Shared Knowledge and Competition forAttention in Information Markets*

Simone Galperti and Isabel Trevino†

UCSD

April 6, 2018

Abstract

Information consumers seek to learn about world events, but often also whatothers know about those events. Capturing consumers’ attention is essential forinformation suppliers to thrive in the market. We find that competition for at-tention leads to homogeneity of information sources—in terms of accuracy andclarity—even when consumers would demand heterogeneous sources. Their equi-librium type exhibits higher clarity as consumers care more about predicting others’beliefs, while this desire does not affect accuracy. We also find that whenever at-tention becomes the “currency” whereby consumers pay for information, it causesnovel market inefficiencies, whose form and size depend on the consumers’ interestin others’ beliefs.

Keywords: information supply, coordination, payoff interdependence, attention,accuracy, clarity, inefficiency.JEL classification: C72, D62, D83, L10.

*We thank for comments and suggestions S. Nageeb Ali, Rahul Deb, John Duffy, Joseph Engelberg,Renato Gomes, Marina Halac, Garrett Johnson, Stephen Morris, Alessandro Pavan, Joel Sobel, PhilippStrack, Xavier Vives, Rakesh Vohra, and Joel Watson and seminar participants at ESSET (Gerzensee),Yale, Princeton, Caltech, and Queen’s University. John Rehbeck provided excellent research assistance.All remaining errors are ours.

†Department of Economics, University of California, San Diego, 9500 Gilman Dr., La Jolla, CA,92093. E-mail addresses: sgalperti@ucsd.edu; itrevino@ucsd.edu. First draft: July 2016

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1 Introduction

People acquire information to learn about world events, but often also what others do(not) know about those events. Political protesters look for mutually known news thatcan catalyze a rally. Stock traders look for information others may not know, so as toanticipate price swings. Shared knowledge matters in many economic and social settingsand mass media contribute to its creation (Chwe (2013)). According to Shiller (2015),“significant market events generally occur only if there is similar thinking among largegroups of people, and the news media are essential vehicles for the spread of ideas.” (p.101) Recognizing that news sources not only help people learn some facts, but also others’(mutual) beliefs about those facts, is then crucial for understanding information markets.

Prominent research has investigated how this desire to learn some facts as well asothers’ beliefs about them affects the demand for information, treating its sources asexogenous. Little is known, however, about that desire’s effects on the supply of infor-mation. To fill this gap, this paper endogenizes the information supply in a full-fledgedcompetitive market. The paper also focuses on another aspect of information, whichsets it apart from other commodities: Information consumes attention while being con-sumed. Since attention can be “harvested” and sold to third parties (usually advertisers),it becomes a key revenue source for information providers, who fiercely compete for it.1

We offer two sets of results. First, we show that competition for attention leadsto a homogeneous supply of information, even when consumers would value accessingheterogeneous sources. We identify how the supplied type depends on the strength ofthe consumers’ fact-learning motive relative to their belief-learning motive. Second, wediscover an inefficiency specific to information markets, which arises despite perfect com-petition and the consumers’ efficient use of information: Whenever attention happens tobecome the “currency” whereby consumers pay for information, the competitive outcomewill be generically inefficient. The inefficiency can involve the volume as well as the typeof supplied information, again depending on the strength of the two learning motives.

To model these motives, we follow the paradigm of games with payoff interdependen-cies that cause externalities in the use of information (Morris and Shin (2002); Angeletosand Pavan (2007); Hellwig and Veldkamp (2009)). Consider a society of identical agentswho have to support some policy within a feasible set. They want to support the bestpolicy, which depends on some unknown state of the world, and also like (or dislike) con-

1The importance of competition for attention when it comes to information sources has been stressedby many scholars across disciplines, including Simon (1971), Benkler (2006), Lanham (2006), Sunstein(2009), Davenport and Beck (2013), and Webster (2014).

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formism, namely, supporting what others support. Each agent then desires to learn thestate (fact-learning motive) as well as the others’ beliefs about it in order to coordinate ornot with them (belief-learning motive). For exposition, let α measure this (primitive) co-ordination motive relative to that of acting on the state: α > 0 (α < 0) captures a desireto coordinate (anti-coordinate); α = 0 if the agents care only about the state. They canattend to multiple sources of information about the state.2 Each is characterized by theaccuracy of its content and the clarity with which it conveys that content (as in Myattand Wallace (2012)). For instance, a source may run a summary or in-depth report ofthe policies’ pros and cons and broadcast it by radio or television, media which differ incommunication clarity and so the return per minute spent attending to them. Clarity istherefore complementary to attention for understanding a given content. Realistically,the agents may interpret each source differently, introducing a risk of miscoordination.

In our model, each information source is produced by a distinct profit-maximizingsupplier, broadly interpreted as traditional news outlets, social media, blogs, and newsaggregators. Attention is their only revenue channel. We shut down the usual pricechannel in order to better understand the competition for attention, which seems under-studied yet important in the Internet era.3 In the spirit of perfect competition, entryis free and suppliers act non-strategically.4 Entrants choose their level of accuracy andclarity, each from a finite set, where higher levels cost more to produce. An accuracy-clarity pair is called a type. After the entrants commit to their types, the consumersobserve them, allocate attention, update beliefs, and choose their actions.

We characterize the competitive-equilibrium supply of information and how it dependson the coordination motive α. Perhaps surprisingly, the supply always involves only onetype of sources. Importantly, this is not because the agents demand only one type—infact, they would pay attention to multiple ones if produced—but because the competitionfor attention renders only one of them profitable.5 This type and its dependence on α

turn out to vary significantly in relation to accuracy and clarity.2Gentzkow (2007), Gentzkow and Shapiro (2011), Webster (2014), and Kennedy and Prat (2017) find

that many people get their news from multiple sources.3Setting a zero price for accessing information may be a deliberate choice for some news outlets.

Gentzkow (2007) estimates that setting zero prices may be optimal for online outlets in the presence ofa strong demand for online advertising. Webster (2014) discusses numerous examples and argues that“advertising now supports much of the world’s media, both online and off.” Newman et al. (2017) findthat an overwhelming majority of people does not pay for their online news.

4Perfect competition is of course an abstraction, which helps derive sharper results. However, newsmarkets—especially in the broad sense used here—tend to be fairly competitive, at least in developedcountries (see Footnote 15).

5Consistent with this result, Boczkowski (2010) argues that “the rise of homogenization in the newshas led to a state of affairs that neither journalists nor consumers like but feel powerless to alter.” (p. 6)

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To illustrate, suppose suppliers compete via accuracy: They choose accuracy, but allshare some fixed clarity. Then, the type with the lowest average cost of accuracy wins themarket, independently of α. Intuitively, having all the same clarity, each supplier offersthe same return to attention and hence receives the same positive amount of attention(i.e., revenue) per unit of accuracy. This amount falls as more entrants—possibly ofdifferent types—compete for attention, so only the most efficient suppliers survive.

Suppose suppliers compete via clarity instead: They choose clarity, but share somefixed accuracy. Then, the equilibrium type becomes less clear as α falls. Imagine claritycan be high or low. High-clarity sources always capture some attention, but possiblyless attention than do low-clarity sources. This is where α matters: As the consumers’conformism weakens and turns into anti-conformism (i.e., α falls), they shift attentionfrom high- to low-clarity sources (Myatt and Wallace (2012); Pavan (2014)). This isbecause easily understandable content tends to be more public among its consumers(i.e., to induce more correlated interpretations). But public information is less worthy ofcostly attention for anti-conformist consumers who want to respond in opposite directionsto the state and to others’ beliefs. This gives low-clarity sources an advantage in terms ofattention revenues as α falls. Importantly, although the size of this advantage depends onthe number of high- and low-clarity entrants, its sign does not. Therefore, one type willalways be more competitive and win the market. In short, we find that the endogeneity ofsupply magnifies the attention reallocation mechanism identified in the literature. Thisresult can overturn the literature prediction that the clearest sources (and possibly others)always receive attention. Here, the demand is indeed biased towards those sources, yetthe competition for attention can drive them out of business.

We consider scenarios where suppliers choose both accuracy and clarity. The equi-librium type continues to primarily depend on clarity, correlation, and the coordinationmotive α, in ways similar to competition via clarity.6 Thus, α can affect the accuracy ofnews only because of production trade-offs with clarity. We apply this insight to pointout that the ongoing transition from hard to soft news may be due to stronger socialconformism and innovations in news production (not to attention becoming scarcer).

We also examine the equilibrium entry and attention allocation. For competitionvia accuracy, each entrant captures the same constant attention, but entry shrinks in α,thereby reducing the total information consumed in equilibrium. For competition viaclarity, the attention captured by each entrant increases in α, which renders informationuniformly more public (in contrast to the case of exogenous supply). Entry is non-

6Chwe (2013) observes that communication aimed at creating common knowledge and facilitate co-ordination often uses simple and repetitive language so as to ensure that the message gets across.

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monotonic, however, rendering the effect of α on the consumed information ambiguous.

Our results have several implications. First, competition for attention need not pro-mote the sources offering the highest return to attention. This challenges the commonwisdom that in the “attention economy” higher clarity is always better (Davenport andBeck (2013)). In fact, being too clear can hurt attention revenues and help less clearsources win the market. The paper highlights how this depends on the consumers’ fact-and belief-learning motives. A second takeaway is to draw a distinction between quantityand diversity of information sources, where competition for attention pushes towards ho-mogeneity. This echoes Webster’s (2014) point that “the digital media marketplace [is]less diverse than its sheer numerical abundance might suggest” (p. 16) and “competitiondoesn’t do much to improve the diversity of news products.” (p. 58) This should notbe confused with the variety of news topics, which can be easily explained by people’sdifferent interests. Our analysis can be viewed as focusing on one topic at a time.

We also investigate the market inefficiencies caused by the very nature of informationas the traded good. This is helped by our perfect-competition assumption. It is wellknown that, when it comes to information, consumers need not use it efficiently due tointerdependencies in their final payoffs (Morris and Shin (2002), Angeletos and Pavan(2007), and Pavan (2014)). We explain how this distorts the supply of information.More importantly, we show that even if the consumers used information efficiently, thecompetitive equilibrium would be inefficient (under a general Pareto criterion to be de-fined below). Compared to the laissez-faire outcome, the efficient supply may be higher orlower, require a different type of sources, or even multiple types. These inefficiencies ariseprecisely from the unique fact that for information—in contrast to other commodities—attention acts as the “currency” whereby consumers pay for it. This enables the existenceof a market. But unlike the frictionless price mechanism, attention does not adjust freelyto allow the trading sides to internalize costs and benefits of producing information. Inshort, when attention is costly and coordination is important, insufficiently correlated in-formation is provided; when attention is cheap, too much information is provided. Thesefindings matter for all real-life situations where we get news “for free” in exchange forour attention. According to Webster (2014), “there is a broad consensus that the waypeople allocate their attention will go a long way toward determining digital media’ssocial impact.” Our results identify one general effect of attention on efficiency.

Related Literature. A rich literature studies information in strategic environmentswith coordination motives. Morris and Shin (2002) highlight the role of informationas a coordination device and show that increasing the accuracy of public information

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can decrease welfare, as agents who strongly value coordination may underweight theirprivate information when choosing actions. Angeletos and Pavan (2007) characterize theuse and social value of information as a function of the coordination motives. In thesepapers the agents do not choose which information to acquire.

Subsequent research has endogenized information acquisition. Hellwig and Veldkamp(2009) show that information acquisition inherits the strategic motives from the underly-ing coordination game, which can lead to multiple equilibria. Colombo et al. (2014) studythe effect of coordination motives on the efficiency in the acquisition and use of informa-tion, and the social value of free public information when agents can access costly privateinformation. Myatt and Wallace (2012) study a framework where information sourcesdiffer in accuracy and clarity and attention is a continuous decision, which can removethe equilibrium multiplicity in Hellwig and Veldkamp (2009). Using this framework, Pa-van (2014) studies the equilibrium and efficient allocation of attention. All these papersassume exogenous and heterogeneous sources. The present paper calls this assumptioninto question by finding that competition for attention pushes towards a homogeneousinformation supply. This literature has shown that decentralized information acquisitionneed not serve the social interest; this paper shows that its supply need not either.

Few papers endogenize information sources in coordination games; none analyze com-petitive information markets and their efficiency. Dewan and Myatt (2008, 2012) examinehow party leaders adjust, at no cost, the clarity of their speeches to gain influence andguide activists concerned about party cohesion. In these papers, the sources’ accuracyand quantity remain exogenous. Cornand and Heinemann (2008) show that increasingthe accuracy of public information improves welfare, as long as one can restrict who canaccess it. Myatt and Wallace (2014) study how central banks design information to sta-bilize output, expectations, and volatility. In Chahrour (2014), a planner may withholdinformation if there is a risk that agents attend to different information sources.7

A vast literature has studied the phenomenon of media bias and its political-economyconsequences.8 Media bias refers to deliberate and systematic distortion of information,so it is orthogonal and complementary to the focus of the present study. The idea of com-petition for attention and the distinction between accuracy and clarity are mostly absentfrom that literature, which asks whether competition—broadly defined—promotes mediaindependence, timeliness, and unbiasedness. An exception is Chen and Suen (2017), whostudy a model of news markets where each outlet’s owner chooses accuracy to attract at-tention and its editor chooses a reporting bias trading off helping readers make informed

7For a review of the broader literature on information design see Bergemann and Morris (2017).8See Gentzkow and Shapiro (2008) and Prat and Stromberg (2013) for literature reviews.

