informs

Vol. 25, No. 4, July–August 2006, pp. 384–391
issn 0732-2399  eissn 1526-548X  06  2504  0384®

doi 10.1287/mksc.1050.0181© 2006 INFORMS

Research Note

Estimating Heterogeneous Price Thresholds
Nobuhiko Terui

Graduate School of Economics and Management, Tohoku University, Sendai 980-8576, Japan, terui@econ.tohoku.ac.jp

Wirawan Dony Dahana

Graduate School of Economics, Osaka University, Osaka 560-0043, Japan, dony@econ.osaka-u.ac.jp

Abrand choice model with heterogeneous price-threshold parameters is used to investigate a three-regime
piecewise-linear stochastic utility function. The model is used to explore the relationships between aspectsof consumer price sensitivity and price thresholds using hierarchical Bayes modeling with the Markov chain
Monte Carlo (MCMC) method. This study contributes to the modeling literature on discontinuous likelihoods
in choice models. The empirical application using our scanner panel data set shows that the reference effectand loss aversion are more marked after price thresholds are taken into heterogeneous price response models.
Furthermore, loss aversion is attenuated by using price thresholds than by an aggregate (homogeneity) model
without price thresholds.
Key words: discontinuous likelihoods; reference effect; price threshold; latitude of price acceptance; brandchoice; Bayesian MCMC; heterogeneity; scanner panel data
History: This paper was received November 11, 2004, and was with the authors 5 months for 2 revisions;processed by Greg M. Allenby.

1.

Introduction

are provided for estimation. The model is illustratedusing a scanner panel data set of instant-coffee purchases.
Our model contributes to the modeling literature
on discontinuous likelihoods. Price thresholds generate unconventional likelihood functions and discreteness at the threshold values that make it impossible
to estimate with gradient-based methods such as
maximum likelihood. As discussed by Gilbride andAllenby (2004) for models with consideration sets,
the Bayesian method of data augmentation is ideallysuited for handling discontinuities in the likelihood
surface. Here, the thresholds are associated with alternative likelihood functions that result in an apportionment of household data to alternative models. Thus,
the model contributes to the marketing literature on
structural heterogeneity (Yang and Allenby 2000) by
allowing for abrupt changes among alternative models within the unit of analysis.
Our empirical application adds to the literature on
the presence of reference prices and loss aversion that
initially began with effects uncovered in scanner data,
for example, as indicated by Winer (1986), Mayhew
and Winer (1992), and Putler (1992), and then Changet al. (1999) and Bell and Lattin (2000) showed thatreference price effects disappear when heterogeneity
is incorporated. Our empirical application shows thatthe reference effect and loss aversion return—at least
for the data used in this study—after price thresholds
are taken into a heterogeneous model. In doing so, theAsymmetric price effects are an active area of researchin marketing. Beginning with the work of Helson(1964) who ﬁrst motivated the usefulness of including reference prices in sales and choice models,
researchers have found asymmetric effects and thepresence of price thresholds that reﬂect the presenceof discontinuities in consumer behavior. The literature includes ﬁndings of asymmetric effect relative tothe reference price (Kahneman and Tversky 1979), thedisappearance of these effects in models that accountfor heterogeneity in price sensitivity (Chang et al.
1999, Bell and Lattin 2000), and models in which pricehas no effect within a price interval, i.e., a latitudeof price acceptance (Gupta and Cooper 1992, Kalwani
and Yim 1992, Kalyanaram and Little 1994, Han et al.
2001). However, models with threshold effects yielding intervals of price acceptance have yet to be estimated heterogeneously.
This paper introduces a heterogeneous choicemodel with reference prices, asymmetric responses,
and price thresholds. It is characterized as a piecewise linear form so that consumers switch their utility
structure accordingly as determined by the relationship between the sticker shock; i.e., the difference
between retail price and reference price, and pricethresholds. Heterogeneity is incorporated through arandom-effects speciﬁcation that allows determination of the relationship between the threshold values and consumer characteristics. MCMC algorithms
384

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Marketing Science 25(4), pp. 384–391, © 2006 INFORMS

degree of loss aversion is attenuated relative to results
obtained using the homogeneity model without pricethresholds.
The remainder of the paper is organized as follows. Section 2 presents our stochastic utility function and its consequent choice model—the threshold
Probit model—and we develop a hierarchical Bayesmodeling for it. Section 3 describes the applicationof scanner panel data to our model. Section 4 concludes this paper. The appendix explains details of themodel estimation including the algorithm for hierarchical Bayes modeling via MCMC.

