Global Games in Macroeconomics

Pau Roldan⇤

This version: November 17, 2014

Abstract

A large variety of macroeconomic phenomena, including debt crises, speculative attacks, bank runs, investment crashes and socio-political instability, can be thought to be the outcome of strategic complementarities in payo↵s and actions that foster coordination among economic agents. In such interpretation, multiple equilibria may emerge. Multiplicity has ambiguous implications from a positive perspective, and an important strand of the macroeconomic literature has focused on providing theoretical reﬁnements that can help overcome this issue. In the last two decades, a stylized and tractable game-theoretic approach building on the global games literature ﬁrst formu- lated by Carlsson and van Damme (1993) has been proposed and explored in depth. This paper reviews the theory of global games, with emphasis on its implications for equilibrium selection, as well as a few of its recent and most relevant applications to macroeconomics.

⇤New York University, Department of Economics. 19 W 4th Street, Oce 620. Contact: [email protected]. Contents

1 Introduction 3

2 Theory summary 5 2.1 Anintroductoryexample ...... 6 2.2 Symmetric binary-action global games ...... 8 2.2.1 Global games with uniform prior and private values ...... 8 2.2.2 Global games with generic prior and common values ...... 10

3 Applications of global games in macroeconomics 11 3.1 Morris and Shin (1998): The baseline game ...... 12 3.2 Theroleofmarket-clearingprices...... 16 3.2.1 Angeletos and Werning (2006): Trade in ﬁnancial markets ...... 16 3.2.2 Hellwig, Mukherji and Tsyvinski (2006): The role of interest rates ...... 20 3.3 The role of equilibrium outcomes as signaling devices ...... 28 3.3.1 Angeletos, Hellwig and Pavan (2006): Policy as a signal for the investors . . . . . 28 3.3.2 Goldstein, Ozdenoren and Yuan (2011): Attacks as a signal for the policy-maker 31 3.4 Dynamic models of global games ...... 34 3.4.1 Morris and Shin (1999): A repeated static global game ...... 35 3.4.2 Angeletos, Hellwig and Pavan (2007): Endogenous dynamics and timing of attacks 37 3.4.3 Steiner (2008): Coordination cycles through the role of participation ...... 43

4 Discussion 49 4.1 A dynamic partial-equilibrium model of coordination-driven growth ...... 50 4.2 A dynamic general-equilibrium model of coordination-driven growth ...... 53

5 Concluding remarks 58

2 1 Introduction

An economy is a large macro-structure that is populated by a variety of agents who interact, form expectations over uncertain future payo↵s and take optimal decisions with respect to their information sets. There exist many situations in which such interactions may give rise to strategic coordination motives because individual payo↵s depend not only on an underlying economic fundamental that shapes the aggregate state of the economy and that the agents might perceive in di↵erent ways, but also on the actions that other players undertake. In these cases, agents must form beliefs about what other agents believe, and di↵erent coordination schemes can be sustained and be self-fulfilling in equilibrium. For example, with the prospect of a devaluation, a great mass of speculators may attack on a currency if the fundamentals are (or are perceived to be) weak, which in turn would drive the policy-maker to devaluate for a suciently strong attack, thereby confirming the ex-ante expectations of the individual agents. Importantly, this may occur even when each individual agent is atomistic and acknowledges that her action alone is insucient for such an attack to yield aggregate success. However, the coordination-driven correlation in actions may push the final outcome to correspond exactly to the overall prior market expectation. From this point of view, many macroeconomic phenomena, ranging from debt and currency crises to investment crashes and political instability, can be understood as the outcome of self-fulfilling expectations, higher-order beliefs and information processing in an environment of strategic uncertainty (i.e, uncertainty about the behavior of other agents) and payo↵complementarities (i.e, actions that individually deliver a higher payo↵when also chosen by others). It is therefore natural to model such scenarios as games in which players interact in coordination and their payo↵s depend on their own actions, the actions of others and the economic fundamentals. If such fundamentals are common knowledge, di↵erent agents may tacitly coordinate into choosing the same actions in equilibrium. There exists a vast body of work in the macroeconomic field in which coordination motives are responsible for the existence of multiple equilibrium outcomes. In currency and balance-of-payment crises, this view has historically been split into two major currents. First generation models, starting with Krugman (1979) and Flood and Garber (1984) and refined by Broner (2008) and others, view crises as arising from inconsistent policies and the central bank’s inability or unwillingness to sustain the costs of high interest rates. Second generation models, starting with Obstfeld (1986), view crises as a coordinated run on the central bank’s reserves of foreign currency. In this latter view, multiple outcomes can be sustained in equilibrium both because informational di↵erences between investors give rise to coordination and because asset markets may clear at di↵erent interest rates that are consistent with an uncovered interest rate parity condition. Chari and Kehoe (2003) show that the investors’ ability to infer other agent’s private information from their actions can account for the unpredictability

3 of crises and for herd-like capital flows. Other models in which debt crises are modeled as coordination games featuring multiple equilibria are Calvo (1988), Obstfeld (1996) and Cole and Kehoe (2000), to name a few relevant examples. Other interpretations include bank runs, asset price crashes, fluctuations in search activity and episodes of revolution and socio-political distress. For bank runs, Diamond and Dybvig (1983) show that even stable banks may be vulnerable to self-fulfilling panics if there is a systemic mismatch of long-maturity assets and short-maturity liabilities. In the spirit of the global games selection mechanism described below, Goldstein and Pauzner (2005) resolve this equilibrium indeterminacy by adding idiosyncratic noise into an otherwise identical economy, which allows the analysis to provide unambiguous statements regarding the probability of runs and the welfare of banks. Within the literature on asset market crashes, Barlevy and Veronesi (2003) show that uninformed stock market investors who panic may cause asset prices to plummet and precipitate financial crises because they react too strongly against fundamentally-driven declining prices. This interpretation again emphasizes the informational friction as being at the core of the main amplification force behind the large impact of seemingly small shocks. For applications to search, Diamond (1982) showed that multiplicity of equilibria can arise from a matching problem with trading frictions in which recessions, as henceforth viewed by Keyne- sians, are associated with “coordination failures”. Based on this result, Steiner (2008) provides a brief but enlightening example of fluctuations in search activity within the framework that we will analyze in section 3.4.3. Finally, for socio-political crises, Edmond (2013), inspired by Atkeson’s (2000) com- ments on Morris and Shin (1998), presents a model of revolution against a political regime in which the information manipulation by autocratic authorities may ensure their survival. Although diverse in style, the models briefly surveyed above all share the idea that information heterogeneities may themselves be at the core of financial crises. While this view is appealing, two rather technical concerns immediately arise from it. First, since coordination requires rationality in the choice of one’s actions as a best response to those of others, it also demands the individual forecast of others’ actions and, in turn, of others’ forecasts about such actions. Since others’ beliefs become individual states, one may need to condition on the entire infinite hierarchy of higher-order beliefs in order to fully describe the equilibrium set of the economy. This dimensionality problem can become intractable if not adequately dealt with1. Second, when economic fundamentals are common knowledge, coordination may give rise to multiplicity. This is an issue for drawing both determinate economic predictions and unambiguous policy implications. Global games o↵er a tractable and stylized solution to both these concerns. A global game is said to be a game of incomplete information within an environment of strategic uncertainty in which

1This is a well-understood problem that goes back to the classic rational expectations revolution literature in macroeconomics, including Muth (1961), Lucas (1975), Townsend (1983), and Sargent (1991), among others.

4 players receive private signals on unknown economic fundamentals. By introducing private noise, the coordination motive is dampened: if the quality of the private signal is suciently high (as measured by the precision of its noise), agents will place enough weight on information others do not share and multiplicity will not to arise in equilibrium. If they expect other agents to behave in a similar fashion, this behavior can aggregate up enough for the model to select a unique outcome. In the words of Chamley (2003), “the problem of multiple equilibria disappears because a contagion process from the agents with extreme beliefs leads all other agents either to action or to inaction”. In short, private information eliminates the multiplicity e↵ects of common knowledge if the former is suciently precise: uniqueness obtains as a perturbation away from perfect information. Moreover, under regularity conditions on payo↵s, the model can be solved by iterated deletion of dominated strategies and reduced into an analysis by threshold strategies, which enable the agents to make use of a more tractable belief inference without loss of generality and reduces the dimensionality of the problem immensely as higher- order expectations drop out of the relevant state of the economy. These insights were first shown by Carlsson and van Damme (1993) and later on applied in macroeconomics, most notably by Morris and Shin (1998). In this paper, I present the basic theory of global coordination games and review a few applications in macroeconomics. To fix ideas, Section 2 o↵ers an introduction to simple global games, as presented by Carlsson and van Damme (1993) and further developed by Morris and Shin (2003). This section presents key technical results that rely on the payo↵structure of the game and are key for the equilibrium selection mechanisms provided later on to be e↵ective. In Section 3, I present several recent applications of the literature to common macroeconomic problems. I start with Morris and Shin (1998), who first introduced a uniqueness result in a model of currency crises. The remainder of the section is devoted to studying di↵erent extensions of the baseline model in relevant dimensions under which the uniqueness result breaks down. The intuition is simple: introducing public information, either exogenously or endogenously through information aggregation in market prices, signaling in public policy or periodical revelation of information in dynamic settings, provides a common source of knowledge that allows agents to coordinate in their actions. In this strand of literature, I will review a few of what I consider to be the most insightful papers. Section 4 includes a discussion and outlines a potential line of further research on the topic. Section 5 concludes.

2 Theory summary

This section reviews the general theory behind the global game approach to economic problems that exhibit complementarities in actions, payo↵s and, possibly, information. In the general framework, the fundamentals of the economy are assumed to be summarized by a single state variable ✓ R 2

5 which enters the payo↵function of the decision maker. If this state is common knowledge, agents can exploit their shared information and coordinate in equilibrium to give rise to a multiplicity of outcomes. However, we additionally suppose that each agent observes a di↵erent signal that is private information to that agent. Assuming that the noise technology is common knowledge, but the fundamental state ✓ is not, agents must form beliefs about the fundamentals, about other players’ beliefs about the fundamentals, about other players’ beliefs about such beliefs, and so on. We next analyze how this can be made tractable.

2.1 An introductory example

Take the following example from Carlsson and van Damme (1993). There are two players, indexed i =1, 2, unilaterally choosing an action a 0, 1 in a one-shot simultaneous game. For example, i 2{ } consider ai = 1 to be investment, and ai = 0 no investment. Their payo↵s are summarized in Table 1. The fundamental ✓ R is drawn by nature before the game starts. 2

a2 =1 a2 =0 a =1 ✓,✓ ✓ 1,0 1 a =0 0,✓ 1 0,0 1 Table 1: Payo↵matrix for the investment game example.

Suppose ﬁrst that ✓ is common knowledge among the players. The game can be solved in three cases. If ✓>1, then each player has a dominant strategy to invest and (1, 1) is the unique Nash equilibrium in pure strategies. If ✓<0, each player has a dominant strategy not to invest and (0, 0) is the unique Nash equilibrium in pure strategies. Finally, for any ✓ (0, 1), both agents investing 2 and both not investing can be sustained as pure Nash equilibria. Overall, the game exhibits multiple equilibria. Now, assume that ✓ is not common knowledge and there is incomplete information. Nature draws ✓ from a uniform over the real line. This can be thought of being a so-called di↵use (or uninformative) common prior belief about ✓. Additionally, each agent i receives a signal xi that is private information, with

xi = ✓ + "i

where " (0, 1/) and 0 is the so-called precision of the private signal. Endowed with this i ⇠N information, agent i’s posterior belief about ✓ is

6 ✓ x (x , 1/) | i ⇠N i Note that, since the information technology is common knowledge, this agent believes his opponent’s signal to be distributed according to x i xi (0, 2/). | ⇠N A strategy for player i is a map x 0, 1 which we call s. The solution can in principle involve i 7! { } the inﬁnite inference of beliefs between the two agents. However, a much simpler solution procedure based on iterative deletion of dominated strategies turns out to be of use here. We will shortly analyze conditions under which such solutions can be applied in more general settings. Conjecture that the solution is in so-called threshold (or monotone) strategies. This is, suppose that there exists a cut-o↵point x⇤ R such that we can write the optimal strategy by 2

s(xi)=1[xi>x⇤]

for either i,where1[ ] is an indicator function. This means that the agent will choose not to invest · if his signal realization conveys information about the fundamental having a bad enough realization, given what the other player’s beliefs are and what her beliefs about the beliefs are, as represented by a realization of the signal that is below the threshold value. This strategy is thus a switching strategy 2 around x⇤, and is monotone in the sense that there is only one such switching point . Under this strategy, the posterior probability assigned by i to her opponent choosing to invest is

P[x i >x⇤]=1 (x⇤ x) r 2 ! where ( ) henceforth denotes the c.d.f. of the standard normal distribution. For example, if the · player has observed a signal that is equal to the cutting point of the opponent, she will always assign probability 1/2 to her opponent choosing to invest.

The conjecture will be confirmed if we can find the value x⇤ that can sustain the proposed strategy profile as an equilibrium outcome. To do this, we will follow an iterated deletion of strictly dominated strategies procedure. Morris and Shin (2003) provide the following insight: let b(x⇤)betheunique value of x solving the equation

x (x⇤ x) =0 r 2 !

This means that if the player’s opponent is following a switching strategy with cuto↵ x⇤,theplayer’s best response is to follow a switching strategy with cuto↵ b(x⇤). It can be argued by induction that if

2The agent is indi↵erent at the switching point, so we should rigorously call this an essentially unique solution. Because this is common in all the models that we will study here, we will henceforth ignore this nomenclature.

7 a strategy s survives n N rounds of iterated deletion of strictly dominated strategies, then we can 2 write

n 1 1ifxi >b (1) si(xi)=8 n 1 <>0ifxi

If the player knew that the opponent would:> choose action a i = 0 if she had observed a signal less n 1 than b (1), then her own best response would always be to choose action ai = 0 if her signal was n 1 n less than b(b (1)) b (1). Moreover, b( ) can be shown to be increasing and to have a unique ﬁxed ⌘ · point at 1/2, so

1 lim bn(0) = =limbn(1) n + n + ! 1 2 ! 1 1 and therefore the equilibrium is in monotone threshold strategies with x⇤ = 2 .

2.2 Symmetric binary-action global games

The previous example illustrated two key results. First, in simultaneous one-shot two-by-two global games in which the prior is di↵use, introducing idiosyncratic noise in the information structure reduces the set of equilibria to a singleton. Second, the analytical convenience of solving for switching strategies with no loss of generality allows the agents to form beliefs that are based on a single layer of rationality, as opposed to an inﬁnite hierarchy. Morris and Shin (2003) show that this insight extends to similar games with more players (for instance a continuum of them), non-binary choices and asymmetric payo↵s. Moreover, when the common prior is not di↵use and there is meaningful public information that partially restores common knowledge, uniqueness can still arise under parametric conditions on the relative signal precisions. In the present paper I will only review models that make use of symmetric-payo↵binary games and a continuum of players3. Therefore, for later reference, the remainder of this section provides general results for such type of games.

2.2.1 Global games with uniform prior and private values

We ﬁrst consider the case of a uniform common prior on the state and each player’s signal being a sucient statistic for how much they care about the state (private information).

