HIGHLY PRELIMINARY – PLEASE DO NOT CITE WITHOUT PERMISSION

ASSISTED SELF-PERSUASION: CHOICE, COMPLEMENTARY

ADJUSTMENT, AND THE ROLE OF ADVERTISING

MATTHEW G. NAGLER* The City College of New York, mnagler@ccny.cuny.edu

The paper presents a theory of imperfect competition à la Hotelling in which consumers choose products and also whether and how much to adjust to (i.e., improve their attitude toward) what they choose. Advertising is modeled as reducing the cost of adjustment. When consumer tastes are distributed symmetrically with respect to products and the marginal costs of getting adjustment are similarly symmetric, advertising raises (lowers) prices when the marginal adjustment costs faced by successive consumers grow lower (higher) as one moves from the position of the indifferent consumer toward positions of extreme preference. Analogous price effects result from advertising-driven translations of the adjustment cost structure that result in a relative reduction (increase) in marginal adjustment costs toward the extremes. While price increases are a necessary condition for advertising to increase profit, firms advertise whenever the adjustment cost reduction due to first dollar of advertising is sufficiently large and the cost reduction function is sufficiently concave, thus prisoners’ dilemma equilibria are possible. *I am grateful to Shay Culpepper for excellent research assistance.

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I. INTRODUCTION

If you are going to do something, you are better off loving it. That is to say,

consumption is naturally complemented by consistent attitude change. This intuitive notion is

embraced by psychology through the framework of cognitive dissonance, which proposes

that when an individual takes an action that is discrepant from his attitude, it creates

displeasure, which in turn motivates him to change his attitude (Festinger 1957). Recent

behavioral and neural evidence across a range of experimental scenarios shows that

individuals routinely undergo a sort of mental re-positioning relative to choices they have

made, changing not only their stated preferences, but also physiological manifestations of

their hedonic responses.1 The mounting evidence supports a paradigm according to which

choices not only reflect, but also create, preferences. (Ariely and Norton 2008).

Why then is attitude adjustment essentially absent from economic models? The

consumer problem in economics is generally conceived as involving choice under imperfect

circumstances, in which options for action do not match perfectly with individuals’

preferences. Utility losses due to imperfect matching are routinely reflected in the modeling

of choice. For example, in Hotelling’s (1929) spatial model of differentiated products,

consumers experience “transportation costs” when their tastes do not align perfectly with

their chosen alternative. Yet, given the standard assumption that tastes are fixed, consumers’

acceptance of the costs or losses associated with imperfect choice (i.e., without adjustment) is

posited as optimizing behavior. Rational consumers assume moreover that they will not

1 Studies offering evidence of preference change based solely on subject ratings of chosen alternatives include Lieberman et al. (2001), Kitayama et al. (2004), Sharot et al. (2010), and Wakslak (2012). Studies that additionally measured changes using functional magnetic resource imaging (fMRI) of subjects’ brains include Sharot et al. (2009), van Veen et al. (2009), Izuma et al. (2010), Jarcho et al. (2011), Qin et al. (2011), Kitayama et al. (2013), and Izuma and Adolphs (2013).

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adjust to their choices, whence, the standard model predicts, they make choices accordingly.2

If, in reality, consumers do adjust, then both the choice predictions and corresponding

hedonic predictions of the models are wrong. Models of consumer choice, updated to account

both for attitude adjustment and for how the adjustment process might be influenced by

relevant market phenomena, could produce superior predictions of behavior and outcomes.

Along these lines, this paper presents a new model of advertising. I posit a model à la

Hotelling in which consumers differ as to their tastes for two competing products. A

consumer can, at a cost, adjust to (i.e., improve his attitude toward) the product he intends to

choose. Advertising is modeled as reducing this cost – in effect facilitating adjustment by

providing fodder for the process of self-persuasion. Neither the cost (or, conversely, the

productivity) of adjustment nor advertising’s impact on it are, in general, the same across all

consumers. The manner in which these vary across consumer “locations” is captured through

the notion of an adjustment map. A minimal set of basic properties are introduced that, it is

argued, all adjustment maps (and advertising-driven translations of the maps) must observe.

A number of important results follow directly from these primitives. Adjustment

lowers prices if the rate of increase of marginal adjustment costs faced by successive

consumers grows higher as one moves from the position of the indifferent consumer toward

positions of extreme preference; it results in higher prices if the rate of increase in these costs

declines toward the extremes. Consequently, when consumer tastes are distributed

symmetrically with respect to the firms and adjustment maps are similarly symmetric,

advertising lowers prices when the rate of increase of marginal adjustment costs increases

2 In the motivated beliefs literature, agents rationally anticipate that their actions will create corresponding beliefs, and their choices reflect this realization (Bodner and Prelec 2003; Benabou and Tirole 2004, 2011; Dal Bo and Tervio 2013). However, none of these papers deals with the possibility that actions might create preferences.

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toward the extremes and increases prices when the rate of increase of these costs declines

toward the extremes. Similar price effects are attributable to advertising-driven translations

of the adjustment map that result in a relative lowering or raising of adjustment productivity

toward the extremes. Outcomes for consumer welfare, firm profits, and overall welfare

depend upon these price effects and the extent to which adjustment cost reductions exceed

the costs of advertising.

Previous economic theory centers on three main views as to how advertising

influences consumers: as persuasion, as information, and as a complement to the advertised

product (Bagwell 2007). The first two of these perspectives are supported by extensive

theoretical and empirical literatures, mostly within the field of industrial organization.3 The

third reflects mostly the contributions of Gary Becker and his collaborators (e.g., Becker and

Murphy 1993). All three perspectives have in common that they presume an essentially

passive consumer: one who is either persuaded or informed by advertising and then, subject

to these influences, makes product decisions. Thus advertising is modeled, for example, as a

demand-shifter (e.g., Bloch and Manceau 1999) or a force that decreases demand elasticity

(e.g., Kaldor 1950). Though a broader variety of roles for advertising have been identified

previously – von der Fehr and Stevik (1998) classify several – there exists to my knowledge

no economic model that posits active consumer engagement with advertising.

And yet analysis by researchers in the fields of marketing and psychology indicates

that often consumers do engage actively with advertising, and that how and whether they do

so has important implications for advertising’s effects. Numerous important contributions in

this area are in the context of the Elaboration Likelihood Model (e.g., Petty 1977), which

3 Important seminal contributions by the persuasion school include Comanor and Wilson (1974) and Dixit and Norman (1978); while among the most important contributions that conceive of advertising as information are Stigler (1961) and Nelson (1974). See Bagwell (2007) for a recent survey.

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considers the conditions under which consumers approach advertising with active thinking as

a part of their decision-making process. Separately, studies of cognitive dissonance reveal

that consumers actively seek persuasive advertising messages that have the potential to

reduce doubts about product choices (Ehrlich et al. 1957, Mills 1965). From the work in

these literatures there emerges a fourth perspective: that advertising is appropriately viewed

as a tool the consumer employs during the purchase decision process. The outcomes of my

model reflect the manner in which viewing advertising as an adjustment-facilitating tool

places its role somewhere between that suggested by the traditional information and

persuasion perspectives.

Section II characterizes adjustment in economic terms. Section III introduces the

model. Section IV examines the model’s equilibrium. Section V discusses welfare

implications. Section VI concludes and discusses opportunities for future research. The

Appendix contains proofs of all lemmas, propositions, and remarks.

II. THE NATURE OF ADJUSTMENT

Let us define adjustment as self-induced attitude change complementary to a

personally-engaged action. The definition distinguishes adjustment from externally-induced

attitude change, such as caused by the persuasive effect of advertising on the passive

consumer attributed by traditional advertising theory. Additionally, it generalizes adjustment

relative to the purest sense of cognitive dissonance-related attitude change in two ways. First,

adjustment can occur with respect to actions the individual has not personally chosen. By

contrast, cognitive dissonance in its original conception arises when “I” take an action that

contradicts my attitudes or values, because it relates critically to my sense of identity as a

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competent and good person (Aronson 2004). This suggests personally choice is a key factor.

Second, construing “complementary to a personally-engaged action” broadly, adjustment

may be viewed as including attempts to mitigate consumption-oriented utility losses of any

kind. Thus adjustment would describe quite broadly consumers’ attempts to make the best of

imperfect situations, whether to mitigate cognitive anguish over being inconsistent or

incompetent (traditional dissonance) or simply to mitigate the displeasure from having or

doing something that is less than ideal.

These generalizations have several justifications. First, adjustment defined in this way

remains conceptually consistent with a broader view of cognitive dissonance according to

which dissonance arises whenever a person commits to a behavior and then assesses that

behavior against some criterion of judgment (Stone and Cooper 2001). Whether or not the

individual chose the activity may not be essentially relevant to whether he feels he

“committed to it,” as forced-compliance experiments demonstrate (Festinger and Carlsmith

1959). Moreover, a consumer’s criterion for judging a consumption experience may be

whatever causes it to fall short of the ideal. Thus any source of emotional disappointment or

let-down that a consumer experiences when taking a less-than-ideal action may be viewed as

cognitive dissonance. For example, a consumer who feels let down by a consumption

experience might feel this way because he begins to doubt his competence as a decision-

maker or think that he is an unlucky person; both sensations are consistent with a broad

construal of cognitive dissonance (Kitayama et al. 2004).

