Loving What You Get: the Price Effects of Consumer Self-Persuasion

Loving What You Get: The Price Eﬀects of Consumer Self-Persuasion

Matthew G. Nagler∗

June 27, 2021

Abstract

The paper considers how consumers’ cognitive efforts at preference adjustment at the time of decision affect prices in competitive markets with differentiated products. Greater ease of self-persuasion implies higher prices when self-persuasion re- inforces first impressions and lower prices when the best opportunities to persuade oneself exist for consumers with weak initial impressions. Exogenous interventions to ease decision-complementing cognition - e.g., advertising - predictably increase or reduce prices, depending upon how they are targeted. While facilitation of consumers’ adjustment always improves welfare in a covered market, firms’ appro- priation of surplus may make consumers worse off even as they learn better to love what they get.

Keywords consumer decision-making; diﬀerentiated products; heterogeneous consumers; market equilibrium; motivated preferences; pricing.

∗Ph.D. Program in Economics, City University of New York, 365 Fifth Avenue, New York, NY 10016-4309. Phone: +1 212.650.6205. Fax: +1 212.650.6341. Email: [email protected]. On rare occasions one does hear of a miraculous case of a married couple falling in love after marriage, but on close examination it will be found that it is a mere adjustment to the inevitable. - Emma Goldman

1 Introduction

This paper explores the effect of decision-complementing cognition by consumers on the competitive price offers that they observe in a differentiated product context. It does so by introducing a form of rational taste change into a spatial model of differentiated product competition à la Hotelling (1929). A continuum of consumers exhibits heterogeneous relative preferences with respect to the offers of two sellers, which are located at either end of the continuum. As is typical in this class of models, a consumer experiences disutility from an imperfect product match as a “transportation cost” that is associated with a product that is not immediately at his location. The special feature of my model is that the consumer can - at a cost - adjust to the product he intends to choose: In essence, the consumer “moves closer” to it, and thereby avoids some of the transportation cost. The model allows these adjustment costs to vary systematically with the strength of the consumer’s initial product preference. The model moreover presumes that consumers anticipate their adjustment and make decisions that take it into account. Thus, induced outcomes are subsumed into “final” preferences such that preferences are, in effect, fixed. Though adjustment implies intensification of preference, prices do not rise unambiguously with increasing intensity of consumer adjustment. The direction and size of adjustment’s effect on equilibrium market prices depend on how consumers’ facility with adjustment varies with the strength of their initial product preference. Exogenous ease of adjustment increases prices when adjustment facility is greater for consumers who initially prefer a product more – that is, when self-persuasion serves to reinforce initial impressions. But it reduces prices if the best opportunities to persuade oneself exist for consumers with weak initial preferences, such that final attitudes exhibit regression to the mean. I relate these differences to variation in a single conceptual variable: visibility over distance in preference space (VoDiPS). VoDiPS represents - with respect to consumers who are “distant” from a product in terms of initial preference - the relative ease or

0 difficulty that they face in attempting to “see” across the preference space so as to align their preferences with the characteristics of the product through self-persuasive thinking. The model’s predictions about the effects of VoDiPS on market prices are empirically relevant: Targeted advertising can influence this variable such that its price effects may be distinguished via field experiments or natural experiments. Conversely, the VoDiPS construct furnishes a previously lacking basis in behavioral primitives that can explain the price effects of advertising that are observed in existing empirical work. The next section offers a motivating parable to help fix ideas. In Section 3, I briefly review the empirical evidence for motivated preference change from the psychological literature, as well as the manner in which the existing economics literature has accounted for preference change conceptually. Section 4 lays out a general model of differentiated product competition with decision-complementing cognition. In Section 5, I derive equilibrium price levels and investigate the effect of a general ease of adjustment on price levels. Section 6 introduces VoDiPS, derives its effect on prices, and examines the empirical relevance of the construct. Section 7 considers the welfare effects of policy interventions to increase general ease of adjustment and VoDiPS - as well as the relative effects of such interventions on the utility of different consumers. Section 8 concludes. All proofs of lemmas, propositions, and corollaries are provided in the Appendix.

2 A Motivating Parable

The typical, traditional economic story about a consumer search process goes something like this: A consumer is shopping for a home. He considers several neighborhoods; he narrows his search to a handful of houses, which he tours, and on which he collects substantial information. Ultimately he chooses the best option from the perspective of his preferences: a house that features an updated kitchen with granite countertops. The consumer moves in with his dog. He is contented - which is to say he obtains utility from his choice. He lives happily ever after. This traditional story omits a key element of the process: While the consumer is searching - particularly toward the end of the process, and just after the process has concluded - he is talking to himself. It is not an informative internal conversation about what to decide, but a persuasive monologue about how good he is going to feel about a particular, presumptive choice - and why he should expect to feel that way. The consumer tells himself how indispensable the featured kitchen will be in his life:

1 those granite countertops will surely matter a lot to him, as he is becoming a more avid cook. Always an aficionado of hardwood floors, he remarks also how truly first-rate the floors are, even though the selling agent did not emphasize them. Functionally, what he is doing is manipulating salience: The consumer is bringing certain aspects of his preferences - and certain characteristics of the house - to the fore in his mind. Such thinking is instrumental: It serves to improve his attitude toward, and consequently enjoyment of, his chosen home. This missing element conjoins a critical omission in the standard model of consumer decision-making. The traditional approach accounts for imperfect matching between choices and preferences and thereby includes utility losses that are deemed unavoidable. Given the assumption that tastes do not change, consumers’ acceptance of the costs or losses that are associated with imperfect choice is posited as optimizing behavior. If, in reality, consumers do not merely accept imperfect matches passively, but “psyche themselves up” with respect to chosen items - and if, as thoughtful consumers, they recognize on some level that they are going to do this and so factor the effort involved and utility obtained into the choice process - then the positive and normative implications of the standard model are wrong. Models of consumer choice that are updated to account for such behavior could produce superior predictions.1

3 Literature Review

Behavioral and neural evidence supports the notion that individuals adjust to what they expect or intend to do. Across a range of experimental scenarios, individuals have been shown routinely to undergo a sort of mental re-positioning relative to choices they make, such that they change their stated preferences and even undergo measurable physiological changes that manifest hedonic shifts.2

1Other sorts of internal monologues are also possible. The old saying that “the grass is always greener” captures the sense in which individuals may sometimes ﬁnd themselves focusing on the positive aspects of what they have foregone - and the inferiority of what they have chosen. While the resulting “buyer’s remorse” may have various resolutions - including returning purchased products - research suggests that in the end individuals quite often resort to dissonance-reducing self-persuasion along the lines of my motivating parable. See, e.g., Ehrlich et al. (1957), Knox & Inkster (1968), and Goetzmann & Peles (1997). 2Studies that oﬀer 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 with the use of functional magnetic resonance imaging (fMRI) of subjects’ brains include: Sharot et al. (2009); Van Veen et al. (2009); Izuma et al. (2010);

2 The evidence suggests the process is not merely, or not always, one of post-hoc justi- fication (e.g., to reduce cognitive dissonance). Preference-related maneuvering has been found to occur prior to commitment to a choice, which gives reason to believe that such maneuvering is integral to the decision process (Simon et al., 2004; Jarcho et al., 2011; Qin et al., 2011; Kitayama et al., 2013). Also, while in some studies subjects have been found to reduce their ratings of non-chosen alternatives - behavior that indicates a clear motivation to justify a decision - in more recent studies subjects are found only to increase ratings of chosen alternatives - behavior that appears focused, rather, on enhancing the anticipatory or future experience of the chosen object (Kitayama et al., 2013; Tompson et al., 2016). Individuals appear to think “episodically” about the future use of prospective chosen objects; they search for positive distinctive features in these objects that might assure a positive experience and seal the choice (Kitayama & Tompson, 2015). It is not en- tirely clear whether this evidence supports a goal of increasing what Thaler (1985) terms acquisition utility – the net value of what one obtains from a decision – as opposed to merely increasing what he terms transaction utility: satisfaction that arises from feeling that one has made a meritorious decision. Nevertheless, there is strong support for the idea that individuals anticipate and seek functional adjustment to chosen objects and use their expectation of such adjustment integrally in the choice process. Theoretically speaking, the idea that “tastes change” has been reflected in the recent behavioral economic literature on reference dependence; and it has long been accepted in the field of psychology: most notably in connection with the framework of cognitive dissonance. The former focuses on recognition of the role of the agent’s reference point in his (potentially variable) interpretation of and response to outcomes (e.g., Kőszegi & Rabin, 2006). The latter focuses on situations in which mental discord – typically due to an individual’s perceiving a discrepancy between an action he has taken and his preferences, beliefs, or identity – provides motivation for a change in mental position to justify the action (Festinger, 1962).3 While the work in both areas has been highly influential, these literatures do not admit the particular proposition - as is suggested by the evidence above - that improving one’s perceived “fit” with a chosen object is a routine and essentially optimizing part of

Jarcho et al. (2011); Qin et al. (2011); Kitayama et al. (2013); and Izuma & Adolphs (2013). 3Meanwhile, Stigler & Becker (1977) have argued that all apparent changes (diﬀerences) in tastes can be explained by the development (existence) of complementary stocks of human or social capital. They contend that, once these are taken into account, there are in fact no diﬀerences in tastes at all.