3Having a continuum of players allows us to invoke the law of large numbers in order to approximate aggregate choices as expected outcomes. It also means that each agent is atomistic, and thus cannot unilaterally inﬂuence the aggregate outcome and, therefore, the action of others (see Corsetti et al. (2004) for a study on large players). Moreover, in all our examples, individual actions will not be observable by others, except for the aggregate action. This means that o↵-equilibrium beliefs will not have to be speciﬁed.

8 There is a continuum of players of unit mass. Each chooses an action a 0, 1 and has utility 2{ } u : 0, 1 [0, 1] R R,whereu(a, A, x) is the payo↵of the individual when taking action a,when { }⇥ ⇥ ! A is the proportion of individuals that take action a = 1 and when the private signal is x.Deﬁne marginal utility ⇡ :[0, 1] R R by ⇥ !

⇡(A, x) u(1, A, x) u(0, A, x) ⌘

Before the game starts, a state ✓ R is drawn from an improper uniform density on the real line, 2 and individual i’s private signal is

" x = ✓ + i i p

where >0 denotes precision and "i has a continuous density f( ) with support R. It can be shown · that an individual with signal x puts (posterior) density pf(p(x ✓)) on state the realization of i i a particular state ✓. We assume that the following properties hold:

A1. Action monotonicity: ⇡(A,✓) is nondecreasing in A. • A2. State monotonicity: ⇡(A,✓) is nondecreasing in ✓. •

A3. Strict Laplacian state monotonicity: There exists a unique ✓⇤ that solves •

1 ⇡(A,✓⇤)dA = 0 (1) Z0 A4. Limit dominance: There exist ✓ R and ✓ R such that (i) ⇡(A, x) < 0 for all A [0, 1] • 2 2 2 and x ✓; and (ii) ⇡(A, x) > 0 for all A [0, 1] and x ✓.  2 A5. Continuity: The function 1 g(A)⇡(A, x)dA is continuous with respect to signal x and • 0 density g( ). R · Assumption A1, which states that each player’s utility function is supermodular in the action proﬁle, implies that there exist strategic complementarities in the game: the individual receives more payo↵ by choosing a = 1 when a larger size of the population is choosing a = 1 as well. Assumption A2 states that each player’s utility function is supermodular in her own action and the state, too. Assumption A3 is a single-crossing property ensuring that there is at most one crossing in the optimal policy function for a player with Laplacian beliefs4. Assumption A4 implies that action 0 (1) is a dominant strategy for suciently low (high) signals. Finally, assumption A5 imposes a weak continuity condition.

4According to the deﬁnition provided by Morris and Shin (2003), an agent has so-called Laplacian beliefs if she applies auniformpriortounknowneventsfromthe“principleofinsucientreason”.Thatis,eachagenthypothesizesthat

9 Refer to the game described above and endowed with assumptions A1 to A5 by G⇤(). Deﬁne a strategy for a player in the incomplete information game G⇤() as a function s : R 0, 1 , !{ } where s(x) 0, 1 is the action chosen if a player observes signal x5. Then, we obtain the following 2{ } generalization of the result obtained in Section 2.1:

Proposition 1 Let ✓⇤ be deﬁned as in equation (1). Then, the unique perfect Bayesian Nash equilibrium (PBE) strategy that survives iterated deletion of strictly dominated strategies in G⇤() is

0 for x ✓⇤  s(x)=8 >1 for x ✓⇤ < :> Proof: In Morris and Shin (2003), page 66. ⌅

This result shows that one can focus on symmetric monotone PBE in this framework without any loss of generality. In particular, assumptions A1 and A2 can be shown to be sucient for equilibria to be monotonic with respect to agent types (the latter here symbolized by the switching threshold). Within this class of monotone equilibria, A3 then ensures symmetry and uniqueness. Although there are ways of relaxing these assumptions (see Morris and Shin (2003), section 2.2.3), assumptions A1, A2 and A3 are key for the uniqueness results in most static binary global games set-ups.

2.2.2 Global games with generic prior and common values

As it turns out, the insights behind Proposition 1 extend to games with an even more generic structure. Suppose now that ✓ is instead drawn from a continuously di↵erentiable strictly positive density p( ) on the real line. Moreover, a player’s utility depends on the realized state ✓, not his · signal x on ✓.Thus,u(a, A, ✓) is the payo↵if the player chooses action a 0, 1 , a proportion 2{ } A [0, 1] of individuals choose action a = 1 and the realized state is ✓. Let, as before, ⇡(A,✓) 2 ⌘ u(1,A,✓) u(0,A,✓). We need the following additional assumptions:

A4*. Uniform limit dominance: There exists ✓ R, ✓ R, and " R++, such that (i) • 2 2 2 ⇡(A, x) " for all A [0, 1] and x ✓; and (ii) ⇡(A, x) >"for all A [0, 1] and x ✓.  2  2 + A6. Integrability: The function 1 zf(z)dz is well-deﬁned. • 1 the proportion of other players who will opt forR each action is a uniform random variable over the unit interval. As a consequence, agents choose an action that is a best response to a uniform belief over the proportion of her opponents choosing each action. The agents in all the global games that I will present here are, in this sense, Laplacian players. 5Note all i subscripts have been dropped, for each individual can now be indexed by the signal she receives.

10 Assumption A4*, which will replace A4, says that the payo↵gain to choosing action 0 (1) is uniformly positive for suciently low (high) values of ✓. Assumption A6 ensures that a well-deﬁned distribution of the noise exists. Let G() denote the incomplete information game with properties A1, A2, A3, A4*, A5 and A6. The following extension of Proposition 1 holds:

Proposition 2 Let ✓⇤ be deﬁned as in equation (1). For any >0, there exists >0 such that for all , if strategy s survives iterated deletion of strictly dominated strategies in the game G(),  then

0 for x ✓⇤  s(x)=8 >1 for x ✓⇤ + < :> Proof: In Morris and Shin (2003), page 68. ⌅

This result therefore says that the uniform prior, private-values game G⇤() is the limit of the general prior, common-values game G() as becomes arbitrarily large. That is, the posterior beliefs when private noise is insigniﬁcant are almost the same as under a uniform prior.

3 Applications of global games in macroeconomics

The previous section has presented a few critical results for generic games of incomplete information. This section puts those results into use and provides a review of the literature of global games in macroeconomics. Many di↵erent interpretations can be provided of the global games approach. In currency crises, Morris and Shin (1998, 1999), Chamley (2003), Corsetti, Dasgupta, Morris and Shin (2004), Hellwig, Mukherji and Tsyvinski (2006), Angeletos, Hellwig and Pavan (2006), and Goldstein, Ozdenoren and Yuan (2011) are some key references. Rochet and Vives (2004) and Corsetti, Guimaraes and Roubini (2006) in debt crises and Goldstein and Pauzner (2005) in bank runs are other classical applications. Morris and Shin (2004) apply the insights of Morris and Shin (1998) to a model of debt pricing. Alternative possible interpretations include liquidity crises, investment crashes, adoption of new technologies and socio-political unrest. I start with a brief description of Morris and Shin (1998), a seminal paper showing equilibrium uniqueness results similar to the ones discussed above within a global game of regime change. I provide a slightly simpliﬁed version of the paper in a way that the model can, but need not, be interpreted as a second-generation model of currency crises. Moreover, the notation is adapted in a way to

11 accommodate almost all the models that will follow, and to which the basic Morris and Shin game will serve as a baseline.

3.1 Morris and Shin (1998): The baseline game

Consider a single-shot game in an economy populated by a measure-one continuum of agents, i [0, 1], and a policy-maker (or central bank). Each agent chooses an action a 0, 1 which we 2 i 2{ } interpret as attacking a status quo (a = 1) or not attacking it (a = 0). There is an opportunity cost c (0, 1) of attacking. The status quo is abandoned (and therefore the attack is successful) if enough 2 agents individually choose to attack it relative to the level fundamental, ✓ R,whichrepresents 2 the strength of the status quo. In particular, there is regime change if, and only if, A>✓,where A 1 a dj is size of the attack. The regime outcome can thus be expressed as ⌘ 0 j R R(✓) 1 ⌘ [A>✓] and the ex-post individual payo↵can compactly be given by

U(a ,A,✓)=a (R(✓) c) i i This payo↵says that no attacking individually earns a payo↵of zero. Attacking earns a payo↵of 1 c>0 if the status quo is abandoned, and c<0 otherwise. This framework is general enough to be interpreted in di↵erent ways. In Morris and Shin’s original description, the status quo is a fixed exchange rate regime, and action a = 1 is interpreted as attacking the peg by exchanging domestic for foreign currency in the central bank. The fundamental ✓ can be viewed to proxy for the policy-maker’s cost of defending the fixed regime, whereas A stands for the amount of reserves of foreign currency that are withdrawn by the population of investors6. Just as in our first simple example, if the fundamental ✓ were common knowledge, the model would exhibit multiple equilibria. Again, the solution is almost trivial for ✓<0 (where nobody attacking is a pure Nash equilibrium) and ✓>1 (where all attacking is a pure Nash equilibrium). Otherwise, there is a nonempty set [✓, ✓] [0, 1] such that all attack if ✓ ✓ and nobody attacks if ✓ ✓.If✓ (✓, ✓), ✓  2 both attack and no attack can be sustained in equilibrium. Suppose, however, that the fundamental ✓ is not common knowledge among the players. We assume

6 More particularly, in their paper, there is a currency pegged to an exchange rate e⇤ which, if the policy-maker decides to abandon, will ﬂoat to a rate ⇣(✓) e⇤,where⇣( )isanincreasingandcontinuousfunction.Acontinuum of speculators with unit measure decides whether or not to· short-sell the currency, and the transaction cost of such an operation is c.Thepolicy-makerdefendsthecurrencyifthemeasureofspeculatorsattackingislowerthansome level k(✓), where k( )iscontinuousandincreasing.Theinstantaneouspayo↵ofattackingiszero,whileattackingearns · apayo↵ofe ⇣(✓) c if k(✓)

12 that ✓ is observed with noise through the following information structure. At the beginning of time, nature draws

✓ (z,1/↵) ⇠N or, equivalently, ✓ = z + " with " (0, 1/↵) and (z,↵) common knowledge, and each agent i ⇠N receives a private signal

xi = ✓ + ⇠i

where ⇠ (0, 1/). Once again, we can then index each individual by the signal this individual i ⇠N receives. Deﬁne again the marginal payo↵as

⇡(A,✓) U(1,A,✓) U(0,A,✓) ⌘ Note that assumptions A1, A2 and A3 hold, and therefore the equilibrium analysis from Section 2 can be used. In particular, we can solve for the monotone Bayesian Nash equilibrium by looking for threshold policies without loss of generality. An informal argument is as follows: note that the c.d.f. of the agent’s posterior about ✓ is decreasing in x, meaning that an agent believes a high signal to convey information about a strong fundamental. Moreover, for xx,wherex solves P[✓ 1 x]=1 c, not attacking | is strictly dominant. Using the monotonicity of beliefs, there must exist a switching point x⇤ [x, x] 2 below which an individual would choose to attack, and above which the individual would refrain from doing so.

Therefore, conjecture once more that there is a level x⇤ R such that we can write the optimal 2 individual strategy as s(x)=1[x>x⇤]. Under this conjecture, the aggregate size of the attack is

A(✓)=P[x x⇤ ✓]= (x⇤ ✓)  | ⇣p ⌘ and thus the status quo is abandoned if and only if ✓ ✓⇤,where✓⇤ solves ✓⇤ = A(✓⇤), that is 

✓⇤ = (x⇤ ✓⇤) (2) ⇣p ⌘ On the other hand, x⇤ is the signal that makes the marginal investor indi↵erent between attacking and not attacking, or P[✓ ✓⇤ x⇤]=c, that is  |

x⇤ + ↵z 1 + ↵ ✓⇤ = c (3) + ↵ ✓ ✓ ◆◆ p

13 Equations (2) and (3) jointly determine the solution for thresholds (✓⇤,x⇤), conﬁrm the conjecture and characterize the equilibrium. Since the common prior is not di↵use, agents can coordinate on the information they share (whose precision is given by ↵), and multiple equilibria may arise if the quality of private knowledge is not high enough. Conversely, a suciently precise private signal ensures uniqueness. The next result summarizes this result:

Proposition 3 The equilibrium is unique if and only if private noise is small relative to public noise, so that

↵2 (4) 2⇡ and is in monotone strategies.

Proof: In Angeletos, Hellwig and Pavan (2007), page 744. ⌅

In other words, provided that the parametric condition (4) holds, Proposition 3 tells us that uniqueness holds as perturbation away from common knowledge. Note that a tiny amount of noise with respect to the perfect information benchmark (the limit of which is =+ ) is sucient to break multiplic- 1 ity insofar as condition (4) continues to hold. This is because private information anchors individual behavior and limits the ability to forecast one another’s actions, and thus non-fundamental volatility vanishes when there is no private noise. Figure 1 illustrates this result by showing an experiment in which increasing enough can place the economy in the uniqueness region.

Figure 1: Morris and Shin (1998) uniqueness result. Notation: x 1/p and z 1/p↵. ⌘ ⌘

14 What is more, for given ↵, when the noise in private information is small, then ✓⇤ converges to ✓ 1 c, and whenever ✓ is in the neighborhood of ✓ , a small variation in ✓ can trigger a large 1 ⌘ 1 and discrete variation in the size of the attack and in the regime outcome:

Proposition 4 For given ↵,as + , the equilibrium size of the attack is described by: ! 1

1 if ✓<✓ A(✓) 1 ! 8 <>0 otherwise

:> Proof: In Angeletos and Werning (2006), page 1732. ⌅

Proposition 4 states that there is a large and discontinuous nonfundamental volatility embedded in the type of crises that a global game can generate: when agents are perfectly privately informed, they are able to globally coordinate on the regime outcome, and a low enough fundamental may trigger an economy-wide attack and a subsequently large devaluation. The simple version of the model just outlined has the advantage of providing clear and testable predictions. Rather than merely o↵ering a mechanism with which to select an equilibrium outcome, in the same spirit as a sunspot variable would do, the global game changes the information structure of the players and, by introducing information that cannot be shared between investors, eliminates all strategic motives that may give rise to multiple coordinated outcomes.

Although unambiguous in its prediction, the model’s main results are however not robust to altering the information structure. Indeed, introducing additional motives for coordination can show how unstable uniqueness becomes. A ﬁrst straightforward way is to introduce exogenous public signals that are precise enough relative to the private signal. This has been studied, for example, by Hellwig (2002) and Hellwig and Veldkamp (2008), among others. Both these papers show that in settings in which there are complementarities in actions, the agents’ optimal choices of information acquisition exhibit complementarities as well. A second well-explored mechanism through which equilibrium uniqueness may break down is the introduction of public information through endogenous equilibrium outcomes. The remainder of this section explores this second channel. First, I will review the role of information aggregation provided by equilibrium prices, be it asset prices (Angeletos and Werning (2006)) or interest rates (Hellwig, Mukherji and Tsyvinski (2006)). Second, I will move on to studying public information revealed by government policies, which act as signaling devices that may a↵ect the outcomes of the Morris and Shin game in critical ways. The papers I will describe in this strand of literature are Angeletos, Hellwig and Pavan (2006) and Goldstein, Ozdenoren and Yuan (2011). Finally, I will focus on public information

15 introduced endogenously through the periodical revelation embedded within past outcomes and the role of coordination cycles within dynamic global games settings. In this literature, I will review three papers: Morris and Shin (1999), Angeletos, Hellwig and Pavan (2007) and Steiner (2008).

