A Global Game with Heterogenous Priors
aa r X i v : . [ q -f i n . T R ] D ec A Global Game with Heterogenous Priors Wolfgang KuhleMax Planck Institute for Research on Collective Goods, Kurt-Schumacher-Str. 10, 53113Bonn, Germany. Email: [email protected].
Abstract:
This paper relaxes the common prior assumption in the public and private informa-tion game of Morris and Shin (2000, 2004). For the generalized game, where the agent’s priorexpectations are heterogenous, it derives a sharp condition for the emergence of unique/multipleequilibria. This condition indicates that unique equilibria are played if player’s public disagree-ment is substantial. If disagreement is small, equilibrium multiplicity depends on the relativeprecisions of private signals and subjective priors. Extensions to environments with public sig-nals of exogenous and endogenous quality show that prior heterogeneity, unlike heterogeneityin private information, provides a robust anchor for unique equilibria. Finally, irrespective ofwhether priors are common or not, we show that public signals can ensure equilibrium unique-ness, rather than multiplicity, if they are sufficiently precise.
Keywords: Global Games, Equilibrium Selection, Heterogenous Priors, Thin outEffectJEL: D53, D82, D83
Heterogeneity in private beliefs is the core component of global coordination games. In the orig-inal two-player games introduced by Carlsson and van Damme (1993) and Rubinstein (1989),as well as in the Morris and Shin (1998) extension with a continuum of players, it is a smallperturbation away from common knowledge, which selects unique equilibria. This pivotal roleof private beliefs was put into perspective by Morris and Shin (2000, 2004), Hellwig (2002), andMetz (2002), who introduce common priors and public signals into the global games model. Insuch extended settings, it turns out that the global game structure, and its inductive equilib- I thank Martin Hellwig for bringing the literature on global games to my attention. I thank Sophie Bade,Nataliya Demchenko, Alia Gizatulina, Olga Gorelkina, Dominik Grafenhofer, Carl Christian von Weizs¨ı¿ cker,and seminar participants in Bonn for helpful and encouraging discussions. heterogeneity in beliefs, whichoriginates from the dispersion of prior expectations µ , contributes to equilibrium uniqueness.Dispersion in private signals, on the contrary, induces equilibrium multiplicity. In the extendedsettings, where we introduce different types of public signals, we find that prior heterogeneitystill unambiguously contributes towards equilibrium uniqueness. The role of private signals,on the contrary, is ambiguous and depends critically on the specification of the public signal.The main finding in the second part of the paper is therefore that heterogeneity in priors,unlike heterogeneity in private information, provides a robust anchor for unique equilibria . Putdifferently, we show that heterogeneous priors robustly select unique equilibria in rather diverseenvironments. In turn, we compare this finding to the literature where the presence of commonpriors induces multiple rather than unique equilibria.Finally, as a byproduct of our analysis, we find that public signals in themselves, irrespectiveof whether priors are heterogenous or common, have an ambiguous effect on equilibrium multi-plicity: increases in the public signal’s precision can ensure equilibrium uniqueness. This result2s of interest in comparison with those of Morris and Shin (2000, 2004), Hellwig (2002), Metz(2002), Angeletos and Werning (2006) who find that increases in the public signal’s precisionunambiguously induce multiple rather than unique equilibria. Compared to the literature, we extend the canonic common prior coordination games byMorris and Shin (2000, 2004) to a setting where agents’ prior distribution’s mean µ , regard-ing the true state of the world, are normally distributed over the economy’s population. Inthe model of Morris and Shin (2000, 2004), equilibrium uniqueness is ensured iff √ α x α p ≥ √ π ;where α x is the private signal’s precision and α p is the prior’s precision. For the general-ized game, where subjective prior expectations µ are distributed with variance σ µ , we obtain q ( √ α x α p ) + σ µ ≥ √ π as an analog condition for equilibrium uniqueness. This formula’s mainstrength is that it isolates how prior heterogeneity σ µ affects the number of possible equilibria.In general, the condition shows that incomplete information games where the population ofplayers is polarized such that there are two large tails have unique equilibria. Or, equivalently,unique equilibria are ensured if there is only a small group of agents which have “intermedi-ate” or “moderate” beliefs. Put differently, increases in the priors dispersion, σ µ , “thin out”the group of agents with “moderate” beliefs which can, potentially, coordinate on multipleequilibria. Technically, equilibrium multiplicity depends on the relative precisions of privateinformation ( α x ) and the subjective prior ( α p ) if the prior’s dispersion is small ( σ µ < √ π ).For sufficiently dispersed priors, ( σ µ > √ π ), equilibria are unique, irrespective of the relativeprecisions of private signals and priors. The original uniqueness condition for the common prioreconomy ( √ α x α p ≥ √ π ) obtains as prior dispersion ( σ µ ) vanishes.In Section 3, we extend the information structure and embed our coordination game intothree different public signal environments. These extensions provide a background to study thefundamental difference between heterogeneity in priors and private signals. The main findingin this section is that only heterogeneity in priors selects unique equilibria reliably . On thecontrary, the role of both private and public signals changes from environment to environment.These findings originate, first of all, from an environment where the public signal reveals thetrue state of the game with exogenous quality; secondly, from an environment that may beseen as a financial markets context, where stock prices aggregate and reveal dispersed private There will be two classes of equilibria in this setting, and the public signal’s precision ensures uniqueness for every given signal realization . However, if the public signal’s precision is sufficiently high, there may bemultiplicity in the public signal’s realization itself. σ µ is the only parameter which can ensure unique equilibria inall environments. The implications of the public and private signals’ precisions, however, varyfrom case to case. In particular, increases in the private signal’s precision ensure uniqueness(multiplicity) when public signals are of exogenous (endogenous) precision.Recently, Steiner and Stewart (2008), Izmalkov and Yildiz (2010), and Mathevet (2012)have introduced heterogenous priors into global games. Steiner and Stewart (2008), Izmalkov and Yildiz(2010), and Mathevet (2012) focus, respectively, on learning, rationalizability of strategies, andapplications to mechanism design. Unlike Steiner and Stewart (2008) and Mathevet (2012),who focus on N-player games, we study games with a continuum of players as in Izmalkov and Yildiz(2010). Contrary to Izmalkov and Yildiz (2010), who focus on the private information limit ofCarlsson and van Damme (1993), which carries over to these more general games, the presentpaper develops a sharp characterization of equilibrium multiplicity at and away from the privateinformation limit. The present paper is therefore the