3 individual decision-making. The present paper seeks to ﬁll this conceptual gap.

4 Model

Consider two differentiated products, which are indexed 0 and 1. Each product is pro- duced by an independent firm that is correspondingly named. Consumers are fully in- formed about both products: their search costs are zero. Suppose that consumers’ preferences over these products have two components: Each consumer is endowed with an exogenous initial preference - which is also known as his non-motivated component. Each consumer’s preference also comprises a motivated component - which is developed through a process of adjustment. Adjustment consists of discretionary efforts by the consumer to increase the satisfaction that he expects to receive from consuming the object that he chooses. These efforts are purely cognitive: I do not assume that the consumer plans to change the product following purchase.4 Rather, the efforts leverage the consumer’s ability to focus thinking in an opportunistic way: on the aspects of his preference that are most aligned with the salient strengths of the product (e.g., the granite countertops in the section 2 parable); and also on characteristics of the product that are most aligned with his exogenous preference (e.g., the hardwood floors). This activity is costly, in that it involves the allocation of scarce attention and the effort of assembling convincing self- persuasive arguments. The consumer knows the costs of adjustment and fully anticipates the benefits to be gained from his adjustment efforts.5 The assumptions are formalized with the use of a spatial model. The following subsection describes the consumers. The subsequent subsection describes the competitive process.

4There are some products - such as a house - for which a prospective buyer might contemplate and plan physical improvements that will increase the utility that he anticipates receiving from the product. Along similar lines, there are products for which the purchase of complementary items - e.g., a trailer hitch for a pickup truck - might result in an increase in the utility that accrues to the principal product. Such physical adjustments do alter final preferences in ways that are to some degree analogous to the process of cognitive adjustment here conceived. In order to focus on the effect of the cognitive type of adjustment, the model here essentially assumes a product for which there is no plausible scope for physical improvement and for which there are no significant complements. 5The standard consumer model, of course, assumes that a consumer anticipates future flows of utility: e.g., from a durable good. My model moves beyond this, specifically, to assume that the consumer anticipates future flows of utility that accrue to cognitive effort that increases the utility that is obtained from a good.

4 4.1 Consumers

As in Hotelling (1929), the two ﬁrms are located at opposite ends of a segment of length 1 that represents the product space. Consumers are assumed to be distributed contin- uously on this segment according to a continuous symmetric density function f and a corresponding distribution function F with full support. The consumer’s location x ∈ [0, 1] identiﬁes his initial (relative) preference over the two products. Consumers buy at most one unit of a single product, so based on the initial component alone the utility of a consumer at x who buys product j is given by

˜ Ux = V − pj − t |x − j| , (1) where V is the common reservation price for the product, pj is the price of product j, and t parameterizes the utility loss that is due to j’s not being the consumer’s ideal choice: the standard Hotelling “transportation” cost, linear in the consumer’s distance from j. All consumers have an outside option of utility zero. I assume that V is suﬃciently large that the market is covered. The formal assumption that guarantees this is introduced later. In section 8, I discuss the consequences for the analysis of relaxing that assumption. Adjustment to a product is represented as relocating on the segment so as to be closer to the product’s location, thereby paying less transportation cost. To keep the modeling simple, I abstract from the particulars of the opportunistic focusing process that enables attitude improvement and the precise mechanism of its eﬀects on utility.6 I represent these instead through a reduced-form marginal cost of relocation that is associated with product j:7 gj (i, x, θ) =g ¯j (i, x) − θ > 0 for θ ≥ 0 . (2)

Here, i is the distance from x as the consumer adjusts and is closer to j’s position, and j ∂gj where g¯ (0, x) = t for all x ∈ [0, 1]. As ∂θ = −1, θ serves to benchmark the ease of adjustment. I assume consumers’ adjustment costs are symmetric with respect to the products: g¯0 (i, x, θ) =g ¯1 (i, 1 − x, θ) everywhere on the domains of x and i. One may view the function g as representing a set of adjustment curves G j :=

6One could conceive of these as manifesting, for instance, through a focus-weighted utility function along the lines of Kőszegi & Szeidl (2013). 7I specify a marginal adjustment cost function, rather than a total adjustment cost function, so as to maintain the analogy to the marginal transportation cost primitive in the standard Hotelling model.

5 Figure 1: Possible Adjustment Map - Product 0

{gj (i) = gj (i, x, θ): x ∈ [0, 1]} that are 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 who is located initially at x. I shall refer to G j as an adjustment map for product j. Figure 1 illustrates a possible adjustment map for product 0. The adjustment curve that corresponds to the consumer who is located at x¯ is displayed in black. For this consumer, the parameter i is as represented in the ﬁgure: a distance i extending toward location 0 from x¯. The corresponding marginal adjustment cost to product 0 for x¯ for this i is given on the black (dark) curve at the corresponding location. The marginal adjustment costs of other consumers for varying levels of i are given by the other (grey, lighter) curves; each one begins at the location of the consumer whose costs the curve represents. A corresponding adjustment map for product 1 could be drawn similar to this di- agram, with adjustment curves for each consumer beginning at the corresponding consumer locations (e.g., x¯) and arcing upward to the right. On that map, the parameter

6 i would represent a distance i extending to the right, toward location 1 from the corresponding consumer’s location. The motivated component of preferences observes the following regularity conditions:

Assumption 1. (Continuity of adjustment in x, i, and θ). g0 (i, x, θ) and g1 (i, x, θ) are deﬁned on [0, x) × [0, 1] × [0, ∞) → R+ and on [0, 1 − x) × [0, 1] × [0, ∞) → R+, respectively, and continuous on their support.

∂gj ∂2gj Assumption 2. (Convexity of adjustment in i). ∂i > 0 and ∂i2 > 0, for j = 0, 1.

0 1 Assumption 3. (Asymptotic adjustment). limi→x g (i, x, θ) = ∞ (and limi→1−x g (i, x, θ) = ∞).

Assumption 1 reflects the notion that adjustment effort is smooth. Assumption 2 reflects the notion that incremental adjustment to a product becomes progressively more difficult as the best opportunities for advantageous focusing are used up. Assumption 3 reflects the notion that no matter how psyched up one is about a product, it is possible to get more psyched up – though it might eventually become quite difficult. Thus a consumer never fully converges to the product’s location even as he continues to adjust to it. Figure 2 illustrates the consumer’s decision with respect to how much to adjust. The consumer adjusts to a product only if he chooses the product. In that event, he expands adjustments to the point where the marginal cost of adjustment equals the marginal transportation cost that is saved: gj (i, x, θ) = t. ∗j This gives rise to an implicit function i (x, t, θ) that is defined on Xj (t, θ)×{t > 0}× + j j ∗j {θ > 0} → R , where Xj (t, θ) := {x: g (0, x, θ) < t}, such that g (i (x, t, θ) , x, θ) = t. Let us call i∗j (x, t, θ) consumer x’s “adjustment productivity given t and θ.” The figure displays the adjustment productivity amounts i∗0 and i∗1 that a consumer at x¯ would attain if he were to choose product 0 or product 1, respectively. ∗j Note that, for x∈ / Xj (t, θ), i (x, t, θ) = 0. In particular - in fact, by design - θ = 0

returns the standard Hotelling case: Xj (t, 0) = ∅. No consumers adjust when the ease of adjustment is zero. Inclusive of adjustment, the utility of a consumer at x who buys product 0 is given by i∗0(x,t,θ) U = V − p − t x − i∗0 (x, t, θ) − g0 (i, x, θ) di , (3) 0 0 ˆ 0

7 Figure 2: The Decision of How Much to Adjust and, for a consumer at x who buys product 1, by

i∗1(x,t,θ) U = V − p − t 1 − x − i∗1 (x, t, θ) − g1 (i, x, θ) di . (4) 1 1 ˆ 0

Utility losses that accrue to choosing a non-ideal product equal the sum of adjustment cost and transportation cost components and are a function of the consumer’s adjustment productivity. Figure 2 displays these losses graphically as areas under the adjustment and transportation cost curves. It is straightforward to see that i∗j (x, t, θ) represents the amount of adjustment that minimizes these losses; this follows naturally from adjustment productivity’s implicit function deﬁnition. Two general regimes characterize the way in which the motivated component of the individual’s preferences can relate to the non-motivated component.8 As one possibility, adjustment might be easier when the individual’s initial preference is stronger (over a range where the initial preference is not too strong9). This might be the case if one’s

