ESADE WORKING PAPER Nº 261 May 2016

The social value of information with an endogenous public signal

Anna Bayona

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The Social Value of Information with an EndogenousPublic Signal

Anna Bayona∗

May 2016

Abstract

This paper analyses the equilibrium and welfare properties of an economy characterisedby 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 aggregateof individual actions and causes an information externality. Agents in equilibrium under-weight private information for a larger payoff parameter region in relation to when publicinformation is exogenous. In addition, the welfare planner gives a higher weight to privateinformation when public information is endogenous rather than exogenous. The welfare effectof increasing the precision of the noise in the public signal has the same sign with endogen-ous or exogenous public information, but its magnitude differs. The social value of privateinformation may be overturned in relation to when public information is exogenous: frompositive to negative if agents in equilibrium coordinate more than is socially optimal withexogenous public information, 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. Email address: anna.bayona@esade.edu. I am grateful to my thesis advisor,Xavier Vives, for the guidance and feedback. I would also like to thank Jose Apesteguia, Ariadna Dumitrescu,Sebastian Harris, Angel L. Lopez, Carolina Manzano, Margaret Meyer, Manuel Mueller-Frank, RosemarieNagel, Morten Olsen and Alessandro Pavan for constructive comments.

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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 increasingly dependenton the public information that these systems aggregate.1 In other words, public informationis endogenous and agents increasingly use strategies that are contingent on noisy statisticsthat aggregate the actions of other agents. In this paper, I study the implications of endogen-ous public information in economies characterised by: uncertainty; a large number of agents;and externalities. The main finding is that endogenous public information can overturn thesocial value of private information in relation to when public information is exogenous andits direction is determined by the interaction of payoff and information externalities.

The introduction of an endogenous public signal in economies with uncertainty and payoff-externalities brings new insights to the existing literature and enables a clear distinctionto be made between environments with endogenous and exogenous public information. Ipresent two arguments that compare and contrast these types of public information andwhich motivate the applications presented in this paper. The first is related to whether thesystems that disclose public information directly monitor fundamentals (exogenous) or trackthe market activity (endogenous). With the prevalence of big data there are nowadays anincreasing number of indicators of economic activity that provide real-time forecasts thatpredict the present (see Choi and Varian (2012)), and therefore, constitute an endogenoussource of public information. The second is related to whether the interaction among agentsoccurs in segmented markets where public information would be exogenous (e.g. in over-the-counter markets), or in centralised markets that facilitate information aggregation wherepublic information is endogenous (e.g. in exchanges).2

An endogenous public signal leads to the following information-contingent strategies:agents use schedules that are contingent on the noisy contemporaneous public forecast aboutthe aggregate action compiled by online platforms, systems, and forums that use, for exam-ple, big data. I concentrate on environments where endogenous public information has aninformation role but not an allocation role since it is not part of an agent’s payoff function.For example, firms that make capacity choices are often uncertain about the shocks thataffect their demand. The increasing prevalence of online platforms facilitates the aggrega-tion of information about the strategies of rivals, and in turn, firms post schedules in whichthe capacity they offer is contingent on noisy public information about the average capacityoffered in the market and on their assessment of local market conditions.

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

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Three research questions emerge. First, how do agents optimally use the different types ofinformation sources in equilibrium? Second, what are the welfare properties of the equilibriumstrategy? Third, what are the effects on welfare of varying the precision of the noise in publicand 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 centre.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 public in-formation aggregates the economy’s dispersed information, the precision of the endogenouspublic signal depends positively on the agents’ response to private information. Agents inequilibrium do not take into account that their response to private information affects theinformativeness of the endogenous public signal, which also provides information about thebehaviour of other players. Hence, endogenous public information generates an informationexternality, which has the following consequences. Firstly, I find that agents place an in-efficiently low weight on private information in a larger payoff parameter region than withexogenous public information. Secondly, the efficient weight given to private informationis larger with endogenous than with exogenous public information since the welfare plannertakes into account the information externality, and correspondingly, its effects on the strategicuncertainty level (uncertainty about other agents’ strategies). The combination of informa-tion and payoff externalities characterise the regions whether agents over-, under-, or equallyweight private information in relation to the efficient strategy. For example, in competitionà la Bertrand with product differentiation, firms may underweight private information inrelation to the efficient level when public information is endogenous, which overturns theconclusions with exogenous public information.

The wedge between equilibrium and efficient strategies generates welfare losses (consist-ing of dispersion, non-fundamental volatility, and a covariance term) which jointly influencethe value of information. Combining the various components of ex-ante utility, I find thenecessary and sufficient conditions which determine whether increasing the noise in the en-

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dogenous public signal has a positive social value. These depend on a combination of payoffrelevant parameters and on the ratio of public to private information precisions. I find thatthe social value of public information always has the same sign with either endogenous orexogenous public information because a higher precision of the noise in the public signal al-ways increases the overall precision of the public signal. This occurs since increasing the noisein the public signal leads to an accuracy effect which is always greater than the crowding-outeffect of exogenous public information on endogenous public information. However, the mag-nitude total social value of public information may differ with endogenous and exogenouspublic information because of the crowding-out effect of increasing the precision of the noisein the public signal on the overall precision of the endogenous public signal. The size ofthe crowding-out effect depends on the extent to which agents respond to fundamentals withcomplete information. In addition, the ratio of public to private information precisions differswith endogenous and exogenous public information. This ratio determines the sign of thewelfare effect in cases when more precise public information is not unambiguously positiveor negative. If the full information response to fundamentals is small, then the informationcontent of the endogenous public signal is smaller than that of the exogenous public signal.In economies where agents coordinate more than is socially optimal with exogenous publicinformation (e.g. in the beauty contest), the result is that, for the same precision of privateinformation, there will be a larger range of precisions of the noise in public informationfor which more precise public information is detrimental. The opposite occurs if agents inequilibrium coordinate less than is socially optimal with exogenous public information.

Turning to the results on the social value of private information, I find that endogen-ous public information may overturn the sign of the social value of private information inrelation to when public information is exogenous (e.g. in competition à la Cournot withproduct differentiation and producer surplus as a welfare benchmark). This is because whenpublic information is endogenous, the value of private information is related to whether pub-lic information is beneficial or detrimental (since changing the precision of the noise in theprivate signal increases the overall precision of the public signal, which has an additionalwelfare effect). I characterise all the possible sign combinations of the various welfare effectsof increasing the precision of the noise in the private signal in relation to the primitives ofthe model. If there is no inefficiency with incomplete information then endogenous publicinformation cannot overturn the sign of the welfare effect with exogenous public informationsince both the social value of public and private information have the same sign (e.g. firmsthat compete in a homogeneous product market with total surplus as a welfare benchmark).However, when agents coordinate less than the efficient level with exogenous public inform-ation then taking into account the effects of the endogenous public signal may overturn the

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total welfare effect from negative to positive in relation to the partial welfare effect. Thisis because the endogenous public signal reduces the difference between the equilibrium andefficient weights given to private information in relation to when public information is exo-genous. The reverse occurs when agents coordinate more than the socially optimal level withexogenous public information, since the endogenous public signal fosters a larger differencebetween the equilibrium and efficient weights to private information, and hence endogenouspublic information may overturn the sign from positive to negative compared to when pub-lic information is exogenous. A consequence of this is that endogenous public informationmay modify the threshold level that defines the regions where the social value of privateinformation is positive or negative.

The paper is organised as follows. Section 2 discusses the related literature. Section 3describes the model. Section 4 studies the equilibrium and efficient strategies. Section 5analyses the comparative statics of equilibrium welfare. Section 6 illustrates the results ofthe paper in several relevant applications. Section 7 concludes. Proofs are derived in theAppendix.

2 Related Literature

This paper studies the role of an endogenous public signal in economies with uncertainty andpayoff externalities. These have been previously studied in contexts where public informationis exogenous in the seminal paper by Morris and Shin (2002) in the beauty contest; Hellwig(2005) in a business cycle framework of monopolistic competition with pricing complemen-tarities among firms; Clark and Polborn (2006) in a binary choice context; Angeletos andPavan (2007) in a general model which encompasses many of the previous applications; Myattand Wallace (2015, 2016) in models of differentiated product oligopolies with rich industryfeatures. This paper adds the following contribution to this literature: I show how payoff andinformation externalities combine in the use of information and in determining the effects onthe social value of public and private information. Furthermore, Ui and Yoshizawa (2015)categorise all of these games into types according to their properties regarding the socialvalue of information. In relation to their paper, I show that introducing an endogenous pub-lic signal may modify the sign of the social value of private information, and consequently, itchanges the threshold value of payoff parameters that define the type of game and also theratio of public to private precisions that determines the social value of information.

The information environment considered in the paper is reminiscent of the literature ofsocial learning such as in Banerjee (1992) and Bikhchandani et al. (1992), which study thewelfare effects of information externalities by focusing on a rational explanation of herd-

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ing where agents insufficiently respond to private information. In contrast to my approach,herding models are typically dynamic and do not consider payoff externalities. Related arethe papers in the rational expectations tradition that typically consider static models wherethe public signal (generally the market price) has both information and allocation roles andagent’s strategies are schedules (e.g. Grossman and Stiglitz (1980), Diamond and Verrechia(1981)). Also connected is the work of Angeletos and Werning (2006) as it focuses on a globalgame applied to crises and analyses the effects of endogenous public information for multi-plicity of equilibria. The framework that I use in this paper proposes a parsimonious modelthat combines elements of rational expectations, herding, and coordination/anti-coordinationgames.

Endogenous information sources have been previously studied in the following papers.Morris and Shin (2005) were among the first to show that in economies that can be describedby the beauty contest, public signals are less informative when central bankers disclose theirforecasts than when they do not disclose anything. With a micro-founded macroeconomicmodel, Amador and Weill (2010) find that more precise public information can be detrimentalwhen there are no payoff externalities, which contrasts the results of this paper. Their modelcontains an additional source of strategic complementarity, generated by the way agentslearn from prices, which is responsible for the negative welfare result. In a related dynamicmodel, Amador and Weill (2012) analyse the social value of information with both publicand private information diffusion channels about the aggregate activity. In terms of policy,Angeletos and Pavan (2009) focus on the optimal design of contingent taxation and alsoconsider that public information is endogenous; and Bond and Goldstein (2015) analyse theimplications of governments relying on prices to guide their decisions. In contrast, this paperfocuses on a general payoff structure that can include many of the previous applications withstraightforward formalisation of information externalities in order to focus on the interactionmechanism between payoff externalities and an endogenous public signal.

This paper complements the results of Colombo et al. (2014) who study the effects ofendogenous private information that agents acquire at a cost. They find that the magnitudeand sign of the social value of public information may be overturned in relation to whenprivate information is exogenous because of the crowding-out effects of public informationon private information. In contrast, this paper focuses on the effects of a costless endogenouspublic signal on the use of 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 en-compasses both 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, qi,to maximise the utility function U(qi, Q, σq, θ), where θ represents the fundamentals thatexogenously affect an agent’s utility function; Q is the average of agents’ actions given byQ =

´qidi; and σ2

q is the variance of the agents’ actions across the population defined asσ2q =

´(qi − Q)2di. Assume that U is a quadratic polynomial in the variables qi, Q, σ, θ.

The dispersion in actions, σ, has only second-order effects, i.e. Uqσ = UQσ = Uθσ = 0

and Uσ(qi, Q, 0, θ) = 0 for all (q,Q, θ) and U is symmetric across agents.3 Furthermore, theutility function is concave with respect to qi. Without loss of generality, let α = −UqQ

Uqqbe the

degree of strategic complementarity.4 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 UQ 6= 0 and with respect tothe dispersion of actions whenever Uσ 6= 0.

