A “Memory-Jamming” Theory of Advertising

Jesse M. Shapiro∗

University of Chicago and NBER

May 19, 2006

Abstract

I present a model in which advertising changes consumers’ recollections of their past experienceswith a product. The model sheds light on several important features of advertising behavior: therelationship between advertising levels and consumer experience, the relationship between ad-vertising and product quality, and the presence of increasing returns in the consumers’ responseto advertising. Importantly for empirical research on advertising, the model’s comparative sta-tics call into question the ability of standard empirical tests to distinguish informative fromnon-informative advertising.

JEL classification: M37, L15

Keywords: advertising, memory, persuasion, signal-jamming

∗The title of this paper parallels Fudenberg and Tirole’s (1986) “A ‘signal-jamming’ theory of predation.” Aprevious version circulated under the title “Fooling Some of the People Some of the Time: Advertising and ConsumerMemory.” I am grateful to Eric Anderson, Gary Becker, Dan Benjamin, Meghan Busse, Raj Chetty, Glenn Ellison,Roland Fryer, Drew Fudenberg, Matthew Gentzkow, Ed Glaeser, Claudia Goldin, Daniel Hojman, Richard Holden,Mark Israel, Emir Kamenica, Larry Katz, David Laibson, Casey Mulligan, Emily Oster, Ariel Pakes, GiacomoPonzetto, Canice Prendergast, Matthew Rabin, Andrei Shleifer, Jeremy Tobacman, Miguel Villas-Boas and seminarparticipants at Harvard University, the Quantitative Marketing and Economics Conference, and the UCLA AndersonSchool for helpful comments. Karen Bernhardt and Fuhito Kojima provided excellent research assistance. I thankthe Institute for Humane Studies, the Center for Basic Research in the Social Sciences, and the National ScienceFoundation for support, and the James B. Kilts Center for Marketing at the University of Chicago Graduate Schoolof Business for data. e-mail: jmshapir@uchicago.edu.

1

1 Introduction

Most economic theories treat advertising either as a means of conveying information about a brand

(Stigler, 1961; Nelson, 1974; Khilstrom and Riordan, 1984; Milgrom and Roberts, 1986) or as a

direct argument in the consumer’s demand or utility function (Dixit and Norman, 1978; Becker and

Murphy, 1993). The first approach provides a highly structured account of how advertising works,

but seems at odds with the persistent advertising of familiar brands,1 and with the intuition that

much advertising does not convey information.2 Because of its generality, the second approach can

avoid these difficulties, but for the same reason it has yielded fewer specific, testable implications.

In this paper I present a theory of advertising based on limited consumer memory. In the

model, consumers learn through experience how much they enjoy consuming a firm’s product.

Each consumer stores in memory the utility he has received from consuming the product during

each past consumption experience. At the point of purchase, the consumer recalls the utility from a

subset of these experiences to memory, and uses this information to infer his tastes for the product.

This inference then guides the consumer in deciding whether or not to buy.

The key assumption in the model is that the firm can use advertising to change the likelihood

that the consumer will remember a favorable consumption experience. This assumption, which is

consistent with a large literature in the psychology of memory (Mitchell and Johnson, 2000), allows

the firm to affect the consumer’s beliefs without possessing any private information about the

quality of the brand. In parallel with the economic literature on “signal jamming” (Fudenberg and

Tirole, 1986; Holmstrom, 1999), I assume that a consumer does not directly observe (or remember)

the amount of advertising to which he was exposed, but that the consumer has correct beliefs about

the firm’s strategy in equilibrium. In other words, although consumers’ memories are bounded, they

are sophisticated in knowing this limitation, and make correct inferences in equilibrium.

I consider first the question of whether more experienced consumers make better or worse

targets for advertising. I show that the influence of the consumer’s level of experience depends

1Milgrom and Roberts (1986) note in their discussion of the signalling hypothesis that their “analysis is strictlyapplicable only to new products whose quality is not generally known. Thus it says little about advertising forestablished brands.” To take an example, Coca-Cola has been the U.S. soft drink market leader for over 80 years(Aaker, 1991 ) and annually sells over 10 gallons per capita (Beverage Aisle, 2001), yet it spends over $200 millionannually on advertising media in the U.S. See Hahn (2003) for an attempt to resolve this issue within the Milgrom-Roberts (1986) framework .

2Becker and Murphy (1993) discuss criticisms of information-based models along these lines.

2

critically on the psychological mechanism by which advertising affects recall. When advertising

works by converting memories of bad experiences into memories of good ones, then more experienced

consumers are less sensitive to advertising, because the likelihood of any one experience being pivotal

is lower the more experiences the consumer has had. In this case, the memory-jamming framework

makes the same prediction as an information-based model: the equilibrium level of advertising falls

as consumers become more experienced with the product. Indeed, I show that advertising will not

occur at all if consumers are sufficiently experienced with the product.

By contrast, when advertising works by selecting which of a consumer’s experiences is most likely

to be recalled–and thus making a consumer’s favorable experiences more likely to be remembered

than his unfavorable ones–advertising levels tend to increase with consumer experience. The

reason is that a more experienced consumer is more likely to have had favorable experiences that

exposure to advertising can induce him to recall. This may help to explain why highly familiar

brands often continue to advertise, why early entrants into some markets advertise more than late

entrants (Bronnenberg, Dhar, and Dubé 2005), or why in some markets advertising levels rise as a

brand matures (Horstmann and MacDonald, 2003).

Next, I examine whether a firm’s level of advertising will be increasing or decreasing with the

quality of its product. I present an extension of the basic model in which all consumers have the

same taste for the product, and this taste parameter is a random variable whose value is known to

the firm. When advertising works by converting memories of unfavorable experiences into memories

of favorable ones, higher product quality is associated with less advertising, since with a high-

quality product there are relatively few negative experiences to replace. When advertising works

by selecting favorable experiences for recall, low-quality products offer too few positive experiences

to draw on, and high-quality ones offer too little scope for increasing the likelihood of a favorable

recollection. Therefore advertising levels tend to be greatest for intermediate values of product

quality, where there are enough favorable experiences available for selection, and enough unfavorable

ones to make advertising worthwhile. This finding suggests that “selection” advertising is most

effective when there is significant variation in consumers’ experiences with a product.

Finally, I turn to the question of increasing returns in consumers’ response to advertising. I

extend the model to allow the firm to make several advertising investments, each of which affects

3

a separate “recall event.” Since advertising increases the likelihood of a favorable recollection,

consumers who recall unpleasant experiences with a product can have high confidence that these

experiences are genuine and not the result of exposure to an advertisement. As a result, negative

recollections carry greater weight than positive ones, and recalling a single negative experience may

be enough to prevent a consumer from buying a product even if all her other recollections are

positive. This in turn means the firm will try to insure that all of the consumer’s memories are of

good experiences. The firm’s desire to “saturate” the consumer with advertising leads to increasing

returns over low levels of advertising. I show that in equilibrium the firm either advertises at all

recall events or not at all, and that there can be equilibria in which the firm randomizes between

advertising and not advertising. These mixed-strategy equilibria resemble the common phenomenon

of “pulsing,” in which a firm oscillates between no advertising and some positive amount (Dubé,

Hitsch, and Manchanda, 2005). Moreover, the model’s prediction of an “S-shaped” advertising

response function is consistent with empirical evidence (Dubé, Hitsch, and Manchanda, 2005).

In addition to providing microfoundations for a number of empirical regularities in advertising

behavior, this paper also shows that caution is needed in interpreting standard tests for informative

advertising. Many of the predictions often used to “test” informative advertising–that advertising

is less effective for more experienced consumers (Ackerberg, 2001), that consumers’ advertising

response will display diminishing returns (Simon and Arndt, 1980), and that advertising levels rise

with the quality of the brand (Caves and Greene, 1996)–are also possible outcomes of the memory-

jamming model, in which advertising is not informative in the traditional sense. This observation

suggests that further work may be needed in order to more convincingly separate informative from

non-informative accounts of advertising.

Formally, the model I present is most closely related to the literature on “signal jamming”

(Fudenberg and Tirole, 1986; Holmstrom, 1999). In the spirit of these papers, I allow the firm

to take an action that affects the information recalled by consumers. Although in equilibrium

consumers have correct beliefs about the firm’s strategy, the possibility of an unobserved deviation

by the firm is sufficient to sustain advertising in equilibrium, and to generate the comparative

statics described above.

Substantively, the paper contributes to the literature on formal models of limited memory

4

(Rubinstein ,1998; Mullainathan, 2002; Wilson, 2004).3 My model differs from these treatments in

that I allow an outside agent to manipulate the formation and recall of a decision-maker’s memory.4

In this sense, the paper draws on recent developments in the economics of persuasion (see, e.g.,

Becker, 2001; DeMarzo, Vayanos and Zwiebel, 2003; Glaeser, 2004 and 2005; Mullainathan and

Shleifer, 2005a and 2005b; Murphy and Shleifer, 2004; Gentzkow and Shapiro, 2004 and 2005).

More broadly, the paper fits into a growing literature on industrial organization with boundedly

rational consumers (DellaVigna and Malmendier, 2004; Gabaix and Laibson, 2004; Spiegler, 2004

and 2005; Eliaz and Spiegler, 2004).

The remainder of the paper is organized as follows. Section 2 briefly reviews psychological

evidence on limited memory, with an emphasis on remembering the context in which a memory

was formed. Section 3 presents the model. Section 4 discusses the effect of consumer experience on

the incentive to advertise. Section 5 provides predictions regarding the relationship of equilibrium

advertising levels to unobserved product quality. Section 6 explores reasons for increasing returns

in the consumer’s response to advertising. Section 7 concludes.

2 Relevant Findings in the Psychology of Memory

This section reviews research in the psychology of memory that motivates the model’s assumptions.5

First, human memory is imperfect. Second, individuals have difficulty separating fact from fiction

in the course of remembering experiences or sensations. Third, advertising and other forms of

suggestion can affect the recall of product experiences, both by converting bad memories into good

ones, and by affecting which evidence consumers attend to in forming their judgments. Finally,

consumers attempt to take efforts to persuade into account when reacting to information they

receive about products.

3The model I present also draws on Gilboa and Schmeidler’s (1995) formulation of memory as a record of theutility associated with past actions. Jehiel (2005) develops a notion of coarse memory applicable in strategic settings.See Mehta, Rajiv and Srinivasan (2004) for an empirical model with imperfect recall of product evaluations.

4Loginova (2005) presents a model of memory for advertising, in which advertising serves to alert consumers to theexistence of a product. Krähmer (2005) considers a signalling model in which advertising helps consumers remembertheir experiences. He also discusses the welfare implications of a model in which advertising distorts memoriesand consumers are non-Bayesian in their response to these distortions. Finally, Brekke and Rege (2005) develop aframework in which consumers cannot distinguish between the recommendations of friends and the recommendationsof advertisers.

5See Schacter (1996) for an overview of the psychology of memory.

5

Memory for facts is limited. As everyday experience suggests, human memory, while impressive

in many respects, is fundamentally limited. Different eyewitnesses to the same events often give

radically divergent testimony (Haber and Haber, 2000). Short-term digit recall is bounded, and

the bounds tighten with age (Dobbs and Rule, 1989). Indeed, the act of remembering one item

may itself interfere with subsequent recall of another (Anderson, Bjork, and Bjork, 1994).

Memory for source is limited. Even when an individual remembers a particular item or sensa-

tion, she may become confused about the origin of the memory. This can in turn lead to confusion

about the meaning of the memory or feeling. Jacoby, Kelley, Brown, and Jasechko (1989) demon-

strated this possibility for the sensation of familiarity. The researchers exposed participants to a

sequence of names of people they were told were famous and some they were told were not famous,

such as the fictitious person “Sebastian Weisdorf.” Participants taking a short quiz immediately

after seeing the sequence of names were generally able to correctly classify names as famous or

non-famous. In a similar quiz given the following day, however, the participants frequently thought

that Sebastian Weisdorf was a famous person, presumably because his name sounded familiar due

to the previous day’s proceedings. These participants were correct in thinking that the name “Se-

bastian Weisdorf” was familiar, but were confused about the source of this familiarity. In this sense,

Jacoby et al were able to make Sebastian Weisdorf a “celebrity overnight” by taking advantage of

participants’ confusion about the origins of their sense of familiarity.6

A related literature in psychology known as “reality monitoring” argues that individuals do

not effortlessly recall whether a given memory or feeling was generated by a real experience or

an imagined one. Rather, the source of a memory must be reconstructed using features of the

memory as well as knowledge of how memories are formed in different contexts. Support for this

view has accumulated from studies of memory for images (Johnson, Kounios, and Reeder, 1994),

autobiographical events (Johnson, Foley, Suengas, and Raye, 1988), words and phrases, and other

types of events and stimuli.7

Exposure to advertising can convert bad experiences into good memories. The fact that imagined

experiences can be remembered as real has obvious implications for persuasion in general, and

6This study relates to a literature on “sleeper effects” in persuasion (Pratkanis, Greenwald, Leippe, and Baum-gardner, 1988).

7See Johnson (1988) for a review and Mitchell and Johnson (2000) and Johnson, Hashtroudi, and Lindsay (1993)for reviews of source monitoring more generally.

6

advertising in particular. Indeed, experimental research supports the view that advertising can

affect the way in which experiences are remembered.8 Braun (1999) had participants taste samples

of a purportedly new brand of orange juice, experimentally manipulated to taste good or bad. An

experimental group was then shown advertisements for the brand, and within one hour of the taste

test participants were asked to describe their experiences. Those who had been exposed to an

advertisement were more likely to describe their experiences positively regardless of the true taste

of the juice. More strikingly, when asked to pick the juice they had tasted out of a juice “lineup,”

participants exposed to advertising systematically identified a superior juice than the one they had

actually tried, even when they had taste-tested obviously bad juice. This finding suggests that the

advertising intervention changed participants’ recollection of the sensory experience of tasting the

juice. A follow-up study (Braun-LaTour and LaTour, 2005) shows that these effects can result from

advertising exposure either before or after the product experience, as long as there is a sufficient

lag between the experience and the evaluation.

