A model for the effects of psychological pricing in Gabor–Granger price studies

A model for the e�ects of psychological pricing inGabor±Granger price studies

Michel Wedel *, Peter S.H. Lee¯ang

Department of Economics, University of Groningen, Groningen, The Netherlands

Received 7 March 1996; received in revised form 7 July 1997; accepted 10 September 1997

Abstract

We present a model of consumers' price sensitivity that explicitly deals with the existence

of so-called psychological price levels or odd prices, i.e. prices ending in an odd number. The

model is formulated in a latent class framework, in which splines are used to model utility as

a function of prices in consumer segments. The knots in the splines represent psychological

prices. Additionally, the model allows for inferences on price expectations and the role of

price as an indicator of quality. The model is tailored to the analysis of so-called Gabor±

Granger price experiments. We provide an empirical application to the analysis of a

Gabor±Granger study, and investigate the performance of our model relative to a competing

model. Ó 1998 Elsevier Science B.V. All rights reserved.

PsycINFO classi®cation: 2240; 3920

JEL classi®cation: C25

Keywords: Pricing; Mixture model; Splines; Gabor±Granger price experiment

Journal of Economic Psychology 19 (1998) 237±260

* Corresponding author. Address: Department of Marketing and Marketing Research, Faculty of

Economics, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands. Tel.: +31 50

3633735; fax: +31 50 3633720.

0167-4870/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved.

PII S 0 1 6 7 - 4 8 7 0 ( 9 8 ) 0 0 0 0 6 - 3

1. Introduction

The practice of setting prices just below the nearest round ®gure is pop-ular among manufacturers and retailers. Such pricing practices are usuallyreferred to as psychological or odd-pricing (see Monroe, 1973; Blattbergand Neslin, 1990, p. 349). Examples of these prices are $4.99 instead of$5.00 or $99.95 rather than $100. The origin of this practice can only beconjectured, but it may for example have arisen as a quantity-discountwhen fresh products were only o�ered by the pound, and the consumers'temptation to buy half the o�ered quantity was reduced when productsare o�ered at such odd prices (see Friedman, 1968). Many retailers believepsychological pricing to be e�ective. In a survey of newspaper advertise-ments Friedman (1968) found prices ending in ``9'' to be by far the mostpopular among food retailers. The reason for the use of these price settingsis that a higher price elasticity is expected at the psychological price, ascompared to those within the surrounding price-interval. The increase insales that results from a price cut from e.g. $1.00 to $0.99 is expected tobe much larger than that from decreasing $0.99 to $0.98 or from $1.01to $1.00 (see Gabor and Granger, 1961, 1966). Monroe (1973) indicatedthat empirical evidence for the existence of discontinuities in demand atpsychological prices is limited, and since then only the work of Blattbergand Wisniewski (1987) has addressed this issue. These authors used retailscanner data to demonstrate that for fast moving consumer goods, the per-centage of increase in sales obtained at psychological price levels was 10%above the e�ect simply due to the price decrease. To this end they devel-oped several alternative models describing the shape of deal-discounts.These models are aggregate-level models.

The purpose of this paper is to develop a stochastic disaggregate-levelchoice model, which may serve to investigate the e�ects of psychological pric-ing within the context of Gabor±Granger price studies (Gabor and Granger,1966). It is a logit model of binary choices of individual consumers, and al-lows for a more detailed representation of the form of price-response func-tions than has been possible until now. Splines are used to modelconsumers' utility as a function of prices. The model is formulated in a latentclass framework and allows for the identi®cation of market segments withdi�erent price-response functions. First we will provide a possible explaininghypothesis for the e�ects of psychological price-settings, using consumer be-havior and economic theories. Then, we will provide a review of additionalissues in the pricing literature, that will be accommodated in the model.

238 M. Wedel, P.S.H. Lee¯ang / Journal of Economic Psychology 19 (1998) 237±260

Section 4 describes the Gabor±Granger procedure, our model and the es-timation procedure. Section 5 provides the results of a study into consumer'sprice sensitivity, where the results are analyzed with the proposed model. InSection 6 our model is compared to an alternative model with respect to pre-dictive validity. An investigation of the external validity is provided as well.Section 7 contains a discussion of the results.

2. Psychological prices

The explaining hypothesis of the e�ect of psychological prices that we willuse, is based upon adaptation level theory (Helson, 1964) and transactionutility theory (Thaler, 1985).

Adaptation level theory states that perceptions of new stimuli areformed relative to a standard or adaptation level. The adaptation levelis determined by previous and current stimuli to which a person has beenexposed. A possible hypothesis for the occurrence of discontinuities inconsumer demand at psychological prices is related to how consumers per-ceive psychological prices, relative to prices perceived as `fair' (see Fried-man, 1968; Schindler, 1991). Consumers may very well evaluate pricesaccording to round monetary units, such as one dollar, pound, franc,mark, etc. Consumers perceive odd prices as being substantially lowerthan even-priced items, even though the real di�erence is perceptually verysmall. Thus, an item priced at $1.99 is thought of as costing about $1rather than $2 (Hanna and Dodge, 1995, p. 28). We conjecture, that aconsumer's reaction to the o�ered psychological price of $1.99 is that(s)he perceives the nearest integer price, e.g. $2.00, to be a `fair' price,and that (s)he perceives that a discount is obtained relative to the `fair'price (see Friedman, 1968).

Brenner and Brenner (1982) suggest that this phenomenon results fromconsumers' limited capacity for storing directly accessible information. Con-sumers exposed to price information store only the more valuable parts of themessage they receive and these are the ®rst digits of a number. When a priceis $1.99, the digit is more important as information than the ®rst and second`9'. Rounding up involves an additional decision compared with storing theinteger part of the number.

