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Assisted Self-Persuasion: A Motivated Preference Theory of Advertising Matthew G. Nagler * February 1, 2020 Abstract I develop a new theory of persuasive advertising, in which consumers seek to improve their attitude toward the products they choose, and advertising facilitates such adjustment. Unlike previous theories, persuasive advertising under the pa- per’s novel framework is consistent with consumer rationality and conventionally fixed tastes. The price effect of advertising has two components, one arising from the easing of adjustment for all consumers, and the other from differential effects on consumers that accrue to targeting. When targeting is more precise, firms are better able to raise prices without sacrificing incremental sales, whence they cap- ture a greater share of the returns to advertising and so opt for higher advertising expenditures. The paper’s findings offer a new perspective on previous empirical evidence relating advertising to consumer price sensitivity. Keywords Persuasive advertising, targeting, competitive strategy, behavioral decision-making * Department of Economics and Business, City College of New York; and Graduate Center, City Uni- versity of New York. Email: [email protected]. Phone: +1 212.650.6205. Fax: +1 212.650.6341. I am grateful to Ben Ho, Edwin Ip, Fahad Khalil, Botond Kőszegi, Jacques Lawarrée, Jidong Zhou, and seminar participants at the University of Toronto Rotman School of Management, Case Western Reserve University Weatherhead School of Management, University of Washington, Vassar College, and INFORMS Marketing Science conference for helpful comments. Shay Culpepper and Ahmed ElKhouly provided excellent research assistance.

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Page 1: Assisted Self-Persuasion: A Motivated Preference Theory of ...mnagler.ccny.cuny.edu/research/adjustment_adv.pdfAssisted Self-Persuasion: A Motivated Preference Theory of Advertising

Assisted Self-Persuasion:A Motivated Preference Theory of Advertising

Matthew G. Nagler∗

February 1, 2020

Abstract

I develop a new theory of persuasive advertising, in which consumers seek toimprove their attitude toward the products they choose, and advertising facilitatessuch adjustment. Unlike previous theories, persuasive advertising under the pa-per’s novel framework is consistent with consumer rationality and conventionallyfixed tastes. The price effect of advertising has two components, one arising fromthe easing of adjustment for all consumers, and the other from differential effectson consumers that accrue to targeting. When targeting is more precise, firms arebetter able to raise prices without sacrificing incremental sales, whence they cap-ture a greater share of the returns to advertising and so opt for higher advertisingexpenditures. The paper’s findings offer a new perspective on previous empiricalevidence relating advertising to consumer price sensitivity.

Keywords Persuasive advertising, targeting, competitive strategy, behavioraldecision-making

∗Department of Economics and Business, City College of New York; and Graduate Center, City Uni-versity of New York. Email: [email protected]. Phone: +1 212.650.6205. Fax: +1 212.650.6341.I am grateful to Ben Ho, Edwin Ip, Fahad Khalil, Botond Kőszegi, Jacques Lawarrée, Jidong Zhou,and seminar participants at the University of Toronto Rotman School of Management, Case WesternReserve University Weatherhead School of Management, University of Washington, Vassar College, andINFORMS Marketing Science conference for helpful comments. Shay Culpepper and Ahmed ElKhoulyprovided excellent research assistance.

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“It’s good to want things.”- Winona Ryder (as Dinky Bossetti) in Welcome Home, Roxy Carmichael.

1 Introduction

A substantial amount of money is spent by firms every year on advertising. In 2018,total US advertising expenditures were approximately $151 billion (Sweeney 2019). Yetwhile advertising clearly provides benefits for firms, how it does so is not entirely clear.Both the two leading economic theories as to why consumers respond to advertising havesignificant limitations.1

One theory holds that advertising is persuasive, altering consumers’ tastes for a firm’sproduct or those of its competitors or spuriously creating product differentiation. Suchadvertising may, in the process, shift demand to the advertising firm; relatedly, it mayalter consumers’ willingness to pay (WTP) for the advertised product or its competitors,to the advantage of the advertising firm.2 But such notions are at odds with the pre-vailing economic view that consumers’ tastes are fixed; and, more broadly speaking, thepersuasion theory offers no explanation as to why advertising should elicit a response atall from a rational consumer.

The other leading theory holds that advertising is informative. According to its con-ception, advertising influences consumers to the extent that it provides useful informationof some kind on the product - its features, price, availability, and so forth.3 Even advertis-ing messages that appear non-substantive (e.g., image-oriented) convey the informationthat the product is advertised and must be of sufficient quality to have elicited a costlyadvertising expenditure (Nelson 1974). But the information theory cannot explain theefforts devoted to the crafting of message and image in ads otherwise devoid of informa-tional content. If the purpose is just to show that money is being spent and the ad isnot intended in some measure to be persuasive, why should such details matter?

This paper presents a new theory of advertising as assisting self-persuasion. Thetheory is based on the notion of a consumer with motivated preferences who rationallyadjusts to the choices he makes. The consumer does not simply make utility-maximizingchoices; he additionally, and simultaneously, invests effort to increase the satisfaction

1For an extensive discussion of the major current advertising theories as well as a literature survey,see Bagwell (2007).

2One recent example from this literature is Chioveanu (2008).3Recent examples from this literature include Anand and Shachar (2009) and Karle and Peitz (2017).

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he obtains conditional on his choice. Indeed, it is common sense that a person shouldwork to get comfortable with the inevitable; and this is something that, introspectionsuggests, we all do to the extent we are able. A rational individual would not only striveto achieve maximum comfort conditional on a given choice; he would also make his choicerecognizing that he is going to adjust to it. A general conception of motivated prefer-ence, therefore, goes beyond the specifically reactive notion of psychological adjustmentreflected in cognitive dissonance theory, to anticipate a range of proactive and reactivemotivations for tweaking one’s consumption attitudes at the margin.4

The behavioral proposition inherent in this theory is based on extensive psycholog-ical research that individuals routinely undergo a sort of mental re-positioning relativeto choices they have made, changing both their stated preferences and the physiolog-ical manifestations of their hedonic responses.5 Bounded rationality provides a usefulcognitive frame for these observations. Existing approaches conceive of the boundedlyrational individual as making decisions subject to limited information processing capa-bilities, whence access to one’s scarce cognitive resources engenders a cost. If attitudestoward consumption objects can be adjusted by the motivated individual, it seems rea-sonable to expect that such adjustment, like decision-making, requires effort, hence theallocation of scarce personal cognitive resources. On this basis, I posit an individual whocan “psyche himself up” about what he is going to do, but at a cost.

Advertising in this context provides persuasive inputs - information, messages, andimages - that enable consumers to improve their attitude toward the product more easily- that is, at lower cost. Consider, for example, a television commercial for a SubaruOutback that describes the car’s high gas mileage and standard all-wheel drive, whileshowing scenes of a family harmoniously going about their errands in the vehicle. Forthe consumer considering the Outback, the commercial supplies fodder for increasing hersatisfaction with the prospective purchase: it provides her the basis in hard facts forarguments that rationalize the choice, while fueling her imagination with visions of howpleasant life will be when her family is someday ensconced in an Outback. Persuasive

4Rationality serves as a point of departure for the motivated preference model; one may reasonablycomplicate the theory consistent with behavioral evidence by assuming that individuals narrowly framethe consequences of their choices and lack perfect foresight.

5Recent studies offering evidence of preference change based on subject ratings of chosen alternativesinclude Lieberman et al. (2001), Kitayama et al. (2004), Sharot et al. (2010), andWakslak (2012). Studiesthat additionally measured changes using functional magnetic resource imaging (fMRI) of subjects’ brainsinclude 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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advertising, as conceived in this scenario, offers a valuable service to the consumer; andon the basis of this it has the potential to improve welfare.

The paper’s main contribution is not to explain previously unexplained, specific em-pirical phenomena - that is, as a function of advertising appropriately conceived. Rather,it is to explain advertising outcomes on the basis of cogent and strongly-supported psy-chological primitives. These primitives align the economic plausibility of persuasion withthe objective evidence that it exists and is effective. Existing theories of persuasive ad-vertising make ad hoc assumptions - for instance, that advertising shifts the demandcurve of the product, that it shifts the distribution of consumers, or that it alters theWTP of individual consumers. Such assumptions have generally not been grounded in apsychological basis. My theory delivers a model that is observationally similar in certainaspects to these existing models, but that departs in some critical aspects, generatingresults that differ in consequential and sometimes surprising ways.

I posit a Hotelling spatial model of differentiated product competition in which con-sumers differ as to their initial tastes for two competing products. A consumer can, ata cost, adjust to the product he intends to choose – in essence, “moving closer” to it,and thereby avoiding some of the “transportation” cost associated with imperfect tastematching. Advertising for a particular product is modeled generally as reducing the costof such adjustment. By incorporating adjustment as a step in a model of rational choice,the theory allows the induced outcomes to be subsumed into “final” preferences suchthat the conventional techniques for analyzing choices (including the axioms of revealedpreference) may be applied to them. In the context in which the consumer operates,choosing simultaneously a product and an amount of complementary adjustment, tastesare effectively fixed. This approach avoids many of the complications associated withprevious efforts to model taste change.

With this framework in hand, I am able to model advertising’s effects on prices withfull generality. Though adjustment implies intensification of preference, prices do not riseunambiguously with advertising that facilitates adjustment. Advertising’s effects accrueto its roles as an overall intensifier of adjustment (i.e., in reducing adjustment costs,advertising causes more adjustment by consumers across the board and so intensifies ad-justment’s own effects on prices) and a shaper of relative adjustment facility levels acrossconsumers (i.e., it has the potential to reduce the relative marginal adjustment costsexperienced by consumers with stronger or weaker initial product preferences, therebyskewing price levels). While the first component may be thought of as the price effect of

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mass-market advertising, the latter may be viewed as the effect of advertising targetedat specific groups.

I apply the model’s pricing results to ascertain how profit-maximizing firms will useadvertising. I demonstrate that it is optimizing for firms to use targeting so as to createa marginal adjustment cost differential favoring just-inframarginal consumers relative tomarginal consumers. This achieves the result of effectively decreasing the elasticity ofdemand, so that the firm may raise prices while simultaneously maintaining sales. Itfollows that a key determinant of both equilibrium prices and the amount of advertisingis the precision of targeting.

Several papers relate closely to the present work. Like the present paper, Bloch andManceau (1999) use a Hotelling model to analyze the effect of persuasive advertising.Advertising in their model shifts the preference distribution of consumers closer to theadvertiser’s product, based mathematically on the principle of stochastic dominance.Thus, consistent with much of the literature on persuasive advertising, they show adver-tising to be as a zero-sum game of demand shifting between competitors. In contrast withthis, advertising by each firm in my model unambiguously creates surplus in the formof reduced adjustment costs for consumers, whence advertising competition results inadditive benefits rather than a wasteful tug-of-war. (Bloch and Manceau do not explainon what basis advertising in their model is predicted to shift consumers’ preferences.)

Becker and Murphy (1993) posit a model in which advertising is a complement tothe advertised product. Advertising increases the marginal utility of the product, whiletastes remain fixed per se. Their approach offers a general framework for advertising’seffect on rational consumers; it generates implications that are consistent with a numberof empirical observations, such as advertising’s ability to influence demand even in per-fectly competitive commodity markets. However, the source of complementarity betweenadvertising and the advertised product in their framework is left unspecified. Becker andMurphy make no behavioral assumptions about consumers (e.g., bounded rationality)and generally do not endeavor to offer a psychological explanation for how advertisingmight work on consumers.

