θm denotethe unique solution to (26). If sZτnðθmÞ, then θ ¼ θm. IfsoτnðθmÞ, then θ is determined as the unique solution to(28). Since θ depends on s and �θ , we denote the equilibriumindifference type by θð �θ ; sÞ.

The equilibrium structure is illustrated in Fig. 1. Priorto the arrival of the shock, as time goes by, low typesabandon their projects: type θ abandons her project attime τbðθÞ, where τbðθÞ is to be determined in the nextsection. This truncates the distribution of existing typesfrom below. When the common shock arrives at date s,the posterior distribution of types is ½ �θðsÞ; θ�, where�θðsÞ ¼ τ�1

b ðsÞ. Upon arrival of the shock, fraction q of alltypes ½ �θðsÞ; θ � is hit by the shock and forced to abandontheir projects. On top of this, there exists an intermediatetype θð �θðsÞ; sÞ, such that each type in ½ �θðsÞ; θð �θðsÞ; sÞ� aban-dons her project even if it is not exposed to the shock.

The results of this section have two interesting implica-tions. First, because θ4 �θ , the common shock leads toabandonment of more projects than are affected by theshock directly. The difference between θ and �θ measuresthe extent of strategic blending in by low-ability agents.We explore the economic implications of this finding inSection 4.

Second, depending on whether the equilibrium belongsto scenario 1 or scenario 2, there can be different time-series properties of abandonments. In the former case, thecommon shock triggers a wave of abandonments, but afterthe shock is over the abandonment rate is comparable tonormal times. By contrast, in the latter case, the commonshock triggers a wave of abandonments, which is followedby a quiet period of no abandonments. We also discussthese implications in more detail in Section 4.

Let θblendðs; �θÞ denote the equilibrium expected typeconditional on blending in and let V csðs; θ; �θÞ denote thevalue to type θwhen the common shock arrives and beforethe agent learns if her project is exposed to the shock.Then,

V csðs; θ; �θÞ ¼ qγθblendðs; �θÞþð1�qÞmaxfγθblendðs; �θÞ;

V aðs; θ; θð �θ; sÞ; sÞg: ð29Þ

11 Conditions (28) and (26) implicitly assume that �θo θoθ . If θ ¼ �θ

or θ ¼ θ , then (28) and (26) may hold as strict inequalities. In the proof ofProposition 3, we establish that θAð �θ ; θÞ indeed.

3.3. Abandonment before the arrival of the shock

To complete the solution of the model, consider thedecision of type θ to abandon the project before the arrivalof the common shock. Similar to Section 3.1, we solve forthe separating equilibrium in two steps. First, we solve forthe agent's abandonment strategy for a given inferencefunction of outsiders. Second, we apply the rational expecta-tions condition that the inference function of outsiders mustbe consistent with the abandonment strategy of the agent.

Suppose that outsiders believe that type θ abandons theproject if it has not paid for time τbðθÞ since inception, whereτbðθÞ is an increasing and differentiable function of θ. Hence,if the agent abandons the project at time τ, outsiders updatetheir belief about the agent's type to ~θðτÞ ¼ τ�1

b ðτÞ. LetVbðt; ~θ; θ; τÞ denote the value to the agent in the range priorto the arrival of the shock, conditional on the project notpaying off for time t, for a fixed belief of outsiders ~θ , wheretrτ is the time since inception of the project, θ is the truetype, and τ is the abandonment timing. Using the standardargument, in the region prior to abandonment, Vbðt; ~θ ; θ; τÞsatisfies the following differential equation:

rþλp tð Þþμð ÞVb ¼ �cþ∂Vb

∂tþλ 1þγð Þθp tð ÞþμV cs t; θ; ~θ tð Þ� �

:

ð30ÞCompared to (18), Eq. (30) includes an additional term thatreflects the possibility that a common shock arrives overthe next instant. In this case, the value function jumps toV csðt; θ; ~θðtÞÞ, given by (29).

Upon abandonment of the project at τ, the payoff to theagent is γ ~θ , implying the boundary condition:

Vbðτ; ~θ ; θ; τÞ ¼ γ ~θ : ð31ÞIf we set ~θ ¼ ~θðτÞ in (31), Eqs. (30)–(31) yield the value tothe agent for a given abandonment timing τ and outsiders'inference function ~θðτÞ. For a given outsiders' inferencefunction, the optimal abandonment timing τ must be suchthat the agent's payoff at the abandonment point satisfiesthe smooth-pasting condition:

∂Vb

∂t t-τ ¼ γ ~θ′ τð Þ: ð32Þ

Plugging (31) and (32) into (30) yields the followingequation for the agent's abandonment timing for a givenoutsiders' inference function:

�c�rγ ~θðτÞþλpðτÞðθþγðθ� ~θðτÞÞÞþγ ~θ′ðτÞ

þμðV csðτ; θ; ~θðτÞÞ�γ ~θðτÞÞ ¼ 0: ð33ÞEq. (33) is similar to the analogous Eq. (22) in Section 3.1. Itreflects the costs and benefits of postponing the abandon-ment for an additional instant. The first two terms reflectthe costs and the next two terms reflect the benefits. Inaddition, Eq. (33) includes an additional term that reflectspotential arrival of the common shock. If the agent delaysabandonment for an additional instant dt, the commonshock will arrive over this instant with probability μ dt.Upon the arrival of the common shock, the value of theproject to the agent jumps to V csðτ; θ; ~θðτÞÞ.

Next, we can apply the rational expectations conditionthat the agent's abandonment strategy must be consistent

Fig. 1. Equilibrium structure upon the arrival of the shock. The figureillustrates the equilibrium abandonment dynamics when the commonshock arrives at time s. Right before the arrival of the shock, types above�θðsÞ have not abandoned their projects yet. When the shock arrives, typesbetween �θðsÞ and θð �θðsÞ; sÞ abandon their projects even if they are notexposed to the shock and types above θð �θðsÞ; sÞ abandon their projectsonly if they are exposed to the shock.

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157146

with outsiders' inference function. This condition impliesthat at the agent's chosen abandonment timing τ , ~θðτÞ ¼ θ.Hence, we can substitute τ¼ τbðθÞ, ~θðτÞ ¼ θ, and ~θ

′ðτÞ ¼1=τ′bðθÞ in (33) and get the equilibrium equation for theabandonment timing τbðθÞ:

�c�rγθþλθp τb θð Þð Þþ γ

τ′bðθÞþμ V cs τb θð Þ; θ; θð Þ�γθ� �¼ 0:

ð34ÞNote that V csðτbðθÞ; θ; θÞ is the value that type θ gets at thearrival of the common shock, if she is the lowest typeremaining at the time of the shock. Because θð �θ ; τbð �θÞÞ4 �θ ,as established in Section 3.2, type θ always chooses toabandon the project in this case, even if the project isnot exposed to the shock. Therefore, V csðτbðθÞ; θ; θÞ ¼γθblendðθ; τbðθÞÞ. Consequently, Eq. (34) reduces to

dτbdθ

¼ γ

cþrγθ�λθpðτbðθÞÞ�μγðθblendðθ; τbðθÞÞ�θÞ; θA θ; θ� �

:

ð35ÞCompared to (23), the denominator of (35) includes anadditional term �μγðθblendðθ; τbðθÞÞ�θÞ. It reflects an addi-tional incentive to delay abandonment due to the potentialarrival of the common shock, which allows the agent toblend in with higher types.

Eq. (35) is solved subject to the appropriate initial valuecondition. Analogous to Section 3.1, the appropriate initialvalue condition is that the lowest type θ abandons theproject as if her abandonment decision did not revealinformation to outsiders:

τb θ� �¼ 1

λln

p01�p0

λθ

cþrγθ�μγðθblendðθ ; τbðθÞÞ�θÞ�1� �� �

:

ð36ÞNote that the right-hand side of (36) is the most preferredthreshold of type θ in the symmetric information frame-work, taking payoff upon arrival of the common shock asgiven. The intuition behind (36) is simple. If the lowesttype did not abandon at this threshold, she would find itoptimal to deviate to it. Such deviation not only wouldincrease the direct payoff from abandonment but alsocould increase the reputation payoff, because the currentoutsiders' inference is already as low as possible. Eqs. (35)–(36) yield the separating equilibrium abandonment timingτbðθÞ. We denote the corresponding equilibrium valuefunction by V bðt; θÞ.