More generally, various authors appear to view attitude change as a general

consequence of the consumer decision process (Jarcho et al. 2011). Consider, for example,

Middlestaedt’s (1969, p. 444) account of re-purchase behavior:

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If the consumer’s experience with the product is satisfactory, the reduction of post-choice dissonance should lead him to increase his preference for the brand he has selected, thus increasing the probability of repurchase … A qualification to this argument is that a high level of dissonance could cause the buyer to reduce dissonance by switching his choice. Therefore, the argument … is restricted to only cases involving slight or moderate post-choice dissonance.

Recasting this process in terms of the initial purchase and initial attitudes toward the product,

one has a fairly general description of a dual process of choice and adjustment, whereby the

consumer decides what product to buy based, in part, on whether he can effectively adjust his

attitude to it. Not incidentally, the general view recognizes adjustment as an optimizing

reaction to costs inherent in the decision and views the nature of those costs as immaterial to

the behavior of the decision-maker.

The rest of this section presents supporting evidence for four economic characteristics

of adjustment that will form the basis for the model presented in the following section.

A. Adjustment increases the utility of the chosen option

This is straightforward economic characterization of the accumulated evidence on

hedonic values and stated preferences from experimental and neural studies on cognitive

dissonance and attitude change. Much of evidence relates to so-called “spreading of

alternatives,” whereby individuals are found to increase their ratings or hedonic responses

with respect to chosen alternatives while decreasing them with respect to non-chosen

alternatives. (See, e.g. Brehm 1956, Festinger 1964, Lieberman et al. 2001, Sharot et al.

2009). Kitayama et al. (2013), in contrast, finds neural evidence only of increased utility

from chosen option, not of decreased utility from non-chosen options.

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B. Adjustment is costly

The traditional conception of attitude change is that it involves a drawn-out and

effortful process of rationalization and rumination (Festinger 1964, Petty and Wagener

1998). More recently, however, Lieberman et al. (2001) found that both amnesiacs and

people under cognitive load engage just as effectively in attitude change, and perhaps more

so, relative to healthy or unburdened individuals. This evidence has cast doubt on whether

significant cognitive effort is in fact involved in attitude change. Van Veen et al.’s (2009)

neural study supports the notion that reflection and memory play a limited role, if any, in

attitude adjustment.

Still more recent evidence suggests that attitude change, though often occurring

quickly in the moment of decision, does involve a characteristic rationalization process

(Jarcho et al. 2011) and clear indication of effort expenditure (Kitayama et al. 2013). It has

moreover been argued that adjustment may occur in different situations – including both

quick in-the-moment adjustments, and also processes invoked at a later time that involve

extended rumination or rationalization (Jarcho et al. 2011).

Another possibility is that adjustment is aversive. “Changing one’s mind” self-signals

personal inconsistency, which may itself cause dissonance. For this reason, individuals might

adjust their attitudes only when a threshold for net benefit is reached, even if the process is

effortless or nearly so. Finally, evidence that various factors moderate attitude change

suggests that adjustment involves costs (see next subsection); otherwise adjustment would

occur independent of such factors.

C. Adjustment is a rational or quasi-rational process

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Introspection suggests that there are situations in which we are aware of having

adjusted to our choices. For example, one may report, “I made the best of it,” or “I have

become fond of her over time.” Whether we are conscious of adjustment in the moment that

it happens is harder to assess, however, and there is no support for conscious attitude change

on the basis of self-reporting. In fact, evidence that attitude change occurs regardless of

whether an individual is experiencing cognitive load or is an amnesiac (Lieberman et al.

2001), and that it is set in motion by the mere act of rejecting an alternative (Izuma et al.

2010), suggest that the process may be practically automatic and perhaps not consciously

attended.

Whether or not adjustment is a conscious process is immaterial for our purposes.

What matters is whether it behaves as if it were being undertaken consciously by a rational

individual; that is, does the individual adjust if and only if the benefits, in terms of increased

utility, outweigh the costs? In this regard, there is indirect supporting evidence from the

effect of moderating factors and the timing of adjustment.

That moderators would impact the likelihood of dissonance-induced attitude change

was proposed from the very genesis of the theory (Festinger 1957). Robust experimental

evidence now supports the truth of this. The likelihood of adjustment may vary, for instance,

depending upon the cultural background of the subject (Kitayama et al. 2004, Qin et al.

2011), whether a choice is perceived to be more self-relevant (Jarcho et al. 2011, Kitayama et

al. 2013), whether the matter is construed as relating to high ideals or trivial secondary

features (Wakslak 2012).

Support for adjustment as a quasi-rational process also comes from van Veen et al.

(2009), whose neural evidence indicates that adjustment occurs concurrent with a perceived

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challenge to the consistency of attitude rather than at a later point in time, as would be

consistent with a reflective process more indicative of self-perception theory (Bem 1972).

The results suggest that adjustment is set in motion when a perceived benefit (the opportunity

to reduce the unpleasant experience of dissonance) arises that outweighs the cost of

adjustment.

D. Adjustment is a rational or quasi-rational process

While Ehrlich et al. (1957) and Mills (1965) provide evidence of subjects using

advertising to facilitate dissonance-reducing attitude change, recent cognitive studies offer

further insights into how the process might work. Kitayama et al. (2013) find that when

individuals experience dissonance they actively direct their attention to seeking attractive

features from their chosen option, linking those features to self-knowledge to establish why

“I” like the item (pp. 206-7). Meanwhile, Dmochowski et al. (2014) find that people seek

stimuli that act as reliable drivers of the brain – that is, they direct attention to stimuli that

can reliably alter attitudes in a way that is predictable across larger groups of individuals like

themselves. Thus, persuasive mass media appeals may offer consumers more promise for

utility-increasing adjustment than information from obscure sources or idiosyncratic

arguments.

III. THE MODEL

Consider two products, indexed 0 and 1, each produced by an independent firm

correspondingly named. The firms are located at opposite ends of a segment of length 1.

Following Hotelling’s model (1929), each consumer is characterized by a location x on the

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segment 0,1[ ] , measuring his relative taste for the two products. Consumers are assumed

distributed on this segment according to an arbitrary distribution function F with full support

and continuous density function f. I assume the baseline utility of a consumer buying product

j is given by

(1) Ux =V − pi − t x − i

where V is the common reservation price for the product, pj is the price of product j, t is a

parameter the cost of buying a product that is not the consumer’s ideal choice – the standard

“transportation” cost.

Suppose that the consumer has the possibility of adjusting to a product, defined as

relocating, in effect, on the segment to be closer to that product and thereby pay less

transportation cost. The process is quite naturally viewed as an incremental one, involving

incremental investment of costly effort (or aversive activity) that pays off with an

incremental improvement in attitude. Consistent with this incremental view, let us posit an

adjustment relocation marginal cost function associated with product j, g j ,0 i, x( ) > 0 , where i

is the distance from x and closer to j’s position – either 0 or 1. One may view this function as

in effect representing a set of adjustment curves Gj ,0 := g j ,0 i( ) = g j ,0 i, x( ) : x ∈ 0,1[ ]{ }

characterized by differing values of x, whereby each curve represents the cost, at each state

of attitude improvement i, of incremental “movement toward” j for the consumer located

initially at x. Let us refer to Gj ,0 as an adjustment map for product j.

Advertising in this framework reduces adjustment costs; one may think of it as

improving the consumer’s technology of adjustment. I model it as a general translation of the

adjustment map defined by

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(2) g j i, x,Aj ,rj( ) ≡ φ Aj( ) rjg j ,1 i, x( ) + 1− rj( )g j ,0 i, x( )⎡⎣ ⎤⎦ + 1−φ Aj( )⎡⎣ ⎤⎦g

j ,0 i, x( )⇔ g j i, x,Aj ,rj( ) = φ Aj( )rj g j ,1 i, x( )− g j ,0 i, x( )⎡⎣ ⎤⎦ + g

j ,0 i, x( )

where φ Aj( ) is a production function for advertising’s improvement of the adjustment

technology; r ∈ 0,1[ ] parameterizes advertising’s productivity; Aj represents firm j’s

advertising expenditure; and g j ,1 i, x( ) is what I shall refer to as the “asymptotic adjustment

function” (AAF).

Within this basic structure, let us introduce a set of assumptions about adjustment and

advertising:

ASSUMPTION 1 (Continuity of adjustment cost in x and i): g0,k i, x( ) and g1,k i, x( ) are

defined on 0, x[ )× 0,1[ ]→ !+ and 0,1− x[ )× 0,1[ ]→ !+ , respectively, and continuous on

their support, for k = 0,1 .

ASSUMPTION 2 (Convexity of adjustment curves): gij ,k > 0 and gii

j ,k > 0 , for j,k = 0,1 .

ASSUMPTION 3 (Continuity of advertising productivity in A): φ Aj( ) is defined on

0,∞[ )→ 0,1[ ] and continuous on its support, for j = 0,1 .

ASSUMPTION 4 (Concavity of advertising productivity): φAj > 0 and φAjAj < 0 , for j = 0,1 .

