Advertising and Perceptual Biases*

Andreas Grunewald, University of Bonn**and

Matthias Krakel, University of Bonn***

Abstract

Consumers are often prone to perceptual biases. These biases (e.g., the opti-mism bias or conservatism) can make advertising less effective, which dampensfirms’ incentives to invest in advertising. If advertising is persuasive and there-fore socially wasteful, biases that make advertising less effective are, at firstsight, welfare enhancing. However, biases also change product market compe-tition, which can induce more advertising. We show under which conditionsconsumers’ perceptual biases lead to a net increase in persuasive advertisingand, hence, to a net welfare loss. Our results show that this outcome will beparticularly likely if consumers rate competitive goods as rather different priorto the firms’ advertising decisions.

Keywords: combative advertising; persuasive advertising; consumer biases; price competi-

tion.

JEL classification: D03; D11; L1; M37.

* Financial support by the DFG, grant SFB/TR 15, is gratefully acknowledged.

** Andreas Grunewald, Department of Economics, University of Bonn, Ade-

nauerallee 24–42, 53113 Bonn, Germany, phone: +49-228-739217, fax: +49-228-

739210, e-mail: gruni@uni-bonn.de.*** Matthias Krakel, Department of Economics, University of Bonn, Adenauer-

allee 24–42, 53113 Bonn, Germany, phone: +49-228-739211, fax: +49-228-739210,e-mail: m.kraekel@uni-bonn.de.

1

1 Introduction

All over the world, companies spend a huge amount of money on advertis-

ing each year. According to the U.S. Census Bureau, total expenditures for

advertising amounted to $158 billion in the United States in 2012.1 Global

companies, in particular those in the high tech industry, invest large amounts

of money in advertising when introducing new products. For example, in 2013

Samsung spent $14 billion on advertising – a sum that exceeds the total GDP of

Iceland.2 To promote its new Galaxy S5 smartphone, Samsung recently rented

the whole Terminal 5 at London’s Heathrow Airport for excessive campaigning

inside, temporarily relabeling it ”Terminal Samsung Galaxy S5.”3

These massive expenditures aim to influence consumers’ quality perceptions

about the advertised good. However, it is a persistent finding that humans

tend to violate Bayes’ rule in various ways when updating their perceptions

(Kahneman et al. 1982). For example, they are prone to perceptual biases

like optimism (e.g., Weinstein 1980, 1982, 1989, Svensson 1981, Camerer and

Lovallo 1999, Van den Steen 2004, Herz et al. 2014) or conservatism (Phillips

and Edwards 1966, Edwards 1968, Eger and Dickhaut 1982, Camerer 1987,

Dohmen et al. 2009). As a consequence, when deciding about a purchase,

consumers may systematically overestimate consumption qualities and be re-

luctant to adapt their quality priors due to advertising. The latter has been

documented for different forms of advertising (Danaher and Rossiter 2011) and

seems to be particularly relevant for innovative products, in which case new

information is sometimes completely discarded (Heiskanen et al. 2007).4 As

the belief formation process bridges the gap between advertising and consumer

choice, these systematic mistakes may substantially impact consumer choices

in response to advertising.5 Consumer perceptual biases are thus bound to be

an important determinant of firms’ advertising policies.

In this paper, we investigate in what way violations of Bayes’ rule shape

1See http://www2.census.gov/services/sas/data/table5.xls.2See http://www.reuters.com/article/2013/11/27/us-samsung-marketing-idUSBRE9A

Q18720131127.3See for example http://time.com/103175/heathrow-airport-samsung-galaxy/4Moreover, conservatism has been extensively discussed as possible explanation for finan-

cial anomalies, see, for example, Barberis et al. (1998), Hirshleifer (2001), Brav and Heaton

(2002), Barberis and Thaler (2003), Brandt et al. (2004).5For example, consumer conservatism can induce overly loyal behavior of consumers as

documented by Prince (2011) and Hortacsu et al. (2015). In consequence, consumer choice

may hardly be affected by advertising.

2

firms’ application of persuasive advertising and analyze the resulting conse-

quences for overall welfare. We focus on persuasive advertising for two rea-

sons.6 First, as persuasive advertising aims at manipulating consumer pref-

erences by influencing their quality perceptions, it is most apparent that the

belief formation process is crucial for its impact. Second, persuasive advertis-

ing is the most problematic form of advertising as it is pure social waste.7

In our model, we consider two firms each supplying one good. The firms

decide on advertising at the first stage and compete in prices at the second

stage. The building blocks we use to model the two stages are well known

in the literature. To address possible manipulation of consumer preferences

at the first stage, we use the Bayesian learning framework with normally dis-

tributed beliefs, which is applied in economics (e.g., Prendergast and Stole

1996, Meyer and Vickers 1997, Holmstrom 1999, Hoffler and Sliwka 2003) as

well as marketing (e.g., Erdem and Keane 1996, Mehta et al. 2003, Janaki-

raman et al. 2009, Goettler and Clay 2011). In this setting, advertising can

influence consumer expectations about the goods’ uncertain perceived quali-

ties. The more a firm invests in advertising, the higher will be the consumers’

posterior perceived quality of the firm’s good. For modeling price competition

at the second stage, we build on the seminal paper by Shaked and Sutton

(1982).

The major novelty in our setting is that we allow for consumers to violate

Bayes’ rule when processing information. Our modelling of perceptual biases

builds on the observation by Koszegi (2014, p. 1085) that decision makers

are typically prone to two general mistakes – ”systematically incorrect priors,

and mistakes in updating beliefs based on information.” We simultaneously

incorporate both kinds of mistakes in our setting. Hence, consumers might be

too optimistic or too pessimistic when assessing a good’s consumption quality

prior to the updating process. At the same time, they might put too much or

too little weight on updating when new information about a good’s consump-

tion quality arrives. We allow for all possible combinations of these perceptual

6Advertising can influence consumer behavior in three different ways (Bagwell 2007,

1708–1724; Belleflamme and Peitz 2010, 135–139). First, it can provide consumers with

useful information about the existence, the price, and specific characteristics of a good.

Second, advertising may complement a good by enhancing its degree of popularity and,

thereby, the consumers’ utility from using or consuming it. Third, advertising may be

neither informative nor complementary but purely persuasive.7See, among others, Braithwaite (1928), Von der Fehr and Stevik (1998), Bloch and

Manceau (1999), and Buehler and Halbheer (2012).

3

biases. However, to avoid redundancies, we will focus on the most prominent

cases (see, for example, Offerman and Sonnemans 1998) – the existence of too

optimistic priors (i.e., consumer optimism) and too little updating (i.e., con-

sumer conservatism). All the remaining biases can be analyzed analogously.

At first sight, one would expect that firms reduce persuasive advertising

if it is less effective due to perceptual biases. For example, if conservatism is

widespread among consumers, i.e., they partially ignore new information and

stick to their prior beliefs about a good’s consumption quality, the influence of

advertising may be severely limited. Firms that anticipate such biases should

choose less persuasive advertising compared to a situation without percep-

tual biases. However, we show that the biases also influence product market

competition, which can result in more advertising.

