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Heterogeneity distributions of willingness-to-pay
in choice models
Garrett Sonnier &Andrew Ainslie &Thomas Otter
Received: 3 January 2006 / Accepted: 8 March 2007 /
Published online: 9 August 2007
#Springer Science + Business Media, LLC 2007
Abstract We investigate direct and indirect specification of the distribution of
consumer willingness-to-pay (WTP) for changes in product attributes in a choice
setting. Typically, choice models identify WTP for an attribute as a ratio of the
estimated attribute and price coefficients. Previous research in marketing and
economics has discussed the problems with allowing for random coefficients on both
attribute and price, especially when the distribution of the price coefficient has mass
near zero. These problems can be avoided by combining a parameterization of the
likelihood function that directly identifies WTP with a normal prior for WTP. We
show that the typical likelihood parameterization in combination with what are
regarded as standard heterogeneity distributions for attribute and price coefficients
results in poorly behaved posterior WTP distributions, especially in small sample
settings. The implied prior for WTP readily allows for substantial mass in the tails of
the distribution and extreme individual-level estimates of WTP. We also demonstrate
the sensitivity of profit maximizing prices to parameterization and priors for WTP.
Keywords Bayesian analysis .Choice modeling .Willingness-to-pay
JEL classification C11 .M31
Quant Market Econ (2007) 5:313–331
DOI 10.1007/s11129-007-9024-6
G. Sonnier
University of Texas at Austin, 1 University Station, Austin, TX 78712, USA
e-mail: garrett.sonnier@mccombs.utexas.edu
A. Ainslie (*)
UCLA, 110 Westwood Plaza, Los Angeles, CA 90095, USA
e-mail: andrew.ainslie@anderson.ucla.edu
T. Otter
Ohio State University, 2100 Neil Avenue, Columbus, OH 43210, USA
e-mail: otter_2@cob.osu.edu
1 Introduction
Markov chain Monte Carlo (MCMC) techniques to simulate from a distribution have
facilitated the exploration of any aspect of the distribution under study, including the
distribution of any transformation of random variables. Once a sample of random
variables from the distribution is available, change-of-variable calculus is no longer
needed to derive the distribution of the transformation. The researcher is free to
empirically explore the posterior distribution of the transformation by simply
computing the transformation on each iteration of the sampler (e.g., Edwards and
Allenby 2003). This technique is referred to as “post-processing”MCMC draws.
Posterior summaries of any transformation of parameters are easily obtained from
the MCMC output. This can be particularly advantageous when the estimation
problem is difficult or intractable in the space of interest, but tractable in another
space. Post processing readily allows the researcher to move between the two spaces.
While the advantages of post-processing techniques are well-documented, less
attention has been paid to the fact that the priors used for the parameters in one space
necessarily form an implied prior on the transformed parameters.
It is well known that in a Bayesian model, a change in the likelihood param-
eterization must be reflected in the prior to leave the posterior predictive density
unchanged. Hierarchical models introduce a prior distribution on the parameters
across the observational units. Changes in the parameterization of the full conditional
likelihood will alter the predictive density of the hierarchical model unless the prior
distribution is adapted accordingly. In applied work, the choice of parameterization is
often viewed in isolation of the prior distribution, which is typically chosen for
analytic convenience (e.g., conjugacy). However, a convenient and diffuse prior in
one space does not necessarily result in an equivalent implied prior in the transformed
space (Rossi et al. 2005).
An interesting example of transforming model parameters with relevance to
marketing and economics occurs when estimating the willingness-to-pay (WTP) for
changes in product attributes using choice data. In this paper, we contrast two
approaches to estimating the distribution of WTP with choice models. In the first
approach WTP is defined as the ratio of attribute and price parameters and the
implied prior distribution for WTP is a function of the priors for these parameters.
The posterior of the WTP distribution is explored empirically via post-processing.
This is consistent with the work of Meijer and Rouwendal (2006), which investigates
the properties of WTP defined as a ratio for different distributions of the numerator
and the denominator. The second approach re-parameterizes the full conditional
likelihood to directly identify WTP. This allows the researcher to directly implement
a prior for WTP. We demonstrate the sensitivity of inferences about WTP to different
parameterizations in combination with what are regarded as standard assumptions
about the hierarchical prior (i.e., the heterogeneity distribution). The sensitivity is
particularly pronounced in small sample settings. We show how a normal prior
directly specified for WTP results in better inferences. Moreover, not only is the
posterior of WTP sensitive to the different parameterization and prior assumptions,
but so are all marketing actions derived from the distribution of WTP, such as the
setting of profit maximizing prices. The results illustrate the practical importance of
paying attention to implied priors.
314 G. Sonnier et al.
Implied priors can be problematic even if there is no interest in the transformed
parameters themselves. In choice-based conjoint (CBC), for example, the analyst
may not be interested in the model coefficients or WTP, per se, but rather in using
the model to analyze demand and pricing policies. The reservation and equalization
prices, which completely characterize incidence and switching behavior in response
to price changes, are a function of attributes and attribute WTPs (Jedidi and Zhang
2002).
1
If the distributions of reservation and equalization prices are impacted by the
implied priors on WTP, demand estimates and price policies will be dramatically
affected, as we will demonstrate. While certain point estimates of the distribution
may be less influenced by the implied prior (e.g., the median versus the mean), any
investigation of the nature of consumer demand will require the researcher to
consider more than just a particular statistic.
The organization of the remainder of the paper is as follows: Section 2presents
two parameterizations of choice models that result in equivalent full conditional
likelihoods. It then discusses how the two parameterizations result in different prior
predictive and posterior densities depending on the choice of the prior, particularly
once one introduces heterogeneity. Section 3illustrates the size of the effect using
simulated data. Section 4presents the results from two CBC studies. Section 5
summarizes and offers a brief discussion on the role of prior information in conjoint
analysis.