8I consider why each of the two regimes might be observed empirically in section 6.2. 9This limitation follows from Assumption 3: since achieving a perfect ﬁt with a product eventually

8 predisposition toward a product is similar to a stock of capital that is complementary with further self-persuasion. I refer to adjustment in such a case as impression-reinforcing. ∂g0 Mathematically, impression-reinforcing adjustment is reflected by ∂x > 0 for product 0 ∂g1 ( ∂x < 0 for product 1); and it results in an adjustment map that looks like the one that is shown in the top panel in Figure 3. In the alternative, adjustment could be easier when the consumer’s initial preference is weaker. This regime might apply if those individuals who were not previously predisposed toward a product benefit from greater unexploited opportunities to convince themselves that it offers a good fit. I refer to adjustment in such a case as mean-regressive. Mathe- ∂g0 ∂g1 matically, mean-regressive adjustment is reflected by ∂x < 0 for product 0 ( ∂x > 0 for product 1); and it results in an adjustment map that looks like the one that is shown in the bottom panel in Figure 3.10 These two regimes admit a range of possibilities in the relationship between non- motivated and motivated preferences. But while adjustment may be mean-regressive, I assume in the extreme that it can only cause full regression to the mean. It can never cause consumers to “leapfrog” positions that are based on initial preferences. Formally:

∂g0 Assumption 4. (Initial preference dominance). For all x ∈ [0, 1] and all i > 0, − ∂x < ∂g0 ∂g1 ∂g1 ∂i (and ∂x < ∂i ). Assumption 4 ensures that, the more preferred is a product initially, the smaller is the marginal adjustment cost at any particular location that is achieved through accumulated adjustment. This implies that an individual who initially prefers a product more than another individual finds it less costly to reach a given level of adjusted preference for that product, as represented by a certain location, than does the other individual. This in turn gives rise to adjustment maps of non-crossing nested contours - which are similar to well-behaved indifference maps. The following lemma offers the intuitive and useful result that ease of adjustment increases adjustment productivity:

∂i∗j Lemma 1. ∂θ > 0 , j = 1, 2. becomes infinitely costly, the individual whose initial preference for a product is sufficiently strong perforce sees his marginal adjustment cost climb quickly. Indeed, one can see that, as a consequence of Assumption 3, there must exist a neighborhood around each product within which increasing initial preference corresponds with increasing marginal adjustment costs. 10Note that Assumption 3 guarantees that adjustment grows increasingly costly within the vicinity of the extremes, irrespective of the regime. Consistent with this, the figure depicts the curves near the extremes identically in the two panels.

9 Figure 3: Two Regimes Relating Motivated and Non-Motivated Preference Components

Note: Panels depict only curves that reﬂect adjustment to good 0 for x ≤ 0.5 and that reﬂect adjustment to good 1 for x ≥ 0.5.

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

∗0 n ∗0 i (x) 0 o Assumption 5. V − t [x − i (x)] − 0 g (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 adjustment:

Lemma 2. Given Assumption 5, the market is covered in equilibrium.

∗ The location xE of the indiﬀerent consumer under adjustment is derived by setting

U0 = U1 using (3) and (4); thus it is deﬁned implicitly by

∗ ∗ ∗1 ∗ ∗0 ∗ Θ(xE, t, p0, p1, θ) ≡ p1 − p0 + t − 2txE − t i (xE, t, θ) − i (xE, t, θ) (5) ∗1 ∗ ∗0 ∗ i (xE ,t,θ) i (xE ,t,θ) + g1 (i, x∗ , θ) di − g0 (i, x∗ , θ) di = 0 . ˆ E ˆ E 0 0

∗ ∗ Market shares for the two products are then deﬁned by D0 = F (xE) and D1 = 1−F (xE).

4.2 Competitive Process

The competitive process proceeds as follows: The firms choose prices, taking each other’s prices as given. Subsequent to that, consumers choose products and contemporaneously adjust to the product that they choose; they anticipate the effects of their adjustment when they make their choice.11 The consumers receive utility, and the firms earn profits. The equilibrium concept used for evaluating the game is subgame perfect Nash. Given demand, profits of the firms are given by

∗ Π0 = p0F (xE); (6) ∗ Π1 = p1 {1 − F (xE)} .

As a final assumption, I employ a variant on a distributional restriction by Caplin & Nalebuff (1991) - which they showed constitutes a sufficient condition for the existence of

11For evidence that supports the contemporaneousness of decision-making and attitude change, and consumers’ rational anticipation of attitude change, see Section 3.

11 a unique equilibrium in a broad class of games. Bloch & Manceau (1999) demonstrated the use of the Caplin-Nalebuﬀ assumption in a Hotelling model of product diﬀerentiation with a generic distribution of consumers. The present variant generalizes that assumption to the model that involves adjustment by imposing restrictions on the consumer distribution f and the adjustment functions gj. In the Appendix, it is demonstrated that the assumption applies to a rather general set of f and g functional form combinations:

Assumption 6. F (· ) is log concave in p0 (and 1 − F (· ) is log concave in p1).

5 Equilibrium

We focus on firm 0’s problem. Differentiating firm 0’s profit equation in (6) with respect to price yields ∗ ∂Π0 ∗ ∗ ∂xE = F (xE) + p0f (xE) . (7) ∂p0 ∂p0 ∂x∗ E is derived by applying Cramer’s rule to (5), ∂pj

∂x∗ ∂x∗ 1 E = − E = < 0 , (8) ∗1 ∗ ∗0 ∗ ∂p0 ∂p1 i (xE ,t) ∂g1 i (xE ,t) ∂g0 −2t + 0 ∂x∗ di − 0 ∂x∗ di ´ E ´ E

dgj j where, as a notational simpliﬁcation, I write ∗ for the ﬁrst derivative of g with respect dxE ∗ to x, evaluated at xE.

Using (7), one obtains ∂Π0 = F (x∗ )| > 0: non-zero demand for product 0 ∂p0 E p0=0 p0=0 ∗ ∂Π0 ∂xE is guaranteed at p0 = 0 by t > θ ≥ 0. Moreover, = p0f (0) < 0, where ∂p0 ∂p0 p0|x∗ =0 E ∗ F (xE ) p0|x∗ =0 > 0. Because, following from Assumption 6, − ∂x∗ must be decreasing in E f x∗ E ( E ) ∂p0 p0, one obtains

∗ ∗ Proposition 1. There exists a unique pure strategy Nash equilibrium in prices (p0, p1) ∗ ∗ ∗ F (xE ) ∗ 1−F (xE ) 12 where the prices are given by p0 = − ∂x∗ and p1 = ∂x∗ . f x∗ E f x∗ E ( E ) ∂p0 ( E ) ∂p1

12This represents essentially the extension of Bloch & Manceau’s (1999) Proposition 2 to the motivated preferences case.

12 Symmetry in the distribution of consumers and of adjustment cost functions yields the following corollary:

∗ 1 Corollary 1. In equilibrium, (i) the market is evenly divided between the ﬁrms: xE = 2 ; ∗ ∗ and (ii) p0 = p1.

∂x∗ Note that the E are functions of θ: the i∗j are functions of θ; and (as (8) indicates) ∂pj ∂x∗ the E are functions of the i∗j. Thus the expressions in Proposition 1 tell us that p∗ and ∂pj 0 ∗ p1 are functions of θ. In the context of the unique equilibrium, let us consider the eﬀect of the exogenous ∗ ease of adjustment θ on prices. Focusing on product 0, recognizing that D0 = F (xE) and using (8), one obtains

∂D f (x∗ ) 0 = E . (9) h i∗1(x∗ ,t,θ) 1 i∗0(x∗ ,t,θ) 0 i ∂p0 E ∂g E ∂g −2t + 0 ∂x∗ di − 0 ∂x∗ di ´ E ´ E Diﬀerentiation with respect to θ leads to the following result:

Proposition 2. The eﬀect of ease of adjustment θ on the price sensitivity of demands ∂g1 ∂g0 takes the sign of ∗ − ∗ . ∂xE ∂xE From this follows:

∂g1 ∂g0 Corollary 2. The effect of ease of adjustment θ on prices takes the sign of − ∗ − ∗ . ∂xE ∂xE Adjustment might appear to amount to a shift in the distribution of consumers, such that its effect would be expected to resemble that of traditionally conceived persuasive advertising (e.g., Bloch & Manceau 1999). The reduction in the concentration of consumers at the center and the increased dispersion toward the extremes that we observe would suggest, then, that adjustment unambiguously softens price competition. In fact, this is not the case: Because consumers incur costs to adjust rather than passively shifting (e.g., under the sway of advertising), adjustment represents (in effect) a nonlinear generalization of transportation costs in the Hotelling model. This can be seen by comparing the utility functions in (3) and (4) with general θ to the same forms restricted to θ = 0. The effects on price - as articulated in Proposition 2 and Corollary 2 - follow from the shape of the particular nonlinear form of those generalized transportation costs. These, in turn, relate specifically to how the motivated and non-motivated components of preferences relate to one another.