3.2 The role of market-clearing prices

If a Morris and Shin (1998) game is augmented to introduce equilibrium prices that must clear the markets, the observation of such prices prior to the coordination game may allow the agents to make inference about the state of the economy. If this information is shared by everyone, coordination may be reinforced, actions may feedback back and forth between market activities and speculative episodes, and multiplicity may once again arise. Indeed, if market activities depend on the information that is privately owned by agents, there may exist interesting interactions between public knowledge and private precision. Angeletos and Werning (2006) provide one such example when there is the possibility of trade.

3.2.1 Angeletos and Werning (2006): Trade in ﬁnancial markets

Consider the Morris and Shin game of Section 3.1 and suppose it is augmented in the following way. Prior to the speculative attack stage, now a second stage within the game, there exists a ﬁrst stage in which each agent i [0, 1] can trade over a risky asset with dividend f(✓) at a price p. Agent i 2 must choose the amount k of her wealth w to allocate to the risky asset in order to maximize a CARA objective:

w v(w)= e where >0 is the coecient of risk aversion and w is total individual wealth, given by

w = w pk + f(✓)k 0 where w 0 is an initial level of wealth. In order to prevent prices from being perfectly revealing, 0 the asset supply is stochastic7 and given by

" Ks(")= p where " (0, 1) is the supply noise and >0 is a commonly observed precision. Once trade has ⇠N ended, prices are observed and individuals play a Morris and Shin (1998) game in the second stage.

7Otherwise, the Grossman and Stiglitz (1980) paradox would emerge: if all information were reﬂected in the price (that is, if markets were informationally ecient), no trader would have the incentive to spend resources in order to gather information and trade on it, which would contradict that all information can be included in the price.

16 This stage is identical to the model described in Section 3.1, and therefore all details are suppressed here. The following describes an equilibrium of this economy:

Deﬁnition 1 An equilibrium is a price function P (✓,"), individual strategies for investment and attacking, k(x, p) and a(x, p), and their corresponding aggregates, K(✓, p) and A(✓, p), such that:

k(x, p) arg max E[v(w0 +(f(✓) p)k) x, p] 2 k | 2R K(✓, p)=E[k(x, p) ✓, p] | K(✓,P(✓,")) = Ks(")

a(x, p) arg max E[U(a, A(✓, p),✓) x, p] 2 a 0,1 | 2{ } A(✓, p)=E[a(x, p) ✓, p] |

This definition says that the first-stage optimal portfolio of the individual maximizes ex-ante expected utility given the budget constraint, where the expectation is taken with respect to the individual’s information set (first equation); the aggregate demand of the asset at any price is the sum of all the individual demands (second equation); asset prices P (✓,") clear the asset market given the asset supply (third equation); the second-stage choice a 0, 1 of whether or not to attack maximizes 2{ } expected utility of attacking (fourth equation); and the aggregate size of the attack is the sum of all the attacks (fifth equation). As before, we define the equilibrium regime outcome as

R(✓,") 1 ⌘ [A(✓,P(✓,"))>✓] For simplicity, suppose that dividends are given by f(✓)=✓. Solving for the equilibrium, we can guess a linear price function that is not perfectly revealing. By Bayes’ rule, the price is a normally distributed public signal with precision p, and the posterior fundamental is

x + p 1 ✓ x, p p , | ⇠N + + ✓ p p ◆ Since agents have CARA utility, their individual asset demand is the ratio of the mean excess posterior return on volatility of posterior return (a Sharpe ratio):

(x p) k(x, p)=

Clearing the asset market, the equilibrium price can be found to be:

17 " P (✓,")=✓ p where p

2 p ⌘ 2 In sum, the price is a signal that the agents use in the second stage during the speculative attack game. Note that this signal is endogenous and public, as it emerges from equilibrium trade. Impor- tantly, public information now improves with private information, as p is an increasing function in . Intuitively, when agents are better privately informed about the fundamental, they are better informed about the dividend that the asset will pay out, which makes trade become less risky and prices to be more informative. This is because when private signals are more precise, asset demands are more sensitive and equilibrium prices react more to fundamental than to non-fundamental variables, conveying more precise information. Crucially, whereas in the Morris and Shin (1998) game of Section 3.1 one could increase the precision of private information and enter the uniqueness region with a ﬁxed quality of public information (recall Figure 1), this is no longer possible as better private information entails now better public information. Furthermore, if private precision increases at a fast enough rate, the economy might never leave the multiplicity region, which may therefore survive even as a negligible perturbation away from perfect information. This last case is illustrated by Figure 2.

Figure 2: Angeletos and Werning (2006) multiplicity result. Notation: x 1/p, z 1/p↵ and p 1/ p. ⌘ ⌘ ⌘ p The next proposition provides the formal statement: in contrast to Proposition 3, uniqueness holds for a suciently small private information precision, conditional on public information precision being

18 small too.

Proposition 5 There are multiple equilibria if either or (or both) are large enough. In particular, if the following parametric condition holds:

>2p2⇡ p Proof: In Angeletos and Werning (2006), page 1732. ⌅

Therefore, multiplicity is ensured when better private information improves public information at a rate that is fast enough. In particular, for the argument to of through, the endogenous public noise must fall at a rate faster than does the square root of the exogenous public noise. In this case, uniqueness can no longer be viewed as a small perturbation away from perfect information, and crises may arise as episodes of non-fundamental volatility (e.g., sunspots that select the equilibrium outcome) in which informative variables are being closely monitored8. A second result refers to the perfect information limit of this economy:

Proposition 6 As either source of noise vanishes ( + for given ,or + for given ), ! 1 ! 1 there exists a passive equilibrium in which R(✓,") 0 for any ✓ (✓, ✓), as well as an aggressive ! 2 equilibrium in which R(✓,") 1,forany✓ (✓, ✓). ! 2

Proof: In Angeletos and Werning (2006), page 1732. ⌅

Therefore, in contrast to the statement of Proposition 4, it is no longer true that there is a unique discontinuity in the regime outcome in the perfect information limit, as both extreme common- knowledge outcomes can be recovered as either source of noise vanishes: the regime can be either abandoned or maintained regardless of the fundamental ✓. In this sense, non-fundamental volatility is non-vanishing even when either source of noise is negligibly small, and multiplicity emerges in the limit. This result is in sharp contrast to the Morris and Shin benchmark of Section 3.1, and turns out to be robust to making the dividend endogenous (a function of the aggregate state A) and to introducing the possibility for agents to observe the actions of each other (see sections III and IV in Angeletos and Werning (2006) for details). In sum, Angeletos and Werning (2006) show that the uniqueness result highlighted in Section 3.1 is fragile to adapting the information structure to a market game that, if played prior to the coordination stage, delivers outcomes that foster otherwise absent coordination within the strategic

8Yet another interpretation comes from using the deﬁnition of and restating the parametric condition as p > p2⇡. p p Thus, multiplicity requires the precision of the price signal to be high enough.

19 uncertainty framework. This channel emphasizes the feedback e↵ects that private information exhibits in relation to the public information that is inferred from ﬁrst-stage equilibrium outcomes, and is shown to be robust enough to survive in the perfect information limit. However, as we show next, multiplicity need not stem from information complementarities. We next examine an example in which multiplicity will emerge from market outcomes directly as opposed to indirectly through the e↵ect of such outcomes on the information set of the agents.

3.2.2 Hellwig, Mukherji and Tsyvinski (2006): The role of interest rates

In the speculative attack interpretation of both Morris and Shin (1998) and Angeletos and Werning (2006), a successful attack is viewed as a coordinated run on the central bank’s foreign reserves when interest rates are exogenously determined and the policy-maker’s decision is mechanically driven by the strength with which the status quo (a peg) can be maintained. Within a similar game in which investors are heterogeneous in their information endowment, Hellwig, Mukherji and Tsyvinski (2006) emphasize the role of market-clearing interest rates and explore the multiplicity channel that is originated in deeper market interactions than those studied by Morris and Shin. By focusing on the role of equilibrium prices, not only they are capable of examining the feedback interactions between endogenous public information and exogenous private information, as in Angeletos and Werning, but also they can disentangle these e↵ects from those that are purely due to the market-clearing conditions in the domestic bond market, and which manifest themselves in the form of non-monotonic demand schedules and, therefore, multiple market-clearing prices even in the absence of informational incompleteness9. Consider the following economy. There is a mass-one continuum of agents i [0, 1] called traders. 2 Each trader is endowed with one unit of domestic currency and must choose only one of two possible actions: buy the domestic bond, which returns a safe market-determined interest rate r>0, or exchange the endowment for foreign currency in the central bank (CB), which earns a net return of one in case of devaluation, and nothing if the CB decides to maintain the peg. The investment returns are summarized in Table 2.

Devaluation No devaluation Currency 1 0 Domestic bond r r

Table 2: Net returns of the investment game.

9To be fair, this channel was also brieﬂy explored in Section III of Angeletos and Werning (2006), though it less depth.

20 The CB acts as in the standard Morris and Shin game in the sense that it mechanically compares the costs of devaluations against the strength of the status quo. Whereas this cost was in our original example equal to the size of the attack A (interpreted as the loss of foreign reserves, or total amount of foreign currency withdrawn by traders), here it may potentially also depend on interest rates, r. Denoting the cost of maintaining the status quo by C(r, A), the CB will choose to devaluate if and only if

✓ C(r, A)  where C( ) is increasing in both arguments. In this way, the model will be able to analyze scenarios · in which devaluations are triggered by high interest rates, as in Obstfeld (1996), as well as cases in which devaluations are the result of unsustainable reserve losses after coordinated runs on CB deposits, as in Morris and Shin (1998) and second generation models of currency crises in the spirit of Obstfeld (1986). The game unfolds in three stages. In the ﬁrst stage, nature draws the fundamental from a uniform with support in the real line (an improper common prior), such that a priori any realization of ✓ R 2 is deemed equally likely. In addition, each trader i receives an unbiased private signal:

xi = ✓ + ⇠i

where ⇠ (0, 1/). In the second stage, the domestic bond market and the CB open, and i ⇠N traders submit contingent bid schedules

a(x ,r) [0, 1] i 2 d(x ,r)=1 a(x ,r) i i

on foreign currency holding and domestic bond holdings, respectively. The supply of bonds is exogenously given by10

1 S(s, r)=(s (r)) where ( ) is the c.d.f. of a standard normal, 0 controls the price elasticity of the bond supply · (with = 0 meaning that the supply is inelastic), and s is a supply shock

10 The functional form of S(s, r) is inessential but tractable. What is essential is that S1( ) > 0andS2( ) 0. · ·

21 s (0, 1/) ⇠N that prevents prices form revealing information. Finally, in the third stage, the CB decides whether or not to maintain the peg after observing ✓, r and A, and a devaluation occurs if and only if ✓  C(r, A). The following deﬁnes an equilibrium of the economy:

Deﬁnition 2 A symmetric Perfect Bayesian Equilibrium consists of a bidding strategy a(x, r),anin- terest rate function R(✓, s), a reserve loss function A(✓, s) and beliefs p(x, r) on the posterior probability of a devaluation, such that:

1. For any x, r, ✓ and s, a(x, r), A(✓, r),andR(✓, s) satisfy

=1 if p(x, r) >r 8 a(x, r) > [0, 1] if p(x, r)=r (5) >2 <> =0 if p(x, r) > A(✓, r)=:> a(x, r) ( (x ✓))dx (6) Z p p 1 A(✓,R(✓, s)) = S(s, R(✓, s)) (7)

2. For all r such that (✓, s):r = R(✓, s) = , p(x ,r) satisﬁes Bayes’ law. { }6 ; i

The equilibrium states, in line (5), that for any pair (x, r), the trader’s optimal bidding strategy is to compare the expected return on domestic bonds, given by the safe net return r,totheexpectedreturn on foreign currency, given by the subjective probability that an attack succeeds, p(x, r). Line (6) says that the total loss of reserves from the CB must be equal to the aggregate amount of foreign currency that is withdrawn by traders for any state ✓ and interest rate r. Line (7) states that an equilibrium interest rate function R(✓, s) must clear the domestic bond market for any state pair (✓, s). This naturally creates a correspondence (✓, s), deﬁned by r (✓, s) if and only if 1 A(✓, r)=S(s, r), R 2R interpreted as the set of market-clearing rates for given state (✓, s). Finally, a consistency condition is imposed: for all market-clearing interest rates, posterior beliefs about the likelihood of devaluation must be consistent from a Bayes’ rule perspective. Let us conjecture that the equilibrium can be constructed in monotone threshold strategies. These are characterized by cut-o↵rules x⇤(r) and ✓⇤(r) for r (0, 1) such that a devaluation occurs if and 2 only if ✓ ✓⇤(r), and traders demand foreign currency whenever their signal satisﬁes x x⇤(r), and   otherwise invest in the domestic bond. The latter implies that we can write

22 p(x, r)=P[✓ ✓⇤(r) x, r]  | The marginal, or indi↵erent, trader is deﬁned as the trader that receives a signal x that makes him indi↵erent between both forms of investments, and therefore is such that x = x⇤(r). From condition (5), then we get

p(x⇤(r),r)=r

Since p(x⇤(r),r) is here the expected gain from exchanging currency as deemed ex-post by the indi↵erent trader, the last equation can be interpreted as an uncovered interest parity condition.Reserve losses are given by A(✓, r)=(p(x⇤(r) ✓)), and therefore ✓⇤(r) solves

✓⇤(r)=C(r, A(✓⇤(r),r)) (8)

Finally, imposing bond market clearing, we need r (✓, s), which under the functional form 2R assumption for S(s, r) can be shown to be written as

1 s x⇤(r) (r)=✓ (9) p p

For a given threshold x⇤(r), an interest rate R(✓, s)=r clears the bond market if and only if the above equation is satisﬁed for all (✓, s) pairs. Noting that the left-hand side is constant in ✓ and s,the variable z ✓ s/p o↵ers a sucient statistic for the interest rate to clear the market, and we can ⌘ compactly re-express the set (✓, s) of market-clearing rates with the alternative correspondence R

1 (z) r [0, 1] : z = x⇤(r) (r) R ⌘ 2 p ⇢ Traders use their own private signal x as well as z to form the posterior

x + z 1 ✓ x, z , | ⇠N 1+ (1 + ) ✓ ◆ and if (z) = , then Bayes’ law dictates that R 6 ;

x + x⇤(r) 1 p(x, r)= (1 + ) ✓⇤(r) + (r) (10) 1+ p(1 + ) ✓ ✓ ◆◆ p To conﬁrm the conjecture, therefore, any monotone strategy equilibrium is characterized by a pair

✓⇤(r),x⇤(r) that solves (8) and (9), a posterior belief p(x, r) given by (10), and an interest rate { } function R(z) such that R(z) (z) for every realization of z. 2R

23 The key result is that even though there exists a unique solution ✓⇤(r),x⇤(r) for the thresholds { } that is continuous in r,thereexistz realizations for which (z) is a multi-valued correspondence. R In short, the equilibrium may exhibit multiple market-clearing rates. Intuitively, this is because the interest rate a↵ects optimal bidding strategies not only directly through a payo↵e↵ect, for it is the interest rate what the investor safely obtains in net gains from investing in the bond, but also indirectly through a devaluation e↵ect, for the interest rate a↵ects the posterior likelihood that a run on CB reserves can be successful. In particular, combining market clearing and the marginal trader condition, it can be shown that r (z) for any z if 2R

1 (r) r = (1 + ) ✓⇤(r) z (11) p(1 + ) ✓ ✓ ◆◆ p There exist three basic components in this equation which identify at least three potential e↵ects of interest rates in equilibrium.