Advertising can induce selective attention to favorable evidence. Even if advertising does not

affect what consumers remember, it can still affect which pieces of information they focus on. Hoch

and Ha (1986) subjected experimental participants to an advertising manipulation, and gave them

an opportunity to inspect the advertised brand as well as a set of competitors. In the case of

a product with ambiguous, hard-to-evaluate quality (polo shirts), the opportunity to inspect the

product made the advertisement more effective in increasing participants’ ratings of target brand

quality. However, in the case of a product with unambiguous, easily testable quality (paper towels),

advertising had less of an effect on the quality assessments of participants who could inspect the

product. The authors interpret these finding as evidence that advertising can induce selective

search and hypothesis-testing, biased towards confirmation of the expectations induced by the

advertisement. When product experience is unambiguous, there is little discretion in interpreting

the facts, but when product experiences are ambiguous, there is ample opportunity for selective

interpretation. Interestingly, they report that advertising seen after product inspection also induced

8There is even some evidence that exposure to advertising can affect whether an individual believes a givenautobiographical experience occurred. Braun, Ellis, and Loftus (2002) show that an advertising manipulation cancause adults to believe that, as children, they shook hands with Mickey Mouse at Disney World. This effect is presenteven when the plausible character (Mickey Mouse) is replaced with an implausible one (Bugs Bunny) who does notappear at Disney World. See Aiken (1999) for an overview of the marketing literature on “manufactured memories.”

7

favorable attitudes toward the target brand, although less so than advertising seen before product

inspection. Although their experiment does not test this hypothesis directly, the authors speculate

that this finding may be due to selective recall of product features favorable to the target brand.

Consumers try to account for persuasion in forming their beliefs. If consumer memory is sub-

ject to manipulation, then we would expect consumers to take this fact into account when forming

their beliefs or attitudes about a product. Indeed, Briñol, Petty and Tormala (2004) show that

consumers have less confidence in their response to advertising from an interested party (a firm

selling a product) than from a disinterested one (a regulatory agency disseminating product in-

formation), and that this difference in confidence translates into a difference in persuasiveness. A

broader literature on “persuasion knowledge” (Wright, 1973; Friestad and Wright, 1994) argues

that consumers’ knowledge of the tactics of persuaders (and the psychological mechanisms through

which persuasion operates) can reduce the impact of an advertisement or other persuasive message.

Although these arguments do not address source memory per se, they do indicate that consumer

awareness of persuasive strategies can play an important role in belief and attitude formation.

In the next section, I present a model of advertising with limited consumer memory that in-

corporates the elements outlined above. In the model, consumers’ recollections of past product

enjoyment can be manipulated by exposure to advertising. Motivated by the psychological evi-

dence, I consider two alternative mechanisms by which this might occur. In the first, advertising

directly converts bad experiences into good memories. In the second, advertising induces selective

memory of the consumer’s most favorable experiences. As I show in greater detail below, these

alternative technologies yield very different predictions about the comparative statics of advertising

levels.

3 A Model of Advertising as Memory Jamming

3.1 Model Setup and Notation

In this section, I present a model in which advertising changes consumers’ recollections of their

past experiences with a product. In the model, a single firm sells a good to a continuum population

of consumers of unit mass. Each consumer has a type θi ∈ [0, 1], and types are distributed in

8

the population according to a probability density function f (), which has full support on the unit

interval (and no mass points).

A consumer’s type determines his likelihood of enjoying the firm’s product. In particular, I

assume that the utility Ui that a consumer i will obtain from consuming the good takes values in

{0, v}, with v > 0, and that

Pr (Ui = v) = θi. (1)

In other words, on a given consumption occasion, a consumer either enjoys the product or he does

not, and the probability of enjoying the product is given by the consumer’s type θi. I will assume

that consumers do not know their types, but that they do know the prior distribution f () from

which their types are drawn. For now, I will also assume that the firm has no private information

about any consumer’s type.9

Although consumers do not initially know their types, they can learn about their types through

experience with the product. In particular, each consumer has N exogenously given experiences

with the product, where each experience n has a value Uin ∈ {0, v}. The probability of a good

experience (an experience with value v) is θi, in parallel with the distribution of consumption utility.

Conditional on the consumer’s type, the values Uin are drawn independently of one another.10

Each consumer i’s experiences comprise a vector (Ui1, Ui2, ..., UiN) that I will call the consumer’s

memory. If Uij > Uik for some j and k, then I will say that the jth experience is more favorable

than the kth experience. In particular, I will describe as favorable any experience j with Uij = v.

Because the elements of this vector are drawn independently, the statistical information in the

consumer’s memory can be fully captured by a single number Ki, equal to the number of favorable

experiences in the consumer’s memory. Note that for a given type θi, E (Ki | θi) = θiN .

Immediately prior to his purchase decision, each consumer has a recollection ri ∈ {0, v} about his

9Because many existing economic models of advertising are based on informational asymmetries (Nelson, 1974;Khilstrom and Riordan, 1984; Milgrom and Roberts, 1986), the assumption that the firm has no private informationis helpful in distinguishing the intuitions I present here from those that have been explored elsewhere. I consider acase with private information in section 5.10 In reality, of course, the number of experiences N will evolve endogenously over the life of the brand and the

consumer, with high-θ consumers tending to consume more on average than low-θ consumers. With an endogenousN , a consumer’s memory will contain information about his past enjoyment of the product as well as about his pastbeliefs. I have not been able to prove comparative statics for this case without imposing very strong restrictions onthe structure of the game.

9

past experiences with the product.11 The distribution of ri depends on whether the consumer has

been exposed to advertising or not. A share d (a) ∈ [0, 1) of consumers are exposed to advertising,

where d () is a strictly increasing function and a is the firm’s level of advertising expenditure. For

simplicity, I will normalize d (0) = 0.

If a consumer is not exposed to advertising, then his recollection ri is a randomly chosen element

of his memory. That is,

Pr (ri = v | Ki) =KiN

(2)

so that the probability that the consumer recalls a good experience is proportional to his number

of good experiences with the product. Note, in particular, that for a given type θi,

E [Pr (ri = v | Ki) | θi] = θi. (3)

For a consumer who has been exposed to advertising, the probability of recalling a good expe-

rience is given by a function m (Ki;N) of his number of good experiences Ki and his total number

of experiences N :

m (Ki;N) ≡ Pr (ri = v | Ki) . (4)

For a given type, I will denote the probability of a positive recollection by

μ (θi;N) ≡ E (m (Ki;N) | θi) (5)

where the expectation is taken over the (binomial) distribution of the number of good memories,

Ki, conditional on the consumer’s type θi. I will assume that m (Ki;N) is nondecreasing in Ki for

a given N , so that the probability of remembering a good experience is at least weakly increasing

in the consumer’s number of good experiences. I will further assume that m (Ki;N) ≥ (Ki/N) for

all Ki ∈ {0, 1, ..., N}, with this inequality strict for at least one value of Ki ∈ {0, 1, ..., N}. In other

words, a given consumer is more likely to recall a favorable experience with the product if she has

11Alternatively, one could think of ri as the consumer’s mood or emotional state when thinking about the product.See Damasio (1995) for evidence that individuals use their emotional states to guide their decision-making in choicesituations. See also Iyer and Kuksov (2005) for a model in which a firm attempts to alter a consumer’s mood throughchoice of lighting, ambient music, etc.

10

been exposed to advertising than if she has not.

At the point of purchase, each consumer evaluates the product based on his expected utility

from consuming it. Importantly, I assume that consumers base their posterior beliefs only on their

recollections ri and on their conjectures about the firm’s behavior. In other words, at the point of

purchase a consumer does not know whether or not he was exposed to advertising. Moreover, he

does not directly observe the firm’s advertising policy, although in equilibrium he will have correct

beliefs about it.

The assumption that consumers base their inferences only on their recollection ri is a simple

way to formalize the psychological evidence described in the previous section, which indicates

that individuals sometimes have difficulty recalling the source of their memories. Although the

assumption that the influence of advertising is undetectable probably goes too far, this extreme

formulation helps to highlight the role of limited source memory. The further restriction that the

consumer bases his decisions on his recollection ri and not on his entire vector of experiences is

necessary in order for advertising to impact behavior: without this assumption, the consumer would

base his decisions exclusively on his memory, ignoring his recollection ri, and thus eliminating any

influence of advertising.

The assumption that the firm’s advertising policy is not directly observable provides the formal

link with the literature on signal jamming (Fudenberg and Tirole, 1986; Holmstrom, 1999). Note

that, because the consumer bases his decision only on his recollection ri, it is unproblematic to allow

consumers to observe the firm’s advertising at some earlier stage. Because consumers’ memories

are limited, all that matters is that the firm’s advertising policy is not directly observable when the

consumer makes his purchase decision.

I will assume that each consumer faces an idiosyncratic transportation cost of purchasing the

product, distributed uniformly in the population on the interval [0, v] (independently of types θi). A

consumer purchases the product if his expected utility from consumption exceeds this transportation

cost. This assumption greatly simplifies analysis of the firm’s problem, because it implies that a

given consumer i’s probability of purchase is equal to his posterior expectation E (θi | ri).12

12To see why, let ti denote consumer i’s transportation cost. Given a recollection ri, the consumer will purchasethe product if

E (θiv | ri) > ti

11

So, to summarize the timing of the model:

Period 0 : Each consumer has N experiences with the product.

Period 1 : The firm chooses its advertising level a, and a share d (a) of consumers are exposed to

advertising.

Period 2 : Each consumer draws a recollection ri from memory.

Period 3 : Consumers form posterior beliefs based on their recollections, and decide whether or not

to purchase the product.

3.2 Two Special Cases

The model so far implies that exposure to advertising increases the likelihood of a favorable rec-

ollection, but it does not tell us how this influence comes about. The discussion of psychological

evidence in section 2 suggests two ways in which advertising might affect consumers’ recollections.

First, seeing an advertisement might change a consumer’s memory, converting bad experiences into

good ones. Second, advertising might affect which experiences are selected from memory, making

consumers more likely to recall their more favorable experiences with the product.

To formalize the first concept, I will consider a conversion technology, in which exposure to

advertising converts some number C, 1 · C · N , of the consumer’s experiences into favorable ones

(value v), regardless of their true value. The experiences to be converted are chosen at random from

the consumer’s memory. I will assume that this technology leaves the recall process unchanged,

so that a consumer’s recollection is still a randomly chosen element of his memory. In this case,

however, the value of the recalled element may have been altered by exposure to advertising.

Formally, the conversion technology can be represented as

m (Ki;N) =

µ1−

C

N

¶µKiN

¶+C

N(6)

This expression has a natural interpretation: the consumer’s recollection is a random draw from

his memory, which, if unconverted, has probability³KiN

´of being favorable. With probability

or, equivalently, if

E (θi | ri) > tiv

Because ti/v is distributed uniformly on [0, 1], the probability of purchase is given by the cdf of the uniform distributionevaluated at E (θi | ri), which is simply equal to E (θi | ri).

12

¡1− C

N

¢, the recalled experience is unconverted, so

³KiN

´is indeed the probability of a favorable

recollection. But with probability CN , the consumer will recall a converted experience, in which case

his recollection is favorable with probability 1. In the limit case of C = 0, advertising has no effect

on the likelihood of a favorable recollection, since advertising will not replace any experiences with

favorable ones. At the other extreme, when C = N , exposure to advertising guarantees a favorable

recollection.

To formalize the second concept, I will consider a selection technology, in which exposure to

advertising changes the vector of experiences from which the consumer’s recollection is drawn.

Under this type of advertising, a consumer’s recollection is a randomly chosen element of his S

most favorable experiences, with 1 · S · N . If Ki · S for some consumer i, then advertising

makes remembering a favorable experience more likely, since it restricts the list of experiences from

which the consumer’s recollection is drawn. If Ki > S , then the consumer is certain to have

a favorable recollection if he is exposed to advertising. Formally, selection advertising will act

according to

m (Ki;N) =1 if Ki ≥ S

KiS if Ki < S

(7)

Observe that in the limit case of S = N , exposure to advertising has no effect on the probability of

a favorable recollection, since advertising will not impose any restriction on the list of experiences

from which the consumer’s recollection is drawn. This case is therefore equivalent to conversion

advertising with C = 0. By contrast, for fixed S as N becomes large the selection case approaches

the case of conversion advertising with C = N , since the consumer will almost certainly have at

least S favorable experiences from which to choose.

Although these two special cases are not the only possible specifications of the advertising tech-

nology, they capture two important mechanisms suggested by the psychological evidence, and turn

out to have very different implications regarding equilibrium advertising levels. I will therefore use

them throughout the paper to highlight the relationship between assumptions about the advertising

technology and predictions about advertising behavior. It is worth noting that, while I will treat

the parameters C and S as exogenous, in reality they may be a function of the firm’s advertising

policies. For example, they may relate to the choice of advertising content, with more effective

13

advertising content resulting in more converted experiences (greater C) or with greater selectivity

among remembered experiences (smaller S).

3.3 Equilibrium Advertising Policies

Given an exogenous unit markup,13 the firm will choose its advertising policy a so as to maximize

the share of consumers who purchase its product. This, in turn, will depend on the share of

consumers whose recollections are positive (have value v). Recall that for a consumer of type θi

who has not been exposed to advertising, the probability of a recollection of value v is simply θi;

for a consumer who has been exposed to advertising, the probability is given by μ (θi;N).

We can write the firm’s profit function as

π (a; a) = Pr (ri = v | a)E (θi | ri = v; a) + Pr (ri = 0 | a)E (θi | ri = 0; a)− a. (8)

Here I abuse notation slightly and use a to denote consumers’ conjectures about the firm’s level of

advertising expenditure, so that expectations conditioned on a are Bayesian posterior expectations

given an assumption about the firm’s strategy. Substituting the advertising technology into the

expression for π (a; a) gives

π (a; a) = [d (a)E (μ (θi;N)) + (1− d (a))E (θi)]E (θi | ri = v; a) (9)

+ [d (a) (1− E (μ (θi;N))) + (1− d (a)) (1− E (θi))]E (θi | ri = 0; a)

−a.