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decisions and pursuing her agenda. Their readers do not exhibit coordination motives,thereby rendering the role of information sources as coordination devices irrelevant.

2 A Model of Information Markets

As we present the model, it helps to keep in mind our introduction story. Agents in alarge society have to choose an action, broadly interpreted as supporting some policy withuncertain effects, forecasting some event, adopting a new technology, or merely pickinga topic for conversation. They want to choose the intrinsically best action, but alsorespond to (anti)conformist pressures—a desire to (not) follow popular trends or groupnorms. This coordination motive may also stem from network externalities—for instance,when adopting products that require a dense user base. Before acting, the agents acquireinformation from news sources, who compete for their attention to maximize profits.

2.1 Demand of Information

This part follows the broadly used setup of Myatt and Wallace (2012) and Pavan (2014).

Consumers, Information Sources, and Attention. There is a unit mass of ex-anteidentical agents, the information consumers, indexed by n. They have a common priorabout the state of the world θ, given by θ ∼ N(0, p−1). A higher precision p meansthat they initially know more about θ. They can acquire information about θ from newssources, indexed by m. If consumer n attends to source m, she privately observes a signal

sn,m = θ + zm + xn,m,

where zm ∼ N(0, σ2zm), xn,m ∼ N(0, σ2

xn,m), and θ, zm, and xn,m are all mutually indepen-dent. The part θ + zm represents the information contained in source m, which consistsof the state plus some “sender noise.” The latter determines source m’s accuracy, definedby am ≡ σ−2

zm . The “receiver noise” xn,m reflects the idiosyncratic meaning each agentgives to a source’s content. This noise depends jointly on how clearly the source conveysinformation and how attentively the consumer listens to it. Think of attention as thetime agent n spends listening to source m, denoted by en,m ≥ 0: The longer the time, thehigher the chances of getting the content right. Given this, let cm be source m’s clarityand σ−2

xn,m ≡ [cm]2en,m.9 A higher cm raises the return of attending to source m in terms9This way of modeling communication is reminiscent of Dewatripont and Tirole (2005), if we rein-

terpret [cm]2 as the sender’s effort and em as the receiver’s effort in their model. We square cm in thedefinition of σ2

xn,m only to simplify some of the expressions below.

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of better understanding of its content (i.e., shrinking σ2xn,m). Choosing en,m = 0 means

ignoring source m, as sn,m is then just noise.

Actions and Payoffs. Agent n’s payoff function is

u(kn, K, σ, θ)− C(en),

where kn ∈ R is her action, K ≡´ 10kndn is the population average action, σ2 ≡

´ 1

0[kn −

K]2dn is the action dispersion, and en is the vector of n’s attention allocations en,m. Thecost function satisfies C(en) = τ

∑m en,m, where τ > 0 is the opportunity cost of the time

spent acquiring information or the shadow value of some time budget constraint. Thesubstantive assumption, also studied by Myatt and Wallace (2012) and Pavan (2014), isthat C depends only on the total attention paid, not how it is divided between sources.That is, in terms of opportunity cost attending one source is the same as attendinganother; their return may differ, but this is captured by clarity. Linearity is mostly atechnical simplification to obtain closed-form attention allocations, which enable a cleaneranalysis of the overall market.10

As is standard in the literature, (i) u is a second-order polynomial, (ii) σ2 has only anon-strategic externality effect, so in terms of partial derivatives ukσ = uKσ = uθσ = 0

and uσ(k,K, 0, θ) = 0 for all (k,K, θ), (iii) ukk < 0, (iv) α = −ukK/ukk < 1, and (v)ukθ = 0. By (i), the agent have linear best responses. By (ii), u is additively separable inσ2, whose coefficient is denoted by uσσ/2. By (iv), the slope of best-response functions isless than 1, delivering uniqueness of equilibrium actions (Angeletos and Pavan (2007)).These assumptions allow for rich payoff externalities. A simple case is the loss function

−(1− γ)(k − θ)2 − γ(k −K)2

for γ ∈ (0, 1), often used in “beauty-contest” games à la Morris and Shin (2002).

Timing. After learning the accuracy and clarity of all sources, the agents simultaneouslyallocate their attention; they then observe their signals and update beliefs; finally, theysimultaneously choose actions, and payoffs realize. Consistent with the literature, weconsider Perfect Bayesian equilibria of this game, called “consumer equilibria” for short.

10Linearity (more generally convexity) of C rules out cost functions based on entropy as in the rational-inattention literature (Sims (2003), Sims et al. (2010)). As showed by Myatt and Wallace (2012),convexity of C ensures that the agents’ behavior is described by a unique equilibrium, a useful propertyto study information supply. We leave the analysis with rational-inattention costs for future work.

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2.2 Supply of Information

We aim to model two key aspects: News sources compete with each other, and capturingattention is a key part of this competition. To do this in the simplest and starkest way,we assume that each supplier cares directly about the attention its source captures andcannot charge a price for accessing it.11 While price competition has been extensivelystudied, competition for attention has not. Yet, many outlets supply news for free andrely on capturing attention to make money, usually by “selling eye balls” to advertisers.12

Production. Each supplier can produce one unit of the good “information,” which takesthe form of a signal with some accuracy and clarity. They can choose among finitely manysignal types, each defined by some at > 0 and ct > 0 where t = 1, . . . , T and T ≥ 2.Think of accuracy as a reporting style (summary vs. in-depth reports) and clarity as acommunication technology (TV vs. newspapers).13 Producing a t-signal involves a fixedcost dt > 0 but zero marginal costs of communicating it—a typical cost structure forinformation goods in the digital age (Hamilton (2004)). We will specify how dt dependson at and ct later. Suppliers can choose to not produce (i.e., set am = cm = 0) at no cost.Note that they have no information about θ when choosing which signal to provide.

Profits. Supplier m’s profit from a t-signal isˆ 1

0

en,mdn− dt.

Suppliers have no preference over which agents attend to which source and how they useinformation. One interpretation is that they sell ad impressions to advertisers (not tobe confused with sn,m). As a first approximation and in the spirit of partial-equilibriumanalysis, revenues are proportional to the amount of attention received with exogenousreturn normalized to 1. Intuitively, the more time agents spend attending a source, themore ads it can show them, and the more impressions it can charge to advertisers.14

11More generally, suppliers may also compete in prices. This can introduce a trade-off between priceand attention revenues (see, for example, Crampes et al. (2009)) and lead to multiplicity of equilibria asin Hellwig and Veldkamp (2009), due to the discreteness of the “price cost” of acquiring an informationsource. We leave the study of this potentially interesting issues for future work.

12These are key aspects of the “attention economy” as argued by Davenport and Beck (2013) andWebster (2014).

13We are agnostic on whether TV or newspapers have higher clarity. This ultimately seems an em-pirical question, whose answer may depend on the type of news considered (for example, financial orbusiness news vs. general-interest news). Broadcast media have a multisensorial approach that enablespotential higher-impact reporting than print media, but the static nature of print media allows for in-definite exposure, as opposed to the fleeting nature of broadcast media. Webster (2014) argues that thetechnology used to deliver media is a key dimension of audience-making strategies. The importance oftechnologies in news provision is also raised by Prat and Stromberg (2013) and Newman et al. (2017).

14Webster (2014), Ch. 4, describes measures that media use to assess their audience which are con-

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Perfect Competition.15 There is an arbitrarily large number of potential suppliers.Entry and exit are free. When making its decisions, each supplier takes the existingsuppliers as given and anticipates the agents’ behavior as described by the consumerequilibrium. Suppliers produce a t-signal if and only if this is profitable. We will usethis zero-profit condition for each type to solve the model. Let the number of t-suppliersbe qt. We say that q∗ = (q∗1, . . . , q

∗T ) is a competitive equilibrium if, given q∗, no active

supplier wants to exit and no new supplier wants to enter for every type of signals.

3 Attention Allocation: A Review

We first review the aspects of the consumer equilibria necessary for our analysis, leavingthe details for Appendix B. Fix q = (q1, . . . , qT ). A consumer equilibrium consists oftwo things for each n: a vector en of attention allocations and a strategy kn(sn; en)

mapping en and signals sn (one for each supplier) into actions. We follow the literatureand focus on the unique equilibrium where en and kn are the same for all n and kn islinear in sn. We therefore drop the superscript n.

Our goal is to understand how the information supply depends on the consumers’desire to learn the state and others’ beliefs about it. This hinges on their payoff motiveof responding to θ relative to that of coordinating with others, which is measured byα = −ukK

ukk. Indeed, as Angeletos and Pavan (2007) show, every equilibrium must satisfy

k(s; e) = E[(1− α)κ+ αK|s, e],

where κ(θ) ≡ κ0+κ1θ is the unique equilibrium strategy under complete information, withκ0 ≡ −uk(0,0,0,0)

ukk+ukKand κ1 ≡ − ukθ

ukk+ukK. Appendix B provides a formula for k(·; ·). Different

α’s can correspond to different societies—some are more conformist than others—or newstopics. Stock-market news may involve α < 0, gossip news α > 0.

We will assume that the demand is sufficiently strong in the following sense: Forevery t = 1, . . . , T , if only t-signals were feasible, their supply would be positive, whichof course also means that they receive positive attention. As we will see, this is implied

sistent with this approximation. Davenport and Beck (2013) describe the “stickiness” of online newsoutlets—namely, their ability to grab and keep attention—as the time spent on a site and the numberof visits and viewed pages per person.

15News markets tend to be fairly competitive, at least in developed countries. Webster (2014) arguesthat “perhaps the most astonishing thing about digital media is their numerical abundance.” Kennedyand Prat (2017) find concentration indexes of news markets that are fairly small relative to usual stan-dards. See also Noam (2009). Finally, recall that we interpret information sources broadly as traditionalnew outlets, social media, blogs, etc..

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by the next condition. Letϕ = |κ1|

√−ukk

2.

Assumption 1 (Strong demand). ϕ > p√τ[

c−1t

1−α+ dtct

at

]for all t = 1, . . . , T .

This condition always holds for sufficiently small p and τ . Intuitively, the demand forinformation is strong if ex ante the agents know little about θ or their attention cost issmall. Assumption 1 is akin to the standard assumption that the intercept of a demandcurve is large enough to induce positive trade in equilibrium.

Consider now the attention allocation. As shown in Appendix B, the agents pay thesame attention to each t-source, denoted by et(q). Of course, et(q) = 0 if qt = 0. Thenext result describes et(q) when qt > 0.16 Assume that ct ≥ ct′ if t > t′ and define

Tq = {t : qt > 0} and Te = {t ∈ Tq : et(q) > 0}.

Lemma 1 (Attention Functions). For every t ∈ Te,

et(q) =atc

−1t

p+∑

t′∈Teat′qt′

ϕ√τ− pc−1

t

1− α+

∑t′∈Te\{t}

at′qt′

1− α(c−1

t′ − c−1t )

.

Let t1 and t2 be the first and second largest types in Tq. When ct1 > ct2 , define

qt1(α) =

[ϕ√τ(1− α)ct2 − p

]ct1a

−1t1

ct1 − ct2.

Corollary 1 (Attention Targets). If ct = c for all t, all active suppliers receive attention:Te = Tq. If ct > ct′ for all t > t′, then Te contains all types above some t ∈ Tq, wheret = t1 if and only if qt1 ≥ qt1(α).

These properties review some key points in Myatt and Wallace (2012) and Pavan(2014). As expected, et(q) decreases in the quantity of competitors of any type.17 Italso decreases if ex ante the agents know more about θ (i.e., p is higher), which weakenstheir incentive to acquire information. Higher clarity has ambiguous effects: By offeringa higher return to attention, it can raise et, but by “consuming” less attention to conveythe same content, it can lower et.18

The coordination motive α has two effects on attention. The first, given by − p1−α

, isnegative. As α rises, the agents’ optimal actions depend less on their signals and more

16Its proof appears in Appendix A together with all the others.17One can show that ∂et(q)

∂qt′< 0 if t′ ∈ Te.

18That agents pay attention to all sources of either type is clearly driven by their symmetry in themodel. Although this is in part at odds with reality, there exists evidence of significant overlap inaudiences across news sources (Gentzkow and Shapiro (2011), Webster (2014), Newman et al. (2017)).

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on their prior, since it captures shared knowledge (see Appendix B). This lowers thevalue of acquiring information in the first place. Thus, suppliers also compete with theagents’ prior to attract attention. The second effect of raising α exists only if ct > ct′ .It is negative for the lower-clarity sources and positive for the higher-clarity ones, whichreflects the latter’s comparative advantage as coordination devices.19 As the agentsattend more to a source, they interpret more similarly its content, which then becomesmore public among them, helping coordination. Clearer sources offer a higher return toattention and so attract more of it when α rises. That less clear sources have an advantagewhen the agents want to anti-coordinate may be counterintuitive, but has a simple logic.Each agent now wants to respond in opposite directions to shared information suggestingthat θ is high and everybody else’s action is high. As a result, such public informationhas little value. It is, however, less likely to arise from unclear and hence more privatesources, which are then more worth attending for the agents.

