Estimating Heterogeneous Price Thresholds

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Estimating Heterogeneous Price Thresholds

Nobuhiko Terui*and

Wirawan Dony Dahana

Graduate School of Economics and ManagementTohoku University

Sendai 980-8576, Japan*E-mail:[email protected]*Voice & Fax:+81-22-217-6311

International Conference at ISM

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General nonlinear stochastic utility function General nonlinear stochastic utility function

( )U f X

1 1 1(1)1

2 2 2(2)1 2

( ) ( )1

if

if

if .m mm mm

X Z r

X r Z rU

X r Z

　　　　　　　　　　… … … … …

Z ： Domain of relevant threshold variable

: Disjoint sub-domains : Disjoint sub-domains

jr ：： Threshold Threshold pointspoints

3

3 Regimes Model

Nonlinear Random Utility Function- Asymmetric Market Response

-

1 2 1 2

1 2

: Consumer 's Reference Price to the Brand at

, Price Thresholds ( 0 )

: Perceived Price

= ( , ]: Latitude of Price Acceptance

jht

h h h h

jht jht

h h

RP h j t

r r r r

P RP

L r r

　　　

　　：　

1 1(1) (1)1

2 2(2) (2)1 2

3(3) (3) (3)2

( ) if

( ) if

( ) if

jhtjh jht h jht jht jht h

jhtjht jh jht h jht h jht jht h

jhtjh jht h jht jht jht h

u X U P RP r

U u X U r P RP r

u X U P RP r

4

Consumer Behavior Theory

1. Reference Price (RP) and its conceptualizations Adaptation-Level Theory

2. Asymmetric response around RP

Prospect Theory

3. The Existence of Price Threshold Assimilation-Contrast Theory

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The Object of This Research

=> Propose “Threshold Probit Model “in the form of incorporating these concepts together

Estimate Price Thresholds 　　　　 Latitude of Price Acceptance” 　 at Household level.

Search for an Efficient Pricing through Customization Strategy

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Tools

・ Threshold Probit Model

・ Hierarchical Bayes Modeling

・ MCMC

　　 =>Gibbs Sampling for Response Parameters

=>Metropolis-Hastings Sampling for Threshold Parameters

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The Meaning of Model: 3 Regimes Model

u

0 Xt-RPtr1 r2

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Threshold Probit Model and Hierarchical Bayes Modeling

1 2 1 1Pr Pr max( , ,..., ) Pr max( ,..., ) 0h jh h h mh jh h m hc j U U U U y y y

●●Choice ProbabilityChoice Probability

(1) (1) (1)1 1 1

(2) (2) (2)1 1 1 2

(3) (3) (3)1 1 2

Pr max( ,..., ) 0 if

Pr Pr max( ,..., ) 0 if

Pr max( ,..., ) 0 if .

jh h m h jht jht h

h jh h m h h jht jht h

jh h m h jht jht h

y y y P RP r

c j y y y r P RP r

y y y P RP r

1 2 1, ,..., 'jh h mh h mh m h mhy U U U U U U

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1 2 1, ,..., 'jh h mh h mh m h mhy U U U U U U

( ) ( ) ( ) ( ) ( ) ( ) ( )0, , 1, , , 1, ,i i i i i i iht ht h ht ht hy X h t ; 　　　　

For 1,2,3,i

( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 2 1; ( ) '; ( ) ';...; ( ) ' 'i i i i i i i

ht m ht mht ht mht m ht mhtX I X X X X X X

●●Within Subject ModelWithin Subject Model

( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 1, ,..., 'i i i i i i iht mht ht mht m ht mhtht

3( )

1

.ih h

i

T T

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(1)

( 2)

1(1) (1) (1) (1) (1) 1 (1) (1) (1)2

( )

1(2) (2) (2) (2) (2) 1 (2) (2) (2)2

( )

1(3) (3) (3)2

1| | exp ( ) ' | | ( )

2

1| | exp ( ) ' | | ( )

2

1| | exp (

2

h

h

ht ht ht ht ht htt R r

ht ht ht ht ht htt R r

ht ht ht

y X y X

y X y X

y X

(3)

(3) (3) 1 (3) (3) (3)

( )

) ' | | ( ) ,h

ht ht htt R r

y X

(1) (2) (3)1 2( , ), and ( ) ( ) ( ) .h h h h h h hr r r R r R r R r T 　

( )

( ) ( )

13( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( )2

1 1 ( )

( | , , )

1| | exp ( ) ' | | ( ) .

