attention economies+ by josef falkinger*

7/27/2019 Attention Economies+ by Josef Falkinger*

1/78

Second draft

December 2002

ATTENTION ECONOMIES+

by

Josef Falkinger*

Abstract

The presented theory explains the basic mechanisms at work in an economy in which

senders compete for attention to increase their impact.

The exogenous fundamentals are the space of receiving subjects with their attention

capacity, and the potential set of competing companies with their radiation technolo-

gies.

The endogenous variables explained by the theory are equilibrium audiences (the

clients belonging to a company), their signal exposure and attention, and the equilib-

rium measure of active companies.

The theory suggests that international integration or range-increasing technical pro-gress decrease global diversity of companies. Local diversity may increase nonethe-

less.

IEL classification: D50, D80, L10

*Socioeconomic Institute, University of Zurich, Rmistrasse 62, 8001 Zrich

[email protected]

+I wish to thank Franz Hackl, Clemens Puppe, Armin Schmutzler and, in particular,

Anke Gerber, for very helpful comments.


2/78

Attention Economies 2

I. Introduction

Why do we try to publish in journals with big impact or to join powerful

scientific networks? Suppose you had an idea but nobody except you is

aware of it. Then, you and your idea don't exist in the scientific commu-

nity. Descartes cogito ergo sum hardly applies to the real world. The

case is only what is perceived. And for being perceived one has to attract

the attention of recipients. The importance of being perceived and having

impact is not only common experience in an economist's life but a basicfeature of the (economic) world in general. To attract the attention of

people is a prerequisite for doing profitable business as well as for win-

ning elections, shaping society or achieving reputation. It follows from

the fact that only those ideas, goods, logos, persons, firms matter which

are perceived.

What determines the set of things we perceive and why does the diver-

sity of perceived objects change? Suppose you are a reader of eco-nomic articles. Apart from your interests, you have individual character-

istics like brain capacity or distractions by non-economic stuff. They put

a limit to your processing of economic literature. Thus, you focus atten-

tion on important sites and outlets of publications. And important is who

produces many powerful papers and gets the attention of the science

community or, more generally, of the ensemble of producers and con-

sumers of scientific work. Range and diversity of the important agents

and places on which this ensemble focuses have changed over time. In

particular, some local heroes have become global heroes and other local

heroes have disappeared at all. Obviously, the circle that producers try to

attract attention and consumers pay attention to producers who send


3/78


powerful signals is not bound to scientific research but applies to many

economic goods and services from simple things like clothes to enter-

tainment and high-tech products. It is the purpose of this paper to workout the equilibrium implied by the logic of this circle.

Teachers (or parents) always have told the children to be attentive. And

the teachers have been told that good teaching requires to attract the au-

dience's attention. The new thing is that managing attention and at-

tracting attention are becoming universal maxims in business and eco-

nomics. An account of the many current trends of attention management

and the practical implications for doing business was recently presented

by Davenport and Beck [2001] under the title "The Attention Econo-

my". The title indicates that the issue of attracting attention really adds a

new dimension to an economy. Herbert A. Simon pointed out that in an

information-rich world a new scarcity problem arises, namely: "a scar-

city of whatever it is that information consumes." According to him,

what information consumes is obvious: "it consumes the attention of its

recipients. Hence a wealth of information creates a poverty of attention"

(Simon [1971], p. 40). Whereas Simon focussed on the question how

economic agents like firms can allocate their attention capacity effi-

ciently among information sources, the focus of this paper is the contest

of information sources for the attention of recipients in anonymous mar-

kets. Under this focus the central question is which and how many in-

formation sources succeed in getting enough attention to be viable. Put

See there for references on the psychological, technological and commercial backgrounds of the

role of attention in business and management. The first version of this paper was written before I

have got knowledge of their book. That I independently chose almost the same title underlines

that the subject is topical. My interest in it was raised by the art journal Kunstforum, whose De-

cember 1999 issue focussed on "Ressource Aufmerksamkeit" (see also Goldhaber [1997] andFrank [1998]. Shapiro and Varian [1999] discuss business strategies for attracting attention by

customizing information.


4/78


differently: Given that the agenda of people is crowded, who and what is

on the agenda if an overabundant set of potential items compete for be-

ing there.

For answering this question, an economy is considered as asystem of subjects (senders) who try to attract the attention of other sub-

jects (receivers) by producing and distributing information packages

(signals). The senders may be firms, news agencies, scientific networks,

political parties etc., promoting products, persons, ideas or views under

some logo. They are addressed also as companies or firms in this paper,

since they typically consist of more than one individual. (However, the

inside structure of companies is not considered.) The receiving subjects,

on which the signals are targeted, are called audiences. The central ques-

tion to be answered is then: What is an equilibrium outcome if each com-

pany tries to attract the attention of subjects by exposing them to the sig-

nal "Look at me" or "Read my message"? As a result, we get characteri-

zations of which companies survive the contest of attention and how big

is their impact.

The ability to attract an audience and the allocation of diverse buyers to

diverse sellers was also the concern of Rosens [1981] "Economics of

Superstars". However, his focus is on product differentiation. Producers

have more or less talent and produce services of different qualities. And

the consumers' willingness to pay for a unit of service is higher if the

service comes from a more talented producer. This leads to large markets

and big rewards for the most talented people. My focus is on the desire

of producers to attract consumers and achieve impact rather than on how

consumers value and choose different products. Before subjects can

Scarcity of attention capacity has not only implications for individual behaviour or business

strategies but also consequences for the (equilibrium) outcome in markets or in the whole econ-omy. To stress this aspect I use the label "attention economies" rather than "economics of atten-

tion" for addressing the subject of this paper.


5/78


value and choose items they must get aware of them. This involves an

interaction between producers and consumers beyond the exchange of

specific products, namely the transmission of information. Regardless ofthe specific content, the essential dimension in this "meta-interaction" is

how many units of the receiver's attention capacity, e.g. time, are ab-

sorbed by information targeted on her or him. Thus, what matters is the

volume or strength of the received signal (measured in absorbed atten-

tion capacity). For instance, if you want to choose the best book in a cer-

tain field, you must first screen all books brought to your attention for

selecting the set of relevant book. Since your time is limited, the atten-

tion devoted to one book will be the lower the more books you have to

screen. Ceteris paribus, the chance that you put a publisher's book into

the basket of relevant books (out of which, later the best one is chosen) is

the higher, the more books this publisher has brought to your attention.

Not only the choice set of books or papers has to be selected before

choosing and consuming. In a similar way, the screening of news and

mails, contacts and events, product information and political issues ab-

sorbs attention capacity. And the time absorbed for processing the sig-

nals of different producers and fields simply adds up, regardless of the

specific content conveyed by the signals.

Although there is much economic jargon when people talk about atten-

tion, closer examination quickly shows that standard economic notions

cannot be simply applied to attention as just another economic good. For

instance, it has become usual to describe attention as a scarce resource

(of the information economy), or to speak of the allocation of attention.

Also trash absorbs attention capacity before you can scrap it. As is well known, the purchase ofinformation can never be based on perfect information. But even recognized noise, destroying the

possibility to focus on other things, cannot always be turned off.


6/78


However, unlike for land or capital we do not observe markets at which

the good "attention" is exchanged among traders at some given price. At-

tentiveness is a characteristic, incorporated in an individual's body andmind, which by its nature can hardly be protected by property rights.

Neither is it possible for the sender of signals to acquire full control of a

person's attentiveness, nor can a person which wants to stay connected

with the outside world fully control the exposure to signals trying to get

her attention. So, although there is rivalry in attracting attention there is

at most partial excludability. Moreover, attention is a non-specific good.

Who- and whatever exposes you to signals absorbs some of your atten-

tion. The signals may come through internet, television, phone or on

printed paper. They may refer to products, politics, science or entertain-

ment. Unlike with advertising single products or signalling private in-

formation in single markets, it is not the processing of specific informa-

tion contents which matters for an individual's attention level but the ag-

gregate volume and strength of signals to which she or he is exposed.

Thus, the interaction of subjects sending signals for attracting the atten-

tion of other subjects has something in common with congestion or

emissions in a public space. But again, there are essential differences.

