Discussion Paper No. 50

A Threshold Model for
Discontinuous Preference Change and Satiation
Nobuhiko Terui
Shohei Hasegawa
Greg M. Allenby
April, 2013
May, 2015
October, 2015

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research
Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku
Sendai 980-8576, JAPANA Threshold Model for
Discontinuous Preference Change and Satiation

byNobuhiko Terui
Graduate School of Economics and Management
Tohoku University
Sendai, Japanterui@econ.tohoku.ac.jp
Shohei Hasegawa
Faculty of Business Administration
Hosei University
Tokyo, Japan
shasegawa@hosei.ac.jp

andGreg M. Allenby
Fisher College of BusinessOhio State University
Columbus, Ohio
allenby.1@osu.edu

April, 2013
May, 2015
October, 2015

1

A Threshold Model for
Discontinuous Preference Change and Satiation

Abstract

We develop a structural model of horizontal and temporal variety seeking using an dynamic factor
model that relates attribute satiation to brand preferences.

The factor model employs a thresholdspecification that triggers preference changes when customer satiation exceeds an admissible level but
does not change otherwise.

The factor model can be applied to high dimensional switching data

often encountered when multiple brands are purchased across multiple time periods.

The model isapplied to two panel datasets, an experimental field study and a traditional scanner panel dataset,
where we find large improvements in model fit that reflect distinct shifts in consumer preferences
over time. The model can identify the product attributes responsible for satiation, and can be used to
produce a dynamic joint space map that displays brand positions and temporal changes in consumer
preferences over time.

Key words and phrases:
Dynamic Factor Model, Horizontal Variety, Product Attributes, Threshold Switching Structure2

A Threshold Model for
Discontinuous Preference Change and Satiation

1. Introduction
Consumers purchase a variety of goods in every product category, and invariably switch brands
at some point in time because of deals or because they get tired of consuming the same products.
Temporary changes in preference have been the subject of a long literature on variety seeking inmarketing, with reasons ranging from the presence of multiple needs, persons and contexts, tochanges in tastes, constraints and available offerings (see McAlister and Pessemier, 1982).

This

paper investigates the relationship between satiation and preference as an explanation for why people
seek variety.

We find that consumer satiation affects preferences in a predictable way up to a point,

after which consumer preferences abruptly change and consumers switch to varieties that are
distinctly different from those consumed in the past.

Our findings have implications for the breadthof products carried by retailers.
Existing models of variety seeking assume that changes in the demand for varieties can be
explained by a continuous model structure with predictable changes in preference.

Models ofhorizontal variety seeking (Kim et al. 2003, Bhat 2005), for example, assume that diminishingmarginal returns (i.e., satiation) explain why people purchase multiple varieties, where marginalutility is a function of quantity purchased.

In these models people are assumed to have stablepreferences that diminish in intensity as consumption quantities increase, leading them to tire of
consuming large quantities.
Variety seeking behavior is also modeled by including lagged purchase variables in models of
discrete choice (Dube, Hitsch and Rossi 2009 , Chintagunta 1998).

A positive coefficient value for

lagged purchase leads to higher repurchase probabilities for the consumed product while leaving thepreference ordering for the remaining varieties unaffected.

Models with lagged purchase variables

make relatively minor changes to the preference ordering unless multiple lagged variables are3included in the model specification.
A problem common to the study of temporal and horizontal variation of brand purchases is withthe dimensionality of switching behavior.

The study of N brands is associated with N2 different

possible "switches" for two time periods if only one brand is purchased.

The dimensionality

increases when considering the temporal variation of horizontal variety, where multiple goods arepurchased in both time periods.

Models of switching behavior are challenged by the large number ofpossible purchase outcomes, even when the number of brands under study is relatively small.
In this paper we propose a model of preference change that distinguishes multiple forms ofvariety seeking and examines the temporal relationship between product satiation and brandpreference.

We find that preference changes are initiated from a latent satiation variable that varies

over time, which is related to brand characteristics that also drive baseline preferences. Consumer
preferences are stable for a period of time, and then abruptly change when the latent satiation variable
exceeds a threshold value.

This behavior is consistent with consumers becoming tired of consuming

the same set of products and then moving on to a new set.

Our threshold model relates productattributes to switching behavior and can identify which product attributes are responsible for thechange, and has implications for the breadth of offerings carried by retailers.
We address the issue of dimensionality by employing a dynamic factor model that relates
preference and satiation.

The factor model allows for continuous variation in product satiation

parameters, and the factor loadings point to product attributes responsible for variation in the satiation
parameters.

The dynamic satiation factor is also used to explain variation in brand preference

parameters through a second set of factor loadings.

We find that the best fitting model has thepreference factors changing only when the dynamic satiation factor exceeds a threshold.
The organization of the paper is as follows.

In Section 2 we introduce our model and contrast

it to existing models of horizontal and temporal variety seeking.

We include a discussion ofalternative models examined in our empirical analysis that relax some of the assumptions of our
4

proposed model.

Section 3 describes two empirical applications – one using experimental data ofcorn chip consumption and the other using a scanner panel dataset of yogurt consumption.
cases we find significant improvement in model fit over existing models.

In bothSection 4 contains a

discussion of the results for the two datasets, illustrating the insights provided by the models.
Concluding remarks are offered in Section 5.

2. ModelModels of discrete behavior, characterized by thresholds and switching regimes, have beenfound to provide an accurate description of many aspects of consumer behavior.

Behavior decision

theory (Einhorn and Hogarth, 1981), for example, describes the effects of framing on consumer
decision processes that reflect discrete differences in how consumers view consumption opportunities.
The most widely known example of this is how consumers react to gains and losses (Thaler, 1985),
but more broadly the similarity, attraction and compromise effects regularly documented in models of
choice (Roe, Busemeyer and Townsend, 2001) point to discrete and discontinuous effects in thedecision process.

The behavioral decision theory literature (e.g., Kahneman and Tversky, 1979)

indicates that human behavior reflects discrete, not continuous changes as choice alternatives aredescribed, framed and presented to consumers.
The modeling literature in marketing has also found that models with discrete thresholds
provide a good description of marketplace behavior.

Switching regression models (Terui and

Dahana, 2006), models of structural heterogeneity (Kamakura et al. 1996), and Markov switchingmodels (Fruhwirth-Schnatter, 2006) all describe different response processes among and within
respondents.

Fong and DeSarbo (2007) propose a model of choice in which consumers can enter apassive state of response once they become fatigued.

Gilbride and Allenby (2004) propose a choicemodel with screening component that sets the choice probability to zero if a brand does not enter aperson's consideration set.

Terui et.al (2011) found that media advertising for mature products5affects brand consideration and choice as long an advertising stock variable is above a thresholdvariable.
We develop our model within the framework of direct utility maximization (Kim et al. 2002,
Consumer h’s utility over j

Bhat 2005 and Hasegawa et al. 2012).

1, … , m varieties at time t

are defined as:

U

x

where

,…,x

,z

0.

