Price Customization Using Price Thresholds
Estimated From Scanner Panel Data

Nobuhiko Terui*andWirawan Dony Dahana
Graduate School of Economics and Management
Tohoku University
Sendai 980-8576, Japan*E-mail:terui@econ.tohoku.ac.jp*Voice & Fax:+81-22-217-6311

November, 2005
Revised April, 2006

Acknowledgement
We would like to thank Russell Winer, the chair of CRM conference at Stern School ofBusiness, New York University, Venky Shankar, the editor of this journal, and the anonymous
reviewer for their careful readings of our manuscript. The comments from the participants atthe conference are also appreciated.

1

Price Customization Using Price Thresholds
Estimated From Scanner Panel Data
Abstract
This study explores a customized pric ing strategy based on heterogeneous price thresholds
estimated from scanner panel data to show that customized pricing could be more efficient than a flat
pricing strategy. A heterogeneous brand choice model with price thresholds is applied to pricecustomization problem and we demonstrate that heterogeneous price thresholds are valuable information
set for the search of efficient pricing levels customized to individual consumers. The expected
incremental profits from customized discounting as well as customized price hike are evaluated by using
hierarchical Bayes modeling with Markov chain Monte Carlo (MCMC) method.

Key Words and Phrases
Price Threshold, Latitude of Price Acceptance, Brand Choice, Bayesian MCMC, Heterogeneity,
Scanner Panel Data, Customized Pricing

1. Introduction
Not only marketers but also marketing researchers have recognized that customizing
marketing activities would be valuable to improving the profitability of direct marketing efforts.
Electronic distributions of coupons through frequent shopper programs and their collectedhousehold’s purchase record have made its potential advantage more realistic in the marketplace.
Price customization or the first degree price discrimination was discussed as target
couponing by Rossi, McCulloch and Allenby (1996). They measure the value of heterogeneous
household information by evaluating incremental profits generated from sales increase byissuing different level of discount coupon to show that there exists a potential for improving the
profitability of direct marketing efforts by fully utilizing household purchase histories. Their
modeling is now established as a benchmark model for consumer marketing to conduct target
marketing.
Terui and Dahana (2004) proposed a choice model with the heterogeneous pricethresholds, which extends conventional heterogeneous choice model with linear utility function
to that with nonlinear utility function. Besides the price thresholds, the model incorporates the2concepts of symmetric market response and reference price discussed in the literature ofnonlinear market response of consumer behavior.
This heterogeneous price threshold model can be used for the search of efficientcustomized pricing. That is, the heterogeneous price thresholds provide the marketer with theprice insensitive region of each consumer. The discount pricing not crossing over lower
threshold leads to the loss because consumers do not respond to this discounting. On thecontrary, in case of price hike, they do not recognize it as far as the price stays below upper
price threshold and the differences from their reference prices produce the profits to retailer. In
other words, the heterogeneous price thresholds contribute toward minimizing the loss incurredfrom discount pricing over their lower price thresholds as well as maximizing the gain obtained
from price hike below their upper price thresholds.
In this study, on the basis of heterogeneous price thresholds, we explore the profitabilityof customized pricing by considering not only different levels of discounting but also price hikestrategy. We set various levels of pricing for each consumer to evaluate expected incremental
sales and profits in the market by heterogeneous price thresholds estimated from scanner panel
data. We show, under a limited simulation study, that optimal levels could occur near pricethresholds and that a customization strategy on pricing could provide marketers with larger
profits than a “non-customized (flat)” pricing strategy.
The organization of the paper is as follows. In section 2, we describe heterogeneous pricethreshold model and define incremental sales and profits generated by customized pricing. We
take up not only discounting but also price hike for customization strategy. Section 3 presents
the application of scanner panel data to our model. We use instant coffee category panel data
because of availability and we intend to see how our methodology is applicable to scanner panel
data. Section 4 concludes this paper. The appendix explains details of the evaluation ofincremental profits including the algorithm for hierarchical Bayes modeling via Markov chain
Monte Carlo method.
3

