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DOI 10.1007/s00291-012-0313-4
REGULAR ARTICLE
Detecting price thresholds in choice models
using a semi-parametric approach
Yasemin Boztu˘g·Lutz Hildebrandt ·
Kalyan Raman
© The Author(s) 2013. This article is published with open access at Springerlink.com
Abstract The semi-parametric methodology, underutilized in marketing, can be
applied to distinguish between competing models of price response and to estimate
the model that most validly describes consumer response to price. The methodology
is robust and flexible, thereby making it applicable to a wide spectrum of models
of consumer response. In the specific context of reference prices, we show that the
semi-parametric methodology helps the manager develop price promotions that most
effectively capitalize on the nature of consumer price response.
Keywords Loss aversion ·Threshold ·Reference price ·
Semi-parametric estimation ·Multinomial logit model
Financial support by the German Research Foundation (DFG) through the Sonderforschungsbereich 649,
and for Boztug by way of research project #B01952/1 is gratefully acknowledged.
Y. Boztu˘g(
B
)
Chair of Marketing with focus on Consumer Behaviour,
Georg-August Universität Göttingen,
Platz der Göttinger Sieben 3,
Göttingen 37073, Germany
e-mail: boztug@wiwi.uni-goettingen.de
L. Hildebrandt
Institute of Marketing, Humboldt University of Berlin,
Spandauer Str. 1, Berlin 10178, Germany
e-mail: hildebr@wiwi.hu-berlin.de
K. Raman
Medill IMC Department, Northwestern University,
1870 Campus Drive, 2nd Floor, Evanston, IL 60208, USA
e-mail: kalyraman@gmail.com
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Y. Boztu˘g et al.
1 Problem motivation and objective of our research
Pricing decisions are a crucial determinant of consumer response and have consider-
able impact on the firm’s profitability. Consequently, mistakes in setting or changing
the price levels are costly to the firm. In particular, price promotions are important deci-
sions that require careful structuring—for example, a price reduction may be achieved
gradually or all at once, but these are two different ways of structuring the price reduc-
tion and they may elicit entirely different kinds of consumer response and different
economic results. Thus, identification of the valid model for a particular market seg-
ment is critical because if we use an incorrect model, the price change may have no
effect or—even worse—have exactly the opposite effect to that intended. Currently,
there are two competing theories of consumer response to price with significantly
different prescriptions for the implementation of price changes—one suggesting rapid
changes in price and the other suggesting gradual changes in price depending upon
whether a price increase or decrease is contemplated. At the heart of both the compet-
ing theories of consumer response to price is the notion of reference prices. Reference
price theory postulates that consumers respond not only to the actual price but also to
the deviation of the actual price from the reference price.
Several studies have shown that ignoring reference prices leads to incorrect con-
clusions about consumer behavior in purchase situations (e.g., Kalyanaram and Winer
1995). The two dominant theories of consumer response to price are built upon the
deviation of the actual price from the reference price. Furthermore, consumer response
may vary asymmetrically to positive and negative deviations. The two theories differ
in how they treat these deviations. Range theory assumes that price variations within
a latitude of acceptance are perceived to be less extreme than they are outside the
latitude of acceptance. That is, within the latitude of acceptance, a price variation is
perceived to be lower (higher) than it actually is. The existence of a latitude of accep-
tance is essentially equivalent to a non-zero threshold effect. Consumers respond more
strongly to those deviations that exceed the threshold, as opposed to deviations below
the threshold. On the other hand, the competing theory—closely related to prospect
theory—assumes that any deviation of the actual price from the reference price, no
matter how small it may be, has an effect on consumer response. Clearly, the latter
theory assumes a zero threshold effect.
Kalyanaram and Winer (1995) urged distinguishing between these competing the-
ories but, until now, this call has been answered in only limited fashion. Only infre-
quently do researchers test the two competing theories to identify the correct one—the
danger here is that the wrong theory often gives significant results and a good fit to
the data, thereby misleading researchers into drawing exactly the wrong conclusions
about consumer response. Our contribution is to offer a non-parametric methodology
that can identify the correct model and estimate consumer response to price more
efficiently compared to existing techniques.
The rest of the paper is organized as follows: In the next section, we discuss the
theoretical background for our research. We then describe our non-parametric and
parametric methodology. Next, we present our empirical results using both simulation
and real data. We conclude with a discussion of our results, managerial implications,
limitations of this paper and suggestions for future research.
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Detecting price thresholds in choice models
2 Theoretical background
There is strong empirical evidence that consumer response is driven not solely by the
price itself, but rather by its relation to the reference price of that product. Over the last
three decades an enormous literature on this topic has been published, a large body
of which is concerned with the subtleties involved in conceptualizing, defining and
measuring reference prices (e.g., Biswas et al. 1993;Gijsbrechts 1993 ;Kalyanaram
and Winer 1995;Mazumdar et al. 2005;Winer 1988).
The significance of the reference price concept for our research is straightforward—
if the price exceeds the reference price, consumers perceive a loss, and if the price
is below the reference price, consumers perceive a gain. The two competing theories
of consumer response to changes in price differ on how they treat losses and gains.
According to both schools of thought, consumers exhibit greater sensitivity to losses
than to gains—a phenomenon known as loss-aversion. But one theory assumes that
every price change is either a loss or gain regardless of how small the change might
be (Kahneman and Tversky 1979;Tversky and Kahneman 1981). The other theory
assumes that there is a latitude of indifference within which consumers are not sensitive
to changes in price (Sherif and Hovland 1961;Janiszewski and Lichtenstein 1999;
Niedrich et al. 2001). The latter theory is often called range theory—we will refer to it as
loss aversion theory with a non-zero threshold (e.g., Kalyanaram and Little 1994;Han
et al. 2001;Raman and Bass 2002;Pauwels et al. 2003). Our description distinguishes
the latter theory from the former one in which there is no zone of indifference—we
will refer to the former theory as loss aversion theory with zero threshold.
