Persuasive Advertising with Sophisticated but

Impressionable Consumers∗

Dominique Olié Lauga†

February 2010

Abstract

I propose a model in which consumers base their purchasing decisions upon their recollections

of product quality, and in which firms can use persuasive advertising to change these recollections.

Although consumers are aware that such advertising has occurred when updating their beliefs, they

cannot prevent their memories from being affected. I show that persuasive advertising may be used

even though consumers are fully aware of it, and that persuasive advertising does not necessarily

signal high-quality products. The model is extended to incorporate informative advertising in which

firms reveal some verifiable information. When firms choose between persuasive and informative

advertising, persuasive advertising may block the full unraveling of information.

Keywords: advertising, persuasion, memory, postexperience advertising

∗I am deeply indebted to Abhijit Banerjee, Glenn Ellison, and Elie Ofek for time, advice, and encouragement. I

thank Antara Dutta, Sara Ellison, Xavier Gabaix, Sergei Izmalkov, Paul Joskow, Nancy Rose, Jean Tirole, and seminar

participants at Babson College, Kellogg School of Management Northwestern University, Massachusetts Institute of

Technology, Paris School of Economics, Rady School of Management University of California San Diego, Simon Graduate

School of Business University of Rochester, University College London, and the 2nd Workshop on the Economics of

Advertising and Marketing (2009) for helpful comments.†Rady School of Management, University of California, San Diego. Email: dlauga@ucsd.edu.

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1 Introduction

The goals of advertising campaigns are diverse; they vary from creating awareness for a new product

to affecting repeated purchases. To achieve these goals, firms decide not only how much to advertise

but also how to advertise their products. Firms may select the content of their advertisements to

convey information about their product quality directly to consumers. Firms may also choose un-

informative advertisements. One explanation behind uninformative advertising is to signal product

quality to consumer by "burning money."1 The message contents of advertisements are, in that case,

irrelevant and their only effect is to reveal information indirectly to consumers. However, the content

of uninformative advertisements might have a direct effect on consumers in itself, through persua-

sion for example. A vast literature documents and uses cognitive psychology to analyze persuasion

in advertising.2 Firms may therefore select uninformative advertising because of its direct effect on

consumers through the persuasive elements of its message. This strategic decision may also convey

information to consumers indirectly and signal the product quality. The focus of this paper is to

address the following questions about persuasive advertising. When consumers understand how per-

suasion works, will persuasive advertising ever be optimal for a firm? If the answer to the previous

question is yes, will persuasive advertising always be associated with high-quality products? Finally,

when firms can select persuasive or informative advertising, does one type of advertising dominate

the other one?

I propose a model in which consumers are uncertain about the quality of a product sold by a

monopolist. At the time of purchase, consumers do not know the exact product quality but try to

assess it using their prior beliefs about quality. These beliefs can be, for instance, the recollections

that consumers have of the product, which are generated by all past interactions between consumers

and the product, including past personal experiences, word-of-mouth, official consumer reports, and of

course advertisements. Different prior beliefs about quality are generated by some consumers having

1Nelson (1974) describes the idea of advertising signaling quality by dissipating part of the profits and Kihlstrom

and Riordan (1984) and Milgrom and Roberts (1986) formalize the money-burning effect.2See Braun (1999) and Braun-Latour and Zaltman (2006) for studies on how memory can be changed by advertising;

and Friestad and Wright (1995) about the similarity of lay people’s and researchers’ beliefs about persuasive advertising.

2

positive recollections and some consumers having negative recollections. The likelihood of having a

positive recollection is an increasing function of quality thus recollections reveal some information

about product quality. Furthermore, the firm may engage in persuasive advertising to increase the

number of positive recollection. Specifically, in this paper, persuasive advertising is assumed to change

the distribution of prior beliefs that consumers have about product quality.3 After being exposed to an

advertisement, consumers are, for example, more likely to remember the positive aspects of their last

experience with the product than they would have remembered without the advertisement (further

examples are provided in Section 2.2). In addition, consumers are sophisticated as they are fully

aware that the firm can choose to engage in persuasive advertising which increases the likelihood of

positive recollections.4 They know whether they have seen persuasive advertisements and that they

might be affected by them. However, they do not know if a positive recollection is the consequence of

persuasive advertising or true experience as, of course, they do not know the recollection they would

have had without persuasive advertising.

This paper shows that consumers cannot fully undo the effects of advertising, and engaging in

persuasive advertising might be optimal even with sophisticated consumers. In addition, I show that

persuasive advertising is not necessarily a signal of high quality. Under some market conditions, the

fact that consumers fully understand the effects of advertising allows firms to signal a high quality

by choosing a high price and by not engaging in advertising. Indeed, if a firm chooses not to engage

in persuasive advertising, consumers with positive signals know that their signals come only from

the intrinsic product quality and were not artificially improved by a marketing campaign. Consumers

with negative signals do not buy and, therefore, the product quality has to be high enough to generate

substantial positive signals and make choosing a high price profitable. As a result, choosing a high

price and not engaging in advertising can help signal a high-quality product. Depending on market

conditions and the degree of advertising persuasion, persuasive advertising can signal an intermediate-

3By contrast, persuasive advertising enters directly in the utility function in Becker and Murphy (1993), creates loyal

customer in Baye and Morgan (2009) and Bala and Bhardwaj (2010), and refers to a shift in the demand of consumers

in Bagwell (2007), Chen, Joshi, Raju, and Zhang (2009), Amaldoss and He (2009), and Wu, Chen, and Wang (2009).4See Mehta, Chen, and Narasimhan (2008) for an empirical study that takes into account that "consumers perceive

ads to be biased, because they realize that ads are coming from a partisan source."

3

quality range, a high-quality range, or even a low-quality range.

Similarly, persuasive advertising is not naturally associated with high prices. Two factors drive the

relationship between persuasive advertising and prices. First, firms use both price and advertising to

signal their quality. therefore, when advertising does not signal high quality, it is not associated with

high prices. Second, keeping everything else constant, exposure to persuasive advertising lowers the

expectation of product quality which reflects the consumers willingness to pay.5 Indeed, a positive

signal with persuasive advertising does not carry as much good news as a positive signal without

persuasive advertising. The latter signal was generated only through the intrinsic product quality

while the former could be spurious and could have been negative without persuasive advertising.

After characterizing, without specifying any functional form, the full set of equilibria and their

properties, I study the case of a uniform distribution of qualities coupled with a particular persuasion

technology. I find that when the degree of persuasion is high, advertising does not signal high quality

as, with a high degree of persuasion, persuasive advertising has a homogenizing effect. Since firms do

not look that different after engaging in persuasive advertising, advertising cannot signal high-quality

firms. Similarly, if the range of quality is small, quality is not signaled through advertising as the

differential effect of advertising on firms is too small to allow quality signaling.

Finally, I introduce informative advertising to study how firms use different types of advertising

strategically. Informative advertising is based on a claim which is a hard/verifiable piece of informa-

tion about the product.6 The monopolist chooses now between persuasive advertising, informative

advertising, or no advertising. I show that the option of engaging in persuasive advertising might

block the full unraveling of information in equilibrium. Some firms, instead of releasing some informa-

tion in informative advertisements, choose persuasive advertising. Moreover, high-quality products

are not always promoted with the same type of advertising. Specifically, persuasive advertising can

signal a higher or lower quality product than a product promoted with informative advertising.

5Anand and Shachar (2005) also find that advertising exposure can have a negative effect when the beliefs of a

consumer are worsened by exposure to an advertisement revealing that the attributes of a promoted product are below

his average match. The negative effect arises from what consumers learn from the content of an advertisement, whereas,

in the current paper, the negative effect comes from the inferences of sophisticated consumers.6See Tirole (1988) for a discussion about hard versus soft information.

4

Literature Review

This paper contributes to the literature on advertising with direct behavioral effects on consumers.

Previous analyses vary in the degree of consumer rationality they assume. Consumers can either be

irrational and not think about the decisions of firms (Schmalensee (1978)), or be rational and form

their beliefs by taking into account that firms behave strategically (Gabaix and Laibson (2006) and

Shapiro (2006)). The current paper falls in the latter category, and is related to Shapiro (2006)

which studies advertising in a limited memory set-up. Specifically, that paper examines how the

level of advertising expenditure is affected by consumer experience and by product quality when

consumers do not remember whether they have been exposed to advertising. By contrast, I analyze

here the situation when consumers are aware of being exposed to persuasive advertising, and use this

information to form their beliefs about product quality. This allows me to examine how a firm choose

strategically its advertising type, as well as its price level.

The current paper also adds to three other lines of research in advertising, namely its signaling

ability, its content, and the different types of advertising. First, my findings that persuasive advertising

does not necessarily signal high quality and is not always associated with high prices are in contrast

with the series of papers in which advertising is simply a money-burning device (Nelson (1974),

Kihlstrom and Riordan (1984), and Milgrom and Roberts (1986)).7 In these papers, the signaling

power of advertising stems from the fact that a firm with a large potential profit is willing to sacrifice

some of it to signal its type, while a firm with a smaller profit would not do so. Recent empirical

studies also disagree on whether advertising signals quality or not. Erdem, Keane, and Sun (2008) find

that firms use both price and advertising to signal quality, while Clark, Doraszelski, and Draganska

(2009) obtain that advertising has no effect on quality perception.