2. Heterogeneous Price Threshold Model and Customized Pricing
There have been many studies for the price thresholds and the latitude of price acceptance.
Related recent studies include Gupta and Cooper (1992), Kalwani and Yim (1992), Kalyanaramand Little (1994), and Han, Gupta and Lehman (2001). However, these previous models withthreshold effects yielding intervals of price acceptance have yet to be estimated heterogeneously.
On the other hand, Terui and Dahana (2004) introduced a three-regime piecewise-linear
stochastic utility function with two price thresholds and proposed a class of brand choice model– heterogeneous price threshold model – under a framework of a continuous mixture modelingfor heterogeneous consumers. This choice model is characterized as a piecewise linear form sothat consumers switch their utility structure according as determined by the relationship between
the sticker shock, i.e., the difference between retail price and the reference price, and the pricethresholds. Price thresholds generate unconventional discontinuous likelihood function in theanalysis and they create difficult ies in estimation. However, the method directly models
thresholds for choice model in a general manner and coherent statistical inference on thethresholds can be done particularly when the number of samples is scarce.
Following empirical application by using scanner data of instant coffee market, ourproposed three-regime heterogeneous reference-price probit model with thresholds was shown
to be superior to other candidate models in the sense of marginal likelihood criterion. The
comparable models included aggregate (homogeneous) reference price probit model without athreshold (Winer (1986), Putler (1992) and Mayhew and Winer (1992)); two regimes
heterogeneous reference price probit model without a threshold (Bell and Lattin (2000), andChang, Siddarth and Weinberg (1999)). This result was robust relative to the selection of thetypes of reference price.
Furthermore, our result added to the literature on the presence of reference prices and loss

4

aversion that initially began with effects uncovered in scanner data, for example, as indicated byWiner (1986), Putler (1992), and Mayhew and Winer (1992), and then Chang, Siddarth andWeinberg (1999) and Bell and Lattin (2000) showed that reference prices’ effects disappear
when heterogeneity was incorporated. It also showed that the reference effect and loss aversion
return, at least for the data used in this study, after price thresholds are taken into aheterogeneous model. The degree of loss aversion is attenuated relative to results obtainedusing the homogeneity model without price thresholds.
Heterogeneous Price Threshold Model
We first define heterogeneous price threshold model proposed by Terui and Dahana(2004). To specify the utility function used in the model, we assume that consumer h ’s utility
to brand j at time t of purchase, U jht , reflects a linear function of k kinds of explanatory
variables. We also suppose that consumer h has a reference price RPjht for brand j , and thattwo 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u (jhg ) + X (jhtg ) β h*( g) +∈(jhtg ) if Pjht − RPjht ≤ r1h( a)( a ) *( a )(a )U jht = u jh + X jht βh + ∈ jht if r1h < Pjht − RPjht ≤ r2 h , ( l)( l ) *( l )(l )if r2 h < Pjht − RPjht ,u jh + X jht β h + ∈ jht(1)

where X jht is the row vector of explanatory variables (price, display, feature and brand(i )royalty) allocated to regime “ i ” according to the level of sticker shock Pjht − RPjht ( Pjht : the* (i )

retail price exposed to consumer h ) at the occasion, βh , i = g , a, l, represent different market
responses around the reference price. Finally, ∈jht , i = g , a, l, respectively represent stochastic(i )error components in the utility for each regime. We assume that they are independent acrossregimes.

The meaning of this stochastic utility function is described in figure 1.

5

Figure 1: Price Threshold Model and Market Response

From that definition, the latitude of price acceptance (LPA hereafte r) for consumer hunder the conditions discussed below can be expressed as Lh = (r1h ,r2h ] . In order for thesecond regime of the utility function (1) to be characterized as the LPA and for r1h and r2h to beinterpreted as price thresholds literally, i.e., for our proposed model to be recognized as a pricethreshold model, we impose the restriction of insensitiveness on the price-response parameter inthe LPA regime as βhp : N (0, σ hp ) , which is an element of βh*( a) .
( a )2( a)

