Information Acquisition in Global Games of Regime Change

Transcription

1 Informaton Acquston n Global Games of Regme Change Mchal Szkup and Isabel Trevno y Abstract We study costly nformaton acquston n global games of regme change (that s, coordnaton games where payo s are dscontnuous n the unobserved state and n the agents average acton). We show that only symmetrc equlbra exst and provde su cent condtons for unqueness. We then characterze the value of nformaton n these games and lnk t to the underlyng parameters of the model. We nvestgate equlbrum e cency, complementartes n nformaton choces, and the trade-o s between publc and prvate nformaton. We show that nformaton acquston can be ne cent and that strategc complementartes n actons do not always translate nto strategc complementartes n nformaton acquston. Fnally, we nd that publc and prvate nformaton can be complements. These results contrast ndngs n lnearquadratc models, where payo s depend contnuously on both the unobserved state and the agents average acton. Key words: global games, nformaton acquston, coordnaton, value of nformaton. JEL class caton: C7, D80 Introducton Global games have been extensvely appled to model economc phenomena featurng coordnaton problems, such as currency crses (Morrs and Shn, 998), bank runs (Goldsten and Pauzner, 005), FDI decsons (Dasgupta, 007), and poltcal revolts (Edmond, 03). In a global game the payo s of agents depend on both the state of the economy and the actons of others. However, agents observe only nosy prvate and publc sgnals about ths state and, n order to choose an optmal acton, they have to make nferences about ts true value and about the belefs that other agents hold. Ths perturbaton of the nformaton structure Correspondng author. mchal.szkupubc.ca, Vancouver School of Economcs, Unversty of Brtsh Columba, 87 East Mall, FL 9, RM 997, Vancouver, V6T Z, Canada. y trevnoucsd.edu, Department of Economcs, Unversty of Calforna San Dego, 9500 Glman Drve #0508 La Jolla, CA 9093, USA.

2 of the game gves rse to a very rch sequence of hgher-order belefs, whch leads agents to coordnate on a unque equlbrum. Ths predcton of a unque equlbrum contrasts the complete nformaton model, whch features multple equlbra. Whle the orgnal models have been extended along many drectons, the precson of prvate sgnals has typcally been exogenously gven and set to be dentcal across agents. In ths paper we ntroduce costly nformaton acquston nto the standard global games framework. Endogenzng nformaton n a global game s a relevant endeavor, not only from a theoretcal pont of vew but also from an appled one. Followng Dasgupta (007), one can thnk of an emergng economy that wants to attract foregn drect nvestment where potental nvestors have to decde whether to nvest or not nvest. For the pro ts to be postve, there has to be enough nvestment so that the lberalzaton program succeeds (due to ncreasng returns to aggregate nvestment), so nvestors wll want to coordnate on ther decsons. The returns of the project depend also on the state of the emergng economy, whch can be uncertan at the tme of the nvestment decson. In ths context, potental nvestors can acqure more precse nformaton about the state of the emergng economy by buyng reports that wll assess the pro tablty of ths nvestment. Introducng costly nformaton acquston nto a global game gves rse to a set of natural questons wth non-trval mplcatons. We focus on the followng questons: Do nvestors acqure the socally e cent amount of prvate nformaton (.e., do they over-acqure or under-acqure nformaton)? Are there strategc complementartes n nformaton choces (.e., do nvestors want to learn what others learn)? What s the trade-o between prvate and publc nformaton n ths context? Does more precse publc nformaton always crowd out prvate nformaton acquston? Does t ncrease the probablty of a successful nvestment? And nally, does t ncrease welfare? In order to answer these questons, we rst characterze an equlbrum n our model. We establsh that only symmetrc equlbra exst, and we nd that under mld condtons on parameters we can guarantee unqueness of equlbrum. We de ne the value of addtonal nformaton n our setup and analyze how t s a ected by pror belefs, the behavor of other players, and the cost of nvestment. We nd that the value of addtonal nformaton s determned by the extent to whch t helps an agent to avod two types of mstakes n the coordnaton game: nvestng when nvestment s not pro table, and not nvestng when nvestment s pro table. Usng these nsghts, we address each of the questons rased above under the assumptons that ensure unqueness. We nd that the unque equlbrum of the game s genercally ne - cent and that, dependng on the characterstcs of the economy, nvestors ether over-acqure or under-acqure nformaton. In terms of strategc motves n nformaton acquston, we nd condtons under whch strategc complementartes n nformaton acquston arse and condtons where ths s not the case, so that the optmal precson choce of an agent s a non-monotonc functon of the precson choces of others. See Hall et al. (986), Hall (987), and Caballero and Lyons (99) for evdence of ncreasng returns to scale n nvestment. Cooper (999) provdes an excellent overvew of the lterature on complementartes n macroeconomcs.