ψ

ln γ

x

1

ln z

,

(1)

is the vector of quantity demanded by consumer h at t , z

represents the outside good, and ψ
γ

∑

, γ

j

1, … , m are parameters restricted as ψis the baseline value of marginal utility for a product j when x0, and γ

0 and
is a

satiation parameter that affects the rate at which marginal utility diminishes.
The utility function in (1) is additively separable in the choice alternatives implying that the
goods are substitutes and the utility generated by one good is not influenced by the amount of anothergood purchased.

Below we re-parameterize the satiation parameter in terms of productcharacteristics and other modeling components that are common across the m goods.

Thus, while

utility is additively separable in terms of the product quantities purchase, changes to product attributesand variables relating to the components can affect multiple goods.

An alternative approach assumes

that utility is generated by a characteristics (Lancasterian) model as in Chan (2006).

However, thisapproach requires the introduction of interaction terms to overcome the separability restrictions, andassumes that there are not unobserved product characteristics.

Our approach instead relies upon theuse of a low-dimensional factor model to relate the goods so that the marginal utility of acharacteristic (c), i.e.,, is not additively separable in the choice alternatives.

A stochastic model is obtained by assuming that the baseline utility parameter has an error, orthat ψ

exp ψ∗

ε

where ψ∗and ε
6are unrestricted and independent errors,respectively.
constraint
E

Then, the likelihood function is obtained by maximizing (1) subject to the budget′z

E , whereis the total expenditure.
Q

U

and

, respectively, mean price and quantity vector, andThis is accomplished by creating the auxiliary equation as follows:
,z

λ

′

z

E

.

(2)

By employing the Kuhn–Tucker conditions of constrained utility maximization, we obtain an
expression that relates the observed demand to the error terms as follows:
ψ∗

ε

ln γ

x

1

p

ln
E

ε

ψ∗

ln γ

x

1

ln

′

if x

0

(3)

if x

0.

(4)

p x

′

Then, the likelihood function is composed of a combination of density and probability mass, arisingfrom the interior and corner solutions, respectively. We assume that it follows independently normal
distribution ε ~N 0, 1 , as developed in Hasegawa et al. (2012).

Baseline and Preference Dynamics with Switching Structure
We assume that the baseline parameters are well projected into a lower-dimensional space, as is donein a choice map when conducting a market structure analysis (Hauser and Shugan 1983, Elrod 1988,
Chintagunta 1994, Wedel and DeSarbo 1996):∗;∼ N 0, V

Each row vector of factor loadings matrix
corresponding factor score vector
will refer todiag v , … , v.

(5)defines the coordinate of brand position and, indicating consumer h’s preference direction at time t.

as the preference direction vector.

We

We assume that the preference direction willchange when consumer satiation level exceeds the admissible level r , but does not change otherwise.
Then, the first dynamic factor model is described as follows:

7

f∗

where
∗

β

,β

′

;

∼ N 0, Ir

if f ∗r ,∼ N β ,1 k

and we set the hierarchical model as β

is described below in our model for satiation dynamics.

if f ∗We set r

(6)
1, 2.

The factor

0 for identification in

the empirical application.
Our formulation is consistent with existing models in the marketing and psychology literatures.
The dynamic attribute satiation (DAS) model of McAlister (1982) and McAlister and Pessemier(1982) predicts that variety seeking occurs as a respondent's consumption history evolves.

Sarigollu(1998) extends the DAS model to include a discrete choice model where preferences are related to an
inventory of past attribute accumulation.

Our formulation includes the discrete choice model as a

special case, and incorporates threshold effects (rh) leading to discrete changes in preference.

The

model explains observed choices through the parameters of the choice model (e.g.,∗through time according to a dynamic factor model specified in equations (5) and (6).

One possible) that vary

motivation for our model is the single peak model by Coombs and Avrunin (1977) in which
consumers reach an optimal level of an attribute and then, because of their satiation, decide toconsume a different attribute on the next purchase occasion.

We investigate changing preferences

for brands through a dynamic factor model with a loading matrix

that, as we see below, helps

visualize spatial patterns of competition.

Satiation Dynamics
We relatesatiation parameters tobrand characteristics

using information

provided to us by the product manufacturer in a linear mapping, similar to that found in conjoint
analysis.

They are used as γ∗c α , and are organized in a matrix form by∗

where γ∗exp γ,

, c is a vector of characteristics for the j

8(7)
product of dimension p, andis the matrix constituted by a row vector.

Thus, product satiation at time t, γ∗ , is assumed to belinearly related to the satiation of the attributes.

We further assume that attribute satiation

follows a dynamic factor model with a time-invariant factor loading matrix

and one-dimensional

factor f :ff
The satiation factor score f

;

f

ν ;

∼ N 0, Σνdiag σ , … , σ(8)∼ N 0,1 .

(9)is specified a priori in (9) as a random walk, which imposes asmoothness prior on changes in the factor over time.

The random walk specification allows for theaccumulating effects of past purchases, where parameter values of high likelihood rationalize theobserved choices.

Equation (9) defines a non-parametric model for temporal dynamics, and it

accommodates a trend component locally linear over time in the non-stationary part worth andsatiation parameters.

This specification has been successfully used in state space modeling (See theliteratures in time series analysis, e.g., Harvey (1989), Kitagawa and Gersh (1984), West and Harrison(1997), Terui et al. (2010), and Terui and Ban (2014)).
The factor score moves rather smoothly when the variance of factor score is smaller than thepart worth’s variance, as is employed and discussed in Hasegawa et al. (2012).

When multiplied by

the factor loading matrix, the result is a vector of attribute satiation coefficients that evolve throughtime with expected valuef .

Thus, the factor loading matrix

can indicate for which of the

product attributes consumers experience temporal variation in satiation.

Alternative Models
We compare our model with six alternative models.

The first model employs a static

preference direction using an ordinary factor model, and is denoted as (Static).
∗;∼ N 0, Vdiag v , … , v

Thus, preferences are assumed to have constant expected value (
9

.

(10)

) that is unrelated to satiationeffects.
The second alternative is a dynamic model that assumes the preference direction vector
follows a random walk:;∼ N 0, I .

(11)This specification is identical to a non-parametric model of a stochastic trend in time series.

Equation (11) has no causal variables or structural parameters, and we refer to this model as a
non-parametric dynamic factor model (NDF).

This specification was successfully employed in

Hasegawa et al. (2012) to capture the locally linear stochastic trend for a non-stationary series.
The third model specifies the preference direction vector

as being related to satiation in theprevious period:f∗h1 ht 1;

∼ N 0, I .

(12)

We call the model represented by (12) as the structured dynamic factor model (SDF).

Both the NDF

and SDF have a common property that preference changes whenever a consumer purchases a product.
The next set of models has a switching mechanism regarding the timing of preference change.
We assume that satiation drives the change when its level exceeds a threshold value, and does not
drive the change otherwise.