First, there is a direct payo↵e↵ect, as seen in the left-hand side of the last equation: all else • equal, a higher interest rate makes the bond pay more, thereby making it more attractive than the foreign currency investment.

Second, r enters in the right-hand side through ✓⇤(r), a devaluation e↵ect: higher r a↵ects the • posterior probability of a devaluation through its e↵ects on the costs of abandoning the currency for the CB, and therefore its direction will depend on C(r, A).

Finally, there is a third e↵ect entering in the last term of the right-hand side which operates • through market clearing: if bond supply is not perfectly inelastic (>0), a higher interest rate means a lower supply of bonds and more traders investing in dollars. By market clearing, this must translate into the indi↵erent investor expecting a higher ✓ and thus becoming less optimistic

about the likelihood of a devaluation (a higher x⇤(r)), which renders the bond more attractive.

While the ﬁrst and third e↵ects are unambiguously positive (higher r increases the incentives for holding the bond), it will be the direction of the second e↵ect what will determine if multiple rates can clear the market. In particular, for such a case to arise, we need the second e↵ect to be strongly negative, that is, for a higher r to increase the probability of a devaluation through the policy-maker’s threshold level ✓⇤(r). Clearly, only if ✓⇤(r) is locally increasing and steep enough in r the devaluation e↵ect can be strong enough. In turn, this critically depends on how the policy-maker reacts to changes in r and, therefore, on the functional form of the cost function C(r, A).

Proposition 7 If the devaluation threshold ✓⇤(r) is continuously di↵erentiable and such that

24 @✓ (r) + p1+ 1 ⇤ > @r p(1 + ) ( 1(r)) ✓ ◆ then there exist multiple market-clearing interest rate functions (that is, (z) is a multi-valued R correspondence).

Proof: In Hellwig, Mukherji and Tsyvinski (2006), page 1777. ⌅

Two examples will clearly illustrate the point. To make sure that the critical e↵ect described above is in action, in both the examples we assume that the supply of bonds is not perfectly inelastic (>0). Moreover, suppose for now that there is common knowledge about ✓.

Example 1: Devaluation triggered by high interest rates First, suppose as in Obstfeld (1996) that the only reason why the CB could choose to devaluate is because of high interest rates. Then,

C(r, A)=r

and devaluation occurs if and only if ✓ r. Furthermore, suppose for now that ✓ is common  knowledge. Figure 3 depicts the domestic bond market supply and demand schedules for this case.

Figure 3: Domestic bond market supply and demand schedules when devaluation is triggered by high interest rates.

The fact that demand is backward-bending delivers multiple equilibrium interest rates in spite of information being perfect. First, for any r (0,✓), the CB does not devaluate, all investors prefer 2

25 the bond and A(✓, r) = 0. For any r (✓, 1), the opposite is true: an attack is successful because 2 interest rates are too high, and no investor chooses to hold the bond for its sure payo↵is lower than the expected net gain from exchanging currency. For r 0,✓,1 , investors are indi↵erent between 2{ } the bond and attacking the currency, and demand is perfectly elastic11. Therefore, since the payo↵ of holding a bond continuously increases as the interest rate increases, but the payo↵of attacking the currency increases discretely only when r passes through the threshold ✓, the devaluation e↵ects locally dominates the payo↵e↵ect, and multiple equilibria emerge. In particular, (z)= 0,✓,1 . R { } Importantly, note that multiplicity arises from the speciﬁc features of the market environment, irrespective of the information structure. As it turns out, the same insights carry out to the case with private information, and multiplicity arises when private signals are suciently precise:

Proposition 8 Suppose that C(r, A)=r. Then, in any monotone strategy equilibrium,

For all r such that z : r = R(z) is nonempty, ✓⇤(r) and x⇤(r) are uniquely characterized by • { }

✓⇤(r)=r

p1+ 1 x⇤(r)=r + (r) ()(1 + ) p r (z) if and only if equation (11) holds. • 2R There are multiple equilibria whenever p(1 + )/(p1+ + ) > p2⇡. •

Proof: In Hellwig, Mukherji and Tsyvinski (2006), page 1779. ⌅

Example 2: Devaluation triggered by reserve losses A similar phenomenon occurs when, as in most second generation models of currency crises, the CB devaluates only if the size of the attack is too large and the loss of reserves is too severe, regardless of the prevailing interest rates. In this case,

C(r, A)=A

and devaluation occurs if and only if ✓ A. As before, start with the assumption that ✓ is common  knowledge. Figure 4 depicts the market for domestic bonds for this scenario.

11For r =0,eventhoughanattackonthecurrencywillnotpayo↵foritwillbeunsuccessful,holdingthebondhas azeronetreturn.Forr =1,attackingwillcauseasuredevaluationandnetgainsofone,butthisisexactlywhatthe investor can obtain by investing in the bond and earning the interest rate out of it for sure. For r = ✓,thecentralbank will devaluate with probability ✓,inturnequaltotheexpectedpayo↵ofattackingthecurrency.Inallcases,thecertain payo↵of holding the bond is equal to the expected payo↵of exchanging the endowment in the CB, and demand is ﬂat.

26 Figure 4: Domestic bond market supply and demand schedules when devaluation is triggered by large reserve losses.

Once again, a non-monotonic demand schedule is responsible for multiple interest rates clearing the bond market. Here, for any r [0, 1], both attacking the currency and not holding the bond (so that 2 A = 1) and conversely (A = 0), can be sustained in a self-fulﬁlling equilibrium. If traders expect no devaluation, they will all hold the bond, the CB will su↵er no attack and no devaluation will indeed be the equilibrium outcome that was ex-ante expected by traders. A similar logic works for the opposite case: if a devaluation is expected, all traders attack the currency and carry out a large withdrawal of reserves, thereby forcing the CB to devaluate, as expected. If r = 0, traders remain indi↵erent between the domestic bonds and the currency as long as ✓>A(✓, r), and thus any demand of bonds that is above 1 ✓ can be sustained in equilibrium. If r = 1, the opposite is true, and any demand lower than 1 ✓ can be part of an equilibrium. Provided that the supply of bonds is not perfectly inelastic, there are again three market-clearing rates in spite of there not being incomplete information in the market. Although omitted here, similar conditions to those of Proposition 8 can be given for the case in which C(r, A)=A and ✓ is observed with idiosyncratic noise. In this paper, therefore, multiplicity arises from the di↵erent roles that interest rates play in the determination of equilibrium outcomes, as there may be multiple market-clearing interest rates that are consistent with an uncovered interest parity condition. The emphasis is not on only on the information- aggregation role played by prices, but also on the payo↵e↵ects that are taken into account by investors even in the absence of information imperfections. This is an important step toward market-based global games that should motivate further research.

27 3.3 The role of equilibrium outcomes as signaling devices

Section 3.2 has outlined examples of global games in which the existence of a Walrasian market provides meaningful information about economic fundamentals that help break down the uniqueness results from Section 3.1. In this subsection we analyze a di↵erent source of endogenous public information: the signaling role of ex-ante public policies and of equilibrium outcomes.

3.3.1 Angeletos, Hellwig and Pavan (2006): Policy as a signal for the investors

Up until this point, we have assumed the decision of the policy-maker to be straightforward: the status quo is abandoned if the cost of the attack (be it in terms of size or interest rates) is too large compared to the gains from maintaining the regime, given by the fundamental ✓. It is reasonable to believe that governments design more sophisticated policies when faced with an economy-wide attack. In the application to currency crises, a central bank can try to prevent an attack by undertaking costly ex-ante policies such as borrowing reserves from abroad, placing taxes on capital outﬂows and so forth. However, if made public before the game is played, such costly policies may in turn signal a high probability of regime fragility, reduce investor’s strategic uncertainty and generate informational feedbacks between public and private policies that may a↵ect equilibrium outcomes in critical ways and, in some cases, force the very policies that were designed to defend the peg to become unwarranted. To gain intuition, consider the following extension of our baseline game from Section 3.1. The population of traders, their objectives and choice sets are identical to those seen above. The only modiﬁcation is on the policy-maker’s (PM) side. For simplicity, consider that nature draws ✓ from a uniform distribution on the real line, and suppose that prior to the Morris and Shin game the PM learns ✓ and chooses a policy, here the opportunity cost c [c, c] (0, 1) that the investors must bear 2 ⇢ in the second stage for attacking the status quo. The payo↵of the PM is

U (R,✓, c, A)=(1 R)(✓ A) (c) PM where as usual R denotes the regime outcome (with R = 1 meaning that the status quo is abandoned), A [0, 1] is the size of the attack and ( ) is a policy cost function, with 0( ) > 0 and 2 · · (c) = 0. Since choosing c = c entails no cost, so we will call this outcome an “inactive policy”. Any choice c>centails an “active policy”. Once the PM has chosen its policy, c becomes common knowledge and the Morris and Shin game starts: each agent i receives a signal x = ✓ + ⇠ with ⇠ (0, 1/) and must choose a 0, 1 to i i i ⇠N i 2{ } maximize U = a (R c). i i Finally, the PM will observe the size of the attack A, and the regime outcome R will be realized. In particular, the status quo will be abandoned if and only if U (1, ) U (0, ), which once again PM · PM ·

28 means that

R(✓)=1[A>✓]

and allows us to write the PM’s ex-post objective as

U (✓, c, A) = max 0,✓ A (c) PM { } The following describes an equilibrium of this economy:

Deﬁnition 3 A symmetric Perfect Bayesian Nash equilibrium consists of a strategy for the PM, c : R [c, c], a (symmetric) strategy for the agents, a : R [c, c] 0, 1 , and a c.d.f., µ : R R [c, c] ! ⇥ !{ } ⇥ ⇥ ! [0, 1], such that

c(✓) arg max UPM(✓, c, A(✓, c)) 2 c [c,c] 2 + 1 a(x, c) arg max a 1[A(✓,c)>✓]dµ(✓ x, c) c 2 a 0,1 | 2{ } Z1 where µ(✓ x, c) is obtained from c( ) using Bayes’ rule and | ·

+ 1 A(✓, c) a(x, c) ( (x ✓))dx ⌘ Z1 R(✓) 1 p p ⌘ [A(✓,c(✓))>✓] are, respectively, the equilibrium size of the attack and regime outcome, for any ✓ R and c [c, c]. 2 2 A ﬁrst point to note is that the Morris and Shin game of Section 3.1 is a particular case of the Angeletos, Hellwig and Pavan (2006) economy in which c = c. That is, our baseline economy can be viewed as one in which policy is not costly, uninformative and, therefore, ine↵ective. Suppose in contrast that c < c, which means that there is a meaningful scope for informationally-relevant policy. Then, the following result is reached:

Proposition 9 Suppose c < c. There are multiple equilibria:

1. There is an equilibrium in which c(✓)=c for all ✓, a(x, c)=1[x

1 (✓⇤(c)) x⇤(c)=✓⇤(c)+ (12) p

✓⇤(c)= (x⇤(c) ✓⇤(c)) =1 c ⇣p ⌘ 29 2. For any c⇤ (c, c], there is an equilibrium in which a(x, c)=1 , R(✓)=1 ˜ 2 [xc otherwise

˜ > where x˜, ✓˜ and ✓˜ are given by :

✓˜ = (c⇤)

˜ 1 1 c 1 ✓˜ = ✓˜ + 1 ✓˜ (✓˜) p 1 c  ✓ ◆ 1 ˜ ˜ (✓) x˜ = ✓˜ + p

Proof: In Angeletos, Hellwig and Pavan (2006), page 470. ⌅

Proposition 9 identiﬁes two classes of equilibria. In the ﬁrst class, the policy remains inactive: the PM avoids the costs of policy by setting c(✓)=c for any ✓ and the same equilibrium outcome as in the baseline game of Section 3.1 surfaces. In the second class of equilibria, however, the PM actively uses the policy at a direct cost (c(✓)) > 0. When doing so, the policy choices convey information to the agents about the economic fundamentals and there is feedback e↵ects that have been ex-ante taken into account by both sides of the economy.

The PM’s side: On the one hand, the PM understands from (12) that, relative to a benchmark • in which the policy remains idle (here, c = c), a higher c means that there is a smaller range of values of ✓ for which the status quo must be abandoned. A myopic PM would use this fact to try to reduce the size of the attack. A sophisticated PM like the one assumed by our deﬁnition of a PBE, however, acknowledges the informational contents of such a policy. In particular, the said policy would render e↵ective only for certain intermediate values of ✓.Indeed,if✓ is too low (in particular, ✓<✓˜), the direct cost (c) of information for deviating from the inactive benchmark toward some c>cwould exceed the indirect payo↵gains from maintaining the status quo and ˜ preventing an attack from being successful. If ✓ is too high (in particular, ✓>✓˜), the size of the attack would otherwise (that is, if the policy were to remain unused) be too small to justify the direct cost (c) of intervention. Therefore, only for intermediate values of ✓ can an active policy ˜ survive. Formally, for any c⇤ (c, c], there is an equilibrium in which c(✓)=c⇤ for any ✓ [✓˜, ✓˜]. 2 2 The investors’ side: From the investors’ point of view, if they agree on a policy a(x, c) that • is insensitive to c, the PM’s policy would remain inactive. However, if they coordinate on a

30 strategy that is decreasing in c (less aggressive attacks when the cost of attacking is higher) and ˜ ✓ [✓˜, ✓˜], then the PM can ex-ante decrease the size of the attack by raising c, as described in 2 the ﬁrst bullet point, and less investors will indeed attack in equilibrium.

Multiplicity here is originated from the feedback e↵ects between the expectations of the PM regarding how sensitive the investor’s policy is to the policy, and reactions of investors to the information embedded in those same policies. There is scope for multiplicity if agents are able to coordinate on di↵erent interpretations of, and di↵erent reactions to, particular sets of policy choices, and if those in turn induce di↵erent incentives to which the PM reacts optimally when choosing those policies.