Whereas polluters simply want to get rid of their emissions somewhere,

senders produce and target signals to achieve impact. As this short dis-

cussion shows, the economics of spending and earning attention should

not be seen as just another instance of standard economic analysis but

requires a new theoretical model. It must provide us with a formal de-

scription of the fundamentals of an attention economy and a characteri-

zation of what is an equilibrium.


7/78


Psychology or neurosciences tell us how people perceive signals and

process information. Moreover, they may give advice how to concentrate

and use our brain capacity optimally. Here the individual's perceptionpsychology and attention capacity are taken as given and constitute one

of the fundamentals of the economy: The attention capacity, formally

represented by a relationship between stimulus exposure and attention

level of an individual, is the relevant characteristic of a receiver. The

other set of fundamentals describes the exogenous properties of potential

senders. Each sender is characterized by its range which measures the

maximal size of the audience that can be connected to it. This allows us

to deal with the fact that more or less local or global companies are com-

peting for the attention of audiences. It is not required that the audience

of a sender is a connected area or neighbourhood of receiving subjects.

This complicates the formal structure of the presented model but is es-

sential for an appropriate analysis. The prototype sender in an attention

economy is not an emittor of electromagnetic waves covering a fixed

geographic area. Direct marketing methods, customer relationship man-

agement or download statistics allow those who want to attract attention

flexible targeting of their signals on attentive individuals. The location of

the targeted receiver is of minor importance, or of none if we think of the

internet.

Given their capacity to address audiences of a certain size, the economic

agents in search for attention can choose the targeted audience and the

strength of the signals they focus in them for attracting their attention.

The purpose of attracting attention is to have an impact. And the impact

of a sender is the bigger, the higher the level of attention of the targeted

audience and the stronger the signal produced by the sender. The send-


8/78


ers' simultaneous pursue of impact together with the individuals' capacity

of attention determine equilibrium signal exposure, equilibrium attention

levels and the equilibrium allocations of audiences to senders. Of course,audiences can overlap and an individual is typically a member of several

audiences. This gives us a measure for the diversity of senders experi-

enced by an individual. I call this the local diversity, since it varies

across individuals with different attention characteristics. By contrast,

the global diversity is given by the total measure of senders which sur-

vive in the equilibrium of the attention economy. The criterion for sur-

vival is derived from what must be the basic axiom of any theory of

competition for attention: Only those firms, political forces, news pro-

ducers, entertainers, scientific networks are viable which are perceived

by some audience and achieve a minimal level of impact.

In sum, an attention economy is modelled as a family ofsenders (com-

panies named by their logo) which employ costly signals to attract the

attention ofaudiences and have an impacton them. An audience is a set

ofreceivers characterized by theirattention capacity. Senders are char-

acterized by theirradiation capacity orrange which limits the measure

of audiences they can address. In competing for the attention of audi-

ences the companies (senders) maximize the value of their impact net of

signal costs required for achieving the impact. A sender is viable if its

net-value of impact is non-negative. An equilibrium is a family of viable

senders which is not contested by entrants. The size of the equilibrium

family of companies is a measure of the diversity of senders perceived in

the whole economy (global diversity). The measure of senders perceived

by a subset of receiving subject is called local diversity. The compara-

tive-static analysis shows how changes in methods of impact generation


9/78


or in the range of radiation affect the measure of viable companies. For

instance, it is shown that an increase in the range of radiation allowing to

companies a wider diffusion of their signals may diminish the equilib-rium set of companies in the economy although each receiving subject

has access to more varieties of senders than before. In other words,

global diversity may decrease, while a higher local diversity is experi-

enced. The reason is that different local audiences turn into more ho-

mogenous global audiences.

Section II presents a simple example. In section III the general structure

of attention economies is formalized. Section IV determines equilibrium

allocations of audiences to given families of senders. Section V analyses

the choice of signal strength in the contest for attention. In section VI the

equilibrium family of senders surviving in the contest for attention is de-

rived and comparative-static results are presented. Section VII summa-

rizes.


10/78


II. A simple example

Consider a set ](0,L L= of potential publishers. They are identical with

respect to access to writing talents and production possibilities. Every

t L can generate publications of equal quality at cost

( ) 21 / 2C = . At some fixed cost 0C , each publication can be sent to a

range ( )t of receivers. (Copying costs and variable mailing costs are

neglected. Think of PDF documents.) Thus, producing publications

and distributing them to ( )t receivers costs

( ) 2 0/ 2c c = + . (2.1)

Publishers are heterogeneous with respect to their feasible range. As-

sume ( )t = for ](0, , 0, 1/ ,t t t t > > and ( ) 1/t t = for

( ],t t L . That means, besides a set of relatively large companies there

are smaller companies ranked according to radiation range.

The set of receivers S has measure ( ) 1/S t > . That means publishers

t t> are small compared to ( )S whereas publishers t t may cover

all receivers. The receivers are reviewers who compile the received pub-lications to surveys. They are endowed with reading time 0 and have

identical competence and taste. Thus, a reviewer who receives publi-

cations spends 0 / = on each. The function 0 / describes the re-

viewers attention capacity. For any , the value 0 / = is the attention

level of the reviewer under signal exposure .

These may be individual authors, scientific research networks, or publishing companies.


11/78


Because of zero marginal distribution costs, a publisher t who produces

( )t publications will send all produced publications to the feasible

range ( )t of reviewers, or to all reviewers if ( ) ( )t S . This means,

( ) ( ){ }min ,tr S t = reviewers receive the ( )t publications of pub-

lisher t.

The chance that publications of t are cited in the survey of a

reviewer S increases with the number of publications received from

t and decreases with the total number s of publications that the re-

viewer receives. Thus, the citation impact on a reviewer who is in the

mailing list of publishert is given by ( ) ( ) 0 / st t = .

Suppose that subset ( ]00,T t= of ( ]0,L with measure ( ) 0T t = is vi-

able. (Of course, we will have to prove that this is a consistent assump-

tion.) Then the total number of publications is given by

( )0

0

t

tX r t dt (2.2)

so that the average number of publications received by a reviewer is

( )/X S = and the impact of publisher ton a reviewer is ( ) 0 /t .

Since taddresses tr reviewers, the total impact oft(in terms of citations)

is given by

( )t tz r t = , 0 / . = (2.3)

If citation impact is rewarded according to a linear scheme

0 ,tR w w z= + 0 0, 0,w w> > (2.4)

the net-profit of t achieved from publishing is given by

Like ( )t also r is a function of t. To save brackets, I write tr instead of ( )( ),r S t .


12/78


( ) 2 / 2 ,t tw r F = (2.5)

where 0 0F c w= . (Combine (2.1), (2.3) and (2.4)). Assume 0 0c w> .

Then, 0t requires 0tz > . That is, no survival without impact (and,

obviously, no impact with zero publications).

Solving ( )max t

we obtain the optimal number of publications

( )* tt w r = . It depends on the reward scheme ( )w , but also on radia-

tion power tr of the publisher and on attention level of the reviewers.

Substituting ( )* t into (2.5), we have

( ) ( )2* / 2 .t tw r F = (2.6)

Thus, the maximal net-profit that can be achieved increases with range tr

of the publisher. The minimum range tr for viability ( )*

0t

is

( )2 /F w . Assuming that for all ,t t ( ){ }min ,tr r S = >

( )2 /F w , we conclude that the marginal publisher is given by

( )0 / 2t w F = . (2.7)

and the set of viable publishers is ( ( )00,T t = , where ( )0t t > . Us-

ing this, ( )* tt w r = and the assumptions tr r= , for , 1/ ,tt t r t = for

t t> , we get from (2.2) for the aggregate publication exposure of all re-

If ( ){ } ( )min , 2 /r S F w , all viable publishers are identical. Each of them produces* w r = publications and achieves net-profit ( )

2* / 2 .w r F = The viability condition is

then: ( )2 /F wr where 0 / = and according to (2.2), ( ) ( )/X S T = = 2r w ,

where ( )T is the measure of viable publishers. Thus, ( ) ( )( )01/ /r w T = and the viability

condition reduces to ( ) ( )0 / 2T w F . Only the measure of the maximally viable set of pub-

lishers ( ) ( )0 / 2T w F = is determined. Their identity doesn't matter.