By verifying that the solution to (30)–(31) satisfiesthe regularity conditions in Mailath (1987), Proposition 4shows that the argument above indeed yields the unique

separating equilibrium threshold τbðθÞ. The proof appearsin the Appendix.

Proposition 4. Let τbðθÞ be the increasing function that solvesdifferential equation (35) subject to the initial value condition(36). Then, τbðθÞ is the abandonment threshold of type θ in theunique (up to the off-equilibrium beliefs) separating equilibriumin the range prior to the arrival of the common shock.

Taken together, Propositions 2–4 imply that there existsa unique equilibrium in which different types separate viadifferent abandonment strategies. At most times thecommon shock does not arrive, so different types abandonprojects at different instances: worse (lower θ) projectsare abandoned earlier, and better (higher θ) projectsare abandoned later. However, on rare occasions when acommon shock triggers abandonment of some projects foran exogenous reason, a wave of abandonments occurs.Importantly, the wave exceeds the fraction of projectsexposed to the shock: agents with bad enough projectschoose to blend in with the crowd and abandon theirprojects together with projects exposed to the shock, eventhough their projects are not affected by the shock directly.

4. Model implications

In this section we analyze the economic implications ofthe model. We first show that even relatively small shockscause many agents to time their abandonment decisions toperiods of common shocks. Then, we show that expecta-tions of a common shock arrival in the future feed back tothe agents’ abandonment strategies today, leading them todelay abandonment. Interestingly, although only low typesbenefit from blending in, common shocks considerablydelay abandonment decisions of all types through theseparation effect. This implies that industries can accumu-late over time a significant number of the “living dead”projects. We quantify the delay caused by common shocksand by signaling and show that the two sources of delayaffect mostly different projects. Finally, we show that smallcommon shocks can actually improve the aggregate pre-sent value of all projects in an industry, even though theycreate additional frictions in abandonment.

4.1. How significant is “blending in”?

First, we quantify the magnitude of the “blending in”effect. Suppose that a fraction q of projects is exposed tothe shock. Out of firms that are not exposed to the shock,fraction ðFðθÞ�Fð �θÞÞ=ð1�Fð �θÞÞ abandons voluntarily, wherethe lowest-quality project that did not abandon by thetime of the shock, �θ , and the higher-quality project thatstrategically blends in, θ , are defined in Section 3.2. Tostudy the importance of strategic abandonment, we intro-duce two measures:

1.

The fraction of projects that blend in out of thosenot exposed to the shock:

ϕ sð Þ ¼ Fðθðτ�1b ðsÞ; sÞÞ�Fðτ�1

b ðsÞÞ1�Fðτ�1

b ðsÞÞ : ð37Þ

0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Arrival time of the shock

ni sdnelb t aht noit carF

0 5 10 151

2

3

4

5

6

7

8

9

10

11

Arrival time of the shock

reilpitlum noit aci fi nga

MFig. 2. The “blending in” effect. The panel plots the fraction of projects that blend in out of those not exposed to the shock, ϕðsÞ, and the magnificationmultiplier, MðsÞ, as a function of the arrival time of the shock. Both ϕðsÞ and MðsÞ are plotted for four different values of q : 0.01 (the solid line), 0.05(the dashed line), 0.1 (the dotted line), and 0.25 (the dash-dotted line). The distribution of θ is uniform on [1, 2]. The other parameters arer¼ c¼ 0:05; λ¼ μ¼ 0:2; γ ¼ p0 ¼ 0:5.

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157 147

By construction, ϕA ½0;1�. If the “ blending in” effectis absent, then ϕ¼ 0 as only projects that areexposed to the shock are abandoned upon its arrival.If the “blending in” effect is present, then ϕ40.

2.

The magnification multiplier, defined as the ratio ofall projects abandoned at the time of the commonshock to projects that are abandoned because theyare exposed to the shock:

M sð Þ ¼ qð1�Fðτ�1b ðsÞÞÞþð1�qÞðFðθðτ�1

b ðsÞ; sÞÞ�Fðτ�1b ðsÞÞÞ

qð1�Fðτ�1b ðsÞÞÞ

¼ 1þ1�qq

ϕ sð Þ: ð38Þ

By construction, MZ1. If the “blending in” effect isabsent, then the magnification multiplier is equal toone: only projects that are exposed to the commonshock are abandoned upon its arrival. If the “blend-ing” in effect is present, then the magnificationmultiplier exceeds one.

12 One exception is a recent model by Nanda and Rhodes-Kropf(2012). They examine “sunspot” equilibria, in which industry-wideinvestment busts arise without any news, because financiers collectivelyswitch from one equilibrium to the other.

Fig. 2 quantifies the “blending in” effect for fourdifferent values of the fraction of firms exposed to theshock, q, for the case of uniform distribution of types. Theleft panel plots the fraction of projects that blend in out ofthose not exposed to the shock, ϕðsÞ. The right panel plotsthe corresponding magnification multiplier, MðsÞ. The “

blending in” effect appears to be quite significant. Forexample, when the common shock affects 5% of outstand-ing projects, 17.5% of projects that are not exposed to theshock abandon strategically at the same time. This impliesthe magnification multiplier of 4.5: every project that gothit by the shock leads to 4.5 projects being abandoned.As the common shock becomes bigger (q is higher), thefraction of projects that are abandoned strategically also

increases. For example, a large shock affecting every fourthproject triggers abandonment of almost 60% of outstand-ing projects. At the same time, larger shocks are associatedwith lower magnification multipliers. Interestingly, evenvery small shocks can lead to large waves of abandon-ments. For example, if the common shock hits the projectwith only 1% probability, the magnification multiplier is10, meaning that 10% of the projects are abandoned at theshock: 1% are forced and the other 9% do so strategically.

Importantly, large waves of abandonments need notfollow bad industry-wide shocks. A neutral or even posi-tive common shock that affects projects differentially maytrigger a wave of abandonments, provided that it forcessome high types to abandon their projects. Recall theexample in the introduction about the impact of thedecline in the price of polysilicon on the US solar energyindustry. Firms that relied heavily on polysilicon as aninput viewed this as a positive shock, while firms whoseproduction technologies depended on non-polysilicon rawmaterials viewed this as a negative shock. Some of thefirms that were forced to liquidate due to their depen-dence on non-polysilicon raw materials could very wellhave been of high quality. Provided that it was difficult foroutsiders to decipher if project abandonment was causedby a shock or not, such common shocks can lead tomassive liquidations, far exceeding the number of projectsnegatively affected by the shock. This implication contrastswith most other models of clustering, in which informa-tion frictions amplify bad news, so clustering alwaysfollows a negative aggregate shock.12

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Fig. 3. Comparison of abandonment policies. The figure plots threeabandonment thresholds: τnðθÞ, abandonment threshold in the friction-less economy (the solid line); τaðθ; θ ;0Þ, abandonment threshold underasymmetric information in the economy without common shocks (thedashed line); τbðθÞ, abandonment threshold under asymmetric informa-tion in the economy with common shocks (the dotted line). The baselinecase has q¼0.05. The distribution of θ is uniform on [1,2]. The otherparameters are r¼ c¼ 0:05, λ¼ μ¼ 0:2, γ ¼ p0 ¼ 0:5.