first to derive closed form expressions forthe emergence of multiple equilibria in large games with heterogeneous priors. Moreover, bymeans of the public signal environments, it provides a framework to interpret the role of priorheterogeneity in the global games structure. Regarding the heterogeneity in priors, we note that Morris and Shin (1998, 2000, 2004)embed their common prior incomplete information coordination games in a currency crises, In the “private information limit”, the private signal’s precision goes to infinity. A crucial consequence ofthis is that the importance of priors for the agent’s decisions vanishes. Izmalkov and Yildiz (2010) discuss prior heterogeneity in a game with a continuum of players.Izmalkov and Yildiz (2010), p. 25, focus on the formation of individual threshold strategies and avoid the“delicate” question of equilibrium multiplicity, which is the focus of the present paper. More precisely,Izmalkov and Yildiz (2010) study games with heterogenous priors (“sentiments”) and characterize for a family ofcoordination games how prior dispersion affects the unique threshold equilibrium which is ensured once privateinformation becomes sufficiently precise. Due to the focus on the private information limit, it remains an openquestion how prior heterogeneity affects the number of equilibria in those cases where private signals are notarbitrarily precise. This question is the focus of the current paper. Finally, while Izmalkov and Yildiz (2010)characterize games with general distribution functions, the present characterization of equilibrium uniquenessand multiplicity, away from the limit where private information is very precise, relies on the afore mentionednormality assumptions. Steiner and Stewart (2008), Izmalkov and Yildiz (2010), and Mathevet (2012) abstract from public signals. µ i ∼ N ( E [ µ ] , σ µ ), which is known to all agents, may be interpreted as a publiclyobservable distribution of exchange rate forecasts or credit ratings. Such a distribution ofconflicting beliefs, may also be seen as consistent with publicly disclosed long and short positionsthat large investors take in a firm’s stock or debt. While Morris (1995) and Sethi and Yildiz(2012), discuss heterogenous priors and their origins in detail, we will take prior heterogeneityas a given. In turn, we focus purely on the implications that public disagreement has for theemergence of multiple equilibria in the global games framework.The rest of the paper is organized as follows. In Section 2.1, we introduce the model. Section2.2 contains the main result. In Section 3, we reflect on our findings in three distinct publicsignal environments. Section 4 concludes. To isolate the implications of prior dispersion, we start with a setting that differs only withregard to the heterogeneity of priors from the canonic framework introduced by Morris and Shin(2000, 2004).
There is a status quo and a unit measure of agents indexed by i ∈ [0 , i can choose between two actions a i ∈ { , } . Choosing a i = 1 means to attack the status quo.Choosing a i = 0 means that the agent does not attack the status quo. An attack on the statusquo is associated with a cost c ∈ (0 , − c >
0. If the attack is not successful, an attackingagent’s net payoff is − c . The payoff for an agent who does not attack is normalized to zero.The status quo is abandoned if the aggregate size of the attack A := R a i di exceeds thestrength of the status quo θ , i.e., if A > θ . Otherwise, if
A < θ , the status quo is maintainedand the attack fails. Regarding the fundamental, θ , each agent i holds a prior belief θ ∼N ( µ i , σ p ). In our setting, we assume that the priors, regarding the fundamental θ , are normallydistributed across the population, i.e., µ i ∼ N ( E [ µ ] , σ µ ). And the distribution of priors iscommon knowledge. In addition to the prior, each agent receives a private signal x i = θ + σ x ξ i .5here signal noise ξ i ∼ N (0 ,
1) is i.i.d. across the population. We parameterize the informationstructure in terms of precisions α x ≡ σ x and α p ≡ σ p . Agents use their information to calculateexpected utility E [ U ( A, θ, c, a i ) | µ i , x i ] = a i (1 × P ( θ < A | µ i , x i ) − c ) . (1)Where P ( θ < A | µ i , x i ) is the probability that the attack is successful given signal x i and priorbelief µ i . Regarding payoffs, (1) reflects that an agent i , who does not attack ( a i = 0) receivesa safe payoff of 0, while an agent who does attack ( a i = 1) receives 1 − c if the attack issuccessful, and − c otherwise. To close the description of agents’ information, we note that thedistributions of x i and µ i are commonly known to all players. With the private choice problemin place, we now characterize equilibrium. As Morris and Shin (1998, 2004), we focus on monotone threshold equilibria. Each thresholdequilibrium will be characterized by a pair ψ ∗ , θ ∗ . Where θ ∗ separates values θ < θ ∗ forwhich the status quo is abandoned from values θ > θ ∗ where it prevails. The threshold level ψ ∗ ≡ α x α x ∗ + α p α µ ∗ summarizes the critical pairs x ∗ , µ ∗ , for which agents are just indifferentbetween attacking and not attacking. The equilibrium pairs, ψ ∗ , θ ∗ , are determined by thesimultaneous evaluation of the payoff indifference condition and the critical mass condition .The payoff indifference condition , PIC, follows directly from the individual choice problem (1).Taking the threshold level θ ∗ as given, we have: P ( θ ≤ θ ∗ | x ∗ , µ ∗ ) = c ⇔ Φ (cid:16) √ α ( θ ∗ − α x α x ∗ − α p α µ ∗ ) (cid:17) = c ; α = α x + α p , (2)where Φ() is the cumulative of the standard normal distribution. The PIC (2) characterizesthose pairs x ∗ , µ ∗ which are such that the agent is indifferent between attacking and not at- In the context of the currency crises model of Morris and Shin (1998), attacking agents would sell shorta country’s currency, and the central bank’s reserves θ are either sufficient ( θ > A ) to defend the peg or not( θ < A ). In another interpretation, agents can run on/sell short a firm’s debt, and if the firm’s financial strengthis insufficient it defaults. See Raiffa and Schlaifer (2000), p. 250, for the standard results on prior and posterior distributions ofnormally distributed variables which are used throughout the paper. Note that we already use the (forthcoming) critical mass condition , (4), which requires that θ ∗ ≡ A ( ψ ∗ , θ ∗ ),and replace A with θ ∗ in (1) to obtain (2). ψ ≡ α x α x + α p α µ such that (2) writes: P ( θ ≤ θ ∗ | ψ ∗ ) = c ⇔ Φ (cid:16) √ α ( θ ∗ − ψ ∗ ) (cid:17) = c ; α = α x + α p . (3)And agents who receive ψ ≤ ψ ∗ , which is evidence of a weak fundamental, attack. Agentswho receive ψ > ψ ∗ do not attack since they believe in a strong fundamental, which makes asuccessful attack unlikely.The critical mass condition , CMC, takes the cutoff value ψ ∗ as given and determines thethreshold θ ∗ , where the attack is just strong enough to overwhelm the status quo. To calculatethe mass of attacking agents, we note that ψ | θ ∼ N ( α x α θ + α p α E [ µ ] , α − ψ ) where α ψ ≡ α α x + α p σ µ and α = α x + α p . The CMC therefore writes: A ( ψ ∗ , θ ∗ ) ≡ P ( ψ < ψ ∗ | θ ∗ ) = θ ∗ ⇔ Φ (cid:16) √ α ψ ( ψ ∗ − α x α θ ∗ − α p α E [ µ ]) (cid:17) = θ ∗ . (4)Again, Φ() is the cumulative of the standard normal distribution. Regarding (4), we note thatit implies that there exists only one ψ ∗ for every θ ∗ . Simultaneous evaluation of (3) and (4)yields threshold equilibria ψ ∗ , θ ∗ : Proposition 1.