Nonetheless, Corollary 1 reflects a bias in the demand favoring higher clarity. Inparticular, the active suppliers of highest-clarity can crowd out attention so that nobodyattends to lower-clarity suppliers (but not vice versa). Note that qt1(α) decreases in α:A larger α raises the value assigned to clarity and so the willingness to attend to thehighest-clarity suppliers, which can now more easily exhaust the agent’s attention.

4 Equilibrium Information Supply

We now turn to the analysis of competitive equilibria. We first derive the zero-profitconditions that characterize them, given the attention functions obtained before. Thinkof these functions as the analog of demand functions in standard markets. Recall thatsuppliers choose signals that can differ in clarity, accuracy, or both. We cover these threecases separately so as to better demonstrate the differences between accuracy and clarity.

4.1 Zero-profit Conditions

Consider the suppliers’ entry and exit decision. Given that q suppliers already enteredacross types, a new t-supplier enters if and only if

et(qt + 1,q−t) ≥ dt, (1)19That clear sources are better coordination devices seems to be a general property that holds beyond

the Gaussian framework considered here (see Chwe (2013)).

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where q−t is the vector of quantities excluding qt. Since all t-suppliers are identical, if (1)holds for the last entrant, it holds for all t-suppliers already in the market, which implies

(qt + 1)et(qt + 1,q−t) ≥ (qt + 1)dt.

Conversely, if this condition holds and only qt suppliers of type t already entered, an extrat-supplier wants to enter. Letting Et(q) = qtet(q), we get the following observation.

Lemma 2. q∗ is a competitive equilibrium if and only if, for all t = 1, . . . , T ,

Et(q∗t ,q∗

−t) ≥ q∗t dt and Et(q∗t + 1,q∗

−t) < (q∗t + 1)dt.

These conditions characterize all equilibria (if any exists), but are inconvenient touse due to the discreteness of qt. This is analogous to the issue that arises in textbookanalysis of long-run competitive equilibria when profits jump from being strictly positiveto being strictly negative if one more firm enters the market. To avoid this, it is typicallyassumed that firms are small so that equilibria can be characterized by a zero-profitcondition. A similar approach can be adopted here by letting qt be a non-negative realnumber for all t. Given this, the equilibrium conditions become

Et(q∗t ,q∗

−t) = q∗t dt, t = 1, . . . , T, (2)qt > q∗t ⇒ Et(qt,q∗

−t) < qtdt, t = 1, . . . , T. (3)To understand the second, consider Figure 1. It represents the total production cost qtdtand the attention function Et(qt,q−t) (when positive), which is strictly increasing andconcave in qt. The quantity qt = 0 always satisfies (2), but can be part of an equilibriumonly if no t-supplier can profitably enter given q−t. In Figure 1, this holds for dt, but notfor dt. In this case, condition (3) rules out qt = 0 and q−t as an equilibrium.

qt0

Et(qt,q−t)

qtdt qtdt

Figure 1: Attention-revenue and production-cost curves

The interpretation of continuous quantities is that each supplier is small relative tothe whole market, which is consistent with the idea of perfect competition. In this model,

13

however, letting the number of information sources grow arbitrarily involves some intri-cacies: Since each provides an independent signal about the state, they could eventuallyallow the agents to fully learn it. We avoid this issue and maintain the spirit of the modelby appropriately rescaling the accuracies at as we let the number of sources grow. Wepresent the details in Appendix C.

We can now see why under Assumption 1 there would be a positive supply of t-signals,if this were the only feasible type. Let qt be the largest solution to Et(qt,q−t) = qtdt forq−t = 0. One can show that

qt =c−1t

dt

[ϕ√τ− pc−1

t

1− α

]− p

at,

which is strictly positive by our assumption.

4.2 Competition via Clarity

Consider the case where suppliers can use clarity as a lever to attract attention, butthey all produce the same content. That is, ct > ct′ for t > t′ and at = a for all t. Forexample, this content may be the narration of some event whose accuracy a stands for thecurrently available facts, and each supplier can communicate this either via a TV newschannel (type t) or an online newspaper (type t′). Arguably, the same content cannot beconveyed in the same way using videos and sounds and using pictures and written words,and the attention required to process it might differ between media.20 It is reasonableto assume that higher clarity costs more: dt = d(ct) is a strictly increasing and weaklyconvex function of ct. For simplicity, we leave the dependence of d on a implicit.

Despite Assumption 1, generically in equilibrium only one type of suppliers is active.To state this, define

αtt′ = 1− a

c−1t′ − c−1

t

ctd(ct)− ct′d(ct′),

with the convention that α10 = −∞ and αT+1

T = 1. The role of αtt′ will become clear

shortly, but it is worth anticipating that αtt′ increases in ct and ct′ , as we show below.

Proposition 1. There exists a competitive equilibrium q∗.Case 1: α = αt

t′ for all t,t′. The equilibrium is unique and satisfies q∗t∗ = qt∗ and q∗t = 0

for all t = t∗, where t∗ is the unique type for which αt∗t∗−1 < α < αt∗+1

t∗ .Case 2: α = αt

t−1 for some t. There is a continuum of equilibria which satisfy q∗t−1 ∈[0, qt−1], q∗t = ct−1d(ct−1)

ctd(ct)

[qt−1 − q∗t−1

], and q∗t′ = 0 for all t′ = t, t− 1.

To see the intuition, think of d(ct)

ac−1t

as the cost per unit of content of a t-signal, taking

20For a similar point, see Dewan and Myatt (2008) and Webster (2014).

14

into account its return to attention for consumers. Similarly, think of et(q)ac−1

t

as the revenueper unit of clarity-adjusted content (the analog of a “price”) earned from a t-signal. Whentypes t and t′ < t attract positive attention, one can show that

et(q)ac−1

t

− et′(q)ac−1

t′=

c−1t′ − c−1

t

1− α. (4)

This positive quantity reflects the demand bias in favor of clearer sources. Using thedefinition of αt

t′ , we have thatc−1t′ − c−1

t

1− α≷ d(ct)

ac−1t

− d(ct′)

ac−1t′

if and only if α ≷ αtt′ . (5)

That is, α > αtt′ (resp. α < αt

t′) implies that the higher-clarity signal is more (resp. less)profitable than the lower-clarity one. This implies that when one type yields zero profits,the other must yield strictly negative profits and so no supplier wants to choose it. Inparticular, when αt

t−1 < α < αt+1t , t-signals are more profitable than any higher and

lower clarity signal and hence wins the entire market. Note that this result follows fromproperties of the attention functions that are derived (not assumed) from more primitiveproperties of how agents use information in a variety of social and economic contexts.

Although based on Section 3 one should expect that high α’s favor clear suppliersand low α’s favor opaque suppliers, it is perhaps surprising that, generically, only onetype is active in equilibrium. Moreover, this is a consequence of the competition forattention, not the lack of demand for one type of sources. Except for large α’s, one typeof suppliers dominates the market not because the agents do not want to attend to theothers at all, but because competition renders these types unprofitable. Indeed, the onlytype that can entirely crowd out the consumers’ attention is T : if qT ≥ qT (α), no othertype can attract positive attention by Corollary 1. The zero-profit quantity qT satisfiesqT ≥ qT (α) if and only if

α ≥ α ≡ 1− ac−1T−1 − c−1

T

cTd(cT ),

where α > αTT−1. Thus, if for instance α < αT

T−1, given the equilibrium q∗ at leastT -signals would receive some attention if provided by some source. Yet, the comparativeadvantage of lower-clarity signals in serving the consumers’ coordination motives rendersentry unprofitable for T -sources.

Another interesting property is that sources of low-clarity (even the lowest one) canwin the competition for attention. This may seem counterintuitive, as such sources offera low return to attention and they can never steal attention entirely from higher-claritysources. But the dependence of the result on the coordination motive being sufficientlyweak helps explain the puzzle and demonstrates that recognizing the role of information

15

sources as coordination devices is important to understand their supply. To see theintuition, suppose the agents’ actions are strong strategic substitutes—so that α < α1

0.We then know that the agents favor the lowest-clarity sources, as they better help themchoose actions far from the population average. With endogenous sources competingfor attention, this predilection for the lowest clarity depresses the profitability of higher-clarity sources, which can drive them out of business.

The thresholds αtt′ have some noteworthy properties. They are independent of the

agents’ payoff function u (which determines κ1 and ukk), prior precision p, and attentioncost τ . These aspects affect how much time the agents are willing to put overall intoacquiring information, but not how they divide it between sources. Accordingly, thesupplied quantity qt∗ is decreasing in p (higher prior knowledge weakens the incentive toacquire information), in |κ1| (a smaller |κ1| weakens the agents’ desire to respond to θ

and hence to acquire information), and in τ .21 The threshold α depends only on aspectsof the supply side of information, as summarized in the next result.

Proposition 2. For every t and t′, αtt′ decreases in the accuracy of information, a, and

increases in the clarity of either type, ct and ct′.

Of course, a affects the production cost d. The question here is what happens ifsources can provide higher accuracy at the same cost. Intuitively, a higher commonaccuracy lowers αt

t′ because a richer content takes more time to be understood and henceis more prone to being interpreted differently, causing mis-coordination. Since this effectis mitigated by higher clarity, consumers assign a premium to it at even lower α’s. TheThe threshold αt

t′ also falls when clarity worsens for the lower-clarity sources, becausethis boosts the extra profitability of the higher-clarity sources. These sources, however,do not benefit from an increase in their clarity: Being clearer can decrease the attentionrevenues and costs more at the margin by convexity of d; hence, it has to be offset by astronger coordination motive for clearer sources to cover their production costs.

It is interesting to consider the consequences of the equilibrium outcome on the con-sumers’ allocation of attention and the overall information they obtain.

Corollary 2. In equilibrium, every active t-supplier receives attention et(q∗) = d(ct).Therefore, as α increases, consumers allocate weakly more attention to each attended

21We conjecture that the equilibrium characterization in Proposition 1 (and 3 below) remains quali-tatively unchanged if the attention cost function is C(e) = C(

∑m em), where the function C satisfies

C′(0) > 0 and C

′′> 0. Indeed, the revenue differential in (4) does not depend on attention costs.

Also, the marginal attention cost enters symmetrically in all equilibrium zero-profit conditions, whichimplies—by the same logic of the proofs—that in equilibrium only one type of signals can win the wholemarket, generically.

16

source of information.

The first property follows from the zero-profit condition for active suppliers. The secondfollows from Proposition 1. As α rises, the equilibrium type of suppliers transitionsfrom lower to higher clarity. Thus, as the coordination motive increases, the consumers’understand better each information source provided in equilibrium, because its clarityincreases and its received attention increases. This also implies that each source becomesmore public, in the sense that its signals become more correlated across consumers.Following Myatt and Wallace (2012), the publicity of source m of type t is defined as thecorrelation between the signals of any pair (n, n′) of its consumers, which satisfies

corr(sn,m, sn′,m|θ) = et(q∗)c2t

a+ et(q∗)c2t. (6)

Note that this conclusion is true only in equilibrium. Myatt and Wallace (2012) showedthat, given a fixed supply of information, as α rises consumers shift attention from less tomore clear sources. Therefore, while the latter become more public, the former becomeless public. So, it is not true that all attended sources become uniformly more public.

The total information consumed by the agents depends on attention as well as thequantity of sources. The latter varies non-monotonically in α. As α rises within eachinterval (αt

t−1, αt+1t ), the quantity qt falls due to the effect of competition with the prior

as a coordination device. The question is what happens when we transition from qt−1 toqt at α = αt

t−1. Straightforward algebra implies that qt−1 > qt at αtt−1 if and only if

d(ct)− d(ct−1)

ct − ct−1

c2t c2t−1 <

aϕ

p√τ. (7)

Note that the right-hand side is strictly increasing in t due to the convexity of d.

Corollary 3. Suppose we refine equilibria so that there exists a unique active type ofsources for every α < 1. The equilibrium quantity of information is strictly decreasingin α if and only if α < αt+1

t, where t is the largest type for which (7) holds.

Thus, if t = T , the supply of information decreases monotonically as α rises. Otherwise,it jumps upward whenever there is a transition to a higher type, provided that α issufficiently large. This is because for low clarity the effect of competition for attentionwith the prior as a coordination device always dominates as α rises. But for sufficientlyhigh clarity, its higher return to attention wins as we transition to higher types. Sincesource-specific attention is monotonically increasing, the overall effect on total attentionis ambiguous. This renders the effect of α on the quality of consumed informationambiguous. One can show that each consumer ultimately conditions his action on the

17

average of the signals he attends, whose equilibrium total variance is 1p+ 1

aqt∗+

c−2t∗

qt∗et∗ (q∗)

(see Appendix C). This variance need not vary monotonically in α.

4.3 Competition via Accuracy

Consider now the case where suppliers use accuracy as a lever to compete for attention,but they all communicate via the same technology. That is, at > at′ for t > t′ and ct = c

for all t. As an illustrative example, consider a stylized scenario where there are onlyonline newspapers and each can choose between running a summary report (at′) or anin-depth analysis (at > at′) of some event or topic of public interest. We assume thathigher accuracy costs more: dt = d(at) is a strictly increasing function of at. Again, weleave the dependence of d on c implicit.