Following the utility function defined as (1), consumer h ’s probability of choosing brandj is written as
Pr { y (jhg ) = max( y1(hg ) ,..., ym( g−)1h ) > 0}
if Pjht − RPjht < r1h
Pr {ch = j} =  Pr { y (jha) = max( y1(ha ) ,..., ym( a−)1h ) > 0 | R} if r1h ≤ Pjht − RPjht ≤ r2 h ,

(l )
(l )
(l )
if Pjht − RPjht ≥ r2h , Pr { y jh = max( y1h ,..., ym−1h ) > 0}(2)where y jht = U jht − U mht represents the relative utility from the last brand andPr { y (jha ) = max( y1(ha ) ,..., ym( a−)1h ) > 0 | R} indicates the choice probability under the restriction on
price response βhp : N (0, σ hp ) in the LPA regime. Consumers’ heterogeneity is incorporated( a)( a )2through random effect specification that allows determination of the relationship between theprice thresholds and consumer characteristics.

r1h = Zhrφ1 + η1h ;r2h = Z hrφ2 + η 2h ; h = 1, L, H ,where Zhr is a vector of d kinds of household specific variables. We assume that r1h < 0 < r2h
for identification and η jh : N (0, σ jη ) for j = 1,2 . We also set a hierarchical structure of2“between subjects model for regime i ” for the market response parameterβh(i ) = ∆( i) Z hβ + υ h( i) ;υ h(i ) : Ν ( 0,Vβ(i ) ) ,
i.i.d

6

h = 1, L, Η. , i = g , a, l ,where Z hβ contains another vector of d ' kinds of household specific varia bles. In particular, wenote that price response βhp in the LPA is assumed a priori to have zero mean.
( a)

As for model calibration, we employ hierarchical Bayes modeling to implement thethreshold probit model (2). Our model includes a threshold variable in the model and it induces
discontinuities in the likelihood relative to thresholds. In general, the proposed model isdifficult to estimate with conventional methods because the likelihood is not differentiable in rh .
Conventional maximum likelihood estimation collapses and classical asymptotic distribution
theory is not operative on this parameter. However, we can apply Metropolis-Hasting samplingalgorithm for price thresholds to obtain conditional posterior“ rh | {I ht },{X ht },{β h }, Λ ,{zh },φ , Ση ”. Prior distributions and MCMC estimation
( i)

(i )

procedures for these hierarchical Bayes models are described in Terui and Dahana(2004). Briefsummary of model estimation algorithm focusing on incremental sales and profit is described in
the appendix.
Next we consider price customization strategy by using this price threshold model.
Incremental Sales and Profits by Customized Pricing Strategy
Based on the knowledge of heterogeneous price thresholds for respective consumer
obtained by proposed model, we consider customized pricing, both of discounting and price
hike, and explore a possible efficient pricing.
[1] Discounting
Conditional on the draw of{( r (< 0), β , β ), h = 1,..., H } , we set discounting level of
1h

(1)
h

(2)
h

( r1h + α ) (>0), ( α = 0, ±1,..., ±15% ) for h = 1,..., H , and we define the expected incremental(−)

sales, IS j (α |{r1h ,β h , h = 1,..., H }) , for “customized discounting” averaged over householdsas7IS (j −) (α |{(r1h , β h(1) ,β h(2) ), h = 1,..., H }) =

(
(

)
)

(2)
 1 ∑ H Pr j β (1)h , Pjh 0 (1 − ( r1h + α )) − Prj ( β h , Pjh 0 ) if α ≥ 0: (Price Gain) (3) H h=1 H 1 H ∑ h=1 Pr j β h(2) , Pjh 0 (1 − ( r1h + α ) ) − Pr j ( β h( 2 ) , Pjh 0 )  if α < 0: (LPA)

Under the assumption of margin M %, the corresponding incremental profit is defined as
IP( − ) j (α |{( r1h , β h(1) , β h( 2 ) ), h = 1,..., H }) =

(
(

)
)

 1 ∑ H  Pr β (1) , P (1 − ( r + α ) ) − Pr ( β (2) , P ) ( M − ( r + α ) ) if α ≥ 0 : (Price Gain)1hjhjh 0 
1h
 H h=1  j h j h0