Psychological pricing can be seen as implicitly presenting a fair price toconsumers, to enhance their notion of a price cut involved in the transaction.Such implicit presentation of a fair price may even lead to higher price

M. Wedel, P.S.H. Lee¯ang / Journal of Economic Psychology 19 (1998) 237±260 239

savings being perceived than in the instance of an explicit presentation of aprice discount (see Liefeld and Heslop, 1985).

Transaction utility theory provides an explanation for the increase in de-mand at the psychological price level. Thaler (1985) proposes that the totalutility of a transaction to a consumer is the sum of the acquisition utilityand the transaction utility. The acquisition utility is derived from the valueof the item to the consumer; it increases monotonically with price acrossthe entire price-range. The transaction utility results from comparing theprice paid to the fair price. We hypothesize that the perceived price cut ata price of, say, $1.99, relative to a perceived fair price of $2.00, increasesthe transaction utility (at $1.99), and results in a discontinuous increase in to-tal utility, and thereby in consumer demand.

A large body of literature supports the hypothesis that the sensitivity toperceived price reductions varies across consumers (see Blattberg et al.,1978; Blattberg and Neslin, 1990, pp. 77±81; GoÈnuÈ l and Srinivasan, 1993).Thereby, the relative importance of transaction utility, and consequentlythe e�ect of psychological prices, may di�er between consumers. It maytherefore be hypothesized that psychological price e�ects occur in some con-sumer segments (market segments) but not in others.

3. Pricing literature review

The purpose of this study is to develop a behavioral pricing model for theinvestigation of consumers' sensitivity to psychological pricing. Such a modelshould not only account for the e�ect of odd prices on consumer demand butalso for a number of other issues that have previously been included in be-havioral pricing models. In this respect the following issues are relevant(see Rao, 1993, pp. 538±543):

1. the formation of price expectations by consumers;2. the role of price as an indicator of quality;3. heterogeneity of consumers with respect to price sensitivity.

(1) Price expectations (the role of reference prices): Consumers have ex-pectations about prices. These expectations are product and individual spe-ci®c; they are known as reference prices. Support for consumers' use ofreference price levels is based on prospect theory and is supported on a largenumber of studies in the pricing literature (see, e.g., Winer, 1986, Kalwaniet al., 1990).


Prospect theory of Kahneman and Tversky (1979) states that consumersreact more strongly to losses than to gains, relative to a reference; a loss/gainmeaning that the observed price is higher/lower than the reference price. Thenotion of asymmetric response above and below the reference price can beexplained by price losses appearing to be larger than gains. Supporting evi-dence for price loss-aversion comes from several studies (see, e.g., Kalwaniet al., 1990; Kalwani and Yim, 1992; Hardie et al., 1993; Bell and Lattin,1993). The notion of reference dependence has been extended to other attri-butes than price by Tversky and Kahneman (1991). Hardie et al. (1993) in-vestigate loss-aversion with respect to, amongst others, quality. Theydemonstrate that consumers display asymmetric response with respect to areference point for quality as well. Also here, quality losses appear largerthan quality gains. In the empirical studies reference price has been opera-tionalized as the average price of similar products (see Emery, 1970), the pricelast paid (see Monroe, 1973; Winer, 1986), the price most frequently paid (seeGabor and Granger, 1961), a weighted average of past prices (see Kalwani etal., 1990), or by direct questioning about the expected price (see Kalwani andYim, 1992).

(2) Price as an indicator of quality: The role of price as a determinant ofbuying intentions and actual choice is complicated by its role as an indicatorof quality. Theories of imperfect information (see e.g. Monroe, 1973; Steen-kamp, 1989, p. 37) explain that in the choice process of consumers who areuncertain about the quality of a product, price may serve as a cue for assess-ing that quality. In that sense, higher prices of a brand have been shown topositively a�ect purchase probabilities (see, e.g., Erickson and Johansson,1985). The associated behavior of consumers has been referred to as price-seeking. The price-perceived quality relationship has been reported to behighly variable across consumers and situations (see Zeithaml, 1988), and ap-pears to be stronger for nondurable than for durable products (see Lichten-stein and Burton, 1989). 1 Thus we conclude that we have to recognize a dualrole of price in consumers' decision-making. The classic economic e�ect ofprice on choice behavior is modeled by the relationship between `actual price'and `perceived monetary sacri®ce'. The higher the price, the more must besacri®ced to purchase the product and the lower the purchase intention.On the other hand higher prices have a positive e�ect on perceived quality

1 There are a vast number of studies on the price quality relationship, which have been reviewed by e.g.

Monroe (1973) and Steenkamp (1989).


and this leads to a higher purchase intention. This trade-o� is known as ``theperceived sacri®ce±perceived quality trade-o�'' (Steenkamp, 1989, p. 195) orthe price±perceived quality trade-o�.

(3) Heterogeneity: A ®nal complexity in the relationship of price to con-sumer choice behavior is that consumers appear to be very heterogeneousin their attention and reaction to price and price promotions (see Dicksonand Sawyer, 1990; GoÈnuÈl and Srinivasan, 1993; Lichtenstein et al., 1993).Thus we may observe di�erent `price segments'. There is heterogeneity be-tween these market segments in terms of price reactions, whereas within aprice segment the consumers' reactions to price changes is homogeneous.In the `early' price-response models heterogeneity has been captured by usingan a priori de®nition of segments on the basis of socio-economic and demo-graphic characteristics (see, e.g., Wildt and McCann, 1980). A post-hoc ap-proach was employed ®rst by Elrod and Winer (1982). They groupedconsumers into segments on the basis of price-elasticities estimated at the in-dividual level. More recently, Kamakura and Russell (1989) proposed a la-tent class model that improves upon the previous approaches bysimultaneously estimating segments and price-response functions within eachsegment. Bell and Lattin (1993) used a related model to investigate heteroge-neity in loss aversion. Our model extends these approaches by including psy-chological price-e�ects.