3.3.2 Goldstein, Ozdenoren and Yuan (2011): Attacks as a signal for the policy-maker

In Angeletos, Hellwig and Pavan (2006), investors’ choices are a↵ected by ex-ante policy from the policy-maker, which acts as a signal that may help reduce the size of speculative attacks in equilibrium. It is reasonable to believe that the interaction can also go in the opposite direction, namely that aggregate trading of speculators may reveal information to the central bank and a↵ect its policy decision. Goldstein, Ozdenoren and Yuan (2011) study this aspect of the informational interaction between market activities and policy decisions by allowing the central bank to learn from the very same market complementarities that its policies generate. The fact that the central bank can learn from market outcomes in order to perform inference about fundamentals, however, feeds right back into the same market outcomes that may arise, for it facilitates further coordination between investors in the same spirit as the ex-ante costly policy signaled the state of the economy in the approach followed by Angeletos, Hellwig and Pavan. However, unlike in that paper, here the policy-maker is oblivious to the realization of the fundamental when undertaking its policy, and observing a large attack on the part of the investors may prove to be useful information about the underlying state of the economy. The following is a concise description of the model. As usual, a continuum of speculators coexist with a single policy-maker (PM). With a single exception noted below, the investors’ side is identical to what we described in Section 3.1: they take a zero-one action, at a cost of c (0, 1) for the active 2 action, upon the observation of unbiased signals about ✓. The PM decides whether or not to maintain the status quo, and the economic fundamental ✓ stands for the value of maintaining it. The PM does not observe the realization of ✓ when making a policy choice. We characterize the decision of the PM as a variable 0, 1 , and its payo↵as 2{ }

UPM = ✓

Note that, unlike in all the settings discussed above, the PM is now not directly a↵ected by the size of the attack. However, the investors’ decisions matter because they may provide information about ✓

31 to the PM. The timing is as follows. First, both the PM and the speculators receive information about ✓.The common prior is di↵use over the real line. The PM receives a private signal:

⇠PM xPM = ✓ + pPM where ⇠ (0, 1). Each speculator i [0, 1] receives two signals. One signal is as in the PM ⇠N 2 benchmark model:

⇠ x = ✓ + i i p with ⇠ (0, 1). A second signal is also private but includes a noise component that is commonly i ⇠N shared by all speculators:

⇠ ⌘ x = ✓ + p + i pi p↵ p⌧

where ⇠ (0, 1) is the common noise component and ⌘ (0, 1) is the private noise component. p ⇠N i ⇠N The terms , PM, ↵ and ⌧ all denote precisions. The precision of xpi is ⌧ 0 = ↵⌧/(↵ + ⌧). All error terms are orthogonal to one another. The commonly shared component in the second signal implies that this signal is correlated across speculators12. Once the agents independently gather these sources of information, speculators simultaneously decide whether or not to attack the status quo. Finally, after observing the size of the attack, the PM decides whether or not to abandon the regime. The payo↵s and information structure are common knowledge, but the true realization of ✓ is not observed at any point of the game. Each individual speculator is not identiﬁed by a pair of signals, and as usual we drop i subscripts. The following deﬁnes the equilibrium.

Deﬁnition 4 A symmetric Perfect Bayesian Nash equilibrium consists of a strategy for the PM,

(A, xPM), a symmetric strategy for the speculators, a(x, xp), a size of the aggregate attack A(✓,⇠p), and posterior probability measures ⌫(✓ A, x ) for the PM and µ(✓ x, x ) for the speculators, such | PM | p that

Decisions are optimal given the information sets: • 12One way of motivating this, as viewed by the authors, is to think of the information generating process as being subject to common random shocks such as market-wide rumors.

32 + 1 (A, xPM) arg max ✓d⌫(✓ A, xPM) 2 0,1 | 2{ } Z1 + + 1 1 a(x, xp) arg max a 1[(A(✓,⇠ ),✓+⇠ /p )=0]dµ(✓ x, xp)(⇠PM)d⇠PM c 2 a 0,1 p PM PM | 2{ } Z1 Z1 + + 1 1 1 A(✓,⇠p)= a(✓ + ⇠/ ,✓ + ⇠p/p↵ + ⌘/p⌧)(⌘)d⌘ (⇠)d⇠ ✓ ✓ ◆ ◆ Z1 Z1 p Beliefs are consistent in the sense that ⌫(✓ A, x ) is obtained using Bayes’ law for any A and • | PM x ,andµ(✓ x, x ) is also obtained using Bayes’ law for any x and x . PM | p p The following proposition states that the PBE is in threshold strategies: there exist cuto↵s levels denoted x⇤(x ),A⇤(x ) such that a speculator attacks the status quo if x x⇤(x ) and the PM { p PM }  p in turn abandons the regime if A A⇤(x ). Moreover, these thresholds are uniquely deﬁned. PM Proposition 10 There is a unique linear threshold equilibrium in which the speculators’ optimal strategy is a(x, xp)=1[x x (x )] and the PM’s optimal strategy is (A, xPM)=1[A A (x )], where  ⇤ p  ⇤ PM

x⇤(x )=x⇤(0) kx p p 1 PM A⇤(xPM)= x⇤(0) + (1 + k) xPM 1 k2 T + ⌧  q k is the unique real root to the cubic equation

⌧k2 (⌧ + ↵)k3 + ↵(1 + k)2(⌧ k)=0 ↵ PM ✓ ◆ and

1 2 ↵ 1+ T ⌘ k ✓ ◆ is the precision of the attack as a signal of the fundamental.

Proof: In Goldstein, Ozdenoren and Yuan (2011), page 270. ⌅

Note that the investors’ signal is decreasing in the correlated signal xp, which o↵ers a source of public information. When xp indicates a sound fundamental, speculators attack only if their own private and independent signal, x, o↵ers particularly pessimistic information about ✓. The weight that the speculators put on the correlated signal is k 0 and it spills over to the PM’s decision, who

33 anticipates the investors’ attacking rule. Indeed, the PM infers the underlying state of the economy from observing attacks, and T , which represents the precision the attack as a signal of the fundamental, is decreasing in k: when investors put more weight on the source of information that is shared by others, an attack becomes less informative for the PM. However, investors anticipate the information transmission e↵ects that their attacks carry over to PM’s policies. In particular, since they internalize that the likelihood of a successful attack is increasing in the precision of the PM’s attack-signal, which is in turn decreasing in the weight k they place on the commonly shared piece of information, investors end up putting “too much” weight on xp, that is, a higher weight than that which would be used in an economy in which the PM could not infer information about ✓ from the size of the attack13. Intuitively, when a single speculator places more weight on xp, other speculators follow because not only this signal provides more information about the true ✓, but also because it a↵ects the likelihood of a large-scale attack being successful through the signaling power of the size of the attack. In this sense, the central conclusion is reminiscent of the Morris and Shin (2002) result regarding the detrimental value of public information, with the only di↵erence that whereas in that set-up informational complementarities emerged from payo↵complementarities (in particular, a Keynesian beauty-contest type of game), in the present framework such complementarities arise endogenously through the feedback e↵ects of policy and the fact that both sides of the economy learn from one another. Overall, the very informational complementarities that are responsible for the investors choosing to overweight the commonly shared signal are the ones that generate large attacks that are not justiﬁed by bad fundamentals14.

3.4 Dynamic models of global games

Section 3.2 has shown that the celebrated uniqueness result from Carlsson and van Damme (1993) and Morris and Shin (1998) is extremely fragile to the information structure of the game. We have outlined two examples, prices as information aggregators and equilibrium actions as signaling devices, under which uniqueness may vanish because the possibility of exploiting additional endogenous strategic complementarities fosters coordination back to multiplicity. This section proposes a third route: while all the models analyzed above are static, one can generate public information by allowing agents to make use of past outcomes and learn from their own actions in dynamic games of regime change. There has been a substantial amount of work in dynamic coordination games over the past few years. Frankel and Pauzner (2000), Burdzy, Frankel and Pauzner (2001) and related papers study coordination

13This is the statement of Proposition 2 in their paper, proved in page 273. 14Anumberofinterestingpolicyimplications,includinge↵ectivenessandtransparency,canbedrawnfromthisresult alone, but I will left these out for the sake of exposition. The interested reader is referred to sections 5 and 6 of the original paper.

34 games in which players are randomly matched into two-player, two-action dynamic games where the fundamental evolves according to a random process and players can revise their actions according to a Poison arrival process. In Levin (2000), at each point in time payo↵s may depend on actions chosen by past players, by future players, and on fundamentals, where the payo↵parameter evolves according to a random walk and is announced publicly15. Matsui (1999) and Oyama (2004) assume complementarities between actions of successive generations in respective OLG models of coordination. In an approach loosely related to the global games view, the herding and social learning literature (e.g., Banerjee (1992) and Bikhchandani, Hirschleifer and Welch (1992)) assumes that players sequentially make discrete choices and partially learn the information of players who choose before them. Strictly within the dynamic global games literature, Chamley (1999, 2003) considers dynamic global games with recurrent incomplete information, Chassang (2010) studies the e↵ects of the fear of misscoordination on the sustainability of cooperation in a dynamic game with exit and Giannitsarou and Toxvaerd (2012) outline a recursive formulation of general dynamic global games and characterize Markov equilibria. Dasgupta (2007) provides a model of private social learning in which the dynamics are augmented by endowing the agents with the option to delay their decisions at a cost, and analyzes the welfare implications of a broad class of games. I have chosen to review three papers that I consider relevant and build up naturally on one another. I will start with Morris and Shin (1999), an arguably straightforward dynamic extension of Morris and Shin (1998) in which the fundamental is revealed at the end of every period. I will then move on to Angeletos, Hellwig and Pavan (2007), who give a characterization of a similar game in which the underlying fundamental is ﬁxed and learned only in the limit of time. Finally, I will review Steiner (2008), who proposes a recursive formulation of a dynamic game in which agents can inﬂuence their own participation in future rounds. All three papers have in common that the threshold equilibrium properties discussed above extend naturally to the dynamic setting, and in some cases the uniqueness result from the stage game is preserved in the multi-stage set-up. Because past outcomes may become public information, which restores common knowledge, multiplicity may resurrect and will be once again a recurrent theme. However, because in some instances players never learn the true realization of the state, the size and number of future possible attacks remains ambiguous and, in some cases, entirely unpredictable.

3.4.1 Morris and Shin (1999): A repeated static global game

To set ﬁrm ground from which to start building dynamics, let us begin with a repeated-game version of Morris and Shin (1998): a stage game played over and over with periodical disclosure of the fundamental of the economy.

15In related work, Levin (2001) studies a similar set-up with overlapping generations of players.

35 Time is continuous and t =, 2, 3,... denotes each instant, where > 0 is negligibly small.

The fundamental of the economy at instant t, called ✓t, evolves according to a Brownian motion, so that

d✓t = zpdt

where z (0, 1). On each period t, the benchmark game of Section 3.2 is played: each speculator ⇠N i [0, 1] receives a signal x = ✓ + ⌘ ,with⌘ (0, /), and chooses an action a 0, 1 .The 2 it t i i ⇠N it 2{ } PM, who observes ✓ noiselessly, will abandon the status quo if and only if A ✓ ,whereA is t t t t the aggregate size of the attack. The true value of the fundamental at instant t becomes common knowledge at the end of the period, when all actions have been realized. Therefore, the information set of the PM at the beginning of period t is ✓ , and the information set of the speculators at the { t} t t 16 beginning of period t is ✓t ,xi ,wherexi xi,,xi,2,...,xit is the history of signals .The { } ⌘{ } deﬁnition of equilibrium is a straightforward extension of Morris and Shin (1998), and therefore is omitted here. The characterization of the main equilibrium dynamics is given by the following proposition:

Proposition 11 For a suciently high , there exists a stochastic process h h(✓ ) with the property t ⌘ t

ht ht if ✓t ✓t 2 

such that the equilibrium regime outcome is given by R(✓t)=1[✓ h ]. t t

Proof: In Morris and Shin (1999). ⌅

This proposition shows that the onset of an successful attack can be succinctly characterized by the fundamentals “tripping over” a hurdle process that moves in the opposite direction as the ﬁrst- lagged underlying state. This implies that the prospect of a currency attack increases more than proportionately to a decrease in the fundamentals. Intuitively, the evolution of beliefs which trigger the change of sentiment and, in turn, precipitate the attack can be again represented by a (time-varying) threshold policy: when is large, private information has a higher quality and speculators put more weight on the last fundamental they observed (call it ✓ ) and more on their contemporaneous signal, xt. If fundamentals are thought to deteriorate, the indi↵erent speculator believes other speculators to be more likely to unilaterally attack, and therefore a coordinated attack to be more likely to succeed.

16 Clearly, since ✓t is commonly observed at the end of each period t,allsignalsthatthetraderhasreceiveduptothat point become uninformative, and are therefore discarded in the individual inference problem.

36 Indeed, the hurdle process then increases and the PM is more likely to abandon the status quo in equilibrium, as expected ex-ante by the speculators.

3.4.2 Angeletos, Hellwig and Pavan (2007): Endogenous dynamics and timing of attacks

An extreme simplification of the approach just laid out is that ✓t is revealed to the private agents at the end of each period, permitting them to discard past signals and, loosely speaking, turning the problem into an infinitely repeated static game rather than a game with dynamic inter-temporal interactions. Angeletos, Hellwig and Pavan (2007) show that dropping the periodical disclosure assumption is crucial for obtaining a much richer understanding of the equilibrium dynamics. Indeed, in this case, since past signal realizations are informative about the fundamental (as long as the latter is somewhat persistent or, in their example, just constant), there is scope for learning over time. Moreover, the game continues for as long as the status quo is in place, which in turns o↵ers an additional source of public information from which agents can learn and form future action profiles. In this model, time is discrete and runs to infinity. Once again, consider a Morris and Shin (1998) game played repeatedly period after period. A fundamental ✓ R is drawn by nature at the beginning 2 of time from ✓ (z,1/↵). The PM observes this fundamental, but the population of traders does ⇠N not. Instead, each trader i [0, 1] receives a signal 2

x˜it = ✓ + ⇠it

each period t,where⇠ (0, 1/⌘ ). The PM abandons the status quo at time t if A >✓,so it ⇠N t t

Rt = 1[At>✓] is the period-t regime outcome. The game ends once and for all when the status quo is ﬁrst abandoned. Speculator i’s payo↵from the entire game is

+ 1 t 1 ⇧ = (1 R )⇡ i t it t=1 X where (0, 1) is a time discount factor and ⇡ is an instantaneous payo↵function, given by 2 it

⇡ = a (R c) it it t+1 where c (0, 1) is the opportunity cost of attacking and a 0, 1 is the action chosen in period 2 it 2{ } t. Note that this specification is a natural dynamic extension of the static payo↵s defined in Section 3.1. t Conditional on the status quo being at place at the beginning of period t,letat : R [0, 1] be ! t + the strategy for period t, a a be the strategy profile up to period t (with a1 a 1 ), t ⌘{ ⌧ }⌧=1 ⌘{ ⌧ }⌧=1 pt(✓; at) be the probability that the status quo is abandoned in period t when all agents follow strategy

37 t t 1 t t a and (✓ x˜ ; a ) be the c.d.f. of posterior beliefs in period t,where˜x x˜ . The following t t | ⌘{ ⌧ }⌧=1 describes an equilibrium:

Deﬁnition 5 A strategy a1 is part of a symmetric monotone Perfect Bayesian Nash equilibrium if t t at(x ) is non increasing in x and independent of past actions, and is such that

+ 1 a1(˜x1) arg max a p1(✓; a1)d 1(✓ x˜1) c 2 a [0,1] | 2 Z1 for all x˜1 R and, at any t 2, 2

+ t 1 t t 1 at(˜x ) arg max a pt(✓; at)d t(✓ x˜ ; a ) c 2 a [0,1] | 2 Z1 for all x˜t Rt. 2

Since traders never observe the fundamental ✓ perfectly, they must keep track of their entire history of signals. This is duly reflected in the definition of equilibrium above. However, it turns out that a convenient recursive representation can be used in this case. To see this, define

t ⌘ t ⌘ ⌧ ⌧=1 X 2 to be the cumulative private precision. It is not dicult to see that if ↵ , the stage game t 2⇡ 17 will have a unique solution . Moreover, assume that limt + t =+ , meaning that the perfect ! 1 1 information benchmark takes place only in the limit of time, but that ✓ is uncertain for the investors in any ﬁnite time. To formulate the problem recursively, deﬁne

t 1xt 1 + ⌘tx˜t xt = and t = t 1 + ⌘t t for any t 2, with = ⌘ and x =˜x . Then, posterior beliefs ✓ x˜t can be conveniently written 1 1 1 1 | in terms of current-period variables as follows:

x + ↵z 1 ✓ x˜t t t , | ⇠N + ↵ + ↵ ✓ t t ◆ t Namely, we have found xt to be a single sucient statistic for the whole history of signalsx ˜ with respect to ✓ and Rt+1. All private information being summarized in a single-dimensional object reduces the dimensionality of the problem immensely, and it is due to the Bayesian-step Gaussian updating and the additive structure of signals.