Examples of preferences that can be modelled with this framework (which are furtherdeveloped in Section 6) include the beauty contest, firms competing in a homogeneous productmarket, and competition à la Bertrand or à la Cournot.

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

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

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

ε ). Second, agents receive a public signal which emerges within the economy andtherefore is endogenous (see motivation in the introduction). The endogenous public signalis a noisy signal about the aggregate action given by

w =

ˆqidi+ u, (1)

where the noise in the endogenous public signal is normally distributed with u ∼ N(0, σ2u),

3U can also be viewed as a second order approximation of a concave function.4α is also refereed to equilibrium degree of coordination.

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which reflects the fact that the aggregation process is imperfect.5 This modelling approachhas the advantage of introducing an endogenous public signal with minimal modificationsto the standard benchmark of exogenous public information, in which the public signal istypically defined as y = θ + v with v ∼ N(0, σ2

v). Therefore, comparisons between the twoenvironments are straightforward.

Fundamentals and error terms of private and public signals are mutually independent andidentically distributed across agents. For ease of interpretation, I shall often work with theprecision of a random variable, which is the inverse of its variance. The precision for anyrandom variable, e.g. x, with a non-zero variance is given by τx = 1/σ2

x. Throughout thetext, I call τε the precision of the noise in the private signal, τu the precision of the noise inthe public signal, and τθ the precision of fundamentals.

Equilibrium Definition Agent i’s strategy q(si, .) 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 q(si, w), that maximises expected utility E[u|si, w] con-ditional on the information set {si, ω} and on knowing w(θ, u), as well as the underlyingdistributions of random variables.

2. The endogenous public signal is formed such that w =´qidi+ u.

The first condition requires that each agent maximises expected utility given his informationset, which involves extracting from w all its informational content. The second conditionenforces the rational expectations consistency requirement, which implies that the endogen-ous public signal aggregates agents’ dispersed actions and simultaneously depends on themthrough its information role. In other words, the conjectured strategy must be self-fulfilling(fixed point of conjectured strategies to actual strategies). Unless the public signal is infin-itely precise, i.e. τu →∞, the REE will be partially revealing.6 Given the linear-quadratic-Gaussian structure of the model, I restrict equilibria in the class of linear strategies, as isusual in the literature.

Discussion of the Equilibrium Concept The main difference between this equilibriumdefinition and the standard REE concept used in the financial economics literature (e.g.Grossman and Stiglitz (1980), Diamond and Verrechia (1981), and further developed by

5I assume that the noise in the endogenous public signal is common for all agents in the economy since itcomes from the aggregation process or measurement error. A different approach, which is not considered inthis paper, would have been to consider that each agent has access to the endogenous signal with idiosyncraticnoise.

6The public signal would then perfectly reveal the fundamentals and the REE would be fully revealing.

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many others) is that, in the classic REE framework the endogenous public signal is typicallyidentified with the price, which has both information and allocation roles. The allocation roleemerges since the price is also part of an agent’s utility function. In contrast, in this paper Ifocus on environments where the endogenous public signal only has a pure information role.A similar approach has been used by Angeletos and Werning (2006) in the context of globalgames where individuals observe one another’s actions.7

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 decidesa schedule by positing a reaction function, q(si, ω), which relates an agent’s strategy tothe realisation of the endogenous public signal. The timing of such game can be specified asfollows. First, fundamentals and errors terms are drawn. Second, agents (who do not observethe fundamentals) receive a private signal and conjecture a reaction function (or schedule)that satisfies maximisation condition (1). Third, the public signal is formed and satisfies theconsistency condition (2), and its precision is determined. Finally, payoffs are collected. Fora discussion on the use and problems of REE in market games refer to Shorish (2010).

4 Equilibrium and Efficiency

Using the equilibrium definition of the previous section, each agent maximises expected util-ity conditional on his own information set maxqiE [U(qi, Q, σq, θ)|si, w] . Given the structureof the information environment, I focus on linear symmetric candidate strategies of the formq(si, w) = b′ + asi + c′w. By substituting a candidate strategy into the definition of theendogenous public statistic w, I find that w =

´qi + u = b′ + aθ + c′w + u, whose informa-

tional content is given by the random variable, z = aθ + u, so that E [θ|si, w] = E [θ|si, z].Given these assumptions, an agent’s strategy can be written as q(si, z) = b+ asi + cE [θ|z] .

Proposition 1 spells out the equilibrium strategy.7Several papers have shown that the main properties and mechanisms of related dynamic models are

preserved in the steady state (or static counterparts). Examples of such models can be found in Morris andShin (2005), with a special focus on the beauty contest application; in Angeletos and Werning (2006) whostudy global games and equilibrium selection; and in Angeletos and Pavan (2009) who focus on contingenttaxation.

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Proposition 1 (Equilibrium)

For finite and non-zero precisions of information such that α < 1, there exists a uniquelinear REE, which is given by q(si, z) = bm + amsi + cmE [θ|z], where am = k1γ

m, bm = k0,cm = k1(1 − γm) and k0, k1 define the full information equilibrium strategy given by k(θ) =

k0 + k1θ, where k0 = −Uq(0,0,0,0)Uqq+UqQ

and k1 =−Uqθ

Uqq+UqQ. The equilibrium weight given to private

information, γm, is the unique real root of the implicit equation γ = (1−α)τε(1−α)τε+τθ+k21τuγ2

, wherethe precision of public information at the equilibrium strategy is: τm = τθ + k21(γm)2τu andγm satisfies 0 < γm < 1.

The equilibrium weight given to the private signal, γm, increases with the precision of thenoise in the private signal, while it decreases with the precision of fundamentals, with the pre-cision of the noise in the public signal, and with a larger degree of strategic complementarity,α.

As in the case of exogenous public information, if actions were strategically independent(α = 0), the weights given to public and private information would correspond to Bayesianweights according to the relative precisions of the signals. When actions are strategic com-plements (α > 0), agents place a higher weight on the public signal since it helps to betterpredict the aggregate action. The converse happens when actions are strategic substitutes(α < 0).8

Endogenous public information modifies the precision of the public signal in relation tothe case of exogenous public information since a higher weight given to private informationmakes the public signal more informative. The precision of the endogenous public signalaffects the size of the Bayesian weight, which modifies the equilibrium weights an agent givesto public and private information. The precision of the endogenous public signal at theequilibrium strategy is: τm = (var [θ|z])−1 = τθ+(k1γ

m)2τu, where τm depends quadraticallyon the equilibrium response an agent gives to private information (k1γ

m)2. The more agentsrespond to private information, the more informative the public signal becomes and thisfeedback effect between an individual strategy and the precision of the endogenous publicsignal drives the main results of the paper. This effect is not present when public informationis exogenous since its precision is: τ exo = τθ + τv.

Furthermore, when the precision of the noise in the private signal tends to zero, agentsdo not give any weight to the private signal. As the precision of this signal increases, agentsincrease the weight given to the private signal. The weight given to the private signal de-creases both with the precision of fundamentals and with the precision of noise in the public

8Throughout the text, am is referred to as the equilibrium response to private information, while γm isthe equilibrium weight given to private information.

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signal. At the limit, when the noise in the public signal is infinitely precise, agents give allthe weight to the public signal and the equilibrium collapses.

Welfare analysis requires a definition of the welfare benchmark which is appropriate forthis setting. I follow the approach of Radner (1962), Vives (1988), and Angeletos and Pavan(2007) that consider that there is a welfare planner that maximises the ex-ante utility subjectto the constraint that information can be neither transferred from one agent to another, nordoes the welfare planner have access to the agent’s private information.

The full information strategy, k∗(θ), satisfies WQ(k∗(θ), 0, θ) = 0. The ex-ante utility fora candidate efficient strategy is then

E [u] = E [W (k∗(θ), 0, θ)] +WQQ

2E[(Q− k∗(θ))2

]+Wσσ

2E[(Q− q)2

], (2)

where W (Q, σq, θ) =´U(qi, Q, σq, θ)di, such that WQQ = Uqq + 2UqQ + UQQ and Wσσ =

Uqq + Uσσ. Assume that WQQ < 0 and Wσσ < 0. The expression of ex-ante utility is thesum of the following terms: (1) the expected aggregate utility at the full information efficientstrategy strategy (E [W (k(θ), 0, θ)]), which only depends on the precision of fundamentals, τθ;(2) welfare losses due to non-fundamental volatility (WQQ

2E [(Q− k(θ))2]); (3) welfare losses

due to the dispersion of actions in the cross-section of the population (Wσσ

2E [(Q− q)2]).

The relative trade-off between volatility and dispersion, WQQ

Wσσ, is denoted by 1 − α∗, where

α∗ was named as the socially optimal degree of strategic complementarity (with exogenouspublic information) by Angeletos and Pavan (2007).9 Welfare losses are defined as L∗ =

E [W (k∗(θ), 0, θ)] − E[u], which are due to dispersion and non-fundamental volatility. Theefficient strategy q∗(si, z) can be found by maximising the ex-ante utility and is given in thenext proposition.

Proposition 2 (Efficiency)

For finite and non-zero precisions of information such that α∗ < 1, the efficient linear strategycan be written as q∗(si, z) = a∗si + b∗+ c∗E [θ | z] , where a∗ = k∗1γ

∗, b∗ = k∗0, c∗ = k∗1(1−γ∗),and the full information strategy is the unique linear function given by k∗(θ) = k∗0 + k∗1θ,where k∗0 =

−WQ(0,0,0)

WQQand k∗1 =

−WQθ

WQQ. The efficient weight to private information γ∗, is the

unique real root of the implicit equation γ = (1−α∗)τε

(1−α∗)τε+τθ+k21τuγ2−(1−α∗)(1−γ)2τuτεk

∗21

(τθ+k21τuγ

2)

,

where the precision of public information at the efficient strategy is τ ∗ = τθ + (k∗1)2(γ∗)2τu,and satisfies 0 < γ∗ < 1.

9Notice that with an information externality α∗ is no longer the optimal degree of strategic complement-arity. α∗ is often referred throughout the text as the efficient or socially optimal degree of coordination withexogenous public information.

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The efficient weight to the private signal increases with the precision of the noise inthe private signal, while it decreases with the precision of fundamentals and with the sociallyoptimal degree of strategic complementarity with exogenous public information, α∗. The com-parative statics of the efficient weight to the private signal with respect to the precision of thenoise in the public signal are ambiguous.

The comparison of the efficient and equilibrium strategies shows the inefficiencies whichmay be present at the equilibrium strategy.10 I note three differences. First, there maybe a full information inefficiency if k∗(θ) 6= k(θ), which can be measured by φ =

k∗1−k1k1

.Second, agents may not have the same individual incentives to align their actions comparedto what is collectively efficient with exogenous public information, which can be summarisedby α∗ − α and was first analysed by Angeletos and Pavan (2007). Third, the endogenouspublic signal causes an information externality. In contrast to the welfare planner, agents inequilibrium do not take into account that a higher weight on private information increasesthe precision of the endogenous public signal. This information externality is reflected inthe denominator of the efficient weight to private information, which includes an additionalterm: −(1− α∗)(1− γ∗)2 τuτεk

∗21

τ∗< 0. Notice that the information externality vanishes if the

noise in the endogenous public signal or the private signal are infinitely volatile (i.e. τu → 0

or τε → 0).Furthermore, I find that as the precision of the noise in the private signal increases from

zero to infinity, the efficient weight given to private information increases from 0 to 1. Theefficient weight given to the private signal is decreasing in the precision of fundamentals and inthe socially optimal degree of strategic complementarity with exogenous public information,α∗. If public information was exogenous, an increase in the precision of the noise in the publicsignal would lead to an unambiguous decrease in the efficient weight to private information. Incontrast, the effects of increasing the precision of the noise in the public signal are ambiguousbecause the welfare planner internalises the information externality and takes into accountthe effects of the change in the overall precision of public information which result from achange in the weight agents give to private information.