To understand the effects of perceptual biases on product market compe-

tition consider the price competition at stage two of the model. The firm that

supplies the good with the higher perceived quality – the posterior market

leader – has a clear competitive advantage. Contrary to the rival firm, it can

charge a higher price without losing too many consumers because the higher

perceived quality of its good leads to greater consumer loyalty. We show that

the posterior market leader earns strictly larger profits than the rival firm, but

the profits of both firms are increasing in the spread between the perceived

qualities of the two goods. At the first stage, a firm can choose advertising

to increase the perceived quality of its good. As both firms’ profits increase

in the perceived quality spread, each firm faces the following trade-off. On

the one hand, advertising will be beneficial if the firm is the posterior market

leader when entering price competition. On the other hand, advertising will

be detrimental ex post if the firm is not the posterior market leader.

These two counteracting effects explain how perceptual biases influence

firms’ marketing strategies at stage one. First suppose consumers have sys-

tematically incorrect priors. In this case, consumer optimism will increase a

firm’s advertising if the perceived quality of its good initially exceeds that of

the rival firm and if the beneficial starting point of the former one is pro-

tected by consumer conservatism. In that situation, the firm with the higher

mean quality at stage one – the prior market leader – will also be the pos-

terior market leader with high probability. This firm has strong incentives

to further increase the perceived quality spread and, hence, expected profits

by advertising. On the other hand, the rival firm is discouraged and chooses

low advertising. Overall welfare will decrease with consumer optimism if the

4

extra persuasive advertising of the prior market leader exceeds the decrease in

advertising of the discouraged rival firm.

Second, consider consumers whose information processing is biased by con-

servatism. Two effects on optimal advertising have to be distinguished. On

the one hand, conservatism hampers Bayesian updating and, thus, makes ad-

vertising less effective (attenuation effect). This effect decreases advertising

incentives. On the other hand, conservatism also influences price competition

at stage two (competition effect): The larger conservatism the more likely the

initially perceived quality ranking of the two goods will be preserved at stage

two. Such preservation of an asymmetric starting point encourages advertising

behavior of the prior market leader and discourages the rival firm, because both

maximize their profits by generating a large perceived quality spread. Alto-

gether, the two effects of conservatism make the rival firm decrease advertising,

whereas advertising of the prior market leader will be increased (decreased)

if the competition effect dominates (is dominated by) the attenuation effect.

Welfare losses due to persuasive advertising are boosted by consumer conser-

vatism, if the competition effect is sufficiently strong so that the additional

advertising of the prior market leader exceeds the reduction in advertising of

the rival firm. This outcome is particularly likely if consumers rate the goods

as rather different prior to the firms’ advertising decision.

Due to its wasteful characteristic there has been an ongoing discussion if

persuasive advertising should be regulated and how to implement regulation.

For example Peltzman (1973) investigates the U.S. drug market and the role

of consumer protection legislation. His starting point is the observation that

pharmaceutical firms often invest in advertising to exaggerate the effectiveness

of new drugs.8 A central topic is the question whether pharmaceutical firms

should be forced by law to prove the efficacy of new drugs. Our findings add

to this discussion as they show under which circumstances perceptual biases

lead to a strong need for regulatory policy and suggest how regulation could

be successful in enhancing welfare. Recall that consumer conservatism will

lead to a net increase in persuasive advertising in situations with a clear mar-

ket leader. Thus, policy measures counteracting a high degree of perceived

heterogeneity of goods can be useful to prevent the negative effects of con-

sumer conservatism. In particular, the prohibition of persuasive comparative

advertising may help to avoid that consumers perceive similar goods as overly

8Comanor (1986) reports that advertising expenditures in the pharmaceutical industry

considerably exceed R&D expenditures.

5

different, which would end up in excessive advertising by the perceived market

leader. Moreover, investment in consumer protection that lowers uncertainty

about the products’ consumption qualities (e.g., mandatory identification of

product characteristics) may make Bayesian updating less important for con-

sumers. Hence, it could also contribute to mitigate detrimental welfare effects

of consumer optimism and consumer conservatism.

Our paper is organized as follows. The next section summarizes the liter-

ature related to our paper. The model is introduced in Section 3. In Sections

4 and 5, we analyze the impact of consumer biases on firms’ advertising deci-

sions in monopoly and duopoly, respectively. Section 6 focuses on the welfare

implications of consumer biases. Section 7 concludes.

2 Related Literature

Our paper is related to two strands in the literature – the part of the behavioral-

economics literature that considers consumer biases, and the economic litera-

ture on advertising. The first strand of literature considers firms’ behavior in

response to consumers with non-standard preferences or cognitive biases (for

an overview, see Koszegi, 2014). In particular, we build upon the work that

investigates selling strategies in competitive markets populated by cognitively

limited consumers. Gabaix and Laibson (2006) and Dahremoller (2013) ex-

plore firms’ decisions if some consumers are not able to observe information

about a product add-on. They argue that, as a consequence of this consumer

myopia, hidden product details are shrouded by firms even in highly competi-

tive markets. Bordalo et al. (2013) go one step further and endogenize which

attribute of the good receives attention by consumers.

More closely related to our analysis are papers considering marketing poli-

cies. Li et al. (2014) investigate the incentives of a monopolist to disclose

potentially adverse product effects if some consumers are unaware of their

existence. They argue that firms will only disclose information through ad-

vertising if the share of unaware consumers is small. Otherwise, mandatory

disclosure policies may enhance welfare. In a similar spirit, Eliaz and Spiegler

(2011) assume that firms can engage in marketing to influence the set of al-

ternatives that a consumer perceives as relevant. Their paper fundamentally

differs from ours in the way how marketing influences consumers. While in

their setting marketing is used to directly manipulate the consideration sets

of consumers, we analyze firms’ persuasive advertising choices that enter the

6

decision process more indirectly through a manipulation of beliefs. The latter

approach is also taken by Shapiro (2006) who builds on the idea that ad-

vertising influences the likelihood of consumers remembering a positive past

experience with the advertised good. Considering only monopolists, the pa-

per argues that advertising expenditures are highest for intermediate product

qualities. Complementary, we find for persuasive advertising that firms with

perceived quality advantages tend to advertise most.

The second strand of literature deals with advertising in economics. Bag-

well (2007) offers a comprehensive overview of the relevant literature. Our

paper is most closely related to the economic literature on combative and per-

suasive advertising, as we assume that total demand by consumers is exoge-

nously given and advertising does not provide additional utility for consumers.

Hence, if a firm attracts additional consumers by investing in persuasive ad-

vertising, rival firms will typically be harmed by losing these consumers.