2.

The Model

2.1.

Threshold Probit Model and Hierarchical
Bayes ModelingTo specify the utility function, we assume that consumer h’s utility to brand j at time t of purchase,
Ujht , reﬂects a linear function of k kinds of explanatory variables. We also suppose that consumer h has areference price RPjht for brand j, and two price thresholds r1h and r2h r1h < 0 < r2h . Consequently, we deﬁne
the three regimes—gain “g ,” price acceptance “a ,”loss “l ”—utility function to brand j as

Ujht =i g
g ∗gg 
g 
ujh + Xjht h + jht ≡Ujht





if Pjht − RPjht ≤ r1h 






ua + X a ∗a + a ≡U a 
jh

jht

h

jht

jht


if r1h < Pjht − RPjht ≤ r2h 






l
l
∗l
l 
l 


ujh + Xjht h + jht ≡Ujht
if r2h < Pjht − RPjht (1)where Xjht is the k-dimensional row vector of explanatory variables allocated to regime i according to
the level of sticker shock Pjht − RPjht Pjht : the retailprice exposed to consumer h) at the occasion, and∗ik-dimensional column vectors h , i = g a l, represent different market responses around the reference
iprice. Finally, let jht , i = g a l, respectively represent stochastic error components in the utility foreach regime, and they are assumed to be independent
across regimes.
Equation (1) is capable of simultaneously reﬂecting
the presence of reference prices (Helson 1964), asymmetric effects (Kahneman and Tversky 1979), andprice thresholds (Sherif et al. 1958). Asymmetries arepresent when the  coefﬁcients in the different regions
take on different values. The price thresholds, rsh 
s = 1 2 , identify different regimes associated with thedata. The model captures a latitude of price acceptance (LPA hereafter) Lh = r1h  r2h  where prices may
not affect choice if price is sufﬁciently small.

Figure 1

Model for Price Thresholds and Market Responses

(g)

βA

LPA

(g)

β

(g)

βBL

r1

Gain

r2

Loss

Model (i)Model (ii)
Model (iii)

(l)

(l)

(l)

βA

β

βBL

Notes. Model (i): aggregate (homogeneous) reference price Probit model
without a threshold; Model (ii): two-regime heterogeneous reference price
Probit model without a threshold (Bell and Lattin 2000); Model (iii): threeregime heterogeneous reference price Probit model with thresholds.

Figure 1 displays our utility function. In orderfor the second regime of the utility function (1) tobe characterized as the LPA, and for r1h and r2h to
be interpreted as price thresholds literally, i.e., forour proposed model to be recognized as a pricethreshold model, we impose the restriction of insensitiveness on the price-response parameter in the LPAaaregime as hp ∼ N 0 hp 2 , which is an element of∗ah . Under the assumption of consumer homogeneity, Kalyanaram and Little (1994) estimated symmetric
LPA that had an insigniﬁcant price-response estimate
over this range. On the other hand, Han et al. (2001)
ﬁnd a signiﬁcant response parameter estimate in the
LPA. We note that their framework differs from ours
particularly in the respect that they use the samedata set throughout regimes. In contrast, our modeling method allocates data into three regimes.
Following the standard multinomial-brand choicemodel, we assume that consumer h is observed tomake a choice at the period t between m alternatives(cht = j . That choice is driven by the relative utility
from the last brand yjht = Ujht − Umht among latent utilities Unht , n = 1 2     m. We consider a panel of H
households observed over Th periods for each household. Conditional on the vector of threshold rh =
r1h  r2h for consumer h, the underlying latent structure at period t when the latent utility induced by the
choice belongs to the regime i, the so-called “within
subject model for regime i” is expressed in the formof m − 1 dimensional multiple regression,
i

i

i

i

yht = Xht h + !ht "i!ht ∼ N 0 #i 
ih = 1     H  t = 1     Th  i = g a l (2)Terui and Dahana: Estimating Heterogeneous Price Thresholds