The externalities discussed above also give rise to an additional consequence: there is adifference between the efficient and equilibrium precisions of the endogenous public signalsince τ ∗ − τm = ((k∗1γ

∗)2 − (k1γm)2)τu. In other words, depending on whether the squared

difference of the efficient and equilibrium responses to private information is positive ornegative, the equilibrium precision of the endogenous public signal may be too high or toolow in relation to the efficient level. The following proposition spells out the comparison

10Throughout the text, a∗ is referred to as the efficient response to private information, while γ∗ is theefficient weight given to private information.

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between the equilibrium and efficient weights for the various benchmarks.

Proposition 3 (Comparison to benchmarks)

i) For non-zero and finite precisions of information, the sign of the difference between theequilibrium and efficient weights given to private information is:

sign (γm − γ∗) = sign

(α∗ − α− (1− α∗)(1− α)(k21τuτε)

((1− α)τε + τm)2

). (3)

Because of the information externality, agents underweight private information for a largerrange of degrees of strategic complementarity (α) with endogenous than with exogenous publicinformation.ii) For non-zero, finite precisions of information and τ ∗ = τθ+τv, the efficient weight given toprivate information is always larger with endogenous public information than with exogenouspublic information: γ∗ > γ∗exo.

The intuition of Proposition 3 is as follows. Beyond payoff externalities due to incompleteinformation (α∗ − α), agents do not take into account that the endogenous public signal notonly provides information about the fundamentals but also about the behaviour of otherplayers. Because of the information externality, Proposition 3 highlights that agents under-weight private information in relation to the efficient level for a larger range of degrees ofstrategic complementarity with endogenous than with exogenous public information.

As a result, if α > α∗ then agents always underweight private information in relation to theefficient level; but the difference in magnitude between the efficient and equilibrium weightsto private information is larger with endogenous than with exogenous public information. Onthe contrary, when α∗ > α, the additional coordination role of the endogenous public signalnarrows the wedge between the equilibrium and efficient weights to private information. Infact, when the information externality is sufficiently large, agents may equally or underweightprivate information in relation to the efficient level, thus overturning the sign of the differencebetween the equilibrium and efficient weights to private information in relation to when publicinformation is exogenous.

In addition, the welfare planner minimises welfare losses and takes into account the in-formation conveyed by the endogenous public signal about the aggregate action (which affectsthe level of strategic uncertainty). The welfare planner balances that an increase in the effi-cient weight to private information has three effects on welfare losses at a candidate efficientstrategy: (1) it increases dispersion; (2) it decreases non-fundamental volatility; (3) it in-creases the overall precision of the endogenous public signal, which reduces non-fundamentalvolatility. When public information is exogenous, the welfare planner balances effects (1)

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and (2). When public information is endogenous, effect (3) is negative and this implies that,for a constant precision of the endogenous public signal, increasing the weight to privateinformation increases welfare losses. Because the welfare loss function is strictly convex, thenthe efficient weight to private information is larger with endogenous than with exogenouspublic information.

5 Social Value of Public and Private Information

In this section, I analyse whether more precise public and private signals have a positive ornegative welfare effect, and focus on the differences in environments with endogenous andexogenous public information. I find the necessary and sufficient conditions for equilibriumwelfare to increase or decrease with an increase in the precision in the noise of the publicsignal, τu, (social value of public information) and private signal, τε, (social value of privateinformation).11

When there is a full information inefficiency, φ 6= 0, equilibrium welfare is E [u] =

E [W (k(θ), 0, θ)]+E[WQ(k(θ), 0, θ)(Q−k(θ))]+WQQ

2E [(Q− k(θ))2]+Wσσ

2E [(Q− q)2]. There

is a new term (E[WQ(k(θ), 0, θ)(Q−k(θ))]) in relation to when the full information strategy isefficient, which is related to the covariance between the social return to the aggregate actionand the difference between the equilibrium aggregate action with incomplete and completeinformation (hereafter it will be abbreviated as the ’covariance term in equilibrium welfarelosses’). Equilibrium welfare losses are then equal to WL = E [W (k(θ), 0, θ)] − E[u] and ata candidate equilibrium strategy the expression is equal to

WL(γ) =| Wσσ | k21

2

(γ2

τε+

(1− α∗)(1− γ)2

τ+

2(1− α∗)φ(1− γ)

τ

), (4)

where the first term corresponds to dispersion, the second to non-fundamental volatility,and the third is the ’covariance term in equilibrium welfare losses’.12 Then I substitute theequilibrium strategy defined by the triplet (k0, k1, γ

m) in the expression for welfare losses andderive comparative statics with respect to the informational parameters.13

11I assume that the welfare planner cannot change the precision of the ex-ante fundamentals, τθ.12Claim 1 in the Appendix derives expression (4).13There are alternative ways to express equilibrium welfare losses. Bergemann and Morris and (2013) and

Ui and Yoshizawa (2015) write welfare losses as a linear combination of dispersion and volatility. Dispersionis the same as defined in this paper but volatility is defined differently and can be expressed as the covarianceof actions (i.e. cov(qi, qj) = var(Q2)).

14

5.1 Social value of public information

When public information is endogenous, the total welfare effect of changing the precision ofthe noise in the public signal can be written as

d(E[u(γm)])

dτu=

(∂(E[u(γm)])

∂τm

)γ cons.

dτm

dτu. (5)

There are two multiplicative effects on equilibrium welfare. The first is the partial welfareeffect of a change in the precision of the noise in the public signal on equilibrium welfare whilstkeeping the weight on the private signal fixed. This effect has the same sign as when publicinformation is exogenous. It depends on the relative weight of dispersion, volatility, and the’covariance term in equilibrium welfare losses’, and can be summarised by the relationshipbetween α, α∗ and on the degree of inefficiency with full information, φ.

The second is the effect on the overall precision of the endogenous public signal due to achange in the precision of the noise in the endogenous public signal. The second effect alwayshas a positive sign because it can be further decomposed into two terms: a) an accuracy effect:for a fixed weight on the private signal, an increase in the precision of the noise in the publicsignal increases the overall precision of the endogenous public signal; b) a crowding-out effectsince increasing the precision of the noise in the public signal reduces the weight that agentsplace on the private signal, which reduces the precision of the endogenous public signal. Byadding the two effects, I find that accuracy effect always dominates the crowding-out effectand hence dτm

dτu> 0. These findings are stated in proposition below.

Proposition 4 (Comparison of the Social Value of Public Information with En-dogenous and Exogenous Public Signals)

i) The sign of the welfare effect of increasing the precision of the noise in the public signal isthe same with exogenous and endogenous public information.ii) If | k1 |≤ 1 then the social value of public information is smaller than the partial welfareeffect, while if | k1 | is sufficiently large then the social value of public information is largerthan the partial welfare effect.

Because the crowding-out effect is always dominated by the accuracy effect, the partialwelfare effect fully determines the sign of the social value of public information. However, themagnitude total social value of public information may differ with endogenous and exogenouspublic information because of the crowding-out effect of increasing the precision of the noisein the public signal on the overall precision of the endogenous public signal. The effect onthe overall precision of the endogenous public signal is bounded above by k21. As a result, if

15

| k1 |≤ 1 then the social value of public information is smaller than the partial welfare effectand the inequality is reversed if the full information response to fundamentals is sufficientlylarge. In the next corollary, I show how endogenous public information modifies each of thethree components of equilibrium welfare losses.

Corollary 1 (Social Value of Public information: Effects on Each the Componentsof Equilibrium Welfare Losses)

Increasing the precision of the noise in the public signal has the following effects on thecomponents of equilibrium welfare losses:

a) decreases dispersion;b) the sign of non-fundamental volatility is equal to sign((1− α)− τm

τε);

c) decreases the ’covariance term in equilibrium welfare losses’ if φ > 0 and increases itif φ < 0.

The intuition of the previous corollary is as follows. First, dispersion decreases withpublic information since when the public signal is more precise agents place less weight onthe private signal, and as a result, the cross-section dispersion of actions decreases. Second,increasing the precision of the public signal has an ambiguous effect on volatility since moreprecise public information decreases non-fundamental volatility, but at the same time, itincreases the weight agents place on public information, which increases non-fundamentalvolatility. Therefore, its overall effect is ambiguous and depends on the degree of strategiccomplementarity and on the ratio of public to private precisions. Third, when the full in-formation strategy is inefficient, then increasing the precision of public information reducesthe ’covariance term in equilibrium welfare losses’ if the full information equilibrium responseto fundamentals is lower than the efficient one, i.e. φ > 0. This is because it reduces thecovariance between the aggregate error due to incomplete information (Q − k(θ)) and thefull information equilibrium strategy (k(θ)). Otherwise, an increase in the precision of publicinformation increases the ’covariance term in equilibrium welfare losses’. Proposition 5 belowfinds the necessary and sufficient conditions for determining whether increasing the noise inthe endogenous public signal has a positive social value.

Proposition 5 (Social Value of Public Information)

For non-zero and finite precisions of information, equilibrium welfare increases with respectto the precision of the noise in the public signal, τu, if

ητε + τmρ < 0, (6)

16

and decreases if ητε + τmρ > 0, where η = (1 − α)(2α − α∗ − 1 − 2φ(1 − α∗)) and ρ =

−(1 + 2φ)(1− α∗).

The proposition shows how the welfare effects of varying the noise in the endogenouspublic signal depend on both the ratio of public to private precisions, ( τm

τε), and also on a

combination of payoff relevant parameters η and ρ (which combine the three components ofwelfare described in Corollary 1). The payoff relevant parameters are invariant to whetherpublic information is exogenous or endogenous (Proposition 4). If both η and ρ have anegative sign then the social value of public information is positive, while if both η and ρ

have a positive sign then the social value of public information is negative. If η and ρ havedifferent signs then there will be a condition which will depend on the ratio of precisions thatdetermines whether more precise public information has a positive social value. Corollaries2 and 3 further develop the implications of Proposition 4 in relation to the primitives of themodel.

Corollary 2 (Social Value of Public Information & the Primitives of the Model)

i) If α∗ = α then the social value of private information is positive if φ > −12, zero if φ = −1

2

and negative if φ < −12.

ii) If α∗ > α anda) φ ≥ −1

2then the social value of public information is always positive.

b) (α−α∗)−(1−α)2(1−α∗) < φ < −1

2then, for a given finite precision of private information, there

exists τ ∗u such that the social value of public information is positive if and only if τu < τ ∗u .c) φ ≤ (α−α∗)−(1−α)

2(1−α∗) then the social value of public information is always negative.iii) α > α∗ and

d) φ ≥ (α−α∗)−(1−α)2(1−α∗) then the social value of public information is always positive.

e) −12< φ < (α−α∗)−(1−α)

2(1−α∗) then, for a given a finite precision of private information, thereexists a τ ∗∗u such that the social value of public information is positive if and only if τu > τ ∗∗u .

f) φ ≤ −12

then the social value of public information is always negative.Note: From (6) define f(τu) = ητε + k21τu(γ

m)2ρ + ρτθ. In case ii.b, η < 0 and ρ > 0 andτ ∗u is the unique root which satisfies f(τ ∗u) = 0. In case iii.e, η > 0 and ρ < 0 and τ ∗∗u is theunique root which satisfies f(τ ∗∗u ) = 0.

