A ROBUST MODEL OF BUBBLES WITH MULTIDIMENSIONAL UNCERTAINTY. ANTONIO DOBLAS-MADRID Michigan State University, East Lansing, MI 48824, U.S.A.

Transcription

1 Econometrica, Vol. 80, No. 5 (September, 2012), A ROBUST MODEL OF BUBBLES WITH MULTIDIMENSIONAL UNCERTAINTY ANTONIO DOBLAS-MADRID Michigan State University, East Lansing, MI 48824, U.S.A. The copyright to this Article is held by the Econometric Society. It may be downloaded, printed and reproduced only for educational or research purposes, including use in course packs. No downloading or copying may be done for any commercial purpose without the explicit permission of the Econometric Society. For such commercial purposes contact the Office of the Econometric Society (contact information may be found at the website or in the back cover of Econometrica). This statement must be included on all copies of this Article that are made available electronically or in any other format.

2 Econometrica, Vol. 80, No. 5 (September, 2012), A ROBUST MODEL OF BUBBLES WITH MULTIDIMENSIONAL UNCERTAINTY BY ANTONIO DOBLAS-MADRID 1 Observers often interpret boom bust episodes in asset markets as speculative frenzies where asymmetrically informed investors buy overvalued assets hoping to sell to a greater fool before the crash. Despite its intuitive appeal, however, this notion of speculative bubbles has proven difficult to reconcile with economic theory. Existing models have been criticized on the basis that they assume irrationality, that prices are somewhat unresponsive to sales, or that they depend on fragile, knife-edge restrictions. To address these issues, I construct a rational version of Abreu and Brunnermeier (2003), where agents invest growing endowments into an asset, fueling appreciation and eventual overvaluation. Riding bubbles is optimal as long as the growth rate of the bubble and the probability of selling before the crash are high enough. This probability increases with the amount of noise in the economy, as random short-term fluctuations make it difficult for agents to infer information from prices. KEYWORDS: Bubbles, coordination, noisy prices. 1. INTRODUCTION OVER THE LAST TWO DECADES, a series of dramatic boom bust episodes in global asset markets have led many economists to question the long dominant efficient market hypothesis and to devote increasing attention to theories of asset price bubbles. Asset price bubbles are often referred to as speculative bubbles, a term that conjures the idea of a market timing game, in which investors buy overvalued assets hoping to sell to a greater fool before the crash. This idea comes up repeatedly, for example, in Kindleberger and Aliber s (2005)famous chronicles of historical boom bust episodes. There is also experimental work (Moinas and Pouget (2009)) documenting the emergence of bubbles in a design with asymmetrically informed participants who ride the bubble knowing that they may earn speculative profits by riding the bubble, but also that they may suffer losses if they get stuck with the asset at the end of the game. Despite its intuitive appeal, however, this notion of a speculative bubble has traditionally been difficult to reconcile with standard economic theory. As Tirole (1982) and Milgrom and Stokey (1982) showed, bubbles are inconsistent with rational expectations equilibrium in a wide range of environments with finite numbers of rational agents, even under asymmetric information. Some approaches that have been taken to circumvent these impossibility results include introducing some form of irrationality, assuming heterogeneous 1 This paper has greatly benefited from comments by Luis Araujo, Braz Camargo, John Conlon, Thomas Jeitschko, Timothy Kehoe, Andreas Park, Francisco Peñaranda, Juan Rubio-Ramírez, and Jean Tirole, as well as by a co-editor and three anonymous referees. I am also indebted to participants in various seminars and conferences for helpful comments and suggestions. All errors are my own The Econometric Society DOI: /ECTA7887

3 1846 ANTONIO DOBLAS-MADRID priors or marginal utilities, and assuming an infinite number of overlapping generations. For example, Harrison and Kreps (1978) and Scheinkman and Xiong (2003) considered agents who are overconfident in the sense that they consider their own information to be superior to that of others, and fail to fully adjust their beliefs as they observe what others believe. In Abreu and Brunnermeier (2003), there are rational agents who ride the bubble and make profits with a certain probability along with behavioral agents who fuel bubble growth and who are doomed to suffer losses in the crash. In Allen, Morris, and Postlewaite (1993)and Conlon (2004), agents are rational, but have either heterogenous priors or heterogenous state-contingent marginal utilities that may give rise to gains from trade. This approach generates speculative bubbles, but has the drawback of relying on fragile, knife-edge parameter restrictions. Another strand of literature (Caballero and Krishnamurthy (2006),Fahri and Tirole (2012), and others) builds on Tirole s (1985) workonrationalbubbles with overlapping generations, where bubbles like money in Samuelson (1958) improve allocations by alleviating a shortage of stores of value. However, the bubbles in these models are less reminiscent of speculation. Trades are typically driven by the life cycle rather than beliefs, and the focus is often on steady states where bubbles grow slowly and never burst. 2 The aim of this paper is to contribute to the theory of speculation by developing a model of a greater fool s bubble that circumvents some of the main critiques of previous studies. To this end, I construct a discrete-time version of Abreu and Brunnermeier (2003; henceforth referred to as AB), where all agents are rational and prices reflect supply and demand at all times. The model inherits from AB the property of being robust to small changes in parameters, and is therefore not subject to the fragility critique of Allen, Morris, and Postlewaite (1993)andConlon (2004). Following AB, I model a bubble as a market overreaction to events that are initially fundamental in nature. An asset price boom is at first justified by fundamentals, but a bubble emerges as growth continues past the point where real gains have been priced in. I also borrow from AB the assumption that the time when the bubble starts is not perfectly observed. Instead, for a number of periods, different agents observe private overvaluation signals, which reveal that a bubble has begun. Since agents do not know when others observe the signal, they make probabilistic assessments about when the bubble and hence the signals may have started. They understand that those who received signals relatively early will make profits, while others will suffer losses in the crash. However, if the chances of being an early-signal agent and the speed at which the bubble grows are high enough, it is optimal to take a risk and ride the bubble. 2 Further approaches to bubbles focus on agency problems (Allen and Gorton (1993), Allen andgale (2000), Barlevy (2008), AcharyaandNaqvi (2012)), solvency constraints (Kocherlakota (2008)), and others. For a survey, see Brunnermeier (2001).