The theories which have been introduced brie¯y in the two preceding sec-tions to explain consumers' reactions to price are summarized in Table 1.

4. Model description

4.1. Gabor±Granger price studies

Our model is tailored to the analysis of Gabor±Granger studies (see Ga-bor and Granger, 1961, 1966). The purpose of Gabor±Granger studies is toestablish a so-called buy-response curve, which depicts the percentages ofconsumers buying a certain brand at various prices. To this end, respon-dents are o�ered the brand at a number of prices, and are required to statewhether they intend to buy the brand or not, at each price. They can alsoindicate not to buy the brand at all regardless of the prices. Because in Ga-bor±Granger studies price-levels can be determined by the researcher, theresulting data have the advantage over observational data of greater pricevariation. In addition, these data are not confounded by simultaneous ef-


fects of e.g. competitive advertising, or the introduction of new brands. Fre-quently, in Gabor±Granger price studies, a bell-shaped price-response func-tion is observed, in which the percentage of consumers buying the brandrises at lower prices, and declines again at higher prices. Gabor and Gran-ger (1966) attributed this e�ect to consumers' use of price as an indicator ofquality. The Gabor±Granger procedure is still frequently used in marketingresearch practice.

4.2. The model

We assume that in a Gabor±Granger study, a sample of n consumers is of-fered a certain brand at, say, J di�erent prices. Among these prices are K psy-chological prices. We assume that the brand we consider is indicated by l� 1,and that the prices of the (L-2) alternative brands are ®xed. One alternative isnot buying. The dependent variable of our model is a 0/1 buy-response ofconsumers.

Establishing the notation used in this study as follows: i � 1; . . . ; n indicateconsumers, j � 1; . . . ; J indicate the prices, k � 1; . . . ;K indicate the psycho-logical prices, l � 1; . . . ;L indicate brands, s � 1; . . . ; S indicate segments,yij� 1 if consumer i buys brand 1 at price level j, and 0 otherwise, pij isthe jth price for consumer i, pi

r, the reference price of consumer i, and tk,the kth psychological price.

Table 1

Theories explaining consumers response to price

Theory Contents Explanation for reactions to price

1. Adaptation level theory Perceptions of new prices are

formed relative to an adaptation

level: round monetary unit

Psychological prices are

compared to `fair' prices

2. Transaction utilily theory Total utility of a transaction is

the sum of the acquisition utility

and the transaction utility

Transaction utility increases

at psychological prices

3. Prospect theory Consumers react more strongly

to losses than to gains relative to

a reference: loss/gain: observed

price higher/lower than reference

price loss/gain: perceived quality

lower/higher than reference

quality

Assymmetric response to

price/quality around reference

price/quality

4. Theory of imperfect

information

Price is cue for assessing quality Higher prices lead to higher pur-

chase probabilities


We start from the assumption that there are a number of (market) seg-ments (or latent classes), S, with relative sizes h1; . . . ; hS; hs � number ofconsumers in segment s divided by bij n, (hs > 0, Rhs� 1). Given that(s)he comes from segment s, the conditional probability that consumer iwishes to buy the brand at price j is equal to the probability that the utilityof buying the brand, Uijjs, is larger than the utility of not buying the brand,UiN js. The latter utility is de®ned as the maximum of the utilities of the sub-set {N} of L-1 brands that are used implicitly as alternatives. The condi-tional probability of buying is

Pijjs � ProbfUijjs > UiN jsg UiN js � max�Uiljs; l � 2; . . . ; L�: �1�As usual, assume the utilities random and divide them into deterministicparts, denoted by respectively Vijjs and ViN js, and random parts, denoted byrespectively �ijjs and �iN js.

2 The latter terms capture the e�ect of unobservedbackground stimuli and uncertainty. It is assumed that the number of alter-natives L-1 in the set {N} and the deterministic parts of their utilities ViN js areconstant in each segment. This assumption of constancy is a rather strongone, but is commonly made in choice models (see Ben-Akiva and Lerman,1985).

The deterministic part of the utility Vijjs will depend on the product fea-tures of the brand, its distribution, its image and among many other variablesits price. In our Gabor±Granger analysis price is the only variable that ¯uc-tuates. Thus Vijjs is related to the actual prices, the psychological price levelsand the respondents' reference prices. For this purpose ®rst order linearsplines are used, which are linear line segments of possibly di�erent slopes,joined at a number of knots (see Smith, 1979). In our model the knots aredetermined a priori, corresponding to the K psychological prices. By choos-ing the knots in such a manner, they represent the discontinuities in utilityassociated with psychological prices.

In addition to the e�ects of the psychological prices, we model the asym-metry in consumers' response relative to the reference price (for which we will

2 According to standard discrete choice theory (cf. e.g. Ben-Akiva and Lerman, 1985, p. 256),UiN js � �ViN js � ln N � ln BiN js � �iN js;where �ViN js � �1=Lÿ 1�PL

l�2 Viljs, is the average (i.e. `deterministic') utility of the L ) 1 implicit alterna-

tives in {N} and

BiN js � 1

Lÿ 1

XL

l�2

eViljsÿ �ViN js ;

is a measure of the heterogeneity of the L ) 1 alternatives in {N}.


also use a spline representation). We start from the assumption that the ref-erence price is determined exogenous to our model.

