Information Acquisition and Transparency in Global Games Michal Szkupy and Isabel Trevino New York University
Abstract We introduce costly information acquisition into the standard global games model. In our setup agents have the opportunity to improve the precision of their private signal, at a cost, before playing the coordination game. We show that only symmetric equilibria exist and we provide su¢ cient conditions for uniqueness. Under these conditions, we …nd that the unique symmetric equilibrium of the game is ine¢ cient. Finally, we study the e¤ects of increased transparency on information acquisition and coordination. We …nd that, depending on the cost of investment and on prior beliefs, an increase in the precision of public information can have a bene…cial or detrimental e¤ect on welfare. Key words: global games, information acquisition, coordination, transparency.
1
Introduction
Global games have been extensively applied to model economic phenomena featuring coordination problems, such as currency crisis (Morris and Shin, 1998), bank runs (Goldstein and Pauzner, 2005), FDI decisions (Dasgupta, 2007) or political revolts (Edmond, 2007). In a global game, the payo¤s of agents depend on the state of the economy and on the actions of others. However, agents only observe a noisy private signal about this state and, in order to choose an optimal action, they have to make inferences about its true value and about the beliefs that other agents hold. This perturbation of the information structure of the game gives rise to a very rich sequence of higher order beliefs, which leads agents to coordinate on a unique equilibrium. This prediction contrasts the complete information model, which features multiple equilibria. The information structure is thus at the heart of the predictions of global games. While the original models have been extended along many directions, the precision of private signals We would like to thank Tomasz Sadzik and Ennio Stacchetti for their constant support and encouragement throughout this project. We also thank David Pearce, Thomas Sargent, Laura Veldkamp, and participants in the Micro Student Lunch Seminar at NYU for their comments. y Corresponding author:
[email protected]
1
has always been exogenously given and typically set to be identical across agents. It remains an open question whether or not these two assumptions are without loss of generality. In this paper we shed light on these issues by studying a two-stage global game where each agent has the possibility to costly improve the precision of his private information about the state of the economy, and, given this precision choice, he receives a signal about the state that will determine his action in the coordination game. In addition to characterizing equilibria, we study the critical role that information plays in the coordination game and carefully analyze the strategic motives in information acquisition. Our …ndings indicate that under mild conditions there exists a unique equilibrium in symmetric strategies. This …nding gives support, in the context of global games, to the commonly made assumption of ex-ante identical agents. We de…ne the value of information in our setup and analyze how it is a¤ected by prior beliefs, the behavior of other players, and the cost of investment. We characterize the critical role that information plays in the coordination game by studying the e¤ect that precision choices have on the incidence of two types of mistakes in the coordination game (investing when investment is not pro…table, or not investing when investment is pro…table). Interestingly, we …nd conditions for strategic complementarities in information acquisition to arise and we provide an example where these conditions are violated, so that the optimal precision choice of an agent is a non-monotonic function of the precision choices of the others. In addition, we contribute to the growing literature that focuses on the importance of government transparency by studying the welfare e¤ects of an increase in the precision of public information. We provide a novel perspective on the issue by analyzing the tradeo¤ between private and public information, since in our model public information a¤ects outcomes not only through agents’actions in the coordination stage, but also by changing their incentives to acquire private information in the …rst stage. We thus characterize the emergence of increasing and decreasing di¤erences in the precision of public and private signals. Our analysis also highlights the di¤erences between global games and the closely related family of beauty contest models (in the spirit of Morris and Shin, 2002).1 First, we …nd that whether an improvement in public information is welfare enhancing or not depends crucially on the ex ante beliefs about the state relative to the cost of investment, while in beauty contest models it depends on the relative informativeness of private and public information (Morris and Shin, 2002). Secondly, in beauty contest models with endogenous private information acquisition an increase in the precision of public information always leads to a decrease in the precision of private information (Tong, 2007), whereas in our context private and public information can be complements. Finally, we provide conditions under which strategic complementarities in actions translate into strategic complementarities in information acquisition. We provide an example where these conditions are violated and 1
Beauty contest models are also coordination games of incomplete information, but di¤er from standard global games in many respects. In particular, in beauty contest models choice sets are continuous and agents have a quadratic utility function that depends on the average action of the other players as well as on the underlying state of the economy.
2
information choices are not strategic complements, which is in contrast to the results of Hellwig and Veldkamp (2009) for a beauty contest game. Endogenizing information in a global games framework is not only an important endeavor from a theoretical point of view but it is also economically relevant. Global games have been applied extensively to analyze di¤erent types of …nancial decisions. It is a well known fact that participants in …nancial markets have the possibility to improve the information they possess, at a cost, and they take advantage of this possibility. Investment groups and individuals pay experts to extract more accurate information about the state of the economy in order to maximize pro…ts, creating a market for information expertise and …nancial advising. Following Dasgupta (2007), one can think of an emerging economy that starts a liberalization program to attract Foreign Direct Investment. Potential investors have to make a discrete decision to invest or not invest (e.g. building a plant to produce goods for the local market). For the pro…ts to be positive there has to be enough investment so that the liberalization program takes o¤ due to increasing returns to aggregate investment.2 This means that a potential international investor will be more likely to undertake investment in this emerging economy if he believes that other investors will do so as well. Moreover, the returns of the project depend also on the state of the emerging economy in the form of aggregate demand, infrastructure, and political environment - all of which can be uncertain at the time of the investment decision. In this context, potential investors can buy reports from international agencies that will assess the pro…tability of this investment decision, i.e. they acquire more precise information about the state of the emerging economy. Alternatively, we can interpret the cost of information as an opportunity cost associated with collecting and analyzing data about the state of the economy. This example illustrates why introducing costly information acquisition is a relevant extension to global games also from an applied perspective. The paper is structured as follows. In section 2 we set up the model with costly information acquisition and explain the assumptions we make to solve the model. In section 3 we solve the model and carefully analyze the role that information plays in our model. We also present three key results: non-existence of asymmetric equilibria, existence of symmetric equilibria, and conditions ensuring uniqueness of the symmetric equilibrium. In section 4 we investigate notions of e¢ ciency of the unique equilibrium. In section 5 we investigate if strategic complementarities in the coordination game translate into strategic complementarities in information choices. In section 6 we ask whether an increase in the precision of public information is welfare enhancing or not. Section 7 summarizes related literature and section 8 concludes. All the proofs (with few exceptions) are relegated to the appendix.
2
The Model
We start with a brief description of the benchmark global games model (a model without information acquisition). We then introduce information acquisition into the model and carefully explain the timing, underlying assumptions, and we highlight the di¤erences between 2
See Farmer and Guo (1994), Hall at al (1986), Hall (1987), and Caballero and Lyons (1992) for evidence of increasing returns to scale in investment.
3
the models with and without information acquisition. There is a continuum of investors in the economy indexed by i, where i 2 [0; 1], who have to decide whether to invest in a risky project or not. The economy is characterized by the strength of its economic fundamentals ; which a¤ects the return of the investment. Each investor faces a binary decision problem: he has to decide whether to invest (I) or not invest (N I). If an investor invests he will incur a cost T 2 (0; 1). The bene…t to investing is uncertain and depends on the state and on p, the proportion of investors who choose to invest. Investment is successful if p > 1 , i.e. if the proportion of investors who invest is high enough. The return of a successful investment for each investor who invests is 1, in which case he will get payo¤ 1 T . Otherwise, if investment is unsuccessful, his payo¤ will be T . The payo¤ to not investing is certain and normalized to 0. The payo¤s are summarized below: u (I; p; ) =
T if p 1 T if p < 1
1
u (N; p; ) = 0
(1a) (1b)
Whether individual investment is successful or not depends on the state of the economy and on the number of individual investments. One can interpret this need for enough aggregate investment as resulting from increasing returns to scale in investment.3 If investors know the state of the economy, , then, depending on the value of the game can have a unique or multiple symmetric pure-strategy Nash equilibria: 1. If
1 everyone invests and investment is successful;
2. If
< 0 no one invests;
3. If
2 [0; 1) there are multiple equilibria characterized by self-ful…lling beliefs.
In contrast, when investors have incomplete information about the state of the economy, it is possible to select a unique equilibrium. The literature on global games started by Carlsson and van Damme (1993) and Morris and Shin (1998) assumes that investors do not observe and believe that N ; 1 . In this case, each investor privately observes xi , a noisy signal of : xi = + 1=2 "i where "i N (0; 1) is i:i:d: across investors and independent of the realization of , and a positive constant. It follows that xi j
N
3
;
is
1
The payo¤s are chosen to make the game analytically tractable. All the qualitative results would still hold if we allowed the bene…t from investing to be an explicit function of both the state and aggregate investment. However, in that case we would need to resort to numerical solutions due to the extra complexity.
