ESADE WORKING PAPER Nº 264 April 2017

The Social Value of Information with an Endogenous Public Signal

Anna Bayona

Supply Function Competition, Private Information, and Market Power: A Laboratory Study

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The Social Value of Information with an EndogenousPublic Signal

Anna Bayona∗

April 2017

Abstract

This paper analyses the equilibrium and welfare properties of an economy characterizedby uncertainty and payoff externalities using a general model that nests several applications.Agents receive a private signal and an endogenous public signal, which is a noisy aggregate ofindividual actions and causes an information externality. Agents in equilibrium underweightprivate information for a larger payoff parameter region in relation to when public informationis exogenous. In addition, the socially optimal endogenous degree of coordination is lowerthan the socially optimal exogenous degree of coordination. The welfare effect of increasingthe precision of the noise in the public signal has the same sign with endogenous or exogenouspublic information, but its magnitude differs. The social value of private information maybe overturned in relation to when public information is exogenous: from positive to negativeif agents in equilibrium coordinate more than is implied by the socially optimal exogenousdegree of coordination, and the opposite if they coordinate less.

Keywords : payoff externalities; information externalities; rational expectations equilib-rium; private information; welfare analysis.

JEL codes: D62; D82; G14.

∗ESADE Business School. Corresponding e-mail address: anna.bayona@esade.edu. I thank the editor, theassociate editor and two anonymous referees for providing very useful comments which have greatly improvedthe paper. I am grateful to my thesis advisor, Xavier Vives, for the guidance and feedback. I would also liketo thank Jose Apesteguia, Ariadna Dumitrescu, Emre Ekinci, Sebastian Harris, Angel L. Lopez, CarolinaManzano, Margaret Meyer, Manuel Mueller-Frank, Rosemarie Nagel, Morten Olsen and Alessandro Pavanfor constructive comments.

1 Introduction

In today’s information age, systems and platforms that collect, aggregate, and process dis-persed data about economic agents’ strategies in nearly real-time are widespread and broadlyaccessible at a negligible cost. At the same time, agents’ strategies are dependent on thepublic information that these systems aggregate.1 In other words, public information is en-dogenous and agents increasingly use strategies that are contingent on noisy statistics thataggregate the actions of other agents.

In this context it is relevant to ask whether providing more precise information is sociallyvaluable. This concerns policymakers in two respects: first, under what conditions doesgreater transparency improves social welfare? Second, is it in fact desirable to facilitatetools that improve the private observation of local market conditions? In this paper, I studythe introduction of an endogenous public signal into economies with uncertainty and payoff-externalities. This brings new insights to the existing literature and enables a clear distinctionto be made between environments with endogenous and exogenous public information.

I present two motivating arguments which compare these two types of public information.The first is related to whether public information either monitors fundamentals directly(exogenous), or whether it tracks market activity (endogenous). With the prevalence ofbig data, there are nowadays many indicators of economic activity that provide real-timeforecasts to predict the present (see Choi and Varian (2012)), and which therefore constitutean endogenous source of public information. In these environments, agents use schedules thatare contingent on the noisy contemporaneous public forecast about the aggregate action.For example, firms that make capacity choices are often uncertain about the shocks thataffect their demand. The increasing prevalence of online platforms facilitates the aggregationof information about the strategies of rivals. In turn, firms post schedules in which thecapacity they offer is contingent on noisy public information about the average capacityoffered in the market and on their own private information. The second motivation relates towhether the interaction among agents occurs in segmented markets, where public informationis exogenous (e.g. in over-the-counter markets), or in centralized markets that facilitateinformation aggregation, where public information is endogenous (e.g. in exchanges).2

Three research questions emerge. First, how do agents make optimal use of the differenttypes of information sources in equilibrium? Second, what are the welfare properties ofthe equilibrium strategy? Third, what are the effects on welfare of varying the precisionof public and private information? The main finding is that endogenous public informationcan overturn the social value of private information in relation to when public information

1See, for example, McAfee et al. (2012) and Einav and Levin (2014).2See, for example, Avdjiev, McGuire and Tarashev (2012).

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is exogenous. The paper finds that the interaction of information and payoff externalitiescharacterize the direction and size of the effects that endogenous public information has onthe social value of private information.

I address these questions by using a static model of the linear-quadratic Gaussian familybased on the payoff structure of Angeletos and Pavan (2007). The economy is populated by alarge number of agents. Each agent’s utility function depends on fundamentals and exhibitspayoff externalities: strategic complementarity or substitutability. Agents have access to aprivate signal and to an endogenous public signal, which is a noisy aggregate of individualstrategies. The equilibrium concept used is the (linear) rational expectations equilibrium.The welfare benchmark is the ex ante utility of a team of agents subject to the constraintthat private information cannot be transferred from one agent to another, or to a center.The model is general and can incorporate a number of applications, such as competition ina homogeneous product market; competition à la Bertrand or à la Cournot; and the beautycontest.

The unique linear equilibrium strategy exhibits the following property. When publicinformation aggregates the economy’s dispersed information, the precision of the endogenouspublic signal depends quadratically on the agents’ response to private information. Agentsin equilibrium do not take into account that their response to private information affectsthe informativeness of the endogenous public signal. Hence, endogenous public informationalways generates an information externality. In addition, there are a further two possibletypes of externalities: first, with full information agents may over- or under- respond to thefundamental; and second, with incomplete information, agents may perceive a higher or lowerdegree of coordination in relation to that perceived by the welfare planner. The combinationof these characterizes whether agents over-, under-, or equally weight private information inrelation to the efficient strategy. For example, in competition à la Bertrand with productdifferentiation, firms may underweight private information in relation to the efficient level,thereby overturning the conclusions with exogenous public information. As a consequence,the socially optimal endogenous degree of coordination is lower than the socially optimalexogenous degree of coordination. In addition, the characteristics of the payoff structurealone are not sufficient to ensure efficiency. The properties of the information structure needto be taken into account.

The wedge between equilibrium and efficient strategies generates a welfare loss consistingof various components, which jointly influence the social value of information. The welfareloss consists of a first-order welfare effect, which is related to the sign and magnitude of thefull-information inefficiency. It also consists of two second order effects due to incompleteinformation generated by the noise in private and public signals, dispersion and volatility,

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respectively. I find the necessary and sufficient conditions which determine the social valueof information with an endogenous public signal.

The social value of private information with an endogenous public signal may changesign and magnitude in relation to when public information is exogenous. This is becauseits social value is related to whether public information is beneficial or detrimental (sincechanging the precision of the private signal increases the overall precision of the endogenouspublic signal, which has an additional welfare effect). The social value of private informationwith an endogenous public signal depends crucially on how the three potential inefficienciesof equilibrium strategy combine.

In contrast, the introduction of an endogenous public signal affects only the magnitudeeffect of the social value of public information. The reason for this is that higher precisionof the noise in the endogenous public signal always increases the overall precision of theendogenous public signal. This is because the positive accuracy effect (the signal is more in-formative) dominates the negative crowding-out effect (which reduces the weight that agentsplace on the private signal thus making the endogenous public signal less informative).

Combining the social value of private and (endogenous) public information, I characterizeall the possible sign combinations of the various welfare effects in relation to the primitives ofthe model. First, I consider the benchmark case where there are no inefficiencies with eithercomplete or incomplete information (e.g. firms that compete in a homogeneous productmarket with total surplus as a welfare benchmark). In this case, with both exogenous andendogenous public information, more precise public or private information is beneficial forwelfare. However, the magnitude of the change in the social value of private information,that results from an increase in the precision of the private signal, is larger with endogenousthan with exogenous public information.

To understand the intuition of the main result, consider an economy, such as monopolisticcompetition à la Bertrand, in which the full information inefficiency is small and firms slightlyunder-respond to the fundamental. With incomplete and exogenous information, firms givetoo much weight to private information since they coordinate less than is efficient. Hence,more precise public information is always welfare improving since it helps agents reduce theexcess equilibrium weight on private information. This is reflected in equilibrium welfarelosses since more precise public information decreases dispersion and the first-order effectmore than the potential increase in volatility. However, more precise private informationmay in fact reduce social welfare. Due to the endogenous public signal, more precise privateinformation will have an additional effect, since it fosters a greater degree of coordination,which is beneficial for welfare. This is because a higher precision of the private signal increasesthe overall precision of the endogenous public signal. Thus agents give greater weight to the

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public signal, which further reduces dispersion. Since dispersion has a large relative weightin welfare loss, the overall effect of the endogenous public signal on the social value of privateinformation is positive. The opposite occurs in economies where agents give too little weightto private information.

In economies where agents with full information over-respond to the fundamental, thenincreasing the precision of either public or private information increases equilibrium welfareloss through the first order effect. Hence, the benefits of more precise either public or privateinformation become smaller in relation to economies that are efficient with full information.At the limit, when the full information inefficiency is very large, more precise private andpublic information always diminish welfare.