In this case, no type of suppliers can crowd out attention (Corollary 1): If qt > 0,

et(q) =atc

−1

p+∑T

t′=1 at′qt′

[ϕ√τ− pc−1

1− α

]> 0, (8)

where the inequality follows from Assumption 1. This already suggests that competitionvia accuracy is qualitatively different from competition via clarity.

Despite Assumption 1, the equilibrium supply of information is again dominated byone type of sources, under the generic property that the average cost of accuracy isminimized by only one type.22

Proposition 3. Suppose {t∗} = argmintd(at)at

. There exists a competitive equilibrium q∗,which is unique and satisfies q∗t∗ = qt∗ and q∗t = 0 for all t = t∗.

To gain intuition, again think of d(at)atc−1 as the per-unit cost of producing information

of type t and et(q)atc−1 as the per-unit revenue (again, “the price”) earned from supplying it.

Because all types have the same clarity, they earn the same price:et(q)atc−1

=1

p+∑T

t=1 atqt

[ϕ√τ− pc−1

1− α

], t = 1, . . . , T.

Moreover, this price falls as more suppliers enter. Free entry then implies that only thesuppliers with the cheapest production can profitably survive in the market.

A few remarks are in order. The result is again driven by the competition for at-tention. By (8), the agents would always pay attention to all active types, and one canfind examples of q where multiple types could make positive profits. Yet, such situa-tions cannot survive the competition for attention. Another interesting aspect is that

22If multiple types have minimal average cost, these will share the market and the others will stay outin equilibrium. Details are available upon request.

18

the equilibrium type of suppliers does not depend on the agents’ coordination motive(even though the signal correlation in (6) depends on a): α only affects—negatively—thesupplied quantity of information (qt∗) due to the aforementioned competition with theagents’ prior. This is in stark contrast with the case of competition via clarity. Theeffects of τ , p, and |κ1| on qt∗ follow the same logic as before. Finally, clarity has anon-monotonic effect on qt∗ :

∂qt∗

∂c=

c−2

d(at∗)

[− ϕ√

τ+

2pc−1

1− α

]> 0 if and only if c <

2p√τ

(1− α)ϕ.

This reflects the non-monotonic effect of clarity on attention explained after Lemma 1:Excessive clarity can depress the attention revenues, thereby reducing market entry.

The consequences of the equilibrium outcome on the consumers’ allocation of atten-tion and the overall information they obtain are as follows.

Corollary 4. In equilibrium, every active t∗-supplier receives attention et∗(q∗) = d(at∗).

Since the market-winning type is independent of α, so is the attention each source re-ceives. However, since qt∗ is strictly decreasing in α, so is the total attention allocation(i.e., Et∗(q∗) = qt∗d(at∗)). Overall, consumers get less information as α increases. Onecan show that each consumer ultimately conditions his action of the average of the sig-nals he attends, whose equilibrium total variance is 1

p+ 1

qt∗at∗+ c−2

qt∗et∗ (q∗)(see Online

Appendix C). This variance then increases as α rises.

4.4 Competition via Accuracy and Clarity

The case where the types of sources differ in both accuracy and clarity turns out tobe qualitatively similar to the case where they differ only in clarity. Clarity remainsthe driver of the competitive outcome. For this reason, we only explain the differencesfrom the previous analysis at an intuitive level for a two-type setting. The formal stepsfollow the same logic of the proof of Proposition 1. We have two possibilities to consider,depending on whether the choice between types involves a trade-off between accuracyand clarity. These possibilities relax the independence between accuracy and clarity thatwas implicit in the previous sections.

High-accuracy-low-clarity vs. low-accuracy-high-clarity. In this case, the feasibletypes are described by (ah, cl) and (al, ch), where ah > al and ch > cl. Without risk ofconfusion, we label the first hl and the second lh. As an intuitive interpretation, supposethat each news source is conveying information on the desirability of some public policy.One option is to publish a short article stating that the policy is desirable in simple words

19

which are easily understandable for everybody (type lh). Another option is to publishan in-depth report explaining the policy effects and all arguments in favor or againstit, which may involve data, graphs, technical jargon, and subtle logics and hence aremore likely to be interpreted in different ways by the agents (type hl). These types maycorrespond to the distinction in the news industry between “soft” and “hard” news.

To explain the outcomes of competition, we need the following notation:

α = 1− c−1l − c−1

hchaldlh − cl

ahdhl

and α = 1− c−1l − c−1

hchaldlh

< 1.

Note that α can exceed 1, in contrast to αtt′ < 1 in Section 4.2. This happens if the

high-accuracy-low-clarity sources cost significantly more: dhl >chahclal

dlh. If α < α, thecharacterization of competitive equilibria is as in Proposition 1 for T = 2, except thatα replaces α2

1. If instead α < 1 < α, for every α < 1 there exists a unique equilibriumwhere low-accuracy-high-clarity suppliers dominate the market. Intuitively, when dhl >chahclal

dlh, a type-hl source costs simply too much relative to the little attention it attractsbecause the agents have to process more content that is also more difficult to understand.Abstracting from details, consider the interpretation of type hl as hard news and typelh as soft news. There has been much discussion that soft news has been replacing hardnews over the recent decades. This model suggests that such a transition may be drivennot by increasing scarcity of consumers’ attention—its cost τ does not affect α—but byrising social conformism or technological changes in news production that lower α.

High-accuracy-high-clarity vs. low-accuracy-low-clarity. In this case, the feasibletype are described by (al, cl) and (ah, ch), where again ah > al and ch > cl. A reasonableassumption here is that dhh > dll. For illustration, consider a stylized situation wherenews providers can hire two types of journalists. Type hh is an expert, hard-working,perceptive journalist with excellent writing skills, who can produce accurate as well asclear articles; type ll is a novice, lazy, dull journalist with poor writing skills, who cannotachieve the same accuracy and clarity of type hh.

Turning to equilibrium analysis, we now need the following notation:

α′ = 1− c−1l − c−1

hchahdhh − cl

aldll

and α′ = 1− c−1l − c−1

hchahdhh

< 1.

Again, it is possible that α′ > 1 > α′. This happens if and only if accuracy differssignificantly between types: ah

al> ch

cl

dhhdll

. If α′ < α, the characterization of competitiveequilibria is again as in Proposition 1 with T = 2, except that α′ replaces α2

1. Note thatin this case, when α < α′, in equilibrium suppliers provide the unambiguously dominatedtype of sources, which is worse in terms of both accuracy and clarity; and this occurs,

20

even though the agents would be willing to pay some attention to the better sources hh.If instead α′ < 1 < α′, we again have a unique equilibrium for every α < 1 where type-hhsuppliers dominate the market. Intuitively, when ah

al> ch

cl

dhhdll

, a type-ll source can simplyattract too little attention to be profitable, due to its extremely low accuracy.

5 Inefficiencies in Information Markets

What inefficiencies can arise in information markets? Some can have familiar causes, suchas monopolistic power. A reasonable starting point is to rule these out by consideringcompetitive settings, so as to focus on inefficiencies specific to information markets.From Morris and Shin (2002), Angeletos and Pavan (2007), and Pavan (2014) (reviewedin the next section) we know that the agents’ payoff interdependencies distort their useand acquisition of information. This can in turn distort the overall market outcomeas we show in Section 5.3. But suppose we switch off these demand-driven distortions.Would the resulting competitive equilibrium be efficient (in some appropriate sense)? Weconsider this more novel question first. For the sake of brevity, we will focus on the moreinteresting case of competition via clarity and only discuss competition via accuracy (forthe generic cases with a unique equilibrium).

5.1 Efficient Demand of Information

Following Angeletos and Pavan (2007) and Pavan (2014), we can identify the inefficien-cies in the demand of information by studying the allocation of attention and use ofinformation that maximize the agents’ ex-ante utility for fixed information sources. Thiscriterion takes as constraints that information is dispersed in the economy and cannot betransferred between agents.23 Their failure to internalize payoff interdependencies leadsto three kinds of inefficiencies:

• complete-information externalities: Even under complete information, each agent mayincorrectly internalize how his action affects everybody else. The equilibrium action rule,κ(θ) = κ0 + κ1θ, can then differ from the first-best action rule, κ∗∗(θ) = κ∗∗

0 + κ∗∗1 θ.24

• socially optimal degree of coordination: Under incomplete information, each agentmay incorrectly internalize how aligning his action with its population average affectsdispersion (i.e., V ar[k − K]) and non-fundamental volatility (i.e., V ar[K − κ∗∗]). The

23Angeletos and Pavan (2009) investigate policies that can achieve efficiency in the use of information.24Angeletos and Pavan (2007) show that κ∗∗

0 = −uk(0,0,0,0)+uK(0,0,0,0)ukk+2ukK+uKK

and κ∗∗1 = − ukθ+uKθ

ukk+2ukK+uKK.

21

efficient action strategy k∗∗ requires to use the coordination motive

α∗∗ = 1− ukk + 2ukK + uKK

ukk + uσσ

,

which can differ from α. A standard assumption is that α∗∗ < 1.25

• social aversion to dispersion: By assumption dispersion, σ2, has a non-strategic effecton the agents’ payoffs, so they do not take it into account when allocating their attention.

Importantly, these inefficiencies arise from the payoff structure of the underlyinggame among the agents. Hence, their existence and effects apply for every configurationof information supply. Removing all them delivers the efficient allocation of attentionand use of information: These are given by similar conditions to those in Lemma 1 byreplacing α with α∗∗, κ1 with κ∗∗

1 , and ukk with ukk + uσσ (see (20) in Appendix B).

5.2 Inefficiency of Competitive Equilibria

Suppose the demand of information is efficient. Given the resulting market equilibrium,would a social planner prefer a different supply of information sources?

We first need to define social welfare. Since we do not have a price, we of course needa more general notion than the usual total surplus generated by a single market. Wedefine welfare as a weighted sum of the consumers’ surplus and the profit of each typeof suppliers. The latter is Et(q)− dtqt. Pavan (2014) shows that, under the efficient useof information (i.e., k∗∗), given attention allocation e the consumers’ surplus is (up to aconstant)

V(e) = L∗(e)− C(e),where

L∗(e) = ukk + 2ukK + uKK

2V ar[K − κ∗∗|e, k∗∗(·; e)] + ukk + uσσ

2V ar[k −K|e, k∗∗(·; e)].

L∗ measures the welfare loss due to volatility (first term) and dispersion (second term).It is worth noting that, although volatility is independent of the attention allocation ofeach agent individually,26 they internalize it through k∗∗(·; e) when choosing e. Recallthat we also assume uσσ = 0. Let γ = (γ1, . . . , γT ), and γc = 1−

∑t γt, where γt > 0 for

all t and γc > 0. We express welfare as

P(q; γ) = γcV(e(q)) +T∑t=1

γt[Et(q)− dtqt]. (9)

25Another standard assumption is that ukk + 2ukK + uKK < 0 and ukk + uσσ < 0, which ensure thatthe first-best attention allocation is unique and bounded (see Pavan (2014)).

26This is because in this class of models the distribution of K is independent of the attention allocationof each agent for every linear k(·; ·).

22

Note three things. The planner can neither transfer information between agents norcontrol how they allocate attention. P assigns the same weight to all agents and allsuppliers of each type, a reasonable simplification since the members of each group areidentical. Even if γt = 1

T+1for all t, P is not equivalent to the total surplus for a single

market, because here we do not have transferability via “money.”

Welfare depends on the information supply in complex ways (see Appendix A.6 fordetails). Changing q affects the total amount of provided content and how agents allocateattention. This allocation in turn determines how they weigh their prior and signalswhen choosing an action, which affects both dispersion and volatility. Attention alsoaffects dispersion by determining how differently the agents interpret each source. Finally,volatility depends on each source’s common noise and hence on the total provided content.

Despite this complexity, we obtain an unambiguous result. Generically, the compet-itive equilibrium does not maximize welfare—even when the demand is efficient.

Proposition 4. Suppose consumers acquire and use information efficiently (i.e., α = α∗∗,κ = κ∗∗, uσσ = 0), and α = αt

t′ for all t, t′. Let q∗∗ be the resulting competitiveequilibrium. For every γ, generically

q∗∗ /∈ argmaxq

P(q; γ).

Thus, generically the competitive equilibrium fails Pareto efficiency, defined by treatingall agents and all suppliers of each type as a single entity.

To gain intuition and add content to Proposition 4, we examine how the welfare effectsof changing the information supply depend on the attention cost.

Proposition 5. Fix γ and all parameters except the attention cost τ . Given α∗∗, letq∗∗ be the equilibrium with active type t∗∗. For every type t of sources that would receiveattention if supplied, there exists τt > 0 such that ∂P(q∗∗)

∂qt≷ 0 if and only if τ ≷ τt. Also,

τt > τt∗∗ if t < t∗∗. If τ = τt∗∗, then ∂P(q∗∗)∂qt∗∗+1

> 0 provided α∗∗ is sufficiently close to αt∗∗+1t∗∗ .