2 Utility and surplus maximization
2.1 Equivalence of likelihood functions
Consider first consumers’discrete choice problem as that of maximizing an indirect
utility function. We have consumers choosing among alternatives on each of
occasions. Let V*
ijt denote consumer i’s indirect utility for alternative jon choice
occasion t. It is assumed that indirect utility can be expressed as a linear function of
the alternative’s non-price attributes, x
ijt
, income y
i
and price p
ijt
.
V*
ijt ¼x0
ijtϕ*þγ*y
ipijt
þ"*ijt with V*
i0t¼"*
i0t:ð1Þ
We assume the error terms are independent and identically distributed according
to a type I extreme value distribution. For exposition, we initially leave the scale
parameter as unknown, "*
ijt EV 0;μðÞ. It is well-known that multiplying the
indirect utility function for each choice by a constant does not change the utility
maximizing alternative. Thus, V*
ijt must be normalized, which is typically
accomplished by standardizing the error distribution to be EV (0, 1) such that
Vijt ¼x0
ijtϕþγyipijt
þ"ijt ¼x0
ijtϕ*
μþγ*
μyipijt
þ"*
ijt
μ:ð2Þ
1
The reservation price is the price that induces indifference between purchase and non-purchase in the
category. The equalization price (Swait et al.1993) is the price that induces indifference between two
choice alternatives within the category.
Heterogeneity distributions of willingness-to-pay 315
The familiar MNL choice probabilities take the form
Pru
ijt ¼
exp x0
ijtϕ*γ*pijt
μ
1þP
J
m¼1
exp x0
imtϕ*γ*pimt
μ
hi
2
6
6
6
4
3
7
7
7
5¼
exp x0
ijtϕγpijt
hi
1þP
J
m¼1
exp x0
imtϕγpimt
½
2
6
6
6
4
3
7
7
7
5ð3Þ
where the superscript udenotes the probability obtained using the utility model.
WTP for an improvement in x
ijkt
, the kth attribute of alternative j, is the price change
that would leave the individual indifferent between the alternative with the new level
and the alternative with the original level. For continuous x, we have @Vijt ¼
ϕk@xijkt g@pijt ¼0 and the change in price that keeps utility constant given a
change in attribute kis ϕk
γ¼ϕ*
k
γ*(Train 2003). Note that the scale parameter μdrops
out of the WTP.
We can re-parameterize the indirect utility function in (1) by dividing through by γ*.
V*
ijt
γ*¼x0
ijt
ϕ*
γ*þyipijt
þ"*
ijt
γ*
Cijt ¼x0
ijt βþyipijt
þηijt
ð4Þ
In this reparameterization, C
ijt
is consumer i’s surplus from good jon purchase
occasion t(Jedidi et al. 2003). Surplus is determined in part by the attributes of the
products in the set, x
ijt
, and the WTP for the attributes, β. Consumers arrive at their
choices by maximizing the surplus (i.e., the difference between the monetary value
of the attribute bundle and the price to acquire the bundle) among the Jalternatives
in a set on occasion t. The MNL choice probability associated with the surplus
model is
Prs
ijt ¼
exp x0
ijtβpijt
μ
1þP
J
m¼1
exp x0
imtβpimt
μ
hi
2
6
6
6
4
3
7
7
7
5ð5Þ
where the superscript sdenotes the surplus model. The probability expressions in
Eqs. (3) and (5) are equivalent over the range of parameters for which the
transformations b¼ϕ
gand m¼1
gare well defined. In the case of maximum
likelihood (ML) estimation, the Invariance Property of the ML estimator ensures that
precisely the same point estimates of WTP will be achieved regardless of whether
the likelihood is based on (3)or(5) (Cameron and James 1987).
2.2 Bayesian analysis, priors, and posterior distributions for WTP
The ML estimator of the WTP ratio, defined as the ratio of the ML estimates of
8
k
and γ, does not possess finite moments and has infinite risk relative to quadratic and
many other loss functions (Zellner 1978). In a Bayesian framework the problems
associated with the ML estimator are alleviated by the introduction of informative
prior distributions. The model thus consists of the full conditional likelihood for the
data and the prior distribution for the model parameters. A hierarchical prior
distribution defined on the positive real line for γsolves the problem of positive
316 G. Sonnier et al.
WTP for a decrease in utility and ensures that the prior and posterior moments of the
ratio are finite. The prior and posterior for the ratio is implied by the same for the
numerator and denominator.
In the context of random coefficient models, Meijer and Rouwendal (2006)
discuss the properties of the WTP ratio for a number of different distributions for
8
k
and γ. Only in special cases (e.g., a log-normal distribution for both coefficients)
does the ratio of coefficients follow the same distribution as the coefficients. This
implies that, generally, the distributional form of the prior used with the likelihood in
(5) will differ from that implied by the prior for
8
k
and γ. Thus, unlike ML estimates
of WTP from the homogenous model, the posterior distribution of WTP formed by
mixing the likelihoods in (3) and (5) with priors for the respective coefficients will
generally result in distinct posterior WTP distributions and distinct characterizations
of demand as a function of price. What discrepancy can we expect from these two
approaches? To the extent that the data overwhelm the prior, the posterior WTP
distributions from the two approaches will converge despite the differences in the
prior. This will generally happen for models that impose homogeneity on the
coefficients. The more interesting case occurs with hierarchical models.
In hierarchical models, we typically encounter many units (e.g., consumers) and
relatively few observations per unit. Thus, the full conditional likelihood of any one
consumer is informed by a limited amount of data and the prior distribution will
generally have much more influence on the posterior compared with a homogenous
model. A hierarchical model may build on either parameterization, using either of
the likelihood functions in Eqs. (3)or(5) as the full conditional likelihood.