3 We then study the e ects of an ncrease n the precson of publc nformaton on welfare. Our analyss provdes a novel perspectve on ths ssue by nvestgatng the trade-o between publc and prvate nformaton acquston. In our model publc nformaton a ects outcomes, not only through agents actons n the coordnaton game but also by changng ther ncentves to acqure prvate nformaton. We provde condtons under whch more precse publc nformaton crowds out prvate nformaton. Surprsngly, we nd cases n whch more precse publc nformaton leads nvestors to acqure more precse prvate nformaton, that s, where prvate and publc nformaton are complements. Fnally, we show that the e ect of more precse publc nformaton on the probablty of successful nvestment and welfare depends on the characterstcs of the economy. Our analyss hghlghts the d erences between global games and the closely related famly of games wth lnear-quadratc payo s (see Angeletos and Pavan, 007). Frst, we nd that whether an mprovement n publc nformaton s welfare enhancng or not depends crucally on the ex-ante belefs about the state of the economy, whle n games wth lnear-quadratc payo s t depends on the relatve nformatveness of prvate and publc nformaton (Morrs and Shn, 00; Colombo et al., 04). Second, n games wth lnear-quadratc payo s strategc complementartes n actons always lead to strategc complementartes n nformaton acquston (Hellwg and Veldkamp, 009; Colombo et al., 04; Myatt and Wallace, 0). In the case of global games, we state condtons under whch strategc complementartes n actons translate nto strategc complementartes n nformaton acquston, and we show that f these condtons are volated then nformaton choces are not strategc complements. Fnally, n games wth lnear-quadratc payo s wth prvate nformaton acquston, an ncrease n the precson of publc nformaton always decreases the ncentves to acqure more precse prvate nformaton (Tong, 007; Colombo et al., 04), whereas n our model prvate and publc nformaton can be complements. We argue that the d erences between our ndngs and the exstng results for games wth lnear-quadratc payo s are due to the fact that the value of addtonal nformaton s very d erent across these two types of models. In global games the value of such nformaton s determned by the tal probabltes of the condtonal jont dstrbuton of the fundamental and the prvate sgnals, whle n games wth lnear-quadratc payo s t s determned by the covarances between nvestors sgnals and the fundamentals. The paper s structured as follows. In Secton we set up the model and explan the assumptons we make to solve t. In Secton 3 we solve the model and present results about the non-exstence of asymmetrc equlbra, the exstence of symmetrc equlbra, and condtons ensurng unqueness of the symmetrc equlbrum. In Secton 4 we nvestgate notons of e cency of the unque equlbrum. In Secton 5 we nvestgate whether strategc complementartes n the coordnaton game translate nto strategc complementartes n Models wth lnear-quadratc payo s are also coordnaton games of ncomplete nformaton but d er from global games n many respects. The choce sets for actons are contnuous (as opposed to bnary, as n global games), and agents have a quadratc utlty functon that depends on both the dstance between an ndvdual acton and the average acton of the other players and the dstance between that ndvdual acton and the underlyng state of the economy. 3

4 nformaton choces. In Secton 6 we ask whether an ncrease n the precson of publc nformaton s welfare enhancng or not. Secton 7 compares our results to prevous results on nformaton acquston n games wth lnear-quadratc payo s. Secton 8 summarzes the related lterature, and Secton 9 concludes. All the proofs are relegated to the appendx. The model We consder a two-perod model where nvestors have to decde rst how much nformaton to purchase and then, gven ths nformaton, whether or not to nvest n a rsky project. The rst perod, where nvestors choose the precson of ther prvate sgnals, consttutes the novel part of the model. The second perod s smlar to a standard global games model, wth the excepton that nvestors observe sgnals wth d erent precsons. There s a contnuum of nvestors n the economy; they are ndexed by, where [0; ]. The economy s characterzed by a parameter that measures the strength of ts economc fundamentals and s unobserved by nvestors. Each nvestor has to make two decsons. Frst, he has to decde how much nformaton to acqure about. Then he has to decde whether to nvest n a rsky project (I) or not nvest (NI). If an nvestor decdes to nvest, he ncurs cost T (0; ). The bene t to nvestng s uncertan and depends on the state and on p, the proporton of nvestors that choose to nvest. Investment s successful f p, that s, f the proporton of nvestors who nvest s hgh enough wth respect to the state. The return on a successful nvestment for each nvestor who nvests s, n whch case he wll get the payo T. If nvestment s unsuccessful, hs payo wll be T. The return from not nvestng s certan and normalzed to 0. The payo s are summarzed below: 3 u (I; p; ) = T f p T f p < u (NI; p; ) = 0 (a) (b) Whether ndvdual nvestment s successful or not depends on the state of the economy and on the number of ndvdual nvestments. One can nterpret ths need for enough aggregate nvestment as resultng from ncreasng returns to scale n nvestment. 4 Investors do not observe the state of the economy. Instead, they share a common pror belef that N ;. In addton, at the begnnng of perod, nvestor observes a nosy prvate sgnal about the realzaton of, gven by x : x = + ", [0; ] where " N (0; ) s an dosyncratc nose, ::d: across nvestors, and ndependent of the realzaton of, and s the precson of nvestor s sgnal. 3 Ths payo structure s standard n the global games lterature (see, for example, Corsett et al., 004; Morrs and Shn, 004; Hellwg et al. 006; and Dasgupta, 007). 4 The payo s are chosen to make the game analytcally tractable. All the qualtatve results stll hold f we allow the bene t from nvestng to be an explct functon of both the state and aggregate nvestment. 4

5 In perod, each nvestor decdes how much nformaton about to acqure by choosng the precson of hs sgnal, [; ). If an nvestor chooses not to acqure nformaton he wll observe a sgnal wth a default precson. The cost assocated wth choosng a precson s gven by C ( ), that s, nvestors face a trade-o between nformatveness and cost of sgnals. After observng ther respectve sgnals, nvestors decde smultaneously whether to nvest n the project or not. The payo s from nvestment decsons, gven by (a) and (b), are realzed at the end of perod.. Assumptons Before solvng the model, we make two sets of assumptons. The rst one consders the underlyng parameters of the game, whle the second one pertans to the cost functon. Assumpton (Concavty) We assume the followng: [ ; ]; 0 < < < > The lower bound for precson choces s set hgh enough to ensure not only that the coordnaton game always has a unque equlbrum but also that the ex-ante utlty functon s concave n the ndvdual precson choce. 5 The detals of determnng can be found n the onlne appendx. Assumpton (Cost functon) We assume that the cost functon C () sats es all of the followng condtons: C () s strctly ncreasng n (C 0 () > 0) C () s strctly convex n (C 00 () > 0) C 0 () = 0 lm C 0 ( ) = These assumptons mply that the cost functon s strctly convex, a common assumpton n the lterature on nformaton acquston. We further assume that an n ntesmal mprovement n precson s costless, to ensure that the problem s non-trval and that nvestors always acqure nformaton. The last assumpton ensures that nvestors wll never choose to acqure perfect nformaton. We consder an addtve Gaussan nformaton structure and model nformaton acquston as a contnuous precson choce. As ponted out by Yang (05), ths s not necessarly the nformaton structure that nvestors would choose f they had the exblty to desgn 5 As ponted out by Radner and Stgltz (984), the margnal value of nformaton can be ncreasng for low levels of nformatveness. We choose to ensure concavty of the ex-ante utlty functon n. 5