13The assumption of exogenous prices is a useful simplification. Allowing the firm to state an observable pricewould open up the possibility of sustaining uninteresting equilibria using pessimistic off-equilibrium beliefs aboutconsumer types that “kick in” whenever an unexpected price is posted. Ruling out these equilibria would require theuse of a refinement or an augmentation of the model in which prices are observed with some noise. I have thereforechosen to put these issues aside by fixing the price of the good exogenously, but I note that prices could be readilyincorporated into the model, subject to the above-mentioned caveat.

14

This expression is algebraically equivalent to:

π (a; a) = d (a) [E (μ (θi;N))− E (θi)] [E (θi | ri = v; a)− E (θi | ri = 0; a)] (10)

+E (θi)E (θi | ri = v; a) + (1− E (θi))E (θi | ri = 0; a)

−a.

Because expressions that do not depend on a are constant from the firm’s perspective, the second

row of this expression is irrelevant to the firm’s behavior. We can therefore treat the firm as though

it is maximizing the following objective function:

π (a; a) = d (a) [E (μ (θi;N))− E (θi)] [E (θi | ri = v; a)− E (θi | ri = 0; a)]− a. (11)

This expression has a natural interpretation. The effect of advertising depends on three components:

the share of consumers who see an ad, the effect of seeing an advertisement on the probability of

a favorable recollection, and the effect of a favorable recollection on the probability of purchase.

The above expression for π (a; a) is simply the product of these three factors, less the cost a of

advertising.

The firm will choose a level of advertising a that maximizes π (a; a), taking as given consumers’

conjecture a. An equilibrium is an advertising policy such that the policy is optimal for the firm,

given that consumers expect the firm to choose it. Note that in any equilibrium, the share of

consumers who will purchase the firm’s product is E (θi), exactly the level of demand that would

obtain with no advertising. (This is true by the law of iterated expectations.) In other words, the

firm would like to commit not to advertise. Although there is evidence that, as a group, firms in

an industry might like to contract to limit their advertising (e.g. Eckard, 1991), this seems unlikely

to hold for a single firm acting on its own. This counterfactual implication of the model depends

on the assumption that consumers know the firm’s strategy in equilibrium; a model with partial

filtering or naïveté on the part of consumers, perhaps in the spirit of Eyster and Rabin (2005),

might lead to firm profits being improved in equilibrium by the option to advertise.

In order to characterize the equilibria of this game, it will be helpful to impose several regu-

15

larity conditions on the function d () that determines the share of consumers who are exposed to

advertising:

Assumption 1 The function d () is everywhere (i) strictly increasing, (ii) thrice differentiable,

and (iii) strictly concave.

Note, in particular, that these assumptions imply that

limx→∞ d (x) = d (12)

for some constant d · 1, with d (x) < d for all x ≥ 0.

It follows from assumption 1 that the firm’s problem is concave, so that there is a uniquely

optimal advertising policy given any conjectures on the part of consumers. This immediately rules

out equilibria in mixed strategies, and allows us to restrict attention to cases in which the firm

plays a single advertising level a. Assumption 1 also implies that a level of advertising a > 0 is

optimal for the firm if and only if it satisfies the firm’s first-order condition. As a result, a level of

advertising a∗ is an equilibrium if and only if either (i) a∗ = 0 and π1 (0; 0) · 0, or (ii) a∗ > 0 and

π1 (a∗; a∗) = 0. (I use π1 () to denote the derivative of π with respect to its first argument.)

If π1 (0; 0) · 0, then there is an equilibrium with no advertising. If not, i.e. if π1 (0, 0) > 0,

then establishing the existence of an equilibrium requires showing that the function π1 (a; a) must

have a positive root. We can write this function as

π1 (a; a) = d0 (a) [E (μ (θi;N))− E (θi)] [E (θi | ri = v; a)− E (θi | ri = 0; a)]− 1. (13)

Because assumption 1 implies that lima→∞ d0 (a) = 0, and because the expression multiplying d0 (a)

is bounded, it is clear that lima→∞ π1 (a; a) < 0. It then follows from the intermediate value theorem

that, if π1 (0, 0) > 0, there is at least one level of advertising a∗ > 0 such that π1 (a∗; a∗) = 0, so

an equilibrium must exist. To summarize:

Lemma 1 Under assumption 1, for all N ≥ 1, there exists at least one equilibrium advertising

policy, and all equilibria are in pure strategies. If π1 (0, 0) > 0, then advertising occurs in any

equilibrium.

16

Because it greatly simplifies the analysis of equilibrium, I will maintain assumption 1 throughout

the paper without explicit reference. In general, uniqueness is not guaranteed by this assumption,

and would require much stronger and less appealing restrictions. To see why, consider that unique-

ness would be established if the equilibrium equation π1 (a; a) were strictly decreasing in a; that

is, if the incentive to advertise given correct consumer beliefs were decreasing in the level of ad-

vertising. The concavity of the function d () implies that this is true for a given set of consumer

beliefs, i.e. that π1 (a, a) is strictly decreasing in a for a given a, since the derivative d0 (a) is strictly

decreasing in a. But to determine how π1 (a; a) varies with a, we must also account for the effect

of a change in a on the difference in expectations between a consumer who has recalled a positive

experience and one who has recalled a negative experience. This difference may actually increase

as the advertising level increases.

For example, suppose advertising works by inducing the consumer to recall his most favorable

experience with the product (which is the selection model described above with S = 1). In this

case, a consumer i with a large number of experiences N and an unfavorable recollection (ri = 0)

knows that, if he has been exposed to advertising, all of his experiences were unfavorable, which

is strong evidence that his type θi is low. By contrast, if he has not been exposed to advertising,

an unfavorable recollection is only moderate evidence of a low type. Therefore, the more likely is

the consumer to have been exposed to advertising, the more an unfavorable recollection will reduce

his posterior expected type, and therefore the greater is the firm’s incentive to induce a favorable

recollection. In such an instance, multiple equilibria are possible, since the incentive to advertise

may be low if consumers expect little advertising, but high if they believe exposure to advertising

is likely.

3.4 Consumer Welfare and Endogenous Advertising Exposure

In the memory-jamming framework, the effect of an advertising exposure on consumer welfare is

ambiguous, and depends on the specific formulation of the advertising technology. Under conversion

advertising, for example, the consumer can always “pretend” to have been exposed to advertising

by replacing his recollection with a favorable one with some probability. This makes it clear that

exposure to conversion advertising cannot make the consumer better off. In general, it will make

17

him strictly worse off, because exposure to conversion advertising reduces the information content

of the consumer’s memory, and therefore reduces his probability of making a correct purchase

decision.

Under selection advertising, the advertising exposure may have positive effects on consumer

welfare. To see why, suppose that S = 1, so that a consumer exposed to selection advertising

will have a favorable recollection unless all his past experiences were unfavorable. In this case, a

consumer exposed to advertising will learn his entire experience history with certainty if he has had

only unfavorable experiences. By contrast, a consumer who has not been exposed to advertising will

never know with certainty whether his unfavorable recollection is an anomaly or is representative of

his other experiences. As a result, it is possible for advertising exposure to improve the consumer’s

information in expectation, although this advantage tends to diminish as the number of experiences

becomes large.

As Becker and Murphy (1993) point out, consumers make a host of choices that affect their

likelihood of exposure to advertising, and firms often compensate consumers for their willingness

to accept advertising messages. In a model with a market for advertising exposure, conversion

advertising would tend to be associated with significant compensation, or with circumstances in

which advertising avoidance is more difficult. On the other hand, selection advertising would tend

to require less compensation and could even be sought out by consumers.

4 Consumer Experience and the Incentive to Advertise

The persistent advertising of highly familiar brands such as Coca-Cola poses a challenge to standard

information-based models of advertising, which typically predict that advertising will decline as

consumers become more experienced with a product (Milgrom and Roberts, 1986; Ackerberg, 2001).

In this section, I ask whether additional product experience inhibits or encourages advertising. I

begin by discussing the two special cases of conversion and selection advertising. In the former case,

advertising is declining in consumer experience and disappears in the limit as consumers become

highly experienced. In the latter case, advertising persists even as consumers become arbitrarily

experienced with the product. I then provide comparative statics results for generalizations of these

18

cases, and show how the nature of the advertising technology determines the effect of experience

on equilibrium advertising levels.

4.1 Advertising as Memory Conversion

Consider, first, the case of conversion advertising, in which advertising works by replacing C ran-

domly chosen experiences with favorable ones. In this case, the probability of a favorable recollec-

tion, conditional on a consumer’s type θi and number of experiences N , can be written as

μ (θi;N) =

µ1−

C

N

¶θi +

C

N. (14)

With probability¡1− C

N

¢, the consumer’s recollection is favorable with probability θi, just as if

he had not been exposed to advertising. With probability¡CN

¢, however, the consumer is certain

to recall a favorable experience, regardless of his type. Exposure to advertising therefore makes

a consumer’s recollection less dependent on his type, and hence less informative. Because the

consumer knows this, the impact of a favorable recollection on his posterior expected type is smaller

the more likely he is to have been exposed to advertising. Consequently, the firm’s incentive to

advertise, π1 (a; a), is strictly decreasing in the level of advertising a, which implies the existence

of a unique equilibrium advertising level a∗ (N).

The effect of an advertising exposure on the probability of a favorable recollection can be

rewritten as

μ (θi;N)− θi =C

N(1− θi) . (15)

The right-hand expression is strictly decreasing in N . On first glance, this observation would seem

sufficient to guarantee that the equilibrium level of advertising a∗ (N) is decreasing in N . It is not

sufficient, however, because the incentive to advertise depends not only on the effect of advertising

on the probability of a favorable recollection, but also on the effect of a favorable recollection on

the consumer’s posterior expected type. As N increases for a given level of advertising, a consumer

becomes more trusting of his recollections, because for fixed C the probability that he has recalled

a “converted” memory is lower. Qualitatively, this effect would tend to lead to greater advertising

as N increases. Quantitatively, however, it is small, and is dominated by the direct effect of a

19

change in N , so that the equilibrium advertising level is declining in consumer experience.

Finally, since the effect of advertising on the likelihood of a favorable recollection tends to 0

as N becomes large, as experience increases there comes a point where advertising does not occur

in equilibrium. Intuitively, when consumers are highly experienced, the probability of recalling a

“converted” experience is extremely small, so that advertising is relatively ineffective. The following

proposition, proved in the appendix, summarizes the characterization of the equilibrium for the case

of conversion advertising.

Proposition 1 Under conversion advertising, there is a unique equilibrium advertising level a∗ (N) ≥

0, which is weakly decreasing in N (strictly whenever a∗ (N) > 0). Moreover, there exists N 0 ≥ 1

such that a∗ (N) = 0 for all N > N 0.

Before concluding this section, it will be helpful to make the following observation about the

limit case of C = N :

Corollary 1 Under conversion advertising, if C = N there is a unique equilibrium advertising

level a∗∗ ≥ 0 whose value is independent of the number of experiences N . Moreover, the advertising

level a∗∗ is strictly positive for sufficiently large d0 (0).

The fact that the equilibrium advertising level a∗∗ ≥ 0 is independent of N follows directly from

the fact that, if C = N , μ (θi;N) = 1, which is independent of N . To see why a∗∗ > 0 for large

enough d0 (0), suppose toward contradiction an equilibrium with a∗∗ = 0. Following arguments in

subsection 3.3, this is an equilibrium if and only if the firm does not want to increase its advertising

at the proposed level, i.e. if

d0 (0) [1− E (θi)] [E (θi | ri = v; a∗∗ = 0)− E (θi | ri = 0; a∗∗ = 0)]− 1 · 0. (16)

Clearly if d0 (0) is large enough, this condition will fail, and the unique equilibrium advertising level

will be on the interior.14

14Note that the expression [E (θi | ri = v; a = 0)−E (θi | ri = 0; a = 0)] is independent of d0 (0), because the ex-pectations are conditional on a = 0. Moreover, this expression must be positive, since the prior distribution on typesθi is non-degenerate, and, given that a = 0, a favorable (unfavorable) recollection is more likely for high (low) types

20

4.2 Advertising as Memory Selection

Recall that in the case of selection advertising, exposure to advertising limits the set of possi-

ble recollections to the consumer’s S most favorable experiences. The probability of a favorable

recollection, conditional on exposure to advertising, is therefore given by

μ (θi;N) = Pr (Ki < S)E

µKiS| Ki < S

¶+Pr (Ki ≥ S) (17)

where Ki ∼ Binomial (θi, N). As N increases, the conditional expectation E (Ki | Ki < S) in-

creases, as does the probability that Ki ≥ S, so that the expression μ (θi;N) is increasing in N for

any θi ∈ [0, 1], strictly if θi ∈ (0, 1). Note also that because E (Ki | θi) = θiN , as N approaches

infinity Pr (Ki ≥ S) tends to 1 for any θi > 0, so that in the limit μ (θi;N) approaches 1 for any

θi > 0.

To build intuition for these properties of μ (θi;N), consider the special case of S = 1, where the

consumer has a favorable recollection if any of his experiences yielded utility v, so that:

μ (θi;N) = 1− (1− θi)N . (18)

Advertising will have a greater effect on the recollections of a more experienced consumer, because

such a consumer offers a longer list of possible experiences to draw on, and therefore a greater

chance that at least one experience was favorable. In the limit as N tends to infinity, even for

small θi the probability of at least one favorable experience tends to one, meaning that in the limit

advertising exposure guarantees a favorable recollection.