2ih

Hi i i i i i i i

ht ht ht ht ht hth i t R r

L y r

y X y X

　　

●●Likelihood for consumer Likelihood for consumer hh

●●Total likelihoodTotal likelihood

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i.i.d

( ) ( ) ( ) ( ) ( ); 0, , 1, , ; 1, 2,3.i i i i ih h h hZ V h i

　　

●●Between Subjects ModelBetween Subjects Model

1 1 1 2 2 2; ; 1, , .r rh h h h h hr Z r Z h 　

(1)Market Response(1)Market Response

(2)Price Threshold(2)Price Threshold

: 1 Household variableshZ d 　

( ) : regression coefficientsi k d 　( ) : error vectorih

1 20h hr r

2(0, ) for 1, 2jh jN j 　　　　

': 1 Household variablesrhZ d 　

1 2, : regression coefficient vectors

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●●Price Threshold ModelsPrice Threshold Models

Note:Note: Model 3 does not assume Model 3 does not assume a prioria priori insensitivity and it can be interpreted insensitivity and it can be interpreted as price threshold model as price threshold model a posterioria posteriori when we observe the insignificant estimate of when we observe the insignificant estimate of or in weaker form or in weaker form

when the relation and is confirmed.when the relation and is confirmed.

(2)hp

(2) (1)| | | |hp hp 　　 (2) (3)| | | |hp hp 　　

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Household “ h’ ”

homogeneity

●HB(Hierarchical Bayes) Model

(2)Homogeneous parameter:

(1)Heterogeneous parameter ： h

V { }hZ : 　 demographi c　 data

homogeneity

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( )

( ) ( ) ( )I. | , , , , Normal Truncatedi

i i iht ht hy I V

　　 : （）

( )

( ) ( ) ( )II. | , , , Normali

i i ih hty V

　　：

( )( )III. | , Inverted Wishartiihy 　　　：

( )

( )( )IV. | Normali

iih V

　 , 　：

( )

( ) ( )V. | Inverted Gammai

i ihV

　 , 　　　：

VI. Threshold Parameter

| Metropolis

| Normal

| Invertied Chi

h

h

h

r

r

r

　　

　　　　 , 　　：（＊）

　　　　 , 　　：

　　　　 , 　　　：

●Conditional Posterior Density for MCMC

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VI. VI. ＝＞＝＞ Metropolis Sampling with Random WalkMetropolis Sampling with Random Walk

21 1 1 1 1; (0, )h h h hr z N 　 ,1,... ,h H

1 1 1 1 1; (0, ).Hr Z N 　

MCMC:MCMC:

Matrix notation:Matrix notation:

HB model for household HB model for household hh::

I.I. ～～ V.V. ＝＞＝＞ Gibbs Sampling by Full Conditional DensitiesGibbs Sampling by Full Conditional Densities　　　　　　　　　　　　　　　　　　 (Rossi, McCulloch and Allenby(1996)) (Rossi, McCulloch and Allenby(1996)) 　　　　　　　　

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Empirical Application

● Data:

・Category：Instant Coffee

・Number of Panels：197（Households）

・Number of Purchases：2840

・ 5 National Brands：

A: 623.5 yen (13.81%) B: 632.9 yen (48.03%) C: 601.3 yen (9.94%) D: 693.2 yen (22.49%) E: 902.4 yen (5.74%).

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●●Marketing Mix VariablesMarketing Mix Variables

X = [Constant, Price, Display, Feature,Brand loyalty ]X = [Constant, Price, Display, Feature,Brand loyalty ]

( Display, Feature: 1 or 0)( Display, Feature: 1 or 0)

Price : log(price)Price : log(price) Display and Feature: binary,Display and Feature: binary, Brand loyalty: smoothing variable over past purchases Brand loyalty: smoothing variable over past purchases proposed by Guadagni and Little(1983)proposed by Guadagni and Little(1983)

Empirical ResultsEmpirical Results

, 1(1 )jht jht jh tGL GL I

1

1

1

1

t

jht jhss

RP Pt

●●Reference PriceReference Price

; Brand Specific RP (Breisch et al. (1997)) ; Brand Specific RP (Breisch et al. (1997))

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●●Household Specific VariablesHousehold Specific Variables

= [Constant, Hsize, Expend, Pfreq]= [Constant, Hsize, Expend, Pfreq]

Hsize: 1-6(Number of Family),Hsize: 1-6(Number of Family), Expend: 9 categories (Shopping Expenditure / Month),Expend: 9 categories (Shopping Expenditure / Month), Pfreq: 3 categories (Shopping Frequency), Pfreq: 3 categories (Shopping Frequency),