386

Marketing Science 25(4), pp. 384–391, © 2006 INFORMS

iwhere yht is the m − 1 dimensional relative-utility
ivector, Xht is the m − 1 × k + m − 1 explanatory
ivariable matrix measured from the last brand, h isik + m − 1 dimensional coefﬁcient vector, !ht is them − 1 dimensional stochastic-error vector, and wegalhave the relation Th + Th + Th = Th . Consumers’
heterogeneity is formulated by way of a hierarchicalregression model, as proposed by Rossi et al. (1996).
Regarding price thresholds, we employ the model
r1h = Zhr %1 + &1h "r2h = Zhr %2 + &2h 
h = 1     H  (3)where Zhr is a vector of d kinds of household speciﬁcvariables. We assume that r1h < 0 < r2h for identiﬁ2
for s = 1 2. We also set a
cation and &sh ∼ N 0 s&hierarchical structure of “between subjects model for
regime i” for the market-response parameterii
i iidi h = (i Zh + )h ")h ∼ N 0 V 
h = 1     H  i = g a l (4)where Zh contains another vector of dkinds of household speciﬁc variables. In particular, we note thataprice response hp in the LPA is assumed a priori tohave zero mean in (4).
Following the utility function deﬁned as (1), consumer h’s probability of choosing brand j is written as  g gg
Pr yjh = max y1h ym−1h > 0






 if Pjht −RPjht ≤ r1h Pry a = maxy a y a  > 0  R
jh
1h
m−1h
Pr+ch = j, =
(5)

if
r
<
P
−RP
≤
r

1h
jht
jht
2h 






 l
 l

l


Pr yjh = max y1h ym−1h > 0
if r2h < Pjht −RPjht 
a

a

a

where Pr+yjh = maxy1h      ym−1h > 0  R, indicatesthe choice probability under the restriction on priceaa
response hp ∼ N 0 hp 2 in the LPA regime.
2.2.

Discontinuous Likelihoods for Thresholds
and Their Modeling
Our model includes a threshold variable of sticker
shock Pjht −RPjht in the model, and it induces discontinuity in the likelihoods, as shown in Figure 1. Conditional on the value of rh = r1h  r2h , the operation generates the latent utility with three different likelihood
functions for consumer h. The independence assumption of stochastic errors across regimes generates theilikelihood function of h  #i for consumer h:i=g a l t∈Ri rh  i
i i #i −1/2 exp − 12 yht − Xht h i
i i · #i −1 yht − Xht hwhere each datum is assigned to one regime Ri rh ,and it holds that Rg rh ∪ Ra rh ∪ Rl rh = Th .
i
In turn, conditional on h  #i , we take the above
as a function of rh to compose the likelihood functionof price thresholds. Under the assumption of independent choice behavior across consumers, we havethe conditional likelihood function of +rh , by taking
products over respective consumers as
iL+rh ," +Iht , +Xht ,  +h , +#i ,
∝

H
h=1

i=g a lt∈Ri rh i
i i #i −1/2 exp − 12 yht − Xht h i
i i · #i −1 yht − Xht h (6)where +Iht , means the personal observed choice data.
Coupled with the prior (3) expressed as hierarchicalstructure, we can apply the Metropolis-Hasting sampling algorithm for price thresholds to obtain condiitional posterior rh  +Iht , +Xht , +h , #i  +Zh , % 1& .
Prior distributions and MCMC estimation proceduresfor these hierarchical Bayes models are described in
the appendix.
The proposed model is difﬁcult to estimate withconventional methods because the likelihood is notdifferentiable in rh . Conventional maximum likelihood estimation collapses and classical asymptotic
distribution theory is not operative on this parameter,
and it must therefore rely on an adaptive approach:conditioned on some speciﬁc value of rh , estimationis conducted through extensive use of goodness-of-ﬁtcriterion, like AIC, in the way of a grid search to ﬁnd
the estimate that minimizes the criterion.
In the price-threshold literature, Kalyanaram and
Little (1994) assume symmetric LPA by setting rh1 =rh2 = r (homogeneous), and they estimate not thethreshold but the length of LPA, where several possible lengths are speciﬁed a priori, and they seek
the LPA that has an insigniﬁcant estimate of priceresponse inside this range. Kalwani and Yim (1992)
take a grid search approach. This approach does notallow statistical inferences such as testing hypotheses and conﬁdence intervals. Another approach is
Bayesian inference. For example, Ferreira (1975),
Geweke and Terui (1993), and Chen and Lee (1995)
provided Bayesian approaches to deal with discontinuous likelihoods in econometrics and nonlinear timeseries models. Our proposed method, using more
updated tools, directly models price thresholds inchoice models in a general way and conducts coherent statistical inference on them under a small-sample
situation. The advantage of our approach is that it
avoids such speciﬁcation search and can accommodate heterogeneity.