H
(2)
(2)
 1 H ∑ h=1  Pr j β h , Pjh0 (1 − ( r1h + α ) ) − Pr j ( βh , Pjh0 )  ( M − ( r1h + α ) ) if α < 0 : (LPA)(Taking expectation of (10) and (11) with respect to posterior distribution of r1h , β h(1) , βh(2)(4))

leads to unconditional incremental sales and profits resectively,

IS ( − ) j (α ) = E( r ,β (1) ,β ( 2 ) )  IS (− ) j (α |{r1h , βh(1) , β h(2) , h = 1,..., H})

(5)

IP ( −) j (α ) = E( r , β (1) ,β ( 2 ) )  IP ( − ) j (α | {r1h , βh(1) , β h(2), h = 1,..., H}) .

(6)

1h

h

1h

h

h

h

These estimates are obtained as by-product of sampling through MCMC iterations.
[2] Price hikeAs for price hike strategy, conditional on{( r (> 0), β , β ) , h = 1,..., H } , we set the hike
2h

(2)
h

(3)
h

rate ( r2h + α ) % (> 0) for h = 1,..., H ( α = 0, ±1,..., ±15% ), then the expected incremental
sales and profits for “customized price hike strategy” averaged over households are respectively
defined as
IS (j +) (α |{(r2 h , βh(2) , β h(3) ), h = 1,..., H }) =

(
(

)
)

 1 ∑ H  Pr β (2) , P (1 + ( r + α ) ) − Pr ( β ( 2 ) , P ) if α < 0: (LPA)
(7)
jh 0
2h
j
h
jh 0 
 H h =1  j hH 1 H ∑h =1  Prj β h(3) , Pjh 0 (1 + ( r2 h + α )) − Prj ( β h(2) , Pjh 0 )  if α ≥ 0: (Price Loss)IP( + ) j (α |{( r2h , β h( 2 ) , β h(3) ), h = 1,..., H }) =

(
(

)
)

 1 ∑ H  Pr β (2) , P (1 − ( r + α ) ) − Pr ( β (2) , P ) ( M − ( r + α ) ) if α < 0 : (LPA)
j h0
2h
j
h
jh 0 
2h
 H h=1  j h

H
(3)
(2)
 1 H ∑ h=1  Pr j βh , Pjh0 (1 − ( r2h + α ) ) − Pr j ( β h , Pjh0 )  ( M − ( r2h + α ) ) if α ≥ 0 : (Price Gain)(8)

The unconditional expected incremental sales and profits are respectively described as

IS ( + ) j (α ) = E( r , β (2) , β (3) )  IS ( + ) j (α | {r2h , βh(2), βh(3) , h = 1,..., H })
2h

h

h

8

(9)

IP ( +) j (α ) = E( r , β (2) ,β ( 3 ) )  IP ( + ) j (α | {r2h , βh(2), βh(3) , h = 1,..., H}) .
1h

h

(10)

h

[3]Difference from non-customized pricing
The average difference of incremental profits between an optimal customized discounting
at the level r1h and a non-customized (flat) discounting at the level d * can be denoted by
DIFj(− ) ( d * | {r1 h , β h(g ) , h = 1,..., H })()= 1 H ∑ Pr j ( β h(1) , Pj 0 (1 − r1h ) ) − Pr j ( β h(2) , Pj 0 ) ( M − r1h )
H

h =1

( (

)

(11)

)

− 1 H ∑ Pr j β h(g ) , Pj 0 (1 − d * ) − Pr j ( β h(2) , Pj 0 ) ( M − d * )H

h =1

where market response βh( g) for non-customized pricing depends on the regime determined bydiscount level d * . Unconditional estimate is obtained by taking expectation

DIFj( − ) ( d * ) = E( r , β ( g) )  DIFj( − ) ( d * | {r1h , βh( g) , h = 1,..., H }) .
1h

(12)

h

The same operation is applied to price hike strategy at an optimal customized price hike at
the level r2h compared with non-customized discounting at the level d * to obtain

DIFj( + ) ( d * ) = E( r ,β ( g ) )  DIFj( + ) (d * | {r2h , β h( g) , h = 1,..., H }) .
2h