2
The
likelihood in Eq. (3) in combination with a (hierarchical) prior for γ
i
that has positive
density arbitrarily close to zero readily accommodates respondents that do not appear
to be sensitive to price. Such respondents can in turn have a tremendous influence on
the posterior WTP distribution implied by the model and thus on any characteriza-
tion of demand as a function of price. Hierarchical models with likelihood functions
built on Eq. (5), measure WTP directly by β
i
. An advantage of this formulation is
that a hierarchical prior for WTP can be specified directly.
3
For example, a normal
prior for β
i
will place less mass on absolutely large WTP values.
The problems we outline with the WTP ratio are neither unique to choice models
nor WTP. They apply to any quantity that can be defined as a ratio of model
parameters. However, estimation of WTP (and the related concept of reservation
price) is a particularly relevant problem in marketing and economics. Recently, the
marketing literature has sharpened its focus on the study of WTP and reservation
prices because of the direct implications for pricing strategy (Jedidi and Zhang 2002;
Jedidi et al. 2003; Shaffer and Zhang 1995,2000). The economics literature has
recognized the potential problems with random coefficient ratio estimates of WTP
(Meijer and Rouwendal 2006; Revelt and Train 1998). Marketing practitioners have
2
We adopt a fully Bayesian approach to inference in this paper. However, the points made apply in the
context of hierarchical models independent of the estimation technique.
3
Another motivation for directly parameterizing the model in WTP terms is that the researcher often has
variables such as demographics that may be useful in the characterization of consumer heterogeneity. In
such instances it seems more reasonable to build hierarchical regression structures for WTP instead of
parameters with less clear interpretation. We do not explore this issue here.
Heterogeneity distributions of willingness-to-pay 317
also recognized the problems, advocating use of the median as a summary of the
posterior WTP distribution (Orme 2001). While the median will likely be a more
robust statistic, Bayesian decision theoretic analyses of demand as a function of price
aimed at identifying optimal actions rely on the entire posterior distribution of WTP.
To the extent that the posterior distribution of WTP is sensitive to the prior
assumptions, so will the optimal action.
2.3 Optimal pricing
Ignoring the implied prior on WTP can adversely impact demand and price analyses.
Firms often use the model coefficients estimated from CBC data to build market
share simulators, which are useful for assessing response to price changes and
optimal pricing. Given a set of non-price attributes, market share (and demand) can
be completely characterized by consumer surplus. Consumers choose the inside
alternative that yields the maximum surplus and forgo a category purchase if the
surplus from the best alternative is less than the surplus generated by the outside
alternative. The price that determines the incidence and choice decisions is the
reservation price, e
pijt, which induces indifference between buying alternative jand
forgoing a category purchase. For the surplus model, e
pijt ¼x0
ijtbiþηijt ηi0t
.
Importantly, any proper indirect utility function implies a function for e
pijt. In our
case, the reservation price from the utility model is e
pijt ¼x0
ijtϕiþ"ijt "i0t
ðÞ
γi
. From this
equation, we can see that the change in the reservation price given a change in an
attribute is given by the WTP for that attribute.
If the no-buy option is not included in the CBC experiments, the reservation price
is not identified. In this case, what we can identify is the equalization price p
ijt which
is the price for good jthat equalizes the surplus generated by goods jand j′. For the
surplus model, p
ijt ¼xijt xij0t
0biþpij0thijt hij0t
. Again, any proper indirect
utility function implies a function for p
ijt. In our case, p
ijt ¼xijtxij 0t
ðÞ
0ϕiþγipij0tþ"ijt "ij 0t
ðÞ
γi.
Consider now using the utility or surplus models to find the profit maximizing
price for firm j, taking the competing firms’prices as given. To the extent that the
utility model a priori puts greater mass on extreme WTP values, the posterior
distribution of reservation and equalization prices will also be thick tailed. This is
especially so in sparse data environments, and implies that the firm could continue to
raise prices and still find consumers willing to purchase. Thus, inference about the
profit maximizing prices based on the posterior distributions of the parameters will
depend on the model.
3 A simulation study
We more closely investigate the properties of the two approaches to WTP estimation
in the following simulation study. We generate four data sets, two each from the
utility and surplus models, which we will refer to as D1, D2, D3, and D4. For the
utility model data sets, D1 and D2, we assume the following population distribution,
ΦiNΦ;ΣΦ
, where Φi¼ϕ0
ilog gi
ðÞ
0. For both D1 and D2, the covariance
matrix Σqis assumed to be diagonal and we choose parameters such that the
distribution of γ
i
is centered near 1. For D1, we allow for some mass of the
318 G. Sonnier et al.
distribution of γ
i
to be near zero by choosing a large value for the variance of log
(γ
i
). For D2, we choose the variance of log (γ
i
) such that γ
i
is tightly distributed
around one, with little to no mass near zero. In the case of the former, some
individual-level WTPs will be extremely large for values of γ
i
→0 while in the latter,
the distribution of WTP should be closer to normal. For the datasets generated by the
surplus model, D3 and D4, we assume qiNq;Σq
where qi¼b0
ilog mi
ðÞ
0.
In this case, the distribution of WTP is specified directly. For D3, we choose
parameters such that μ
i
is, on average, larger and the deterministic component of
surplus has relatively lower explanatory power. For D4, we choose parameters such
that μ
i
is, on average, smaller, translating into more extreme choice probabilities.
For all models, we assume 300 individuals choosing amongst three alternatives
and an outside good on each of 15 choice occasions. The covariates include three
alternative specific constants, a discrete attribute with four levels, and a price. Each
alternative is created by randomly choosing a level of the discrete attribute and a
price from the range [1.5–2.5] (in increments of 0.1). Tables 1and 2contain the
parameters of the distributions used to generate the four data sets. We retain the last
choice of each simulated respondent to create a holdout sample. Using MCMC
methods, we estimate the utility and surplus models on each of the four data sets, for
a total of eight sets of results. The details of the sampler have been reported
elsewhere (e.g., Allenby and Lenk 1994; Arora et al. 1998; Train 2003). We use a
normal-inverted Wishart hyper-prior structure for the population distribution
parameters qand Σq. The prior on qis set to N0KðÞ
;106IKKðÞ
. The prior on
Σqis set to IW K þ1;IKKðÞ
. These are proper but diffuse priors. We use identical
priors for Φand ΣΦin the utility model.