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Marketing Science 25(4), pp. 384–391, © 2006 INFORMS

3.

Empirical Application to Scanner
Panel Data

3.1.

Data, Variables, and Model Speciﬁcation

Data and Variables. Video Research Inc., Japan,
supplied scanner panel data for the instant-coffee category. In all, 2,840 records for 197 panels during 1990–
1992 were available. We assume that ﬁve national
brands existed in the market during the tracking
period. We deal with ﬁve primary brands in the market: A, B, C, D, and E. Table 1 provides descriptive information about the data. Brand B has the
maximum share—over 48.03%; the minimum share—approximately 5.74%—is for brand E. We rescale all
prices as yen/100g to equalize quantitative differences for each package of the ﬁve brands.
Variables included in the analysis are:
• Explanatory Variables: X = [Constant Price
Display Feature Brand Loyalty], where Price is thelog(price); Display and Feature are binary values; andBrand Loyalty is a smoothing variable over past purchases proposed by Guadagni and Little (1983) as
GLjht = 2GLjht + 1 − 2 Ijh t−1 , where a grid search(Keane 1997) is applied to ﬁx the smoothing parameter as 0.75 based on the criterion of minimum
marginal likelihood.
• Household Speciﬁc Variables: Z r = [ConstantDprone Pfreq ARP BL], where Dprone is the deal
proneness deﬁned as the proportion of purchase (ofany ﬁve brands) made on promotion (Bucklin and
Gupta 1992, Han et al. 2001); Pfreq is the shopping frequency (three categories; H3 of Kalyanaramand Little 1994: shopping frequency is a factor toaffect the price thresholds); ARP represents the measure of average reference-price level as deﬁned byTh
ARPh = m
j=1  t=1 logRPjht /Th /m; and BL represents the brand loyalty measure deﬁned as BLh =Th
maxj  t=1
GLjht /Th , both of which are used by
Kalyanaram and Little (1994) and Han et al. (2001).
We further deﬁne• Household Speciﬁc Variables: Z  = [Constant
Hsize Expend], where Hsize is 1–6 (number of household members) and Expend is nine categories (shopping expenditure/month) used in Rossi et al. (1996).
Table 1

Descriptive Statistics for Data

Alternative

Choiceshare

AveragepriceTime
displayed (%)

Timefeatured (%)

Brand A
Brand B
Brand C
Brand D
Brand E

0.138
0.480
0.099
0.225
0.057

623.5
632.9
601.3
693.2
902.4

0.264
0.135
0.317
0.182
0.191

0.423
0.294
0.405
0.344
0.286

Speciﬁcation of Reference Price. The literature presents some conceptualizations for the reference price
RPjht . Following Briesch et al. (1997), we employ abrand-speciﬁc reference price. We also consider fourkinds of reference prices of two categories: memorybased and stimulus-based. That is,(1) Memory-based as (A) RPjht = Pjh t−1 , the price at
its last purchase and (B) RPjht = 2RPjh t−1 + 1 − 2 ·Pjh t−1 , the smoothed price over previous purchases;(2) Stimulus-based as (C) RPjht = Pkht , where kmeans the brand at the last purchase, and (D) RPjht =Prht , where r indicates the price of a brand chosenrandomly at the time of purchase.
Model Speciﬁcation. We consider three models
for comparison with other candidates: (i) aggregate(homogeneous) reference-price Probit model withouta threshold (Winer 1986, Mayhew and Winer 1992,
Putler 1992); (ii) two regimes’ heterogeneous reference price Probit model without a threshold (Changet al. 1999, Bell and Lattin 2000); and our proposed(iii) three-regime heterogeneous reference-price Probit
model with thresholds. With four types of reference
prices for each model above, summing to 12 models
are considered for comparison.
Table 2 describes logs of marginal likelihood formodel comparison. The model (C)(iii): stimulus-based
RP(C) three regime heterogeneous Probit with thresholds is most supported by marginal likelihood criterion. Consequently, some evidence supports the
hypothesis of price-threshold existence that was discussed by Kalyanaram and Little (1994) even by using
the models incorporating heterogeneous consumers.
The summary statistics of parameter estimates for thatmodel are provided in Table 3. To save space, estimates of constant terms are not listed here.
Our chosen stimulus-based RP is compatible withprevious studies of Hardie et al. (1993), stating thatit seems likely that characteristics of the brand lastpurchased will be more available in memory in particular with consumer package goods, and moreover
Rajendran and Tellis (1994) discuss the reasons ofappropriateness for stimulus-based RP in detail.
Table 2Log of Marginal Likelihood
(i)