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ASSUMPTION 5 (Asymptotic convergence of advertising productivity): limAj→∞

φ Aj( ) = 1 ,

whence limAj→∞

φAj Aj( ) = 0 for j = 0,1 .

ASSUMPTION 6 (Advertising-driven dominance): g j ,1 i, x( ) < g j ,0 i, x( ) ∀i > 0 , x ∈ 0,1[ ] , for

j = 0,1 .

It follows that incremental adjustment costs are non-increasing in advertising, that is,

∀i > 0, x ∈ 0,1[ ] , g j i, x,Aj ,r( ) ≤ g j ,0 i, x( ) for all r ∈ 0,1[ ],Aj ≥ 0 , with strict inequality

holding for r ∈ 0,1( ],Aj > 0 . This allows us to establish the following:

LEMMA 1: For all i > 0, x ∈ 0,1[ ] , for any Aj > 0 , ∂2g j ∂i∂Aj ≤ −∂g j ∂Aj .

The precise effect of advertising reflected in (2) is to reduce adjustment costs asymptotically

toward a bound determined by r. At one extreme, when r = 0 , we have nested the case in

which advertising has no effect on adjustment. At the other extreme, when r = 1 , increases in

advertising expenditure move consumers toward the AAF, which may be thought of as the

ultimate limit of consumers’ adjustment technology. Over the intermediate range, r ∈ 0,1( ) ,

the consumer’s cost of adjusting converges toward some weighting of g j ,0 i, x( ) and the AAF

as advertising expenditure increases.

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Assumption 6 in effect sets forth an intuitive presumption that advertising generally

makes consumers better at adjusting – or at least no worse. Let us make a similarly intuitive

presumption about relative preference for products:

ASSUMPTION 7 (Preference-based dominance): For all x ∈ 0,1[ ] , − ∂g0,k

∂x ≤ ∂g0,k

∂i (and

∂g1,k

∂x ≤ ∂g1,k

∂i ).

That is, the more preferred a product is initially, the lower the cost of incremental adjustment

at every particular level of attitude improvement achieved through accumulated adjustment.

This “dominance” condition implies that an individual who initially prefers a product more

than another individual finds it less costly to achieve a given attitude toward that product

through adjustment than the other individual, and it gives rise to adjustment maps of non-

crossing nested contours, similar to well-behaved indifference maps. It follows that

adjustment results in greater achieved attitude the greater the individual’s initial proximity to

a product. Because this property applies both to the pre-advertising adjustment function and

the AAF, it applies also to the post-advertising adjustment function g j .

Extending Assumption 2, we assume that adjusting to the point where a product is

viewed as ideal, whether with advertising assistance or without it, is infinitely costly. Thus,

while someone might get quite comfortable with a product at finite cost, one cannot get

perfectly comfortable. I adopt this both because it is sensible and also helps avoid corner

cases:

ASSUMPTION 8 (Asymptotic satiability): limi→x

g0,k i, x( ) = ∞ (and limi→1−x

g1,k i, x( ) = ∞ ).

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For consumers for whom adjustment is preferred with respect to a given product

relative to accepting the product “on spec,” it is possible to define the notion of adjustment

productivity: how much attitude improvement with respect to the product the consumer will

choose to attain, given his preferences and his particular capabilities at adjusting to it. Define

the set Xj t( ) := x :g j ,0 0, x( ) < t{ } ; since g j ,0 0, x( ) is not required to be monotonic in x,

Xj t( ) need not be compact. Then one may define the implicit function i* j ,0 x,t( ) on

Xj t( )× t > 0{ }→ !+ such that g j ,0 i* j ,0 x,t( ), x( ) = t as consumer x’s “adjustment

productivity given t.” One may similarly define i* j ,1 x,t( ) and i* j x,t,Aj ,r( ) based on

g j ,1 i* j ,1 x,t( ), x( ) = t and g j i* j x,t,Aj ,r( ), x,Aj ,r( ) = t , respectively. Note that, for x ∉Xj t( ) ,

i* j ,0 x,t( ) = 0 . Thus the adjustment model nests non-adjustment as a sub-case.

The following lemma advances several useful results that follow from the definition

of adjustment productivity:

LEMMA 2: (i) ix*0 ≤1 (and ix

*1 ≥ −1); (ii) iA0*0 ≥ 0 (and iA1

*1 ≥ 0 ); (iii) limA0→∞

iA0*0 = 0 (and

limA1→∞

iA1*1 = 0 ); and (iv) ir

*0 ≥ 0 (and ir*1 ≥ 0 ).

The utility of a consumer buying product 0 is given by

(3) U0 =V − p0 − t x − i*0 x,A0,r( )⎡⎣ ⎤⎦ − g0,0 i, x,A0,r( )di0

i*0 x,A0 ,r( )

∫

and, for a consumer buying product 1, by

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(4) U1 =V − p1 − t 1− x − i*1 x,A1,r( )⎡⎣ ⎤⎦ − g1,0 i, x,A1,r( )di

0

i*1 x,A1,r( )

∫

Notice that (3) reduces to the traditional Hotelling utility when i*0 = 0 .4 Following Bloch and

Manceau (1999), but extended to the adjustment case, I impose what is in effect a restriction

on the size of V relative to t and to the rate of change of g0,0 with respect to x:

ASSUMPTION 9: V − t x − i*0,0 x( )⎡⎣ ⎤⎦ − g0,0 i, x( )di0

i*0,0 x( )

∫⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪F x( ) is increasing for all x ∈ 0,1[ ] .

The assumption is a sufficient condition for the market to be covered under adjustment, as

shown by the following lemma:

LEMMA 3: Given Assumption 9, the market is covered in equilibrium.

To simplify the analysis and avoid corner cases, the analysis of advertising will be

performed in the context of the following minimally-restrictive assumption about consumers’

adjustment productivity:

ASSUMPTION 10 (Strong adjustment feasibility): Let x* be the location of the indifferent

consumer when there is no adjustment.5 Then x*,1⎡⎣ ⎤⎦ ⊂ X0 t( ) and, correspondingly,

0, x*⎡⎣ ⎤⎦ ⊂ X1 t( ) .

4 See, for example, Bloch and Manceau (1999, p. 560). 5 The existence of an x* is guaranteed by Lemma 3.

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This assumption provides that all consumers have the incentive to adjust to at least one

product, and moreover that all consumers that a firm might target with its advertising (i.e.,

marginal, or else inframarginal to its rival) have positive adjustment productivity with respect

to its product. Given this assumption, advertising does not need to be expended to bring

consumers to the point where adjustment becomes feasible; rather its purpose is focused on

improving the adjustment productivity of consumers who are already at least minimally

productive at adjustment.

The location xE* of the indifferent consumer under adjustment can be derived by

setting U0 =U1 . Thus it is defined implicitly by

(5)

Θ xE* , p0, p1,A0,A1,r( ) ≡ p1 − p0 + t − 2txE

* − t i*1 xE* ,A1,r( )− i*0 xE

* ,A0,r( )⎡⎣ ⎤⎦

+ g1 i, xE* ,A1,r( )di

0

i*1 xE* ,A1,r( )∫ − g0 i, xE

* ,A0,r( )di0

i*0 xE* ,A0 ,r( )∫ = 0

Based on this, one may define market shares for the two products as D0 = F xE*( ) and

D1 = 1− F xE*( ) .

The game consists of two periods. In the first, firms choose advertising expenditures,

taking each other’s advertising expenditure choices as given. In the second, they choose

prices, taking each other’s prices and their previous advertising choices as given. Firms

recognize in the first period that their own prices will depend on their prior advertising

choices and so treat these strategically with respect to their advertising decisions; however,

they treat each other’s pricing decisions as given. Given demand, profits of the firms are

given by

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(6) Π0 = p0F xE

*( )− A0Π1 = p1 1− F xE

*( ){ }− A1

As a final assumption, I employ a variant on a distributional restriction by Caplin and

Nalebuff (1991), which they showed constitutes a sufficient condition for the existence of a

unique equilibrium in a broad class of games. Bloch and Manceau (1999) demonstrated the

use of the Caplin-Nalebuff assumption in a model of persuasive advertising. The present

variant generalizes it to the model involving adjustment by imposing a set of complementary

restrictions on the consumer distribution f and the adjustment functions g j ,k . In the

Appendix, it is demonstrated that the assumption applies to a rather general set of functional

form combinations with an intuitive basis.

ASSUMPTION 11: F .( ) is log concave in p0 (and 1− F .( ) is log concave in p1 ).