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157148

4.2. Delay due to “blending in” and the “living dead”

Possible arrival of the common shock in the futurecreates incentives for the agent to delay abandonmenttoday. This is evident from Section 3.3: the equilibriumabandonment timing τbðθÞ increases in the shock arrivalintensity, μ. Intuitively, bad types have incentives to delayabandonment in an effort to pool with better types whenthe common shock hits. Interestingly, abandonments of allprojects are delayed, even projects with very high qualityθ. This might seem counter-intuitive, because agentsmanaging projects with very high θ have little incentiveto wait for the common shock and pool. Intuitively, theseagents delay abandonment of their projects because thetiming of abandonment is a signaling device about quality.Because lower types delay their abandonments, highertypes are forced to delay their abandonments even moreto separate from lower types. Hence, higher μ delaysabandonment of all types.

Fig. 3 plots three abandonment thresholds: the thresh-old in the case of symmetric information, τnðθÞ, the thresh-old in the case of asymmetric information when there areno common shocks, τaðθ; θ;0Þ, and the threshold in thecase of asymmetric information when there is a possibilityof a common shock in the future, τbðθÞ. Fig. 3 illustratestwo sources of delay compared to the case of symmetricinformation. The first source of delay is costly signaling:higher types delay abandonment of their projects toseparate from earlier abandonments by lower types andthereby get a higher reputation payoff. This effect drivesthe change of the abandonment trigger from τnðθÞ toτaðθ; θ ;0Þ. The second source of delay is “blending in.”Lower types have incentives to wait for the common shock

in expectation of pooling with higher types exposed to theshock. This allows lower types to get a higher reputationpayoff at the expense of higher types and leads to anadditional delay compared to τaðθ; θ ;0Þ.

These effects are illustrated in more detail in Fig. 4.It shows the extent of delay caused by each of the twomechanisms. Signaling incentives lead to a delay in aban-donment ranging from no delay (for the lowest type) to160% (for the highest type) relative to the first-bestabandonment timing. The additional effect of “blendingin” is more uniform for all types. For the case of q¼0.01, “blending in” leads to a delay in abandonment timingranging from 40% (for the lowest type) to 20% (for thehighest type). As the magnitude of the shock increases, theextent of delay caused by “blending in” increases as well.For example, if q¼0.05, a typical project is delayed by 60%due to “blending in.”

An interesting consequence of delaying behavior is thatmany projects are kept alive even though in the first-bestworld they would have been abandoned a long time ago.To study this phenomenon, we measure the fraction ofthese firms and refer to them as the “living dead.” In thefirst-best setting, the agent managing a type θ project willchoose to abandon it if it has not paid off for time τnðθÞ,determined in Section 2. Let θnðτÞ equal the inverse of τnðθÞ.Then, all projects whose types are less than θnðtÞ will beabandoned by time t. Similarly, in the private informationsetting prior to the arrival of the common shock, the agentmanaging a type θ project will choose to abandon it if ithas not paid off for time τbðθÞ. Let θbðτÞ equal the inverse ofτbðθÞ. All projects whose types are less than θbðtÞ will beabandoned by time t. Let LD(t) denote the fraction ofoutstanding projects at time t that are among the livingdead. We can write this as

LD tð Þ ¼ F½θnðtÞ��F½θbðtÞ�1�F½θbðtÞ�

: ð39Þ

In Fig. 5 we see that for the first 2.5 years there areno living dead because the likelihood of success is highenough to make abandonment suboptimal even in thefirst-best case. However, beyond that point, agents mana-ging bad enough projects begin abandoning them in thefirst-best case, while a substantially smaller number ofprojects are abandoned in the private information case.After about five years, given the small posterior likelihoodof success, all types will have abandoned in the first-bestcase. This implies that all of the non-abandoned projects inthe private information setting are among the living dead.

This result implies that in many industries, negativeNPV projects are likely to be the norm rather than anexception, especially after a long streak of stable times,during which the industry has experienced few shocks andaccumulated “bad cholesterol” projects. It also explainsat an intuitive level why small shocks can lead to theavalanche of industry abandonments. By the time theshock hits it is not unlikely that most outstanding projectshave negative NPV. Thus, when we observe what appearsto be a major collapse in which large numbers of firms exitan industry, we may actually be seeing the public exit offirms that were essentially dead for some time; they weresimply waiting for an opportune time to leave in unison.

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Fig. 4. Delay caused by signaling vs. delay caused by “blending in.” The figure plots the relative delay caused by signaling (the solid line),ðτaðθ; θ ;0Þ�τnðθÞÞ=τnðθÞ, and caused by “blending in” (the dashed line), ðτbðθÞ�τaðθ; θ ;0ÞÞ=τnðθÞ, for different θ. The top-left panel plots the case ofq¼0.01. The top-right panel plots the case of q¼0.05. The bottom panel plots the case of q¼0.25. The distribution of θ is uniform on [1, 2]. The otherparameters are r¼ c¼ 0:05, λ¼ μ¼ 0:2, γ ¼ p0 ¼ 0:5.

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157 149

4.3. The value of common shocks

Although one might think that agents of all typeswould find the possibility of being hit by a common shockharmful, we find that virtually all types can benefit fromthe existence of such shocks. Consider the competingimpacts of market shocks for different quality types. Fora lower-quality agent, having the possibility of a commonshock provides the real opportunity for blending in, andthus avoiding recognition of her low type. On the otherhand, if the likelihood of a common shock becomes toohigh, then the agent faces the detrimental event of havinga potentially good project exogenously eliminated. Weshould thus expect lower-quality types to prefer moderatelikelihoods of common shocks, at the point at which thesetrade-offs are equated. For a higher-quality agent, there is

still some moderate benefit from having a possibility ofa common shock. While the ability to blend in is ratherunimportant for a high type, the possibility of the shockreducing the number of firms means that the cost ofsignaling its high type is lower, and it can abandon atcloser to the first-best time. This moderate benefit istraded off against the cost of being hit by the shock andlosing a potentially valuable project. We should thusexpect higher-quality types to prefer low, but positivelikelihoods of common shocks.

Fig. 6 illustrates this argument. The left panel showshow the expected value of the average agent changes withthe intensity of the shock arrival, μ. It suggests that theaverage agent benefits the most from negative commonshocks of moderate size. Under this set of parametervalues, she would prefer that the shock have an expected

0.25

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157150

arrival time of around two years ðμ¼ 0:51Þ. The figureshowing the average expected value of agents as a functionof the intensity of the shock arrival is similar. The rightpanel of Fig. 6 shows the value-maximizing intensity of thecommon shock for several different types, ranging fromthe lowest to the highest. For the lowest type, the optimalexpected arrival time of a shock is the quickest, at around1.5 years (μ¼0.68), given that they benefit most from the

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daed gnivil eht f o noit car F

Fig. 5. The “living dead.” The figure plots the fraction LDðtÞ of outstandingprojects at time t that are among the living dead, meaning that theywould have already been abandoned if information were symmetric.The distribution of θ is uniform on [1, 2]. The other parameters arer ¼ c¼ 0:05, λ¼ μ¼ 0:2, γ ¼ p0 ¼ 0:5.

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0 etad ta eul aV

Fig. 6. The effect of the common shock on the agents' expected values. The lefθ¼ 1:5, as a function of intensity of the common shock arrival, μ. The right panvalue at date 0 for five different types: 1, 1.25, 1.5, 1.75, and 2. The distribution oγ ¼ p0 ¼ 0:5.

opportunity to blend in. Higher types usually prefer lessfrequent common shocks. In the extreme case, we see thatfor the highest type, the costs of the shock outweigh anyof the benefits, and they prefer no possibility of a shockoccurring.

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t panel plots the expected value at date 0, V bð0; θÞ, for the average typeel plots the intensity of the common shock that maximizes the expectedf θ is uniform on [1, 2]. The other parameters are r¼ c¼ 0:05, λ¼ μ¼ 0:2,

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Total abandonments at the shock

Non-strategic abandonments at the shock

Strategic abandonments at the shock

Fig. 7. The effect of reputation on abandonments at the common shock.The figure plots the expected fraction of projects abandoned at the arrivalof the shock, as of date 0, as a function of the reputation parameter γ.The solid line shows this fraction under the full-information equilibriumof Section 2.2. The dashed line shows this fraction under the asymmetric-information equilibrium of Section 3. It equals the sum of the expectedfraction of projects abandoned at the arrival of the shock because they areexposed to the shock (the dotted line) and because they are not exposedto the shock but choose to blend in (the dash-dotted line). The distribu-tion of θ is uniform on [1, 2]. The other parameters are r¼ c¼ 0:05,λ¼ μ¼ 0:2, γ ¼ p0 ¼ 0:5.