Indeed, for general S ≥ 1, in the limit as N becomes large there is a unique equilibrium

advertising level a∗ (N), whose value approaches a finite positive limit whenever d0 (0) is sufficiently

large. As an intuition for this result, note that in the conversion model, when C = N all experiences

than for low (high) types. Formally, the posterior expectations are given by

E (θi | ri = v; a = 0) =E¡θ2i¢

E (θi)

E (θi | ri = 0; a = 0) =E (θi (1− θi))E (1− θi)

from which it follows that E (θi | ri = v; a = 0) > E (θi | ri = 0; a = 0) if and only if E¡θ2i¢> E (θi)

2. This lastinequality must hold for any non-degenerate distribution of θi.

21

are favorable, so that μ (θi;N) = 1. In other words, as N becomes large, the selection model

approximates the conversion model with C = N , which corollary 1 above shows to have a unique

equilibrium advertising level a∗∗ that is strictly positive for d0 (0) sufficiently large. Importantly, and

in contrast to the conversion mechanism, this means that selection advertising need not disappear

in the limit as consumers become highly experienced with the product.

Although these arguments characterize the equilibrium in the limit as N becomes large, they

do not tell us how the equilibrium advertising level changes as N increases. The fact that μ (θi;N)

is increasing in N suggests that the effect of additional consumer experience should be to raise the

equilibrium advertising level, because it makes advertising more effective in producing favorable

recollections. Completing this argument requires addressing two complications: the possibility of

multiple equilibria, and the fact that a change in N will also impact how much the consumer’s

posterior expected type is affected by his recollection. In the next subsection, I treat these issues

in greater generality, and therefore focus here only on the limit properties of the selection model:

Proposition 2 In the case of selection advertising, for large enough N there is a unique equilibrium

advertising level a∗ (N), with

limN→∞

a∗ (N) = a∗∗.

Recall that a∗∗ > 0 if d0 (0) is sufficiently large, so that proposition 2 implies that selection adver-

tising may persist in the limit even as N becomes arbitrarily large.

4.3 Advertising Technology and the Role of Experience

In the case of conversion advertising, additional experiences crowd out the “converted” experiences

and hence reduce the effectiveness of advertising. The equilibrium advertising level is therefore

decreasing in the number of experiences N . In the limit as consumers become highly experienced,

advertising does not occur in equilibrium. By contrast, under selection advertising, more experi-

enced consumers are more likely to have had favorable experiences that the advertising can call

to mind, so that additional experiences increase the effectiveness of advertising. Advertising can

therefore occur even in the limit as consumers become arbitrarily experienced.

These two special cases suggest that the properties of the function μ (θi;N), which relates the

22

consumer’s type and number of experiences to his likelihood of a favorable recollection (conditional

on exposure to advertising), play an important role in determining both the limit properties of the

equilibria as N →∞ and the comparative statics with respect to N . In this subsection, I formalize

this intuition, and characterize how equilibrium advertising changes with consumer experience in

two classes of models that generalize the special cases above. To do so, it will be helpful to introduce

some vocabulary:

Definition 1 Consider an advertising technology that yields a probability μ (θi;N) of a favorable

recollection for a consumer of type θi. Then advertising is a

1. complement for experience if μ (θi;N) is strictly increasing in N for all N ≥ 1 and for almost

all θi ∈ [0, 1]

2. strong complement for experience if advertising is a complement for experience and

limN→∞

μ (θi;N) = 1

for almost all θi ∈ [0, 1].

3. substitute for experience if μ (θi;N) is strictly decreasing in N for all N ≥ 1 and for almost

all θi ∈ [0, 1].

4. strong substitute for experience if advertising is a substitute for experience and

limN→∞

μ (θi;N) = θi

for almost all θi ∈ [0, 1].

It should be immediately clear from the discussion in subsections 4.1 and 4.2 that conversion

advertising is a strong substitute for experience and selection advertising is a strong complement

for experience. In this sense, the concepts defined above generalize these two special cases. Unsur-

prisingly, technologies with strong substitutability or complementarity inherit the limit properties

of conversion and selection advertising, respectively, as the number of experiences N grows large.

For the remainder of this section, I focus on the question of comparative statics with respect to N .

23

Because the advertising game may have multiple equilibria, I will characterize the local com-

parative statics of the stable interior equilibria. Following convention, we may define an interior

equilibrium as stable if a small perturbation to the advertising level around the equilibrium does

not create an incentive for the firm to move further away from the equilibrium policy. Formally, an

interior equilibrium a∗ > 0 is stable if and only if

d

daπ1 (a; a) |a=a∗< 0. (19)

Given this definition, if the incentive to advertise at some stable interior equilibrium a∗ > 0 is

increasing in the number of experiences N ; that is, if

∂

∂Nπ1 (a; a) |a=a∗> 0, (20)

then the level of advertising in that equilibrium is locally increasing in N .

To sign the effect of a change in N , we must therefore evaluate the sign of the partial derivative

∂

∂Nπ1 (a; a) = d0 (a)

µ∂

∂NE (μ (θi;N))

¶(E (θi | ri = v; a)− E (θi | ri = 0; a)) (21)

+d0 (a) (E (μ (θi;N))− E (θi))µ

∂

∂NE (θi | ri = v; a)−

∂

∂NE (θi | ri = 0; a)

¶

The first of these terms is signed by the effect of a change in N on the probability E (μ (θi;N)) that

a consumer exposed to advertising will have a favorable recollection. The sign of this expression

depends on whether advertising is complement or a substitute for experience.

The second expression is more difficult to sign, because it depends on how experience affects

the information content of the consumer’s recollection. Heuristically, if increasing N tightens the

relationship between the probability of a favorable recollection and the consumer’s type, then this

expression will be positive; if increasing N weakens that relationship it will be negative. Note,

however, that for a consumer i who has not been exposed to advertising, the probability of a

favorable recollection is θi regardless of his number of experiences N . In other words, the number

of experiences N only matters for the information content of the consumer’s recollection if the

consumer has been exposed to advertising. Therefore as the probability of advertising exposure

24

becomes small, so does the value of the second expression, regardless of its sign. When advertising

exposure is rare, then, the dominant force is whether additional experiences enhance or detract

from the effect of advertising on the likelihood of a favorable recollection. This conclusion can be

formalized as follows:

Proposition 3 At any level of experience, if the maximum probability of advertising exposure d is

sufficiently small, then the level of advertising in any stable interior equilibrium is (locally) strictly

increasing in the number of experiences N if advertising is a complement for experience, and strictly

decreasing in N if advertising is a substitute for experience. Moreover:

1. If advertising is a strong substitute for experience, then there exists N 0 such that for all

N > N 0 there is no equilibrium with a positive advertising level.

2. If advertising is a strong complement for experience, then there exists N 0 such that for all

N > N 0 there is a unique equilibrium advertising level a∗ (N), with

limN→∞

a∗ (N) = a∗∗.

Recall that a∗∗ > 0 if d0 (0) is sufficiently large, so that the last part of the proposition implies

that equilibrium advertising may reach a positive limit as consumers become highly experienced, if

advertising is a strong complement for experience.

Figure 1 illustrates the relationship between consumer experience N and the equilibrium level

of advertising for conversion and selection advertising, which serve as canonical examples of the

concepts of substitutability and complementarity treated in proposition 3.

4.4 Evidence on the Relationship Between Advertising Levels and Consumer

Experience

Ackerberg (2001) finds that consumers who have already purchased Yoplait 150 yogurt are less

affected by exposure to advertising for that product. Erdem, Keane and Sun (2004) find that

advertising for many products has its greatest effect on marginal consumers, with the interesting

exception of advertising for Heinz ketchup, where inframarginal consumers are most heavily affected.

25

These observations seem most consistent with the case of advertising as a substitute for experience

described in proposition 3 above.

In contrast, Bronnenberg, Dhar, and Dubé (2005) find that, in some consumer package good

industries, brands that entered a given local market earlier tend to achieve persistently higher sales,

even decades after the original entry occurred. Perhaps surprisingly, they further show that these

first-movers advertise more than their competitors. Relatedly, Horstmann and MacDonald (2003)

show that, in the market for compact disc players, product advertising rises over the lifetime of

the product at a decreasing rate, displaying a pattern not unlike the selection advertising curve

in figure 1. Although a number of authors (e.g., Telser, 1961; Stigler, 1961) have argued that the

advertising “stock” must be constantly refreshed, these models by themselves cannot account for a

tendency to advertise more as a brand ages. While it is by no means the only way to account for

the patterns in the data, part 2 of proposition 3 provides a reason why greater consumer experience

may heighten the effectiveness of advertising, and hence the incentive to advertise.

Of course, the proposition also has negative implications for empirical attempts to infer the

informativeness of advertising by estimating the relationship between consumer experience and

advertising effectiveness (or intensity). Conversion advertising is not informative, and indeed it

strictly reduces the information content of consumers’ memories, but its behavior with respect to

changes in experience is qualitatively identical to the predictions of traditional informational models.

This finding suggests that there may be fundamental difficulties in identifying the informativeness

of advertising using variation in consumer experience.

5 Private Information and the Relationship of Advertising to Prod-

uct Quality

Up to now, the model has assumed that the firm has no private information about any consumer’s

type. This may be a reasonable assumption in markets where the uncertainty over product quality

is largely idiosyncratic, i.e., where individual consumers know the average quality of the good but

not how much they themselves will enjoy it (such as packaged foods). In some markets, there is

likely also to be aggregate uncertainty–uncertainty about the average consumer’s taste for the

26

product (as might be the case, for example, with consumer electronics). In such cases, it seems

reasonable that a firm may have private information about the product’s quality, and that this

information might be used to determine its level of advertising expenditure.

In signalling theories of advertising (Khilstrom and Riordan, 1984; Milgrom and Roberts, 1986),

the possibility of repeat business makes a firm more inclined to advertise the greater is its confidence

in its product, which in turn reinforces consumers’ tendency to purchase heavily advertised goods.

In other words, such models predict that a firm will advertise more the better is its product. In this

section, I provide a general result on the quality-advertising relationship in the memory-jamming

model, and highlight its implications for the two special cases defined above.

To treat the simplest case of private information, I will assume that all consumers have an

identical type θ, drawn from a prior distribution with density function f (). As in section 3, I will

assume that f () is common knowledge and has full support on the unit interval. Finally, I will

assume that the firm knows the realization of θ when setting its advertising policy. Other than the

assumption of an aggregate type known to the firm, I will retain all other assumptions from section

3. Importantly, I retain the assumption that at the time of purchase each consumer has access only

to his recollection; he does not remember whether he was exposed to advertising, or the firm’s level

of advertising expenditure.

With private information, the firm’s advertising strategy can be represented as a function a (θ)

of the aggregate type θ ∈ [0, 1]. To characterize the equilibria, it will be helpful to think in terms

of the expression

J = E (θ | ri = v; a)− E (θ | ri = 0; a) , (22)

which measures how much a consumer’s recollection affects his posterior expectation of the true

type. Here, I abuse notation slightly and let a represent consumers’ beliefs about the firm’s adver-

tising strategy. Given some beliefs on the part of consumers, and hence some value J ≥ 0, the firm

has an optimal optimal advertising level for each θ, defined by

a∗ (θ, J) = argmaxa[d (a) (μ (θ;N)− θ)J − a] . (23)

(The uniqueness of a∗ (θ, J) is guaranteed by the concavity of d (a).) An equilibrium is then a value

27

J such that

J − [E (θ | ri = v; a = a∗ (θ, J))− E (θ | ri = 0; a = a∗ (θ, J))] = 0. (24)

Note first that in any equilibrium J > 0; that is, consumers must have a higher expectation of

the product’s quality when their recollection is favorable than when it is not. If this is not so, i.e.

if J = 0, then because advertising will make no difference to consumers’ posterior expectations, the

firm will set a∗ (θ, 0) = 0 for all θ. Given that the firm does not advertise, however, any consumer

recollection will be genuine–and, hence, informative–and therefore (following arguments in section

4) a consumer’s posterior expectation will be higher conditional on a favorable recollection than on

an unfavorable one. This, in turn, implies that J > 0, which is a contradiction.

Next observe that the difference in posterior expectations caused by having a positive recollec-

tion instead of a negative one cannot exceed 1, so that when J = 1 the left-hand side of expression

(24) must be weakly positive. Then since the left-hand side of expression (24) is negative when

J = 0 and positive when J = 1, equation (24) must have at least one solution, which guarantees

the existence of an equilibrium in the private-information game.

Having established the existence of an equilibrium, the next question of interest is how the

level of equilibrium advertising depends on the quality θ of the firm’s product. Let J∗ denote an

equilibrium value of the difference in posterior expectations between a favorable recollection and an

unfavorable one. It follows from equation (23) above and from the concavity of the function d (a)

that whenever the equilibrium level of advertising a∗ (θ, J∗) is positive given some quality level θ,

it is uniquely defined by the first-order condition

d0 (a∗ (θ, J∗)) [μ (θ;N)− θ]J∗ − 1 = 0. (25)

The effect of a marginal increase in product quality θ (given an equilibrium) will then depend on how

the effect of advertising on the probability of a favorable recollection, (μ (θ;N)− θ), changes with

θ. If (μ (θ;N)− θ) is increasing in θ at a given θ, then a higher realization of θ will be associated

with greater equilibrium advertising expenditure, and vice versa if (μ (θ;N)− θ) is decreasing in θ.

These conclusions are formalized as follows:

Proposition 4 There exists an equilibrium of the advertising game with private information.

28

Given an equilibrium advertising policy a∗ (θ, J∗) such that a∗¡θ0, J∗

¢> 0 for some realization

θ0 ∈ (0, 1) :d

dθa∗ (θ, J∗) |θ=θ0≷ 0 ⇐⇒

d

dθ(μ (θ;N)− θ) |θ=θ0≷ 0.

That is, advertising is increasing in the realization of product quality if and only if the effect of

advertising on the probability of a favorable recollection is increasing in the realization of product

quality.