(13)

h

We note that the marketer taking non-customized pricing strategy does not know the pricethresholds and therefore he/she has to try constant pricing to every consumer at possible levels.
We also note that the assumption of constant margin (M %) imply that the price discounts are
optimized and this is consistent with some other papers in the area, for example, Kopalle, Mela,and Marsh (1999).
3. Empirical Application to Scanner Panel Data
3-1. Data, Variables and Model Specification
Video Research Ltd., Japan, supplied scanner panel data for the instant coffee (regular)
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. There are 11 brands in9the data set and we deal with five primary brands A, B, C, D, and E, which totally account for
75.6% market share.
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/100 g to equalize quantitative differences for each package of the five brands. The
aggregated price in table 1 was calculated by averaging the prices normalized by 100 gm over
the whole purchase occasions in the data. We found that the price correlations between different
volumes of SKU’s are positively higher (0.794 max, 0.740 min).
Table 1 Descriptive Statistics for Data
Variables for our model are:g Explanatory Variables: X = [Constant, Price, Display, Feature, GL],
where Price is the log(price); Display and Feature are binary values; and GL is state dependent
variable defined as a smoothing variable over past purchases proposed by Guadagni and Little(1983) as GL jht = α GL jht + (1 − α ) I j h, t−1 , where a grid search (Keane(1997)) is applied to fix thesmoothing parameter as 0.75 based on the criterion of minimum marginal likelihood.

g Household Specific Variables: Z r = [Constant, Dprone, Pfreq, RP, BL],
where Dprone is the deal proneness defined as the proportion of purchase (of any the fivebrands) made on promotion (Bucklin and Gupta (1992), Han et al. (2001))); Pfreq is the
shopping frequency (three categories); RP represents the measure of average reference pricelevel as defined by RPh = ∑ j =1 ( ∑ t h=1 log( RPjht ) / Th ) / m and BL represents the brand loyaltymTmeasure defined as BLh = max j ( ∑ t =h 1 GL jht / Th ) . These are used in Kalyanaram and LittleT(1994) and Han et al. (2001).

g Household Specific Variables: Z β = [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).
10

As for the choice of reference price, Terui and Dahana (2004) employed four kinds ofreference prices of two categories. That is, (A) RPjht = Pj h,t −1 , the price at its last purchase(B) RPjht = α RPj h, t −1 + (1 − α ) Pj h,t −1 , the smoothed price over previous purchases;(C) RPjht = Pkht , where k means the brand at the last purchase and (D) RPjht = Prht , where r
indicates the price of a brand chosen randomly at the time of purchase. As model specification,
they showed that the marginal likelihood for each model suggests the model (C): stimulus-based
RP (C)-3 regimes heterogeneous probit with thresholds and we use it as the estimated model forcustomized pricing in this paper.
3-2. Customized Pricing
Heterogeneous Distribution of Price Thresholds
Figure 2 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 of threshold parameters forhousehold h . Both distributions exhibit skewness showing distinct features each other. The
average distances from zero are different: -0.113 for the lower threshold rˆ1h and 0.138 for theupper threshold rˆ2h . For that reason, the symmetric LPA around zero could fail to reflect
heterogeneity in the brand choice study.

Figure 2: Heterogeneous Distribution of Price Thresholds

We utilize the magnitude of price thresholds and their uncertainty expressed by
distribution in figure 2 to explore the profitability of price customization effort in what follows.
Customized Discounting and Price hike
The price thresholds say that the discount pricing below lower threshold does not effecton sales because of non-response to this discounting. However, in case of price hike, consumer
does not take a negative attitude as far as the price stays below upper price threshold and the11differences from their reference prices produce the profits to retailer. Since the heterogeneous
price thresholds contribute toward minimizing the loss incurred from discount pricing over theirlower price thresholds as well as maximizing the gain obtained from price hike below their
upper price thresholds, we expect that the price customization accommodating heterogeneous
insensitive region makes it possible to identify an efficient pricing.
In order to evaluate the efficiency of price customization, we consider several levels of
pricing for each consumer to evaluate expected incremental sales and profits in the market byusing the knowledge obtained by heterogeneous price thresholds estimated from scanner panel
data. We set the discounting level of ( r1h + α ) (>0)and the hike rate ( r2h + α ) % (> 0) for

h = 1,..., H , where α = 0, ±1,..., ±15% .