Second, the current paper is related to the literature on advertising content. Anand and Shachar

(2009) study how informative advertising can convey indirect information by signaling product at-

tributes in addition to the direct information included in the messages. They found that consumers

use the content of advertising to form their out-of-equilibrium beliefs, but ignore the ad content on

the equilibrium path. In my paper, however, the persuasive elements of the advertisements matter

7For a review of the literature on advertising as a signaling device, see Bagwell (2007).

5

on and off the equilibrium path. Anderson and Renault (2006) also study ad content and find that

a monopolist might not reveal all its information when consumers face a search cost.8 The current

paper suggests a different explanation: firms may not reveal all their information as they choose to

engage in another type of advertising. Finally, Shaffer and Zettelmeyer (2009) analyze comparative

advertisements and how in-store displays can reinforce their messages, while Anderson and Renault

(2009) show that welfare goes down due to comparative advertising.

Third, empirical studies have tried to disentangle the different effects of advertising. Studying the

yogurt industry, Ackerberg (2001) and Ackerberg (2003) find that consumers are primarily affected by

the informative elements of the advertisements over the persuasive effects. Finally, the current paper

focuses on the persuasive and informative elements in advertising even if an advertising campaign

entails more than that, for example targeting (Iyer, Soberman, and Villas-Boas (2005)).

The rest of the paper proceeds as follows. Section 2 describes the set-up of the model and provides

some support for its assumptions. Section 3 derives the equilibrium when the firm cannot advertise.

Section 4 characterizes the equilibrium of the model with persuasive advertising. Section 5 studies

the case of a uniform distribution of qualities and a particular persuasion technology. Section 6

characterizes the equilibrium with persuasive and informative advertising. Section 7 concludes the

paper.

2 The model

In this section, I first present the model set-up and then explain in more detail what quality represents,

how signals are generated, and how an advertising campaign can be persuasive.

2.1 Model Set-up

A monopolist sells a product of quality q, drawn from a distribution with probability distribution

function f on the support [q, q] ⊂ (0, 1]. There is a unit-mass of consumers who demand at most

one unit of the product. If consumers buy a product of quality q and price p, their utility is q − p,8See Koessler and Renault (2009) for a condition about full disclosure when firms do not know the consumers’ tastes.

6

their outside option being normalized to zero. Consumers do not know the product quality, but

they know the quality distribution f and, furthermore, each consumer receives a signal s about the

product quality. These signals are either positive, s = 1, or neutral/negative, s = 0. The probability

of positive signals, α(q) ≡ Pr(s = 1| q), is an increasing function of the true product quality. Asmore consumers have positive signals about the product when the product quality is high, signals

reveal information about the underlying product quality. In addition, the worst quality is assumed

to generate few positive signals. Specifically, the worst-quality firm prefers selling to all consumers at

the lowest price p = q over selling only to consumers with positive signals (i.e., α(q) consumers get a

positive signal) at p = E[q|s = 1], the expected quality when consumers have a positive signal.

Assumption 1 The distribution of qualities satisfies q > α(q)E[q|s = 1].

The firm cannot communicate its product quality directly to consumers. It can, however, run a

persuasive advertising campaign to increase the probability that consumers receive positive signals

about the product. After a persuasive advertising campaign, the probability of positive signals is

ϕ(q) ≡ Pr(s = 1|P, q), which is larger than α(q), the probability of positive signals without advertising(ϕ(q) ≥ α(q)).9 ϕ is also an increasing function of q so that better products generate more positive

signals even after an advertising campaign. For example, if persuasive advertising converts a share

θ of negative signals into positive signals, then the probability of positive signals becomes ϕ(q) =

α(q) + θ(1 − α(q)). As the focus of this paper is on the demand-side effects, i.e., how firms can use

the direct effect of advertising on consumers to signal their quality, a persuasive advertising campaign

covers the full market at no cost.10 Similarly, marginal costs are assumed to be constant and without

loss of generality, equal to zero.

Finally, consumers are fully aware of the effects of persuasive advertising. They observe whether

the firm engages in persuasive advertising and form Bayesian beliefs about the product quality.

9 In the conditional probabilities, P means conditional on the firm engaged in persuasive advertising while ∅ meansconditional on no advertising (α(q) = Pr(s = 1|∅, q)).10 Introducing small fixed costs would not change the nature of the results. A more detailed discussion about costs

can be found at the end of Section 4.

7

The timing of the model is illustrated in Figure 1. After learning its product quality, the firm

chooses its advertising campaign and its price. Consumers see the advertising campaign if there

is one, then they receive their signal, observe the price and decide whether to buy. I analyze pure-

strategy perfect Bayesian Nash equilibria, with a tie-breaking rule such that a firm indifferent between

advertising and no advertising picks no advertising.

Before analyzing the model, I give more detail and provide some support for the different assump-

tions of the model in the next section.

time

The monopolistlearns q.

The monopolistchooses

advertisingand price.

Consumers seeadvertising(if any).

Consumers learntheir signals, the price,

and decide whether to buy.

time

The monopolistlearns q.

The monopolistchooses

advertisingand price.

Consumers seeadvertising(if any).

Consumers learntheir signals, the price,

and decide whether to buy.

Figure 1: Time line of the model.

2.2 Motivation

In this paper, product quality refers to the probability that the good delivers a satisfactory experience

when it is consumed. The level of satisfaction may vary for several reasons. First, consumers do

not always use the product in the same situations. In the case of cough medicine, for example,

non-drowsiness is not crucial if the medicine is taken before going to bed whereas it involves safety

issues if taken before driving or operating machinery. Secondly, consumers have heterogeneous needs.

For example, some people take allergy drugs to treat a specific allergy, while others need help for all

possible sources of allergy. Finally, the quality of some goods, such as restaurant meals, varies from

one production batch to another.

Another assumption of this paper is that firms cannot communicate their product quality directly.

This could be the case because the information is too difficult to process for a non-expert consumer

and/or the information would not fit into the format of an advertisement or on the package of the

good. On the other hand, consumers might have used the product in the past or might have talked

to someone who has some experience with it. They have a prior about the product at the time of

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purchase, which is captured in the model by the signal consumers have about the product quality.11

One example based on a limited-memory assumption, is that consumers remember whether their last

experience with the product was satisfactory, in that case α(q) = q. The origin of the signal can be

more complex. The signal could be the memory of one experience randomly drawn from all the past

experiences, weighting them differently depending on whether the experience was good or bad, how far

in the past it happened, and on their ordering (first and last experiences might be more salient). The

effects of confirmation bias, namely the fact that consumers put more weight on experiences confirming

their first experience, can be incorporated.12 In addition, the firm might be able to interfere with

these signals by engaging in persuasive advertising, which increases the probability that a consumer

receives a positive signal about the product.

The characterization of how advertising affects consumers is consistent with behavioral research

done in marketing and psychology. Braun (1999) and Braun, Ellis, and Loftus (2002) demonstrate

and cite many studies that support the effectiveness of post-experience advertising, namely, exposure

to advertising after the last time a product was consumed. Their papers show that the memory of

consumers can be altered by advertisements viewed after they have consumed a product, even after

an unpleasant experience. In these studies, recollections of products are closer to what advertisements

claim products are than to the true experiences as remembered by the control group. Post-experience

advertising might create suggested experiences. For example, if consumers mostly remember whether

their last experience with a product was positive, their experience does not have to stem from actually

using the product themselves. It could simply be a suggested experience as consumers cannot distin-

guish between memories arising out of suggested experiences from memories generated by personal

experiences. Marketers can therefore generate more positive memories than what products would

generate by themselves. Another way advertising works is mood congruence memory, which means

that people more easily remember memories that match their current mood (Teasdale and Russell

(1983)). Product advertising may activate mood elements, and from that may also activate memories

11See Boatwright, Kalra, and Zhang (2008) about consumers forming product evaluations in the case of experience

attributes.12 In Iyer and Kuksov (2010), the signals that consumers get about the product quality depend on the true product

quality and also on the affects of the consumers at the time of purchase.

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of the product sharing the same mood elements. For example, when Nescafé advertisements were

centered around "Making time for friends...," positive memories about drinking coffee while spending

time with friends might have been retrieved more frequently. Finally, state-dependent learning could

also play a role in persuasive advertising. Eich, Weingartner, Stillman, and Gillin (1975) show that a

memory is more easily retrieved if one is in the same emotional or physical state as when the memory

was formed. The memories about a drug treating depression or migraine can be very different de-

pending on whether people are currently depressed or having a migraine. Therefore, marketers could

change recollections about a drug by helping potential consumers remember how they feel when they

are sick.

Finally, I briefly present a model providing some micro-foundations for post-experience advertising.

First, consumers use the product and their experience is either satisfactory or neutral, and q is the

probability of having a satisfactory experience. Then time elapses and consumers might forget about

their satisfactory experience. At the time of the next purchase, α(q) consumers remember a positive

experience, while 1− α(q) consumers have a neutral memory as they either had a neutral experience

or they forgot about their satisfactory experience. In that context, persuasive advertising serves as

a reminder for consumers. It reminds them why their experience was satisfactory and decreases the

number of consumers forgetting.13

The next section describes the effects of persuasive advertising in more detail.

2.3 Positive and Negative Effects of Persuasive Advertising

At this point, one might ask why a firm would forgo persuasive advertising as it is free and increases

the number of consumers with a positive prior about the product. Before answering that point, I first

present an assumption about the persuasive advertising technology and discuss its implications.