From the above result, it is easy to see that conversion and selection advertising make very

different predictions about the relationship between a firm’s advertising level and its (unobserved)

product quality. Under conversion advertising, a higher draw on quality means that fewer of a

consumer’s experiences are likely to be negative, so the value of replacing a negative experience

with a positive one is smaller when quality is higher. Algebraically, it is easy to see that under

conversion advertising the expression

μ (θ;N)− θ =C

N(1− θ) (26)

is strictly decreasing in θ, so that the effect of advertising on the probability of a favorable recol-

lection is declining in product quality.

In the case of selection advertising, when quality is extremely high (θ is close to 1) the ability to

manipulate which of a consumer’s experiences is recalled is relatively unimportant, since the con-

sumer is very likely to remember a favorable experience in any case. Likewise, when product quality

is extremely low (θ is close to 0), most consumers will not have had a favorable experience with

the product, so that the firm will generally not be able to induce favorable recollections. Therefore

advertising levels will tend to be very low at the extremes of the quality distribution. Advertising

levels will tend to be greatest in the interior of the quality distribution, where a technology that

causes the consumer to remember his most favorable experiences will have the greatest impact.

This is easiest to see formally in the case of S = 1, where exposure to selection advertising leads

the consumer to have a favorable recollection if at least one of his experiences is positive. Here,

μ (θ;N)− θ = (1− θ)³1− (1− θ)N−1

´. (27)

29

The first factor is decreasing in θ, but the second is increasing in θ; the overall expression takes

on an inverted-U shape, peaking at intermediate values of quality θ. Intuitively, this result shows

that selection advertising is most effective when there is significant variation in each consumer’s

experience with the product. This variation is greatest for intermediate values of θ, where a given

consumer is likely to have both positive and negative experiences with the product.

Formally, the relationship between advertising levels and product quality for replacement and

selection advertising follows directly from proposition 4:

Corollary 2 Under conversion advertising, any equilibrium advertising policy a∗ (θ, J∗) is decreas-

ing in the realization of θ, strictly whenever a∗ (θ, J∗) > 0. Under selection advertising, any equi-

librium advertising policy a∗ (θ, J∗) is increasing in θ for θ sufficiently low and decreasing in θ for

θ sufficiently high, strictly whenever a∗ (θ, J∗) > 0.

Figure 2 illustrates the relationship between equilibrium advertising levels and product quality for

representative numerical examples of conversion and selection advertising.

A literature in marketing and economics has examined the empirical relationship between mea-

sures of product quality and advertising expenditures, with mixed results. Rotfeld and Rotzoll

(1976) and Caves and Greene (1996), for example, find that the relationship is positive in some

product categories and negative in others, with little relationship on average. Archibald, Halman,

and Moody (1983) and Nichols (1988) find a positive relationship between advertising and measured

quality, which these authors interpret as evidence in favor of a signalling theory of advertising.

Corollary 4 suggests that the variation across product categories in the advertising-quality

relationship may be informative about the underlying mechanism by which advertising affects

consumers’ purchase behavior. It also indicates that caution is needed in inferring a signaling

motive from the fact that advertising and product quality are positively related in some product

categories. Indeed, this result raises similar identification issues to those highlighted by proposition

3 in the context of consumer experience, and suggests that many previous tests of signalling based

on product quality data may not be able to deeply distinguish signalling theories from theories

based on memory-jamming mechanisms.

30

6 Increasing Returns, Pulsing, and Multiple Memories

At high frequencies, firms’ advertising levels often follow a “pulsing” or “flighting” pattern, in which

a firm’s advertising oscillates between low and high levels (Dubé, Hitsch, and Manchanda, 2005).

The tendency to prefer fluctuating advertising levels to a constant intermediate level suggests the

presence of increasing returns in consumers’ response to advertising, at least over some range. In

this section, I discuss why such increasing returns might arise from a model in which advertising

can enhance consumers’ memories of product experiences. I first show that, in the presence of

advertising, an unfavorable recollection will tend to have a greater impact on a consumer’s posterior

expected valuation than a favorable recollection. I then show that, when the firm’s advertising can

affect the memories of multiple experiences, this force can create a strong incentive to ensure that

all memories are of good experiences. This provides a natural source of increasing returns, and can

lead in equilibrium to mixed strategies not unlike the pulsing strategies observed in high-frequency

advertising data.

6.1 Asymmetric Effects of Positive and Negative Recollections

I begin this section by demonstrating that, in the presence of advertising, unfavorable recollections

(i.e., recollections of unfavorable experiences with the product) will tend to have a greater impact

on consumer beliefs than favorable recollections. This is because advertising tends to increase the

likelihood of a positive recollection for a given consumer type, thus “diluting” the information

content of a favorable recollection.

Consider a case in which the prior distribution of types f () is symmetric about 12 , so that there

are no inherent asymmetries in the effects of favorable or unfavorable recollections. First, suppose

that there is no advertising (a = 0). If we measure the impact of a recollection as its effect on the

consumer’s posterior expectation relative to his prior expectation, then it is easy to see that both

favorable and unfavorable recollections have the same effect. Formally, we wish to show that:

E (θi | ri = v)−1

2=1

2− E (θi | ri = 0) (28)

since a favorable recollection results in a posterior expectation greater than 12 , and an unfavorable

31

one results in an expectation below 12 . Given the assumption of a symmetric prior, equation (28)

can be rewritten as

Pr (ri = 0)E (θi | ri = 0) + Pr (ri = v)E (θi | ri = v) = E (θi) . (29)

The law of iterated expectations implies that this equation must hold, which means that favorable

and unfavorable recollections have identical effects on posterior expectations when there is no

advertising.

In the presence of advertising (a > 0), a negative recollection will impact beliefs more than a

positive one:

E (θi | ri = v)−1

2<1

2− E (θi | ri = 0) (30)

To see why, observe that by the law of iterated expectations the above inequality is equivalent to

µ1

2− Pr (ri = 0)

¶E (θi | ri = 0) <

µPr (ri = v)−

1

2

¶E (θi | ri = v) . (31)

Because a > 0 implies that E (θi | ri = v) > E (θi | ri = 0) and Pr (ri = v) > Pr (ri = 0), this

inequality must hold. Intuitively, since advertising raises the likelihood of a positive recollection

for any type, it must correspondingly decrease the effect of a positive recollection on posterior

expectations, because in equilibrium consumer beliefs must be consistent with Bayes’ rule. This

result is reminiscent of Israel’s (2005) study of customer learning in the market for automobile

insurance, in which consumers respond very strongly to negative information.

6.2 An Extension with Increasing Returns

Having established the general principle that, in the presence of advertising, unfavorable recol-

lections will affect consumers’ posteriors more than favorable ones, I turn in this subsection to an

application of this intuition to dynamic advertising policy. To make the notion of dynamic advertis-

ing policy meaningful, I consider a reformulation of the basic model in which advertising can affect

several sequential recollections. I show that the asymmetries between favorable and unfavorable

recollections described in the previous subsection have potentially important implications for the

32

consumer’s response to advertising and the firm’s advertising policy.

In particular, I will consider a reformulate the general model of section 3 to allow each consumer’s

recollection −→r i to be a vector with T elements, each of which comes from a separate process that

I will refer to as a “recall event.” (The parameter T can be thought of as an index of the richness

of the consumer’s recollection.) Advertising can influence each recall event separately, perhaps

by reaching the consumer on different days in the period leading up to his purchase decision. In

particular, the probability that a recall event t is influenced by advertising is given by d (at), where

at is the firm’s level of advertising expenditure on recall event t. Following the basic model, if a

consumer is not influenced by advertising at recall event t, then his tth recollection is simply a

random element of his memory. If he is influenced by advertising at recall event t, his probability of

a favorable recollection will be governed by the function m (Ki;N). The consumer uses the entire

recollection vector to infer his type. Note that, up to this point, the general model of section 3 is

just a special case of the current model setup, with the restriction that T = 1.

The current framework allows the firm to affect the probabilities of 2T possible memory states,

which makes analysis of the general case difficult. In order to make this richer model more tractable,

I will make the strong assumption that types are distributed according to θi ∼ Bernoulli (η).

Under this assumption, there is no noise in consumers’ experiences, so consumers with high types

will always enjoy the product, and consumers with low types never will. As a result, consumers’

experiences are very informative about their types, which makes the distinction between favorable

and unfavorable recollections highlighted in the previous subsection even starker. In this sense, the

results in this section are most relevant for environments in which individual experiences reveal a

lot about underlying tastes.

Given the assumption of Bernoulli types, it follows that Ki ∈ {0, N}; that is, a given consumer’s

experiences are either all favorable or all unfavorable. Consequently, in order for exposure to

advertising to weakly increase the probability of a favorable recollection for all types, and strictly

increase it for at least some types, we must assume

m (N ;N) = 1 (32)

m (0;N) > 0.

33

Note, in particular, that selection advertising is ruled out by these restrictions. Indeed, selection ad-

vertising would have no effect in the Bernoulli case, since all experiences would have the same value

and therefore greater selectivity toward favorable experiences would not impact recall probabilities.

The fact thatKi ∈ {0, N} , together with the above restrictions onm (), imply that if a consumer

expects the firm to advertise at all recall events (i.e. if the consumer believes that at > 0 for all

t), then a consumer i whose recollection includes even a single unfavorable experience must have

type θi = 0. In other words, since advertising increases the probability of favorable memories, the

only way a consumer can remember a bad experience is if his type is low; i.e., if θi = 0. Figure 3

shows the implication of this fact for consumer posteriors: as long as consumers expect the firm to

advertise with positive probability, the only consumers who assign a positive expected value to the

product are those for whom all memories of the product are favorable; i.e., for whom all elements

of the recollection vector −→r i have value v. This is an extreme version of the intuition developed in

the preceding subsection.

The firm’s objective is then to maximize, net of costs, the share of consumers whose recollections

are exclusively favorable; that is, the firm maximizes

π (a1, ..., aT ; a) =

"η + (1− η)

TYt=1

(d (at)μ (0;N))

#E (θi |

−→r i = {v, v, ..., v} ; a)−TXt=1

at. (33)

Here, I continue to abuse notation and let a denote consumers’ beliefs about the firm’s advertising

strategy. (Technically, consumers’ beliefs will describe a distribution over the vector of advertising

levels at different recall events.) The interpretation of the above profit function is as follows. If

consumers expect the firm to advertise at all recall events, then a consumer will only purchase the

product with positive probability if his recollections are exclusively favorable, in which case he will

purchase with probability E (θi |−→r i = {v, v, ..., v} ; a). If the consumer’s type is high, i.e. if θi = 1

(which occurs with probability η), then his recollections will be exclusively favorable with proba-

bility 1. If his type is low, i.e. if θi = 0 (which occurs with probability (1− η)), then he will have

a favorable recollection only if each recall event is affected by an advertisement (probability d (at))

and each advertisement is successful in inducing a favorable recollection (probability μ (0;N)).

We can further simplify the firm’s problem by noting that the firm will never set different

34

(positive) advertising levels at different recall events. To see why, consider that the firm’s first-

order conditions imply that, for any two recall events t and t0,

d0 (at)d (at)

=d0 (at0)d (at0)

(34)

Under assumption 1, this can only be true if at = at0 , which means that there exists a such that

at = a for all t, 1 · t · T . In other words, if the firm advertises, it maintains a constant level

of advertising across all recall events. The firm’s objective function, up to a constant, is then

proportional to:

π (a; a) = d (a)T E (θi |−→r i = {v, v, ..., v} ; a)− φa (35)

where

φ =T

(1− η)μ (0;N)T(36)

is a constant indexing the relative cost of advertising.

Because the firm’s goal is to maximize the probability of an entirely favorable recollection,

advertising at one recall event is complementary with advertising at another recall event, a fact

that can result in increasing returns in the firm’s profit function over some levels of advertising.

To see this, it will be helpful to introduce the notion of an advertising response function z (a, a),

defined as:

z (a, a;T ) ≡ d (a)T E (θi |−→r i = {v, v, ..., v} ; a) . (37)

This function, which is strictly increasing in a, describes how consumer demand (i.e., the share of

consumers purchasing the product) responds to the level of advertising for a given consumer belief

about the firm’s strategy. We can then define the concept of an S-shaped advertising response

function as follows:

Definition 2 The advertising response function z (a, a;T ) is S-shaped if, given that consumers

expect the firm to advertise (a > 0), then there exists a constant a > 0 independent of consumer

beliefs a such that the advertising response function z (a; a;T ) is strictly concave in a for a > a and

strictly convex in a for a < a.

35

Figure 4 provides a graphical example of an S-shaped advertising response function, illustrating

the curvature of the function as well as the location of the inflection point a.

Algebraically, the second derivative of the profit function π (a; a) (for a given set of consumer

beliefs) is positive at some value a > 0 if and only if

d00 (a)d0 (a)

+ (T − 1)d0 (a)d (a)

> 0 (38)

For T = 1, this condition is never satisfied, so the profit function is globally concave, just as in

the model in section 3. However, since the second term is positive and strictly increasing in T , for

any value a > 0 the profit function is convex in advertising at a for sufficiently large T . Moreover,

because d (a) is strictly concave in a, the second term will be positive and decreasing in a. Finally,

if d00 (a) does not change too much with a, then the first term will be negative and decreasing in a.

The profit function therefore has a tendency to be S-shaped :

Lemma 2 If the second derivative d00 (a) of the advertising exposure function does not vary too

much with a, i.e. if

d000 (a) <(d00 (a))2

d0 (a)(39)

for all a ≥ 0, then the advertising response function z (a, a;T ) is S-shaped for all T > 1.

The presence of an S-shaped advertising response function seems consistent with empirical evidence

of non-concavities in consumer advertising response over low levels of exposure (Bemmaor, 1984;

Vakratsas, Feinberg, Bass, and Kalyanaram, 2004). As we will see, it also leads to mixed-strategy

equilibria that closely resemble empirically observed patterns in firm’s dynamic advertising policies.

Motivated by lemma 2, I will begin by adopting the following assumption:

Assumption 2 The function d () satisfies

d000 (a) <(d00 (a))2

d0 (a)

for all a ≥ 0.