To develop the spline representation, the total price range is partitionedinto K + 1 price ranges de®ned by the K psychological prices. Utility is as-sumed to depend linearly on price, in between the K psychological prices.The sequence of knots that forms the basis of the spline function, represent-ing the psychological prices, is de®ned as ft0; t1; . . . ; tK ; tK�1g; t0 denotes thelowest, tK�1 the highest price o�ered, and tk (k � 1; . . . ;K) are the K psycho-logical prices. For consumer i, the spline function contains one additionalknot at pi

r, his/her reference price level.We introduce the function Sq(.):

Sq�Z� � Zq if Z > 0

� 0 if Z6 0;

where the exponent q is a nonnegative integer. This function is commonlyused to formulate splines (see Smith, 1979). A simple example follows. Twointervals of a variable x, say [)1, t1); [t1, 1] are distinguished, where t1

denotes a knot. Now, S1(x ) t1)� x ) t1, for x > t1. The spline functionis de®ned in this case as: y� b1x + b2S1(x ) t1). This function has regres-sion coe�cients b1 for x < t1, and b1 + b2 for x > t1. The line segmentsare connected at t1. In the presence of more intervals, similarly, the coe�-cients are cumulated (b1 + b2 + b3, etc.) across the preceding intervals, inthe calculation of the e�ect of x in a certain interval. We will use q� 0,1,implying linear splines. The deterministic part of the utility function for aconsumer of market segment s is written as the sum of three e�ects (eachcomprising two terms): a linear price e�ect + reference price e�ect + psy-chological price e�ect (the interpretation of the terms in the model willbe further explained below):

Vijjs � b00s � b10spij � b2spri � b3sS

1�pij ÿ pri � �

XK

k�1

b0ksS0�pij ÿ tk�

�XK

k�1

b1ksS1�pij ÿ tk�: �2�

For an individual consumer in market segment s:, b00s denotes the intercept(INTCPT), b10s represents the linear price e�ect (PRICE0), b2s representsthe overall e�ect of the reference price (REFERP), b3s represents the addi-tional e�ect of price `losses', where a price loss is de®ned as the di�erencebetween the actual/observed price and the reference price if the actual price


is higher than the reference price (PRLOSS), b0ks represents the e�ect ofthe kth psychological price level (PSYPRk), b1ks represents the additionallinear price e�ect in the kth price interval, relative to the (k ) 1)th interval(PRICEk).

The intercept term in Eq. (2) represents that part of the deterministic partof the utility which is not related to price. It represents the `overall' probabil-ity of choosing the brand (l� 1) by consumers in market segment s. The termPRICE0 captures the overall linear price e�ect. The term PRLOSS representsthe additional e�ect of a price `loss', relative to price `gains' (see Kahnemanand Tversky, 1979; Tversky and Kahneman, 1991; Hardie et al., 1993). Notethat S1(pij ) pi

r)� 0 for pij < pir, and S1(pij ) pi

r)� (pij ) pir), otherwise.

For example, the partial e�ect of a price gain (pij6 pir) equals b10spij. The

partial e�ect of a price loss (pij > pir), can then be represented as

(b10s + b3s)pij. Thus b3s represents the additional e�ect of price losses (onthe price pij). The formulation used enables a test of loss aversion: ifb3s� 0, price e�ects above and below the reference price level are symmetric.

The term REFERP represents the absolute e�ect of the reference price onthe choice probabilities. Compare in this respect Tversky and Simonson(1993), who have shown that in addition to relative evaluations, absoluteevaluations play a role in consumer choice processes.

The coe�cients b0ks of the terms PSYPRk (for k � 1; . . . ;K) represent thechanges in intercept at the psychological price levels, and thus the discontinu-ities in utility at these price levels. The coe�cients b1ks of the terms PRICEk(k � 1; . . . ;K) represent the changes in the slope of price at the psychologicalprice levels, relative to the slope in the next lower price range. These coe�-cients are used to investigate the shape of the price-response function.

At this point, we formulate an expression for the conditional choice prob-abilities Pijjs. We use the standard assumption of a logistic distribution of thedi�erence �ijjs ) �iN js. The following expression for the probability that sub-ject i, coming from market segment s, wishes to buy the brand at price levelpij, is obtained:

Pijjs � 1

1� exp�ViN js ÿ Vijjs� �1

1� exp�ÿV �ijjs�: �3�

Since the prices of the other brands and thus ViN js is assumed to be con-stant in segment s during the experiment, this term may be absorbed in theconstant b�00s in V �ijjs. A given consumer i has the probability hs that it belongsto segment s. The unconditional probability that subject i chooses the brandat price j is:


Pij �XS

s�1

hsPijjs �XS

s�1

hs1

1� exp�ÿV �ijjs�: �4�

4.3. Estimation

The parameters of the model are estimated by the method of maximumlikelihood. Using the binomial distribution for the choice probabilities, thelikelihood can be formulated as:

l �Yn

i�1

XS

s�1

hs

YJ

j�1

�Pijjs�yij�1ÿ Pijjs��1ÿyij�: �5�

The likelihood, or equivalently, the log-likelihood is maximized using anEM-algorithm (see Wedel and DeSarbo, 1994) to obtain estimates of the pa-rameters of the model. The EM algorithm has the advantages of being easy toimplement, while convergence of the iterative procedure is ensured.