13 In this example, advertising helps consumers remember their own experiential level of satisfaction. Depending on

one’s interpretation of the results on post-experience advertising, it could also suggest a positive memory to someone

who had a negative experience. It is easy to imagine that a consumer had a negative experience because the last time

he used the product he was under a lot of stress and nothing seemed pleasant. An ad depicting a nice experience might

show him what he could experience when consuming the product. In the end, that consumer might remember the

positive suggested memory over the negative experienced one.

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Assumption 2 ϕ(q)α(q) and

1−ϕ(q)1−α(q) are non-increasing.

The first part of Assumption 2 means that the share of positive signals after persuasive advertis-

ing relative to the positive signals generated solely by the product does not become larger as quality

increases. Similarly, the second part means that the share of remaining negative signals after per-

suasive advertising relative to the negative signals generated by the product does not become larger

as quality increases. In particular, this assumption is satisfied in the previous example in which ad-

vertising converts a fixed share of negative signals into positive signals. Assumption 2 implies that

holding the signal constant, persuasive advertising decreases the average consumer willingness to pay,

i.e. E[q|P, s, q1 ≤ q ≤ q2] ≤ E[q|∅, s, q1 ≤ q ≤ q2].14 I will refer this as the negative direct effect

of persuasive advertising. Sophisticated consumers understand that getting a positive signal with

persuasive advertising conveys less good information about the quality than getting a positive signal

without persuasive advertising as some positive signals result from the marketing strategy and not the

product quality. Overall, the negative direct effect of persuasive advertising has to be compared with

its positive direct effect, which is the generation of more consumers with positive signals, translating

into a demand expansion if the firm sells only to consumers with positive signals. These two effects are

illustrated in Figure 2. Firstly, as consumers respond to persuasive advertising, the share of consumers

with positive signals increase with persuasive advertising (arrow pointing to the right in Figure 2).

Secondly, consumers are sophisticated and take into account that their signals might be changed by

persuasive advertising. Thus, their willingness to pay decreases with persuasive advertising (arrows

pointing to the left in Figure 2). In the end, a firm may choose to engage in persuasive advertising if

the positive effect dominates, i.e., selling to more consumers at a lower price is more profitable than

selling to fewer consumers at a higher price.

Furthermore, advertising does not change consumers’ preferences as, with or without persuasive

advertising, consumers have the same ex-ante expected quality E[q] = Es[E[q|∅, s]] = Es[E[q|P, s]].Persuasive advertising changes the dispersion around the mean, i.e. the variance. In the above exam-

ple, where persuasive advertising converts a fixed share of negative memories into positive memories,

14Conditional on signal s and q ∈ [q1, q2], the distribution of q with no persuasive advertising first-order stochasticallydominates the distribution with persuasive advertising: F (q|∅, s, q1 ≤ q ≤ q2) ≤ F (q|P, s, q1 ≤ q ≤ q2).

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persuasive advertising reduces the variance.

Share of Consumers

E[q|P, s = 0] E[q|∅, s = 1]E[q|∅, s = 0] E[q] E[q|P, s = 1]Expected Quality

Impressionable

Sophisticated

Sophisticated

With Persuasive Advertising

With No Advertising

Share of Consumers

E[q|P, s = 0] E[q|∅, s = 1]E[q|∅, s = 0] E[q] E[q|P, s = 1]Expected Quality

Impressionable

Sophisticated

Sophisticated

With Persuasive Advertising

With No Advertising

Figure 2: Distribution of the expected quality with and without persuasive advertising. The difference

in the two distributions can be analyzed in two steps. First, the share of consumers with s = 1 increases as a direct

effect of persuasive advertising. Secondly, sophisticated consumers understand that a positive signal with persuasive

advertising could have been generated by persuasion and not because of the true product quality. Hence the decrease

in the expected quality given a positive signal.

3 No advertising case

I start by analyzing a simpler model where firms cannot advertise. This provides some useful insight

into the role played by prices in this model.15 In this section, a pure-strategy equilibrium is charac-

terized by a triplet (p(q), b(s, p), µ(q|s, p)). p(q) is the price chosen by the firm as a function of its

quality q. b(s, p) is the decision of the consumers to buy or not to buy and µ(q|s, p) are their beliefswhen they receive signal s and observe price p. The beliefs are a probability distribution over the set

of qualities.

The support of µ(q|s, p) when p is on the equilibrium path does not depend on s as the choice of the15See Wolinsky (1983) and Bagwell and Riordan (1991) for studies about how prices signal product quality in other

contexts.

12

firm p(q) is independent of the signals received by consumers, and as each signal occurs with a positive

probability for every quality. Specifically, when p is chosen in equilibrium, consumers with neutral

signals and consumers with positive signals believe the same set of firms could have chosen that price.

The beliefs with a neutral signal put however more weight on the low-quality firms than the beliefs

with a positive signal. This implies that on the equilibrium path, if consumers with neutral signals

are willing to buy at p, consumers with positive signals are also willing to buy, namely b(0, p) =buy

implies b(1, p) =buy. On the equilibrium path, firms either sell to everybody or to consumers with

positive signals only. In addition, as consumers always buy if the price is equal to the worst quality,

the profits of the firms are larger than q in equilibrium.

The next lemma shows formally that in equilibrium two prices at most are chosen. It proves that

when exactly two prices are on the equilibrium path, every consumer buys at the lower price while

only consumers with positive signals buy at the higher price. It also pins down the ordering in terms

of quality: a low-quality firm chooses a low price and a high-quality firm chooses a high price.

Lemma 1 In the model with no advertising,

i. At most two prices are chosen on the equilibrium path.

ii. When exactly two prices (pL, pH) are on the equilibrium path, there exists qL such that

- p(q) = pL if q ≤ qL and pH if qL < q,

- b(s, pL) = buy for any s and b(s, pH) = buy iff s = 1.

Intuitively, firms choose between the highest price where all consumers buy and the highest price

where only consumers with positive signals buy as firms always want to increase their prices as long as

demand remains constant. The trade-off is between selling at a low price to the full market or selling

at a high price to one segment. This effect is the usual monopoly trade-off between increasing the

price and reducing the demand. Setting up a high price allows firms to signal their high quality by

forgoing some demand. Indeed, when consumers with positive signals observe a high price, they know

that consumers with negative signals never buy at that price. It is profitable for a firm to choose that

13

price only if the firm generates enough positive signals. Therefore, only high-quality firms choose the

high price over the low price and consumers with positive signals buy at the high price.

Lemma 1 states that any equilibrium has exactly one or two prices on the equilibrium path.

They are referred to as pooling equilibrium or as semi-separating equilibrium, respectively. A pooling

equilibrium is characterized by a price pL such that the strategies and beliefs along the equilibrium

path are p(q) = pL, b(pL, s) = buy, and µ(q|pL, s) = g∅(q|s)R qq g∅(y|s)dy

.16 Moreover, a semi-separating

equilibrium is characterized by two prices (pL, pH) and one quality cutoff qL such that the strategies

and beliefs along the equilibrium path are p(q) =©pL if q≤qLpH if qL<q

, b(s, pL) = buy, b(s, pH) = buy iff s = 1,

and µ(q|s, pL) and µ(q|s, pH) are derived using Bayes’ rule.17

Figure 3 illustrates graphically these two types of equilibria while Proposition 1 states the necessary

conditions that have to be satisfied by each equilibrium. Proposition 1 is general as it is stated with

general functional forms of the distribution of qualities f and the signal function α. E[q|s] representsthe expected quality conditional on having received signal s.

Proposition 1 In the model with no advertising, any pure-strategy equilibrium is either:

i. a pooling equilibrium characterized by pL such that pL ∈ [q, E[q|0]];

ii. a semi-separating equilibrium characterized by (pL, pH) and qL such that

pL = α(qL)pH

with pL ∈ [q, E[q|0, q ≤ qL]] and pH ∈ (E[q|0, qL < q], E[q|1, qL < q]].

These necessary conditions are also sufficient in the case of the worst out-of-equilibrium beliefs.

The next two corollaries present some additional results related to which equilibrium has the

highest profit or the highest welfare. Corollary 1 shows that the pooling equilibrium in which firms

extract the full surplus of consumers with neutral signals has the highest profits among all the pooling

16g∅(q|s) = α(q)s(1 − α(q))1−sf(q) is, up to a multiplicative constant, the density function over the qualities given

that signal s was received.17µ(q|pL, s) = g∅(q|s)R qL

q g∅(y|s)dyif q ≤ qL, 0 otherwise. µ(q|pH , s) = g∅(q|s)R q

qLg∅(y|s)dy

if qL < q, 0 otherwise.

14

qqq qL

pL

pH

p

qqq

pL

pbuy if s = 1buy if s = 0

qqq qL

pL

pH

p

qqq

pL

pbuy if s = 1buy if s = 1buy if s = 0buy if s = 0

Figure 3: No advertising case: The product quality varies along the horizontal axis and equilibrium prices along

the vertical axis. The purchasing decision on the equilibrium path is represented by a dashed line for consumers with

negative signals and by a solid line for consumers with positive signals. The left graph illustrates a semi-separating

equilibrium in which high-quality firms signal themselves by choosing a high price and forgoing selling to consumers

with a negative signal. The right graph illustrates a pooling equilibrium.

equilibria. This pooling equilibrium has the highest profits if the profit achieved by the highest-quality

firm when selling only to consumers with positive signals is lower than the expected quality after

receiving a negative signal. Furthermore, the profits of any semi-separating equilibrium are smaller

than the profits of a semi-separating equilibrium in which low-quality firms extract the full surplus of

consumers with negative signals. These results give some support in favor of studying the equilibria

with binding upper constraints on prices.