13 They find that CEOs with shorter tenures and greater equity-linkedcompensation schemes are more likely to engage in layoffs duringrecession months.

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157 151

4.4. The role of reputation

An important driver of the incentive to blend in is themagnitude of the reputation component. Higher levelsof γ correspond to greater importance of reputation. As γincreases, agents of all types have greater incentives todelay abandonment: low-quality types have added incen-tives to protect their reputations by blending in, and high-quality types have added incentives to signal their quality.This effect of higher γ leading to greater delay in abandon-ments implies that higher γ leads to more projects beingabandoned at the time of the shock. As the reputationcomponent increases, the act of delaying abandonmentleads to more projects being alive at the time of the shock.Thus, there are more projects alive that get hit by theshock, as well as a greater proportion of surviving projectsthat are voluntarily abandoned.

Fig. 7 displays the impact of reputation on the expectedproportion of projects that are abandoned upon the arrivalof the shock. In this figure, 5% of projects alive at the timeof the shock experience forced abandonment. As a base-line, we see that for the case of full information, a smallpercentage of projects are expected to be abandoned at thetime of the shock. This is depicted in the solid line. Sincewith full information higher γ leads to earlier abandon-ment, we see that increasing γ leads to a smaller fraction ofprojects remaining alive at the time a shock hits. For γ¼0.1,under full information, 3.8% of projects are expected to beabandoned at the time of the shock, while the correspond-ing value for γ¼0.8 is less than 2%. Thus, with fullinformation and no incentive to signal or pool, higherreputation concerns lead to diminishing chances ofwitnessing clusters of abandonments.

The dotted and dash-dotted curves on Fig. 7 show theexpected fraction of projects that are hit by the shock andare abandoned voluntarily, respectively, in the equilibriumwith private information. Their sum, the dashed curve,shows the total expected fraction of projects abandoned atthe time of the shock. Now, we find that the strength ofthe reputation effect greatly impacts the fraction of pro-jects that are expected to be abandoned at the moment ofa shock. For example, if γ¼0.10, as of time zero we expect10% of the industry's projects to be abandoned at the timeof the shock. This means that for a low level of thereputation effect, the shock results in roughly the samenumber of projects that are voluntarily abandoned as thenumber of projects that are actually hit by the shock. Moreprecisely, 4.2% of projects are abandoned because they arehit by the shock, while 5.8% blend in. As we increase γ to0.6, we find that we expect around 20% of the industry toabandon at the time of the shock. This means that roughlythree times as many projects are voluntarily abandoned asare actually hit by the shock. More specifically, 4.5% ofprojects will be hit by the shock, while 15% will blend in.

Consider these effects in the context of strategic defaultson home mortgages. As evidenced in Guiso, Sapienza, andZingales (2013), the reputation impact of voluntary defaultson home mortgages depends on the moral and social stigmaperceived by mortgagors. Thus, in an environment in whichsuch stigma is considerable, the model predicts that a sizableproportion of defaults that occur at the time of a dramatic

shock to home prices or incomes will be voluntary rather thanforced. To the extent that the social stigma of strategic defaultmay have now declined, we should anticipate a smaller“blending in with the crowd” effect.

The model implies that what on the surface appearsto have been a large negative common shock could be arelatively small (and not necessarily negative) shock com-bined with voluntary blending in. Thus, it would beimportant to see how one could empirically distinguishbetween a standard large industry shock (large q), and asmall industry shock (small q) combined with blending in.The fact that it is likely that projects differ in theirreputation incentives (i.e., γ), may allow to identify thetrue size of q. For example, suppose that there are twogroups of projects. Group 1 contains firms with γ¼0.50.Group 2 contains firms with γ¼0. Thus, no firm in Group 2has an incentive to blend in, so that by looking at whatfraction of Group 2 abandoned in a cluster, we can identifyq. Then, from observing what fraction of Group 1 aban-doned in the cluster, one can identify the amplificationmultiple. Similarly, if firms (or industries) differ in thelikelihoods of shock arrivals (i.e., μ), one could comparehow different the firms are in delaying abandonment (i.e.,τbðθÞ) to estimate the degree of “blending in.”

Empirical estimates of cross-sectional variation in repu-tation concerns may be helpful in this regard. Chung,Sensoy, Stern, and Weisbach (2012) provide an interestingempirical approach to measuring the strength of reputa-tion concerns in the private equity industry. While privateequity agreements typically provide direct compensationin the form of a management fee and earned interest,there is also a reputation component of compensation.General partner lifetime incomes depend on their abilityto raise capital in the future. Chung et al. find that theindirect pay for performance from future fund-raising isof the same order of magnitude as direct pay for perfor-mance. Importantly, they find that this reputation measureis heterogeneous across the industry. In particular, it isstronger for buyout funds than for venture capital funds. Inthe study of corporate layoffs, Agarwal and Kolev (2012)identify the tenure of the chief executive officer (CEO) inthe firm and the proportion of CEO compensation obtainedthrough equity-linked instruments as measures of reputa-tion concerns.13 Continuing the example from the preced-ing paragraph, one might obtain an estimate of q from thelayoff decisions of long-tenured CEOs with low pay-for-performance compensation. Then, one could estimatethe magnification multiple due to blending in from thelayoff decisions of short-tenured CEOs with high pay-for-performance compensation.

5. Pooling equilibria

So far our paper focused on the separating equilibriumoutside of the common shock. As a consequence, clustering

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157152

of abandonments could only be triggered by the arrival ofthe common shock. However, there may also exist poolingand semi-pooling equilibria, in which abandonments areclustered at times different from the common shock simplybecause agents believe that only low-quality agents aban-don projects at different times.14 Here, we construct anexample of a pooling equilibrium, in which all typesabandon projects at time τpool outside of the common shock.Importantly, the “blending in with the crowd” effect existsin this case too: If a common shock arrives at time soτpool,a low enough type will abandon her project even ifunaffected by the shock. This suggests that the focus onthe separating equilibrium is not critical.

Let τpool and s be the conjectured abandonment thresh-old outside of the common shock and the stochastic timeat which the common shock arrives, respectively. To allowfor the largest set of equilibrium thresholds, we imposethe harshest off-equilibrium beliefs. Specifically, if out-siders observe that the agent abandons the project at anytime τ=2fτpool; sg, then they believe that the agent's type is θ .The following proposition states a sufficient condition forτpool to be the pooling equilibrium threshold and showsthat the arrival of the common shock can trigger abandon-ment of projects not exposed to the shock:

Proposition 5. Suppose that τpool satisfies

Yðθ; τdðθÞ; γθÞrYðθ; τpool; γE½θ�Þ 8θA ½θ ; θ�; ð40Þ

Y ðθ; τdðθÞ; γθ; τpoolÞr Y ðθ; τpool; γE½θ�; τpoolÞ 8θA ½θ ; θ �; ð41Þwhere Yðθ; τ;RÞ and Y ðθ; τ;R; τpoolÞ are given by (70) and (77)in the Appendix, respectively, and τdðθÞ ¼ arg maxτZ0

Yðθ; τ; γθÞ and τdðθÞ ¼ arg maxτZ0Y ðθ; τ; γθ ; τpoolÞ. Then, thereexists an equilibrium in which:

�

sep

outside of the common shock (both before and after), eachtype θ abandons the project that has not paid off yet attime τpool;

� if the common shock arrives at time soτpool, typesθo θðsÞ, where θðsÞ is defined in the Appendix, abandontheir projects even if they are not impacted by the shockand types θZ θðsÞ abandon their projects only if they areimpacted by the shock.