Figure 3: Expected Incremental Sales
Figure 4: Expected Incremental Profits

The negative part each graph in figure 3 shows calculated expected incremental sales forthe case of customized discounting. We observe that the large difference at the boundary r1h
between price gain and LPA regimes and the largest change happens for brand E having highest
price, and for brand D with second highest price. Similarly, the negative part of figure 4 shows
those expected incremental profits, where the margin was set as M =30%. We have an optimal
discount level at the lower threshold r1h for every brand, as we could expect. The positive
region of each graph in figures 3 and 4 respectively shows expected incremental sales andprofits for the case of customized price hike. We observe that maximum profit happens at theupper threshold r2h for every brand.

Figure 5: Difference of Profit Between Customized and Non-customized Pricing

12Next we compare the performance of customized pricing with that of flat pricing. We
note that marketer taking a non-customized pricing strategy is oblivious to respective
consumers’ price thresholds, and thus marketer applies flat (constant ) pricing to every
consumer. The negative and positive regions of each graph of figure 5 respectively show the
(− )

(+)

plots of DIFj ( d ) and DIFj ( d ) for d * = 1,2,...,15% . Different margins produce**

slightly different graph shapes, but the findings described below do not change.
First, this graph shows that two kinds of optimal customized pricing – discount at r1h andhike at r2h – dominate every level of non-customized pricing d * = 1,2,...,15% . In the negative
LPA region, non-customized pricing does not generate so large sales increase as expected
because of price change insensitivity over this region. In contrast, customized pricing does nottake a discount strategy over this region. Identical logic applies to the case of price hike asshown in the pos itive part of LPA. Those gains for customized pricing stem from pricethreshold information. In the price gain regime as well as in the price loss regime, it would bereasonable to consider that efficiency can occur at those limits. Almost identical sit uations
apply to other brands, but the change is largest for the highest-price brand E.

4. Concluding Remarks
In this study, we applied price thresholds model by Terui and Dahana (2004) to the
exploration of customized pricing strategy by using scanner panel data.
We demonstrated that optimal customized pricing levels are provided at the lower pricethresholds for discounting, and at the upper price thresholds for a price hike. Moreover, ourperformance-comparison exercises with possible non-customized pricing strategies showed that
customized pricing could yield greater profits than flat pricing. These results are limited to oursimulation study under specified conditions , although it would be plausible .

13

We used instant coffee category panel data because of availability and this could not be
always appropriate data set for the price hike strategy. However our purpose of this application
was to see how our methodology was applicable to panel data and we showed the possibility of
application to other panel data set where customized price hike could be appropriate.
In the literatures of optimal pricing, for example, Greenleaf (1995), Kopalle, Rao, andAssuncao (1996) and Kopalle and Winer (1996) examine optimal pricing policies which are notcustomized to each individual but are allowed to vary over time, i.e. heterogeneity in the time
horizon. On the other hand, our model incorporates only cross sectional heterogeneity in theanalysis. The temporary discounting could lead to consumer stockpiling and it affects
negatively to the purchase in the future. By contrary, temporary price hike could makerestrained purchasing to hold it out to next purchase. However, we expect that customized
pricing by using heterogeneous price thresholds keeps these reactions as small as possible,
because it utilizes the limits of the range in which consumer does not recognize the price change.
We also note that the reference price (RP) by itself could be affected by pricing strategy. The
frequent price promotions can make consumer with memory based RP lower their level of RP,
on the other hand, they might not affect the consumers with stimulus based RP. To incorporate
the effect of pricing on RP into the analysis, we need to model heterogeneous consumer withseveral types of RP formation discussed by, for example, Mazumdar and Papatla (2000).
Full discussion regarding these dynamic effects, in addition to the papers above, including
recent related work by Van Heerde et al. (2004) and others which explores factor decompositionof price promotion and its long term effects, is left for future investigation.

