New product pricing strategy under customer asymmetric anchoring

Intern. J. of Research in Marketing 28 (2011) 309–318

Contents lists available at SciVerse ScienceDirect

Intern. J. of Research in Marketing journal homepage: www.elsevier.com/locate/ijresmar

New product pricing strategy under customer asymmetric anchoring Joo Heon Park a, 1, Douglas L. MacLachlan b,⁎, Edwin Love c, 2 a b c

Dept. of Economics, 23-1 Wolgok-Dong, Sungbuk-Gu, Dongduk Women's University, Seoul 136-714, Republic of Korea Dept. of Marketing & International Business, Michael G. Foster School of Business, University of Washington, Seattle, WA 98195-3226, United States Department of Finance & Marketing, Western Washington University, Bellingham, WA 98225, United States

a r t i c l e

i n f o

Article history: First received in 20, November 2009 and was under review for 10 months Available online 12 September 2011 Area Editor: Russell S. Winer

a b s t r a c t Potential customers' willingness to pay (WTP) for a new product can be affected by their observing a posted price and this can be modeled in terms of an anchoring mechanism. A theoretical argument and mathematical proof are developed, showing that if customers use an asymmetric WTP anchoring mechanism, it will normally be optimal for firms to price higher than otherwise. Experimental evidence is provided supporting the notion that an asymmetric anchoring mechanism can be involved in purchase decisions. © 2011 Elsevier B.V. All rights reserved.

Keywords: New product pricing Willingness to pay Anchoring mechanisms Price strategy

1. Introduction Marketers often tell the story of a Native American jewelry shop that had trouble selling an allotment of turquoise jewelry to its customers, who were mainly tourists in Arizona. After trying several other promotions, the owner of the store left her assistant a note with instructions stating, “Mark down the items by 1/2.” Misunderstanding the note, the assistant instead doubled the price of each piece…and sold out (Cialdini, 1985). A critical factor in this story is that the vacationing customers who bought the assortment were unfamiliar with turquoise jewelry. If they had a well-formed idea of how much this jewelry was actually worth, then they never would have responded as they did. Because they were unfamiliar with the product, they were strongly influenced by its price. Based on their behavior, we infer that these customers must employ a rationale for their purchasing decision different from the traditional rationale in which customers make their subjective valuation of a product independently of the product's listed price. In these situations, customers are forced to make purchasing decisions regarding unfamiliar new products. In such cases, the listed price can be an important determinant of the customer's willingness-to-pay (WTP), which is defined as the maximum amount that a customer is willing to pay for a product. Just as the jewelry shop owner did not consider the influence of the price on her customers' WTP, the most

⁎ Corresponding author. Tel.: + 1 206 543 4562; fax: + 1 206 543 7472. E-mail addresses: [email protected] (J.H. Park), [email protected] (D.L. MacLachlan), [email protected] (E. Love). 1 Tel.: + 82 2 940 4435; fax: + 82 2 940 4192. 2 Tel.: + 1 360 650 4614. 0167-8116/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ijresmar.2011.04.003

common new product pricing strategies rarely consider the possibility that price setting can affect customers' WTP. For new products in which WTP plays a key price-setting role, the two main pricing strategies are skimming and penetration. Businesses that employ a skimming strategy price their products higher than the average customer's WTP. By doing so, the businesses first attract early adopters, who presumably have relatively inelastic demand. After setting the initial price at a higher level, the businesses could attract later customers with lower WTP by reducing their prices. Businesses that utilize the penetration strategy price their new products low enough that the initial offer price is less than the WTP of a large number of potential customers. Both of these strategies set prices for products by assuming that customer WTP is fixed. This assumption may be based on a wellestablished method for estimating WTP, such as conjoint modeling or the contingent valuation method (CVM). However, these methods will fail to provide an accurate estimate of customer WTP if a customer's WTP is not fixed but influenced by the listed price. We contend that customers are normally unsure about their original WTP for a new product and are likely to modify this value after observing the listed price. This claim leads us to our central contribution to the literature. Specifically, we develop and evaluate new pricing models that show how a customer's awareness of the listed price affects his or her WTP. In addition, we provide companies with a suggestion regarding their price-setting strategies for their new products. Another key feature of our approach is that we treat a customer's uncertainty regarding WTP as a probability distribution from which the customer draws a random WTP at the time of the purchase. Furthermore, the customer can consider the observed market price as one indicator of the probability distribution of the other customers' WTP.

310

J.H. Park et al. / Intern. J. of Research in Marketing 28 (2011) 309–318

Our proposed models assume that a customer's WTP is semiendogenous in that the customer's original WTP is formed independently of the price but is influenced and changed by the price at the final purchasing stage. More specifically, in our model, customers do not simply compare the original WTP with the listed price but also adjust their original WTP to accommodate the additional information presented by the observed price. Then, they compare the adjusted WTP with the price in the final purchasing decision. Our ideas are similar to those of Wathieu and Bertini's (2007) study, where the listed price is not simply compared with the WTP but also stimulates the customers to reconsider their WTP. However, our study utilizes a different assumption with regard to whether the WTP is affected by the observed price. Our model assumes that the listed price affects the customers' WTP while they reconsider their decision, whereas in their model, the WTP remains unaffected by the observed price at this point. We postulate that customers go through one of two adjustment mechanisms that yield different purchasing decisions in which their WTPs are presumed to be modified by the listed price. Whereas in the first adjustment mechanism (i.e., the One-step Anchoring Mechanism (AM1)), the customers adjust their WTP symmetrically in both upward and downward directions; in the second mechanism (i.e., the Two-step Anchoring Mechanism (AM2)), the customers only adjust their WTP in an upward direction. Under the AM2, companies can justifiably set prices higher than they would if the WTP were determined exogenously (i.e., without considering the reaction of the customers to the listed market price). According to the AM2, if customers see a listed price that is lower than their WTP, then they automatically purchase the product. However, if the listed price is higher than their WTP, then they modify their WTP instead of simply forgoing the purchase, as suggested by the conventional purchasing mechanism. Essentially, we argue that a higher-than-WTP price provides a customer with an opportunity to revalue his or her WTP such that the new WTP reflects a higher level of quality or scarcity, which the customer would have missed at the first evaluation stage because of risk aversion but which the other customers had recognized. 3 Based on this new information, the customer draws a new WTP from the revised WTP probability distribution. This distribution is asymmetrically modified, which means that the ex post distribution is shifted upward, increasing the probability that the new WTP drawn from this distribution will be higher than the listed price. The remainder of the paper is organized as follows. Section two reviews the relevant literature, section three introduces our theoretical model, section four provides some empirical support for our model, section five describes the managerial implications, and section six provides our conclusions, the practical implications, the limitations of the paper, and our proposals for future research. 2. Related literature In this section, we review prior studies on the uncertainty surrounding customers' willingness to pay, the relevance of reference pricing, and the psychological process of anchoring and adjustment. We compare this paper with the prior research using price as a stimulus to reconsider the WTP. Each area supports our argument that customer WTP is dynamic and influenced by external stimuli, such as the listed price. Our models assume that the customer WTP is drawn from a probability distribution, which reflects a degree of uncertainty regarding the WTP. Closely related to uncertainty in WTP is the concept of reference pricing. Customers utilize reference prices to form their WTP 3 Prices that are both higher and lower than a customer's WTP are indications of the other customers' evaluations. However, we claim that the customer disregards lower prices but responds to higher prices because of risk aversion.