The intuition behind Proposition 5 is as follows. For τpoolto be the pooling equilibrium threshold, no type canbenefit from deviating from it. Condition (40) ensures thatno type wants to deviate from abandoning the project atτpool in the period after the arrival of the common shock.Similarly, condition (41) ensures that no type wants todeviate in the period prior to the arrival of the commonshock. For every type, the decision to deviate entails thetrade-off between a better choice of the time to abandonthe project and a lower reputation payoff upon abandon-ment. Conditions (40)–(41) impose bounds on τpool, ensur-ing that the cost of deviation always exceeds the benefit.

14 Bustamante (2012) and Morellec and Schürhoff (2011) derive botharating and pooling equilibria in a real-options setting.

When a common shock hits, the agent gets an attractiveopportunity to abandon her project without harmingreputation and having to wait until τpool. Lower types findit costlier to wait until τpool to abandon their projects, sothey choose to abandon their projects at the commonshock strategically, blending in with projects impacted bythe shock.

For the parameter set used to illustrate the equilibriumin Sections 3 and 4,15 conditions (40)–(41) reduce to τpoolbeing between 2.6 years and 9.6 years. Fig. 8 illustratesthe degree of strategic abandonment upon the arrival ofthe common shock for the case of τpool ¼ 8. Each projectimpacted by the shock triggers additional abandonmentsof up to 1.8 projects that are not impacted by the shock.Interestingly, the degree of “blending in” has an invertedU-shape relationship with the timing of the arrival of theshock. Intuitively, abandoning at the common shock veryclose to the inception of the project is not very attractive,since the project has a high chance of paying off later.Abandoning at the common shock very late is also not veryattractive, since the agent can wait slightly longer untilτpool and get a higher reputation payoff from abandoningthe project then. It is the middle interval when “blendingin” at the common shock is the most attractive.

6. Conclusion

In this paper, we provide a model of an industry inwhich agents (corporate managers or entrepreneurs) stra-tegically decide when to stop investing in their projects.We show that when a public common shock forces a smallsubset of agents to abandon their projects irrespective oftheir quality, many agents choose to abandon their pro-jects strategically, even if they are unaffected by the shock.Importantly, this “blending in with the crowd” effect feedsback to the agents' investment strategy even outside theshock. Because low-ability agents know that a shock mayarrive in the future and offer them an opportunity to hidetheir private information, they choose to delay abandon-ment today, before the shock arrives. Delaying abandon-ment by low-ability agents, in turn, forces high-abilityagents to delay abandonment even more to separate fromlow-ability agents.

These effects lead to the accumulation of “living dead”projects, implying that in many industries negative badactive projects may be the norm rather than the exception,especially during long calm periods. In such times, a smallcommon shock that forces some high-ability agents toabandon their projects will trigger a large wave of aban-donments. Such a common shock may even be neutral orpositive on aggregate, rationalizing investment bustsfollowing neutral or even good news. The key requirementis that a shock negatively affects at least some high-abilityagents forcing them to abandon their projects andproviding a shelter for other agents to hide their privateinformation by “blending in.”

15 θ is uniformly distributed over [1,2], r¼0.05, c¼0.05, λ¼ 0:2,μ¼ 0:2, γ ¼ 0:5, and p0 ¼ 0:5.

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MFig. 8. The “blending in” effect in the pooling equilibrium. The panel plots the fraction of projects that blend in out of those not exposed to the shock, ϕðsÞ,and the magnification multiplier, MðsÞ, as a function of the arrival time of the shock in the pooling equilibrium. The pooling equilibrium abandonmentthreshold is τpool ¼ 8. The distribution of θ is uniform on [1, 2]. The other parameters are r ¼ c¼ 0:05, λ¼ μ¼ 0:2, γ ¼ p0 ¼ 0:5.

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157 153

We also briefly explore the possibility of “blending inwith the crowd” in investment decisions. Although wesuggest that a common shock needs to be very different totrigger an investment wave than an abandonment wave,a formal analysis of this issue could prove useful. Otherpotentially interesting avenues would be incorporatingmultiple common shocks, multiple projects, and fire salediscounts to the setting.

Appendix A. Proofs

Proof of Proposition 2. We prove the proposition by apply-ing Theorems 1–3 from Mailath (1987). To do this, weshow that Uð ~θ ; θ; τÞ satisfies regularity conditions fromMailath (1987).

Condition 1 (Smoothness). Uð ~θ; θ; τÞ is C2 on ½θ ; θ�2 � ½s;1Þ.It is straightforward to see that this condition is satisfied.

Condition 2 (Belief monotonicity). U ~θ ð ~θ ; θ; τÞ never equalszero, and so is either positive or negative. Differentiating(21) with respect to ~θ ,

U ~θ~θ ; θ; τ� �¼ 1�p0

p0e� rτγþe�ðrþ λÞτγ40: ð42Þ

Hence, the belief monotonicity condition is satisfied.

Condition 3 (Type monotonicity). Uθτð ~θ; θ; τÞ never equalszero, and so is either positive or negative. Differentiating(21) with respect to θ,

Uθ~θ; θ; τ� �¼ �e�ðrþ λÞτλð1þγÞ

rþλ: ð43Þ

Differentiating (43) with respect to τ,

Uθτð ~θ; θ; τÞ ¼ λð1þγÞe�ðrþ λÞτ40: ð44ÞHence, the type monotonicity condition is satisfied.

Condition 4 (“Strict” quasiconcavity). Uτð ~θ ; θ; τÞ ¼ 0 has aunique solution in τ, which maximizes Uð ~θ; θ; τÞ, andUττð ~θ ; θ; pÞo0 at this solution. Consider the derivative of(21) with respect to τ when ~θ ¼ θ:

Uτ θ; θ; τð Þ ¼ �1�p0p0

e� rτ cþrγθð Þ�e�ðrþ λÞτ cþrγθ�λθð Þ:

ð45ÞEquation Uτðθ; θ; τÞ ¼ 0 has a unique solution in τ, given byτnðθÞ, determined in Section 2.2. The second derivative is

Uττ θ; θ; τn θð Þ� �¼ �1�p0p0

e� rτnðθÞλ cþrγθð Þo0: ð46Þ

Hence, the “strict” quasiconcavity condition is satisfied.

Condition 5 (Boundedness). There exists δ40 such that forall ðθ; τÞA ½θ ; θ � � ½s;1ÞUττðθ; θ; τÞZ0 implies jUτðθ; θ; τÞj4δ.To ensure that the boundedness condition is satisfied, werestrict the set of potential abandonment times to bebounded by k from above, where k can be arbitrarily large.We will later show that extending the set of times toτA ½s;1Þ neither destroys the separating equilibrium norcreates additional separating equilibria. DifferentiatingUτðθ; θ; τÞ with respect to τ,

Uττ θ; θ; τð Þ ¼ 1�p0p0

e� rτr cþrγθð Þþe�ðrþ λÞτ rþλð Þ cþrγθ�λθð Þ:

ð47ÞCondition Uττðθ; θ; τÞZ0 is equivalent to τZτnðθÞþð1=λÞlnð1þλ=rÞ. Hence, for any ðτ; θÞA ½θ ; θ �A ½s;1Þ,

Uτ θ; θ; τð Þ ¼ e� rτ 1�p0

p0cþrγθð Þ�e� λτ λθ�c�rγθð Þ

Ze� rτ 1�p0p0

cþrγθð Þ� rrþλ

e� λτnðθÞ λθ�c�rγθð Þ

¼ e� rτ1�p0

p0

λðcþrγθÞrþλ

Ze� rk1�p0p0

λðcþrγθÞrþλ

40; ð48Þ

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157154

for any arbitrarily large k. Thus, the boundedness conditionis satisfied.