6 ther own type of nformaton structure. Yang shows that, n a smlar setup, nvestors would typcally prefer to observe bnary dscrete sgnals for a gven. An advantage of Yang s approach s that nvestors can choose not only how much nformaton to acqure, but also the type of sgnal they observe and ts nformatveness for any value of the fundamentals. Ths allows nvestors to coordnate on ther sgnal structures, and not only on ther nformatveness or precson, as s typcally assumed n the lterature and n our model. Despte ths lmtaton, assumng an addtve nformaton structure has several advantages n the context of our model. Frst, allowng for exble nformaton acquston as n Yang (05) ntroduces multplcty of equlbra nto the model, whch makes t d cult to establsh comparatve statcs results. By choosng an addtve structure we can guarantee a unque equlbrum. Second, an addtve Gaussan nformaton structure s more tractable and allows us to analyze the resultng equlbrum n greater detal, whch would not be possble under exble nformaton acquston. Fnally, usng an addtve nformaton structure allows us to compare our results wth the exstng lterature, both on global games wth exogenous nformaton structures and on nformaton acquston n games wth lnear-quadratc payo s. 6 3 Solvng the model We solve the model usng backward nducton. We start n perod, takng as gven the precson choces made by nvestors n perod. Once we characterze the equlbrum outcome at t = ; we move to the rst stage of the game to determne optmal nformaton choces. 3. Solvng the model: t = Let be a dstrbuton of precson choces, that s, () s the proporton of nvestors who choose precson n the rst perod. To make hs decson, nvestor has to take nto account the dstrbuton of s n the economy ( ), hs own precson level ( ), hs sgnal (x ), and hs pror belef about. Followng the lterature, we show that there exsts a unque equlbrum n monotone strateges and that ths s the only type of equlbrum n the coordnaton game. Assume that all nvestors follow monotone strateges, and let a (x ; ; ) be nvestor s strategy. 7 Then a () s monotone f there exsts x ( ; ) such that I f x x ( ; ) a (x ; ; ) = NI f x < x ( ; ) 6 Note that an addtve nformaton structure s a common modellng devce, not only n the context of global games or games wth lnear-quadratc payo s but also n the broad lterature on costly nformaton acquston. See Veldkamp (0) for examples n macroeconomcs and nance, or Hwang (993) and Hauk and Hurkens (00) for examples n ndustral organzaton. 7 In what follows, we assume that each nvestor condtons hs strategy on the dstrbuton of precson choces, rather than on each nvestor j s precson choce j, j 6=. Ths assumpton s wthout loss of generalty, snce nvestors do not care about the dentty of a partcular nvestor j who chooses precson j, they care only about the proporton of nvestors that choose a gven precson level. 6

7 Note that the thresholds can d er across nvestors wth d erent precson levels and that they also depend on. We assume that all nvestors wth the same precson level,, have the same threshold x ( ; ). As n the standard global games models, the equlbrum n monotone strateges s characterzed by two equatons: a payo nd erence (PI) condton and a crtcal mass (CM) condton. The d erence wth respect to the standard setup s that n our model each type has a d erent PI condton. 8 Consder rst the CM condton, whch requres that at state the mass of nvestors that nvest be equal to the mass of nvestors needed for nvestment to succeed: Z Pr (x x ( ; ) j ) d ( ) = Next, consder nvestor, whose precson level s. The PI condton states that when observng sgnal x ( ; ), nvestor s nd erent between nvestng and not nvestng: Pr ( > ( ) jx ( ; )) T = 0 () An equlbrum n monotone strateges s characterzed by a set of sgnal thresholds fx ( ; )g [0;] and a threshold level for the fundamentals, ( ), that solve the PI and CM equatons smultaneously. In the case of a normal dstrbuton, ths system of equatons can be smpl ed to one equaton n one unknown, ( ): Z ( ( ) ) + ( + ) (T ) d ( ) = ( ) (3) Each ( ) that solves Equaton (3) s then assocated wth a d erent equlbrum. The next proposton spec es condtons for ths equaton to have a unque soluton and for no other non-monotone equlbra to exst. Proposton Under Assumpton A, for any the coordnaton game has a unque equlbrum n whch all nvestors use threshold strateges fx ( ; ) ; [0; ]g and where nvestment s successful f and only f ( ). Note that Proposton s a generalzaton of the standard unqueness result n global games (as establshed by Hellwg, 00, and Morrs and Shn, 004) to the settng where nvestors are heterogenous wth respect to the precson of ther nformaton. 9 Armed wth ths result, we move on to the rst perod to analyze nvestors optmal choces of precson. 3. Solvng the model: t = We now consder the rst stage of the game, n whch nvestors choose the precson of the sgnal they wll observe at the begnnng of the second stage. We assume that each nvestor wll act optmally n the second perod and that he beleves that all other nvestors wll act optmally as well. 8 See Hellwg (00) for a detaled dervaton of PI and CM condtons n the model where nvestors share the same precson. 9 Assumpton A s stronger than necessary. The concluson of Proposton holds as long as nf (supp( )) >. 7

