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1
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
2
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
11
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
13
Household “ h’ ”
homogeneity
●HB(Hierarchical Bayes) Model
(2)Homogeneous parameter:
(1)Heterogeneous parameter : h
V { }hZ : demographi c data
homogeneity
14
( )
( ) ( ) ( )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
15
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))
16
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%).
17
●●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))
18
●●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;
;;
19
.
20
21
22
●●Distribution of Price ThresholdsDistribution of Price Thresholds
23
●●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
24
●●LPA v.s. Household CharacteristicsLPA v.s. Household Characteristics
““Dprone” increasesDprone” increases=> LPA is getting => LPA is getting NarrowerNarrower
““Pfreq” increasesPfreq” increases=> LPA is getting => LPA is getting WiderWider
25
●●LPA v.s. Household CharacteristicsLPA v.s. Household Characteristics
““RP” increasesRP” increases=> LPA is getting => LPA is getting WiderWider
““BL” increasesBL” increases=> LPA is getting => LPA is getting WiderWider
26
● ● Effectiveness of Marketing Mix: Beta coefficientsEffectiveness of Marketing Mix: Beta coefficients
27
○ Price Gain (Regime 1) & Price Loss (Regime 3): :
“ Price” > “Display” > (”Brand Loyalty”)
“Feature”,
○ LPA (Regime 2):
“Display” > (”Brand Loyalty”>) “Feature”.
●Marketing Mix Effectiveness
28
Figure 2: Expected Incremental Sales and Profits
● Customized Pricing
29
[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
30
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
31
[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
32
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
33
●●Large Difference of SalesLarge Difference of Sales
・・ Between (LPA and LOSS)Between (LPA and LOSS)・・ Between (LPA and GAIN)Between (LPA and GAIN)
34
●●Maximal Profits happen atMaximal Profits happen at
・ ・ rr1h 1h (Discounting)(Discounting)
・ ・ rr2h2h (Hike) (Hike)
35
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
36
[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
37
38
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
39
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
40
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)
41
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.