4
so that is the precision level of the private signals. Note that while investors receive signals of di¤erent values, they share the same precision. In this setup, Morris and Shin (1998) show that the equilibrium is characterized by two objects, (x ; ), where x is the threshold value for signals such that if an investor observes a signal xi x he will invest, and not invest if xi < x . Likewise, corresponds to the critical level of fundamentals such that if then investment will be successful, and if < investment will fail. It has been shown that both x and depend crucially on the precision of the information that investors receive ( and ). We present the following result, due to Hellwig (2002) and Morris and Shin (2004), characterizing the unique equilibrium of the game: 1
Theorem 1 If 2 > p12 then there is a unique equilibrium of the above game. This equilibrium is in monotone strategies and is characterized by a pair of thresholds (x ; ). The above result assumes that all investors have access to information with the same precision ( is the same for all investors). In what follows we focus on relaxing this assumption by allowing investors to purchase the precision of their private signal. This introduces strategic motives in information acquisition and generalizes the model to a case where the precision of signals di¤ers across investors.
2.1
Information Acquisition
We consider now a two-period model where investors …rst simultaneously and privately choose the precision of their private signal, and then move on to play the coordination game, as described above. Investors are initially endowed with a signal with precision 0 and they have the opportunity to improve the precision of their signal by choosing i 2 [ 0 ; 1). This improvement of precision is costly, which results in a trade-o¤ between informativeness and cost of signals. Once investors have chosen i , we move on to period 2. At the beginning of period 2 all investors receive private signals: xi = +
1=2 i
"i , 8i 2 [0; 1]
where i is the precision level chosen by investor i in the …rst stage. As before, "i N (0; 1) is i:i:d: across investors and independent of the realization of . Note that in contrast to the model without information acquisition, we now allow investors to observe signals with di¤erent precisions. After observing their signals investors simultaneously decide whether to invest or not. The payo¤s in the coordination game are the same as in the benchmark model, given by equations (1a) and (1b). Once all investors make their investment decisions payo¤s are realized.
2.2
Assumptions
The …rst set of assumptions considers the underlying parameters of the game. 5
Assumption 1 (Concavity) We assume the following: 2[ ;
];
1
(see appendix, claim A:4). Therefore, we conclude that i ( ) is a continuous function mapping [ 0 ; ] into itself. By Brouwer’s Fixed Point theorem we know that i ( ) has a …xed point, call it . This …xed point of i ( ) is a symmetric equilibrium since if an investor believes that all other investors choose , his best response is to choose himself. The above theorem and proposition 1 establish existence of a symmetric equilibrium in which all investors choose the same precision in the …rst stage, , they follow a symmetric monotone strategy with threshold x ( ), and where investment is successful if and only if ( ). The following theorem establishes su¢ cient conditions for the existence of a unique equilibrium in symmetric strategies. Theorem 4 There exists
< 1 such that if
> , then there is a unique equilibrium in the
information acquisition game, that is there is a unique
2 [ ; 1) that solves
=
i(
).
Proof (Sketch). Denote by i ( ) the best response function of investor i. This best response function is de…ned implicitly by the …rst order condition of investor i’s decision problem @ i @ B ( i ( ); ) C ( i ( )) = 0 @ i @ i
13
De…ne the set E as the set of all equilibrium choices of : E=f j =
i
( )g
The full proof in the appendix shows that there exists @
such that
0
>
implies that for
i( ) j i= @
any 2 E we have < 1; that is, the slope of the best-response function at any symmetric equilibrium is less than 1. This immediately implies uniqueness of a symmetric equilibrium since existence of multiple equilibria requires that there exists at least one 2 E @ ( ) such that @i j = 1. Notice that the condition we impose on 0 , i.e. that signals are precise enough, is in the same spirit as the standard condition to ensure uniqueness of equilibrium in global games.7 In what follows we assume that this condition for uniqueness is satis…ed and we investigate the properties of the unique equilibrium.
4
Spillover E¤ects and the Ine¢ ciency of Equilibrium
The game above exhibits spillover e¤ects, in the sense that investors’ choices of precision a¤ect the decision of each individual investor in a way that is not taken into account by them. In particular, an increase in the precision of all but one investor a¤ects this investor’s utility through the impact of these precision choices on . However, since all investors take as given they ignore this e¤ect when choosing their individual level of precision. This, as we show below, leads to ine¢ ciency of the unique equilibrium. We de…ne an e¢ cient symmetric precision choice as the one that maximizes the ex-ante expected utility taking into account these spillover e¤ects. De…nition 2 (E¢ ciency) Precision choice 2 max[
is e¢ cient if
;1) B
i
( ; )
C( )
The above de…nition says that the precision choice is e¢ cient if it allows investors to achieve the highest ex-ante utility when they coordinate their actions in the …rst stage. The di¤erence between the equilibrium precision and the e¢ cient precision is that the …rst one is chosen in a non-cooperative fashion, that is = i ( ), while the other one is chosen in a cooperative fashion and hence is not necessarily a best-response to all other investors choosing . Indeed, we will show that generically, 6= i ( ). Under assumption (A2) we show in claim A:6 in the appendix that the set of arguments that maximizes the above expression is non-empty and that < 1. The solution to the above problem is either a corner solution, = 0 , or it satis…es the following necessary …rst order condition: B1i (
;
) + B2i (
;
)
C0 (
)=0
Therefore, a necessary condition for the equilibrium precision choice to be e¢ cient is that at , B2i ( ; ) = 0. The following proposition establishes conditions under which this is indeed the case. 7
See Theorem 1.
14
Proposition 2 (Ine¢ ciency) Let s K( ( );
;T)
( ) ( )+
1
!
(T )
+p
1
1
( )+
1. A necessary condition for the unique equilibrium to be e¢ cient is that K( ( ); ; T ). 2. For any pair ( ; T ), such that T 6= K( ( ); ; T )
1 , 2
there exists a unique
(T ) solves
=
that solves
=
It follows that generically the unique equilibrium is ine¢ cient. 3. For T =
1 2
the unique solution to
= K( ( );
= 21 .
; T ) is
The …rst part of the above proposition provides conditions for the following to hold at the equilibrium precision choice : 0 = B1i ( ;
)
C 0 ( ) = B1i ( ;
) + B2i ( ;
)
C0 ( )
(10)
This condition is necessary for e¢ ciency but not su¢ cient for two reasons. First of all B i ( ; ) C ( ) may not be a concave function of , hence the above condition is only necessary for to be the utility maximizing choice of precision. Secondly, the e¢ cient precision level might be 0 . This can happen because a higher precision level chosen by investors can lead to a narrowing of the region where investment is successful (increase in ), therefore leading to a decrease in ex-ante utility. In that case, since we showed that the non-cooperative equilibrium precision is always greater than 0 , it follows that the equilibrium is ine¢ cient. The second part of the proposition states that when T 6= 21 then for any combination of and T there is a unique for which equation (10) holds. Because the condition in the above proposition is very speci…c, it follows that in general the unique equilibrium of the game is ine¢ cient. An example where this necessary condition is satis…ed, for any , corresponds to the very particular case where T = 12 and = 12 . Above we explained why the game features spillover e¤ects. Formally, the spillover e¤ects are captured by the fact that the partial derivative of the ex-ante bene…t function with respect to the precision choice of others is di¤erent from zero, i.e. B2i ( i ( ) ; ) 6= 0. We now de…ne our notion of global spillover e¤ects. De…nition 3 Consider investor i and let
be the precision choice of all other investors.
1. If 8 >
we have
@B i @
> 0 then we say that the game exhibits global positive spillovers
2. If 8 >
we have
@B i @
< 0 then we say that the game exhibits global negative spillovers
15
Global spillover e¤ects are of interest because their presence means that the equilibrium of the game is necessarily ine¢ cient (see Cooper and John, 1988). Moreover the presence of global negative spillovers implies that investors over-acquire information while the presence of global positive spillovers means that investors under-acquire information compared to the e¢ cient information choice . The result below provides su¢ cient conditions for the existence of global spillover e¤ects. Lemma 3 Consider the information acquisition game. 1. If the cost of investment is high ( T > 21 ) and investors are ex-ante optimistic ( then the game exhibits global positive spillovers
< 12 )
2. If the cost of investment is low ( T < 21 ) and investors are ex-ante pessimistic ( then the game exhibits global negative spillovers
> 12 )
3. If cost of investment is “neutral” (T = 12 ) and investors are neither optimistic nor pessimistic ( = 21 ) then the game exhibits no spillover e¤ects. The above lemma provides conditions under which investors will over-acquire information (part 1) or under-acquire information (part 2). Under these conditions the unique equilibrium of the game is ine¢ cient. On the other hand, part 3 of the above lemma, provides an example where the unique equilibrium is e¢ cient. While the above result may seem contradictory to the proposition above it is actually complementary. In particular one can show that when that solves = K( ( ); ; T ) the cost of investment is high ( T > 21 ) then the unique 1 1 that solves = K( ( ); ; T ) is is larger than 2 . Similarly, when T < 2 the unique smaller than 21 .
