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The influence of individuals in forming collective household
preferences for water quality
Cam Rungie
a
, Riccardo Scarpa
b,c,
n
, Mara Thiene
d
a
Business School, University of Queensland, Australia
b
Gibson Institute for Land, Food and Environment, and Institute for Global Food Security, Queens University Belfast, Northern Ireland
c
Waikato Management School, University of Waikato, New Zealand
d
Department of Land, Environment, Agriculture and Forestry University of Padua, Italy
article info
Article history:
Received 26 November 2012
Keywords:
Structural choice model
Household preference
Tap water
Preference heterogeneity
abstract
Preference for water quality and its nonmarket valuation can be used to inform the
development of pricing policies and long term supply strategies. Tap water quality is a
household concern. The objective status quo of water provision varies between house-
holds and not between individuals within households, while charges are levied on
households not individuals. Individual preferences differ from collective preferences. In
households where there are two adults, we examine the preferences of each separately
and then as a couple in collective decisions. We show the level of influence each has in
developing the collective decision process. We use discrete choice experiments to model
preference heterogeneity across three experiments on women, men and on both. We
propose a random utility model which decomposes the error structure in the utility of
alternatives so as to identify the individual influence in collective decisions. This approach
to choice data analysis is new to environmental economics.
&2014 Elsevier Inc. All rights reserved.
Introduction
Tap water is a typical complex good that is provided at the household level and which can be decomposed into a number
of attributes. While tap water is certainly a good familiar to all members of households, each member may display
substantially different tastes for its attributes. Because of the composite nature of welfare changes in household water
supply, due to this intra-household heterogeneity of taste, conducting stated surveys based on a representative of the
household might lead to misleading results. This is an important issue from the empirical viewpoint and motivates our
study.
The theoretical and applied literature on household economics has made substantial progress in modelling joint
preferences in marketing and transport (Arora and Allenby, 1999;Adamowicz et al., 2005;Hensher et al., 2008;Marcucci
et al., 2010), whereas with few exceptions (Dosman and Adamowicz, 2006;Bateman and Munro, 2005;Strand 2007;
Beharry et al., 2009), less progress has been made in terms of empirical applications in the field of non-market valuation.
Investigating preferences from choice data coming from group decisions, rather than individual decisions, requires the
ability to handle latent correlations amongst individual and joint choices in a structured manner. In the context of tap water,
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/jeem
Journal of
Environmental Economics and Management
http://dx.doi.org/10.1016/j.jeem.2014.04.005
0095-0696/&2014 Elsevier Inc. All rights reserved.
n
Corresponding author.
E-mail address: r.scarpa@qub.ac.uk (R. Scarpa).
Journal of Environmental Economics and Management ](]]]])]]]–]]]
Please cite this article as: Rungie, C., et al., The influence of individuals in forming collective household preferences for
water quality. Journal of Environmental Economics and Management (2014), http://dx.doi.org/10.1016/j.
jeem.2014.04.005i
results obtained from disentangling individual preferences in group decisions have important implications for both policy
and survey practice. These implications are of particular salience when preference surveys are designed to inform the
process of definition or/and negotiation of water tariff between water utilities and regulatory bodies in charge of evaluating
the adequacy of the tariffs and the economic management of investment by water utilities. At the time of data collection for
this study this was of particular relevance in Italy, where recent legislation was intended to shift the control of water supply
to newly constituted local water network utilities, with the intent of directing water management to be more responsive to
market forces. The debate over this legislation proposal has been relegated to backstage after the results of a national
referendum (12–13 June 2011 on the composition of water tariffs), but the focus on cost efficiency and social benefits is still
driving the debate in Italy as much as elsewhere.
In this study we use data from a widely employed form of stated preference survey for multi-attribute goods, choice
experiments (Adamowicz et al., 1998). The salient feature of the data collection is that members of households have
provided choice responses first as individuals, and then jointly as a family. To adequately investigate preference
heterogeneity of household members for tap water one of the main issues is how to empirically measure such differences,
considering that results can be quite sensitive to choice of model specification. Previous work usefully employed power
function approaches based on the concept that the household’s indirect utility is determined by a convex combination (a
power function) of the indirect utility of man and woman (Dosman and Adamowicz, 2006). This was later extended to
power functions at the single attribute level. That is, the contribution of each attribute to the household’s utility function
was modelled as a convex combination (Beharry et al., 2009), with the power parameter specified as a household specific
random component.
Within this context, we now explore the use of an innovative modelling approach, that we call structural choice
modelling (hereafter SCM). SCM is an alternative econometric framework for modelling choice data using latent variables,
by combining data generated from separate but related surveys and thereby simultaneously modelling choice outcomes
from several DCEs (Rungie, 2011;Rungie et al., 2011;2012;Coote et al., 2011). With respect to previous applications in
environmental economics this approach allows two advantages: (i) the incorporation of latencies and (ii) the simultaneous
estimation of structural causal factors from individual and joint choice.
SCM is designed to incorporate latent variables and structural equations into the analyses of DCEs and, more generally,
into choice processes (McFadden, 1974,2001). There are indeed several important precursors to SCM. Firstly, factor analytic
models have been used to study brands in a product category. This is as if “brand”is an attribute and the individual brands
are levels. Factors have been applied across brands and other attributes by Elrod (1988);Elrod and Keane (1995);Keane
(1997) and Walker (2001). Secondly, factor analytic models have also been applied to the characteristics of respondents by
using indicator variables (Walker, 2001,Ashok et al., 2002,Morikawa et al., 2002,Temme et al., 2008,Bolduc and Alvarez-
Daziano, 2010;Yáñez et al., 2010;Hess and Stathopoulos, 2011). Thirdly, methods using latent variables have been
developed for the analysis of combined RP and SP data (Ben-Akiva and Morikawa, 1990;Hensher et al., 1999; Louvier et al.,
1999; Ben-Akiva et al., 2002;Louviere et al., 2002;Morikawa et al., 2002). The various approaches differ in the nature of the
covariates employed; in the first the covariates are the attributes of the alternatives and in the second the characteristics of
the respondents. However, all approaches rely on similar mathematics.