If loss aversion occurs with a non-zero threshold, the managerial implications are
very different than if loss aversion occurs with zero threshold. A threshold effect is
equivalent to a range of prices—a zone of indifference—within which people are
indifferent to changes in price. In the absence of a threshold effect, there is no zone of
indifference—therefore the price should not increase even in a gradual way but may be
lowered in a gradual way. On the other hand, in the presence of a threshold effect, there
is a zone of indifference within which price changes will be ineffectual—under these
circumstances, price increases may be done gradually, but price reductions should not
be made in a gradual way. This is because any price reduction within the zone of
indifference will lower revenue without stimulating demand. Since price deviations
require measurement of the reference price, we briefly discuss alternative methods
reported in the literature for measuring reference prices. In Fig. 1,wegiveanoverview
how gains and losses influence the utility for a model of loss aversion with and without
a threshold effect. Special emphasis should be given on the zone around zero [price
equals the reference price (P−Pref =0)], which shows how differently these two
theories view consumer response to changes in price.
Because the reference price is a latent construct, it cannot be measured directly.
One way of measuring it is to conduct experiments or to ask consumers to state their
expected price but each of these methods have their own problems, as discussed in
Estelami et al. (2001). Scanner data provide other information that can be used to
estimate a reference price. There are two ways of accomplishing this. One method
uses prices paid in the past and the other uses current prices. In the first approach, it is
assumed that consumers use prices paid in the past to construct their reference price;
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Y. Boztu˘g et al.
Fig. 1 General figure for loss aversion model with zero threshold and loss aversion model with non-zero
threshold
this is called an “internal reference price” (e.g., Winer 1986). In the second approach,
it is assumed that consumers use prices at the point of sale, which are actual prices;
this is called an “external reference price” (e.g., Hardie et al. 1993). For an empirical
comparison of both methods, see, for example, Briesch et al. (1997).
Clearly, the internal reference price applies only to consumers who are able to
recall past prices. These consumers are likely to be either highly involved shoppers
or price-sensitive consumers (e.g., Rajendran and Tellis 1994). They have an internal
reference price for each brand. There are various formulations for computation of an
internal reference price, but none of them have become standard practice (e.g., Wricke
et al. 2000).
The external reference price is based on the current prices at the point of purchase.
It is not brand-specific, but only category-specific. It is assumed that the consumer
does not use any information on past purchases, perhaps because he cannot remember
past prices paid or does not trust his memory of them. Therefore, this type of consumer
uses only information that is available during the current shopping trip. As with the
internal reference price, the literature provides no consensus about the best way to
measure external reference price.
3 Description of the methodology
3.1 General discussion on parametric and non-parametric estimation methods
Researchers have traditionally relied heavily upon parametric estimation methods. We
supplement the traditional methodology with a sophisticated non-parametric technique
to identify the correct theory rather than impose one theory or the other upon the data
as parametric techniques do. Non-parametric techniques do not commit themselves
to specific functional forms to describe the input–output relationship. Consequently,
they are flexible and very well suited to let “the data speak for themselves.” However,
non-parametric techniques are not the same as ‘data mining’ which is devoid of theory;
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Detecting price thresholds in choice models
in fact non-parametric techniques can identify from competing theories that one which
best describes the data.
Our methodology is best described as semi-parametric because we combine both
non-parametric and parametric approaches. The semi-parametric approach has a num-
ber of attractive features. For example, when the correct theory turns out to be loss aver-
sion with a threshold, the methodology automatically yields estimates of the threshold
points. Another advantage of the semi-parametric technique is that it does not assume
symmetric threshold effects as parametric methods do. Finally, the semi-parametric
method offers a more efficient approach to dealing with consumer heterogeneity issues.
For example, one source of heterogeneity is that some consumers are brand loyal
and others are not. Suppose the market is segmented by loyalty. Then within each
segment—loyal and non-loyal—there are two possible models: one with and another
without a threshold effect. Consequently, using parametric techniques, we would have
to estimate four possible models. The semi-parametric method could potentially reduce
the estimation complexity because if it identifies the correct model to be loss aversion
with no threshold, then we need not deal with the estimation of threshold effects,
which is typically a complex matter.
While the non-parametric approach is powerful precisely because it assumes no spe-
cific functional forms, that very strength is also a weakness in that the non-parametric
model has no controllable parameters that the manager can use to fine-tune the mar-
keting program. However, this issue may be addressed in the following manner. We
can compute the slopes of the utility function at all the price points in the gain regime,
repeat the procedure in the loss regime, and then determine the average slope. This can
then be used as the imputed beta in the model to find optimal prices and promotions
in the three regimes under consideration—gain, loss, and indifference. Admittedly,
this is a purely numerical recipe and lacks the elegance of an analytical closed-form
result. While the lack of analytical optimality results may seem dissatisfying from an
academic point of view, we get an easy implementable pricing algorithm that adds
value from a managerial perspective.
We propose the following two-stage procedure. First, estimate a reference price
model with a non-parametric price response function without pre-supposing either loss
aversion with zero threshold or loss aversion with non-zero threshold. One of the results
of the non-parametric estimation is the discovery of threshold levels. Second, given the
identified theory from the non-parametric phase, estimate a parametric model repre-
senting that theory, and, if necessary, use the non-parametrically determined width of
the indifference zone for the model. Note that the model of loss aversion with a thresh-
old subsumes the model of loss aversion without a threshold. If no zone of indifference
exists, then the model of loss aversion with a threshold will estimate the threshold to be
zero. The semi-parametric approach facilitates detection of the threshold—it combines
identification of the true model and its estimation in one fell swoop.