The equilibrium ψ ∗ , θ ∗ is unique, for all parameter pairs, if and only if q ( √ α x α p ) + σ µ ≥ √ π .Proof. We solve (3) for the threshold level ψ ∗ = − Φ − ( c ) √ α + θ ∗ , and substitute ψ ∗ into (4)to obtain a one-dimensional equation in θ ∗ :Φ (cid:16) √ α ψ (cid:16) − Φ − ( c ) 1 √ α + θ ∗ − α x α θ ∗ − α p α E [ µ ] (cid:17)(cid:17) = θ ∗ . (5)The sufficient condition for uniqueness of the threshold θ ∗ is therefore: √ α ψ α p α φ (cid:16) √ α ψ (cid:16) − Φ − ( c ) 1 √ α + α p α θ ∗ − α p α E [ µ ] (cid:17)(cid:17) ≦ ⇔ √ α ψ α p α √ π e − (cid:16) √ α ψ (cid:16) − Φ − ( c ) √ α + αpα θ ∗ − αpα E [ µ ] (cid:17)(cid:17) ≦ . (6)Finally, we recall that α ψ = α √ α x + α p σ µ and take logarithms to obtain: ln (cid:16) √ π q ( √ α x α p ) + σ µ (cid:17) ≦ (cid:16) √ α ψ (cid:16) − Φ − ( c ) 1 √ α + α p α θ ∗ − α p α E [ µ ] (cid:17)(cid:17) . ln (cid:16) √ π q ( √ α x α p ) + σ µ (cid:17) ≤ ⇔ s(cid:16) √ α x α p (cid:17) + σ µ ≥ √ π . (7)In one interpretation, the role of the prior’s dispersion σ µ in Condition (7) indicates thatpolarized economies, with two large groups (tails) which believe that the status quo is goingto be maintained (abandoned), have unique equilibria. Put differently, increases in the prior’sdispersion σ µ “thin out” the group of agents, around the mean E [ µ ] who hold “moderate”beliefs. And it is this group which can potentially coordinate on multiple equilibria. Regardingthe prior’s weight, α p , we find that, for every given prior expectation µ , increases in α p makeactions more predictable. This allows agents to coordinate, as in the common prior economyof Morris and Shin (2004), which contributes towards equilibrium multiplicity. Comparison ofthe two origins of belief heterogeneity indicates that increased dispersion of private signalscontributes towards equilibrium multiplicity, while increases in the dispersion of prior beliefs µ contributes towards uniqueness. In the next section, we show that the thinning-out effect isrobust to various changes in the informational environment. And it is the only avenue that canensure unique equilibria in all three public signal environments that follow.From a technical perspective, we note that Proposition 1 has two corollaries: Corollary 1.
If the prior µ is sufficiently dispersed, such that σ µ > √ π , equilibria are uniqueirrespective of the relative precision √ α x α p of private signal and prior. And in the case where theprivate signal is uninformative, α x = 0 , unique (multiple) equilibria exist if σ µ > ( < ) √ π . Corollary 2.
The uniqueness condition q ( √ α x α p ) + σ µ ≥ √ π converges smoothly to the unique-ness condition, √ α x α p ≥ √ π , of the Morris and Shin (2004) common prior game, as σ µ → . To reflect on the role of prior heterogeneity, we introduce three different types of public signalsinto our baseline setting. Each of these public signals is chosen to isolate particular differences That is, if (7) holds, then (6) never holds with equality for real-valued θ ∗ ’s. Put differently, the polynomialwhich characterizes those values θ ∗ , for which (6) holds with equality, has two complex roots. σ µ , is substantial, then they play unique equilibria irrespectiveof the particular public signal context. That is, the result from the previous section, i.e., thatprior heterogeneity induces equilibrium uniqueness through the thinning-out effect, is robustto the introduction of public signals. On the contrary, the role the private signal will changefrom environment to environment.The first public signal, which we introduce into our baseline model from Section 2, is ofexogenous quality. That is, it reveals the true fundamental of the game with exogenous precision α z . In such an extended setting, we find that the comparative statics of the subjective prior’sprecision α p change: contrary to the baseline setting, where increases in the prior weight alwayscontribute towards multiplicity, we find that increases in the prior weight can now shift themodified economy from multiplicity towards uniqueness. In a second step, we endogenize thequality of the public signal. To do so, we embed our coordination game into a financial marketscontext, where a stock price aggregates and reveals dispersed private information. In thissetting, we find that the role of private information, with respect to equilibrium multiplicity,is reversed. Namely, equilibrium multiplicity is ensured in the limit where private informationbecomes arbitrarily precise. On the contrary, the role of prior dispersion is robust to suchchanges in the model structure.In the last Section 3.1, we introduce a public signal which partially reveals the aggregateattack A . This signal provides an environment where changes in the prior’s dispersion may, forintermediate values, induce multiplicity rather than uniqueness. However, in the limit whereprior dispersion grows large, it still ensures unique equilibria. Finally, as a byproduct of ouranalysis in Section 3.1, we find that public signals in themselves have an ambiguous effect onequilibrium multiplicity: sufficiently precise public signals can ensure unique threshold equilib-ria. This finding is of independent interest in comparison with the games of Morris and Shin(2000, 2004), Hellwig (2002), Metz (2002), Angeletos and Werning (2006), where increases inthe public signal’s precision unambiguously induce multiple rather than unique threshold equi-libria. 9 ) Public Signal with Exogenous Precision The public signal Z = θ + σ z ε, ε ∼ N (0 , , (8)allows agents to forecast the true state of the fundamental with precision α z = σ z . Agents cantherefore use Z , in addition to x and µ , to calculate the probability with which the aggregateattack overwhelms the status quo. In Appendix A, we show that, if this signal is used as anadditional source of information in the coordination game of Section 2, we have: Proposition 2.