SCM adapts this mathematics to extend the analysis of the attributes. In particular, it adds to the factor analytics the
capacity to specify simultaneous equations and correlations (Jöreskog, 1970;Bollen, 1989;Jöreskog and Sörbom, 1996,
Jöreskog, 1970) and it exploits the potential relationships between uses and choice outcomes (Rungie, 2011;Rungie et al.,
2011,2012;Coote et al., 2011).
In the traditional random coefficient model (e.g. Ben-Akiva et al., 1997;McFadden and Train, 2000;Dube et al., 2002;
Train, 2009), the coefficients for each covariate are independent random variables with means and variances estimated
from the data; i.e. the variance covariance matrix, denoted by Σ, is either diagonal or with off-diagonal elements that refer
to only covariances between random coefficients. In SCM the coefficients have a multivariate distribution where, through
the parsimonious use of factor analytics in the form of simultaneous equations and correlations, Σcan be significantly more
complex, yet structured. Although to be practical, the number of parameters must not be excessive. In addition competing
models, i.e. competing specifications for the structure of Σ, can be empirically evaluated. In other words, the factor
analytics are used to bring testable correlation structures to the error component nature of mixed logit models. The
contribution of SCM is in its capacity to specify and evaluate competing models for how preferences for attributes are
related. Error component models, of the type explored to define flexible substitution patterns between alternatives
(Brownstone and Train, 1999;Herriges and Phaneuf, 2002;Thiene and Scarpa, 2008) can also be seen as special cases of
SCM specifications.
The present study adds to the existing literature in several ways. First, it is one of the few existing applications of
structural choice models to investigate latency in preference heterogeneity. Second, to our knowledge this is the first
empirical study using this approach in the field of environmental and resource economics. Ultimately, it is one of the few
contributions using data from more than two choice experiments that are simultaneously modelled within a natural group,
such as the couple.
The rest of the paper is organized as follows. The next section discusses the formation of joint preferences from
the preferences of individuals. Then follows a section illustrating the methodology. Survey and data are described in
Section 4, whereas Section 5 defines model specifications and provides a discussion of result estimates. The last section
concludes.
Please cite this article as: Rungie, C., et al., The influence of individuals in forming collective household preferences for
water quality. Journal of Environmental Economics and Management (2014), http://dx.doi.org/10.1016/j.
jeem.2014.04.005i
C. Rungie et al. / Journal of Environmental Economics and Management ](]]]])]]]–]]]2
Joint preference formation
Traditional random utility theory (McFadden, 1974,2001;Train, 2009) states that alternative iis perceived to deliver
utility u
i
. This is composed of a systematic component v
i
, and an error term, ε
i
, which may be GEV or Gumbel distributed.
1
u
i
¼v
i
þε
i
ð1Þ
The probability of an alternative being selected from choice set Cis a function of the systematic utility, v.
p
i
¼exp v
i
ðÞ
∑
jAC
exp v
j
ð2Þ
This functional form for choice probabilities is known as the logit kernel. The random utility theory that supports (2) is often
applied, but not as an explicit assumption, to the choices made by individuals; e.g. in a commercial environment the
decision makers may be a manager or teams of managers. In the research described below the primitive choices of the man
and woman in a couple are recorded independently and then their joint collaborative choices are recorded. The models used
to analyze the observed choices are based on random utility theory. As a result the logit kernel is our functional form of
choice for both the primitive and joint choices. Yet, the expressed preferences differ between choices by man, woman and
joint. Thus, the logit kernel is used throughout but with different vectors of parameter estimates for each set of choices
collected within the couple.
Next, we discuss the nature of joint preference. In the primitive choice each individual makes a selection for him or
herself. In the joint choice the selection is both ‘for’and ‘by’the couple. As the choice task changes from ‘for me’to ‘for us’
then preferences will change. Even in the case where one individual were to make both choices, the systematic utility will
differ between the choice for him/herself as compared to the choice on behalf of the couple. But the joint choice is not made
by an individual, it is made ‘by’the couple. There will be collaboration between man and woman and new type of
preferences will form. Thus, joint preference reflects (i) the different choice task; ‘for’the couple, and (ii) the different
decision maker; ‘by’the couple. Primitive preference will differ from this joint preference but they will nevertheless have an
impact on them.
Next, we explore the role of collaboration between the couple in joint preference formation. Any of several scenarios may
apply. For the primitives, the systematic utility in the logit kernel of each alternative is implicit and it can be derived from
observational research, as is customarily done in most standard applications. Yet, the exact process of joint preference
formation may not be something that the individual is overly aware of or might be able to explicitly communicate, even if
asked. In the joint, the utilities may be made explicit if there was extensive and complete communication between the man
and the woman. However, this condition is extremely difficult to realize in the operating conditions of the fieldwork. We
hence do not assume that joint preference formation is through complete communication. As with primitive utilities, the
joint utilities are assumed to be implicit. As an aside we note that there are other forms of collaboration, with less than
complete communication, that can lead to joint preference formation. One individual may be dominant in a couple and,
with no or minimal communication, his/her primitive systematic utilities become the joint utilities. Alternatively,
negotiations may take place. Or, the couple may each have substantial knowledge and consideration of the other and so
the implicit joint preferences can be quickly formed, albeit possibly inaccurately, with the use of cryptic, efficient or non
verbal communications. Furthermore, the individuals may know each other well enough to read levels of comfort and
discomfort reflecting the implicit systematic utilities. Thus, through any of several latent processes of collaboration, the joint
preferences are formed.
It might be argued that joint choice is based on the primitive choices rather than the primitive systematic utilities. It is as
if the joint choice set contains only two alternatives being the primitive selection of the man and of the woman. This choice
based proposition for joint preference formation is rejected on several grounds. (i) In such a proposition the functional form
for the joint is no longer as similar to the primitives as in the model developed below. (ii) The man and woman may each
have a strong preference against the other’s selection. But a third alternative may have been a close second for both the man
and the woman and so that would be selected. Thus, the joint choice set can easily contain more than two alternatives.
(iii) Under random utility theory the primitive choice reflects a stochastic component, ε, that is unique to the specific
alternative and does not influence the joint choice. It is the primitive systematic utility, not choice, that influences the joint
preference. (iv) Unlike in our purpose-specific data, in empirical data collection, the choice sets used in the primitive will
often differ from those in the joint. So, the proposition may frequently be untestable.