5
Strategic Complementarities in Information Acquisition
We now investigate whether the strategic complementarities present in the coordination game translate into strategic complementarities in information acquisition.8 In the context of beauty contest games, Hellwig and Veldkamp (2009) have shown that in this is indeed the case. It turns out that in our model this is not always the case. We will provide conditions under which strategic complementarities in information acquisition arise and we provide an example of a situation where they do not. There is a set of pairs of underlying parameters of the game ( ; T ) for which strategic complementarities in actions indeed translate into strategic complementarities in information acquisition. For other pairs of ( ; T ) we might have regions where investor i’s incentives to acquire a higher level of precision decrease in the precision choice of his opponents, resulting in a non-monotone best-response function. Recall from section (3:2:1) that the bene…t of increasing the precision of private signals is equal to the reduction in the expected cost of mistakes. From equation (9), it follows that 8
As shown by Vives (2005), global games are games of strategic complementarities.
16
the game has strategic complementarities in information choices if show that: @ 2M ( i; ) ) (x ) = M (xi @ i@
@2M ( i; ) @ i@
< 0. One can (11)
where M is a negative constant, xi is investor i’s threshold and x is the common threshold of all other investors. Equation (11) implies that whenever (xi ) and (x ) are of the same sign precision choices are strategic complements. In general, nothing guarantees ) and (x ) are both positive or negative. However, when investors are that (xi ex-ante optimistic (high mean of the prior) and the investment cost is low (small T ), or when investors are ex-ante pessimistic and the investment cost is high, we have strategic complementarities in information choices. Proposition 3 Suppose that one of the following holds: 1. The cost of investment is high (T > 21 ) and the investors are ex-ante pessimistic ( 1 ) 2 2. The cost of investment is low (T < 12 ) and the investors are ex-ante optimistic (
< > 12 )
Then there are strategic complementarities in the information acquisition game. 3. There exist pairs of (T;
) such that there is lack of strategic complementarities.
The fact that strategic complementarities do not arise for all combinations of parameters is due to the fact that when making his choice of precision, an investor cares about the reduction in the expected cost of mistakes that his precision choice will lead to. For some parameters, the rate at which the increase in precision decreases the total expected cost of mistakes is a decreasing function of the other investors’precision choices. Since information is costly, an increase in the precision of all other investors will lead to a lower precision choice by investor i, in which case, there are no strategic complementarities in precision choices. In particular, in the Appendix we show that we can …nd pairs of (T; ) for which equation (11) is positive, implying that the rate at which the expected cost of mistakes decreases due to an increase in the investor’s own precision is decreasing in the precision choice of others. This implies that there exist regions for which a higher precision acquired by other investors decreases the incentives to acquire more precise information by a single investor, making precision choices strategic substitutes over these regions. Finally, we provide a numerical example that supports this intuition. We set = 0, T = 0:22, = 3 and we consider precision choices in the interval [3; 20]. Figure 2 depicts the best response function obtained from simulations performed with these parameters. We see that the best response function is non-monotonic, con…rming the lack of global strategic complementarities.
17
i
1 4.1 7
1 4.1 4 5
1 4.1 2 5
10
15
20
Figure 2: Best-Response Function
6
Transparency and Welfare
In recent years, the e¤ect of transparency on welfare has attracted a lot of attention (see Morris and Shin, 2002, and the literature that followed their paper). This motivates us to study notions of transparency and welfare in the context of our model. Going back to our example in the introduction, consider a government that tries to encourage foreign direct investment. We interpret prior beliefs as public information released by a governmental agency and an increase in government transparency as an increase in the precision of the prior, . We investigate the impact that an increase in transparency has on the incentives to acquire private information, on coordination in the investment game, and on ex-ante social welfare. We …rst focus on the interaction between public and private information and provide conditions under which they are substitutes. We also provide intuition via an example of a situation where public and private information are complements. We then turn our attention to the e¤ect that an increase in transparency has on coordination among investors. Finally, we investigate welfare implications of a change in the informativeness of public information. Changes in the precision of public information determine the outcome of the game through a direct and an indirect e¤ect. First, holding individual precision choices constant, it a¤ects the probability of a successful investment in the second stage through its e¤ect on (direct e¤ect). Second, it a¤ects the optimal choice of precision of private signals in the …rst stage, which will have a further e¤ect on (indirect e¤ect).
6.1
Trade-o¤ between Public and Private Information
Public information is di¤erent from private information. More precise private signals increase the amount of information that an investor has in hand in the second stage of the game, allowing him to make a better decision and avoid costly mistakes (see section 3:2:1). On the other hand, more precise public signals not only increase the amount of information available to all investors when making their investment decision (“information e¤ect”) but also change the probability of successful coordination on investment through its impact on (“coordination e¤ect”). While the …rst e¤ect always leads to substitution between private and public information, this additional “coordination e¤ect” can make public and private 18
information complimentary. Proposition 4 Suppose that one of the following holds: 1. The cost of investment is high (T > ( < 21 p1 ) or,
1 ) 2
and investors are ex-ante very pessimistic
2. The cost of investment is low (T < 12 ) and investors are ex-ante very optimistic ( 1 + p1 ) 2
>
Then private and public information are substitutes. The intuition for the above result is simple. Fix i for all investors at some arbitrary level. At the extreme values of and T stated in the above proposition, investors correctly assign a small probability to the realizations of for which they miscoordinate. Because of this, an increase in the precision of public information has a small impact on , leading to a small “coordination e¤ect”. On the other hand, because investors are coordinating heavily on one of the two actions (investing when is high and T is low, or not investing when is low and T is high), further increases in the precision of any type of information are of little value, compared to cases with less extreme values of and T . Therefore, any increase in the precision of public information has a strong negative e¤ect on the choice of precision of private information, leading to a strong “information e¤ect”. This implies strategic substitutability between private and public information. Interestingly, it is also possible for private and public information to be complements. This arises when the “coordination e¤ect”dominates the “information e¤ect”. In particular, this can happen when both and T are either high or low. Best Response Functions - µ=1; T=0.3
Best Response Functions - µ=0; T=0.1
7
6
6.5 6 5.5
τ (τ)
5
* i
* i
τ (τ)
5.5
4.5 5 4
45° τ =1
45° τ =1
τ =1 .12 5
τ =1 .12 5
τ =1 .25
τ =1 .25
θ
3.5
θ
θ
θ
θ
3
3
4
5
τ
6
7
θ
4.5
8
Figure 3: E¤ect of a change in
3
4
5
τ
6
7
8
on the best-response function
Figure 3 summarizes this discussion by showing on the left panel a situation of strategic substitutabilities, and on the right panel one of strategic complementarities. In both panels 19
of …gure 3 we plot the e¤ect of an increase in the precision of public information on the best response functions i ( ; ). The left panel of …gure 3 shows that a higher precision of the prior leads to a downward shift of the best response function (substitutabilities). On the other hand, the right panel depicts how a higher shifts the best response function upwards (complementarities).
6.2
E¤ects of Increasing Public Information on Coordination
In the previous subsection we analyzed the relationship between an increase in and investors’behavior in the …rst stage of the game, i.e. the impact of on . In this subsection we …rst analyze the e¤ect of an increase in on , keeping …xed (the direct e¤ect). Then we compute the total e¤ect of an increase in the precision of public information on the probability of a successful investment (this encompasses both the direct and indirect e¤ects). The next result shows that, depending on parameters of the model, an increase in has 9 di¤erent e¤ects on . Lemma 4 Assume that the condition for a unique equilibrium is satis…ed. Then: 1 1 1. If the cost of investment is high (T ) and investors are ex-ante pessimistic ( ), 2 2 then an increase in the precision of the prior leads to a decrease in the probability of a successful investment, holding constant ( @@ 0). 1 1 ) and investors are ex-ante optimistic ( ), 2. If the cost of investment is low (T 2 2 then an increase in the precision of the prior leads to an increase in the probability of 0). a successful investment, holding constant ( @@
These conditions hold with strict inequalities if either
or T are di¤erent from 21 .
Combining lemma 3 with proposition 4 we have the following result on the probability of a successful investment: Proposition 5 Suppose that 1. If T < 12 and increases ( #). 2. If T > 21 and decreases ( ")
>
p21 + 12 . We established that an increase in the precision of the public signal for these values of parameters leads to a decrease in , holding …xed. Moreover, for this set of parameters public and private information are substitutes, so an increase in decreases , which further decreases . Hence the e¤ect on 9
This result is a direct corollary of a more general result established in lemma A:3 in the appendix.
20
is unambiguously negative. Since the probability of a successful investment is simply the probability that , we conclude that in this case a higher precision of public information leads to an increase in . The opposite is true when T > 12 and < p21 + 12 . In that case both the direct and indirect e¤ects of an increase in lead to an increase and hence the ex-ante probability of a successful investment decreases.