17This is a direct extension of Proposition 3 in Section 3.1.

38 The following proposition shows that an equilibrium may still be represented by (time-varying) thresholds of attack and devaluation:

+ Proposition 12 Any monotone equilibrium is characterized by a sequence x⇤,✓⇤ 1 with x⇤ R { t t }t=1 t 2 [ , ✓t⇤ (0, 1) and ✓t⇤ ✓t⇤ 1, such that: {1} 2

1. At any t, an agent attacks if and only if xt

2. The status quo is in place in period t if and only if ✓>✓t⇤ 1. Otherwise, the game ends.

Proof: In Angeletos, Hellwig and Pavan (2007), page 722. ⌅

As in the static version of the game, each period ✓⇤ and x⇤ X(✓⇤, ) solve ✓⇤ = A (✓⇤)=P[x t t ⌘ t t t t t t  x⇤ ✓⇤], that is, t | t

✓⇤ = (x⇤ ✓ ) t t t t p and the indi↵erent condition for the marginal trader:

P[✓ ✓t⇤ xt⇤,✓>✓t⇤ 1]=c  | Clearly, since by construction of the game the agents can choose an action only if the status quo has never been abandoned in past periods, ✓⇤ is a non-decreasing sequence in time: the status quo { t } cannot be in place in a certain period without it also being in place in the previous period. The evolution of xt⇤ will critically determine, however, the size of the attacks in equilibrium. Interestingly,

x⇤ need not be monotonic, and therefore periods in which some agents might attack (x⇤ > ) { t } t 1 may alternate with periods in which nobody attacks (x⇤ = ). t 1 To understand more deeply the dynamics of attack, the authors propose the following alternative specification by shifting the focus of the study to the conditions of the marginal indi↵erent trader, MG i.e. the agent with signal xt = X(✓t⇤,t). Let U(✓t⇤,✓t⇤ 1,t,↵,z), sometimes Ut for brevity, be the net payo↵from attacking in period t for this agent. To construct this function, first define 2 u : R [0, 1] R R R [ c, 1 c] and X :[0, 1] R+ R by ⇥ ⇥ ⇥ + ⇥ ! ⇥ !

x+↵z (p+↵( ✓⇤)) 1 +↵ if ✓ >✓ p x+↵z ⇤ ⇤ 1 ( +↵( +↵ ✓⇤ 1)) u(x, ✓⇤,✓⇤ 1,,↵,z) ⌘ 8 <> c if ✓⇤ ✓⇤ 1  1 > (✓⇤) X(✓⇤,) ✓:⇤ + ⌘ p

39 respectively, and so the marginal trader’s utility function U :[0, 1] R R2 R [ c, 1 c]is ⇥ ⇥ + ⇥ !

limx u(x, ✓⇤,✓⇤ 1,,↵,z)if✓⇤ =0 !1 8 U(✓⇤,✓⇤ ,,↵,z) 1 >u(X(✓⇤,),✓⇤,✓⇤ 1,,↵,z)if✓⇤ (0, 1) ⌘ > 2 <> limx + u(x, ✓⇤,✓⇤ 1,,↵,z)if✓⇤ =1 > ! 1 > :> Here, u(x, ✓⇤,✓⇤ 1,,↵,z) is the net payo↵from attacking in period t if the agent receives signal x, when it is known that the status quo is still in place (✓>✓⇤ 1) and that the regime will change if and only if ✓ ✓⇤; X(✓⇤, ) is the threshold x⇤ such that, if agents attack in period t if and only if x x⇤,  t t  then At(✓) ✓ if and only if ✓ ✓⇤; U(✓⇤,✓⇤ 1,t↵, z) is again the net payo↵from attacking for the  marginal agent with signal x⇤ = X(✓⇤,t) when it is known that ✓>✓⇤ 1. Clearly, since X(✓⇤,)is the solution to ✓⇤ = A(✓⇤) (that is, to ✓⇤ =(p(x⇤ ✓⇤)) for x⇤ being the individual attack cuto↵ level), then (✓t⇤ 1,✓t⇤) belongs to the equilibrium path if and only if

U(✓t⇤,✓t⇤ 1,t,↵,z)=0

This representation allows us to draw several important results, which I summarize in the next three central propositions. The ﬁrst one, Proposition 13, presents existence of the threshold paths and o↵ers a recursive algorithm to ﬁnd them.

Proposition 13 The strategy a1 is a monotone equilibrium if and only if there exists a sequence of + thresholds x⇤,✓⇤ 1 such that { t t }t=1

t t 1. For all t, at(˜x )=1if xt xt⇤.

2. For t =1, ✓⇤ solves U(✓⇤, , ,↵,z)=0and x⇤ = X(✓⇤, ). 1 1 1 1 1 1 1

3. For any t 2, either ✓t⇤ = ✓t⇤ 1 > 0 and xt⇤ = ,or✓t⇤ >✓t⇤ 1 > 0 is a solution to 1 U(✓t⇤,✓t⇤ 1,t,↵,z)=0and xt⇤ = X(✓t⇤,t). A monotone equilibrium always exists.

Proof: In Angeletos, Hellwig and Pavan (2007), page 745. ⌅

What this proposition states is that learning takes the form of a truncation in the support of beliefs about ✓. Upon the observation that the regime has survived past attacks, the agents acquire the knowledge that ✓ is above a threshold ✓t⇤ 1. Moreover, the results provides a recursive algorithm to compute equilibria. Note that U(✓⇤, , ,↵,z) = 0 admits a solution, and there is at least one 1 1 1 monotone equilibrium in this economy. One way of constructing all other equilibria is to initialize the

40 algorithm at this ﬁrst-period solution, ✓1⇤, and move forward in time. At t = 2, if U(✓2,✓1⇤,2,↵,z)=0 admits no solution for ✓ ,weset✓⇤ = ✓⇤, and if a solution ✓⇤ exists, then ✓ ✓⇤,✓⇤ . Moving forward, 2 2 1 2 2 2{ 1 2} + one can construct potentially all the equilibrium paths ✓⇤ 1, together with the corresponding { t }t=1 + thresholds x⇤ 1. { t }t=1 Proposition 14 states more formally that there may be multiplicity of equilibria in this environment:

Proposition 14 The following are features of the equilibrium set, for any t Z+: 2

1. U(✓t⇤,✓t⇤ 1,t,↵,z) is continuous in all its arguments, non-monotonic in ✓t⇤ when ✓t⇤ 1 (0, 1), 2 and strictly decreasing in ✓t⇤ 1 and z for ✓t⇤ 1 <✓t⇤. ˆ ˆ 2. Let ✓t be the unique solution to U(✓t, ,t↵, z)=0.AsolutiontoUt(✓t⇤,✓t⇤ 1,t,↵,z)=0 1 ˆ ˆ exists only if ✓t⇤ 1 < ✓t, and is bounded from above by ✓t.

3. If ✓t⇤ 1 > 1 c,asolutiontoUt(✓t⇤,✓t⇤ 1,t,↵,z)=0does not exist for t suciently high.

4. If ✓t⇤ 1 < 1 c,asolutiontoUt(✓t⇤,✓t⇤ 1,t,↵,z)=0necessarily exists for t suciently high.

5. If ✓t⇤ 1 is the highest solution to Ut 1(✓t⇤ 1,✓t⇤ 2,t 1,↵,z)=0, there exists a >t 1 such that Ut(✓t⇤,✓t⇤ 1,t ↵, z)=0admits no solution for any ⌧ t such that ⌧ <.

MG An implication of points 1 and 2 of this proposition is that U is ﬂat at c for ✓t⇤ <✓t⇤ 1,then increases with ✓⇤, and eventually decreases with ✓⇤ by converging again to c as ✓⇤ goes up. Every t t t intersection of the function with the horizontal axis corresponds to a solution. The non-monotonicity therefore leaves open the possibility for multiplicity of equilibria. Parts 3, 4, and 5 highlight that agents must have accumulated precise enough private information up to a given period for further attacks to be possible after that period. Only under such a parameter condition future attacks may emerge in equilibrium. If this does not hold for any period, a ﬁrst-period attack is the only equilibrium of the economy. Intuitively, on the one hand, the knowledge that the regime has survived past attacks (inferred from the observation that the game has not ended) operates as a deterrent motive for agents to attack, and even more so if they expect other agents not to attack, either. Whether or not future attacks can take place also then depends in critical ways on the mean common prior, z.

If the common prior z is low, this motive is reinforced, for the perceived probability of regime • change is low and the net payo↵from attacking for the marginal agent is low. In that case, a unique equilibrium of the entire game exists in which an attack occurs only in the ﬁrst period.

In contrast, if the accumulated private information is precise enough, a suciently high z can • o↵set said deterrent motive: a positive prior belief about the fundamental raises optimism over

41 and above adverse incentives stemming from the knowledge that past attacks failed to succeed, and make the economy prone to further attacks.

Figure 5 illustrates a particular example in which this second case holds. The figure depicts the utility of the marginal agent for period 1 (dashed line), period 2 (dotted line) and period 3 (solid line). Note that the dashed line crosses the horizontal axis once, meaning that there is always an attack in the first period. The parametrization is such that no attacks are possible in period 2, due to a combination of a low 2 relative to z: agents have not yet acquired enough private information about the fundamental, and z is not high enough to convince them that an attack is worthwhile even if the first-period one failed to succeed. In period 3, however, the private signal is precise enough to make a second attack possible. Moreover, by Proposition 14, a third attack is possible at some future date.

In terms of the evolution of thresholds ✓t⇤, three possible sequences can be constructed by means of the algorithm outlined in Proposition 13: either ✓⇤ = ✓⇤, t 2; or ✓⇤ = ✓⇤ and ✓⇤ = ✓0 , t 3; or t 1 8 2 1 t 3 8 ✓⇤ = ✓⇤ and ✓⇤ = ✓00, t 3. 2 1 t 3 8

Figure 5: An example of equilibria with multiple attacks. The dashed line is the payo↵ for the marginal agent in period 1, the dotted line is his payo↵in period 2, and the solid line is the payo↵in period 3. Notation: ✓ 1 c. 1 ⌘

The next proposition formalizes these intuitions regarding the mean prior.

Proposition 15 There exist thresholds z z z such that   1. If z

2. If z (z, z), there are at most ﬁnitely many monotone equilibria and t<+ such that, in any 2 9 1 of these equilibria, no attack occurs after period t.

42 3. If z>z, there are inﬁnitely many equilibria and if, in addition, z>z,forany(t, N) Z2 there 2 + is an equilibrium in which N attacks occur after period t.

Finally, z = z = z when c 1/2, whereas z z1/2.  

Proof: In Angeletos, Hellwig and Pavan (2007), page 728. ⌅

At least two important corollaries can be drawn from this result. First, if ✓>1 c and z>z, the status quo survives in any equilibrium. However, for any t>tand some t Z+, an attack (or 2 any number of them, for that matter) can occur yet does not necessarily take place. An important consequence of this is that neither the timing nor the number of attacks can be predicted. Furthermore, after the most aggressive attack for a given period takes place, the game enters a phase of tranquility, during which no attack is possible, and the length of which is decreasing in the speed of arrival of private information. Hence, in equilibrium, there may exist cycles of repeated succession of phases of tranquility, during which no attack occurs, and phases of distress, which conclude after a strong though failed attack on the status quo. This theme will reappear in section 3.4.3. I conclude this exposition with two ﬁnal remarks. First, note that coordination cycles emerge purely from the interaction between coordination and information, and not from any change in the underlying fundamental. This dimension was completely ignored by Morris and Shin (1999), who could not disentangle fundamental e↵ects from informational e↵ects. Second, observe that the source of the dynamics is extremely sensitive to the cumulative private precision t, for a small change in this parameter from one period to another can trigger the switch from tranquility to distress. This discontinuity in outcomes with respect to changes in information is reminiscent of nonfundamental volatility as a source of regime instability, as for instance in Angeletos and Werning (2006) (recall our discussion of Proposition 4).

3.4.3 Steiner (2008): Coordination cycles through the role of participation

Angeletos, Hellwig and Pavan (2007) is a stylized example of how global games settings can generate cycles of coordination that can lead to di↵erent patterns of equilibria even in the absence of exogenous ﬂuctuations of economic fundamentals. In that example, the link between periods is encoded in the ability of players to transform the knowledge that the regime has survived in past rounds into knowledge about the soundness of the fundamental. This endogenous and inter-temporal transmission of information explains why agents may consider worthwhile to postpone their attack decisions. But because di↵erent agents can agree on di↵erent interpretations and di↵erent reactions to the same equilibrium outcomes, coordination cycles of uncertain length and strength emerge eventually as new private information arrives exogenously.

43 Steiner (2008) refines this idea18. In his example, the dynamic link between periods is that each player influences, with the choice of her current action, not only her instantaneous payo↵but also her own future participation in the game. The risky action (attack) is risky not only because it pays o↵if and only if suciently many agents undertake it, but also because it can otherwise lead to bankruptcy and prevent the agent to participate in future projects. Individual and multilateral fear of bankruptcy introduce additional strategic risk into the problem: future coordination undermines current coordination by making attacks more risky in the present round, which incorporates an additional feedback e↵ect between adjacent periods that can lead to chaotic cycles and no steady-state convergence. Here is a description of the model, once again adapted to the nomenclature introduced above. Time 19 is discrete and finite , t =1,...,T. The fundamental is now a stochastic i.i.d. process ✓t with twice continuously di↵erentiable c.d.f. ( ) on the real line. The realization ✓ is revealed at the end of every · t period t, but unknown through period t. Once more, each agent i receives an unbiased private signal

⇠i xit = ✓t + p

where the idiosyncratic errors ⇠i are independent across players and rounds and drawn from a continuous p.d.f. f( ) with support in the real line20. The focus will be on the perfect information · limit, where + . ! 1 Each agent in the continuum has to choose an action a 0, 1 . Agent i’s payo↵of the whole it 2{ } game is

T t 1 ⇧i = ⇡it t=1 X where (0, 1) is a factor of time discount and, in a slight abuse of notation, ⇡ is an instantaneous 2 it payo↵function given by

⇡ a ⌧ ⇡(✓ ,A ) it ⌘ it it t t

where At is the measure of players choosing to attack, the marginal payo↵ ⇡(✓t,At) satisﬁes properties A1, A2, A3, A4, A5 and A6 listed in section 2.2.1 above21, and, as a novelty, ⌧ 0 is the degree it of agent i’s involvement into the game. Note that, unlike in Angeletos, Hellwig and Pavan (2007), the

18See also related work by Mathevet and Steiner (2013). 19As we will see later on, ﬁniteness is not essential except for the equilibrium uniqueness result. See Section 3.3 of Steiner (2008) for a more detailed discussion. 20 We can think of the case ✓t iid (z, 1/↵), as in Section 3.4.2, though the exposition is general enough to accommodate any twice continuously⇠ di↵erentiableN c.d.f. on the real line, ( ). Similarly, the idiosyncratic errors may, but need not, be normally distributed. · 21 Note that the speciﬁcation ⇡(✓t,At)=1[At>✓t] c for c (0, 1), which we have used in all the models above, is a particular example of one such function. Later on we will specialize2 this framework to the Morris and Shin game that we have repeatedly analyzed in this paper.