This result reveals a novel and general cause of inefficiency for information markets: thefact that attention becomes the “currency” whereby consumers pay for information. Thisis a key difference from standard markets where consumers pay with “money” throughthe frictionless price mechanism.

Two aspects of attention drive the result: (1) its cost τ to the consumer and (2) thefact that attention contributes to determining her understanding and hence her value forthe acquired information. We can see this with an analogy. A high τ acts analogouslyto a tax on the price in standard markets—with the important difference that τ is part

23

of the consumers’ preference! As a result, consumers pay little attention to informationsources, giving suppliers weak incentives to enter and causing inefficient under-provisionof information. A low τ instead acts analogously to a subsidy on a standard price.In this case, the consumers pay lots of attention to information sources, incentivizingexcessive entry and leading to inefficient over-provision of information. This possibilityindicates that the kind of inefficiency discovered here differs from the usual under-supplyinefficiency in the IO literature, caused by consumers’ not internalizing the full benefitsof firms’ entry. Continuing our analogy to understand (2), consider a situation where theprice paid for a good enters directly into the utility for that good (as in the case of luxuryproducts for some people). In this case, the efficient supply—which takes into accountthe effect of the resulting market price on the marginal utility from the good—can easilydiffer from the laissez-faire supply.27

Crucially, here the effect of attention on the value of information depends on thecoordination motive, which is therefore central to the welfare effects of changing thesupply of each signal type. This is because α∗∗ (but not τ) determines not only theequilibrium type t∗∗, but also which other types would receive attention if added tothe supply and which attention cost would render this welfare improving. Interestingly,if there is over-supply of the equilibrium type t∗∗, adding lower-clarity sources neverimproves welfare. This is because the consumers allocate too much attention due toits cheap cost; hence, welfare can improve only by adding sources with a higher returnto attention, that is, higher clarity. For instance, the proposition identifies cases whereadding sources of type t∗∗ + 1 strictly improves on q∗∗.

To iterate on the uniqueness of information markets, note that in our model entrantsof each type t essentially generate a horizontal supply curve of level dt (which is akin toa constant marginal cost of production for society). Therefore, if they faced a standarddownward-sloping demand curve, the competitive equilibria would be efficient, becausethe price mechanism would convey to the trading sides the social marginal cost andbenefit of the traded good. The ability of information suppliers to monetize attention,however, can be seen as giving rise to a non-standard demand schedule, which we showcauses inefficiency. Granting suppliers more freedom of adjusting clarity might allow thisdimension to play the role of a price. But in reality clarity need not be as adjustable asprices—due to granularity, technological constraints, or dependence on content—so thatthe inefficiencies found here may persist.

27That (2) contributes to the inefficiency can also be seen by noting that in general τt = 1 in Proposi-tion 5. Thus, even if the attention marginal cost for consumers equals its marginal revenue for suppliers(i.e., τ = 1), the equilibrium outcome is not guaranteed to be efficient.

24

Proposition 5 shows the effects of changing the equilibrium supply of information.But what is the welfare-maximizing supply? Are equilibria inefficient only in terms ofquantities or also in terms of the composition of source types? Answering these questionsanalytically is hard due to the intricate dependence of P(q; γ) on q. Turning to numericalmethods, we show that the efficient q can involve a positive quantity of multiple sourcetypes. Thus, while competition for attention pushes towards an homogeneous supplyof information, welfare maximization calls for an heterogeneous supply. To see why,suppose there are two feasible types (T = 2). Consider a situation where there areonly high-clarity sources, which cost less for the agents to acquire but more to produce.Since the marginal value of information is diminishing, at some point it equals the costof the marginal high-clarity source (d2). At this point, however, the marginal value ofinformation can still exceed the cost of the cheaper low-clarity sources (d1 < d2). Hence,adding such sources increases welfare. On the other hand, a supply with only low-clarity sources cannot be efficient—at least for large α’s—because high-clarity sourcesare superior coordination devices.

α

d2d1

= 1.2

d2d1

= 5

d2d1

= 10

d2d1

= 20

α

d2d1

= 20

d2d1

= 10

d2d1

= 5

d2d1

= 1.2

(a) Beauty-contest payoff (b) General payoff

Figure 2: Efficient supply

Figure 2 represents the efficient supply of information for different specifications ofthe agents’ payoffs and the production costs. The curves describe the efficient share ofhigh-clarity sources (i.e., q2

q2+q1) as a function of α ∈ (−1, 1) for different values of d2/d1.

Figure 2(a) is for the beauty-contest payoff specification of Myatt and Wallace (2012),which can be written as

u(k,K, σ, θ) = −k2 + 2αkK − αK2 + 2(1− α)kθ − (1− α)θ2

and by construction satisfies κ = κ∗∗, α = α∗∗, and uσσ = 0. Figure 2(b) is for a more

25

general payoff function that satisfies

u(k,K, σ, θ) =ukk

2k2 + ukKkK +

uKK

2K2 + ukθkθ,

where we impose that ukK = −ukkα and uKK = −ukK to ensure that α∗∗ = α. Toease the comparison with the beauty-contest case, we fix ukk = −2. Also, we selectedukθ = 0.3.28 Remarkably, the share of high-clarity sources is often interior and far fromthe equilibrium outcome (it is never one!). Also, the efficient supply can be qualitativelydifferent depending on the nature of the agents’ interaction. One important aspect isthat in the beauty-contest case α controls directly the agents’ desire to respond to thestate as measured by ukθ = 2(1−α), which shrinks to zero as α → 1. By contrast, in themore general payoff ukθ is independent of α. This explains the stark difference betweenthe two panels in Figure 2.

5.3 Inefficient Use of Information and Supply of Information

Efficiency in the use and acquisition of information cannot be taken for granted. Howdo these demand-driven inefficiencies affect information supply? Recall that the efficientattention allocation is given by replacing α with α∗∗, κ1 with κ∗∗

1 , and ukk with ukk +uσσ

in the consumer equilibrium. Combining this with Proposition 1, we obtain the following.

Proposition 6. Suppose α = αtt′ for all t, t′.

• Inefficiencies caused by complete-information externalities or social aversion to dis-persion do not affect the equilibrium type of suppliers t∗. Considering its equilibriumquantity as a function of (κ′, u′

σσ, α′), we have

qt∗(κ∗∗, 0, α) > qt∗(κ, 0, α) = q∗t∗ if and only if |κ∗∗

1 | > |κ1|,qt∗(κ, uσσ, α) > qt∗(κ, 0, α) = q∗t∗ if and only if uσσ < 0.

• Differences between private and socially optimal coordination motives can affect theequilibrium type of suppliers: If α < αt∗+1

t∗ < α∗∗ (resp. α > αt∗t∗−1 > α∗∗), this inefficiency

causes the equilibrium type t∗ to be too opaque (resp. clear).• If the equilibrium type is t∗ under both α and α∗∗, we have

qt∗(κ, 0, α∗∗) < qt∗(κ, 0, α) = q∗t∗ if and only if α < α∗∗.

Intuitively, any shortfall in the agents’ willingness to acquire information lowers theattention revenues for suppliers and ultimately their equilibrium quantity. Such short-falls can result from not internalizing the complete-information externalities, if this lowers

28We chose this value simply because it delivers a clear graph in Figure 2. Other simulations suggestthat its qualitative properties are robust to changing ukθ and allowing for uk = 0, uK = 0, or uKθ = 0.

26

the agents’ desire to respond to the state (i.e., |κ1| < |κ∗∗1 |). Similarly, not internaliz-

ing society’s aversion to dispersion renders the agents less willing to pay attention toinformation sources, as doing so curbs the heterogeneity in beliefs that causes dispersion.An excessive coordination motive (i.e., α > α∗∗) leads the agents to overweight theirprior when choosing their actions, which reduces the value of acquiring new information.When competition is via clarity, differences between the private and socially optimalcoordination motive can also change the equilibrium type of suppliers, as that motivedrives the agents’ preference for clarity.

When competition is via accuracy, inefficiencies in the information demand affect onlythe supplied quantity of information, not its type. As Proposition 3 shows, α is irrelevantfor which type of sources dominates the market. The distortions in the supplied quantitiesare qualitatively the same as in Proposition 6.

Finally, Proposition 6 shows how the supply of information changes if we removeinefficiencies in the use and acquisition of information. While doing so improves theagents’ welfare when information sources are fixed (by definition), this need not be thecase once their supply is endogenous.

6 Concluding Remarks

Information sources allow agents to learn about relevant events as well as about others’knowledge of those events, which is essential for coordination to occur in social and eco-nomic contexts. We examine how the dual role of information sources as learning andcoordination devices affect how information markets work, in contexts where supplierscompete for consumers’ attention. The paper offers two main takeaways. First, competi-tion for attention pushes towards a homogenization of information sources, whose type isdriven by the consumers’ desires to coordinate in combination with the sources’ clarities.Second, by becoming the “currency” whereby consumers pay for information, attentioncauses information markets to work inefficiently. This novel source of inefficiency is ontop of those due to inefficient uses of information and suppliers’ market power.

We close with three remarks. First, to focus on the consequences of competition forattention, this paper intentionally rules out price competition. Though in many settingsinformation providers seem to optimally set zero prices for accessing their sources, thisis not always the case. In future research, we plan to investigate how adding pricecompetition affects information markets. On the one hand, it would be interesting toexamine the trade-off that suppliers face between attention and price revenues and its

27

solution under competition. On the other hand, it would be interesting to understand ifand when the price mechanism can restore efficiency in competitive information markets.We conjecture that as long as attention provides a share of the revenues of informationsources—which is often the case in reality—the inefficiencies we found in this papershould persist to some degree.

Second, a promising line for future research is to investigate policies that can ad-dress the market inefficiencies caused by the fact that the traded good is information.Angeletos and Pavan (2009) propose policies that can address inefficiencies originatingin the demand side of the market. It would be interesting to consider policies directedat the supply side (e.g., entry regulations) that aim to address both the demand-sideinefficiencies and the novel kind of inefficiencies found in this paper.

Finally, one could extend the model analyzed in this paper in several directions to offera more realistic description of information markets. Real-world consumers of informationare not identical. They differ in terms of conformism, topics of interest, and abilities oravailable time for acquiring information. Some of these aspects can be easily incorporatedin the framework of this paper without changing its thrust. For instance, as suggested byDewan and Myatt (2008, 2012), one can interpret the state θ not as the same ideal actionfor all agents in society, but as the average ideal action. Angeletos and Pavan (2009) allowfor the possibility of multidimensional states with private and common components. Insome cases, different topics and social groups give rise to different markets, where eachcan be studied using our model. Other extensions are more intricate. For instance,allowing for different coordination motives or attention costs within a single market havedeep consequences already for the demand of information, which remain unexplored.Either way, allowing for heterogeneity of consumers is likely to result in heterogeneityof information sources. This should not be surprising. Yet, the point remains that,within market segments, competition for attention creates a force towards homogeneityand generic inefficiencies.

28

A Appendix: Proofs

A.1 Proof of Lemma 1

For t ∈ Te, let δt = etc2t , bt(δt) = 1−α

(1−α)a−1t +δ−1

t

, and B = p +∑

t∈Teqtbt, where we omit the

dependence on δt for simplicity. Given this, equation (18) in Appendix B becomes

δt =ϕ√τ

btBct. (10)

Substituting (10) in the expression for bt yields

bt = at −Bat

√τ

(1− α)ϕct. (11)

It follows thatB − p =

∑t∈Te

qtbt =∑t∈Te

atqt −B√τ

(1− α)ϕ

∑t∈Te

atqtct

,

which yields

B = (p+∑t∈Te

atqt)

[1 +

√τ

(1− α)ϕ

∑t∈Te

atqtct

]−1

. (12)

Now use (11) and (12) to replace in the expression of δt to obtain

δt =

1 +

√τ

(1− α)ϕ

∑t′∈Te

at′qt′

ct′

atp+

∑t′∈Te

atqt− at

√τ

(1− α)ϕct

ϕ√τct

=

1 + √τ

(1− α)ϕ

∑t′∈Te

at′qt′

ct′−

p+∑

t′∈Teat′qt′

(1− α)ϕct/√τ

atϕct/√τ

p+∑

t′∈Teat′qt′

=

1 + √τ

ϕ

∑t′∈Te\{t}

at′qt′

1− α(c−1

t′ − c−1t )− p

√τ

(1− α)ϕct

atϕct/√τ

p+∑

t′∈Teat′qt′

=

ϕ√τ+

∑t′∈Te\{t}

at′qt′

1− α(c−1

t′ − c−1t )− p

(1− α)ct

atctp+

∑t′∈Te

at′qt′.

Finally, using the definition of δt and ϕ, we obtain

et(q) =atc

−1t

p+∑

t′∈Teat′qt′

ϕ√τ+

∑t′∈Te\{t}

at′qt′

1− α(c−1

t′ − c−1t )− pc−1

t

1− α

. (13)

29

A.2 Proof of Corollary 1

The first part of the corollary is immediate from (13) and Assumption 1. Now, suppose ct > ct′

for all t > t′. Assumption 1 and part 5 of Lemma 4 imply that et1(q) > 0 whenever t1 is thelargest type in Tq. Part 5 of Lemma 4 also implies that if t ∈ Te, than all higher types belongto Te. We now show when Te = {t1}.