Corollary 1 In the model with no advertising,

i. among all pooling equilibria, the one with pL = E[q|0] has the highest profits ,

ii. if α(q)q ≤ E[q|0], the pooling equilibrium with pL = E[q|0] has the highest profits among allequilibria,

iii. among the semi-separating equilibria, the highest profits are achieved in a semi-separating equi-

librium with pL = E[q|0, q ≤ qL].

It is intuitive that the pooling equilibrium with the full consumer surplus being extracted leads

to the highest profit among all the pooling equilibria. How do these profits compare with the profits

15

of a semi-separating equilibrium? Assume that the highest-quality product generates so few positive

signals that, even if consumers with positive signals believed the product to have the highest quality,

the firm would still prefer to sell to all consumers at p = E[q|0]. It must be the case that this poolingequilibrium has the highest profits. Finally, if the monopolist does not extract the full surplus of the

consumers with neutral signals when its quality is low, then there is some scope for increasing the

profit by increasing the price so another equilibrium with higher profits exists in which the price has

been increased.

In addition, the highest welfare is achieved by any pooling equilibrium as all consumers buy in

these equilibria. The highest welfare among all semi-separating equilibria is achieved by a semi-

separating equilibrium in which low-quality firms extract the full surplus of consumers with negative

signals. These results are presented in the next corollary.

Corollary 2 In the model with no advertising,

i. a pooling equilibrium leads to higher welfare than a semi-separating equilibrium,

ii. all pooling equilibria have the same welfare,

iii. the highest welfare among all semi-separating equilibria is achieved by a semi-separating equi-

librium satisfying pL = E[q|0, q ≤ qL].

This section has illustrated how firms use price to signal their quality. A high-quality firm can

signal its type by choosing a price high enough to forgo the demand of consumers with neutral signals

while a low-quality firm would never find it profitable. In the next section, I analyze how firms signal

their quality when they can use not only their price but also their advertising decision to communicate

with their customers.

4 Persuasive advertising case

In this section, I analyze the full model where firms choose both their advertising and their price.

Two new insights about persuasive advertising are developed. First, persuasive advertising does not

16

naturally signal high quality. When the price is high, a firm doing no persuasive advertising relies only

on its intrinsic product quality to create positive signals. Consequently, if the share of positive signals

has to be high to make that choice optimal, consumers understand that the quality is high. Second,

persuasive advertising helps the monopolist extract rents. In the extreme case where the expected

quality conditional on a signal does not depend on the value of the signal, the firm can extract the

full surplus simply by charging a price equal to the expected quality. Nevertheless, when the expected

quality conditional on a signal has some variance, the firm cannot extract the full surplus. The firm

either sells to every consumer at a low price, leaving some surplus to the consumers with positive

signals, or it does not sell to consumers with neutral signals, extracting only the surplus of consumers

with positive signals. When it is more profitable to sell only to consumers with positive signals,

persuasive advertising helps extract the surplus from additional consumers as more of them get a

positive signal after persuasive advertising.

An equilibrium is ((p(q), a(q)), b(s, p, a), µ(q|s, p, a)). (p(q), a(q)) ∈ R+×{∅, P} is the price and theadvertising chosen by the firm as a function of its quality.18 b(s, p, a) is the decision of the consumers

to buy or not to buy and µ(q|s, p, a) their beliefs when they receive signal s and observe price p andadvertising a. The beliefs are a probability distribution over the set of qualities. The support of

µ(q|s, p, a) when (p, a) is on the equilibrium path does not depend on s as the choice of the firm p(q)

is independent of the signals received by consumers and each signal occurs with a positive probability

for every quality. Therefore, using a similar reasoning as the one in the no-advertising case, on the

equilibrium path, firms either sell to everybody or only to consumers with positive signals.

Intuitively, firms face three kinds of demand: sell to all consumers D = 1; sell to consumers

with positive signals with persuasive advertising D = ϕ(q); or sell to consumers with positive signals

without persuasive advertising D = α(q). Obviously, the highest price is always chosen when demand

is unchanged. The trade-off is thus between increasing the price and decreasing the demand. If these

three kinds of demand occur in equilibrium, as they are ordered from the largest to the smallest (1 ≥ϕ(q) ≥ α(q)), they must be associated with prices with reverse ordering. The following lemma proves

formally that a firm selling only to consumers with positive signals without engaging in persuasive

18a = ∅ represents no advertising while a = P represents persuasive advertising.

17

advertising chooses the highest price, and that a firm selling to all consumers chooses the lowest price.

This lemma also shows that a firm selling only to consumers with positive signals without engaging

in persuasive advertising is part of the high-quality range and that a firm selling to all consumers

belongs to the low-quality range.

Lemma 2 In the model with persuasive advertising, at most three different prices are on the equilib-

rium path.

If there exists q0 such that (p(q0), a(q0)) = (p0, ∅), then

i. if b(s, p0, ∅) =buy iff s = 1,

then for any q > q0, (p(q), a(q)) = (p0, ∅), and for any q, p(q) ≤ p0,

ii. if b(s, p0, ∅) =buy for any s,

then for any q < q0, (p(q), a(q)) = (p0, ∅), and for any q, p(q) ≥ p0.

How do firms use persuasive advertising to signal their types? Assume for the moment that a firm

can choose to sell to consumers with positive signals either at a high price without advertising (price

pH and α(q) consumers) or at a low price with persuasive advertising (price pL and ϕ(q) consumers).

The trade-off is once again between price and quantity. The firm chooses no advertising and a high

price if the proportional increase in quantity ϕ(q)α(q) is smaller than the proportional increase in price

pHpL.

Per Assumption 2, ϕ(q)α(q) , the share of positive signals with persuasive advertising relative to the share of

positive signals without persuasive advertising, becomes smaller when the product quality increases.

Therefore by not engaging in persuasive advertising and choosing a high price, high-quality firms can

be separated from firms engaging in persuasive advertising and selling at a low price. Specifically,

forgoing the extra demand created by persuasive advertising allows firms to signal their high quality.

The exact nature of the equilibria is presented in the rest of the section.

According to Lemma 2, in an equilibrium with three different prices, low-quality firms choose a

low price and no advertising, intermediate-quality firms choose an intermediate price and persuasive

advertising while high-quality firms choose a high price and no advertising. In addition, low-quality

firms sell to all consumers whereas intermediate and high-quality firms sell only to consumers with

18

positive signals. That equilibrium is characterized by three prices (pL, pM , pH) and two quality

cutoffs (qL, qM ) such that the strategies and beliefs along the equilibrium path are a(q) = ∅ if q ≤ qLor qM ≤ q and = P if qL < q ≤ qH , p(q) = pL if q ≤ qL, = pM if qL < q < qM , = pH if qM ≤ q,

b(pM , P, s) = b(pH , ∅, s) = buy iff s = 1, µ(q|pL, ∅, s), µ(q|pM , P, s) and µ(q|pH , ∅, s) are derived usingBayes’ rule.19

Proposition 2 characterizes a set of necessary conditions of any equilibrium where persuasive

advertising signals intermediate quality. The cutoff qualities are given by firms being indifferent

between two strategies. Moreover, prices are such that the purchasing decisions are rational. This

equilibrium is shown graphically in Figure 4.

Proposition 2 In the model with persuasive advertising, any equilibrium with three different prices

is characterized by (pL, pM , pH) and two quality cutoffs (qL, qM ) such that

pL = ϕ(qL)pM and ϕ(qM )pM = α(qM )pH

with pL ∈ [q, E[q|∅, 0, q ≤ qL]], pM ∈ (E[q|P, 0, qL < q < qM ], E[q|P, 1, qL < q < qM ]], and pH ∈(E[q|∅, 0, qM ≤ q], E[q|∅, 1, qM ≤ q]].20

These necessary conditions are also sufficient in the case of the worst out-of-equilibrium beliefs.

In the equilibrium of Proposition 2, persuasive advertising signals intermediate-quality products.

Firms have a choice between a high price with a regular demand or a medium price with an expanded

demand. Since the demand expansion expressed as a share of the regular demand becomes smaller as

quality increases,21 the demand expansion created by persuasive advertising for intermediate-quality

firms is large enough to overcome the price decrease from a high price to a medium price, while it

is too small for high-quality firms. As for the low quality firms, they generate too many consumers

19Define g∅(q|s) = α(q)s(1 − α(q))1−sf(q) and gP (q|s) = ϕ(q)s(1 − ϕ(q))1−sf(q). µ(q|pL, ∅, s) = g∅(q|s)R qLq g∅(y|s)dy

if

q ≤ qL, 0 otherwise. µ(q|pM , P, s) = gP (q|s)R qMqL

gP (y|s)dyif qL < q < qM , 0 otherwise. µ(q|pH , ∅, s) = g∅(q|s)R q

qMg∅(y|s)dy

if qM ≤ q, 0otherwise20To be fully exhaustive, if pL ≤ E[q|P, 0, q ≤ qL], there exists another equilibrium identical to the previous one except

when q ≤ qL, a(q) = P , and µ(q|pL, P, s) is derived using Bayes’ rule.21For example if the regular demand is α(q) = .5, the demand expansion can go up to 200% while if the regular

demand is α(q) = .8, the demand expansion cannot exceed 125%.