3.2 Description of algorithms
Our algorithm uses individual level data. The general model structure is an additive
random utility framework (e.g., McFadden 1974), as described below, where Uin is
the overall utility of brand ifor consumer n,Vin is his systematic utility component
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Y. Boztu˘g et al.
and εin is the random component.
Uin =Vin
systematic utility
+εin
random component
.
Different formulations—either parametric or non-parametric—can be specified for
the systematic and random components. In one of our formulations, both components
are described in a parametric way and the model can be formulated as a multinomial
logit model (e.g., McFadden 1974). The other model is a semi-parametric one resulting
from a non-parametric formulation of the systematic part and parametric modeling of
the random component. A suitable model description for this type is a generalized
additive model (e.g., Hastie and Tibshirani 1990).
3.2.1 Parametric approach
A well-known disaggregated choice model is the multinomial logit model (MNL),
which was introduced in the marketing context by Guadagni and Little (1983). Here,
both the systematic component and the random component are modeled parametrically.
For the difference of the errors, we assume a Gumbel distribution as is standard practice
in the literature on MNL models. The model structure is linear-additive. A popular
MNL model is:
Prn(i)=exp(βxin)
j∈Cnexp(βxjn)(1)
where Prn(i)is the probability of individual nto buy product i,xin the explanatory
variables of product ifor consumer n, and βthe parameter vector to be estimated. We
use maximum-likelihood estimation.
3.2.2 Semi-parametric approach
Our semi-parametric method provides an alternative to the MNL model. Here, we focus
on a generalized additive model (GAM) (Hastie and Tibshirani 1990).1This method
has several advantages. The error structure can be modeled as in the MNL framework,
thus facilitating comparison of the estimation results. The systematic component of
a GAM has an additive structure, as does the MNL. The main difference from the
MNL is a one-dimensional non-parametric function for each explanatory variable, in
contrast to the linear modeling of the explanatory variables in a MNL. Consider the
conditional expectation of Y, given a specific value xof X, denoted by the notation
E[Y|X=x]. We use conditional expectations, because this is the standard way to
describe models in a generalized way (also see the description for generalized linear
models (McCullagh and Nelder 1989)). The formal model of a GAM is presented in
1An alternative approach using P-Splines is used by Steiner et al. (2007), but their main focus is on
estimating price response functions rather than incorporating any reference price.
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Detecting price thresholds in choice models
Eq. (2).
E[Y|X=x]=G
p
fp(xp)with G=1
1+exp(−x)(2)
Here, the fpare one-dimensional non-parametric functions to be estimated and xp
are the explanatory variables; Gis called a link function. By specifying Gas in Eq. (2),
the error term distribution of the GAM is the same as in the MNL approach.
The standard GAM is defined only for continuous explanatory variables. In market-
ing applications, there are usually also categorical explanatory variables, for example,
display or feature. To incorporate these variables, the common GAM is extended to
include a linear-additive component, as in the MNL model. Our extended GAM is
defined in Eq. (3).
E[Y|X=x]=G
p
fp(xp)+βx(3)
In Eq. (3), βxis the linear component, and xcontains both continuous and cat-
egorical variables. Two different algorithms exist for estimating a GAM. The most
common approach is backfitting, introduced by Friedman and Stuetzle (1981). The
method is based on a variance decomposition of the total variance into the variance
components accounted for all explanatory variables specified in the model. An alter-
native method is called marginal integration, where the marginal influences of the
explanatory variables to the response variable are estimated. We do not describe the
mathematical structure of the estimation procedures in detail because the reader will
find excellent treatments of the topic in Hastie and Tibshirani (1990)orLinton and
Nielsen (1995).
3.2.3 Integration of the reference price concept in a parametric model
The integration of the reference price into the standard choice model is achieved by
defining two components which describe the loss or gain perceived by the consumer.
The pure price component should be included only in a model with an internal reference
price formulation. In using an external reference price, the price component has to be
excluded due to strong correlations with the gain and/or loss components (e.g., Hardie
et al. 1993). The resulting model describing the utility Uint of consumer ifor product
nat time tis shown in Eq. (4).
Uint =β0+β1Pint +β2(Pint −IRPint)IPint>IRPint
Loss
+β3(IRPint −Pint)IIRPint>Pint
Gain
+β4LOYint +β5DISPLint +β6FEATint +εint (4)
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Y. Boztu˘g et al.
In Eq. (4)IRP is the internal reference price. The variable ‘I’ is an indicator vari-
able which takes on the value when the condition indicated in its subscript is satis-
fied and is zero otherwise. For example, the variable IPint>IRPint is equal to 1 when
Pint >IRPint—thus is equal to 1 only when the price exceeds the internal reference
price. Consequently, the price deviation (Pint −IRPint)will influence the utility only
when the price exceeds the internal reference price. The corresponding parameter β2
measures the sensitivity of the utility to increases in price above the internal reference
price. The gain and loss components come into play only when the internal reference
price is larger or smaller, respectively, than the actual price. All other explanatory
variables are included as in a standard logit model, for example, Pfor price, LOY for
loyalty, as specified by Guadagni and Little (1983), DISPL for the binary variable for
the existence of display, and, in the same manner, FEAT for feature. The parameter
vector βcontains the parameters to be estimated.
In a loss aversion model with non-zero threshold, a mid range, also known as a
range of indifference, must be included. Here, we assume again that we have loss and
gain components (Kalyanaram and Little 1994).2In the formal representation of a loss
aversion model with non-zero threshold (Eq. (5)), in addition to the common variables,
as specified for Eq. (4), two additional parameters, δGain and δLoss, need to be specified.