According to Leiter (1950), the problem of combative and persuasive adver-

tising was first addressed by Marshall (1919), who emphasizes the social waste

of redistributing consumers from rival firms. In his empirical study, Lambin

(1976) uses data on advertising, sales, quality, and prices from 107 individual

brands to analyze how an increase in advertising expenditures by one firm in-

fluences own sales as well as the sales of rival firms. His results support the

notion that advertising is combative. Further evidence comes from Metwally

(1975, 1976) and Kelton and Kelton (1982). Similarly, Cubbin and Domberger

(1988) and Thomas (1999) show that firms respond to entry by rival firms with

increased combative advertising. More recently, Chen et al. (2009) theoret-

ically and experimentally investigated combative advertising. Theory shows

that while in some situations combative advertising leads to a more intense

price competition, there are other situations in which combative advertising

dampens price competition. The experimental results strongly support the

theoretical findings. Chioveanu (2008) argues that persuasive advertising may

not only influence the intensity of price competition but also lead to asym-

metric advertising and price choices by firms. This provides an alternative

explanation for price dispersion phenomena. Contrary to ours, none of these

papers offer an analysis of the impact of consumer biases on combative adver-

tising.

7

3 The Model

We consider a two-stage market game in which two risk-neutral and profit-

maximizing firms decide on advertising at stage one and, thereafter, compete

in prices at stage two. Each firm i (i = A,B) offers a good i whose consump-

tion quality is uncertain in advance, because it depends on the specific match

of individual consumer preferences and the characteristics of the good. For

example, suppose that each firm produces an innovative consumption good

(e.g., a gadget). In this case, some individuals may enjoy consumption of the

good, whereas others may be less enthusiastic after consumption. Concerning

the Samsung example from the introduction, on the one hand the consump-

tion quality crucially depends on the technical features of the new smartphone

whose usefulness is uncertain for a consumer. On the other hand, consumption

quality also depends on the specific preferences of an individual consumer (e.g.,

whether an individual extensively uses all new features). This is uncertain for

the firms. Altogether, neither consumers nor firms know the consumption

quality of a good ex ante with certainty.

Our analysis of the price competition at stage two is based on the model of

Shaked and Sutton (1982).9 Production costs are normalized to zero and the

mass of risk-neutral consumers is assumed to be one. Consumer types θ are

uniformly distributed over [¯θ, θ] with

¯θ ≥ 0, θ ≥ 2

¯θ and θ =

¯θ+1 such that the

density is 1. A consumer of type θ receives utility θ · qi − pi from purchasing

one unit of good i (i = A,B) at price pi. The variable qi denotes the ex

ante uncertain consumption quality of good i and is normally distributed with

positive mean qi0 and variance σ2i0, i.e., qi ∼ N (qi0, σ

2i0). Consumer type θ

indicates the importance a consumer attaches to the consumption of the given

good. While consumers with high levels of θ enjoy the consumption of a good

with positive qi more intensely than their counterparts with lower θ, they will

also suffer more intensely if qi is negative.

At stage one, firms choose the intensity of advertising. Subsequently, con-

sumers and firms observe signals sA and sB about the consumption quality

of each good in the market. The value of each signal is determined by the

underlying true consumption quality of the good and the previously chosen

9The following model set-up with vertical differentiation is frequently used in the in-

dustrial organization literature. It can also be found in the marketing literature. See, for

example, Mehta et al. (2003), who use the same consumer utility function with normally

distributed product qualities to discuss consumer search under price uncertainty.

8

advertising by the firm:

si = qi + ai + εi (i = A,B). (1)

Here, εi ∼ N (0, σ2ε) denotes exogenous noise and ai ≥ 0 firm i’s advertis-

ing choice. Advertising ai leads to costs c (ai) with c (0) = c′ (0) = 0 and

c′ (ai) , c′′ (ai) > 0 for ai > 0. Moreover, we assume that c′′ is bounded from

below and above with c′′ ∈ [¯c, c]. To avoid technical problems, we assume

that ai has a finite upper bound. All random variables are assumed to be

statistically independent.

After having observed si, consumers update their prior beliefs about the

distribution of the consumption qualities, qi ∼ N (qi0, σ2i0), and form a pos-

terior distribution. From DeGroot (1970) we know that Bayesian updating

conditional on si would lead to a posterior distribution qi ∼ N (qi1, σ2i1) with

qi1 = qi0 +σ2i0 (si − ai − qi0)

σ2ε + σ2

i0

and σ2i1 =

σ2i0σ

2ε

σ2ε + σ2

i0

,

where ai denotes consumers’ belief about firm i’s advertising decision. Hence,

consumers’ posterior mean of good i’s consumption quality will be larger than

the prior mean, if and only if the realized quality signal exceeds its expected

value.

However, we intend to study the consequences of perceptual biases on the

advertising choices of firms. In general, consumers may be prone to two dif-

ferent kinds of mistakes (see also Koszegi 2014, p. 1085). First, empirical

evidence shows that consumers have systematically incorrect priors. In partic-

ular, they are overly optimistic about future life events (Weinstein 1980, 1982,

1989). Second, consumers make systematic mistakes when updating their pri-

ors according to new information. For example, they may firmly stick to their

priors and partially ignore new information (Philips and Edwards 1966). We

simultaneously incorporate both kinds of mistakes into our setup in a stylized

way.10 For this purpose, we assume that consumers deviate from Bayes’ rule

and form their posterior beliefs according to

qi1 = α · qi0 + β ·σ2i0 (si − ai − α · qi0)

σ2ε + σ2

i0

(2)

with i = A,B. Each kind of bias is described by one parameter (α > 0 and

β > 0, respectively). Here, α indicates consumers’ misperception of the prior.

10Brav and Heaton (2002) and Brandt et al. (2004) model conservatism by overweighting

the prior mean and underweighting Bayesian updating at the same time. However, we follow

the suggestion of Koszegi (2014) and disentangle both effects.

9

If α > 1 for example, consumers will be optimistic about future consumption

utilities. If in contrast α < 1, consumers are pessimistic. We name the first

part of the updating formula (αqi0) the biased prior mean of the consumption

quality of good i.

The parameter β describes the weight consumers attach to new information.

The smaller β, the fewer consumers update their prior information. If β < 1

for example, consumers will tend to stick to their priors and update these more

conservatively than a Bayesian consumer would do. In this case, we speak of

consumer conservatism. If, in contrast, β > 1, consumers will put excessive

weight on new information about consumption quality.

The timing of the game is the following. At the beginning of stage one, the

two firms simultaneously choose their advertising intensities aA and aB, leading

to publicly observable signals sA and sB, respectively. At the end of this stage,

consumers update their beliefs according to (2). At the beginning of stage

two, firms simultaneously choose prices pA and pB. Thereafter, consumers

make their purchasing decisions.

As a solution concept we apply perfect Bayesian Nash-equilibrium, with

the only distinction that consumers form their quality perceptions according

to (2).11 Thus, an equilibrium of the game consists of a strategy profile in-

corporating the strategies of both firms and all consumers and a belief system

such that the following three statements hold. First, both firms play mutually

best responses, anticipating consumer behavior. Second, on the equilibrium

path consumers derive their quality perceptions from firms’ advertising choices

according to (2). Third, consumers’ choices maximize their perceived utility.