The equilibrium in the public and private information game with heterogenouspriors is unique if s(cid:16) √ α x α p + α z (cid:17) + σ µ α z α p ) ≥ √ π . (9) In particular, if the prior’s dispersion is large, such that σ µ αzαp > √ π , the equilibrium isunique independently of the private signal’s precision α x . Compared to the uniqueness condition (7) from the baseline model, we find once again thatthe modified condition (9) has two elements. The first, √ α x α p + α z , reflects the trade-off betweenprivate information α x and prior α p , described by Morris and Shin (2000, 2004) in an economywithout public signals, or respectively, the trade-off between private information and publicsignals α z which was emphasized by Metz (2002) and Hellwig (2002) in an economy with auniform uninformative prior. Regarding this first term, we find that public information andprior are perfect substitutes, and both contribute to equilibrium multiplicity. The second term σ µ αzαp ) , however, shows that increases in the public signal’s precision α z reduce the effectof the prior’s dispersion, while the prior weight α p increases it. Condition (9) therefore showsthat the public signal’s precision unambiguously contributes towards multiplicity. Increasesin the prior’s weight α p on the contrary have an ambiguous consequences as they shift theeconomy towards uniqueness (multiplicity) if σ µ > ( < ) α x α p α z . Finally, (9) reflects that equilibriaare unique in the private information limit where α x → ∞ .
2) Public Signal with Endogenous Precision
To highlight the different implicationsof prior dispersion and the dispersion of private signals, we discuss an environment where theglobal game is embedded in a financial market setting. Following, Atkeson (2000), Angeletos and Werning(2006), and Hellwig et al. (2006), we introduce a financial market which aggregates dispersed10rivate information on the unknown fundamental θ , through its publicly observable stock price,as in Grossman and Stiglitz (1976, 1980), and Hellwig (1980). In one interpretation, the ex-tended model may describe a situation, where bond investors use a firm’s stock price to infer itsdefault probability, which is of importance for a coordination game that concerns a potentialrun on the firm’s debt. We show in Appendix B that it is possible to specify the financialmarket such that the public stock price signal, Z , partially reveals the true fundamental θ : Z = θ − γσ ε σ x ε, ε ∼ N (0 , . (10)Thus, the signal’s precision α z = γσ ε ) α x is an increasing function of the private signal’sprecision α x = σ x . That is, the stock price’s informativeness increases once the stock investors’information becomes more informative. In the current context, it is important that the precisionwith which this financial market publicly reveals the true state of the world θ is increasingfaster (in the private signal’s precision) than the private signal’s precision α x itself. To performthe equilibrium analysis which concerns the coordination game, we recall (9) and note that α z = α z ( α x ). This yields Proposition 3.
The equilibrium is unique if vuut(cid:16) √ α x α p + α z ( α x ) (cid:17) + σ µ α z ( α x ) α p ) ≥ √ π , α z := 1( γσ ε ) α x . (11) Multiple equilibria exist in the private information limit where α x → ∞ . The equilibrium isunique in the limit where σ µ → ∞ .Proof. Follows from (11) with α z ( α x ) = γσ ε ) α x .Proposition 3 establishes that the finding of Angeletos and Werning (2006) carries over toan economy with heterogenous priors. Namely, if stock prices aggregate private informationrapidly as in (10), then it is precise private information which ensures equilibrium multiplic-ity. Moreover, as α z ( α x ) becomes large, it marginalizes the influence of prior heterogeneity That is, we assume that agents trade stocks prior to the coordination game. These stocks are traded ata market price P and pay an unknown amount θ . This market price will, in equilibrium, aggregate dispersedprivate information and reveal the true fundamental θ partially. Where the partial revelation is due to aggregatenoise-trader activity, σ ε ε , ε ∼ N (0 , µ αzαp ) . Finally, if private noise becomes large as α x → α z ( α x ) →
0, equilibriummultiplicity depends on prior dispersion, σ µ , alone. Concerning the different implications of heterogenous priors and heterogenous private sig-nals, the key insight is that the endogeneity of public information inverts the original findingsof Morris and Shin (2000, 2004), Metz (2002), and Hellwig (2002), where increases in privateinformation induce equilibrium uniqueness as in (9), where the public signal’s precision is ex-ogenous. The same is not true for the role of prior dispersion, which is, contrary to the privatesignal’s dispersion, robust to the introduction of an endogenous public price signal and unam-biguously contributes towards equilibrium uniqueness.
That is, heterogeneity in priors, unlikeheterogeneity in private information, provides a robust anchor for unique equilibria.
In this section, agents can observe the size of the aggregate attack through a noisy publicsignal S = Φ − ( A ) + σ ε ε where ε ∼ N (0 , α p = 0. Compared to the analysis inAngeletos and Werning (2006) we note that public signals of high precision can ensure uniquethreshold equilibria in our specification if priors are informative . That is, the role of the publicsignal in Angeletos and Werning (2006), pp. 1733-1734, depends critically on the absence ofan informative prior. This finding naturally differs from Angeletos and Werning (2006), where σ µ = 0, such that noisy pri-vate signals unambiguously induce unique equilibria when public information is endogenous once α x → α z ( α x ) →
0. Related to this observation, we note that among all parameters, α x , α p , σ µ , the prior’s dispersion σ µ is the only parameter which can ensure equilibrium uniqueness regardless of the values of the remainingparameters. More precisely, for a priorless game, Angeletos and Werning (2006) show that threshold equilibria are alwaysunique, but there may exist multiple equilibria in “strategies”. In the present model, which includes informative(possibly unique) priors, we show that multiple threshold equilibria may exist. However, if the public signal,over others’ actions A , is sufficiently precise then threshold equilibria are always unique. The observationthat public signals of high quality can ensure unique rather than multiple threshold equilibria is of interest S carries two types of information.First, similar to signals (8) and (10), the signal S allows agents to make inference on the truefundamental θ since A = A ( θ, ψ ∗ ). Second, unlike signals (8) and (10), the particular signalrealization S is endogenous in the sense that S is implicitly defined by S = Φ − ( A ( θ, ψ ∗ ( S ))) + σ ε ε . And there may exist multiple signal values S for every given pair θ, ε . We examinethese potential sources of multiplicity, namely equilibrium multiplicity in thresholds θ ∗ , ψ ∗ and equilibrium multiplicity in strategies S, ψ ∗ ( S ) in separate steps.