Haan and Moraga-González (2011) and Astorne-Figari et al. (2019) conceive of con-sumers as boundedly rational, whereby firms use advertising to vie for the consumer’slimited attention and powers of recall. Unlike the present paper, they do not conceiveof advertising as persuasive, but effectively informative: advertising reminds a consumerwho is forgetful - in the sense of not having essential information accessible at top of

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mind - of the availability of certain products. Eliaz and Spiegler (2011) also propose thatconsumers are boundedly rational in the traditional sense, but conceive of advertising aspersuading the consumer to include the advertised product in his limited considerationset. Thus their theory of “advertising as persuasion” is novel in that it maintains thatpreferences are stable. But while the goal of the advertiser in Eliaz and Spiegler’s theorymay be ostensibly to persuade, what advertising boils down to is information to be usedto make a decision as to whether the product should be considered. Thus their paper ismost properly grouped with the informative advertising literature.

Anand and Shachar (2009) propose a theory of informative advertising in which, asin the present paper, targeting plays a key role. Targeting in their model benefits firmsby ensuring advertisements reach the most appropriate (heterogeneous) consumers forthe firm’s product so that the expenditures are not wasted. Derivative of this, targetingserves as a credible signaling tool: consumers infer that the product is a good fit for thembecause they observe that they were targeted. In contrast, targeting’s primary purposein my paper is the manipulation of differences in WTP across consumers, in such a waythat the market price may be elevated; this result emerges endogenously from my assistedself-persuasion framework.

The idea of advertising as a tool for the consumer has several precedents in theliterature. It is consistent with the Elaboration Likelihood Model, which considers theconditions under which consumers approach advertising with active thinking as a part oftheir decision-making process (Petty and Cacioppo 1986); and it is a feature of the usesand gratifications literature (O’Donohoe 1994, Ko et al. 2005, Aitken et al. 2008, Phillipsand McQuarrie 2010). Separately, studies of cognitive dissonance reveal that consumersactively seek persuasive advertising messages that have the potential to reduce doubtsabout product choices (Ehrlich et al. 1957, Mills 1965). Advertising may be viewed asa stand-in for other forms of costly marketing communication, particularly salespersoncommunications to customers, which have been discussed as facilitating post-purchasedissonance reduction (Hunt 1970, Milliman and Decker 1990, Grewal and Sharma 1991).

The rest of this paper is structured as follows. Section 2 introduces a model, in whichthe pattern of advertising’s effects across the mass of consumers is presumed exogenousand firms decide how to price and how much to advertise given the pattern. Section 3derives the equilibrium for this model. Section 4 allows the pattern of effects to be set bythe firm, whereby the firm’s optimizing targeting strategy emerges as an outcome of themodel, and market outcomes follow on the basis of exogenous targeting precision. Section

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5 re-examines the empirical literature on advertising and prices from the perspective of themotivated preference model’s findings. Section 6 concludes and discusses opportunitiesfor future research. The Appendix contains proofs and derivations of all lemmas andpropositions.

2 A Model with Generically-Targeted Advertising

Consider two differentiated products, indexed 0 and 1, each produced by an independentfirm correspondingly named. Suppose that consumers’ preferences over these productshave two components. Each consumer is endowed with an initial preference, also knownas his non-motivated component. This may be viewed partly as an exogenous character-istic of the individual, according to which we have the old saying, “Different strokes fordifferent folks”; but also, in part, as exogenously determined based on random exposures.Though I presume no particular dynamic process of preference learning, one may conceiveof the initial preference as being discovered over the course of the consumer’s lifetimeleading up to the time of decision; that discovery process may include a focused search tosupport the decision, in the traditional sense. Each consumer’s preference also comprisesa motivated component, which is developed through a process of adjustment, consistingof costly efforts by the consumer to increase instrumentally the satisfaction he expects toreceive from consuming the object he has chosen. Adjustment may involve instrumentalseeking of external information, wracking one’s brain for information already in memory,or constructive rationalization, all with the purpose of being more satisfied with one’spresumed choice. The consumer knows the costs of adjustment and fully anticipates thebenefits to be gained from his adjustment efforts.

The assumptions are formalized using a spatial model. Following Hotelling (1929), thetwo firms are located at opposite ends of a segment of length 1 representing the productspace. Consumers are assumed distributed continuously on this segment according toan arbitrary distribution function F with full support and continuous density functionf . The consumer’s location x ∈ [0, 1] identifies his initial relative preference over thetwo products. Consumers buy at most one unit of a single product, so based on thenon-motivated component alone the utility of a consumer at x buying product j is givenby

Ux = V − pj − t |x− j| (1)

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Figure 1: An Adjustment Map

where V is the common reservation price for the product, pj is the price of product j,and t parameterizes the utility loss due to j’s not being the consumer’s ideal choice – thestandard “transportation” cost, linear in the consumer’s distance from j. All consumershave an outside option of utility zero.

Adjustment to a product is represented as relocating on the segment to be closer to itslocation, thereby paying less transportation cost. The process is quite naturally viewedas an incremental one, involving incremental investment of costly or aversive adjustmenteffort that pays off with an incremental improvement in attitude toward the product.Accordingly, let us posit a marginal adjustment cost function associated with product j,gj (i, x) > 0, where i is the distance from x and closer to j’s position. One may view thisfunction as representing a set of adjustment curves G j := {gj (i) = gj (i, x) : x ∈ [0, 1]}characterized by differing values of x, whereby each curve represents the cost, at eachstate of attitude improvement i, of incremental “movement toward” j for the consumerlocated initially at x. I shall refer to G j as an adjustment map for product j. Figure 1illustrates a possible adjustment map for product 0.

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The motivated component of preferences observes the following regularity conditions:

Assumption 1. (Continuity of adjustment in x and i). g0 (i, x) and g1 (i, x) are definedon [0, x)× [0, 1]→ R+ and [0, 1− x)× [0, 1]→ R+, respectively, and continuous on theirsupport.

Assumption 2. (Convexity of adjustment in i). ∂gj

∂i> 0 and ∂2gj

∂i2> 0, for j = 0, 1.

Assumption 3. (Asymptotic adjustment). limi→x g0 (i, x) =∞ (and limi→1−x g

1 (i, x) =

∞).

Assumption 1 implies adjustment effort is smooth for a given consumer and its vari-ation from one consumer to the next is smooth. Assumption 2 reflects the notion thatincremental adjustment to a product becomes progressively more difficult as the bestopportunities for taking a more positive perspective on it are used up. Assumption 3indicates that no matter how psyched up one is about a product, it is possible to getmore psyched up – though it might eventually become quite difficult. Thus a consumernever fully converges to the product’s location even as he continues to adjust to it.

Two general regimes characterize the way in which the motivated component of theindividual’s preferences may relate to the non-motivated component. As one possibility,adjustment might be easier the stronger the individual’s initial preference – at least, asconstrained by Assumption 3, over a range where initial preference is not too strong. Thiswould occur if, say, a consumer’s initial impressions play a large role in how easy it is forhim to rationalize and assemble arguments supporting a product. I refer to adjustment insuch a case as impression-reinforcing. Mathematically, impression-reinforcing adjustmentis reflected by ∂g0

∂x> 0 for product 0 (∂g

1

∂x< 0 for product 1); and it results in an

adjustment map that looks like the one shown in the top panel in Figure 2. In thealternative, adjustment could be easier the weaker the consumer’s initial preference.This would occur if, say, random differences account for a significant portion of the non-motivated component of preferences and if initial impressions do not play much of a rolein making adjustment easier. Then adjustment would tend to result in an attenuationof the random differences in initial preference across consumers such that, in the end,preferences across consumers would tend to converge. I refer to adjustment in such a caseas mean-regressive. Mathematically, mean-regressive adjustment is reflected by ∂g0

∂x< 0

for product 0 (∂g1

∂x> 0 for product 1); and it results in an adjustment map that looks

like the one shown in the bottom panel in Figure 2.

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Figure 2: Two Regimes Relating Motivated and Non-Motivated Preference Components

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These two regimes admit a range of possibilities in the relationship between non-motivated and motivated preferences, and even more possibilities could follow by blendingthe two to varying degrees; but the concept of adjustment, as doing the best one canconditional on a choice that was based on one’s known initial preference, imposes a cruciallimit. While adjustment may be mean-regressive, I assume that in the extreme it can onlycause full regression to the mean. It can never cause consumers to “leapfrog” positionsbased on initial preferences; to admit such a case would require a level of ineptitudeat adjustment by individuals with strong initial preferences for which is no reasonableconceptual justification. Formally:

Assumption 4. (Initial preference dominance). For all x ∈ [0, 1], −∂g0

∂x< ∂g0

∂i(and

∂g1

∂x< ∂g1

∂i).

Assumption 4 tells us that, the more preferred a product is initially, the lower themarginal adjustment cost at any particular location achieved through accumulated ad-justment. This “dominance” condition implies that an individual who initially prefersa product more than another individual finds it less costly to achieve a given attitudetoward that product than the other individual. This in turn gives rise to adjustmentmaps of non-crossing nested contours, similar to well-behaved indifference maps.

In this context, let us conceive of advertising for a given product as making adjustmenteasier with respect to that product, that is, improving the consumer’s product-specifictechnology of adjustment so that the same amount of attitude improvement toward theproduct may be achieved at lower cost. This concept is represented by modifying theadjustment marginal cost function to involve weighting of two component functions,

gj (i, x, Aj) = φ (Aj) gj,1 (i, x) + [1− φ (Aj)] g

j,0 (i, x) (2)

where now gj,k (i, x) > 0 ∀x, i for j, k = 0, 1, and Aj represents firm j’s advertising ex-penditure. Thus we have, in effect, pre- and post-advertising marginal adjustment costfunctions. Here I let the gj,k (.) observe Assumptions 1 through 4; impression-reinforcingor mean-regressive characterizations can apply to either the pre- or post-advertisingmarginal adjustment cost functions, or to both. I make the following assumptions aboutadvertising’s effects on the marginal cost of adjustment:

Assumption 5. φ (.) is continuous on its support.

Assumption 6. φ (.) is increasing and strictly concave.

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Figure 3: Pre- and Post-Advertising Adjustment Maps

Assumption 7. limAj→∞ φ (Aj) = 1.

Assumption 8. (Advertising-driven dominance). gj,1 (i, x) < gj,0 (i, x) ∀i > 0, x ∈ [0, 1],for j = 0, 1.

Notice the generality of the formulation. Advertising is conceived as reducing product-specific adjustment costs asymptotically from a zero-advertising level given by gj,0 (i, x)

toward a lower bound given by gj,1 (i, x). Other than Assumption 8, there is no restrictionon the shape of gj,1 (i, x) relative to gj,0 (i, x). Figure 3 provides an illustration of thecorresponding adjustment maps.