14

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Terui, N. and W.D.Dahana(2004), “Estimating Heterogeneous Price Thresholds”, Proceedingsof “International Conference on Recent Development of Statistical Modeling ofMarketing”, Institute of Statistical Mathematics, Tokyo, 113-152.(Forthcoming inMarketing Science)
Van Heerde, H.J., P.S.H. Leeflang, D.R.Wittink(2004),”Decomposing the sales promotion
bump with store data,” Marketing Science, 23,317-334.
Winer, R.S. (1986), "A reference price model of brand choice for frequently purchased
products," Journal of Consumer Research, 13, 250-256.

16

Appendix
Markov chain Monte Carlo Algorithm for Incremental Sales and Profit-Hierarchical Bayes Modeling of Threshold Probit ModelEstimation of the Model
As for model calibration, we employ hierarchical Bayes modeling to imple ment thethreshold Probit model (2). Given the value of rh , according to the level of consumer h's
sticker shock Pjht − RPjht , we first assign data { X ht } of the explanatory variable to make{X (jhti) ,i = g , a, l} at each purchase occasion. The corresponding latent utility vector{ yht(i ) , i = g , a, l} is generated based on personal choice data {I ht } (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
posterior distributions: ” yht( i ) | { Iht },{ X ht }, βh( i ) , Λ( i ) , rh ”,−1“ βh( i ) | { yht(i )},{ X ht(i ) }, Λ (i ) , ∆( i) ,Vβ( i) , zh , rh ”, “ Λ (i ) |{ yht(i ) },{ X ht(i ) },{βh(i )},{rh } ”,
“∆

(i )

−1

| {βh(i ) }, Vβ(i ) ,{z h },{rh} ”, and “Vβ( i) |{βh( i) }, ∆( i) ,{z h},{rh } ” for i = g , a , l . The variance(a )σ hp( a) for βhp( a) : N (0, σ hp
) in the LPA was fixed as 0.01.
2

2

As for the posterior for the threshold parameters, conditional on ( β h( i) , Λ (i ) ) , under theassumption of independent choice behavior across consumers, Terui and Dahana(2004) showthat we have the conditional likelihood function of {rh } by taking products over respective
consumers as

L({rh};{I ht },{X ht }|{ βh(i )},{Λ( i ) }) ∝
1


 1

( i) − 2|Λ|exp − ( yht(i ) − X ht(i ) βh(i ) ) ' Λ( i)−1 ( yht( i) − X ht(i ) βh(i ) ) .
∏  ∏∏
 2
 
h =1  i =g ,a ,l 
t∈ R(i ) ( rh )Hand jointly with (4) expressed as hierarchical structure, we can apply MetropolisHasting sampling with random walk algorithm for price thresholds to obtain
conditional posterior “ rh | {I ht },{X ht },{β h }, Λ ,{zh },φ , Ση ”. Prior distributions and
( i)

(i )

MCMC estimation procedures for these hierarchical Bayes models are described in
Terui and Dahana(2004). Thus we have necessary conditional posterior

distributions

φ | {rh },{zh }, Ση : Nomal distribution(A-1)

Ση−1 | {rh },{zh },φ : Inverted Wishart doistribution{rh}|{I ht },{ X ht },{β h(i ) }, Λ ( i) ,{z h}, φ, Ση : Metropolis-Hasting sampling

17

(

)

Finally, we denote by f(i) {yht },{βh }, Λ , ∆ ,Vβ ,{rh},φ, Ση | {Iht }, {Xht },{zh} the joint
(i)

(i)

(i)

( i)

(i)

posterior density for the regime i , and under the assumption of uncorrelated errors for latentutility equations of each regime, overall joint posterior density across regimes can be expressedas∏ f ({ y },{β }, Λ , ∆ , V , r , φ , Σ | {I }, { X } ,{z }) . In terms of samplingi = g ,a ,l

( i)
ht

( i)

(i )
h

(i )

( i)

(i )
β

η

h

ht

ht

h

algorithms (A-1) for Markov chain Monte Carlo, we can constitute the posterior distribution ofeach regime respectively to get overall joint posterior density across regimes.
Incremental Sales and Profit{}