44 game does not end after a successful attack, and therefore regime switches constitute a dimension the agents can learn from.

When taking ⌧it as a ﬁxed parameter, Propositions 1 in section 2.2.1 above state that each one of the T static games of which the model is composed is a global game with a unique equilibrium that can be solved by threshold strategies: for each t =1, 2,...,thereexistsa✓t⇤ such that the optimal individual strategy at stage t is a(xit)=1[x ✓ ]. it t⇤ The dynamic link between the di↵erent stage games is that the involvement is endogenous. In particular, ⌧it is deﬁned recursively by

⌧i,t+1 = ⌧itb(ait,✓t,At)

with b(a ,✓ ,A ) 0, and ⌧ = 1 is a normalization. If b = 0, so that ⌧ = 0, then the player it t t i1 it i,t+1 is not involved in the play of stage t + 1. From the perspective of period t, the function b(ait,✓t,At) can thus be interpreted as a continuation probability, and ⌧i,t+1 as expected participation next period. Note that b( ) need not be bounded above by one, a case in which the player’s participation would · wear out over time. In contrast, high involvement benefits players because it allows them to take advantage of accessing potentially higher payo↵s by waiting (i.e, choosing a = 1) while having the zero-payo↵outside option still available every period. Thus, withholding embeds a double reward: a direct payo↵e↵ect in the case of successful attack, and an augmented payo↵e↵ect stemming from the benefit of not having participated in rounds in which attacks were not successful. To complete the description of the environment, we need a few more technical assumptions. Where we define ⇢(✓ ,A ) b(1,✓ ,A ) b(0,✓ ,A ), we assume that ⇢(✓ ,A ) is weakly increasing in both t t ⌘ t t t t t t ✓ and A , and 1 g(A)⇢(✓ ,A)dA is continuous with respect to ✓ , for any t [T ], and densities g( ). t t 0 t t 2 · Moreover, we assumeR that the function

⇡˜(✓ ,A ) ⇡(✓ ,A )+⇢(✓ ,A )V t t ⌘ t t t t t+1 satisﬁes the uniform limit dominance condition (condition A4 in section 2.2.1) for any number

Vt+1 > 0. Finally, in order to keep the measure of players constant, it is assumed that the measure of players who disappear from play at each period t (that is, the set i : ⌧ =0 [0, 1]) is replaced by { it }✓ an equal measure of entrants.

Let us formalize notation and derive the solution. We denote by Vt the value of the game conditional on participation at stage t. Under this notation,⇡ ˜(✓ ,A ) ⇡(✓ ,A )+⇢(✓ ,A )V is interpreted t t ⌘ t t t t t+1 as the period payo↵di↵erential between attacking and refraining22. The information set of each agent is xt,✓t,at as of stage t [T ], where superscripts denote histories. We call a pure strategy a Iit ⌘{ i i} 2 22 Of course,⇡ ˜(✓T ,AT )=⇡(✓T ,AT )asthegameendsafterperiodT ,thatis,VT +1 =0.

45 sequence s =(s ,...,s ) of functions that assign to each pair of information sets ( ,..., ) a path 1 T Ii1 IiT of actions (s ( ),...,s ( )). 1 Ii1 T IiT The solution method is by backward induction. At stage T , we know by Proposition 2 in section

2.2.1 above that the unique equilibrium of the stage game can be characterized by a threshold ✓T⇤ such that sT⇤ (x)=1[x ✓⇤ ]. As seen before, the threshold ✓T⇤ is such that sT⇤ ( ) is the best response to the T · belief that a measure AT of participating players is distributed uniformly on the unit interval. That is, ✓T⇤ is as deﬁned by equation (1) in condition A3 (strict Laplacian state monotonicity), i.e. it is the unique solution of

1 ⇡(✓T⇤ ,A)dA =0 Z0

Hence, all players attack (i.e, a = 1, i [0, 1]) if and only if ✓ ✓⇤ , in which case A =1 iT 8 2 T T T and the period expected payo↵is ⇡(✓T , 1) + b(1,✓T , 1)VT +1.Else,if✓T <✓T⇤ , all players wait (i.e, a = 0, i [0, 1]), so A = 0 and all players receive b(0,✓ , 0)V . Therefore, the expected proﬁt iT 8 2 T T T +1 in the last round for individual i is ⌧iT VT ,where

✓⇤ + T 1 VT = b(0,✓,0)VT +1d(✓)+ ⇡(✓, 1) + b(1,✓,1)VT +1 d(✓) ✓ Z1 Z T⇤ + 1 Since VT +1 0 as the game ends after round T ,thenVT = ✓ ⇡(✓, 1)d(✓). At stage T 1, ⌘ T⇤ players anticipate the continuation payo↵ VT they get from waitingR in addition to the direct payo↵ of immediate participation. Waiting (i.e, ai,T 1 = 0 because ✓T 1 <✓T⇤ 1) gives them a payo↵of b(0,✓T 1, 0)VT , whereas attacking (i.e, ai,T 1 = 1 because ✓T 1 ✓T⇤ 1) gives them a payo↵of ⇡(✓T 1, 1) + b(1,✓T 1, 1)VT , so the value of playing in period T 1is

✓T⇤ 1 + 1 VT 1 = b(0,✓,0)VT d(✓)+ ⇡(✓, 1) + b(1,✓,1)VT d(✓) ✓ Z1 Z T⇤ 1 + 1 where VT = ✓ ⇡(✓, 1)d(✓) and ✓T⇤ 1 is given by the indi↵erence condition T⇤ R 1 ⇡˜(✓T⇤ ,A)dA =0 Z0 Following backward, we can continue ﬁnding threshold for earlier rounds and o↵er the following recursive characterization:

Proposition 16 There is a unique symmetric monotone Perfect Bayesian Nash equilibrium described by the following optimal strategy: s (x )=0for all x <✓⇤ and s (x )=1for all x ✓⇤, where t it it t t it it t ✓t⇤ = #(Vt+1) is the unique solution to

46 1 ⇡(✓t⇤,A)+⇢(✓t⇤,A)Vt+1 dA =0 Z0 and the value Vt is deﬁned recursively by Vt = G(Vt+1), where

#(V ) + t+1 1 G(Vt+1) b(0,✓,0)Vt+1d(✓)+ ⇡(✓, 1) + b(1,✓,1)Vt+1 d(✓) ⌘ #(V ) Z1 Z t+1 with the boundary condition VT +1 =0.

Proof: Follows from the text above and from the main results of Morris and Shin (2003) outlined in section 2 above. ⌅

Note that, in all equilibria, the regime outcome is extreme in the sense that, as already illustrated by Proposition 2 in section 2.2.1 and here by Proposition 16, A 0, 1 . This is reflected by the fact t 2{ } that the individuals’ final decision at stage t depends on a signal-independent threshold ✓t⇤ as opposed to a common threshold, say xt⇤, that triggers attack only for certain realizations of the private signal. In this sense, this model can speak nothing about partial attacks (those in which some people attack and some others do not), as all attacks render success. Consequently, coordination cycles will emerge solely from participation in the game (the, so to speak, extensive margin) and not from coordination on the intensity of the aggregate attack (the intensive margin). Since thresholds depend on the continuation values of the game, the equilibrium is fully described by the law of motion of such value functions, given by the map G( ). The system converges to a · stable steady state if G( ) has a unique and stable fixed point V = G(V ). For many setups, however, · G( ) has a unique, unstable fixed point, in which case the system may exhibit chaotic dynamics. · Equilibrium uniqueness, however, holds for any finite T , and if the system has an unstable steady state, the thresholds fluctuate permanently even for large, but finite, T . The existence of fluctuations is preserved when letting T go to infinity, though uniqueness may be lost as the boundary condition

VT +1 = 0 is absent. To derive results that we can compare with, for instance, Angeletos, Hellwig and Pavan (2007), let us specialize the environment to our familiar example23. Suppose

⇡(✓ ,A ) 1 c t t ⌘ [At>✓t] for some c (0, 1). Moreover, assume that an unsuccessful attack results into bankruptcy with 2 probability p [0, 1]. The continuation probabilities b(a ,✓ ,A ) are then as in Table 3. 2 it t t 23Steiner (2008) and Mathevet and Steiner (2013) provide other enlightening examples, including debt crises in emerg- ing economies, ﬂuctuations in search intensity and the e↵ects of taxes on investment reversal decisions.

47 A >✓ A ✓ t t t  t Attack 0 1 p Wait 0 1

Table 3: Continuation probabilities b(ait,✓t,At).

Note that the standard game is a special case with p = 1. If p<1, some cycles may emerge due to coordination in the extensive margin. In particular, we obtain the following specialization of Proposition 16:

Proposition 17 Proposition 16 applies with thresholds ✓t⇤ = #(Vt+1) deﬁned by

1 c #(Vt+1)= 1+pVt+1

The evolution of expected future payo↵s is given by Vt = G(Vt+1), where

G(V ) (1 c) (#(V )) + V [1 (#(V ))] t+1 ⌘ ⇥ t+1 t+1 ⇥ t+1

Proof: Follows directly from specializing the formulas in Proposition 16 to ⇡(✓ ,A ) 1 c t t ⌘ [At>✓t] and to the continuation probabilities given in Table 3. ⌅

An illustration of this economy is given by Figure 6, which shows, for a particular simulation, the evolution of thresholds below which the speculators attack. Periods with high thresholds correspond to periods in which attacks are more likely because the agents correctly infer that others will attack even if the fundamentals are quite sound. The endogenous fluctuations of beliefs create cycles of coordination which manifest themselves in infrequent though successful large-scale attacks in which all agents choose to attack and no agent chooses to wait. The equilibrium dynamics are reminiscent of Angeletos, Hellwig and Pavan (2007) in the sense that there exist periods of tranquility in which no attacks are observed. However, in the Steiner (2008) economy, agents will not disagree in their actions: either all agents attack or none does, so that all attacks are successful. This means that the phases of distress are short-lived and carry little information inter-temporally, as the fundamental is revealed at the end of every period (like in Morris and Shin (1999)). However, both Angeletos, Hellwig and Pavan (2007) and Steiner (2008) have in common that dynamics emerge endogenously from self-fulfilling learning and coordination upon past actions. Moreover, cycles are purely behavioral in the sense that they may flow independently of the realizations

48 ? Figure 6: Evolution of thresholds ✓t below which the speculators attack the status quo. A successful attack occurs in the period denoted by the symbol ⌅ for a particular realization of the fundamental. of the economic fundamentals. This result helps bridging the theoretical gap between the insights from the static global games literature, in which economic outcomes change only if fundamentals change, and the inherently dynamic self-fulﬁlling properties of the classic macroeconomic models of coordination listed in the introduction, such as Cole and Kehoe (2000), Obstfeld (1986, 1995) and Diamond and Dybvig (1983).

4 Discussion

The main results derived in section 3.4 should persuade us that the dynamic global games approach is well suited for studying how informational frictions can propagate into real fluctuations of the business cycle. Above we have presented a few examples in which the evolution of beliefs helps accounting for the evolution of equilibrium outcomes over and above the fundamentally-driven forces of the economy. According to this view, an economy can grow irrespectively of the structural processes that drive their directly payo↵-relevant outcomes, for agents can find it optimal to coordinate in actions that may have a real aggregate impact provided that there is a suciently optimistic and widespread prospect that such actions will deliver the ex-ante expected e↵ects when taken collectively. In this framework, coordination failures may arise: agents may fail to choose those actions that would be in their collective interest if they believe that others will fail to choose them as well. This problem is recurrent in the literature on coordination problems in macroeconomics (see Diamond (1982) for a seminal example), and it may operate as an amplification or multiplier mechanism in the

49 business cycle: when agents perceive economic fundamentals to be weak and believe other agents to possess similar beliefs about the fundamentals and about the beliefs of others, they may coordinate in those actions that pay o↵only in those events that are most correlated with economic distress, for instance a regime change. A coordinated attack on the status quo in turn debilitates the real economy even further, thereby making it prone to stronger future attacks, amplifying the cycle and turning recessions into more persistent, or even frequent, phenomena. If fundamentals themselves have a certain degree of persistence, beliefs will adapt slowly and recessions may be long-lasting due to repeated coordination on pessimistic outcomes. In this way, business cycles are self-reinforcing in the sense that when aggregate activity is perceived to be low, agents take precautionary actions that further depress the level of activity. Arguably, this view of behavioral-driven fluctuations is suggestive of the theory of so-called sentiments that has been recently proposed by Angeletos and La’O (2013). This paper shows in an otherwise standard unique-equilibrium, rational-expectations Walrasian economy that, when limiting the communication protocols that are embedded within a neoclassical economy by allowing trading to be random and decentralized, the business cycle may be driven by a certain type of extrinsic shocks, akin to sunspots, that shift expectations of economic activity without shifting the underlying pref- erences and technologies. Communication may further help propagate shocks in a similar way that market sentiments, animal spirits and rumors do in contagion-based models and alter the amplitude of the cycles in a mechanism similar to the one described in the last paragraph. Yet another theory in which information-driven mechanisms can produce aggregate fluctuations even in the absence of large shocks is the literature on so-called uncertainty shocks, initiated by Bloom (2009), on how exogenous shocks to the volatility of productivity shocks can a↵ect aggregate economic activity. Fajgelbaum, Schaal and Taschereau-Dumouchel (2014) propose an example of an economy in which uncertainty is endogenous, as it is associated with the variance of firms’ beliefs about their own investment. In that economy, times of low activity produce high uncertainty, which are in turn linked with slow learning and low investment incentives, further depressing the economy and, in some scenarios, tying it to uncertainty traps that lengthen the duration of recessions. I the next section I propose a global-games-based approach that I would like to study further. Please note that what I present is, for now, only a first example of such an economy.

4.1 A dynamic partial-equilibrium model of coordination-driven growth

Environment and payo↵s Time is discrete and inﬁnite, indexed by t =1, 2,... There is a measure- one continuum of agents indexed by i [0, 1]. Each agent is endowed with 1 unit of the single good in 2 each period (whose price is normalized to one).

50 Each period, each individual must choose how to allocate this unit of good between two mutually exclusive investment opportunities. First, the agent can attack the status quo, which pays one-for-one if the status quo is abandoned. This investment is risky because its return depends on the realization of an economic fundamental, ✓t R, and on the action of others. Else, the agent can invest into a 2 project that returns g (0, 1), which is ﬁxed and exogenous and accumulates wealth, denoted k .The 2 t payo↵s are described in Table 4.

A ✓ A <✓ t t t t Attack 1 0 Project g g

Table 4: Net returns for investing attacking the status quo and investing into the project, with g (0, 1). 2

The status quo is abandoned if and only if ✓ A ,whereA is the size of the attack given by t  t t 1 At = ajtdj . The regime outcome is Rt+1 1[✓ A ],withRt+1 = 1 meaning that there is a regime 0 ⌘ t t switchR from periods t to t + 1, and next-period wealth is

k =(1 A )g t+1 t We label wealth as k (for capital) because, in the general equilibrium version of the model, the safe investment will be associated to a productive opportunity run by entrepreneurs, while speculation will not produce direct wealth accumulation or growth.