13 Again consider Figure 3: In the top panel of the figure, the curves at first grow flatter ∗ 1 as one moves from xE = 2 toward positions of stronger initial preference. This reflects ∗ 1 ∂g0 ∂g1 ∂g1 ∂g0 a sub-case in which, evaluating at xE = 2 , ∂x > 0 and ∂x < 0, whence ∂x − ∂x is negative.

Consider the effect of a small price increase for product 0 in this context. The locus ∗ 1 of indifference, xE, would move slightly to the left of 2 . The new marginal consumer faces a flatter adjustment curve for product 0 and then finds adjustment to product 0 1 more productive than did the consumer at 2 . He also finds adjustment to product 1 1 less productive than did the consumer at 2 . Thus this consumer has a reduced incentive 1 to switch relative to the consumer at 2 . It follows that an adjustment map with this particular shape sets up an decreased incentive for switching for consumers who are faced with price increases, which causes demand to be less price-sensitive and prices to be correspondingly higher, relative to the standard Hotelling case (θ = 0). The higher is θ, the more intensively do consumers engage in adjustment, and therefore the larger are these effects.

In contrast, the bottom panel of Figure 3 shows a case where the curves grow steeper ∗ 1 as one moves from xE = 2 toward positions of stronger initial preference. This reflects ∗ 1 ∂g0 ∂g1 ∂g1 ∂g0 a sub-case in which, evaluating at xE = 2 , ∂x < 0 and ∂x > 0, whence ∂x − ∂x is positive. Now consider the effect of a small price increase for product 0 in this context. 1 The new marginal consumer, just to the left of 2 , faces a steeper adjustment curve for product 0 and so finds adjustment to product 0 less productive than did the consumer at 1 1 2 . He also finds adjustment to product 1 more productive than did the consumer at 2 . It follows that this consumer has an increased incentive to switch relative to the consumer 1 at 2 . An adjustment map with this particular shape sets up an increased incentive for switching for consumers who are faced with price increases, which causes demand to be more price-sensitive and the prices correspondingly lower, relative to the standard Hotelling case (θ = 0). As with the flattening map, the higher is θ, the more intensively do consumers engage in adjustment, and therefore the larger are these effects.

14 6 Visibility over Distance in Preference Space

6.1 Parameterization and Comparative Statics

To be able to harness the framework in section 4 as a means to connect measurable characteristics of a market with observable price diﬀerences, we need to specify the model further: Consider the following parameterization of the marginal adjustment cost functions, where for simplicity each function is deﬁned over only the relevant half of the 1 1 consumer continuum. Here, σ ∈ − 2 , 2 represents a new concept, visibility over distance in preference space (VoDiPS):

 t−θ 1 1 1  i for x ∈ 4 , 2 , i ∈ 0, 2σx + (1 − 2σ) · 4 0  1− 1 g˜ (i, x, θ; σ) = 2σx+(1−2σ)· 4 (10)  t−θ 1  i for x ∈ 0, 4 , i ∈ [0, x); 1− x  t−θ for x ∈ 1 , 3 , i ∈ 0, 2σ (1 − x) + (1 − 2σ) · 1  1− i 2 4 4 1  1 g˜ (i, x, θ; σ) = 2σ(1−x)+(1−2σ)· 4  t−θ 3  i for x ∈ 4 , 1 , i ∈ [0, 1 − x) . 1− 1−x

It may be observed that the g˜j are standard hyperbolic functions of the form g (h (i)) = t−θ j h(i) ; h (i) ranges between 0 and 1. Consistent with the general form of g that was introduced in (2), they take a minimum value of t − θ at i = 0. The functions in (10) are parameterized in such a way that σ < 0 induces an impression-reinforcing pattern, with decreasingly sloped hyperbolas that result as one 1 1 3 moves from x = 2 toward either x = 4 or x = 4 ; while σ > 0 induces a mean-regressive 1 pattern, with increasingly steeply-sloped parabolas that result as one moves from x = 2 1 3 toward either extreme. (Beyond 4 and 4 , toward 0 and 1, respectively, the curves steepen progressively under both regimes.13) Moreover, one may verify that higher values of σ 1 3 result in strictly lower levels of marginal adjustment cost for consumers x ∈ 4 , 4 : for relatively indifferent consumers. Figure 4 displays adjustment maps for these functions 1 for five values of the parameter σ, with t − θ fixed at 2 . Conceptually, VoDiPS captures the relative facility with which consumers who are initially relatively indifferent between products adjust cognitively to their chosen product. The higher is the value of σ, the easier - less costly - it is for these “distant” consumers to “see” across the preference space and thereby focus opportunistically so as to align the

13See footnote 9.

15 Figure 4: How Adjustment Varies with VoDiPS, σ Adjustment curves for product 0 arc up to the left. Adjustment curves for product 1 arc up to the right. product and their preferences. (To be clear, “distance” refers to the initial separation between a consumer’s ideal preference and the Hotelling location of the product.) As a relative adjustment ease measure for distant consumers as compared to those who are close to their chosen product, σ is distinguished from θ, which measures the general ease of adjustment that is common to all consumers. The following proposition provides the key comparative static result for the eﬀect of σ on prices:

Proposition 3. Assume adjustment costs are given by (10), and ﬁx a level of θ ∈ (0, t). ∂pj Then, ∂σ < 0: Prices are lower when the VoDiPS is greater.

16 6.2 Empirical Relevance

Proposition 3 provides a basis for linking consumer and product characteristics to predicted price effects in the market. There is reason to believe that ease of “seeing” over distance in preference space is a consumer trait - which relates positively to consumers’ intelligence and domain- specific experience. Psychological studies have reported a positive relationship between openness to experience and performance on intelligence tests (Ackerman & Heggestad, 1997; Gignac et al., 2004). Additional evidence connects familiarity with increased liking for objects (Cox & Cox, 1988; Reis et al., 2011). It seems likely also that such ease of seeing is related negatively to a consumer’s cognitive load in a particular situation: Individuals who have a lot on their minds would find it harder to adjust their preferences flexibly. Predictions of lower prices would therefore follow for more intelligent, more experienced, and less cognitively-loaded consumers. There is also reason to hypothesize that markets for products and services with complicated or hard-to-evaluate arrays of attributes will exhibit low VoDiPS. When attribute- based evaluation of options is difficult, consumers tend to fall back on heuristics or summary impressions (Hutchinson & Alba, 1991; Mantel & Kardes, 1999; Dempsey & Mitchell, 2010). It then becomes critically important to one’s assessment of an option whether one has a strong initial impression or not; the individual has no other way to arrive at a strongly-felt position. On this basis, the motivated preference theory predicts higher prices for complex products. For these predicted effects to be empirically relevant, it must also be the case that they can be distinguished from other observable effects. It is not clear that they meet this criterion. The same consumer characteristics that give rise to greater VoDiPS also imply lower search and decision costs (e.g., Salop & Stiglitz, 1977) and therefore lower prices (e.g., Stigler, 1961; Butters, 1977). It is therefore not possible to distinguish the extent to which the price levels one observes in the presence of these characteristics follow from differing VoDiPS as opposed to differing search and decision costs. Similarly, complex products that would tend to be associated with lower VoDiPS would also likely engender higher search and decision costs (see, e.g., Mothersbaugh & Hawkins, 2016), which leads to predictions of higher prices on that basis. One area in which VoDiPS does seem to offer an empirically-relevant contribution is in explaining the price effects of advertising. A motivated preference conceptualization

17 of advertising sees it as a facilitator of consumer self-persuasion.14 Advertising targeted at marginal consumers would facilitate the self-persuasion of these consumers relative to inframarginal consumers; thus, according to the model, it would increase VoDiPS and result in lower prices. Advertising that is targeted at inframarginal consumers would have the opposite effect. The literature in marketing has long referenced two types of advertising: one that strengthens brand loyalty and results in higher prices; and another that attempts to gener- ate new purchases and that results in lower prices. (See, for example, Popkowski Leszczyc & Rao, 1990.) Recently, Erdem et al. (2008) have empirically distinguished these two types of advertising: They observe that advertising that decreases price sensitivity does so by increasing the willingness-to-pay of inframarginal consumers; meanwhile, advertising that increases price sensitivity increases the willingness-to-pay of marginal consumers, who thereby increase the elasticity of demand. Erdem et al.’s approach - they estimate a heterogeneous logit model of consumer demand for branded products in which consumers’ individual responses to brand advertising are allowed to vary - permits them to identify cases in which marginal and inframarginal consumers’ willingness-to-pay for brands vary with advertising differentially. Erdem et al. do not offer their own theory as to the mechanism by which advertising influences consumers’ willingness-to-pay; instead, they posit this influence as a maintained assumption about what advertising does. The contribution of the present theory is to explain both types of price effects of advertising with a single psychological primitive that is also consistent with the detailed empirical findings of Erdem et al. (2008). Further work - which would involve controlled field experiments to manipulate the targeting of advertising and estimation of its effects along the lines of Erdem et al.’s approach - could provide direct validation of the role of motivated preferences in determining prices in competitive markets.