By Theorems 1 and 2 from Mailath (1987), any separat-ing equilibrium abandonment threshold τaðθÞ is continu-ous, differentiable, satisfies Eq. (23), and dτaðθÞ=dθ hasthe same sign as Uθτ . Because Uθτ40, the abandonmentthreshold τaðθÞ is increasing in θ. To ensure that theincreasing solution to Eq. (23) subject to the initial valuecondition (24) is indeed the unique separating equili-brium, we check the single-crossing condition:Single-crossing condition: Uτð ~θ ; θ; τÞ=U ~θ ð ~θ ; θ; τÞ is a strictly

monotonic function of θ. The ratio of the two derivatives isequal to

Uτð ~θ ; θ; τÞU ~θ ð ~θ ; θ; τÞ

¼ �cþrγ ~θγ

�p0e� λτλðγ ~θ�θ�γθÞ

γðp0e� λτþ1�p0Þ: ð49Þ

Consider the derivative of Uτð ~θ ; θ; τÞ=U ~θ ð ~θ ; θ; τÞ with respectto θ:

p0e� λτλð1þγÞ

γðp0e� λτþ1�p0Þ40: ð50Þ

Hence, the single-crossing condition is verified. By Theo-rem 3 in Mailath (1987), the unique solution to (23)–(24),τaðθÞ, is indeed the separating equilibrium.Finally, it remains to prove that considering the set of

abandonment times bounded by k is not restrictive. First,let us prove that τaðθÞ is a separating equilibrium in aproblem with τA ½s;1Þ. Note that the single-crossingcondition holds for all τA ½s;1Þ. Thus, local incentivecompatibility guarantees global incentive compatibilityfor all τA ½s;1Þ. Hence, τaðθÞ is also a separating equili-brium in a problem with τA ½s;1Þ. Second, let us provethat no other separating equilibrium exists in a problemwith τA ½s;1Þ. By contradiction, suppose that there is anadditional separating equilibrium τa2ðθÞ, other than τaðθÞ.Then, τa2ðθÞ must be infinite for some θ: otherwise, itwould be a separating equilibrium in a problem withτA ½s; k� for high enough k. However, an infinite abandon-ment threshold cannot be optimal for any θA ½θ ; θ �:over infinite time, pðtÞ approaches zero, so it would beoptimal for this type to deviate to any finite and highenough threshold at which no type abandons. Thus, thereis no other separating equilibrium in a problem withτA ½s;1Þ. □

Proof of Proposition 3. Let ϕðθÞ denote the left-hand side of(26) and (28). Note that ϕðθÞ, ½ ~θ ; θ � is a continuous functionthat takes values in ½ ~θ ; θ �. Hence, by Brouwer's fixed pointtheorem, it has a fixed point. Taking the derivative of ϕðθÞ,ϕ′ θ�

¼ ð1�qÞf ðθÞðθðqþð1�qÞFðθÞ�Fð ~θÞÞ� R θ~θ θf ðθÞ dθ�qR θθ θf ðθÞ dθÞ

ðqþð1�qÞFðθÞ�Fð ~θÞÞ2:

ð51ÞHence, ϕ′ðθÞ has the same sign as θ�ϕðθÞ. Thus, at anyfixed point, ϕ′ðθÞ ¼ 0. The fixed point is unique, becauseany point at which ϕ′ðθÞ ¼ 0 must be a local minimum.To see this, note that according to (51) we can writeϕ′ðθÞ ¼ ψðθÞðθ�ϕðθÞÞ for some strictly positive and differ-entiable function ψðθÞ. Hence,ϕ″ðθÞ ¼ ψ ′ðθÞðθ�ϕðθÞÞþψðθÞð1�ϕ′ðθÞÞ: ð52Þ

Around point θ : ϕ′ðθÞ ¼ 0, ϕ″ðθÞ ¼ ψðθÞ40. Hence, anypoint ϕ′ðθÞ ¼ 0 must be a local minimum. Therefore, thefixed point is unique. In particular, this result implies thatEq. (25) has a unique solution in θ , which is also a point atwhich ϕðθÞ reaches its minimum.Next, consider any p. It must be the case that

ð1=γÞVaðp; θÞZ θ with the strict equality if and only ifprpnðθÞ. To the left of θm : ϕðθmÞ ¼ 0, ϕðθÞ is a decreasingfunction of θ . Because Vaðp; θÞ is a strictly increasingfunction in θ for any p taking values from Vaðp; ~θÞto Vaðp; θmÞZ θm, Eq. (28) has a unique solution onθA ½ ~θ ; θm�, provided that Vaðp; ~θÞrE ~θ ½θ�. If p4pnðθmÞ, thissolution is θo θm. If prpnðθmÞ, this solution is θ ¼ θm. Thecase Vaðp; ~θÞ4E ~θ ½θ� corresponds to the case when no typewants to blend in with the crowd, because the option towait is too valuable to abandon immediately even for themost optimistic belief possible.Finally, it remains to show that Eq. (28) has no root in

the range θAðθm; θ �. Note that because ϕ′ðθÞ40 for anyθ4 θm, θ4ϕðθÞ for any θ4 θm. However, ð1=γÞVaðp; θÞZ θby definition of the option. Hence, Eq. (28) has no root inthe range θAðθm; θ �. □

Proof of Proposition 4. The general solution to (30) is

Vb t; ~θ ; θ; τ� �¼ C

eðrþμþ λÞt

ð1�p0Þeλtþp0� crþμ

þ λpðtÞrþλþμ

crþμ

þθþγθ

� �

�μeðrþμþ λÞt

ð1�p0Þeλtþp0

Z t

0

ð1�p0Þeλxþp0eðrþμþ λÞx V cs x; θ; ~θ xð Þ� �

dx:

ð53ÞTo determine constant C, we use boundary (31). This givesus the following value to the agent for a given abandon-ment timing and outsiders' inference:

Vb t; ~θ; θ; τ� �¼ eðrþμþ λÞt

ð1�p0Þeλtþp0U ~θ ; θ; τ� �� c

rþμ

þ λpðtÞrþλþμ

crþμ

þθþγθ

� �

�μeðrþμþ λÞt

ð1�p0Þeλtþp0

Z t

0

ð1�p0Þeλxþp0eðrþμþ λÞx V cs x; θ; ~θ xð Þ� �

dx;

ð54Þwhere

U ~θ ; θ; τ� �¼ 1�p0

� �e�ðrþμÞτ c

rþμþγ ~θ

� �

þp0e�ðrþμþ λÞτ c�λð1þγÞθ

rþλþμþγ ~θ

� �

þμ

Z τ

0

ð1�p0Þeλxþp0eðrþμþ λÞx V cs x; θ; ~θ xð Þ� �

dx: ð55Þ

Function U ð ~θ ; θ; τÞ is analogous to function Uð ~θ; θ; τÞ in thepost-shock case, but includes an additional term. In (55),~θðxÞ is the inference function of outsiders and is taken asgiven, while ~θ is a particular point in ½θ ; θ �, which is givenby the argument of the function.Differentiating U ð ~θ; θ; τÞ with respect to ~θ:

U ~θ ð ~θ ; θ; τÞ ¼ ð1�p0Þe�ðrþμÞτγþp0e�ðrþμþ λÞτγ40: ð56Þ

Hence, jU ~θ ðθ; θ; τÞjrγ, i.e., jU ~θ ðθ; θ; τÞj is bounded above. ByProposition 5 in Mailath (1987), the restricted initial valueproblem (35), (36), and dτb=dθ40 has a unique solution

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157 155

on ½θ ; θ�. Let τbðθÞ denote this unique solution. Below weverify that it corresponds to the unique separating equili-brium threshold.To verify this, we apply Theorems 1–3 from Mailath

(1987). We do this in two steps. First, we restrict the spaceof admissible abandonment times τ from ½0;1Þ to ½0; k�,where k is any finite number above τbðθÞ.16 We show thatin this case U ð ~θ ; θ; τÞ satisfies regularity conditions fromMailath (1987). Hence, we can apply Theorems 1–3 fromMailath (1987) to this problem and conclude that τbðθÞ isthe unique separating equilibrium if τ is restricted to bein [0, k]. Second, we extend this to show that τbðθÞ is theunique separating equilibrium in the unrestricted pro-blem, where τA ½0;1Þ.First, we restrict τA ½0; k� and check that U ð ~θ ; θ; τÞ satis-

fies regularity conditions from Mailath (1987).