The following analysis investigates how perceptual biases – i.e., variations

of α and β – affect equilibrium advertising and overall welfare. We analyze

variations of α and β for their entire domain. However, to ensure readability of

the analysis, our wording is oriented to the prominent case in which consumers

are overly optimistic regarding consumption quality (α > 1) and characterized

11We are interested in how systematic mistakes in information processing changes firms’

advertising decision. Hence, we apply a solution concept that is as closely related as possible

to Nash-equilibrium and at the same time is adequate to incorporate perceptual biases. This

ensures comparability to the case of rational consumers, which is indeed incorporated in our

setting. The approach to use Nash-equilibrium, despite agents having cognitive limitations

or non-standard preferences is also common to the literature (see, for example, Gabaix and

Laibson 2006, Eliaz and Spiegler 2011 and Bordalo et al. 2013). An interesting alterna-

tive to model perceptual biases would be to adopt an equilibrium notion that incorporates

systematically wrong beliefs of the consumers about firms’ advertising intensity.

10

by consumer conservatism (β < 1).12

4 The Monopoly Case

We start our analysis with the benchmark case of a monopoly. Hence, only

one firm, say A, is active whereas the other firm stays outside the market. We

assume that each consumer buys at most one unit of good A. A consumer of

type θ will buy one unit if θqA1 − pA ≥ 0 and has zero utility otherwise.

When deciding on the unit price at stage two, firm A anticipates that the

fraction θ − pA/qA1 of consumers will purchase good A if it chooses price pA.

The firm, thus, maximizes

pA ·

(

θ −pAqA1

)

,

which yields the solution p∗A = θqA1/2 and the optimal profit θ2qA1/4. Since

qA1 positively depends on sA (see (2)) and, hence, on aA, advertising increases

the posterior mean of the uncertain consumption quality, leading to a higher

monopoly price and higher monopoly profits.

At stage one, firm A decides on the level of advertising and solves

maxaA

E

[

θ2qA1

4

]

− c(aA)

=maxaA

θ2

4

(

αqA0 + βσ2A0 (qA0 + aA − aA − αqA0)

σ2ε + σ2

A0

)

− c(aA)

with E denoting the expectation operator with respect to qA and εA. Straight-

forward computations lead to the following result:

Proposition 1 In the monopoly case, firm A’s optimal advertising, a∗A, is

described byθ2

4β

σ2A0

σ2ε + σ2

A0

= c′(a∗A) . (3)

Proposition 1 shows that the stronger consumer conservatism (i.e., the

lower β), the less effective is advertising. As a consequence, a monopolist

decides to advertise less. Since this deterrent effect on advertising choices is

due to an attenuated impact of advertising on consumer decisions, we call it the

attenuation effect of consumer conservatism. In contrast, consumer optimism

does not impact advertising choices in the monopoly case. Although optimism

12See, e.g., the experimental findings by Offerman and Sonnemans (1998).

11

leads to a higher biased prior mean and therefore also a higher posterior mean,

it does not influence the (marginal) impact of advertising.

Furthermore, the proposition shows that optimal advertising decreases in

the magnitude of noise. Intuitively, with a less precise quality signal sA (i.e.,

a low value of 1/σ2ε), it becomes unattractive for the monopolist to invest in

advertising since consumers react little to the signal. Finally, the higher the

prior uncertainty about consumption quality (i.e., the higher σ2A0), the higher

is the optimal level of advertising.13 This effect is also intuitive. If, initially, the

uncertainty about the unknown consumption quality is large, by advertising

the firm can shift a lot of probability mass under the density of qA to the right.

5 The Competition Case

5.1 Equilibrium Behavior

In this section, both firms, A and B, are active and compete for consumers

via advertising at stage one and setting prices at stage two. For simplicity,

we concentrate on the case of combative advertising where total demand is

fixed and advertising only influences the allocation of consumers across firms.

Hence, we assume that each consumer buys exactly one unit of either good

A or B.14 We will see that a key determinant of firms’ advertising choices at

stage one and price choices at stage two is the relative perceived quality of the

goods. At stage one, the relative perceived quality, α(qi0 − qj0), depends on

consumer optimism and the expected true consumption qualities; at stage two,

it is given by qi1− qj1. For an easy understanding of the driving forces, we will

say that firm i is the prior market leader if it provides higher mean quality at

stage one (qi0 > qj0). Depending on advertising choices and resulting quality

signals, consumers update their quality perceptions before they purchase a

good. If a firm achieves a higher perceived quality after the advertising stage

(qi1 > qj1), we will call this firm the posterior market leader.

To determine the equilibria, we use backward induction and start by solving

the price competition game at stage two. Suppose, w.l.o.g., that firm i is the

posterior market leader. The resulting equilibrium prices are derived in the

13The same outcome does not necessarily hold for the competition case. See the note by

Grunewald and Krakel (2015).14In our setting, this corresponds to assuming that the utility of consuming no good is

negative infinity as done for example in Heidhues and Koszegi (2008).

12

appendix.15 They are given by

p∗i =(qi1 − qj1)(θ + 1)

3and p∗j =

(qi1 − qj1)(2− θ)

3, (4)

leading to equilibrium profits

π∗

i = Θ · (qi1 − qj1) and π∗

j =¯Θ · (qi1 − qj1) (5)

with Θ := (θ + 1)2/9 > (2 − θ)2/9 =:¯Θ. Due to its higher perceived quality,

firm i can charge a higher product price than firm j without losing too much

market share. The derivations in the appendix show that prices are strategic

complements, which is a typical finding for price competition. Altogether,

firm i chooses a high price in equilibrium and firm j reacts by choosing a high

price as well. Firm i’s comparative advantage yields higher expected profits

relative to j. However, both firms profit from a high spread between perceived

qualities, qi1 − qj1. Thus, a firm benefits from being the posterior market

leader, but if it is in the inferior position at stage two, it will prefer to be as

weak as possible.

At stage one, both firms decide on advertising. In the analysis, we first

derive the objective functions of the firms. In a second step, we character-

ize optimal advertising. Consider, for this purpose, the optimization problem

of firm A (we immediately face the problem of firm B by interchanging sub-

scripts). Firm A chooses aA to maximize

E[Θ · (qA1 − qB1)|qA1 > qB1] · P (qA1 > qB1) (6)

+E[¯Θ · (qB1 − qA1)|qA1 < qB1] · P (qA1 < qB1)− c(aA) ,

where P (qA1 > qB1) denotes the probability that A will be the posterior market

leader and E the expectation operator with respect to qi and εi. Moreover, we

define

Σi :=σ2i0

σ2ε + σ2

i0

(7)

as the weighted uncertainty about consumption quality i. Hence, Σi is a

measure of how strongly a consumer reacts to the value of the quality signals. If

initial quality uncertainty is small or noise is large, consumers will not respond

much to new information.

As explained above, the key variable characterizing advertising incentives

for firms is the difference in perceived qualities. This difference is determined

by stochastic and deterministic variables. For an easier comprehension of the

15See also Tirole (1988), 296–297.

13

problem, we disentangle the two parts of the spread such that qA1 − qB1 =

β · [δA − ΩA], with δA summarizing the stochastic elements and being defined

by

δA := ΣA · (qA + εA)− ΣB · (qB + εB) .