6.3
Welfare Consequences of a Higher
Since all investors are ex-ante identical and play a symmetric equilibrium, it is enough to analyze the ex-ante utility of a single investor to determine welfare consequences of an increase in the precision of . Given that the investor plays a symmetric equilibrium with precision choice ; his exante utility is given by i
U ( ; )=
Z
1
Z
1
x (
Z
T dF (xj ) dG ( ) ) Z 1 (1 T ) dG ( ) +
1
Z
x (
)
(1
T ) dF (xj ) dG ( )
1
C( )
The total impact of a change in the precision of public information is given by: d i U ( ; )= d
Z 1Z x d d T (1 T ) (f (xi j ) g ( )) dxd (f (xj ) g ( )) dxd d d 1 x 1 Z 1 d d d (1 T ) C0 ( ) + (g ( )) d g ( ) Pr (x x j ) (12) d d d Z
Z
1
Equation (12) implies that the e¤ect of a change in on welfare depends on four factors: (1) the change in the ex-ante expected cost of Type I mistake (…rst term), (2) the change in the ex-ante expected cost of Type II mistake (second term), (3) the change in the probability of a successful investment (third term), and (4) the change in the cost of the equilibrium precision choice (fourth term). Recall that in section 6:1 when analyzing incentives of investors to acquire private information we introduced the direct and indirect e¤ects of an increase in on the outcome of the game, and we further decomposed the indirect e¤ect into a “coordination e¤ect”and an “information e¤ect”. We can see from equation (12) that the same e¤ects play a role when analyzing the total e¤ect of a change in the precision of public information on the ex-ante utility function. Because of the complexity of our model we are not able to determine analytically the total welfare e¤ect of an increase in transparency Hence, in the next section, we focus on numerical exercises. 6.3.1
Numerical Results
In this section we evaluate numerically the e¤ect of an increase in on the ex-ante utility for di¤erent parameter values. For the purposes of this exercise we assume that 0 = 3, 21
1=2
1=2
11
1 10 C ( i ) = 100 ( i 2 [1; 2].10 0 ) , and We investigate six di¤erent cases. The …rst case is ( ; T ) = ( 0:5; 0:6) and corresponds to the situation where an increase in leads to a reduction in the probability of a successful investment. The second case is ( ; T ) = (1:5; 0:4) where an increase in leads to an increase in the probability of a successful investment. The other four cases consider situations where either both and T are greater than 12 or both are smaller than 12 . Figure 4 below depicts the …rst two cases. In each panel we observe the ex-ante net utility as well as the expected cost of mistakes as changes. In both cases the cost of mistakes decreases with the precision of the public signal. -3
8
x 10
0.61
0.025
0.022
0.6
0.02
C o s t o f Mis ta k e s
4
0.015
2
Ex-Ante Net Utility
0.02
Cost of Mistake
Ex-Ante Net Utility
6
0.018
0.58
0.016
0.57
0.014
0.56
0.012
0.55
0.01
Ex - An te N e t U tility
0.59
Cost of Mistake
Cost of Mistakes
0.01 Ex-Ante Net Utility
0.54
0 1
1.1
1.2
1.3
1.4
1.5
τ
1.6
1.7
1.8
1.9
0.53 1
0.005 2
θ
Case 1
0.008
1.1
1.2
1.3
1.4
1.5
τ
1.6
1.7
1.8
1.9
0.006 2
θ
Case 2 Figure 4
Consider the …rst case. By proposition 4, for this set of parameters private and public information are strategic substitutes, that is, an increase in will lead to a lower equilibrium precision . While in general one cannot predict the e¤ect of these changes in precisions on the total expected cost of mistakes, we see in the left panel of …gure 4 that for the parameters chosen the cost of mistakes decreases due to a strong “information e¤ect” of an increase in the precision of public information. At the same time, from proposition 5, we know that a higher leads to a negative “coordination e¤ect”, i.e. as the public signal becomes more precise, the probability of successful investment decreases, via an increase in . This strong negative “coordination e¤ect” outweighs the bene…t of the decrease in the total expected cost of mistakes and of the reduction of the cost associated to a lower equilibrium precision. Therefore, the total e¤ect of an increase in the precision of public information leads to a decrease in the ex-ante utility. In contrast, in the second case (right panel of …gure 4), we see that an increase in the precision of the public signal leads to an increase in welfare even if, by proposition 4, we know that private and public information are strategic substitutes. This is due to the strong 10
The cost function is chosen in such a way that it satis…es assumption (A2) and ensures that agents will choose to acquire a high precision of private signals.
22
positive “coordination e¤ect” that arises from the increase in the probability of successful investment, via a decrease in (see proposition 5). The above numerical examples considered economies in which investors are ex-ante pessimistic (low ) and face a high cost of investment (high T ), or the opposite, ex-ante optimistic (high ) facing a low cost of investment (low T ). The next four examples consider other interesting cases. The …rst two cases depict situations with low but also a low cost of investment T ( = 0:4 and T 2 f0:2; 0:4g), while the other two cases consider a high but also a high T ( = 0:8 and T 2 f0:6; 0:8g). 0 .2 7
0 .0 5
0 .0 5 6
0 .2 6 8
0 .0 5 4
E x-A nte N et U tility
0 .4 6
0 .0 4
C ost of Mistakes
Ex-Ante Net Utility
0 .2 6 6
Cost of Mistake
Ex-Ante Net Utility
E x-A nte N et U tility
0 .0 5 2
0 .2 6 4
0 .0 5
0 .2 6 2
0 .0 4 8
0 .2 6
Cost of Mistake
0 .4 8
0 .0 4 6
C ost of Mistakes 0 .2 5 8
0 .4 4 1
1 .1
1 .2
1 .3
1 .4
1 .5
τ
1 .6
1 .7
1 .8
1 .9
2
0 .0 3
0 .0 4 4
0 .2 5 6 1
1 .1
1 .2
= 0:4 and T = 0:2
1 .4
1 .5
τ
1 .6
1 .7
1 .8
1 .9
2
0 .0 4 2
θ
= 0:4 and T = 0:4 0 .0 4 5
0 .2 2 6
0 .2 2 4
0 .0 7
0 .0 5
0 .0 4 4
C ost of Mistakes
E x-A nte N et U tility
0 .0 4 2
0 .2 1 8
0 .0 4 1
0 .0 4 5
Cost of Mistake
E x-A nte N et U tility
0 .2 2
0 .0 6
Ex-Ante Net Utility
0 .0 4 3
Cost of Mistake
0 .2 2 2
Ex-Ante Net Utility
1 .3
θ
0 .0 5
0 .0 4
C ost of Mistakes 0 .2 1 6
0 .2 1 4 1
0 .0 4
1 .1
1 .2
1 .3
1 .4
1 .5
τ
1 .6
1 .7
1 .8
1 .9
2
0 .0 3 9
0 .0 4 1
θ
= 0:8 and T = 0:6
1 .1
1 .2
1 .3
1 .4
1 .5
τ
1 .6
1 .7
1 .8
1 .9
2
0 .0 3 5
θ
= 0:8 and T = 0:8 Figure 5
Note that if the cost of investment is su¢ ciently high then more precise public information has a detrimental e¤ect on ex-ante utility. This is due to the fact that public information allows investors to coordinate better in their actions - in the case of a high cost of investment investors coordinate on not investing. The opposite is true when the cost of investment is low with respect to the mean of the prior. Figure (5) depicts these four cases. From the above examples we conclude that public information is welfare improving if the cost of investment is su¢ ciently small compared to the expected value of fundamentals.
7
Related Literature
Our work is related to three strands of literature: global games, information acquisition and transparency. Global games were introduced by Carlsson and van Damme (1993) in their 23
seminal work as an equilibrium re…nement and further extended by Frankel et al (2003). This technique was …rst applied by Morris and Shin (1998) to the context of currency crises and since then it has been extensively used to model economic phenomena featuring coordination problems (see Dasgupta, 2007, Edmond, 2007, Goldstein and Pauzner, 2005, or Morris and Shin, 2004, among others). While the original global games models were static, that is, they featured only one-shot coordination games, several authors extended these models to multi-stage games (Angeletos et al, 2007, and Dasgupta, 2007, among others). We contribute to this literature by considering a model in which investors have the choice to acquire more precise information before playing the standard one-shot global game. Unlike these papers, in our model investors make choices in the …rst period that in‡uence the structure of the game they play in the second period, whereas in the above papers investors play repeatedly a static global game. In this respect our work is most closely related to Angeletos and Werning (2005) and Chassang (2008). However, none of these studies considers costly information acquisition and its impact on the coordination game. Finally, Szkup and Trevino (2012) consider a discrete version of our model and test its predictions experimentally. Our model is also related to the literature on the role of information in beauty contest models. In this type of games, investors’ payo¤s depend on how closely their action is to the average action taken by others and to the unknown state. In the context of incomplete information games with private and public signals, these models were …rst analyzed by Morris and Shin (2002), who study the trade-o¤s between public and private information. They show that public information may be detrimental to welfare if investors have access to very precise private information. In contrast, in a global games setup, we …nd that whether public information is welfare enhancing or reducing depends on the ex-ante optimism or pessimism of investors and on the cost of investment (see section 5:3). In a beauty contest game, Tong (2007) analyzes the impact of an increase in the disclosure of public information on the incentives to acquire private information. He shows that an increase in the precision of public information always decreases investors’incentives to acquire private information and leads to a lower precision of private information in equilibrium. In contrast, we …nd that in our model public and private information can be complements (see section 5:1). This follows from the fact that in the beauty contest models there is no clear distinction between the “information”and “coordination”e¤ects. More precisely, due to the continuous nature of the beauty contest model, investors do not make mistakes (there is no “wrong”action). Finally, Hellwig and Veldkamp (2009) show that in a beauty contest model complementarities in actions translate into complementarities in information acquisition. While this is true for a wide range of parameters in our model, we provide a numerical example in which this result does not hold (see section 5). Hence, our work emphasizes important di¤erences between global games and beauty contest models. Finally, the notion of transparency in the context of speculative attack models has been addressed in the global games literature by Heinemann and Illing (2002) and Bannier and Heinemann (2005). There are several key di¤erences between these studies and our work. First of all, their notion of transparency is very di¤erent from ours. They interpret an 24
increase in transparency as an increase in the precision of private signals. In contrast, we follow closely the literature on transparency in beauty contest models and interpret higher transparency as a higher precision of the public signal. Secondly, in the above papers players only make decisions in the one-stage coordination game.11 In contrast, in our model investors have the option to choose the precision of the private signal they receive before playing the coordination game.