The sum of δGain and δLoss describes the width of the range of indifference. In the
model presented here, we assume a non-symmetric indifference zone around the price
gap (price minus reference price). This is a generalization of the model presented by
Kalyanaram and Little (1994).
Uint =β0+β1Pint +β2(Pint −IRPint)IPint>IRPint+δLoss
Loss
+β3(Pint −IRPint)IIRPint−δGain<Pint<IRPint+δLoss
Mid
+β4(IRPint −Pint)IPint<IRPint−δGain
Gain
+β5LOYint +β6DISPLint +β7FEATint +εint (5)
In a parametric approach, it is nearly impossible to detect δGain and δLoss if these are
assumed to be unequal. Even in the symmetric case, a grid search for δGain and δLoss is
laborious. This is where the non-parametric approach enjoys a decisive advantage—it
facilitates much easier estimation of unequal δGain and δLoss.
2In alternative approaches, Han et al. (2001), Raman and Bass (2002), and Pauwels et al. (2003)use
models with price threshold components. The price threshold models are not clearly based on assimilation
contrast theory; they seem to be more data driven (Raman and Bass present some theory, but their model
is at the aggregate level and they use only a reference price and not the difference between price and the
reference price).
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Detecting price thresholds in choice models
3.2.4 Integration of the reference price concept in a semi-parametric approach
The semi-parametric estimation combines non-parametric and parametric terms in
the specification of the utility function. All continuous explanatory variables can be
modeled with a one-dimensional non-parametric function. In a reference price context,
these variables are loyalty, price gap, and price. Price gap is defined as price minus
reference price. It is negative for a gain and positive for a loss. Equation (6) does not
include a non-parametric function for price because the price component is used only in
models with an internal reference price effect (see the section on loss aversion above).
Uint =β0+β1Pint +f1(PRICEGAPint)+f2(LOYint)
+β2DISPLint +β3FEAT +εint (6)
The fi(i=1,2) is a one-dimensional non-parametric function [see Eq. (2)formore
explanation]. In many models, using a non-parametric function for loyalty leads to
an approximately linear functional form (see e.g., van Heerde et al. 2001). The key
non-parametric component is the price gap. Therefore, our semi-parametric approach
reduces to an equation with only one non-parametric function for the price gap. By
inspecting the estimated functional form of the price gap, the researcher can determine
whether loss aversion with zero or non-zero threshold provides the best description
for the data.3
4 Empirical application
4.1 Simulation study
We conducted a set of simulations with two goals in mind. First, we wanted to show
the consequences of estimating a model representing one theory, for example, a loss
aversion model with non-zero threshold, when the data in fact follow the other theory—
namely, a loss aversion model with zero threshold. Second, we wanted to illustrate the
power of the semi-parametric approach through its ability to detect the correct theory
underlying the data.
We generated data sets based on a sample size of 5,000 purchases for 200
households—the consumer choices in one data set reflected loss aversion with a non-
zero threshold, and the consumer choices in the other data set reflected loss aversion
with a zero threshold. This leads to four possible scenarios corresponding to estimat-
ing each of the two possible models using data generated by the same or by the other
model. Two additional scenarios correspond to estimating the semi-parametric model
on each of the two data sets—if the semi-parametric method works, it should correctly
identify the non-zero (zero) threshold for the data set that was generated according to
a loss aversion model with non-zero (zero) threshold. In Table 2,wegiveanoverview
showing which tables and figures display our estimation results. Thus, we have esti-
3For example, Pauwels et al. (2003) remark that fully non-parametric models have high data requirements,
which is not the case in our approach because we model only one component non-parametrically.
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Y. Boztu˘g et al.
Tab l e 1 Variable specification for simulation study
Brand 1 Brand 2 Brand 3
Price Uniform [0.75; 1.00] Uniform [0.65; 0.90] Uniform [0.60; 0.70]
Promoted price Uniform [0.50; 0.70] Uniform [0.40; 0.55] Uniform [0.45; 0.55]
Promotional
frequency
Randomly 1 out of 4
weeks
Randomly 1 out of 4
weeks
Randomly 1 out of 8
weeks
Feature frequency Randomly 1 out of 16
weeks
Randomly 1 out of 16
weeks
Randomly 1 out of 36
weeks
Tab l e 2 Description of which table displays the estimation results for each of the six scenarios correspond-
ing to three estimation methods used on two data sets
Estimation approach Data generating approach
Loss aversion model with zero
threshold
Loss aversion model with non-zero
threshold
Loss aversion model with
zero threshold
Tab l e 3, 2nd column and Fig. 2a Table 4, 4th column and Fig. 3a
Loss aversion model with
non-zero threshold
Tab l e 3, 3rd column and Fig. 2b Table 4, 2nd (assumption of
asymmetric threshold) and 3rd
(assumption of symmetric
threshold) column and Fig. 3b
Semi-parametric approach Table 3, 4th column and Fig. 2c Table 4, 5th column and Fig. 3c
mation results for a total of six possible scenarios and these are reported in Tables
3and 4. The data sets contain three brands and three explanatory variables—price,
promotional price, and feature. The parameter values for the βs are chosen as can be
seen in Table 3, respectively, in Table 4in the column named “true model”.
The research design is comparable to that of Chang et al. (1999); however, in our
study, we included an internal reference price and used a loss aversion model with zero
threshold to generate one data set, and a loss aversion model with non-zero threshold
to generate the other data set. The error term was created based on the standard MNL
assumption. Thus, we calculated the utility of each brand for each observation based
on the model with the underlying theory. Following random utility theory, the brand
with the highest utility was “chosen” by the consumers. Variable specifications are laid
out in Table 1. In the interest of improving the clarity of the tables with the estimation
results (Tables 3,4,5,6)—which already contain a lot of information—we report
only the key results. However, the first author will be happy to supply the rest of the
empirical results available to interested readers.