Considering an exogenous information structure, Ui and Yoshizawa (2015) categorize thedifferent welfare effects of information into eight types of games. I study how endogenouspublic information changes their categorization. There are three differences in the socialvalue of information due to the endogenous public signal. First, the payoff parameters alonedo not always characterize the social value of information since the characteristics of theinformation structure matter. Second, some of the cutoffs between types of game change(those that depend on the parameters related to the social value of private information).Third, there is a change in cutoffs within the types of game where the sign of the social valueof information depends on the ratio of public to private precisions. The social value of privateinformation may be overturned from positive to negative if agents in equilibrium coordinatemore than is efficient, and the opposite may occur if they coordinate less.

The paper is organized as follows. Section 2 discusses the related literature. Section 3describes the model. Section 4 studies the equilibrium and efficient strategies. Section 5presents the results of the social value of information. Section 6 illustrates the main resultsof the paper in two applications. Section 7 concludes. Proofs are derived in the Appendix.

2 Related Literature

This paper is the first to show how payoff inefficiencies combine with the information exter-nality through studying the social value of information in a general type of quadratic payoffstructures with incomplete information. There are two relevant strands of literature. Thefirst concerns the efficiency in the use and the social value of exogenous information sources.The second deals with the role of endogenous information structures in a variety of settings.

The social value of information with exogenous information in linear-quadratic-normalgames has been studied previously by Morris and Shin (2002), in the beauty contest; byHellwig (2005), in a business cycle framework; by Clark and Polborn (2006), in a binary

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choice context; by Angeletos and Pavan (2007), henceforth AP, in a general model whichencompasses many of the previous applications; and by Myatt and Wallace (2015, 2016), inmodels of differentiated product oligopolies with rich industry features. Ui and Yoshizawa(2015), henceforth UY, categorize all of these games mentioned previously into types accord-ing to the properties regarding the social value of information. In relation to this literature,I first show how payoff and information externalities combine in the use of information. Sec-ond, I show how introducing an endogenous public signal may modify the sign of the socialvalue of private information. In consequence, this changes both the threshold value of payoffparameters that define the type of game and also the ratio of public to private precisionsthat determines the sign of the social value of information.

The information environment is reminiscent of the literature of social learning such asin Banerjee (1992) and Bikhchandani et al. (1992), which study the welfare effects of pureinformation externalities where agents respond insufficiently to private signals. Papers inthe rational expectations tradition are also related, which typically consider static modelswhere the public signal has both information and allocation roles and agent’s strategies areschedules (e.g. Grossman and Stiglitz 1980; Diamond and Verrechia 1981). In contrast, Iconcentrate on environments where the endogenous public signal has an information rolebut not an allocation role since it does not directly affect an agent’s payoff function. Theframework that I use in this paper proposes the simplest model that combines elements ofrational expectations, herding, and games of strategic complementarity and substitutability.

The results presented here are related to several papers that focus on the implicationsfor policy and that also consider an endogenous public signal. Morris and Shin (2005) werethe first to show that public signals are less informative when central bankers disclose theirforecasts than when they do not disclose anything. Angeletos and Pavan (2009) also considerthat public information is endogenous and focus on the optimal design of contingent taxation.Amador and Weill (2010) find that more precise public information can be detrimental ina model with no payoff externalities, which contrasts with the results of this paper. Theirmodel contains an additional source of strategic complementarity, generated by the wayagents learn from prices, which is responsible for the negative welfare result. Bond andGoldstein (2015) analyze the implications of governments relying on prices to guide theirdecisions. This paper also complements the results of Colombo et al. (2014) who study theeffects of costly endogenous private information. They find that the crowding-out effectsof public information on private information may overturn the magnitude and sign of thesocial value of public information in relation to when private information is exogenous. Incontrast, this paper focuses on the effects of a costless endogenous public signal on the useof information and on the social value of private information.

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3 The model

I present a static linear-quadratic-Gaussian model with an information structure that includesboth a private signal and an endogenous public signal.

Preferences The economy is composed of a continuum of agents distributed uniformlyover the unit interval and indexed by i. Simultaneously, each agent chooses a strategy, ki,to maximize the utility function U(ki, K, σk, θ), where θ represents the fundamentals thatexogenously affect an agent’s utility function; K is the average of agents’ actions given byK =

´kidi; and σ2

k is the variance of the agents’ actions across the population defined asσ2k =´

(ki − K)2di. I assume that U is a quadratic polynomial in the variables ki, K, σ, θthat is symmetric across agents. The dispersion in actions has only second-order effects, i.e.Ukσ = UKσ = Uθσ = 0 and Uσ(ki, K, 0, θ) = 0 for all (k,K, θ). Furthermore, the utilityfunction is concave with respect to ki. Without loss of generality, let α = −UkK

Ukkbe the

degree of strategic complementarity.3 Actions are strategic complements whenever α > 0,strategic substitutes whenever α < 0, and strategically independent whenever α = 0. Toensure uniqueness, assume that α < 1. Note that the payoff function described above admitspayoff externalities with respect to the mean action whenever UK 6= 0 and with respect tothe dispersion of actions whenever Uσ 6= 0.

Examples of preferences that can be modeled with this framework from the literatureare as follows. First consider an economy that is composed of a continuum of households,each consisting of producers and consumers, who make production choices with quadraticproduction costs in a homogeneous product market. Suppose that agents are uncertainabout common fundamentals, θ, which represent the intercept of the marginal cost. Vives(1988) and AP show that households’ utility function can be written as U(k,K, σk, θ) =

(δ − βK)ki + βK2

2− θki − λk2i

2, such that β + λ > 0 and 2β + λ > 0. Hence α = −β

λ.

A second application is the Keynesian beauty contest formalized by Morris and Shin(2002), where agents are not only concerned about predicting fundamentals but also aboutoutguessing the likely actions of others. This has been used as a metaphor on how financialmarkets work. An agent’s utility function can be expressed as U(k,K, σk, θ) = −(1− r)(ki−θ)2 − r(ki −K)2 + rσ2

k, where α = r ∈ (0, 1).Two other applications are developed in Section 6, which focus on monopolistic competi-

tion à la Bertrand and à la Cournot with product differentiation.

Uncertainty and Information The common demand shock, θ, follows a normal distri-bution with mean θ and variance σ2

θ (i.e. θ ∼ N(0, σ2θ)). To form expectations about the

3AP refer to α as the equilibrium degree of coordination.

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common shock, each agent receives two types of information. First, agents have access toa noisy private signal about the common shock, xi = θ + ξi, with the noise distributed asξi ∼ N(0, σ2

ξ ). Second, agents receive a public signal about the aggregate action, whichemerges within the economy, and therefore is endogenous (see motivation in the introduc-tion). The endogenous public signal is given by

w =

ˆkidi+ v, (1)

where the noise in the endogenous public signal is normally distributed with v ∼ N(0, σv2),

which reflects the fact that the aggregation process is imperfect.4 Fundamentals and errorterms of private and public signals are mutually independent and identically distributedacross agents. For ease of interpretation, I shall often work with the precision of a randomvariable, which is the inverse of its variance. The precision for any random variable, e.g. S,with a non-zero variance is given by τs = 1/σ2

s . Throughout the text, τξ is the precision ofthe noise in the private signal, τv the precision of the noise in the public signal, and τθ theprecision of fundamentals.This modeling approach has the advantage of introducing an endogenous public signal withminimal modifications to the standard benchmark of exogenous public information, in whichthe public signal is typically defined as y = θ+ ε with ε ∼ N(0, σ2

ε). Therefore, comparisonsbetween the two environments are straightforward.

Equilibrium Definition Agent i’s strategy k(xi, .) is a mapping from the signal space tothe action space of each agent. A rational expectations equilibrium (REE) satisfies:

1. Each agent i chooses a strategy k(xi, w) that maximizes expected utility E[u|xi, w] con-ditional on the information set {xi, ω} and on knowing w(θ, v), as well as the underlyingdistributions of random variables.

2. The endogenous public signal is formed so that w =´kidi+ v.

The first condition requires that each agent maximizes expected utility given his informationset, which involves extracting from w all its informational content. The second conditionenforces the required consistency of rational expectations. This implies that the endogenouspublic signal both aggregates agents’ dispersed actions and at the same time depends onthe information which these provide. In other words, the conjectured strategy must be self-fulfilling (fixed point of conjectured strategies to actual strategies). Unless the public signal is

4This is not the only way in which the public signal can be made endogenous. Refer to the Section 2 fora discussion on the alternative formulations used in the literature.

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infinitely precise, the REE will be partially revealing.5 Given the linear-quadratic-Gaussianstructure of the model, I restrict equilibria in the class of linear strategies, as is usual in theliterature.