Table 1 Data generating parameters, utility model data sets
Data set D1 D2
Mean Variance Mean Variance
ϕ
i1
−0.5 1 −0.5 1
ϕ
i2
11 11
ϕ
i3
1.5 1 1.5 1
ϕ
i4
0.5 1 0.5 1
ϕ
i5
0.75 1 0.75 1
ϕ
i6
11 11
Log (γ
i
)−1 2 0 0.2
Table 2 Data generating parameters, surplus model data sets
Data set D3 D4
Mean Variance Mean Variance
β
i1
−0.5 1 −0.5 1
β
i2
11 11
β
i3
1.5 1 1.5 1
β
i4
0.5 1 0.5 1
β
i5
0.75 1 0.75 1
β
i6
11 11
log (μ
i
) 1 0.1 −1 0.5
Heterogeneity distributions of willingness-to-pay 319
For the utility models, we compute the individual-level WTPs as ϕi
gion each
iteration of the sampler. For the surplus models, draws of the individual-level WTPs
are directly available. We compute the mean absolute error (MAE) and the root
mean-squared error (RMSE) between the true and estimated WTPs on each iteration
of the sampler and report the means over iterations. Using the harmonic mean
estimator (Newton and Raftery 1994), we compute the log marginal density (LMD)
statistic for each model. We also report the deviance information criteria (DIC)
(Spiegelhalter et al. 2002) and the log predictive density (LPD) of the holdout data.
Tables 3,4and 5presents the results of our simulation study. D1 and D2 are
generated according to the heterogeneous utility model. D1 contains individuals with
price coefficients near zero and thus extremely large WTPs. Relative to the other
conditions, the error statistics are quite high in this setting. As evidenced by smaller
RMSE and MAE , note that the surplus model has more accurate recovery of the true
WTPs, even though the utility model is consistent with the data generating process.
In terms of fit statistics, the LMD, DIC and LPD all favor the utility model. In D2,
the distribution of the price coefficient has most of its mass away from zero. Again,
the surplus model has lower RMSE and MAE. The LMD and LPD favor the utility
model, while the DIC favors the surplus model. Thus, even when the true WTPs are
a ratio of random coefficients, the surplus model more accurately recovers the true
WTPs compared with the utility model under a range of population distribution
parameters.
Table 3 Root mean squared error
Data set D1 D2 D3 D4
Data generation Utility Utility Surplus Surplus
Model Utility Surplus Utility Surplus Utility Surplus Utility Surplus
WTP
1
48.54 45.68 2.43 1.55 26.06 1.60 1.52 1.19
WTP
2
67.61 64.82 1.53 1.30 16.98 1.36 1.04 0.85
WTP
3
91.83 91.25 1.42 1.19 16.31 1.36 1.06 0.72
WTP
4
26.78 24.82 1.74 1.28 16.90 1.60 1.33 0.95
WTP
5
41.58 38.96 1.68 1.33 16.97 1.46 1.37 0.95
WTP
6
33.42 29.87 1.65 1.27 19.16 1.73 1.13 0.83
Average RMSE 51.63 49.23 1.74 1.32 18.73 1.52 1.24 0.92
Table 4 Mean absolute error
Data Set D1 D2 D3 D4
Data generation Utility Utility Surplus Surplus
Model Utility Surplus Utility Surplus Utility Surplus Utility Surplus
WTP
1
11.88 10.25 1.52 1.21 14.64 1.29 1.10 0.94
WTP
2
14.85 12.32 1.05 1.01 9.15 1.08 0.75 0.66
WTP
3
18.01 15.92 0.97 0.89 8.44 1.08 0.73 0.56
WTP
4
8.58 7.10 1.17 1.00 9.05 1.27 0.94 0.74
WTP
5
10.55 8.64 1.16 1.03 9.14 1.16 0.97 0.75
WTP
6
10.91 8.95 1.13 0.95 10.49 1.38 0.82 0.66
Average MAE 12.46 10.53 1.17 1.01 10.15 1.21 0.89 0.72
320 G. Sonnier et al.
D3 and D4 are generated with the heterogeneous surplus model. In D3, the true scale
parameter μ
i
is, on average, larger. In this setting, the utility model estimates of WTP
are particularly error-prone. Once more, the surplus model is better at recovering the
true WTP parameters. Interestingly, the LMD statistic favors the utility model, despite
the lack of recovery of the true WTPs.
4
The DIC and LPD favor the surplus model. In
D4, the scale parameter is, on average, smaller. Relative to D3, the utility model does
a better job of recovering the WTPs here, but again, the surplus model has more
accurate WTP recovery. All three of the fit statistics favor the surplus model.
In summary, the surplus models always recover the true WTPs with more
accuracy, regardless of the data generating mechanism. Even when the true WTPs
are distributed as a ratio of random coefficients, the ratio estimator does not recover
the true WTPs as accurately as simply directly specifying a prior on WTP. We
attribute this to the fact that the surplus model employs a more sensible prior
distribution for WTP.
4 Two CBC studies
Using CBC data sets provided to us by firms in the camera and automotive
categories, we replicate the findings from our simulation study in the sense that the
posterior of WTP from the utility model is rather different from the posterior
obtained from the surplus model. Moreover, inferences obtained with the utility
model lack face validity. Table 6presents the attributes and levels involved in the
design of each study.