13/78


viewers together ( )X w = ( )( )0

2 21/

t

t

r t t dt

+ =

( )2 01/ 1/w r t t t + . After substitution of (2.7), this reduces to

( ) ( )2 1/ 2 ,X w r t t F = + which implies an average publication

exposure ( ) ( ) ( )/X S = for each reviewer. By assumption, the at-

tention capacity of a reviewer was given by the function 0 / . Thus, in

an equilibrium the attention level must fulfil the equation ( )0 / = .

In sum, the equilibrium attention level * is given by the (positive) solu-

tion of the equation

( ) ( )2 2 01/ 2w r t t F S + = + . (2.8)

This equation defines a unique positive solution * . Straightforward

comparative static analysis by implicit differentiation of (2.8) shows that

the attention level of reviewers decreases if t or r increase or ifF de-

creases, i.e. if there is progress in radiation technology, so that either

more publishers ( )t have access to the radiation technology with wide

range , or range r of the dominant publishers ( )t t is extended, or

diffusion cost F decline. The equilibrium attention level of reviewers

also declines, if a steeper incentive w induces publishers to increase out-

put. Finally, * increases with 0 , i.e. if there is progress in attention

capacity of reviewers by mental training or new techniques like mail fil-

tering programs.

Note that 1/r t> by assumption. Moreover, ( ){ }min ,r S = . Thus, for ( )S < , an in-

crease in r means an increase of for a given ( )S at the right side of equation (2.8). If

( )S , i.e. if large publishers cover already all reviewers, then an increase in radiation range

is useless.


14/78


According to (2.7), the equilibrium diversity of publishers is

* * / 2t w F= . Whereas w and F affect *t directly and indirectly

through * in opposite way, changes in ,t r and 0 have definite ef-

fects. An increase in radiation power due to an increase in t or , i.e.

more publishers with large range or an extension of the range of domi-

nant publishers, increases the time pressure of the reviewers. Therefore,

small publishers escape their attention and are no longer viable. By the

same mechanism, an increase in attention capacity 0 works in the op-

posite direction.

Equilibrium diversity *t is relevant from the global perspective of an

outside observer. The diversity experienced by a single reviewer (local

diversity) is given by the set of publishers from which (s)he receives

publications. If ( )S > , each publisher t t addresses reviewers

and each publisher ( *,t t t addresses 1/ t reviewers. Thus, the total

number of reviewers addressed is ( )

*

*1/ ln ln

t

t

t t dt t t t + = + .

Since there are ( )S reviewers, the average number of publishers from

which a single reviewer receives publications is

( )* *ln ln /m t t t S = + . (2.9)

If ( ){ } ( )min , 2 /r S F w so that only a subset T if ](0, t survives in equilibrium and

( ) ( )* 0 / 2t T w F = = (see preceding footnote), then ( )( )* 0 / 2m r w F S = . If in addition( )S , then ( )r S= and * *m t= .


15/78


Local diversity *m may increase with radiation power, , t although *t

declines. Thus, the decline in diversity of publishers may escape to any

single reviewer. To the contrary, from their local perspective they may

have the impression that more publishers are around. The reason is that

an expansion of the radiation range of dominant publishers implies more

overlap in the sets of publications reviewed by the different reviewers.

So every reviewer compiles more, but more of the same sources. Obvi-

ously, if only dominant companies with ( )S survive so that all re-

viewers are covered by all publishers, then * *m t= and each single re-

viewers experiences the changes in global diversity.

The example should only be taken as an illustration. Identical attention

capacities and simple functional forms were assumed. Moreover, there

was no explicit analysis of the mechanisms governing the assignment of

publications to reviewers. Implicitly it was assumed that publishers, al-

though not able to manipulate the reviewers, are picking reviewers so

that they can expect maximal impact. As the further analysis will show

the example illustrates a fundamental mechanism. Clearly, this mecha-

nism not only applies to publications but is generally at work when per-

ception capacities are bounded and impact is important for survival. Be-

ing mentioned in the talk of the town, references in newspapers, TV-presence or internet visibility involve impact mechanisms for persons,

products, events or ideas, which are similar to the citation impact for

publications. Moreover, as everybody realizes every day, rivalry in the

absorption of one's attention capacity ultimately time is not restricted

to signals of the same market or field. There is cross-over competition

for the attention of subjects between all sorts of impact-chasing agents

from the different business braches to politics.


16/78


For instance, substitute internet portals for publishers in the above exam-

ple. These are sites like search engines which gather information from

different sources together into an integrated website. The volume ofgathered information distributed through the portal corresponds to .

Think of an average surfer, instead of a reviewer, and use download sta-

tistics as impact measure. Portals with a large range have an advantage in

achieving impact, and information channelled through them attract more

attention by surfers than information in portals of small size. Information

which is not visible in a search engine will hardly have any impact on a

positive mass of surfers and therefore is irrelevant for business in

anonymous markets. Indeed, as Graham [2001] reports, large portals

dominate the internet, and the number of sites that attract large shares of

the time people spent online substantially dropped from 1999 to 2001.

The reduced diversity of portals doesn't imply a reduction in the volume

of information received by the surfers. To the contrary, as the above cal-

culations have illustrated, the average receiver is exposed to more signals

if the range of dominant senders is extended. But the portals control

which information is gathered and a reduced diversity of portals may

lead to a reduced diversity of information.

Usual economic analysis concentrates on the specific choices of subjects

out of the varieties that are available. Under that focus, diversity is nar-

rowly related to interpersonal differences in tastes and talents as empha-

sized in Rosen [2002]. The above example has illustrated, and the fol-

lowing analysis will prove, that there is another important aspect of di-

versity. Although typically only a small subset of the available varieties

is relevant for an individual, the vast varieties that are offered may be

judged as poor. Perhaps the most preferred varieties are not viable in the


17/78


given economic environment.

In an attention economy those varieties

(of products, ideas etc.) are viable which survive the contest for atten-

tion. The resulting equilibrium diversity is not determined by the qualityof things, or the preferences over things, but depends on attention capaci-

ties and radiation power.

Even ideas or products that would be appreciated by all consumers if they were aware of them

may not be viable since consumer attention is distracted to more powerful signals. In personalrelationships you may be able to force others to listen, but in an anonymous market an audience

of positive mass must be reached. This is impossible with radiation limited to personal

relationships.


18/78


III. The structure of an attention economy

An attention economy consists of two types of agents: The companiessending signals to earn attention and the receivers exposed to the signals.

The exogenous fundamentals of the economy are the space of receivers

together with theirattention capacity and thespace of potential senders

together with thefeasible audiences. I present first the space of receivers

and the space of senders. Then the formal structure of possible

assignments of receivers to senders (audience allocations) are describedand the notions attention, signal exposure and impact are defined.

III.1 Senders and audiences

Let the total population of receivers be given by a set S, for instance the

set of consumers. The potential audiences for senders are then given by

subsets S .

Examples of potential audiences are the set of all

internet users with certain surfing characteristics, the subjects on the

mailing list of a marketing firm, the set of subscribers to a broadcasting

company or to the NBER working paper series. For a theatre company,

the audience is located in a geographical space with a clear measure for

the distances between actors and visitors. If less traditional media are

used for attracting attention, the potential audiences have more complex

structures. Visitors of web sites are scattered all over the world and

companies may be interested in targeting their signals at isolated islands.

Therefore, the size of an audience cannot be characterized by the length

or the radius of some neighbourhood around a sender. A natural way to

denotes weak inclusion.


19/78


deal with this problem is to represent thespace of receivers by a measure

space (S, A, ), where A is a class of measurable subsets ofSand is

a measure on A. Every element of A is a subset S representing a

potential audience. The size of the audience is given by the measure

( ) . It is assumed that is finite. The following analysis focuses on

contests for attention in competitive environments. One requirement for

competition is that there are many potential audiences for whose

attention the companies can compete. In particular, it should be possible

to pick also small sets of receivers. Formally, the following divisibility

assumption is made: For any A with ( ) 0A > and any constant

( )0 c A< < there exist B Aso that ( )B c = .