7 Welfare

It is intuitive that the ability of consumers to improve their attitudes toward their options should have relevance for welfare. Consider a policy intervention to inﬂuence this ability, either by increasing the general ease of adjustment - θ - or by increasing VoDiPS: σ. Advertising that is focused on facilitating self-persuasion, for example, and is projected

14For an extended analysis of the role and eﬀects of advertising under motivated preferences, see Nagler (2020).

18 to all consumers equally - the mass market - would conceivably increase θ. The same sort of advertising, when targeted at marginal consumers, would be expected to increase σ, as was discussed in the previous section. Such interventions would function specifically to reduce the disutility that targeted individuals experience when they consume an option that, while perhaps their best choice, is not their ideal. Intuitively, so long as consumers positively adjust in equilibrium, this should improve total welfare in the context of a covered market of fixed size, as any change in prices in such a circumstance would entail a pure transfer of surplus between firms and consumers without the loss or gain of total surplus. The following propositions formalize results in this regard:

∂W Proposition 4. Increased ease of adjustment improves welfare - ∂θ > 0 - provided that θ > 0.

∂W Proposition 5. Increased VoDiPS improves welfare - that is, ∂σ > 0 - provided that θ > 0.

Perhaps more surprising are the effects of the different interventions on consumer welfare - and, more specifically, on the utility of different individual consumers. Let us suppose that marginal adjustment cost functions are as in (10), and fix t = 1. Addition- 1 1 F ( ) f 1 ally, fix 2 / ( 2 ) = 4 , which is approximately consistent with f distributed Beta (3, 3) as proposed in the Appendix. The first panel of Figure 5 displays the effects on the utility that is enjoyed by consumers at different locations x ∈ [0, 1] that arises from an increase 1 3 1 in θ from 2 to 4 , with σ alternately fixed at 0 and at − 4 . The second panel of the figure 1 1 shows the corresponding effects of an increase in σ from 0 to 2 , with θ fixed at 2 . While one might broadly expect an intervention that facilitates adjustment to increase consumers’ utility, this is not always what happens. As the first panel of Figure 5 illustrates, in the case of low σ, an increase in the ease of adjustment may cause all consumers net harm. The reason for this is that - consistent with Corollary 2 - a greater intensity of impression-reinforcing adjustment decreases consumer price sensitivity and so makes it easier for firms to increase prices and appropriate consumer surplus. The resultant loss of utility - borne equally by all consumers - has the potential to outstrip the benefits from eased adjustment - even for those consumers who benefit most. In the example that 1 1 involves σ = − 4 that is illustrated in the figure, consumers in the vicinity of x = 4 and

19 Figure 5: Eﬀects of Changes in θ and σ on Utility of Consumer at x

3 15 x = 4 adjust the most and so they beneﬁt the most from a general easing of adjustment. But even these consumers experience no net gain in utility. One might also expect mass-market advertising that broadly eases adjustment to beneﬁt brand loyalists more than advertising that is targeted only at marginal consumers. Figure 5 shows that the opposite may be true, however. When σ is set initially to zero, the increase in individual consumer utility that arises from an increase in the general ease of adjustment θ dwindles to zero as one approaches the extremes: x = 0 and x = 1. Meanwhile, an increase in VoDiPS in the same situation provides brand loyalists with a

15This is an artifact of the impression-reinforcing adjustment pattern, according to which adjustment curves ﬂatten as one moves from x = 1/2 toward these locations, but then steepen again as one continues toward the extremes. See footnote 9.

20 substantial increase in utility.16 The reason for this is that the value of an increase in the ease of adjustment to consumers with initially strong preferences is small, because those consumers adjust little in any event. But a decrease in price - that is engendered by an increase in σ - beneﬁts all consumers equally - independent of the strength of their initial preference.

8 Conclusion

This paper has presented a new theory of competitive markets in which costly attitude adjustment complements consumer choice. I have laid out a model whose core construct – the adjustment map – provides a general setup for analyzing how differences in the distribution of the ease of adjustment across consumers who are endowed with different initial preferences leads to different market outcomes. As a general matter, I find that when initially indifferent consumers find it relatively challenging to acclimate to their chosen product, being unable to see clearly the arguments or rationalizations that might allow post-choice self-persuasion, higher prices result for all consumers in the market. The analysis assumed a covered market. If that assumption - specifically Assumption 5 - is relaxed, a symmetric, uncovered interval emerges at the center of the unit segment that represents the product space. The effects of adjustment on price for this case require evaluation of the price elasticity of demand at the boundaries of the interval, ∗ ∗ which may be notated {x0E, 1 − x0E} - as opposed to our previous evaluation at the ∗ locus of indifference between the two products in the covered case, xE. Because these 1 boundary points are not located at 2 , but symmetrically on either side, the demand elasticity analysis is complicated by the need to account for f 0 (x) at the boundaries. 0 ∗ (f (xE) = 0 due to symmetry, which ensures that the rate of change of the consumer density drops out of key expressions in the covered market case, which simplifies the analysis.) Subject to assuming a uniform density, or else relevant bounding conditions on 0 ∗ 0 ∗ f (x0E) = −f (1 − x0E), the results of the model should carry over, in slightly modified form, to the uncovered market case. One significant change under uncovered markets is that price changes affect welfare; Propositions 4 and 5 must be altered accordingly. Governments have traditionally intervened in product markets under a range of cir- cumstances to help individuals understand the options that they face. More up-to-date

16 Still, consumers at or near x = 1/2 do substantially better, because the increase in VoDiPS selectively reduces their cost of adjustment the most.

21 interventions - “nudges”17 - have been proposed in numerous decision-making contexts to offer targeted help where behavioral research suggests that the individuals might benefit. For example, forcing individuals to opt out of contributing a minimum percentage of their income to a retirement plan rather than giving them the option to opt in can increase the chance that those individuals engage in personally-beneficial saving. My model points to a new type of motivation for the idea of nudges. The model’s findings suggest that helping boundedly rational consumers - specifically, those in a competitive market who do not have strong preferences and who find adjusting to their options to be difficult - may have market-wide effects that affect the welfare of consumers who are not themselves so challenged. To suggest, more broadly, that the public sector should get out of the business of pro- viding information to help with decision-making because the internet makes search easier than was once the case seems premature in light of my findings. Even when information search in the physical environment is easy, the cognitive obscurity of information might conceivably persist, which forestalls adjustment and then motivates market intervention on that particular basis. Future research should examine directly the role of advertising. I have discussed how advertising could increase visibility in preference space, but I have not modeled this advertising explicitly in the current framework - nor considered directly the private incentives to provide it. It is straightforward to see that firms could benefit from reducing the adjustment costs of marginal consumers, if the consumers might be induced to switch brands. On the other hand, firms might benefit from employing advertising to reduce the adjustment costs of inframarginal consumers, as such advertising in effect may decrease the elasticity of demand and increase market prices. A formal analysis of adjustment- facilitating advertising could uncover the relevant incentives and could provide guidance as to which factors might cause it to be under- or over-provided by the market.

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

17See Thaler & Sunstein (2009).

22 pointwise-symmetric adjustment map pairs for any symmetric distribution f, and (2) for an example of a non-pointwise-symmetric adjustment map pair when f is Beta distributed with 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 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 result whenever the incentive to de-stabilize such an equilibrium is mitigated by pointwise 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.