These models have various forms, or types.

The first is an SDF modelwith threshold switching, called a switching non-parametric dynamic model (SNDF):;∼ N 0, I

if f ∗

r

if f ∗r(13)

The second model is our proposed switching structured dynamic factor model (SSDF1) shown in (6).
The third model is composed of the two previous models, called a hybrid dynamic factor model(SSDF2):f∗h1 ht 1;

∼ N 0, I

if f ∗
if f ∗

10

r
r

(14)

This model provides a flexible specification of the preference vector updating equation, allowing thecurrent preference vector to be influenced by both the satiation variable and the past preference vector.
It allows the expected preference vector to the informed by the satiation variable but not entirely
dependent on it.
The fourth model includes an autoregressive term (flexibility relative to equation (14).
f∗h1 ht 1) in the model to allow for greaterThis model is referred to as SSDF3:;

h2∼ N 0, I

if f ∗
if f ∗

r
r(15)

These models provide a comprehensive set for assessing the benefit of the proposed dynamic modelfor describing preference change.

Table A provides a summary of the alternative model

specifications:== Table A ==3. Empirical Application
We apply the model to two datasets, an experimental dataset of corn chip consumption ofundergraduate students at a large U.S. university, and a scanner panel dataset of yogurt consumption.
3.1 Field Experiment Data
Data and Variables
Students were recruited for the experiment if they frequently purchased salty snacks for
personal consumption.

Students were allocated a $2.00 weekly budget and asked to purchase among

eight varieties of corn chips.
six packets each week.

The offerings were priced at $0.33, allowing the students to select up to

The regular price of a corn chips packet was $0.99. The students were toldthat any unused budget allocation would be paid in cash at the end of the experiment.

By offering

the chips at reduced prices, we hoped to induce higher levels of consumption, which might provide
useful information about satiation.

Students were instructed to purchase the chips for their own
11consumption, not for the consumption of others.

These data were previously analyzed by Kim et al.

(2009) using a subset of product characteristics and a stationary demand model.
Table 1 lists the offerings and associated product characteristics that were provided by the cornchips manufacturer.

The chip varieties and characteristics are disguised for proprietary purposes, butreflect summary taste characteristics such as “citrus,” “red pepper,” and “treated corn” that aremeaningful to the manufacturer.

The experiment was conducted over a seven-week period, resulting

in a total of 634 observations for 101 subjects.

The data for each purchase occasion is composed ofa vector of purchase quantities of each of the eight corn chip varieties, and the quantity of the outside
good that was set equal to the unspent budget allocation.

Previous analysis of a portion of thecharacteristics reported in Table 1 indicated that product characteristics could successfully be related
to baseline utility in a static model of a choice model.
== Table 1 ==Summary statistics of the data are reported in Table 2.

Very few of the purchase occasions

resulted in a corner solution where just one of the varieties were selected.

Purchase incidence of thevarieties ranged from 168 to 244, indicating that no variety was dominant in the data.

The

prevalence of interior solutions points to the need for a demand model that can accommodate interior
solutions.
== Table 2 ==Model Comparison
We employ Bayesian MCMC methods to evaluate the joint posterior density for these models.
Algorithms for model estimation are provided in the appendix. Models converged relatively quickly
and were estimated on the basis of 20,000 iterations of the Markov chain after 10,000 burn-in samples.
The interpretation of satiation parameters and the number of factors are robust throughout the models.
Table 3 reports two measures of model plausibility for each model: the log marginal density(ML) and Bayesian deviance information criterion (DIC).
12

The DIC is a measure of modelcomparison that explicitly penalizes a model for its number of parameters (Spiegelhalter et al., 2002).
We use the DIC instead of calculating model performance on a holdout set of data because our student
panel experienced a fair amount of attrition toward the end of the study, particularly in the sixth andseventh weeks.

The loss of data toward the end of the panel makes it difficult to compare theout-of-sample predictions, particularly with dynamic models.

The results indicate that the models

differ greatly in their fit to the data.
== Table 3 ==First, we find that incorporating dynamics into the baseline parameters ψleads to adramatic improvement in model fit in terms of criteria by observing dramatic improvement between
the static and dynamic models.

The static model fit shows DIC = 11067.6, and log ML = −5088.8.

On the other hand, the dynamic models have approximately 80% lesser DIC and 60% greater log ML
values.

The fit results indicate that consumer variety seeking behavior is episodic, with regimechanges in tastes as predicted by our model and explained by increased levels of satiation.

Second, a

comparison among the dynamic models shows that the switching models dominate the steadychanging models in both criteria.

That is, preferences change discretely in relation to the level ofsatiation in the previous period.
The best-fitting model is our proposed model (SSDF1).

In this model, preference changes are

represented by discrete switches related to previous levels of satiation.

The next best model is theswitching structured dynamic factor autoregressive model (SSDF3), indicating that a parametricspecification is better than a non-parametric local trend specification (SNDF and SSDF2).

Parameter Estimates
Table 4 summarizes the posterior distributions of parameters for the model SSDF1, thebest-fitting model.
== Table 4 ==13

The top portion of the table, Table 4(a), reports estimates of parameters in the dynamic factor
model for the baseline parameters across households and over time.

Since the posterior distributionsof some of these quantities are skewed, we also report the posterior median when needed.
∗

observe that the estimated baselines
purchase records.

Wefor products are almost proportional to their shares in

The factor loading matrix, as well as the variance matrix V, are almost

significantly estimated.
The middle portion of the table, Table 4(b), reports the posterior mean and median forcoefficient parameters in the switching equation.

The satiation level fpreference in the first dimension as E β

1.323 standard deviation S. D. ∶ 0.885 .

the other hand, it does not affect the second dimension as E β

explains the direction ofOn

0.112(S.D.:0.854).

The bottom portion of the table, Table 4(c), reports estimates of parameters in the dynamic
factor model for brand satiation, i.e., product attribute part-worth on satiation, factor loadings, andvariances.

The estimates forandhave opposite signs, implying that f

represents an

excitement (anti-satiation) factor score, the same as that shown in Hasegawa et al. (2012).
estimates of part worth
satiation.

Themean the importance weights of product characteristics c1-c12 on the

The characteristics (c4, c5, c10, c12) have positive large numbers of estimates, indicatingthat these characteristics contribute to brand satiation.

In contract, (c1, c8, c11) with negative large

values are characteristics that reduce consumer satiation. Finally, (c2, c3, c6, c7, c9) have almost zero
impact on satiation.
== Figure 1 ==Figure 1 depicts a histogram of individual consumer propensity score kpreference.

The score is defined bykwhere kto change their

∑

k

k

T

(20)

⁄R is the posterior probability of consumer h’s changing preference at period14

t across R iterations of MCMC, and kis the indicator taking binary according to the switchingmechanism
k

1

if f

∗

0

0

if f

∗

0

(21)

The figure shows that many consumers change their preference, as E k
consumers with k

0.8 is 71.3%.