Senders are the economic agents, for instance firms, who want to attract

the attention of audiences. Let ](0,L be the index set of potential senders

in the economy. Thus, t ](0,L is the name ("logo") under which a

company, a scientific network or any other attention-seeking agent

conveys information (signals) to receivers. In general, not all potential

senders will be active. Actually, it is the purpose of the following

analysis to determine the subset ](0,T L

of companies surviving the

competition for attention. It is assumed that the potential sets of active

senders are Lebesgue-measurable. That is, thespace of senders is given

by ](( )0, , ,L B where B, are the class of Borel sets and the Lebesgue

This is for instance fulfilled ifAis the class of Borel sets and is the Lebesgue measure.


20/78


measure, respectively. Every element of B is a subset ]( )0,T L

representing a possible family of active senders. ( )T measures its size.

Each potential sender ](0,t L is endowed with a radiation capacity.

The radiation capacity is characterized by a real integrable function

](: 0,L + + R . ( )t describes the maximal size of audiences that can

be addressed by company t. (I will call ( )t also radiation range oft.) It

is assumed that the senders are ordered according to their size, that is,

( ) ( )t t for t t< . Radiation capacity determines the set of

feasible audiences. Formally, this will be made precise in the next

section. In reality, the maximal size of audiences that a company can

handle depends on two factors: the available media and the resources

attributed to the management of receiver relationship. Obviously, new

media have dramatically changed the range of attention-seeking senders.

Ceteris paribus, a larger audience can be addressed through the internet

than by phone calls. In general, such technological changes are

exogenous to the single company. Also the means for entertaining the

communication channels to the receivers, for instance the organizational

infrastructure for maintaining mailing lists or evaluating the impact of

distributed information on the addressed audience, are fixed in the short-

run. The following analysis assumes that capacity ( )t is an

exogenously given characteristic of sender t. The comparative-static

analysis of the impacts of changes in on the diversity of companies

Use , ,R R R+ ++

denote the sets of real numbers, non-negative real numbers, positive real

numbers, respectively.


21/78


and their impact on the receiving subjects will be an important part of the

equilibrium analysis in section VI.

Whereas range is exogenous, senders can choose theirsignal strength.

This is the means by which a company attracts attention and which

absorbs attention capacity of receivers. For a set T B of active senders,

signal strength is represented by real integrable function : RT T + .

For every ( ), Tt T t describes volume and intensity of the

information sent to the members of the audience of t, for instance, the

number of mails or calls of clients per period, or the loudness or

conspicuousness of the transmitted signals. Production of signal strength

is costly. Companies incur this cost for earning attention and achieving

impact. This optimisation problem and the implied equilibrium signal

exposure is analysed in section V.

III.2 Audience allocations

The interaction of senders and receivers depends on which subject are

attracted by which companies. For characterizing the possible outcomes

in an attention economy we must describe the assignment of receivers tocompanies. From the perspective of a signal sending company the

question is which audience is reached by its signals. From the side of the

receiving subjects the question is to which set of companies they pay

attention. I call the assignment of audiences to senders an audience

allocation. Formally, an audience allocation for a set of senders, T B,

is given by a relation aT T S , where aT is measurable B A. If a


22/78


pair ( ),t s is element of aT, then sender t addresses subject s under

audience allocation aT. Thus, for any t T , the section

( ) ( ){ }: ,a aT Tt s S t s= of aT defines the audience of receiving the

signals sent by t. From the perspective of a receiver S , the section

( ) ( ){ }, : ,a aT TM s t T t s= of aT describes the set of perceived

companies. I call ( ),aTM s the membership ofs under aT. Note that

aT B A implies that ( ),aTM s and ( )aT t are measurable B and A,

respectively. Not all audience allocations are feasible. The audienceassigned to a company must have positive measure and cannot be larger

than the company's radiation capacity ( )t . Nor can it be larger than

( )S , the measure of all subjects in the economy. Since modern

technologies like the internet have a very large range I don't want to

exclude the case that senders have the capacity to cover the whole

economy (i.e. ( ) ( )t S ). In sum, an audience allocation aT for

T B is feasible under radiation capacity if for all t T :

( )( )0 aT tt r< , where ( ) ( ){ }: min ,tr S t = . The fact that only the

size of audiences is limited by the radiation technology has far-reaching

implications. It means that the audiences of companies need not be

connected areas or neighbourhoods in a geographical space. This would

be the case if, for instance, a sender would have a fixed focus like a cone

of light. In contrast to this, according to the given notion of feasibility,

companies can target their signals fully flexibly, provided the targeted

audiences are not larger than what they can handle by their given

capacity. I think this is the adequate modelling of the media channels

through which modern companies disseminate and promote information

to anonymous audiences.


23/78


Membership ( ),aTM s gives us a measure forlocal diversity underaT.

In an economy in which audience allocation aT is realised,

( )( ), aTM s measures the variety of companies perceived by subject s.

By extending the notion to sets of subjects AA, we can describe the

variety of companies perceived in A by ( )T, :aM A =

( ){ }T, for someat T t s s A . Then, ( ),aTM S describes the total

variety of companies perceived somewhere in the economy. By

definition, ( ),aTM S T= . I call ( ) ( )( ),aTT M S = the global

diversity underaT.

It puts an upper limit to local diversity. If for many

subjects ( )( ),aTM s is small compared to ( )T , then the variety of

companies perceived by different subjects is relatively heterogeneous

across subjects. If for alls local diversity coincides with global diversity,

every subject experiences the same set of companies. For instance, ifTis

the set of active scientific networks in the world and each subject of the

scientific community is also a member of all t T , then local diversity is

equal to global diversity. There is a uniform international scientific

community paying attention to the same set of working paper series etc.

If many networks t T have only local membership, then local diversity

perceived by a member of a local scientific community is small

compared to the global diversity which is experienced by a subject

moving around through all local communities.

Note that ( )( ),aTM A can be seen as kind of a diversity function in the sense of Nehring and

Puppe [2002], since ( ) ( ){ },a aT TM A t t A= and ( )( ),aTM

M A d = .


24/78


III.3 Attention, signal exposure and impact

The basic characteristic of subjects in their role as receiver in anattention economy is not taste or choice behaviour but the attention

capacity. An agent may be in a more or less attentive state. If (s)he is

more attentive (s)he processes the received information more carefully.

On the one side, this capacity depends on individual psychological

factors which can vary from individual to individual. On the other side,

the attentiveness of an individual with given psychological

characteristics is also influenced by the strength and volume of signals to

which she or he is exposed. People may almost fall into a doze if there is

little stimulus. And they typically loose concentration when signal

exposure is above a certain level. More generally, the attention level of a

subject S depends on subjective characteristics but also on the

environment of , i.e. on the signal strength T chosen by the existing

companies and the prevailing audience allocation aT. This can be

represented by a positive real function ( ),aT T on the space of

receivers. It is assumed that for all ,aT T function is measurable and

bounded (see lemma 1 for a characterization of such functions). The

function models the attention capacity of subjects. It assigns to every

subject S the attention level ( ),aT Ts = ofs when living in an

economy with audience allocation aT and signal strength T .

Attention capacities can vary across individuals since people are

heterogeneous with respect to their abilities to concentrate. This ability

may be the result of some self-management regarding a person's mental

resources or time. In this analysis, it is taken as an exogenous individual


25/78


characteristic. For a given s, the attention level depends on the

environment given by aT and T . However not all variations in the

audience allocation or of signal strength matter. Any given aT, T

induces for S a certain total signal exposure given by the

accumulated signal strength

( ),aT

T

M s

d over all senders to whichs is

exposed. This total signal exposure is the relevant factor for a subject's

attention level under aT, T . Therefore, the following assumption is

made with respect to the properties of the attention capacity.

Assumption AN (Anonymity). For all S , if , ,aT TT and , ,aT TT

induce the same signal exposure, i.e. if

( ),aT

T

M s

d =( ),aT

T

M s

d

,

then also the induced attention level is equal, i.e. ( ),aT Ts =

( ),aT Ts .