because these prices help the customers develop the shape and the locations of their WTP distributions. In addition, reference prices are relevant to our understanding of the factors that may cause customers to update their WTP to accommodate the new information. Our review of the psychological process of anchoring and adjustment cites the findings of prior studies in which pricing stimuli had shifted WTP. These studies provide support for our mechanism. Finally, we compare our approach with Wathieu and Bertini's (2007) work on price as a stimulus for reconsidering the WTP. 2.1. Uncertainty of willingness to pay (WTP) In traditional economic theory, a customer's WTP for a product can be interpreted as the money amount by which he or she subjectively evaluates the incremental utility added from consuming the product. By definition, if a listed price is below a customer's WTP, then the consumer purchases the product; otherwise, he or she will either forego the purchase or seek a more affordable alternative, such as a private label brand in the case of consumer packaged goods (Sinha & Batra, 1999). However, in reality, customers rarely know for certain how much incremental utility they can expect from consuming a product at the time of the purchase. Customers can only determine this utility at the consumption stage, and consumption usually does not coincide with purchase. Customers generally do not know how to value a new product, especially one that they have never purchased or used. Additionally, assuming that customers' price knowledge varies depending on macroeconomic factors (Estelami, Lehmann & Holden, 2001), we can infer that customers are unlikely to have a single, immutable and accurate valuation for a new product, even in the most optimal circumstances. At best, they may hold a probability distribution of the WTP from which they randomly draw their WTP for their purchasing decisions. Cameron and Quiggin (1994) advanced a similar argument regarding consumer behavior when they surveyed their subjects on their WTP for an environmental resource. 2.2. Reference pricing Prior research on reference pricing has advanced our understanding of customer WTP. A substantial body of research has shown that customers compare a store's price of a good to their reference price for that good (c.f., Degeratu, Rangaswamy & Wu, 2000; Della Bitta & Monroe, 1974; Klapper, Ebling, & Temme, 2005; Thaler, 1985; Urbany, Bearden, & Weilbaker, 1988). This reference price is based on a variety of factors, such as previously observed prices and the store's environment. Reference price is closely related to WTP in that a relatively high reference price for a good will generally result in a high WTP. In the context of new products, reference prices are particularly influential in purchasing decisions because customers will be uncertain of the actual value of the good. Reference prices are typically generated by the prices of analogous products or products that solve similar problems, but these prices only provide guidelines to customers who exhibit considerable uncertainty about their WTP. Previous marketing studies have shown that reference prices influence the customers' evaluations of price attractiveness (Niedrich, Sharma, & Wedell, 2001). Many researchers have suggested that certain stimuli, such as the listed price, can change the customers' WTP by influencing their reference prices. Although some reference pricing models assume that a reference price is the weighted average of the past prices of related goods (Briesch, Krishnamurthi, Mazumdar, & Raj, 1997; Kalyanaram & Winer, 1995; Klapper, et al., 2005), other models show that, because consumers have poor memories of past prices, consumers tend to form reference prices based on the prices they observe at the time of the purchase (Briesch et al., 1997; Hardie, Johnson, & Fader, 1993; Rajendran & Tellis, 1994).

J.H. Park et al. / Intern. J. of Research in Marketing 28 (2011) 309–318

311

2.3. Anchoring and adjustment

3. Theory

Prior empirical research has supported the proposition that customers anchor their purchasing behavior to certain kinds of market information, such as the listed price. Simonson and Drolet (2004) asked their respondents to express their WTP for four products (i.e., toasters, cordless phones, backpacks, and headphone radios), some of which were associated with randomly generated prices ranging from $0 to $100. The researchers reported that the median WTP of the respondents who are given prices lower than $50 is $13 less than the WTP of those who are given prices higher than $50. Similar results have been found in experiments employing the contingent valuation method (CVM), which surveys the respondents' WTP (Frykblom & Shogren, 2000; Green, Jacowitz, Kahneman, & McFadden, 1998). Boyle, MacDonald, Cheng, and McCollum (1998), Holmes and Kramer (1995), Monroe (2003), and Thaler (1985) have all argued that WTP is likely to be influenced not only by a person's subjective valuation but also by relevant market information, such as the market price of the product. A customer's subjective (i.e., internal) valuation (i.e., WTP) will shift in relation to the purchase context (Kamins, Dreze, & Folkes, 2004). In this sense, both the internal valuation and the purchase context affect the WTP. However, the influence of each factor depends on the product to be purchased. According to Kalwani, Yim, Rinne, and Sugita (1990), a customer's past experience and present context help form what they refer to as the customer's adaptation level. These researchers argue that an internal valuation will have less influence and purchase context will have greater influence on the decision to buy rarely purchased items. We doubt that customers symmetrically anchor on both high and low prices. Instead, many experimental studies have shown evidence of asymmetrical anchoring effects. For example, customers have been shown to react more strongly to prices above their reference level than to prices below it. This behavior is likely due to loss aversion (Kalyanaram & Winer, 1995). Comparing the deviations of the observed price from the reference price, Kalwani et al. (1990) also showed that the impacts of the positive deviations are asymmetrical to those of the negative deviations.

In this section, we propose a theory that describes how customers might adjust their WTP and arrive at their purchasing decision when given a posted price. In Section 2.1, we proposed that customers have distributions of possible WTP values in mind. However, we emphasize that customers are also unsure about the nature of these distributions because they know that these distributions reflect only their own ill-formed subjective valuations. If customers subjectively evaluate the product, then they presumably use as much of the available information as possible. However, this information is still incomplete, which means that they must be ready to revise their evaluations as new information becomes available. We propose that a customer wants to know how other customers value the product to complement the incompleteness of his or her information. If customers become aware of the market price, then they may revise their WTP to incorporate that new information. Our approach makes the following key assumptions: (1) customers are homogeneous in all respects except in their subjective evaluations of the product, and (2) each customer believes that the firm has researched the customers' valuations and set the price at the mean value 4 of the customers' WTP such that the other customers' WTP are distributed around the listed price. We will call the WTP drawn from a customer's own internal distribution the subjective WTP and the WTP drawn from the distribution around the listed price the market WTP. Each customer's subjective WTP is weakly held because it is based on his or her own ill-formed subjective valuation. Thus, one may expect a process of anchoring and adjustment to occur. However, if the product is well established, then an anchoring process cannot be expected because, unlike in the case of a totally new product, the customers are familiar with the product and confident in their valuations. To summarize the anchoring model proposed in this paper, we hypothesize that customers integrate their subjective WTP with the market WTP to determine their final WTP from which they make their actual purchasing decision. This final WTP is also called the anchored WTP. Next, we describe the traditional mechanism used to predict consumer purchasing behavior. We subsequently propose two alternative anchoring mechanisms for purchasing decisions by making different assumptions about how each customer integrates his or her subjective WTP with the market WTP.