For consumers for whom adjustment is preferred with respect to a given productrelative to leaving one’s attitude fixed, it is possible to define the notion of adjustmentproductivity : how much attitude improvement the consumer will attain, given his pref-erences, his particular capabilities at adjusting to the product, and the transportationcost (i.e., his opportunity cost of adjusting). Define the set Xj,0 (t) := {x : gj,0 (0, x) < t};since gj,0 (0, x), while continuous, is not required to be monotonic in x, Xj,0 (t) may con-tain (compact) gaps. One may then define the implicit function i∗j,0 (x, t) on Xj,0 (t) ×

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{t > 0} → R+ such that gj,0 (i∗j,0 (x, t) , x) = t as consumer x’s “adjustment productiv-ity given t.” One may similarly define i∗j,1 (x, t) and i∗j (x, t, Aj); on Xj,1 (t) and Xj (t),respectively; based on gj,1 (i∗j,1 (x, t) , x) = t and gj (i∗j (x, t, Aj) , x, Aj) = t, respectively.Note that, say, for x /∈ Xj (t), i∗j (x, t) = 0. The adjustment model thus nests non-adjustment as a sub-case (i.e., Xj (t) = ∅).

The following lemma advances some useful results that follow from the definition ofadjustment productivity and the assumptions made thus far:

Lemma 1. (i) i∗0x < 1 (and i∗1x > −1); (ii) i∗0A0≥ 0 (and i∗1A1

≥ 0); (iii) limA0→∞ i∗0A0

= 0

(and limA1→∞ i∗1A1

= 0); and (iv) i∗0t = 1/ ∂g0

∂i∗0> 0 (and i∗1t = 1/ ∂g

1

∂i∗1> 0).

Accounting for adjustment, the utility of a consumer at x buying product 0 is givenby

U0 = V − p0 − t[x− i∗0 (x,A0)

]−

i∗0(x,A0)ˆ

0

g0 (i, x, A0) di (3)

and, for a consumer at x buying product 1, by

U1 = V − p1 − t[1− x− i∗1 (x,A1)

]−

i∗1(x,A1)ˆ

0

g1 (i, x, A1) di (4)

Utility losses accruing to choosing a non-ideal product equal the sum of adjustment costand transportation cost components and are a function of the consumer’s adjustmentproductivity. Figure 4 displays these losses graphically as areas under the adjustmentand transportation cost curves.6

Following Bloch and Manceau (1999), but extended to the adjustment case, I imposewhat is in effect a restriction that V be sufficiently large relative to t:

Assumption 9.{V − t [x− i∗0 (x)]−

´ i∗0(x)

0g0 (i, x) di

}F (x) is increasing for all x ∈

[0, 1].

The assumption is a sufficient condition for the market to be covered under adjust-ment:

6Note that the setup in (3) and (4) is isomorphic to a traditional Hotelling model with nonlineartransportation costs. The rationale for layering adjustment into the model explicitly (i.e., by means ofthe “adjustment map”) is so that its effects may be seen distinctly.

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Lemma 2. Given Assumption 9, the market is covered in equilibrium.

To avoid corner solutions and make the welfare effects of advertising more easily mea-surable, I will assume that all consumers find adjustment productive for both productsbefore advertising is introduced. This provides that advertising does not need to be ex-pended to bring consumers to the point where adjustment becomes feasible; rather itspurpose is to improve the adjustment productivity of consumers who are already at leastminimally productive at adjustment.

Assumption 10. (Strong Adjustment Feasibility). Xj,0 (t) = [0, 1], j = 0, 1.

The location x∗E of the indifferent consumer under adjustment is derived by settingU0 = U1, thus defined implicitly by

Θ (x∗E, t, p0, p1, A0, A1) ≡ p1 − p0 + t− 2tx∗E − t[i∗1 (x∗E, t, A1)− i∗0 (x∗E, t, A0)

](5)

+

i∗1(x∗E ,A1)ˆ

0

g1 (i, x∗E, A1) di−

i∗0(x∗E ,A0)ˆ

0

g0 (i, x∗E, A0) di = 0

Market shares for the two products are then defined byD0 = F (x∗E) andD1 = 1−F (x∗E).

There are two periods. In t = 1, firms choose advertising expenditures, taking eachother’s advertising expenditure choices as given. In t = 2, they choose prices, takingeach other’s prices and their previous advertising choices as given. Firms recognize thattheir prices and their rivals’ prices will depend on their prior advertising choices andso treat these strategically with respect to their advertising decisions in t = 1. At theend of t = 2, consumers choose products and contemporaneously adjust to the productthey choose, accounting for the effects of their adjustment when making their choice.The consumers receive utility, and the firms earn profits. I seek subgame perfect Nashequilibria to this game.

Given demand, profits of the firms are given by

Π0 = p0 (A0, A1)F (x∗E)− αA0

Π1 = p1 (A0, A1) {1− F (x∗E)} − αA1 (6)

where α is the unit cost of advertising.

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Figure 4: Components of Utility Loss from Selecting Non-ideal Product 0

As a final assumption, I employ a variant on a distributional restriction by Caplin andNalebuff (1991), which they showed constitutes a sufficient condition for the existence ofa unique equilibrium in a broad class of games. Bloch and Manceau (1999) demonstratedthe use of the Caplin-Nalebuff assumption in a Hotelling model of product differentiationwith a generic distribution of consumers. The present variant generalizes that assumptionto the model involving adjustment by imposing a set of complementary restrictions onthe consumer distribution f and the adjustment functions gj. In the Appendix, it isdemonstrated that the assumption applies to a rather general set of f and g functionalform combinations.

Assumption 11. F (· ) is log concave in p0 (and 1− F (· ) is log concave in p1).

3 Equilibrium

I begin by identifying a set of sufficient conditions for a unique Nash equilibrium withpositive advertising for the advertising-and-pricing game played by the firms, and I derivethe equilibrium prices. After that, I derive the effect of advertising on prices as a com-parative static result that characterizes the equilibrium. Equilibrium in the two-stagegame is solved by backward induction: one first must determine the Nash equilibrium

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price-setting strategies of the firms in t = 2 , then the t = 1 advertising strategies thattake account of the t = 2 decisions. I obtain the following:

Proposition 1. Assume the following hold:(i) φ is sufficiently concave;(ii) the initial marginal productivity of advertising φ (Aj)|Aj=0 is sufficiently large

(j = 0, 1);(iii)

∂x∗E∂A0

> −1− F (x∗E)

f (x∗E)

∂x∗E∂p1

∂g0

∂x∗Ei∗0A0

+

i∗0(x∗E ,A0)ˆ

0

∂2g0

∂x∗E∂A0

di

; and

(iv)

∂x∗E∂A1

>F (x∗E)

f (x∗E)

∂x∗E∂p0

∂g1

∂x∗Ei∗1A1

+

i∗1(x∗E ,A1)ˆ

0

∂2g1

∂x∗E∂A1

di

Then, there exists a unique pure strategy Nash equilibrium in which (A∗0, A

∗1) � 0, p∗0 =

−F(x∗E)/f(x∗E)∂x∗E∂p0

and p∗1 = [1−F(x∗E)]/f(x∗E)∂x∗E∂p1

.

Condition (i) ensures the uniqueness of each firm’s optimizing advertising level. Con-ditions (ii) and (iii) together are sufficient to guarantee at least some advertising by firm0 is profitable in equilibrium, given advertising’s positive incremental cost; (iii) in par-ticular assures that the marginal revenue product of advertising for firm 0 is positive.(Conditions (ii) and (iv) together play an analogous role for the advertising of firm 1.)The bracketed expression in (iii) reflects how advertising, via the structure of the adjust-ment map, influences the price sensitivity of demand. As we will demonstrate shortly (inProposition 2), advertising increases firm 0’s own price when the bracketed expression ispositive and decreases it when the expression is negative. (The reverse is true for (iv) andfirm 1.) Since advertising’s effect on the location of the indifferent consumer is always

nonnegative for firm 0 - that is, ∂x∗E∂A0≥ 07 - and noting that p∗1 =

1−F(x∗E)

f(x∗E)∂x∗E

∂p1

, (iii) holds

if advertising raises firm 0’s price; or if, in the alternative, its negative effect on price is

7This is intuitive, but is established formally by (A.18), using the expression for p∗0 in the Propositionand observing that p∗0 is necessarily nonnegative.

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not large, and an advertising-induced price cut is made worthwhile by the rival firm’srelatively low price and a relatively low elasticity of demand to the rival firm’s price.Otherwise, advertising in any amount will not be profitable to the firm.

Now let us turn to advertising’s price effect in equilibrium. It is sufficient to con-sider the effect of varying the exogenous unit cost of adjustment-facilitating advertising,α. I eliminate sectoral effects by confining the analysis to a context in which both de-mand and adjustment are symmetric with respect to products. Taking the price functionpj (A0 (α) , A1 (α)), the price effects of α through advertising are given by (for j = 0):8

∂p0

∂α=∂p0

∂A0

∂A0

∂α+∂p0

∂A1

∂A1

∂α+∂p0

∂p1

∂p1

∂A0

∂A0

∂α+∂p0

∂p1

∂p1

∂A1

∂A1

∂α(7)

Signing this expression, one obtains the following result:

Proposition 2. Assume f (x) = f (1− x) and g0,k (i, x) = g1,k (i, 1− x) ∀i ≥ 0, x ∈[0, 1] , k = 0, 1. Then advertising increases prices if

∂g0

∂x∗Ei∗0A0

+

i∗0(x∗E ,A0)ˆ

0

∂2g0

∂x∗E∂A0

di > 0

whence, by symmetry,

∂g1

∂x∗Ei∗1A1

+

i∗1(x∗E ,A1)ˆ

0

∂2g1

∂x∗E∂A1

di < 0

and decreases them if the inequalities are reversed.

Proposition 2 shows advertising affects prices via two component effects: what weshall call an intensification effect, and a shaping effect. The intensification effect, reflectedin the first term in each of the two expressions in Proposition 2, arises from advertis-ing’s amplification of adjustment’s existing effect on price, as reflected in the existingadjustment map. The effect reflects advertising’s role in making all consumers equallybetter at - and therefore more intensely involved in - adjustment; it therefore causes theexisting price-influencing tendencies of adjustment across the mass of consumers to beexpressed more fully. In particular, the intensification effect will imply an increase inprices if the existing map is impression-reinforcing; and, more specifically, if it shows an

8It is straightforward to show that α affects pj only through advertising (i.e., there is no direct effect).

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impression-reinforcing pattern in the vicinity of the indifferent consumer’s location, x∗E.It will instead imply a decrease in prices if the existing map is mean-regressive, or moreparticularly mean-regressive near x∗E.

The shaping effect, reflected in the second term in the two expressions given in theProposition, arises from how advertising alters the shape of the adjustment map, that is,how advertising affects the manner in which consumers’ adjustment facility varies withthe strength of initial product preference. The shaping effect component will imply anincrease in prices if advertising causes adjustment to become more impression-reinforcingin the vicinity of x∗E, and a decrease in prices if advertising instead causes adjustment tobecome more mean-regressive in the vicinity of x∗E.

Figure 5 illustrates an example of the component effects from the Proposition. Adver-tising induces movement from the solid red to dashed blue adjustment map; this involvestwo distinct steps. The first - movement from the solid red map to the solid blue map- constitutes the shaping effect. The displayed example shows movement to a mean-regressive pattern of adjustment from a neutral pattern; this would imply lower prices.The second step - movement from the solid blue to the dashed blue map - constitutesan intensification effect. The displayed example shows a mean-regressive map becomingmore fully expressed; this implies that prices fall further.