The assumption stipulates an anonymity property. The identity of the

signal sending source is irrelevant for the concentration of a subject

exposed to signals. Only their strength matters for the absorption of

attention capacity. Therefore, the attention level is invariant with respect

to audience allocations leading to the same signal exposure. Note

however that the impact of a given signal exposure on attention may

vary across s. The attention capacity is an individual characteristic of

receivers. Assumption AN allows us the following characterization of

attention capacities.


26/78


Lemma 1. Any measurable bounded function v : S+ ++

! !

represents attention capacities satisfying AN, namely ( ),aT Ts : =

( ), sv s ,( ),aT

s T

M s

d = .

Proof. According to lemma A1(b), for any ,aT T , the integral

( ),aT

T

M s

d is a measurable function of . Thus, the composition

( ), sv s is measurable. AN holds by construction.

The crucial mathematical point in the proof of lemma 1 is that signal

exposure s is a measurable function. (Again, I use s instead of ( ) to

save brackets.) The important economic assumption is AN. It allows the

characterization of attention capacities by a function of signal exposure

only. The attention level resulting when a subject with attention capacity

v is exposed to an environment s , is denoted by the variable . In the

further analysis, AN is always assumed and, as a rule, representation

( ), sv s is used for describing attention capacities.

For various results of the paper, a further property of the attention

capacity is required. It gives a precise meaning to the vague idea of

attention as a scarce resource.

Assumption DA (Declining attention). Attention capacities represented

by ( ), sv s satisfy DA if for all S there is a threshold 0s+

so that

( ) ( ), ,

s sv s v s < if

s s + > .


27/78


This assumption expresses the fact that too much signal exposure

stresses people and destroys their concentration. The threshold ( )s+

above which increasing signal exposure is harmful for a subject's level of

attention can be different for different subjects.

There remains one element to be specified. By definition, in an attention

economy companies send signals to earn the attention of receivers and to

have an impacton them. Obviously a receiver can pay more attention if

(s)he is concentrated, i.e. if her or his attention level is high. And a given

attention level will more likely be focussed on a sender if the sender

increases its signal strength, whereas a company of which no signal is

perceived has no impact at all. Thus, the impact of t T with strength

( )T t on a receiver with attention level is given by a non-negative

bounded real function ( )( ),t Tz t , which is zero at ( ) 0T t = ,

increasing in ( )T t and, for ( ) 0T t > , also increasing in . (Later, for

some results, additional concavity and differentiability properties will be

imposed.) tz describes how signal strength and attention generate

impact. Note that impact function tz can vary across companies. The

same signal strength may generate different impact for different types of

senders. ( T comprises only that dimension of transmitted informationwhich absorbs attention capacity. Any other aspect which may be

relevant for impact, for instance the used medium or the clarity of t's

presentation, are captured by tz .) Whereas the input variable ( )T t in

impact function tz is chosen by company t, the attention level

( ), sv s = is exogenous to t. It is determined by total signal exposure,


28/78


( ),aT

s T

M s

d = , on which single companies have no influence since

they have zero measure. Since the signals of company t reach all

members of audience ( )aT t , the total impact of t T under ,aT T is

given by ( ) ( ) ( )( ) ( )( )

, : , ,

a

a

T

t T T t T s

t

V z t v s d s = , where

( ),aT

s T

M s

d = . Since v and tz are non-negative, measurable and

bounded, the integral exists (note

( )S < ).

III.4 Behaviour and equilibrium

What is an equilibrium of an attention economy with the described

psychological and technological fundamentals? How do changes in these

fundamentals, in particular an increase in radiation range , affect

equilibrium outcomes?

The receivers react to signal exposure according to their exogenously

given attention capacity v . The individual psychological processes

behind v are taken as a black box. However, v is a rich box. Only very

general properties are imposed by assumption AN and DA so that a wide

variety of behavioural phenomena can be covered. For instance, that v

declines very sharply if changes in ,aT T lead to an increase in signal

exposure. Subjects possibly concentrate on a certain amount of

information and neglect the rest more or less. In the contest for attention,

the active role is played by the senders. They choose signal strength

and select the audience of subjects at which their signals are targeted. It


29/78


is assumed that the production of signal strength is costly whereas the

diffusion of the signals is costless within the radiation capacity defined

by range . Setting up and running the radiation technology may implyfixed costs which limits the set of active companies. The decision of an

active sender about signal strength is a standard cost-benefit calculation.

The selection of the audience is based on the following reasoning: As

long as there is a choice between more or less attentive receivers, the

impact of information can be increased by targeting it on subjects with

higher attention. This requires that companies are able to observe the

attention level of subjects. In a strict sense this is only possible with

direct attention-monitoring methods evaluating brain waves or eye-ball

movements. Although such methods exist and are further developed

Davenport and Beck [2001, p. 49] speak of the "Wire'em up principle"

observable proxies for the attention of audiences are more realistic.

Media watch, download statistics, speed and frequency of response, self-

reported data on attentiveness are examples.

For a given set of active companies, an equilibrium is reached if all

companies choose their optimal signal strength and no company has an

incentive to retarget its signals to more attentive audiences. The

equilibrium set of active firms is determined by a viability condition.

This condition will be derived from the premise that impact is the central

determinant of the value of a sender. Therefore, only those companies

survive which achieve a non-negative value of impact (net of the cost for

generating the impact). By characterizing the properties of equilibrium

sets of active companies T and of equilibrium allocations of audiences,

A recent example for measuring the attention paid by different scientific audiences to economic

journals is the analysis of citation flows by Pieters and Baumgartner [2002].


30/78


aT, we are able to assess the global and local diversity experienced in an

attention economy by looking at ( )T and ( )( ),aTM s , respectively.

Moreover, we can provide explanations for changes in equilibrium signal

exposure and equilibrium attention levels.

The further analysis proceeds as follows. In a first step, for a given set of

companies T with signal strength T , the properties of equilibrium

audience allocations are determined. It will be shown that under quite

general conditions the attention level of subjects is uniquely determined

by aggregate signal imission ( ): t TT

X r t dt= , ( ) ( ){min ,tr S t = . In

a second step, I analyse, for a given set of senders T with aggregate

signal level X, the optimal choice of signal strength *t of a single

sender t T . Then I determine the equilibrium values of , TX which

are consistent with this choice. In the third step, I determine the size of

equilibrium set *T of companies which achieve sufficient impact to

survive in an attention economy and describe the characteristics of

companies in *T . The results are used for characterizing the effects of

changes in the technology of impact production or in the feasible range

of radiation on sender diversity and attention level of receivers.


31/78


IV. Equilibrium allocation of audiences

Given a set of senders, TB, with signal strength T . Each company t T

can reach an audience of measure ( )( ) ( ) ( ){min ,aT tt r S t = .

Depending on which feasible audience allocation is actually realised, total

impact of t is ( ),at T TV . Suppose that there is another feasible audience

allocation aT under which company twould achieve a higher impact. Then

t clearly would prefer aT to aT. However, a single company hasn't the

power to decide about which audience allocation is realised in the

economy. It can only chose its own audience, i.e. the set of receivers

AA on which the produced volume of signals, ( )T t , is targeted. Thus,

for t T , a deviation from an audience allocation aT to another audience

allocation aT is feasible if aT results from aT by exchanging ( )aT t

through ( )a

T t

and leaving all other audience assignments unchanged. This

leads to the following definition of an equilibrium allocation of audiences.

Definition 1. ForTBwith T

(i) An audience allocation aT is afeasible deviation from aT for t T , if

aT is a feasible audience allocation and ( ) ( )a aT Tt t = for all

{t T t .

(ii) A feasible audience allocation is an equilibrium audience allocation if

for no t T there is a feasible deviation aT from aT with

( ) ( ), ,a at T T t T T V V > .