4 A ROBUST MODEL OF BUBBLES 1847 While these core ideas are the same as in AB, I modify the environment so as to address the two critiques of irrationality and of partial disconnect between sales and prices. In AB, bubble growth is fueled by behavioral agents who invest growing amounts into a risky asset and are willing to do so indefinitely. These agents are doomed to get caught in the crash, which occurs when a critical mass of rational agents exit the market. By contrast, in this paper, there are no behavioral agents. The bubble is fueled by rational agents who invest in it as long as it is optimal to do so. Agents receive growing endowments, which they can use to buy the risky asset. Importantly, these endowments cannot be pledged as collateral, that is, agents cannot borrow against their time-t endowment at some earlier date s<t. Borrowing constraints limit the amounts that agents can use to bid up the risky asset. Thus, the price does not reflect agents private valuations given by their best guesses about fundamentals plus the value of the chance to sell at the peak of the bubble. Instead, the price simply reflects the maximum resources that agents are able to invest at each date. This also has the important implication that the boom, instead of consisting of a one-time jump in the price, takes place gradually over time as investors access larger endowments. I also assume that, every period, preference shocks force a fraction θ t of agents to sell for reasons such as life events or liquidity needs unrelated to price expectations. This ensures that a positive mass of shares is sold every period, even when nobody expects an imminent crash. 3 Preference shocks also serve another function, adding noise to the economy. Because θ t is subject to random variability, prices are noisy. If the variability of θ t is high enough, prices can hide sales, as late-signal agents cannot distinguish price slowdowns due to sales by early-signal agents from those due to high realization of θ t.thus, the likelihood that an agent can sell before the crash tends to increase with the variability of θ t. The second critique is that prices in AB are somewhat independent of sales. During the time interval when a growing mass of early-signal agents is leaving the market, the price is assumed to continue to grow at the same rate as if nobody was selling. 4 To address this, I explicitly model market clearing and price formation. I model the asset market as a Shapley Shubik trading post. In a first stage, agents submit orders to buy the risky asset (bidding a risk-free asset) or submit shares of the risky asset for sale. In a second stage, all orders are combined, and the price emerges as the ratio of the risk-free asset bid to the amount of the risky asset sold. 3 Note that the preference shock is not a substitute for irrational agents, since it does not force agents to stay in the market during the crash. On the contrary, it forces some agents to sell before they otherwise would. 4 AB conjectured, in a remark, that adding noise to the price process would allow prices to respond to supply and demand at all times without revealing all private information.

5 1848 ANTONIO DOBLAS-MADRID To solve the model, I first consider the case with so little noise that as soon as one type sells (a type includes all those who observe the overvaluation signal in the same period), all uncertainty is revealed, triggering a crash in the next period. In this case without noise, the effect of the first type s sales on the price is always greater than the effect of any possible random fluctuation in θ t. I show that in this effectively noiseless environment, a no-bubble equilibrium in which agents sell as soon as they observe the signal always exists. The no-bubble equilibrium is unique if G/R, the growth rate of the bubble net of the riskfree rate, is below a threshold Γ. This threshold depends on λ, the parameter governing the relative likelihoods of early versus late overvaluation signals. As G/R rises above Γ, the set of equilibria expands to include, in addition to the no-bubble equilibrium, equilibria with bubbles. The duration of the bubbles ranges from zero to a maximum that is a function of G/R and λ. I continue the analysis by increasing the amount of noise so that it can conceal sales of one type, but not more. This allows multiple types to sell before the crash, since sales of the first type can be confused with noise and thus may fail to burst the bubble. In this case, prices reflect selling pressure monotonically, but reveal information only imperfectly. When noise can hide sales by one type, but not two, prices fall into one of three categories. High prices reveal with certainty that nobody has sold, medium prices reveal that sales may or may not have begun, and low prices reveal with certainty that sales have begun, thereby triggering the crash. If the number of types is large, the analysis of equilibria with Markov strategies is simple enough to be analytically tractable. The strategies I consider are Markovian, in the sense that agents sell-or-wait choices depend only on how much time has passed since observing the signal and on whether the last price observed was high, medium, or low. Restricting attention to this class of strategies, I show that there are two key ways in which noise helps generate bubbles. First, in the noisy case, it is possible to rule out equilibria without bubbles for high enough G/R. Second, there exist parameters such that, with noise, (arbitrarily) long bubbles may arise, even if G/R is below Γ. In other words, there exist parameters such that in the noiseless case, the only equilibrium is the one without bubbles, while in the noisy case, arbitrarily long bubbles may arise. Finally, I relax the assumption made in the basic analysis for simplicity that agents cannot reenter the market after selling and show that although some equilibria vanish, the overall picture remains unchanged, and bubbles with Markovian strategies still arise. The paper is organized as follows. In Sections 2 and 3, respectively, I describe the model and define equilibrium. In Section 4, I illustrate how bubbles arise in equilibrium. In Section 5, I consider the extension where agents may reenter the market after selling. Section 6 concludes.