The role of the impression-reinforcing and mean-regressive adjustment patterns ineffect of advertising on price levels has an intuitive explanation. Consider again the twopanels of Figure 2. The first panel shows a symmetric adjustment map in which thecurves grow flatter, at first, as one moves from x∗E (which equals 0.5 in the symmetriccase pictured) toward positions of stronger initial preference. In such a case, a priceincrease for one of the products would move to the margin previously-inframarginalconsumers who find adjustment more productive at improving their attitude than theconsumer at x∗E. These consumers, if they switched products, would forgo lower totalcosts (i.e., transportation plus adjustment costs) from the product they left than wouldthe consumer at x∗E. Given symmetry, they would also incur greater total costs fromtheir new product relative to the consumer at x∗E. Thus demand is less price elastic,all else equal. Advertising that intensifies adjustment uniformly across consumers wouldamplify the effect borne out by this particular adjustment map pattern, reducing theprice elasticity of demand and causing prices to rise, all else equal. Advertising thatreshapes adjustment to be more consistent with this pattern and less consistent with amean-regressive pattern will similarly intensify the associated price tendency.

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Figure 5: Intensification and Shaping Effects

In the case displayed in the lower panel of Figure 2, a price increase moves to themargin previously-inframarginal consumers who find adjustment less productive thanthe consumer at x∗E. These consumers, if they switched products, would forgo greatertotal costs from the product they left than would the consumer at x∗E. An adjustmentmap with this particular shape sets up an increased incentive for switching, whencedemand becomes more elastic. Advertising that uniformly intensifies adjustment wouldincrease the price elasticity of demand in this case, causing prices to fall, all else equal.9

Advertising that reshapes adjustment to be more consistent with this pattern and lessconsistent with an impression-reinforcing pattern will similarly intensify the associatedprice tendency.

9Note that what is critical to our result in both cases is what happens near x∗E ; in both cases, thecurves must eventually become increasingly steep as one approaches x = 0 or x = 1, as this follows fromour limiting assumptions on the gj,k.

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4 Endogenous Targeting

4.1 Setup

Sections 2 and 3, respectively, proposed a model of generically-targeted adjustment-facilitating advertising, and derived its effect on prices. I imposed no substantive restric-tions on the advertising other than to have it be non-increasing to adjustment costs forall consumers at all points in their adjustment process. Any pattern of effects meetingthis criterion is covered by the model. For the purpose of developing the advertising’spricing implications, the pattern of effects was treated essentially as exogenous, withfirms having the option to decide how much advertising to provide, given the pattern.

Now let us allow the pattern of advertising effects to be determined by the firmsimultaneously with the amount of advertising. I will refer to “mass market advertising”as advertising that translates downward the adjustment curves of all consumers by thesame amount, in such a way that there is no change in the shape of the adjustment map.One may conceive of this as being the result of advertising that lands on all consumersequally. In terms of the pricing effects derived in Proposition 2, mass market advertisingis pure intensification effect; that is, it has a zero shaping effect. In contrast, “targetedadvertising” causes a greater intensity of advertising to land on some consumers relativeto others. Such advertising changes the shape of the adjustment map. Without loss ofgenerality, targeted advertising may be viewed as attenuating the extent to which theadvertising lands on certain consumers relative to a baseline of mass market advertising.

Formally, let us define targeted advertising as consisting of advertising that lands ona set of compact intervals of consumers within [0, 1] at full strength, while landing on theremaining consumers at less than full strength. For the purposes of further discussion,I will refer to the endpoints of the fully-targeted intervals as targeting endpoints. Con-sumers on whom the advertising lands at full strength are fully targeted - or more simply,targeted - consumers, while those on whom advertising lands at something less than fullstrength are partially targeted. Consumers on whom the advertising does not land at allare not targeted.

I assume that mass market advertising can be purchased at a unit cost of α. At-tenuating the advertising in order to target consumers selectively can be accomplishedat no incremental cost, but neither does it reduce the cost. One could make argumentsalternately that targeting requires more precise efforts and is therefore more costly; orthat not having to place ads everywhere, but only in select outlets for select groups of

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targeted recipients, saves the firm money. The model splits the difference.Consistent with the observed realities of advertising practice, I assume that perfect

targeting, whereby specific consumers may be fully targeted while immediately adjacentconsumers are not targeted, is not possible. Rather, targeting is governed by an exogenousprecision parameter, as follows:10

Assumption 12. (Targeting precision). Suppose firm j targets its advertising to a com-pact interval whose right endpoint is x ≤ 3

4and whose left endpoint is x ≥ 1

4; and suppose

the advertising reduces consumers’ adjustment costs by an amount k along the targetedinterval. Define ε ∈

(0, 1

4

)as the targeting precision of the firms’ advertising, such

that advertising reduces consumers’ adjustment costs by, at minimum,(1− x−x

ε

)k for

any consumer located at x ∈ (x, x+ ε) and, at minimum,(1− x−x

ε

)k for any consumer

located at x ∈ (x− ε, x).

The assumption embodies the recognition that advertising intentionally targeted ata specific geographic location or demographic or lifestyle profile will also tend to landon people with similar though not identical characteristics. The greater the precision oftargeting, the smaller such spillovers will be - i.e., the less the extent to which consumerswho are farther afield from those for whom the advertising is intended will benefit fromthe advertising. The assumption allows that an advertiser might choose intentionallyto partially target consumers off the fully-targeted interval, attenuating the advertisingby less than what targeting precision allows; for this reason, the modifier “at minimum”is used. In the case in which the advertiser makes full use of targeting precision, letus refer to the interval over which the effect of advertising gradually attenuates (e.g.,x ∈ (x, x+ ε) in Assumption 12) as the phaseout interval, and the point at which thephaseout ends (e.g., x+ ε in Assumption 12) as the phaseout endpoint.

4.2 Firm Strategy

The benefit to the firm of using targeted advertising instead of mass market advertisingfollows from the shaping effect identified in Proposition 2. To wit, advertising targeted insuch a way as to increase the adjustment cost differential between the marginal consumer

10Instead of assuming targeting precision exogenous, one could alternatively assume it is chosen by thefirm, whereby advertising costs more per unit when the firm targets it more precisely. This representsone of numerous alternative targeting assumptions of which one could explore the implications withinthe motivated preference framework. See section 6.

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and the immediate inframarginal consumer to the advantage of the latter allows the firmto profitably increase its price. The effect occurs because such advertising makes it lesslikely that the inframarginal consumer will switch when price is increased, decreasing theelasticity of demand with respect to price.

Based on the preceding setup, the following result establishes the firm’s profit-maximizingtargeting strategy:

Proposition 3. A firm’s targeting strategy is profit-maximizing if and only if it targetsthe pre-advertising marginal consumer and attenuates advertising immediately beyondthat consumer to the full extent allowed by the targeting precision parameter ε.

The proposition implies that firms are indifferent as to whether they target con-sumers inframarginal either to themselves or their rival. Only the marginal consumermatters when it comes to increasing one’s sales. By targeting the marginal consumerand attenuating the advertising maximally beyond that consumer, the firm garners allthe sales-increasing and price-elasticity-reducing benefits it can hope to obtain from ad-vertising. Note that, paradoxically, fully targeting the marginal consumer constitutesoptimal use of the strategy of increasing the adjustment cost differential between themarginal consumer and immediate inframarginal consumer. Given that it has reducedthe adjustment cost of the inframarginal consumer relative to the marginal consumer, thefirm would prefer to decrease the adjustment cost of the marginal consumer as much aspossible, as doing so increases its sales at the margin. In the limit, the marginal consumeris fully targeted.

4.3 Market Outcomes

Proposition 3 revealed that variations in inframarginal targeting are irrelevant to pricesand sales. For the purposes of investigating market outcomes, I will assume, withoutloss of generality, that the firm’s targeting strategy includes fully targeting all of its owninframarginal consumers and not targeting any of its rival’s inframarginal consumers -with the exception of those of its rival’s consumers that are partially targeted out ofnecessity due to the limits of targeting precision.

Assuming, again without loss of generality, that advertising’s effect on fully-targeted

consumers is to reduce adjustment costs in the limit by some κ ∈(

0, minj,x∈[0,1]

gj,0 (0, x)

),

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we can write the adjustment marginal cost function for firm 0 specifically as

g0T (x, i) =

g0,0 − κ x0−x

εφ (A0) for x ∈ [x0 − ε, x0]

g0,0 − κφ (A0) for x ∈ [0, x0 − ε]

g0,0 elsewhere

(8)

and for firm 1,

g1T (x, i) =

g1,0 − κx−x1

εφ (A1) for x ∈ [x1, x1 + ε]

g1,0 − κφ (A1) for x ∈ [x1 + ε, 1]

g1,0 elsewhere

(9)

where xj is the phaseout endpoint for the advertising. I keep xj explicit in the equationto allow for x∗E to be determined endogenously by the system.

Profit functions for the firms remain as given in (6). To simplify the analysis andease exposition, I will assume in what follows that α = 1 in (6) and that consumers aredistributed uniformly, that is, F (x) = x. Based on (8) and (9), adjustment productivityi∗j is defined implicitly for j = 1, 2 by

g0,0(i∗0 (x∗E, t, A0) , x∗E

)= t+ κ

x0 − x∗Eε

φ (A0)

g1,0(i∗1 (x∗E, t, A1) , x∗E

)= t+ κ

x∗E − x1

εφ (A1) (10)

where the positioning of x∗E relative to xj in the formula follows from Proposition 3. Theposition of the indifferent consumer x∗E is defined implicitly by

p1 − p0 + t− 2tx∗E − t[i∗1 (x∗E, t, A1)− i∗0 (x∗E, t, A0)

]+

i∗1(x∗E ,t,A1)ˆ

0

[g1,0 − κx

∗E − x1

εφ (A1)

]di (11)

i∗0(x∗E ,t,A0)ˆ

0

[g0,0 − κx0 − x∗E

εφ (A0)

]di = 0

We obtain the following comparative static results:

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Proposition 4. Given each firm’s profit-maximizing strategy, prices increase with tar-geting precision, that is, ∂pj

∂ε< 0, for j = 1, 2.

Proposition 5. Given each firm’s profit-maximizing strategy, advertising expendituresincrease with targeting precision, that is, ∂Aj

∂ε< 0, for j = 1, 2.

Given the way in which targeting is used, precision is a critical parameter for marketoutcomes. The goals of raising price and increasing sales are at cross purposes and en-gender a tradeoff for advertising: one either advertises more aggressively to the marginalconsumer to increase sales, or else ignores to some degree the marginal consumer in favorof his inframarginal neighbor in order to decrease the price elasticity of demand. Butas price is increased by creating a differential in adjustment cost with respect to theimmediate inframarginal neighbor, the opportunity cost of increasing sales is minimizedthe more precise targeting is. In turn, as this opportunity cost diminishes, advertisingbecomes more valuable to the firm.

5 Empirical Evidence: A Motivated Preference Per-

spective

Let us turn to considering the empirical relevance of the theory articulated in the pre-vious sections. An extensive empirical literature examines the effects of advertising onmarket prices and the price sensitivity of consumers. It offers conflicting results (Kauland Wittink 1995) that have been critiqued as providing an incomplete picture of howadvertising affects demand (Erdem et al. 2008). The motivated preference theory pro-vides a new perspective on this literature and its limitations. In particular, it providesnew insights into what data would give a more complete picture of advertising’s effects.