If there are no inefficiencies with exogenous public information then increasing the preci-sion of the public signal has a positive social value. Similarly, if α∗ = α and agents insuffi-ciently respond to fundamentals with full information then more precise public informationdecreases the ’covariance term in equilibrium welfare losses’ and has a positive social value.

17

If φ > −12

then increasing the precision of the noise in the public signal fosters a decrease indispersion which is always greater than the increase in non-fundamental volatility and thepotential increase of ’covariance term in equilibrium welfare losses’.

In environments with α∗ > α or α > α∗ not all the cases a-c or d-f, respectively, need toapply since the parameter restrictions of the game may make some of the regions infeasible. Inthese economies, if the full information response to fundamentals is sufficiently large then thesocial value of public information is unambiguously positive since the decrease in dispersionand the ’covariance term in equilibrium welfare losses’ outweighs the potential increase innon-fundamental volatility. In contrast, if the full information response to fundamentals issufficiently small then the social value of public information is unambiguously negative. Thethreshold value of φ that determines whether the social value of public information is alwayspositive or negative is different for economies where agents in equilibrium coordinate more orif they coordinate less than is efficient with exogenous public information. Note that if φ = 0

and α∗ > α then more precise public information always has a positive social value since ithelps agents coordinate and reduce the gap between the equilibrium and efficient strategies(case a in Corollary 2). However, if φ = 0 and α > α∗ then more precise public informationmay be detrimental since it increases the gap between the equilibrium and efficient strategies(case d or e in Corollary 2).

The social value of public information in the intermediate regions depends on a conditionwhich depends on the primitives of the model and on the relative precisions of public toprivate information. If α∗ > α then more precise public information has a positive socialvalue if the precision of the noise in the public signal is sufficiently small in relation to a givenprecision of the private signal (i.e. in cases ii.b of Corollary 2). This is because increasingthe precision of the noise in the public signal reduces the social value of public informationsince the increase of the ’covariance term in equilibrium welfare losses’ and the potentialincrease in non-fundamental volatility are larger than the reduction in dispersion. However,if α > α∗ then the social value of public information is positive if the precision of the noisein the public signal is sufficiently large in relation to a given precision of private information.The next corollary outlines further differences that require specific assumptions.

Corollary 3 (Differences between the Social Value of Public Information withEndogenous and Exogenous Public Signals)

For a given precision of private information, if τu = τv and | k1 |≤ 1 and:

• α∗ > α and (α−α∗)−(1−α)2(1−α∗) < φ < −1

2(case b of Corollary 2) then the social value of

public information is positive for a larger range of precisions of the noise in the public

18

signal when public information is endogenous compared to when public information isexogenous.

• α > α∗ and −12< φ < (α−α∗)−(1−α)

2(1−α∗) (case e of Corollary 2) then the social value of publicinformation is positive for a smaller range of precisions of the noise in the publicsignal when public information is endogenous compared to when public information isexogenous.

With endogenous public information and if | k1 |≤ 1 and τu = τv then the informationalcontent of the endogenous public signal is smaller than that of the exogenous public signal.If agents coordinate more than is socially optimal with exogenous public information, agentsin equilibrium place too much weight on the public signal, which implies that more precisepublic information is detrimental if private information is sufficiently more precise than publicinformation. This means that for the same precision of private information, there will bea smaller range of precisions where the social value of public information is positive. Theopposite occurs when agents in equilibrium coordinate less than is socially optimal withexogenous public information.

5.2 Social value of private information

When the public signal is endogenous, the total welfare effect of changing the precision ofthe private signal is as follows:

d(E[u(γm)])

dτε=

(∂(E[u(γm)])

∂τε

)τ cons.

+

(∂(E[u(γm)])

∂τm

)γ cons.

dτm

dτε. (7)

The are two additive effects which result from changing the precision of the noise in theprivate signal on welfare. First, the partial welfare effect : the effect of changing the precisionof the noise in the private signal on equilibrium welfare, whilst keeping the precision of thenoise in the public signal fixed. The sign of this effect is the same as when public informationis exogenous and depends on the relative weights of volatility, dispersion, and the ’covarianceterm in equilibrium welfare losses’.

The second effect, the welfare effect on the overall precision of the endogenous publicsignal due to a change in the precision of the noise in the private signal is as follows: if theprecision of the noise in the private signal increases then the precision of the endogenouspublic signal also increases, which has a welfare effect composed of two parts. The first partis that a higher precision of the noise in the private signal increases the overall precision ofthe endogenous public signal (dτm

dτε> 0) and is related to the crowding-out effect of exogenous

public information on the overall precision of the endogenous public signal discussed in 5.1.

19

The second part is the welfare effect of increasing the overall precision of the endogenouspublic signal on the social value of private information (its sign is given in Proposition 5).The next corollary describes the welfare effects of increasing the precision of the noise in theprivate signal on each of the components of equilibrium welfare losses.

Corollary 4 (Social Value of Private information: Effects on Each the Compon-ents of Equilibrium Welfare Losses)

Increasing the precision of the noise in the private signal has the following effects on thecomponents of equilibrium welfare losses:

a) has an ambiguous effect on dispersion. Increasing the precision of the private signalreduces dispersion in relation to when public information is exogenous and the sign of thedispersion effect may be overturned from a positive to a negative sign;

b) has an ambiguous effect on non-fundamental volatility. If (1−α) < τm

τεthen increasing

the precision of the private signal reduces non-fundamental volatility in relation to when publicinformation is exogenous, while if (1−α) > τm

τεthen non-fundamental volatility increases and

the sign may be overturned from negative to positive in relation to when public informationis exogenous;

c) decreases the ’covariance term in equilibrium welfare losses’ if φ > 0 and increases itif φ < 0.

The interpretation is as follows. First, when public information is exogenous, an increasein the precision of the private signal increases dispersion if and only if (1 − α) < τm

τε. With

an endogenous public signal, an increase in the precision of private signal will foster thatthe overall precision of the public signal is more informative, which makes agents place moreweight on the public signal and less on the private signal. This further reduces the dispersionin actions in relation to when public information is exogenous, and may overturn the signof the dispersion effect from positive to negative compared to when public information isexogenous. Second, the effect on non-fundamental volatility is ambiguous. When publicinformation is exogenous, an increase in the precision of the private signal always reduces non-fundamental volatility. However, there is now an additional effect: increasing the precision ofthe private signal increases the precision of the public signal, which increases non-fundamentalvolatility if (1 − α) > τm

τε. The total effect on non-fundamental volatility will depend on

which effect dominates. Third, when the full information strategy is not efficient, the signof the ’covariance term in equilibrium welfare losses’ follows from the same arguments asthose discussed in Corollary 1. The next proposition summarises how the effects combine indetermining the social value of private information.

20

Proposition 6 (Social Value of Private Information)

For non-zero and finite precisions of information, equilibrium welfare increases with the pre-cision of the noise in the private signal if

(ϕτε + χτm) + υ(ητε + ρτm) < 0, (8)

and decreases if (ϕτε + χτm) + υ(ητε + ρτm) > 0, where ϕ = −(1− α)2(1− α+ 2φ(1− α∗)),χ = (1 − α)((2α∗ − α − 1) − 2φ(1 − α∗)), η = (1 − α)(2α − α∗ − 1 − 2φ(1 − α∗)), ρ =

−(1 + 2φ)(1− α∗) and ν = dτm

dτε> 0.

The proposition spells out the necessary and sufficient conditions for equilibrium welfareto increase with respect to the precision of the noise in the private signal. Note that an endo-genous public signal changes the payoff parameter combination that determines whether thesocial value of private information is positive or negative since the social value of private in-formation is related to whether public information is beneficial or detrimental. There are fourparameters, η, ρ, ϕ and χ, which depend on a combination of dispersion, non-fundamentalvolatility, and the ’covariance term in equilibrium welfare losses’ that determine whether moreprecise private information has a positive social value. If the four parameters are negativethat means that more precise private information will have a positive social value, while ifthey are all positive then the social value of private information will be negative. If someparameters are positive and others are negative then there will be a condition which will de-pend on the ratio of public to private precisions that determines whether more precise privateinformation has a positive social value. Corollary 5 combines these effects and characterisesall the possible cases in terms of the primitives of the model.

Corollary 5 (Social Value of Private Information & the Primitives of the Model)

i) For α∗ = α the social value of private information is positive if φ > −12, zero if φ = −1

2,

and negative if φ < −12.

ii) For α∗ > α there are five possibilities:a) If φ ≥ (α∗−α)−(1−α∗)

2(1−α∗) then social value of private information is positive.b) If −1

2≤ φ < (α∗−α)−(1−α∗)

2(1−α∗) then the partial welfare effect has a positive social value ifand only if τε > τ ∗ε , while the welfare effect on the overall precision of the endogenous publicsignal always has a positive social value.

c) If −(1−α)2(1−α∗) < φ < −1

2then the partial welfare effect has a positive social value if and only

if τε > τ ∗ε , while the welfare effect on the overall precision of the endogenous public signal hasa positive social value if and only if τu < τ ∗u .

21

d) If (α−α∗)−(1−α)2(1−α∗) < φ ≤ −(1−α)

2(1−α∗) then the partial welfare effect always has a negative socialvalue, while the endogenous public precision effect has a positive social value if and only ifτu < τ ∗u .

e) If φ ≤ (α−α∗)−(1−α)2(1−α∗) then the social value of private information is negative.

iii) For α > α∗ there are five possibilities:f) If φ ≥ (α−α∗)−(1−α)

2(1−α∗) then the social value of private information is positive.g) If −(1−α)

2(1−α∗) ≤ φ < (α−α∗)−(1−α)2(1−α∗) then the partial welfare effect always has a positive social

value while the welfare effect on the overall precision of the endogenous public signal has apositive social value if and only if τu > τ ∗∗u .

h) If −12< φ < −(1−α)

2(1−α∗) then the partial welfare effect has a positive social value if and onlyif τε < τ ∗∗ε while the welfare effect on the overall precision of the endogenous public signalhas a positive social value if and only if τu > τ ∗∗u .

i) If (α∗−α)−(1−α∗)2(1−α∗) < φ ≤ −1

2then the partial welfare effect has a positive social value if

and only if τε < τ ∗∗ε while the welfare effect on the overall precision of the endogenous publicsignal always has a negative social value.

j) If φ ≤ (α∗−α)−(1−α∗)2(1−α∗) then the social value of private information is negative.

Note: From (8) define g(τε) = ϕτε + χτm, which determines the sign of the partial welfareeffect. In case ii.b when χ > 0 and ϕ < 0 then τ ∗ε is the unique root that satisfies g(τ ∗ε ) = 0,where the partial welfare effect has a positive social value if and only if τε > τ ∗ε for a givenprecision of the noise in the public signal. In case iii.e when χ < 0 and ϕ > 0 and τ ∗∗ε is theunique root which satisfies g(τ ∗∗ε ) = 0 where partial welfare effect has a positive social valueif and only if τε < τ ∗∗ε for a given precision of the noise in the public signal.

The previous corollary shows how all the possible sign combinations depend on the rela-tionship between the full information inefficiency, φ, and the inefficiency in coordination dueto incomplete information (α∗−α). In environments with α∗ > α or α > α∗ not all the casesa-e or f-j, respectively, need to apply since the parameter restrictions of the game may makesome of the regions infeasible.14

When there are no inefficiencies with incomplete information (α∗ = α), the social value ofprivate information is positive if agents under- or equally-respond to the fundamentals whenthey have complete information, or if the full information over-response to the fundamentalsis sufficiently small (0 > φ > −1

2). When there are inefficiencies with incomplete information

(α∗ 6= α), if the full information inefficiency is sufficiently large then more precise privateinformation has a positive social value since it sufficiently decreases the ’covariance term inequilibrium welfare losses’ to compensate any potential detrimental effect. In contrast, when

14When there are no inefficiencies with full information: i) If α∗ > α then only cases a or b (or both) ofCorollary 5 may apply; ii) If α > α∗ then only cases f or g (or both) of Corollary 5 may apply.