IV. EQUILIBRIUM

E. Equilibrium in prices

We consider the second-period problem of price-setting first, where firms take each

other’s prices and advertising levels as given. Differentiating firm 0’s profit equation in (6)

with respect to price yields

(7) ∂Π0

∂p0= F xE

*( ) + p0 f xE*( ) ∂xE

*

∂p0

To derive ∂xE* ∂p0 , one applies Cramer’s rule to (5), which yields

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(8) ∂xE

*

∂p0= 1

−2t + dg1

dxE* di

0

i*1 xE* ,A1( )∫ − dg0

dxE* di

0

i*0 xE* ,A0( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

< 0

Note that this expression reduces to the constant − 12t in the no-adjustment case. The nature

of the departure of (8) from that case depends on two things: the marginal consumer’s

adjustment productivity with respect to both products, and the rate with which the adjustment

functions change with xE* . We may therefore state:

PROPOSITION 1: (i) The greater the rate at which the adjustment curves for either product

flatten (steepen) as one moves from the marginal consumer toward the extremes, the lower

(higher) the responsiveness of demand for both products to changes in price. (ii) The

responsiveness of demand for both products to changes in price falls (rises) with increases in

the marginal consumer’s adjustment productivity of either product when adjustment curves

for that product grow flatter (steeper) as one moves toward the extremes.

Proposition 1 provides our first glimpse into the forces that govern competitive

equilibrium under adjustment-facilitating advertising. The core effects have to do with the

shape of the adjustment map. Consider Figure 1. The first panel of the figure shows a map in

which the adjustment curves grow flatter, at first, as one moves from xE* toward the

extremes. The second panel shows a map in which the curves grow steeper as one moves

toward the extremes. In both cases, the curves must eventually become increasingly steep as

one approaches x = 0 or x = 1 . This follows from Assumption 8. What is critical to our

result, however, is what happens near xE* . If the curves grow flatter toward the extremes in

19

the vicinity of xE* , a price increase for a product moves to the margin previously-

inframarginal consumers who find adjustment more productive at improving their attitude

than the consumer at xE* . These consumers, if they switched products, would forgo lower

total costs (i.e., transportation plus adjustment costs) from the product they left and incur

greater total costs from their new product relative to the previously-marginal consumer. Thus

demand is per se less price-sensitive, all else equal, when the adjustment map has this

particular shape.

< INSERT FIGURE 1 APPROXIMATELY HERE >

Consider, on the other hand, what occurs when the adjustment map has the shape

displayed in the second panel of Figure 1. Then, a price increase moves to the margin

previously-inframarginal consumers who find adjustment less productive than the consumer

at xE* . These consumers, if they switched products, would forgo higher costs from the

product they left and incur lower costs from their new product. An adjustment map with this

particular shape sets up an increased incentive for switching, whence demand becomes more

price-sensitive, all else equal.

A second component also influences the price sensitivity of demand for both

products: the magnitude of adjustment productivity. Because the effects we have described

are due to the role of adjustment in consumer decision-making, greater adjustment

productivity simply intensifies the price-sensitivity effect that holds sway for a given product

due to the steepening or flattening shape of that product’s map, all else equal. That accounts

for the second part of the proposition.

An important recognition is that the relative flatness of either product’s adjustment

map affects the price sensitivity of both products’ demands. This is evident from inspection

20

of (8). Thus an adjustment map for product 0 that flattens more quickly toward the extremes

imparts lower price sensitivity to both products’ demands. Similarly, the adjustment

productivity levels of the marginal consumer with respect to both products play a role in the

price sensitivity of each product’s demand. If, for example, the marginal consumer’s

adjustment productivity with respect to product 0 is greater in absolute terms, all else equal,

this will impart greater price sensitivity to both products’ demands if the adjustment map for

product 0 steepens moving toward the extremes, while it would impart lower price

sensitivity, again, to both products’ demands if the map for product 0 flattens toward the

extremes.

Using (7), one obtains ∂Π0 ∂p0( )p0=0

= F xE*( )

p0=0> 0 : non-zero demand for product

0 is guaranteed at p0 = 0 by t > 0 and g j ,0 i, x( ) > 0 . Moreover,

∂Π0 ∂p0( )p0 xE* =0

= p0 f 0( ) ∂xE* ∂p0( ) < 0 , where p0 xE

* =0 > 0 . Because, following from

Assumption 11, −F xE*( ) f xE

*( ) ∂xE*

∂ p0 must be decreasing in p0 , it follows that:6

PROPOSITION 2: There exists a unique pure strategy Nash equilibrium in prices p0*, p1

*( )

where the prices are given by p0* = −F xE

*( ) f xE*( ) ∂xE

*

∂ p0 and p1* = 1− F xE

*( )⎡⎣ ⎤⎦ f xE*( ) ∂xE

*

∂ p1 .

The equilibrium prices exhibit some intuitive characteristics. When the demand is

skewed toward one of the firms (i.e., the distribution f is lopsided such that F xE*( ) and

6 The proposition represents a variation on Bloch and Manceau’s (1999) Proposition 2. We are essentially extending it to the adjustment case.

21

1− F xE*( ) diverge), prices are higher for the favored firm. Importantly, in view of

Proposition 2, it is possible to say something about the effect of adjustment on prices:7

PROPOSITION 3: (i) The greater the rate at which the adjustment curves for either product

flatten (steepen) as one moves toward the extremes, the higher (lower) equilibrium prices are

for both products. (ii) Equilibrium prices for both products are increasing (decreasing) in the

marginal consumer’s adjustment productivity for either product when adjustment curves

grow flatter (steeper) moving toward the extremes for that product.

The price outcomes follow from the effects that a differing shape to the adjustment map

imparts to price sensitivity. Again, the intuition has to do with switching. Since price

increases encourage more rapid switching if adjustment maps steepen, whence the new

marginal consumers find adjustment of the increased product less productive and adjustment

of the other product more productive than did the previously marginal consumers, firms will

naturally find it less profitable to increase prices under such circumstances. The opposite is

true when adjustment maps flatten toward the extremes. Since these effects are driven by

adjustment, the magnitude of adjustment productivity, which conditions the relative size of

the role adjustment costs play in the consumer’s decision, also mediates the magnitude of the

price effects. Where a relatively rapid rate of flattening of a product’s map implies higher

prices for both products, the magnitude of adjustment productivity for the product in question

determines to what degree those effects express themselves.

7 The proposition follows directly from the price equations given in Proposition 2 and (8): lower (higher) responsiveness of demand to price implies a higher (lower) price.

22

F. Equilibrium in advertising

Let us now turn to the first-period advertising decision of the firms. We re-write (6) to

reflect the dependence of price on the choice of advertising levels:

(9) Π0 = p0 A0,A1( )F xE

*( )− A0Π1 = p1 A0,A1( ) 1− F xE

*( ){ }− A1

Differentiating firm 0’s profit equation in (9) with respect to A0 yields

(10) ∂Π0

∂A0= p0 f xE

*( ) ∂xE*

∂A0+ ∂xE

*

∂p0

∂p0∂A0

⎡

⎣⎢

⎤

⎦⎥ +

∂p0∂A0

F xE*( )−1

We derive ∂xE* ∂A0 by applying Cramer’s rule to (5), yielding

(11) ∂xE*

∂A0= φA0r0

g0,1 − g0,0⎡⎣ ⎤⎦di0

i*0 xE* ,A0( )∫

−2t + dg1

dxE* di

0

i*1 xE* ,A1( )∫ − dg0

dxE* di

0

i*0 xE* ,A0( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

> 0

Substituting expressions for p0 from Proposition 2, and then expressions for ∂xE* ∂p0 and

∂xE* ∂A0 from (8) and (11), respectively, yields the convenient expression

(12) ∂Π0

∂A0= −F xE

*( )φA0r g0,1 − g0,0⎡⎣ ⎤⎦di0

i*0 xE* ,A0 ,r( )∫ −1

Notice what happens at A0 = 0 : for r > 0 - that is, any situation in which advertising

is productive at all – it is only necessary that the first unit of advertising be sufficiently

productive (i.e., for φA0 A0=0 sufficiently large) for ∂Π0 ∂A0 > 0 . Specifically, it must be that

23

(13) φA0 A0=0r > 1

F xE* ,0( ) g0,0 − g0,1⎡⎣ ⎤⎦di

0

i*0 xE* ,0,r( )∫

which is the minimally sufficient condition for advertising to be deemed profitable.

Meanwhile, by Assumption 5 and Lemma 2, respectively, as A0 grows large, φA0 and iA0*0

approach zero. Thus g0,0 − g0,1⎡⎣ ⎤⎦di0

i*0 xE* ,A0 ,r( )∫ is bounded above with respect to increases in

A0 > 0 , implying that there exists A0 such that, for A0 > A0 , ∂Π0 ∂A0 < 0 . It remains only

to demonstrate the conditions under which ∂2Π0 ∂A02 < 0 to show8

PROPOSITION 4: There exists a unique pure strategy Nash equilibrium in advertising

choices A0*,A1

*( ) with A0*,A1

* > 0 if and only if (i) the initial marginal productivity of

advertising φA0 A0=0 is large enough, and (ii) φ is concave enough.

With Proposition 4 as a baseline, let us now consider variations in the exogenous

advertising productivity parameter, r. These allow us to study effects of the intensity of

adjustment-facilitating advertising on prices and related market outcomes. Our main focus

will be on the overall impacts that occur in a symmetric market context.

Taking the functional form pj A0 r( ),A1 r( )( ) and differentiating with respect to r, one

obtains (for j = 0 ):

(14) ∂p0∂r T

= ∂p0∂A0

∂A0∂r

+ ∂p0∂A1

∂A1∂r

+ ∂p0∂p1

∂p1∂A0

∂A0∂r

+ ∂p0∂p1

∂p1∂A1

∂A1∂r

+ ∂p0∂r D

+ ∂p0∂p1

∂p1∂r D

8 See Appendix.

24

This expression comprises direct effects of r on price and indirect effects through advertising.