Once the parameters have been estimated, the posterior probabilities thatsubject i comes from segment s, his, can be calculated using Bayes' rule (seeWedel and DeSarbo, 1994). The asymptotic variances of the estimatedspline-function coe�cients are obtained from the Fisher information matrix.These variances allow for signi®cance testing of the coe�cients.

In practical applications, the number of segments S is unknown. We usethe ICOMP criterion proposed by Bozdogan (1993) to determine the numberof segments, where the number of segments that yields the minimum value ofICOMP is selected (the usual likelihood-ratio tests are invalid, because cer-tain regularity conditions are not satis®ed). ICOMP is an information theo-retic measure, that improves upon the traditional Akaike (and ConsistentAkaike) Information criterion by adding a correction for model complexity,and thus controlling for the risks of over-parameterising the model. ICOMPis de®ned for our model as:

ICOMP � ÿ2 ln l� Q ln �trace�R�� ÿ det�R�; �6�where Q is the number of parameters estimated and R is the estimated co-variance matrix of the parameters. Additionally, the percentage of varianceexplained, R2, and the entropy Es, are used to evaluate the models. R2 isde®ned as 1 minus the ratio of the likelihoods of the current model andthe null-model, where the latter model includes only an intercept (Vij� b0).Es is a measure that indicates the separation of classes (Es� 1


indicates complete separation; Es� 0 indicates complete overlap), and isde®ned as:

Es � 1ÿXn

i�1

XS

s�1

his ln his ln log�S�: �7�

5. Application: Parameterization

5.1. Data collection

In this section we will provide an application of our model to a Gabor±Granger study for a brand in a category of products for personal care. Thebrand name cannot be revealed because of the con®dential nature of the study.

In the study, data were collected from a mall-intercept sample of 377 fe-male shoppers in The Netherlands. The interview procedure was as follows.Subjects were not informed beforehand about the purpose of the study. Theywere told that some questions were to be posed on their preferences with res-pect to brands in the category of products for personal care. Nine di�erentprice levels, varying from D¯ 2.19 to 4.80 were called out to each of the re-spondents. In order to obtain substantial price variation while minimizing thenumber of prices o�ered to each respondent, six di�erent sequences of nineprices were used. Respondents were randomly assigned to one of those six se-quences. The lowest prices in these six sequences were, respectively, D¯ 2.19,2.20, 2.29, 2.30, 2.39 and 2.40. The prices in a sequence were obtained by suc-cessively adding D¯ 0.30 to the lowest price. The ®rst price called out to arespondent was in the middle of the price sequence, while subsequent priceswere presented in random order to prevent order e�ects. Monroe (1990),p. 125, describes such randomization as the appropriate procedure to over-come order e�ects in price-research methods. At each price called out, re-spondents were required to respond with a statement whether they intendto buy the brand (1) at that price or not (0). The question asking consumersfor their purchase intent was conditioned on a purchase of the category beingmade in the subsequent month. Respondents could indicate not to buy thebrand at any of the prices called out.

Whereas speci®c theory as to exactly what constitutes a psychological pricelevel is currently lacking, we de®ned prices ending in D¯ 0.99 as potentialpsychological prices (cf. Friedman, 1968). Therefore, the prices called outto the respondents included two psychological prices: D¯ 2.99 and D¯ 3.99.


In a small audit, conducted among 31 supermarkets in the region wherethe Gabor±Granger study was done, we investigated the occurrence of pricesetting at the two psychological price levels mentioned. More than 50% of the(482) items in the personal care products were priced at levels of D¯ 2.99 andD¯ 3.99. These results illustrate the frequency with which psychological pricesetting is used in practice.

Reference price levels were operationalized in this study as the last pricepaid for a brand within the product class, assessed through direct question-ing. 3

A number of demographic, socio-economic and usage characteristics ofthe respondents were collected: age in years, monthly income in D¯ 1000, us-age frequency of the product in times per week, and whether respondents hadpreviously bought the brand.

5.2. Expected e�ects

The expectations of the direction of most of the e�ects captured in themodel depend upon whether consumers see price as a monetary sacri®ce,or as an indicator of quality, since in the presence of price-seeking behavior,the price-response relation is reversed. Table 2 presents a summary of the ex-pected e�ects.

The coe�cients b1ks (PRICEk, k � 0; . . . ;K), presenting the price-e�ects inthe di�erent price ranges, are hypothesized to be negative, if price is perceivedas a monetary sacri®ce. If price is perceived as an indicator of quality, how-ever, these coe�cients are positive, indicating that utility increases with price.

3 The asymmetric e�ect relative to the price last paid could not be estimated for those consumers who

did not remember the price last paid, or for whom the remembered price last paid was outside of the price

range in the study.

Table 2

Expectations of model coe�cients

Term Coe�cient Price as monetary sacri®ce Price as quality indicator

PRICEk b1ks ) +

REFERP b2s + )PRLOSS b3s ) +

PSYPRk b0ks ) �


Manufacturers bene®t from high reference or expected price levels. Highreference prices (REFERP) ensure that price decreases appear more attrac-tive to consumers, and regular prices do not seem so unattractive (cf. Blatt-berg and Neslin, 1990, p. 41). We therefore expect the sign for b2s to bepositive. If price is used as an indicator of quality, however, manufacturersbene®t from low reference price levels. Here, such low price-expectations willlead to higher prices being perceived as indicating higher quality across theentire price range. The sign for b2s is therefore expected to be negative in suchsituations.