Claim 1. Let t2 be the second largest type in Tq. For every α < 1, we have et(q) = 0 for allt ∈ Tq \ {t1} if and only if qt1 ≥ qt1(α) where

qt1(α) =

[ϕ√τ(1− α)ct2 − p

]ct1a

−1t1

ct1 − ct2.

Proof. (Only if) Suppose et = 0 for all t ∈ Tq \ {t1}. Then, by part 2 of Lemma 4, et1 mustsatisfy

et1 =at1c

−1t1

p+ at1qt1

[ϕ√τ−

pc−1t1

1− α

]. (14)

Also, by part 3 of Lemma 4, it must be that

τ

ϕ2c2t≥

[p

1− α+

1

(1− α)a−1t1

q−1t1

+ c−2t1

e−1t1

q−1t1

]−2

=

[p

1− α+

(1− α

at1qt1+

1

qt1et1c2t1

)−1]−2

=

p

1− α+ at1qt1

ϕ√τct1 −

p1−α

(1− α) ϕ√τct1 + at1qt1

−2

,

where we used the expression for et1 in (14). Let qt(α) be defined as the unique value of qt suchthat the previous inequality holds with equality:

qt(α) =

[ϕ√τ(1− α)ct − p

]ct1a

−1t1

ct1 − ct> 0, (15)

where the inequality follows from Assumption 1. It is then immediate that et = 0 for all t < t1

implies qt1 ≥ maxt<t1 qt(α) = qt1(α). Note that qt1(α) is strictly decreasing in α and reacheszero for some α < 1.

(If) Let qt1 ≥ qt1(α). We reason by induction. Let t be the smallest t ∈ Tq and supposeet > 0. Since et1 > 0, by expression (13) et must satisfy

et =atc

−2t

p+∑

t′∈Teat′qt′

ϕ√τct +

1

1− α

∑t′∈Te\{t}

at′qt′

(ctct′

− 1

)− p

1− α

≤

atc−2t

p+∑

t′∈Teat′qt′

[ϕ√τct +

1

1− αat1qt(α)

(ctct1

− 1

)− p

1− α

]

30

=atc

−2t

p+∑

t′∈Teat′qt′

[ϕ√τct −

ϕ√τct

]= 0,

where qt(α) is defined in (15). This contradicts et > 0. Now suppose that et′′ = 0 for all t′′ < t

in Tq and et > 0. Then, et must satisfy

et =atc

−2t

p+∑

t′∈Teat′qt′

ϕ√τct +

1

1− α

∑t′∈Te\{t}

at′qt′

(c1

ct′− 1

)− p

1− α

≤ atc

−2t

p+∑

t′∈Teat′qt′

[ϕ√τct +

1

1− αat1qt(α)

(ctct1

− 1

)− p

1− α

]=

atc−2t

p+∑

t′∈Teat′qt′

[ϕ√τct −

ϕ√τct

]= 0,

where the inequality follows because Te contains only types larger than t and qt(α) < qt1(α).

A.3 Proof of Proposition 1

Given q, let Q =∑T

t=1 qt and for all t define

µt(q) =1

p+ aQ

ϕ√τ− pc−1

t

1− α+

a

1− α

∑t′ =t

qt′(c−1t′ − c−1

t )

− ctd(ct)

a.

Note that µt(qt,q−t) is strictly decreasing in qt for all q−t. Also, µt(q) > µt′(q) if and only if

1

p+ aQ

a

1− α

∑t =t

qt(c−1t

− c−1t )−

∑t=t′

qt(c−1t

− c−1t′ )

+p

1− α(c−1

t′ − c−1t )

is strictly larger than ctd(ct)−ct′d(ct′ )

a . The quantity in square brackets equals∑t =t,t′

qt(c−1t

− c−1t ) + qt′(c

−1t′ − c−1

t )−∑t =t′,t

qt(c−1t

− c−1t′ )− qt(c

−1t − c−1

t′ )

=∑t =t,t′

qt(c−1t′ − c−1

t ) + qt′(c−1t′ − c−1

t ) + qt(c−1t′ − c−1

t ) = (c−1t′ − c−1

t )Q.

Therefore, µt(q) > µt′(q) if and only if

c−1t′ − c−1

t

1− α>

ctd(ct)− ct′d(ct′)

a,

which holds if and only if α > αtt′ .

31

Note that the zero-profit condition (2) is equivalent to q∗t µt(q∗t ,q∗

−t) = 0 and the no-entrycondition (3) is equivalent to qtµt(qt,q∗

−t) < 0 if qt > q∗t .

Case 1: α = αtt′ for all t and t′.

Claim 2. If q∗ is an equilibrium, then Tq∗ = {t : q∗t > 0} is a singleton.

Proof. We know from Assumption 1 that q = 0 cannot be an equilibrium. Suppose that|Tq∗ | ≥ 2. Then, for every t ∈ Tq∗ we must have µt(q∗) = 0. Take any t, t′ ∈ Tq∗ witht > t′. If α > αt

t′ , then µt(q∗) > µt′(q∗) = 0, which is a contradiction. If α < αtt′ , then

µt(q∗) < µt′(q∗) = 0, which is also a contradiction.

Claim 3. If q∗ is an equilibrium and {t∗} = Tq∗ , then q∗t∗ = qt∗ and αt∗t∗−1 < α < αt∗+1

t∗ .

Proof. Since {t∗} = Tq∗ means that q∗t∗ > 0, it must be that q∗t∗ = qt∗ by the definition ofthe latter as the unique positive solution to Et∗(qt∗ ,0) = qt∗d(ct∗). This also implies thatµt∗(qt∗ ,0) = 0. Suppose t∗ < T , so that suppliers of type t∗+1 would receive positive attentionupon entering by Corollary 1. By condition (3), it must be that qt∗+1µt∗+1(qt∗ , qt∗+1,0) < 0 forall qt∗+1 > 0, where 0 has T − 2 elements. Therefore, µt∗+1(qt∗ , qt∗+1,0) < 0 for all qt∗+1 > 0.This holds if and only if µt∗+1(qt∗ , 0,0) ≤ 0 by the continuity and monotonicity of µt∗+1 inqt∗+1. By the previous calculations we know that µt∗+1(qt∗ ,0) ≤ 0 = µt∗(qt∗ ,0) if and only ifα ≤ αt∗+1

t∗ . Since α = αt∗+1t∗ , we established one inequality. If t∗ = 1, we are done.

Now consider t∗ − 1 (if any). There are two possibilities. First, suppliers of type t∗ − 1

cannot attract attention, which requires that qt∗ ≥ qt∗(α) by Corollary 1. This holds if andonly if

α ≥ αt∗ ≡ 1− ac−1t∗−1 − c−1

t∗

ct∗d(ct∗).

Second, suppliers of type t∗ − 1 would attract attention upon entering. In this case, by thesame argument as before, it must be that µt∗−1(qt∗ , 0,0) ≤ 0. Substituting qt∗ , we get thatthis condition is equivalent to α ≥ αt∗

t∗−1. Since αt∗ > αt∗t∗−1 and α = αt∗

t∗−1, we established theother inequality.

Claim 4. If αt∗t∗−1 < α < αt∗+1

t∗ , q∗ with {t∗} = Tq∗ and q∗t∗ = qt∗ is the unique equilibrium.

Proof. As shown in Proposition 2 (whose proof is next), αtt′ is increasing in both ct and ct′ .

Therefore, αt∗t∗−1 < α < αt∗+1

t∗ implies αt∗t < α < αt′

t∗ for all t < t∗ < t′. Consequently,µt(qt∗ ,0) < 0 for all t = t∗. By the monotonicity property of µt in qt, it follows thatµt(qt∗ , qt,0) < 0 for all qt > 0 and t = t∗. Thus, for every t, t-suppliers strictly prefer not

32

to enter because, even if they could attract attention, they would make strictly negative prof-its. By definition, qt∗ is the unique solution to qt∗µt∗(qt∗ ,0) = 0; hence, q∗ = (qt∗ ,0) is theunique equilibrium.

Case 2: If α = αtt−1 for some t.

Claim 5. If q∗ is an equilibrium, t′ /∈ Tq∗ for all t′ = t, t− 1.

Proof. If α = αtt−1 for some t (which cannot be t = 1), µt(q) = µt−1(q) for all q. Also, since

αtt−1 < αt′

t−1 for all t′ > t by Proposition 2, µt′(q) < µt(q) for all q and t′ > t. Similarly, sinceαtt−1 > αt

t′ for all t′ < t−1 by Proposition 2, µt′(q) < µt−1(q) for all q and t′ < t−1. Therefore,if t′ < t − 1 belonged to Tq∗ , we would have to have 0 = µt′(q∗) < µt−1(q∗). If q∗t−1 > 0, thisimmediately contradicts the zero-profit condition for type t− 1. If q∗t−1 = 0, continuity of µt−1

in qt−1 implies that there exists qt−1 > 0 such that µt−1(qt−1,q∗−(t−1)) > 0, now contradicting

the condition (3) for type t− 1. A similar argument rules out that t′ > t belongs to Tq∗ .

Claim 6. For every qt−1 ∈ [0, qt−1], there exists an equilibrium with q∗t−1 = qt−1 and q∗t =ct−1d(ct−1)

ctd(ct)[qt−1 − qt−1].

Proof. Consider qt−1 = qt−1. By definition of qt−1, µt−1(qt−1,0) = 0. By α = αtt−1,

µt(qt−1,0) = 0. By monotonicity of µt in qt, µt(qt−1, qt,0) < 0 for all qt > 0. Therefore,no t-suppliers want to enter. Consider qt−1 = 0. In this case,

qt =ct−1d(ct−1)

ctd(ct)

[1

ct−1d(ct−1)

(ϕ√τ−

pc−1t−1

1− α

)− p

a

]

=1

ctd(ct)

(ϕ√τ−

pc−1t−1

1− α+

pc−1t

1− α− pc−1

t

1− α

)− ct−1d(ct−1)

ctd(ct)

p

a

=1

ctd(ct)

(ϕ√τ− pc−1

t

1− α

)− p

actd(ct)

[c−1t−1 − c−1

t

1− α− ct−1d(ct−1)

]

=1

ctd(ct)

(ϕ√τ− pc−1

t

1− α

)− p

a= qt,

where the forth equality uses α = αtt−1. By the same argument as before, given this qt, no

supplier of type t − 1 wants to enter. Consider now qt−1 ∈ (0, qt). By definition of qt−1, weknow that µt−1(qt−1,0) > 0. Therefore, to have an equilibrium, qt must be positive and induceµt−1(qt, qt−1,0) = µt(qt, qt−1,0) = 0. This is equivalent to

ctd(ct)[pa+ qt + qt−1

]=

ϕ√τ− pc−1

t

1− α+

c−1t−1 − c−1

t

1− αaqt−1,

33

orctd(ct)

[pa+ qt + qt−1

]=

ϕ√τ−

pc−1t−1

1− α+ a

[pa+ qt−1

] c−1t−1 − c−1

t

1− α.

Using α = αtt−1, this becomes

ctd(ct)[pa+ qt + qt−1

]=

ϕ√τ−

pc−1t−1

1− α+[pa+ qt−1

][ctd(ct)− ct−1d(ct−1)],

or

qt =ct−1d(ct−1)

ctd(ct)

[1

ct−1d(ct−1)

(ϕ√τ−

pc−1t−1

1− α

)− p

a− qt−1

]=

ct−1d(ct−1)

ctd(ct)

[qt−1 − qt−1

].

Finally, given the suggested quantities of types t and t− 1, no other type of suppliers wantto enter the market. Since µt(qt, qt−1, 0) = µt−1(qt, qt−1, 0) = 0 and α = αt

t−1, every typet′ = t, t− 1 will make negative profits after entering.

A.4 Proof of Proposition 2

The first part of the proposition is immediate. For the second part, fix t > t′. Supposect > ct > ct′ . Then, αt

t′ > αtt′ if and only if

c−1t′ − c−1

t

ctd(ct)− ct′d(ct′)<

c−1t′ − c−1

t

ctd(ct)− ct′d(ct′),

or equivalently

ct

[d(ct)− d(ct′)

ct − ct′ct + d(ct′)

]< ct

[d(ct)− d(ct′)

ct − ct′ct + d(ct′)

].

By convexity of d, we have d(ct)−d(ct′ )ct−ct′

≤ d(ct)−d(ct′ )ct−ct′

. Since ct > ct, the inequality holds. Now,suppose ct > ct′ > ct′ . Then, αt

t′ > αtt′ if and only if

c−1t′ − c−1

t

ctd(ct)− ct′d(ct′)<

c−1t′ − c−1

t

ctd(ct)− ct′d(ct′),

or equivalently

ct′

[ct′

d(ct)− d(ct′)

ct − ct′+ d(ct)

]< ct′

[ct′

d(ct)− d(ct′)

ct − ct′+ d(ct)

].

Using convexity of d, the same argument as before establishes that this inequality holds.

34

A.5 Proof of Proposition 3

We can write the equilibrium equations (2)–(3) as q∗t µt(q∗t ,q∗

−t) = 0 and qtµt(qt,q∗−t) < 0 if

qt > q∗t , where now

µt(q) =1

p+∑T

t=1 atqt

[ϕ√τ− pc−1

1− α

]− c

d(at)

at.