For generating the data set based on a loss aversion model with non-zero threshold,
we assumed an asymmetric range of indifference with δGain =0.2 and δLoss =0.1.
This asymmetric assumption follows from Kalyanaram and Little (1994), who pro-
posed that an asymmetric range is much more plausible than a symmetric one. Accord-
ing to a loss aversion model with zero threshold, consumers will react more strongly
to a loss than to a gain, which leads to the assumption that the indifference zone should
be smaller on the loss side than it is on the gain side.
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Detecting price thresholds in choice models
Tab l e 3 Estimation results for simulated data set based on a loss aversion model with zero threshold
True model Loss aversion model
with zero threshold
Loss aversion model
with non-zero threshold
Semi-parametric
approach
P−7−7.42a−7.42a−7.40a
FEAT 22.16a2.16a2.14a
Gain 44.33a4.33a
Plot
Loss –6.5 −6.28a−6.28a
δ0n.a. 0 Plot
−2,263 −2,263 −2,262
¯ρ20.495 0.495 0.495
aTrue parameter value lies within the 95 % confidence interval.Bolded values indicate significant parameter
estimates
Tab l e 4 Estimation results for simulated data set based on a loss aversion model with non-zero threshold
True
model
Loss aversion model
with non-zero threshold,
δAsymmetric
Loss aversion model
with non-zero threshold,
δsymmetric
Loss aversion
model with
zero threshold
Semi-
parametric
approach
P−7−7.42a−7.85a−7.60a−7.26a
FEAT 22.02a1.87a1.83a2.03a
Gain 43.92a3.21a2.96
Mid 0.05 0.52 −0.68 n. a. Plot
Loss −6.5 −6.24a−5.98a−5.30
δ0.2/0.1 0.2/0.1 0.1/0.1 n. a. Plot
−2,383 −2,409 −2,434 −2,386
¯ρ20.469 0.463 0.458 0.468
aTrue parameter value lies within the 95 % confidence interval.Bolded values indicate significant parameter
estimates
4.1.1 Data generated by loss aversion with zero threshold
The first simulation study describes the results for the data set generated from a loss
aversion model with zero threshold. In Table 3and Fig. 2, we present the parametric
and semi-parametric results, respectively.
In Table 3, we show the main estimation results for the simulated data set based
on a loss aversion model with zero threshold. In the table, we present in the first
column the true values for the βs and the size of the indifference zone as denoted of
the generated data set. In the second column, we present the estimation results for a
model which follows a loss aversion model with zero threshold. It can be shown that
when estimating a model for loss aversion with zero threshold using the data set based
on a loss aversion model with zero threshold, all true parameter values lie within the
approximate 95 % confidence interval and are significant.
When estimating a loss aversion model with non-zero threshold on this data set,
as presented in column three, the grid search for δGain +δLoss leads to an “optimal”
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Y. Boztu˘g et al.
a
0.1 0.2 0.3
-0.3 -0.2 -0.1
Loss av. model with
zero threshold
true values
2
1
-1
-2
0
Utility
Losses
Gains
Utility
Losses
Gains
Loss av. model with
non-zero threshold
true values
2
1
-1
-2
b
GAM approach
true values
2
1
-1
-2
c
GAM approach
Loss av. model with
zero threshold
Loss av. model with
non-zero threshold
true values
2
1
-1
-2
d
Utility
Losses
Gains
Utility
Losses
Gains
0.1 0.2 0.3
-0.3 -0.2 -0.1 0
0.1 0.2 0.3
-0.3 -0.2 -0.1 0
0.1 0.2 0.3
-0.3 -0.2 -0.1 0
Fig. 2 Semi-parametric and parametric estimation for the price gap in the simulation study with data
based on a loss aversion model with zero threshold. aThe estimation for a model based on loss aversion
with zero threshold. bThe estimation for a model based on loss aversion with non-zero threshold. cThe
semi-parametric estimation, and da combination of a–c
value of 0, so the model reduces to one regarding a loss aversion model with zero
threshold. Therefore, all estimated values for βare equal to those based on the loss
aversion model with zero threshold (as shown in columns 2 and 3, respectively).
For the semi-parametric model, we specify price and feature in a parametric man-
ner, and the price gap (price minus reference price) in a non-parametric way. The
results for the non-parametric estimation are given in column 4 and they are displayed
graphically in Fig. 2c, d. The horizontal axis in Fig. 2represents the price gap and the
vertical axis denotes the effect of the price gap on the utility function (see Fig. 1for a
general picture as well). The parameter estimates are all significant and the approxi-
mate 95 % confidence interval includes the true parameter values. The semi-parametric
model correctly identifies the model that generated these data. This results hold for the
parametrically estimated sss for price and feature, as well as for the non-parametric
estimation of the price gap as presented in Fig. 2. In addition, we give the values for the
log-likelihood value (l)and the corrected likelihood ratio index ( ¯ρ2), which accounts
for the number of additional explanatory variables.
4.1.2 Data generated by a loss aversion model with non-zero threshold
In the second simulation study, the results for the data set generated from a loss aversion
model with non-zero threshold are described. The indifference zone for the simulated
data is asymmetric. We estimate two different models based on a loss aversion model
with non-zero threshold: one with the true asymmetric values for δGain and δLoss and
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Detecting price thresholds in choice models
one with symmetric δGain and δLoss values, as suggested by Kalyanaram and Little
(1994). All parametric estimation results for a loss aversion model with non-zero
threshold are presented in columns 2 and 3 in Table 4.