8
Conclusions
In this paper we analyze the role of endogenous information in a standard static global games model. We show that in these games investors are prone to making two types of mistakes: investing when investment is not pro…table, or not investing when investment is pro…table. We study the e¤ect that precision choices have on the incidence of these two types of mistakes in the coordination game and we carefully analyze how the value of more precise information is a¤ected by prior beliefs, the behavior of other players, and the cost of investment. We characterize conditions under which our game has a unique equilibrium and analyze several aspects of it. First, we show that in general the choice of precision made by investors in the unique equilibrium is ine¢ cient. Depending on the parameters of the model investors acquire too much or too little information. We also show that even though there are strategic complementarities in actions in the second stage, in contrast to the …ndings of Hellwig and Veldkamp (2009) for beauty contest games, the strategic complementarities in actions not always translate into strategic complementarities in information acquisition. We provide numerical examples where the best response function is non-monotonic in information choices of other investors and provide intuition for this result. Finally, we consider the e¤ect of an increase in the precision of the prior on ex-ante utility. We interpret this result as an illustration of the e¤ect of an increase in transparency on exante welfare. We …nd that, depending on the ex-ante optimism or pessimism of investors and on the cost of investment, an increase in transparency may lead to a decrease or an increase in welfare. More precisely, when investors are optimistic about the fundamentals (high mean of the prior) and the cost of investment is low, more precise public information enables better coordination on investing. The opposite is true when investors are pessimistic about fundamentals and the cost of investment is high. Hence, when the economy is considered to be relatively strong, higher transparency is welfare enhancing, while if it is considered to be relatively weak, higher transparency is welfare reducing. Our model abstracts from considerations of a strategic government who can choose the precision of public information based on its own signal. A strategic government in a global games setup has been analyzed in a standard speculative attack game by Angeletos et al (2006), Angeletos and Pavan (2011), and Goldstein et al (2011). Exploring the issue of strategic release of information in the model with endogenous information acquisition is an important direction for further research. A short coming of the global games results 11
Bannier and Heinemann (2005) present a two-stage model in which a governmental agency chooses …rst the precision of private signals, and then agents play a global game in the second stage with an exogenously given precision.
25
is that they restrict the precision of public information, relative to private information, to ensure uniqueness of equilibria. It would be interesting to investigate whether endogenous information acquisition can mitigate this critique. This issue is also left for future research.
References [1] G.-M. Angeletos, C. Hellwig, A. Pavan (2006), “Signalling in a Global Game: Coordination and Policy Traps”, Journal of Political Economy, 114(3), pp. 452-484. [2] G.-M. Angeletos and A. Pavan (2011), “Preempting Speculative Currency Attacks: Robust Predictions in a Global Game with Multiple Equilibria”, working paper. [3] G.-M. Angeletos, C. Hellwig, A. Pavan (2007), “Dynamic Global Games of Regime Change: Learning, Multiplicity and Timing of Attacks”, Econometrica, 75(3), pp. 711756. [4] G.-M. Angeletos and I. Werning (2006), “Crises and Prices: Information Aggregation, Multiplicity, and Volatility”, American Economic Review, 96(5), pp. 1720–1736. [5] A. Atkeson (2000), “Rethinking Multiple Equilibria in Macroeconomic Modeling: Comment”, NBER Macroeconomics Annual 2000, Vol. 15, pp. 162-171. [6] C. Bannier and F. Heinemann (2005), “Optimal Transparency and Risk-Taking to Avoid Currency Crises”, Journal of Institutional and Theoretical Economics, 161(3), pp.374391. [7] R. Caballero and R. Lyons (1992), “External E¤ects in U.S. Procyclical Activity”, Journal of Monetary Economics, 29, pp. 209-226. [8] H. Carlsson and E. van Damme (1993), “Global Games and Equilibrium Selection”, Econometrica, 61(5), pp. 989-1018. [9] S. Chassang (2008), “Uniform Selection in Global Games”, Journal of Economic Theory, 139, pp. 222 –241. [10] R. Cooper and A. John (1988),“Coordinating Coordination Failures in Keynesian Models”, The Quarterly Journal of Economics, 103(3), pp. 441-463. [11] A. Dasgupta (2007), “Coordination and Delay in Global Games”, Journal of Economic Theory, 134, pp. 195 –225. [12] C. Edmond (2011), “Information Manipulation, Coordination and Regime Change”, working paper, NYU Stern. [13] R. Farmer and J.-T. Guo (1994), “Real Business Cycles and the Animal Spirit Hypothesis”, Journal of Economic Theory, 63(1), pp. 42-72. [14] D. Frankel, S. Morris, and A. Pauzner (2003), “Equilibrium Selection in Global Games with Strategic Complementarities”, Journal of Economic Theory, 108, pp. 1–44. [15] I. Goldstein and A. Pauzner (2005), “Demand-Deposit Contracts and the Probability if Bank Runs”, Journal of Finance, 60(3), pp. 1293-1327. [16] R. Hall, O. Blanchard, and G. Hubbard (1986), “Market Structure and Macroeconomic Fluctuations”, Brookings papers on economic activity, vol. 1986(2), pp. 285-338. [17] R. Hall (1987), “Productivity and the Business Cycle”, Carnegie-Rochester Conf. Ser. Public Policy, 27, pp. 421-444. [18] F. Heinemann and G. Illing (2002), “Speculative Attacks: Unique Equilibrium and 26
[19] [20] [21] [22] [23] [24] [25] [26]
Transparency”, Journal of International Economics, 58(2), 2002, 429-450. C. Hellwig, Public Information, Private Information, and the Multiplicity of Equilibria in Coordination Games, Journal of Economic Theory 107, pp. 191–222. C. Hellwig and L. Veldkamp (2009), “Knowing What Others Know: Coordination Motives in Information Acquisition”, Review of Economic Studies, 76, 223–251. S. Morris and H. Shin (1998), “Unique Equilibrium in a Model of Self-Ful…lling Currency Attacks”, The American Economic Review, 88(3), pp. 587-597. S. Morris and H. Shin (2002), “Social Value of Public Information”, The American Economic Review, 92(5), pp. 1521-1534. S. Morris and H. Shin (2004), “Coordination Risk and Price of Debt”, European Economic Review, 48, pp. 133–153. M. Szkup and I. Trevino (2012), “Costly Information Acquisition in a Speculative Attack Model: Theory and Experiments”, working paper, NYU. H. Tong (2007), “Disclosure Standards and Market E¢ ciency: Evidence from Analysts’ Forecasts”, Journal of International Economics, 72, pp. 222–241. Vives, X. (2005), “Complementarities and Games: New Developments”, Journal of Economic Literature, 43(2), pp. 437-479.
A
Appendix
In this appendix we provide all the proofs and derivations that have been omitted in the main body of the paper. The appendix is divided into …ve sections. In section A:1 we prove proposition 1 which establishes the existence of a unique equilibrium in the second stage of the game. Section A:2 contains the derivation of equation (7), as well as the implications of assumptions (A1) and (A2) and the proofs of lemmas 1 and 2. Section A:3 contains the proof of the uniqueness theorem for the two-stage game. Section A:4 contains the proofs of the claims stated when analyzing global spillover e¤ects, ine¢ ciency, and strategic complementarities. Finally, section A:5 contains the proofs of the results regarding welfare and transparency.