= 0.791 and the share ofThus, while about 20% of the consumers exhibit stablepreferences, the remainder exhibit changes in baseline preferences over the course of their purchasehistory.
== Figure 2 ==Figure 2 shows histograms of estimated coefficients on the switching equation.
and right figures are the estimates of βof β

for the first dimension of the preference direction and thosefor the second dimension, respectively.

The heterogeneous distribution across consumers is

relatively stable, although slightly skewed to the right for β
0.112.

g
Considering the relation g

The left

β
β

f∗

ω
ω

.

Eβ

1.323 and E β, this means that the satiation level

affects the first dimension more than the second dimension.

3.2 Market Data- Scanner Panel Data
Data and Variables
The second dataset is the scanner panel data of yogurt category, which was used in Kim etal.(2002).

It contains five popular flavors of Dannon yogurt - blueberry, mixed berry, pin a colada,

plain, and strawberry- in eight ounce size, and their purchase records are restricted to only thosehouseholds with more than one purchase occasion for at least one of our five varieties.
data set with 332 households and 2,380 purchase observations.

This yields a

However, we extracted 127

households with more than five purchase records in order to keep holdout samples for evaluating
RMSE of forecasts.

The data is summarized in Table 5.

15

We note that total counts and proportions in the bottom row are calculated based on the numberof brand purchases (not purchase incidences) in conformity with experimental data in section 3.1.
We observe that this dataset has 41% interior solutions among 2,109 brand purchases, and it suggests
weaker satiation on the category than the case of experimental data.

We also denote that five flavors

have an identical price which is changing over time.
== Table 5 ==Model Estimation
This dataset does not contain the precise information on consumer’s budget, i.e., the amount ofoutside goods, and this situation leaves the equilibrium level of λ undetermined.

In this case, weconstitute the likelihood based on relative likelihood by taking difference of respective brand’s
conditions from that of a specified brand, as is done by Kim et al. (2002).
ψ∗That is, let D
(4)

ln γ

x

1

ln p

, then the Kuhn-Tucker conditions (3) andlead toεDln λ if x0εD

ln λ if x

0

and we take difference from a purchased brand k at t for h in order to get rid of the term ln λ fromthese equality and inequality equations respectively, then we have

In the above, we define e

ε

e

V

if x

0

(22)

e

V

if x

0

(23)

and V

D

ε

D

, where k means the index for apurchased brand at t for h.
Then if we assume that εee′sfollowe,ethereduced,…,e～matrix are V efollows a standard normal distribution, the derived error terms
dimensional

multivariate

normaldistribution

with0, Σ , where the diagonal elements of variance covariance2, and offdiagonal elements are Cov e
16

,e

1.

In this case, theprobability mass for non-purchase occasions after arranging terms for non-purchase brands into latter
positions ordering by

1, … ,

1,

0,

1, … ,

1

Pr x
⋯

e

,e

,…,e

de

de

… de

(24)

can be evaluated via GHK simulator.
The dataset does not contain product characteristic information, and we directly apply dynamic
factor model to satiation parameters∗.

That is, we define the product characteristics matrixin(7) as the identity matrix in this analysis.
== Table 6 ==Model Comparison
Table 6 shows the results of model comparison.

In addition to log ML as in-sample criterion,

we directly evaluated the RMSE of forecasts as out of sample criteria because we could keep holdout
samples.

The same conclusions are derived by these criteria as those obtained in experimental data.

That is, incorporating dynamics improves model fit significantly, and the dynamic structured models
with switching structure perform substantially better.

Both criteria suggested SSDF1 as the bestmodel.
== Table 6 ==Parameter Estimates
The parameter estimates are shown in Table 7.

The panel (a) shows that the estimated baseline

parameters are significant in the sense of posterior t ratio from grand means and S.D.’s, and they areproportional to purchase shares which justify our estimates. The factor loadings as well as theirvariances are also significantly estimated.
for identification.

We note that the baseline parameter for brand E is set as 0

The panel (b) denotes the coefficient estimates on the previous satiation level.

We observe that the previous level of satiation positively affects first dimension of preference vector
17and it has a lesser influence on the second dimension in opposite direction.
== Table 7 ==== Figure 3 ==== Figure 4 ==Figure 3 illustrates the heterogeneity distribution of individual propensity score of preferencechange.

The heterogeneity distribution is similar to that found for the experimental data.

Figure 4shows the distribution of heterogeneity on the coefficients of switching equations.

4. Discussion
In this section, we investigate model implications using the experimental data.
the scanner panel dataset are similar and are available upon request.

Results fromFirst, we investigate theimplication of our model of preference dynamics by comparing the compensating values (CV)implied by the dynamic factor model to those from the traditional static model.

The results indicatethat varieties are more highly valued with the proposed dynamic model, implying that consumer
highly value wide assortment. We also examine the dynamics of preference change by considering
three panelists with three preference patterns that exhibit frequent changes over time, moderately
frequent change, and non-changing preferences, and consider how estimates for individual consumers
are related to observed purchase behavior and switching structure.

The dynamic joint space maps

are depicted.

4.1 Compensating Values
Following Kim et al.(2002), we compute the compensating value (CV) of the offerings thatallows us to assess the value of an assortment.

The CV is computed by first computing the utility ofthe observed choices from our utility model:

18

max

V

ln

1

ln
(24)

s. t.

px

z

E

The posterior mean of parameters ψ, γ and the observed values of p, E are inserted in (24) to obtainx and V.

We then delete brand i and search for the value of CV so that V

V

p ,E

V

and V

p ,

are equal:(25)

Table 8 provides a comparison of the posterior means of CV for the static and dynamic models, where
the latter model takes grand mean over time, and table 9 expresses CV as a share of budget∑.

Reported in Tables 8 and 9 is the mean and standard deviation of heterogeneity of thecompensating value statistics.

We also report the median of the distribution because of the

distribution of values is highly skewed. The results imply that the static model severely
underestimates CV for every brand and, as a result, undervalues product assortment.

Our model of

discontinuous preference change leads to larger estimates of the value due to the occasions for which
there is strong preference.

Compensating values calculated for the static model are lower because

preference estimates are averaged over the entire purchase history of the consumer, resulting invarieties that appear to be closer substitutes. Our dynamic model supports the existence of wide
assortments, whereas the static model does not.
== Table 8 ==== Table 9 ==4.2 Preference Dynamics
Figure 5(a) shows that the preference direction associated with the purchase occasions of panelist

19

#97 in the experimental dataset.

The purchase records indicate multiple purchases with a broad

range, a satiation score (minus f value) that remains at a high level, and then the switching mechanism
works such that the coordinates in the attributed space move all the time.
characterized as a variety seeker.

This consumer can be

The satiation (excitement) level affects both coordinates positively,

and this impact is much greater for the first dimension.
== Figure 5 ==Figure 5(b) provides the map and tables for panelist #35, showing moderately frequent change.
Preference changes up to the third time purchase, but does not change anymore after this period.