Our focus here in on the sign of the indirect effects, which represent the effects of exogenous

changes in the amount of advertising, all else equal. Therefore, we proceed by dropping the

last two terms and signing the rest of the expression. Expressions for ∂p0 ∂p1 and all four

price-advertising partial derivatives may be derived by applying Cramer’s rule to the first-

order condition for price F xE*( ) + p0 f xE

*( ) ∂xE* ∂p0( ) = 0 . Meanwhile, the ∂Aj ∂r come

from the application of Cramer’s rule to the first-order condition for advertising

p0 f xE*( ) ∂xE

*

∂A0+ ∂xE

*

∂p0

∂p0∂A0

⎡

⎣⎢

⎤

⎦⎥ +

∂p0∂A0

F xE*( ) = 1 . We are thus able to derive

PROPOSITION 5: Assume a symmetric case, that is, (1) f symmetric, i.e., f x( ) = f 1− x( ) ;

(2) g0,k i, xE*( ) = g1,k i, xE*( )∀i , gx

0,k i, xE*( ) = −gx

1,k i, xE*( )∀i , and gxx

0,k i, xE*( ) = gxx1,k i, xE*( )∀i .

Then, advertising’s effect on prices is the sum of two effects:

(i) an intensification effect – the effect of increasing the impact of adjustment based

on the existing map, which either flattens or steepens toward the extremes; and

(ii) a shaping effect – the effect of making the adjustment map for each product flatter

or steeper toward the extremes.

Specifically, prices will increase with advertising if dg0

dxE* iA0*0 + ∂2 g0

∂xE* ∂A0

di0

i*0 xE* ,A0( )∫ > 0 (whence, by

symmetry, dg1

dxE* iA1*1 + ∂2 g0

∂xE* ∂A1

di0

i*1 xE* ,A1( )∫ < 0 ) and decrease otherwise.

25

Proposition 5 embodies the intuitive result that advertising’s effect on prices arises

from two component effects that advertising, in general, has on the consumers’ collective

technology of product-attitude improvement as reflected in the adjustment map. First,

advertising moves the existing adjustment map downward, that is, it decreases the cost to

every consumer of adjusting to the advertised product – or, equivalently, increases

adjustment productivity – holding variation in the cost of incremental adjustment (i.e.,

attitude improvement) across consumers constant. We refer to this effect as the

“intensification effect”; it may be thought of as analogous to the income effect of a change in

the price of a product, in that it represents a “parallel” shift in the consumers’ collective

utility-generating technology. This component represents the notion that advertising

intensifies adjustment’s existing effects on prices by making everyone better at adjustment.

Second, advertising changes the shape of the adjustment map, that is, it alters the structure of

how incremental adjustment costs vary across consumers, holding constant adjustment

productivity. We refer to this effect as the “shaping effect,” and it may be thought of as

analogous to the substitution effect of a price change. This component represents the notion

that advertising may change adjustment’s effect on prices by changing how some consumers

go about adjusting relative to others.

Figure 2 illustrates. The solid red lines show the initial adjustment maps for both

products. The dashed red lines show the intensification effect. The solid blue lines, which

illustrate the final post-advertising adjustment map, allow us to observe the shaping effect

when compared to the dashed red lines.

< INSERT FIGURE 2 APPROXIMATELY HERE >

26

The components may be either positive or negatively signed. Because iA0*0 > 0 (and

iA1*1 > 0 ), the sign of the intensification effect depends upon dg j

dxE* , which is positive in the case

of product 0 (or, equivalently, negative in the case of product 1) for an adjustment map that

grows flatter toward the extremes, and the reverse for a map that grows steeper toward the

extremes. Meanwhile, ∂2 g0

∂xE* ∂A0

> 0 (and ∂2 g1

∂xE* ∂A1

< 0 ) implies advertising causes the adjustment

map to flatten increasingly toward the extremes, whence the sign of the shaping effect is

positive. Reversing the signs causes the opposite effect.

V. WELFARE

Let us now turn to the welfare implications of adjustment-facilitating advertising that

flow from the equilibrium results.

REMARK 1: Profits to the firms increase under advertising if and only if the increase in

price exceeds 2A0* , that is, if 3

2F xE

*( )f xE

*( )dg0

dxE* iA0*0 + ∂2 g0

∂xE* ∂A0

di0

i*0 xE* ,A0

*( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪0

A0*

∫ dA > 2A0* .

At the center of this expression (in square brackets) is the determinant from Proposition 5;

clearly, a positive sign on that component is a necessary, but not sufficient, condition for the

firms to enjoy increased profits. Relatedly, we observe

27

REMARK 2: Whenever prices decrease or increase too little, advertising represents a

prisoners’ dilemma for firms: advertising dominates not advertising for each firm, but profits

are reduced when both firms advertise.

This follows from Proposition 4’s result that the advertising level is positive in equilibrium

under the conditions set within the model.

The last two remarks relate to consumer surplus and overall social welfare:

REMARK 3: When prices drop, there is an unambiguous improvement in consumer surplus.

When they do not, consumer surplus improves if and only if the sum of adjustment and

transportation cost reductions exceeds the increase in prices.

REMARK 4: Advertising improves overall welfare if and only if the sum of adjustment and

transportation cost reductions across all consumers exceeds 2A0* .

Because the market is covered in equilibrium and there are no outside goods, prices do not

affect total market demand and so do not affect social welfare. They only affect how welfare

is distributed between the consumers and firms.

VI. CONCLUSION

This paper has presented a new economic theory of advertising as assisted self-

persuasion. This model is consistent with the Elaboration Likelihood Model from the

28

marketing literature, which brings to bear the possibility that, in high-motivation or high-

ability situations, individuals will engage actively with advertising, effectively using it as a

tool. Its main proposition – that individuals through costly activity may increase the utility

they obtain from a product – is supported by evidence from psychology on reactions to

cognitive dissonance.

The results of the theoretical model reveal similarities to both the traditional

advertising-as-information and advertising-as-persuasion views, as well as differences from

both. Similar to advertising under the information perspective, advertising in our model

reduces consumer costs. It therefore tends naturally to lead to increased welfare, and indeed

welfare increases in the model whenever the reduction in adjustment and transportation costs

across consumers exceeds the costs of the advertising. But, quite unlike advertising as

usually viewed from information perspective, the purpose of advertising in our model is not

the reduction of uncertainty. Both the price and welfare effects of adjustment-facilitating

advertising thus have nothing to do with search- or decision-cost reduction in any

conventional sense. Similar to advertising under the persuasion perspective, the main

mechanism of advertising in our model is to bring about shifts in the mass of consumers in

preference space. The effects on prices that we observe in the model are rightly viewed as

accruing to the “taste-altering” effects of persuasion. But the nature of the price effects –

which occur because of advertising’s intensification and restructuring of the adjustment

process – differs categorically from that of the usual demand-shifting competitive effects

observed in traditional persuasion models.

This paper has only initiated the process of understanding adjustment-facilitating

advertising, and there are many opportunities for future work. First, it would be useful to

29

extend the welfare analysis to examine whether there is too much or too little advertising in

equilibrium, both from the perspective of social welfare and joint-profit maximization. It

would also be helpful not just to know the general welfare effects of this advertising, but the

determinants of which consumers benefit. Second, while we have focused on the symmetric

case with respect to price effects, it would be useful additionally to study how advertising

effects pricing and competition in the case where the market is biased in favor of one of the

firms. How does a dominant firm use adjustment-facilitating advertising versus how an

upstart firm might use it? What are the implications for equilibrium pricing and welfare when

the market is lopsided?

Third, it would be informative to extend the model to consider firms having a choice

among advertising strategies that would affect the adjustment map in different ways. This

conception is realistic and could be revealing with respect to a number of welfare-relevant

phenomena. Consider, for example, a case in which advertising reduces adjustment costs and

flattens adjustment curves. The firm faces three strategic options: (i) target all consumers

with the advertising, (ii) target only consumers near the inframarginal extreme to it, and (iii)

target only consumers away from the inframarginal extreme (i.e., nearer the margin and

extramarginal extreme) to it. Figure 3 illustrates the three cases and their effects on the

adjustment map: the dashed red curves show the pre-advertising map, while the solid blue

curves show the post-advertising map. Which strategy would the firms choose? Would the

choice depend upon certain parameter values? What is interesting in this scenario is the

possibility that the firms might prefer (ii) to (i). We know from Proposition 5 that the

adjustment map resulting from (ii) entails higher prices. On this basis, it would not be

surprising if the firms favor it. If they do, one is faced a “shrouding equilibrium” in which the

30

firms and some consumers benefit substantially from advertising, while a group of

consumers (those who are indifferent between products or nearly so) are “left in the dark”

and experience far less benefit. Note that, if (ii) is preferred by firms in Nash equilibrium, it

is a dominant strategy, meaning neither firm would have an independent incentive to

enlighten the median consumers. Thus, as in other scenarios involving shrouding equilibria

(e.g., Gabaix and Laibson 2006), a suboptimal market outcome results under competition.