8 3.. Ex-ante utlty Denote by G () the pror belef of nvestors regardng, and by F (xj) the condtonal dstrbuton of x gven and gven that the sgnal x has precson. Recall that all nvestors are ex-ante dentcal, that s, they have the same ex-ante utlty. For any ( ; ; T ) ; the ex-ante utlty of nvestor who chooses precson and faces a dstrbuton of precson choces can be wrtten as 0 U ( ; ) = Z Z x T df (xj) dg () Z Z x ( T ) df (xj) dg () Z + ( T ) dg () C ( ) (4) The expresson on the RHS of Equaton (4) has an ntutve nterpretaton. The last term s the cost assocated wth the precson choce. Recall that nvestment s successful f and only f ( ), n whch case an nvestor s payo s T f he nvests. Hence, the thrd term s the expected payo at tme t = for an nvestor who can perfectly observe n the second perod. However, for any < an nvestor s nformaton at t = s nosy. Ths means that the nvestor wll sometmes make mstakes, ether nvestng when nvestment s unsuccessful (Type I mstake) or not nvestng when nvestment s successful (Type II mstake). The rst two terms capture the expected costs of these two mstakes, respectvely. We denote by by M ( ; ) the total expected cost of these mstakes for an nvestor wth precson who faces a dstrbuton of precson choces M ( ; ) = Z Z x T df (xj) dg () + Z Z x ( T ) F (xj) dg () To better understand how a hgher precson s bene cal to nvestors, we abstract from the cost of precson and focus on ts bene t, whch s captured n the rst three terms on the RHS of Equaton (4). We de ne ths bene t as B ( ; ): B ( ; ) M ( ; ) + ( T ) dg () From the above equaton, we see that more precse prvate nformaton s valued by an nvestor to the extent that t allows hm to avod commttng costly mstakes. The spec c mechansm s formalzed n the followng lemma. 0 See Secton A:3 of the appendx for dervatons. A hgher precson of prvate sgnals changes the expected cost of mstakes n two ways. Frst, a hgher changes the ex-ante jont dstrbuton of (; x ) by better algnng the realzaton of the sgnal x to the state. Second, t a ects the threshold x. A decrease n x, holdng everythng else constant, leads to a hgher expected cost of a Type I mstake and a lower expected cost of a Type II mstake, snce nvestors now nvest more aggressvely. However, snce x s chosen to equalze the bene t from a successful nvestment to the potental cost of an unsuccessful nvestment, the margnal change n x due to a change n has no e ect on expected utlty. Therefore, the margnal bene t of a hgher precson comes from the change n the ex-ante jont dstrbuton of (; x ) that better algns sgnals x wth the fundamentals. 8 Z

9 Lemma The margnal bene t of an ncrease n the precson of prvate sgnals s equal to the reducton n the expected cost of mstakes due to a change n the ex-ante jont dstrbuton of (; x ) mpled by ths ncrease, and s gven by B ( ; ) = + x Equaton (5) shows that, for a Gaussan nose structure, the value of addtonal nformaton depends on the dstance between x and and on the dstance between and (wth larger dstances decreasng the value of addtonal nformaton), but t does not depend on the relatve cost of mstakes. ;3 To provde ntuton for ths result, we rst focus on the dstance between and. Consder the case when the d erence between and s large and postve (the case when the d erence s negatve s analogous). In ths case, an nvestor assgns a low ex-ante probablty to a successful nvestment, snce pror belefs ndcate that s unlkely to take a value greater than. As such, he assgns a low probablty to commttng a Type II mstake. Thus, n equlbrum he rarely chooses to nvest (he sets a hgh x ) and expects ths acton to be correct most of the tme. In ths case, therefore, the value of addtonal nformaton s low. The opposte s true when and le close to each other. In ths case, from an nvestor s perspectve, both nvestment outcomes are almost equally lkely. Therefore, he assgns relatvely hgh probabltes to commttng the two types of mstakes and thus attaches a hgh value to addtonal nformaton. To analyze how the value of addtonal nformaton vares wth the dstance between x and, we need to understand why n equlbrum x mght be far away from. Consder the case where x s hgher than. Ths occurs n equlbrum when T s hgh and s low. In ths case, an nvestor s mostly concerned about makng a Type I mstake, snce he expects nvestment to be unsuccessful (low ) and nvestng s costly (hgh T ). Therefore, n equlbrum he chooses a hgh x n order to mnmze a Type I mstake. An ncrease n the precson of hs prvate sgnal allows the nvestor to reduce the total expected cost of mstakes. However, snce he was already avodng the mstake that he cares relatvely more about, the reducton n the expected cost of mstakes that accompanes the ncrease n hs precson s not very valuable. Ths s n contrast to the case when x s close to, whch happens only f the nvestor ntally cares about avodng both types of mstakes. As a result, an ncrease n precson allows hm to reduce the probabltes of commttng the two types of mstakes at a smlar pace. It follows that n ths case the value of addtonal nformaton s hgher than n the case where x s far away from. To summarze, n our setup the value of addtonal nformaton depends on the relatve dstance between x, and, whch n equlbrum are determned by the cost of nvestment, T, and the mean of the pror belef,. Therefore, as we wll see n the followng Ths surprsng result s a consequence of the equlbrum condton T Pr ( < jx ) = ( T ) Pr ( > jx ) and the propertes of the normal dstrbuton. 3 In the appendx (Secton A:3), we provde an expresson for the reducton n the expected cost of each type of mstake. Equaton (5) s obtaned by addng those two expressons. 9 (5)