32/78


Whenever a company has an opportunity to increase its impact by retarget-

ing its signals, the company will use the opportunity. It will skip less atten-

tive receivers and use the radiation capacity to address subjects on which

the produced signals have a higher impact. If audiences are allocated to

senders in a way that offers no such opportunities, no company is interested

in changing the audience allocation. If subjects have very heterogeneous

attention capacities, it cannot be excluded that some of them are not

covered at all, that is, no company targets its signals on them. Others may

be fully covered by all companies. Let ( ){ }: ,aTU s M s= = , and

( ){ }: ,aTFC s M s T= = be the (possibly empty) sets of uncovered and

fully covered subjects, respectively. The following proposition

characterizes the attention levels of covered and uncovered subjects in an

equilibrium audience allocation.

Proposition 1. For TB with 0T > , let aT be a feasible audience

allocation.

IfaT is an equilibrium audience allocation, the following conditions hold:

(a) For all t T : ( ) ( ), ,s sv s v s for almost all ( ), aTU s t .

(b) ( ) ( ), ,s sv s v s , for almost all ,S CF s CF .

(c) For ++

! , let ( ){ }: , sS s v s

= < , ( ){ }: , sS s v s +

= . For all

t T : If ( )( ) 0aT t S , then ( )( ) 0.aTS t + =

(d) For all t T : ( )( )aT tt r = .

If aT satisfies conditions (c) and (d) then aT is an equilibrium audience

allocation.

Proof. Appendix


33/78


The proposition provides necessary and sufficient conditions for an

equilibrium. An immediate consequence of the sufficient conditions is that

any audience allocation with equalized attention levels and fully utilized

radiation range is an equilibrium.

Corollary 1. aT is an equilibrium audience allocation if ( ) ( ), ,s sv s v s = ,

for almost all , s S , and ( )( ) ( ) ( ){min ,aT t S t = .

Proof. Equal attention levels imply condition (c), and (d) holds by

assumption.

The necessary conditions imply that, apart from extreme cases, attention

levels must be equal, as the following discussion of conditions (a) (d)

shows.

Condition (a) says that only the least attentive subjects are possibly not ad-

dressed by any company, that is, are uncovered by T. Since they would pay

less attention than other audiences, no company would wish to retarget its

signals on them. In contrast, condition (b) deals with subjects which are in

the focus of all companies. Such receivers, if there are any, must have a

very high attention capacity. Despite full exposure (to all t T ), their atten-

tion level is at least as high as the attention level of subjects which are not

fully covered by T (i.e. for which ( ),aTM s is a proper subset of T).

Condition (c) is the most interesting one. Any company which has people

with relatively low attention level among its receivers (i.e. for which

( )( ) 0aT t S has certainly also almost all receivers with higher atten-

tion in its audience ( )( )( )0aTS t + = . Only a company that has ex-


34/78


hausted all subjects with relatively high attention is willing to target its sig-

nals also on less attentive receivers. The alternative would be to leave part

of the radiation capacity unused. But this would be no equilibrium, as

stated by (d), since additional receivers, if they are feasible, always

increase the total impact of a sender. This has two important implications:

First, only if receivers with sufficiently heterogeneous attention capacities

are in S, equilibrium attention levels can differ. According to (a) (c), if

different attention levels are realised in an equilibrium, precisely those

subjects which are exposed to more signals have the higher attention level.

Thus, they must have a much higher attention capacity than those who are

exposed to less senders and have nonetheless a lower attention level. Put

the other way round, if attention capacities are not too heterogeneous, then

all attention levels are equal in an equilibrium audience allocation.

The second implication of proposition 1 concerns size and heterogeneity ofthe radiation capacity of companies in T. If T contains enough small

companies (i.e. ( )t is small relative to the measure ( )S of receivers),

their chase after attentive audiences tends to equalize attention levels also if

receivers are very heterogeneous. To see this consider an audience alloca-

tion with two subject pools ,A A A. Some companies target their signals

on A , others on or on both sets. Small companies with

( ) ( ) ( ){ }min ,t A A < can fully utilize their capacity by concentrating

on one set. As long as attention levels are high in one set and low in the

other, small companies can always switch to the more attractive audience.

If there are many such companies, signal exposure and attention levels

change. The retargeting process only stops when attention levels in the two

sets are equal or if all companies have switched to the more attentive


35/78


audience. But the latter implies that this audience consists of an elite whose

attention capacity is so high that they process any volume of signals better

than others. The following corollary proves this intuitive argument more

generally.

Corollary 2. Suppose that aT is an equilibrium audience allocation for T

with 0T > . For any + +! let ,S S + be defined as in proposition 1.

Choose a constant + so that ( ) 0S + = for +> and ( ) 0S + for

+ . Define ( ){ }* : , sS s v s += = .

(a) If ( ) ( )*t S for t T , then ( )( )* 0aTS t = , i.e. *S is covered

by t.

(b) If ( ) ( )*t S for all t T , then almost all subjects which are

covered (i.e. for which ( ), TM s a ) have attention level + .

Proof. Suppose ( ) ( )*t S and ( )( )* 0aTS t . Then ( )aT t contains

subjects with attention levels below + which implies ( )( )* 0aTS t =

according to proposition 1(c). A similar argument leads to (b).

The corollary makes no restriction concerning heterogeneity of attention or

radiation capacities. However, the more subjects have similar attention

capacities, the more likely becomes ( ) ( )*S t so that case (b) becomes

relevant. And if senders have similar radiation capacities, either all of them

cover *S (case (a)) or all covered subjects have equal attention levels (case

(b)). In sum, in an attention economy with senders and receivers whose


36/78


radiation and attention capacities are not so different that some receivers

are addressed by all companies whereas others are ignored by all, almost all

attention levels are equal in an equilibrium allocation of audiences. The

reason is analogous to competitive markets. Like exploitation of arbitrage

opportunities by traders leads to an equilibrium price, the senders' chase

after impact equalizes attention levels. The further analysis concentrates on

audience allocations with equalized attention levels.

The following example describes a class of economies for which an

equilibrium audience allocation with equal attention levels can be con-

structed explicitly.

Example 1. For , 0TT > , let heterogeneity of senders be restricted by the

assumption that there is a finite partition ( ), 1, , , 0Bk kT k K T =


37/78


In the proof of fact 1 a concrete audience allocation is constructed

satisfying the properties of an equilibrium. Other audience allocations can

fulfil these properties as well. However, the construction of the equilibrium

audience allocation suggests that in any equilibrium audience allocation

with equal attention levels, the equilibrium attention level is uniquely deter-

mined by average signal imission ( ) ( )/k k kk

r T S . (Note that k kr is

signal imission implied by sender k who spreads k over range kr . And

( )k k kk

r T is aggregate signal imission.) The following proposition

shows that under DA this is generally the case for equilibrium audience

allocations with equalized attention levels, i.e. for allocations aT which

satisfy ( )( )aT tt r = , ,t T and ( ) ( ), ,s sv s v s = for almost all , s .

Proposition 2. For , 0TT >B define ( )t TT

X r t dt= . Under DA, there

exists a decreasing real function ( )X so that for any equilibrium audience

allocation aT with s s +

for almost all the following holds: If attention

levels are equal under aT, then they are given by ( )X (i.e.

( ) ( ), sv s X = almost everywhere).

Proof. Appendix

Like in all other statements ( )S is fixed throughout the analysis. If ( )S changes a different

function ( )X is relevant. In general, the comparative-static effects of changes in ( )S are

ambiguous. But if attention capacities are identical as in example 1, equal attention levels imply

( )/s X S = = . Then the equilibrium attention level is a function of ( )/X S . At the end of this

paper this fact is used to illustrate possible consequences of international integration in an attentioneconomy (see example 3).


38/78


Proposition 1 and its corollaries have shown that equalized attention levels

can be considered as the normal case in an equilibrium with sufficient

competition (i.e. if there are enough senders with similar radiation capacity

and if receivers' attention capacities are not too different). Proposition 2

shows that in an attention economy with given sender set T and signal

strength T , in any equilibrium audience allocation with equalized

attention levels the equilibrium attention level of almost all receivers is

uniquely determined, provided that assumption DA holds and s s +

. In

other words, there may be multiple equilibrium audience allocations. But

the equilibrium attention level of receivers is given by ( )X which is

independent of the signal strength ( )T t of a single sender { }( )( )0t = .