(ii)

(iii)

Memory-basedA
B−36700434−36711345−34981349−35836148−33829653−34176325

Stimulus-basedCD−36048875−36140055−34716514−35372912−33658890−34103315(i) Conventional Probit without both heterogeneity and thresholds.
(ii) Two-regime Probit with heterogeneity and without thresholds.
(iii) Three-regime Probit with both heterogeneity and thresholds.

Terui and Dahana: Estimating Heterogeneous Price Thresholds

388
Table 3Marketing Science 25(4), pp. 384–391, © 2006 INFORMS

Model Speciﬁcation and Parameter Estimates RPjht = Pkht
With heterogeneity(i) Without
heterogeneity(ii) Without
threshold(iii) With
thresholds

Gain regimePrice−4251∗ [0.556] −1.637 (−3.875) −3.142 (−4.749)
0.744 (2.793)
1.155 (2.251)
Display
0.526∗ [0.184]
1.665 (7.830)
1.773 (4.021)
Feature1.370∗ [0.353]
Brand Loyalty
1.112∗ [0.426]
1.199 (5.161)
0.413 (1.342)
LPA
Price
Display
FeatureBrand Loyalty−0.907 (−1.711)
0.646 (2.969)
1.595 (7.361)
3.232 (12.601)

Loss regimePrice−7021∗ [1.594] −2.622 (−3.034) −4.518 (−4.725)
1.046 (3.135)
1.086 (2.972)
Display
0.688∗ [0.212]
3.057 (8.044)
2.679 (7.366)
Feature1.720∗ [0.389]
Brand Loyalty
1.293∗ [0.460]
3.483 (8.805)
5.427 (12.846)
Thresholds
LML−36,041.732−34,716.514(−0.113, 0.138]−33,658.890(i) Conventional Probit without both heterogeneity and thresholds.
(ii) Two-regime Probit with heterogeneity and without thresholds.
(iii) Three-regime Probit with both heterogeneity and thresholds.
∗
Signiﬁcant at 0.95 HPD region. The number inside [ ] in the model(i) means standard deviation, and ( ) for (ii) and (iii) means the quasi t-value
calculated from the frequency distribution of household’s estimates.

3.2.