With a non-zero threshold, the slope of the utility function is zero in the mid-range
(the latitude of indifference). Consequently, we would expect the parameter estimates
to be insignificant in the mid-range. This result can be found in columns 2 and 3 at
the “Mid” row. As you can see, for both approaches the parameters are insignificant.
All other parameter estimates are significant (for price, feature, gain and loss), and all
the approximate confidence intervals include the true parameter values. In the model
with the true δGain and δLoss values, the estimated parameters are closer to the real
values than the estimates obtained using the symmetric δGain and δLoss values; the
log likelihood is also better in this model than in the one with symmetric values for
δGain and δLoss. In the model with a symmetric indifference zone, we find an optimal
width of δGain =δLoss =0.1, so that, δGain +δLoss =0.2, which is smaller than the
original value. But using an indifference zone than its true value still leads to much
better estimation results than using a wrong approach, as we will demonstrate next.
Using the wrong model for this data set corresponds to estimating a loss aversion
model with zero threshold. Doing so yields the following findings, which are presented
in Table 4in column 4. First, every parameter estimate is still significant. Second, the
approximate confidence intervals of loss and gain do not include the true parameter
values, and this is something that would be undiscovered with real data. Although
the estimates are significant, they are actually not even close to the true underlying
parameters. Therefore, parametric estimation with the wrong model may still generate
significant parameters—despite misspecification, thereby misleading the researcher.
We now turn to the results of the semi-parametric approach. Again, both parametri-
cally estimated parameters are significant and their approximate confidence intervals
include the true values. The non-parametric estimates are shown in Fig. 3c, d.
The functional form of the non-parametric part of the semi-parametric model is
close to the real values, as is the asymmetric indifference zone; clearly, a loss aversion
model with non-zero threshold is at work. Figure 3c, d illustrates the boundaries of the
indifference zone. From the figure, it is obvious that the estimates of a loss aversion
model with zero threshold are wrong, especially in the zone around the price gap,
which is the most relevant area for price discounts.
Several conclusions follow from the simulation study results. Estimating the correct
model for the data yields significant parameter estimates whose approximate 95 %
confidence intervals include the true parameters. Using the “wrong” model for the data
can produce three different outcomes. First, and most important, using a loss aversion
model with zero threshold on a data set that came from a loss aversion model with non-
zero threshold leads to significant parameter estimates, but the approximate confidence
interval will not include the true parameters. This fact would remain undiscovered for
a real data set. Second, if a loss aversion model with zero threshold is the correct
description of the data, estimating the “wrong” model—a loss aversion model with
non-zero threshold— will still produce correct results because the estimated “optimal”
width for the indifference zone will be zero. Third, regardless of which theory is the
“correct” one for a data set, the semi-parametric estimates will clearly indicate which
model should be used. If it turns out that the semi-parametric estimates recommend the
123
Y. Boztu˘g et al.
ab
cd
0.1 0.2 0.3
-0.3 -0.2 -0.1
2
1
-1
-2
0
Utility
Losses
Gains
0.1 0.2 0.3
-0.3 -0.2 -0.1
2
1
-1
-2
0
Utility
Losses
Gains
0.1 0.2 0.3
-0.3 -0.2 -0.1
2
1
-1
-2
0
Utility
Losses
Gains
0.1 0.2 0.3
-0.3 -0.2 -0.1
2
1
-1
-2
0
Utility
Losses
Gains
Loss av. model with
zero threshold
true values
Loss av. model with
non-zero threshold
true values
GAM approach
true values
GAM approach
Loss av. model with
zero threshold
Loss av. model with
non-zero threshold
true values
Fig. 3 Semi-parametric and parametric estimation for the price gap in the simulation study with data based
on a loss aversion model with non-zero threshold. aThe estimation for a model based on loss aversion
with zero threshold. bThe estimation for a model based on loss aversion with non-zero threshold. cThe
semi-parametric estimation. dA combination of a–c
use of a loss aversion model with non-zero threshold, non-parametric estimates can
be applied to provide guidance on the width and even the symmetry (or lack thereof)
of the indifference zone.
4.2 Real data set
The database consists of purchases of one product category in a micro test market.
Confidentiality restrictions prevent us from revealing the brand and category names.
Subject to these confidentiality constraints, we provide the following limited informa-
tion about the data set. The data span 104 weeks and 7 brands of merchandise from a
major retail store. The product category belongs to daily care products. The category
is a typical fast moving consumer good, which is bought frequently by the consumers.
All major brands in the category are included in our data set. During the specified
time period, 876 households made 3,647 purchases. For calibration purposes, we use
the data for the first 80 weeks aside. The remaining 24 weeks are used for validation.
The data consisted of purchase information about the actual price paid and display
and feature levels. From these data, we constructed the loyalty variable proposed by
Guadagni and Little (1983). The literature suggests many different specifications for
reference price concept (internal and external). In this paper, we use the reference
price specification that leads to the best model fit as determined by the log-likelihood
value. Based on this criterion, we present the results for the weighted mean of last
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Detecting price thresholds in choice models
prices as internal reference price and the actual price of last brand bought as external
reference price.
In the weighted mean of last prices (e.g., Winer 1985), not all former prices have
the same weight in the calculation, and these weights must be estimated, a task which
increases the estimation complexity. This form of the internal reference price is similar
to the loyalty measure introduced by Guadagni and Little (1983) and can be modeled
as IRPint =ζIRPint−1+(1−ζ)Pint−1−IRPint−1.
The logic of the Guadagni and Little (1983) specification for IRPint is straightfor-
ward. The current internal reference price is a weighted linear (convex) combination
of the previous period’s internal reference price and its deviation from the previous
period’s actual price. The linear combination is called convex because the weights add
up to one.