Hence, each sender t T can take the attention level of its receivers as

given when deciding about signal strength. This decision and the implied

equilibrium signal strength are analysed in the next section.


39/78


V. Contest for attention and equilibrium signal exposure

The question solved so far was: Given a set of companies T with signal

strength T , what are the properties of equilibrium audience allocations

under , TT . This leaves open the questions: (i) Which sets of companies

are equilibrium sets, and (ii) which signal strength do companies choose to

have an optimal impact on their audience? In this section the second ques-

tion is answered. Given an audience allocation aT for TB, what are

equilibrium choices ( ) ,T t t T ?

In an attention economy, the essential performance measure is impact. Im-

pact is achieved by attracting the attention of audiences. Thus, companies

compete for the attention of subjects by sending them signals of a certain

strength. Formally, the impact of tsending with strength R+

on a re-

ceiver with attention level was modelled as ( ),tz . The attention level

of a subject depends on the environment to which it is exposed. It is

given by ( ), sv s where signal exposure s depends on the given audience

allocation aT and on signal strength T . The total impact achieved by t

sending with strength is given by ( )( ) ( )( )

, ,

aT

t t s

t

V z v s d s = . In

principle, a company choosing signal strength would have to reflect also

the consequences of its choice on s . However, a single company has no

influence on since it has measure zero. Thus, any t T can take ( ), sv s

as given when deciding about its signal strength.

Formally,

( )( ) ( )( ), ,a aT Ts T T

M s M s

t dt t dt

= =

for

{ }T T t = .


40/78


Impact has economic value since only those who earn attention of

audiences are in the trade, i.e. have a market, make money or earn

reputation. Suppose that tz is some physical measure of t's impact like

frequency of citations or time spent on signals coming from t. Let the

economic value of tz be given by some function ( )t tz . If t is bounded

and monotonously increasing starting at ( ) ( )0,t t t t t z z z = =! satisfies all

properties that have been required from impact functions in section III.

Thus, provided one accepts the idea that economic performance is

positively related to impact, it can be assumed without loss of generality

that tz and thus tV measure the economic value of impact rather than

impact in the physical sense.

A company tcan increase its value of impact in an equilibrium allocation of

audiences by choosing a higher signal strength . The other determinants

of tV are exogenous to the single company. Higher signal strength can

mean everything that induces receivers to spend more attention on the

signal sender, from volume and content of signals to eye-catching

attributes. This strength may but need not be related to qualities which are

valued by the receiving subjects. The important characteristic described by

is the extent to which a company t absorbs attention capacity and

thereby achieves impact. For company t strength is valuable but also

The class of functions : R R Rtz + ++ + with ( ), 0tz = for 0, tz = bounded, increasing in

and, for 0 > , increasing in is invariant under transformations : R Rt t t with

( )0 0,t t = bounded and increasing.

The first aspect of this double role of signal strength is reflected by the fact that ( )T t contributes to

a subject's signal exposure ( )( )( ),aT

s T

M s t

t dt = , which influences the attention level ofs, and to

aggregate signal imission Xwhich determines the equilibrium attention level. The second aspect is

reflected by the fact that ( )T

t enters impact functiont

as an argument determining how much a

given attention level is devoted to company t, i.e. how deep is the trace left by information from t.


41/78


costly. It is assumed that the production of strength requires resources.

But the dissemination of to an audience is free provided that the

audience can be handled with t's radiation capacity ( )t . In view of

modern technologies like the internet, this seems a reasonable assumption.

Producing papers costs effort, sending them to a given mailing list is easy.

In sum, for a company t the cost of exposing an audience A of measure

( ) ( )t to signal strength 0 are given by a function

( )( ) ( ),t tC A c = , (5.1)

where tc is differentiable with 0, 0,t tc c > starting at ( )0 0tc . This al-

lows for possible fixed costs ( )0 0tc > . They may arise from setting up

signal production or distribution capacity, i.e. the capacity to address

audiences of size ( )t . It is assumed that tc represents signal costs in units

of the economic value of impact.

In an equilibrium, each company chooses signal strength such that the net-

value of its impact cannot be improved by choosing another strength. Let

( ) ( )( )( )

( ) ( ), ,aT

nt t s t

t

V z v s d s c = be the net-value of impact of

. Signal strength is optimalfortunder ,aT T if ( ) ( )n nt tV V for

There may be information flows from senders to receivers which attract attention and have impactwithout absorbing attention capacity, for instance, a good idea presented so clearly that it is grasped

immediately. This poses no problem, as long as impact functions tz are allowed to be t-specific. A

"good" sender can have relatively high impact with low signal strength. Moreover, also signal costs

tc are allowed to be t-specific. An alternative way to account for the possibility that a signal

generates impact without absorbing attention capacity would be to extend the model by adding a

second costly instrument of senders, say quality ( )Tq t , which enters tz as an argument in addition

to ( )T

t but is irrelevant for the level of . Obviously, such an extension would not change the

analysis presented so far.


42/78


. (Since thas measure zero no strategic aspects are involved in this

choice.)

Definition 2 (Equilibrium signal strength). For TB, let aT be a feasible

audience allocation.

(i) T is an equilibrium under aT if, for all ( ), Tt T t is optimal under

,aT T .

(ii) ,aT T is an equilibrium forT, ifaT is an equilibrium audience alloca-

tion for , TT , and T is an equilibrium underaT.

As shown in section IV under quite general assumptions, in an equilibrium

audience allocation aT with signal strength T attention levels of subjects

are equalized and given by a decreasing function ( ) ( ), sv s X = of

aggregate signal imission ( )t TT

X r t dt= , ( ) ( ){ }min ,tr S t = . In such

an audience allocation sender t T has impact ( ) ( )( ),t Tz t X on all

subjects belonging to its audience ( )aT t . Since ( )( )aT tt r = in an

equilibrium audience allocation, the value of total impact ( ),at T TV which

is achieved by t T is given by

( ) ( )( ),t t t T V r z t X = . (5.2)

According to (5.2), expanding signal strength ( )T t has a positive effect

on company t's reception for any given X. But if a set of companies with

positive measure increases signal strength, then X increases which has an

external effect on other companies. Under the assumptions of proposition 2,

( )X decreases in X so that the external effect is negative. Since single t


43/78


have measure zero, they ignore this effect when deciding about their

optimal ( )T t . Determining the equilibrium signal strength therefore

requires two steps. First, one has to clarify which ( )T t is optimal for

individual companies t T if aggregate signal imission is X. Second, one

has to look for an equilibrium which satisfies ( )t TT

X r t dt= .

Given (5.1) and (5.2), for each t T , the choice of optimal strength is a

standard optimization problem [ ]max t tV c

. Since the innovation of this

paper lies elsewhere, I want keep this problem simple enough to apply

differential calculus for deriving comparative-static results about the choice

of signal strength. So far only boundedness of impact function ( ),tz

was required. Moreover, ( )0, 0,t tz z = increasing in and, for 0 > ,

increasing in . From now on it is assumed that tz is differentiable and

satisfies the following concavity property:

2 2

20, 0t t

z z

<

(5.3)

Moreover,0

lim tz

=

and lim 0t

z

=

.

With this specification the senders' optimal choices of signal strength andthe achieved net-value of impact nt t tV V c= can be derived in a straight-

forward way if attention levels of receivers are equalized. The following

proposition shows that the outcome of the optimisation of signal strength is

a monotonous function of audience size and attention level.

These are sufficient conditions guaranteeing generally the existence of a positive optimal signal

strength with comparative-static properties as required for an equilibrium. They need not hold inconcrete examples, in which the solution can be calculated explicitly.


44/78


Proposition 3. For , TT let aT be a feasible audience allocation with

( ), sv s = almost everywhere. If cost and impact functions satisfy (5.1)

and (5.3), respectively, then:

(a) A company's optimal signal strength is given by a differentiable

function ( )( )( )* ,at T t with * / 0t > and * / 0t .

(b) For all t T , the maximal net-value of impact is given by a

differentiable increasing function of ( )( )aT t and .