Returning to the discussion of preference formation, the joint systematic utilities are a combination or amalgamation of
the primitive utilities. In its simplest form this will be linear with
v
j
i
¼c
w
v
w
i
þc
m
v
m
i
ð3Þ
The joint systematic utility is a weighted linear sum of the primitive systematic utilities with weights c
w
and c
m
. It is unlikely
that the joint utility will be as high as the sum of the primitives where c
w
¼c
m
¼1 and c
w
þc
m
¼2. Some duplication of utility
between the man, woman and joint is to be expected. Similarly, in most cases and with functional couples, the contribution
1
For simplicity the subscripts for the individual, the choice set and alternative within the choice set are omitted.
Please cite this article as: Rungie, C., et al., The influence of individuals in forming collective household preferences for
water quality. Journal of Environmental Economics and Management (2014), http://dx.doi.org/10.1016/j.
jeem.2014.04.005i
C. Rungie et al. / Journal of Environmental Economics and Management ](]]]])]]]–]]] 3
of the primitive utilities is unlikely to be negative, i.e. it is unlikely that c
w
o0orc
m
o0. Thus, it is proposed that there are
approximate upper and lower limits to the values of c
w
and c
m
.
In the power model the linear combination is assumed to be convex, there is a constrained sharing of primitive utility
and c
w
þc
m
¼1. There is no argument to support this. Utility is not a fixed resource and joint utility is not a zero sum game. A
cake is not being divided, where his gains are her losses and vice versa. On the contrary, the collaboration of the couple
might lead to higher levels of utility than the primitives such as in ‘I get what you’re saying’or ‘Doing what you want pleases
me even more’etc. Collaboration and joint consumption can create extra utility to the one available in the primitive.
The very nature of the formation of couples reflects the reciprocally recognised potential for greater utility. Consequently, in
the models below a convex combination implying tradeoffs between agents is not assumed, nor warranted. However,
the constraint of the Power model is still a special case, and so is not ruled out. It can be empirically evaluated by testing the
hypothesis that c
w
þc
m
is 1 on the data.
Finally, we have discussed joint preference as a property of the systematic utility of the alternative. Throughout this
paper, and as is often the case in RU analyses of discrete choice experiments data, the systematic utility is assumed as
linearly additive in the attributes of the alternative. Thus, the arguments above are extended to include joint preference for
each attribute, separately. There are unique weights for each attribute.
In conclusion, we have argued that joint preference formation should be operationalised using the mathematical
convenience of a logit kernel. Joint systematic utility should be a linear combination of the primitive systematic utilities
using positive weights unique for each attribute. Summing of the weights over the man and the woman generally will be
less than 2 and not constrained to be 1.
Methods
In this section we start by laying out a notation that we then use to move from the conventional and by now quite
familiar mixed logit model to what we call a structural (equation) choice model or SCM. In the latter latent variables are
brought to bear so as to develop a plausible structure of correlation across the determinants of choices. In our application we
focus on a plausible structure between choice by members of the same residential unit (man and woman) and their joint
deliberations. Specifically, we try to account for influences of individual taste coefficients of single respondents in a
household as latent determinants of choice in the joint household decisions. Following Rungie et al., (2011;2012) it is
conceptually desirable to cast the approach around the familiar random utility framework.
Traditional random utility theory states that for each alternative the systematic components of utility, v, are specified to
be linear combinations of the mcovariates in the vector xwith random coefficients grouped in the vector β. To illustrate the
structural choice model proposed here Rungie et al., (2011;2012) used a notation and approach that is borrowed from the
conventions employed in the broad literature of structural equation modelling and adapted to choice modelling. However,
this would not be a familiar notation for those who, such as this audience, have been exposed to the conventional mixed
logit notation. So, in order to facilitate the understanding of the proposed notation we proceed as follows. We note that in
random parameter logit with a continuous mixture of taste the individual taste coefficient for a given attribute x
k
is
composed of two additive terms: the mean value of the taste parameter for the kth covariate β
k
and its random idiosyncratic
component s
k
~
β
kn
, where ~
β
kn
is the random component drawn by some distribution (perhaps standard normal) for the nth
individual and s
k
is the dispersion parameter for this random element to be estimated. So, omitting the subscript ifor the
single choice selection and nfor the respondent, the conventional mixed logit notation for the systematic component of the
utility is given by
v¼S
k
ðβ
k
þs
k
~
β
k
Þx
k
ð4Þ
Rather than being a single random entity, in the SCM ~
β
k
can be expressed as a structural equation:
~
β
k
¼a
k;1
~
β
1
þ⋯þa
k;m
~
β
m
þδ
k
ð5Þ
where the ais element from a matrix of regression parameters and the δis element from a vector of random components,
from which after estimation measures of fit, such as the classic R-squares, can be derived. These help to evaluate the overall
model and the suitability of the proposed constructs.
From the above equations, it can be seen that the variance–covariance matrix of ~
βis considerably more structured than a
simple diagonal matrix. In other words, specific correlation structures can be imposed on the coefficients for the covariates.
In a way, structural choice modeling (SCM) can be seen as an extension of error component modeling of the mixed logit
model as described in Brownstone and Train (1999), Train (2009) and Herriges and Phaneuf (2002) in the context of flexible
substitution patterns.
Typically, the random components δin Eq. (5) are specified to have Gaussian distributions, but other distributions can
also be assumed. In estimation via simulated maximum likelihood the expectation of mixtures of choice probabilities is
obtained via variance reduction techniques based on quasi-random draws. In this application we use Halton draws for their
well-known equidispersion properties, but others can be used (Baiocchi, 2005).
From the above it should be apparent that two special cases of the utility structure underlying observed choice that we
presented so far—the traditional fixed and random coefficient models—need not be addressed by means of SCM. Indeed
standard software packages can be used and results from identical models on the same data will differ slightly due to
Please cite this article as: Rungie, C., et al., The influence of individuals in forming collective household preferences for
water quality. Journal of Environmental Economics and Management (2014), http://dx.doi.org/10.1016/j.
jeem.2014.04.005i
C. Rungie et al. / Journal of Environmental Economics and Management ](]]]])]]]–]]]4
differences in maximization algorithms and features of simulation techniques. In what follows we use SCM to create an
‘Influence Model’, which is designed to uncover the latent structure of correlated choices in couples. Specifically, we focus
on the influences between men and women individual preferences and their joint choices as couples. In the process we
highlight some stylized identification issues that are typical of SCM.