∂x∗ f(x∗ ) E ∗ Let us deﬁne the log concavity of F (·) in p0 as E ∂p0 /F (xE ) being decreasing in p0. This gives rise to the following necessary and suﬃcient condition:

∂2x∗ E ∗ 0 ∗ ∂p2 f (x ) f (x ) 0 < E − E (A.1) ∗ 2 ∗ ∗ ∂xE F (xE) f (xE) ∂p0

Using (8),

 ∗1 ∗ ∗0 ∗ −2 i (xE ) i (xE ) 2 ∗ 1 0 ∂ xE  ∂g ∂g  = − −2t + di − di · 2 ˆ ∗ ˆ ∗ ∂p0  ∂xE ∂xE  0 0

 ∗1 ∗ ∗0 ∗  i (xE ) i (xE ) 2 1 ∗ 2 0 ∗ 1 ∗1 ∗ 0 ∗0 ∗  ∂ g ∂xE ∂ g ∂xE ∂g ∂i ∂xE ∂g ∂i ∂xE   di − di + −  ˆ ∗2 ˆ ∗2 ∗ ∗  ∂xE ∂p0 ∂xE ∂p0 ∂xE ∂x ∂p0 ∂xE ∂x ∂p0  0 0

 ∗1 ∗ ∗0 ∗  i (xE ) i (xE ) ∗ 2 2 1 ∗ 2 0 ∗ 1 ∗1 ∗ 0 ∗0 ∗ ∂xE  ∂ g ∂xE ∂ g ∂xE ∂g ∂i ∂xE ∂g ∂i ∂xE  = −  di − di + −  ˆ ∗2 ˆ ∗2 ∗ ∗ ∂p0  ∂xE ∂p0 ∂xE ∂p0 ∂xE ∂x ∂p0 ∂xE ∂x ∂p0  0 0

d2gj j ∗ where ∗2 represents the second derivative of the g with respect to x, evaluated at xE. dxE

23 Hence (A.1) may be re-written

 ∗1 ∗ ∗0 ∗  i (xE ) i (xE ) ∂x∗ ∂2g1 ∂2g0 ∂g1 ∂i∗1 ∂g0 ∂i∗0 f (x∗ ) f 0 (x∗ ) − E  di − di + −  < E − E (A.2)  ∗2 ∗2 ∗ ∗  ∗ ∗ ∂p0 ˆ ∂xE ˆ ∂xE ∂xE ∂x ∂xE ∂x F (xE) f (xE) 0 0

∂x∗ where E , which is not a function of i, has been pulled out of the integrals. ∂p0

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 adjustment curve in G¯0 corresponding to a given location x ∈ (0, 1), the corresponding curve in G¯1 would be its mirror image about x.18 A subset of such pairs, for j = 0, 1 and ρ = [−1, 1], is given by   x for x ∈ 0, 1  2(x−i) 4   ρx + 1−ρ for x ∈ 1 , 1 gj = 2(x−i) 2−8i 4 2 (A.3) ρ(1−x) 1−ρ 1 3  2(1−x−i) + 2−8i for x ∈ 2 , 4   1−x 3  2(1−x−i) for x ∈ 4 , 1

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, the ∂g0 ∂g1 ∂2g0 ∂2g1 ∗0 ∗1 ∂i∗0 ∂i∗1 following conditions hold: (i) ∗ = ∗ , (ii) ∗2 = ∗2 , (iii) i = i , (iv) = , ∂xE ∂xE ∂xE ∂xE ∂x ∂x 0 ∗ and (v) f (xE) = 0. It may be veriﬁed, based on these, that the necessary and suﬃcient

condition (A.2) above for log concavity is met, whence log concavity of F (x (p0)) holds for any symmetric distribution f.

Consider now an example of a non-pointwise-symmetric adjustment map pair, given 0 x 1 1−x by g (i, x) ≡ 2(x−i) and g (i, x) ≡ 2(1−x−i) for x ∈ [0, 1]. These functions have the 0 1 1 ∗0 2t−1 property that g (0, x) = g (0, x) = 2 . Observe further that i (x, t) = 2t x is deﬁned 1 ∗ ∗1 2t−1 ∗ for t ≥ 2 , whence i < x; similarly i (x, t) = 2t (1 − x), whence i < 1 − x. We ∂i∗0 2t−1 ∂i∗1 2t−1 also have ∂x = 2t and ∂x = − 2t . We evaluate the left-hand side of (A.2) at i = 0 (i.e., the position at which the indiﬀerent consumer evaluates his decision between

18Note that pointwise-symmetry of adjustment maps, which requires curve-by-curve symmetry at each location, is not the same as symmetry across products, which requires that the maps be symmetric about 1 x = 2 .

24 Figure 6: Pointwise-Symmetric Adjustment Map Pairs Per (A.3)

product options) for all x ∈ [0, 1], using integration by parts:

∗1 ∗ ∗0 ∗ i (xE ) i (xE ) ∂2g1 ∂2g0 ∂g1 ∂i∗1 ∂g0 ∂i∗0 di − di + − ˆ ∗2 ˆ ∗2 ∗ ∗ ∂xE ∂xE ∂xE ∂x ∂xE ∂x 0 0 2t−1 2t−1 2t (1−x) 2t x i i i 2t − 1 −i 2t − 1 = di − di + − − (A.4) ˆ (1 − x − i)3 ˆ (x − i)3 2 (1 − x − i)2 2t 2 (x − i)2 2t 0 0 2t−1 (1−x) 2t−1 x 2 i 1 2t i 1 2t 4t − 4t + 1 (2x − 1) = 2 − − 2 − = 2 (1 − x − i) 2 (1 − x − i) 0 2 (x − i) 2 (x − i) 0 2x (1 − x)

0 2 0 1 2 1 where ∂g = −i ≤ 0, ∂ g = i ≥ 0, ∂g = i ≥ 0, and ∂ g = i ≥ 0. ∂x 2(x−i)2 ∂x2 (x−i)3 ∂x 2(1−x−i)2 ∂x2 (1−x−i)3 ∗ ∂xE 1 Substituting into (8) for our example functions we obtain ∂p = − 1 , whence we 0 1−ln 2t may re-write the left-hand side of (A.2) as

 ∗1 ∗ ∗0 ∗  i (xE ) i (xE ) ∂x∗ ∂2g1 ∂2g0 ∂g1 ∂i∗1 ∂g0 ∂i∗0 4t2 − 4t + 1 (2x − 1) − E  di − di + −  = (A.5)  ˆ ∗2 ˆ ∗2 ∗ ∗  1 ∂p0 ∂xE ∂xE ∂xE ∂x ∂xE ∂x 2x (1 − x) 1 − ln 2t 0 0

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

x [x (1 − x)]2 [u (1 − u)]2 du f (x) = ; F (x) = 0 1 2 ´ 1 2 0 [u (1 − u)] du 0 [u (1 − u)] du ´ ´ whereby

0 2x (1 − x) (1 − 2x) f (x) = 1 2 0 [u (1 − u)] du ´ Thus,

f 0 (x∗ ) f (x∗ ) 2x (1 − x) (1 − 2x) [x (1 − x)]2 E − E = − (A.6) f (x∗ ) F (x∗ ) 2 x 2 E E [x (1 − x)] 0 [u (1 − u)] du 2 (1 − 2x) 30 [1 ´− x]2 = − x (1 − x) x [6x2 − 15x + 10]

1 One may verify using (A.5) and (A.6) that (A.2) holds for all x ∈ (0, 1), t > 2 .

Proof of Lemma 1

Beginning with the expression gj (i∗j (x, t, θ) , x, θ) = t which implicitly deﬁnes i∗j, and totally diﬀerentiating (here, for j = 0),

∂g0 ∂g0 di∗0 = − dθ ∂i∗0 ∂θ

Using Cramer’s rule, one obtains, given gj (i, x, θ) =g ¯j (i, x) − θ,

0 ∂i∗0 ∂g −1 = − ∂θ = − > 0 ∂θ ∂g0 ∂g0 ∂i∗0 ∂i∗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 & Manceau’s (1999) Lemma 1. Sup- ∗ ∗ pose that the market is not covered, that is, at equilibrium prices (p0, p1) there exists a

26 consumer x for whom

i∗0(x,t,θ) V − p∗ − t x − i∗0 (x, t, θ) − g0 (i, x, θ) di < 0 0 ˆ 0 and i∗1(x,t,θ) V − p∗ − t 1 − x − i∗1 (x, t, θ) − g1 (i, x, θ) di < 0 1 ˆ 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

 i∗0(x ,t,θ)   0  ∗  ∗0 0  Π0 = p0 (x0) F (x0) ≡ V − t x0 − i (x0, t, θ) − g (i, x0, θ) di F (x0) (A.7)  ˆ   0 

∗ ∗ where x0 is the position of the consumer who, at prices (p0, p1), is just indiﬀerent between ∂Π buying product 0 and buying nothing. By assumption, 0/∂x0 > 0. Now note that

∗0 i (x0,t,θ) ∗0 ∗0 0 ∂p0 ∂i 0 ∗0 ∂i ∂g = −t + t − g i , x0, θ − di ∂x0 ∂x ∂x ˆ ∂x0 0 ∗0 ∗0 i (x0,t,θ) i (x0,t,θ) ∂g0 ∂g0 = −t − di < −t + di ˆ ∂x0 ˆ ∂i 0 0 ∗0 = −t + g i (x0) , x0, θ − g (0, x0, θ) = −g (0, x0, θ) < 0

∂Π ∂Π ∂p which follows from Assumption 4. Since 0/∂x0 = ( 0/∂p0)( 0/∂x0), it follows that ∂Π ∗ 0/∂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) constitutes an equilibrium.