Discussion of the Equilibrium Concept The REE concept may be problematic; itslimitations for Bayesian games have been discussed extensively in the literature. Specifically,in the set-up of this paper, strategies and the endogenous public statistic are determinedsimultaneously. In essence, this is the same problem that has been studied in the financialeconomics literature where agents use the informational content of the price (the endogenouspublic statistic) but they do not consider the influence of their strategies on market clearing(in the model considered here, w, is not part of an agent’s utility function).6 This equilibriumconcept is questionable when the number of agents is finite. However, a noisy REE with acontinuum of agents, as in this paper, can be shown to overcome many of its problems. Fora further discussion of this issue refer to Hellwig (1980), Kovalenkov and Vives (2014), andVives (2014). An alternative approach would be to consider a dynamic model. The mainadvantage of the static model over a dynamic one, is its greater simplicity which allowsthe main insights to be demonstrated more clearly. A similar approach has been used byAngeletos and Werning (2006) in the context of global games where individuals observe oneanother’s actions.

Vives (2014) discusses the equivalence between an implementable REE and a BayesianNash equilibrium of a competitive game where each agent submits a schedule and has ademand or supply function interpretation (see Klemperer and Meyer 1989). In the contextof this paper, the REE is implementable by considering a game where each agent decides aschedule by positing a reaction function, which relates an agent’s strategy to the realizationof the endogenous public signal. The timing of such game can be specified as follows. First,fundamentals and errors terms are drawn. Second, agents (who do not observe the funda-mentals) receive a private signal and conjecture a schedule that satisfies the maximizationcondition (1). Third, the public signal is formed and satisfies the consistency condition (2),and its precision is determined. Finally, payoffs are collected. For a further discussion on theuse of REE in market games refer to Shorish (2010).

5If τv →∞ then the REE would be fully revealing.6Refer to Section 2 for further details of this literature.

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4 Equilibrium and Efficient strategies

In this section, I present the equilibrium and efficient strategies, describe their properties,and compare them to the benchmark with exogenous public information.

4.1 Equilibrium strategy

To derive the equilibrium with an endogenous public signal, I use the REE definition fromthe previous section. Focusing on linear arbitrary strategies of the form k(xi, z) = b′ +

axi + c′E [θ | z], where a is the response to private information. Given the structure of theinformation environment, I show that the informational content of the endogenous publicsignal is equivalent to that of z = aθ+ v. Hence, a strategy can be more concisely written ask(xi, z) = b+axi+cE [θ|z]. Owing to the properties of normal distribution: E [θ|z] = τvaz+τθ θ

τθ+a2τv

and (var [θ|z])−1 = τ(a) = τθ + a2τv, the latter of which is the overall precision of theendogenous public signal.

With full information, AP show that there is a unique equilibrium which satisfies ki =

K = κ(θ) = κ0 + κ1θ for all i, where κ0 = −Uk(0,0,0,0)Ukk+UkK

and κ1 = −UkθUkk+UkK

. Using theseconsiderations, Proposition 1 spells out the equilibrium strategy when public information isendogenous. In what follows, the precisions of any exogenous noise are always non-zero andfinite.

Proposition 1

There exists a unique linear REE strategy which is given by

km(xi, z) = κ0 + κ1(γmxi + (1− γm)E [θ|z]), (2)

where the equilibrium weight to private information, γm ∈ (0, 1), is the unique real root ofthe implicit equation γ =

(1−α)τξ(1−α)τξ+τθ+κ21γ

2τv.

Let us define the precision of the endogenous public signal at the equilibrium strategyτm = τ(κ1γ

m) = τθ+(κ1γm)2τv. Endogenous public information modifies the precision of the

public signal in relation to the case of exogenous public information since as agents respondmore to private information, the more informative the public signal becomes. This feedbackeffect between an individual strategy and the precision of the endogenous public signal drivesthe main results of the paper. The precision of the endogenous public signal affects the sizeof the Bayesian weight, which modifies the equilibrium weights an agent gives to public andprivate information. This effect is not present when public information is exogenous since

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the equilibrium strategy derived by AP is kexo(xi, z) = κ0 + κ1(γexoxi + (1 − γexo)y), whereγexo =

(1−α)τξ(1−α)τξ+τθ+τε

, and the overall precision of the exogenous public signal is τθ + τε.7

The following corollary states comparative statics of the equilibrium weight to privateinformation with respect to information and payoff relevant parameters.

Corollary 1

The comparative statics of the equilibrium weight given to the private signal, γm, are

∂γm

∂τξ> 0,

∂γm

∂τθ< 0,

∂γm

∂τv< 0 and

∂γm

∂α< 0. (3)

When the precision of the noise in the private signal tends to zero, agents do not giveany weight to this signal. As the precision of the private signal increases, agents increasethe weight given to the private signal. As τθ, τv increase, the public signal becomes moreinformative and agents shift their weight from the private to the public signal. At the limit,when the noise in the public signal is infinitely precise, agents give all the weight to thepublic signal and the equilibrium collapses. As in the case of exogenous public information,if actions were strategically independent (α = 0), the weights given to public and privateinformation would correspond to Bayesian weights according to the relative precisions of thesignals. When actions are strategic complements (α > 0) agents place greater weight on thepublic signal since this helps to better predict the aggregate action. The converse happenswhen actions are strategic substitutes (α < 0). These comparative statics have importantimplications on how τm changes with the information parameters as shown below.

Corollary 2

The effects of changing τξ, τθ, τv on the overall precision of the public signal at the equilibriumstrategy, τm, are

dτm

dτξ> 0,

dτm

dτθ> 0,

dτm

dτv> 0. (4)

Increasing τξ makes agents place more weight on the private signal, γm, which increasesthe overall informativeness of the endogenous public signal, τm.

7The paper closely follows AP’s notation, except from the weight to private information. In this paper, theweight to private information with an endogenous public signal is γm, while the weight to private informationwith an exogenous public signal is γexo. AP use 1 − γ to denote the weight to private information withan exogenous public signal. The reason for this alternative notation is that the mathematical expressionsbecome significantly more compact since most of the results throughout the paper depend primarily uponthe weight given to private information.

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However, increasing τv has two effects on τm, which are of opposite sign. First, for afixed weight on the private signal, increasing the precision of the noise in the public signalincreases the overall precision of the endogenous public signal since information is moreaccurate (accuracy effect). Second, it reduces the weight that agents place on the privatesignal, which decreases the informativeness of the endogenous public signal (crowding-outeffect). I find that the accuracy effect always dominates the crowding-out effect, and henceand dτm

dτv> 0.8

4.2 Efficient strategy

Welfare analysis requires a definition of the welfare benchmark appropriate for this setting.I follow the approach of Radner (1962), Vives (1988), and AP who consider that there is awelfare planner that maximizes the ex ante utility subject to the constraint that informationcannot be transferred from one agent to another, and that the welfare planner does nothave access to the agent’s private information. Let us define social welfare as W (K, σk, θ) =´U(ki, K, σk, θ)di such that WKK < 0 and Wσσ < 0.AP show that with full information there is a unique efficient strategy which satisfies

k∗i = K∗ = κ∗(θ) = κ∗0 + κ∗1θ, where κ∗0 = −WK(0,0,0)WKK

and κ∗1 = −WKθ

WKK. With incomplete

information, the ex ante utility for a candidate efficient strategy is then

E [u] = E [W (κ∗(θ), 0, θ)] +WKK

2E[(K − κ∗(θ))2

]+Wσσ

2E[(K − ki)2

], (5)

which is the sum of the following terms: (1) the expected aggregate utility at the full informa-tion efficient strategy, which only depends only on τθ, and is the first best allocation; (2) non-fundamental volatility; (3) dispersion. Expected welfare loss is L∗ = E [W (κ∗(θ), 0, θ)]−E[u],and the trade-off between volatility and dispersion is WKK

Wσσ= 1−α∗. AP define α∗ < 1 as the

socially optimal degree of coordination. Here after I shall refer to α∗ as the socially optimalexogenous degree of coordination (if α∗ > 0 then it is efficient that agents perceive strategiesas strategic complements; if α∗ < 0 as strategic substitutes; and if α∗ = 0 as independent).9When public information is endogenous, the efficient strategy k∗(xi, z) can be found bymaximizing the ex ante utility and is given in the next proposition.

8This discussion also applies to the logic of the comparative statics with respect to τθ.9I have added the word exogenous to “socially optimal exogenous degree of strategic coordination” since

α∗ is no longer optimal with an endogenous public signal. Notice that α∗ only depends on the parameters ofthe utility function. For further details refer to Proposition 4.