Survey and data
The study is based on survey data collected with face-to-face DCEs interviews of 80 couples. One group of 20 couples was
sampled in the city of Torino in the North-West. A second group of 60 couples was obtained in the city of Vicenza, in the
North-East. The two locations in terms of water quality are similar for a variety of reasons not discussed here, but mainly
linked to their proximity to the Alps. The motivation for investigating preferences for residential tap-water is to be found in
the recently debated reforms of the national legislation regulating water utilities, which considered shifting the control of
water supply to newly constituted authorities with the intention to make water supply more market driven. This would turn
out to be challenging for municipalities, because it will force them to implement a series of changes in water utility
management by merging water management utilities across local authorities and creating new locally regulated commercial
entities. Therefore, local water authorities (Integrated Water Services) are interested in investigating preference hetero-
geneity for tap water quality attributes to strategically define water tariffs across city locations to collect the necessary
capital for investment and operations.
Data
The data used here come from an explorative and preliminary survey specifically designed to prepare a more complex
data collection, which will be the subject of another application. The application provided here is for the purpose of proof of
concept. As mentioned above, reported results are based on interviews of 80 couples, which in total provided 1920 choice
responses from 8 choice tasks with four alternatives each. Choices were expressed by 160 respondents individually (80 men
and 80 women) which then also provided 80 sets of joint decisions. The gross sample size is thus 80 households which for
some surveys is small enough to lead to large sampling errors and non significant parameter estimates with small
tstatistics. However, the repeated measures aspect of the data, with 24 observations per household leads to small sampling
errors. In the results below almost all parameters estimates are significant. The sample is large enough.
In the survey, respondents were asked to choose among alternatives described using the same attribute structure, which
differed on the basis of four quality attributes relating to drinking water characteristics plus the cost (Chlorine Odour,
Chlorine Taste, Water Turbidity, Calcium Carbonate Stains and Cost). Cost was described as an additional amount of money
people would pay in the water bill over a year. In particular, respondents were asked to choose among water service supply
contracts displaying different levels of water supply characteristics or “water service factors”to use a term commonly
employed in similar utility studies (Willis et al., 2005) and commonplace in the UK water industry. The attributes and their
relative levels are reported in Table 1. Respondents were asked to choose between the frequencies of events in which they
could smell (odour) and/or taste chlorine (once a day, once a week, once a month, never or always). Turbidity due to fine air
bubbles was also considered. Its levels included its absence, and its presence in a mild, medium and extreme form. Due to
the hardness of water in this area calcium carbonate staining in pipes is quite a concern and the effect of presence/absence
of staining was also investigated. In the survey respondents faced four alternatives in each choice set, where one alternative
was always the status quo and involved no additional cost. An example of choice set is reported in Table 2.
The design of the survey was finalized by contacting and interviewing experts employed by local utilities supplying
Integrated Water Services (water supply as well as water treatment services). These provided specific and technical
information which turned out to be valuable in the selection of the attributes levels. This information was supplemented
with suggestions provided by technicians from public institutions involved in the management of such water services. The
combined information was then used to conduct repeated focus groups, the results fromwhich were then used to design the
choice experiments. The complete questionnaire was then tested in the field in a pilot survey, which also provided priors for
the coefficient values to be used in the Bayesian design.
The choice data from each household were collected first with man and woman conducting individual experiments and
being asked their individual preferences. Then, it proceeded by asking man and woman to join together in a choice exercise
to select favourite alternatives for the household. In this way for each household we collected 3 sets of choices, one for the
man, one for the woman and one for the household.
Experimental design
The survey employed a sequentially experimental design and one of the aims of the research was to use the information
collected with the first design as a prior to inform the subsequent ones. In particular, in the survey was used a sequential
efficient Bayesian design. The purpose was to ensure a high accuracy of the estimates despite the relatively small sample
size affordable. One of the main advantages of such an approach is that as more responses are collected during the course of
the survey, gradually more accurate information becomes available on the priors of the population, thereby increasing the
Please cite this article as: Rungie, C., et al., The influence of individuals in forming collective household preferences for
water quality. Journal of Environmental Economics and Management (2014), http://dx.doi.org/10.1016/j.
jeem.2014.04.005i
C. Rungie et al. / Journal of Environmental Economics and Management ](]]]])]]]–]]] 5
efficiency of the final estimates and decreasing the potential for mis-specification (Kanninen, 2002;Scarpa et al., 2007;
Ferrini and Scarpa, 2007;Scarpa and Rose, 2008;Kerr and Sharp, 2010;Vermeulen et al., 2011).
In the Turin sample, the overall survey design was articulated in subsequent phases, as additional information was
sequentially collected in six waves of sampling. Each sample wave used a different WTP
b
-efficient design
2
developed using
Bayesian priors (as indicated by the subscript “b”), derived by combining the information collected in all previous waves.
The initial prior information was gathered from the pre-test and the pilot survey; the first wave of interviews then informed
in turn the design of the following waves. At the end of waves 1–6 basic multinomial logit models were estimated so as to
provide priors for the efficient design of the subsequent sample wave. Each respondent tackled 8 choice tasks.
For the second group of respondents in Vicenza we employed a Bayesian D-efficient design (Sandor and Wedel, 2001;
Ferrini and Scarpa, 2007;Rose and Bliemer, 2009), derived on the basis of existing information on parameter estimates
previously obtained from the previous study. The point estimates from the earlier Turin study were used to inform the prior
distribution on the Bayesian design for Vicenza, while the standard errors were used to define the variances of the
distributions of priors. The probabilities in the derivation of the design were obtained via simulation using 200
Halton draws.
Sampling
The survey focussed on couples and the preferences of their two members. As a consequence it focussed on modelling
joint choices as functions of primitive individual preferences of the two members of the couple (man and woman).