27 Proof of Proposition 2

Diﬀerentiating (9) with respect to θ yields

∂2D f (x∗ ) 0 = − E h ∗1 ∗ 1 ∗0 ∗ 0 i2 ∂p0∂θ i (xE t,θ) ∂g i (xE t,θ) ∂g −2t + ∗ di − ∗ di 0 ∂xE 0 ∂xE " ´ ´ ∗1 ∗ 1 ∗1 ∗1 ∗ i (xE t,θ) 2 1 2 1 ∗ ∂g ∂i ∂i ∂xE ∂ g ∂ g ∂xE · ∗ + + ∗ + 2 di ∂xE ∂θ ∂x ∂θ ˆ0 ∂xE∂θ ∂x ∂θ ∗0 ∗ # 0 ∗0 ∗0 ∗ i (xE t,θ) 2 0 2 0 ∗ ∂g ∂i ∂i ∂xE ∂ g ∂ g ∂xE − ∗ + − ∗ + 2 di ∂xE ∂θ ∂x ∂θ ˆ0 ∂xE∂θ ∂x ∂θ

2 0 2 1 ∗ ∂ g ∂ g ∂xE Inspection of (2) reveals ∗ = ∗ = 0. Given symmetry, = 0. Therefore, ∂xE ∂θ ∂xE ∂θ ∂θ simplifying, ∗ ∂g0 ∂g1 ∂i∗0 ∂2D f (xE) ∂x∗ − ∂x∗ ∂θ 0 = E E h ∗1 ∗ 1 ∗0 ∗ 0 i2 ∂p0∂θ i (xE t,θ) ∂g i (xE t,θ) ∂g −2t + 0 ∂x∗ di − 0 ∂x∗ di ´ E ´ E ∗0 2 0 1 ∂i ∂ D0 ∂g ∂g ∂D0 Per Lemma 1, > 0, whence takes the sign of ∗ − ∗ . Given < 0, the ∂θ ∂p0∂θ ∂xE ∂xE ∂p0 ∂g1 ∂g0 eﬀect of ease of adjustment on price sensitivity of demand takes the sign of ∗ − ∗ . ∂xE ∂xE

Proof of Corollary 2

∗ F (x ) ∗ ∗ E F (x ) ∗ Using p0 = − ∂x∗ , and noting that symmetry makes E /f(xE ) constant in θ: f x∗ E ( E ) ∂p0

∂g0 ∂g1 ∂i∗0 ∗ 2 ∗ ∗ ∂p∗ F x ∂ x F x ∂x∗ − ∂x∗ ∂θ 0 = E E = E · E E ∗ 2 ∗ 2 2 ∂θ ∂x ∂p0∂θ ∂x i∗1(x∗ t,θ) 1 i∗0(x∗ t,θ) 0 f x∗ E f x∗ E E ∂g E ∂g E ∂p0 E ∂p0 −2t + 0 ∂x∗ di − 0 ∂x∗ di ´ E ´ E

∂g1 ∂g0 which takes the sign of − ∗ − ∗ . ∂xE ∂xE

Proof of Proposition 3

Begin by deriving adjustment productivity for g˜j for generic x:

t − θ g˜0 i∗0 (x, t) , x = t ⇔ = t (A.8) i∗0 1 − 1 2σx+(1−2σ)· 4 t − θ 1 1 ⇒ 2σx + (1 − 2σ) · = 2σx + (1 − 2σ) · − i∗0 t 4 4 θ 1 1 ⇒ i∗0 = 2σx + − σ (A.9) t 4 2

28 and by symmetry,

θ 1 1 i∗1 = 2σ (1 − x) + − σ (A.10) t 4 2 θ 1 3 = −2σx + + σ t 4 2

∗1 ∗0 2σθ whence one can obtain the expression i (x, t, θ) − i (x, t, θ) = t (1 − 2x). Using ∗ (5) and (10), one may then derive xE explicitly. Substitution of our expressions for i∗1 (x, t, θ) − i∗0 (x, t, θ) and g˜j into (5) yield

∗1 ∗ i (xE ,t,θ) ∗ t − θ p1 − p0 + (t − 2σθ) (1 − 2xE) + i di ˆ 1 − ∗ 1 0 2σ(1−xE )+(1−2σ)· 4 ∗0 ∗ i (xE ,t,θ) t − θ − di = 0 ˆ i 1 − 2σx∗ +(1−2σ)· 1 0 E 4

Integration, followed by substitution of (A.8) and (A.10), leads to

∗ p1 − p0 + (t − 2σθ) (1 − 2xE) 1 3 t − θ 1 1 t − θ + (t − θ) − + 2σx∗ − σ ln + (t − θ) 2σx∗ + − σ ln = 0 4 E 2 t E 4 2 t t − θ ⇔p − p + (t − 2σθ) (1 − 2x∗ ) − 2σ (t − θ) (1 − 2x∗ ) ln = 0 1 0 E E t

∂x∗ Simplifying, one obtains x∗ = 1 − p0−p1 , whence E = − 1 = E 2 t−θ ∂p0 t−θ 2[t−2σθ−2σ(t−θ) ln t ] 2[t−2σθ−2σ(t−θ) ln t ] ∂x∗ − E . Now substitution into the expressions provided for equilibrium price levels in ∂p1 Proposition 1 yields

F (x∗ ) t − θ ∗ E ∗ ∗ F (xE ) f(x ) p0 = − ∂x∗ = 2 t − 2σθ − 2σ (t − θ) ln / E f (x∗ ) E t E ∂p0 1 − F (x∗ ) t − θ ∗ E ∗ ∗ [1−F (xE )] f(x ) p1 = ∂x∗ = 2 t − 2σθ − 2σ (t − θ) ln / E f (x∗ ) E t E ∂p1

∂p0 t−θ ∗ ∗ ∂p1 t−θ ∗ ∗ F (xE ) f(x ) [1−F (xE )] f(x ) whence ∂σ = 2 −2θ − 2 (t − θ) ln t / E and ∂σ = 2 −2θ − 2 (t − θ) ln t / E . Both of these expressions are negative for all t > θ ≥ 0.

Proof of Proposition 4

Firms incur no costs of production, and based on Corollary 1 we know that there is a common market price. Welfare may therefore be calculated as the integral of utility over

29 the mass of consumers, plus the market price. Given that the market is evenly divided (again, Corollary 1), one may express welfare using (3) and (4) as

1  ∗0  2 i (x,t,θ)   W = V − t x − i∗0 (x, t, θ) + g0 (i, x, θ) di dx ˆ  ˆ  0  0 

1  i∗1(x,t,θ)    − t 1 − x − i∗1 (x, t, θ) + g1 (i, x, θ) di dx ˆ  ˆ  1  0  2 which can be re-written again, using symmetry,

1  ∗0  2 i (x,t,θ)   W = V − 2 t x − i∗0 (x, t, θ) + g0 (i, x, θ) di dx (A.11) ˆ  ˆ  0  0 

Diﬀerentiating,

1  ∗0  2 i (x,t,θ) ∂W  ∂i∗0 ∂g0 ∂i∗0  = −2 −t + di + g0 i∗0, x, θ dx ∂θ ˆ  ∂θ ˆ ∂θ ∂θ  0  0 

0 ∗0 ∂g0 and, recognizing that g (i , x, θ) ≡ t and ∂θ = −1,

1 ∗0 1 2 i (x,t,θ) 2 ∂W ∂g0 = −2 didx = 2 i∗0 (x, t, θ) dx ∂θ ˆ ˆ ∂θ ˆ 0 0 0 which is positive for θ > 0 (whence i∗0 > 0 ∀x ∈ (0, 1), by Lemma 1).