A.1
Solving the model: t = 2
Proposition A.1 For any given , suppose that inf (supp( )) > 21 2 . Then the coordination stage has a unique equilibrium in which all investors use threshold strategies and investment is successful if and only if . Proof. Suppose that in the second stage the distribution of precision choices among investors is given by some distribution function ( ) with bounded support. The bounded support assumption follows from assumptions (A1), (A2), and claim A:1 (see also the proof of theorem 2). To show uniqueness of equilibrium for any ( ) we use iterative deletion of dominated strategies - a standard approach in this literature (see Carlsson and van Damme, 1993). Suppose …rst that investor i expects no one to invest. That is, he expects investment to be successful if and only if > 0 = 1. Then, even though no one else chooses to invest,
27
investor i will choose to invest if his signal is high enough, i.e. he will invest if Pr ( > 1jx ( i )) > T Denote by xi1 ( i ) the value of the signal that makes investor i indi¤erent between attacking and not attacking (xi1 ( i ) solves the above equation with equality). Since the above equation xi1 ( i ) and not invest is monotone in x ( i ) it follows that the investor will invest if xi otherwise. But this implies that each investor i who receives a signal with precision i will …nd it optimal to invest if the signal he receives is higher than xi1 ( i ) ; where +
i
xi1 ( i ) =
+
0 i
(
i
)1=2
+
i
1
(T )
i
Note that all investors with the same precision level will have the same xi1 ( i ). Therefore, in what follows we drop the superscript i. The fact that all investors follow a monotone strategy implies that investment will be successful whenever Z Pr (xi x1 ( i ) j ) d > 1 De…ne
1
as the value of
or
Note that when
1
R
Z
1=2 i 1=2 i
that solves the above equation with equality, i.e.: Z Pr xi x1 ( i ) j 1 d = 1 1 i
+
+
0 i
x1 ( i )
(
i
)1=2
+
i
dF > 0 when
1
1
1
= 0 and
= 1. Moreover,
=
@ @ Z1
n 1 holds. Hence, investor i who receives a signal with precision i will invest whenever his signal is higher than xn ( i ), where xn ( i ) solves Pr
n 1 jxn
28
( i) = T
or xn ( i ) =
i
+
+
n 1 i
(
i
)1=2
+
i
1
(T )
i
Note that since n 1 < n 2 it follows that 8 i 2supp( ) ; xn ( i ) < xn 1 ( i ). Consider again the CM condition. The fact that each investor i invests whenever his signal is higher than xn ( i ) implies that investment will be successful for all n where n solves the following equation: Z Pr xi xn ( i ) j n d = 1 n or
Z
1=2 i
i
+
(
+
n 1 i
i
)1=2
+
i
By the previous argument we know that there exists that since 8 i 2supp( ) ; xn ( i ) < xn 1 ( i ) we have Pr xi
xn ( i ) j
n 1
1
(T )
n
i n
< Pr xi
!!
d =
n
that solves the above equation. Note xn
1
( i) j
n 1
Suppose now that n = n 1 . Then the RHS of the CM condition is now greater than the LHS: But note that the RHS is increasing in n , while the LHS is decreasing in n . Hence it must be the case, by continuity of the LHS and RHS expressions, that n < n 1 and n 2 [0; 1]. This implies a new threshold signal for each investor, xn+1 ( i ) such that xn+1 ( i ) < xn ( i ) < xn 1 ( i ). 1 By induction we obtain decreasing sequences n n=1 and fxn ( i )g1 n=1 for each i . Since 1 n n=1 is bounded from below by 0 it follows that this sequence is convergent. This implies that fxn ( i )g1 and limn!1 xn ( i ) = n=1 is also a convergent sequence. De…ne limn!1 n = x ( i ). This concludes the iterative deletion of strictly dominated strategies. Consider now the case when the investor expects that everyone will invest regardless of their signal and apply again the iterative deletion of dominated strategies. Following exactly 1 the same argument, we obtain increasing and bounded sequences f n g1 n=1 and fxn ( i )gn=1 . De…ne limn!1 n = and limn!1 xn ( i ) = x ( i ). To show that there is a unique equilibrium we need to show that 8 i 2 supp (F ) ; x ( i ) = x ( i ) and = . Note that both limits have to satisfy simultaneously the following system of equations jx ( i )) = T ,8 x ( i) j ) = 1
Pr ( Pr (xi
i
2 supp ( )
But we can use the equation that determines x ( i ) (see equation 2 in the main text) to express it in terms of : x ( i) =
i
+ i
+ i
29
(
i
)1=2
+ i
1
(T )
Substituting x ( i ) into the CM condition and re-arranging, we get ! Z ( i + )1=2 1 + (T ) d = 1=2 1=2 1=2 i
i
i
We want to show that there exists a unique that solves the above equation. It is su¢ cient to show that the slope of the RHS is always strictly less than 1. Note that ! Z @ ( i + )1=2 1 + (T ) d 1=2 1=2 1=2 @ i i i ! Z ( i + )1=2 1 = + (T ) d 1=2 1=2 1=2 1=2 i
Z
i
i
i
1 p d 2 1=2 i 1 p 2 1=2 0 g T xi ) i ( We di¤erentiate with respect to Z @x @B i = 1f > g T @ i @ i
1
(1
1 2
1 2
T) > 0
(
) d
i: 1 2
i
+
1 2
1 2
i
xi )
(
32
1 2
i
(
xi )
1 2
1 2
(
) d
Using the PI condition, the fact that
i xi + i+
1
( i+
)
!!
1 2
normal distribution we arrive at the expression: 0 1 p i @B 1 1 i @x A qi = i @ i 2 i+ 2 i+
1
= T , and by properties of the
(1
T) > 0
i
Furthermore, from the above expression we can …nd an upper bound to this derivative: 1 0 p i @B 1 i 1 A @x qi (1 T ) = i1 + @ i 2 + i 2 i i p 1 1 1 i i 2 2 i+ p 1 1=2 4 i i+ p 1 1=2 4 0 0+
Claim A.2 lim
i
@B i !1 @ i
=0
Proof. Recall that i
1 @B = @ i 2 Note that
(
1
1 i
p i
0
@x qi
i
+
T )) is independent of 0
(1
lim
i
Hence it follows that 1 !1 2
lim i
1 i
p
i
i+
!1
2
i
while 1
i
i+
2
@x qi
A
i+
2
@x qi
0
1
i
i+ i
A
1 A
1
(1
T)
1
1
(1
T) = 0
Claim A.3 Suppose that assumption (A1) holds. Then 9 < 1 such that 8 @2Bi < 0. @ 2 i
33
we have
Proof. The proof of this result is lengthy and not illuminating and can be obtained from the authors by request. It requires showing that as increases several terms that include endogenous variables like and x converge to zero. assumption (A1) is su¢ cient for that result. 1. B i ( i ; ) is strictly increasing in
Lemma A.2 2. lim
i !1
3. For
@B i @ i
= 0; 2
i
i;
i
> ; @@ B2 < 0. i
Proof. Follows from claims A:1, A:2, and A:3. Claim A.4 9 < 1 such that investor i will never …nd it optimal to choose i
1=2
1 Proof. Note that from claim A:1, @B 0 @ i 4 (A2) lim i !1 C 0 ( i ) = 1. Hence there exists
@B i @ i
p
C 0 ( i) < 0
It follows that no investor will ever choose Claim A.5 lim
!1
C 0 ( 0) > 0
i= 0
i
> .
B2i ( ; ) < 1
Proof. We take the derivative of B i ( i ; ) with respect to : !! @B i ( i ; ) @ xi 1=2 = 1 1=2 @ @ i @B i ( i ; ) j @
@ @
=
i=
x
1
@ @ Since x
=
(
=
+
3=2
1 2
1 2
3=2
2
1 1=2
)+ lim
!1
p
1
+
1=2
1
(
1(
1=2
q
1 +
))
(T ), it follows that x 1=2
34
!
1=2
1=2
We show in claim A:8 that 1 2
> .
. On the other hand, by assumption < 1 such that 8 i > 0+
On the other hand we know that @B i j @ i
i
=1
T
(T )
!
Moreover lim
!
1=2 1=2
!1
=
!
T
1=2
1=2
Finally, @ !1 @
lim Therefore,
=0
lim B2i ( ; ) = 0 !1
A.3
Uniqueness of the Symmetric Equilibrium
Theorem A.1 There exists
< 1 such that if
> , then there is a unique equilibrium in
the information acquisition game, that is there is a unique
2 [ ; 1) that solves
=
i(
).
Proof. Note that the best response function is de…ned by investor i’s …rst-order condition @ i B ( @ i
i
@ C( @ i
( ); )
( )) = 0
i
Then the derivative of the best-response function is: @
i
( )
@
@
=
@2 Bi @( i )2
(
@2 i@ i
@
@ i@
Bi (
i
( ); )=
1 2
1 i
1 2 If
p i
0
i
+ 2
1=2
i
( ); ) @2 C @( i )2
( ); )
where 2
Bi (
( +
@x qi )
2
h
(
i
1 A
i+ i
(xi
( ))
1
) (x
(1
T) )
1 1=2
(
1(
))
i
= then xi = x and the above expression is necessarily positive. Let E = f j = i ( )g be the set of all symmetric equilibrium precision choices. Then @ ( ) the numerator of @i is positive for all 2 E since i = . Under assumption (A1) we know that the denominator is negative and hence it follows that i
@
i
@ Note that 8 2 E:
@
i
@
( )
j
i=
( )
> 0; 8 2 E @2 Bi ( i; @ i@ @2 Bi ( i ( @( i )2
35
)j
i=
); )j
i=
since
@2 C @( i )2
( i ) > 0 by convexity of the cost function (assumption (A2)). De…ne 1 2
N
1
p
i
0
@x qi
i
i+
Then @
( )
i
@
N
or @
i
( )
h
@
h
)
2
1=2 (
i
1
(x
+
3 + 2 ( +
)
2( +
1=2 2
2( +
)
)
(x
i )
!1
)
2( +
)
(x
where the last expression follows from the fact that lim clude that @ i( ) j i= 0 lim !1 @ By continuity of that 8
@
i(
@
we have
)
j
@
it follows that for each ( ;
i= i(
@
)
j
i=
) 1 1=2
1(
(
))
)=0 1 1
(T ))
n: De…ne a sequence f n ; n g1 ; n ), ( n ; n ) > n. Since n=1 such that at ( h i f n ; n g1 [ ; ] ; , which is compact, there exists a subsequence f nk ; nk g1 n=1 k=1 h i nk nk such that ( ; ) > k and it converges to ( ; ) 2 [ ; ] ; . As k ! 1, nk nk nk nk ! , ! , and ( ; ) ! 1, which implies that 9 ( ; ) such that for all h i @ i( ) ; , there exists 1. But we argued that for any ( ; ) 2 [ ; ] 2 R, @
that ( ;
@
( )
9 < 1 such that 8 we have @i j i = < 1. This leads to a contradiction. Hence, if 0 then there is necessarily a unique symmetric equilibrium.