= [Constant, Hsize, Expend, Pfreq, Dprone, RP, BL]= [Constant, Hsize, Expend, Pfreq, Dprone, RP, BL] Dprone: deal proneness (Bucklin and Gupta(1992))Dprone: deal proneness (Bucklin and Gupta(1992)) RP: reference price level (Kalyanaram and Little(1994)) RP: reference price level (Kalyanaram and Little(1994)) BL: brand loyalty level (Kalyanaram and Little(1994)) BL: brand loyalty level (Kalyanaram and Little(1994))

rZ

Z

1 1

log( ) / /hTm

h jht hj t

RP RP T m

1max /

hT

h jht hj t

BL GL T

・・ Proportion of purchase (of any the five brands) made on promotion;

;;

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●●Distribution of Price ThresholdsDistribution of Price Thresholds

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●●LPA v.s. Household CharacteristicsLPA v.s. Household Characteristics

““Hsize” increasesHsize” increases=> LPA is getting => LPA is getting NarrowerNarrower

“ “Expend” increases Expend” increases =>LPA is getting =>LPA is getting WiderWider

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““Dprone” increasesDprone” increases=> LPA is getting => LPA is getting NarrowerNarrower

““Pfreq” increasesPfreq” increases=> LPA is getting => LPA is getting WiderWider

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““RP” increasesRP” increases=> LPA is getting => LPA is getting WiderWider

““BL” increasesBL” increases=> LPA is getting => LPA is getting WiderWider

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● ● Effectiveness of Marketing Mix: Beta coefficientsEffectiveness of Marketing Mix: Beta coefficients

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○ Price Gain (Regime 1) & Price Loss (Regime 3): :

　 “ Price” > “Display” > (”Brand Loyalty”)

“Feature”,

○ LPA (Regime 2):

“Display” > (”Brand Loyalty”>) “Feature”.

●Marketing Mix Effectiveness

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Figure 2: Expected Incremental Sales and Profits

● 　 Customized Pricing

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[1] Customized Discounting

(2)Expected Incremental Profit : (M: margin 0.3 assumption)

(1)Expected Incremental Sales (Discount Promotion)

I. αI. α% discount from individual (lower) price threshold% discount from individual (lower) price threshold 1̂hr

( ) (1) (2)1

(1) (2)0 1 01

(2) (2)0 1 01

( |{( , , ), 1,..., })

1 Pr , 1 Pr , if 0 : (Price Gain)

1 Pr , 1 Pr , if 0 : (LPA)

j h h h

H

j h jh h j h jhh

H

j h jh h j h jhh

IS r h H

P r PH

P r PH

　　　　

( ) (1) (2)1

(1) (2)0 1 0 11

(2) (2)0 1 0 11

( |{( , , ), 1,..., })

1 Pr , 1 Pr , if 0 : (Price Gain)

1 Pr , 1 Pr , if 0 : (LPA)

j h h h

H

j h jh h j h jh hh

H

j h jh h j h jh hh

IP r h H

P r P M rH

P r P M rH

(1) (2)1 ( 0), , , 1,...,h h hr h H Conditional on Conditional on

1 0, 0, 1,..., 5%hr 　

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Unconditional EstimatesUnconditional Estimates

(1) ( 2)1

( ) ( ) (1) (2)1( , , )

( ) ( |{ , , , 1,..., })h h h

j j h h hrIS E IS r h H

(1) ( 2)1

( ) ( ) (1) (2)1( , , )

( ) ( |{ , , , 1,..., })h h h

j j h h hrIP E IP r h H

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[2] Customized Price Hike Strategy

II. αII. α% price hike from individual (upper) price threshold% price hike from individual (upper) price threshold

(1)Expected Incremental Sales

(2)Expected Incremental Profit : (M: margin 0.3 assumption)

2̂hr

(2) (3)2 ( 0), , , 1,...,h h hr h H Conditional onConditional on

( ) (2) (3)2

(2) (2)0 2 01

(3) (2)0 2 01

( |{( , , ), 1,..., })

1 Pr , 1 Pr , if 0 : (LPA)

1 Pr , 1 Pr , if 0 : (Price Loss)

j h h h

H

j h jh h j h jhh

H

j h jh h j h jhh

IS r h H

P r PH

P r PH

　　　　

( ) (2) (3)2

(2) (2)0 2 0 21

(3) (2)0 2 0 21

( |{( , , ), 1,..., })