2.4. Listed price as a stimulus to reconsider the WTP Our mechanism is related to the ideas proposed by Wathieu and Bertini (2007). They assumed that a product's price plays two roles: an incentive to buy and a stimulus to reconsider the purchasing intention. The study assumes that customers decide whether to reconsider their WTP by comparing their initial WTP with the price. If they observe a difference in the range between the initial WTP and the observed price that is smaller than a pre-assigned threshold, then they will reconsider their WTP. Otherwise, they decide whether to purchase by following a conventional rule; they purchase if the WTP is greater than the price but do not purchase otherwise. The proximity of the price to the initial WTP increases the incentive to reconsider the WTP. Wathieu and Bertini argue that if the listed price is far from the WTP, the purchase will be a “no-brainer” (i.e., the customers will either purchase or reject the product without thinking further). The Wathieu and Bertini model analyzes cases in which a new benefit is added to a familiar product. Because of their familiarity with the product, the customers have prior beliefs about the product's value. Wathieu and Bertini model the change in price as a stimulus to think about the value of the improved product. Their model includes a “cost of thinking” parameter, but unlike our model, their model does not include a weighting factor that accounts for how much the initial WTP should be modified by the listed price. Unlike our mechanisms, their model presumes that such a weighting factor is unnecessary because it assumes that the customers' WTP is not affected by the observed price during the rethinking stage.

3.1. Conventional mechanism As a basis for comparison, we define the conventional mechanism (CM) (see Fig. 1.a). In the conventional mechanism, a customer makes his or her final purchasing decision simply by comparing his or her initial (i.e., subjective) WTP with the listed price, as shown by the following: CM (conventional mechanism) D0 ¼ 1 if yS ≥ p D0 ¼ 0 otherwise yS e dF S ð:Þ

ð1Þ

where D0 indicates whether the customer buys the product (i.e., the customer buys if D0 = 1 but does not buy if D0 = 0). In addition, p is the listed price, subscript 0 denotes the conventional mechanism and y S denotes the consumer's subjective WTP, which is drawn from the distribution F S(∙). Note that the degree of uncertainty is implicitly 4 We assume that customers believe firms set prices by estimating the customers' WTP distribution. The listed price is the only information that the customers can observe and believe to be related to other customer' evaluations. We only assume that the listed price lies at the mean of the other customers’ WTP for convenience's sake. In other words, this assumption is not essential to our study.

312

J.H. Park et al. / Intern. J. of Research in Marketing 28 (2011) 309–318

a) CM

Note that the weight w can be interpreted as a measurement indicating the degree of anchoring, which increases as w → 1. As w increases, more weight is given to the market WTP, and the subjective WTP becomes less important. Given the proposed anchoring process, a customer can use two possible mechanisms to determine whether to buy the product.

Question: S

>

?

yes

no

Decide to purchase

Decide not to purchase

b) AM1

3.2.1. One-step Anchoring Mechanism (AM1) First, we propose an anchoring mechanism in which a representative customer directly compares his or her anchored WTP with the price. This mechanism is called the One-step Anchoring Mechanism (i.e., AM1) (see Fig. 1.b). We can model the AM1 as follows: AM1 (One-step Anchoring Mechanism)

Question: A

>

?

yes

no

Decide to purchase

D1 ¼ 1 if yA ¼ wyM þ ð1−wÞyS ≥ p D1 ¼ 0 otherwise yS e dF S ð⋅Þ; yM e dF M ð⋅Þ

Decide not to purchase

where subscript 1 denotes the AM1. In the AM1, a customer never solely depends on the subjective WTP. In this version of the model, the customer uses the subjective WTP only in conjunction with the market WTP to set the anchored WTP. In addition, the anchored WTP is a random variable that is distributed between the market WTP distribution and the subjective WTP distributions because 0 b w b 1. Although purchasing behavior cannot be deterministically predicted with accuracy, it may be predicted stochastically. In our model, we only consider the purchasing probabilities Pr[D1 = 0] or Pr[D1 = 1]. Therefore, there is some probability that a customer will purchase the product if he or she has an anchored WTP distribution partially above the listed price.

c) AM2 Question: S

>

1st Step

? no Question: A

yes

>

yes Decide to purchase

2nd Step

? no Decide not to purchase

Fig. 1. Flow charts of proposed mechanisms.

reflected by the variance of the distribution F S(∙). A larger variance means that a customer is less certain of his or her subjective WTP. The variance of F S(∙) will depend partly on the novelty of the product and the customer's experiences with similar or substitute products. 3.2. Anchoring mechanisms We assume that each customer believes that other customers' WTPs are distributed around the listed price. The customer becomes aware of how differently other customers evaluate the product by seeing the difference between his or her own subjective WTP and the listed price, which reflects the other customers' WTPs. As a result, the customer adjusts his or her subjective WTP toward the listed price before making the final decision. In this paper, we assume that the anchoring mechanism is a weighted average of the market WTP and the subjective WTP. The anchored WTP is defined in Eq. (2). yA ¼ wyM þ ð1−wÞyS yS e dF S ð⋅Þ; yM e dF M ð⋅Þ

ð3Þ

ð2Þ

where y A denotes the anchored WTP, and w is the weight. In addition, y M denotes the market WTP that is randomly drawn from the distribution F M (·), which we assume is symmetric around the listed price. 5

5 A symmetric distribution is not essential. We may use any kind of distribution for FS(∙) and FM(∙). However, in Section 4, we assume that these distributions are normal for convenience's sake.

3.2.2. Two-step Anchoring Mechanism (AM2) As noted in Section 2.3, the anchoring process is likely to be asymmetric. After observing the listed price, a customer tends to increase his or her anchored WTP rather than lowering it. Keeping the possibility of asymmetric anchoring in mind, we argue that customers make purchasing decisions in two steps (see Fig. 1.c). In the first step, customers use the CM to decide whether they buy the product or reevaluate their WTP. Note that this preliminary decision is tentative in that it could be changed in the next step. If the subjective WTP exceeds the price, then we expect the customers to buy the product. However, if the subjective WTP is lower than the listed price, then we assume that the customer reconsiders his or her decision (Wathieu & Bertini, 2007). Unlike the conventional mechanism, our model suggests that the customer checks whether the product generates any unperceived benefits instead of making an instant decision to forego the purchase. The customer further evaluates the benefits of the product by considering the additional information presented by the listed price. In other words, the customer moves to the second step, where he or she makes the final purchasing decision, by comparing the listed price with his or her reevaluated (i.e., anchored) WTP, which is drawn from the distribution of the anchored WTP. The most salient feature of the AM2 is that it operates under the assumption that the customers exhibit asymmetrical behavior with regard to the low and high prices. If a listed price is lower than the subjective WTP, then a customer immediately decides to buy the product. Otherwise, the customer reconsiders whether his or her WTP is actually lower than the listed price. This assumption of asymmetrical behavior is defensible. Customers naturally expect a surplus from a purchase, and risk-averse customers tend to conservatively evaluate the benefits of a new product. Whereas purchasing a product can provide either a surplus or a loss, not purchasing a product provides neither. As a result, customers seeking a surplus will search for more information rather than simply foregoing the purchasing opportunity.