36
A.4
E¢ ciency and Strategic Complementarities
In this section we …rst prove that there exists an e¢ cient choice of precision. We then provide a proof of proposition 2 on e¢ ciency of equilibrium. We do it in steps. First we show that @U i is proportional to @@ . Then we determine the sign of the @@ and we prove the main @ claims: 1. A necessary condition for the unique equilibrium to be e¢ cient is that
=K(
2. For any
; T ).
; T there exists a unique
that satis…es
=K(
( );
( );
Finally, we prove the proposition on strategic complementarities. Claim A.6 There exists an e¢ cient choice of precision. Proof. The e¢ cient choice of precision, if exists, is a solution to the following problem max B i ( ; )
[
C( )
0 ;1)
The …rst derivative of the above equation is given by B1i ( ; ) + B2i ( ; ) i
C0 ( ) 1=2
p
1 From claims A:1 and A:5 we know that @B and that lim !1 B2i ( ; ) = 0. 0 @ i 4 0+ Finally, by assumption (A2) lim !1 C 0 ( ) = 1. Hence there exists b such that no > b can be solution to the above maximization problem. But this implies that we are looking for a maximum of a continuous function over a compact subset of R, hence B i ( ; ) C ( ) must attain a maximum in [ 0 ; b]. Since B i ( ; ) C ( ) is di¤erentiable it has to be the case that either the e¢ cient precision choice satis…es the …rst order condition or = 0.
Claim A.7 The derivative of investor i’s ex-ante utility function with respect to the precision choice of other investors is given by ! ! @U i @ xi 1=2 = 1=2 1=2 @ @ i Proof. Di¤erentiating the ex-ante utility yields:
@xi @
Z1
1=2 i
@U i ( i ; ) @ = @ @ ! xi 1=2 1=2
i
xi
1
1=2
1=2
!
i
@xi T @
d
!! Z1 1
37
1=2 1=2
1=2 i
xi 1=2 i
!
! 1=2 1=2
!
;T);
Using the PI condition one can show that the last two terms cancel out and hence @ @
xi
1
@U i ( i ; ) @
=
1=2 1=2
1=2
i
The above claim tells us that the e¤ect of an increase in the precision of other investors’ signals on investor i’s ex-ante utility is inversely related to @@ . The next claim identi…es the sign of this derivative. Claim A.8 Consider
@ @
:
1. If
0
q
+
where K( ;
;T) =
Proof. Recall that
1
(T ) +
p 1 +
1
(T )
is determined by 1 1=2
( )=
1=2
Using the implicit function theorem we can …nd @ @
=
1 2
3=2
+
1 2
r @ @
2
1 1=2
(
1
(T )
: 1 2
3=2
+
1(
q
1 +
(T )
))
Since 1=2 ( 11 ( )) < 0 the sign of this derivative depends only on the sign of the numerator. We now consider the case in which T 2 (0; 1). We want to …nd conditions under which the above numerator is negative, i.e. conditions under which: r 1 1 1 1 + (T ) < 0 2 3=2 2 3=2 2 2 + Using the CM condition one can show that this is the case when r 1 > (T ) + Using the CM condition we can determine when this is true. Recall that the CM condition is given by r + 1 1 ( ) = 1=2 (T ) 1=2 38
For
q
>
1 +
(T ) it must be the case that at
is strictly greater than the LHS: r 1 (T ) +
1=2
After rearranging further we get r
r
1
+
q
=
1=2 1
+
(T ) + p
(T ) >
1 +
1
r
1 +
(T ) the RHS
+
1
(T )
(T ) >
The claim follows from that last inequality. Claim A.9 There exists a unique Proof. Recall that K(
( );
s
;T) =
that satis…es
( ) ( )+
=K(
1
!
(T )
( );
; T ).
1
+p
( )+
1
(T )
We want to show that as a function of , K ( ) has unique …xed point. So …x T and . 1 1 Note that K ( ) is bounded from below by 0 if 1 > T or by p1 (T ) if 0 < T < 12 2 1 1 (T ) if T or by 1 if T < 12 . and it is bounded from above by 1 + p1 2 1 Consider the case when 1 > T . Then we know that there exists 1 such that 1 < 2 r r ( 1) ( 2) 1 1 1 1 2 2 (T ) + q (T ) + (T ) and such that > 1 + ( ) ( 2 )+ ( 1 )+ 1 1 q (T ). Since K ( ( ) ; ; T ) is continuous in it follows that there exists 2 + ( ) such that = K ( ( ); ;T) We now show that at any such the derivative of K ( ( ) ; ; T ) with respect to less than 1 implying that K ( ( ) ; ; T ) has a unique …xed point. Taking the derivative of K ( ( ) ; ; T ) with respect to we get: @K ( Consider …rst satis…es:
@ @
( ); @
and recall that
;T)
=
( ); @
;T) @ @
is a …xed point of the best response function, that is it =
where Hence
@K (
is
( ;
)
denotes the investor’s best response implicitly de…ned by the …rst-order condition. @ @
@
=
1
( ; ) @ @ ( ; ) @
@2U i @ @
=
@2U i @ 2
39
( ;
( ; ;
; )
) C 00 ( )
Consider now q
1
= K ( ( ) ; ; T ) : Using the CM condition one can show that = 1 (T ) and this implies that x ( ( ) ; ( ) ; ) = . Using this fact it
( )+ is straightforward to show that
@2U i ( ; ; @ @ @K( ( ); ;T ) j @
) = 0 and so
@ @
j
= 0. Since
=
@K(
( @
);
;T )
is always …nite it follows that = 0 and so at any …xed point of K ( ) its = derivative is strictly less than one, implying that K ( ) has a unique …xed point. The same argument holds when 0 < T < 21 . This concludes the proof. Claim A.10 = K ( ( ) ; ; T ) is a necessary (but not su¢ cient) condition for e¢ ciency of the unique equilibrium. Proof. The e¢ cient precision choice is the solution to the following maximization problem: max B i ( ; )
2[
C( )
0 ;1)
By claim A:6, the solution to the above problem (call it ) always exists and it either satis…es = 0 , or it solves the following …rst-order condition: B1i (
) + B2i (
;
;
C0 (
)
)=0
Therefore, if the unique equilibrium is to be e¢ cient we require unique equilibrium investors always choose a higher precision than equilibrium precision ( ) solves: B1i (
( );
( ))
C0 (
( )) = 0
If = K ( ( ) ; ; T ) then by claim A:8 we know that 0, implying that ( ) solves: B1i (
( );
( )) + B2i (
( );
> 0 , since in the 0 . Recall that the
@ @
( ))
= 0 and so B2i ( C0 (
( );
( )) = (13)
( )) = 0
Therefore, the very strict condition that we have imposed on is required for ( ) to be a critical point of the maximization problem above, however this condition does not guarantee that = ( ). Hence, we conclude that the unique symmetric equilibrium of the game is generically ine¢ cient. Claim A.11 For T =
1 2
the unique solution to
= K( ( );
; T ) is
= 21 .
Proof. Recall that K( ( );
;T)
s
( ) ( )+
1
40
!
(T )
+p
1 ( )+
1
(T )
Setting T = K
1 2
yields ( );
s
1 ; 2 =
Proposition A.2 Let K( ( );
;T)
(0) =
s
( ) ( )+
!
1 2
1
1 2
( ) ( )+
1
!
(T )
+p
+p
1
1
( )+
1
1
( )+
1. A necessary condition for the unique equilibrium to be e¢ cient is that K( ( ); ; T ). 2. For any pair ( ; T ), such that T 6= K( ( ); ; T ) 3. For T =
1 2
the unique solution to
1 , 2
there exists a unique
= K( ( );
; T ) is
1 2
(T ) solves
that solves
= =
= 21 .