To characterize the firm’s equilibrium advertising behavior, observe that, because advertising costs

36

are linear, the objective function π (a; a) inherits the second-order properties of the advertising

response function z (a, a). It is then easy to see that, in any equilibrium, the firm will play at

most one advertising level in the concave region with positive probability. Similarly, the firm will

never play a positive level of advertising in the convex region. Therefore in any equilibrium the

firm plays at most two levels of advertising with positive probability: a∗ = 0 and some a∗ ≥ a,

where a is the inflection point above which the profit function becomes concave. Thus, there are

then three possible types of equilibria: (i) a pure-strategy equilibrium in which the firm chooses

some advertising level greater than a, (ii) a pure-strategy equilibrium in which the firm does not

advertise, (iii) and a mixed-strategy equilibrium in which the firm mixes between advertising at

some level greater than a and not advertising at all.

In fact, the game has a unique equilibrium advertising policy, with only one type of equilibrium

possible for any given set of parameter values. I will focus here on the intuition for the presence of

a mixed-strategy equilibrium, leaving the full characterization of the equilibrium to proposition 5

below and its associated proof. To see why, over some range of parameters, the unique equilibrium

may involve mixing between high and low advertising levels, consider a region of the parameter

space in which

maxa≥a

π (a; a = 0) > 0 (40a)

maxa≥a

π (a; a = a) < 0. (40b)

(Because the profit function π (a; a) is strictly decreasing in the level a of advertising that consumers

expect, these two inequalities can hold simultaneously.) Under these conditions, if the firm plays

advertising level a = 0 with certainty, then there will be a profitable deviation to some level of

advertising a ≥ a. But if the firm plays any advertising level a ≥ a with certainty, then the second

inequality implies that there is a profitable deviation to advertising level a = 0. In other words,

the firm will not be willing to play any single level of advertising with certainty in equilibrium.

Note, however, that in this case there is a unique value a ∈ (0, a) such that

maxa≥a

π (a; a = a) = 0. (41)

37

In other words, if consumers expect the firm to play a = a, then the firm will be indifferent between

playing some (unique) optimal advertising level a∗∗∗ > a and playing a = 0, and will therefore be

willing to mix between these two advertising levels.

Suppose, next, that the firm plays advertising level a∗∗∗ with probability σ∗∗∗ and advertising

level 0 with probability (1− σ∗∗∗), where σ∗∗∗ is given by

σ∗∗∗ =d (a)T

d (a∗∗∗)T. (42)

That is, σ∗∗∗ is chosen so that, given that the firm plays advertising level a∗∗∗ with probability

σ∗∗∗, the probability of an entirely favorable recollection is exactly equal to d (a)T . In this case,

consumers’ beliefs will be as if the firm were playing advertising level a with certainty, and the

firm will therefore be willing to play the proposed strategy. Playing advertising level a∗∗∗ with

probability σ∗∗∗, and advertising level 0 with probability (1− σ∗∗∗), then constitutes an equilibrium

advertising policy, and it is straightforward to verify that this can be the only equilibrium policy

under conditions (40).

Figure 5 further illustrates the determination of the mixed-strategy equilibrium. In order to be

willing to mix, the firm must be indifferent between advertising at some positive level greater than

a and not advertising at all. Because π (0; a) = 0, this in turn means that demand z (a, a;T ) and

advertising costs φa must be equal at the proposed positive advertising level. In other words, the

cost and demand curves must cross. Additionally, because the firm’s objective function is concave

above a, the firm must be indifferent to a marginal change in advertising at the proposed positive

level, or it will prefer to deviate to a higher or lower amount of advertising. This means that the

cost curve must also be tangent to the demand curve at the positive advertising level. These two

conditions can only occur simultaneously at advertising level a∗∗∗. For a mixed-strategy equilibrium

to exist, consumers’ expectation of the firm’s advertising level must be sufficiently low to guarantee

that the two curves do actually have a tangency at a∗∗∗. The probability of positive advertising

that ensures this tangency is the equilibrium mixing probability σ∗∗∗.

The following proposition, proved formally in the appendix, characterizes the equilibrium for

the remaining regions of the parameter space:

38

Proposition 5 Under assumption 2, there is a unique equilibrium advertising policy. When T = 1,

the firm plays some advertising level a∗ with certainty. When T > 1, there is a unique equilibrium

advertising policy, which can have one of three forms:

1. The firm chooses some advertising level a∗ ≥ a with certainty.

2. The firm chooses some advertising level a∗∗∗ ≥ a with some probability σ∗∗∗ and advertising

level 0 with probability (1− σ∗∗∗).

3. The firm chooses advertising level a = 0 with certainty.

Figure 6 depicts the equilibrium advertising policies for a numerical example. For low adver-

tising costs, the firm always advertises, but the extent of advertising is decreasing in costs φ. For

intermediate costs, the firm randomizes between not advertising and advertising at some positive

level greater than the “inflection point” a. Finally, with high enough costs the firm never advertises.

Note that, regardless of the costs, the firm never chooses a positive advertising level below a. In

this sense the firm’s choice of advertising level has a “threshold” below which it never falls, except

when the firm does no advertising at all.

The mixed-strategy equilibria characterized in proposition 5 bear a close resemblance to the

empirical phenomenon of “pulsing,” in which advertising levels oscillate from high to low at weekly

frequencies. This phenomenon has been documented in several markets, most recently in Dubé,

Hitsch, and Manchanda’s (2005) study of the market for frozen meals. In addition to demonstrating

the presence of pulsing in the frozen meal market, Dubé, Hitsch, and Manchanda (2005) estimate

consumers’ advertising response function and find evidence of greater marginal effects of advertising

at higher levels of exposure. Their findings suggest that pulsing may indeed result from an “S-

shaped” response function of the sort characterized in lemma 2 above.

Figure 7 provides an illustration of the “pulsing” phenomenon in the context of Heinz ketchup

advertising in an American town in the late 1980s (see appendix B for data description), based on

household-level data on advertising exposure. In the early weeks of advertising measurement, Heinz

advertising oscillated between reaching households several times per week and hardly reaching them

at all.15 Because Heinz advertising accounted for over 90% of recorded television advertising expo-15As the standard error bars on the figure suggest, these patterns are not simply due to sampling variability. A

39

sures, motivations for pulsing based on competitive interactions (Villas-Boas, 1993) seem unlikely

to play a major role in this market.16

Although the implications of threshold and S-shaped advertising response functions for dynamic

advertising policy have been studied extensively,17 the memory-jamming framework provides, to

my knowledge, the first complete microfoundation for S-shaped advertising response functions.18

Increasing returns arise in the memory-jamming model because consumers who have even a single

bad memory about the product will not buy. This means that firms will try to “saturate” such a

consumer with good memories about the product by advertising in bursts or “pulses.”

7 Conclusions

Measured US advertising expenditures across all media totalled $128.3 billion in 2003 (TNS, 2004),

more than one percent of total US GDP that year. When direct mail, corporate sponsorships,

and other marketing expenditures are taken into account, persuasive communication is involved in

a substantial fraction of economic exchange.19 In this paper, I have shown that a set of simple,

psychologically realistic assumptions about consumer memory can shed light on several interesting

features of firm advertising behavior.

random effects model (not reported) of advertising exposure confirms that a significant share of the variability inhousehold exposure probability occurs at the level of the market-week. Additional tests (not reported) indicate thatthe “pulses” do not correspond strongly to changes in average price. Additionally, evidence from Dubé, Hitsch, andManchanda (2005) on the behavior of multiple firms in the same market shows that seasonality alone cannot accountfor the observed pulsing patterns.16To see how the results in proposition 5 might lead to advertising dynamics similar to those observed in figure

7, consider an infinite-horizon extension in which consumers make a purchase decision every T periods, and inwhich consumers’ memories do not persist across purchase cycles. This could be motivated, for example, by amodel in which successive generations of consumers spend T periods acquiring information, and then after makinga purchase decision exit the market for a significant period of time. It is easy to see that in the mixed-strategyregion of proposition 5 this game will have an equilibrium in which the firm randomizes its advertising behaviorindependently across purchase cycles. Advertising dynamics in such an equilibrium would resemble those oftenobserved in practice: consecutive periods of high advertising followed by consecutive periods of low advertising. Ofcourse, without additional assumptions we cannot say that this equilibrium is unique, since the firm could chooseto correlate its randomizations across purchase cycles. But since correlated randomization would yield no additionalprofit to the firm, if even an arbitrarily small fraction of consumers observe the history of advertising levels this couldpush the firm to randomize independently.17Studies of optimal high-frequency advertising policies allowing for non-concavities in consumer response include

Mahajan and Muller (1986), Mesak and Darrat (1982), Villas-Boas (1993) and Mesak (2002). Nerlove and Arrow(1962) provide a more general treatment of the economics of dynamic advertising policy.18See Feichtinger, Hartl, and Sethi (1994) for a review of research on optimal dynamic advertising policies. Berg

(2001) mentions that pulsing can result from reference-dependent preferences but does not appear to provide acharacterization of advertising dynamics in his model.19 Indeed, McCloskey and Klamer (1995) go so far as to argue that “One quarter of GDP is persuasion.”

40

The foundation of the model is the assumption that advertising can affect the likelihood that

consumers recall a favorable experience with the product. This assumption is supported by an

abundance of psychological evidence on “source memory” (Mitchell and Johnson, 2000) and may

provide a microfoundation for concepts such as “goodwill” (Nerlove and Arrow, 1962) and “fa-

vorable notice” (Becker and Murphy, 1993), which are frequently used in motivating taste-based

models of the consumer response to advertising. The approach here is to treat these effects not as

exogenous components of consumer preferences but as the results of an explicit internal dynamic

in which advertising affects recollections of product enjoyment.

An important issue that I have not explored here is the role of competition. The memory-

jamming framework provides a natural vocabulary for thinking about negative advertising, in the

sense that this can be modeled as advertising that reduces the probability that the consumer recalls

a favorable experience with the competitor brand. This effect would tend to lessen the asymme-

try between favorable and unfavorable recollections highlighted in the previous section. As the

number of firms grows large, however, the incentive for any one firm to advertise negatively about

another tends to shrink, which suggests that the asymmetries between favorable and unfavorable

recollections might actually intensify in highly competitive markets.

41

References

[1] David A. Aaker. Managing brand equity: Capitalizing on the value of a brand name. FreePress, New York, 1991.

[2] Daniel A. Ackerberg. Empirically distinguishing informative and prestige effects of advertising.RAND Journal of Economics, 32(2):100—118, Summer 2001.

[3] K. Damon Aiken. Manufactured memories and nostalgia advertising. American MarketingAssociation Conference Proceedings, 10:44—51, Summer 1999.

[4] Michael C. Anderson, Robert A. Bjork, and Elizabeth L. Bjork. Remembering can causeforgetting: Retrieval dynamics in long-term memory. Journal of Experimental Psychology:Learning, Memory, and Cognition, 20(5):1063—1087, 1994.

[5] Robert B. Archibald, Clyde A. Haulman, and Carlisle E. Moody, Jr. Quality, price, advertising,and published quality ratings. Journal of Consumer Research, 9(4):347—356, March 1983.

[6] Gary S. Becker and Kevin M. Murphy. A simple theory of advertising as a good or bad.Quarterly Journal of Economics, 108(4):941—964, November 1993.

[7] Albert C. Bemmaor. Testing alternative econometric models on the existence of advertisingthreshold effect. Journal of Marketing Research, 21(3):298—308, August 1984.

[8] Nathan Berg. Persuasive advertising and reference-point dependent preferences. University ofTexas at Dallas Working Paper, November 2001.

[9] Kathryn A. Braun. Postexperience advertising effects on consumer memory. Journal of Con-sumer Research, 25(4):319—334, March 1999.

[10] Kathryn A. Braun, Rhiannon Ellis, and Elizabeth F. Loftus. Make my memory: How adver-tising can change our memories of the past. Psychology and Marketing, 19(1):1—23, January2002.

[11] Kathryn A. Braun-LaTour and Michael S. LaTour. Transforming consumer experience: Whentiming matters. Journal of Advertising, 34(3):19—30, Fall 2005.

[12] Kjell Arne Brekke and Mari Rege. Advertising as a distortion of social learning. Case WesternReserve University Mimeograph, June 2005.

[13] Pablo Briñol, Richard E. Petty, and Zakary L. Tormala. Self-validation of cognitive responsesto advertisements. Journal of Consumer Research, 30:559—573, March 2004.

[14] Bart J. Bronnenberg, Sanjay K. Dhar, and Jean-Pierre Dubé. Market structure and thegeographic distribution of brand shares in consumer package goods industries. University ofChicago Mimeograph, September 2005.

[15] Richard E. Caves and David P. Greene. Brands’ quality levels, prices, and advertising out-lays: Empirical evidence on signals and information costs. International Journal of IndustrialOrganization, 14:29—52, 1996.

[16] Antonio R. Damasio. Descartes’ Error : Emotion, Reason, and the Human Brain. HarperPerennial, New York, November 1995.

42

[17] Stefano DellaVigna and Ulrike Malmendier. Contract design and self-control: Theory andevidence. Quarterly Journal of Economics, 119(2):353—402, May 2004.

[18] Peter M. DeMarzo, Dimitri Vayanos, and Jeffrey Zwiebel. Persuasion bias, social influence,and unidimensional opinions. Quarterly Journal of Economics, 118(3):909—968, August 2003.

[19] Avinash Dixit and Victor Norman. Advertising and welfare. Bell Journal of Economics,9(1):1—17, Spring 1978.

[20] Allen R. Dobbs and Brendan Gail Rule. Adult age differences in working memory. Psychologyand Aging, 4(4):500—503, 1989.

[21] Jean-Pierre Dubé, Günter J. Hitsch, and Puneet Manchanda. An empirical model of advertisingdynamics. Quantitative Marketing and Economics, 2005. Forthcoming.

[22] E. Woodrow Eckard, Jr. Competition and the cigarette TV advertising ban. Economic Inquiry,29:119—133, January 1991.