ΩA embraces the deterministic elements, with

ΩA :=

(

1

β− ΣB

)

αqB0 −

(

1

β− ΣA

)

αqA0

+ ΣB · (aB − aB)− ΣA · (aA − aA) .

Since any convolution of two normal densities again yields a normal density

(e.g., Ross 2010, pp. 35, 67–68), the composed random variable δA is normally

distributed: δA ∼ N(

µδA , σ2δA

)

with

µδA := ΣAqA0 − ΣB qB0 and σ2δA

:= ΣAσ2A0 + ΣBσ

2B0.

Let gA denote the density of δA and GA the corresponding cumulative distribu-

tion function. Consequently, firm A’s objective function (6) can be rewritten

as

Θ ·

∫

∞

ΩA

β (δA − ΩA) gA(δA) dδA −¯Θ ·

∫ ΩA

−∞

β (δA − ΩA) gA(δA) dδA − c(aA) .

For the characterization of the optimal advertising of the firms, note, first,

that there may exist interior solutions a∗A > 0 as well as corner solutions a∗A = 0.

Suppose that an interior equilibrium in pure strategies exists such that optimal

advertising a∗A is described by firms’ first-order conditions. Applying Leibniz’s

formula, the first-order condition of firm A can be computed as

βΣA ·[

Θ−(

Θ +¯Θ)

GA(ΩA)]

= c′(a∗A) . (8)

In any perfect Bayesian equilibrium, consumers derive their quality perceptions

from firms’ actual advertising intensity (ai = a∗i ). Nevertheless, as soon as a

firm deviates from the equilibrium level of advertising, the perceived quality

of the corresponding good is distorted. Overall, we obtain the following result:

Proposition 2 There will exist a unique pure strategy equilibrium (a∗A, a∗

B) if

for both firms

βΣ2i

(

Θ +¯Θ)

gi(µδi) < ¯c. (9)

The equilibrium is characterized by a corner solution a∗i = 0 if and only if

Gi(Ωi) ≥ ΘΘ+

¯Θ. Otherwise, firms advertise their goods with a∗i > 0 being

described by

βΣi

[

Θ−(

Θ +¯Θ)

Gi

((

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

)]

= c′(a∗i ) (10)

with i, j = A,B and i 6= j.

14

Proof. See Appendix.

Proposition 2 shows how firms’ advertising strategy depends on their prob-

ability of becoming the posterior market leader, 1 − Gi(·). If the probability

that firm i will be the posterior market leader is low enough (i.e., Gi(·) is large),

such that the condition for a corner solution is satisfied, firm i will prefer not

to advertise its good. By this behavior, firm i minimizes advertising costs at

stage one and – given that qi1 < qj1 is indeed realized – maximizes expected

profits from the price competition at stage two (see the above discussion of

(5)). However, firm i will engage in advertising if the probability of becoming

the posterior market leader is high. The optimal level of advertising is then

described by equation (10).

5.2 Consumer Optimism

We are interested in how the level of consumers’ perceptual biases influence

firms’ advertising behavior in equilibrium. In the following discussion, we

focus on interior solutions with a∗i > 0, which directly yields an intuition for

the corner solution as well.

As introduced above, α > 1 characterizes the level of consumer optimism

regarding uncertain consumption quality. Recall that optimism does not have

any influence on equilibrium advertising in monopoly. Equation (10) shows

that this is not true for the competition case, where α influences firm i’s

probability of becoming the posterior market leader. We immediately obtain

the following result:

Proposition 3 In any interior equilibrium, advertising of firm i increases

with consumer optimism if and only if (1− βΣi) qi0 > (1− βΣj) qj0.

According to (2), βΣm (m = i, j) describes how eagerly a consumer takes

newly arriving information into account.16 The counterpart 1−βΣm then cor-

responds to the stickiness of beliefs already held by consumers. Thus, the term

(1− βΣm) qm0 indicates how likely it is that firm m will be the posterior mar-

ket leader. If this value is larger for i than for j, stronger consumer optimism

will give firm i an even higher chance to outperform its competitor at stage

one (i.e., Gi(Ωi) decreases). As we can see from equation (5), if a firm antic-

ipates that it will become the posterior market leader, it is optimal to choose

high advertising to maximize the difference between the perceived qualities.

16The posterior mean can be written as qi1 = αqi0 + βΣi · (si − ai − α · qi0).

15

In other words, consumer optimism further encourages firm i in the given un-

even situation, thus boosting its motivation for advertising. The opposite will

be true for firm j. It anticipates a low probability of becoming the posterior

market leader (i.e., a high value of Gj(Ωj)). Consequently, stronger consumer

optimism further weakens j’s unfavorable position, which discourages j when

deciding on advertising. Note that, since α only affects Gj(Ωj), the intuition

for the influence of optimism on the possibility of a corner solution is exactly

the same as just described.

5.3 Consumer Conservatism

Next, we consider the impact of consumers’ perceptual biases on updating,

indicated by β. Recall that β < 1 describes the level of consumer conservatism,

i.e., consumers’ tendency to ignore newly arriving information that differs from

their prior beliefs about the consumption quality of the goods. In our modeling

approach, a high impact of conservatism is characterized by small values of β.

Importantly, equation (10) shows that in the competition case – contrary to

the monopoly case – changes in consumer conservatism involve a trade-off for

firms’ advertising decisions.

On the one hand, there is the attenuation effect as in the monopoly case.

Stronger consumer conservatism leads to less effective advertising and thus to

weaker incentives for firms to advertise their goods. This effect is described by

βΣi outside the expression in square brackets in (10). Since βΣi increases in

β, firm i will again reduce advertising in equilibrium if conservatism becomes

stronger (i.e., β decreases) and, hence, advertising less effective.

On the other hand, there is an additional competition effect since conser-

vatism influences the likelihood for each player to become the posterior market

leader. Firm i’s equilibrium condition (10) contains the probability that its

opponent becomes the posterior market leader, Gi. The competition effect is

described by the influence of β on this probability:

∂

∂βGi

((

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

)

(11)

=gi

((

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

)

α

β2(qi0 − qj0) .

The sign of this derivative, and thus the competition effect, will be positive if

and only if qi0 > qj0, that is, if and only if firm i is the prior market leader. In

this case, a higher degree of conservatism increases equilibrium advertising via

16

an enhancement of the probability that player i will also become the posterior

market leader.

The intuition for the competition effect of conservatism is the following. If

firm i is the prior market leader, a higher degree of conservatism will preserve

firm i’s favorable starting position more effectively. As a consequence, it is

more likely that firm i will also become the posterior market leader, which,

in turn, boosts firm i’s advertising incentives at stage one to maximize profits

(see (5)). If the competition effect is sufficiently strong and dominates the

familiar attenuation effect, firm i’s advertising will increase with conservatism.

The following result characterizes under which circumstances this is the case.

Proposition 4 Suppose qi0 > qj0. Firm i’s advertising will be increasing with

consumer conservatism (i.e., da∗i /dβ < 0) over some range of β if and only if

(α− 1)(Σiqi0 − Σj qj0)√

2π[

Σiσ2i0 + Σjσ2

j0

]

>1

2

Θ−¯Θ

Θ +¯Θ. (12)

Proof. See Appendix.