3) Equilibrium Multiplicity in Thresholds
In the augmented game, with heterogeneouspriors, where agents observe the aggregate attack through signal S , we have: S = Φ − ( A ) + σ ε ε, ε ∼ N (0 ,
1) (12) A ≡ P ( ψ ≤ ψ ∗ ( S ) | θ ∗ ) = θ ∗ (13) P ( θ ≤ θ ∗ | ψ ∗ , S ) = c. (14)Evaluation of (12)-(14) yields: Proposition 4.
For every given signal realization S, threshold equilibria θ ∗ , ψ ∗ are unique if √ π ≤ ( α x + α z ) α x r(cid:16) √ α x α p (cid:17) + σ µ where α z = α x α ψ σ ε α = α x σ ε ( α x + α p σ µ ) . And equilibria are unique whenpriors are either sufficiently dispersed or when the public signal is sufficiently precise.Proof. See Appendix CWith regard to the role of prior dispersion, Proposition 4 shows that prior dispersion con-tributes towards equilibrium uniqueness in the generalized setting where agents can observeeach other’s actions. The more significant finding, however, is that the public signal’s preci-sion induces equilibrium uniqueness rather than multiplicity. That is, in the present framework,we find that the public signal allows agents to coordinate on one particular equilibrium ratherthan multiple equilibria as in Morris and Shin (2000, 2004), Metz (2002), Hellwig (2002), andAngeletos and Werning (2006). Moreover, the comparative statics with regard to the public in comparison with Morris and Shin (2000, 2004), Metz (2002), and Hellwig (2002), who show that multiplethreshold equilibria emerge once public signals are of high quality. Note that increases in the prior’s dispersion reduce the public signal’s precision α z = α x α ψ σ ε α = α x σ ε ( α x + α p σ µ ) ;for intermediate values of σ µ , it is therefore not necessarily true that increases in σ µ contribute towards unique-ness. σ µ = 0. Finally, for an uninformative priorwhere α p = 0, we find that the uniqueness result of Angeletos and Werning (2006) obtains asa special case.
4) Equilibrium Multiplicity in Strategies
In this paragraph, we study the uniquenessof the equilibrium with respect to the signal S . In Appendix C, we show that signal S is,in equilibrium, equivalent to a signal Z ( S ) = αα x ψ ∗ ( S ) − αα x √ α ψ S = θ + α p α x E [ µ ] − σ ε αα x √ α ψ ε .And multiple equilibria can emerge in the sense that there may exist several signal values S ,and thus several values ψ ∗ ( S ), which satisfy Z ( S ) = ¯ Z . Concerning this potential source ofequilibrium multiplicity we note Proposition 5.
Equilibria in strategies ψ ∗ ( S ) are unique if √ π ≤ r(cid:16) √ α x α p + α z (cid:17) + σ µ αzαp , with α z = α x σ ε ( α x + α p σ µ ) . In the limit, where σ µ → ∞ there exists a unique equilibrium. In the limitwhere α z → ∞ , there exist multiple equilibria in strategies.Proof. See Appendix CComparison of propositions 4 and 5 with regard to the prior’s dispersion yields an importantcorollary:
Corollary 3.
The overall equilibrium is unique in the limit where σ µ → ∞ . Corollary 3 underscores the main result of the paper, i.e., it confirms that sufficiently dis-persed priors ensure unique equilibria. The public signal’s precision has a more differentiatedinfluence on equilibria: it ensures uniqueness in thresholds if it is sufficiently precise, but at thesame time it opens the door to multiple equilibria in strategies. In a setting with a uniform uninformative prior, Angeletos and Werning (2006), pp. 1733-1734, prove thatthreshold equilibria are always unique irrespective of the precisions α z and α x . To obtain this result, one caneither repeat the calculations in Appendix C with α p = 0. Alternatively, one can observe that the uniquenesscondition in Proposition 4 is always satisfied once a sufficiently small value α p is chosen. At this point we do not discuss the conceptual validity of this alternative type/source of equilibriummultiplicity, which implies that the particular signal S is endogenous, i.e., depends on the ψ ∗ chosen.Angeletos and Werning (2006), p.1730, provide a brief discussion and further references regarding this fun-damental problem. Conclusion
We have introduced heterogenous priors into the canonic global games model of Morris and Shin(2000, 2004), Metz (2002), and Hellwig (2002). The analysis of the baseline model indicates thatheterogeneity in priors, unlike heterogeneity in private signals, makes it more difficult for agentsto coordinate on multiple equilibria. That is, the origins of belief heterogeneity are of crucialimportance to the global games approach: heterogeneity in beliefs, which originates from thevariance σ µ in prior expectations, contributes to equilibrium uniqueness. Dispersion in privatesignals, on the contrary, induces equilibrium multiplicity. In general, the prior’s dispersion canensure unique equilibria as it “thins-out” the group of agents who hold “moderate” beliefs.That is, it reduces the mass of agents with moderate beliefs, and it is this group which canpotentially coordinate on multiple equilibria. Equivalently, our results indicate that if player’sdisagreement, as measured by σ µ , is substantial, then they play unique equilibria.More precisely, we find that if prior dispersion is small, ( σ µ < √ π ), equilibrium multiplicitydepends on the relative precisions of private information ( α x ) and the subjective prior ( α p ). Ifpriors are sufficiently dispersed, ( σ µ > √ π ), equilibria are unique irrespective of the relativeweights that players assign to private signals and priors. If prior dispersion ( σ µ ) vanishes, theoriginal uniqueness condition for the common prior economy ( √ α x α p ≥ √ π ) obtains.To compare the implications of prior dispersion and dispersion in private information, wehave discussed a modified game in which a financial market aggregates private information into apublic price signal. Such a modified environment inverts the original findings of Morris and Shin(2000, 2004), Metz (2002), and Hellwig (2002): increases in private information now induce equi-librium multiplicity instead of uniqueness. The same is not true for the role of prior dispersion,which is robust to such a change in the modelling environment and contributes unambiguouslytowards equilibrium uniqueness. Put differently, the extended model indicates that prior dis-persion, rather than arbitrarily precise private information, anchors unique equilibria reliably.In general, we found that sufficiently dispersed priors ensure unique equilibria across all threepublic signal environments. Regarding these public signals, it turned out that their implicationsin themselves varied significantly from case to case: increases in the public signal’s precisionensure multiple threshold equilibria in the first two environments, where signals only containinformation on the unknown fundamental. The opposite can be true in the third environment,where public signals allow agents to observe each other’s actions. If such signals are of high15uality, they can enable agents to coordinate on one unique threshold equilibrium .Unlike previous studies, which have introduced heterogenous priors into the global gamesframework, we have given explicit conditions in terms of means and variances, which allow tostudy equilibrium multiplicity for a large economy at and away from the private informationlimit . That is, the present framework facilitates comparative statics in the information structureitself, which allows to characterize and compare the different implications of belief heterogeneitywhich originate from priors and private signals, respectively. Moreover, these comparativestatics are useful in those applications of the global games framework where it is interesting, ornecessary, to study the interaction of private and public information away from the limit whereprivate signals are infinitely precise. 16 Game with Exogenous Public Information