19

buy if s = 1buy if s = 0

qqq qL

pL

pH

p

qM

pM

PersuasiveAdvertising

buy if s = 1buy if s = 0buy if s = 1buy if s = 1buy if s = 0buy if s = 0

qqq qL

pL

pH

p

qM

pM

PersuasiveAdvertising

Figure 4: Persuasive advertising signals intermediate quality. The product quality varies along the horizontal

axis and the equilibrium prices along the vertical axis. The purchasing decision on the equilibrium path is represented by

a dashed line for consumers with neutral signals and by a solid line for consumers with positive signals. The equilibrium

advertising choice is shown on top of the price level.

with neutral signals to be able to forgo them by selling at a higher price with or without persuasive

advertising.

The result that persuasive advertising does not signal high quality is in contrast with the previous

literature describing advertising as a means of burning money. If the advertising content is irrelevant

and advertising matters only because it is costly, firms with higher profits can signal themselves by

burning some of these profits as firms with lower profits cannot afford to burn the same amount of

money. In that case, only high-quality firms engage in advertising, whereas Proposition 2 shows that

if the direct effect of advertising is taken into account, advertising can signal intermediate quality

while high-quality firms signal themselves by not engaging in persuasive advertising. The burning-

money story also implies that the relationship between price and advertising is monotonic; advertising

is associated with the highest price while Proposition 2 shows that a non-monotonic relationship

between price and advertising can arise. No advertising is coupled with prices that are either higher

or lower than the price associated with persuasive advertising.22

22Organic milk can be purchased in most supermarkets, not just the ones specializing in organic food. It is not

associated with any advertising and it is sold at a higher price than its closest substitute which is non-organic milk.

20

Furthermore, persuasive advertising does not necessarily signal intermediate-quality products.

First, it signals high quality when high-quality firms choose persuasive advertising while low-quality

firms opt out of advertising. Such an equilibrium is characterized by two prices (pL, pM ) and one

quality cutoff qL such that the strategies and beliefs along the equilibrium path are a(q) =©∅ if q≤qLP if qL<q

,

p(q) =©pL if q≤qLpM if qL<q

, b(pL, ∅, s) = buy, b(pM , P, s) = buy iff s = 1, and µ(q|pL, ∅, s), and µ(q|pM , P, s)are derived using Bayes’ rule. Second, persuasive advertising signals low-quality products when the

ordering of the firms engaging in persuasive advertising is reversed. In that case, only consumers

with positive signals buy, even for the worst-quality firm. That equilibrium is characterized by two

prices (pM , pH) and one quality cutoff qM such that the strategies and beliefs along the equilibrium

path a(q) =©P if q<qM∅ if qM≤q , p(q) =

©pM if q<qMpH if qM≤q , b(pM , P, s) = b(pH , ∅, s) = buy iff s = 1, and µ(q|pM , P, s)

and µ(q|pH , ∅, s) are derived using Bayes’ rule. Finally, in some cases persuasive advertising has nosignaling power if all firms choose it. Such an equilibrium is characterized by one price pM such that,

on the equilibrium path, a(q) = P , p(q) = pM , b(pM , P, s) = buy iff s = 1, µ(q|pM , P, s) = gP (q|s).Proposition 3 gives some necessary conditions for any equilibrium with one or two different prices

on the equilibrium path. In part (i), persuasive advertising signals high quality while it signals low

quality in part (ii). Part (iii) describes a pooling equilibrium in which all firms engage in persuasive

advertising. Finally the equilibria in part (iv) have no advertising occurring in equilibrium. In the

last two cases, advertising has no signaling value either because all firms run a persuasive campaign

or because none of them do.

Proposition 3 In the model with persuasive advertising, any equilibrium with less than three different

prices is characterized by either:

i. two prices (pL, pM ) and one quality cutoff qL such that pL = ϕ(qL)pM

with pL ∈ [q, E[q|∅, 0, q ≤ qL]] and pM ∈ (E[q|P, 0, qL < q], E[q|P, 1, qL < q]];

ii. two prices (pM , pH) and one quality cutoff qM such that ϕ(qM )pM = α(qM )pH and

ϕ(q)pM ≥ q with pM ∈ (E[q|P, 0, q < qM ], E[q|P, 1, q < qM ]] andpH ∈ (E[q|∅, 0, qM ≤ q], E[q|∅, 1, qM ≤ q]];

21

iii. one price pM such that ϕ(q)pM ≥ q with pM ∈ (E[q|P, 0], E[q|P, 1]];

iv. the conditions described in Proposition 1.23

These necessary conditions are also sufficient in the case of the worst out-of-equilibrium beliefs.

In the equilibrium drawn on the left of Figure 5, persuasive advertising signals high-quality firms

while low-quality firms opt out of advertising and sell to the full market at a lower price. In the

equilibrium drawn on the right of Figure 5, the reverse takes place in terms of advertising signaling

quality and pricing. Persuasive advertising signals low-quality firms while high-quality firms choose

a higher price and no advertising. Consumers with neutral signals never buy in this equilibrium.

qqq

pH

p

qM

pMPersuasiveAdvertising

buy if s = 1buy if s = 0

qqq qL

pL

p

pM

PersuasiveAdvertising

qqq

pH

p

qM

pMPersuasiveAdvertising

qqq

pH

p

qM

pMPersuasiveAdvertising

buy if s = 1buy if s = 0

qqq qL

pL

p

pM

PersuasiveAdvertising

buy if s = 1buy if s = 0buy if s = 1buy if s = 1buy if s = 0buy if s = 0

qqq qL

pL

p

pM

PersuasiveAdvertising

Figure 5: Persuasive advertising signals high or low quality. The left graph illustrates an equilibrium in

which persuasive advertising signals high quality while the right graph illustrates an equilibrium in which persuasive

advertising signals low quality. The product quality varies along the horizontal axis and the equilibrium prices along

the vertical axis. The purchasing decision on the equilibrium path is represented by a dashed line for consumers with

negative signals and by a solid line for consumers with positive signals. The equilibrium advertising choice is shown on

top of the price level.

I now discuss three important assumptions of the model, namely, that in addition to advertising

decisions firms also choose their price, that advertising has no cost in this model, and that consumers

are ex-ante homogeneous.

23To be fully rigorous, b(p, s) should be replaced by b(p, ∅, s) and the expectations should also be conditional on noadvertising.

22

Variable price

Assuming that firms choose both their advertising and their price is a key assumption and reflects

the fact that firms do use several instruments to communicate with their consumers. To illustrate this

point, assume in this paragraph that prices are fixed and that the firm chooses only its advertising

campaign. If the price is low, p ≤ E[q|∅, 0], all firms sell to all consumers and do not engage inpersuasive advertising. If the price is intermediate, E[q|∅, 0] < p ≤ E[q|P, 1], all firms engage inpersuasive advertising and sell to consumers with positive signals. Persuasive advertising has no

signaling power in that case; it is used only to take advantage of its demand expansion effect. Finally,

if the price is high, E[q|P, 1] ≤ p < E[q|∅, 1], all firms opt out of persuasive advertising and are able tosell only to consumers with positive signals. A firm would like to expand its demand with persuasive

advertising and have more consumers with positive signals but it is not possible due to the negative

direct effect of persuasive advertising. Recall that consumers are sophisticated and that persuasive

advertising lowers their willingness to pay. If a firm engages in persuasive advertising, the share of

consumers will certainly be larger but none of them are willing to buy at that price. Therefore, with

a fixed price, advertising loses all its signaling power. It is important to study how firms use both

advertising and price when communicating with consumers.

Advertising costs

Advertising is assumed to come at no cost for the firms. If a small fixed cost for advertising is

introduced, the results are qualitatively unchanged. More specifically, in a given type of equilibrium,

the quality cutoffs would become larger and the prices would also be larger. By small cost, I mean

that the cost does not completely overshadow the direct effect of advertising. If the cost becomes

large, then the main effect is a burning-money effect and the equilibria in which persuasive advertising

signals an intermediate quality or a low quality disappear. Nevertheless, advertising has to be more

than a means of burning money since some advertising campaigns are famous for failing. They did

not fail because consumers did not understand that they were expensive. They failed because of the

message they conveyed. Hence the advertising message is a strategic choice for firms.

Consumer Homogeneity

Consumers are supposed to be ex-ante identical. Once they receive their signal, the firm faces two

23

types of consumers: the ones with a positive prior about the product and the ones with a negative

prior. If all consumers are still assumed to value quality, and if their disutilities of price were drawn

from a continuous distribution, the monopolist would be facing a continuous distribution of consumers

instead of facing a distribution of consumers with two types. The drawback would be to introduce

some equilibria where prices would be a continuous function of quality. The consumers’ beliefs would

be extremely sensitive to slight variations in prices. However, the results presented above are still

valid when the beliefs of consumers are the limit of beliefs when firms make small errors when setting

prices.

5 Application to a uniform distribution

In this section, I focus on the case of a uniform distribution of qualities coupled with a particular

persuasive technology. More results are given about the sustainability of the different equilibria

depending on the market characteristics. In particular, I analyze how the existence of an equilibrium

depends on the degree of persuasion of the advertising technology.