Wenote that the preference direction is not heading to any product during the first period, and four
varieties of corn chips of a single quantity were purchased at this time.

This could imply that she

was unfamiliar with this product category and getting excited as she purchased them.

The satiation(excitement) level negatively and positively affects the first and the second coordinates to change,
respectively.
Figure 5(c) provides the map and tables for panelist #15, showing no preference changes.
record for purchasing D is consistent with the preference direction.

The

The satiation (excitement) level

negatively and positively affects the first and second coordinates to change, respectively, and it ismuch greater for the first coordinate.

Figures 5(a)-(c) demonstrate the flexibility of the proposed

dynamic model for studying preferences.

5. Concluding Remarks
In this paper we develop a parsimonious model of dynamic variety seeking that relates
preference changes to brand satiation.

Two dynamic factor models are developed for baseline andsatiation parameters in a direct utility model of horizontal variety that are integrated so that preference
can change abruptly when the satiation factor score exceeds a threshold level.

We find strong

empirical support for our model in two datasets, and show how the model can be used to identify
20

product characteristics responsible for satiation.

Our model is motivated by theories of adoption andchange similar to McAlister (1982) and McAlister and Pessemier (1982), who proposed a framework
for relating satiation to preference.

The presence of different operating regimes also has a long

history in the marketing and psychology literatures (e.g., Coombs and Avrunin, 1977).
We compare our model to alternative specifications, including a static model implying that
preference does not change at all, a dynamic model without a switching structure on preferencechange, and dynamic models with switching structures.

The models in the last category arecomposed of non-parametric local linear trend, parametric regression, and their hybrid models.

The

measures of model fit, log of ML and DIC support the model with a switching structure andparametric regression.

This means that preference will change occasionally after a consumer issatiated enough, and that it stays the same until the critical level.
The empirical applications demonstrate that abrupt preference changes are common acrossconsumers, with 70% or more changing their preferences over the course of their purchase history.
The results indicate that the standard modeling assumption of static preferences in problematic,
particularly in light of the large increase in model fit for our proposed dynamic models.

We findthat consumers with wider varieties of purchases tend to change their preferences more often – that is,through periodic shifts in preference as described by the model.

This finding is consistent with

consumers having well defined tastes that vary through time, as opposed to broad tastes for which
many brands will suffice.

Our results are consistent with emerging evidence of binge behavior, or"clumpliness" in consumer demand that is not consistent with the notion of stable preferences (Zhang,
Bradlow and Small, 2015).
Future research is needed to understand the context of purchase and consumption.

Our results

indicate that the unit of analysis for heterogeneity is not the respondent, but instead the respondent at
a specific purchase occasion who is influenced by their past decisions and other factors.

Additional

work is needed to identify and integrate these factors and past events into models of consumer
21

decision making that allow marketers to anticipate shifts in preference.

We believe that the temporal

study of satiation, and other triggers of preference change, is a fruitful area for future research.
Appendix – Identification Condition and MCMC Algorithm
We explain the identification condition for the dynamic factor model, and summarize the prior andconditional posterior distribution used for our proposed one-factor model below.

1. Identification Condition on and Priors for Factor Models
For a two-factor model applied to baseline parameters, we restrict the loadings to achieve
statistical identification:
1
a
a
⋮
a

0
1
a
⋮
a

(A.1)

This restriction due to the factor model being applied to parameters of a latent utility is stronger than
Geweke and Zhou’s (1996) condition for the conventional factor model.
We then define prior distribution of factor model as:σ ∼ IG n /2, s /2a

∼N a , A ;as suggested in Lee (2007).

a ,a

′

(A.2)

∼N

for 3

,

The first column of (A.1) is set on factor loadings

model applied to satiation parameter.

p,

(A.3)

for a one-factor

The following same prior distributions are employedvbk∼ IG n /2, s /2

∼N b , B

for 2

(A.4)
k

p,

(A.5)

2. Prior Distributions on Hyper Parameters
Prior

Settinga ∼ N a ,A

a

0,

∼N

,

,

∼N

,

,

A

100
100

β0

10

v ∼ IG n ⁄2 , s ⁄2

n

2,

s

2

σ ∼ IG nσ ⁄2 , sσ ⁄2

nσ

2,

sσ

2

22

3. Conditional Posterior Distributions for MCMC
As for the experimental data, we run 20,000 MCMC iterations for all models, and we used last
10,000 iterations to calculate posterior distribution of model parameters.

The analysis for scanner

panel data needs 40,000 MCMC iterations for all models, and we used last 20,000 iterations to
calculate posterior distribution of model parameters.
(1)

∗

|

,

, ,
∗

p

,
|

,

, ,

,
⁄

∝ det| |
∗

t

∗

′

⁄2

(A.6)

∗

L
The term L∗

exp

is the likelihood function for consumer h

1, … , H at purchase time

1, … , T , where the likelihood function is composed of a combination of density and mass,

arising from the interior and corner solutions, respectively, and is defined for experimental data as
||

,…,

⋯

,…,

⋯

where the probability mass is evaluated in computation by calling function of univariate normal
distribution function.

Fore scanner data, the likelihood is defined similarly as (24), where multiple

integrals are evaluated by GHK simulator.
Setting r

1, … , R to MCMC iterations, we use Metropolis–Hastings with a random walkalgorithm, each h

1, … , H and t

1, … , T .

∗

∗

ψ;

(A.7)

ψ ∼ N 0,

The tuning parameter k was chosen as 0.5 for experimental data, and 0.8 for scanner panel data.
The acceptance probability is
min
|

(2)

,

∗

∗

p
p

,

∗

, ,
,

,

, ,

,1

,

(A.8)

, ,f ,
p|,∗, ,f ,∝ det| |

⁄

exp

f

′

f

⁄2

(A.9)

L
As fort∗, we use Metropolis–Hastings with a random walk algorithm, each h

1, … , H and1, … , T .
α;α ∼ N 0,′(A.10)The tuning parameter k’ was chosen as 0.01 for experimental data, and 0.6 for scanner panel data.
The acceptance probability is

23

min

(3)

∗

,

p
p

, ,
∗

,

,

, ,

,1

,

(A.11),f ,|Under the assumption of uncorrelated a ’s, we define′

and then make downward stacking over h,,ααwe define

′

,…,α

: T

′

f ,f ,⋯,f, ′ ,⋯; ′

′

: ∏
′

1, and

,

T
′

′

,⋯,

: T

1 matrix,

1 matrix.

Similarly,

′

: ∏

T

1.

and explanatory matrix .

Then, we have the regression equation with coefficient parameter vectora ∼ N a ,A ,(A.12)

where
A

A

′

σ

,

a

The identification condition is considered when k
(4)

|

∗

,

∗

σ

′

1.

