1 Estimating Heterogeneous Price Thresholds Nobuhiko Terui* and Wirawan Dony Dahana Graduate School of Economics and Management Tohoku University Sendai 980-8576, Japan *E-mail:[email protected]*Voice & Fax:+81-22-217-6311 International Conference at ISM
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1 Estimating Heterogeneous Price Thresholds Nobuhiko Terui* and Wirawan Dony Dahana Graduate School of Economics and Management Tohoku University Sendai.
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1
Estimating Heterogeneous Price Thresholds
Nobuhiko Terui*and
Wirawan Dony Dahana
Graduate School of Economics and ManagementTohoku University
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
5
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
6
Tools
・ Threshold Probit Model
・ Hierarchical Bayes Modeling
・ MCMC
=>Gibbs Sampling for Response Parameters
=>Metropolis-Hastings Sampling for Threshold Parameters
7
The Meaning of Model: 3 Regimes Model
u
0 Xt-RPtr1 r2
8
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
9
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
10
(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
12
●●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%).
( 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
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
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)