6 A ROBUST MODEL OF BUBBLES THE MODEL 2.1. The Environment Time is discrete and infinite with periods labeled t = There are two assets: a risk-free asset with exogenous gross return R>1andarisky asset. At all times, the risk-free asset can be turned into consumption at a one to one rate. The supply of the risky asset is fixed at 1, and its price at time t is p t units of the risk-free asset. As in AB, the risky asset s fundamental value f t is based on dividends to be paid in the distant future, with current dividends set equal to zero for convenience. While t 0, the expected value of future dividends, discounted at the risk-free rate, is αr t. The fundamental value f t and price p t also equal αr t in expectation. At t = 1, fundamental shocks increase the risky asset s expected future dividends. The fundamental value f t and price p t begin to grow, on average, at the rate G>R. Booming prices are justified by fundamentals until period t = t 0 1, but starting at time t 0 1, the average f t /f t 1 falls back to R, andifp t continues to grow on average at the rate G, p t starts to diverge from f t, that is, a bubble arises. The bubble inflates until period T t 0 and bursts at T + 1, at which point equality between price and fundamental value is restored. Thus, as in AB, bubbles arise as markets overreact to events that are at first fundamental. 5 The first period of overvaluation t 0 is geometrically distributed with probability function ϕ given by (1) ϕ(t 0 ) = ( e λ 1 ) e λt 0 for all t 0 = 1 2 where λ>0. The expected value of t 0 is given by 1/(1 e λ ). There is a unit mass of rational agents indexed by i [0 1]. Theydonot observe t 0 perfectly. Instead, every period from t 0 to t 0 + N 1 a mass 1/N of them observe a signal revealing that the risky asset is overvalued. In other words, the signal reveals that f t is no longer growing at the rate G. Signals define N types, n = t 0 t 0 + N 1. Formally, ν : [0 1] {t 0 t 0 + N 1} assigns a type to each agent, where ν(i) = n denotes that agent i is of type n or, in other words, that agent i observes the signal at time n. As in AB, agents observe n, butnott 0. Once an agent observes her signal at time n, sheknows that t 0 may have been as early as n (N 1) or as late as n. (Except for the special case with t 0 <N, where types with n<n know that t 0 must be above 5 According to Kindleberger and Aliber (2005), bubbles often occur in the aftermath of major displacements, which cause large shifts in prices. Price movements that are justified by fundamentals for some time turn into bubbles if markets overshoot. In keeping with this idea, AB mentioned episodes in stock markets after the arrival of new technologies (e.g., the Internet in the 1990s, the radio in the 1920s) as examples of bubbles.

7 1850 ANTONIO DOBLAS-MADRID n (N 1), sincen (N 1) 0.) Conditional on n, the distribution of t 0 becomes e λt 0 ϕ(t 0 n) = e λ(max{1 n (N 1)}) + +e λn (2) if max { 1 n (N 1) } t 0 n 0 otherwise Sequential arrival of signals places agents along a line, but they are uncertain about their relative order in the line. This plays a key role in generating bubbles, as all agents even those late in the line assign positive probability to the event that they could be early in the line. 6 As we will see later, signals are the key reference points on which agents condition their equilibrium selling strategies. In the absence of noise, type-n agents will plan to ride the bubble for τ 0 periods and sell at time n + τ 0.When prices are noisy, strategies will be augmented to allow agents to wait longer if they observe higher prices. Figure 1 summarizes the assumptions made thus far. The boom starting at t = 1 is at first fundamental, but turns into a bubble at the imperfectly observed time t 0, with signals arriving at t = t 0 t 0 + N 1. Bubble duration T t 0 will be endogenously determined in equilibrium. Preferences are characterized by risk neutrality and preference shocks à la Diamond and Dybvig (1983), which force agents to liquidate assets and consume. At time t, a randomly chosen mass θ t (0 1) of agents are hit by a shock FIGURE 1. Timeline of events. 6 One could also assume that some agents receive no signal. These agents would find themselves in a situation similar, but not identical, to that of agents who know that they will observe a signal but have not yet observed it. As we will see when solving the model, agents who have yet to observe the signal are less inclined to sell preemptively than agents who have observed it. In fact, the agents whose choices set limits on bubble duration are those whose signal arrives just one period after the agents who sell at the peak. Given this, introducing agents who do not observe signals would add complications to the analysis without affecting results.

8 A ROBUST MODEL OF BUBBLES 1851 that sets their discount factor δ i t equal to zero. The remaining mass 1 θ t have δ i t = 1/R.Agenti s expected utility is defined as (3) E i t [ U ( {ci τ } τ=t )] = Ei t [ c i t + ( τ 1 τ=t+1 s=t ) δ i s c i τ ] where c i t denotes agent i s time-t consumption, U denotes utility, and E i t denotes expectation given information available to agent i in period t. This information includes whether δ i t is zero, in which case (3) reduces to E i t c i t. Preference shocks induce an urgent need to consume, but do not represent the agent s death. After being hit, agents continue to receive endowments and can be hit again. To simplify, shocks are assumed to be independent and identically distributed (i.i.d.), so that the probability that δ i t = 0 is independent of past values δ i t 2 δ i t 1 ; that is, agents who have not been hit for a long time are no more likely to be hit than those who were recently hit. Shocks are also type-independent, in the sense that for all t, and within any type, the fraction of agents hit by the shock is θ t. Since θ t is unobservable, agent i knows whether she has been hit by the shock, but not how many agents have been hit. Moreover, θ t varies over time as (4) θ t = θ + ε t where θ (0 1) is a constant and ε t is an i.i.d. random variable uniformly distributed over [ ε ε], with0< ε <min{ θ 1 θ}. The term ε t serves an important function in the model by generating random price fluctuations. If θ t is constant, as soon as the first agents sell in anticipation of the crash, the price reveals these sales, precipitating a crash. In a noisy environment, by contrast, agents cannot distinguish whether a price deceleration is due to a high ε t or the start of the crash. It is important to note that the role of preference shocks is precisely to generate a positive and noisy amount of sales. The role of the shock is not to make speculation a positive sum game by forcing some agents to stay in the market and get caught in the crash. On the contrary, the shock saves some agents from the crash by forcing them to sell. The boom is fueled by agents investing endowments into the risky asset. Every period, agents receive e t > 0 units of the risk-free asset. As long as they do not anticipate an impending crash and are not hit by the shock, they invest the endowment into the risky asset. Endowments cannot be capitalized, that is, an