The survey developed in several stages. The first stage aimed at selecting households that could be considered as
“couples”into a sampling frame. These were subjects living in a stable relationship with a partner. Then the sampling was
randomly executed on this frame.
Table 2
Example of choice-set.
Which of the following alternative would you choose? AB C D
Chlorine odour: Always 1 day per week 1 day per month None
Chlorine taste: Always 1 day per week Never
Turbidity: Absent Medium Extreme
Calcium carbonate staining: No Yes Yes
Additional WTP in the bill per year 18€5€6€
Choice
Table 1
Description of the qualitative attributes.
Variable name Attribute Description of attribute and level
O_ALWAYS OD Odour Chlorine odour always
O_MONTH OD Odour Chlorine odour once a month
O_WEEK OD Odour Chlorine odour once a week
O_NEVER OD Odour Chlorine odour never
T_ALWAYS TS Taste Chlorine taste always
T_MONTH TS Taste Chlorine taste once a month
T_WEEK TS Taste Chlorine taste once a week
T_NEVER TS Taste Chlorine taste never
NO_TURB TR Turbidity No turbidity from fine air bubbles
MILD_TURB TR Turbidity Mild turbidity from fine air bubbles
MED_TURB TR Turbidity Medium turbidity from fine air bubbles
EXTR_TURB TR Turbidity Extreme turbidity from fine air bubbles
STAIN ST Stain Presence of calcium carbonate staining in pipes
2
Specifically, the WTP
b
-efficient criterion was adopted to select the fraction of the full factorial to be used as a design in the sequence of sub-samples.
This is based on the minimization of the expected variance of some non-linear functions of the utility coefficients, namely the sum of the variances of the
marginal willingness to pay estimates. Considering that different attributes can be described in different units, as in the case at hand, Scarpa and Rose
(2008) point out that the minimisation process of variance sum across marginal WTPs with uneven unit of measurement may result in an unsatisfactory
outcome. To overcome such a limitation, they suggest the adoption of a criterion that maximizes the minimum t-value for the marginal WTP. This choice
places more emphasis on the attribute whose WTP was estimated with least accuracy, as measured by the t-value. We note in passing that Bayesian WTP-
efficiency has also been found to provide designs with higher robustness to outliers and less prone to producing extreme WTP estimates (Vermeulen
et al., 2011).
Please cite this article as: Rungie, C., et al., The influence of individuals in forming collective household preferences for
water quality. Journal of Environmental Economics and Management (2014), http://dx.doi.org/10.1016/j.
jeem.2014.04.005i
C. Rungie et al. / Journal of Environmental Economics and Management ](]]]])]]]–]]]6
During the second stage, respondents were asked whether they would be willing to participate in the survey. They were
contacted by mail first and then by telephone. Once both partners agreed on participation, the interviewer would fix an
appointment to visit the couple. At the household's house, they were debriefed jointly and given the stated preference tasks.
Importantly, in order to avoid that any difference in choice across individuals of the same household could be due to
differences in choice tasks, each respondent within a given household unit was given the same sequence of choice tasks.
These tasks were performed first individually, so as to derive individual preferences, and then jointly. When performed
individually, respondents were asked their individual preferences. When performed jointly, they were asked to negotiate a
mutually satisfying outcome for the couple.
Model specifications and estimates
In what follows we illustrate the specifications of indirect utility for the preference space model.
Model specifications and rationale
The choice data is made of responses to three identical discrete choice experiments (DCEs) conducted separately. With y
w
we denote the responses by women (DCE 1), with y
m
those by men (DCE 2) and with y
j
the joint responses provided as a
couple (DCE 3). To simultaneously model choice probabilities for the separate DCEs the three data matrices were combined
at the household level. In each DCE the alternatives were described by using five attributes, three of which had 4 levels
defined as unimproved and 3 levels of improvement. In this study these were then each aggregated into a single dummy-
coded variables denoting extreme improvement. The fourth attribute (stain) had two levels and was also coded as a single
dummy variable denoting the “presence”of stains. The fifth attribute was the cost (tariff) which was coded numerically in
Euros. Because of dummy coding with each attribute (except cost) and the alternative specific constant for the status-quo in
total there were six covariates and six identifiable coefficients for the indirect utility function.
To evaluate the identification power of the SCM influence model in explaining unobserved heterogeneity we compare it with
two standard logit specifications. In total, three logit probability models have been specified and estimated for the three data sets:
(i) the fixed coefficient model, (ii) the random coefficient model, and (iii) the influence model.First,thefixedcoefficientlogitmodelwas
estimated, from which a mean value estimate (β
k
) for each attribute coefficient is obtained. Next, the well-known restrictive
assumptions of the fixed coefficient logit model were relaxed by estimating a random coefficient panel model; this, besides mean
values (β
k
), provided estimates of the dispersion parameter (s
k
) for the random coefficients of each covariate.
Ultimately and more importantly, we pose the following question: is there a structural link in the heterogeneity within the joint
DCE and the heterogeneity in the separate DCEs by men and women? The influence model specifies these links, in that the utilities
for the joint decisions are also a function of the individual utilities for women and men. Each utility in the joint DCE model is
simultaneously specified to be a linear function of the equivalent utilities in the women and men DCEs. By doing so we wish to
investigate if and, in case, to what extent, the joint decision making process of couples is influenced by individuals. Within this
exploration, as we will show, we can also answer the question of whether women or men are most affecting joint decisions.
In the equations and model specifications below the attributes are referred to as follow: odour¼OD, taste¼TS,
turbidity¼TR, stain¼ST, cost¼CO and status quo¼SQ.
The random coefficient model
In this model, the four water factor services—odour (OD), taste (TS), turbidity (TR) and stain (ST)—are assumed to have
random coefficients. The other two attributes—cost and status quo—are given fixed coefficients.
For women's individual choices the random coefficient model involves the following indirect utilities:
v
OD;w
¼β
OD;w
þs
OD;w
~
β
OD;w
x
OD;w
v
TS;w
¼β
TS;w
þs
TS;w
~
β
TS;w
x
TS;w
v
TR;w
¼β
TR;w
þs
TR;w
~
β
TR;w
x
TR;w
v
ST;w
¼β
ST;w
þs
ST;w
~
β
ST;w
x
ST;w
v
CO;w
¼β
CO;w
x
CO;w
v
SQ;w
¼β
SQ;w
x
SQ;w
ð6Þ
and, for alternative i,
μ
w
i
¼v
OD;w
i
þv
TS;w
i
þv
TR;w
i
þv
ST;w
i
þv
CO;w
i
þv
SQ;w
i
þε
w
i
ð7Þ
For men's individual choices and the joint decisions the random coefficient model repeats the same structure.