4.1 Data and models
The first data set is CBC data on midsize sedans. The data were provided by a major
automotive manufacturer. Respondents qualified for participation in the study on the
basis of the vehicle they currently own, their intention to purchase a midsize sedan,
and other socio-economic information. A total of 333 respondents participated in the
study. Each respondent completed 15 choice tasks, with each task consisting of three
sedans. The no-buy option was not included in this study. The second data set is
Table 5 Model fit statistics
Data set D1 D2 D3 D4
Data generation Utility Utility Surplus Surplus
Model Utility Surplus Utility Surplus Utility Surplus Utility Surplus
LMD −3532.20 −3553.00 −3942.80 −3946.30 −5309.60 −5428.90 −2493.40 −2340.50
DIC 3859.20 3871.10 4324.80 4318.30 5570.70 5497.90 2913.60 2751.50
LPD −287.47 −296.64 −322.79 −331.15 −410.41 −403.82 −236.34 −231.89
4
This is similar to previous simulation studies in the literature that find the harmonic mean estimator of
the LMD sometimes favors models with relatively poor parameter recovery (Andrews et al. 2002; Liechty
et al. 2005).
Heterogeneity distributions of willingness-to-pay 321
CBC data on cameras. The study was conducted by the Eastman Kodak Company to
assess the market for a new camera format, the Advanced Photo System (APS). A
detailed description of the data is given by Gilbride and Allenby (2004). A total of
302 respondents participated in the study. Each respondent completed 14 choice
tasks, with each task consisting of three 35 mm cameras, three APS cameras, and a
no-buy option. Some attributes were available only on the APS camera, and price
was nested within camera type.
Table 6 Attributes and levels
Camera data Sedan data
Attribute Levels Attribute Levels
Body Style Low Make/Model Ford Taurus
Medium Toyota Camry
High Nissan Maxima
Honda Accord
VW Passat
Mid-roll change
a
None Engine 4 cylinder; 1.8 L;
150 HP
Manual 4 cylinder; 2.4 L;
160 HP
Automatic 6 cylinder; 3.0 L;
155 HP
6 cylinder; 3.0 L;
222 HP
Annotation
a
None Audio and navigation Standard Audio
Pre-set List Premium Audio
Customized List Premium Audio
with Navigation
Custom Input
Method 1
Custom Input
Method 2
Custom Input
Method 3
Camera operation
feedback
a
No Antilock Brakes (ABS) No
Yes Yes
Zoom None Side Door/Window Curtain
Airbags(CAB)
No
2X Yes
4X
Viewfinder Regular Vehicle Skid Control (VSC) No
Large Yes
Camera settings
feedback
None
LCD
Viewfinder
LCD and Viewfinder
Price (nested within
camera type)
from $41 to $499 Price $17,400
$18,900
$20,400
$21,900
$23,400
$24,900
$26,400
a
feature only available on APS
322 G. Sonnier et al.
For both data sets, we model consumer i’s surplus for alternative jat choice
occasion tas a linear function of non-price attributes, attribute WTPs, and price
Cijt ¼x0
ijtβipijt þ"ijt "ijt EV 0;μi
ðÞ
θiNθ;Σθ
θi¼β0
ilog μi
ðÞ
hi
0:ð6Þ
For the camera data, we set the deterministic component of the surplus for the no-
buy option to zero. For identification, the lowest level of each attribute is dropped
(with the exception of the body type attribute since the baseline is the “no-buy”
option). We use the negative of price (in $100s) in the likelihood. The coding
scheme is the same as that employed by Gilbride and Allenby (2004), and results in
a total of K=18 parameters. For the sedan data, the make/model “VW Passat”is
dropped, as are the lowest level of each of the remaining non-price attributes. This
results in a total of K=13 parameters. We use the negative of price (in $1,000s) in
the likelihood.
For both data sets, we compare estimates of the distribution of WTP from the
surplus model with that of the linear utility model, where θ
i
is replaced with
Φi¼ϕ0
ilog gi
ðÞ
0. Here, the choice probabilities are based on (3). The same
normal-inverted Wishart hyper-prior structure used for qand Σqis used for hyper-
priors on the population parameters Φand ΣΦ. The linear utility model requires we
calculate the WTP from the model parameters using the ratio transformation. On each
iteration of the sampler, we compute the ratio ϕi
giusing the draws of the individual level
parameters. We then compute the mean, median, and standard deviation over
individuals, and report the mean of these quantities over iterations of the sampler.
For both data sets, the samplers are run for 20,000 iterations. We keep the last 5,000
iterations for posterior inference. Parameter estimates are calculated with T−1choice
tasks. We keep the last task for each individual to assess holdout performance via
LPD. To assess in-sample performance, we compute the LMD and DIC statistic for
each model.
4.2 Results
In Tables 7and 8we report the mean and standard deviation of the distribution of
WTP for the utility and surplus models. Posterior standard deviations of the reported
statistics are in parentheses. Table 7contains the results from the sedan data while
Table 8contains the results from the camera data. The mean and standard deviation
of the population distribution of WTP are dramatically affected by the priors. For the
sedan data, the means of the utility model estimates are two to three times the
magnitude of the surplus model. The WTP distributions are also far more dispersed,
with standard deviations that are five to six times larger. For the camera data, the
means are also much larger for the utility model. However, most of the standard
deviation estimates for the utility model have large posterior standard deviations. For
both data sets, the median of the population distribution is much less sensitive to the
prior than the mean or standard deviation.
The in-sample fit statistics are somewhat mixed. In both data sets, the LMD
strongly favors the utility model. This result echoes that of the third synthetic data
Heterogeneity distributions of willingness-to-pay 323
set, D3, which was generated by the surplus model. In this case, the LMD strongly
favored the utility model despite its inconsistency with the data generating process
and its extremely poor parameter recovery. The DIC favors the utility model in the
sedan data and the surplus model in the camera data. In contrast to the in-sample fit
measures, the LPD favors the surplus model in both the sedan data and the camera
data, indicating the surplus model has superior out-of-sample performance. We will
now examine more closely the distribution of WTP and optimal prices implied by
the two models. From this vantage point, the differences across the two models are
less ambiguous.