Consider the model of the previous section with a uniform distribution of qualities between q

and q, i.e. f(q) = 1q−q . The signal without persuasive advertising is the memory of the outcome of

the last experience with the product, namely α(q) = q. Moreover, persuasive advertising transforms

a fixed share of negative experiential memories into positive suggested memories, which leads to

ϕ(q) = q+θ(1−q). It is easily checked that the assumptions of the model are satisfied as α(q)E[q|s =1] = qE[q|s = 1] < q, ϕ(q)

α(q) = (1− θ) + θq and

1−ϕ(q)1−α(q) = 1− θ are non-increasing. The model has three

parameters which are the range of qualities, q and q, and the degree of advertising persuasion, θ.

In this section, I focus on the equilibria where the upper constraints on prices are binding. The

prices are simply equal to the expected quality of the marginal consumers. Furthermore, in order to

determine when persuasive advertising cannot signal high-quality firms, the worst out-of-equilibrium

beliefs are used, that is, if an out-of-equilibrium action is observed, consumers believe they are facing

the worst-quality firm. If an equilibrium in which persuasive advertising signals high quality firms

does not exist with the worst out-of-equilibrium beliefs, such an equilibrium will not exist with any

24

other beliefs system. The next proposition shows that if the degree of persuasion is high, persuasive

advertising cannot signal high quality. If a high share of negative signals is transformed into positive

signals, persuasive advertising has a homogenizing effect on firms. After engaging in advertising, firms

do not look that different as they generate mainly positive signals. Therefore low-quality firms and

high-quality firms have similar profits when engaging in persuasive advertising and as a consequence,

persuasive advertising cannot signal high quality. Likewise when the range of quality is small and

the degree of persuasion is small, persuasive advertising cannot signal high quality. Indeed, when

the range of quality is small, all firms generate a similar number of positive signals. If persuasive

advertising has a small impact, even after engaging in persuasive advertising, all firms still have a

similar share of positive signals. Hence, persuasive advertising does not help signal high quality.

Proposition 4 Consider the model with persuasive advertising with f(q) = 1q−q , α(q) = q and ϕ(q) =

q + θ(1− q). There exist θ∗ ∈ (0, 1) and θ∗∗ ∈ [0, θ∗) such that persuasive advertising never signalshigh quality if

θ < θ∗∗ or θ∗ < θ

There exists q∗(q) such that θ∗∗ = 0 iff 4−√73 ≤ q and q < q∗(q).

The proposition states that if the degree of persuasion is large, persuasive advertising cannot

signal high quality in equilibrium. Besides, when the range of quality is large, persuasive advertising

might signal high-quality products for any low degree of persuasion. But if the range of quality is

small, persuasive advertising cannot signal high quality in equilibrium when the degree of persuasion

is too small. In that case, persuasive advertising might signal high-quality products only if the

degree of persuasion is in between two cutoff values. Figure 6 illustrates when the equilibrium in

which persuasive advertising signals high quality does not exist depending on the value of the highest

quality q and the degree of persuasion θ.

Similarly, Figure 7 illustrates when the equilibrium in which persuasive advertising signals low

quality does not exist depending on the value of the highest quality q and the degree of persuasion

θ. Recall that, in this equilibrium, all firms sell only to consumers with positive signals. Therefore,

when the degree of persuasion is low, by engaging in persuasive advertising, low-quality firms do not

25

q

θ

q = 0.25 q = 0.75

θ

q

Persuasive advertisingsignals high quality

Persuasive advertisingsignals high quality

q

θ

q = 0.25 q = 0.75

θ

q

Persuasive advertisingsignals high quality

Persuasive advertisingsignals high quality

Figure 6: Persuasive advertising signals high quality. The upper range of quality q ∈ (q, 1) varies along the

horizontal axis and the level of persuasion θ along the vertical axis. The left figure is drawn for q = 0.25 and the right

one for q = 0.75. The region where persuasive advertising signals high quality, is located in between the two curves.

get a large demand expansion. Instead they prefer to mimic high-quality firms by selling at a higher

price with no advertising. This explains why persuasive advertising does not signal low quality when

the degree of persuasion is low. On the other hand, when the range of qualities is small, the share

of positive signals is similar across firms. In this case, high-quality firms cannot signal themselves by

not engaging in persuasive advertising.

The findings of this section are in sharp contrast to the results in the previous literature. When

advertising has no direct effect and is only a means of burning money, only the high-quality firms

signal their type with advertising and advertising is coupled with the highest price. If, however,

the direct effect of advertising is taken into account and if consumers are sophisticated, high quality

cannot be signaled with persuasive advertising when the degree of persuasion is high or when the

range of quality is small. Furthermore, persuasive advertising does not necessarily signal low quality

either. That is, when the degree of persuasion is low or when the range of quality is small, persuasive

advertising cannot signal low-quality products.

26

Persuasive advertisingsignals low quality

Persuasive advertisingsignals low quality

q

θ

q = 0.25 q = 0.75

θ

q

Persuasive advertisingsignals low quality

Persuasive advertisingsignals low quality

q

θ

q = 0.25 q = 0.75

θ

q

Figure 7: Persuasive advertising signals low quality. The upper range of quality q ∈ (q, 1) varies along the

horizontal axis and the level of persuasion θ along the vertical axis. The left figure is drawn for q = 0.25 and the right

one for q = 0.75. The region where persuasive advertising signals low quality, is located in between the two curves.

6 Two types of advertising

After studying how firms choose between persuasive advertising and no advertising, I analyze how

firms choose between two types of advertising and no advertising. Informative advertising is intro-

duced as a second type of advertising with a different direct effect on consumers from that of persuasive

advertising.

I analyze the choice of a monopolist when it chooses between two types of advertising that differ

in how they affect consumers. Persuasive advertising interacts with the private signal received by

consumers while informative advertising is a way to convey a piece of hard/verifiable information to

consumers. Firms are assumed to choose only one type of advertising or no advertising at all.24 I

start by presenting how informative advertising works, then I analyze the equilibrium focusing on the

interaction between persuasive advertising and informative advertising.

24This assumption is consistent with the fact that advertisements compete to grab the attention of consumers; there-

fore, an ad conveys principally one message.

27

6.1 Informative Advertising

Suppose that prior to deciding on the advertising campaign, the monopolist performs marketing tests

which might deliver a claim about the product that can fit in the format of an advertisement. An

example of such a claim is “this toothpaste was clinically proven to fight germs for twelve hours.”

The probability η(q) ≡ Pr(σ = 1|q) that a firm gets a positive claim is an increasing function of the

product quality, where σ = 1 represents the fact that a claim was produced and σ = 0 means no

claim was created. The firm can engage in informative advertising only the marketing tests delivered

a positive claim. Informative advertising is thus noisy in this paper (see Anand and Shachar (2009)

for stylized facts about formulating informative advertising as a noisy process).

Note that for both types of advertising, the effect of advertising depends on the product quality.

The share of extra positive signals created by persuasive advertising varies with the underlying product

quality. Similarly, the probability to generate a positive claim about the product also depends on

the quality. The crucial point is that they affect consumers differently. Informative advertising adds

an element to consumers’ information set while persuasive advertising changes the information set

without allowing consumers to know what their information would have been without persuasive

advertising. Hence informative advertising has only a positive direct effect—increase in willingness to

pay—while persuasive advertising has both a positive and a negative direct effect–demand expansion

and decrease in willingness to pay.

The only difference with Section 4 is that firms with a positive claim choose between informative,

persuasive advertising, or no advertising while firms with no claim chooses between persuasive adver-

tising or no advertising. The timing of the model is unchanged and I analyze pure-strategy perfect

Bayesian Nash equilibrium.

6.2 Equilibrium Analysis

An equilibrium is ((p(q,σ), a(q,σ)), b(s, p, a), µ(q|s, p, a)). (p(q,σ), a(q,σ)) ∈ R+×{∅, P, I} is the priceand the advertising chosen by the firm as a function of its quality and its claim status.25 b(s, p, a)

is the decision of the consumers to buy or not to buy and µ(q|s, p, a) their beliefs when they receive25a = I represents informative advertising.

28

message s and observe price p and advertising a. Using the same reasoning as in the previous model,

on the equilibrium path, firms either sell to all the consumers or only to the consumers with positive

signals.

In equilibrium, only five situations can arise. The firm chooses a high price associated with either

informative advertising or no advertising and sells only to consumers with positive signals. Conversely,

the firm chooses a low price while choosing either informative advertising or no advertising, and sells

to all consumers. Finally, the firm chooses persuasive advertising and sells only to consumers with

positive signals. These results are described formally in the next lemma.

Lemma 3 In the model with persuasive and informative advertising, at most five different prices are

on the equilibrium path.

If there exists (q0,σ0) such that (p(q0,σ0), a(q0,σ0)) = (p0, I), then

i. if b(s, p0, I) =buy iff s = 1,

then for any q > q0 (p(q, 1), a(q, 1)) = (p0, I), and for any (q,σ) p(q,σ) ≤ p0,

ii. if b(s, p0, I) =buy for any s,

then for any q < q0 (p(q, 1), a(q, I)) = (p0, 1), and if p = p(q,σ) < p0 then a(q,σ) = ∅ andb(s, p, ∅) =buy for any s.