According to the Tversky and Kahneman (1991) prospect theory, consum-ers should react more strongly to price losses, which gives us reason to expectb3s, the additional e�ect of a price loss (PRLOSS) to be negative. If consum-ers use price as an indicator of quality (cf. Hardie et al., 1993), loss-aversionmay occur as well for perceived quality. Consequently, it may be expectedthat for prices above the reference price (price losses) a gain in quality is per-ceived. Therefore the b3s are expected to be positive in situations where priceis used as an indicator of quality.

The terms PSYPRk (for k � 1; . . . ;K) the discontinuity in utility at thepsychological price levels. If the kth psychological price causes such discon-tinuities, b0ks is hypothesized to be negative, otherwise b0ks is zero. It is not apriori clear what the e�ects of psychological prices are in the presence of aprice-perceived quality relationship.

5.3. Results

Our model was applied to the above Gabor±Granger data of 377 consum-ers. We speci®ed from S� 1 to S� 6 segments. Two knots were included inthe spline function, at the psychological price levels of D¯ 2.99 and D¯ 3.99,respectively. Thus we have two knots and three price ranges: price6 2.99;3.00 < price < 3.99 and price P 4.00. Table 3 shows the number of itera-tions, the log-likelihood, ICOMP, R2 and entropy (Es) criteria for each ofthe S� 1 to S� 6 solutions. The ICOMP criterion indicated the S� 5 seg-ment solution to be optimal. However, the di�erences between the S� 4and S� 5 segment solutions were small. In the latter solution one segmentwas split into two segments with similar interpretation and managerial impli-cations. We report the S� 4 segment solution, because it is more parsimoni-ous. The four segments were pro®led by regressing the logit-transformedposterior probabilities on the consumer descriptor variables.


R2 is 0.435 for the S� 4 segment solution. The value of E4 of 0.913 indi-cates that the segments were very well separated. The estimates of the param-eters of the four segment solution are contained in Table 4. The parametersof the four segments are well interpretable, and conform to our expectationsto a large extent, as will be detailed below. Fig. 1 displays the form of theprice-utility function in the four segments. (The ®gure was calculated fromthe PRICE# and PSYPR# coe�cients, holding the other e�ects constant).

Segment 1 contains 30.6% of the sample and displays a negative price re-sponse function (Fig. 1). In the lower price range demand is inelastic(PRICE0). Above D¯ 2.99 utility decreases faster with price (PRICE1,p < 0.10). This decrease is enhanced in the highest price range (PRICE2,p < 0.10). To this segment of consumers, price represents the amount ofmoney that must be sacri®ced in order to obtain the brand, and therefore

Table 4

Parameter estimates for the S� 4 solution

Parameter /Segment 1 2 3 4

INTCPT 7.240 b 3.880 b )20.960 b )11.190 b

PRICE0 )0.012 )0.020 b 0.068 b 0.056 b

PSYPR1 (2.99) )0.022 1.987 b 0.794 0.520

PRICE1 )0.024 a )0.032 b )0.058 b )0.028

PSYPR2 (3.99) )0.783 a 1.390 0.075 )3.710 b

PRICE2 )0.020 a )0.009 )0.014 a )0.072 b

PRLOSS )0.042 b )0.025 b 0.143 b 0.035 b

REFERP 0.009 b 0.015 b )0.011 b )0.017 b

hs (relative size of segment s) 0.306 0.314 0.205 0.175

a p < 0.10.b p < 0.05.

Table 3

Statistics of the S� 1 to S� 6 solutions

No. of Classes No of Iterations Log L ICOMP R2 Entropy Es

1 2 )2184.320 4447.256 0.094 )2 14 )1801.777 3778.856 0.277 0.903

3 49 )1549.788 3354.552 0.376 0.910

4 28 )1383.251 3130.700 b 0.435 0.913

5 19 )1290.894 3052.293 a 0.465 0.911

6 33 )1242.535 3071.440 0.480 0.912

a Denotes minimum ICOMP.b Denotes the solution selected.


prices a�ect purchase probabilities negatively. At D¯ 2.99 there is no signi®-cant psychological price e�ect. Apparently at prices upto D¯ 3.00 consumershardly perceive price-changes to be important. At D¯ 3.99, there is a signi®-cant psychological price e�ect. Apparently, D¯ 3.99 is perceived as a reducedprice relative to a possible fair price of D¯ 4.00. At prices above D¯ 4.00,consumers are more sensitive to price changes, since the perceived sacri®ceincreases sharply. The e�ect of the reference price (REFERP) is positive, in-dicating that higher expected prices lead to higher intentions of buying thebrand. This was expected to occur in the presence of negative linear price ef-fects (PRICEk). If the reference price is higher, the whole price-sequence of-

Fig. 1. The price-utility function in segments 1±4, personal care brand.


fered appears to be less of a monetary sacri®ce. The coe�cient of PRLOSS isnegative, which indicates that prices above the reference price more negative-ly a�ect purchase probabilities than prices below the reference price. This wasexpected on the basis of prospect theory, according to which consumers reactmore strongly to price losses (higher prices) than to price gains. Membershipin segment 1 is associated with lower income, and not previously having pur-chased the brand.

In Segment 2, comprising 31.4% of the sample, utility also decreases as afunction of price. The decrease is not monotonic: in the second price range(PRICE1) it is stronger than in the ®rst price range (PRICE0). The priceresponse function decreases at the same rate in the third price range(PRICE2). To these consumers price represents a monetary sacri®ce. Forthis segment the psychological price e�ects are quite di�erent from thosein segment 1. A signi®cant psychological price e�ect is present at D¯2.99 (PSYPR1). There is no psychological price e�ect at D¯ 3.99(PSYPR2). Apparently, at D¯ 2.99 price is perceived to be reduced relativeto a fair price of D¯ 3.00. The coe�cient of REFERP is positive, indicatingthat higher reference price leads to a higher probability of indicating a buy,which is consistent with expectations and the ®ndings in the ®rst segment.Just as in segment 1, we ®nd a negative coe�cient for PRLOSS. Note, how-ever, that loss aversion is less strong in this segment as compared to seg-ment 1. Consumers that have previously purchased the brand have ahigher probability of belonging to this segment.