If firm i is the prior market leader, a higher degree of consumer conser-

vatism can lead to an increase in its advertising. As condition (12) shows,

this will be the case, if firm i’s position in the market is particularly strong.

This can either be the case due to sufficiently optimistic (α > 1) consumers or

sufficiently heterogeneous mean qualities qi0 > qj0. As stronger conservatism

preserves this leadership in the market it increases i’s advertising incentives.

Moreover, condition (12) shows that conservatism tends to have a positive im-

pact on advertising, the larger Θ +¯Θ or, conversely, the smaller Θ −

¯Θ. The

intuition behind this finding can best be seen by the expression −(

Θ +¯Θ)

Gi(·)

in equation (10), which characterizes optimal advertising in case of an interior

equilibrium: Recall that conservatism will have a positive influence on i’s ad-

vertising decision if it enhances the probability that firm i will be the posterior

market leader. If i wins the advertising competition at stage one, it will gain

Θ. If, however, i does not win it, additional advertising will yield a loss that

increases in¯Θ. Hence, increasing the probability of becoming the posterior

market leader has a double advantage for i, indicated by Θ +¯Θ.

The previous findings for the interior equilibrium directly reveal the impact

of conservatism on the corner solution. According to (11), ∂Gi/∂β will be

positive if and only if firm i is the prior market leader, or in other words:

Corollary 1 If qi0 > qj0 holds (does not hold), a higher degree of conservatism

17

will make the condition for a corner solution a∗i = 0 more difficult (easier) to

be satisfied.

The intuition for the result of Corollary 1 is similar to that for the compe-

tition effect of consumer conservatism. If firm i is the prior market leader, it

will benefit from a higher impact of conservatism since this ensures its lead.

Consequently, firm i has stronger incentives to choose positive advertising.

If, however, firm i starts from an inferior position at stage one, conservatism

will make it more difficult for it to get ahead of j such that firm i has fewer

incentives to become active in the advertising competition.

6 Welfare Analysis

In this section, we analyze the welfare implications of perceptual biases. Recall

that production costs are zero and advertising is purely combative, i.e., the

total demand is fixed and advertising does not influence the true consumption

utility from a consumed good. Perceptual biases influence the equilibrium

in two ways. First, they induce a different level of advertising and, second,

they change the market price. Since, in our setting, prices correspond to

utility transfers from consumers to firms, the overall welfare is not affected

when prices change. The welfare implications of changes in consumer biases

are thus solely determined by the corresponding changes in advertising. If

advertising increases, firms will bear higher costs and welfare decreases. The

welfare implications in the monopoly case thus directly follow from the effects

of perceptual biases on the monopolist’s equilibrium advertising:

Corollary 2 If there is a monopoly, consumer optimism will not affect wel-

fare, while stronger consumer conservatism will increase welfare.

For the competition case, welfare implications are more subtle. As argued

before, welfare consequences of perceptual biases correspond to the changes in

overall advertising. We start with the welfare analysis of consumer optimism,

given an interior equilibrium. If, initially, firm i is in an advantageous position

in the sense that (1− βΣi) qi0 > (1− βΣj) qj0, stronger consumer optimism

will increase i’s equilibrium advertising but decrease advertising of its oppo-

nent j (Proposition 3). The overall welfare implication of consumer optimism

depends on which of the two effects is dominant:

da∗idα

+da∗jdα

= −Ωβ

α

(

Θ +¯Θ)

[

Σigi(Ω)

c′′(a∗i )−

Σjgj(−Ω)

c′′(

a∗j)

]

18

with Ω :=(

1β− Σj

)

αqj0 −(

1β− Σi

)

αqi0. As the convolutions gi and gj have

identical variances and their means satisfy µδi = −µδj , we obtain gi(Ω) =

gj(−Ω) such that

sign

(

da∗idα

+da∗jdα

)

= sign

(

−Ω ·

[

Σi

c′′(a∗i )−

Σj

c′′(

a∗j)

])

.

Since c′′ is bounded, the following result holds:

Proposition 5 Suppose Σi¯c > Σj c. Welfare will decrease with consumer op-

timism if and only if (1− βΣi) qi0 > (1− βΣj) qj0.

The proposition shows that the effect of consumer optimism on welfare

depends on the market structure and the competing products. Consider, for

example, the case where (1− βΣi) qi0 > (1− βΣj) qj0. According to Proposi-

tion 3, i’s advertising increases and j’s advertising decreases with optimism. If,

additionally, the difference between the weighted uncertainties about product

qualities – as defined in (7) – is sufficiently large, i.e., Σi > Σj , advertising of

firm i will have a stronger impact on consumer behavior than that of firm j.

As a consequence, the increase in i’s advertising – as a response to consumer

optimism – exceeds the decrease in j’s advertising and welfare decreases. As

the condition Σi¯c > Σj c states, the result holds as long as marginal costs are

not too sensitive to increased advertising. In case of greatly changing marginal

costs, firm i might be deterred from increasing advertising simply because of

cost reasons. This effect works against the argumentation explained above.

If, however, advertising costs are quadratic, the counteracting cost effect will

vanish (as¯c = c).

Next, we consider the welfare implications of consumer conservatism. Sup-

pose that firm i is the prior market leader. From Section 5.3 we know that

the attenuation effect and the competition effect are both negative for firm j.

Hence, a stronger conservatism bias (smaller β) leads to less advertising by firm

j. Whether firm i behaves accordingly, depends on the strength of i’s initial

position. Whenever consumers rate both goods rather homogeneous, condition

(12) in Proposition 4 does not hold for firm i. In that situation, the competi-

tion effect is negligible so that the attenuation effect remains and makes firm

i also reduce advertising. As a consequence, overall welfare is unambiguously

enhanced. Thus, contrary to consumer optimism, consumer conservatism can

drive both firms’ equilibrium advertising in the same direction.

However, Proposition 4 also establishes that the two firms may react differ-

ently to increased conservatism if the competition effect is positive for the prior

19

market leader. In this case, the overall effect on welfare can even be negative.

To see this, consider an interior equilibrium with firm i being the prior market

leader and condition (12) being satisfied. Firm i thus increases advertising as

a response to stronger conservatism, whereas firm j reduces its advertising. By

using equation (10), we can derive the overall effect of a stronger conservatism

bias on advertising:

da∗idβ

+da∗jdβ

= Σi

Θ−(

Θ +¯Θ)

Gi(Ω)−(

Θ +¯Θ)

gi(Ω)αβ(qi0 − qj0)

c′′(a∗i )

+ Σj

Θ−(

Θ +¯Θ)

Gj(−Ω)−(

Θ +¯Θ)

gj(−Ω) αβ(qj0 − qi0)

c′′(

a∗j)

with Ω being defined as above. Recall that da∗i /dβ < 0 and da∗j/dβ > 0.