,

In the same way as
′

A A a

,

′

,⋯;

∗ ′

,

∗ ′

′

′

,⋯,

,

, we define
: ∏
∗ ′

′

T
: ∏

∗

2 matrix and
T

′

,⋯,

: T

2 matrix,ψ∗ , ψ∗ , … , ψ∗

′

: T

1,

1.
∼N

,

,

(A.13)

where
′

v

,

a

The identification condition is considered when j(5) f ,

|

,

∗

∗

′

v
2.

, ,

We reformulate measurement equation (Equations (8) and (13)) and system equation (Equations(9) and (14)).
Measurement equation:
∗

0

f

;

0

System equation:24∼ N 0,

00(A.14)

1

f

K

where

1
′

,β

β

ν

0
K

f

K

1, if f ∗r

K0, if f ∗

r ,

ν

;

∼ N 0,

1
0

0
K

,

(A.15)

andwhere we define f ∗f(A.16)

by interpreting the factor of anti-satiation for f.

We useCarter and Kohn (1994) for a time-varying coefficient in state space model expressed as Equation(A.14) and Equation (A.15)(6) SSDF1:

β

′

,β

|f ,

,
,

∗′ ∗

∗′ ∗

1

∼N β

β

1

k

1,2

(A.17)

where
β
and
f

∗

∗

r

.

and

or K

1 .

∗′

β

are the data matrix and vector respectively collected in case of regime
∅ (K

If

0 at all t), posterior is β

∼ N β , 1 by thehomogeneity.
′β ,β(7) SSDF1:|∼N β , HβH

where ββ(8) SSFD3:
,

,

∑

νβ

and

,β

β
,

k

νβ

1,2

(A.18)

νβ β .
β

,β

|f ,

,

,

,

,

|

,

are sampled by MCMC procedure of general hierarchical Bayes

regression model similarly as equation (A.17) and (A.18).
∼N
We use the data
∅ (K

and

,

(A.19), which are collected in case of regime f

0 at all t), the posterior density is defined by

25

∗

∼N

0 or K
, .

1 .

If

References[1] Bhat, C. R. (2005), “A Multiple Discrete-Continuous Extreme Value Model: Formulation andApplication to Discretionary Time-Use Decisions,” Transportation Research, 39 (8) 679-707.
[2] Carter, C. K. and Kohn, R. C. (1994), “On Gibbs sampling for state space models,” Biometrika,
81, 541-553.
[3] Chan, T.Y. (2006) "Estimating a continuous hedonic-choice model with an application to
demand for soft drinks," RAND Journal of Economics, 37, 2, 466-482.
[4] Chintagunta, P. (1994), “Heterogeneous Logit Model Implications for Brand Positioning,”
Journal of Marketing Research, 32, 304-311.
[5] Chintagunta, P. (1998), “Inertia and Variety Seeking in a Model of Brand-Purchase Timing,”
Marketing Science, 17(3), 253-270.
[6] Coombs, C.H. and G.S. Avrunin (1977), “Single-peaked functions and the theory of preference,”
Psychological Review, 84(2), 216-230[7] Dubé, J.P., J.H. Günter and P. E. Rossi (2010), “State dependence and alternative explanations for
consumer inertia,” The RAND Journal of Economics, Vol. 41(3), 417-445.
[8] Elrod, T. (1988), “Choice Map: Inferring a Product Market Map from Panel Data,” Marketing
Science, 7, 21-40.
[9] Einhorn, H.J. and R. M. Hogarth (1981), “Behavioral Decision Theory: Processes of Judgment
and Choice,” Journal of Accounting Research, 19(1), 1-31
[10] Fong, D.K.H. and W.S. DeSarbo (2007), “A Bayesian methodology for simultaneously detecting
and estimating regime change points and variable selection in multiple regression models formarketing research,” Quantitative Marketing and Economics, 5, 427-453.
[11] Frühwirth-Schnatter, S. (2006), Finite Mixture and Markov Switching Models, Springer, NewYork.
[12] Geweke, J. and G. Zhou (1996), “Measuring the Pricing Error of the Arbitrage Pricing Theory,”
The Review of Financial Studies, 9, 557-587.
[13] Gilbride, T. and G. Allenby (2004), “A Choice Model with Conjunctive, Disjunctive, andCompensatory Screening Rules,” Marketing Science, 23 (3), 391-406.
[14] Hasegawa, S., N. Terui and G. Allenby (2012), “Dynamic Brand Satiation,” Journal ofMarketing Research, vol. XLIX, 842-853.
[15] Hauser, J. R. and S. M. Shugan (1983), “Defensive Marketing Strategies,” Marketing Science, 2(4), 319-360.
[16] Harvey, A. (1989), Forecasting Structural Time Series Models and the Kalman Filter,
Cambridge University Press, London.
[17] Kahneman, D. and A. Tversky (1979), “Prospect theory: an analysis of decision under risk,”
Econometrica, 47, 263-291.
[18] Kamakura, W.A., B Kim and J. Lee (1996), “Modeling Preference and Structural Heterogeneity26

in Consumer Choice,” Marketing Science, 15 (2), 152-172.
[19] Kim, J., G. Allenby and P. Rossi (2002), “Modeling Consumer Demand for Variety,” Marketing
Science, 21, 229-250.
[20] Kim, J., G. Allenby and P. Rossi (2009), “Product attribute and models of multiple discreteness,”
Journal of Econometrics, 138, 208-230.
[21] Kitagawa, G. and J. Gersh (1984), “A smoothness priors-state space modeling of time series withtrends and seasonalities,” Journal of the American Statistical Association, vol.79, 378–389.
[22] McAlister, L. and E. Pessemier (1982), Variety Seeking Behavior: An Interdisciplinary Review,
Journal of Consumer Research, 9, 311-322.
[23] McAlister, L. (1982), “A Dynamic Attribute Satiation Model of Variety-Seeking Behavior,”
Journal of Consumer Research, 9, 141-150.
[24] Roe, R. M., J.R. Busemeyer, J.T. Townsend (2001), “Multialternative decision field theory: A
dynamic connectionist model of decision making,“ Psychological Review, 108(2), 370-392.
[25] Sarigollu, E. and D.C. Schmittlein (1998), “The effect of variety seeking behaviour on optimal
product positioning,” Applied Stochastic Models and Data Analysis, 12(1), 27–44.
[26] Spiegelhalter, D. J., N. Best, B. P. Carlin, and A. van der Linde (2002). “Bayesian measures ofmodel complexity and fit,” Journal of the Royal Statistical Society, Series B, 64 (4), 583-616.
[27] Terui, N. and W. D. Dahana (2006), “Estimating Heterogeneous Price Thresholds,” Marketing
Science, 25 (4), 384-391.
[28] Terui, N., M. Ban and T. Maki (2010), “Finding Market Structure by Sales Count Dynamics—
Multivariate Structural Time Series Models with Hierarchical Structure for Count Data—,”Annals of the Institute of Statistical Mathematics, 62, 92-107.
[29] Terui, N., M. Ban and G. Allenby (2011), “The Effect of Media Advertising on Brand
Consideration and Choice,” Marketing Science, 30 (1), 74-91.
[30] Terui, N. and M. Ban (2014), “Multivariate Structural Time Series Models with Hierarchical
Structure for Over-dispersed Discrete Outcome,” Journal of Forecasting, 33, 376-390.
[31] Thaler, R. (1985), “Mental Accounting and Consumer Choice,” Marketing Science, 4(3),199-214.
[32] Wedel, M. and W.S. Desarbo (1996), “An Exponential Family Multidimensional Scaling Mixture
Methodology,” Journal of Business and Economics Statistics, 14(4), 447-459[33] West, M and J. Harrison (1997), Bayesian Forecasting and Dynamic Models(2nd ed.), Springer,
New York.
[34] Yang, S., G. Allenby and G. Fennel (2002), “Modeling Variation in Brand Preference: The Rolesof Objective Environment and Motivating Conditions,” Marketing Science, 21 (1), 14-31.
[35] Zhang Y, E.T. Bradlow and D.S.Small (2015), “Predicting customer value using clumpiness:
From RFM to RFMC,” Marketing Science, 34(2), 195–208.