10 sectons T and wll play an mportant role when characterzng the propertes of an equlbrum (see Secton 7: for a summary of the role played by T and for our results). Moreover, note that, as explaned above, when x,, and are all close to one another an nvestor s equally lkely to commt the two types of mstakes. Equaton (5) mples that, from an nvestor s perspectve, the value of addtonal nformaton s hgh when the two types of mstakes are equally lkely, and t s low otherwse. 3.3 Equlbrum at t = In perod ; nvestors choose the precson of ther sgnals. The expected payo to nvestor from choosng precson when he faces a dstrbuton of precson choces and beleves that all nvestors wll behave optmally at t = s gven by where ( ) solves Z U ( ; ) = B ( ; ) C ( ) Pr (x x ( ; ) j ( )) d = ( ) and x ( ; ) = + ( ) + ( + ) (T ) Wth the above descrpton of the nvestor s problem at tme t =, we can now de ne a Perfect Bayesan Nash Equlbrum of the two-stage game. De nton A pure strategy Perfect Bayesan Nash Equlbrum s a set of precson choces f ; [0; ]g, together wth a set of decson rules for the second perod fa (x ; ; ); [0; ]g and a dstrbuton of precson choces such that all the followng hold:. Each nvestor s choce of precson s optmal, gven : B ( ; ) C( ) B (b ; ) C(b ) 8b [; ). The dstrbuton mpled by the nvestors choces s almost surely equal to the dstrbuton ; 3. All nvestors behave optmally n the second stage: where a (x ; ; ) = ( I f x x ( ; ) NI f x < x ( ; ) x ( ; ) = + ( ) 0 + ( + ) (T )

11 and ( ) solves Z ( ( ) ) + ( + ) (T ) d ( ) = ( ) The rst condton requres nvestors to choose the precson of ther prvate sgnals optmally. The second condton s a standard consstency requrement. Fnally, the thrd condton requres nvestors to follow equlbrum strateges n the second stage, gven ther choce of precson and ther belefs about the equlbrum precson choces of others,. In partcular, ths condton requres an nvestor to behave optmally n the second perod, even n the case of an ndvdual devaton n precson choces. Wth the above de nton, we can now state our man exstence result. 4 Theorem Suppose that Assumptons A and A hold. Then we have the followng:. There are no asymmetrc equlbra n whch nvestors choose d erent precson levels n the rst stage.. There exsts a symmetrc equlbrum of the nformaton acquston game where all nvestors choose the same precson n perod and equlbrum n perod s characterzed by a par of thresholds f ( ) ; x ( )g. 3. There exsts < such that f >, then there s a unque equlbrum n the nformaton acquston game. Theorem establshes the exstence of symmetrc equlbra and rules out the exstence of asymmetrc equlbra. 5 Moreover, f the default precson level s hgh enough, there s a unque symmetrc equlbrum. Notce that the condton we mpose on, that s, that the default precson of sgnals be hgh enough, s n the same sprt as the standard condton to ensure unqueness of equlbrum n global games. In what follows, we assume that the above condton for unqueness of the two-stage game s sats ed and denote the unque equlbrum precson choce by. 6 Snce n equlbrum all nvestors choose the same precson, wth a slght abuse of notaton we express the bene t functon as B ( ; ) and the ex-ante utlty functon as U ( ; ) ; rather than B ( ; ) and U ( ; ), respectvely. In the remander of the paper, we nvestgate the propertes of the unque equlbrum. 4 For ths result to be true, we need quas-concavty of the ex-ante utlty functon, net of the precson cost, and a unque equlbrum n the second stage. The assumptons made n Secton ensure that these condtons are met (see the onlne appendx). 5 Snce n a symmetrc equlbrum all nvestors choose the same precson, we abuse notaton slghtly and wrte x ( ) and ( ) nstead of x ( ; ) and ( ) ; where =. 6 To be more precse, we assume that s not only hgh enough to ensure unqueness of equlbrum but also hgh enough to mply that the slope of the best-response functon s lower than 5 6 (the unqueness argument requres ths slope to be less than ). Snce the slope of the best response functon converges to 0 for all > as, such a lower bound exsts. We need ths addtonal condton to prove Proposton 6.

12 4 Spllover e ects and the ne cency of equlbrum The nformaton acquston game exhbts spllover e ects, snce nvestors do not take nto account the mpact of ther precson choces on the equlbrum nvestment outcome. In partcular, an ncrease n the precson of all nvestors a ects ther utlty through ts mpact on. However, snce all nvestors take as gven, they gnore ths e ect when choosng ther ndvdual level of precson. As we show below, ths leads to the unque equlbrum of the game beng ne cent. We de ne an e cent symmetrc precson choce as one that maxmzes the ex-ante expected utlty, takng nto account these spllover e ects. De nton We say that a precson choce s e cent f arg max B (; ) C() [;) That s, a precson choce s e cent f t allows nvestors to acheve the hghest ex-ante utlty when they coordnate ther precson choces. Let () be nvestor s best-response functon. The d erence between the equlbrum precson and the e cent precson s that the former s chosen n a non-cooperatve fashon, that s, = ( ), whle the latter s chosen n a cooperatve fashon. Hence, s not necessarly a best-response to all other nvestors choosng. Indeed, we show that genercally 6= ( ). 7 A precson choce s e cent f ether =, or t sats es the followng necessary rst-order condton: B ( ; ) + B ( ; ) C 0 ( ) = 0 Ths condton s necessary, but not su cent, for the equlbrum to be e cent, snce n some cases B (; ) C () s not a quas-concave functon of. We dscuss ths ssue n more detal below. 8 We rst show that the unque equlbrum s typcally ne cent. To state our result, we de ne E (T ) as the unque soluton to s = ( ) (T ) + p (T ) ( ) + ( ) + where ( ) s the equlbrum choce of precson, gven that the mean of the pror s. We show n the appendx (proof of Proposton ) that n equlbrum B ( ; ) = 0 f and only f = E. Usng ths observaton, we arrve at the followng result: Proposton Consder the equlbrum precson choce. For any T (0; ), f 6= E (T ) then the equlbrum precson choce s ne cent. 7 We show n the appendx that the set of arguments that maxmzes B (; ) C() s non-empty and that they are all nte. 8 See also Secton 3: of the onlne appendx.