In this appendix, we derive the uniqueness condition that obtains once we augment our baselinemodel of Section 2 with a public signal: Z = θ + σ z ε, ε ∼ N (0 , . (15)This signal allows agents to improve their forecast of the probability with which the aggregateattack overwhelms the status quo. The modified payoff indifference condition therefore reads: P ( θ ≤ θ ∗ | x ∗ , µ, Z ) = c ⇔ Φ (cid:16) √ α ( θ ∗ − α x α x ∗ − α p α µ − α z α Z ) (cid:17) = c ; α = α x + α p + α z , (16)where Φ() is the cumulative of the standard normal distribution. Again, we define ψ ≡ α x α x + α p α µ and rewrite (16) as:Φ (cid:16) √ α ( θ ∗ − ψ ∗ − α z α Z ) (cid:17) = c ; α = α x + α p + α z . (17)The PIC in (17) locates a critical ψ ∗ such that agents attack if ψ > ψ ∗ and do not attack if ψ <ψ ∗ . To calculate the mass of attacking agents, we note that ψ | θ ∼ N ( α x α θ + α p α E [ µ ] , ( α α x + α p σ µ ) − ).Once we define α ψ ≡ α α x + α p σ µ , the CMC can be written as: P ( ψ < ψ ∗ | θ ∗ ) = θ ∗ ⇔ Φ (cid:16) √ α ψ ( ψ ∗ − α x α θ ∗ − α p α E [ µ ]) (cid:17) = θ ∗ . (18)Substitution of (17) into (18) again yields a one-dimensional equation in the threshold level θ ∗ :Φ (cid:16) √ α ψ ( α z + α p α θ ∗ − α z α Z − α p α E [ µ ] − Φ − ( c ) 1 √ α ) (cid:17) = θ ∗ . (19)Accordingly, equilibria are unique if: √ α ψ α z + α p α √ π ≦ ⇔ s(cid:16) √ α x α p + α z (cid:17) + σ µ α z α p ) ≥ √ π (20)which is what we needed to show. B Financial Market and Information Aggregation
In this appendix, we present a financial market that aggregates and reveals dispersed privateinformation, on the fundamental θ , through the stock price. For the present purpose, it is Recall that α ψ ≡ α α x + α p σ µ . P and pay an unknown amount θ , which represents thefirm’s fundamental strength. This market price will, in equilibrium, aggregate dispersed privateinformation and reveal the true fundamental θ partially. Where the partial revelation is dueto aggregate noise-trader activity, σ ε ε , ε ∼ N (0 , P = η θ + η ε + c . Regarding θ , this function is informationally equivalent to asignal Z ≡ P − cη = θ + η η ε , which reveals the true fundamental with precision α z = η η . Second,given this price function, we characterize individual demands based on the information x, µ, Z ,and calculate the market equilibrium. Finally, we determine the ratio η η as α x γ σ ε . That is,price signal Z indeed carries information α z = α x γ σ ε as claimed in (10) in the main text.
5) Demand
Agents choose their optimal demands k i for the asset to maximize expectedCARA utility: k i = arg max k i { E [ − e − γ ( θ − P ) k i | x i , µ, Z ] } = arg max k i { γ E [( θ − P ) k i | x i , µ, Z ] − γ V ar [( θ − P ) k i | x i , µ, Z ] } = arg max k i { γ ( α x α x i + α p α µ + α z α Z − P ) k i − γ k i α } , α = α x + α p + α z and the individual demand function writes: k di = α x α x i + α p α µ + α z α Z − Pα − γ . (21)
6) Equilibrium
Aggregate supply K S = σ ε ε , is unobservable and distorted by noise-traderactivity ε ∼ N (0 , K D is: K D = Z [0 , Z k i ( µ ) φ ( µ ) dµdi = α x α θ + α p α E [ µ ] + α z α Z − Pα − γ . (22)Equilibrium requires that: K D = K S ⇔ α x α θ + α p α E [ µ ] + α z α Z − Pα − γ = σ ε ε. (23)To close the argument, we now resubstitute Z = P − cη and calculate the ratio η η . First, we solve(23) for P to obtain: P = α x α − α z η θ − γσ ε α − α z η ε + α p E [ µ ] − α z cη α − α z η . (24)18omparison of (24) with our initial guess, P = η θ + η ε + c , indicates that η , η must satisfy: η = α x α − α z η , η = − γσ ε α − α z η ; α = α x + α p + α z . (25)We quickly determine η = α x + α z α , η = − ( α x + α z ) √ α z α to calculate η η = − α x γσ ε . At the sametime, it follows from the definition of Z that Z = P − cη = θ + η η ε . Hence, agents who observe P ,and know the model’s coefficients, receive a signal Z = θ − α − x γσ ε ε , as claimed in (10) in themain text. C Proof of Propositions 4 and 5
In this appendix, we start by laying out the equations that describe equilibria. In turn, wecharacterize the possible equilibria described in propositions 4 and 5 in two separate paragraphs.We recall the model from the main text S = Φ − ( A ) + σ ε ε, ε ∼ N (0 ,
1) (26) A ≡ P ( ψ ≤ ψ ∗ ( S ) | θ ) = θ (27) P ( θ ≤ θ ∗ | x, µ, S ) = c (28)To calculate equilibria, we recall that agents act on x = θ + σ x ξ , with ξ ∼ N (0 ,
1) and θ | µ ∼ N ( µ, σ p ), where the prior µ is distributed over the population as µ ∼ N ( E [ µ ] , σ µ ).Moreover, we define ψ ≡ α x α x + α p α µ with α = α x + α p + α z . The PIC (28) now writes as:Φ (cid:16) √ α ( θ ∗ − ψ ∗ − α z α Z ) (cid:17) = c ; α = α x + α p + α z . (29)Again, (29) defines a critical ψ ∗ ( Z ) such that agents attack if ψ ≤ ψ ∗ and do not attack if ψ >ψ ∗ . To calculate the mass of attacking agents, we note that ψ | θ ∼ N ( α x α θ + α p α E [ µ ] , ( α α x + α p σ µ ) − ).Once we define α ψ ≡ α α x + α p σ µ , the CMC (27) can be written as: A = P ( ψ < ψ ∗ | θ ∗ ) = θ ∗ ⇔ Φ (cid:16) √ α ψ ( ψ ∗ − α x α θ ∗ − α p α E [ µ ]) (cid:17) = θ ∗ . (30)Using this expression for the aggregate attack A in condition (30), we can return to the publicsignal S in (26) and write: S = √ α ψ ( ψ ∗ − α x α θ − α p α E [ µ ]) + σ ε ε. (31)19here S in (31) is informationally equivalent to a signal Z ( S ) ≡ αα x ψ ∗ ( S ) − αα x √ α ψ S = θ + α p α x E [ µ ] − σ ε αα x √ α ψ ε. (32)Regarding (32), we note that Z ( S ) contains two aspects (i) Z is a noisy public signal whichreveals the true state of the economy θ with precision α z = α x α ψ σ ε α = α x σ ε ( α x + α p σ µ ) and (ii) thesignal S allows agents to align their strategies ψ ∗ ( S ). That is, for every given ¯ Z , there may beseveral S such that Z ( S ) = ¯ Z . That is, there is a potential source of equilibrium multiplicity,concerning S , to which we turn in Paragraph 2) of this appendix. For now, we take S as givenand study the threshold equilibria θ ∗ ( S ) , ψ ∗ ( S ).