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The influence model
The random coefficient model introduces heterogeneity across the panel of choices. But it does not uncover any latent
structure of choice between members of the same household. In particular, no relation exists between the primitive of the
utility function of the individuals in their choices and their joint choice. Behaviourally this is clearly counter-intuitive and
contrary to empirical findings reporting corroborating evidence in favour of such correlation (Dosman and Adamowicz,
2006;Beharry et al., 2009;Scarpa et al., 2012). To account for this we propose an SCM that elaborates further on the random
coefficient model by imposing structure in the correlation of the ~
βs, but only for the joint choices. As in the random
coefficient model the primitive of the utility for women individual choices are expressed as independent random
coefficients.
~
β
OD;w
¼δ
OD;w
~
β
TS;w
¼δ
TS;w
~
β
TR;w
¼δ
TR;w
~
β
ST;w
¼δ
ST;w
ð8Þ
The four random components δin (8) have independent standard Gaussian distributions leading to a model for the women’s
individual choices identical to the random coefficient model in (6). For men's individual choices the influence model repeats
the same structure.
Things are different for the joint decisions, which have random components specified as linear combinations applied to
the primitive utilities:
~
β
OD;j
¼a
OD;w
~
β
OD;w
þa
OD;m
~
β
OD;m
þδ
OD;j
~
β
TS;j
¼a
TS;w
~
β
TS;w
þa
TS;m
~
β
TS;m
þδ
TS;j
~
β
TR;j
¼a
TR;w
~
β
TR;w
þa
TR;m
~
β
TR;m
þδ
TR;j
~
β
ST;j
¼a
ST;w
~
β
ST;w
þa
ST;m
~
β
ST;m
þδ
ST;j
ð9Þ
where adenotes the regression coefficients. The four random components δin (9) have independent Gaussian distributions with
means zero but with standard deviations to be estimated from the data (error components). Then, the indirect utilities are:
v
OD;j
¼β
OD;j
þs
OD;j
a
OD;w
~
β
OD;w
þs
OD;j
a
OD;m
~
β
OD;m
þs
OD;j
δ
OD;j
x
OD;j
v
TS;j
¼β
TS;j
þs
TS;j
a
TS;w
~
β
TS;w
þs
TS;j
a
TS;m
~
β
TS;m
þs
TS;j
δ
TS;j
x
TS;j
v
TR;j
¼β
TR;j
þs
TR;j
a
TR;w
~
β
TR;w
þs
TR;j
a
TR;m
~
β
TR;m
þs
TR;j
δ
TR;j
x
TR;j
v
ST;j
¼β
ST;j
þs
ST;j
a
ST;w
~
β
ST;w
þs
ST;j
a
ST;m
~
β
ST;m
þs
ST;j
δ
ST;j
x
ST;j
v
CO;j
¼β
CO;j
x
CO;j
v
SQ;j
¼β
SQ;j
x
SQ;j
ð10Þ
The heterogeneity of the women's individual choices is exogenous, specified by the independent coefficient δin (8).Sotoo
is the heterogeneity of the men’s individual choices. However, in (9) the heterogeneity for the joint decisions is now a
combination of an exogenous effect, specified as the δ, and an endogenous effect, specified by including the ~
β
:;w
and ~
β
:;m
terms.
As discussed below, the influence model was fitted to the data in two similar forms, the full model (Full) and a slightly
simplified model (S) without redundancies, which in the empirical analysis shows to fit the data just as well.
The random components, δ, were all specified to have Gaussian distributions which is appropriate as a default based on
the Central Limit Theorem. But in addition these entities represent components of latent utilities for qualitative variables,
such as taste, that are unbounded, and are associated with manifest binary covariates.
Model estimates
All models were estimated by using DiSCos (Rungie, 2011).
3
The estimates of the mean values of the model with fixed
taste coefficients, reported in Table 3, show expected signs and high significance for all attributes. All coefficients for the
“never”smell and taste for chlorine and the no turbidity display positive intensities of taste. Women show less inclination to
adhere to the status-quo than men and what emerges from joint decisions.
Table 4 reports the statistics for the fit of the various preference space models. As it can be noted by comparing the log-
likelihood values, the random coefficient model (see Table 5 for result estimates) performs better than the fixed model, as
one would expect. Nevertheless the influence model gives the best fit. The improvement in terms of performance is
3
Structural choice models were estimated by means of a software program called DiSCos (Rungie, 2011) and written in MatLab by using 10,000 Halton
draws. Estimation of each model with relatively good starting values took about a day in a Dell M6500 quad core 64 bit computer.
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substantial, with more than 70 points, thereby supporting our hypothesis of existence of a latent structure in the
unobserved heterogeneity. Information criteria that penalize for over parameterization, such as AIC, AIC3 and BIC, are
concordant to indicate this model to provide best fit.
Identification—location, order and rank
The formal evaluation of identification—location, normalization, and stability of estimates—of latent variable models is
described by Walker (2001, p. 94) and we adopt a similar approach.
In the usual manner for discrete choice experiments, location issues are resolved by modeling utility differences. Each
attribute, except cost, has two levels (binary) and the model estimates the change in utility between the levels. Cost is coded
as a continuous variable, with one covariate for each DCE, and the model estimates the change in utility for a unit change
in cost.
Order and rank identification conditions, that assess the impact of choice set size on identification, are of less concern in
the influence model because in this unlabeled experiment for all alternatives except the status quo (which is effectively the
no choice) there are no alternative specific constants. The latent variables and all their associated model parameters affect
utility by being first transformed through the observed covariates (the attributes). Apart from status quo, none affects utility
of an alternative without this transformation and so do not have the identification properties of alternative specific
constants. Furthermore, being a random coefficient model, all 24 choices made by each household are simultaneously
estimated; i.e. the eight choices of the woman, the eight choices of the man and the eight joint are all entered as a single
Table 3
Fixed model.