22

the full information inefficiency is sufficiently small then more precise private information hasan unambiguously negative social value. The threshold values that determine whether thesign of the social value of private information is unambiguously positive or negative differ ifpublic information is endogenous compared to when public information is exogenous.

The intermediate cases are as follows. Due to the partial welfare effect, if α∗ > α thenmore precise private information may be detrimental since it fosters a larger equilibriuminefficiency. In particular, the greater the precision of public information in relation toprivate information, the more likely it is that the provision of more precise private informationdecreases welfare. But the social value of public information needs to be taken into accountbecause of the welfare effect on the overall precision of the endogenous public signal. Infact, I show that if more precise public information has a negative social value then privateinformation also has a negative social value. In contrast, due to the partial welfare effect, ifagents coordinate more than is socially optimal with exogenous public information (α > α∗)then more precise private information is beneficial since it further aligns actions with theefficient strategy. If, however, the full information inefficiency is sufficiently negative then thegreater the precision of private information in relation to public information, the more likelyit is that the provision of more precise private information decreases welfare. However, takinginto account the welfare effect on the overall precision of the endogenous public signal reducesthe social value of private information since I show that if more precise private informationhas a negative social value then more precise public information also has a negative socialvalue.

Remark 1: Endogenous public information modifies the ratio of public to private pre-cision that determines whether the partial welfare effect is positive or negative in relation towhen public information is exogenous. For a given precision of public information such thatτu = τv, | k1 |≤ 1, and if α∗ > α then the range of precisions of the noise in the private signalwhere the partial welfare effect has a positive social value is larger with endogenous publicinformation compared to when public information is exogenous. Conversely, if α > α∗ thenthe partial welfare effect has a positive social value for a smaller range of precisions of thenoise in the private signal in environments where public information is endogenous comparedto when public information is exogenous.

Remark 2: The threshold value of φ that determines whether the social value of privateinformation is unambiguously negative is smaller with endogenous than with exogenous publicinformation if α∗ > α. If α > α∗ then the threshold value of φ that determines whether thesocial value of private information is unambiguously positive is larger with endogenous than

23

with exogenous public information.

Proposition 7 (Social Value of Private Information with an Endogenous PublicSignal)

i) Endogenous public information changes the sign and the magnitude of the total welfareeffect that results from increasing the precision of the noise in the private signal in relationto the partial welfare effect:

-it augments the social value of private information if the social value of public informationis positive;

-it decreases the social value of private information if the social value of public informationis negative.

ii) When the welfare effect on the overall precision of the endogenous public signal andthe partial welfare effect have different signs it may be that endogenous public informationoverturns the sign of the total welfare effect in relation to the partial welfare effect:

-will not overturn the sign of the social value of private information if α∗ = α;-may overturn the social value of private information from negative to positive if α∗ >

α and if (α−α∗)−(1−α)2(1−α∗) < φ < (α∗−α)−(1−α∗)

2(1−α∗) ;-may overturn the social value of private information from positive to negative if α >

α∗ and if (α∗−α)−(1−α∗)2(1−α∗) < φ < (α−α∗)−(1−α)

2(1−α∗) .

In relation to when public information is exogenous, increasing the precision of the privatesignal augments the overall informativeness of the public signal, which has an additionalpositive (negative) effect on the social value of private information if the social value of publicinformation is positive (negative). When both the partial welfare effect and the welfare effecton the overall precision of the endogenous public signal have the same sign then there is onlya magnitude effect of the endogenous public signal on the social value of private information.This occurs for example when there are no payoff externalities with incomplete information(α∗ = α), or in cases where the social value of private information is unambiguously positive(cases a and f of Corollary 5) or negative (cases e and j of Corollary 5). When the partialwelfare effect and welfare effect on the overall precision of the endogenous public signal havedifferent signs then it is possible that endogenous public information overturns the sign ofthe total welfare effect in relation to the partial welfare effect.

If α∗ > α then endogenous public signal reduces the difference between the equilibriumand efficient weights given to private information in relation to when public information isexogenous (Proposition 3). Because of this gain in efficiency the sign of the social value ofprivate information may be overturned from negative to positive when public information isexogenous (and not the other way round). This may occur in regions b, c or d of Corollary 5.

24

The opposite occurs if α > α∗ since the information externality magnifies the extent to whichagents underweight private information in relation to when public information is exogenous.As a result, because of this loss in efficiency the sign of the social value of private informationmay be overturned from positive to negative when public information is exogenous (and notthe other way round). This may occur in regions g, h or i of Corollary 5.

As a consequence, endogenous public information may modify the cutoff degree of stra-tegic complementarity that determines whether the total welfare effect of increasing the pre-cision of the noise in private information is positive or negative. Since the endogenous publicinformation may overturn the social value of private information from negative to positive, itthen follows that the threshold value of φ̂ such that if φ > φ̂ then the social value of privateinformation is positive is smaller with endogenous than with exogenous public information,i.e. φ̂ ≤ φ̂exo. In other words, the social value of private information will be positive fora larger range of φ. The converse occurs if α > α∗ for the reasons outline throughout thissection. In addition, this threshold value φ̂ will not only depend on the parameters of thepayoff function but also on the information structure.

6 Applications

In this section, I illustrate the main findings of the paper with specific well-known applicationswhere endogenous public information is relevant and compare them to the related literature(e.g. exogenous public information; information acquisition literature; when the endogenouspublic signal is the market price).

6.1 Firms Competing in a Homogeneous Product Market

Suppose that an economy is composed of a continuum of households, each consisting ofproducers and consumers who make production choices with quadratic production costs.Agents are uncertain about common fundamentals, θ, which represent the intercept of themarginal cost. Each household chooses quantities q1i, q2i of the two goods in the economy bymaximising the utility given by: ui = δq1i+q2i, subject to the budget constraint lq1i+q2i = πi,where l is the price of good 1, good 2 is the numeraire, and πi are the profits of producer igiven by: πi = (l− θ)qi− λq2i

2, where qi is the quantity produced by household i. The inverse

demand for good 1 is l = δ − βQ, where Q is the total amount produced by all householdsin the economy. By imposing symmetry of production choices and market clearing, themaximisation problem subject to the budget constraint is equivalent to:

25

U(q,Q, σq, θ) = (δ − βQ)qi +βQ2

2− θqi −

λq2i2. (9)

Regularity conditions require that β + λ > 0 and 2β + λ > 0. An agent’s information setconsists of a private forecast about common fundamentals and a noisy public forecast aboutthe total amount produced by all households in the economy. The public forecast is compiledby an information agency that produces real-time forecasts about the economic activity andis therefore endogenous. An agent’s strategy is contingent on the private forecast and on thecontemporaneous endogenous noisy public forecast about aggregate production.

The degree of strategic complementarity is α = −βλ

and total surplus is W (Q, σq, θ) =

TS(Q, σq, θ) =´U(q,Q, σq, θ)di = δQ − (β+λ)Q2

2− θQ − λσ2

q

2. With full information the

economy is efficient since k(θ) = k∗(θ) = δ−θλ+β

, and hence φ = 0. Furthermore, α∗ = α = −βλ.

Next, I apply the main results of the paper to this economy.

Corollary 6

i) Agents in equilibrium give an inefficiently low weight to private information for all degreesof strategic complementarity.ii) Welfare increases with the precision of the noise in public and private signals.

First, in an economy with no externalities with complete information (φ = 0) nor withincomplete information (α∗ = α), with an endogenous public signal agents in equilibriumalways give an inefficiently low weight to private information since they do not take intoaccount how their individual actions affect the precision of the endogenous public signal (in-formation externality). In contrast, if public information was exogenous then the equilibriumstrategy would be efficient.

Second, welfare increases with the precision of the noise in the public and private signalssince there are no payoff externalities with complete nor incomplete information (Corollaries2 and 5). In fact, with exogenous public information, it is also the case that welfare increaseswith both the precision of the noise in the public and private signals. However, the differencein the two environments is that endogenous public information changes the magnitude of thewelfare effect. If λ + β ≥ 1 then a change in the precision of the noise in the public signalfosters a smaller increase in total surplus in relation to when public information is exogenous.

This model was first studied by Vives (1988) and further developed by Angeletos andPavan (2007) with exogenous public information, and Bru and Vives (2002) with endogenouspublic information in a pure prediction with k(θ) = k∗(θ) and α = α∗. Models that considerthe same payoff structure but in which price has an additional allocation role, such as inVives (2015), and where agents use price contingent strategies, lead to different results as

26

those presented in Corollary 6. For example, agents in equilibrium can overweight privateinformation whenever actions are strategic substitutes (since the market price has dual in-formation and allocation roles). In addition, the utility function presented in this section canbe viewed as a reduced form of Amador and Weill (2010) who consider that prices are anendogenous source of public information. Their results contrast with those of Corollary 6.iisince they find that more precise public information can be detrimental. The difference isdue to an additional source of strategic complementarity generated by the way agents learnfrom prices, which amplifies the negative effect of changing the precision of the noise in thepublic signal on the overall precision of the endogenous public signal.

6.2 Beauty Contest

Morris and Shin (2002) considered a utility function that formalises Keynes’ beauty contestmetaphor on how financial markets work: agents are not only concerned about predictingfundamentals but also about outguessing the likely actions of others. An agent’s utilityfunction can be expressed as

U(q,Q, σq, θ) = −(1− r)(qi − θ)2 − r(qi −Q)2 + rσ2q , (10)

where the degree of strategic complementarity is α = r and is assumed to be r ∈ (0, 1).Each agent has access to a private signal and an endogenous public signal that emergesbecause trades occur in centralised markets, such as in exchanges that aggregate dispersedmarket information. Agents submit a schedule that is contingent on the private signal andon the endogenous public signal about the aggregate action. Social welfare is W (Q, σq, θ) =

−(1− r)(Q− θ)2 + rσ2q with α∗ = 0. With full information, the equilibrium action is efficient

and given by: k(θ) = k∗(θ) = θ. The next corollary applies the main results of the paper tothe beauty contest.

Corollary 7

i) Agents in equilibrium always underweight private information in relation to the efficientlevel.ii) Social value of information:

1. if α ≤ 12then, for a given precision of the private signal, welfare increases with the

precision of the noise in public and private signals.2. if α > 1

2then, for a given precision of the private signal, there are two regions15:

15τ∗∗u is defined in Corollary 2.

27

-region 1: welfare increases with the precision of the noise in public and private signals ifτu > τ ∗∗u .

-region 2: welfare increases with the precision of the noise in the private signal anddecreases with the precision of the noise in the public signal if τu < τ ∗∗u .

The corollary shows that agents in equilibrium overweight public information in relationto the efficient level since public information helps agents coordinate more than is sociallyoptimal with exogenous public information (α = r > 0 = α∗). The size of difference betweenthe efficient and equilibrium weights to private information is larger with endogenous thanexogenous public information due to the information externality.

With respect to the social value of information, I find the following differences with respectto the well-known result for the beauty contest (Morris and Shin (2002)) that if the degreeof strategic complementarity is large and if the private signal is sufficiently precise then moreprecise public information may be welfare-reducing. First, I note that for a given precisionof the private signal, the social value of public information is positive for a smaller rangeof precisions of the noise in the public signal with endogenous than with exogenous publicinformation (Corollary 3). Second, the magnitude of the total welfare effect of changing theprecision of the noise in the endogenous public signal is smaller than the partial welfare effect(which is the only effect that is present when there is exogenous public information). Theimplication for public policy is that in order to achieve a certain welfare increase, the precisionof the noise in public information needs to be larger with endogenous than exogenous publicinformation.16

In addition, the results of Corollary 7 can be related to models that consider thatagents can acquire costly private information and hence, private information is endogen-ous. Colombo et al. (2014) find that the crowding-out effect of public information on privateinformation may turn the social value of public information from negative (with exogenousprivate information) to positive (with endogenous private information) in the beauty contest.