13 We now nvestgate whether nvestors over-acqure or under-acqure nformaton. We say that nvestors globally over-acqure nformaton f ( ) > ( ). On the other hand, nvestors locally over-acqure nformaton f a small decrease n precson from the equlbrum level would lead to an ncrease n welfare. The de ntons for the under-acquston of nformaton are analogous. The followng proposton fully characterzes the condtons under whch nvestors locally under- or over-acqure nformaton n equlbrum. Proposton 3 Consder the nvestors equlbrum precson choces.. If > E (T ) ; then nvestors locally over-acqure nformaton.. If = E (T ) ; (and T ) then nvestors choose the locally e cent level of nformaton. 3. If < E (T ) ; then nvestors locally under-acqure nformaton. To understand the ntuton behnd Proposton 3, recall that nvestors take nto consderaton only ther prvate bene t and cost when choosng ther precson. In partcular, they choose a precson takng as gven the equlbrum threshold, gnorng the e ect ther collectve decsons have on the equlbrum probablty of a successful nvestment. Thus, the socal bene t of addtonal nformaton tends to d er from the prvate bene t of a hgher precson, snce the former also takes nto account the e ect of precson choces on. When the nvestment threshold s decreasng n the precson of all nvestors n the neghborhood of the equlbrum precson choce, whch happens when < E (T ), then the margnal prvate bene t of extra nformaton s lower than the margnal socal bene t. Ths s because the margnal socal bene t takes nto account the postve e ect of a hgher prvate precson on nvestment. Snce at the equlbrum precson the margnal prvate bene t s equal to the margnal cost of extra nformaton, the socal bene t of more precse nformaton s strctly hgher than ts margnal cost. Thus, n ths case t would be welfare mprovng f all nvestors acqured more nformaton, that s, nvestors are locally underacqurng nformaton n equlbrum. The opposte s true f the nvestment threshold s ncreasng n nvestors precson choces n the neghborhood of the equlbrum precson choce, whch happens when > E (T ). In ths case, the margnal prvate bene t s hgher than the margnal socal bene t and nvestors locally over-acqure nformaton. 9 Fnally, as s shown n Proposton, f = E (T ) then the prvate and socal margnal bene ts of addtonal nformaton are equal at the equlbrum precson level, and hence s an extremum pont of the welfare functon. However, ths s not enough to conclude that agents acqure the locally e cent amount of nformaton. In partcular, t can be shown that f T <, then corresponds to a local mnmzer of the welfare functon, whle f T, then corresponds to a local maxmzer of the welfare functon. 9 See Iachan and Nenov (04) for an analyss of the e ects n equlbrum of changes n the precson of prvate nformaton n a general class of games of regme change. 3

14 The above ntuton can also be used to understand when nvestors globally over-acqure or under-acqure nformaton. In partcular, f the nvestment threshold s monotone n, then the local results translate drectly nto global results. In ths case, the margnal prvate bene t of addtonal nformaton s ether always lower (when s a decreasng functon of ) or always hgher (when s an ncreasng functon of ) than the socal margnal bene t of nformaton. The d culty of fully characterzng global results s due to the fact that can be a non-monotone functon of prvate precson choces. 0 In the onlne appendx (Proposton 9), we show that the local results translate drectly nto global results except for the case when T < and (b (T; ; ) ; T ), where r b (T; ; ) = + (T ) + p + (T ) In ths case, t s possble for nvestors to locally under-acqure but globally over-acqure nformaton. Ths s because for these parameters s rst ncreasng and then decreasng n the nvestors precson choces. Thus, f the equlbrum precson s hgh, a small ncrease n nvestors precson choces from the equlbrum level s welfare mprovng, snce t leads to a lower nvestment threshold. At the same tme, t s possble that from the planner s perspectve t s optmal to acqure no nformaton, snce t s costly and t leads to a hgher. Verfyng ths analytcally, however, s d cult because the welfare functon may not be quas-concave. Secton 3 of the onlne appendx explores these ssues n more detal. 5 Strategc complementartes n nformaton acquston We now nvestgate whether strategc complementartes n the coordnaton game translate nto strategc complementartes n nformaton acquston. In the context of games wth lnear-quadratc payo s, Hellwg and Veldkamp (009) have shown that ths s ndeed the case. In our model ths s not always true. De nton 3 Let be nvestor s precson choce, whle s the precson choce of all the other nvestors. We say that nformaton choces are strategc complements f for all 0 See Szkup (05) for a complete characterzaton of condtons under whch s non-monotone n global games. To understand why the welfare functon may not be quas-concave, note that a hgher precson has three separate e ects on the welfare functon. Frst, a hgher precson allows nvestors to avod costly mstakes. Second, a hgher precson, through ts e ect on nvestment choces, a ects the threshold. Fnally, a hgher s assocated wth a hgher cost. If T < and (b (T; ; ) ; T ) ; then s ntally ncreasng and then decreasng n. Thus, not only s a small ncrease n costly, but t also lowers the probablty of a successful nvestment. These two negatve e ects tend to reduce welfare. However, as keeps on ncreasng, the negatve e ect of a hgher on nvestment decreases sharply. For ntermedate values of (precson choces near the pont where acheves the global maxmum), the negatve e ect of a hgher on nvestment becomes neglgble. At ths pont, t s possble that the reducton n the expected cost of mstakes becomes the domnant e ect and, as a result, the welfare functon becomes ncreasng n. However, as ncreases further, the reducton n the expected cost of mstakes becomes smaller and smaller. Intutvely, f nvestors already have precse nformaton, then they are able to avod commttng mstakes to a large extent, and there s lttle value to addtonal nformaton. As a result, the welfare functon agan becomes decreasng n, drven by the ncreasng cost of a hgher precson. 4