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Proposition 2

The efficient linear strategy under incomplete information is

k∗(xi, z) = κ∗0 + κ∗1(γ∗xi + (1− γ∗)E [θ | z]), (6)

where the efficient weight to private information, γ∗ ∈ (0, 1), is the unique real root of theimplicit equation γ =

(1−α∗)τξ

(1−α∗)τξ+τθ+κ∗21 τvγ2−(1−α∗)(1−γ)2τvτξκ

∗21

(τθ+κ∗21 τvγ2)

.

The welfare planner balances the effects of an increase in the weight to private informationon the components of expected welfare loss since: (1) it increases dispersion; (2) it decreasesnon-fundamental volatility and, (3) due to endogenous public information, it increases theoverall precision of the endogenous public signal (which affects the level of strategic uncer-tainty) and further reduces volatility. Taking these three effects into account, the welfareplanner finds the unique γ∗ which minimizes expected welfare loss.

Comparison of the efficient and equilibrium strategies shows the three types of inefficien-cies which may be present at the equilibrium strategy. First, there may be a full informationinefficiency if κ∗(θ) 6= κ(θ). Second, keeping the precision of the public signal fixed, dif-ferences between α and α∗ mean that agents do not have the same individual incentives tocoordinate their actions in relation to the efficient level. These first two inefficiencies werefirst analyzed by AP and are reflected in the efficient strategy with exogenous public infor-mation given by k∗exo(xi, z) = κ∗0 + κ∗1(γ∗exoxi + (1 − γ∗exo)y), where γ∗exo =

(1−α∗)τξ(1−α∗)τξ+τθ+τε

suchthat 0 < γ∗exo < 1.

The third inefficiency, which is the core of this paper, is due to the endogeneity of thepublic signal. It is reflected in the denominator of the efficient weight to private information,which includes an additional term in relation to when public information is exogenous, andcaptures effect (3) described above. In contrast to the welfare planner, agents in equilibriumdo not take into account that a higher weight on the private signal increases the precision ofthe endogenous public signal, thus generating an information externality. This informationexternality is always present and depends on the information parameters. This is a majordifference with respect to the first and second sources of externalities, which only dependon the payoff structure, and not on the superimposed information structure. Notice thatthe information externality vanishes if the noise in the endogenous public signal or the pri-vate signal are infinitely volatile (i.e. τv → 0 or τξ → 0). The next corollary derives thecomparative statics of the γ∗ with respect to information and payoff parameters.

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Corollary 3

The comparative statics of the efficient weight given to the private signal, γ∗, are

∂γ∗

∂τξ> 0,

∂γ∗

∂τθ< 0, and

∂γ∗

∂α∗< 0. (7)

The comparative statics of γ∗ with respect to τv are ambiguous.

The efficient weight on the private signal increases with τξ, whereas it decreases with τθand with α∗ (for similar reasons to the ones given in Corollary 2). However, the comparativestatics of γ∗ with respect to the precision of the noise in the endogenous public signal areambiguous and different from the case of exogenous public information (where a higherprecision of the noise in the exogenous public signal always decreases the efficient weighton xi). In addition to the effect with exogenous public information, the planner takes intoaccount that, due to the information externality, increasing τv has a crowding-out effect whichrequires a higher weight on the private signal. Hence the ambiguous comparative statics ofγ∗ with respect to τv follow.

Proposition 3 shows how the information externality combines with payoff externalitiesdue to incomplete information in establishing if agents over- or under- weight private infor-mation in relation to the efficient weight strategy.

Proposition 3

Define α = α∗ − (1−α∗)(1−α)(κ21τvτξ)

((1−α)τξ+τm)2. The sign of the difference between the equilibrium and

efficient weights given to private information is

sign (γm − γ∗) = sign (α− α) . (8)

Endogenous public information breaks the alignment between α∗−α and γm−γ∗ that ex-ists when public information is exogenous (AP show that: sign (γexo − γ∗exo) = sign (α∗ − α)).The comparison of the efficient weights to private information with endogenous and exoge-nous public signals is as follows. First, denote the precision of the endogenous public signalat the efficient strategy found in Proposition 2 as τ ∗ = τ(κ∗1γ

∗) = τθ + τv(κ∗1γ∗)2. Take an

exogenous public signal with noise precision τε = τ ∗−τθ. Then the efficient weight on privateinformation under such exogenous public signal is always smaller than γ∗.

Economies with α = α∗ represent an important benchmark. If public information wasexogenous then the equilibrium weight to private information would always be efficient.However, with endogenous public information, agents give an inefficiently low weight toprivate information for all the possible parameter combinations of the payoff function and

13

endogenous information structures. This is because agents do not take into account howtheir individual actions affect the precision of the endogenous public signal. An exampleof an economy with such structure is given when firms compete in a homogeneous productmarket with total surplus as welfare benchmark (described in Section 3) since α∗ = α andκ = κ∗.10

Due to the additional coordination role of the endogenous public signal, if α∗ > α thenthe there is a smaller distance between the equilibrium and efficient weights to private infor-mation. In fact, when the information externality is sufficiently large, agents may equally orunderweight private information in relation to the efficient level. Thus, the information exter-nality may overturn the sign of the difference between the equilibrium and efficient weights toprivate information in relation to when public information is exogenous. As a result, agentsin equilibrium may underweight private information for a larger payoff parameter region inrelation to when public information is exogenous. In Section 6, I show how this may occurin competition à la Bertrand with product differentiation.

In contrast, if α > α∗ then agents always give too little weight to private information inrelation to the efficient level. However the difference in magnitude between the efficient andequilibrium weights to private information is larger with endogenous than with exogenouspublic information.11 For example, this occurs in the beauty contest described in Section3 since its payoff structure satisfies α = r > 0 = α∗. The discussion has illustrated that,with an endogenous public signal, the conditions α∗ = α and κ = κ∗ are not sufficient forefficiency as is further developed in the next proposition.12

Proposition 4

The economy characterized by U(k,K, σk, θ) and an information structure (τξ, τθ, τv), wherew =

´kidi+ v, are jointly efficient if and only if

κ(θ) = κ∗(θ) ∀ θ, andα = α < α∗.

Proposition 4 emphasizes that efficiency condition with an endogenous public signal re-10This result has also been found in Bru and Vives (2002) focus on a pure prediction (α∗ = α = 0) with

endogenous public information. Models that consider a payoff structure with α = α∗ but in which pricehas an additional allocation role, such as in Vives (2015), and where agents use price contingent strategies,lead to different results as those presented in Proposition 3. For example, agents in equilibrium can give toomuch weight to private information whenever actions are strategic substitutes (since the market price hasdual information and allocation roles).

11An implication is that the equilibrium precision of the endogenous public signal, τm, may be too high ortoo low in relation to the efficient level, τ∗.

12I thank a referee for suggesting that this result should be discussed.

14

quires the planner to take into account both the payoff and information structures. Thiscontrasts with the efficiency result with exogenous public information which only depends onthe characteristics of the payoff structure: AP find that an economy is efficient if and only ifκ(θ) = κ∗(θ) for all θ and α = α∗.

Furthermore, the socially optimal endogenous degree of coordination, α, is lower than thesocially optimal exogenous degree of coordination, α∗.13 This is because the information ex-ternality provides additional information about the strategies of other players, thus fosteringan excessive degree of coordination, which is corrected by the welfare planner in the sociallyoptimal endogenous degree of coordination.

5 Social Value of Public and Private Information

In this section, I analyze whether more precise public and private signals have positive ornegative welfare effects, and focus on the differences in environments with endogenous andexogenous public information.

Ex ante utility can be expressed as E[u] = E [W (κ(θ), 0, θ)] − WL, where WL =

−Wσσ

2E [(K − k)2]− WKK

2E [(K − κ(θ))2]−E[WK(κ(θ), 0, θ)(K−κ(θ))]. The first two terms

are second-order effects: the first term is dispersion and the second is non-fundamentalvolatility. The third term is a first order effect, which is related to the full informationinefficiency, and is the covariance between marginal social welfare, WK(κ(θ), 0, θ), and theaggregate error due to incomplete information, K − κ(θ). AP show that substituting for theequilibrium strategy in WL gives

WL =| Wσσ | κ2

1

2

((γm)2

τξ+

(1− α∗)(1− γm)2

τm+

2φ(1− α∗)(1− γm)

τm

), (9)

where φ =κ∗1−κ1κ1

is the full information inefficiency ratio defined by AP.14 This last term of(9) will be henceforth abbreviated as first order effect in WL.

UY use the same measure of welfare but represent it differently, expressing it as a linearcombination of idiosyncratic variance and common variance. The idiosyncratic variance cor-responds to the dispersion term, while the common variance is different and is defined as thecovariance of actions. This paper, however, follows the same form of welfare representationused by AP. The advantage of the welfare formulation expressed in (9) is that the social value

13With endogenous public information, if α > 0 then it is efficient that strategies are strategic complements;if α < 0 are strategic substitutes; and if α = 0 are independent.