<INSERT FIGURE 3 APPROXIMATELY HERE>

APPENDIX

A. Applicability of log concavity of F .( ) in pj

In this section, I will show that log concavity of F .( ) in p0 – a critical condition for

the existence of an interior equilibrium in prices – may be met (1) for the general class of

symmetric adjustment map pairs for any log concave distribution f, and (2) for an example of

a non-symmetric adjustment map pair when f is Beta distributed with shape parameters

α ,β( ) = 3,3( ) . The main issue in the case of non-symmetric map pairs is that, approaching

the extreme locations x = 0 and x = 1 , consumers’ marginal adjustment costs approach

infinity for the nearby product. Thus, unless marginal adjustment costs for the distant product

similarly grow without limit, sensitivity of demand to price drops precipitously at the

extremes, making it potentially profitable for firms to attempt to drop price from any

candidate interior maximum to a low enough level to take the whole market. This situation is

avoided if the density of consumers at the extremes is sufficiently low, as with some log-

concave distributions such as the Beta. So, to summarize, an interior price equilibrium will

31

result whenever the incentive to de-stabilize an interior price equilibrium is nonexistent due

to adjustment symmetry; or when there are not enough consumers with extreme tastes for

firms to want to de-stabilize an interior price equilibrium, despite non-symmetry.

We may define the log concavity of F .( ) in p0 as f xE*( ) ∂xE

*

∂ p0 F xE*( ) being decreasing

in p0 or, equivalently, −F xE*( ) f xE

*( ) ∂xE*

∂ p0 decreasing in p0 . Suppose first that F .( ) is log

concave in its own argument x; then ∂2 xE* ∂p0

2 < 0 is then a sufficient condition for log

concavity of F .( ) in p0 . Now, from (8) we derive

∂2 xE*

∂p02 = − −2t + dg1,0

dxE* di

0

i*1,0 xE*( )∫ − dg0,0

dxE* di

0

i*0,0 xE*( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−2

⋅

d 2g1,0

dxE*2

∂xE*

∂p0di

0

i*1,0 xE*( )∫ − d 2g0,0

dxE*2

∂xE*

∂p0di + dg

1,0

dxE* ix

*1,0 ∂xE*

∂p0− dg

0,0

dxE* ix

*0,0 ∂xE*

∂p00

i*0,0 xE*( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= − ∂xE*

∂p0

⎛⎝⎜

⎞⎠⎟

2d 2g1,0

dxE*2

∂xE*

∂p0di

0

i*1,0 xE*( )∫ − d 2g0,0

dxE*2

∂xE*

∂p0di + dg

1,0

dxE* ix

*1,0 ∂xE*

∂p0− dg

0,0

dxE* ix

*0,0 ∂xE*

∂p00

i*0,0 xE*( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

We note in (8) that ∂xE*

∂ p0 is not a function of i, so it may be pulled outside the integral,

allowing us to write the sufficient condition as

(A1) −∂2 xE

*

∂ p02

∂xE*

∂ p0( )3= d 2g1,0

dxE*2 di

0

i*1,0 xE*( )∫ − d 2g0,0

dxE*2 di + dg

1,0

dxE* ix

*1,0 − dg0,0

dxE* ix

*0,0

0

i*0,0 xE*( )∫ ≤ 0

Consider now the set of pairs of symmetric adjustment maps, G 0,0 ,G 1,0{ } . For these

pairs, for each adjustment curve in G0,0 corresponding to a given location x ∈ 0,1( ) , the

corresponding curve in G1,0 would be its mirror image about x. Two examples of symmetric

adjustment map pairs are displayed in Figure 4. In the first, adjustment curves in both maps

flatten monotonely as one moves toward the midline x = 12 . These maps conceive of

32

consumers quite intuitively as being more facile with adjustment to both options if they are

initially more indifferent between their options; consumers who feel strongly about an option

initially are less able to find good opportunities to adjust to either their preferred option,

because they are already almost perfectly satisfied with it; or its alternative, because they

simply find it hard to conceive of how they might get comfortable with that distant option. In

the second, adjustment curves flatten at first toward the midline, but then steepen to reach a

local maximum steepness at x = 12 . These maps conceive of the most indifferent consumers

as being more rigid than those just a bit closer to one option or another; perhaps consumers

the most initially dissatisfied with their options become pig-headed or embittered, hence

inflexible.

<INSERT FIGURE 4 APPROXIMATELY HERE>

With symmetric adjustment map pairs, the following conditions clearly hold: (i)

dg0,0

dxE* = dg1,0

dxE* , (ii) d

2g0,0

dxE*2 = d2g1,0

dxE*2 , (iii) i*0,0 = i*1,0 , and (iv) ix

*0,0 = ix*1,0 . It may be verified, based on

these, that the sufficient condition (A1) above for log concavity is met. That is, we have log

concavity of F x p0( )( ) for any log concave distribution f.

More generally, if f is not log concave, it must be the case that

(A2) ∂2 xE

*

∂ p02

∂xE*

∂ p0( )2<f xE

*( )F xE

*( ) −f ' xE

*( )f xE

*( )

for log concavity of F .( ) in p0 . Consider now an example of a non-symmetric adjustment

map pair, given by g0,0 i, x( ) ≡ x 2 x − i( ) and g1,0 i, x( ) ≡ 1− x( ) 2 1− x − i( ) . These functions

have the property that g0,0 0, x( ) = g1,0 0, x( ) = 1 2 . Observe further that i*0,0 x,t( ) = 2t −12t

x is

33

defined for t ≥ 12 , whence i* < x ; similarly i*1,0 x,t( ) = 2t −1

2t1− x( ) , whence i* <1− x . We

also have ix*0,0 = 2t −1

2t and ix

*1,0 = − 2t −12t

. Let us now take the first and second derivatives

with respect to x:

∂g0,0

∂x= −i

2 x − i( )2 ≤ 0 ; ∂2g0,0

∂x2 = ix − i( )3 ≥ 0

and

∂g1,0

∂x= i

2 1− x − i( )2 ≥ 0 ; ∂2g1,0

∂x2 = i1− x − i( )3 ≥ 0

Now we evaluate (A1) at i = 0 (i.e., the position at which the indifferent consumer evaluates

his decision between product options), for all possible x (i.e., x ∈ 0,1[ ]). Using integration by

parts:

(A3)

d 2g1,0

dxE*2 di

0

i*1,0 xE*( )∫ − d 2g0,0

dxE*2 di + dg

1,0

dxE* ix

*1,0 − dg0,0

dxE* ix

*0,0

0

i*0,0 xE*( )∫

= i1− x − i( )30

2t−12t

1−x( )

∫ di − ix − i( )3

di + i2 1− x − i( )2

− 2t −12t

⎛⎝⎜

⎞⎠⎟ −

−i2 x − i( )2

2t −12t

⎛⎝⎜

⎞⎠⎟

0

2t−12t

x

∫

= i2 1− x − i( )2

− 12 1− x − i( )

⎡

⎣⎢

⎤

⎦⎥0

2t−12t

1−x( )

− i2 x − i( )2

− 12 x − i( )

⎡

⎣⎢

⎤

⎦⎥0

2t−12t

x

=4t 2 − 4t +1( ) 2x −1( )

2x 1− x( )

Substituting into (8) for our example functions we obtain ∂xE*

∂ p0 = −1 1− ln 12t( ) .

Now assume f is distributed Beta with shape parameters α ,β( ) = 3,3( ) . We have:

34

f x( ) = x 1− x( )⎡⎣ ⎤⎦2

u 1− u( )⎡⎣ ⎤⎦2du

0

1

∫; F x( ) =

u 1− u( )⎡⎣ ⎤⎦2du

0

x

∫

u 1− u( )⎡⎣ ⎤⎦2du

0

1

∫

⇒ f ' x( ) = 2x 1− x( ) 1− 2x( )u 1− u( )⎡⎣ ⎤⎦

2du

0

1

∫

Thus,

(A4)

f ' xE*( )

f xE*( ) −

f xE*( )

F xE*( ) =

2x 1− x( ) 1− 2x( )x 1− x( )⎡⎣ ⎤⎦

2 −x 1− x( )⎡⎣ ⎤⎦

2

u 1− u( )⎡⎣ ⎤⎦2du

0

x

∫

=2 1− 2x( )x 1− x( ) −

30 1− x[ ]2x 6x2 −15x +10⎡⎣ ⎤⎦

One may verify, using (A3), (A4), and ∂xE*

∂ p0 = −1 1− ln 12t( ) , that (A2) holds for all x ∈ 0,1( ) ,

and for any t > 12 .

B. Proofs and Derivations of Lemmas, Propositions, and Remarks

B1. PROOF OF LEMMA 1.