6.3 Competition à la Cournot and à la Bertrand with product dif-

ferentiation

Consider a large market (monopolistic competition) where firms compete either on pricesor on quantity strategies, and sell differentiated products. Each firm is uncertain abouta common shock that affects its demand, θ. A firm that competes à la Cournot faces thefollowing linear inverse demand: pi = θ−(1−δ)κi−δK, where (pi, κi) are price-quantity pairs

16Furthermore, even though when α > 12 the welfare effect on the overall precision of the endogenous

public signal may potentially overturn the partial welfare effect in determining the social value of privateinformation, this does not occur in the beauty contest because of the specific payoff parameter configuration.

28

for firm i, K is the aggregate quantity of all firms in the economy, and δ can be interpretedas the degree of product differentiation with δ ∈ (0, 1). At the limit, when δ → 0 firms areisolated monopolies and there is perfect competition when δ → 1. A firm that competes à laBertrand faces the following linear demand: ki = θ− (1 + ς)pi + ςP , where ς = 0 and (pi, κi)

are price-quantity pairs for firm i and P is the aggregate price of all firms in the economy.17

This payoff structure is from Vives (1990).To form expectations about the common shock that affects its demand, a firm has access

to a private signal as a result of the observation of local market conditions and a publicsignal, which all other firms can access without cost. The public signal is a contemporaneousforecast of the aggregate strategy of all other firms in the market. This aggregate strategysignal contains noise since the aggregation process is imperfect, or there is measurementerror. Firms condition their price or quantity strategies on both the private signal andthe public contemporaneous forecast, which is endogenous, about the aggregate quantity orprice, respectively. In contrast, if the system or platform tracked and disclosed fundamentals(instead of the market activity) then public information would be exogenous.

Competition à la Cournot The profit of firm i that competes à la Cournot can bewritten as: πi = θκi − (1− δ)κ2i − δqiK. Setting κi = qi and K = Q, the profit of each firmis:

U(q,Q, σq, θ) = θqi − (1− δ)q2i − δqiQ. (11)

I note that if firms had complete information then the profit maximising quantity would beequal to k1(θ) = θ

2−δ . Notice that firms strategies are strategic substitutes since α = −δ2(1−δ) <

0. Total producer surplus is as follows: PS = W (Q, σq, θ) = θQ−Q2−(1−δ)σ2q , which implies

that α∗ = 2α < 0. Furthermore, with full information firms over-respond to fundamentalssince k1 > k∗1 = 1

2, which implies that the full-information inefficiency is φ = −δ

2= α

1−α∗ < 0.The next corollary applies the main results of the paper to the competition à la Cournot.

Corollary 8

i) Firms that compete à la Cournot underweight private information in relation to the efficientlevel.ii) Social value of information. Depending on the degree of substitutability there are threecases for producer surplus18:

1. if −12≤ α < 0 then producer surplus increases with the precision of the noise in public

and private signals.17It can be shown that one equation in the demand system is the inverse of the other if δ = ς

1+ς .18τ∗∗u is defined in Corollary 2 and τ∗∗ε is defined in Corollary 5.

29

2. if ϑ ≤ α < −12, where −1 < ϑ < −1

2, then there are two regions:

-Region 1. Producer surplus increases with the precision of the noise in public and privatesignals if, for a given precision of the private signal, τu > τ ∗∗u .

-Region 2. Producer surplus decreases with the precision of the noise in the public signaland increases with the precision of the noise in the private signal if, for a given precision ofprivate signal, τu < τ ∗∗u .

3. if α < ϑ, where −1 < ϑ < −12, then there are three regions:

-Region 3. Producer surplus increases with the precision of the noise in public and privatesignals if, for a given finite precision of the private signal, τu > τ ∗∗u .

-Region 4. Producer surplus decreases with the precision of the noise in the public signaland increases with the precision of the noise in the private signal. This region is defined if,for a given finite precision of the private signal, τu < τ ∗∗u and if, for a given precision of thenoise in the public signal, τε < τ ∗∗ε .

-Region 5. Producer surplus decreases with the precision of both the noises in public andprivate signals, if for a given precision of the private signal, τu < τ ∗∗u , and for a given precisionof the noise in the public signal, τε > τ ∗∗ε .

Both with exogenous and endogenous public information, firms that compete à la Cournotunderweight private information since strategies are strategic substitutes but less than issocially optimal from the perspective of a planner who maximises producer surplus. Themagnitude of the difference between the equilibrium and efficient weights is larger with en-dogenous than with exogenous public information due to the information externality.

Next, I apply Corollaries 2 and 5 to this economy. When considering that public inform-ation is endogenous, there is one similarity and two differences in relation to when publicinformation is exogenous (e.g. Angeletos and Pavan (2007), Myatt and Wallace (2015), Uiand Yoshizawa (2015)).19 The similarity is that the Cournot game defined above with pro-ducer surplus as welfare benchmark has three cases depending on the combination of thecomponents of welfare losses (Corollary 8), which correspond to a game of Type I, Type II,or Type III, respectively, in the terminology of Ui and Yoshizawa (2015). Hence, the type ofgame depends on the structure of the payoff function and not the structure of the informationset.

The two differences are as follows. First, endogenous public information modifies thecut-off degree of strategic complementarity that defines cases 2 and 3 in Corollary 8. Whenpublic information is exogenous, the cut-off degree of strategic complementarity is -1 whilewith endogenous public information it is ϑ, which satisfies −1 ≤ ϑ < −1

2. In addition, ϑ

19With total surplus as welfare benchmark the social value of information would be positive both withexogenous (such as in Vives (1988)) and with endogenous public information.

30

depends not only on the parameters of the payoff function but also on the characteristicsof the information structure. The cut-off ϑ changes since endogenous public informationmay overturn the social value of private information from positive to negative in this region(Corollary 7). This means that endogenous public information reduces the range of strategiccomplementarities that fall into case 2 and increases the ones that fall into case 3. The seconddifference is that endogenous public information modifies the threshold values τ ∗∗u and τ ∗∗ε

that define the regions within the cases. In particular, the social value of public informationis positive for a smaller range of precisions of the noise in public information with endogenousthan with exogenous public information.

Competition à la Bertrand The profit of a firm that competes à la Bertrand can bewritten as: πi = θpi − (1 + ς)p2i + ςPpi. Setting pi = qi and P = Q, the profit is:

U(q,Q, σq, θ) = θqi − (1 + ς)q2i + ςQqi. (12)

With full information, the profit maximising price is equal to k(θ) = θ2+ς

and the degree ofstrategic substitutability is α = ς

2(1+ς)> 0. In Bertrand competition, W , is also equal to

producer surplus and PS = W (Q, σq, θ) = θQ − (1 + ς)σ2q − ςQ2, which implies that α∗ =

2α = ς(1+ς)

> 0. In other words, actions are strategic complements but less than is optimalfrom a collective perspective. With full information, firms under-respond to fundamentalssince k∗1 = 1

2> k1, and therefore the full-information inefficiency is φ = ς

2> 0. Next, I apply

Propositions 1, 2 and 3 to this framework.

Corollary 9

i) There exists an αo > 0 such that γm(αo) = γ∗(αo) and sign(γm − γ∗) = sign(α− αo).ii) Social value of information. Producer surplus always increases with the precision of bothpublic and private information.

When public information is exogenous, firms that compete à la Bertrand always give aninefficiently high weight to private information because strategies are strategic complementsbut less than is optimal from a producer surplus perspective. 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), Angeletos andPavan (2007), Colombo et al. (2014)).

The intuition of the social value of information results follows from the application ofCorollaries 2 and 5. More precise public information always increases producer surplus since

31

α∗ > α and φ > 0 and, as a result, equilibrium welfare losses decrease since the reduction indispersion and in the ’covariance term in equilibrium welfare losses’ outweigh the potentialincrease in non-fundamental volatility. Furthermore, more precise private information alsoincreases producer surplus since both the partial welfare effect and the endogenous publicprecision effect have a positive social value.

7 Concluding Remarks

I have investigated the social value of information with an endogenous public signal in eco-nomies that are characterised by payoff externalities and heterogeneous information aboutfundamentals. The main contribution in relation to the previous literature is to analysethe effects on the use and on the social value of information when considering that the en-dogenous public signal is a noisy statistic regarding the aggregate action. This has beenmotivated both by the observation that non-price systems for aggregating information arenowadays ubiquitous, and also by the theoretical need to study information externalities thatare independent of the market structure in the simplest form.

The main results of the paper are driven by the interaction of payoff and information ex-ternalities. The endogenous public signal not only provides information about fundamentalsbut also about the aggregate action of agents. As a result, agents in equilibrium underweightprivate information for a larger payoff parameter region in relation to when public inform-ation is exogenous, and in addition, the welfare planner gives a greater weight to privateinformation with endogenous than with exogenous public information. Endogenous publicinformation has a magnitude effect on the social value of public information and importantlymay overturn the sign of the social value of private information: from a positive to a negativesign if agents coordinate more than is socially optimal with exogenous public informationand in the opposite direction if agents coordinate less.

This paper suggests a few open questions for future research. Future work could explorehow robust these results are to more general information structures, such as in Amador andWeill (2012), Myatt and Wallace (2012, 2015, 2016) or Bergemann and Morris (2013). Inaddition, one could investigate how different mechanisms for aggregating information affectthe use and the social value of public and private information in relation to models of therational inattention literature (e.g. Sims (2006) and Veldkamp (2011)).

32

Appendix

Proof of Proposition 1 The equilibrium strategy with full information satisfiesUq(q, q, 0, θ) = 0. Note that Uqq < 0 and therefore the second order condition is satisfied.With full information qi = Q = k(θ), and since U is quadratic, the full information strategy isa linear function of fundamentals k(θ) = k0 +k1θ. A Taylor expansion of Uq(q, q, 0, θ) aroundq = 0 and θ = 0 gives Uq(k, k, 0, θ) = Uq(0, 0, 0, 0) + Uqqk + UqQk + Uqθθ. Therefore, k(θ) =

−Uq(0,0,0,0)

Uqq+UqQ− Uqθθ

Uqq+UqQ. Notice that Uqq + UqQ 6= 0 since α < 1. Second, I find the equilibrium

strategy with incomplete information and an endogenous public signal. A best response is astrategy q′, satisfies E [Uq(q

′, Q, σq, θ) | si, z] = 0, which implies that the first order conditionis q′ = E[(1 − α)k(θ) + αQ|si, w] and the second order condition is satisfied since Uqq < 0.Focusing on linear arbitrary strategies of the form q(si, z) = b+asi + cE [θ | z] and using thedefinition of the public signal, I obtain: w =

´qi+u = b′+aθ+ c′w+u, whose informational

content is given by the random variable, z = aθ + u, so that E [θ|si, w] = E [θ|si, z] and itsprecision is τ = (var [θ|z])−1 = τθ +a2τu. Then, E [θ|si, z] = τε

τε+τsi+

ττε+τ

E [θ | z]. Therefore,using the definition of α, the best response strategy is q′(si, z) = E [(1− α)k(θ) + αQ | si, z] .I then substitute this expression into the best response strategy, match the coefficients tofind the fixed point to obtain: q(si, z) = k0 + k1(γsi + (1 − γ)E [θ | z]) , where γ can bedefined implicitly by the following cubic expression: γ = k1(1−α)τε

(1−α)τε+k21τuγ2+τθ. The equilibrium

weight to private information, γm, is a solution of: π(γm) = 0, where π(γ) = γ3k21τu +

γ((1− α)τε + τθ)− (1− α)τε. By the Descartes’ Rule of signs, I note that there is only onesign change and, therefore, there will be only a positive real root. Checking π(−γ), I noticethat there are no sign changes, and therefore, there are no negative real roots. Additionally,π(0) = −(1−α)τε < 0 and π(1) = k21τu + τθ > 0, and, therefore, the positive real root, whichis the equilibrium weight to private information, will be between 0 and 1.Comparative statics of γm with respect to the precisions of information and the degree ofstrategic complementarity are as follows:Differentiating π(γm) with respect to τε, I obtain that ∂γm

∂τε= (1−α)(1−γm)

(1−α)τε+τθ+3(γm)2k21τu> 0, since

0 < γm < 1; and with respect to τθ, ∂γm

∂τθ= −γm

(1−α)τε+τθ+3(γm)2k21τu< 0, since 0 < γm < 1.