Timing and information Within each period t, there are three stages. In the ﬁrst stage, nature chooses a new realization ✓t R of the fundamental. It is common knowledge that the fundamental 2 follows an AR(1) process

✓ ✓t = ⇢✓✓t 1 + "t

where ⇢ (0, 1), "✓ (0, (1 ⇢2)/↵) and the initial fundamental is known to be drawn from ✓ 2 t ⇠N ✓ ✓ (z,(1 ⇢2)/↵). The realization of ✓ is unknown to the agents, who receive a private signal 0 ⇠N ✓ t

xit = ✓t + ut + ⇠it

where ⇠ (0, 1/) is an idiosyncratic component and u (0, 1/) is a common component. it ⇠N t ⇠N In the second stage, markets open and individuals privately choose whether to attack, by choosing

51 a( ) 0, 1 , or refrain and invest, k( )=1 a( ), where denotes agent i’s information set at Iit 2{ } Iit Iit Iit date t. In the third stage, At is revealed publicly (although ✓t remains unknown) and the status quo is abandoned if and only if ✓ A . Note that the fact that A is revealed at the end of the period t  t t means that all within-period heterogeneity is wiped out and only the component y ✓ + u . t ⌘ t t We suppose that no agent can observe the individual action of others (so that o↵-equilibrium beliefs are not relevant). At period t, the information set of any agent i [0, 1] is therefore = 2 Iit t t 1 t 1 t 1 x ,a ,k ; A , where superscripts denote histories. Since all relevant information is made { i i i } public at the end of the period (in particular, agents do not use their private signals to update because these become uninformative after the revelation of At), the period-t individual posterior beliefs can be summarized by common beliefs:

1 ✓ µ , t|Iit ⇠N t t for all i [0, 1]. This means that all relevant information can be summarized by the single state 2 vector (µt,t) at the beginning of period t + 1. Therefore, the law of motion for common beliefs is

tµt + yt µt+1 = ⇢✓ t + 1 ⇢2 1 ⇢2 = ✓ + ✓ t+1 + ↵ ✓ t ◆

for all t =1, 2,...,withµ = ⇢ z and = ↵/(1 ⇢2) as the initial state vector. Note that posterior 0 ✓ 0 ✓ mean and precision are common across agents (no need to keep track of the belief distribution) and sucient statistics for beliefs.

Equilibrium Adopting the convention that agents attack in case of indi↵erence, the optimal bidding strategy at period t is

1ifp(µ , ) g t t a(µt,t)= 8 <>0ifp(µt,t) where p(µt,t) is the posterior belief of a regime: switch (thus also the expected payo↵for attacking), and the optimal investment strategy is k(µ , )=1 a(µ , ). t t t t Conjecture that the model can be solved by monotone strategies. Then, there exist cuto↵points

✓⇤ ✓⇤(µ , ) and x⇤ x⇤(µ , ) such that the status quo is abandoned if and only if ✓ ✓⇤, and t ⌘ t t t ⌘ t t t  t an agent attacks if and only if her private signal satisﬁes x x⇤, and otherwise invests in the project. t  t

52 1 Since ✓ µ , (µ , ), the latter implies that we can write the posterior belief of a regime switch t| t t ⇠N t t as

p(µ , )=P[✓ ✓⇤ µ , ]= p ✓⇤ µ t t t  t | t t t t t ⇣ ⌘ 1 1 Since x = ✓ + u + ⇠ ,thenx ✓ (✓ , + ), and therefore the size of the attack is given t t t t t| t ⇠N t by

A(✓t)=P[xt xt⇤ ✓t]= (xt⇤ ✓t)  | s + !

and (✓ ) 1 A(✓ ) is the fraction of agents who invest into the project at time t. The threshold t ⌘ t level ✓t⇤ that determines a regime switch solves ✓t⇤ = A(✓t⇤), that is

✓t⇤ = (xt⇤ ✓t⇤) (13) s + !

Finally, the indi↵erence condition for the marginal trader says that p(µt,t)=g, or

p ✓⇤ µ = g (14) t t t ⇣ ⌘ and the thresholds (xt⇤,✓t⇤) are deﬁned as the unique joint solution to (13) and (14), given the state

(µt,t). Figure 7 is an example of a simulated economy. The calibration is g =0.5, ↵ = 1, = 5, = 20,

⇢✓ =0.98 and z =0.5, for T = 100 periods. Note that, over time, At converges toward zero and thus kt does toward g. Intuitively, as time goes by, agents learn about ✓t and coordinate into the safe and “productive” investment. However, in the transit to this steady state, failures in coordination may occur which may lead to sustained episodes of repeated attacks and extended economic cycles.

4.2 A dynamic general-equilibrium model of coordination-driven growth

The last subsection has proposed a preliminary example to show through simulation a simple mechanism that hints that episodes of sustained coordination can survive over and above fundamentally- driven cycles of wealth. Admittedly, this point has been made with an extremely stylized partial- equilibrium dynamic model of regime change. In this subsection I outline a general-equilibrium coun- terpart of the same economy and present a deﬁnition for its equilibrium.

Demographics and production Time is discrete, runs forever and is indexed by t =1, 2,....There is a measure-one continuum of inﬁnitely-lived agents, indexed by i [0, 1], and a single consumption 2

53 Thresholds Fundamentals and public signals 1 0.8 θ∗ x∗ θ y t t t t 0.5 0.6

0 t 0.4 and X t

−0.5 θ 0.2

Evolution of thresholds −1 0

−1.5 −0.2 20 40 60 80 100 20 40 60 80 100 Time Time

Investment levels Posterior mean and actual fundamental 0.7 A k µ θ 0.7 t t t t 0.6 0.6 0.5 0.5 t t θ 0.4 0.4 and k and t 0.3 t 0.3 µ A 0.2 0.2 0.1 0.1 0 0 0 20 40 60 80 100 20 40 60 80 100 Time Time

Figure 7: Simulation of the partial-equilibrium economy. Calibration: g =0.5, ↵ =1, =5, =20,⇢✓ =0.98 and z =0.5, for T = 100 periods.

54 good (with unit price) produced with a constant-returns-to-scale production function Yt = F (Kt,Lt), where Kt R+ is the capital stock of the economy and Lt [0, 1] is hours of labor, with K0 given and 2 2 nonnegative. Labor is supplied inelastically, or Lt = 1, so that we can write production in intensive form as Y /L = f(k ), where f(k ) F (k , 1) and k K /L . We assume that f 0( ) > 0 >f00( ), t t t t ⌘ t t ⌘ t t · · and the standard Inada conditions hold. Factor markets are competitive.

Investment opportunities In order to consume, each individual must allocate his wealth (i.e, wage plus the returns on past investments) into either one of two investment technologies.

1. First, an individual can become a speculator. Turning into a speculator means choosing an action a 0, 1 within a Morris and Shin game, that is, investing into a project that pays o↵ it 2{ } one-for-one only if a suciently large fraction of agents undertake it. This is risky in the sense that it pays o↵depending on the action chosen by others, and pays more in expected terms the more people undertake it (a payo↵complementarity). This does not accumulate wealth for the economy: the wage is stored at zero net return for next period.

2. Else, the individual can become an entrepreneur. Turning into an entrepreneur means investing into a project that will turn into capital next period. This is a safe investment in the sense that its outcome does not depend directly on the action of other entrepreneurs. The investment technology is described by a function g(.), in the sense that investing w units of the good returns

g(w) units of capital, where g0( ) > 0 >g00( ). This capital is then rented out to the ﬁrm at a · · rental rate of Rk.

As usual, the status quo is dropped in period t if ✓t At, and we let ⇡t+1 1[✓ A ] be the regime  ⌘ t t outcome, with ⇡t+1 = 1 meaning that there is a regime switch from period t to t + 1. Table 5 describes the payo↵s.

A ✓ A <✓ t t t t Speculator 1 0 Entrepreneur Rkg(w) Rkg(w)

Table 5: Net returns for investing attacking the status quo and investing w into the capital-accumulation technology.

The law of motion of per-capita capital is

k = g(w )+(1 )k (15) t+1 t t k t

55 where [0, 1] is the depreciation rate of capital and [0, 1] is the equilibrium mass of agents k 2 t 2 who invest into capital (that is, the measure of speculators in the economy). Equation (15) describes the growth pattern of the economy, and for now it is enough to note that ﬂuctuations in aggregate activity stem from two di↵erent sources: exogenously through capital depreciation, and endogenously through changes in t (the extensive margin of investment).

Timing and information Timing and information are identical to the partial-equilibrium version of the model. At each period t, it is common knowledge that the fundamental evolves according to an auto-regressive process

✓ ✓t = ⇢✓✓t 1 + "t

where ⇢ (0, 1), "✓ (0, (1 ⇢2)/↵) and the initial fundamental level is known to be drawn ✓ 2 t ⇠N ✓ from ✓ (z,(1 ⇢2)/↵). The realization ✓ is unknown throughout period t and at any point in 0 ⇠N ✓ t the future.

Within each period, there are three stages. In the ﬁrst stage, nature draws ✓t and each consumer i observes an unbiased private signal xit on it, given by:

xit = ✓t + ut + ⇠it

where ⇠ (0, 1/) and u (0, 1/). In the second stage, all markets open, individuals it ⇠N t ⇠N choose whether to become speculators or entrepreneurs, and each type of agent submits demands for non-capital and capital investments, respectively, all of which are placed privately24.Inthethird stage, A is revealed publicly (so that y ✓ + u serves as a public signal for the agents) and the t t ⌘ t t status quo is abandoned if A ✓ . Payments for both investment technologies are realized only at t t the beginning of the next period. 1 Once again, posterior beliefs are common by the end of the period, such that ✓ x (µ , ), t| t ⇠N t t where posterior mean and precision evolve according to:

tµt + yt µt+1 ⇢✓ ⌘ t + 1 ⇢2 1 ⇢2 ✓ + ✓ t+1 ⌘ + ↵ ✓ t ◆

for all t =1, 2,...,withµ = ⇢ z and = ↵/(1 ⇢2). . 0 ✓ 0 ✓ 24Once more, the fact that individual actions are not observable by others allows us to ignore the role of o↵-equilibrium beliefs.

56 Deﬁnition of a recursive equilibrium Each period t, an agent i with wage w 0mustdecide it whether to become a speculator or an entrepreneur. Let V S(µ, , k) be the value of becoming a speculator and V E(µ, , k) the value of becoming an entrepreneur for an agent that starts a period with a belief system (µ, ) and the prevailing capital stock is k. In each period, given beliefs about the fundamental, an agent can either invest in capital or choose whether or not to attack the status quo. The following deﬁnes a recursive equilibrium:

Deﬁnition 6 A symmetric monotone recursive PBE are value functions V (µ, , k), V E(µ, , k) and V S(µ, , k) for the agent, the entrepreneur and the speculator, respectively; policy rules z(µ, , k) 0 and a(µ, , k) 0, 1 ; factor prices Rk(k) and w(k) of capital and labor; measures A(k) [0, 1] and 2{ } 2 (k) [0, 1] of speculators and entrepreneurs; a perceived probability of regime switch P (✓; k) [0, 1]; 2 2 and a cdf (✓ µ, , k) [0, 1], such that, for any state (µ, , k), | 2 1. Consumers’ optimality: Given prices Rk(k) and w(k), the value function satisﬁes

V (µ, , k) = max V S(µ, , k),VE(µ, , k) ⇢ where

+ S 1 V (µ, , k) = max a0 P (✓; k)d (✓ µ, , k) + ⇢Eµ0,0 V (µ0,0,k0) µ, , k a0 0,1 | 2{ } ✓ Z1 ◆  and

E k V (µ, , k) = max R (k)g(z0)+⇢Eµ , V (µ0,0,k0) µ, , k z (k) 0 0 02 k ⇥ ⇤ (k) z0 :0 z0 w(k)+R (k)g(z(µ, , k)) + 1[A(✓,k)>✓]a(µ, , k) ⌘   where ⇢ (0, 1) is the time discount factor, and the Bellman equations have solutions at the 2 policy functions, namely a0 = a(µ, , k) and z0 = z(µ, , k).

1 2. Belief consistency: (✓ µ, , k) is obtained by Bayes’ rule. In particular, ✓0 µ, (µ, ), | | ⇠N with

1 2 2 µ + y ⇢✓ 1 ⇢✓ µ0 = ⇢ 0 = + ✓ + + ↵ ✓ ◆

The perceived probability of a regime switch is given by P (✓; k)=1[A(✓,k) ✓], for each ✓ R. 2

57 3. Equilibrium measures: The equilibrium measure of speculators is A(k)=✓?(k), where ✓?(k) is the solution to the joint system for (x?(k),✓?(k)):

✓?(k)= (x?(k) ✓?(k)) s + ! Rk(k)g(w(k)) = (p(✓?(k) ))

The equilibrium measure of entrepreneurs is (k)=1 A(k).

4. Market clearing conditions: Competitive factor markets clear, so that w(k)=f(k) kf0(k) k and R (k)=f 0(k). Next-period capital is given by k0 = (k)g(w(k)) + (1 )k. k

Note that, conditional on choosing to become a speculator, not attacking is never optimal, for a weakly greater payo↵could have been obtained by investing in capital as the continuation value is the same in both cases. Therefore, all speculators attack the status quo. This is already imposed in the deﬁnitions above, as A(✓, k) stands both for the aggregate size of the attack and the measure of agents that become speculators. ! Figure 8 is an example of a simulated economy. The calibration is f(kt)=kt with ! =0.3, # k g(wt)=wt with # =0.3, ↵ = 3, = 500, = 50, k =0.16, ⇢✓ =0.99, k0 =0.6 and z = R0 g(w0), for T = 200 periods. Note that, over time, At converges toward zero and thus kt does toward g. Once again, in periods in which the fundamental is perceived to be low, there is coordination into speculation, which lengthens the crisis in the real sector through a slower accumulation of capital.

My attention is now focused on the two models just outlined, which need to be made more tractable and a little less stylized. This is very much work in progress.

5 Concluding remarks

This paper has surveyed theory and applications of the global games literature in macroeconomic problems. The main justiﬁcation for the use of this theory is in problems of coordination, in which the commonality of certain payo↵-relevant information may lead to multiplicity of equilibria. We have analyzed how the global games approach can select one such equilibrium within a particular class of monotone strategy Bayesian-Nash equilibria. The main insights are not robust to the introduction of public information, but it is still possible to derive uniqueness results for a large class of models that allow us to draw determinate policy implications.

58 Factor prices Fundamentals, public signals and beliefs 0.7 0.5 θ y µ 0.65 t t t 0.4 0.6 t 0.55 µ 0.3

0.5 and t , y 0.45 t 0.2 θ Factor prices k 0.4 w R t t 0.1 0.35

0 50 100 150 200 50 100 150 200 Time Time

Equilibrium measures Investment and production 1 1 0.95 0.8 0.9 t

t 0.6 0.85 χ , t ) and k A 0.4 t 0.8 f(k A χ t t 0.75 0.2 f(k ) k 0.7 t t 0 0.65 50 100 150 200 50 100 150 200 Time Time

! # Figure 8: Simulation of the general-equilibrium economy. Calibration: f(kt)=kt with ! =0.3, g(wt)=wt k with # =0.3, ↵ =3, =500, =50,k =0.16, ⇢✓ =0.99, k0 =0.6 and z = R0 g(w0), for T = 200 periods.

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