Condition 1 (Smoothness). U ð ~θ; θ; τÞ is C2 on ½θ ; θ �2 � ½0;1Þ.The condition is satisfied provided that the last term of(55) is smooth.

Condition 2 (Belief monotonicity). U ~θ ð ~θ ; θ; τÞ never equalszero, and so is either positive or negative. This condition isa direct consequence of (56).

Condition 3 (Type monotonicity). U θτð ~θ ; θ; τÞ never equalszero, and so is either positive or negative. Differentiating(55) with respect to θ,

U θ~θ ; θ; τ� �¼ �p0e

�ðrþμþ λÞτ λð1þγÞrþλþμ

þμ

Z τ

0

ð1�p0Þeλxþp0eðrþμþ λÞx V csθ x; θ; ~θ xð Þ� �

dx: ð57Þ

Differentiating (57) with respect to τ and noting that~θðτÞ ¼ ~θ by construction,

U θτð ~θ ; θ; τÞ ¼ e�ðrþ λþμÞτ½p0λð1þγÞþμðð1�p0Þeλτþp0ÞV csθðτ; θ; ~θÞ�: ð58Þ

Because V csðτ; θ; ~θÞ is non-decreasing in θ, U θτð ~θ ; θ; τÞ40.Hence, the type monotonicity condition is satisfied.

Condition 4 (“Strict” quasiconcavity). U τðθ; θ; τÞ ¼ 0 has aunique solution in τ, which maximizes U ðθ; θ; τÞ, andU ττðθ; θ; τÞo0 at this solution. Consider the derivative of(55) with respect to τ when ~θ ¼ θ:

U τðθ; θ; τÞ ¼ �ðð1�p0Þe�ðrþμÞτþp0e�ðrþ λþμÞτÞ

�ðcþrγθ�μðV csðτ; θ; θÞ�γθÞ�λθpðτÞÞ: ð59ÞNote that V csðτ; θ; θÞ is weakly decreasing in τ. Therefore,the term on the lower line of (59) is strictly increasing in τ.Hence, equation Uτðθ; θ; τÞ ¼ 0 has the unique solution in τ.Because V csðτ; θ; θÞ�γθZ0, the solution of Uτðθ; θ; τÞ ¼ 0must weakly exceed τnðθÞ. However, at any point ðτ; θ; θÞsuch that τZτnðθÞ: V csðτ; θ; θÞ ¼ γθðθÞ, where θðθÞ is thesolution of (26). Thus, the unique solution to Uτðθ; θ; τÞ ¼ 0in τ, defined by τnðθÞ, satisfiescþrγθ�μγðθðθÞ�θÞ�λθpðτnðθÞÞ ¼ 0; ð60Þ

16 The fact that τbðθÞ is finite is a direct consequence of Proposition 5from Mailath (1987).

which gives

τn θð Þ ¼ 1λln

p01�p0

λθ

cþrγθ�μγðθðθÞ�θÞ�1� �� �

: ð61Þ

The second derivative of U ðθ; θ; τÞ with respect to τ at τnðθÞis given by

U ττ θ; θ; τn θð Þ� �¼ � 1�p0� �

e�ðrþμÞτnðθÞ þp0e�ðrþ λþμÞτnðθÞ

�

� �μV csτ τn θð Þ; θ; θ� �þ λ2θp0ð1�p0ÞeλτnðθÞ

ð1�p0ÞeλτnðθÞ þp0� �2

!o0; ð62Þ

because V csðτ; θ; θÞ is weakly decreasing in τ. Hence, the“strict” quasiconcavity condition is satisfied.

Condition 5 (Boundedness). There exists δ40 suchthat for all ðθ; τÞA ½θ; θ � � ½0;1Þ U ττðθ; θ; τÞZ0 impliesjU τðθ; θ; τÞj4δ. Differentiating (59) with respect to τ:

U ττ θ; θ; τð Þ ¼ 1�p0� �

rþμð Þþp0 rþλþμð Þ� ��e�ðrþμÞτ cþrγθ�μ V cs τ; θ; θð Þ�γθ

� ��λθp τð Þ� �� 1�p0þp0e

� λτ� �

e�ðrþμÞτ �μV csτ τ; θ; θð Þþ λ2θp0ð1�p0Þeλτð1�p0Þeλτþp0� �2

!:

ð63ÞConsider any τr τnðθÞ. The top line of (63) is non-positiveby monotonicity of the term in the last brackets. Thebottom line of (63) is negative, because V csðτ; θ; θÞ is non-increasing in τ. Therefore, U ττðθ; θ; τÞo0 for any τr τnðθÞ.By continuity of U ττðθ; θ; τÞ at τnðθÞ, U ττðθ; θ; τÞZ0 is onlypossible if τ4 τnðθÞþɛ for some ɛ40. Hence,8ðθ; τÞA ½θ; θ � � ½0;1Þ : U ττðθ; θ; τÞZ0:

jU τðθ; θ; τÞj ¼ e�ðrþμÞτð1�p0þp0e� λτÞðcþrγθ

�μðV csðτ; θ; θÞ�γθÞ�λθpðτÞÞ4e�ðrþμÞkð1�p0Þðcþrγθ�μγðθblendðθ; τÞ�θÞ

�λθpðτnðθÞþɛÞÞZδ; ð64Þ

where

δ� e�ðrþμÞkð1�p0Þ � minθA ½θ ;θ �

fcþrγθ�μγðθblendðθ; τÞ�θÞ

�λθpðτnðθÞþɛÞg: ð65ÞNote that δ40, because ɛ40 and δ¼ 0 if ɛ¼ 0. Thus, theboundedness condition is satisfied.

By Theorems 1 and 2 from Mailath (1987), any separat-ing equilibrium abandonment threshold τbðθÞ is continu-ous, differentiable, satisfies Eq. (35), subject to (36), anddτbðθÞ=dθ40. Because τbðθÞ is the unique solution to thisproblem, there is at most one separating equilibrium in therestricted problem. To ensure that τbðθÞ is indeed theunique separating equilibrium in the restricted problem,we check the single-crossing condition:Single-crossing condition: U τð ~θ ; θ; τÞ=U ~θ ð ~θ ; θ; τÞ is a strictly

monotonic function of θ. The ratio of the two derivatives isequal to

U τð ~θ ; θ; τÞU ~θ ð ~θ ; θ; τÞ

¼ �cþrγ ~θ�μðV csðτ; θ; ~θÞ�γ ~θÞγ

�p0e� λτλðγ ~θ�θ�γθÞ

γðp0e� λτþ1�p0Þ:

ð66Þ

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157156

Consider the derivative of U τð ~θ ; θ; τÞ=U ~θ ð ~θ ; θ; τÞwith respectto θ:

μ

γV csθ τ; θ; ~θ

� �þ p0e� λτλð1þγÞ

γðp0e� λτþ1�p0Þ40; ð67Þ

because V csðτ; θ; ~θÞ is non-decreasing in θ. Hence, thesingle-crossing condition is verified. By Theorem 3 inMailath (1987), τbðθÞ is indeed the separating equilibriumin the restricted problem.Finally, it remains to prove that τbðθÞ is also the unique

separating equilibrium in the unrestricted problem. Thesingle-crossing condition holds for all τA ½0;1Þ. Hence,local incentive compatibility guarantees global incentivecompatibility for all τA ½0;1Þ. Consequently, τbðθÞ is also aseparating equilibrium in the unrestricted problem. Sec-ond, we prove that no other separating equilibrium existsin a problem with τA ½0;1Þ. By contradiction, suppose thatthere is another separating equilibrium τb2ðθÞaτbðθÞ. Then,τb2ðθÞ ¼1 for some θ, as otherwise, it would be a separat-ing equilibrium in a problem with τA ½0; k� for a highenough k. However, an infinite abandonment thresholdcannot be optimal for any θA ½θ ; θ �: over infinite time, pðtÞapproaches zero, so it would be optimal for this type todeviate to any finite and high enough threshold at whichno type abandons. Thus, there is no other separatingequilibrium in a problem with τA ½0;1Þ. □