It is evident that the utility and surplus models result in dramatically different
estimates of the distribution of WTP. The utility model estimates seem to be
implausible and not reflective of consumers’monetary valuation of product
attributes. Figure 1presents boxplots of the individual-level make/model WTP
estimates for both the utility and surplus models. These are measured relative to the
VW Passat and can be interpreted as equalization prices; the relative price difference
that equalizes the utility of comparably equipped competitive sedans and the Passat.
The median of the utility model’s individual level WTP estimates for the three
Table 7 WTP estimates for sedan data (standard errors in parentheses)
WTP ($1,000’s) Utility model Surplus model
ϕi
giβ
i
Attribute Level Mean Median Std dev Mean Median Std dev
Make/Model Ford Taurus −5.02 −2.20 46.54 −2.23 −2.27 10.06
(2.76) (0.52) (12.80) (0.32) (0.43) (0.29)
Toyota Camry 16.23 6.76 47.87 7.00 6.74 9.73
(3.43) (0.58) (15.86) (0.23) (0.36) (0.37)
Nissan Maxima 12.42 4.82 46.88 5.13 5.12 9.35
(3.46) (0.52) (15.92) (0.27) (0.42) (0.40)
Honda Accord 9.25 3.70 42.10 3.96 3.81 8.48
(3.06) (0.48) (13.68) (0.26) (0.31) (0.25)
Engine 4-cyl; 2.4L; 160HP 4.88 1.98 14.27 1.82 1.82 2.00
(1.12) (0.39) (5.50) (0.37) (0.37) (0.25)
6-cyl; 3.0L; 155HP 7.89 3.78 16.61 3.67 3.60 3.48
(1.33) (0.49) (5.65) (0.34) (0.33) (0.36)
6-cyl; 3.0L; 222HP 9.89 5.21 20.25 4.87 4.78 3.95
(1.39) (0.51) (6.85) (0.23) (0.24) (0.38)
Audio Premium Audio 2.52 1.10 9.29 1.34 1.31 1.56
(1.02) (0.26) (3.61) (0.21) (0.20) (0.21)
Premium Audio w/Navi 3.54 1.57 10.54 1.61 1.62 2.02
(1.12) (0.26) (4.20) (0.24) (0.25) (0.21)
Safety features Antilock Brakes 4.78 2.11 12.23 2.08 2.07 2.21
(1.05) (0.26) (4.20) (0.20) (0.20) (0.18)
Side Curtain Airbags 2.89 1.22 9.36 1.18 1.17 1.52
(0.88) (0.26) (3.46) (0.19) (0.21) (0.27)
Vehicle Skid Control 2.83 1.28 9.64 1.29 1.28 1.42
(0.82) (0.23) (3.59) (0.26) (0.26) (0.24)
Fit statistics
LMD −2220.4 −2665.8
DIC 2,966.4 3,044.3
LPD −340.7 −333.5
324 G. Sonnier et al.
Japanese make/models are near or in excess of the range of prices shown to
respondents.
5
This is not being caused by a just a handful of respondents with
estimates of γ
i
near zero. The 75th percentiles for the individual-level equalization
price between Toyota Camry vs. VW Passat and Nissan Maxima vs. VW Passat are
approximately $30,428 and $24,684, respectively. The retail price of the Passat is
about $23,000. This implies that a quarter of the respondents would require Passat to
Table 8 WTP estimates for camera data (standard errors in parentheses)
Attribute WTP ($100’s) Utility model Surplus model
ϕi
giβ
i
Level Mean Median Std dev Mean Median Std dev
Body style Low −25.78 −4.67 84.40 −5.47 −5.51 4.90
(6.74) (0.82) (54.41) (0.38) (0.43) (0.36)
Medium −10.15 0.49 56.60 1.07 1.14 3.59
(4.52) (0.28) (44.88) (0.22) (0.25) (0.38)
High −7.82 0.33 48.09 0.85 0.80 3.78
(3.83) (0.27) (36.43) (0.27) (0.31) (0.64)
Mid-roll change Manual 1.96 0.44 14.02 −0.36 −0.36 2.84
(1.55) (0.24) (6.12) (0.50) (0.48) (0.26)
Automatic 3.93 0.52 16.99 0.80 0.71 2.40
(1.21) (0.12) (10.99) (0.12) (0.14) (0.14)
Annotation Pre-set list 1.51 0.38 7.32 0.34 0.33 1.01
(0.60) (0.13) (3.39) (0.08) (0.09) (0.12)
Customized list 3.25 0.98 9.47 1.21 1.19 1.09
(0.76) (0.16) (4.80) (0.11) (0.12) (0.14)
Custom Input Method 1 4.10 −0.28 22.63 −1.44 −1.48 3.37
(1.62) (0.17) (16.02) (0.24) (0.28) (0.24)
Custom Input Method 2 4.64 1.06 14.71 1.23 1.22 1.73
(1.12) (0.16) (8.49) (0.11) (0.13) (0.18)
Custom Input Method 3 2.09 −0.78 19.75 −0.99 −1.03 2.58
(1.35) (0.15) (11.85) (0.31) (0.32) (0.28)
Operation feedback Feedback 2.65 0.58 10.50 0.85 0.82 1.72
(0.86) (0.12) (4.85) (0.25) (0.25) (0.10)
Zoom ×2 Zoom 5.47 1.95 14.96 2.70 2.70 2.09
(1.52) (0.25) (8.04) (0.18) (0.19) (0.15)
×4 Zoom 7.66 2.50 20.70 3.11 3.13 3.13
(2.12) (0.41) (10.42) (0.24) (0.26) (0.33)
Viewfinder Large viewfinder 0.40 −0.18 8.91 −0.47 −0.46 1.76
(0.75) (0.10) (4.23) (0.17) (0.16) (0.22)
Settings feedback LCD 4.30 1.12 13.27 −0.23 −0.23 1.30
(1.19) (0.20) (8.30) (0.16) (0.16) (0.16)
Viewfinder 4.49 1.20 13.42 −0.16 −0.15 1.59
(1.07) (0.19) (7.99) (0.16) (0.16) (0.19)
LCD and Viewfinder 5.33 1.51 14.92 0.21 0.21 1.53
(1.24) (0.24) (9.23) (0.17) (0.18) (0.16)
Fit statistics
LMD −3879.7 −4100.4
DIC 4,633.6 4,620.4
LPD −438.5 −434.7
5
Note that the median of the posterior mean of the individual-level estimates will not be the same as the
posterior mean of the median of the population distribution.