Women Men Joint
Μ|t-Value|μ|t-Value|μ|t-Value|
Odour 0.85 9.30 0.79 8.30 0.89 8.35
Taste 0.28 2.86 0.27 2.66 0.31 2.73
Turbidity 0.85 8.95 0.79 8.58 1.04 10.57
Stain 1.90 9.04 1.63 7.87 1.90 8.12
Cost 0.06 4.78 0.06 5.54 0.04 3.53
Status Quo 0.06 0.29 0.32 1.94 0.48 2.97
Table 4
Summary of model statistics.
Model Number of parameters Log likelihood BIC AIC AIC3
Fixed coefficient 18 1343.42 2778 2723 2741
Random coefficient 30 1267.18 2687 2594 2624
Influence (Full) 38 1200.36 2594 2477 2515
Influence (S) 34 1200.36 2573 2469 2503
Table 5
Random coefficient model.
Means Women Men Joint
μ|t-Value|μ|t-Value|μ|t-Value|
Odour 1.01 7.93 0.98 7.26 1.02 8.18
Taste 0.30 2.38 0.30 2.24 0.34 2.29
Turbidity 1.00 7.31 1.04 7.20 1.19 9.24
Stain 3.69 6.00 3.04 5.71 3.14 2.60
Cost 0.05 3.15 0.07 5.41 0.04 3.55
Status Quo 0.10 0.39 0.43 2.23 0.52 2.83
Dispersions s|t-Value|s|t-Value|s|t-Value|
Odour 0.42 2.84 0.64 3.02 0.00 0.00
Taste 0.44 2.22 0.48 2.69 0.66 3.90
Turbidity 0.58 4.27 0.71 4.46 0.46 2.82
Stain 2.48 2.19 1.88 4.84 1.94 1.72
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probability in the log-likelihood function. Consequently, the number of alternatives in any one choice set is of less concern
(Walker, 2001, p. 38). Positive semi-definiteness (PSD) at convergence of the asymptotic covariance matrices was examined
for all models, as could be expected.
Identification—normalization of the structural model
“Recall that the reason that an identifying restriction is necessary is that there are an infinite number of solutions (i.e.,
parameter estimates) to match the given model structure. The point of an identifying restriction is to establish the existence
of a single unique solution, but not change the underlying model in any way”(Walker, 2001, p. 35). This is a particularly
salient point for SCM in that, because just like in structural equation models, there is an array of parameters. Different
parameters can have the same impact on a model, creating the potential for parameter redundancy. When this occurs, the
model is normalized by fixing some parameters to a value, often zero or one. The aim is to remove redundancy without
changing the nature of the model. This process for structural equation modeling is described in detail by Bollen (1989).
The first step in examining normalization of an SCM is to view the structural equations as a set of simultaneous equations
for variables, with means, variances and covariances, as if there is full knowledge of the distribution of the random
coefficients, over the population. This can help determine if parameters are redundant, as well as an appropriate form of
normalization to eliminate the problem.
SCM can estimate quite complex structures for covariances but not for the means, β. The number of identifiable means is
limited to the number of covariates. Consequently, in the influence model, as is often the case for SCM, the latent variables,
and so all variables in the structural component of the model, are specified to have means of zero. In the influence model
this was operationalized as the random components δall having means that were fixed to zero.
In the women and men DCEs, for any one attribute, (9) indicates a potential confounding between two parameters; the
dispersion parameter sand the standard deviation of δ. To avoid such a problem the standard deviation of δwas fixed
to one.
The influence model, as it is described above, has 42 parameters. Embedded in the model there are three DCEs, women,
men and joint, each with six attributes creating a total of 18 mean estimates in β. In each DCE four attributes, OD, TS, TR and
ST, have random coefficients with dispersion parameters screating a total of 12. The same four attributes in the women and
men experiments influence the preferences in the joint experiment creating a total of 8 regression parameters a. Finally, as
in (9), each of the four δin the joint experiment has a standard deviation to be estimated. Thus, there are in total 42
parameters to estimate from the data. Not all are identified and additional normalization is required.
In the joint DCE, for any one attribute, (9) again indicates a confounding. This time between three parameters; the
regression coefficients a, the dispersion parameter sand standard deviation of δ. One of the three is not identified. As a
comparison, in the women and men DCEs the standard deviation of δwas fixed to one leading to the remaining dispersion
parameter, s, being identified. Exploratory data analysis indicated that for the joint experiment a similar approach of fixing
the standard deviation of δto one was not appropriate as it reduced the ability to estimate and interpret the regressions
parameters a. As an alternative, the standard deviation of δwere free to be estimated from the data, and some regression
parameters, a, were fixed. Specifically, for the influence of the women on the joint DCE the regression parameters were all
fixed to one; i.e. a
OD;w
¼a
TS;w
¼a
TR;w
¼a
ST;w
¼1. Thus the influence of the women is normalized. The influence of the men on
the joint experiment is then evaluated by comparing the equivalent regression parameters, a
OD;m
,a
TS;m
,a
TR;m
and a
ST;m
, to the
standard of one. This led to the (full) influence model having 38 identified parameters. The estimates are reported in Table 4.
The model does not assume there is influence, nor does it impose it. In the joint DCE the combined roles of the dispersion
parameter, s, and the standard deviation of the random component δdetermine the relative exogenous and endogenous
effects on heterogeneity. The degree of influence is determined by the data. The results are discussed below.
Simplifying the influence model
The estimates of the standard deviations for the four δin (9) are all close to zero. The result does not indicate that the
decision making in the joint experiment is deterministic when conditioned on the women and men experiments. Rather,
the result indicates that all the heterogeneity in the joint experiment can be accounted for by heterogeneity from the
separate women and men experiments and that the expressions δ
OD;j
,δ
TS;j
,δ
TR;j
and δ
ST;j
in (9) do not contribute to the fit of
the model. This is a strong result, but it is unsurprising. The DCEs for women and men were conducted first. Then the joint
DCE was conducted immediately after. Apart from the heterogeneity influencing the women and men DCEs, there was no
opportunity for a new exogenous source of heterogeneity to influence the joint DCE.