A number of studies have suggested that advertising decreases price sensitivity inthe market. Krishnamurthi and Raj (1985) use a “split-cable” television experiment inwhich half the households were exposed to higher levels of advertising for one brand of afrequently purchased consumer product during the second half of the sample period. Theprice sensitivity for that brand dropped for those consumers who received the increasedad exposure. Mela et al. (1997) use scanner data on purchases within a product categorymade by a panel of consumers to estimate the effect of quarterly advertising expenditureson the derivatives of consumers’ brand choice probabilities with respect to price. The

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authors find that a reduction in advertising expenditures over time is associated withincreased price sensitivity.

Other studies have suggested that advertising increases price sensitivity. An exampleof this category is Kanetkar et al. (1992), who use supermarket scanner data on purchasesin two nondurable product categories (dog food and aluminum foil) and correspondingtelevision ad exposure data for the same households. They observe a negative interactionbetween advertising and price in a mulitnomial logit model of choice among brands,which they interpret as evidence that increased advertising exposures led to increasedprice sensitivity.

On the basis of motivated preferences, there are multiple possible explanations forthese findings. The product categories examined in studies that found decreased pricesensitivity might have been conducive to highly precise targeting, while those in studiesthat found increased sensitivity might not have lent themselves to precise targeting. Theresults may also have been critically influenced by the underlying adjustment patternof the consumers in the market. If one supposes mass market ads were used uniformlyacross the studies, a finding that advertising increases price sensitivity may reflect animpression-reinforcing adjustment pattern, whereas the opposite finding might be indica-tive of a mean-regressive pattern. To understand the exact adjustment-relevant factors atwork in such scenarios requires their empirical measurement. Cross-category differencesin targeting precision might be estimated based on advertising spillovers, which havebeen measured previously in numerous studies (see, e.g., Erdem and Sun 2002). Nagler(2019) proposes methods by which cross-consumer adjustment patterns might be divinedempirically.

Additional studies in the literature have delivered mixed, or nuanced, findings onprice sensitivity. Popkowski Leszczyc and Rao (1990) use bimonthly data from a con-sumer nondurable product category on share of advertising expenditures, market shares,and prices for the three firms accounting for 90% of the category sales. They find thatnational advertising decreases the price sensitivity of consumers, while local advertisingincreases it. The authors theorize that national advertising’s purpose is to affect pref-erence, in effect eliciting cognitive and affective responses from consumers. Meanwhile,local advertising’s intent is to deliver a terminal behavioral result (sales). The differentobserved effects of each on price sensitivity, they contend, are the direct result of twotypes of advertising engineered to achieve different goals. From the perspective of moti-vated preference theory, however, the notion of national advertising’s role as focused on

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cognitive and affective responses is not very helpful in terms of understanding the signof its price effect, which, even so, can vary with who is targeted, their level of facilitywith adjustment to the product, and how precise the targeting is. These dimensionsmust be explicitly measured, the theory tells us, in order to predict with accuracy whatadvertising that assists self-persuasion will do to price levels.

In a controlled experiment involving consumer response to advertising for 12 brandsof candy bars, Mitra and Lynch (1995) use advertisements to manipulate separately thesize of the consumer’s consideration set (i.e., by inducing recall of incremental brands)and the relative strength of preference for individual brands. They find that the consid-eration set-influencing component of the advertising was responsible for increased priceelasticity, while the component focused on relative preference decreased consumers’ priceresponse. The authors interpret their findings as indicating that different price effectsfollow from the different roles played by the different advertisements. One possible mo-tivated preference critique of this conclusion points to the single-category focus of thestudy. Candy bars might present unique opportunities for precisely-targeted advertising,such that advertising in this market focused on facilitating self-persuasion will lead to agreater reduction in price sensitivity than could be achieved in the average market. Forexample, an ad used in Mitra and Lynch (1995) that characterized one brand of candybar as “deliciously thick yet amazingly light” seems intended to assist the self-persuasionof those particular consumers who value thickness and lightness in a candy bar. Equallygood opportunities for targeting consumers based on such specific tastes may not existin many categories. The conclusion that advertising that “differentiates” leads to higherprices might represent an inappropriate generalization based on the results of a singlecategory. The hypothesis should be explored further by replicating Mitra and Lynch’sexperiment for different product categories.

Most recently, Dave and Saffer (2012) examine direct-to-consumer advertising in thepharmaceutical market, separately considering the price effects of broadcast and non-broadcast advertising. They find a large positive impact of broadcast advertising on price,while non-broadcast has a smaller, albeit still positive, impact. A motivated preferenceinterpretation of their results considers both special characteristics of the prescriptiondrug market, and differences in the nature of broadcast and non-broadcast advertising.Pharmaceutical drugs as a category involve inter-product differences that may be hardfor most consumers to fathom. Processing the attributes necessary for drug comparisonsmay be difficult, such that consumers fall back on heuristics or summary impressions to

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form relative preferences (Hutchinson and Alba 1991, Mantel and Kardes 1999, Dempseyand Mitchell 2010). Such product categories may demonstrate an impression-reinforcingadjustment pattern (Nagler 2019); whereby mass-market advertising, most typically as-sociated with broadcast, would raise prices through the intensification effect. Meanwhile,non-broadcast media, which tend to be more localized, might facilitate self-persuasionmore for marginal consumers in particular. Such a shaping effect would make the ad-justment pattern more mean-regressive, causing the positive price effect of the resultantadvertising to be more muted.

6 Conclusion

The paper has offered a theory of persuasive advertising as distinct from informativeadvertising and with its own efficient purpose. The updated conception of advertising’srole arises from a new model of consumer behavior that recognizes the phenomenonof motivated preference. When rational consumers desire to adjust to their choices, itbecomes possible to conceive of how persuasion might influence them. That influence mayindeed be consensual. Thus the motivated preference proposition solves an importantpuzzle as regards persuasive advertising primitives. Beyond this, it offers a complexframework that cannot only make sense of prior empirical findings, but also provide asupporting basis for future empirical investigation.

Recent empirical efforts have sought a more complex understanding of advertising’spricing effects, motivated by the conflicting predictions of the traditional advertising the-ories, as well as the conflicting results obtained by traditional empirical approaches. Forexample, Erdem et al. (2008) focus on how advertising differentially influences the WTPof different groups of consumers, using innovative “mixed” multinomial logit demandmodels that accommodate several sources of unobserved heterogeneity. Methodologi-cal advances along these lines have created an opportunity for rich theories to advancehypotheses of effects that can more accurately describe reality than past conceptions.

Here the present model has offered two notable advances. First, it provides a frame-work, based on the conception of advertising as assisting self-persuasion, for predictingan array of differential price effects attributable to consumer heterogeneity. These effectsaccrue to different response tendencies to the same advertising by different consumers- in particular, consumers with different levels of initial preference for a brand. Theimpression-reinforcing and mean-regressive adjustment patterns suggested in section 2

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constitute two models for how response tendencies may vary in such way. Product-baseddifferences and consumer-based differences may exist that can explain why in some casesone would observe impression-reinforcing adjustment while in others one would observemean-regressive adjustment.11 Such theories of differential effects may be taken to data.With new approaches, including neural data, it may be possible to observe sources ofheterogeneity in consumers’ responses and so test the theory’s predictions in ways notpreviously possible.

Second, the model offers a general structure for looking at how advertising that assistsadjustment can shape the heterogeneous responses of different consumers differentially.In doing so, it permits the advancement of testable hypotheses for the price effects ofcomplex advertising strategies that target different messages at different audiences. Posit-ing a simple assumption of exogenous targeting precision, I explored optimal targetingstrategy and its implications in section 4; but that exploration represents just a begin-ning. Other targeting assumptions, informed by empirical observation, could allow foralternative investigations within the framework of the motivated preference model. Thuswhereas Erdem et al. (2008) consider the special implications of Heinz’ “horizontal” strat-egy, future empirical studies might use the present model as a basis for examining theimpacts of various types of campaigns.

The theory has noteworthy limitations. Advertising quite clearly serves informativepurposes, in addition to facilitating self-persuasion. The findings the paper has set forthin regard to advertising’s persuasive function may well be consistent with advertising alsofunctioning in a supplementary informative role, though future work should consider howthe roles interact. Moreover, certain features of persuasion have not be modeled in mysimple framework. Price likely plays a role in persuading, rather than just being a resultof it. This might have distinct implications for prices and for the use of advertising, andit should be considered further.

This paper has only initiated the process of understanding the implications of moti-vated preferences, and there are numerous opportunities for further research in this vein.I will recommend here two additional interesting avenues relating to advertising’s role.First, it would be helpful to understand differences between the adjustment effects of me-dia advertising and price promotions. While promotions encourage trial and so facilitateself-persuasion, they also create heterogeneity in the prices paid by different consumers

11For a extended discussion of product-based and consumer-based explanations of different patternsof variation between the motivated and non-motivated components of preferences, see Nagler (2019).

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in ways that may interact predictably with motivated preference effects. Second, whileI have focused on the symmetric case with respect to price effects, it would be useful tostudy how advertising affects pricing and competition in the case where the market isbiased in favor of one of the firms. How does a dominant firm use adjustment-facilitatingadvertising versus how an upstart firm might use it? What are the implications forequilibrium pricing when the market is lopsided?

A Appendix

Applicability of Log Concavity of F (·) in pj

In this section, I show that log concavity of F (·) in p0 – a critical condition for theexistence of an interior equilibrium in prices – may be met (1) for the general class ofpointwise-symmetric adjustment map pairs for any symmetric distribution f , and (2) foran example of a non-pointwise-symmetric adjustment map pair when f is Beta distributedwith shape parameters (α, β) = (3, 3). The main issue in the case of non-pointwise-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 rises precipitously at the extremes, making it potentiallyprofitable for firms to attempt to drop price from any candidate interior maximum toa low enough level to take the whole market. This situation is avoided if the densityof consumers at the extremes is sufficiently low, as with some log-concave distributionssuch as the Beta. So, to summarize, an interior price equilibrium will result wheneverthe incentive to de-stabilize such an equilibrium is mitigated by pointwise adjustmentsymmetry; or when there are not enough consumers with extreme tastes for firms towant to de-stabilize an interior price equilibrium despite non-symmetry.

We may define the log concavity of F (·) in p0 as f(x∗E)∂x∗E∂p0/F(x∗E) being decreasing in

p0 or, equivalently, −F(x∗E)/f(x∗E)∂x∗E∂p0

decreasing in p0. This gives rise to the followingnecessary and sufficient condition:

∂2x∗E∂p2

0(∂x∗E∂p0

)2 < f (x∗E)

F (x∗E)− f ′ (x∗E)

f (x∗E)(A.1)

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Derive ∂x∗E∂p0

by applying Cramer’s rule to (5),12

∂x∗E∂p0

= −∂x∗E

∂p1=

1[−2t+

´ i∗1(x∗E ,t)0

∂g1

∂x∗Edi−

´ i∗0(x∗E ,t)0

∂g0

∂x∗Edi

] < 0 (A.2)

Now differentiating,

∂2x∗E∂p2

0

= −

−2t+

i∗1(x∗E ,A1)ˆ

0

∂g1

∂x∗Edi−

i∗0(x∗E ,A0)ˆ

0

∂g0

∂x∗Edi

−2

·

i∗1(x∗E ,A1)ˆ

0

∂2g1

∂x∗2E

∂x∗E∂p0

di−

i∗0(x∗E ,A0)ˆ

0

∂2g0

∂x∗2E

∂x∗E∂p0

di+∂g1

∂x∗Ei∗1x

∂x∗E∂p0

−∂g0

∂x∗Ei∗0x

∂x∗E∂p0

= −(∂x∗E∂p0

)2

i∗1(x∗E ,A1)ˆ

0

∂2g1

∂x∗2E

∂x∗E∂p0

di−

i∗0(x∗E ,A0)ˆ

0

∂2g0

∂x∗2E

∂x∗E∂p0

di+∂g1

∂x∗Ei∗1x

∂x∗E∂p0

−∂g0

∂x∗Ei∗0x

∂x∗E∂p0

=∂2x∗E∂p2

1

(A.3)

whence (A.1) may be re-written

−∂x∗E

∂p0

i∗1(x∗E ,A1)ˆ

0

∂2g1

∂x∗2Edi−

i∗0(x∗E ,A0)ˆ

0

∂2g0

∂x∗2Edi+

∂g1

∂x∗Ei∗1x −

∂g0

∂x∗Ei∗0x

< f (x∗E)

F (x∗E)− f ′ (x∗E)

f (x∗E)(A.4)

where ∂x∗E∂p0

, which is not a function of i, has been pulled out of the integrals.