Proof of Proposition 5

1 3 It follows from (10) that, over the range x = 0 to x = 4 (and x = 4 to x = 1), all the terms in (A.11) are invariant to σ. Diﬀerentiation of (A.11) with respect to σ thus yields

1  ∗0  2 i (x,t,θ) ∂W  ∂i∗0 ∂g0 ∂i∗0  = −2 −t + di + g0 i∗0, x, θ dx ∂σ ˆ  ∂σ ˆ ∂σ ∂σ  1  0  4

30 whence g0 (i∗0, x, θ) = t allows us to write

1 ∗0 2 i (x,t,θ) ∂W ∂g0 = −2 didx (A.12) ∂σ ˆ ˆ ∂σ 1 0 4

0 1 1 Now using the expression for g˜ in (10), for x ∈ 4 , 2 ,

1 ∂g˜0 2i (t − θ) x − = − 4 ∂σ 1 2 2σx + (1 − 2σ) · 4 − i

Using integration by parts and the expression for i∗0 in (A.8) ,

i∗0(x,t,θ) ∗0 i∗0(x,t,θ) " 1 #i (x,t,θ) 1 ∂g0 2i (t − θ) x − −2 (t − θ) x − di di = − 4 − 4 ˆ ∂σ 2σx + (1 − 2σ) · 1 − i ˆ 2σx + (1 − 2σ) · 1 − i 0 4 0 0 4   t−θ 1 1 1 i∗0(x,t,θ) 2θ t x − 4 2σx + 4 − 2 σ 1 1 = −  − 2 (t − θ) x − ln 2σx + (1 − 2σ) · − i t−θ 1 1 4 4 0 2σx + 4 − 2 σ t 1 1 t − θ 1 t − θ = −2θ x − − 2 (t − θ) x − ln = − x − 2θ + (2t − 2θ) ln 4 4 t 4 t

Substituting back into (A.12),

1 2 ∂W t − θ 1 = 2 2θ + (2t − 2θ) ln x − dx ∂σ t ˆ 4 1 4 1 t − θ x2 x 2 = 4 θ + (t − θ) ln − t 2 4 1 4 t − θ 1 1 = 4 θ + (t − θ) ln − + t 32 16 1 t − θ = θ + (t − θ) ln 8 t which is positive for θ > 0. —————–

Acknowledgements

I am grateful to Ben Ho, Fahad Khalil, Botond Kőszegi, Jacques Lawarrée, and seminar participants at the University of Toronto, University of Washington, Vassar College, and the ISMS Marketing Science conference for helpful comments. Shay Culpepper provided excellent research assistance.

31 References

Ackerman, P. L. & Heggestad, E. D. (1997). Intelligence, personality, and interests: Evidence for overlapping traits. Psychological Bulletin, 121(2), 219–245.

Bloch, F. & Manceau, D. (1999). Persuasive advertising in Hotelling’s model of product diﬀerentiation. International Journal of Industrial Organization, 17(4), 557–574.

Butters, G. R. (1977). Equilibrium distributions of sales and advertising prices. The Review of Economic Studies, 44(3), 465–491.

Caplin, A. & Nalebuﬀ, B. (1991). Aggregation and imperfect competition: On the existence of equilibrium. Econometrica, 59(1), 25–59.

Cox, D. S. & Cox, A. D. (1988). What does familiarity breed? Complexity as a moderator of repetition eﬀects in advertisement evaluation. Journal of Consumer Research, 15(1), 111–116.

Dempsey, M. A. & Mitchell, A. A. (2010). The inﬂuence of implicit attitudes on choice when consumers are confronted with conﬂicting attribute information. Journal of Consumer Research, 37(4), 614–625.

Ehrlich, D., Guttman, I., Schönbach, P., & Mills, J. (1957). Postdecision exposure to relevant information. Journal of Abnormal and Social Psychology, 54(1), 98–102.

Erdem, T., Keane, M. P., & Sun, B. (2008). The impact of advertising on consumer price sensitivity in experience goods markets. Quantitative Marketing and Economics, 6(2), 139–176.

Festinger, L. (1962). A theory of cognitive dissonance, volume 2. Stanford, CA: Stanford University Press.

Gignac, G. E., Stough, C., & Loukomitis, S. (2004). Openness, intelligence, and self- report intelligence. Intelligence, 32(2), 133–143.

Goetzmann, W. N. & Peles, N. (1997). Cognitive dissonance and mutual fund investors. Journal of Financial Research, 20(2), 145–158.

Hotelling, H. (1929). Stability in competition. Economic Journal, 39(153), 41–57.

32 Hutchinson, J. W. & Alba, J. W. (1991). Ignoring irrelevant information: Situational determinants of consumer learning. Journal of Consumer Research, 18(3), 325–345.

Izuma, K. & Adolphs, R. (2013). Social manipulation of preference in the human brain. Neuron, 78(3), 563–573.

Izuma, K., Matsumoto, M., Murayama, K., Samejima, K., Sadato, N., & Matsumoto, K. (2010). Neural correlates of cognitive dissonance and choice-induced preference change. Proceedings of the National Academy of Sciences, 107(51), 22014–22019.

Jarcho, J. M., Berkman, E. T., & Lieberman, M. D. (2011). The neural basis of rational- ization: Cognitive dissonance reduction during decision-making. Social Cognitive and Aﬀective Neuroscience, 6(4), 460–467.

Kitayama, S., Chua, H. F., Tompson, S., & Han, S. (2013). Neural mechanisms of dissonance: An fMRI investigation of choice justiﬁcation. NeuroImage, 69, 206–212.

Kitayama, S., Snibbe, A. C., Markus, H. R., & Suzuki, T. (2004). Is there any ‘free’ choice? Self and dissonance in two cultures. Psychological Science, 15(8), 527–533.

Kitayama, S. & Tompson, S. (2015). A biosocial model of aﬀective decision making: Implications for dissonance, motivation, and culture. In J. M. Olson & M. P. Zanna (Eds.), Advances in Experimental Social Psychology, volume 52 (pp. 71–137). Burling- ton: Academic Press.

Knox, R. E. & Inkster, J. A. (1968). Postdecision dissonance at post time. Journal of Personality and Social Psychology, 8(4), 319–323.

Kőszegi, B. & Rabin, M. (2006). A model of reference-dependent preferences. Quarterly Journal of Economics, 121(4), 1133–1165.

Kőszegi, B. & Szeidl, A. (2013). A model of focusing in economic choice. The Quarterly Journal of Economics, 128(1), 53–104.

Lieberman, M. D., Ochsner, K. N., Gilbert, D. T., & Schacter, D. L. (2001). Do amnesics exhibit cognitive dissonance reduction? The role of explicit memory and attention in attitude change. Psychological Science, 12(2), 135–140.

33 Mantel, S. P. & Kardes, F. R. (1999). The role of direction of comparison, attribute- based processing, and attitude-based processing in consumer preference. Journal of Consumer Research, 25(4), 335–352.

Mothersbaugh, D. L. & Hawkins, D. I. (2016). Consumer behavior: Building marketing strategy. New York: McGraw-Hill Education, 13th edition.

Nagler, M. G. (2020). Assisted self-persuasion: A motivated preference theory of advertising. City College of New York working paper.

Popkowski Leszczyc, P. T. & Rao, R. C. (1990). An empirical analysis of national and local advertising eﬀect on price elasticity. Marketing Letters, 1(2), 149–160.

Qin, J., Kimel, S., Kitayama, S., Wang, X., Yang, X., & Han, S. (2011). How choice modiﬁes preference: Neural correlates of choice justiﬁcation. NeuroImage, 55(1), 240– 246.

Reis, H. T., Maniaci, M. R., Caprariello, P. A., Eastwick, P. W., & Finkel, E. J. (2011). Familiarity does indeed promote attraction in live interaction. Journal of Personality and Social Psychology, 101(3), 557–570.

Salop, S. & Stiglitz, J. (1977). Bargains and ripoﬀs: A model of monopolistically competitive price dispersion. The Review of Economic Studies, 44(3), 493–510.

Sharot, T., De Martino, B., & Dolan, R. J. (2009). How choice reveals and shapes expected hedonic outcome. Journal of Neuroscience, 29(12), 3760–3765.

Sharot, T., Velasquez, C. M., & Dolan, R. J. (2010). Do decisions shape preference? Evidence from blind choice. Psychological Science, 21(9), 1231–1235.

Simon, D., Krawczyk, D. C., & Holyoak, K. J. (2004). Construction of preferences by constraint satisfaction. Psychological Science, 15(5), 331–336.

Stigler, G. J. (1961). The economics of information. Journal of Political Economy, 69(3), 213–225.

Stigler, G. J. & Becker, G. S. (1977). De gustibus non est disputandum. American Economic Review, 67(2), 76–90.

34 Thaler, R. (1985). Mental accounting and consumer choice. Marketing Science, 4(3), 199–214.

Thaler, R. H. & Sunstein, C. R. (2009). Nudge: Improving decisions about health, wealth, and happiness. New Haven: Yale Univ. Press.

Tompson, S., Chua, H. F., & Kitayama, S. (2016). Connectivity between mPFC and PCC predicts post-choice attitude change: The self-referential processing hypothesis of choice justiﬁcation. Human Brain Mapping, 37(11), 3810–3820.

Van Veen, V., Krug, M. K., Schooler, J. W., & Carter, C. S. (2009). Neural activity predicts attitude change in cognitive dissonance. Nature Neuroscience, 12(11), 1469– 1474.

Wakslak, C. J. (2012). The experience of cognitive dissonance in important and trivial domains: A construal-level theory approach. Journal of Experimental Social Psychology, 48(6), 1361–1364.