7) Proof of Proposition 4: Multiplicity in Thresholds θ ∗ For every given signal S ,we rewrite (30) as: ψ ∗ = Φ − ( θ ∗ ) 1 √ α ψ + α x α θ ∗ + α p α E [ µ ] . (33)To obtain an equation in θ ∗ only, we substitute Z ( S ) = αα x ψ ∗ ( S ) − αα x √ α ψ S and (33) into (29).Rearranging then yields:Φ (cid:16) √ α ( α p α θ ∗ − √ α ψ α x + α z α x Φ − ( θ ∗ ) − ( α x + α z ) α p α x α E [ µ ] + α z α x √ α ψ S ) (cid:17) = c (34)To derive the uniqueness condition, which ensures that there exist only one θ ∗ ( S ) for everygiven signal S , we differentiate (34) with respect to θ ∗ : φ (Φ( θ ∗ ) − ) ≤ ( α x + α z ) α x s(cid:16) √ α x α p (cid:17) + σ µ , (35)and hence, threshold equilibria are always unique iff √ π ≤ ( α x + α z ) α x r(cid:16) √ α x α p (cid:17) + σ µ . Otherwise,if √ π ≥ ( α x + α z ) α x r(cid:16) √ α x α p (cid:17) + σ µ , there may exist up to three threshold equilibria θ ∗ ( S ) , ψ ∗ ( S ); θ ∗ ( S ) , ψ ∗ ( S ); θ ∗ ( S ) , ψ ∗ ( S ) for every given signal value S .
8) Proof of Proposition 5: Multiplicity in Strategies ψ ∗ ( S ) To preclude multiplesolutions S ( ¯ Z ) to the equation Z ( S ) = ¯ Z , where Z ( S ) = αα x ψ ∗ ( S ) − αα x √ α ψ S , it will suffice Note that for y = Φ − ( θ ∗ ), we have dθ ∗ dy = φ ( y ) and thus dydθ ∗ = φ ( y ) = φ (Φ − ( θ ∗ )) . The existence of at least one solution is ensured. It follows from (33) that lim S →∞ Φ( θ ∗ ( S )) − S and lim S →−∞ Φ( θ ∗ ( S )) − S are constants. Rewriting Z ( S ) = αα x ψ ∗ ( S ) − αα x √ α ψ S as Z ( S ) = αα x S ( ψ ∗ ( S ) S − αα x √ α ψ ) andrecalling ψ ( θ ∗ ( S )) as given in (33) one can show that Z ( S ) varies with S between ∞ and −∞ .
20o show that ∂Z ( S ) ∂S | ( ) = αα x ∂ψ ∗ ∂S − αα x √ α ψ ≤
0. To calculate the derivative ∂ψ ( θ ∗ ( S )) ∂S = ∂ψ ∗ ∂θ ∗ ∂θ ∗ ∂S ,defined by (33) and (34), we differentiate (34) which yields ∂θ ∗ ∂S = − αzαx √ αψαpα − √ αψ αx + αzαx φ (Φ( θ ∗ ) − and(33), (which is a 1 : 1 mapping between ψ ∗ and θ ∗ ), to obtain ∂ψ ∗ ∂θ ∗ = √ α ψ φ (Φ( θ ∗ ) − ) + α x α . Hence,we have ∂Z ( S ) ∂S = αα x ∂ψ ∗ ∂S − αα x √ α ψ = αα x ∂ψ ∗ ∂θ ∗ ∂θ ∗ ∂S − αα x √ α ψ = − αα x √ α ψ | {z } − + (cid:16) √ α ψ φ (Φ( θ ∗ ) − ) + α x α (cid:17)| {z } + − α z α x √ α ψ α p α − √ α ψ α x + α z α x φ (Φ( θ ∗ ) − ) | {z } + / − αα x . (36)Once we recall that α ψ ≡ α α x + α p σ µ , rearranging (36) gives: ∂Z ( S ) ∂S = − p α x + α p σ µ α x + (cid:16)q α x + α p σ µ + α x φ (Φ( θ ∗ ) − ) (cid:17) − α z α x φ (Φ( θ ∗ ) − ) α p α x √ α x + α p σ µ − ( α x + α z )= 1 α x h − q α x + α p σ µ + (cid:16)q α x + α p σ µ + α x φ (Φ( θ ∗ ) − ) (cid:17) − α zφ (Φ( θ ∗ ) − ) α p α x √ α x + α p σ µ − ( α x + α z ) i = 1 α x h − α x ( α p + α z ) φ (Φ( θ ∗ ) − ) + α x q α x + α p σ µ i φ (Φ( θ ∗ ) − ) α p α x √ α x + α p σ µ − ( α x + α z )= h − ( α p + α z ) φ (Φ( θ ∗ ) − ) + q α x + α p σ µ i φ (Φ( θ ∗ ) − ) α p α x √ α x + α p σ µ − ( α x + α z ) (37)From (37), and the fact that θ ∗ ∈ (0 , √ π ≤ s(cid:16) √ α x α p + α z (cid:17) + σ µ α z α p , α z = α x σ ε ( α x + α p σ µ ) (38)and 1 √ π ≤ s ( α x + α z ) α x α p + σ µ ( α x + α z ) α x . (39)That is, once (38) and (39) hold, we have ∂Z ( S ) ∂S <
0, which ensures unique solutions S ( ¯ Z ) tothe equation Z ( S ) = ¯ Z . Comparison shows that inequality (39) is less restrictive than (38). Evaluation of (38) therefore yields:1. In the limit where σ µ → ∞ , equilibria in strategies are unique. r α x ( α p + α z ) + σ µ αzαp ≤ r ( α x + α z ) α x α p + σ µ ( α x + α z ) α x follows from the inequalities α x ( α p + α z ) ≤ ( α x + α z ) α x α p and σ µ αzαp ≤ σ µ ( α x + α z ) α x , which are easy to verify.