Consider first the set of pairs of pointwise-symmetric adjustment maps,{G 0,G 1 : G 0 = −G 1}. Considering the exemplar pair G 0 and G 1, for each adjustmentcurve in G 0 corresponding to a given location x ∈ (0, 1), the corresponding curve in G 1

would be its mirror image about x.13 A subset of such pairs, for j = 0, 1 and ρ = [−1, 1],is given by

gj =

x2(x−i) for x ∈

[0, 1

4

]ρx

2(x−i) + 1−ρ2−8i for x ∈

[14 ,

12

]ρ(1−x)

2(1−x−i) + 1−ρ2−8i for x ∈

[12 ,

34

]1−x

2(1−x−i) for x ∈[

34 , 1]

(A.5)

12As a notational simplification, I write throughout dgj

dx∗Efor dgj

dx (i, x∗E) and d2gj

dx∗2Efor d2gj

dx2 (i, x∗E), thefirst and second derivatives of gj with respect to x evaluated at x∗E .

13Note that pointwise-symmetry of adjustment maps, which requires curve-by-curve symmetry at eachlocation, is not the same as symmetry across products (discussed earlier), which requires that the mapsbe symmetric about x = 1

2 .

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Map pairs corresponding to the values ρ = 1 and ρ = −1 are shown in Figure 6.

With pointwise-symmetric adjustment map pairs, and symmetric distribution f , thefollowing conditions hold: (i) ∂g0

∂x∗E= ∂g1

∂x∗E, (ii) ∂2g0

∂x∗2E= ∂2g1

∂x∗2E, (iii) i∗0 = i∗1, (iv) i∗0x = i∗1x ,

and (v) f ′ (x∗E) = 0. It may be verified, based on these, that the necessary and sufficientcondition (A.4) above for log concavity is met, whence log concavity of F (x (p0)) holdsfor any symmetric distribution f .

Consider now an example of a non-pointwise-symmetric adjustment map pair, givenby g0 (i, x) ≡ x

2(x−i) and g1 (i, x) ≡ 1−x2(1−x−i) for x ∈ [0, 1]. These functions have the

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

2tx is defined

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

2t(1− x), whence i∗ < 1 − x. We also

have ∂i∗0

∂x= 2t−1

2tand ∂i∗1

∂x= 2t−1

2t. We evaluate the left-hand side of (A.4) at i = 0 (i.e.,

the position at which the indifferent consumer evaluates his decision between productoptions) for all x ∈ [0, 1], using integration by parts:

i∗1(x∗E)ˆ

0

∂2g1

∂x∗2Edi−

i∗0(x∗E)ˆ

0

∂2g0

∂x∗2Edi+

∂g1

∂x∗Ei∗1x −

∂g0

∂x∗Ei∗0x

=

2t−12t

(1−x)ˆ

0

i

(1− x− i)3di−

2t−12t

0

i

(x− i)3di+

i

2 (1− x− i)2

(−

2t− 1

2t

)−

−i2 (x− i)2

(2t− 1

2t

)(A.6)

=

[i

2 (1− x− i)2−

1

2 (1− x− i)

] 2t−12t

(1−x)

0

−[

i

2 (x− i)2−

1

2 (x− i)

] 2t−12t

x

0

=

(4t2 − 4t+ 1

)(2x− 1)

2x (1− x)

where ∂g0

∂x= −i

2(x−i)2 ≤ 0, ∂2g0

∂x2 = i(x−i)3 ≥ 0, ∂g1

∂x= i

2(1−x−i)2 ≥ 0, and ∂2g1

∂x2 = i(1−x−i)3 ≥ 0.

Substituting into (A.2) for our example functions we obtain ∂x∗E∂p0

= − 11−ln 1

2t

, whence wemay re-write the left-hand side of (A.4) as

−∂x∗E∂p0

i∗1(x∗E)ˆ

0

∂2g1

∂x∗2Edi−

i∗0(x∗E)ˆ

0

∂2g0

∂x∗2Edi+

∂g1

∂x∗Ei∗1x −

∂g0

∂x∗Ei∗0x

=

(4t2 − 4t+ 1

)(2x− 1)

2x (1− x)(1− ln 1

2t

) (A.7)

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

f (x) =[x (1− x)]2´ 1

0[u (1− u)]2 du

; F (x) =

´ x0

[u (1− u)]2 du´ 1

0[u (1− u)]2 du

⇒ f ′ (x) =2x (1− x) (1− 2x)´ 1

0[u (1− u)]2 du

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Figure 6: Pointwise-Symmetric Adjustment Map Pairs

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Thus,

f ′ (x∗E)

f (x∗E)− f (x∗E)

F (x∗E)=

2x (1− x) (1− 2x)

[x (1− x)]2 − [x (1− x)]

2

´ 10

[u (1− u)]2du

(A.8)

=2 (1− 2x)

x (1− x)− 30 [1− x]

2

x [6x2 − 15x+ 10]

One may verify using (A.7) and (A.8) that (A.4) holds for all x ∈ (0, 1), and for anyt > 1

2.

Proof of Lemma 1

Begin with the expression gj (i∗j (x,Aj) , x, Aj) = t which implicitly defines i∗j, andexpand (here, shown for j = 0, with the arguments of i∗0 suppressed,

φ (Aj) g0,1(i∗0, x

)+ [1− φ (Aj)] g

0,0(i∗0, x

)= t

→ φ (Aj)[g0,1

(i∗0, x

)− g0,0

(i∗0, x

)]+ g0,0

(i∗0, x

)− t = 0

Totally differentiating,

∂g0

∂i∗0di∗0 = −∂g

0

∂xdx− ∂g0

∂A0

dA0 + dt⇔ ∂g0

∂i∗0di∗0 = −∂g

0

∂xdx− φA0

(g0,1 − g0,0

)+ dt

Using Cramer’s rule, it follows from Assumption 4 that

i∗0x (x, t, θ) = −∂g0

∂x∂g0

∂i∗0

<∂g0

∂i∗0

∂g0

∂i∗0

= 1

Also using Cramer’s rule one obtains

i∗0t (x, t, θ) =1∂g0

∂i∗0

> 0 and i∗0A0=φA0 (g0,0 − g0,1)

∂g0

∂i∗0

≥ 0

Finally, limA0→∞ φA0 = 0 follows from the concavity of φ and Assumption 7, whencelimA0→∞ i

∗0A0

= 0. Corresponding results can be derived along the same lines for j = 1.

Proof of Lemma 2

The proof is an extension of the proof of Bloch and Manceau’s (1999) Lemma 1. Sup-pose that the market is not covered, that is, at equilibrium prices (p∗0, p

∗1) there exists a

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consumer x for whom

V − p∗0 − t[x− i∗0 (x)

]−

i∗0(x)ˆ

0

g0 (i, x) di < 0

and

V − p∗1 − t[1− x− i∗1 (x)

]−

i∗1(x)ˆ

0

g1 (i, x) di < 0

One can show these prices do not constitute a Nash equilibrium, in that firm 0 canincrease its profit by lowering its price p0 without altering the profit, hence strategy,of firm 1. Begin by noting that, under (p∗0, p

∗1), because there is a consumer for whom

neither good provides nonnegative utility somewhere between the firms, the profit of firm0 can be written

Π0 = p∗0 (x0)F (x0) ≡

V − t [x0 − i∗0 (x0)]−

i∗0(x)ˆ

0

g0 (i, x0) di

F (x0) (A.9)

where x0 is the position of the consumer who, at prices (p∗0, p∗1), is just indifferent between

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

∂p0

∂x0

= −t+ t∂i∗0

∂x− g0

(i∗0, x0

) ∂i∗0∂x−

i∗0(x0)ˆ

0

∂g0

∂x0

di

= −t−i∗0(x0)ˆ

0

∂g0

∂x0

di < −t+

i∗0(x0)ˆ

0

∂g0

∂idi

= −t+ g(i∗0 (x0) , x0

)− g (0, x0) = −g (0, x0) < 0

which follows from Assumption 4. Since ∂Π0/∂x0 = (∂Π0/∂p0) (∂p0/∂x0), it follows that∂Π0/∂p0 < 0. Therefore a small downward deviation in the price p0 from p∗0 increasesfirm 0’s profits while not affecting firm 1’s profits. This contradicts the assertion that(p∗0, p

∗1) constitutes an equilibrium.

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Proof of Proposition 1

Let us begin by considering the firms’ Nash price-setting strategies at t = 2. Differenti-ating firm 0’s profit equation in (6) with respect to price yields

∂Π0

∂p0

= F (x∗E) + p0f (x∗E)∂x∗E∂p0

(A.10)

Using (A.10), one obtains ∂Π0

∂p0

∣∣∣p0=0

= F (x∗E)|p0=0 > 0: non-zero demand for product 0

is guaranteed at p0 = 0 by t > 0 and gj (i, x) > 0. Moreover, ∂Π0

∂p0

∣∣∣p0|x∗

E=0

= p0f (0)∂x∗E∂p0

<

0, where p0|x∗E=0 > 0. Because, following from Assumption 11, − F(x∗E)

f(x∗E)∂x∗E

∂p0

must be

decreasing in p0, it follows that there exists a unique solution to the first-order condition∂Π0

∂p0= 0 and that it is p∗0 = −F(x∗E)/f(x∗E)

∂x∗E∂p0

. A corresponding analysis of firm 1’s problemyields the unique solution to the first-order condition ∂Π1

∂p1= 0 at p∗1 = [1−F(x∗E)]/f(x∗E)

∂x∗E∂p1

.

Now consider the firms’ advertising choices at t = 1. We focus on analyzing firm 0’sproblem. Differentiating firm 0’s profit equation in (6) with respect to A0 yields

∂Π0

∂A0

= p0f (x∗E)

[∂x∗E∂A0

+∂x∗E∂p0

∂p0

∂A0

+∂x∗E∂p1

∂p1

∂A0

]+∂p0

∂A0

F (x∗E)− α (A.11)

Firm 0 knows it will choose price optimally in t = 2 taking its advertising choice as given;accordingly the first-order condition for firm 0’s advertising decision involves substitutingthe first-order condition for price and the expression p∗0 = −F(x∗E)/f(x∗E)

∂x∗E∂p0

into (A.11)and setting the resulting expression equal to zero. Thus (A.11) reduces to

∂Π0

∂A0

= F (x∗E)

[− ∂x∗E

∂A0/∂x∗E∂p0

+∂p1

∂A0

]− α (A.12)

which is the sum of advertising’s effect on revenue through the prices of both firms. (Thedirect effect of advertising on 0’s sales drops out due to the envelope theorem.)