Heterogeneity distributions of willingness-to-pay 325
Make/Model WTP Estimates: Utility Model
-$175
-$125
-$75
-$25
$25
$75
$125
$175
Taurus Camry Maxima Accord
($1,000's)
Make/Model WTP Estimates: Surplus Model
-$30
-$20
-$10
$0
$10
$20
$30
Taurus Camry Maxima Accord
($1,000's)
Fig. 1 Boxplots of individual-level make/model WTP estimates
Table 9 Attributes and levels for optimal pricing, sedan data
Attribute Levels Attributes and levels for
alternatives
12345
Make/Model Ford Taurus X
Toyota Camry X
Nissan Maxima X
Honda Accord X
VW Passat X
Engine 4 cylinder; 1.8 L; 150 HP X
4 cylinder; 2.4 L; 160 HP X X
6 cylinder; 3.0 L; 155 HP X
6 cylinder; 3.0 L; 222 HP X
Audio and navigation Standard Audio X X X X X
Premium Audio
Premium Audio with Navigation
Antilock brakes No X
Yes X X X X
Side door/Window curtain airbags No
Yes X X X X X
Vehicle skid control No X
Yes X X X X
Price ($1,000) $20.1 $20.7 $24 $20.5 $22.5
326 G. Sonnier et al.
have a zero price as well as a cash subsidy to induce indifference with a similarly
equipped Camry or Maxima, which does not seem credible.
In the camera study, the individual-level estimates of WTP implied by the utility
models also seem lacking in face validity. For example, the surplus model estimate
of the median of the individual-level WTP estimates for a 2× zoom lens is $295,
with demand essentially zero at prices exceeding $550. According to the utility
model, the median of the individual level WTPs is $322. At a price of $550, 32% of
respondents are still in the market. A quarter of respondents have WTP estimates in
excess of $750. Demand does not reach zero until prices exceed $3,000. These
estimates of WTP seem unreasonably high. Furthermore, any analysis of demand
should take into account the uncertainty in the individual-level estimates. We now
turn our attention to such an analysis.
Table 10 Attributes and levels for optimal pricing, camera data
Attributes and levels for alternatives
Attribute Levels 1 2 3
Body style Low X
Medium X
High X
Mid-roll change None
Manual X
Automatic X X
Annotation None
Pre-Set List X
Customized List
Custom Input Method 1 X
Custom Input Method 2 X
Custom Input Method 3
Camera operation feedback No X
Yes X X
Zoom None
2× X
4× X X
Viewfinder Regular X X
Large X
Camera settings feedback None
LCD
Viewfinder
LCD & Viewfinder X X X
Price (nested within camera type) from $41 to $499 $100 $225 $400
Table 11 Market shares, sedan scenario
Taurus Camry Maxima Accord Passat
Price $20.1 $20.7 $24 $20.5 $22.5
Utility (%) 16 36 19 21 8
Surplus (%) 16 36 18 22 9
Heterogeneity distributions of willingness-to-pay 327
4.3 An optimal pricing exercise
For the utility model, profits from alternative jin scenario zcan be written as
puz
jΦi;xz
j;pj¼Puz
jΦi;xz
j;pj
pjcj
ð7Þ
We seek the price puz*
jthat maximizes the firm’s expected profit,
EΦπuz
jΦi;xz
j;pj
hi
. The expected profit in scenario zis easily calculated with the
output of the Gibbs sampler. For a given price, we simply average the profits
calculated over the draws of Φ
i
. Using routine optimization procedures, it is
straightforward to find the optimal price. For the surplus model, profits from
alternative jin scenario zcan be written as
psz
jqi;xz
j;pj¼Psz
jqi;xz
j;pj
pjcj
:ð8Þ
As with the utility model, we seek the price psz*
jthat maximizes the firm’s expected
profit, Eqpsz
jqi;xz
j;pj
hi
.
Our goal is to compare puz*
jand psz*
j. Tables 9and 10 present the attributes and
levels used to construct the competitive scenarios for our pricing exercise. In the
sedan data, we consider a competitive set consisting of five sedans. In the camera
data, we consider a competitive set consisting of three cameras. Tables 11 and 12
present the prices and market shares for each alternative for the sedan and camera
scenarios. On each iteration of the sampler, we compute P
j
and report the mean over
iterations. For the sedan data, the two models predict practically the same shares. For
the camera data, there is some disagreement, with the utility model predicting higher
shares for Cameras 1 and 2 and lower shares for Camera 3 and the No-Buy alternative.