9 1852 ANTONIO DOBLAS-MADRID agent receiving e t at t cannot borrow against it at earlier dates s<t. After time 0, endowment growth accelerates as 7 (5) { Rt if t 0, e t = G t if t>0. Three remarks are in order. First, the assumption that e t grows at the rate G forever should not be interpreted literally. In the long run, endowment growth must eventually slow down. Limits to endowment growth are not modeled, however, because the focus of the paper is on endogenous crashes, where agents sales burst the bubble before growth decelerates for exogenous reasons. 8 Second, AB also assumed that rational agents are constrained and fully invested in the bubble. 9 However, in AB the inflow of funds that fuels bubble growth is dumb money from behavioral agents who are doomed to suffer losses. Here, rational agents invest in the bubble only as long as it is optimal to do so. The third remark is that an alternative specification with constant θ t and noisy aggregate endowment would also generate fluctuations in price, but notintradingvolume. The within-period timing of shocks and actions is as follows. Agent i starts period t with nonnegative holdings b i t and h i t of the risk-free and risky assets, respectively. The period proceeds in two steps. In Step 1, agent i receives e t, learns whether δ i t is zero or 1/R, and,ifν(i) = t, observes her signal. Also in Step 1, agent i knowing δ i t, p t 1 ={ p t 2 p t 1 } and, if ν(i) t, the 7 Endowment growth captures the idea that as a bubble grows, the availability of funds that can be invested into it also grows. Two interpretations of this increasing availability of resources were suggested by Kindleberger and Aliber (2005), who saw expanding credit and the arrival of new investors as usual sources of bubble fuel. Regarding the first interpretation, risky assets such as stocks and especially real estate are typically more pledgeable as collateral than labor income/endowments, because unlike human capital, risky assets can be seized and sold by lenders. If the risky asset serves as collateral, price growth loosens credit constraints, allowing investors to borrow more and bid prices even higher. The other interpretation gradual arrival of investors may reflect liberalization of capital flows into a country or industry, or it may reflect technological factors that prevent agents from investing all their wealth into the risky asset at time 1. For example, in the short run, some wealth may be tied up in projects that can only be liquidated at a heavy loss. In such a setting, funds would become progressively available as projects matured. 8 A similar issue arises in AB, where behavioral agents are assumed able to purchase a given number of shares of the risky asset no matter how high the price becomes. The price in AB may become arbitrarily large because, although there is an exogenous cap on bubble duration, the distribution of t 0 is not bounded. 9 Note that borrowing constraints depress fundamental values for t {1 t 0 2}. During these periods, expected fundamental value f t is αg t, since agents cannot afford to pay α(g/r) t0 1 R t, the present value of future dividends discounted at the rate R.Fromperiodt 0 1 onward, agents can afford to pay this, and the fundamental value becomes α(g/r) t0 1 R t ;that is, borrowing constraints depress fundamental values in the early phase of the boom. However, when agents start to observe signals and knowingly ride the bubble, borrowing constraints continue to bind, since agents view the risky asset as valuable, not only because of its dividends, but also because of the chance to sell at the top of the bubble.

10 A ROBUST MODEL OF BUBBLES 1853 signal chooses actions a i t = (m i t s i t χ i t ).Thepair(m i t s i t ) captures asset market choices, and χ i t [0 1] captures the consumption choice. (Although agents consume in Step 2, the decision to consume is driven by the preference shock, which is known in Step 1.) The asset market is modeled as a Shapley Shubik trading post, where each agent i bids m i t units of the risk-free asset and offers s i t shares of the risky asset for sale. Due to borrowing/short sales constraints, agent i s choices must satisfy (6) 0 m i t b i t + e t and (7) 0 s i t h i t Agent i chooses (m i t s i t ) before knowing the price p t, which will be determined in Step 2 when all bids and offers are combined. 10 Preference shocks and risk neutrality greatly simplify decision problems. Agents with δ i t = 0 sell everything so as to consume in Step 2, that is, they set (m i t s i t ) = (0 h i t ). Agents with δ i t = 1/R sell, setting (m i t s i t ) = (0 h i t ), if they expect the risky asset s return p t+1 /p t to be less than R; they buy, setting (m i t s i t ) = (b i t + e t 0), if the expected return is greater than R, and they are indifferent between buying, selling, and doing nothing in the knife-edge case. 11 Agent i comes out of the asset market holding (8) and h i t+1 = h i t + m i t p t s i t (9) b i t = b i t + e t m i t + p t s i t where b i t denotes agent i s within-period or interim risk-free asset holdings. In Step 2, bids and offers are combined and the price is determined by market clearing: (10) h i t di = 1 i [0 1] 10 The assumption that agents submit orders before observing others orders or the price is similar to Kyle (1985) and also to models à la Cournot. In microstructure terms, agents are placing market orders, which they know will be executed, but they do not know at what price. 11 Although individual agents within a type hold different amounts of the risky asset, they all buy/sell it in lockstep (except for agents who are forced by the shock to sell). This is a consequence of risk neutrality and of the fact that all agents within a type observe the same signal.

11 1854 ANTONIO DOBLAS-MADRID Substituting (8) into (10), noting that (10) alsoholdsattimet + 1, and solving for the price yields p t = M t (11) S t where for all t, (12) M t i [0 1] m i t di and S t s i t di; i [0 1] that is, the price is the ratio of the amount of the riskless asset bid to the number of shares of the risky asset offered for sale. Since there is always a positive mass of shock-induced sellers, S t is always positive and p t is well defined. Finally, agent i consumes a fraction χ i t [0 1] of b i t, (13) c i t = χ i t bi t and saves the rest, so that next period s risk-free asset holdings b i t+1 are given by (14) b i t+1 = R(1 χ i t ) b i t Figure 2 summarizes within-period timing. Having described market clearing, we can now fill in details about the preboom, boom and post-crash phases. Before proceeding, however, it will be useful to assume that even if the boom is not anticipated the risky asset is valuable enough to absorb agents entire wealth in the pre-boom phase. This makes it (weakly) optimal for agents to hold b i t = 0whilet 0. In turn, this implies FIGURE 2. Within-period timing.