Reference Effects, Loss Aversion, andPrice Thresholds
Our analysis indicates the presence of asymmetric reference effects, loss aversion, and heterogeneity. Thisﬁnding is different from Chang et al. (1999) and Belland Lattin (2000) where asymmetric effects were negligible when heterogeneity and price thresholds wereincorporated into the model.
Corresponding to each type of RP, (A)–(D), we havefour sets of models: (A)(i), (A)(ii), (A)(iii) through(D)(i), (D)(ii), (D)(iii). Based on the most supported
RP of (C), price-response estimates at the aggregatedlevel show some loss aversion because, in the case of(C)(iii), we have the estimate of regime (g)—gain—
which is smaller than that of regime (l)—loss—as(gain: −3142, loss: −4518). The same applies to themodel (i) (gain: −4251, loss: −7021) and (ii) (gain:−1637, loss: −2622). We further observe the following: (i) aggregate (homogeneous) Probit without a threshold shows loss aversion most clearly at
the aggregated level; (ii) two-regime heterogeneous
Probit without a threshold is the most vague; and(iii) our proposed three-regime heterogeneous Probit with thresholds yields performance between (i)and (ii).
We found the presence of asymmetric reference
effects and loss aversion after price thresholds wereincorporated into the model for this data set. Thisobservation is robust relative to other types of reference price, except for (A)(ii), (B)(ii), and (B)(iii).
Following Bell and Lattin (2000, p. 190, Figure 1),
we present intuitive reasoning for our results. First,
Bell and Lattin (2000) discuss the relation between
models (i) and (ii) relative to loss aversion: model (i)llhas steeper slopes in the loss regime (A  > BL  asg
g
well as the gain regime (A  > BL  than model (ii),llhowever, the difference (A  − BL  is larger in theggloss regime than (A  − BL  is in the gain regime.
Therefore, loss aversion would decrease or disappear
if heterogeneity were incorporated into the model.
Next, we examine the relation between models (ii)and (iii). The response functions assumed in eachmodel are shown in Figure 1; we note that the
response function of model (ii), corresponding to themodel of Bell and Lattin (2000), has two regimes
without price thresholds. Two response functions arekinked at the zero of sticker shock. Our model (iii),
however, has three regimes. We suppose that the true
response function has LPA (insensitive range) limitedby the price thresholds, as shown in Figure 1, and thatthe data are observed along with three lines.
If we ﬁt only two response functions, as employedin Bell and Lattin (2000), to data over positive andnegative sides of sticker-shock domain separately,
glthe ﬁtted slopes (BL  BL will be less steep than
the slopes g  l of the three-regime model in theglgain-and-loss regimes. That is, BL  < g , BL  <l  because the ﬁtted slopes are calculated so as tocatch up with the data inside LPA, to which a ﬂatterline is applied by deﬁnition. For that reason, a modelwith price thresholds is likely to be more responsive
than a model without a threshold for both loss andgain regimes. Table 3 presents empirical evidence tosupport this conjecture, which is robust with respect
to the choice of reference price.
On the other hand, that discussion indicates no tendency for loss aversion between models with andwithout thresholds. However, relying on Kahneman
and Tversky (1979), it would be likely for g − l(we deﬁne it as the degree of loss aversion) to be positive in most cases because we are trying to model
individual consumers’ responses. In fact, we used ourdata to calculate the posterior probability Pr+Gain −Loss  data, at the aggregated level for models (i),(ii), and (iii). Figure 2 displays the posterior density ofthe difference Gain − Loss . The distributions agreewith Prospect theory (Kahneman and Tversky 1979),
and the degree of loss aversion is strongest for model(i) in the sense that the posterior probability of loss
aversion Pr+Gain − Loss > 0  data, is largest; it is
weakest for model (ii). Model (iii) is located between
them. The relation between (i) and (ii) is consistent
with the results of Bell and Lattin (2000). The relation
between (ii) and (iii) implies that the degree of loss

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Marketing Science 25(4), pp. 384–391, © 2006 INFORMS

Figure 2

Posterior Distribution for Loss Aversion

r1

35
30

Frequency

Relative frequency

Loss aversion

Model (ii)
Model (iii)

0.04

Heterogeneous Distribution of Price Thresholds

40

Model (i)0.05

Figure 3

0.03
0.02

25
20
15
10
5

0.01

0
–0.25

0.00
–5

–4

–3

–2

–1

0

1

2

(1)

(3)

3

4

5

6

7

–0.20

–0.15

–0.05

Next

0.15

0.20

Next

r2

8
40

βp –βp

–0.10

35

–5

30

1.0
0.9

Frequency

Cumulative relative frequency

Model (i)Model (ii)
Model (iii)

0.8
0.7
0.6
0.5

–3

–2

–1

0.0

15

5
0

0

1

2

(1)

(3)

3

4

5

6

7

8

βp –βpaversion would increase after incorporation of heterogeneous price thresholds into the models. These
results are robust to the selection of reference price,
except for (B).
The relationship between household-speciﬁc variables and their market-response parameters was signiﬁcantly estimated. However, they are not reported
here to save space.
3.3.