For the external reference price (ERPnt),theactual price of the last brand bought
(e.g., Hardie et al. 1993) is used in cases when consumers cannot remember the last
price paid, but can remember the last brand bought. It can be modeled as ERPnt =Pit
if iwas bought in t–1. Raman and Bass (2002) show that a rational expectations
approach, operationalized through Box and Jenkins’ ARIMA technology, also leads
to the actual price of the last brand bought.
4.2.1 Consideration of heterogeneity
Ignoring consumer heterogeneity could lead to incorrect parameter estimates (e.g.,
Chang et al. 1999).4To take consumer heterogeneity into account, we follow the
lead taken by Mazumdar and Papatla (1995,2000), Krishnamurthi and Raj (1991),
and Krishnamurthi et al. (1992). These authors create a priori segments of loyal and
non-loyal consumers.5Assuming that loyal consumers are not very price sensitive, it
follows that they use an external reference price. Non-loyal consumers are much more
price-conscious and thus are able to use an internal reference price.
Following the lead of these authors, we account for heterogeneity of preference
through the loyalty variable. Heterogeneity of response is captured by brand intercepts;
structural heterogeneity is accounted for through segmentation of consumers.
In Table 5, we first look at the parametrically estimated results, which are given
in columns 1–4. We estimate one approach based on a loss aversion model with zero
threshold and one approach based on a loss aversion model with non-zero threshold.
As described before, we estimate one part for non-loyal consumers using IRP and
the other part for loyal consumers using ERP. Display and feature have the expected
magnitudes and signs. The model based on a zero threshold effect shows different
4For a brief presentation of different heterogeneity concepts, the reader is referred to DeSarbo et al. (1999).
5We also used a latent class approach (Kamakura and Russell 1989;Jain et al. 1994) limited to two
segments, but the results are close to the a priori segmentation. The results of the a priori segmentation can
be interpreted more easily so only those results are used in the following. Another reason for not using the
latent class approach is that in that method, group membership is only weakly related to secondary drivers
data (DeSarbo et al. 1997). Erdem et al. (2001) explicitly demonstrated the problems that arise when using
a latent class approach for a reference price model.
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Y. Boztu˘g et al.
Tab l e 5 Estimation results for real data set incorporating heterogeneity
Loss aversion model
with zero threshold
Loss aversion model
with non-zero threshold
Semi-parametric approach
IRP ERP IRP ERP IRP ERP
P−3.55 n.a. −2.04 n. a. −0.87 n. a.
DISPL 0.69 0.58 0.67 0.58 0.61 0.57
FEAT 1.27 1.01 1.24 1.06 1.10 0.98
Gain 3.22 0.20 4.08 0.05
Mid n. a. n. a. −14.39 −5.22 Plot Plot
Loss −3.99 −7.64 −3.47 −6.75
LOY 2.55 7.34 2.72 8.41 2.50 7.51
δn.a. 0.1 Plot
−3,589 −3,581 −3,545
¯ρ20.317 0.318 0.324
Bolded values indicate significant parameter estimates
Tab l e 6 Estimation results for real data set incorporating heterogeneity based on loss aversion model with
non-zero threshold, with parametrically and non-parametrically produced δvalues
Loss aversion model with
non-zero threshold and δsymmetric
(suboptimal)
Loss aversion model with non-zero
threshold and δdetected from
non-parametric estimation results
(optimal)
IRP ERP IRP ERP
P−2.04 n. a. −1.80 n. a.
DISPL 0.67 0.58 0.64 0.60
FEAT 1.24 1.06 1.24 1.03
Gain 4.08 0.05 4.34 0.77
Mid −14.39 −5.22 −13.41 2.61
Loss −3.47 −6.75 −1.94 −7.21
LOY 2.72 8.41 2.84 8.44
δ0.1 0.1 0.14 0.1/0.04
−3,581 −3,575
¯ρ20.318 0.319
Bolded values indicate significant parameter estimates
parameter values for both consumer segments, particularly for all price components
(e.g., price, loss, and gain) and for the loyalty variable.
The estimation results for the model based on a non-zero threshold effect are similar
to those from the zero threshold model except that all price component values are
larger. However, for loyal consumers, the gain and the mid range components are not
significant. The optimal width of the indifference zone is 0.1. The width is found by
a grid search, as explained in Sect. 3.
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Detecting price thresholds in choice models
Loss av. model with
non-zero threshold
Loss av. model with
zero threshold
Utility
GAM approach GAM approach
Loss av. model with
zero threshold
Loss av. model with
non-zero threshold
0.1 0.2 0.3
-0.3 -0.2 -0.1
2
1
-1
-2
0
Losses
Gains
Utility
0.1 0.2 0.3
-0.3 -0.2 -0.1
2
1
-1
-2
0
Losses
Gains
Utility
0.1 0.2 0.3
-0.3 -0.2 -0.1
2
1
-1
-2
0
Losses
Gains
ab
cd
Fig. 4 Semi-parametric and parametric estimation for the price gap in the real data set for loyal consumers
incorporating heterogeneity. aThe estimation for a model based on loss aversion with zero threshold.
bThe estimation for a model based on loss aversion with non-zero threshold. cThe semi-parametric
estimation. dA combination of a–c
The most interesting discovery of the semi-parametric approach is that loyal and
non-loyal customers are described by entirely different perceptual processes. Loyal
customers are clearly following a non-zero threshold model (see Fig. 4) whereas non-
loyal customers are described best by a zero threshold model (see Fig. 5). This is
intuitively sensible since we would expect that loyal customers would tolerate small
price changes whereas non-loyal customers would react to even small price changes.
The parameter estimates for the variables not related to price (display, feature, and
loyalty) are very close to the results from the parametric approaches.
As pointed out previously, the non-parametric estimates are used in two ways.