27

Table A : Summary of Alternative Models1

1

Model

Dynamic

Switching

Equation

Static

No

No

(10)

NDF

Yes

No

(11)

SDF

Yes

No

(12)

SNDF

Yes

Yes

(13)

SSDF1

Yes

Yes

(6)

SSDF2

Yes

Yes

SSDF3

Yes

Yes

∗

;

∼ N 0, V

Specificationf∗
if f ∗

rf∗

if f ∗

r

(14)

f∗

if f ∗

r

(15)

f∗

if f ∗rdiag v , … , v

28

Table 1 Product Varieties and Characteristics
Varieties
Variety A
Variery B
Variety C
Variety D
Variety E
Variety F
Variety G
Variety H

c1
1.00
0.00
1.00
0.00
1.00
0.00
0.00
0.00

c2
0.00
0.00
0.00
0.00
0.00
0.00
0.00
2.17

c3
0.00
0.00
3.17
0.00
0.00
0.00
3.17
0.67

c4
0.00
0.00
0.00
0.00
0.50
0.00
0.00
0.00

c5
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.50

c6
0.00
1.00
0.00
1.33
1.83
0.00
0.00
0.00

c7
4.00
3.00
0.00
0.88
4.25
4.00
0.00
3.38

c8
1.50
0.67
1.33
3.67
0.00
1.50
1.33
0.00

c9
4.00
3.83
4.17
3.33
4.17
4.00
4.17
3.83

c10
2.00
4.25
4.00
5.50
5.00
2.00
4.00
0.00

c11
0.00
2.00
0.00
2.25
1.00
0.00
0.00
0.00

Table 2 Purchase Summary-Experimental Data
Varieties

Purchaseincidence

Variety A
Variery B
Variety C
Variety D
Variety E
Variety F
Variety G
Variety H
Total

168
177
188
180
190
244
235
218
1600

Total
purchasequantity
224
262
231
235
295
446
338
277
2308

CornersolutionInterior
solution

4 (.02)2 (.01)
6 (.02)12 (0.01)

168 (1.00)
173 (0.98)
188 (1.00)
180 (1.00)
188 (0.99)
238 (0.98)
235 (1.00)
218 (1.00)
1588 (0.99)

Kim et al. (2007)
Table 3 Model Comparison-Experimental Data

Model
StaticS
NDF(Nonparametric)
SteadySDF(Structured)
SNDF
Dynamic
SSDF1
Switching
SSDF2
SSDF3

29

ML
-5088.8-3124.5-3171.8-2829.4-2754.1-2829.5-2758.1

DIC
11067.6
8649.08756.78435.08170.08425.98283.1

c12
1.75
0.00
0.00
0.00
0.00
1.75
0.00
0.00

Table 4 Parameter Estimate-Experimental Data(a) Baseline Parameters
∗

A
B
C
D
E
F
G
H

Mean
-1.763-1.668-1.782-1.552-2.127-0.841-0.836-1.074V

Median-2.307-2.291-1.870-1.900-2.316-0.685-0.385-0.846

S.D(3.026)(3.339)(3.146)(3.065)(3.782)(3.007)(3.129)(2.716)

Mean
1.000
0.000
1.110
1.000
1.066
0.008
1.045
0.772
1.184
-0.353
0.683
1.108
0.690
1.395
0.806
0.863

S.D
(0.107)(0.084)(0.098)(0.085)(0.110)(0.129)(0.093)(0.118)(0.105)(0.129)(0.092)(0.144)(0.093)

Mean
3.140
4.528
3.003
3.304
3.773
4.722
2.479
3.183

S.D
(0.567)(0.673)(0.545)(0.576)(0.642)(0.653)(0.478)(0.534)

These numbers show the grand mean and median of panelist’s estimates over time across panel members. S.D.
refers to the standard deviation of heterogeneity for ∗ . Numbers for b and V show the posterior mean andthe posterior standard deviation in parentheses.

(b) Switching Equation: Baseline ParametersMean-1.3230.112

ββMedian-1.4280.162

S.D(0.885)(0.854)Mean-1.322
0.110

S.D(0.186)(0.189)These numbers show the grand mean of panelist’s estimates across panel members.

(c) Satiation Parameters: Characteristic Level
Σ
c1
c2
c3
c4
c5
c6
c7
c8
c9
c10
c11
c12

Mean
-1.230
0.007
0.081
0.547
0.421
0.052
0.012
-0.224-0.0330.220-0.1290.283

Median-1.396
0.006
0.094
0.603
0.469
0.054
0.016
-0.246-0.0330.253-0.143
0.325

S.D(1.455)(0.033)(0.096)(0.658)(0.498)(0.076)(0.035)(0.262)(0.056)(0.253)(0.151)(0.332)Mean
1.000-0.003-0.064-0.441-0.336-0.042-0.0100.180
0.029-0.1740.102-0.226

S.D(0.029)(0.043)(0.049)(0.042)(0.035)(0.026)(0.033)(0.020)(0.037)(0.037)(0.053)

Mean
0.187
0.105
0.063
0.461
0.322
0.136
0.040
0.069
0.034
0.034
0.094
0.120

S.D
(0.058)(0.026)(0.012)(0.107)(0.056)(0.024)(0.006)(0.010)(0.004)(0.004)(0.019)(0.028)

These numbers show the grand mean and median of panelist’s estimates over time across panel members.
S.D. refers to the standard deviation of heterogeneity for. Number forand Σ show the posteriormean and the posterior standard deviation in parentheses.