15 and all we have B ( ; ) > 0 The above de nton states that nformaton choces are strategc complements f and only f the value of addtonal nformaton to nvestor s ncreasng n the precson choces of the other nvestors for all pars f ; g. Recall from Secton 3: that the value of addtonal nformaton to nvestor s determned by the dstance between x and, as well as the dstance between and. A change n the precson choce of the other nvestors,, a ects nvestor s ncentves to acqure nformaton by a ectng these dstances, and hence the value of addtonal nformaton to nvestor. As shown n the next proposton, there s no guarantee that strategc complementartes n nformaton choces arse n our model. Proposton 4 Consder nvestors nformaton choces.. For T 6= ; nformaton choces are strategc complements f = mn T; SC (; ; T ) ; max T; SC (; ; T ) where SC (; ; T ) T + p + (T ) Otherwse, there s a lack of strategc complementartes.. For T = ; nformaton choces are always strategc complements. Proposton 4 ndcates that when T 6=, for extreme values of the pror mean, nformaton choces are strategc complements, whle for ntermedate values they are not. To see ths, x T and consder the case when s low, so that s hgh and the dstance between the two s large (the case for a hgh s analogous). In ths case, an nvestor cares manly about Type I mstakes, snce he assgns a low ex-ante probablty to a successful nvestment, so he attaches relatvely low value to addtonal nformaton. An ncrease n, the precson choce of other nvestors, leads to a decrease n. Ths s because when s low, an ncrease n mples that nvestors assgn a lower weght to the unfavorable nformaton, represented by low ; and thus nvest more often. However, a decrease n ncreases the expected probablty of a successful nvestment. As a result, nvestor shfts hs concern from avodng manly a Type I mstake to avodng both types of mstakes more evenly. Ths ncreases hs demand for nformaton. By lack of strategc complementartes we refer to the stuaton where there exst pars f ; g such that B ( ; ) < 0, that s, where an ncrease n the other nvestors precson choces leads to lower ncentves for nvestor to further ncrease hs own precson. Ths s d erent from strategc substtutabltes, whch would correspond to the stuaton where for all and all we have B ( ; ) < 0. It can be ver ed that n our model nformaton choces cannot be strategc substtutes (see the proof of Theorem ). 5

16 To see why nformaton choces mght not be strategc complements, consder the case when T > and T; SC (; ; T ) ; and assume that nvestor has a low precson,, and that the precson of the rest of the nvestors,, s hgh. When s low, nvestor wll care slghtly more about a Type I mstake than the rest of the nvestors, snce a hgh T mples that ths mstake s relatvely more costly, and hs nformaton s not as precse as that of the rest of the nvestors. When both and T are hgh, an addtonal ncrease n wll ncrease (see the proof of Proposton ), thus decreasng the probablty of a successful nvestment. Ths, n turn, wll make nvestor shft hs concern even further towards avodng a Type I mstake, thus becomng less concerned about a Type II mstake. Snce the value of addtonal nformaton s hgher when an nvestor cares about both types of mstakes, ths adjustment n nvestor s behavor makes hm value addtonal nformaton even less, whch decreases hs ncentves to acqure nformaton. An analogous argument holds when T < and takes a value n SC (; ; T ) ; T. 6 Publc nformaton and welfare In recent years, the e ect of publc nformaton on welfare has attracted a lot of attenton (see Morrs and Shn, 00, and the lterature that followed). Ths motvates us to study, n the context of our model, the e ects of the precson of publc nformaton on welfare. Gong back to our example n the ntroducton, consder a government that, n order to encourage foregn drect nvestment, decdes to provde nvestors wth detaled nformaton about the current state of the economy. Ths ntal report provded by the government shapes the nvestors pror belefs about the state of the economy. In addton to ths nformaton, nvestors have the possblty to gather more nformaton prvately. It s of nterest to understand the e ect of the publc nformaton ntally released by the government on nvestors ncentves to acqure prvate nformaton, on the probablty of successful nvestment, and on ex-ante socal welfare. We nterpret pror belefs as publc nformaton and study how changes n the precson of ths type of publc nformaton a ect equlbrum strateges and outcomes. 3 Gven our nterpretaton, we rst nvestgate how an ncrease n the precson of publc nformaton a ects nvestors ncentves to acqure prvate nformaton. We then turn our attenton to the e ects on coordnaton among nvestors, and nally on the welfare mplcatons of changes n the nformatveness of the pror. In what follows, we assume that T =. Ths assumpton mples that Type I and Type II mstakes are equally costly, and that nvestors care equally about coordnatng wth other nvestors on nvestng and on not nvestng. Whle not wthout loss of generalty, ths assumpton smpl es the analyss substantally, allowng us to completely characterze the mpact of an ncrease n the precson of publc nformaton on prvate nformaton acquston and on the probablty of a successful nvestment. The case when T 6= s dscussed n detal n the onlne appendx. One should note that our modellng of publc nformaton s d erent from the typcal 3 Ths nterpretaton of publc nformaton s smlar to Metz (00) and Morrs and Shn (004). 6

17 approach n the lterature. Publc nformaton s commonly modelled as a separate publc sgnal that s observed smultaneously wth the prvate sgnal, and an ncrease n publc nformaton s modelled as an ncrease n the precson of ths sgnal (see Morrs and Shn, 00, and Colombo et al., 04, among others). In those setups agents choose the precson of prvate nformaton before the publc sgnal s realzed. In our approach, nvestors condton ther prvate nformaton choces on the realzaton of the publc sgnal, captured by. 4 In order to facltate the comparson of our model wth the exstng lterature on games wth lnear-quadratc payo s, n Secton 7 we compare our results to a verson of the model wth lnear-quadratc payo s wth a proper pror, but wthout an explct publc sgnal. 6. Trade-o between publc and prvate nformaton To analyze the trade-o between publc and prvate nformaton, notce that more precse publc nformaton a ects the value of acqurng prvate nformaton through three d erent channels. Frst, more precse publc nformaton changes the jont densty of f; x g. Snce ths e ect s ndependent of nvestors behavor, we call ths the passve nformaton e ect. Second, a change n, by changng the nformatveness of the pror, a ects an ndvdual nvestor s nvestment strategy for any gven precson choce. Snce ths e ect nvolves a change n the nvestor s behavor, we call t the actve nformaton e ect. Fnally, a change n a ects the equlbrum threshold through a change n the other nvestors nvestment strateges. We call ths the coordnaton e ect. In comparson, more precse prvate sgnals a ect the value of acqurng more prvate nformaton only through the passve and actve nformaton e ects. Not only s the coordnaton e ect not present, but the passve nformaton e ect s also d erent. In partcular, more precse prvate nformaton better algns the sgnals wth the realzaton of the fundamental. In contrast, more precse publc nformaton ncreases the lkelhood of the fundamentals takng values closer to ther mean. Ths subtle d erence n the passve nformaton e ect can lead to complementartes between publc and prvate nformaton. Proposton 5 Let T =. There exst cuto s b and b + such that b < < b + and the followng holds:. If = (b ; b + ), then prvate and publc nformaton are substtutes.. If (b ; b + ), then prvate and publc nformaton are complements. To understand the ntuton behnd Proposton 5, we consder rst the passve nformaton e ect (.e., we keep and x constant). An ncrease n ncreases the lkelhood of the fundamental takng a value near. If les near ; ths leads to a hgher probablty that the realzaton of wll be close to the crtcal threshold. For a gven precson of prvate nformaton, such a change n the dstrbuton of ncreases the ex-ante probablty that an 4 Ths has the dsadvantage of ntroducng senstvty to the pror mean when studyng the e ects of changes n the precson of the pror. Unfortunately, ntroducng a separate publc sgnal makes the analyss ntractable. 7