If there exists (q0,σ0) such that (p(q0,σ0), a(q0,σ0)) = (p0, ∅), then

i. if b(s, p0, ∅) =buy iff s = 1,

then for any q > q0 (p(q, 0), a(q, 0)) = (p0, ∅), and if p = p(q,σ) > p0 then a(q,σ) = I and

b(s, p, I) =buy iff s = 1,

ii. if b(s, p0, ∅) =buy for any s,

then for any q < q0 (p(q, 0), a(q, 0)) = (p0, ∅), and for any (q,σ) p(q,σ) ≥ p0.

Lemma 3 shows that in an equilibrium with five different prices on the equilibrium path, low-

quality firms sell to all consumers at a low price. If the firm has a claim, it reveals it through informa-

29

tive advertising and is able to charge a higher price than firms with no claim who chooses not to adver-

tise. Intermediate-quality firms choose an intermediate price, engage in persuasive advertising and sell

only to consumers with positive signals. Finally, high-quality firms with a claim choose an informative

campaign and charge the highest price, while high-quality firms with no claim signal their type with a

high price and no advertising. That equilibrium is characterized by five prices (p0L, p1L, pM , p

0H , p

1H)

and four quality cutoffs (q0L, q1L, q

0M , q

1M ) such that the strategies and beliefs on the equilibrium path

are

a(q, 0) =

∅ if q≤q0LP if q0L<q<q

0M

∅ if q0M≤q, a(q, 1) =

I if q≤q1LP if q1L<q<q

1M

I if q1M≤q, p(q,σ) =

pσL if q≤qσLpM if qσL<q<q

σM

pσH if qσM≤q

,

b(p0L, ∅, s) = b(p1L, I, s) = buy, b(pM , P, s) = b(p0H , ∅, s) = b(p1H , I, s) = buy iff s = 1, and µ on the

equilibrium path is derived using Bayes’ rule.26

The next proposition gives a set of necessary conditions for an equilibrium with five prices on the

equilibrium path. The cutoff qualities are given by the indifferent firms, while the prices are such

that the purchasing decisions are rational. That equilibrium is also depicted graphically in Figure 8.

Proposition 5 In the model with persuasive or informative advertising, any equilibrium with five

different prices is characterized by (p0L, p1L, pM , p

0H , p

1H) and four quality cutoffs (q

0L, q

1L, q

0M , q

1M )

such that

pσL = ϕ(qσL)pM and ϕ(qσM )pM = α(qσM )pσH

with pσL ∈ [q, E[q|∅,σ, 0, q ≤ qσL]], pM ∈ (E[q|{P, 0, 0, q0L < q < q0M} ∪ {P, 1, 0, q1L < q < q1M}],E[q|{P, 0, 1, q0L < q < q0M} ∪ {P, 1, 1, q1L < q < q1M}]], and pσH ∈ (E[q|∅,σ, 0, qM ≤ q], E[q|∅,σ, 1, qM ≤q]].27

26Define g∅(q|s,σ) = α(q)s(1− α(q))1−sη(q)σ(1− η(q))1−σf(q) , gP (q|s, 0) = ϕ(q)s(1− ϕ(q))1−s(1− η(q))f(q), and

gP (q|s) = ϕ(q)s(1−ϕ(q))1−sf(q). Let aσ = I if σ = 1 and ∅ if σ = 0. µ(q|pσL, aσ, s) = g∅(q|s,σ)R qσL

q g∅(y|s,σ)dyif q ≤ qσL, 0 other-

wise. µ(q|pσH , aσ, s) = g∅(q|s,σ)R qqM

g∅(y|s,σ)dyif qσM < q, 0 otherwise. µ(q|pM , P, s) = gP (q|s,0)R

(q0L,q0M)∪(q1

M,q1L)gP (y|s,0)dy+

R q1M

q1L

gP (y|s)dy

if q0L < q < q1L or q1M < q < q0M , =gP (q|s)R

(q0L,q0M)∪(q1

M,q1L)gP (y|s,0)dy+

R q1M

q1L

gP (y|s)dyif q1L < q < q1M , 0 otherwise.

27The ordering of the elements in the conditional expectations is a,σ, s, q. For example E[q|∅, 0, 0, q ≤ q0L] representsthe expected quality when a = ∅, σ = 0, s = 0 and q ≤ q0L.

30

These necessary conditions are also sufficient in the case of the worst out-of-equilibrium beliefs.

Firms: s=0

Consumers:buy if s=1buy if s=0

Persuasive Adv

p

qqq q0L q1L q1M q0M

p0L

pM

p0H

p1L

p1H

Firms: s=0

Consumers:buy if s=1buy if s=0buy if s=1buy if s=0

Persuasive Adv

p

qqq q0L q1L q1M q0M

p0L

pM

p0H

p1L

p1H

Firms: s=1

Consumers:

Persuasive Adv

buy if s=1buy if s=0

Informative Adv

Informative Adv

qq q0L q1L q1M q0M

p

q

p1L

pM

p1H

p0L

p0H

Firms: s=1

Consumers:

Persuasive Adv

buy if s=1buy if s=0buy if s=1buy if s=0

Informative Adv

Informative Adv

qq q0L q1L q1M q0M

p

q

p1L

pM

p1H

p0L

p0H

Figure 8: Informative and persuasive advertising case. On the top graph, the pricing and advertising choices

of a firm without a claim are drawn with respect to its quality. On the bottom graph, the pricing and advertising

choices of a firm with a claim are drawn with respect to its quality. The purchasing decision on the equilibrium path, is

represented by a dashed line for consumers with neutral signals and by a solid line for consumers with positive signals.

Low-quality firms choose a low price p0L if they choose no advertising or p1L if they reveal a positive

claim through informative advertising. These firms sell to all consumers. Intermediate-quality firms

run a persuasive campaign and choose an intermediate price pM . They sell only to consumers with

positive signals. High-quality firms also sell only to consumers with positive signals. They choose a

high price p0H and no advertising if they have no claim to reveal and they choose the highest price p1H

and informative advertising if they can.

In addition, the set of qualities of firms with a claim choosing persuasive advertising is strictly

included in the set of qualities of firms with no claim engaging in persuasive advertising. In terms

31

of quality cutoffs this translates into q0L < q1L < q1M < q0M . The low price for a firm without a

claim is lower than the low price for a firm with a claim as the firm with a claim can always forgo

informative advertising and mimic the firm without the claim. Therefore, the profit of the firm

without a claim indifferent between no advertising with the lowest price and persuasive advertising

is equal to p0L which is smaller than p1L. On the other hand, with or without a claim there is a single

price associated with persuasive advertising. Hence the quality of the indifferent firm with a claim is

higher than the quality of the indifferent firm without a claim. The intuition is similar in the case

of the cutoff qualities between persuasive advertising with the intermediate price and no advertising

with the high price.

It is striking that some firms with a claim choose persuasive advertising; these firms have a claim

to reveal but choose not to reveal it to consumers. In this equilibrium, the information does not fully

unravel. Consumers do not know whether a firm that engages in persuasive advertising has a claim.

On the other hand, consumers rationally infer that a firm engaging in no advertising has no positive

claim to reveal.

This equilibrium also illustrates the important role played by prices. When consumers observe an

informative campaign, they cannot distinguish between low and high quality. They need to check the

price to be able to form their final beliefs. Depending on the price level, they think the product is

from the bottom quality group or from the top quality group.

Finally, these results show that no clear ordering exists between the two types of advertising.

Persuasive advertising can signal a higher or a lower quality than informative advertising.

7 Conclusion

Manufacturers of personal care products spent more than 5 billion dollars in 2006 on advertising in

the United States.28 Total advertising spending across all categories and media was estimated at 285

billion dollars. Firms have large advertising budgets and their choice of advertising is an important

strategic decision.

28See AdvertisingAge (2007).

32

In this paper, I analyze how firms can signal their product quality when consumers are sophis-

ticated and understand that they can be affected by the persuasive side of advertising. I show that

some firms engage in persuasive advertising even if consumers are fully aware of their motivation for

choosing this form of advertising. In addition, firms can strategically signal their product quality by

launching or not launching a persuasive advertising campaign. I show that neither the relationship

between advertising and quality nor the one between advertising and price are monotonic. Specif-

ically, persuasive advertising does not naturally signal high-quality products and is not necessarily

associated with the highest price. Depending on the market conditions, persuasive advertising can

signal low-quality firms, high-quality firms, or even an intermediate range of qualities. In particular,

if the persuasion technology is very effective or if the range of qualities is small, persuasive advertising

cannot signal high quality since firms engaging in persuasive advertising look very similar from the

consumers point of view. Finally, when choosing between persuasive or informative advertising, firms

may prefer to withhold their information and instead engage in persuasive advertising.

These results were obtained by developing a model where persuasive advertising has a direct

effect on consumers. Unlike in the previous literature where persuasive advertising was included

as a direct element in the utility of consumers or where it shifted their preferences, in this paper

persuasive advertising affects the memory of consumers in a way that is consistent with results found

in the psychology and marketing literature on post-experience advertising. Furthermore, the results

of this paper could also be applied to other strategic choices made by a firm as long as they affect

the distribution of signals that consumers get about the product (see Brown, Camerer, and Lovallo

(2009) for the effects of movie cold on box office revenue and Milgrom (2008) for a review on disclosure

models).29

The effects of competition has yet to be studied in this context. It will raise some interesting

issues about the effect of mentioning a competitive product in an advertisement. In particular, is this

always a signal that the competitive product has a higher quality? Will it depend on the advertising

type chosen by the competitor?