Similarly,∂γm

∂τu=

−(γm)3k21(1−α)τε+τθ+3(γm)2k21τu

< 0 and ∂γm

∂α= −(1−γm)τε

(1−α)τε+τθ+3(γm)2k21τu< 0, since 0 < γm <

1. �

Proof of Proposition 2 First, the efficient strategy with full information, k∗(θ), sat-isfies WQ(k∗, 0, θ) = 0. The second order condition is satisfied since WQQ < 0. A Taylorexpansion ofWQ around k∗(θ) = 0 givesWQ(k∗, 0, θ) = WQ(0, 0, 0)+WQQk

∗+WQθθ = 0. By

33

the same argument as in Proposition 1, k∗(θ) will be a linear function of θ, which can be writ-ten as k∗(θ) = k∗0 + k∗1θ. Therefore, k∗(θ) = −WQ(0,0,0)

WQQ− WQθθ

WQQ. Second, Angeletos and Pavan

(2007) the expression of E [u] for a candidate efficient strategy is E [u] = E [W (k∗(θ), 0, θ)] +WQQ

2E [(Q− k∗(θ))2] + Wσσ

2E [(Q− q)2], where Q − k(θ) = k1(1 − γ)(E [θ | z] − θ)) and

q −Q = k1γεi. Third, with an endogenous public signal, the welfare planner internalizes allexternalities and maximises ex-ante utility, or equivalently minimises welfare losses, definedas L∗ = E [W (k∗(θ), 0, θ)] − E[u]. The first order condition satisfies dL∗

dγ|γ=γ∗ = 0 where

dL∗

dγ=

(∂L∗

∂γ

)τ const

+ ∂L∗

∂τ∂τ∂γ, and hence a candidate efficient strategy solves:(

γ

τε− (1− α∗)(1− γ)

τ− (k∗1)2τuγ(1− α∗)(1− γ)2

(τ)2

)= 0, (A1)

which can be written implicitly as γ = (1−α∗)τε

(1−α∗)τε+τθ+k21τuγ2−(1−α∗)(1−γ)2τuτεk

∗21

(τθ+k21τuγ

2)

. Then, γ∗ is a

solution of a quintic equation which can be written as ψ(γ∗) = 0, whereψ(γ) = (γ)5τ 2uk

41+2(γ)3k21τuτθ+(γ)2k21τuτε(1−α∗)+γ(τ 2θ +τε(1−α∗)(τθ−τuk21))−τετθ(1−α∗).

(A2)I apply Descartes’ Rule of signs to find out the number of real roots. There is only one changein sign, and therefore, this polynomial will have only one real positive root. Furthermore,ψ(0) = −τετθ(1 − α∗) < 0, while ψ(1) = τ 2uk

41 + 2k21τuτθ + τ 2θ > 0 and the unique positive

real root will be between 0 and 1. The SOC guarantees that the denominator of γ∗ =(1−α∗)τε

(1−α∗)τε+τ∗−(1−α∗)(1−γ∗)2τuτε(k

∗1)

2

τ∗is positive and hence γ∗ > 0.

Next, I conduct comparative statics of the efficient weight to private information in re-lation to the information parameters and the degree of strategic complementarity. Usingχ(γ), I find that: ∂γ∗

∂τε=

γ∗(1−γ∗)(1−α∗)(τuk21)5(γ∗)4τ2uk

41+6(γ∗)2k21τuτθ+2γ∗k21τuτε(1−α∗)

> 0, since 0 < γ∗ < 1. Second,∂γ∗

∂τθ= −2γ∗τ∗+τε(1−α∗)(1−γ∗)

5(γ∗)4τ2uk41+6(γ∗)2k21τuτθ+2γ∗k21τuτε(1−α∗)

< 0, which can be easily shown to be negative bythe definition of γ∗. Third, I find that

∂γ∗

∂τu= k21γ

∗( −2(γ∗)2τ∗+τε(1−α∗)(1−γ∗)5(γ∗)4τ2uk

41+6(γ∗)2k21τuτθ+2γ∗k21τuτε(1−α∗)

), whose sign is ambiguous. Finally, ∂γ∂α∗

=−(γτετuk21+τετθ)(1−γ∗)

5(γ∗)4τ2uk41+6(γ∗)2k21τuτθ+2γ∗k21τuτε(1−α∗)

< 0, since 0 < γ∗ < 1. �

Proof of Proposition 3 i) For non-zero precisions of information, the efficient strategysatisfies q∗i = k∗0+k∗1(γ∗si+(1−γ∗)E[θ|z]), such that k∗0, k∗1 are defined in Proposition 2 and γ∗

is defined such that dL∗

dγ|γ=γ∗ = 0, where L∗(γ) =

|Wσσ |(k∗1)22

(γ2

τε+ (1−α∗)(1−γ)2

τ

). Furthermore

L∗(γ) is a strictly convex function of γ and note that dL∗

dγ|γ=γm will determine the sign

of γm − γ∗. Hence, dL∗

dγ= 2( γ

τε− (1−α∗)(1−γ)

τ− (1−α∗)(1−γ)2

τ2(k21τuγ)). Substituting at the

equilibrium strategy, I obtain that dL∗

dγ|γ=γm= 2( (1−α)−(1−α

∗)(1−α)τε+τm −

(1−α∗)(1−α)k21τuτε((1−α)τε+τm)3

). Therefore,

sign(dL∗

dγ|γ=γm

)= sign (γm − γ∗) , which implies that

34

sign (γm − γ∗) = sign(α∗ − α− (1−α∗)(1−α)(k21τuτε)

((1−α)τε+τm)2

), where the information externality at

the equilibrium strategy is 4m =−(1−α∗)(1−α)(k21τuτε)

((1−α)τε+τm)2< 0.

ii) The efficient weight to private information with an endogenous public signal, γ∗, is the solu-tion of dL∗

dγ|γ=γ∗ =

(∂L∗

∂γ|γ=γ∗

)τ const

+ ∂L∗

∂τ∂τ∂γ|γ=γ∗ = 0, which implies that

(∂L∗

∂γ|γ=γ∗

)τ const

> 0

since ∂L∗

∂τ∂τ∂γ|γ=γ∗ < 0 (because ∂L∗

∂τ< 0 and ∂τ

∂γ> 0). Also note that γ∗exo is the solution

of(∂L∗

∂γ|γ=γ∗,exo

)τ const

= 0. Therefore, because L∗ is strictly convex, then it follows thatγ∗ > γ∗exo. �

Claim 1 The expression for equilibrium welfare losses, WL = E [W (k(θ), 0, θ)]−E[u],at a candidate equilibrium strategy is: WL(γ) =

|Wσσ |k212

(γ2

τε+ (1−α∗)(1−γ)2

τ+ 2(1−α∗)φ(1−γ)

τ).

Proof of claim 1 When there is a full information inefficiency, φ 6= 0, then equilibriumwelfare is E [u] = E [W (k(θ), 0, θ)] + E[WQ(k(θ), 0, θ)(Q − k(θ))] +

WQQ

2E [(Q− k(θ))2] +

Wσσ

2E [(Q− q)2]. From the equilibrium of welfare losses,WL = E [W (k(θ), 0, θ)]−E[u], I have

already developed the first two terms in the Proof of Proposition 2. I now show that the termE[WQ(k(θ), 0, θ)(Q − k(θ))] can be found as follows: a Taylor expansion of WQ(k(θ), 0, θ))

around k(θ) = k∗(θ) is equivalent to WQ(k(θ), 0, θ)) = WQ(k∗(θ), 0, θ)) +WQQ(k(θ)− k∗(θ)),which is equal toWQQ(k(θ)−k∗(θ)) sinceWQ(k∗(θ), 0, θ)) = 0. Since E[(E [θ | z]−θ), θ] = −1

τ

and WQQ < 0, the expectation term is equal to E[WQ(k(θ), 0, θ)(Q− k(θ))] =−|WQQ|k21φ(1−γ)

τ,

where the full information inefficiency is φ =k∗1−k1k1

. Therefore, ex-ante utility at a candidateequilibrium strategy is equivalent to minimising equilibrium welfare losses as given in theexpression of Claim 1. �

Proof of Proposition 4 i) and ii) Follow from (5) and by noting that that 0 < dτm

dτu=

k21γ2(

(1−α)τε+τθ+(γm)2k21τu(1−α)τε+τθ+3(γm)2k21τu

) < k21. �

Proof of Corollary 1 The derivatives of the components of equilibrium welfare losseswith respect to τu are as follows:a) dispersion: d

dτu( (γ

m)2

τε) = −2(1−α)2τε

((1−α)τε+τm)3dτm

dτu< 0;

b) non-fundamental volatility: ddτu

( (1−α∗)(1−γm)2

τm) =

((1−α∗)((1−α)τε−τm)

((1−α)τε+τm)3

)dτm

dτu, whose sign de-

pends on sign( ddτu

( (1−α∗)(1−γm)2

τm)) = sign((1− α)− τm

τε);

c) ’covariance term in equilibrium welfare losses’: ddτu

(2(1−α∗)φ(1−γm)τm

) = −2(1−α∗)φ((1−α)τε+τm)2

dτm

dτuwhose

sign is sign(2(1−α∗)φ(1−γm)τm

) = sign(−φ). �

Proof of Proposition 5 Comparative statics of ex-ante utility with respect to τu areequivalent to the opposite comparative statics of equilibrium welfare losses:

35

(∂WL(γm)

∂τm)γ cons. =

| Wσσ | (k1)2

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

(τε(1− α) + τm)3),

(A3)and 0 < dτm

dτu. Therefore,d(E[u(γm)])

dτu> 0 if and only if ητε + τmρ < 0, where η = (1− α)(2α−

α∗ − 1− 2φ(1− α∗)) and ρ = −(1− α∗)(1 + 2φ).�

Proof of Corollary 2 The signs of the parameters are as follows: ρ < 0 is equivalentto φ > −1

2and η < 0 is equivalent to φ > (α−α∗)−(1−α)

2(1−α∗) .i) If α∗ = α then the sign of both ρ and η depend on 1 + 2φ. It then follows that ρ < 0 andη < 0 if and only if φ > −1

2.

ii) If α∗ > α then −12> (α−α∗)−(1−α)

2(1−α∗) from which a) and c) follow immediately. If (α−α∗)−(1−α)2(1−α∗) <

φ < −12

then η < 0 and ρ > 0, which means that f(τu) = ητε + k21τu(γm)2ρ+ ρτθ is a strictly

increasing function of τu and limτu→∞f(τu) → +∞. This means that for a given finite τε,there will be a solution of f(τ ∗u) = 0 such that the social value of public information is positiveif τu < τ ∗u ; zero if τu = τ ∗u ; and negative if τu > τ ∗u .iii) If α > α∗ then (α−α∗)−(1−α)

2(1−α∗) > −12

from which c) and e) follow immediately. If −12< φ <

(α−α∗)−(1−α)2(1−α∗) then η > 0 and ρ < 0, which means that f(τu) = ητε + k21τu(γ

m)2ρ + ρτθ is astrictly decreasing function of τu and limτu→∞f(τu) → +∞. This means that for a givenfinite τε, there will be a solution of f(τ ∗∗u ) = 0 such that the social value of public informationis negative if τu < τ ∗∗u ; zero if τu = τ ∗∗u ; and positive if τu > τ ∗∗u . �

Proof of Corollary 3 Suppose τu = τv and that there is a given precision of privateinformation. If α∗ > α then region b of Corollary 2 is characterised by ρ > 0 and η < 0 whichmeans that f(τu) = ητε + k21τu(γ

m)2ρ+ ρτθ is an increasing function of τu but it is less steepthan f exo(τu) = ητε+ρτu+ρτθ since ∂fexo(τu)

∂τv= ρ, while ∂fexo(τu)

∂τu= γ2ρk21(

(1−α)τε+τθ+(γm)2k21τu(1−α)τε+τθ+3(γm)2k21τu

).