[23] Kfir Eliaz and Ran Spiegler. Contracting with diversely naive agents. Tel Aviv UniversityMimeograph, August 2004.

[24] Tülin Erdem, Michael Keane, and Baohong Sun. The impact of advertising on consumer pricesensitivity in experience good markets. Yale University Mimeograph, February 2004.

[25] Erik Eyster and Matthew Rabin. Cursed equilibrium. Econometrica, 73(5):1623—1672, Sep-tember 2005.

[26] Gustav Feichtinger, Richard F. Hartl, and Suresh P. Sethi. Dynamic optimal control modelsin advertising: Recent developments. Management Science, 40(2):195—226, February 1994.

[27] Marian Friestad and Peter Wright. The persuasion knowledge model: How people cope withpersuasion attempts. Journal of Consumer Research, 21(1):1—31, June 1994.

[28] Drew Fudenberg and Jean Tirole. A "signal-jamming" theory of predation. The RAND Journalof Economics, 17(3):366—376, Autumn 1986.

[29] Xavier Gabaix and David Laibson. Shrouded attributes and information suppression in com-petitive markets. Harvard University Mimeograph, May 2004.

[30] Matthew A. Gentzkow and Jesse M. Shapiro. Media, education, and anti-americanism in themuslim world. Journal of Economic Perspectives, 18(3):117—133, Summer 2004.

[31] Matthew A. Gentzkow and Jesse M. Shapiro. Media bias and reputation. Journal of PoliticalEconomy, 114(2):280—316, April 2006.

[32] Itzhak Gilboa and David Schmeidler. Case-based decision theory. Quarterly Journal of Eco-nomics, 110(3):605—639, August 1995.

[33] Edward L. Glaeser. Psychology and the market. American Economic Review, 94(2):408—413,May 2004.

[34] Edward L. Glaeser. The political economy of hatred. Quarterly Journal of Economics,120(1):45—86, February 2005.

43

[35] Ralph Norman Haber and Lyn Haber. Experiencing, remembering, and reporting events.Psychology, Public Policy, and Law, 6(4):1057—1097, 2000.

[36] Sunku Hahn. The advertisement of a product with a well-known quality. Hitotsubashi Journalof Economics, 44(2), October 2003.

[37] Stephen J. Hoch and Young-Won Ha. Consumer learning: Advertising and the ambiguity ofproduct experience. Journal of Consumer Research, 13(2):221—233, September 1986.

[38] Bengt Holmstrom. Managerial incentive problems: A dynamic perspective. Review of Eco-nomic Studies, 66:169—182, 1999.

[39] Ignatius Horstmann and Glenn MacDonald. Is advertising a signal of product quality? ev-idence from the compact disc player market, 1983-1992. International Journal of IndustrialOrganization, 21:317—345, 2003.

[40] TNS Media Intelligence/CMR. US advertising market exhibits strong growth in 2003. Pressrelease, March 2004.

[41] Mark Israel. Services as experience goods: An empirical examination of consumer learning inautomobile insurance. American Economic Review, 95(5):1444—1463, December 2005.

[42] Ganesh Iyer and Dmitri Kuksov. Feelings about quality and thinking about feelings. Universityof California, Berkeley Mimeograph, August 2005.

[43] Larry L. Jacoby, Colleen Kelley, Judith Brown, and Jennifer Jasechko. Becoming famousovernight: Limits of the ability to avoid unconscious influences of the past. Journal of Per-sonality and Social Psychology, 56(3):326—338, 1989.

[44] Philippe Jehiel. Analogy-based expectation equilibrium. Journal of Economic Theory, 123:81—104, 2005.

[45] Marcia K. Johnson. Reality monitoring: An experimental phenomenological approach. Journalof Experimental Psychology: General, 117(4):390—394, December 1988.

[46] Marcia K. Johnson, Mary A. Foley, Aurora G. Suengas, and Carol L. Raye. Phenomenalcharacteristics of memories for perceived and imagined autobiographical events. Journal ofExperimental Psychology: General, 117(4):371—376, December 1988.

[47] Marcia K. Johnson, Shahin Hashtroudi, and D. Stephen Lindsay. Source monitoring. Psycho-logical Bulletin, 114(1):3—28, 1993.

[48] Marcia K. Johnson, John Kounios, and John A. Reeder. Time-course studies of reality monitor-ing and recognition. Journal of Experimental Psychology: Learning, Memory, and Cognition,20(6):1409—1419, 1994.

[49] Richard E. Kihlstrom and Michael H. Riordan. Advertising as a signal. Journal of PoliticalEconomy, 92(3):427—450, June 1984.

[50] Daniel Krähmer. Advertising and consumer memory. University College London Mimeograph,March 2004.

[51] Oksana Loginova. Competing for consumers’ attention: Advertising when consumers haveimperfect memory. University of Missouri-Columbia Mimeograph, March 2005.

44

[52] Donald McCloskey and Arjo Klamer. One quarter of GDP is persuasion. American EconomicReview, 85(2):191—195, May 1995.

[53] Nitin Mehta, Surendra Rajiv, and Kannan Srinivasan. Role of forgetting in memory-basedchoice decisions: A structural model. Quantitative Marketing and Economics, 2(2):107—140,June 2004.

[54] Hani I. Mesak. On the impact of initial performance on the effectiveness of advertising pulsationpolicies. Journal of the Operational Research Society, 53:1247—1255, 2002.

[55] Hani I. Mesak and Ali F. Darrat. On comparing alternative advertising policies of pulsation.Decision Sciences, 23(3):541—564, May/June 1992.

[56] Paul Milgrom and John Roberts. Price and advertising signals of product quality. Journal ofPolitical Economy, 94(4):796—821, August 1986.

[57] Karen J. Mitchell and Marcia K. Johnson. Source monitoring: Attributing mental experiences.In Endel Tulving and Fergus I.M. Craik, editors, The Oxford Handbook of Memory, chapter 12,pages 179—185. Oxford University Press, New York, 2000.

[58] Sendhil Mullainathan. A memory-based model of bounded rationality. Quarterly Journal ofEconomics, 117(3):735—774, August 2002.

[59] Sendhil Mullainathan and Andrei Shleifer. The market for news. American Economic Review,95(4):1031—1053, September 2005.

[60] Sendhil Mullainathan and Andrei Shleifer. Persuasion in finance. Harvard University Mimeo-graph, July 2005.

[61] Kevin M. Murphy and Andrei Shleifer. Persuasion in politics. American Economic Review,94(2):435—439, May 2004.

[62] Phillip Nelson. Advertising as information. Journal of Political Economy, 82(4):729—754,July-August 1974.

[63] Marc Nerlove and Kenneth J. Arrow. Optimal advertising policy under dynamic conditions.Economica, 29(114):129—142, May 1962.

[64] Mark W. Nichols. Advertising and quality in the u.s. market for automobiles. SouthernEconomic Journal, 64(4):922—939, April 1988.

[65] Anthony R. Pratkanis, Anthony G. Greenwald, Michael R. Leippe, and Michael H. Baumgard-ner. In search of reliable persuasion effects: III. the sleeper effect is dead. long live the sleepereffect. Journal of Personality and Social Psychology, 54(2):203—218, 1988.

[66] Peter E. Rossi, Robert E. McCulloch, and Greg M. Allenby. The value of purchase historydata in target marketing. Marketing Science, 15(4):321—340, 1996.

[67] Herbert J. Rotfeld and Kim B. Rotzoll. Advertising and product quality: Are heavily adver-tised products better? Journal of Consumer Affairs, 10(1):33—47, Summer 1976.

[68] Ariel Rubinstein. Modeling Bounded Rationality. MIT Press, Cambridge, MA, 1998.

45

[69] Daniel L. Schacter. Searching for memory: The brain, the mind, and the past. Basic Books,New York, 1996.

[70] Julian L. Simon and Johan Arndt. The shape of the advertising response function. Journal ofAdvertising Research, 20(4):11—28, August 1980.

[71] Ran Spiegler. The market for quacks. Tel Aviv University Mimeograph, December 2004.

[72] Ran Spiegler. Competition over agents with boundedly rational expectations. Tel Aviv Uni-versity Mimeograph, January 2005.

[73] George J. Stigler. The economics of information. Journal of Political Economy, 69(3):213—225,June 1961.

[74] Lester G. Telser. How much does it pay whom to advertise? American Economic Review,51(2):194—205, May 1961.

[75] Demetrios Vakratsas, Fred M. Feinberg, Frank M. Bass, and Gurumurthy Kalyanaram. Theshape of advertising response functions revisited: A model of dynamic probabilistic thresholds.Marketing Science, 23(1):109—119, Winter 2004.

[76] J. Miguel Villas-Boas. Predicting advertising pulsing policies in an oligopoly: A model andempirical test. Marketing Science, 12(1):88—102, Winter 1993.

[77] Andrea Wilson. Bounded memory and biases in information processing. University of ChicagoMimeograph, May 2004.

[78] Peter L. Wright. The cognitive processes mediating acceptance of advertising. Journal ofMarketing Research, 10:53—62, February 1973.

46

Figure 1 The effect of consumer experience on equilibrium advertising levels

0 20 40 60 80 100 120 140

Level of consumer experience (N )

Adv

ertis

ing

leve

l (a*)

ConversionSelection

47

Figure 2 Equilibrium advertising and product quality

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Product quality (θ )

Adv

ertis

ing

leve

l (a*

)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

SelectionConversion

48

Figure 3 The effect of recalled experiences on consumers’ beliefs

0

0 1 2 T-1 T ….

Con

sum

er’s

exp

ecte

d va

luat

ion

Number of favorable recollections

49

Figure 4 Example of an S-shaped advertising response function

0

1

Level of advertising (a )

Prob

abili

ty o

f pur

rcha

se ( z

)

a

50

Figure 5 Illustrating the mixed-strategy equilibriumD

eman

d/C

ost

Cost = φa

Demand = E( θ i |r i ={1,..,1})(d(a)) T

Amount of advertising (a )

a ***a

51

Figure 6 Equilibrium advertising policies with multiple recall events

0

Lev

el o

f adv

ertis

ing

(a)

Advertising cost (φ)

a

“Pulsing” Never advertise Always advertise

0

Prob

abili

ty o

f adv

ertis

ing

(s)

0

1

Equilibrium advertising probability

52

Figure 7 Advertising for Heinz ketchup in Sioux Falls, ND from September 1987 through June1988

0

0.5

1

1.5

2

2.5

3

3.5

Week

Ave

rage

num

ber o

f Hei

nz T

V co

mm

erci

al e

xpos

ures

Notes: Data are from the ERIM database. (See appendix B for details.) Figure shows average number ofHeinz television advertising exposures, per week per household, among households with working telemetersin Sioux Falls market during the given week. Error bars represent ±1 standard error.

53

A Proofs

PROOF OF PROPOSITIONS 1-3

Claim 1: Under conversion advertising, there is a unique equilibrium advertising level a∗ (N) ≥ 0,which is weakly decreasing in N (strictly whenever a∗ (N) > 0).

Following arguments in the text, to establish uniqueness, it is sufficient to show that the functionπ1 (a; a) is strictly decreasing in a. In the case of conversion advertising,

π1 (a; a) = d0 (a)

C

NE (1− θi) [E (θi | ri = v; a)− E (θi | ri = 0; a)]− 1. (43)

The derivative d0 (a) is strictly decreasing in a by the strict concavity of the function d (). Thereforeit is sufficient to show that the expression

[E (θi | ri = v; a)− E (θi | ri = 0; a)] (44)

is strictly decreasing in a. By Bayes’ rule, we can write:

E (θi | ri = v; a) =

£1− C

N d (a)¤E (θi)£

1− CN d (a)

¤E (θi) + d (a)

CN

E¡θ2i¢

E (θi)(45)

+d (a) CN£

1− CN d (a)

¤E (θi) + d (a)

CN

E (θi)

E (θi | ri = 0; a) =E [θi (1− θi)]

E (1− θi).

The second expression is invariant to changes in a. The first expression is a weighted average ofE(θ2i )E(θi)

and E (θi), with the weight on E (θi) increasing in a.20 Because

E¡θ2i¢

E (θi)> E (θi) (46)

must hold for any non-degenerate distribution of θi, the posterior expectation E (θi | ri = v; a) isstrictly decreasing in a. Therefore π1 (a; a) is strictly decreasing in a, which guarantees the existenceof a unique equilibrium advertising policy a∗ (N).

The desired comparative static with respect to N is established if π1 (a; a) is strictly decreasingin N . The function π1 (a; a) can be rewritten as:

π1 (a; a) = d0 (a)

"E¡θ2i¢− E (θi)

2

NCE (θi) + d (a)E (1− θi)

#− 1 (47)

20For further interpretation of the expression for E (θi | ri = v; a), note that

E (θi | ri = v; a = 0) = E¡θ2i¢

E (θi)

so that the expression for E (θi | ri = v; a) is a weighted average of the posterior expectation given an “unconverted”favorable memory, and the posterior expectation given no new information. Intuitively, given a favorable recollection,the consumer assigns some probability to its being genuine, in which case he would update as though there were noadvertising, and some probability to its being the result of conversion, in which case he would not update at all.

54

from which it is clear by inspection that π1 (a; a) is strictly decreasing in N for all a.¥

Claim 2: At any level of experience, if the maximum probability of advertising exposure d is suf-ficiently small, then the level of advertising in any stable interior equilibrium is (locally) strictlyincreasing in the number of experiences N if advertising is a complement for experience, and strictlydecreasing in N if advertising is a substitute for experience.