30

Table 5 Purchase Summary-Scanner Panel Data

Total
Purchase
Varieties
purchaseincidencequantity
A
712
1138
B
366
583
C
392
612
D
226
323
E
413
708
Total
2109
3364

Corner solution

Interior solution

403 (0.57)
205 (0.56)
186 (0.47)
209 (0.92)240 (0.58)
1243 (0.59)

309 (0.43)
161 (0.44)
206 (0.53)
17 (0.08)
173 (0.42)866 (0.41)

A: Strawberry; B: Blueberry; C: Piña Colada; D: Plain; E: Mixed berry

Table 6 Model Comparison -Scanner Panel Data

Model
StaticSteadyDynamic

Switching

NDF
SNDF
SSDF1
SSDF2
SSDF3

31

ML
-2668.32-1838.09-1697.95-1626.99-1748.68-1759.50

RMSE
101.94
96.92
96.85
90.11
102.68107.59

Table 7 Parameter Estimate-Scanner Panel Data(a) Baseline Parameters
∗

A
B
C
D
E

Mean
4.235
4.334
7.044
-16.438-Median
3.430
3.493
5.640
-13.807-

V
S.D(4.884)(8.067)(7.649)(17.718)
-

Mean
1.000
1.887
1.000
1.003
-0.761-2.7591.308-

S.D(0.102)(0.078)(0.251)-(0.113)(0.234)
-

Mean
2.188
3.675
2.234
14.296-

S.D(0.429)(0.867)(0.507)(3.805)-

These numbers show the grand mean and median of panelist’s estimates over time across panel members. S.D.
refers to the standard deviation of heterogeneity for ∗ . S.D. refers to the standard deviation of heterogeneityfor ∗ . Numbers for b and V show the posterior mean and the posterior standard deviation in parentheses.

(b) Switching Equation

Mean
1.214
-0.876ββMedian1.380-0.845

S.D(0.979)(0.887)Mean
1.213-0.875

S.D(0.120)(0.205)(c) Satiation Parameter
∗

A
B
C
D
E

Mean
-3.665-9.017-8.931-1.649-0.362

Median-3.274-7.883-7.789-1.435-0.363

Σ
S.D(2.923)(7.212)(7.173)(1.350)(0.400)Mean
1.000 2.466
2.446
0.453
0.097

S.D
(0.121)(0.190)(0.221)(0.063)

Mean
1.638
1.640
1.665
2.695
2.256

S.D
(0.356)(0.405)(0.428)(0.885)(0.610)

These numbers show the grand mean and median of panelist’s estimates over time across panel members. S.D.
refers to the standard deviation of heterogeneity for ∗ . Numbers forand Σ show the posterior mean andthe posterior standard deviation in parentheses.

32

Table 8 Compensating Value (CV)

StaticDynamic

mean

std

medianmean

std

median

A

0.077

0.198

0.000

0.478

0.627

0.243

B

0.232

1.166

0.000

1.560

6.023

0.328

C

0.060

0.126

0.000

0.499

0.686

0.249

D

0.086

0.251

0.000

0.723

1.723

0.190

E

0.204

0.761

0.001

6.158

44.794

0.353

F

0.410

0.880

0.001

9.132

49.874

0.650

G

0.217

0.613

0.007

4.089

14.403

0.534

H

0.071

0.136

0.007

0.635

0.818

0.333

Table 9 Percent Compensating Value (PCV) as a Function of Expenditure

StaticDynamic

mean

std

medianmean

std

median

A

0.039

0.099

0.000

0.041

0.054

0.020

B

0.116

0.583

0.000

0.142

0.525

0.028

C

0.030

0.063

0.000

0.043

0.061

0.023

D

0.043

0.125

0.000

0.062

0.145

0.016

E

0.102

0.380

0.001

0.509

3.728

0.036

F

0.205

0.440

0.001

0.846

4.958

0.050

G

0.108

0.307

0.004

0.341

1.217

0.052

H

0.035

0.068

0.004

0.052

0.062

0.028

33

Figure 1 Preference Change

30
0

10

20

Frequency

40

50

60

-Experimental Data-

0.0

0.2

0.4

0.6

0.8

1.0

Figure 2 Parameter Estimates in Switching Equation-Experimental Data -

25
20
0

5

10

15

Frequency

15
10
5
0

Frequency

20

25

30

2

30

1

−3

−2

−1

0

1

−2

34

−1

0

1

2

Figure 3 Preference Change

0

20

40

60

80

- Scanner Panel Data -

0.2

0.4

0.6

0.8

1.0

Figure 4 Parameter Estimates in Switching Equation-Scanner Panel Dataβ

0

0

5

10

10

15

20

20

30

25

30

40

β

−1

0

1

2

3

4

−3

35

−2

−1

0

1

Figure 5 Preference Dynamics- Experimental Data(a) ID#97 (frequently changing)
97
−0.5

0.0

0.5

1.0

B
D

23

2

1.0

−1.0

H

5
6

0

AC

0.0

1

F

0.5

G

E

−1.0

−2

−0.5

−1

4
1

−2

1
2
3
4
5
6

β
β

A
0
0
1
3
1
0

B
1
0
0
0
0
2

C
1
1
0
3
1
1

D
1
1
1
0
1
2

E
1
1
1
0
1
1

−1

F
1
1
1
0
1
0

0

G
0
1
1
0
0
0

H
0
1
1
0
1
0

1.237
0.226

36

1

f
-1.443-1.462-1.374-1.693-1.331-1.309

2

g
0.887
0.915
0.929
0.960
0.997
1.000

g
-0.4630.404
0.370-0.281
0.081
0.008

k
1.000
0.981
0.984
0.988
0.994
0.961

(b) ID#35 (occasionaly changing)
35
−1

0

1

2

2

2

−2

1

F
H

1

G
B
D
3

0

0

2
AC

−1

−1

E

−2

−2

1

−2

1
2
3
4
5
6

β
β

A
0
0
0
0
1
0

B
1
0
1
2
0
0

C
1
0
0
1
2
0

D
1
2
0
2
2
2

E
1
3
2
0
0
0

−1

F
0
0
0
0
0
2

0

G
0
1
2
0
0
0

H
0
0
1
1
1
2

-0.462
0.258

37

1

f
-0.485-0.421-0.068
0.413
1.066
1.583

2

g
-0.162
0.977
0.876
0.933
0.992
0.917

g
-0.987
0.212
0.483
0.359
0.130
0.400

k
1.000
0.561
0.506
0.462
0.433
0.297

(c) ID#15 (never changing)
15
−1

0

1

2

2

2

−2

1

G
1

F
H

B
D
1
0

0

AC

−2

−2

−1

−1

E

−2

1
2
3
4
5
6

β
β

A
0
0
0
0
0
0

B
1
1
0
0
2
0

C
1
0
0
0
0
0

D
1
1
3
3
4
4

E
1
1
0
0
0
0

−1

F
0
0
0
0
0
0

0

1

H
0
0
0
0
0
0

f
0.865
0.743
0.548
0.810
0.783
0.666

G
2
3
3
3
0
2

-1.147
0.278

38

2

g
0.967
0.942
0.840
0.768
0.815
0.723

g
0.254
0.335
0.542
0.640
0.579
0.690

k
1.000
0.156
0.198
0.204
0.173
0.152