20

10

0.4
0.3
0.2
0.1
–4

25

Price Thresholds and Their Hierarchical
Structure
Figure 3 shows the frequency distribution of Bayes
estimates +rˆ1h  h = 1     H , and +rˆ2h  h = 1     H ,,where rˆ•h is the mean of posterior distribution ofthreshold parameters for household h. Both distributions exhibit skewness showing distinct features ofeach other. The average distances from zero are different: −0113 for the lower threshold rˆ1h and 0.138 forthe upper threshold rˆ2h . For that reason, the symmetric LPA around zero could fail to reﬂect heterogeneity
in the brand choice study.
Table 4 shows that all hierarchical Bayes estimates of regression coefﬁcients %1 and %2 of (3)on household-speciﬁc variables are signiﬁcant in thesense that the 95% HPD region does not include zero.
From these estimates, we ﬁrst note that LPAs relative to households are not symmetric around thezero levels of their respective sticker shocks. We also

0.00

0.05

0.10

observe the following: (1) For Pfreq versus LPA, themean level of LPA increases with the purchase frequency, which is consistent with previous empiricalﬁndings; e.g., on the hypothesis H3 of Kalyanaramand Little (1994): “Pfreq is a factor to affect pricethresholds.” (2) For Dprone versus LPA, the meanlevel of LPAs decreases accordingly as the increase indeal proneness, which is compatible with Han et al.
(2001). (3) For ARP versus LPA, the estimates showthat the household with high ARP value has wider
LPA, which is consistent with the result to the hypothesis H2 of Kalyanaram and Little (1994): “ARP is afactor to affect price thresholds.” (4) For BL versus
Table 4

Hierarchical Regression Coefﬁcients on
Household Data (Price Thresholds)(C)(iii)

Variable

r1

r2

Const

−0292∗0053
0028∗
0004
0042∗
0006−0049∗0006−0149∗0017

0321∗
0041

Pfreq
DproneARP
BL−0010∗0005−0033∗0006
0026∗
0006
0165∗
0017∗
Signiﬁcant at 0.95 HPD region. The number inside [ ]means posterior standard deviation.

Terui and Dahana: Estimating Heterogeneous Price Thresholds

390

Marketing Science 25(4), pp. 384–391, © 2006 INFORMS

LPA, we observe that the high brand-loyalty household has wider LPA, which is consistent with resultsby hypothesis H4 of Kalyanaram and Little (1994): “BL
is a factor to explain price thresholds.”

Acknowledgments

4.

Appendix. Estimation of the Model: Markov Chain
Monte Carlo Algorithm for Hierarchical Bayes
Modeling of Threshold Probit Model

Concluding Remarks

This study introduced a three-regime piecewise-linear
stochastic utility function with two price thresholds, and proposed a class of brand-choice model—
threshold Probit model—under a framework of acontinuous mixture modeling for heterogeneous consumers. Price thresholds generate unconventional discontinuous likelihood functions in the analysis, andthey create difﬁculties in estimation. However, ourmethod directly models thresholds for choice models
in a general manner, and coherent statistical inference
on the thresholds can be done particularly when thenumber of samples is scarce.
Our empirical application, as far as our data set
is concerned, shows that our proposed model issuperior to a conventional linear-utility-based Probit
model without price thresholds as well as aggregate Probit models; the reference effect and loss
aversion are observed after price thresholds are incorporated into a heterogeneous price-response model,
but that loss aversion is attenuated then by an aggregate (homogeneous) model without price thresholds.
Moreover, through hierarchical modeling, we investigated explanatory factors of heterogeneous pricethresholds. Some are consistent with previous studies
that assumed consumer homogeneity.
Individual heterogeneity makes it possible to design a customized strategy, which has been demanded
recently as we can see, for example, in Gal-Or and
Gal-Or (2005) for customized advertising. As an application of our model, we explored a possibility of customized pricing strategy based on the knowledge ofprice thresholds for respective households. Our limited simulation study showed that customized pricing could yield greater proﬁts than ﬂat pricing, andoptimal pricing levels are provided at the lower pricethresholds for discounting, and at the upper pricethresholds for a price hike. However, this is notreported here to save space.
Our analysis can be generalized further in several
aspects. For example, heterogeneous modeling could
be applied to smoothing parameters of the reference price derived from exponentially weighted past
prices. Moreover, it can be extended to accommodate the dynamic relation on a utility function, forexample, such as Leichty et al. (2005) recently proposed. Together with a comprehensive discussion ofcustomized pricing mentioned above, we leave these
problems for future research.