Sufficient conditions for a unique pure strategy Nash equilibrium in advertising with(A∗0, A

∗1)� 0 are conditions that can guarantee three outcomes: that ∂Π0

∂A0> 0 at A0 = 0;

that there exists A0 such that, for A0 > A0, ∂Π0

∂A0< 0; and that ∂2Π0

∂A20< 0 everywhere.

We will first establish what is required for the first outcome. It can be shown easilythat the marginal revenue product of firm 0’s advertising – the first term in (A.12) -is a linear function of φA0 . Thus it is sufficient that the first unit of advertising be

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sufficiently productive (i.e., φA0|A0=0 sufficiently large, as per condition (ii)) and themarginal revenue product of advertising be positive for ∂Π0

∂A0> 0 at A0 = 0 . Our task,

then, is to establish the equivalence of condition (iii) of the proposition to a positivemarginal revenue product of advertising for firm 0 (and, by extension, the equivalence of(iv) to a positive marginal revenue product of advertising for firm 1).

We may derive an expression for ∂p1

∂Ajby totally differentiating the first-order condition

in price for firm 1 and applying Cramer’s rule:

∂p1∂Aj

= −f (x∗E)

∂x∗E∂Aj

+ p1f′ (x∗E)

∂x∗E∂p1

∂x∗E∂Aj

+ p1f (x∗E)∂2x∗E

∂p1∂Aj

∂x∗E∂p1

{2f (x∗E) + p1f ′ (x∗E)

∂x∗E∂p1

}+ p1f (x∗E)

∂2x∗E∂p2

1

(A.13)

Using this, and ∂x∗E∂p0

= −∂x∗E∂p1

, it can be shown that

− ∂x∗E∂A0/

∂x∗E∂p0

+∂p1∂A0

= −f (x∗E)

{−∂x∗E

∂A0+ p1

∂2x∗E∂p1∂A0

− ∂x∗E∂A0

p1

(∂2x∗E∂p21

/∂x∗E∂p1

)}∂x∗E∂p1

{2f (x∗E) + p1f ′ (x∗E)

∂x∗E∂p1

}+ p1f (x∗E)

∂2x∗E∂p2

1

(A.14)

Because the denominator is known to be positive by the second-order conditions for amaximum in price, signing this expression positive is equivalent to signing the curly-bracketed expression negative.

Recalling that p∗1 = −[1−F(x∗E)]/f(x∗E)∂x∗E∂p1

, the curly bracketed expression in (A.14) maybe written

−∂x∗E

∂A0

+[1− F (x∗E)]

∂2x∗E∂p1∂A0

f (x∗E)∂x∗E∂p1

−[1− F (x∗E)]

∂x∗E∂A0

∂2x∗E∂p2

1

f (x∗E)(∂x∗E∂p1

)2 (A.15)

Using (A.2), we derive

∂2x∗E∂p1∂A0

=

−2t+

i∗1(x∗E ,A1)ˆ

0

∂g1

∂x∗Edi−

i∗0(x∗E ,A0)ˆ

0

∂g0

∂x∗Edi

−2

·

i∗1(x∗E ,A1)ˆ

0

∂2g1

∂x∗2E

∂x∗E∂A0

di−

i∗0(x∗E ,A0)ˆ

0

[∂2g0

∂x∗E∂A0+∂2g0

∂x∗2E

∂x∗E∂A0

]di+

∂g1

∂x∗Ei∗1x

∂x∗E∂A0

−∂g0

∂x∗E

(i∗0A0

+ i∗0x∂x∗E∂A0

) (A.16)

This and (A.3) allow us to relate ∂2x∗E∂p1∂A0

and ∂2x∗E∂p2

1by the following expression:

∂2x∗E∂p1∂A0

=∂2x∗E∂p21

(∂x∗E∂A0/

∂x∗E∂p1

)−

∂g0∂x∗E

i∗0A0+

i∗0(x∗E ,A0)ˆ

0

∂2g0

∂x∗E∂A0di

(∂x∗E∂p1

)2

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Substituting into (A.15) we obtain

−∂x∗E

∂A0−

[1− F (x∗E)][

∂g0

∂x∗Ei∗0A0

+´ i∗0(x∗E ,A0)

0∂2g0

∂x∗E∂A0di]

∂x∗E∂p1

f (x∗E)(A.17)

which yields a necessary condition for an interior equilibrium in advertising

∂x∗E∂A0

> −1− F (x∗E)

f (x∗E)

∂x∗E∂p1

∂g0

∂x∗Ei∗0A0

+

i∗0(x∗E ,A0)ˆ

0

∂2g0

∂x∗E∂A0

di

Analogous analytics show that another necessary condition arises from firm 1’s optimiza-tion, to wit,

∂x∗E∂A1

>F (x∗E)

f (x∗E)

∂x∗E∂p0

∂g1

∂x∗Ei∗1A1

+

i∗1(x∗E ,A1)ˆ

0

∂2g1

∂x∗E∂A1

di

It remains to show that the proposition follows from pre-condition (i). That is, we

endeavor to show that (i) is sufficient to establish that there exists A0 such that, forA0 > A0, ∂Π0

∂A0< 0; and that ∂2Π0

∂A20< 0 everywhere. By Assumption 7 and Lemma 1,

respectively, as A0 grows large, φA0 and i∗0A0approach zero. Applying Cramer’s rule to

(5), we can obtain

∂x∗E∂A0

=∂x∗E∂p0

φA0

i∗0(x∗E ,A0)ˆ

0

[g0,1 − g0,0

]di (A.18)

Inspection of (A.13) and (A.18) reveals that the components of the marginal revenueproduct of advertising that multiply φA0 are bounded above with respect to increases inA0 > 0, implying that there exists A0 such that, for A0 > A0, ∂Π0

∂A0< 0. Because the

marginal revenue product of advertising is a linear function of φA0 , this term will declinemonotonely with A0, if φA0 declines quickly enough with A0. Thus, for φ (.) sufficientlyconcave, ∂2Π0

∂A20< 0 is guaranteed, whence, given parallel results for firm 1, the existence

of a unique pure strategy Nash equilibrium with (A∗0, A∗1)� 0.

Proof of Proposition 2

To sign the effect of α on price, begin by observing that in the symmetric case ∂A0

∂α=

∂A1

∂α< 0 . The other components of (7) are obtained by totally differentiating the first-

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order condition in price for firm 0, which is obtained by setting (A.10) equal to zero, andusing Cramer’s rule:

∂p0

∂p1

= −f (x∗E)

∂x∗E∂p1

+ p0f′ (x∗E)

∂x∗E∂p0

∂x∗E∂p1

+ p0f (x∗E)∂2x∗E∂p0∂p1

∂x∗E∂p0

{2f (x∗E) + p0f ′ (x∗E)

∂x∗E∂p0

}+ p0f (x∗E)

∂2x∗E∂p2

0

(A.19)

and, for j = 0, 1,

∂p0

∂Aj= −

f (x∗E)∂x∗E∂Aj

+ p0f′ (x∗E)

∂x∗E∂p0

∂x∗E∂Aj

+ p0f (x∗E)∂2x∗E∂p0∂Aj

∂x∗E∂p0

{2f (x∗E) + p0f ′ (x∗E)

∂x∗E∂p0

}+ p0f (x∗E)

∂2x∗E∂p2

0

(A.20)

(A.13) provides an analogous expression for firm 1’s pricing decision with respect tochanges in advertising.

When incorporating these expressions in the evaluation of (7), certain other facts willbe useful. From (A.3) and (A.16), we observe ∂2x∗E

∂p20

=∂2x∗E∂p2

1=

∂2x∗E∂p0∂p1

and ∂2x∗E∂p0∂A0

= − ∂2x∗E∂p1∂A0

,respectively. Moreover, consistent with (A.16), we have

∂2x∗E∂p0∂A1

=

−2t+

i∗1(x∗E ,A1)ˆ

0

∂g1

∂x∗Edi−

i∗0(x∗E ,A0)ˆ

0

∂g0

∂x∗Edi

−2

·

i∗1(x∗E ,A1)ˆ

0

[∂2g1

∂x∗E∂A1+∂2g1

∂x∗2E

∂x∗E∂A1

]di−

i∗0(x∗E ,A0)ˆ

0

∂2g0

∂x∗2E

∂x∗E∂A1

di+∂g1

∂x∗E

(i∗1A1

+ i∗1x∂x∗E∂A1

)−∂g0

∂x∗Ei∗0x

∂x∗E∂A1

(A.21)

Additionally, under the symmetric case, x∗E = 12, whence F (x∗E) = 1 − F (x∗E) = 1

2,

f ′ (x∗E) = 0, and ∂x∗E∂A0

= −∂x∗E∂A1

(using (A.18)). Using these facts and (A.19), ∂p0

∂p1= 1

2.

Using (A.20), we may write

∂p0

∂A0+∂p0

∂A1= −

[f(x∗E)

+ p0f ′(x∗E) ∂x∗E∂p0

] (∂x∗E∂A0

+∂x∗E∂A1

)+ p0f

(x∗E)( ∂2x∗E

∂p0∂A0+

∂2x∗E∂p0∂A1

)∂x∗

E∂p0

{2f(x∗E)

+ p0f ′(x∗E) ∂x∗

E∂p0

}+ p0f

(x∗E) ∂2x∗

E

∂p20

(A.22)

= −p0f

(x∗E)( ∂2x∗E

∂p0∂A0+

∂2x∗E∂p0∂A1

)∂x∗

E∂p0

{2f(x∗E)

+ p0f ′(x∗E) ∂x∗

E∂p0

}+ p0f

(x∗E) ∂2x∗

E

∂p20

It may be observed from (A.16) and (A.21) that when ∂g0

∂x∗Ei∗0A0

+´ i∗0(x∗E ,A0)

0∂2g0

∂x∗E∂A0di =

0 (and, by symmetry, ∂g1

∂x∗Ei∗1A1

+´ i∗1(x∗E ,A1)

0∂2g1

∂x∗E∂A1di = 0), ∂2x∗E

∂p0∂A1= − ∂2x∗E

∂p0∂A0, whence

∂p0

∂A0+ ∂p0

∂A1= 0. Moreover, ∂g0

∂x∗Ei∗0A0

+´ i∗0(x∗E ,A0)

0∂2g0

∂x∗E∂A0di > 0 (and, by symmetry, ∂g1

∂x∗Ei∗1A1

+

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´ i∗1(x∗E ,A1)0

∂2g1

∂x∗E∂A1di < 0) imply the numerators of ∂2x∗E

∂p0∂A1and ∂2x∗E

∂p0∂A0are both increased by

the same positive quantity, whereby ∂2x∗E∂p0∂A0

+∂2x∗E∂p0∂A1

> 0, whence it follows from (A.22)that ∂p0

∂A0+ ∂p0

∂A1> 0. (The denominator is negative, based on the profit second-order

condition for price.) By symmetry, ∂p1

∂A1> − ∂p1

∂A0. Thus the second and third pair of

terms in (7) are both positive in this case, whereby it follows that prices increase with

advertising if ∂g0

∂x∗Ei∗0A0

+´ i∗0(x∗E ,A0)

0∂2g0

∂x∗E∂A0di > 0 (and ∂g1

∂x∗Ei∗1A1

+´ i∗1(x∗E ,A1)

0∂2g1

∂x∗E∂A1di < 0).