Which of the two dominates, depends on how strong firm i’s position as the

prior market leader is and on how strongly consumers attend to the advertising

signal of the two firms.

Proposition 6 Suppose qi0 > qj0 and Σi¯c > Σj c. If

(α− 1)Σiqi0√

2πΣiσ2i0

>1

2

Θ−¯Θ

Θ +¯Θ

(13)

holds, there exists a cutoff σj0 such that for all σj0 ≤ σj0 stronger conservatism

(i.e., smaller β) decreases overall welfare for some range of β.

Proof. See Appendix.

At first sight, the finding of Proposition 6 is surprising since conservatism

makes advertising less effective, which should lead to less advertising by both

firms in equilibrium and, therefore, an increase in welfare. The proposition

characterizes the conditions for the opposite outcome.17 Inequality (13) holds

if firm i’s position in the market is sufficiently strong, i.e., optimism is strong

or the quality of firm i’s good is large. For this case, Proposition 4 implies

that firm i’s advertising increases and firm j’s advertising decreases with con-

servatism. If, moreover, consumers tend to disregard new information about

firm j’s good (i.e., σj0 is sufficiently small), the impact of firm j’s advertising

on the perceived quality will be small. Since firm j’s influence is small any-

how, its advertising does not adapt as strongly given changes in conservatism.

Hence, the positive impact on welfare implied by the decrease in firm j’s ad-

vertising is dominated by the negative welfare consequences of the increase in

firm i’s advertising. This dominance results in an overall welfare decrease due

to stronger consumer conservatism.

17The intuition for condition Σi¯c > Σj c is the same as in Proposition 5.

20

7 Conclusion

In this paper, we consider a two-stage model in which two firms decide on

persuasive advertising for their goods with uncertain consumption qualities at

stage one and set product prices at stage two. We analyze how perceptual

biases influence firms’ advertising decisions and overall welfare. Consumer op-

timism has an impact on the posterior mean of a good’s uncertain consumption

quality and, thus, on a firm’s price policy. However, it will not influence adver-

tising if firms operate as monopolists in separate markets. Therefore, welfare

is not influenced by optimism in the monopoly case. If firms compete in the

same market, advertising behavior will crucially depend on consumers’ qual-

ity perceptions. A firm’s equilibrium advertising will increase with consumer

optimism if and only if the firm’s good is favored by the consumers ex ante.

The opponent’s advertising will decrease with optimism and welfare will be

enhanced if this decrease exceeds the increase in advertising of the favored

firm.

Consumer conservatism influences advertising behavior in both the monopoly

and the competition case. The impact of conservatism is twofold. On the one

hand, conservatism makes advertising less effective, which reduces a firm’s ad-

vertising incentives and increases welfare due to saved advertising expenses.

On the other hand, conservatism preserves the initially advantageous market

position of a firm, boosting the advertising incentives of that firm and, hence,

decreasing welfare. We show that the second effect can become dominant if

consumers associate especially high quality with one particular firm.

At first glance, the finding that conservatism may yield excessive advertis-

ing if there is a clear market leader seems puzzling against the background of

both the insights from the advertising literature and the work on contests. In-

tuitively, one would expect that firms would strongly invest in artificial product

differentiation via advertising if goods do not differ greatly from the consumers’

perspective and advertising is rather effective.18 Moreover, from contest the-

ory we know that competition is strong (weak) if the contestants are quite

homogeneous (heterogeneous) and use an effective (ineffective) contest success

technology.19

In our setting, the relative success of advertising at stage one determines

18Bagwell (2007, p. 1724): ”If the real differences between brands are modest, then

combative advertising may be excessive.”19See the early contest survey by McLaughlin (1988), 243–247, and Schmalensee (1976,

1992) on advertising contests.

21

who becomes the posterior market leader. Thus, the advertising competition

in our model is similar to a contest, with the winner prize (loser prize) being

given by the expected profit of the posterior market leader (the other firm).

However, our setting crucially differs from a pure contest, in which prizes are

exogenously given at the time players enter the contest. Typically, they have

either been fixed by the contest designer or they are completely exogenous as,

e.g., in a rent-seeking contest. In our model, prizes endogenously depend on

the actions taken by the firms during the contest so that large heterogeneity

and a low degree of effectiveness of advertising do not contradict high powered

advertising incentives. In particular, the winner prize will be larger the more a

firm invests. Hence, if there is a clear prior market leader and if its position is

persistent due to conservative consumers, it will engage in intense advertising.

Appendix

Derivation of the market equilibrium

We start by constructing the demand functions of the consumers for given

prices pi and pj and given perceived qualities qi1 and qj1. Suppose w.l.o.g that

the perceived qualities satisfy qi1 > qj1. The expected utility of a consumer k

of type θk buying good i is thus

uk = θkqi1 − pi.

Since we assume that the market is fully covered, consumer k will buy good i

if and only if θkqi1 − pi ≥ θkqj1 − pj. Hence, the demand for the good of firm

i is given by

Di(pi, pj) =

θ −pi−pjqi1−qj1

if pi ≥ pj + (θ − 1)qi1 − qj1)

1 otherwise.

The maximizer of the profit function piDi(pi, pj), given pi ≥ pj + (θ− 1)(qi1 −

qj1), is

pi =pj + (qi1 − qj1)θ

2.

This in turn will be larger than the cutoff for the demand function if

pj + (qi1 − qj1)θ

2> pj + (θ − 1)(qi1 − qj1) ⇔ (qi1 − qj1)(2− θ) > pj.

Since the objective function is concave, the best response of firm i is given by

pi =

pj+(qi1−qj1)θ

2if (qi1 − qj1)(2− θ) > pj

pj + (θ − 1)(qi1 − qj1) otherwise.

22

The demand for the good of firm j is given by

Dj(pj, pi) =

pi−pjqi1−qj1

− (θ − 1) if pj ≤ pi − (θ − 1)(qi1 − qj1)

0 otherwise.

The maximizer of firm j’s profit function for the situation where pj ≤ pi− (θ−

1)(qi1 − qj1) is

pj =pi − (qi1 − qj1)(θ − 1)

2.

This in turn will be smaller than the cutoff for the corresponding demand

function if

pi − (qi1 − qj1)(θ − 1)

2≤ pi− (θ−1)(qi1− qj1) ⇔ pi ≥ (θ−1)(qi1− qj1).

If the maximizer is larger than zero, firm j will price its good accordingly. For

pi ≤ (θ− 1)(qi1 − qj1), firm j does not get any share of the market for positive

prices. The best response of firm j is, hence,

pj =

pi−(qi1−qj1)(θ−1)

2if pi ≥ (θ − 1)(qi1 − qj1)

pj ∈ ℜ+ otherwise.