14Specifically, E[WK(κ(θ), 0, θ)(K − κ(θ))] =| WKK | E[(K − κ(θ))κ(θ)](E[κ(θ)(κ∗(θ)−κ(θ))]

E[(κ(θ))2]

), where the

term inside the brackets is equal to φ.

15

of information results can be explicitly tied to each of the inefficiencies of the equilibriumallocation making its interpretation more intuitive.

5.1 Social value of public information

The total welfare effect of changing the precision of the noise in the public signal can bewritten as

d(E[u])

dτv=

(∂(E[u])

∂τm

)γm cons.

dτm

dτv. (10)

There are two effects on equilibrium welfare. The first is related to how, whilst keeping theweight fixed on the private signal, a change in the precision of the noise in the public signalaffects equilibrium welfare. The first effect has the same sign as when public information isexogenous. The second concerns how the overall precision of the endogenous public signalincreases as a result of an increase in τv (i.e. 0 < dτm

dτvfrom Corollary 2). Consequently,

the first effect fully determines the sign of the social value of public information. However,the magnitude of the social value of public information may differ between endogenous andexogenous public information because dτm

dτv≤ κ2

1. If | κ1 |≤ 1 then the social value of public

information is smaller than(∂(E[u])∂τm

)γm cons.

. The inequality is reversed if the full information

response to fundamentals is sufficiently large.The main result shows the necessary and sufficient conditions for determining whether

increasing τv has positive or negative social value.15

Proposition 5

Define ι = (1−α)(2α−α∗− 1− 2φ(1−α∗)) and ρ = −(1 + 2φ)(1−α∗). It then follows that

d(E[u])

dτv≷ 0⇐⇒ ιτξ + ρτm ≶ 0. (11)

Define the ratio of payoff parameters Y = −ιρ

such that

sign(ι) = sign(I − φ), sign(ρ) = sign(R− φ), (12)

where

I =2α− α∗ − 1

2(1− α∗)and R =

−1

2. (13)

15I assume that the welfare planner cannot change the precision of the ex ante fundamentals, τθ.

16

Proposition 5 shows that the ratio of payoff parameters that determines the social valueof public information, Y , is the same both with exogenous and endogenous public informa-tion. Hence, given (10) and Corollary 2, Proposition 5 can be implied from the results ofUY. However, since the welfare representation of (9) emphasizes on the inefficiencies of theequilibrium allocation, Proposition 5 allows us to intuitively interpret how the social valueof public information depends on the combination of sign, size and relative weight of eachof the different components of equilibrium welfare loss (dispersion, volatility, and first ordereffect) summarized by ι, ρ.

Specifically, increasing τv: (1a) decreases dispersion since agents place less weight on theprivate signal; (2a) reduces agents’ forecast error about θ, which decreases volatility, but (2b)also increases the weight that agents place on the public signal, which implies that K − κ(θ)

is larger, and hence that volatility increases; (3a) if φ 6= 0 then there is a first order effect.Increasing τv lowers an agent’s forecast error based on public information, hence K becomescloser to κ(θ), and if φ > 0, the distance between K and κ∗(θ) becomes smaller which isimproves welfare; if φ < 0 then increasing τv is reduces welfare since the length between Kand κ∗(θ) becomes larger and the first order effect increases.

Combining the effects of increasing τv on the three components of WL, I obtain thefollowing results. First, ρ < 0 if and only if (2a) and (3a) are jointly beneficial for welfare,and R is the threshold level of the full information inefficiency such that the first order effectoffsets the positive welfare effect (2a). Second, ι < 0 if and only if (1a), (2b) and (3a) arebeneficial for welfare, and I is the cutoff level of φ that balances the combination of (1a) and(2b) with the first order effect.

5.2 Social value of private information

When public information is endogenous, the total welfare effect of changing the precision ofthe private signal is

d(E[u])

dτξ=

(∂(E[u])

∂τξ

)τm cons.

+

(∂(E[u])

∂τm

)γm cons.

dτm

dτξ. (14)

The social value of private information with an endogenous public signal is the sum oftwo effects. First, the direct effect that changing τξ has on equilibrium welfare, whilst theprecision of the noise in the public signal remains fixed. The sign is the same as when publicinformation is exogenous. Second, there is an indirect effect since dτm

dτξ> 0 (as shown in

Corollary 2), and therefore affects total welfare. The sign of this indirect effect is determinedby

(∂(E[u])∂τm

).

Due to the endogenous public signal, the social value of private information is related

17

to whether public information is beneficial or detrimental. Adding the two effects, I noticethat endogenous public information may change both the sign and the magnitude of thetotal welfare effect that results from changing τξ in relation to the direct welfare effect. Thisaugments the social value of private information if the social value of public information ispositive, and it decreases the social value of private information if the social value of publicinformation is negative.

The main proposition spells out the necessary and sufficient conditions for equilibriumwelfare to increase or decrease with respect to τξ.

Proposition 6

Define ϕ = −(1− α)2(1− α + 2φ(1− α∗)) and χ = (1− α)((2α∗ − α− 1)− 2φ(1− α∗)). Itthen follows that

d(E[u])

dτξ≷ 0⇐⇒ (ϕτξ + χτm) +

dτm

dτξ(ιτξ + ρτm) ≶ 0. (15)

and define the ratio X =−(ϕ+ι dτ

m

dτξ

)χ+ρ dτ

m

dτξ

such that

sign

(ϕ+ ι

dτm

dτξ

)= sign(F − φ), sign

(χ+ ρ

dτm

dτξ

)= sign(C − φ), (16)

where

F =−(1− α)2 + (2α− α∗ − 1)dτ

m

dτξ

2(1− α∗)((1− α) + dτm

dτξ)

, and C =(1− α)(2α∗ − α− 1)− (1− α∗)dτm

dτξ

2(1− α∗)((1− α) + dτm

dτξ)

. (17)

Proposition 6 shows how the social value of private and public information combine inestablishing the sign of the social value of private information. Notice that the ratio of payoffand informational parameters X is different with exogenous and endogenous public informa-tion, since it depends on the strength of the information externality. Using Proposition 5,(14) and Corollary 2, the result can be implied by the work of UY. As mentioned earlier, theadvantage of the formulation of Proposition 6 is that the social value of private informationcan be explicitly tied to the potential inefficiencies of the equilibrium allocation. The effectsof increasing τξ on the components of equilibrium welfare loss are the following.

Dispersion: (1A) keeping τm fixed, dispersion diminishes since information is more ac-curate but, (1B) it also increases dispersion since agents place more weight on the privatesignal. Due to endogenous public information, there is also an additional effect (1C) which

18

reduces dispersion since dτm

dτξ> 0 and agents place more weight on the public signal.

Volatility : (2A) when τm remains fixed, volatility reduces since agents place less weighton public information. Due to endogenous public information, (2B) volatility is lower sincea higher τm increases the accuracy of information, but (2C) volatility increases since agentsplace more weight on public information, thus making overall volatility larger;

First order effect : (3A) if φ 6= 0 and keeping τm fixed, there is a reduction in the first-order effect in WL if φ > 0 since agents place less weight on the public signal thus reducingK − κ(θ). The opposite occurs if φ < 0. Because of endogenous public information, the risein τm causes an additional effect (3B) that if φ > 0 the first order effect in WL decreases,whereas if φ < 0 then the first order effect increases.

The sign and magnitude of these effects are summarized by ϕ, χ, ι, ρ and by a characteristicof the endogenous information structure, which jointly determine the social value of privateinformation. Parameter ϕ contains effects (1A) and (3A), while parameter χ combines (1B),(2A) and (3A): χ < 0 and ϕ < 0 if and only if the combined effects are beneficial for welfare.The effects of the social value of private information also depend on ι, ρ, which have beendescribed in section 5.1, and are weighted by dτm

dτξ. In addition, F and C correspond to the

threshold level of full information inefficiency, φ, such that the first order effect offsets secondorder effects in equilibrium welfare loss.

5.3 Combining the social value of public and private information

In symmetric quadratic payoff games of incomplete and exogenous information, UY showhow changing the precision of information cause different welfare effects which form eighttypes of possible games:−I,−II,−III,−IV,+I,+II,+III,+IV . Due to the endogenouspublic signal, the differences are: (i) payoff parameters alone do not always determine thetype of game since the characteristics of the information structure matter; (ii) some of thecutoffs between types change (those that depend on the parameters related to the social valueof private information); (iii) some of the cutoffs within types change (those that depend onX).

To illustrate these differences, Table 1 presents the classification that is obtained withan endogenous information structure. The advantage of the welfare representation used hereis that the types of game are defined according to the sign and size of the full informationexternality (φ) in relation to α − α∗. Therefore, the type of game is directly related to thedifferent types of externalities.