18 nvestor s sgnal wll le on the wrong sde of, leadng hm to take the ncorrect acton. 5 In ths case, an ncrease n the precson of publc nformaton ncreases the expected cost of mstakes, whch ncreases the value of addtonal nformaton. Therefore, when les near the passve nformaton e ect encourages nvestors to acqure more prvate nformaton. Note that ths e ect s strongest when =, whch happens exactly when =. The opposte s true when s far from, whch happens when s far from. In ths case, an ncrease n shfts the probablty mass away from the values of at whch nvestors are partcularly susceptble to takng the ncorrect acton. In ths case, the passve nformaton e ect dscourages prvate nformaton acquston. Ths e ect s partcularly strong when s far from, whch happens when s far from. The above argument explans why the passve nformaton e ect encourages nformaton acquston when s close to and dscourages t otherwse. What about the other e ects? When T = ; the coordnaton e ect wll always dscourage prvate nformaton acquston by ncreasng the gap between and. Intutvely, when s hgh (so that > ), an ncrease n reassures nvestors that the fundamentals are strong, whch encourages nvestment and leads to a decrease n. Snce les now further away from, the probablty that the actual realzaton of the fundamental wll be close to the crtcal threshold s lower. For a gven precson of prvate nformaton, such a change n decreases the ex-ante probablty that an nvestor wll take the ncorrect acton, reducng hs ncentves to acqure more prvate nformaton. Analogous ntuton apples to the case when s low. By smlar logc, the actve nformaton e ect pushes the nvestors threshold, x, away from (and away from ), thus decreasng the probablty that an nvestor wll take the ncorrect acton that he wants to avod the most. 6 Note that the actve nformaton and coordnaton e ects become stronger as moves away from. In partcular, the actve nformaton e ect s strong when a small change n leads to a large change n x. In turn, the change n x s large when x les far from ; snce a small change n the precson of the pror has a large e ect on nvestors posteror belefs, evaluated at the sgnal threshold. Snce nvestors choose the threshold sgnal to be close to when s close to ; the actve nformaton e ect s strong when s far from and weak when s close to. Fnally, snce the change n s drven by a change n x, the same ntuton apples to the coordnaton e ect. 6.. The case T 6= One may wonder whether the above ntuton extends to the case when T 6=. In partcular, are there values of such that prvate and publc nformaton are complements when T 6=? In the onlne appendx (Secton 4), we show that we can use the same ntuton to understand the case T 6=, snce there exst T L and T H, 0 < T L < < T H <, such 5 The probablty of takng the ncorrect acton s hghest when les close to, snce an nvestor faces the hghest lkelhood of recevng a sgnal x <, whle n realty >, and vce versa. 6 When T = ; the value of determnes whch mstake nvestors care more about. When >, nvestors want to make sure that they nvest when nvestment s successful, and they choose x < <. The opposte s true when <, n whch case nvestors prefer to coordnate on not nvestng and set x > >. 8

19 that for all T (T L ; T H ) there are values of for whch prvate and publc nformaton are complements. Moreover, as shown n Fgure, numercal smulatons suggest that ths result extends to all T (0; ) µ θ Substtutes Complements Passve=0 Actve= Fgure : Relaton between prvate and publc nformaton Understandng whch e ects drve the results when T 6= s more d cult, snce t requres comparng the absolute magntudes of the three e ects. However, our analytcal results reported n the onlne appendx, suggest that, unless T takes extreme values, the passve nformaton e ect stll plays an mportant role n drvng the complementarty between publc and prvate nformaton. Ths s because the e ect of the passve nformaton on the ncentves to acqure prvate nformaton s the same regardless of the value of T 6=. In partcular, t s stll true that whenever s close to the passve nformaton e ect encourages nformaton acquston, whle the opposte s true when les far from. Fgure further supports ths clam. In the gure the area between the two dashed curves corresponds to the regon where the passve nformaton e ect s postve, and the area between the two dash-dotted curves corresponds to the regon where the actve nformaton e ect s postve. 8 We can see that, unless T takes extreme values, the regon where prvate and publc nformaton are complements les n the nteror of the regon where the passve nformaton e ect s postve. Ths suggests that, unless T s very small or very large, the passve nformaton e ect s the key force drvng the complementarty between publc and prvate nformaton. When T takes on extreme values, the complementarty between prvate and publc nformaton can be drven by the actve nformaton e ect. To understand why ths s the case, 7 See Secton 4: n the onlne appendx for numercal robustness checks. 8 The two dash-dotted curves ntersect at T =, snce n ths case the actve nformaton e ect s always non-postve (and strctly negatve when 6= ). T 9