Conditional on the draw of r1h[ s] and

(

)

{β

(1)[ s ]h, β h(2)[s ]} in the s-th iteration of MCMC,

we set discounting level of r1h[ s] + α (>0), ( α = 0, ±1,..., ±5% ) for h = 1,..., H . Then, we
evaluate the incremental sale for the consumer h(
Pr ( β(, P (1 − ( r)) (+ α ) ) ) − Pr ( β) if α ≥ 0 for price gain
, P ) if α < 0 for LPA

Pr j βh(1) , Pjh0 1 − ( r1h[ s] + α ) − Pr j β h(2) , Pjh0

(A-2)

j

[s ]

(2)[ s ]
h

jh 0

[s ]

1h

j

[s ]

(2) [ s ]hjh 0

and these amounts are averaged over consumers to get the average in the market,

IS (j − ) (α |{( r1h[ s] , βh(1) , βh(2) ), h = 1,..., H }) .
[s ]

(A-3)

[s ]

This is iterated through S times to get the estimate of unconditional expected incremental
sales
(−)

IS j (α ) = 1

(A-4)

S

 IS (α |{( r
S∑
s =1

(−)
j

1h

[ s]

(1)[ s ], βh(2) [ s ], βh), h = 1,..., H })  .
The same operation is applied to the case of price hike to obtain the estimate of positive
( +)

(− )

(+ )

side IS j (α ) , and also those of expected profits IPj (α ) , IPj (α ) and their difference

DIFj(− ) ( d * ) , DIFj( + ) ( d * ) .

18

Figure 1: Price Threshold Model and Market Response

u

r1

r2

0

Pt‑RPtTable 1: Descriptive Statistics for Data
Alternative

ChoiceShare

AveragePrice% of Time
Displayed% of Time
Featured

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

Figure 2: Heterogeneous Distribution of Price Thresholds
r2

40

40

35

35

30

Frequency

Frequency

r1

25
20
15
10

30
25
20
15
10

5

5

0
-0.25

-0.20

-0.15

-0.10

-0.05

0

Next

0.00

19

0.05

0.10

0.15

0.20

Next

Figure 3: Expected Incremental Sales

Brand B

Brand A
15%

15%

10%

10%
5%

5%

0%

0%

-5%

-5%

-10%

-10%

-15%

r1

0

-15%

r2

r1

Brand C

15%

10%

10%

5%

5%

0%

0%

-5%

-5%

-10%

-10%
r1

0

-15%

r2

Brand E
15%
10%
5%
0%
-5%
-10%
-15%

r1

0

r2

Brand D

15%

-15%

0

r2

20

r1

0

r2

Figure 4: Expected Incremental Profits

Brand A

Brand B

10%

10%

5%

5%

0%

0%

-5%

-5%

-10%

r1

0

-10%

r2

r1

Brand C
10%

5%

5%

0%

0%

-5%

-5%

r1

0

-10%

r2

Brand E
10%
5%
0%
-5%
-10%

r1

0

r2

Brand D

10%

-10%

0

r2

21

r1

0

r2

Figure 5: Difference of Incremental Profits
between Optimal Customized Pricing at ( r1h , r2h ) and Non-customized Pricing Strategies

Brand A
r1

10%

Brand B

0

r2

10%

8%

DIF ( − )DIF ( + )

8%

6%

6%

4%

4%

2%

2%

0%-15%-10%-5%-1% 1%

5%

10%

15%

0%
-15%

r1

DIF ( − )

-10%

8%

r1

r2

10%
8%

DIF ( + )

6%

6%

4%

4%

2%

2%

0%-15%-10%-5% -1% 1%

5%

10%

15%

Brand E
10%
8%

r1

0

DIF ( − )r2

DIF ( + )

6%
4%
2%
0%
-15%

-10%

-5% -1% 1%

5%

10%

15%

Brand D0

DIF ( − )DIF ( + )-5% -1% 1%

Brand C
10%

r2

0

5%

10%

15%

21

0%-15%

r10

r2

DIF ( − )-10%

-5% -1% 1%

DIF ( + )

5%

10%

15%