12 A ROBUST MODEL OF BUBBLES 1855 that price growth accelerates at t = 1, but there is no additional one-time jump in price. 12 With this assumption in place, consider now a pre-boom period t 0, and let agents start with b i t = 0 units of the risk-free asset. Preference shocks force a mass θ t of agents to sell S t = θ t shares of the risky asset, while all other agents use their endowments to bid for these shares. Since b i t is zero, M t = (1 θ t )e t and thus (15) p t = ( θ 1 t 1 ) e t Since the expected p t+1 /p t is R, agents who are not hit by the shock find it (weakly) optimal to invest only in the risky asset, letting b i t+1 = 0. Agents who are hit by the shock consume c i t = b i t = e t + p t h i t and save nothing, setting (b i t+1 h i t+1 ) = (0 0). Given(15) and the assumption that p t αe t in the preboom and boom phases, it must be that (16) α = E [ θ 1 t 1 ] = 1 2 ε ε ε [ ( θ + ε t ) 1 1 ] dε t = ln(( θ + ε)/( θ ε)) 1 2 ε At t = 1, endowment and price growth accelerate. For a while, the only sales are the ones that are forced by shocks. Agents who are not hit remain fully invested in the risky asset, and p t is given by (15) withe t = G t.sinceshocks are type-independent, in the aggregate, each type holds h n t = 1/N shares of the risky asset, where for all n {t 0 t 0 + N 1} and for all t, (17) h n t h i t di {i ν(i)=n} All N types hold h n t = 1/N shares until the last few periods of the boom, when some start to sell in anticipation of the crash. When the first z t > 0 types sell at t, the mass of shares for sale becomes S t = z t /N + θ t (1 z t /N),whereamass z t /N of agents sell, anticipating a crash, and a mass θ t (1 z t /N) of agents sell because of preference shocks. Total bids M t amount to (1 θ t )(1 z t /N)G t, 12 If the risky asset during the pre-boom phase is not valuable enough to absorb all of the agents wealth, agents accumulate shares of the risk-free asset before the boom starts. They pour these holdings into the risky asset once the boom begins, causing a price jump at time 1, followed by some time where the price grows at the rate R. Once holdings of risk-free asset reach zero, borrowing constraints bind again and the price grows at the rate G. As long as the time it takes for the risk-free holdings to reach zero is not too long, relative to the expected duration of the boom, it is possible to incorporate these additional dynamics into the analysis without affecting results.

13 1856 ANTONIO DOBLAS-MADRID as only agents who are not hit by the shock and are not of the exiting types buy. Consequently, the price becomes ([ ( zt p t = N + θ t 1 z )] 1 t (18) 1) G t N After trade, h n t+1 is 0 for the z t types who have sold and 1/(1 z t /N) for the other types. Agents hit by the shock consume the proceeds from selling the risky asset. Those who sold without being hit store their wealth in the risk-free asset, setting b i t+1 = R b i t. The likelihood that p t reveals the exit of these z t types depends on the relative magnitudes of ε and z t /N.If ε<(1 θ)z t /(2N z t ), sales will surely be revealed, because (( θ+ ε) 1 1)G t the lowest possible price if z t = 0 exceeds ([z t /N + ( θ ε)(1 z t /N)] 1 1)G t the highest possible price if z t types sell. However, if ε (1 θ)z t /(2N z t ), p t may be greater than or equal to (( θ + ε) 1 1)G t, in which case the bubble will continue until period t + 1. If the bubble survives period t and another z t+1 0typessellatt + 1, the aggregate bid becomes M t+1 = (1 θ t+1 )(1 (z t + z t+1 )/N)G t On the selling side, z t+1 /N sellers anticipate a crash and θ t+1 (1 (z t + z t+1 )/N) sellers are strictly shock-induced. Since risky-asset holdings across sellers average 1/(1 z t /N), the total mass of shares for sale equals [ S t+1 = 1 z ] 1 ( ( t zt+1 N N + θ t+1 1 z )) t + z t+1 N Rearranging terms, the equilibrium price can be written as ( p t+1 = 1 z ) t 1 z t N (19) N ( z t+1 N + θ t+1 1 z ) 1 t + z t+1 Gt+1 N The likelihood that p t+1 /G t+1 falls below (( θ + ε) 1 1) now depends on ε, z t, and z t+1.ifp t+1 /G t+1 falls below this threshold, sales will be revealed, causing a crash; otherwise, the bubble will last until time t + 2 or later. Equation (19) can be generalized to allow sales over more than two periods. 14 However, since in the equilibria analyzed later, sales burst the bubble in one or two periods, (19) lays all the groundwork necessary for our purposes. 13 M t+1 and S t+1 would be different for negative z t+1 (i.e., if some types reentered the market after selling). I restrict attention to z t+1 0, because although reentry is allowed in Section 5, it does not actually happen in equilibrium. 14 If sales start at t 0andz t z t+h 0 types sell at times t t+ h,thepricep t+h is given by (19), replacing (θ t+1 G t+1 z t+1 ) with (θ t+h G t+h z t+h ) and z t with z t + +z t+h 1.

14 A ROBUST MODEL OF BUBBLES 1857 The post-crash phase starts at time T + 1, where T is the first period in which p t /G t falls below (( θ + ε) 1 1). Assuming that t 0 is revealed at T, the expected fundamental value α(g/r) t 0 1 R t also becomes known. 15 From time T + 1 onward, agents who are not hit by the shock invest a decreasing fraction (R/G) t (t 0 1) of their endowments in the risky asset, and the rest in the risk-free asset. This is weakly optimal since, throughout the post-crash phase, the expected ratio p t+1 /p t equals R. Moreover, equality between p t and f t is preserved. 3. EQUILIBRIUM I next define equilibrium under the restriction to be relaxed in Section 5 that once a type has sold in anticipation of the crash, agents of that type stay out of the market until the bubble bursts. (Note that this does not preclude agents who are forced to sell by shocks from investing their endowments in the risky asset in later periods.) RESTRICTION I No Reentry: For any i and any t T, if b i t > 0, then h i τ = 0 τ {t + 1 T}. The equilibrium concept is perfect Bayesian equilibrium (PBE), consisting of strategies and beliefs {a i μ i } i [0 1].Agenti s strategy a i is a sequence {a i t } t Z, where a i t is a triplet (m i t s i t χ i t ).Agenti s belief μ i t (t 0 ) is a probability distribution over values of t 0.Botha i t and μ i t (t 0 ) are contingent on information available to agent i in Step 1 of date t. This includes the discount factor δ i t, past prices p t 1 ={ p t 2 p t 1 },and,ifν(i) t, the signal ν(i). Sinceδ i t does not inform about t 0, and all agents within a type observe the same prices and signal, they have the same common belief, defined as μ n t (t 0 ) μ i t (t 0 ) for all i with ν(i) = n. In equilibrium, for all i, a i t is optimal given agent i s shock realization δ i t and the belief μ i t (t 0 ),andμ i t (t 0 ) is consistent with the equilibrium strategy profile. To be consistent with a strategy profile, a belief μ i t (t 0 ) must assign positive probability only to values of t 0 that are not ruled out by strategies, given past prices and, if t ν(i), the signal. The set of values of t 0 that are not ruled out is the support of t 0, denoted by supp i t (t 0 ). Since beliefs are the same for all agents within a type, one can define supp n t (t 0 ) supp i t (t 0 ) for all i with ν(i) = n. Toseehowsupp n t (t 0 ) evolves in equilibrium, recall that the signal n implies that supp n t (t 0 ) {max{1 n (N 1)} n}.moreover, 15 In the equilibria presented later, prices p 1 p T often reveal t 0 exactly. However, in some instances, this will hold only approximately, and for some agents a couple of values of t 0 will be consistent with prices. In these cases, it would take some time after the crash for agents to learn the exact value of t 0. Modeling these details explicitly, however, would add complications without significantly affecting results or providing much insight.