Note that, for every q, µt(q) > µt′(q) if and only if d(at)at

<d(at′ )at′

. Also, for all t, µt(qt′ ,q−t′) isstrictly decreasing in qt′ for every t′.

By Assumption 1, we know that there does not exist an equilibrium with q∗ = 0. Byassumption arg mint

d(at)at

= {t∗}. If q∗ is an equilibrium, we must have {t∗} = Tq∗ . If not,q∗t > 0 for some t = t∗ and qtµt(q∗) = 0, which implies that µt(q∗) = 0. But then, bycontinuity, there exists qt∗ > 0 such that µt∗(qt∗ ,q∗

−t∗) > 0 and hence qt∗µt∗(qt∗ ,q∗−t∗) > 0,

which contradicts condition (3). This shows that t∗ ∈ Tq∗ . Given this, we must have µt∗(q∗) = 0

by condition (2). Since then µt(q∗) < 0 for every other t, it follows that t /∈ Tq∗ for all t = t∗.

It follows immediately that q∗t∗ = qt∗ and q∗t = 0 for all t = t∗ is an equilibrium. By theprevious points, this is also the unique equilibrium.

A.6 Proof of Propositions 4 and 5

We start by unpacking the components of the planner’s objective function. Recall that thesymmetric consumer equilibria considered in this paper ultimately depend on the total attentionallocated to each type of sources, that is, E = (E1, . . . , ET ).

Lemma 3. Given q and E, let W ∗∗t (E) = qtw

∗∗t

(E1q1, . . . , ET

qT

)where w∗∗

t is the weight assignedto each t-signal by the efficient strategy k∗∗(·; ·).29 Then, we have

V ar[k −K|E, k∗∗(·;E)] = [κ∗∗1 ]2T∑t=1

[W ∗∗t (E)]2

c2tEt

V ar[K − κ∗∗|E, k∗∗(·;E)] = [κ∗∗1 ]2

{T∑t=1

[W ∗∗t (E)]2

aqt+

[W ∗∗p (E)]2

p

},

where W ∗∗p (E) = 1−

∑Tt=1W

∗∗t (E).

Proof. Suppose first that, for all t, qt is a fixed non-negative integer. When each consumer n

follows the efficient strategy k∗∗(sn;E), we have

k∗∗(sn;E) = κ∗∗0 + κ∗∗1

T∑t=1

∑m∈qt

w∗∗t

(E1

q1, . . . ,

ET

qT

)sn,m = κ∗∗0 + κ∗∗1

T∑t=1

W ∗∗t (E)Sn

t ,

29In this lemma, we adopt the convention that 00 = 0 to simplify its statement.

35

where W ∗∗t (E) = qtw

∗∗t

(E1q1, . . . , ET

qT

)and (if qi > 0)

Snt =

1

qt

∑m∈qt

sn,m =1

qt

∑m∈qt

(θ + ym + xn,m) = θ +1

qt

∑m∈qt

(ym + xn,m).

Given this, the average action satisfies

K∗∗((sn)n∈[0,1];E) =

ˆ 1

0k∗∗(sn;E)dn = κ∗∗0 + κ∗∗1

T∑t=1

W ∗∗t (E)

ˆ 1

0Snt dn,

where (again, if qt > 0)ˆ 1

0Snt dn = θ +

1

qt

∑m∈qt

(ym +

ˆ 1

0xn,mdn

)= θ +

1

qt

∑m∈qt

ym,

since with i.i.d. error terms and a continuum of agents´ 10 xn,mdn equals the mean of each xn,m,

which is zero. Thus, letting Xnt = 1

qt

∑m∈qt x

n,m, we have

[k∗∗ −K∗∗]((sn)n∈[0,1];E) = κ∗∗1

[T∑t=1

W ∗∗t (E)

(Snt −ˆ 1

0Snt dn

)]= κ∗∗1

T∑t=1

W ∗∗t (E)Xn

t .

Since for each t the variable Xnt is normally distributed with mean zero and variance [c2tEt]

−1

and Xnt is independent of Xn

t′ for t = t′, we obtain

V ar[k −K|E, k∗∗(·;E)] = [κ∗∗1 ]2T∑t=1

[W ∗∗t (E)]2

c2tEt.

Now letting Yt =1qt

∑m∈qt y

m, we have

[K∗∗ − κ∗∗]((sn)n∈[0,1];E) = κ∗∗1

[T∑t=1

W ∗∗t (E)

(ˆ 1

0Snt dn

)− θ

]

= κ∗∗1

[T∑t=1

W ∗∗t (E) (θ + Yt)− θ

]

= κ∗∗1

[T∑t=1

W ∗∗t (E)Yt −W ∗∗

p (E)θ

],

where W ∗∗p (E) = 1 −

∑Tt=1W

∗∗t (E). Since Yt ∼ N(0, a−1q−1

t ) for each t, θ ∼ N(0, p−1), and θ

and the Yt’s are mutually independent, we obtain

V ar[K − κ∗∗|E, k∗∗(·;E)] = [κ∗∗1 ]2

{T∑t=1

[W ∗∗t (E)]2

aqt+

[W ∗∗p (E)]2

p

}.

36

Thus, we proved the result for every vector q of non-negative integers. As argued in Appendix C,it is possible to allow qt to be any non-negative real number for all t, thereby simplifying theanalysis while maintaining the interpretation of the above expressions.

We now proceed with the proof of the proposition. To simplify notation, hereafter letχ = ukk + 2ukK + uKK and γc = 1 −

∑Tt=1 γt. Note that χ < 0 because α∗∗ = 1 − χ

ukk< 1

and ukk < 0 by assumption. Using Lemma 3 and letting dt = d(ct), we can write the planner’sobjective function as

P(q) =T∑t=1

γt(Et − dtqt) + γcχ[κ∗∗1 ]2

2

{T∑t=1

[W ∗∗t (E)]2

aqt+

[W ∗∗p (E)]2

p

}(16)

+γc

{ukk2

[κ∗∗1 ]2T∑t=1

[W ∗∗t (E)]2

Etc2t− τ

T∑t=1

Et

},

where we will leave the dependence of each Et on q implicit. Note that for every q the derivativesof the last line in (16) are always equal to zero by a standard envelope argument, becauseconsumers always choose E optimally. Moreover, when consumers pay attention to t-sources,by Lemma 4 (adapted to the case of α = α∗∗, κ1 = κ∗∗1 , and uσσ = 0) we have that

W ∗∗t (E) =

ct√τ

ϕEt.

Combining these observations, we can reduce the object of study to

P(q) = γcχ[κ∗∗1 ]2

2

T∑t=1

τc2tE2t

ϕ2aqt+

1

p

[1−

√τ

ϕ

T∑t=1

ctEt

]2+T∑t=1

γt(Et − dtqt).

Using this, we have

∂P∂qt

= γcχ[κ∗∗1 ]2

∑t′ =t

[τc2t′Et′

ϕ2aqt′

]∂Et′

∂qt−

√τ

pϕ

[1−

√τ

ϕ

T∑t′′=1

ct′′Et′′

][T∑

t′′=1

ct′′∂Et′′

∂qt

] (17)

+γcχ[κ∗∗1 ]2{τc2tEt

ϕ2aqt

[∂Et

∂qt− Et

2qt

]}+∑t′ =t

γt′∂Et′

∂qt+ γt

[∂Et

∂qt− dt

].

To simplify this, we use the formula of Et to calculate ∂Et∂qt

and ∂Et′∂qt

for t′ = t:

∂Et

∂qt=

∂

∂qt

qtc−1t

pa +

∑t′∈Te

qt′

ϕ√τ+

∑t′′∈Te\{t}

aqt′′(c−1t′′ − c−1

t )

1− α∗∗ − pc−1t

1− α∗∗

37

=

pa +

∑t′∈Te\{t} qt′( p

a +∑

t′∈Teqt′)2

ct

ϕ√τ+

∑t′′∈Te\{t}

aqt′′(c−1t′′ − c−1

t )

1− α∗∗ − pc−1t

1− α∗∗

,

∂Et′

∂qt=

∂

∂qt

qt′c−1t′

pa +

∑t′′∈Te

qt′′

ϕ√τ+

∑t′′∈Te\{t′}

aqt′′(c−1t′′ − c−1

t′ )

1− α∗∗ −pc−1

t′

1− α∗∗

=

−qt′c−1t′( p

a +∑

t′′∈Teqt′′)2 ϕ√

τ+

∑t′′∈Te\{t′}

aqt′′(c−1t′′ − c−1

t′ )

1− α∗∗ −pc−1

t′

1− α∗∗

+

qt′c−1t′

pa +

∑t′′∈Te

qt′′

[a(c−1

t − c−1t′ )

1− α∗∗

]

=−qt′c

−1t′( p

a +∑

t′′∈Teqt′′)2 ϕ√

τ− pc−1

t

1− α∗∗ +∑t′′∈Te

aqt′′(c−1t′′ − c−1

t )

1− α∗∗

.

Given α∗∗, let t∗∗ be the type of sources supplied in the ensuing equilibrium q∗∗. We will firstevaluate (17) with respect to qt∗∗ at the equilibrium (0, qt∗∗) for qt∗∗ = 1

dt∗∗ct∗∗

[ϕ√τ− pc−1

t∗∗1−α∗∗

]−

pa > 0. We have,

∂Et∗∗(0, qt∗∗)

∂qt∗∗=

pac

−1t∗∗( p

a + qt∗∗)2 [ ϕ√

τ− pc−1

t∗∗

1− α∗∗

]=

pact∗∗d

2t∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

,

and clearly ∂Et(0,qt∗∗ )∂qt∗∗

= 0 for t = t∗∗. Recall that Et∗∗(0, qt∗∗) = qt∗∗dt∗∗ , Et(0, qt∗∗) = 0 fort = t∗∗. Combining terms, we obtain

∂P(0, qt∗∗)

∂qt∗∗= −γcχ[κ∗∗1 ]2

√τ

pϕ

[1−

√τ

ϕct∗∗qt∗∗dt∗∗

] pac

2t∗∗d

2t∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

+γcχ[κ∗∗1 ]2

τc2t∗∗dt∗∗

ϕ2a

pact∗∗d

2t∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

− dt∗∗

2

+ γt∗∗

pact∗∗d

2t∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

− dt∗∗

= −γcχ[κ∗∗1 ]2

τ

ϕ2

[c−1t∗∗

1− α∗∗ +ct∗∗dt∗∗

a

] pac

2t∗∗d

2t∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

+γcχ[κ∗∗1 ]2

τc2t∗∗dt∗∗

ϕ2a

pact∗∗d

2t∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

− dt∗∗

2

+ γt∗∗

pact∗∗d

2t∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

− dt∗∗

= −γcχ[κ∗∗1 ]2

τ

ϕ2

1

1− α∗∗

pact∗∗d

2t∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

+c2t∗∗d

2t∗∗

2a

38

+γt∗∗dt∗∗

pact∗∗dt∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

− 1

.

Note thatlimτ→0

∂P(0, qt∗∗)

∂qt∗∗= −γt∗∗dt∗∗ < 0.

On the other hand, since χ < 0, ∂P(0,qt∗∗ )∂qt∗∗

is strictly increasing in τ and is unbounded fromabove. Therefore, there exists τ is sufficiently large (and p sufficiently small so that qt∗∗ remainspositive) that renders ∂P(0,qt∗∗ )

∂qt∗∗strictly positive. Thus, ∂P(0,qt∗∗ )

∂qt∗∗> 0 if and only if τ > τt∗∗(γ)

for some unique τt∗∗(γ) > 0.

We now evaluate (17) with respect to qt at the equilibrium (0, qt∗∗) for types t = t∗∗ that,given α∗∗ and qt∗∗ , would receive attention if available to consumers (recall Corollary 1). Firstcalculate

∂Et(0, qt∗∗)

∂qt=

c−1t

pa + qt∗∗

[ϕ√τ+

aqt∗∗(c−1t∗∗ − c−1

t )

1− α∗∗ − pc−1t

1− α∗∗

]= c−1

t

[dt∗∗ct∗∗ +

a(c−1t∗∗ − c−1

t )

1− α∗∗

]

∂Et∗∗(0, qt∗∗)

∂qt=

−qt∗∗c−1t∗∗( p

a + qt∗∗)2 [ ϕ√

τ− pc−1

t

1− α∗∗ +aqt∗∗(c

−1t∗∗ − c−1

t )

1− α∗∗

]=

−qt∗∗c−1t∗∗

pa + qt∗∗

[dt∗∗ct∗∗ +

a(c−1t∗∗ − c−1

t )

1− α∗∗

]

and of course ∂Et′ (0,qt∗∗ )∂qt

= 0 for t′ = t, t∗∗. Also, recall that

Et(0, qt∗∗)

qt=

c−1t

pa +

∑t′′∈Te

qt′′

ϕ√τ+

∑t′∈Te\{t}

aqt′(c−1t′ − c−1

t )

1− α∗∗ − pc−1t

1− α∗∗

∣∣∣∣∣∣(0,qt∗∗ )

= c−1t

[dt∗∗ct∗∗ +

a(c−1t∗∗ − c−1

t )

1− α∗∗

].