19

Table 1: Types of games with an endogenous public signal.

Note. The table assumes that R and C cannot be simultaneously equal to zero.

Proposition 7 explores how modifications to the cutoffs depend on the relationship betweenα and α∗.

Proposition 7

Endogenous public information modifies the following cutoffs:(i) Cutoff between types of games: If α > α∗ (α∗ > α) then there is a larger range of φ thatsatisfy the conditions for types +III,−I (−III,+I), and a smaller range range of φ thatsatisfy the conditions for types +II,+IV (−II,−IV ). This is because

20

sign(α− α∗) = sign(F − Fexo) = sign(C − Cexo), (18)

where Fexo = −(1−α)2(1−α∗)

and Cexo = (2α∗−α−1)2(1−α∗)

are the corresponding cutoffs with exogenous publicinformation.(ii) Cutoffs within types of games: The range of τm

τξthat satisfies the conditions for a positive

social value of private information is smaller in games of types +III and +IV , while it islarger in games of types −III and −IV since X > Xexo = −ϕ

χ.

Let us illustrate the main intuition of the proposition with an example. From Table 1,we observe that if α∗ > α then the game can be of types −I,−II,−III,−IV,+I. Supposethat φ = 0. Then the game can be of types +I and −IV (since R < 0). More precise publicinformation is always welfare improving because it helps agents reduce the excess equilibriumweight on private information, thus reducing dispersion more than the potential increase innon-fundamental volatility. However, more precise private information may be detrimental.The endogenous public signal fosters greater coordination because higher precision of theprivate signal increases γm, which rises the τm, and agents will give higher weight to thepublic signal. This results in a further decrease in dispersion, which improves welfare. For agiven α, a higher α∗ implies that dispersion has a higher weight inWL in relation to volatility.As a result, a positive social value of private information always implies a positive social valueof public information. Hence, C < Cexo, and the range of φ that satisfies the conditions fora −IV game is smaller than when public information is exogenous. In contrast, the rangeof φ that satisfies the conditions for a +I game is larger than when public information isexogenous. In addition X > Xexo. To sum up, the overall effect of an endogenous publicsignal on the social value of private information is positive, as reflected in the changes incutoffs between and within types, and may overturn its sign from negative to positive. Theopposite occurs when α > α∗.

If with full information agents sufficiently under-respond to the fundamental (i.e. φ ispositive and sufficiently large) then the game is of type +I because benefits of more preciseinformation through the first order effect are greater than the potential negative second ordereffects. However, if with full information agents over-respond to the fundamental (i.e. as φbecomes negative) then more precise public or private information increase WL through thefirst order effect. Hence, the benefits of more precise private information or public informationare smaller (if φ is slightly negative the game will be of type −III, if φ is moderately negativethe game will be of type −II). In the limit, when φ has a large absolute size and is negative,the benefits of either more precise public or private information will disappear and the game

21

will be of type −I. 16 The same logic applies when α > α∗.In contrast, economies that satisfy α = α∗ can only be games of types −I and +I.

An example is when firms compete in a homogeneous product market and total surplusis the welfare benchmark (described in section 3.1) since there are no payoff inefficiencies(α = α∗ and φ = 0). This can be described as a game of type +I both with endogenousand exogenous public information. However, the magnitude of the change in the social valueof private information, that results from an increase in the precision of the private signal, islarger with endogenous than with exogenous public information.

An example in which the endogenous public signal does not change the main conclusionsof the social value of information is the beauty contest (e.g. Morris and Shin 2002, UY). Thisgame’s payoff structure satisfies 0 = α∗ < α and φ = 0. With exogenous public informationthe game is of types +I and +II. With an endogenous public signal, one might wonder if thegame could be of type +III. However, the payoff parameters of the application ensure thatthis does not occur since F < 0 = φ, and hence cutoffs within and between types of game donot change.17 In the beauty contest, endogenous public information has only a magnitudeeffect and does not change the main conclusions of the social value of information. To sumup, the interaction of inefficiencies (α − α∗, φ, and the information externality) determinewhich of the effects on WL dominate, and these specify the type of game.

6 Applications: Competition à la Cournot and à la Bertrand

with product differentiation

In this section, I illustrate the results of two environments where endogenous public infor-mation has the opposite effect on the social value of information.

Consider a large market (with monopolistic competition) where firms compete either inquantity (à la Cournot) or price (à la Bertrand) strategies, and sell differentiated products.Each firm is uncertain about a common shock, θ, that affects its demand. To form expecta-tions about θ, a firm has access to both a private signal, as a result of the observation of localmarket conditions, and to a public signal, which all other firms can access without cost. Thepublic signal is a contemporaneous forecast of the aggregate strategy of all other firms in themarket. This aggregate strategy signal contains noise since the aggregation process is imper-fect, and due to measurement error. Firms base their strategies on both the private signaland on the public contemporaneous forecast about the aggregate quantity or price, which

16Specific applications often have restrictions on the payoff parameters, only a few types may apply.17This is because dτm

dτξis small.

22

is endogenous. In contrast, if the information agency tracked and disclosed fundamentals(instead of the market activity) then public information would be exogenous.

Competition à la Cournot A firm faces the linear inverse demand pi = θ−(1−δ)qi−δQ, where (pi, qi) are price-quantity pairs for firm i, Q is the aggregate quantity producedby all firms, and δ ∈ (0, 1) can be interpreted as the degree of product differentiation. Atthe limit, when δ → 0 firms are isolated monopolies, and if δ → 1 then there is perfectcompetition. This payoff structure is from Vives (1990). Firm i’s profit is: πi = θqi − (1 −δ)q2

i − δqiQ. Set qi = ki and Q = K. Then the profit can be re-written asU(k,K, σk, θ) = θki − (1− δ)k2

i − δkiK. (19)

It can be shown thatW (K, σk, θ) is equal to total producer surplus, PS = θK−K2−(1−δ)σ2k.

The characteristics of the payoff structure are as follows: φ < 0 (i.e. with full informationfirms over-respond to the fundamental since κ1 = 1

2−δ > κ∗1 = 12); and α∗ = 2α < α < 0,

where α = −δ2(1−δ) (i.e. firms perceive an inefficiently low degree of strategic substitutability

in relation to the monopoly benchmark). The next corollary applies the main results tocompetition à la Cournot.

Corollary 4

i) Equilibrium and efficiency: Firms that compete à la Cournot always give too little weightto private information in relation to the efficient level.ii) Social value of information. The game is of:

• type +I if −12≤ α < 0;

• type +II if ϑ ≤ α < −12, where −1 < ϑ < −1

2;

• type +III if α < ϑ: where the cutoff that defines whether the social value of privateinformation is positive satisfies: − (1+α)

(1−α)= Xexo < X.

Both with exogenous and endogenous public information, firms that compete à la Cournotunderweight private information since α > α∗. The size of the difference between the equi-librium and efficient weights to private information is larger with endogenous than withexogenous public information due to the information externality.

The social value of information with an endogenous public signal has the following dif-ferences in relation to when public information is exogenous (e.g. AP, UY and Myatt andWallace (2015)). First, both the payoff and information structures are needed to fully de-termine the social value of information. Second, the endogenous public signal modifies the

23

cutoff degree of strategic complementarity that distinguishes games of type +II and +III.In particular, the range of α that defines type +II is smaller than compared to when publicinformation is exogenous, whereas the range of α that delimits type +III is larger (if infor-mation was exogenous, the cutoff α that delimits the two cases is equal to −1, while withendogenous public information it is equal ϑ such that −1 ≤ ϑ < −1

2). If α < ϑ, so that the

game is of type +III, for a given τm, there is a smaller range of τξ for which the social valueof private information is positive. Overall, the social value of private information may beoverturned from positive to negative due to the endogenous public signal, as reflected in thechanges of cutoffs between and within types.

Competition à la Bertrand Firms face the following linear demand qi = (θ−pi+bP ),where (pi, qi) are price-quantity pairs for firm i and P is the market’s aggregate price. Afirm’s profit is πi = piqi−cq2

i . Setting pi = ki and P = K, the firm i’s profit can be expressedas

U(k,K, σk, θ) = (θ − ki + bK)ki − c(θ − ki + bK)2, (20)

where c > 0 and 0 < b < 1. Producer surplus is equal to W (K, σk, θ) = (θ − (1− b)K)K −c(θ− (1− b)K)2 − (1 + c)σ2. The payoff structure satisfies: φ > 0 (i.e. with full informationfirms under-respond to the fundamental since κ∗1 = 1+2c(1−b)

2(1−b)(1+c(1−b)) > κ1 = (2c+1)(2−b)+2c(1−b) > 0),

and α∗ = b(1+c(2−b))(1+c)

> α = b(2c+1)2(1+c)

> 0 (i.e. firms perceive a lower degree of strategic comple-mentarity in relation to the monopoly level). Next, I summarize the results for competitionà la Bertrand.