15 1858 ANTONIO DOBLAS-MADRID prices p t 1 and strategies rule out values of t 0 as follows. If t 0 takes on the value τ 0, given the price history p t 1, there are discount-factor contingent implied values of a i τ for all i and all τ<t. These implied actions and the price p τ can be substituted into (19) to compute the implied ε τ = θ τ θ. The value τ 0 is excluded from supp n t (t 0 ) if it implies ε τ > ε for some τ. After discarding all the values of t 0 that are ruled out by this process, the probabilities that μ n t (t 0 ) assigns to each remaining value in supp n t (t 0 ) are obtained using Bayes rule as follows 16 : (20) μ n t (t 0 ) = ϕ(t 0 ) ϕ(τ 0 ) τ 0 supp n t (t 0 ) For all agents and at all times, given b i t and h i t, the equilibrium strategy a i t solves the problem [ ( τ 1 ) ] (21) max E i t c i t + δ i τ c i τ m i t s i t χ i t τ=t+1 s=t subject to (6) (9), (13), (14), 0 χ i t 1 and Restriction I. Note that equilibrium beliefs are embedded in the expectations operator E i t. As previously stated, preference shocks and risk neutrality greatly simplify this decision problem. Agents hit by the shock set a i t = (0 h i t 1), that is, they sell and consume everything. Agents with δ i t = 1/R set χ i t = 0 and, depending on whether E i t [p t+1 /p t ] is above, below, or equal to R, they choose to be, respectively, fully invested in the risky asset, fully invested in the risk-free asset, or indifferent between any mix of the two. 4. EQUILIBRIA WITH BUBBLES: BASIC ANALYSIS For ease of exposition, I will carry out the analysis in Sections 4 and 5 under the assumption that N is large, which implies that throughout the boom including the last few periods the price p t approximates αg t ; that is, I assume that price fluctuations matter because of their informational content, but are otherwise too small to have any sizable revenue effects. This assumption keeps formulas simple enough to convey intuition and tractable enough to allow for analytical equilibrium characterization. In Appendix D, I derive the formulas for general values of N. 16 In a more general expression, ϕ would be multiplied by the likelihood of observed prices for each value of t 0.Butin(20), this likelihood is simplified away, because in the coming analysis, it is equal for all values in supp n t (t 0 ). This does hold exactly in most instances, although in a few cases it is only true under an approximating assumption. In Appendix D, I show how the analysis is modified when that assumption is removed.

16 A ROBUST MODEL OF BUBBLES 1859 I will begin the analysis in Section 4.1 with the case in which ε <(1 θ)/(2n 1). In this case, which for brevity I will refer to as the noiseless case, the price is certain to reveal sales as soon as one type (z t = 1) exits the market. I examine the possibility of bubbly equilibria with simple trigger strategies akin to those studied by AB. These strategies dictate that after observing the signal at t = ν(i), agenti shall unless forced to sell by the preference shock ride the bubble for τ periods and sell at t = ν(i) + 0 τ 0. Although noise cannot hide sales of the first type, the discreteness of the model, together with the withinperiod timing, allows a mass 1/N of agents to sell at the peak of the bubble. Since there is positive probability of being among these sellers, there are equilibria with bubbles if G/R is high enough. 17 More precisely, in Proposition 1, I will show that to support equilibria with τ 0 = 1 G/R must surpass a threshold Γ = Γ(λ). More generally, I derive an (increasing) relationship between the highest τ 0 that can be supported and G/R. The proposition also establishes that τ 0 = 0 is always an equilibrium, no matter how high we set G/R. This is because if an agent knows that others of her same type are selling, she knows that sales will be revealed for sure and, therefore, she sells. In Section 4.2, I increase the amount of noise so that it can hide sales by one type. In this case, which I will refer to as the noisy case, the inference from prices is often ambiguous, with investors unable to distinguish the beginning of the crash from a temporary price dip due to noise. For tractability, I restrict attention to the case where noise can hide sales of one type, but not two, so that prices can be categorized as high, medium, or low. High prices reveal that no types have left the market, medium prices are consistent with either no sales or with sales by one type, and low prices reveal with certainty that sales have begun. While restrictive, this assumption simplifies the analysis and makes it possible to focus on Markovian strategies, which condition behavior in addition to the signal and preference shock only on the most recent price. In Proposition 2, I establish conditions under which the equilibrium without bubbles can be ruled out. Proposition 2 also establishes conditions under which there exist equilibria with long bubbles. By long bubbles, I mean bubbles that are so large relative to fundamental value that the fraction of the price lost in the crash is close to 100 percent. Finally, in Proposition 3, I contrast Propositions 1 and 2 to compare bubble formation with and without noise. 17 In AB, since time and types are continuous, in the benchmark case, sales begin gradually and the mass of agents selling at any given instant is zero. If the price reflected sales, any positive mass of sales would be detected, and thus the probability of selling before the crash would be zero. To avoid this, AB assumed that as long as sales do not surpass a threshold κ, the price simply does not reveal these sales. The price continues to grow as if sales had not started, and it reacts only when total sales reach a threshold κ. As far as the ability of agents to exit the market is concerned, assuming discrete periods and types is in a sense similar to having κ = 1/N in AB s continuous model.