Suppose not. Specifically, for some !i > 0, !x ∈ 0,1[ ] let

∂2g j ∂i∂Aj

!i , !x( ) > −∂g j ∂Aj!i , !x( ) . This implies φA0r0 gi

0,1 − gi0,0( ) > −φA0r0 g

0,1 − g0,0( ) , thus

gi0,1 − gi

0,0 > g0,0 − g0,1 . This means, for some Δ!i > 0 , we have

g0,1 !i + Δ!i , !x( )− g0,1 !i , !x( )− g0,0 !i + Δ!i , !x( )− g0,0 !i , !x( )⎡⎣ ⎤⎦ > g0,0 !i , !x( )− g0,1 !i , !x( )

⇔ g0,1 !i + Δ!i , !x( ) > g0,0 !i + Δ!i , !x( )

which is a contradiction of adjustment costs being non-increasing in advertising.

B2. DERIVATION OF LEMMA 2.

35

Begin with the expression g j i* j x,t,Aj ,rj( ), x,Aj ,rj( ) = t which implicitly defines i* j

and expand (here, shown for j = 0 ):

φ A0( )r g0,1 i*0 x,A0,r( ), x( )− g0,0 i*0 x,A0,r( ), x( )⎡⎣ ⎤⎦ + g0,0 i*0 x,A0,r( ), x( ) = t

Totally differentiating yields

gi*00 di*0 = −gA0

0 dA0 − gr0dr − gx

0dx⇔

gi*00 di*0 = − φA0r g

0,1 − g0,0( ){ }dA0 − φ A0( ) g0,1 − g0,0( ){ }dr − gx0dx

Using Cramer’s rule, it follows from Assumption 7 that

ix*0 x,A0( ) = − gx

0

gi*00 ≤

gi*00

gi*00 = 1

Also using Cramer’s rule one obtains

iA0*0 x,A0( ) = −

gA00

gi*00 =

φA0r g0,0 − g0,1( )gi*00 ≥ 0

ir*0 x,A0,r( ) = − gr

0

gi*00 =

φ A0( ) g0,0 − g0,1( )gi*00 ≥ 0

whence it follows from Assumption 5 that limA0→∞

iA0*0 = 0 . Corresponding results can be derived

along the same lines for j = 1 .

B3. PROOF OF LEMMA 3.

The proof is an extension of the proof of Bloch and Manceau’s (1999) Lemma 1.

Assume no advertising, and suppose that the market is not covered, that is, at equilibrium

prices p0*, p1

*( ) there exists a consumer x for whom

36

V − p0

* − t x − i*0,0 x( )⎡⎣ ⎤⎦ − g0,0 i, x( )di0

i*0,0 x( )

∫ < 0 and

V − p1* − t 1− x − i*1,0 x( )⎡⎣ ⎤⎦ − g1,0 i, x( )di

0

i*1,0 x( )

∫ < 0

One can show these prices do not constitute a Nash equilibrium, in that firm 0 can increase

its profit by lowering its price p0 without altering the profit, hence strategy, of firm 1. Begin

by noting that, under p0*, p1

*( ) , because there is a consumer for whom neither good provides

nonnegative utility somewhere between the firms, the profit of firm 0 can be written

Π0 = p0* x0( )F x0( ) ≡ V − t x0 − i

*0,0 x0( )⎡⎣ ⎤⎦ − g0,0 i, x0( )di0

i*0,0 x0( )

∫⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪F x0( )

where x0 is the position of the consumer who, at prices p0*, p1

*( ) , is just indifferent between

buying product 0 and buying nothing. By assumption, ∂Π0 ∂x0 > 0 . Now note that

∂p0 ∂x0 = −t + tix*0,0 − g0,0 i*0,0 , x0( )ix*0,0 − dg0,0

dx0di

0

i*0,0 x0( )

∫

= −t − dg0,0

dx0di

0

i*0,0 x0( )

∫ < −t + dg0,0

didi

0

i*0,0 x0( )

∫= −t + g i*0,0 x0( ), x0( )− g 0, x0( ) = −g 0, x0( ) < 0

which follows from Assumption 7. Since ∂Π0 ∂x0 = ∂Π0 ∂p0( ) ∂p0 ∂x0( ) , it follows that

∂Π0 ∂p0 < 0 . Therefore a small downward deviation in the price p0 from p0* increases firm

0’s profits while not affecting firm 1’s profits. This contradicts the assertion that p0*, p1

*( )

constitute an equilibrium.

Since the market is covered when there is no advertising, it follows that it is also

covered when there is advertising.

37

B4. PROOF OF PROPOSITION 4.

Let us differentiate (12) with respect to A0 :

∂2Π0

∂A02 = F xE

*( )φA0A0r0 g0,0 − g0,1⎡⎣ ⎤⎦di0

i*0 xE* ,A0( )∫

+ f xE*( ) ∂xE

*

∂A0φA0r0 g0,0 − g0,1⎡⎣ ⎤⎦di

0

i*0 xE* ,A0( )∫ + F xE

*( )φA0r0 g0,0 − g0,1⎡⎣ ⎤⎦ iA0*0

= r0 g0,0 − g0,1⎡⎣ ⎤⎦di0

i*0 xE* ,A0( )∫ F xE

*( )φA0A0 + f xE*( ) ∂xE

*

∂A0φA0

⎧⎨⎩

⎫⎬⎭

+F xE*( )φA0r0 g0,0 − g0,1⎡⎣ ⎤⎦ iA0

*0

= r0 g0,0 − g0,1⎡⎣ ⎤⎦di0

i*0 xE* ,A0( )∫ F xE

*( )φA0A0 + f xE*( ) ∂xE

*

∂A0φA0

⎧⎨⎩

⎫⎬⎭

+F xE

*( )φA02 r02 g0,1 − g0,0( )2φ A0( ) r0gi*00,1 + 1− r0( )gi*00,0⎡⎣ ⎤⎦ + 1−φ A0( )⎡⎣ ⎤⎦gi*0

0,0

One can see that for φA0A0 sufficiently negative – that is, for φ sufficiently concave –

∂2Π0 ∂A02 < 0 is guaranteed, hence the existence of a unique pure strategy Nash equilibrium

A0* p0, p1( ),A1* p0, p1( )( ) .

B5. PROOF OF PROPOSITION 5.

As our interest is in the effects of exogenous changes in advertising expenditure on

price, we consider only the indirect effects portion of (14) (i.e., all but the last two terms).

Begin by totally differentiating the first-order condition in price for firm 0. Using Cramer’s

rule, one obtains

∂p0∂p1

= −f xE

*( ) ∂xE*

∂ p1 + p0 f ' xE*( ) ∂xE

*

∂ p0∂xE

*

∂ p1 + p0 f xE*( ) ∂2 xE

*

∂ p0 ∂ p1∂xE

*

∂ p0 2 f xE*( ) + p0 f ' xE*( ) ∂xE

*

∂ p0{ }+ p0 f xE*( ) ∂2 xE

*

∂ p02

and, for j = 0,1 ,

38

∂p0∂Aj

= −f xE

*( ) ∂xE*

∂Aj + p0 f ' xE*( ) ∂xE

*

∂ p0∂xE

*

∂Aj + p0 f xE*( ) ∂2 xE

*

∂ p0 ∂Aj∂xE

*

∂ p0 2 f xE*( ) + p0 f ' xE*( ) ∂xE

*

∂ p0{ }+ p0 f xE*( ) ∂2 xE

*

∂ p02

Analogously, using the first-order condition for price for firm 1, one obtains

∂p1∂Aj

= −f xE

*( ) ∂xE*

∂Aj + p1 f ' xE*( ) ∂xE

*

∂ p1∂xE

*

∂Aj + p1 f xE*( ) ∂2 xE

*

∂ p1∂Aj∂xE

*

∂ p1 2 f xE*( ) + p1 f ' xE*( ) ∂xE

*

∂ p1{ }+ p1 f xE*( ) ∂2 xE

*

∂ p12

Totally differentiating the first-order conditions for advertising for the firms, based on setting

(12) equal to zero, and using Cramer’s rule, one obtains

∂A0∂r

=

− F xE*( ) + f xE

*( ) ∂xE*

∂rr⎡

⎣⎢⎤⎦⎥φA0 g0,0 − g0,1⎡⎣ ⎤⎦di

0

i*0 xE* ,A0 ,r( )∫

−F xE*( )φA0φ A0( )r g0,0 − g0,1⎡⎣ ⎤⎦

2gi*00

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

F xE*( )φA0A0 + f xE

*( ) ∂xE*

∂A0φA0

⎡

⎣⎢

⎤

⎦⎥r g0,0 − g0,1⎡⎣ ⎤⎦di

0

i*0 xE* ,A0 ,r( )∫

+F xE*( )φA02 r2 g0,0 − g0,1⎡⎣ ⎤⎦

2gi*00

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

and

∂A1∂r

=

− 1− F xE*( ){ }− f xE

*( ) ∂xE*

∂rr⎡

⎣⎢⎤⎦⎥φA1 g1,0 − g1,1⎡⎣ ⎤⎦di

0

i*1 xE* ,A1,r( )∫

− 1− F xE*( ){ }φA1φ A1( )r g1,0 − g1,1⎡⎣ ⎤⎦

2gi*11

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

1− F xE*( ){ }φA1A1 − f xE

*( ) ∂xE*

∂A1φA1

⎡

⎣⎢

⎤

⎦⎥r g1,0 − g1,1⎡⎣ ⎤⎦di

0

i*1 xE* ,A1,r( )∫

+ 1− F xE*( ){ }φA12 r2 g1,0 − g1,1⎡⎣ ⎤⎦

2gi*11

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

where, applying Cramer’s rule to (5) one may obtain

∂xE*

∂r=

φ A0( ) g0,1 − g0,0⎡⎣ ⎤⎦di0

i*0 xE* ,A0 ,r( )∫ − φ A1( ) g1,1 − g1,0⎡⎣ ⎤⎦di

0

i*0 xE* ,A0 ,r( )∫

−2t + dg1

dxE* di

0

i*1 xE* ,A1,r( )∫ − dg0

dxE* di

0

i*0 xE* ,A0 ,r( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

39

When incorporating these expressions in the evaluation of (14), certain other facts

that arise generally or from the symmetric case will be useful. It is clear from examination of