First, we use it to detect which theory best represents the underlying data set. Second,
if the first step shows non-zero threshold to be the best description of the data, we
additionally use the non-parametric estimates to obtain the optimal δGain and δLoss
for the width of the zone of indifference.
Figures 4and 5show that, for the segment of non-loyal consumers using an internal
reference price, the zone of indifference is a symmetric interval with a width of δGain =
δLoss =0.07, respectively, δGain +δLoss =0.14. Loyal consumers using an external
reference price have an asymmetric zone, with a width of δGain =0.1 on the gain side
and δLoss =0.04 on the loss side. With these new values for δGain and δLoss,were-
estimate our parametric model incorporating heterogeneity by separating consumers
into loyal and non-loyal segments.
Table 6compares the estimation based on a loss aversion model with non-zero
threshold with symmetric δ(as already presented in Table 5) and, due to the parametric
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Y. Boztu˘g et al.
Loss av. model with
zero threshold
2
1
-1
-2
Utility
Loss av. model with
non-zero threshold
GAM approach GAM approach
Loss av. model with
zero threshold
Loss av. model with
non-zero threshold
0.1 0.2 0.3
-0.3 -0.2 -0.1
Losses
Gains
0
2
1
-1
-2
Utility
0.1 0.2 0.3
-0.3 -0.2 -0.1
Losses
Gains
0
2
1
-1
-2
Utility
0.1 0.2 0.3
-0.3 -0.2 -0.1
Losses
Gains
0
2
1
-1
-2
Utility
0.1 0.2 0.3
-0.3 -0.2 -0.1
Losses
Gains
0
ab
cd
Fig. 5 Semi-parametric and parametric estimation for the price gap in the real data set for non-loyal
consumers incorporating heterogeneity. aThe estimation for a model based on loss aversion with zero
threshold. bThe estimation for a model based on loss aversion with non-zero threshold. cThe semi-
parametric estimation. dA combination of a–c
model restriction, “suboptimal” δGain and δLoss with the model using δGain and δLoss
values extracted from Figs. 4and 5. Due to a slightly larger zone of indifference in
the model with δGain and δLoss from the non-parametric estimation, all price-related
values change slightly, the largest changes being to the gain and mid range components
in the external reference price model. Most important is the significant improvement
in model fit, which is also reflected in the predictive accuracy for the holdout samples.
5 Managerial implications
It has long been known that the reference price used by consumers is one of the
key determinants of their purchase behavior. Our semi-parametric method identifies
whether zero or non-zero thresholds best describe consumer behavior based on the
observed data. For the most part, published studies of reference price effects have
assumed a model with zero threshold so that every deviation of the price from the refer-
ence price, no matter how small, has a substantial influence on consumer behavior—a
view that is clearly strongly driven by prospect theory. However, we find that both
zero and non-zero thresholds are possible—a finding that is managerially significant
because of the dramatically different implications of zero and non-zero thresholds for
managing promotions.
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Detecting price thresholds in choice models
We find that whether zero or non-zero thresholds best describe consumer purchase
behavior is contingent upon loyalty characteristics. Loyal and non-loyal consumers
respond differently to price promotions. Loyal consumers behave according to a non-
zero threshold model and furthermore, the indifference zone is asymmetric, being
longer on the gains side and shorter on the loss side. The promotional implications
of this finding are noteworthy—for one thing, it suggests that small promotions will
not be worthwhile to the firm because they will result in a loss of revenue since
consumers will not respond to price reductions within the zone of indifference. Unless
the promotion is big enough to exceed the threshold, it will not have a negligible effect.
The behavior of non-loyal consumers is consistent with a zero threshold model and
this finding has significantly different implications for promotions compared to the
non-zero threshold model. For the non-loyal segment of consumers, we found that
response is very steep around the zero region of the gap between price and reference
price. Clearly, what this finding suggests is that non-loyal customers will respond even
to promotions of small magnitude. Thus a large promotional reduction may not only
be unnecessary—it could be seriously suboptimal from a profitability perspective.
To sum it up, the implications of zero and non-zero thresholds are not just of
theoretical importance—rather they have important managerial implications. Thus,
managers should first find out which of these situations best describes their customer
segments and then tailor their price promotions in the segment accordingly. Our semi-
parametric method facilitates distinguishing between the zero and non-zero threshold
models.
6 Conclusions
We developed a semi-parametric technique, which when used in conjunction with
traditional parametric methods in a two-stage fashion, enables distinguishing between
two important competing theories of consumer behavior in the first stage and estimat-
ing the relevant model of price response in the second stage. By doing so, managers can
structure their price promotions more effectively by basing them on the correct model
of price response. Our simulations showed very clearly that the wrong model often
gives significant results and a good fit to the data, thereby misleading researchers into
exactly the wrong conclusions about consumer response and consequently, designing
promotions sub-optimally. Our contribution is to offer a methodology that can identify
the correct model and estimate consumer response to price more efficiently compared
to existing techniques.
We note the following limitations of our research, which could serve as motivations
for future research. Out of the extensive literature on reference prices, we considered
only two models of the reference price effect. Our rationale for doing so was that
these two are the predominantly used models in the literature. Another limitation is
that our estimation method works in two stages—it would be desirable to have an
integrated estimation framework that combines these stages seamlessly. A potentially
interesting area for future research would be to disentangle different processes that
underlie reference price formation from the consumer’s point of view so that we
can understand the different processes that may explain loss versus gain asymmetry
123
Y. Boztu˘g et al.
(prospect theory offers one explanation). We see these as promising areas for future
work and hope that our paper encourages other researchers to pursue these directions.
Open Access This article is distributed under the terms of the Creative Commons Attribution License
which permits any use, distribution, and reproduction in any medium, provided the original author(s) and
the source are credited.
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