Segment 3 is smaller than the segments 1 and 2 and contains 20.5% ofthe sample. Contrary to segments 1 and 2, in this segment utility increaseswith price in the ®rst price range (PRICE0). The price responses in the sec-ond and third price range are almost inelastic. Consumers in this segmentuse price as an indicator of quality. Note that consumers apparentlytrade-o� perceived-quality and perceived-sacri®ce. The price±perceivedquality relationship is primarily found in the lowest price range, but inthe higher price ranges the monetary sacri®ce becomes more important,so that the positive price e�ect levels o�. No signi®cant psychological pricee�ects are observed in this segment. When the price-response is positive,psychological prices may not be perceived as price reductions relative toa fair price, so that transaction utility is not increased at such price levels.Consistent with the positive price relation, the coe�cient of reference price(REFERP) is negative, which was expected. This indicates that low priceexpectations (i.e. a low price last paid) will lead to prices across the entireo�ered price sequence being perceived as indicating higher quality, which


results in higher purchase probabilities. The coe�cient of PRLOSS is pos-itive, which con®rms our expectations on the basis of prospect theory. Sinceprice is an indicator of quality in this segment, higher prices losses corres-pond to perceived quality gains, so that the term PRLOSS, representinghigher prices relative to the reference price, indicates perceived qualitygains. The coe�cient for prices above the reference price (PRLOSS) is pos-itive, indicating a stronger price±perceived quality relation for these prices.This ®nding supports prospect theory, which predicts the e�ect of a per-ceived quality loss to be greater than that of a corresponding quality gain.The large negative intercept indicates that consumers in this segment have amuch lower overall probability of choosing the brand, as compared to seg-ments 1 and 2. Segment membership does not display strong associationswith consumer descriptors, although consumers in this segment tend tohave a somewhat higher (p < 0.10) income. This corresponds to the ®nd-ings of e.g. Steenkamp (1989), p. 205, that consumers with higher incomesare more quality conscious.

Segment 4 is the smallest segment and contains 17.5 percent of thesample. Here a clear bell-shaped price response function is observed, aswas reported to occur in the original studies of Gabor and Granger(1966). At lower price levels consumers' utility signi®cantly increases withprice (PRICE0). Since PRICE1 is not signi®cant, the increase appears tocontinue in the middle price range. In the highest price range the price-response is clearly negative. This segment of consumers use price as anindicator of quality in the lower and middle price range. For prices aboveD¯ 3.00 the importance of the perceived sacri®ce is apparently greaterthan that of perceived quality, and the price response function decreases.At D¯ 2.99, where the price-response function is positive, there is no sig-ni®cant psychological price e�ect. This ®nding is consistent with the re-sults for segment 3, where also no psychological price e�ects werefound for prices that were used as an indicator of quality. At D¯ 3.99there is a signi®cant psychological price e�ect. Note that the psychologicalprice e�ect is very large indeed, and that this e�ect occurs at the onset ofthe negative price-response function. Consistent with our expectations thecoe�cient of REFERP is negative. Just as in segment 3, and consistentwith prospect theory, the coe�cient for PRLOSS is positive, indicatingthat prices below the price last paid are less strong indicators of quality.Judged by the magnitude of the coe�cients, the loss aversion is muchgreater in segment 3 than in segment 4. Again, as evidenced by the inter-cept, the overall probability of choosing the brand is low as compared to


segments 1 and 2. Consumers that have not previously purchased thebrand have a higher probability of belonging to this segment. This ®ndingmay indicate that these consumers are less sure of the quality of thebrand, and therefore depend more heavily on price as a quality cue(Steenkamp, 1989, p. 95).

6. Validation

In this section the predictive validity of our model is compared to the la-tent class logit model with linear price e�ects. The deterministic part of theutility function of this model has the following, simple structure

Vijjs � c00s � c10spij; �8�where for an individual consumer in segment s, c00s denotes the intercept andc10s represents the linear price e�ect. This is the model proposed by Kama-kura and Russell (1989) in a binary context. Two validation studies were car-ried out, establishing internal and external validity, respectively.

First, as there were no holdout data available, we randomly eliminated10% of the data of the Gabor±Granger study, estimated the models on theremaining data, and predicted choices for the eliminated price levels. To thisend, Eq. (5) was used. This procedure was repeated ®ve times for each model.To evaluate the predictive accuracy we calculated the correlation betweenpredictions and holdout data. This dependent measure was analyzed by uni-variate Analysis of Variance to test di�erences among the methods (the AN-OVA has 90% power to detect e�ects that account for approximately 15% ofthe variance, Cohen, 1991).

The predictive ®t of the latent class spline model was signi®cantly betterthan that of the latent class linear model: the validation correlation for theformer was 0.636, the validation correlation for the latter 0.603 (Standard er-ror of the di�erence, SED� 0.005, p < 0.001). The results support the inter-nal validity of our results. The proposed spline model is superior to the modelwith linear price e�ects only, although the di�erence is modest. The improve-ment is about 5% in the present application, and will depend on the extent towhich the price response functions are nonlinear.