Corollary 5

i) Equilibrium and efficiency: When the information externality is small, firms overweightprivate information in relation to the efficient level. However, when the information exter-nality is large, there exists an αo > 0 such that γm(αo) = γ∗(αo) and sign(γm − γ∗) =

sign(α− αo).ii) Social value of information. The game is of:

• type +I if C ≤ φ;

• type −IV if C > φ, where the cutoff that defines whether the social value of privateinformation is positive satisfies: X > Xexo = −ϕ

χ,

such that C < Cexo.

24

When public information is exogenous, firms that compete à la Bertrand always give aninefficiently high weight to private information because α∗ > α. In contrast, I find that whenpublic information is endogenous, firms that compete à la Bertrand may underweight privateinformation in relation to the efficient level if αo > α > 0, thus overturning one of the mainconclusion of models of price setting complementarities (e.g. Hellwig (2005), AP, Colomboet al. (2014)). This occurs when the information externality is large.

The intuition of the social value of information results is as follows. More precise publicinformation always increases producer surplus since α∗ > α and φ > 0. However, moreprecise private information may be detrimental. In relation to when public information isexogenous, endogenous public information generates two differences in the social value ofinformation. First, there is a smaller range of φ that satisfy the conditions for type −IVgame, while a larger range of φ that satisfies the conditions for a +I game.18 When thegame is of type −IV , for a given τm, there is a larger range of τξ for which the social valueof private information is positive. Overall, the social value of private information may beoverturned from negative to positive due to the endogenous public signal, as reflected in thechanges of cutoffs between and within types.

7 Concluding Remarks

I have investigated the effects of an endogenous public signal on the use of information andon the social value of information in economies that are characterized by payoff externalitiesand heterogeneous information about fundamentals. This has been motivated on the onehand, by the observation that non-price systems for aggregating information are nowadaysubiquitous, and on the other, by the theoretical need to study information externalities thatare independent of the market structure.

The endogenous public signal provides information not only about fundamentals, but alsoabout the aggregate action of agents. As a result, agents in equilibrium underweight privateinformation for a larger payoff parameter region in relation to when public information isexogenous. In addition, the welfare planner would like agents to perceive a lower degree ofcoordination with endogenous, rather than exogenous, public information. The combinationof payoff and information externalities characterizes whether agents over-, under-, or equallyweight private information in relation to the efficient strategy.

The main contribution in relation to the previous literature is to show that endogenouspublic information may overturn the sign of the social value of private information. This may

18Due to the complexity of the mathematical conditions, a classification of types based on the primitivesof the model has not been found.

25

be from a positive to a negative sign if agents coordinate more than is efficient, due to payoffexternalities; or from a negative to a positive sign if the reverse occurs. However, endogenouspublic information has only a magnitude effect on the social value of public information. Animplication for policy is that information and payoff structures cannot be separated whenevaluating the welfare effects of providing more precise public and private information.

A limitation of this paper is that the model considered is static. The main insights shouldbe robust in a dynamic specification of the model. Because the essence of the informationexternality is the same in either static or dynamic models, where agents respond insufficientlyto private information, one might conjecture that the main results would be unaffected bythe dynamic element. Nevertheless, a multi-period model would be an interesting extensionto develop.

This paper suggests several additional open questions for future research. Future workcould explore how robust these results are to more general endogenous information structures,and how different mechanisms for aggregating information affect the use and the social valueof public and private information.

Appendix

Proof of Proposition 1Focusing on linear arbitrary strategies of the form k(xi, z) = b′ + axi + c′E [θ | z] and

using the definition of the public signal, I obtain: w =´ki + v = b′ + aθ + c′w + v, whose

informational content can be seen to the same as that of z = aθ + v. From the propertiesof the normal distribution: E [θ|z] = τvaz+τθ θ

τθ+a2τvand (var [θ|z])−1 = τ(a) = τθ + a2τv. Hence

E [θ|xi, z] =τξ

τξ+τ(a)xi + τ(a)

τξ+τ(a)E [θ | z].

Using the maximization condition in the REE definition, I find that the best response is astrategy k′, satisfies E [Uk(k

′, K, σk, θ) | xi, z] = 0 and hence k′ = E[(1− α)κ(θ) + αK|xi, w].The second order condition is satisfied since Ukk < 0. Equating coefficients, I obtain: b = κ0,a =

κ1(1−α)τξ(1−α)τξ+τ(a)

and c = κ1τ(a)(1−α)τξ+τ(a)

. Hence, k(xi, z) = κ0 + κ1(γxi + (1 − γ)E [θ|z]), where

γ =(1−α)τξ

(1−α)τξ+τθ+κ21γ2τv

. The equilibrium weight to private information, γm, is then a solutionof Γ(γm) = 0, where

Γ(γ) = γ3κ21τv + γ((1− α)τξ + τθ)− (1− α)τξ. (A1)

By the Descartes’ Rule of signs, I note that there is only one sign change and, therefore,there is only a positive real root. Checking Γ(−γ), I notice that there are no sign changes, andtherefore, there are no negative real roots. Additionally, Γ(0) = −(1− α)τξ < 0 and Γ(1) =

26

κ21τv + τθ > 0, and, therefore, the positive real root is between 0 and 1. The precision of the

endogenous public signal at the equilibrium strategy is then τm = τ(κ1γm) = τθ + (κ1γ

m)2τv.�

Proof of Corollary 1Differentiating Γ(γm) = 0 with respect to the various information and payoff parameters, Iobtain that:∂γm

∂τξ= (1−α)(1−γm)

(1−α)τξ+τθ+3(γm)2κ21τv> 0;

∂γm

∂τθ= −γm

(1−α)τξ+τθ+3(γm)2κ21τv< 0;

∂γm

∂τv=

−(γm)3κ21(1−α)τξ+τθ+3(γm)2κ21τv

< 0;

and ∂γm

∂α=

−(1−γm)τξ(1−α)τξ+τθ+3(γm)2κ21τv

< 0 since 0 < γm < 1. �

Proof of Corollary 2The proof follows from: dτm

dτξ= (2κ2

1τvγm) (1−α)(1−γm)

(1−α)τξ+τθ+3(γm)2κ21τv> 0;

0 < dτm

dτv= κ2

1γ2(

(1−α)τξ+τθ+(γm)2κ21τv(1−α)τξ+τθ+3(γm)2κ21τv

) < κ21;

and 0 < dτm

dτθ=

(1−α)τξ+τm

(1−α)τξ+τθ+3(γm)2κ21τv< 1. �

Proof of Proposition 2The welfare planner internalizes all externalities and maximizes ex ante utility, or equivalentlyminimizes welfare loss. Focusing on a linear arbitrary efficient strategy of the form k∗(xi, z) =

b∗ + a∗xi + c∗E [θ | z]. By a similar argument as the one used in Proposition 1, b∗ = κ∗0 andk∗(xi, z) = κ∗0 + κ∗1(γxi + (1− γ)E [θ|z]), where γ is an arbitrary weight to the private signal.The first order condition satisfies dL∗

dγ|γ=γ∗ = 0. Hence a candidate efficient strategy with

weight to private information γ and precision given by τ(γ) = τθ + (κ∗1γ)2τv solvesγ

τξ− (1− α∗)(1− γ)

τ(γ)− (κ∗1)2τvγ(1− α∗)(1− γ)2

τ(γ)2= 0, (A2)

where the last term is due to the endogeneity of public information. The previous expressioncan be rewritten as

γ =(1− α∗)τξ

(1− α∗)τξ + τθ + κ∗21 τvγ2 − (1− α∗)(1− γ)2 τvτξκ

∗21

(τθ+(κ∗1)2γ2τv)

. (A3)

Then, γ∗ is a solution of a quintic equation ψ(γ∗) = 0, whereψ(γ) = (γ)5τv

2(κ∗1)4+2(γ)3(κ∗1)2τvτθ+(γ)2(κ∗1)2τvτξ(1−α∗)+γ(τ 2θ+τξ(1−α∗)(τθ−τv(κ∗1)2))−τξτθ(1−α∗).