J.H. Park et al. / Intern. J. of Research in Marketing 28 (2011) 309–318

While evaluating the benefits involved in purchasing a new product, a customer may commit two types of errors: unperceived benefits, where the customer fails to perceive certain benefits that are yielded by the product, and misperceived benefits, where the customer perceives benefits that are not actually yielded by the product. A riskaverse customer rarely commits the misperceived benefits error because he or she is wary of conferring benefits upon products that are not apparent at the time of purchase. A risk-averse customer is rather apt to commit unperceived benefits errors. Hence, based on the perception of the expected benefit, a risk-averse customer's subjective WTP must be lower than the WTP that reflects all of the actual benefits that the customer feels at the point of consumption. Therefore, if the subjective WTP exceeds the listed price, then the customers will purchase the product because they are sure that the real WTP is greater than the price. However, in the opposite case, the customer will be reluctant to forego the purchasing opportunity because there is a chance that the real WTP is actually greater than the price. In this latter case, the customers will reconsider their own subjective WTP. We refer to this process as a two-step anchoring mechanism, which is modeled as follows: Two-step Anchoring Mechanism (AM2) D2 ¼ 1 if yS1 ≥ p or yA ¼ wyM þ ð1−wÞyS2 ≥ p D2 ¼ 0 otherwise

ð4Þ

where y1S and y2S indicate the subjective WTP drawn at the first step and at the second step, respectively. Because the subjective WTP (y2S) used to calculate the anchored WTP in the second step is drawn independently of the initial WTP (y1S), the anchored WTP is statistically independent of the initial WTP, even though y1S and y2S are drawn from the same distribution. 6 In sum, in the first step of the AM2, customers compare the listed price (p) with their initial subjective WTP (y1S) to decide whether to instantly buy the product. If they do not buy the product, then they move on to the second step, where they make their final purchasing decision by comparing the listed price with the anchored WTP (y A), which is formed by integrating the second subjective WTP (y2S) with the market WTP. 4. Empirical evidence in support of anchoring mechanisms Although our theory can be applied to any product in which a potential customer has incomplete information, our empirical illustration employs a product that is not yet available on the market. A totally new product is a non-market good, and by definition, its market information is not available because it has not yet been traded on the market. The WTP for a non-market good should be estimated in a different manner from the WTP for a market good for which market information is available. We use the contingent valuation method (CVM) to estimate the WTP for a non-market good. Although mainly developed and used in the economics field (c.f., Boyle et al., 1996; Cameron & Quiggin, 1994; Green et al., 1998), the CVM is becoming more common in the marketing literature as well (e.g., Park and MacLachlan (2008)). Many CVM studies have shown that customers would utilize prices as anchoring points. With the exception of Boyle et al. (1996), contingent valuation studies have consistently obtained significantly higher WTP estimates by using the dichotomous choice CVM rather than by using the open-ended CVM. 7 Through a CVM experiment, Green et al. (1998) found that the distribution of affirmative responses to 6 We can relax the assumption of independence, but doing so will add considerable complexity to our model and distract from the main purpose of the study. 7 In a closed-ended CVM, the respondents are given a bid price and are required to answer either yes or no to the question: “Would they buy at that price?” However, the respondents in an open-ended CVM are asked to directly state their WTP without being offered a bid price.

313

dichotomous questions is systematically higher than those to openended questions concerning the same underlying value. O'Conor, Johannesson and Johansson (1999) also found similar and more specific evidence suggesting that dichotomous questions result in an estimated mean approximately twice as high as that of the open-ended questions. However, an anchoring effect cannot exist in an open-ended CVM because a bid price is not given to the respondents in an open-ended CVM. Note that the bid price can be treated as the market-listed price. In contrast, an anchoring effect can exist in a close-ended CVM because each respondent is asked to provide a dichotomous answer (i.e., yes if they buy a product and no if they do not) for each given bid price. If no anchoring process exists, then both of the WTP estimates obtained from the open- and close-ended CVMs will be statistically indistinguishable from one another. We conducted a CVM study to shed light on our proposed mechanisms. The CVM is not incentive-compatible and can cause a bias because it uses survey data that are collected by asking respondents to value a non-market good in a hypothetical situation for which no actual transaction is required. Incentive-compatible methods, such as Vickrey auctions, have been proposed to reduce such bias. However, the Vickrey auction is inappropriate for analyzing anchoring effects because it is an open-ended process that provides no prices. More importantly, the potential bias of the CVM is not a problem here because we are not interested in the level of the WTP but rather in the differences between the WTPs estimated by the different CVM formats. 8 4.1. Data We conducted a CVM survey to obtain information about the customers' WTP for a new cell-phone-related service to be provided by a mobile telecommunication company. Six trained interviewers conducted the survey in Seoul, Korea and its suburbs for two weeks during the spring of 2008. The respondents were asked about their WTP for the new service in two different formats: an open-ended format and a close-ended format. We divided the potential respondents into nine groups. The subjects in the first group were given the open-ended question, which asked how much they were willing to pay for the service. The subjects in the other eight groups were asked close-ended questions in which a different price that was randomly selected from eight predetermined bid prices was given to each group. Then, the groups were asked whether they were willing to pay the given price for the service. In this case, the new product is a customized service in which a mobile company determines the best rate plan for each customer by analyzing the data provided by the customers. The company calculates the amounts due for three months under all rate plans available, finds the lowest amount due, and informs the customer about the associated rate plan. A trained interviewer verbally described this new service to each respondent for no longer than 3 min in a careful manner. The respondents were allowed to ask questions about the new service before answering the WTP questions. Because the service was completely new and had never been offered before, the customers were uncertain how much benefit they could obtain from the service. Thus, this situation was similar to those in which our proposed anchoring mechanisms are likely to operate. We assigned each interviewer 100 interviews with randomly selected potential customers. Of the 600 individuals contacted, we succeeded in acquiring information from 562 respondents, for a response rate of 93.7%. Deleting the surveys in which no answer was given for 8 Scholars have often reported that the WTP tends to be overestimated by the CVM and have called this effect the exaggeration bias (Park & MacLachlan, 2008). This bias should be considered if the level of WTP is of interest, but the bias is inconsequential in this study.

314

J.H. Park et al. / Intern. J. of Research in Marketing 28 (2011) 309–318

Table 1 Number of samples in different formats. Open-ended Number of respondents

Close-ended 167

Bid price No. of respondents Bid Price No. of respondents

2000 45 6000 32

3000 41 7000 27

4000 40 8000 40

5000 40 9000 44

one of the WTP questions, we obtained 476 responses from which we estimated the customers' probabilities of purchasing the products. Table 1 shows the number of respondents in each group.