For the sedan data, we will find the optimal price for Ford Taurus, assuming the
competitive vehicle prices remain at their current levels. For the camera data, we will
Table 12 Market shares, camera scenario
Camera 1 Camera 2 Camera 3 No Buy
Price $100 $225 $400 $0
Utility (%) 17 35 32 16
Surplus (%) 14 28 35 23
Table 13 Taurus optimal price, sedan scenario
Utility model
Taurus* Camry Maxima Accord Passat
Price($1,000) $33.2 $20.7 $24 $20.5 $22.5
Share (%) 3 41 22 25 9
Surplus model
Taurus* Camry Maxima Accord Passat
Price $25.8 $20.7 $24 $20.5 $22.5
Share (%) 6 40 20 24 10
* denotes optimized product
328 G. Sonnier et al.
find the optimal price for Camera 3, assuming the competitive camera prices remain at
their current levels. To conduct the exercise, we need to make some assumptions on
costs. For simplicity, we assume the sedans are all built at a variable cost of $18,000.
For the cameras, we assume variable costs of $50, $60, and $70 for Cameras 1, 2, and
3, respectively. Similar results were obtained using other sedans and cameras in the
competitive scenarios, as well as other cost assumptions.
Tables 13 and 14 present the findings from the optimal pricing exercise. We
present the optimal price for Ford Taurus and Camera 3 along with the new market
shares. For the sedan data, using the utility model coefficients in the optimization
results in an optimal price for Taurus of $33,200. At this price, the largest relative
price difference is $12,500, observed between Taurus and Camry. The largest
relative price difference shown in the experiments is $9,000. The prior implied by
the utility model supports excessive equalization prices, leading to optimized prices
beyond the empirical range of prices in the data. In contrast, optimization based on
the surplus model leads to an optimal price for Taurus of $25,800. The largest
relative price difference is well within the range of experimental prices. We obtain
similar results from the camera data. Using the utility model, the optimal price for
Camera 3 is over $1,500. The maximum price shown to respondents in the study
was $499. For the camera data, using the surplus model results in an optimal price of
about $520. While this is slightly in excess of the maximum price, it is much more
reasonable.
5 Summary and conclusions
Researchers in marketing and economics have recognized the problems associated
with using random coefficient choice models derived from linear indirect utility
functions to estimate WTP for product attributes. In this setting, WTP is estimated
via the ratio of attribute and price coefficients. We illustrate that the prior implied for
WTP by seemingly reasonable priors for the attribute and price coefficients results in
posterior WTP distributions with extremely fat tails. This also affects the model’s
characterization of demand which has implications for pricing analyses. A number of
ad-hoc solutions have been proposed, including constraining the price coefficient to
be homogenous, or using the median as a measure of central tendency of the WTP
Table 14 Camera 3 optimal price, camera scenario
Utility model
Camera 1 Camera 2 Camera 3* No Buy
Price $100 $225 $1,525 $0
Share (%) 19 35 10 21
Surplus model
Camera 1 Camera 2 Camera 3* No Buy
Price $100 $200 $522 $0
Share 15% 33% 27% 25%
* denotes optimized product
Heterogeneity distributions of willingness-to-pay 329
distribution. In this paper, we present a straightforward solution to the problems
caused by the implied prior for WTP. Parameterizing the choice model in the space
of consumer surplus allows for direct specification of a prior distribution for WTP.
Such a direct specification is especially advantageous in the context of hierarchical
models where the aforementioned solutions conflict with the purpose and value of
quantifying consumer heterogeneity.
Using both simulated data and CBC data sets from the automotive and camera
categories, we document the influence of the implied prior for WTP. Commonly
employed diffuse priors for the attribute and price coefficients put too much prior
mass on extreme WTP values to render reasonable posterior WTP distributions in
small sample settings. Some posterior summaries are less sensitive to the assumed
prior (e.g. median versus mean). However, marketing actions, such as setting profit
maximizing prices, depend on the entire posterior distribution of WTP and thus will
be sensitive to the implied prior. In the surplus parameterization a hierarchical prior
for WTP can be directly specified. We found a hierarchical normal prior to be useful
in controlling the tails of the WTP distribution. The relatively thinner tails of the
normal result in more reasonable estimates of the WTP distribution and, in turn,
profit-maximizing prices.
The surplus model results in more reasonable estimates of the distribution of
WTP and profit-maximizing prices as well as superior out-of-sample performance.
However, the in-sample fit statistics across the two parameterizations are ambiguous,
even with simulated data. We leave this issue, specifically the performance of the
Newton-Raftery estimator of the LMD and the DIC statistic as criteria for model
choice, to future research. We acknowledge the existence of data generating
mechanisms that leave respondent WTP for a particular attribute level inestimable.
Among these are non-compensatory processing, price based quality inferences or the
simple ignorance of the price attribute in the conjoint exercise. The utility model
with standard priors will readily accommodate respondents who are, for whatever
reason, insensitive to price in the conjoint task. The modeling question then becomes
one of how and whether to implement prior knowledge about the range of likely
WTP values. We have demonstrated that the surplus model is very effective in terms
of how to implement such prior knowledge because it allows the researcher to put a
prior directly on WTP.
Whether to implement prior knowledge about WTP in conjoint studies, especially
when the data are better fit with arbitrarily large WTP values, touches upon the core
of the inferential problems associated with conjoint experiments in marketing.
Conjoint data are collected with the implicit goal of characterizing market demand.
To the extent that the conjoint likelihood differs from the likelihood that generates
choices in the market place, this generalization calls for the diligent use of prior
knowledge held by the researcher about market behavior. That is, the prior should
preserve certain well-known aspects of the target environment in the posterior and
still be informed by the conjoint likelihood in other respects. We acknowledge that
our argument here is limited to forming prior-predictive distributions given the
conjoint data and other prior knowledge. In the long run, only a better understanding
of the actual data generating mechanism underlying the conjoint data will enable
researchers to develop the necessary procedural modifications to move it closer to
the likelihood that generates choices in the market.
330 G. Sonnier et al.
Acknowledgement The authors would like to thank Peter Rossi, JP Dubé, Jordan Louviere, Kenneth
Train, and Greg Allenby for helpful insights. We also thank seminar participants at The Ohio State
University, Duke University, University of Michigan and the University of Chicago for providing useful
comments.
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