So, if | k1 |≤ 1 then ∂fexo(τu)∂τv

> ∂fexo(τu)∂τu

, which implies that τ ∗u > τ ∗,exou and the result follows.If α∗ > α then ρ < 0 and η > 0 and it is also the case that τ ∗u > τ ∗,exou . Given the result inCorollary 2.iii then it is immediate that the social value of public information is positive fora smaller region than if public information were exogenous. �

Proof of Corollary 4 The derivatives of the components of equilibrium welfare losseswith respect to τε are as follows:

a) dispersion: ddτε

( (γm)2

τε) = (1−α)2(

−(1−α)τε+τm−2 dτm

dτε

((1−α)τε+τm)3), whose sign depends on sign( d

dτε( (γ

m)2

τε)) =

sign(τm − (1− α)τε − 2dτm

dτε). Due to the endogeneity of public information there is the new

term −2dτm

dτε, which further decreases dispersion in relation to when public information is

exogenous;

36

b) non-fundamental volatility: ddτε

( (1−α∗)(1−γm)2

τm) = (1 − α∗)(

−2(1−α)τm+ dτm

dτε((1−α)τε−τm)

((1−α)τε+τm)3).

The new term in relation to when public information is exogenous is dτm

dτε((1 − α)τε − τm),

whose sign depends on the sign of (1− α)τε − τm.c) ’covariance term in equilibrium welfare losses’:ddτε

(2(1−α∗)φ(1−γm)τm

) = −((1−α)+dτm

dτε) 2(1−α∗)φ((1−α)τε+τm)2

, whose sign is sign( ddτε

(2(1−α∗)φ(1−γm)τm

)) =

sign(−φ). �

Proof of Proposition 6 Comparative statics of ex-ante utility with respect to τε, asdescribed in equation (7), are equivalent to the opposite comparative statics of equilibriumwelfare losses. The partial welfare effect is:

(∂WL(γm)

∂τε)τ cons. = Λ(

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

(τε(1− α) + τm)3),

(A4)where Λ = (1 − α) |Wσσ |(k1)2

2and note that ν = dτm

dτε= 2k21τuγ

m∂γm

∂τε> 0. Combining the

partial welfare effect with the welfare effect on the overall precision of the endogenous publicsignal of equation (A3), I obtain that d(E[u(γm)])

dτε> 0 ⇐⇒ (ϕτε + χτm) + υ(ητε + ρτm) < 0,

where ϕ = −(1 − α)2(1 − α + 2φ(1 − α∗)), χ = (1 − α)((2α∗ − α − 1) − 2φ(1 − α∗)),

η = (1− α)(2α− α∗ − 1− 2φ(1− α∗)), ρ = −(1 + 2φ)(1− α∗) and ν = dτm

dτε> 0. �

Proof of Corollary 5 The signs of the parameters are as follows: ρ < 0 is equivalentto φ > −1

2; η < 0 to φ > (α−α∗)−(1−α)

2(1−α∗) ; χ < 0 to φ > (α∗−α)−(1−α∗)2(1−α∗) ; and ϕ < 0 to φ > −(1−α)

2(1−α∗) .i) If α∗ = α then the sign of ρ, η, χ, ϕ depend on 1 + 2φ. It then follows that social value ofprivate information is positive if φ > −1

2, zero if φ = −1

2, and negative if φ < −1

2.

ii) If α∗ > α then (α∗−α)−(1−α∗)2(1−α∗) > −1

2> −(1−α)

2(1−α∗) >(α−α∗)−(1−α)

2(1−α∗) and there are five possibilities:a) if φ ≥ (α∗−α)−(1−α∗)

2(1−α∗) then χ < 0; ρ < 0; ϕ < 0 and η < 0, and the result follows; b)if −1

2≤ φ < (α∗−α)−(1−α∗)

2(1−α∗) then χ > 0; ρ < 0; ϕ < 0 and η < 0. The partial welfare effecthas a positive social value if and only if τε > τ ∗ε , while welfare effect on the overall precisionof the endogenous public signal always has a positive social value; c) if −(1−α)

2(1−α∗) < φ < −12

then χ > 0; ρ > 0; ϕ < 0 and η < 0. The partial welfare effect has a positive social value ifand only if τε > τ ∗ε , while the welfare effect on the overall precision of the endogenous publicsignal has a positive social value if and only if τu < τ ∗u ; d) if

(α−α∗)−(1−α)2(1−α∗) < φ ≤ −(1−α)

2(1−α∗) thenχ > 0; ρ > 0; ϕ > 0 and η < 0. The partial welfare effect always has a negative social value,while the endogenous public precision effect has a positive social value if and only if τu < τ ∗u ;e) if φ ≤ (α−α∗)−(1−α)

2(1−α∗) then χ > 0; ρ > 0; ϕ > 0 and η > 0 and the result follows.Note: τ ∗ε is the root of g(τε) = ϕτε+χτ

m such that g(τ ∗ε ) = 0 which exists since limτε→0g(τε) =

χτθ > 0 and limτε→+∞g(τε)→ −∞. Given the properties of g(τε) there will be a unique root

37

in the interval. It then follows that, for a given precision of the noise in the public signal,the partial welfare effect is negative if τε < τ ∗ε , zero if τε = τ ∗ε , and positive if τε > τ ∗ε . Thecut-offs for the social value of public information are found in Corollary 2.Note: In d. the social value of private information can turn from negative to positive (andnot from positive to negative) since ϕτε + χτm < 0, or equivalently τm

τε< −ϕ

χ, implies that

ητε + ρτm < 0, or equivalently τm

τε< −η

ρ. It then follows that −ϕ

χ< −η

ρsince −ϕρ < −ηχ.

iii) If α > α∗ then (α−α∗)−(1−α)2(1−α∗) > −(1−α)

2(1−α∗) >−12> (α∗−α)−(1−α∗)

2(1−α∗) and there are five possibilities:f) If φ ≥ (α−α∗)−(1−α)

2(1−α∗) then η < 0 ; ϕ < 0; ρ < 0 and χ < 0 and the result follows; g)if −(1−α)

2(1−α∗) ≤ φ < (α−α∗)−(1−α)2(1−α∗) then η > 0 ; ϕ < 0; ρ < 0 and χ < 0. The partial welfare

effect always has a positive social value while the welfare effect on the overall precisionof the endogenous public signal has a positive social value if and only if τu > τ ∗∗u ; h) If−12< φ < −(1−α)

2(1−α∗) then η > 0 ; ϕ > 0; ρ < 0 and χ < 0. The partial welfare effect has apositive social value if and only if τε < τ ∗∗ε while the welfare effect on the overall precisionof the endogenous public signal has a positive social value if and only if τu > τ ∗∗u ; i) If(α∗−α)−(1−α∗)

2(1−α∗) < φ ≤ −12

then η > 0 ; ϕ > 0; ρ > 0 and χ < 0. The partial welfare effect has apositive social value if and only if τε < τ ∗∗ε while the welfare effect on the overall precision ofthe endogenous public signal always has a negative social value; j) If φ ≤ (α∗−α)−(1−α∗)

2(1−α∗) thenη > 0 ; ϕ > 0; ρ > 0 and χ > 0, and the result follows.Note: τ ∗∗ε > 0 is the root of g(τε) = ϕτε + χτm such that g(τ ∗∗ε ) = 0 which exists sincelimτε→0g(τε) = χτθ < 0 and limτε→+∞g(τε) → +∞. Given the properties of g(τε) there willbe a unique root in the interval. It then follows that, for a given precision of the noise inthe public signal, the partial welfare effect will be positive if τε < τ ∗∗ε , zero if τε = τ ∗ε , andnegative if τε > τ ∗∗ε . The cut-offs for the social value of public information are found inCorollary 2.Note: In h. the social value of private information can turn from positive to negative (andnot from positive to negative) since ϕτε + χτm > 0 implies that ητε + ρτm > 0 becauseϕ−χ <

η−ρ , or equivalently, −ϕρ < −ηχ. �

Proof of Proposition 7 i) Follows from d(E[u(γm)])dτε

= (∂(E[u(γm)])∂τε

)τ cons.+∂(E[u(γm)])

∂τm γ cons.dτm

dτε

and by noting that dτm

dτε= (2k21τuγ

m∂γm

∂τε) > 0. The proof of the proposition follows from the

arguments given in the text.ii) It follows immediately from considering combination of the signs of the partial welfareeffect and the welfare effect on the overall precision of the endogenous public signal for thevarious cases presented in Corollary 5. �

38

Proof of Corollaries 6, 7, 8 & 9 Corollary 6: Direct application of Proposition 3,Corollaries 2 and 5.Corollary 7: (i) Direct application of Proposition 3. ii) If 0 < r ≤ 1

2then Corollary 2.e

applies, while if r > 12then Corollary 2.e applies. The results of the social value of private

information follow from the application of Corollary 5.f and g. The other conditions do notapply given the parameter restrictions.Corollary 8: i) It follows from Proposition 3. ii) First note that φ = α

1−2α < 0 and α∗ =

2α < α < 0. Case 1 follows the application of Corollaries 2.d and 5.f.; Case 2 follows fromthe application of Corollaries 2.e and 5.g.; Case 3 follows from the applications of Corollaries2.e and 5.h. The other conditions do not apply given the parameter restrictions. Notethat for the Cournot application, if more precise private information is detrimental thenpublic information will also be detrimental. I can also show that the threshold value whichdifferentiates Case 2 from Case 3 satisfies: ϑ = υ −

√v2 + v + 1 > −1 since v > 0.

Corollary 9: i) α∗(α) = 2α. The equilibrium and efficient weights to private information,γ(α) and γ∗(α∗(α)), respectively, are both strictly decreasing function of α, which rangefrom 1 to 0 as α → −∞ to α → 1 and α∗ → 1, respectively. Using the intermediate valuetheorem, there exists a unique intersection between the two curves, αo, which is equal toαo =

(1−α∗)(1−α)(k21τuτε)((1−α)τε+τ)2 > 0 and satisfies 0 < αo < 1. This implies that sign (γm − γ∗) =

sign(α − αo). ii) φ = α1−2α > 0 and α∗ = 2α > α > 0. Results follow from applying

Corollaries 2.a. and 5.a since φ ≥ (α∗−α)−(1−α∗)2(1−α∗) . �

39

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