For each mechanism, the purchase probabilities can be concretely calculated by assuming particular distributions for the subjective WTP and the market WTP. For example, if we assume the following normal distributions 9:

H1. Pr[y ≥ p|open − ended] = Pr[y ≥ p|close − ended] for all p H2. Pr[y ≥ p|open − ended] ≥ Pr[y ≥ p|close − ended] for p ≤ μ

ð6Þ

Then, we can compute the purchase probabilities associated with each mechanism as follows: p−μ  h i S CM : Pr½D0 ¼ 1 ¼ Pr yi ≥ p ¼ 1−Φ 0σ 1 h i B ð1−wÞðp−μ Þ C A AM1 : Pr½D1 ¼ 1 ¼ Pr yi ≥ p ¼ 1−Φ@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA σ w2 þ ð1−wÞ2 h i h i S A AM2 : Pr½D2 ¼ 1 ¼ 1−Pr yi b p Pr yi b p 0 1 p−μ  B ð1−wÞðp−μ Þ C Φ@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA ¼ 1−Φ σ σ w2 þ ð1−wÞ2

Proof. See Appendix A. Proposition 1 suggests the following hypotheses:

ð5Þ

where μ is the mean of the subjective WTP and p is the price set by the seller, then we can obtain a normal distribution of yiA, as shown by the following:  n o A 2 2 2 yi e N wp þ ð1−wÞμ; σ w þ ð1−wÞ :

Proposition 1. Pr½D1 ¼ 1 b Pr½D0 ¼ 1 b Pr½D2 ¼ 1 for p b μ Pr½D0 ¼ 1 b Pr½D1 ¼ 1 b Pr½D2 ¼ 1 for p N μ

4.2. Hypotheses

    S 2 M 2 yi e N μ; σ ; yi e N p; σ

is closer to the observed price. If the price is lower than the subjective WTP, then the adjusted WTP should be lower than the subjective WTP. This downward adjustment makes the purchase probability lower with the AM1 than with the CM, which is based on the subjective WTP. By the same reasoning, the AM1 yields a higher purchase probability than that suggested by the CM over the region in which the observed price is higher than the mean of the subjective WTP. Proposition 1 summarizes the relationships between the purchase probabilities under the assumptions in Eq. (5), and Fig. 2 plots these relationships while assuming that μ = 10, σ = 3, and w = 0.5.

ð7Þ

We observe that the purchase probabilities produced by the twostep anchoring mechanism are higher than those of the conventional and one-step anchoring mechanisms over all of the suggested price levels. Previously, we mentioned that at the first step of AM2, where the decision is actually based on the CM, a decision to buy the product is once and for all, whereas a decision to not buy the product is reconsidered and might be changed at the second step. Based on these assumptions, we can clearly see why our mechanisms generate the abovementioned results. In addition, we find that the AM1 yields a purchase probability that is lower than that suggested by the CM over the region in which the observed price is less than the mean of the subjective WTP. Because the distribution of the anchored WTP shifts to the left and shrinks around its mean (i.e., a decrease in variance occurs), where the price is lower than the mean of the subjective WTP, the probability that the anchored WTP exceeds the price should be less than that of the subjective WTP. To support this claim, we note that the AM1 is a symmetric anchoring mechanism in which consumers adjust their subjective WTP symmetrically for both the high and the low price. Relative to the original subjective WTP, the adjusted WTP 9 We assume that both yS and yM have the same variance σ2 for exposition's sake. Assuming different variances for yS and yM will make the analysis more complicated without changing the core implications.

H3. Pr[y ≥ p|open − ended] ≤ Pr[y ≥ p|close − ended] for p ≥ μwhere y is the WTP that the customers compare with the price to make their final purchasing decisions. Additionally, Pr[y ≥ p] is the purchase probability when the price is set at p. Because a bid price that plays an anchoring role is not given in the open-ended CVM but given in the close-ended CVM, Pr[y ≥ p|openended] can be regarded as the purchase probability under the CM without any anchoring effect, whereas Pr[y ≥ p|close-ended] can be regarded as the purchase probability under the AM1 or AM2 that includes an anchoring effect. If H1 is not rejected, then the consumer's WTP will not be influenced by the listed market price and an anchoring mechanism (e.g., AM1 and AM2) cannot explain the customer's purchasing behavior. However, if H1 is rejected, then the customer's purchasing behavior is affected by the price that is determined through an anchoring mechanism. Thus, customers will form their final anchored WTP by utilizing an anchored mechanism (e.g., AM1 or AM2) that integrates the observed price and their initial subjective WTP. Focusing only on the AM1 and the AM2, we can determine which mechanism will be appropriate by testing H2 and H3 on the condition that H1 is rejected. We may think of two cases in which either AM1 or AM2 will be used. If neither H2 nor H3 is rejected, then the AM1 will be considered a more appropriate mechanism than the AM2 because the AM1 implies that the purchase probabilities are lower than those generated by the

Fig. 2. Purchase probability functions of various mechanisms.

J.H. Park et al. / Intern. J. of Research in Marketing 28 (2011) 309–318 Table 2 Estimated purchasing probabilities for bid prices.

2000 3000 4000 5000 6000 7000 8000 9000

Won Won Won Won Won Won Won Won

Open-ended

Close-ended

z-value

0.53 0.47 0.35 0.35 0.20 0.20 0.20 0.17

0.62 0.66 0.55 0.43 0.31 0.41 0.28 0.36

1.0690 2.1288⁎ 2.2896⁎ 0.9182 1.3600 2.3280⁎ 1.0747 2.8467⁎

⁎ Statistically significant (significance level: 0.05; critical value: 1.96).

315

ended data. Note that the mean WTP collected from the open-ended question is 3582 Won ( μˆ ¼ 3; 582). The purchase probability generated with the open-ended data is not greater than that derived from the close-ended data at the 3000 Won bid price, which is less than the mean value of WTP. In contrast, the purchase probabilities derived from the close-ended data are significantly higher than those derived from the open-ended data at the three bid prices that were set higher than the mean (i.e., 4000, 6000, and 9000 Won). From these findings, we conclude that H2 is rejected but that H3 is not rejected. Thus, in this case, we infer that the respondents are more likely to use the AM2 than either the CM or the AM1 for purchasing decisions. 5. Managerial implications

CM for p ≤ μ (H2) and higher than those generated by the CM for p ≥ μ (H3), whereas the AM2 implies that the purchase probabilities are higher than those generated by the CM for all of the price levels. These results are discussed in proposition 1. The AM2 will be a more appropriate conclusion than the AM1 if H2 is rejected and if H3 is not rejected. 4.3. Test results The estimated purchase probabilities for the bid prices that are predetermined with the open-ended and close-ended data are presented in Table 2. The probability estimates with the open-ended data are presented in the second column of Table 2 and are calculated by the following equation: h i no   o ˆ y ≥ pj jopen − ended ¼ 1 ∑ 1 yi ≥pj Pˆ j ≡ Pr o n i¼1

ð8Þ

o where Pˆ j is the probability of WTP exceeding the jth bid price (pj), which is estimated by using the open-ended data; n o is the number of open-ended samples; 1[.] is the indicator function that takes a value of one if the expression inside of the bracket is correct and a value of zero otherwise; and yi is the WTP of the ith respondent in the open-ended sample. The third column of Table 2 shows the probability estimates derived from the close-ended data. These estimates are obtained in a similar way, as shown by the following equation: c

nj   c ˆ y ≥ pj jclose−ended ¼ 1 ∑ Iij Pˆ j ≡ Pr ncj i¼1

ð9Þ

c where Pˆ j is the probability that WTP exceeds the jth bid price (pj), which is derived from the close-ended data; njc is the number of the jth close-ended sample group to which the jth bid price pj was given; Iij is the observed binary response that takes a value of one if the ith respondent in the jth group answers “Yes, I will buy it at the bid price pj” and zero otherwise. Both of the probabilities estimated by Eqs. (8) and (9) represent the proportions of respondents who indicated that their WTP is larger than the bid price. If the proportion in a close-ended sample is the same as that in the open-ended sample with the same bid price, then we cannot claim that there is evidence of an anchoring effect. However, if the proportions are significantly different, then we have evidence supporting the existence of an anchoring effect. We test the significance of the observed difference with a two-proportion ztest. We conclude that the purchase probabilities are not the same for most of the bid prices. If the bid prices are 3000 Won, 4000 Won, 6000 Won and 9000 Won, then we reject H1, which states that the purchase probabilities under both the open- and close-ended formats are the same. The purchase probabilities estimated with the close-ended data are all consistently higher than those estimated with the open-