Applying Descartes’ Rule of signs to find out the number of real roots. There is only onechange in sign, and therefore, this polynomial has only one real positive root. Furthermore,ψ(0) = −τξτθ(1 − α∗) < 0, while ψ(1) = τv

2(κ∗1)4 + 2(κ∗1)2τvτθ + τ 2θ > 0 and the unique

positive real root is between 0 and 1. The SOC guarantees that the denominator of γ∗ =

27

(1−α∗)τξ

(1−α∗)τξ+τ∗−(1−α∗)(1−γ∗)2τvτξ(κ

∗1)

2

(τθ+(κ∗1)2γ2τv)

is positive and hence γ∗ > 0. �

Proof of Corollary 3Differentiating ψ(γ) = 0 with respect to the various information and payoff parameters, Iobtain that:∂γ∗

∂τξ=

γ∗(1−γ∗)(1−α∗)(τvκ21)

5(γ∗)4τv2κ41+6(γ∗)2κ21τvτθ+2γ∗κ21τvτξ(1−α∗)> 0;

∂γ∗

∂τθ=

−2γ∗τ∗+τξ(1−α∗)(1−γ∗)

5(γ∗)4τv2κ41+6(γ∗)2κ21τvτθ+2γ∗κ21τvτξ(1−α∗)< 0;

∂γ∂α∗ =

−(γτξτvκ21+τξτθ)(1−γ∗)

5(γ∗)4τv2κ41+6(γ∗)2κ21τvτθ+2γ∗κ21τvτξ(1−α∗)< 0 which follow since 0 < γ∗ < 1.

Notice that the sign of ∂γ∗

∂τv= κ2

1γ∗(

−2(γ∗)2τ∗+τξ(1−α∗)(1−γ∗)

5(γ∗)4τv2κ41+6(γ∗)2κ21τvτθ+2γ∗κ21τvτξ(1−α∗)) is ambiguous. �

Proof of Proposition 3The efficient strategy γ∗ satisfies dL∗

dγ|γ=γ∗ = 0, where L∗(γ) is a strictly convex function of

γ. Hence sign (γm − γ∗) can be found from inspecting the sign of dL∗

dγ|γ=γm . Substituting

for the equilibrium strategy in dL∗

dγ|γ=γm , I obtain that

sign

(dL∗

dγ|γ=γm

)= sign (γm − γ∗) = sign

(α∗ − α− (1− α∗)(1− α)(κ2

1τvτξ)

((1− α)τξ + τm)2

), (A4)

which implies that sign (γm − γ∗) = sign (α− α) , where α = α∗ − (1−α∗)(1−α)(κ21τvτξ)

((1−α)τξ+τm)2. �

Proof of Proposition 4Follows immediately from Propositions 2 and 3 since κ(θ) = κ∗(θ) ∀θ and α = α

⇔k(xi, z) = k∗(xi, z). �Proof of Proposition 5Substituting γm into (9), I obtain:

WL =| Wσσ | κ2

1

2

((1− α)2τξ + (1− α∗)τm

((1− α)τξ + τm)2+

2φ(1− α∗)τm

(1− α)τξ + τm

). (A5)

Comparative statics of ex ante utility with respect to τv are equivalent to the opposite com-parative statics of WL. Since dWL

dτv=

(∂WL∂τm

)γ cons.

dτm

dτvthen(

∂WL

∂τm

)γm cons.

=| Wσσ | (κ1)2

2

((1− α)((2α− α∗ − 1)− 2φ(1− α∗))τξ − (1− α∗)(1 + 2φ)τm

(τξ(1− α) + τm)3

),

(A6)and 0 < dτm

dτvby Corollary 2. Define the coefficient on τξ as ι = (1−α)(2α−α∗−1−2φ(1−α∗)),

and the coefficient on τm as ρ = −(1 + 2φ)(1− α∗). Notice that ι ≷ 0⇐⇒ I = 2α−α∗−12(1−α∗)

≷ φ

and ρ ≷ 0⇐⇒ R = −12≷ φ. �

Proof of Proposition 6Noting that dWL

dτξ=

(∂WL∂τξ

)τm cons.

+(∂WL∂τm

)γ cons.

dτm

dτξ, I obtain: (13)(

∂WL

∂τξ

)τm cons.

= Λ

(−(1− α)2(1− α + 2φ(1− α∗))τξ + (1− α)(2α∗ − α− 1− 2φ(1− α∗))τm

(τξ(1− α) + τm)3

),

(A7)

28

where Λ = |Wσσ |(κ1)2

2. Define the coefficient on τξ as ϕ = −(1− α)2(1− α+ 2φ(1− α∗)), and

the coefficient on τm as χ = (1−α)((2α∗−α−1)−2φ(1−α∗)). The result is then obtained.Now rearranging the condition for the social value of information, I obtain d(E[u])

dτξ≷ 0 ⇐⇒

(ϕ+ ιdτm

dτξ)τξ + (χ+ ρdτ

m

dτξ)τm ≶ 0.

Notice that (ϕ+ ιdτm

dτξ) ≷ 0⇐⇒ F =

−(1−α)2+(2α−α∗−1) dτm

dτξ

2(1−α∗)((1−α)+ dτm

dτξ)≷ φ, and

(χ+ ρdτm

dτξ) ≷ 0⇐⇒ C =

(1−α)(2α∗−α−1)−(1−α∗) dτm

dτξ

2(1−α∗)((1−α)+ dτm

dτξ)

≷ φ. �

Proof of Proposition 7i) Using the definitions of the cutoffs, I can obtain that: If α > α∗ then I > F > R > C;

if α∗ > α then C > R > F > I; and if α∗ = α then C = R = F = I. Manipulating theexpressions previously defined, we can obtain that F > Fexo, C > Cexo ⇔(α − α∗)dτm

dτξ> 0.

Hence, the result obtains by checking the types of game which have C and F as cutoffs.ii) Games of type +III have: ι > 0, ϕ + ιdτ

m

dτξ> 0 and ρ < 0, χ + ρdτ

m

dτξ< 0, χ < 0.

Hence X =−(ϕ+ι dτ

m

dτξ

)χ+ρ dτ

m

dτξ

> −ϕχ

since (ϕρ − ιχ)dτm

dτξ> 0 because 0 < ϕρ − ιχ. Games of

type +IV have: ι > 0, ϕ + ιdτm

dτξ> 0 and ρ > 0, χ + ρdτ

m

dτξ< 0, χ < 0 and by the previous

argument the result follows. A similar argument applies to games of type −III that have:ι < 0, ϕ + ιdτ

m

dτξ< 0 and ρ > 0, χ + ρdτ

m

dτξ> 0, χ > 0 and −IV that have ι < 0, ϕ + ιdτ

m

dτξ< 0

and ρ < 0, χ+ ρdτm

dτξ> 0, χ > 0. Hence the result follows. �

Proof of Corollary 4i) Follows from Proposition 3 by noting that γm < γ∗ will always hold since α > α∗.ii) The parameters which determine the social value of information are as follows: ρ = −1 < 0,ι = −(1 − α)(1 + 2α), ϕ = −(1 − α)(1 + α), χ = −(1 − α)2 < 0, since φ = α

1−2αand

α∗ = 2α < α < 0. Note that R = −12< φ < 0. By Table 1, note that the game can

be of types +I,+II,+III. If α > −12

then ρ < 0, ι < 0, ϕ < 0, χ < 0 and the gameis of type +I. The cutoff between types +II and +III changes with endogenous public

information and is given by sign(ϕ+ ιdτ

m

dτξ

)= sign(F − φ), where F =

−(1−α)2− dτm

dτξ

2(1−2α)((1−α)+ dτm

dτξ).

Then φ > F⇔∆(α) = (1 − α2) + dτm

dτξ(1 + 2α) > 0. Note that ∆(−1) < 0, ∆(−1

2) > 0

and ∆(α) is a strictly increasing function of α since some regularity conditions regardingdτm

dτξare satisfied. Hence, there a exists a unique ϑ such that ∆(ϑ) = 0, where ϑ ≤ α < −1

2.

Then, if ϑ ≤ α < −12

the game is of type +II and if α < ϑ then the game is of type +III.From Proposition 7, when the game is of type +III, for a given τm, there exists a largerrange of τm

τξsuch that the social value of private information is negative. This occurs since

ϕτξ +χτm > 0 implies that ιτξ + ρτm > 0 because ϕ−χ <

ι−ρ , or equivalently, −ϕρ < −ιχ. �

29

Proof of Corollary 5

i) Follows from Proposition 3 and by noting that α∗ > α. When the information externality islarge there will exist an αo = α∗− (1−α∗)(1−α)(κ21τvτξ)

((1−α)τξ+τm)2> 0 because γ(α) and γ∗(α∗), respectively,

are both strictly decreasing function of α, which range from 1 to 0 as α→ −∞ to α→ 1 andα∗ → 1, respectively. Using the intermediate value theorem and Proposition 3, there existsa unique intersection between the two curves, αo, which is equal to and satisfies 0 < αo < 1.Hence, sign (γm − γ∗) = sign(α−αo). When the information externality is small, it is alwaysthe case that γm − γ∗ > 0.ii) φ > 0 and α∗ > α > 0 then the game has ρ < 0, ι < 0, ϕ < 0, and χ + ρdτ

m

dτξmay be

positive or negative depending on the relationship between φ and C. Because of Proposition7, C < Cexo and hence the result follows. Similarly for Xexo < X. �

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