21. In the limit where α z → ∞ , there exist multiple equilibria in strategies.3. Multiple equilibria are ensured in the limit where α x → ∞ .4. Finally, in the special case where σ µ = 0 and α p = 0, condition (38) collapses into theuniqueness condition √ π ≤ √ α x α z , of Angeletos and Werning (2006) pp. 1733-1734, whichis nested in the present framework. D Alternative Derivation of (7)
This appendix contains a derivation of (7) which “explicitly” accounts for the influence whichthe prior’s distribution has on the critical mass condition. Recalling the PIC, we have: P ( θ ≤ θ ∗ | x ∗ , µ ) = c ⇔ Φ (cid:16) √ α ( θ ∗ − α x α x ∗ − α p α µ ) (cid:17) = c ; α = α x + α p . (40)Regarding the CMC, we now account explicitly for the prior’s distribution and write: A ( x ∗ ( µ ) , θ ∗ ) = P ( x ≤ x ∗ | θ ∗ ) = θ ∗ ⇔ Z ∞−∞ Φ (cid:16) √ α x (cid:16) x ∗ ( µ ) − θ ∗ (cid:17)(cid:17) φ ( µ ) dµ = θ ∗ , (41)where Φ() represents the cumulative of the standard normal distribution and φ ( µ ) is the normaldensity of the prior. To show that the pairs x ∗ ( µ ) , θ ∗ , which solve (40) and (41) are unique if q ( √ α x α p ) + σ µ ≥ √ π , we solve (40) for the threshold level x ∗ = − Φ − ( c ) √ αα x + θ ∗ αα x − α p α x µ , andsubstitute x ∗ into (41) to obtain a one-dimensional equation in θ ∗ alone: Z ∞−∞ Φ (cid:16) √ α x (cid:16) − Φ − ( c ) √ αα x + θ ∗ αα x − α p α x µ − θ ∗ (cid:17)(cid:17) φ ( µ ) dµ = θ ∗ . The sufficient condition for uniqueness of the threshold θ ∗ is therefore: α p √ α x √ π Z ∞−∞ e − (cid:16) √ α x (cid:16) − Φ − ( c ) √ ααx + θ ∗ αpαx − αpαx µ (cid:17)(cid:17) φ ( µ ) dµ ≦ . (42)To obtain the final condition, we recall that µ ∼ N ( E [ µ ] , σ E [ µ ] ), and use the moment generatingfunction for the non-central χ distribution in Paragraph 1) to rewrite (42), as: α p √ α x √ π q σ µ α p √ α x ) e − (cid:16) √ α x (cid:16) − Φ − ( c ) √ ααx + θ ∗ αpαx − αpαx E [ µ ] (cid:17)(cid:17) σµ αp √ αx )2 ≤ . (43)Taking logarithms yields the uniqueness condition q ( √ α x α p ) + σ µ ≥ √ π . Note that the public signal’s precision is endogenous, i.e., given by α z = α x σ ε ( α x + α p σ µ ) . However, the presentobservation is informative in the sense that the public signal’s precision can be varied through σ ε which isindependent of the other parameters. ) Moment Generating Function Regarding (42), we recall our assumption that µ ∼N ( E [ µ ] , σ E [ µ ] ). We can therefore define y = − Φ − ( c ) √ αα x + θ ∗ α p α x − α p α x µ , where y ∼ N ( − Φ − ( c ) √ αα x + θ ∗ α p α x − α p α x E [ µ ] , σ µ α p α x ). If a variable y is normally distributed with mean E [ y ] and variance σ y ,then z = y σ y is non-centrally χ distributed. We can therefore use the moment generatingfunction for the non-central χ distribution (Rao (1965), p. 181): E [ e − tz ] = 1 √ t e − E [ z ]2 t t , t > , (44)to rewrite (42) as (43). To do so, we set t = α x σ y = σ µ α p α x , and substitute z = y σ y , with y = − Φ − ( c ) √ αα x + θ ∗ α p α x − α p α x µ , into (42), to obtain: α p √ α x √ π Z ∞−∞ e − tz φ ( µ ) dµ = | ( ) α p √ α x √ π p α x σ y e − E [ y ]2 σ y α x σ y αxσ y (45)= α p √ α x √ π q σ µ α p √ α x ) e − (cid:16) √ α x (cid:16) − Φ − ( c ) √ ααx + θ ∗ αpαx − αpαx E [ µ ] (cid:17)(cid:17) σµ αp √ αx )2 . (46)Where the final step from (45) to (46) involves cancelling and resubstitution of α x σ y = ( σ µ α p √ α x ) and E [ y ] = − Φ − ( c ) √ αα x + θ ∗ α p α x − α p α x E [ µ ]. E e tz = √ π R ∞−∞ e tz e − ( z − E [ z ])22 dz = √ π R ∞−∞ e − (1 − t ) z − E [ z ] z + E [ z ]22 dz = √ π R ∞−∞ e − ( √ − tz − E [ z ] ) − E [ z ]21 − t + E [ z ]22 dz = √ − t √ π R ∞−∞ e − E [ z ]2 t − t e − ( √ − tz − E [ z ] ) d (cid:0) √ − tz (cid:1) = √ − t e − E [ z ]2 t − t . Regarding the equality in (45), we note that α p √ α x √ π R ∞−∞ e − tz φ ( µ ) dµ = α p √ α x √ π R ∞−∞ e − tz √ πσ µ e − ( µ − E [ µ ])22 σ µ dµ = α p √ α x √ π R ∞−∞ e − tz √ πσ µ e − ( z − E [ z ])22 σ µ dz = α p √ α x √ π R ∞−∞ e − tz √ π e − ( z − E [ z ])22 dz . It now follows from the steps in footnote 21, equation (44) respec-tively, that the equality in (45) holds. eferences Angeletos, G.-M. and Werning, I. (2006). Crises and prices: Information aggregation, multi-plicity, and volatility.
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