We next consider a firm's pricing problem while assuming that its customers' purchasing decisions are determined by an anchoring mechanism. A firm usually expects its customers' WTP to be predetermined while pricing a product. Believing that its customers have a stable WTP distribution unaffected by the listed market price, the firm selects a point from the anticipated WTP distribution as the price that will maximize its expected profit. However, if the customers shift their WTP according to the listed price and anchor their WTP toward this price, then the firm should consider the changes in its customers' WTP while setting its price. 5.1. Expected profit maximizing pricing In this section, we describe the pricing conditions that enable a firm to maximize its expected profit under the purchase decision mechanisms proposed in Section 3. A firm confronting a customer's randomly drawn WTP tries to maximize its expected profit that is defined as in Eq. (10). E½π ¼ ðp−cÞE½I ðy ≥pÞ ¼ ðp−cÞf1−GðpÞg

ð10Þ

where y⁎ denotes the WTP that is the basis for the final purchasing decision; G(∙) is a cumulative distribution function (cdf) of the WTP; and p and c denote the price and a constant marginal cost, 10 respectively. To maximize its expected profit, the firm must set its price at the level in which the following first-order condition is satisfied:        

dE½π ¼ −g p p −c þ 1−G p ¼0 dp

⇕

  1 g ðp Þ ¼ ¼λ p  p −c 1−Gðp Þ

ð11Þ

where g(·) and λ(·) are the probability density function (pdf) and the hazard function of the WTP, respectively. Eq. (11) tells us how the optimal price is defined, but the equation does not represent an explicit solution for determining the optimal price because the equation cannot be expressed as a closed-form solution. Furthermore, an analytical solution to determining the optimal price is still unknown. Nevertheless, we can determine how the optimal price is affected by analyzing the behavior of the hazard function λ(p). The solution in Eq. (11) depends on G(·), the distribution of the final WTP. If we employ the same distributional assumptions for the subjective WTP and the market WTP as we did in Eq. (5), then the

10

A constant marginal cost is assumed here for the sake of simplicity.

316

J.H. Park et al. / Intern. J. of Research in Marketing 28 (2011) 309–318

pricing rule (11) takes on the more concrete forms presented in Eq. (12) in accordance with the hypothesized decision mechanisms. CM :

  1 ϕ p0 −μ 1 ¼ σ pσ0 −μ  ≡ λ0 ðp0 Þ p0 −c 1−Φ σ

 p1 −μ p1ffiffiffiffiffiffiffiffi φ p ffiffiffiffiffiffiffiffi 2 2 1 σ 1þθ σ 1þθ  ≡ λ1 ðp1 Þ ¼ AM1 : p1 −c p1 −μ 1−Φ pffiffiffiffiffiffiffiffi2 σ

ð12aÞ

ð12bÞ

1þθ

1 ¼ p2 −c ! ! p −μ  1 p2 −μ  p2 −μ 1 p2 −μ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Φ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ Φ φ σ σ σ σ 1 þ θ2 σ 1 þ θ2 σ 1 þ θ2 ! AM2 : p −μ  p −μ p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1−Φ 2 Φ σ σ 1 þ θ2 ≡ λ2 ðp2 Þ

ð12cÞ w where θ ¼ 1−w : Previous scholars have shown that the hazard function of the normal distribution increases monotonically. Thus, if we can order the hazard functions by magnitude, then we can also order the profitmaximizing prices (i.e., p0, p1, p2) because the functions in the lefthand side of Eq. (12), 1/(p − c), are monotonically decreasing in p. Proposition 2 below summarizes the order of the profit-maximizing prices under the three mechanisms.

Proposition 2. Let p0, p1 and p2 be the profit-maximizing prices for the conventional mechanism (CM), the One-step Anchoring Mechanism (AM1) and the Two-step Anchoring Mechanism (AM2), respectively. Additionally, let p⁎ be the price level at which λ0(p*) = λ1(p*). Then, there are inequality relations among p0, p1 and p2, as shown by the following: 1 λ0 ðp Þ 1 : p1 b p0 if c b p − λ0 ðp Þ p2 N p0

p1 N p0 if c ≥ p −

Proof. See Appendix B. From proposition 2, we conclude that the firm should set a higher price if its customers use the AM2 rather than the CM as their purchasing decision mechanism. The firm can maximize its expected profit by setting a higher or lower price, depending on the cost and the WTP distribution, if its customers utilize the AM1 rather than the CM. Fig. 3 demonstrates a special case in which p2 N p1 N p0. The pricing solutions in Proposition 2 suggest that the optimal prices suggested by the anchoring mechanisms depend on the degree of anchoring that is measured by the weight as well as the uncertainty of the WTP that is measured by the variance of the distribution, because the hazard function on the right-hand side of Eq. (12) contains the weight w and the variance σ 2. It is evident that increasing the degree of anchoring toward the price (i.e., increasing the weight w) increases the optimal price under AM2, i.e., p2 is positively related to w. Because increasing the price should decrease a product's demand to a lesser degree if the WTP is anchored than if the WTP is not anchored, the firm should set a higher price if its consumers adjust their WTP with the AM2. Fig. 4 compares the optimal prices associated with the different degrees of anchoring under the AM2. These degrees of anchoring

Fig. 3. Expected profit maximizing price determination under different mechanisms — CM, AM1 and AM2.

are measured by weight to determine whether the customers' WTP are skewed toward the market WTP (w = 0.7), neutral (w = 0.5), or skewed toward the subjective WTP (w = 0.3). As expected, the higher the degree to which the customer's WTP is anchored (i.e., the higher w is), the higher the optimal price. A firm's decision regarding its introductory price should also consider the amount of uncertainty that the customers have about their subjective WTP and the market WTP. We undertook a simulation to see how the optimal price is affected by the uncertainty. The simulation was conducted by employing different assumptions regarding the variance of the distributions, but the results were more equivocal than with the above simulations involving the weights. Additional research should be directed toward investigating this effect.

Fig. 4. Optimal prices with different anchoring effects under AM2.