Letting ξt = dt∗∗ct∗∗ +a(c−1

t∗∗−c−1t )

1−α∗∗ and combining terms, we obtain

∂P(0, qt∗∗)

∂qt= γcχ[κ∗∗1 ]2

{[τc2t∗∗Et∗∗

ϕ2aqt∗∗

]∂Et∗∗

∂qt−

√τ

pϕ

[1−

√τ

ϕct∗∗Et∗∗

] [ct∗∗

∂Et∗∗

∂qt+ ct

∂Et

∂qt

]}+γcχ[κ∗∗1 ]2

τc2tEt

ϕ2aqt

[∂Et

∂qt− Et

2qt

]+ γt∗∗

∂Et∗∗

∂qt+ γt

[∂Et

∂qt− dt

]= γcχ[κ∗∗1 ]2

{[τc2t∗∗dt∗∗

ϕ2a

]−qt∗∗c

−1t∗∗

pa + qt∗∗

−√τ

pϕ

[1−

√τ

ϕct∗∗dt∗∗qt∗∗

] [−qt∗∗

pa + qt∗∗

+ 1

]}ξt

39

+γcχ[κ∗∗1 ]2τξ2t2ϕ2a

+ γt∗∗−qt∗∗c

−1t∗∗

pa + qt∗∗

ξt + γt

[ξtct

− dt

]= γcχ[κ∗∗1 ]2

τ

ϕ2

{− c−1

t∗∗

1− α∗∗

[−qt∗∗

pa + qt∗∗

]−[

c−1t∗∗

1− α∗∗ +ct∗∗dt∗∗

a

]+

ξt2a

}ξt

+

[γt∗∗

ct∗∗

−qt∗∗pa + qt∗∗

+γtct

]ξt − γtdt

= −γcχ[κ∗∗1 ]2τ

aϕ2

pc−1

t∗∗1−α∗∗

pa + qt∗∗

+ξt2

ξt +γt∗∗ξtct∗∗

[ pa

pa + qt∗∗

− 1

]+ γt

[ξtct

− dt

]

= −γcχ[κ∗∗1 ]2τ

aϕ2

dt∗∗ct∗∗

12+

pc−1t∗∗

1−α∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

+a(c−1

t∗∗ − c−1t )

2(1− α∗∗)

ξt

+γt∗∗ξtct∗∗

padt∗∗ct∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

− 1

+γtct(ξt − dtct).

We need to consider whether t > t∗∗ or t < t∗∗. If t > t∗∗, it means that α∗∗ < αt∗∗t and

hence 0 < ξt < ctdt. If t < t∗∗, it means that α∗∗ > αt∗∗t and hence again ξt < ctdt, but also

that α∗∗ < αt∗∗ as defined in the proof of Proposition 1.30 The latter condition implies thatξt > 0. Therefore, either way, ∂P(0,qt∗∗ )

∂qtis strictly increasing in τ and unbounded from above,

and limτ→0∂P(0,qt∗∗ )

∂qt= −γt∗∗

ct∗∗ξt+

γtct(ξt− dtct) < 0. Therefore, there exists a unique τt > 0 such

that ∂P(0,qt∗∗ )∂qt

> 0 if and only if τ > τt.

We now prove that τt > τt∗∗ if t < t∗∗. This follows from showing that, if ∂P(0,qt∗∗ )∂qt∗∗

= 0 at

τt∗∗ , then ∂P(0,qt∗∗ )∂qt

< 0 at τt∗∗ for t < t∗∗, and by using the monotonicity property of ∂P(0,qt∗∗ )∂qt

in τ . If ∂P(0,qt∗∗ )∂qt∗∗

= 0, then

γcχ[κ∗∗1 ]2τ

aϕ2=

γt∗∗

pact∗∗dt∗∗

ϕ√τ−

pc−1t∗∗

1−α∗∗

− 1

1

1−α∗∗

pct∗∗dt∗∗

ϕ√τ−

pc−1t∗∗

1−α∗∗

+c2t∗∗dt∗∗

2

.

30If α∗∗ ≥ αt∗∗ , it means that qt∗∗ ≥ qt∗∗(α) and hence no t < t∗∗ can attract attention byCorollary 1.

40

Therefore, ∂P(0,qt∗∗ )∂qt

< 0 if and only if

γt∗∗

pact∗∗dt∗∗

ϕ√τ−

pc−1t∗∗

1−α∗∗

− 1

dt∗∗ct∗∗

pc−1t∗∗

1−α∗∗

ϕ√τ−

pc−1t∗∗

1−α∗∗

+ ξt2

1

1−α∗∗

pct∗∗dt∗∗

ϕ√τ−

pc−1t∗∗

1−α∗∗

+c2t∗∗dt∗∗

2

>γt∗∗

ct∗∗

padt∗∗ct∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

− 1

+γtctξt

(ξt − dtct),

or equivalently

γt∗∗

−ct∗∗dt∗∗qt∗∗

ϕ√τ−

pc−1t∗∗

1−α∗∗

pc−1t∗∗

1−α∗∗

ϕ√τ−

pc−1t∗∗

1−α∗∗

+ 12 +

a(c−1t∗∗−c−1

t )

2ct∗∗dt∗∗ (1−α∗∗)

ct∗∗

pc−1t∗∗

1−α∗∗

ϕ√τ−

pc−1t∗∗

1−α∗∗

+ 12

+ γt∗∗

dt∗∗qt∗∗

ϕ√τ− pc−1

t∗∗1−α∗∗

>γtctξt

(ξt − dtct),

or equivalently

−qt∗∗γt∗∗a(c

−1t∗∗ − c−1

t )

ct∗∗(1− α∗∗)>

γt(ξt − dtct)

ctξt

[ϕ√τ+

pc−1t∗∗

1− α∗∗

].

The last inequality holds because the left-hand side is positive and the right-hand side is negativegiven ξt < dtct.

Finally, suppose that τ = τt∗∗ so that ∂P(0,qt∗∗ )∂qt∗∗

= 0. Then, by the previous calculations,∂P(0,qt∗∗ )

∂qt> 0 if and only if

qt∗∗γt∗∗a(c

−1t − c−1

t∗∗)

ct∗∗(1− α∗∗)<

γt(ξt − dtct)

ctξt

[ϕ√τ+

pc−1t∗∗

1− α∗∗

].

For t = t∗∗ + 1, the left-hand side is strictly negative. The right-hand side is negative andapproaches zero as ξt∗∗+1 → ct∗∗+1dt∗∗+1. Given the definition of ξt∗∗+1, this happens whenα∗∗ → αt∗∗+1

t∗∗ .

B Appendix: Consumer Equilibrium

In order to describe the equilibrium k(·; ·), using Myatt and Wallace’s (2012) language andsymmetry, define the total precision of source m of type t as

πmt (em) = [V ar(zm + xn,m)]−1 =

atem[ct]

2

at + em[ct]2

41

and its publicity as

ρmt (em) = corr(zm + xn,m, zm + xn′,m) =

em[ct]2

at + em[ct]2.

We drop the subscript m when πmt and ρmt are the same across all t-sources. Abusing notation,

let qt denote the set of active t-suppliers.

Lemma 4 (Consumer behavior). Given Tq = {t : qt > 0}, there exists a unique symmetricequilibrium e and k(·; ·), which satisfies the following properties:1. em = et for all m ∈ qt and all t ∈ Tq;2. if et > 0, then

−ukk2

[κ1wt(e)ctet

]2= τ, (18)

2. where

wt(e) =(1− α)πt(et)

1− αρt(et)

[p+

T∑t′=1

qt′(1− α)πt′(et′)

1− αρt′(et′)

]−1

;

3. if et = 0, then

−ukk2

[ctκ1]2

p

1− α+∑t′ =t

qt′(1− α)πt′(et′)

1− αρt′(et′)

−2

≤ τ ; (19)

4. for all n ∈ [0, 1],kn = k(sn; e) = κ0 + κ1

∑t∈Tq

wt(e)∑m∈qt

sn,m; (20)

5. there exists r > 0 such that, if et > 0 then ct > r√τ , and if et = 0 then ct ≤ r

√τ .

Lemma 4 follows from Proposition 1 and Corollary 1 in Pavan (2014) and the property that allt-suppliers offer identical sources of information. We omit its proof.

To gain intuition, note that given k(·; ·) in (20) each consumer’s expected payoff can bewritten as (see Pavan (2014))

E[u(K,K, σk, θ)|e, k(·; e)] +ukk2

V ar[k −K|e, k(·; e)]− C(e) (21)

where the variation of individual actions around their mean—called dispersion—satisfies

V ar[k −K|e, k(·; e)] = κ21

T∑t=1

∑m∈qt

[wmt (e)]2

em[ct]2,

42

with

wmt (e) = (1− α)πm

t (em)

1− αρmt (em)

p+ T∑t′=1

∑m′∈qt′

(1− α)πm′t′ (em

′)

1− αρm′

t′ (em′)

−1

.

Importantly, when all consumers follow strategy k(·; ·), the distribution of the average actionK is independent of their attention allocation—hence, consumers treat the first term in (21) asa constant. Keeping k fixed by the envelope theorem, we get that the right-hand side of (18)measures the private benefit of marginally increasing the attention to source m. To obtain part1 of the lemma, note that by Corollary 1 in Pavan (2014) if one t-supplier receives positiveattention, then all t-suppliers must receive positive attention. Since condition (18) must thenhold for all of them, using wm

t (e), we get that

(1− α)πmt (em)

em(1− αρmt (em))=

(1− α)πm′t (em

′)

em′(1− αρm′

t (em′))

for every two t-suppliers m and m′. This condition leads to em = em′ .

C Appendix: Continuum of Suppliers

From the characterization of the unique equilibrium in the continuation game between con-sumers (Lemma 4), it is easy to see that what ultimately matters for each consumer is howmuch attention she allocates to the group of t-suppliers as a whole—this amount is then di-vided evenly among all its members. Of course, this simplification follows from the propertythat in the model, all t-suppliers are identical sources of information. Indeed, we can writecondition (18) as

τ =−ukk2

κ21

[qtwt(e)ctqtet

]2,

where

qtwt(e) =1− α

(1− α)[atqt]−1 + ([ct]2qtet]−1

p+ ∑t′∈Te

1− α

(1− α)[at′qt′ ]−1 + ([ct′ ]2qt′et′)−1

−1

;

similarly, condition (19) becomes

τ ≥ −ukk2

[ctκ1]2

p

1− α+∑t′∈Te

1− α

(1− α)[at′qt′ ]−1 + ([ct′ ]2qt′et′)−1

−2

,

43

and the optimal action policy becomes

k(sn; e) = κ0 + κ1∑t∈Te

qtwt(e)Snt ,

where we let Snt = 1

qt

∑m∈qt s

n,m. Letting Et = qtet for all t, we see that all these conditionsdepend only on the group quantities Et. So, for every q, the consumers’ behavior is characterizeby

τ =−ukk2

κ21

[Wt(E)

ctEt

]2,

where

Wt(E) =1− α

(1− α)[atqt]−1 + ([ct]2Et)−1

p+ ∑t′∈Te

1− α

(1− α)[at′qt′ ]−1 + ([ct′ ]2Et′)−1

−1

;

similarly, condition (19) becomes

τ ≥ −ukk2

[ct′κ1]2

p

1− α+∑t′∈Te

1− α

(1− α)[at′qt′ ]−1 + ([ct′ ]2Et′)−1

−2

,

and the optimal action policy becomes

k(sn;E) = κ0 + κ1∑t∈Te

Wt(E)Snt .

Also, note that the random variable Snt is normally distributed with mean zero and variance

1p +

1qtat

+ 1[ct]2Et

. Thus, we can interpret the consumers as basing their actions on the sufficientstatistic St, which summarizes all the information they will get from the group of t-suppliers;given this, each consumer can be viewed as choosing how much attention to allocate to thesource of St.

We now want to argue that we can let qt be any positive real number and continue to use theabove equations to characterize the consumers’ behavior, while preserving the economic meaningof the model. To this end, first note that if qt is a positive rational number (that is, qt ∈ Q+),then we can write atqt as at

ζtχt where χt, ζt ∈ Z+. Moreover, if we take ζ ∈ Z+ sufficiently large,

we can approximate all qt in Q+ with χt

ζ for some χt ∈ Z+ for all t.31 In this case, atζ χ

t can beinterpreted as a situation where there are χt suppliers (which is a positive integer) of type t inthe market and the informational content of each potential supplier is at

ζ . Note that atζ decreases

to zero as ζ increases, which can be interpreted as saying that the informational content of eachsupplier becomes arbitrarily small when there is a large number of potential suppliers. This is

31This extra step takes care of the case in which the types of signals have the same accuracy,but different clarities.

44

consistent with a notion of perfect competition in information markets defined as the propertythat each supplier is “small” in terms of the amount of information it can provide. Since Q+ isdense in R+, by letting ζ become arbitrarily large and χt adjust correspondingly across t’s, inthis way we can approximate every q ∈ RT

+ and hence every level of atqt across t’s. In the limit,the interpretation of at is that it measures the rate at which t-suppliers entering the marketcontribute to the total amount of information that they provide to the consumers.

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