(5) and (8) that ∂xE* ∂p0 = −∂xE

* ∂p1 . Moreover,

∂2 xE*

∂p12 = −2t + dg1

dxE* di

0

i*1 xE* ,A1( )∫ − dg0

dxE* di

0

i*0 xE* ,A0( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−2

⋅

d 2g1

dxE*2∂xE

*

∂p1di

0

i*1 xE* ,A1( )∫ − d 2g0,0

dxE*2

∂xE*

∂p1di + dg

1

dxE* ix

*1 ∂xE*

∂p1− dg

0

dxE* ix

*0 ∂xE*

∂p10

i*0 xE* ,A0( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

= − −2t + dg1

dxE* di

0

i*1 xE* ,A1( )∫ − dg0

dxE* di

0

i*0 xE* ,A0( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−2

⋅

d 2g1

dxE*2

∂xE*

∂p0di

0

i*1 xE* ,A1( )∫ − d 2g0,0

dxE*2

∂xE*

∂p0di + dg

1

dxE* ix

*1 ∂xE*

∂p0− dg

0

dxE* ix

*0 ∂xE*

∂p00

i*0 xE* ,A0( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥= ∂2 xE

*

∂p02 = − ∂2 xE

*

∂p0 ∂p1

Also,

(A5)

∂2 xE*

∂ p0 ∂A0 = − −2t + dg1

dxE* di

0

i*1 xE* ,A1( )∫ − dg0

dxE* di

0

i*0 xE* ,A0( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−2

⋅

∂2 g1

∂xE*2

∂xE*

∂A0 di0

i*1 xE* ,A1( )∫ − ∂2 g0

∂xE* ∂A0

+ ∂2 g0

∂xE*2

∂xE*

∂A0⎡⎣

⎤⎦di

0

i*0 xE* ,A0( )∫ + dg1

dxE* ix*1 ∂xE

*

∂A0 −dg0

dxE* iA0

*0 + ix*0 ∂xE

*

∂A0( )⎡

⎣⎢⎢

⎤

⎦⎥⎥

∂2 xE*

∂ p0 ∂A1 = − −2t + dg1

dxE* di

0

i*1 xE* ,A1( )∫ − dg0

dxE* di

0

i*0 xE* ,A0( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

−2

⋅

∂2 g1

∂xE* ∂A1

+ ∂2 g1

∂xE*2

∂xE*

∂A1⎡⎣

⎤⎦di

0

i*1 xE* ,A1( )∫ − ∂2 g0

∂xE*2

∂xE*

∂A1 di0

i*0 xE* ,A0( )∫ + dg1

dxE* iA1

*1 + ix*1 ∂xE

*

∂A1( )− dg0

dxE* ix*0 ∂xE

*

∂A1

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Under the symmetric case, f ' x( ) = 0 , F x( ) = 1− F x( ) = 12 , and ∂xE

* ∂A0 = −∂xE* ∂A1 ; and

∂xE* ∂r = 0 , whence it follows that ∂A0 ∂r = ∂A1 ∂r . Finally, it follows straightforwardly

40

under the conditions of the symmetric case given in the Proposition that

∂2 xE* ∂p0

2 = ∂2 xE* ∂p1

2 = ∂2 xE* ∂p0 ∂p1 = 0 .

Using these facts, ∂p0 ∂p1 = 12 . Also using these facts, it may be observed from (A5)

that when dg0

dxE* iA0*0 + ∂2 g0

∂xE* ∂A0

di0

i*0 xE* ,A0( )∫ = 0 (and, following from symmetry,

dg1

dxE* iA1*1 + ∂2 g0

∂xE* ∂A1

di0

i*1 xE* ,A1( )∫ = 0 ), ∂2 xE

*

∂ p0 ∂A0 = − ∂2 xE*

∂ p0 ∂A1 , whence ∂p0 ∂A0 = −∂p0 ∂A1 . Now, as is

evident from (A5), when dg0

dxE* iA0*0 + ∂2 g0

∂xE* ∂A0

di0

i*0 xE* ,A0( )∫ > 0 (and, by symmetry,

dg1

dxE* iA1*1 + ∂2 g0

∂xE* ∂A1

di0

i*1 xE* ,A1( )∫ < 0 ), the numerators of ∂2 xE

*

∂ p0 ∂A0 and ∂2 xE*

∂ p0 ∂A1 are both increased by the

same positive quantity, whereby it follows that ∂p0 ∂A0 > −∂p0 ∂A1 . By symmetry,

∂p1 ∂A1 > −∂p1 ∂A0 . Thus the first and second pair of terms in (14) are both positive in this

case, thus prices increase with advertising if dg0

dxE* iA0*0 + ∂2 g0

∂xE* ∂A0

di0

i*0 xE* ,A0( )∫ > 0 (and

dg1

dxE* iA1*1 + ∂2 g0

∂xE* ∂A1

di0

i*1 xE* ,A1( )∫ < 0 ).

Now suppose instead dg0

dxE* iA0*0 + ∂2 g0

∂xE* ∂A0

di0

i*0 xE* ,A0( )∫ < 0 (and by symmetry

dg1

dxE* iA1*1 + ∂2 g0

∂xE* ∂A1

di0

i*1 xE* ,A1( )∫ > 0 ). Then the numerators of ∂2 xE

*

∂ p0 ∂A0 and ∂2 xE*

∂ p0 ∂A1 are both decreased by

the same positive quantity, whereby ∂p0 ∂A0 < −∂p0 ∂A1 and, by symmetry,

41

∂p1 ∂A1 < −∂p1 ∂A0 . It follows that prices decrease with advertising if

dg0

dxE* iA0*0 + ∂2 g0

∂xE* ∂A0

di0

i*0 xE* ,A0( )∫ < 0 (and dg1

dxE* iA1*1 + ∂2 g0

∂xE* ∂A1

di0

i*1 xE* ,A1( )∫ > 0 ).

B6. DERIVATION OF REMARK 1.

In the symmetric case, the price increase due to an exogenous increase in advertising

ΔA by both firms – Δ2A in total – equals:

(A6) Δp0 =∂p0∂A0

+ ∂p0∂A1

+ ∂p0∂p1

∂p1∂A0

+ ∂p1∂A1

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ΔA

The price increase for product 0 and for product 1 are identical in the symmetric case, so let

us define ∂p ∂A ≡ ∂p1 ∂A0 + ∂p1 ∂A1 = ∂p0 ∂A0 + ∂p0 ∂A1 . This allows us to re-write (A6)

as

Δp0 = 1+ ∂p0∂p1

⎛⎝⎜

⎞⎠⎟∂p∂A

ΔA

As noted in the proof of Proposition 5, under the symmetric case ∂p0 ∂p1 = 12 . Moreover,

using (A5) and the conditions of the symmetric case articulated in the proof of Proposition 5,

42

∂p∂A

= −f xE

*( ) ∂xE*

∂A0 + p0 f ' xE*( ) ∂xE

*

∂ p0∂xE

*

∂A0 + p0 f xE*( ) ∂2 xE

*

∂ p0 ∂A0∂xE

*

∂ p0 2 f xE*( ) + p0 f ' xE*( ) ∂xE

*

∂ p0{ }+ p0 f xE*( ) ∂2 xE

*

∂ p02

+ −f xE

*( ) ∂xE*

∂A1 + p0 f ' xE*( ) ∂xE

*

∂ p0∂xE

*

∂A1 + p0 f xE*( ) ∂2 xE

*

∂ p0 ∂A1∂xE

*

∂ p0 2 f xE*( ) + p0 f ' xE*( ) ∂xE

*

∂ p0{ }+ p0 f xE*( ) ∂2 xE

*

∂ p02

= −

2p0 f xE*( ) ∂xE

*

∂ p0

2 dg0

dxE* iA0*0 + ∂2 g0

∂xE* ∂A0

di0

i*0 xE* ,A0( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

2 ∂xE*

∂ p0 f xE*( )

= − p0∂xE

*

∂ p0dg0

dxE* iA0*0 + ∂2 g0

∂xE* ∂A0

di0

i*0 xE* ,A0( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥=F xE

*( )f xE

*( )dg0

dxE* iA0*0 + ∂2 g0

∂xE* ∂A0

di0

i*0 xE* ,A0( )∫

⎡

⎣⎢⎢

⎤

⎦⎥⎥

Thus the price increase attributable to advertising an amount A0* is as specified in the remark.

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