Proof. Appendix

The next task is to characterize for the case of equal attention levels the

properties of equilibrium signal strength. For this purpose two further

purely technical restrictions have to be imposed on the functions

representing attention capacities, signal impacts and costs. First, it is as-

sumed that the space of receivers ( ), ,AS is a real interval withBorelsets

and Lebesgue measure. Moreover, it is assumed that for all s the function

representing attention capacities, ( ), sv s , is differentiable with respect to

. Secondly, it is assumed that impact and cost functions vary across tin a

measurable way. Formally, for all , , the derivatives ( ), /tz and

( )tc are measurable functions of t. (Note that this is trivially fulfilled if

,t tz c are identical for all t.) These properties guarantee the following facts

which are useful for the derivation of equilibrium signal strength.

Lemma 2. Function ( )X in proposition 2 is differentiable ( )/ 0d dX < .

Function ( ) ( )( )( )*: ,aT t Tt t = of optimal signal strength, derived in

proposition 3, is measurable.

Proof. Appendix


45/78


With this preparation equilibrium signal strength can be determined as

follows: In an equilibrium audience allocation ( )( )aT tt r = . Thus,

according to proposition 3, optimal signal strength of t T is given by

( )* ,t tr . Since tr is an exogenous characteristic oft, argument tr can be

omitted without loss of information. Thus, in the further exposition I write

( )*t instead of ( )* ,t tr . Equilibrium signal strength T must satisfy

two conditions: It must be consistent with the optimal choice of individual

companies, i.e.:

( ) ( )*

T tt = (5.4)

for all t T . And T must be consistent with attention level . If the

assumptions of proposition 2 are fulfilled, this attention level is given by a

decreasing function ( )X , i.e.:

( )X = with ( )t TT

X r t dt= . (5.5)

(Lemma 2 guarantees that *t is measurable so that the integral definingX

exists.) Combining the two conditions (5.4) and (5.5), we obtain the

equation

( ), , TX Z X T r= (5.6)

with ( ) ( )( )*, , :T t tT

Z X T r r X dt = and Tr denoting the function on T

assigning to each t T range tr. Lemma 2 guarantees that is differenti-

able and decreasing in X. Since *t is differentiable and non-increasing in

, / 0Z X so that equation (5.6) defines for each , TT r a unique

equilibrium level [ ]* , TX T r of aggregate signal imission. Together with

(5.4) and (5.5), this defines also a unique attention level


46/78


[ ]( )* *: ,T TX T r = and a unique equilibrium signal strength

( ) ( )* * *:T tt = . The following theorem summarizes this and presents a

first main result about the comparative-static properties of equilibria in an

attention economy.

Theorem 1. Under (5.1), (5.3), the assumptions for lemma 2 and DA with

s+ sufficiently small for all s, in any equilibrium ,aT T for BT in

which attention levels are equalized the following properties hold:

(a) Equilibrium signal strength *T and equilibrium attention level*T are

uniquely determined by Tand tr.

(b) If ( ) ( )t S < and ( )t increases to ( ) ( )t t >! on a measurable

subset ( )0 0, 0T T T < , then equilibrium attention*T declines.

Proof. Appendix

Assumption DA with " s+ sufficiently small" means that receivers are

strained by the prevailing signal exposure, so that their attention capacity is

decreasing in the relevant range. An increase in the range of radiation

allowing a wider diffusion of signals induces companies to increase their

signal strength in competing for attention. The reason is that diffusion of

signals has zero marginal cost, so that the production of impact by

addressing signals on audiences is subject to economies of scale. At the

same time, there is a second effect leading to higher signal exposure of

subjects even without such economies of scale at the company level. Any

given signal strength reaches more subjects when the radiation range of a

company is extended. As long as the set of companies doesn't change, this

necessarily means more overlap among audiences. For instance, if a


47/78


national scientific network extends its range to the international level it has

to address subjects belonging to other national or international

communities. As a consequence, these subjects will be exposed to more

signals, say publications. If signal exposure was high before, relative to the

threshold s+ defined by DA, then the increased volume of signals will be

perceived with less attentiveness. Of course, the described phenomenon

isn't specific to scientific communities. Info-stress and attention deficit are

not uncommon. According to theorem 1, a responsible economic

fundamental is the possibility to address larger audiences.

The extension of the range of one company may wipe out other companies

so that T, which up to now has been taken as given, changes. This brings us

to the question of how equilibrium diversity is determined in an attention

economy. This question is answered in section VI. Before turning to this

section, I want to illustrate theorem 1 by the following example.

Example 2. Suppose that T can be partitioned in K measurable sets

1, , KT T" with ( ) kt = for kt T , where the divisibility condition of

example 1 is satisfied. Like in example 1 let ( ), sv s be given by a non-

increasing function ( )sf . Moreover, for kt T , ( ) 1 0k k

tc c c = + for

some constants 10

k

kc r<

, ( ),tz ( ) ( )lnk kg h = +

,

where ( ) ( )1, 0k kg h for 0> and 0, 0k kg h with one

inequality holding strictly. (For 1 , ( ), 0tz = is assumed.)

Fact 2. For any Tsatisfying the properties of example 2, (i) an equilibrium

,aT T with equalized attention levels exists, and (ii) in any such


48/78


equilibrium ( )* * 1/ kk k kr g c = , ( )* *f = where signal exposure of

subjects is given by a function ( ) ( )( )* 1 1, , , , ,k KT T with

( )* / 0kT > and * / 0k > if ( ), 1, , .k S k K < =

Proof. Appendix

As a result of the analysis in this section we know how equilibrium signal

exposure, attention level and signal strength depend on ( ) , ,t t T for a

given T. This leaves us with the question of what is an equilibrium set of

companies T.


49/78


VI. Viability and equilibrium diversity in an attention

economy

Two characteristics are basic for the notion of a competitive environment:

The agents must be viable, and free entry is allowed. In an attention econ-

omy, in which it is vital to attract attention, a natural notion of viability is

that companies achieve a non-negative net-value 0ntV from sending

costly signals for having impact on audiences. Free entry means that com-

panies can participate in the contest for attention if they want.

Since single companies have zero measure, no strategic considerations are

involved in the entry decision. Any individual sender t T chooses on the

basis of the given attention levels whether to enter or not. More formally,

for any single company the entrance problem looks as follows: Given a set

T B of active companies, a feasible audience allocation aT and some

signal strength 0T > , the attention levels of subjects are ( ), sv s . If t T

enters by targeting ++

! on an audience AA , the set of active

companies becomes {T T t = . Since {( ) 0t = , this doesn't change the

attention levels of subjects. Thus, in case of entry, t achieves maximal

impact when picking an audience * with ( ) ( ) ( ){ }* min ,S t = in

such a way that ( ) ( ), ,s sv s v s for all* *,A s S A , and maximal

net-value of impact when choosing optimal signal strength

( )( ) ( ) ( )*

* arg max , ,t t s t A

z v s d s c = . Denote by [ ]*

,at T TV the

maximal net-value of impact which t achieves when entering in an

optimal way. Company t is viable if [ ]* , 0at T T

V . It definitely has an


50/78


interest to enter if the achieved value is strictly positive. If [ ]* , 0at T TV = ,

it is indifferent with respect to entry.

Whereas the entry decision for a single company is straightforward, the

aggregate consequences of simultaneous entry decisions of many firms are

more subtle. Suppose that there is a set BB of companies so that every

t B decides to enter. Then the set of active companies expands to

T T B = . If ( ) 0B > , attention levels of receivers and thus sender

decisions are affected. In particular, the function ( )nt

V , describing the

net-value of impact oftwhen sending with strength , isn't the same under

T and T . For a proper treatment of this problem, let for , , ,aT TT

( ), :aT T TV t = ( ) ( )( )( )

( ) ( )( ), ,aT

t T s t T

t

z t v s d s c t be the net-value

of impact achieved by t T under , ,aT TT . For ,aBB B feasible and

0,T>

call , ,aT TT

an expansion , ,aB BB of , ,aT TT if T B=

,

T T B = , and ,aT T and ,aB B are the restrictions of ,aT T on Tand

attention economies+ by josef falkinger*

Documents