1 Pr , 1 Pr , if 0 : (LPA)

1 Pr , 1 Pr , if 0 : (Price Loss).

j h h h

H

j h jh h j h jh hh

H

j h jh h j h jh hh

IP r h H

P r P M rH

P r P M rH

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Unconditional EstimatesUnconditional Estimates

( 2) (3)2

( ) ( ) (2) (3)2( , , )

( ) ( |{ , , , 1,..., })h h h

j j h h hrIS E IS r h H

( 2) (3)2

( ) ( ) (2) (3)2( , , )

( ) ( |{ , , , 1,..., })h h h

j j h h hrIP E IP r h H

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●●Large Difference of SalesLarge Difference of Sales

・・ Between (LPA and LOSS)Between (LPA and LOSS)・・ Between (LPA and GAIN)Between (LPA and GAIN)

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●●Maximal Profits happen atMaximal Profits happen at

・・ rr1h 1h (Discounting)(Discounting)

・・ rr2h2h (Hike) (Hike)

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Non-customized Pricing (flat pricing)Non-customized Pricing (flat pricing)

Manager does not now know the price thresholdsManager does not now know the price thresholds

＝＞＝＞ has to try possible levels f pricing.has to try possible levels f pricing.

＝＞＝＞ d*d* = 0,±1, ±2, … , ±15% = 0,±1, ±2, … , ±15%

　　 Compare their incremental profits with those of optimal Compare their incremental profits with those of optimal customized pricing at customized pricing at rr1h1h and and rr2h2h

1

2

( ) ( ) *1

( ) ( ) *2

( ) ( ) : Discounting

( ) ( ) : Hike

h

h

j jr h

j jr h

E IP r IP d

E IP r IP d

　

　

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[3] Difference from non-customized pricing[3] Difference from non-customized pricing

( ) * ( )1

(1) (2)0 1 0 1

1

( ) * (2) *0 0

1

( |{ , , 1,..., })

1 Pr , 1 Pr ,

1 Pr , 1 Pr ,

j h h

H

j h j h j h j hh

H

j h j j h jh

DIF d r h H

P r P M rH

P d P M dH

　　　

　　　　　　　　　

depends on the regime determined bydepends on the regime determined by

discount leveldiscount level *d

ConditionalConditional on on ( )

1{ , , 1,..., }h hr h H

UnconditionalUnconditional

( )1

( ) * ( ) * ( )1( , )

( ) ( |{ , , 1,..., })h h

j j h hrDIF d E DIF d r h H

( )2

( ) * ( ) * ( )2( , )

( ) ( |{ , , 1,..., })h h

j j h hrDIF d E DIF d r h H

(ii) Price Hike(ii) Price Hike

(i) Discounting(i) Discounting

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A. Customized Discount Strategy (customized couponing)A. Customized Discount Strategy (customized couponing)

Sales:Sales:(1)(1) For every brand, there is For every brand, there is a great differencea great difference of sales of sales increase between price gain regime and (negative)increase between price gain regime and (negative) LPA at LPA at (2) The sales of most expensive (2) The sales of most expensive brand Ebrand E change most. change most.

Profit:Profit:(1)(1) Optimal discount levels happen at the lower priceOptimal discount levels happen at the lower price threshold for every brand. threshold for every brand.

1̂ %hr

●●Empirical Implications Empirical Implications

1̂ %hr

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B. Customized Price Hike StrategyB. Customized Price Hike Strategy

Sales:Sales:(1)(1)Large difference between inside and outside of the Large difference between inside and outside of the upper price threshold for every brand. upper price threshold for every brand.

Profit:Profit:(2) The price hike at the level of makes (2) The price hike at the level of makes the incremental profits most. the incremental profits most.

2̂hr

2̂hr

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Summary

1. Modeling

(i) Non-linear(piecewise linear) Random Utility

Price Threshold,

=> Latitude of Price Acceptance

Asymmetric Market Response

(ii) Continuous Mixture Model (HB to Threshold Probit Model)

=> Heterogeneous Consumers

(iii)

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２． Empirical Findings and Implications

(i) Price threshold models dominates Linear model

=>Existence of Heterogeneous Price Threshold

(ii) Marketing Mix Effectiveness

○ Price Gain &Loss Regimes: 　 “Price” > “Display” > (”Brand Loyalty”) “Feature”

○ LPA Regime: “Display” > (”Brand Loyalty”) “Feature”

(iii) Estimated “Heterogeneous Price Thresholds”

=> Incremental Profit

r1h => Discounting(Target Couponing)

rr2h2h => Price Hike => Price Hike

=> => Important information for customization strategyImportant information for customization strategy of of

pricing.pricing.

Estimating Heterogeneous Price Thresholds

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