Second, the external validity of our model for the e�ects of psycholog-ical pricing in Gabor±Granger price studies was investigated. This impliesthe validation of the data collection procedure which measures purchaseintentions and the validation of the model which has been calibrated using


these data. Market shares and prices were obtained for the brand in 12two-monthly periods for a representative sample of stores in The Nether-lands, from A.C. Nielsen. The period concerns two years, covering the pe-riod in which the data for the Gabor±Granger study were collected. Themarket shares were adjusted for price-e�ects of competing brands using re-gression methods, and one extreme observation was eliminated from thedata. The two models were used to predict the brand's market shares inthe (remaining) 11 periods in the basis of aggregate level prices for eachsegment.

The correlations of observed and predicted shares were 0.500 for the latentclass linear logit model, and 0.517 for the latent class spline model. The cor-relation of market shares and prices calculated directly from the two-monthlydata was 0.479. The predictive validity of both latent class models thusoutperform the predictive ®t of a regression model ®tted on the market sharesthemselves. The correlations con®rm the somewhat better predictive validityof the latent class spline model. The improvement of the spline model is of thesame order of magnitude as found above. The results support the external va-lidity of the Gabor±Granger procedure in conjunction with the proposedmodel as a procedure for assessing price sensitivity.

7. Discussion and conclusions

Below we discuss the substantive ®ndings derived from our model.

7.1. Price±perceived quality

Our study has yielded supportive evidence of a number of hypotheses andprevious empirical ®ndings. It has provided additional support for consum-ers' use of price as an indicator of quality, in two out of four segments ofthe market. One of these segments displayed a bell-shaped price responsefunction. Such a response curve was already observed by Gabor and Granger(1966) in their original studies (see also Monroe, 1990, p. 114). The bell-shaped curve can be explained as a situation where at lower price levels theimportance of perceived quality, as inferred from price, is higher than the im-portance of perceived sacri®ce, whereas the reverse holds at higher pricelevels.

In our application consumers appeared to be very heterogeneous in theirreaction to price. This is in accordance with the ®ndings of e.g. Zeithaml


(1988), and Dickson and Sawyer (1990). The allocative (price as perceivedmonetary sacri®ce) and informative (price as an indicator of quality) aspectsof price were found to operate simultaneously in two consumer segments,while the allocative aspects clearly dominated in two other segments. This®nding corresponds with the ®ndings of Lichtenstein and Burton (1989)who reported both price±perceived-quality groups of consumers and no-price±perceived-quality groups of consumers.

7.2. Loss aversion

The application of our model has provided additional support for the ex-istence of loss aversion with respect to price (see Kahneman and Tversky,1979). For all four segments the parameter estimates of PRLOSS(b̂3s; s � 1; . . . ; 4) are signi®cantly di�erent from zero. We have providedsupportive evidence for the ®ndings of Hardie et al. (1993), that consumersdisplay loss aversion with respect to quality as well. In the segments with apositive price response function, emanating from the use of price as an indi-cator of quality, such loss aversion with respect to quality was inferred. Con-sumers in these segments relied less on prices below the reference price(indicating quality loss) as indicators of quality. Research is needed to furthersubstantiate these ®ndings (see Bell and Lattin, 1993). Segments of consum-ers di�ered in the extent to which they displayed such loss aversion, both withrespect to price and with respect to quality.

7.3. Psychological price e�ects

Finally, and most importantly, the results have demonstrated that psycho-logical price setting may cause discontinuities in demand (see Monroe, 1973).Whereas Blattberg and Wisniewski (1987) already demonstrated such e�ectson the aggregate level using scanner data, our study includes psychologicalprices in a consumer choice model and demonstrates the e�ects at the disag-gregate level. Although the existence of discontinuous e�ects on demand issupported by our results, it appears from our study that they do not occurat all psychological price levels and for all market segments. As has been es-tablished in the literature on deal-proneness (cf. Blattberg and Neslin, 1990,pp. 77±81), consumers may vary in the extent to which they are sensitive toprice reductions. Segments exist in which the perceived price reduction at thepsychological price level does not lead to an increase in transaction utility.Also, it appeared that consumers may perceive a price reduction at one


psychological price level, but not at another. From these ®ndings, it may beconcluded that the importance of psychological price setting may be overes-timated in pricing decisions in the market place.

7.4. Limitations and future research

We conclude that the approach proposed here is a valuable tool in the inv-estigation of consumers' price sensitivity and we argue that the bene®ts of ourmodel accrue from its use in conjunction with the Gabor±Granger methodand related methods used in pricing research (see Monroe, 1990, pp. 106±137). A limitation of the Gabor±Granger procedure is that the e�ects of pric-es of competitive brands are not considered explicitly. Our model can in fu-ture research be extended to multibrand situations. The external validity ofthe proposed procedure was supported by our validation study on store-levelmarket shares, although the gain in predictive validity over competing proce-dures was modest. Future research should corroborate these ®ndings. It isclear that the external validity of the proposed procedure can yet be furtherenhanced by designing in-store experiments in which actual choices are ob-served in response to price changes, e.g. through check-out scanning devices.The analyses of such experimental scanner data may provide evidence of theexistence of bell-shaped price-response curves, loss aversion, and psycholog-ical pricing e�ects in actual market situations. We leave this for future re-search.

Acknowledgements

We wish to acknowledge A.C. Nielsen, Netherlands for providingaggregate-level sales data. We also thank the reviewers and especiallyAlan J. Mac Fadyen for helpful suggestions on an earlier version of thispaper.

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A model for the effects of psychological pricing in Gabor–Granger price studies

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