17 1860 ANTONIO DOBLAS-MADRID Strategies and Equilibria 4.1. The Noiseless Case In the noiseless case, the amount of noise ε is so low that, as soon as the first type sells, the price is certain to reveal these sales, because ([ θ + ε] 1 1)G t the lowest possible price while all types are still in the market exceeds ([1/N + ( θ ε)(1 1/N)] 1 1)G t the highest possible price when one type sells; that is, in the noiseless case, ε is below a threshold ε 0 1 given by (22) ε 0 1 (1 θ)/(2n 1) where the subscript 0 1 denotes zero types selling before time t and one type selling at t. The strategy of agent i is the following: if hit by the shock, she sells and consumes; otherwise, she does not consume. Pre-crash, she invests into the bubble before period ν(i) + τ 0 and exits the market at time ν(i) + τ 0.Postcrash, she invests a fraction (R/G) t (t 0 1) of her endowment into the risky asset and the rest into the risk-free asset. Put more formally, we have the following profile. STRATEGY PROFILE 1: For any i [0 1], the strategy of agent i is defined as follows: If δ i t = 0, then a i t = (0 h i t 1) for any t. If δ i t = 1/R, then χ i t = 0forallt and the choice of (m i t s i t ) is as follows: (a) If t T, then (23) { (bi t + e t 0) if t<ν(i)+ τ 0 (m i t s i t ) =, (0 h i t ) if t ν(i) + τ, 0 with τ 0 0. (b) If t T + 1, then (m i t s i t ) = ((R/G) t (t 0 1) e t 0). When agents follow these strategies, only type-t 0 agents succeed in riding the bubble. They sell at t 0 + τ, p 0 t 0 +τ 0 reveals their sales, and the crash happens at T +1 = t 0 + τ In Proposition 1, I characterize the set of possible equilibria in different regions of the parameter space. I show that if e λ <G/R<Γ,where Γ e λ ( e λ )/2, then agents follow (23) ifandonlyifτ 0 = 0; that is, if e λ <G/R<Γ, there is a unique no-bubble equilibrium where agents sell as soon as they observe the signal. We will later use this case as a benchmark for comparison with the noisy case. If Γ G/R < 1 + e λ, equilibrium can be supported for any τ 0 between zero and a positive upper bound, and if G/R

18 A ROBUST MODEL OF BUBBLES e λ, any integer τ 0 can be supported in 0 equilibrium.18 Note that even when bubbly equilibria exist, τ 0 = 0 is always an equilibrium. Before proceeding to the proposition and proof, it may be useful to sketch the main ideas behind the results. In an equilibrium with τ 0 0, type-n agents must be willing to (i) sell at time n + τ and (ii) not sell before n + 0 τ.for 0 any τ 0, a type-n agent will always sell at n + 0 τ 0, since other type-n agents are selling and p n+τ 0 will reveal the sales, causing a crash. The key to (ii) is to focus on the time when a type-n agent is most tempted to deviate from the strategy by selling early. This crucial time is t = n + τ 0 1, one period before she is supposed to sell. Since the bubble has not burst, she knows that type n must have been either first or second to observe the signal. Thus, supp n t (t 0 ) = {n 1 n} with μ n t (n 1) = 1/(1 + e λ ) and μ n t (n) = e λ /(1 + e λ ).Ifshe waits, she will receive the discounted post-crash price αg n 2 R τ 0 +1 if t 0 = n 1; if t 0 = n, she will ride the bubble for one more period and earn the discounted price αg n+τ 0/R. If she sells, she will earn the expected time-t price αg n+τ 0 1. In sum, waiting is preferable if (24) ( ) (τ 1 G 0 +1) 1 + e λ G 1 + e λ R 1 + e λ R Since the right-hand side of (24) is decreasing in τ,if(24) 0 failsforτ 0 = 1, it fails for all τ > 1. In Appendix A, I show that e λ >(G/R) 2 + e λ G/R holds if 1 <G/R<Γ; that is, if G/R < Γ, there are no equilibria with τ 0 1. If G/R Γ, equilibrium can be sustained for any integer below an upper bound 1 ln(1 + (1 G/R)e λ )/ ln(g/r), which is obtained solving (24) forτ. 0 Finally, note that if G/R 1 + e λ, then (24)holdsforanyτ 0. 0 To see why type-n agents are most tempted to sell at t = n + τ 0 1, consider, for instance, period n + τ 0 2. Two periods before agents are supposed to sell, supp n t (t 0 ) ={n 2 n 1 n} and the crash probability is μ n t (n 2) = 1/(1 + e λ + e 2λ ),lessthanin(24). The crash probability only falls further as we consider type-n agents choices at times n + τ 0 s for s>2. PROPOSITION 1: Assuming that 1/N 0 and letting Γ e λ ( e λ )/2, the values of τ 0 that can be supported in equilibrium depend on parameters as follows: (a) If e λ <G/R<Γ, only τ 0 = 0 can be supported in equilibrium. (b) If Γ G/R < 1 + e λ, equilibrium can be supported for any integer τ between zero and an upper bound 1 ln(1 + (1 G/R)e λ )/ ln(g/r). 0 (c) If 1 + e λ G/R, any integer τ 0 0 can be supported in equilibrium. 18 While infinite bubbles are technically possible for some parameter values, this possibility is not of interest, since as remarked in Section 2, the focus of the paper is on bubbles that burst endogenously in finite time.