J.H. Park et al. / Intern. J. of Research in Marketing 28 (2011) 309–318

6. Conclusions and future research In this study, we developed two mechanisms for describing how customers adjust their WTP for new products. We presume that customers anchor their WTP by calculating the weighted average of their subjective WTP and the market WTP. In a One-step Anchoring Mechanism (AM1), the customer directly compares the observed price with his or her anchored WTP. In a Two-step Anchoring Mechanism (AM2), the customer first compares his or her subjective WTP with the observed price and then (if necessary) compares his or her anchored WTP with the listed price. We provided logical arguments for these mechanisms, showed how they relate to the findings of prior research, and then developed a series of propositions that followed from the mechanisms. Then, we provided experimental evidence in support of AM2. Lastly, we examined the pricing implications for firms that desire to maximize profits. As our study and prior research on anchoring and adjustment have shown, a firm that uses well-proven methods to research its customers' subjective WTP before it sets the price for new products may still price its offering below the profit-maximizing level because these methods do not usually account for the anchoring and adjustment effect described in this paper. Note that we are not suggesting that such methods be dropped altogether. However, we do suggest that prices based on such methods be adjusted upwards and retested in realistic environments that allow for anchoring and adjustment effects to be observed. These findings will be of interest to firms that develop or sell innovative products. Pricing these types of products is a major concern for these firms, and our findings suggest that firms using “skim” pricing strategies stand to benefit from the upward adjustment in the WTP that their customers experience at the time of the purchase. To develop our model, we imposed many distributional assumptions, such as the assumption that both the subjective WTP and the market WTP have the same type of distributions. In addition, our model can be developed in many ways by relaxing the distributional assumptions. For example, by relaxing the assumption of identical variance, we can develop a model that captures the relationship between pricing and the degree of uncertainty associated with the subjective WTP. Our logic suggests that if customers have more ill-formed notions of their WTP for new products, then we should find higher prices in the marketplace, but we know of no research that has evaluated the relationship between pricing and WTP variance. Moreover, future scholars can probably find evidence suggesting that different types of customers have different WTP distributions, which may lead to certain opportunities for legal price discrimination across the isolated customer segments. Finally, our model can also be expanded to a multi-period model that explains a customer's repeated purchasing behavior. A multi-period model based on our models will help explain how customer WTP evolves as new products become established in the marketplace and will also help account for the differences in the price elasticities between repeat purchasers and first-time buyers.

Appendix A

Since

w2 þ ð1−wÞ2 2

ð1−wÞ

¼

317

w2 ð1−wÞ2

þ 1N1,

0 1 p−μ  ð 1−w Þ ð p−μ Þ B C N1  Φ@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA Pr½D0 ¼ 1 ¼ 1  Φ σ σ w2 þ ð1−wÞ2 ¼ Pr½D1 ¼ 1 for p b μ; 0 1 p−μ  ð 1−w Þ ð p−μ Þ B C b1  Φ@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA Pr½D0 ¼ 1 ¼ 1  Φ σ σ w2 þ ð1−wÞ2 ¼ Pr½D1 ¼ 1 for p N μ: ■ Appendix B Proof of Proposition 2. We first prove two lemmas and then prove the proposition with the lemmas. Lemma 1. There exists a price level p⁎ less than μ such that λ0(p) ≤ λ1(p) over (−∞, p⁎) and λ0(p) ≥ λ1(p) over [p⁎, ∞), where λ0(p) and λ1(p) are defined as in Eqs. (12a)–(12c). Proof of Lemma 1. Step 1: A normal density has a monotonically increasing hazard rate, λ′(p) N 0. ′

By letting ηðt Þ ¼ − ff ððttÞÞ, where f(t) and f′(t) are pdf and its first derivative, respectively, Glaser (1980) proved that if η(t) is increasing,   f ðt Þ then the hazard rate, λðt Þ ≡ 1−F ðt Þ , also monotonically increases with minimal assumptions (i.e., η′(t) N 0 ⇒ λ′(t) N 0). Applying the theorem to a normal distribution, we can easily prove that a normal density has a monotonically increasing failure rate, λ′(p) N 0 because a normal distribution with mean μ and variance σ 2 has its η(p) and η′(p) as ηðpÞ ¼ p−μ and η′ ðpÞ ¼ σ12 N0, respectively. σ2 Step 2: λ0(p) is increasing faster than λ1(p) for all p (i.e., λ′0(p) N λ′1(p)). Letting g ðpÞ ¼ λð1pÞ, Glaser (1980) showed that ∞

g′ðpÞ ¼ ∫p ½ f ðyÞ=f ðpÞ½ηðpÞ−ηðyÞdy:

ðB1Þ

Additionally, applying (B1) to the normal distribution yields g′0 ðpÞ−g′1 ðpÞ o1 0n 2 3 ðp−μ Þ2 −ðy−μ Þ2 1 @ A 6 7 6 σ 2 exp 7 2σ 2 7 ∞6 n o1 7½p−ydy b 0 ðB2Þ 0 ¼ ∫p 6 6 7 2 2 ð p−μ Þ − ð y−μ Þ 6 7 4−  1 @ A5    exp 2 2 2 2 σ 1þθ 2σ 1 þ θ 1 Because σ12 N σ 2 ð1þθ 2 Þ and y N p. Therefore, λ′0(p) N λ′1(p). Step 3: λ0(μ) N λ1(μ)  1 p1ffiffiffiffiffiffiffiffi φð0Þ φð0Þ σ 1þθ2 : N λ1 ðμ Þ ¼ λ0 ðμ Þ ¼ σ 1−Φð0Þ 1−Φð0Þ

ðB3Þ

Proof of Proposition 1. Since 0 1 p−μ  B ð1−wÞðp−μ Þ C 0≤Φ ≤ 1 and 0 ≤ Φ@ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA≤ 1; σ σ w2 þ ð1−wÞ2 Pr½D2 ¼ 1 ≥ Pr½D0 ¼ 1 and Pr½D2 ¼ 1 ≥ Pr½D1 ¼ 1:

From the results of the above three steps, there must exist a price level p* less than μ at which λ0(p) intersects λ1(p) from below such that λ0(p) ≤ λ1(p) over (−∞, p*) and that λ0(p) ≥ λ1(p) over [p*, ∞). ■ Lemma 2. The hazard function of AM2 is always less than or equal to that of CM (i.e., λ2(p) ≤ λ0(p)).

318

J.H. Park et al. / Intern. J. of Research in Marketing 28 (2011) 309–318

  1 pffiffiffiffiffiffiffiffiffiffi ffi Proof of Lemma 2. Let φ0 ≡ φ0 ðpÞ ≡ σ1 φ p−μ σ ; φ1 ≡ φ1 ðpÞ ≡ σ ð1þθ2 Þ   p−μ  p−μ p−μ ffi and Φ0 ≡ Φ0 ðpÞ ≡ Φ σ ; Φ1 ≡ Φ1 ðpÞ ≡ Φ pffiffiffiffiffiffiffiffiffiffi ffi φ pffiffiffiffiffiffiffiffiffiffi 2 2 σ

ð1þθ Þ

σ

ð1þθ Þ

for notational convenience. λ0 ðpÞ−λ2 ðpÞ φ0 φ Φ þ Φ0 φ1 ¼ − 0 1 1−Φ0 1−Φ0 Φ1 n o 1 2 φ0 −φ0 Φ0 Φ1 −φ0 Φ1 −Φ0 φ1 þ φ0 Φ0 Φ1 þ Φ0 φ1 ¼ ð1−Φ0 Þð1−Φ0 Φ1 Þ 1 ¼ fφ ð1−Φ1 Þ−φ1 Φ0 ð1−Φ0 Þg: ð1−Φ0 Þð1−Φ0 Φ1 Þ 0

ðB4Þ Lemma 2 can be proven if φ0(1 − Φ1) − φ1Φ0(1 − Φ0) ≥ 0. As shown in Lemma 1, we find this result by dividing the support of p into two parts: {p|p ≥p* ⇔λ0(p) ≥λ1(p)} and {p|p b p*⇔λ0(p) b λ1(p)}. For the interval of {p|p ≥ p* ⇔ λ0(p) ≥ λ1(p)}, φ0 ð1−Φ1 Þ−φ1 Φ0 ð1−Φ0 Þ ≥ φ0 ð1−Φ1 Þ−φ1 ð 1−Φ0 Þ  φ0 φ1 − ¼ ð1−Φ0 Þð1−Φ1 Þ ð1−Φ0 Þ ð1−Φ1 Þ