It follows that generically the unique equilibrium is ine¢ cient. Proof. Follows immediately from the above claims. Lemma A.3 Consider the information acquisition game. 1. If the cost of investment is high ( T > 12 ) and investors are ex-ante optimistic ( then the game exhibits global positive spillovers
< 12 )
2. If the cost of investment is low ( T < 21 ) and investors are ex-ante pessimistic ( then the game exhibits global negative spillovers
> 12 )
3. If cost of investment is “neutral” (T = 12 ) and investors are neither optimistic nor pessimistic ( = 21 ) then the game exhibits no spillover e¤ects. Proof. Recall from claim A:5 that @U i ( i ; ) = @
@ @
xi
1
1=2 i
!!
1=2 1=2
!
Suppose that T > 12 and < 12 . Then, K( ( ); ; T ) > 21 , hence by claim A:8, @@ < 0, i regardless of the other parameters and equilibrium play. This implies that @U @( i ; ) > 0, i.e. the game exhibits global positive spillovers. Using the same argument one can show that when T < 21 and > 12 the game exhibits 1 1 global negative spillovers, and when T = 2 and = 2 , then the game exhibits no spillover e¤ects. 41
Proposition A.3 Suppose that one of the following holds: 1 ) 2
1. The costs of investment is high (T > ( < 12 ) or
and the investors are ex-ante pessimistic
2. The costs of investment is low (T < 12 ) and the investors are ex-ante optimistic ( Then there are strategic complementarities in the information acquisition game. 3. There exist pairs of (T;
) such that there is lack of strategic complementarities.
Proof. We consider the cross-partial derivative of U i ( i ; ) with respect to i and . 2 0 1 3 p i @ 41 1 @U ( i ; ) i 1 @x A qi = (1 T ) 5 i @ i@ @ 2 i+ 2 i+ i 8 2 0 139 p < = @ 4 @ xi 1 i 1 A5 q (1 T ) = i1 :@ ; 2 i+ 2 i+ i
Using the fact that
@ 1 @x
x
x
=
x
1 2
Then i
@U ( i ; ) == @ i@
1 2
1 i
p i
De…ne K2 =
1 2
1 i
i
1
+
p i
(1
i
1
+
0
@x qi
T)
(1
Recall that xi
=
i+
(
i
@ @
=
1 2
3=2
h
(
i
+
1
)+
( +
42
(
1(
)
)1=2 ))
1 A
i+ i
1 1 2
1
1 1=2
2
1
)+
(
i
@x qi
T)
@ @
A
i+
2
0
1
!
(T )
i (T )
(xi
)
> 12 )
so @U i ( i ; ) = K2 @ i@ 1 K2 2
=
i
(
1
)+ (
i
i 3=2
+
+ i
"
(
1
+
i
)
1 2
1
)+ (
i
1
+
)
1 2
!
1 2
(T )
#0
(T ) @
3=2
h
(
)+
1 )1=2
( + 1
1=2
(
1
)+
(
1(
))
1
(T )
1 ( +
)
1=2
1 1=2
1(
(
))
i (T )
1 A
where K2 < 0. For increasing di¤erences to arise the above derivative should be positive. In general, it is not possible to determine the sign of this derivative for general parameters. h i This is due to 1 1 the fact that when i 6= it is possible that ( ) + ( + )1=2 (T ) has a di¤erent i h i 1 1 sign than ( ) + ( + 1 )1=2 1 (T ) . However, note that if T and then 2 2 1 2
@U i ( i ; ) @ i@ 1 and T 2
and it follows that
0 so that there are strategic complementarities in i
1 we have @U@ (i @i ; ) 0. This proves the …rst precision choices. Similarly, if 2 two parts of the proposition. For the last part of the proposition, assume that investor i chose precision i and all other investors chose a symmetric precision . Then investor i’s threshold and the threshold of all other investors are given by p + i+ 1 xi ( )+ (T ) (14) = i i p + + 1 x = ( )+ (T ) (15)
From the CM condition we have 1 1=2
( )=
1=2 1=2
= We plug this in equation (15) " " 1=2 + x = So from equation(15) we have " + x = 1=2
1
1
"
1
r
( )
( )
r
r
+
43
+
1
r
( )
+
1
1
(T )
+
1
##
(T )
#
(T ) +
+
p
#
(T )
p
+
+
1
1
(T )
(T )
Which simpli…es to x We now …nd a
=
p
+
1
, in terms of T , for which x x
+
(T ) +
1
1=2
( )
= 0:
= 0 1=2
() We now …x T , , and , we get p
(T )
r
1=2 1
+
1
+
. Using the CM condition and substituting the above expression for
1=2 1=2
p
=
(T )
p
1
+
(T ) =
1=2
From above we see that for the …xed parameters T , , and 1=2 1 p = (T ) . For such a we have that + =
1
(T ) p
+
1
there exists a
(T ) such that
1 +
Substituting this expression into equation (14) we get p i+ 1 xi = (T ) 1 i
p + pi +
such that x = 0. Since is continuous and Set T > 12 . Then we can …nd decreasing in , for a very small increase in , then x > 0. But for T > 12 and < 0. By continuity, a small change in will such that x = 0, if i > , then xi < 0. Therefore, (x ) (xi ) < 0 implying a lack of not change the sign of xi strategic complementarities in this region of parameters.
A.5 A.5.1
Welfare and Transparency Trade-o¤ between public and private information
Proposition A.4 Suppose that one of the following holds: 1. The cost of investment is high (T > ( < 21 p1 ) or,
1 ) 2
and investors are ex-ante very pessimistic
2. The cost of investment is low (T < 12 ) and investors are ex-ante very optimistic ( 1 + p1 ) 2 Then private and public information are substitutes. 44
>
Proof. The strategy of the proof is simple: we calculate the cross-partial derivative of U i and …nd conditions that ensure it is negative. This is a tedious but otherwise straightforward exercise. i Di¤erentiating @U with respect to we get: @ i 1 0 p 1 2 i ( i ) @ U 1 i 1 2 A @x qi (1 T ) = i1 + @ i@ 2 + ( + ) i 2 i i i 1 ! 0 p 1 1 xi ) i 1 A @xi + i (xi @x qi (1 T ) i i+ 2 @ 2 ( i+ ) i+ 2 i+ i i
Rearranging yields 2
@ U @ i@
i
=
1 2
1
p
i 1 2
i
( (
0
+
) ) i+
i
Note that
@ @ i
1 2( (
xi
i
1 i
+
@xi @
i+
1 3=2
)
)+(
+ (
(T )
i
2
)
1 2
i
xi
( (
) ) i+
i
(xi
)
( i+
)
i+
(T )
1
T)
@xi + @ 2
+
=
(1
1 ( i+
)2
i
(xi ( i+
) )
(16)
>0
is given by
)@ @
+
(2
+
i)
i 2
1
A
i
i i+
2
i+
i
1 2
@ @ i while
@x qi
i
1
2 i
@ @
+
2 2 i
(
(
)+
+
2 i( i+ i+2 ) + 2p 4 i i+
i
1
)
1 2
1
!
(T )
(T )
i:
1 2
This is true when ( ) @@ > 0 and ( )2 > 1 . Since it is not possible to calculate ) @@ analytically, we only require this term to be positive and focus the magnitude of ( on the latter condition. Note …rst that T > 21 and < 12 implies that ( ) > 0. Moreover, by lemma A:3 presented below, we know that under these conditions we also have @@ > 0. The opposite is true when T < 21 and > 21 ; in that case we have @@ < 0 and ( ) < 0. 2 1 To investigate when ( ) > , we look at two cases: ( ) > p1 or ( )< p1 .
46
Consider …rst ( ) > p1 . By the CM condition for the case with symmetric precision choices (degenerate ), this implies that ! r 1 2 + 1 1 1 ( ) (T ) > p 1
r
( )
p p
>
+
1
r
+
+
p (T ) > p ! 1
(T )
We now …nd a condition, using the CM condition, such that the above inequality holds. Recall that under the uniqueness condition we have LHS of the CM condition decreasing in q . So for
p p
>
1=2
>
1=2
+
+
p p
r
r
+
1
(T ) , we need the following to hold:
+
+
1
1
!
p p
(T )
+
r
+
(T )
Simplifying we get: 1=2
0 if 1
1
( )
r
+
1
!
(T ) 48
11 + 2
r
1
+
(T ) > 0
implying that
r
> q
1
+
1 2
+ +
1
!
(T )
+
+ 2 1 (T ) we use the CM condition again. Since the To determine when > + CM condition is decreasing in it has to be true that ! r r r 1 + + 12 + + + 1 1 1 2 (T ) (T ) > 1=2 (T ) 1=2 + +
After rearranging we get < De…ne K
( ; ;T)
r
+
r
1 2
+ +
+
1 2
1
+ +
!
(T )
1
+
1 1 p 2 + !
(T )
+
1
1 1 p 2 +
(T )
1
(T )
= Then if < K ( ; ; T ) we have @@ > 0. The same argument implies that if @ @ K ( ; ; T ) then @ = 0 and if > K ( ; ; T ) then @ < 0. This completes the proof.
49