To characterize the local effect of a change in the level of experience, fix some level of experienceN 0 and suppose that advertising is complementary to experience. (The results for the case whereadvertising is a substitute for experience follow symmetrically.) It is sufficient to show that

∂

∂Nπ1 (a; a) |N=N 0> 0 (48)

for all a > 0. This is true if

0 <

µ∂

∂NE (μ (θi;N)) |N=N 0

¶[E (θi | ri = v; a)− E (θi | ri = 0; a)] (49)

+ [E (μ (θi;N))− E (θi)]

µ∂

∂NE (θi | ri = v; a) |N=N 0 −

∂

∂NE (θi | ri = 0; a) |N=N 0

¶for all a > 0. By Bayes’ rule, we have that

E (θi | ri = v; a) =d (a)E [θiμ (θi;N)] + (1− d (a))E

¡θ2i¢

d (a)E [μ (θi;N)] + (1− d (a))E (θi)(50)

E (θi | ri = 0; a) =d (a)E [θi (1− μ (θi;N))] + (1− d (a))E [θi (1− θi)]

d (a)E [1− μ (θi;N)] + (1− d (a))E (1− θi)

As d becomes small, d (a) becomes close to 0 for all a. Condition (49) therefore approaches

0 <

µ∂

∂NE (μ (θi;N)) |N=N 0

¶ÃE¡θ2i¢

E (θi)−E [θi (1− θi)]

E (1− θi)

!(51)

Since both terms on the right-hand side are strictly positive, this condition must be satisfied. Forsmall enough d, then, the effect of a change in N on consumer expectations becomes small relativeto its effect on the probability of a favorable recollection, and the latter term dominates.¥

Claim 3: If advertising is a strong substitute for experience, then there exists N 0 such that for allN > N 0 there is no equilibrium with a positive advertising level.

A necessary condition for the existence of an equilibrium with positive advertising is thatπ1 (a; a) > 0 for some positive level of advertising a. Since

π1 (a; a) = d0 (a) [E (μ (θi;N))− E (θi)] [E (θi | ri = v; a)− E (θi | ri = 0; a)]− 1, (52)

it is immediate thatmaxa

π1 (a; a) · d0 (0) [E (μ (θi;N))− E (θi)]− 1. (53)

However, since (E (μ (θi;N))− E (θi)) approaches 0 as N grows large, there exists N 0 such thatthe right-hand side of the above expression is negative for all N > N 0. But then for N > N 0 therecannot be a level of advertising a such that π1 (a; a) > 0, which means there can be no equilibrium

55

with a positive advertising level. The limit result in proposition 1 follows from this argument, sinceconversion advertising displays strong substitutability with experience.¥

Claim 4: If advertising is a strong complement for experience, then there exists N 0 such that for allN > N 0 there is a unique equilibrium advertising level a∗ (N), with

limN→∞

a∗ (N) = a∗∗.

To guarantee uniqueness for a given level of experience N , it is sufficient to show that π1 (a; a)is everywhere strictly decreasing in a at experience level N , where

π1 (a; a) = d0 (a) (E (μ (θi;N))− E (θi)) [E (θi | ri = v; a)− E (θi | ri = 0; a)]− 1. (54)

The first factor in this expression is strictly decreasing in a by the concavity of d (). The secondfactor is independent of a. The third factor is strictly decreasing in a if

0 >d

da[E (θi | ri = v; a)− E (θi | ri = 0; a)] . (55)

From Bayes’ rule we have that

E (θi | ri = v; a) =d (a)E [θiμ (θi;N)] + (1− d (a))E

¡θ2i¢

d (a)E [μ (θi;N)] + (1− d (a))E (θi)(56)

E (θi | ri = 0; a) =d (a)E [θi (1− μ (θi;N))] + (1− d (a))E [θi (1− θi)]

d (a)E [1− μ (θi;N)] + (1− d (a))E (1− θi)

so that the desired condition on the derivative can be written as

0 >

⎡⎢⎢⎢⎣µ

(d (a)E [μ (θi;N)] + (1− d (a))E (θi))¡E [θiμ (θi;N)]− E

¡θ2i¢¢

−¡d (a)E [θiμ (θi;N)] + (1− d (a))E

¡θ2i¢¢(E [μ (θi;N)]− E (θi))

¶(d (a)E [μ (θi;N)] + (1− d (a))E (θi))

2

⎤⎥⎥⎥⎦ (57)

−

⎡⎢⎢⎣µ

(d (a)E [1− μ (θi;N)] + (1− d (a))E (1− θi)) (E [θi (1− μ (θi;N))]− E [θi (1− θi)])− (d (a)E [θi (1− μ (θi;N))] + (1− d (a))E [θi (1− θi)]) (E [1− μ (θi;N)]− E (1− θi))

¶(d (a)E [1− μ (θi;N)] + (1− d (a))E (1− θi))

2

⎤⎥⎥⎦As N grows large, the right-hand side of this expression approaches

−

³E¡θ2i¢− E (θi)

2´

(d (a) + (1− d (a))E (θi))2 (58)

which is strictly negative for all a ≥ 0. Therefore there is an N 0 such that for N > N 0, thefunction π1 (a; a) is everywhere strictly decreasing in a, so that there is a unique equilibriumadvertising policy a∗ (N) for N > N 0. That limN→∞ a∗ (N) = a∗∗ then follows from the continuityof π1 (a; a). Proposition 2 is also proved by this argument, since selection advertising displaysstrong complementarity with experience.¥

56

PROOF OF LEMMA 2Suppose that T > 1. The advertising response function is convex at some advertising level a

(for a given number of recall events T ) if and only if

d00 (a)d0 (a)

+ (T − 1)d0 (a)d (a)

> 0. (59)

The existence of a a satisfying the necessary conditions is guaranteed if (i) condition (59) is satisfiedfor a sufficiently small, (ii) condition (59) fails for a sufficiently large, and (iii) the expression onthe left-hand side of condition (59) is strictly decreasing in a. To show point (i), note that as abecomes small the second term on the left-hand side of condition (59) becomes arbitrarily large(and positive) but the first term remains bounded, so that the condition is satisfied for a sufficientlysmall. On point (ii), observe that as a grows large the first term on the left-hand side becomesarbitrarily large but the second term approaches 0, so that the condition fails for large enough a.Finally, note that the second term is strictly decreasing in a by the concavity of d (), and undercondition (2) the first term is strictly decreasing in a.¥

PROOF OF PROPOSITION 5If T = 1, then the firm’s problem is concave, so that the firm plays at most one level of

advertising with positive probability in any equilibrium. Moreover, by Bayes’ rule a consumer’sposterior expectation conditional on a favorable recollection, E (θi |

−→r i = {v} ; a), is given by

E (θi |−→r i = {v} ; a) =

1

η + (1− η) d (a), (60)

which is strictly decreasing in the expected level of advertising a. Therefore the function π1 (a; a)is strictly decreasing in a. Additionally, because d0 (a) approaches 0 as a grows large, π1 (a; a) isnegative for a sufficiently large. Therefore if π1 (0; 0) > 0 there is a unique equilibrium advertisinglevel a∗ > 0, and if π1 (0; 0) · 0, then there is a unique equilibrium advertising level a∗ = 0.

Consider next the case of T > 1. To simplify the firm’s problem, note that in equilibrium thefirm will either advertise at all recall events or it will not advertise at all. To see this, supposethat in equilibrium the firm advertises at some recall event t0, but that it does not advertise atsome other recall event t00. Then a consumer’s type will be fully revealed by the t00th element of hisrecollection. Consequently, demand will be unchanged if the firm ceases to advertise at recall eventt0, which means that there is a profitable deviation for the firm. Therefore if the firm advertises atsome recall event t0 in equilibrium, it must advertise at all recall events t. Moreover, the argumentin the text establishes that if the firm advertises at all recall events, it must choose equal advertisinginvestments at all events, which implies that we can represent the firm’s equilibrium advertisingpolicy by a single advertising level a. If consumers expect the firm to advertise in equilibrium, thenthe firm’s objective function can be written as

π (a; a) = d (a)T E (θi |−→r i = {v, v, ..., v} ; a)− φa (61)

which, by lemma 2, is S-shaped.Therefore, if consumers expect the firm to advertise with positive probability in equilibrium,

the firm’s objective function is strictly concave for a < a and strictly convex for a > a. As a result,in any equilibrium, the firm will play at most one positive level of advertising, and that level ofadvertising must be greater than a. Therefore, in any equilibrium, either (i) the firm does notadvertise, (ii) the firm randomizes between not advertising and advertising at some positive levelgreater than a, or (iii) the firm chooses some level of advertising greater than a with probability

57

1. The remainder of the proof characterizes the regions of the parameter space in which theseequilibria are possible.

I begin by establishing conditions for the existence of an equilibrium with no advertising. Suchan equilibrium exists if the firm does not wish to switch to a positive advertising level given thatconsumers expect it not to advertise. If consumers do not expect advertising, then many possiblerecollections are off the equilibrium path, and therefore we must specify beliefs in these cases. Tofind the weakest possible conditions for the existence of an equilibrium with no advertising, I willsuppose that if −→r i 6= {v, v, ..., v}, consumers believe that they are low types, i.e. that

E (θi |−→r i 6= {v, v, ..., v} ; a = 0) = 0. (62)

In this case, the firm will not want to deviate from the proposed equilibrium as long as

maxa≥a

π (a; a = 0) · 0. (63)

(Note that, because the firm’s objective function is S-shaped, we can restrict attention to deviationsto advertising levels greater than a, since no deviation to an advertising level below a can beoptimal.) If the above condition holds, then the firm will be willing to sustain the proposedequilibrium, and there will exist an equilibrium with no advertising. Moreover, this equilibriumis unique, since if consumers expect the firm to advertise, then the profits from playing a positiveadvertising level must be negative, since the profit function is strictly decreasing in the level ofadvertising a that consumers expect the firm to play.

Turn next to the case wheremaxa≥a

π (a; a = 0) > 0. (64)

In this case, advertising must occur with positive probability in equilibrium, because if consumersexpect the firm not to advertise, a profitable deviation will exist. We can consider two sub-cases.First, suppose that

lima0→∞

maxa≥a

π¡a; a = a0

¢≥ 0. (65)

In this case, the firm will always obtain positive profits from playing the optimal positive level ofadvertising. As a result, the firm cannot be indifferent between playing its optimal positive levelof advertising and playing advertising level 0, because advertising level 0 always delivers profits ofzero. Therefore an equilibrium in mixed strategies does not exist in this case. There does, however,exist a unique pure-strategy equilibrium (in which the firm advertises with certainty). To see this,observe that, because the profit function is concave above a, there is a unique level of advertisinga∗ ≥ a such that

a∗ = argmaxa≥a

π (a; a = a∗) . (66)

Moreover, condition (65) implies that π (a∗; a = a∗) > 0, so that if consumers expect the firm toadvertise at level a∗ > 0 with certainty, the firm will not wish to deviate to an advertising level of0. Advertising level a∗ therefore constitutes a pure-strategy equilibrium advertising policy, and isunique since it is the only solution to equation (66).

Finally, consider the sub-case in which

lima0→∞

maxa≥a

π¡a; a = a0

¢< 0. (67)

58

There must then exist a unique advertising level a > 0 such that

maxa≥a

π (a; a = a) = 0. (68)

The concavity of the profit function above a then implies the existence of a unique advertising levela0 that solves

a0 = argmaxa≥a

π (a; a = a) . (69)

If a0 · a, then because the optimal level of advertising is decreasing in the expected levelof advertising a, there exists a unique level of advertising a∗ ∈ [a0, a] satisfying equation (66),with π (a∗; a = a∗) ≥ 0 (since a∗ · a). Playing advertising level a∗ with certainty constitutes apure-strategy equilibrium advertising policy, since given that a = a∗, the optimal positive levelof advertising is a∗, and playing a = a∗ is at least weakly preferred to playing a = 0. Since a∗

is the unique solution to (66), it is the only possible pure-strategy equilibrium advertising policy.Moreover, a mixed-strategy equilibrium is not possible, since this would require the firm to beindifferent between playing advertising level 0 and advertising level a0. But there is no way torandomize between these two advertising levels such that consumer demand would be as if thefirm were playing advertising level a with certainty, which means that any mixed strategy wouldresult in non-zero profits for playing the optimal positive level of advertising, making indifferenceimpossible.

If a0 > a, then the unique solution to equation (66) occurs on the range [a, a0], meaning thatif the firm plays this level of advertising with certainty, its profits are negative, and it will preferto deviate to playing a = 0. Therefore there cannot exist a pure-strategy equilibrium in this case,and the only possible equilibrium is in mixed strategies. To see that there is such an equilibriumand that it is unique, define σ to be the unique solution to:

σd¡a0¢T= d (a)T (70)

and observe that by Bayes’ rule if consumers expect the firm to play advertising level a0 withprobability σ and advertising level 0 with probability (1− σ), then their posterior expectations,and hence the firm’s objective function π, will be as if the firm were playing advertising levela with certainty. As a result, it is an equilibrium for the firm to play advertising level a0 withprobability σ and advertising level 0 with probability (1− σ), since given that consumers expectthis strategy, the firm will obtain 0 profits from both equilibrium advertising levels, and will notbe able to achieve higher profits from a deviation.¥

B Data

To create figure 7, I use data from the ERIM scanner panel database, which is available publiclythrough the James B. Kilts Center for Marketing at the University of Chicago Graduate School ofBusiness.21 To construct the database, the ERIM division of A. C. Nielsen tracked the grocery pur-chases of 2500 households in each of two local markets–Sioux Falls, South Dakota and Springfield,Missouri–from 1985 through 1988. In the final year of data collection (roughly from August 1987to August 1988), approximately 60% of households had “telemeters” attached to their television

21The data and documentation can be downloaded at http://gsbwww.uchicago.edu/kilts/research/db/erim/. SeeRossi, McCulloch, and Allenby (1996); Ackerberg (2001); and Erdem, Keane and Sun (2004) for examples of researchusing the ERIM database.

59

sets that allowed ERIM to track the household’s television viewing and exposure to television com-mercials. Because telemeters malfunctioned in some cases, I treat a household as having missingadvertising exposure data for a given week anytime its telemeter is reported as non-working, orif the household is recorded as not having been exposed to advertising in a given week and totalrecorded television viewing by the household for that week is zero. To construct the figure, I usedata on the average number of weekly exposures to a Heinz ketchup commercial during the sampleperiod among households with non-missing exposure data.

60