ðB5Þ

¼ ð1−Φ0 Þð1−Φ1 Þfλ0 ðpÞ−λ1 ðpÞg≥0: For the interval of {p|p b p* ⇔ λ0(p) b λ1(p)}, φ0 ð1−Φ1 Þ−φ1 Φ0 ð1−Φ0 Þ ≥ φ0 Φ1 −φ1 Φ 0  φ0 ðpÞ φ1 ðpÞ − ¼ Φ0 ðpÞΦ1 ðpÞ Φ0 ðpÞ Φ1 ðpÞ

 φ0 ð2μ−pÞ φ1 ð2μ−pÞ − ¼ Φ0 Φ1 1−Φ0 ð2μ−pÞ 1−Φ1 ð2μ−pÞ

ðB6Þ

¼ Φ0 Φ1 fλ0 ð2μ−pÞ−λ1 ð2μ−pÞg≥0: For the above derivation, we know that φ(p) = φ(2μ − p) and Φ(p) = 1 − Φ(2μ − p) for the symmetry of the normal distribution. It is obvious that 2 μ –p N p* for p b p* because μ N p* N p, which directly means λ0(2μ − p) − λ1(2μ − p) ≥ 0. ■ 1 Let wðpÞ ¼ p−c . If c≥p − λ0 ð1p Þ, then w(p*) ≥ λ0(p*) = λ1(p*). w(p) is decreasing in p, but λ(p)is increasing in p. Thus, the gap between w(p*)and λ0(p*)(i.e., λ1(p*)) should shrink as p increases from p⁎. Therefore, there must exist a p0 where w(p0) = λ0(p0). However, at p = p0, w(p0) N λ1(p0) because p0 belongs to the interval of {p|p ≥ p* ⇔ λ0(p) ≥ λ1(p)}. As a result, we can find a higher price (p1) than p0 at which point w(p1) = λ1(p1). Therefore, p1 N p0. By the same reasoning, we not only show that p1 b p0 if cbp − λ0 ð1p Þ but also show that p2 N p0 is always true because λ2(p) ≤ λ0(p) for all p, as shown by Lemma 2.

References Boyle, K. J., Johnson, F. R., McCollum, D. W., Desvousges, W. H., Dunford, R. W., & Hudson, S. P. (1996). Valuing public goods: Discrete versus continuous contingent-valuation responses. Land Economics, 72(3), 381–396. Boyle, K. J., MacDonald, H. F., Cheng, H. T., & McCollum, D. W. (1998). Bid design and yea saying in single-bounded, dichotomous-choice questions. Land Economics, 74(1), 49–64. Briesch, R. A., Krishnamurthi, L., Mazumdar, T., & Raj, S. P. (1997). A comparative analysis of reference price models. Journal of Consumer Research, 24(2), 202–214. Cameron, T. A., & Quiggin, J. (1994). Estimation using contingent valuation data from a dichotomous choice with follow-up questionnaire. Journal of Environmental Economics and Management, 27(3), 218–234. Cialdini, R. B. (1985). Influence: Science and practice. Glenview, Illinois: Scott, Foresman and Company. Degeratu, A. M., Rangaswamy, A., & Wu, J. N. (2000). Consumer choice behavior in online and traditional supermarkets: The effects of brand name, price, and other search attributes. International Journal of Research in Marketing, 17(1), 55–78. Della Bitta, A. J., & Monroe, K. B. (1974). The influence of adaptation levels on subjective price perceptions. Advances in Consumer Research, 1(1), 359–369. Estelami, H., Lehmann, D. R., & Holden, A. C. (2001). Macro-economic determinants of consumer price knowledge: A meta-analysis of four decades of research. International Journal of Research in Marketing, 18(4), 341–355. Frykblom, P., & Shogren, J. F. (2000). An experimental testing of anchoring effects in discrete choice questions. Environmental & Resource Economics, 16(3), 329–341. Glaser, R. E. (1980). Bathtub and related failure rate characterizations. Journal of the American Statistical Association, 75(371), 667–672. Green, D., Jacowitz, K. E., Kahneman, D., & McFadden, D. (1998). Referendum contingent valuation, anchoring, and willingness to pay for public goods. Resource and Energy Economics, 20(2), 85–116. Hardie, B. G. S., Johnson, E. J., & Fader, P. S. (1993). Modeling loss aversion and reference dependence effects on brand choice. Marketing Science, 12(4), 378–394. Holmes, T. P., & Kramer, R. A. (1995). An independent sample test of yea-saying and starting point bias in dichotomous-choice contingent valuation. Journal of Environmental Economics and Management, 29(1), 121–132. Kalwani, M. U., Yim, C. K., Rinne, H. J., & Sugita, Y. (1990). A price expectations model of customer brand choice. Journal of Marketing Research, 27(3), 251–262. Kalyanaram, G., & Winer, R. S. (1995). Empirical generalizations from reference price research. Marketing Science, 14(3), G161–G169. Kamins, M. A., Dreze, X., & Folkes, V. S. (2004). Effects of seller-supplied prices on buyers' product evaluations: Reference prices in an Internet auction context. Journal of Consumer Research, 30(4), 622–628. Klapper, D., Ebling, C., & Temme, J. (2005). Another look at loss aversion in brand choice data: Can we characterize the loss averse consumer? International Journal of Research in Marketing, 22(3), 239–254. Monroe, K. B. (2003). Pricing making profitable decisions. Boston: McGraw-Hill/Irwin. Niedrich, R. W., Sharma, S., & Wedell, D. H. (2001). Reference price and price perceptions: A comparison of alternative models. Journal of Consumer Research, 28(3), 339–354. O'Conor, R. M., Johannesson, M., & Johansson, P. O. (1999). Stated preferences, real behaviour and anchoring: Some empirical evidence. Environmental & Resource Economics, 13(2), 235–248. Park, J. H., & MacLachlan, D. L. (2008). Estimating willingness to pay with exaggeration bias-corrected contingent valuation method. Marketing Science, 27(4), 691–698. Rajendran, K. N., & Tellis, G. J. (1994). Contextual and temporal components of reference price. Journal of Marketing, 58(1), 22–34. Simonson, I., & Drolet, A. (2004). Anchoring effects on consumers' willingness-to-pay and willingness-to-accept. Journal of Consumer Research, 31(3), 681–690. Sinha, I., & Batra, R. (1999). The effect of consumer price consciousness on private label purchase. International Journal of Research in Marketing, 16(3), 237–251. Thaler, R. (1985). Mental accounting and consumer choice. Marketing Science, 4(3), 199–214. Urbany, J. E., Bearden, W. O., & Weilbaker, D. C. (1988). The effect of plausible and exaggerated reference prices on consumer perceptions and price search. Journal of Consumer Research, 15(1), 95–110. Wathieu, L., & Bertini, M. (2007). Price as a stimulus to think: The case for willful overpricing. Marketing Science, 26(1), 118–129.