Suppose instead ∂g0

∂x∗Ei∗0A0

+´ i∗0(x∗E ,A0)

0∂2g0

∂x∗E∂A0di < 0 (and, by symmetry,

∂g1

∂x∗Ei∗1A1

+´ i∗1(x∗E ,A1)

0∂2g1

∂x∗E∂A1di > 0). Then the numerators of ∂2x∗E

∂p0∂A1and ∂2x∗E

∂p0∂A0are both

decreased by the same positive quantity, whereby ∂p0

∂A0< − ∂p0

∂A1and, by symmetry, ∂p1

∂A1<

− ∂p1

∂A0. It follows that prices decrease with advertising if ∂g0

∂x∗Ei∗0A0

+´ i∗0(x∗E ,A0)

0∂2g0

∂x∗E∂A0di < 0

(and ∂g1

∂x∗Ei∗1A1

+´ i∗1(x∗E ,A1)

0∂2g1

∂x∗E∂A1di > 0).

Proof of Proposition 3

Without loss of generality, consider the problem of firm 0. We begin by proving that allstrategies that target the (pre-advertising) marginal consumer and attenuate advertisingimmediately beyond that consumer to the full extent allowed by ε (henceforth, the “pro-posed set”) are equivalent from the perspective of firm 0’s profit. We then prove thesestrategies strictly dominate all others available to firm 0.

The proposed set comprises strategies that vary on the basis of two characteristics:the extent to which they target consumers between the firm’s position and the marginalconsumer (i.e., “inframarginal targeting”); and the extent to which they target consumersbeyond the marginal consumer, with the exception of immediate neighborhood of thatconsumer (i.e., “extramarginal targeting”). As (5) indicates, the location of the marginalconsumer depends on advertising only when targeted at the marginal consumer; thussales are independent of variations in both inframarginal and extramarginal targeting.Meanwhile, following from Assumption 12, all strategies in the proposed set share thecharacteristic that ∂2g0

∂x∗E∂A0= − ∂g0/∂A0

ε, that is, they possess identical shaping effects (i.e.,

the second term in the critical expression in Proposition 2). Moreover, because all strate-gies in the proposed set identically target the marginal consumer, they possess identicalintensification effects (i.e., the first term in the expression in Proposition 2). It followsthat both sales and price effects are invariant across strategies within the proposed set;thus we have proven that profits to firm 0 do not vary across these strategies.

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It remains to prove that strategies in the proposed set strictly dominate all othertargeting strategies available to firm 0. Consider first a generic strategy that does nottarget the marginal consumer. Referring to (A.18), it is clear that ∂x∗E

∂A0= 0 for such

a strategy because g0,1 = g0,0 (where these functions are evaluated at x∗E), whereas∂x∗E∂A0

> 0 holds for any strategy that targets the marginal consumer. Meanwhile, for such a

strategy, ∂2g0

∂x∗E∂A0≤ 0, whereas for any strategy in the proposed set ∂2g0

∂x∗E∂A0= − ∂g0/∂A0

ε> 0.

This implies a shaping effect that is strictly larger for strategies in the proposed set.Moreover, because i∗0A0

is evaluated at x∗E, the intensification effect is also strictly greaterfor strategies in the proposed set. Thus both firm 0’s sales and price are strictly greateras a consequence of strategies in the proposed set as compared to any strategy that doesnot target the marginal consumer, whence it follows that firm 0’s profits are strictlygreater.

Next, consider a generic strategy that targets the marginal consumer, but attenuatesadvertising beyond that consumer at less than the full extent allowed by ε. Such astrategy would characterized in effect by an alternative, “looser” targeting parameter∞ > ε > ε. Intensification effects are identical to those of strategies in the proposed set,because both the target the marginal consumer identically. But whereas strategies in theproposed set result in ∂2g0

∂x∗E∂A0= − ∂g0/∂A0

ε> 0, the alternative under consideration results

in ∂2g0

∂x∗E∂A0= − ∂g0/∂A0

ε< − ∂g0/∂A0

ε. Thus the effect of such an advertising strategy on price

is strictly smaller. Because both strategies target the marginal consumer identically,their effects on sales are identical. It follows that firm 0’s profits are strictly greaterunder strategies in the proposed set.

Proof of Proposition 4

We begin with (11), which can be simplified to

p1 − p0 + t− 2tx∗E − t[i∗1 (x∗E , t, A1)− i∗0 (x∗E , t, A0)

]+

i∗1(x∗E ,t,A1)ˆ

0

g1,0di− κx∗E − x1

εφ (A1) i∗1 (x∗E , t, A1)

i∗0(x∗E ,t,A0)ˆ

0

g0,0di+ κx0 − x∗E

εφ (A0) i∗0 (x∗E , t, A0) = 0

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Suppressing the arguments of the i∗j and totally differentiating, we obtain

dx∗E

−2t− t[i∗1x − i∗0x

]+

i∗1(x∗E ,t,A1)ˆ

0

∂g1,0

∂x∗Edi−

i∗0(x∗E ,t,A0)ˆ

0

∂g0,0

∂x∗Edi+ g1,0i∗1x − g0,0i∗0x

−κ[φ (A1)

εi∗1 +

x∗E − x1

εφ (A1) i∗1x +

φ (A0)

εi∗0 −

x0 − x∗Eε

φ (A0) i∗0x

]=dp0 − dp1 +

{−ti∗0A0

+ g0,0i∗0A0− κ

x0 − x∗Eε

φ′(A0) i∗0 − κ

x0 − x∗Eε

φ (A0) i∗0A0

}dA0

+

{ti∗1A1

− g1,0i∗1A1+ κ

x∗E − x1

εφ′(A1) i∗1 + κ

x∗E − x1

εφ (A1) i∗1A1

}dA1

+

{κx1 − x∗E

ε2φ (A1) i∗1 − κ

x∗E − x0

ε2φ (A0) i∗0

}dε

Recognizing from (10) that g1,0−t−κx∗E−x1

εφ (A1) = 0 and g0,0−t−κ x0−x∗E

εφ (A0) = 0,

this simplifies to

dx∗E

−2t+

i∗1(x∗E ,t,A1)ˆ

0

∂g1,0

∂x∗Edi−

i∗0(x∗E ,t,A0)ˆ

0

∂g0,0

∂x∗Edi− κ

[φ (A1)

εi∗1 +

φ (A0)

εi∗0] (A.23)

=dp0 − dp1 − κx0 − x∗E

εφ′(A0) i∗0dA0 + κ

x∗E − x1

εφ′(A1) i∗1dA1

+

{κx1 − x∗E

ε2φ (A1) i∗1 − κ

x∗E − x0

ε2φ (A0) i∗0

}dε

Defining the expression in large curly brackets in (A.23) as “Dx∗E,” we may write

∂x∗E∂p0

=1

Dx∗E= −∂x

∗E

∂p1(A.24)

Using Proposition 1 and F (x) = x (whence f (x) = 1), we obtain specific explicitexpressions for the prices of both products,

p0 = −x∗E/∂x∗E∂p0

p1 = (1−x∗E)/∂x∗E

∂p1(A.25)

Thus,

∂p0

∂ε= −

(∂x∗E∂p0

∂x∗E∂ε−x∗E

∂2x∗E∂p0∂ε

)/(∂x∗E∂p0

)2

(A.26)

where ∂x∗E∂ε

is evaluated by applying Cramer’s rule to (A.23). Invoking symmetry (i.e.,A0 = A1, and x∗E = 1

2) and reflecting the optimal target location x0 − x∗E = x∗E − x1 = ε,

we obtain ∂x∗E∂ε

= 0, whereby

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∂p0

∂ε=(x∗E

∂2x∗E∂p0∂ε

)/(∂x∗E∂p0

)2

= −(p0

∂2x∗E∂p0∂ε

)/∂x

∗E

∂p0

This takes the sign of ∂2x∗E∂p0∂ε

. Evaluating that, using (A.23) and (A.24),

∂2x∗E∂p0∂ε

= − 1(Dx∗E

)2 · ∂ (Dx∗E)

∂ε= − 1(

Dx∗E)2 · [κφ (A1)

ε2i∗1 + κ

φ (A0)

ε2i∗0]< 0

where I have dropped terms containing ∂x∗E∂ε

, which equals 0; and where, given symmetry(i.e., A0 = A1, and x∗E = 1

2) and reflecting the optimal target location x0−x∗E = x∗E−x1 =

ε, I note also that i∗0ε = i∗1ε = 0. Thus ∂p0

∂ε< 0. It follows analogously that ∂p1

∂ε< 0.

Proof of Proposition 5

Using (A.12) with α = 1, and substituting F (x) = x, I obtain

x∗E

[− ∂x

∗E

∂A0/∂x∗E∂p0

+∂p1

∂A0

]= 1 (A.27)

I obtain an expression for ∂x∗E∂A0

from (A.23), again defining the expression in large curlybrackets in (A.23) as “Dx∗E,”

∂x∗E∂A0

= −κx0−x∗E

εφ′(A0) i∗0

Dx∗E

whence, using (A.24),− ∂x

∗E

∂A0/∂x∗E∂p0

= κx0 − x∗E

εφ′(A0) i∗0

and∂x∗E∂A0

∂x∗E∂p1

=κx0−x∗E

εφ′(A0) i∗0

(Dx∗E)2

Now, using (A.25),

∂p1∂A0

= − (x0 − x∗E)κφ′(A0)

εi∗0 + (1− x∗E)κ

φ′(A0)

ε

[(∂g0,0

∂x∗E+ κ

φ (A0)

ε

)x0 − x∗Eg0,0i

+ i∗0

]

so,

− ∂x∗E∂A0/

∂x∗E∂p0

+∂p1∂A0

= (1− x∗E)κφ′(A0)

ε

[(∂g0,0

∂x∗E+ κ

φ (A0)

ε

)x0 − x∗Eg0,0i

+ i∗0

]

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Thus we rewrite (A.27) as

(1− x∗E)x∗Eκφ′(A0)

εK = 1 (A.28)

where we define

K ≡(∂g0,0

∂x∗E+ κ

φ (A0)

ε

)x0 − x∗Eg0,0i

+ i∗0

Totally differentiating (A.28) yields

dA0 {DA0} = dε

x∗E [1− x∗E ]κφ′(A0)

ε2K − x∗E [1− x∗E ]κ

φ′(A0)

ε

−κφ(A0)

ε2

g0,0i

(x0 − x∗E)−κx0−x∗Eε2

φ (A0)

g0,0i

(A.29)

where

DA0 ≡∂{

(1− x∗E)x∗Eκφ′(A0)ε

[(∂g0,0

∂x∗E+ κφ(A0)

ε

)x0−x∗Eg0,0i

+ i∗0]}

∂A0

< 0

which is signed based on the second-order condition for a maximum. Moreover, basedon (A.28), it is straightforward to sign K > 0. It follows the curly bracketed expressionon the right-hand side of (A.29) is positive. Therefore, using Cramer’s rule, ∂A0

∂ε< 0. It

follows analogously that ∂A1

∂ε< 0.

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