Proof of Proposition 5. We solve for the equilibrium bybackward induction. First, consider any time t4s and thedecision of type θ to abandon the project at time τZt. LetWaðt; θ; τ;RÞ denote the value to this agent at time t, if shefollows the abandonment strategy τ, and gets the reputa-tion payoff of R upon abandonment. By analogy withSection 3.1, it satisfies

rþλp tð Þð ÞWa ¼ �cþ∂Wa

∂tþλp tð Þ 1þγð Þθ; ð68Þ

subject to the boundary condition Waðτ; θ; τ;RÞ ¼ R. Thesolution is

Wa t; θ; τ;Rð Þ ¼ eðrþ λÞtp tð ÞY θ; τ;Rð Þ�crþ λ

rþλ

crþθþγθ

� p tð Þ;

ð69Þwhere

Y θ; τ;Rð Þ ¼ 1�p0p0

e� rτ crþR

� þe�ðrþ λÞτ c�λð1þγÞθ

rþλþR

� �:

ð70ÞThe beliefs of outsiders imply

R¼γE½θjθZ θ � if τ¼ τpool;

γθ otherwise;

(ð71Þ

where θ is the lowest possible type on the equilibriumpath in the post-shock case. Type θ has no incentives todeviate from τpool if and only if Yðθ; τ; γθÞrYðθ; τpool; γE½θjθZ θ �Þ 8τZs. Then, a sufficient condition for incentivecompatibility of τpool is that (40) is satisfied for all θA ½θ ; θ �.Second, consider time soτpool at which the shock arrives

and the decision of type θ to abandon the project. If it isexposed to the shock (with probability q), it gets aban-doned irrespectively of its type θ. Otherwise, the agentmay either abandon the project strategically or postpone

the abandonment. If the agent abandons the project, shegets the reputation payoff, which is independent of θ.If she postpones, she gets the highest feasible continuationpayoff. Because Waðs; θ; τ;RÞ is strictly increasing in θ forsoτ, the highest feasible continuation payoff is alsostrictly increasing in θ. Hence, like in Section 3.2, thereexists type θðsÞZθ , such that the non-exposed project isabandoned if and only if θo θðsÞ. If Waðs; θ ; τpool; γE½θ�ÞZγE½θ�, i.e., the continuation payoff is higher even for thelowest type θ , then θðsÞ ¼ θ . Otherwise, θðsÞ is given as aunique solution to

γ

R θðsÞθ θ dFðθÞþq

R θθ ðsÞ θ dFðθÞ

FðθðsÞÞþqð1�FðθðsÞÞÞ¼Wa s; θ sð Þ; τpool; γE θ θZ θ sð Þ

ih :

�ð72Þ

The argument from the proof of Proposition 3 appliesto show that θðsÞAðθ ; E½θ�Þ. Let Wcsðs; θ; τpoolÞ denote theequilibrium value to type θ at the time of arrival of thecommon shock but before the agent learns if her project isexposed to the shock:

Wcs s; θ; τpool� �¼ qγ

R θ ðsÞθ θ dFðθÞþq

R θθ ðsÞ θ dFðθÞ

FðθðsÞÞþqð1�FðθðsÞÞÞ

þ 1�qð Þmax γ

R θ ðsÞθ θ dFðθÞþq

R θθ ðsÞ θ dFðθÞ

FðθðsÞÞþqð1�FðθðsÞÞÞ;

8<:

Waðs; θ; τpool; γE½θjθZ θðsÞ�Þo: ð73Þ

If type θ does not abandon at τpool and the commonshock arrives at time s4τpool, then she abandons theproject at the time of the shock if and only if her projectis exposed to the shock, as outsiders’ off-equilibrium beliefis always θ . Let W csðs; θ; τpoolÞ denote the highest continua-tion payoff to type θ at the arrival of the common shock attime s. Then:

W csðs; θ; τpoolÞ ¼Wcsðs; θ; τpoolÞ if soτpool;

qγθþð1�qÞWaðs; θ;maxfs; τdðθÞg; γθÞ if sZτpool:

(

ð74Þ

Finally, consider the decision of type θ to abandon theproject prior to the arrival of the common shock. LetWbðt; θ; τ;R; τpoolÞ denote the value to this agent at time tbefore the common shock arrives, where upon abandon-ment at time τ, the agent gets a reputation payoff of R, andother agents follow the abandonment policy τpool. In theregion prior to abandonment, the expected payoff to theagent satisfies:

rþλp tð Þþμð ÞWb ¼ �cþ∂Wb

∂tþλp tð Þ 1þγð ÞθþμW cs s; θ; τpool

� �;

ð75Þwhich is solved subject to the boundary conditionWbðτ; θ; τ;R; τpoolÞ ¼ R. The solution is

Wb t; θ; τ;R; τpool� �¼ Y θ; τ;R; τpool

� � eðrþμþ λÞt

ð1�p0Þeλtþp0

� crþμ

þ λpðtÞrþλþμ

crþμ

þθþγθ

� �

S.R. Grenadier et al. / Journal of Financial Economics 111 (2014) 137–157 157

�μeðrþμþ λÞt

ð1�p0Þeλtþp0

Z t

0

ð1�p0Þeλxþp0eðrþμþ λÞx W cs x; θ; τpool

� �dx;

ð76Þwhere

Y θ; τ;R; τpool� �¼ 1�p0

� �e�ðrþμÞτ c

rþμþR

� �

þp0e�ðrþμþ λÞτ c�λð1þγÞθ

rþλþμþR

� �

þμ

Z τ

0

1�p0þp0e� λs

eðrþμÞs W cs s; θ; τpool� �

ds: ð77Þ

Type θ has no incentives to deviate from τpool if and only if(41) is satisfied for all τZ0.Therefore, any τpool such that (40) holds for all θA ½θ ; θ �

and (41) holds for all θA ½θ ; θ� and τZ0 yields a poolingequilibrium, in which all agents abandon their projects attime τpool. If the common shock arrives at time soτpool, alltypes θo θðsÞ abandon their projects, even if they are notexposed to the shock. □

A.1. Off-the-equilibrium-path beliefs

There are many off-the-equilibrium-path beliefs thatsupport the equilibrium of Section 3. Below, we providethe beliefs that satisfy the D1 refinement. These beliefs aremonotone: As time goes by, outsiders become weaklymore optimistic about the type of the agent.

�

Prior to the arrival of the shock case: If abandonmentoccurs before τbðθÞ, then outsiders believe that theagent's type is the worst possible, i.e., θ . If abandon-ment occurs after τbðθÞ, then outsiders believe that theagent's type is θ . The differential equation ensures thatno type wants to deviate to an abandonment strategythat is marginally different from the equilibrium one.The single-crossing condition further ensures that notype wants to deviate to any threshold in the interval½τbðθÞ; τbðθÞ�. The off-equilibrium-path beliefs ensurethat no type wants to deviate to any threshold thatdoes not belong to ½τbðθÞ; τbðθÞ�. � After the arrival of the common shock case: Suppose thecommon shock arrives at date s, and the lowest type inthe post-shock history (according to outsiders' beliefs)is θ . If abandonment occurs before τaðθ ; θ; sÞ, thenoutsiders believe that the agent's type is θ , i.e., theworst possible type in the post-shock history. Ifabandonment occurs after τaðθ ; θ ; sÞ, then outsidersbelieve that the agent's type is θ . The differentialequation and the single-crossing condition ensure thatno type wants to deviate to any threshold in theinterval ½τaðθ ; θ; sÞ; τaðθ ; θ; sÞ�. The off-equilibrium-pathbeliefs ensure that no type wants to deviate to anythreshold that does not belong to ½τaðθ ; θ; sÞ; τaðθ ; θ; sÞ�.

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