In equilibrium, the best-response functions intersect. Note that there is

no equilibrium in which firm j is indifferent between prices. Firm j will be

indifferent only if pi ≤ (qi1 − qj1)(θ − 1). However, the best-response function

of player i is above this value for any positive price pj. Moreover, the best-

response functions can only intersect once, since the slope of pi in pj is less

than 1 and the slope of pj in pi is12(i.e., it is 2 in the (pi, pj)-plane). Hence,

we are looking for an interior equilibrium in which

2pj + (qi1 − qj1)(θ − 1) =pj + (qi1 − qj1)θ

2

holds. As a result, we get equilibrium prices and profits for both firms:

pi =(qi1 − qj1)(θ + 1)

3pj =

(qi1 − qj1)(2− θ)

3

πi =(qi1 − qj1)(θ + 1)2

9πj =

(qi1 − qj1)(2− θ)2

9.

Proof of Proposition 2

Inequality (9) describes a sufficient condition for strict concavity of i’s objective

function, which guarantees existence of pure-strategy equilibria. It is obtained

23

from i’s second-order condition assuming that it holds in the worst case. Here,

we use the fact that the density gi attains its maximum at µδi .

Firm i will choose the corner solution a∗i = 0 if the derivative of its objective

function is negative at ai = 0. From (8) we know that this is the case exactly

if

Θ−(

Θ +¯Θ)

Gi(Ωi) < 0 (14)

with

Ωi =

(

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

since aA = a∗A and aB = a∗B in equilibrium. Hence, if Gi(Ωi) exceeds ΘΘ+

¯Θ,

i.e., if Ωi becomes large, inequality (14) will be satisfied. Note that Ωi can

indeed become arbitrarily large, for example if β is small and qj0 > qi0. If the

condition for a corner solution does not hold, an interior solution will exist

being described by equation (10).

Proof of Proposition 4

We show that inequality (12) is a sufficient and necessary condition for adver-

tising to increase with conservatism in any equilibrium. An interior equilibrium

is given by

βΣi

[

Θ−(

Θ +¯Θ)

Gi

((

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

)]

= c′(a∗i ) . (15)

There is a corner solution whenever the left-hand side of this equation is neg-

ative. Differentiating the left-hand side of equation (15) with respect to β

yields

Σi

[

Θ−(

Θ +¯Θ)

Gi

((

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

)

(16)

−(

Θ +¯Θ)

gi

((

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

)

α

β(qi0 − qj0)

]

.

Taking the second derivative with respect to β, we get

Σi

(

Θ +¯Θ)

gi

((

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

)

(qj0 − qi0)α

β2

+Σi

(

Θ +¯Θ)

gi

((

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

)

(qi0 − qj0)α

β2

−Σi

(

Θ +¯Θ)

g′i

((

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

)

(qi0 − qj0)2 α

2

β3

= −Σi

(

Θ +¯Θ)

g′i

((

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

)

(qi0 − qj0)2 α

2

β3.

24

This expression will be larger than zero if and only if

g′i

((

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0

)

< 0

⇔

(

1

β− Σj

)

αqj0 −

(

1

β− Σi

)

αqi0 > qi0Σi − qj0Σj

⇔α

β(qi0 − qj0) < (α− 1) (Σiqi0 − Σj qj0) . (17)

To show that (12) is a sufficient condition, suppose that it is fulfilled. Hence,

the following is given:

(α− 1)(Σiqi0 − Σj qj0)√

2π[

Σiσ2i0 + Σjσ2

j0

]

>1

2

Θ−¯Θ

Θ +¯Θ.

Consequently, (α− 1)(Σiqi0−Σj qj0) > 0. Since qi0 > qj0, there exists a β such

that for all β < β inequality (17) is not fulfilled and for all β > β it is fulfilled.

At β the term inside Gi, gi and g′i is exactly the mean µδi . Thus, the left-hand

side of equation (15) is positive at β and the equilibrium is indeed interior at

this point. Hence, da∗i /dβ is falling up to some level and increasing afterward.

We conclude that da∗i /dβ is quasi-convex in β with minimum at β. The value

of (16) at β is given by

Σi

1

2

(

Θ−¯Θ)

−(

Θ +¯Θ)

α

β(qi0 − qj0)

√

2π[

Σiσ2i0 + Σjσ2

j0

]

. (18)

This minimum will be smaller than zero if and only if

(α− 1)(Σiqi0 − Σj qj0)√

2π[

Σiσ2i0 + Σjσ2

j0

]

>1

2

Θ−¯Θ

Θ +¯Θ,

which is given by assumption.

Next we show that inequality (12) is also a necessary condition. For this ar-

gument suppose that inequality (12) is not fulfilled. First, inequality (12) may

not be fulfilled since (α− 1)(Σiqi0 − Σj qj0) < 0. As a consequence, inequality

(17) implies that da∗i /dβ is falling for each β. However, for β → ∞ equation

(16) implies that da∗i /dβ is positive for the case of (α − 1)(Σiqi0 − Σj qj0) < 0

and thus it has to be positive throughout. Second, (α− 1)(Σiqi0 −Σj qj0) may

be positive but condition (12) is not satisfied. In this case, da∗i /dβ has an

interior minimum. This minimum must be positive though since its sign is

given by the sign of (18). Finally, da∗i /dβ is clearly zero in any corner solution.

Hence, there can neither exist a corner solution nor an interior equilibrium

with da∗i /dβ > 0, which concludes the proof.

25

Proof of Proposition 6

By assumption, (13) is satisfied, which implies that α > 1. Recall that

Σj = σ2j0/(σ

2ε + σ2

j0), which is increasing in σj0. As the left-hand side of (12)

is monotonically decreasing with σj0, (13) also implies that (12) holds for all

σj0 up to some threshold σj0. Hence, we know from the proof of Proposition

4 that for all σj0 < σj0, there exists β = β such that (17) is fulfilled with

equality. At this point, da∗i /dβ is negative and da∗j/dβ is positive. The overall

welfare effect at β = β is given by

da∗idβ

+da∗jdβ

=

Σi

c′′(a∗i )

1

2(Θ−

¯Θ)−

(

Θ +¯Θ) (α− 1)(Σiqi0 − Σj qj0)√

2π[

Σiσ2i0 + Σjσ2

j0

]

+Σj

c′′(

a∗j)

1

2(Θ−

¯Θ) +

(

Θ +¯Θ) (α− 1)(Σiqi0 − Σj qj0)√

2π[

Σiσ2i0 + Σjσ2

j0

]

.

Since c′′ is bounded from below and above with c′′ ∈ [¯c, c] this expression will

be definitely smaller than zero if

Σi¯c

1

2(Θ−

¯Θ)−

(

Θ +¯Θ) (α− 1)(Σiqi0 − Σj qj0)√

2π[

Σiσ2i0 + Σjσ2

j0

]

+ Σj c

1

2(Θ−

¯Θ) +

(

Θ +¯Θ) (α− 1)(Σiqi0 − Σj qj0)√

2π[

Σiσ2i0 + Σjσ2

j0

]

< 0. (19)

Moreover, inequality (19) also holds if σj0 becomes small. As (19) is monotone

in σj0, there exists a cutoff σj0 such that for all σj0 ≤ σj0 inequality (19) is sat-

isfied. Overall (12) and (13) are jointly satisfied for all σj0 < minσj0, σj0 =:

σj0. In this case, overall welfare decreases with conservatism at β = β, which

also holds in a surrounding of β.

26

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