Nobuhiko Terui acknowledges the ﬁnancial support fromthe Japanese Ministry of Education Scientiﬁc Research
Grant (C)15530137. The authors appreciate helpful comments from the area editor and three anonymous reviewers.

As for model calibration, extending the framework of Rossi
et al. (1996), we employ hierarchical Bayes modeling to
implement the threshold Probit model (6). Given the value ofrh , according to the level of consumer h’s sticker shock Pjht −RPjht , we ﬁrst allocate data +Xht , of the explanatory variablei
to make +Xjht  i = g a l, at each purchase occasion. The cori
responding latent utility vector +yht  i = g a l, is generated
based on personal choice data +Iht , (the index of observed
choices) using the algorithm of the Bayesian Probit model(Rossi et al. 1996, pp. 338–339) applied to each regime. Then,
except for price threshold rh  −, we can use conditional posi
i
i
i
terior distributions: yht  +Iht , +Xht , h  #i  rh , h  +yht ,
i
i
i
i
i
+Xht , #i  (i  V  zh  rh , #i −1  +yht , +Xht , +h , +rh ,,
i
i
i −1
i
i
i
(  +h , V  +zh , +rh ,, and V +h , (  +zh , +rh ,aaa
for i = g a l. The variance hp 2 for hp ∼ N 0 hp 2 inthe LPA was ﬁxed as 0.01.
As for hierarchical modeling for the price threshold,the lower (negative) threshold is modeled as r1h = zh %1 +2&1h ; &1h ∼ N 0 1&
and stacking over all households h =1     H, leads to matrix notation r1 = Z%1 + &1 ; &1 ∼NH 0 11& . We ﬁrst set prior distributions on % 1& as%1  r1  11& ∼ N %10  11% "1−1
1& ∼ Wishart51&0  V1&0 "%10 = 0 11% = 001Id

51&0 = d + 4 V1&0 = 51&0 IH 

Then we have conditional posterior distributions
 Z Z + 1−1 −1 
%1  +r1h , +zh , 11& ∼ N %
1%
where
−1
6Z Z %1 +1−1
%1 = Z Z +1−1
1%
1% %10  and%1 = Z Z −1 Z r1 

1−1
1&  +r1h ,+zh ,%1
∼ Wishart 51&0 +HV1&0 +t

r1t −Zt %1 rt −Zt %1The same formulations apply to upper (positive) pricethreshold r2h .
iNext, as for conditional posterior +rh ,  +Iht , +Xht , +h ,i#  +zh , % 1& , we use the likelihood function L+rh ," +Iht ,i+Xht ,  +h , +#i , of (8) in §2 jointly with the priorp+rh ,  % 1& ∼ N Z% 1& deﬁned above, and we employ
Metropolis-Hastings sampling with random walk algorithm
as the following steps:
Denote +rh , = r; then(1) i = 0. Set r 0 .
2(2) i > 1" z ∼ N 0 RW
IH and set r = r i−1 + z.
(3)2r i−1 " r = minpr  % 1& Lr" +Iht , +Xht ,1 pr i−1  % 1& Lr i−1 " +Iht , +Xht ,Terui and Dahana: Estimating Heterogeneous Price Thresholds
Marketing Science 25(4), pp. 384–391, © 2006 INFORMS(4) Sample u ∼ U60 1 ,if u ≤ 2r i−1 " rthen r i = rotherwise r i = r i−1 Thus, we have necessary conditional posterior distributions%  +rh , +zh , 1& 8 Normal distribution






1−1  +r , +z , %8 Inverted Wishart distribution
h
h
&
(A-1)


i


+rh ,  +Iht , +Xht , +h , #i  +zh , % 1& 8
Metropolis-Hasting sampling
i

i

i

Finally, we denote by fi +yht , +h , #i  (i  V  +rh ,
% 1&  +Iht , +Xht , +zh , the joint posterior density for the
regime i, and under the assumption of uncorrelated errors
for latent utility equations of each regime, overall joint posterior density across regimes can be expressed as
(A-2)

 i
i
i
fi +yht , +h , #i  (i  V  rh  %i=g a l
1&  +Iht , +Xht , +zh , In terms of sampling algorithms (A-1) and (A-2) for Markov
chain Monte Carlo, we can constitute the posterior distribution of each regime, respectively, to get overall joint posterior density across regimes.

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