Lastly, there remains a general issue on how to precisely classify the different types of advertising.

29 I thank for pointing out this interesting remark.

33

Johnson and Myatt (2006) suggest to use a different taxonomy: hype versus real information. Their

distinction comes from a vertical versus horizontal differentiation, while the distinction that I used

is based on the notion of soft versus hard information. The more appropriate taxonomy depends of

course on the context, but the important point is that the type of advertising matters and affects

consumers differently. It is therefore a strategic tool that firms can use to communicate with their

customers.

Appendix

Proof of Lemma 1. Consider two firms q and eq choosing p and ep in equilibrium, respectively.First, assume that consumers buy only when they receive a positive signal for both prices or that all

consumers buy when they observe either price. It must be the case that the prices are equal. Other-

wise, the firm with the lower price deviates and chooses the higher price. Its demand is unchanged

but it now sells at a higher price. As firms either sell to everybody or to consumers with positive

signals only, at most two prices are on the equilibrium path. Now assume that consumers buy only

when they receive a positive signal when they see one price and buy irrespectively of the value of their

signal when they observe the other price. Wlog assume that b(s, p) =buy iff s = 1 and b(s, ep) =buyfor any s. The profits of firm q after choosing p must be at least as large as its profits would have

been if it had chosen ep. Its no-deviation condition is α(q)p ≥ ep which implies that p ≥ ep. Similarly,the no-deviation condition for firm eq is ep ≥ α(eq)p. As the signal function α is non-decreasing, these

conditions imply that q ≥ eq.Proof of Proposition 1. Lemma 1 states that an equilibrium has at most 2 prices on the

equilibrium path. Firms can always choose to price at q and sell to everybody, hence any equilibrium

price is larger than q. Consider first a pooling equilibrium around pL. Assume that consumers buy

at pL only if their signal is positive, which is the case iff E[q|0] < pL ≤ E[q|1]. Thus, the profitof the worst-quality firm is π(q) = α(q)pL ≤ α(q)E[q|1] < q.30 This firm can deviate to p = q and

increase its profit. Therefore, in a pooling equilibrium all consumers buy irrespectively of their signals.

30The last inequality is Assumption 1.

34

Bayes’ rule pins down their beliefs when they observe pL and consumers with a negative signal are

willing to buy iff pL ≤ E[q|0]. Consider now an equilibrium with two prices. Lemma 1 shows that

p(q) =©pL if q≤qLpH if qL<q

, b(s, pL) = buy, b(s, pH) = buy iff s = 1. The condition pL = α(qL)pH simply states

that the firm with the cutoff quality is indifferent between the two prices. Bayes’ rule delivers the

beliefs of the consumers on the equilibrium path. Finally, consumers with negative signals buy at pL

iff pL ≤ E[q|0, q ≤ qL] while they do not buy at pH iff E[q|0, qL < q] < pH . Consumers with positivesignals buy at pH iff pH ≤ E[q|1, qL < q].Proof of Corollary 1. In a pooling equilibrium, the profits are π(q) = pL implying the first

result. In a semi-separating equilibrium, profits are non-decreasing with respect to quality and π(q) =

α(q)pH ≤ α(q)q. Hence, if α(q)q ≤ E[q|0], the equilibrium with the highest profits is the pooling

equilibrium with pL = E[q|0]. Consider a semi-separating equilibrium such that pL < E[q|0, q ≤ qL]and pH < E[q|1, qL < q], then there exists a semi-separating equilibrium with (pL+δpL, pH+δpH , qL)

such that δpL > 0, δpH > 0, δpL = α(qL)δpH and pH + δpH ≤ E[q|1, qL < q] with strictly higher

profits for any quality. Consider a semi-separating equilibrium such that pL < E[q|0, q ≤ qL] and

pH = E[q|1, qL < q], then there exists a semi-separating equilibrium with (pL + δpL, pH , qL + δqL)

such that δpL > 0, δqL > 0, pL + δpL = α(qL + δqL)pH and E[q|0, qL + δqL < q] < pH with higher

profits, strictly higher for the low qualities. Therefore, the semi-separating equilibrium with the

highest profits among all semi-separating equilibria satisfies pL = E[q|0, q ≤ qL].Proof of Corollary 2. The total surplus in a pooling equilibrium is simply E[q] as all the

consumers buy. In a semi-separating equilibrium the total surplus isR qLq qf(q)dq+

R qqL

α(q)qf(q)dq ≤R qLq qf(q)dq +

R qqLqf(q)dq = E[q]. Note that the inequality is an equality iff all the consumers of the

high-quality firms receive positive signals, i.e., q > qL ⇒ α(q) = 1. Part iii. is immediately derived

from the last part of the proof of the previous corollary.

Proof of Lemma 2. Consider two firms q and eq choosing (p, a) and (ep,ea) in equilibrium. Ifall consumers buy whenever they observe (p, a) or (ep,ea) then the prices must be equal. Otherwise,the firm with the lower price could increase its profit by choosing the same price and advertising

as the other firm. Now assume that only consumers with positive signals buy when they observe

(p, a) or (ep,ea). If the advertising choices are equal, the prices must be equal, i.e., a = ea ⇒ p = ep.35

If the advertising choices are different, wlog assume that a = ∅ and ea = P . The profits of firm q

after choosing (p, a) must be at least as large as if it had chosen (ep,ea); its no-deviation condition isα(q)p ≥ ϕ(q)ep. Similarly for firm eq, the no-deviation condition is ϕ(eq)ep ≥ α(eq)p. Recall that thesignal function ϕ/α is non-increasing and larger than 1. Therefore these conditions imply that q ≥ eqand p ≥ ep. Finally, consider the case when only consumers with positive signals buy if they observe(p, a) and when all consumers buy if they observe (ep,ea). Firm q can choose (ep,ea) and sell to allconsumers. As firm q prefers to sell to a smaller share of consumers, it must sell at a higher price:

p ≥ ep. Furthermore, the share of consumers with positive signals of firm eq is too small to make thedeviation to a higher price profitable. In particular, the no-deviation condition implies q ≥ eq. In theend, at most one price corresponds to firms selling to all consumers, and at most two prices correspond

to firms selling only to consumers with positive signals with or without persuasive advertising.

Proof of Proposition 2. Lemma 2 implies that in an equilibrium with three prices, pL < pM < pH ,

high-quality firms (qH ≤ q) choose pH , no advertising and sell to consumers with positive signals

while low-quality firms (q ≤ qL) choose pL, no advertising and sell to every consumer. Hence, the

intermediate price pM and persuasive advertising is chosen by intermediate-quality firms (qL < q <

qM) and only consumers with positive signals buy. pL = ϕ(qL)pM and ϕ(qM )pM = α(qM )pH are

the indifference conditions for cutoff firms qL and qM , respectively. The beliefs of consumers are

determined by Bayes’ rule on the equilibrium path. The inequalities on prices are derived from the

rationality of the purchasing decision by consumers depending on their private signals and from the

fact that prices have to be larger than q.

The proof of Proposition 3 is derived by using a reasoning similar to the proof of Proposition 2.

Proof of Proposition 4. Equilibrium (i) in Proposition 3 is the only equilibrium where per-

suasive advertising signals high-quality firms. This equilibrium is sustainable iff there exist two

prices (pL, pM ) and one quality cutoff qL such that pL = ϕ(qL)pM and pL = E[q|∅, 0, q ≤ qL] and

pM = E[q|P, 1, qL < q]. Thus, this equilibrium exists iff E[q|∅, 0, q ≤ qL] = ϕ(qL)E[q|P, 1, qL < q] hasa solution qL in (q, q). The LHS is an increasing function of qL and it is concave as

∂2E[q|∅,0,q≤qL]∂q2L

=

− 4(1−q)23(1−q+1−qL)3 . The RHS is an increasing function of qL and it is convex.

31 Note that the LHS is

31 ∂2ϕ(qL)E[q|P,1,qL<q]∂q2

L= num

3((1−θ)(q+qL)+2θ)3 where num is a polynomial function of degree 2 in q with a positive

36

actually independent of θ and the RHS is increasing in θ.32 Besides when θ = 1, no solution exists

as the LHS is strictly smaller than the RHS. Indeed LHS = E[q|∅, 0, q ≤ qL] < qL < E[q|P, 1, qL <q] = RHS|θ=1. Let’s analyze what happens when θ = 0. When qL tends to q, the LHS tends to q

and the RHS tends to qE[q|∅, 1] which is smaller. When qL tends to q, the LHS tends to E[q|∅, 0]and the RHS tends to qq. Hence the LHS is larger than the RHS when θ = 0 and when qL tends

to q iff E[q|∅, 0] > q2 ⇔ q+q

2 − q2+qq+q2

3 − q2(1 − q+q

2 ) > 0 which is a polynomial function of degree

2 in q. Let’s define q∗(q) = 14

³3q2 − 2q + 3−

p9q4 + 12q3 − 42q2 + 12q + 9

´.When

³q < 4−√7

3

´or³

4−√73 ≤ q and q∗(q) ≤ q

´, E[q|∅, 0] ≥ q2 which implies that no solution exists when θ = 0. And

when 4−√73 ≤ q and q < q∗(q), E[q|∅, 0] < q2 which implies that a unique solution exists when θ = 1.

Hence the results of the proposition.

The proofs of Lemma 3 and Proposition 5 are derived by using a reasoning similar to the ones of

Lemma 2 and Proposition 2.

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40