19 1862 ANTONIO DOBLAS-MADRID PROOF: Choices by agents who are hit by the shock, as well as the choices of agents who are not hit in pre-boom and post-crash periods, are rather trivial. Agents hit by the shock do not value the future, and thus willingly sell and consume everything. Pre-boom and post-crash, agents have no reason to deviate from strategies, since the expected price growth rate is R. Thus, for the rest of the proof, we will focus on the choices of agents who have not been hit by the shock in boom periods t {1 T}. Moreover, the assumption that 1/N is small will allow us to neglect revenue effects of the sales by the first type and make the convenient approximation p T αg T. To support a given τ 0 0inequilibrium,itmustbethatforanyn and any boom period t {1 T}, type-n agents are willing to (i) sell at time n + τ 0 and (ii) not sell before time n + τ. Condition (i) holds for any n and 0 τ 0. 0 To see why, note that at t = n + τ,atype-n agent knows that t 0 0 = n, that other type-n agents are selling, and that p n+τ 0 will reveal these sales, causing acrashatn + τ + 1. Selling is optimal as the expected time-t price 0 αgn+τ 0 exceeds the payoff from waiting, given by the expected discounted post-crash price αg n 1 /R. 19 Regarding (ii), consider the sell-or-wait choice of a type-n agentattimet = n + τ s, withs>0. First, focus on cases with s 0 τ 0, so that the agent has observed the signal as of time t. The agent may be motivated to deviate from the strategy and sell preemptively if it is possible that t 0 = n s, in which case type-t 0 agents will sell at t and precipitate a crash at t + 1. Clearly, the greater the probability that t 0 = n s, denoted by μ n t (n s), the greater the agent s incentive to deviate. Since t 0 must be positive and greater than n N, the probability μ n t (n s) is zero if s n or s N. Ifs<min{n N}, however, supp n t (t 0 ) is given by {n s n} and the likelihood of a crash at t + 1is μ n t (n s) = 1/(1 + +e λs ). Clearly, this likelihood is highest for s = 1, as in (24), and, therefore, if agents choose not to sell preemptively when s = 1, they will not choose to do so when s>1 either. Next, consider cases with s>τ 0, which implies that t<n. Before observing the signal, the type-n agent only knows that t 0 must be positive, greater than t (N 1), since otherwise she would have observed the signal, and greater than t τ 1, since otherwise sales would have started. Thus, supp (t 0 n t 0) = {τ 0 Z τ 0 max{1 t N + 2 t τ 0 }} and the likelihood of a crash at t + 1 is μ n t (t τ ).Ift 0 τ 0 max{1 t N + 2}, as is the case in the equilibrium with τ = 0, μ 0 n t(t τ ) = 1 0 e λ.(ift τ < max{1 t N + 2}, then μ 0 n t(t τ ) = 0.) An agent in this situation can sell preemptively at a price 0 αgt or wait, in which case with probability e λ she will sell at t + 1 at a higher (discounted) price αg t+1 /R and with probability 1 e λ obtain the post-crash price. Even if 19 Note that the expected payoff to the agent is she waits is αg n 1, regardless of whether she is hit by the shock at t + 1 or not. If she is hit, she will have a strict preference for selling at t + 1; if not, she will be indifferent between selling and not selling. In either case, the utility she expects fromoneunitoftheriskyassetisαg n 1.

20 A ROBUST MODEL OF BUBBLES 1863 the post-crash price is zero, if 1 <e λ G/R, waiting is optimal. Thus, the mild condition e λ <G/Rsuffices to rule out preemptive sales if t<n. Note that since 1 e λ < 1/(1 + e λ ), agents are less tempted to sell preemptively before observing the signal than in the situation captured by (24). We have now established that of all situations a type-n agent may face before n + τ, the situation considered in (24), with t = n + 0 τ 1, n 0 2andτ 0 1, is the one where she is most inclined to sell preemptively. Thus, if (24) precludes preemptive sales when they are most tempting, it also precludes such sales in allpossiblecaseswitht<n+ τ 0. Parts (a) (c) follow directly from (24). As I show in Appendix A,ifG/R < Γ, agents are not even willing to wait for τ = 0 1 period after observing the signal. Part (a) follows from here. If G/R > Γ, equilibrium can be sustained for any integer τ 0 between zero and an upper bound. If G/R Γ, equilibrium can be sustained for any integer below an upper bound 1 ln(1 + (1 G/R)e λ )/ ln(g/r), which is obtained solving (24) forτ 0. Part (b) follows from here. To establish (c), simply note that if G/R 1 + e λ, then (24)holdsforanyτ, including 0 τ =. Q.E.D Discussion Equilibria with positive and finite τ 0 are examples of speculative bubbles that do not last forever and are not supported by irrational buyers. To better understand what makes such bubbles possible, it is useful to recall previous well known results. Tirole (1982) famously provided three arguments to explain why bubbles cannot occur in a finite horizon model. The first is that if there is a final date T, then nobody would pay more than the discounted present value of the asset at T 1 and similarly for all dates before. The second argument is that if the probability of selling the asset goes to zero as the horizon approaches, then the price must go to infinity. But this is not possible because there is a finite amount of wealth. The third argument is that since a bubble is a zero-sum game, risk-averse investors cannot all be better off by participating in the bubble and thus refuse to participate. Risk-neutral investors would be indifferent between participating and not, since they expect zero profits. Regarding the first argument, it is important to note that while the model is infinite, bubbles are always finite in the equilibria of interest. 20 In a typical equilibrium, the bubble lasts for a finite number of periods τ 0 until period 20 Given the distribution in (1), the model as presented is infinite. However, one could construct a finite version by truncating the distribution at a maximum value, and assuming that there is a date when dividends are paid and the model ends. In choosing equilibrium strategies, one would have to be mindful of the fact that some agents may observe signals that reveal that they are late in the line. Clearly, these agents would be unwilling to ride the bubble. However, agents observing signals earlier would face a situation more similar to agents in the infinite model, not knowing where they are in the line. To generate bubbles in such a finite model, strategies would have to specify longer waiting times for agents receiving earlier signals, with waiting times gradually falling and being equal to zero for agents who observe the latest possible signals. While the