Consequently, the model in (9) is simplified as in (11); the regression parameters, a, for women are again fixed to one and
in addition the standard deviations for the δin the joint experiment are fixed to zero, giving rise to the following latent
structure:
~
β
OD;j
¼
~
β
OD;w
þa
OD;m
~
β
OD;m
~
β
TS;j
¼
~
β
TS;w
þa
TS;m
~
β
TS;m
~
β
TR;j
¼
~
β
TR;w
þa
TR;m
~
β
TR;m
~
β
ST;j
¼
~
β
ST;w
þa
ST;m
~
β
ST;m
ð11Þ
Please cite this article as: Rungie, C., et al., The influence of individuals in forming collective household preferences for
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Table 4 shows this simpler form of the influence model (denoted by (S) from “simplified”) fits the data just as well,
empirically confirming that four random components, the δin the joint DCE, are redundant and do not contribute to the full
influence model. The reduced form (S) is the model we use to evaluate the influence. Further output for the influence model
(S), is given in Tables 6 and 7.
Evaluating influence
The estimates of the regression coefficients, a, are shown in Table 7. The influence of men on the attribute of odour in
joint choices is greater than the influence of women. Conversely, the influence of women in joint choices is greater on the
other three qualitative attributes, taste, turbidity and stain.
Aggregating over (the square of) the regression parameters for the four attributes identifies that women provide 58% of
the heterogeneity in the joint experiment and men 42%. This conclusion, that women have greater influence, is further
demonstrated by applying two constraints to the influence model (S). First, only women are specified as influencing the
heterogeneity in the joint experiment, and second, only men. The results in Table 8 for these two models again show clearly
that women have greater influence on the heterogeneity in the joint DCE than men.
Identification—empirical evidence
The Hessian matrix provides useful empirical diagnostics on identification (Walker, p. 136). A Hessian matrix that is
singular, or near singular, is strong evidence of poor identification. Furthermore, once the Hessian is inverted, if the diagonal
elements imply that the variance of some parameter estimates is actually, and incorrectly, negative, then again there is
evidence of poor identification where the negative variances indicate the parameters that are likely to be not identified. In
this manner the Hessian indicates that either there are redundant parameters or the data is of insufficient quality to
estimate all the parameters. If there are redundant parameters then the log-likelihood will not have a unique global maxima
and instead will have a ridge or plateau with the Hessian being singular. If there is multicolinearity amongst the covariates
in the data then the Hessian will be singular or near singular. In the Influence (S) model the Hessian was ‘well behaved’;it
Table 6
Result estimates of the regression coefficient, a, for the influence model (S).
Attribute Women Men
a
w
a
m
|t-Value|
Odour Fixed to 1 1.34 1.89
Taste Fixed to 1 0.47 1.69
Turbidity Fixed to 1 0.81 2.43
Stain Fixed to 1 0.42 2.34
Table 7
Reduced form of the influence model (S).
Means Women Men Joint
μ|t-Value|μ|t-Value|μ|t-Value|
Odour 1.07 8.22 0.94 6.64 1.23 7.59
Taste 0.26 2.08 0.25 1.95 0.39 2.39
Turbidity 1.07 7.68 0.97 7.36 1.48 8.91
Stain 4.70 5.24 3.39 5.35 5.23 4.85
Cost 0.05 3.35 0.07 5.29 0.04 3.68
Status Quo 0.15 0.59 0.45 2.37 0.63 3.29
Dispersions s|t-Value|s|t-Value|s|t-Value|
Odour 0.52 4.45 0.75 5.20 0.37 2.72
Taste 0.50 3.70 0.43 2.47 0.65 4.46
Turbidity 0.63 5.06 0.64 4.13 0.61 4.26
Stain 3.08 5.28 2.04 4.01 2.83 3.67
Table 8
Summary of constrained model statistics.
Model Number of parameters Log likelihood BIC AIC AIC3
Women influence only 30 1214.52 2581 2489 2519
Men influence only 30 1227.23 2607 2514 2544
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was negative semi-definite, invertible, with a determinant value of 6.9E57, and the associated estimates of the variance of
the parameters were all positive. The properties of the Hessian provide useful identification diagnostics.
The empirical evidence further supports that the Influence (S) model is identified. Of the 34 parameters all but 3 had
absolute tvalues greater than 2 indicating that the log likelihood function relied on identified coefficient estimates at the
maximum. Of particular importance in considering identification are the regression parameters, a. The statistical fit values of
the models reported in Table 8 show that when the four as are constrained to zero for either of the sexes then the model
does not fit as well as the Influence (S) model, the fit statistics of which are reported in Table 4. Likelihood ratio tests show
the regression parameters to be significant (po0.0001) for both sexes. This is further empirical evidence of identification for
the Influence (S) model.
All models were fitted to the data using simulated maximum likelihood, with 10,000 draws and the Nelder–Mead
optimizer, which is a standard approach but with known limitations. (i) The optimizer may find a local maximum rather
than the global maximum. Given that most of the attributes are coded as having two levels only, then local maxima are less
likely to exist but still an optimization strategy was adopted to further minimize the risk. The optimizer allows the analyst to
specify the step size. For the initial runs this was large and then brought down to a finer resolution. The optimization was
run several times in sequence starting each time where the previous run ended. Each time this happened the optimizer
experimented by itself with different step sizes increasing the chance of avoiding a local maxima and removing the chance
of the optimizer stopping prematurely. Also the model was fitted to the data from scratch using different initial values. (ii)
The re-sample, even though the number of draws was large, can influence the results (Walker, p. 59). Consequently, the
model was run using different re-samples to assess the stability of results. The correlation in the parameter estimates
between re-samples was 0.987.
Identification—scale
The final identification issue to be discussed is that of scale. In all three DCEs, women, men and joint, the original random
utility model was applied in which the error term εin (1) is normalized. The true and un-estimated error variance may differ
between the three DCEs. If so, then the parameter estimates for each DCE will be affected, proportionately. The final
outcome of the influence model is a partitioning of the variance of the latent variable in the joint DCE as operationalized
with the aparameter. It is a proportional result; a given portion of the variance is attributable to the women DCE and the
remaining portion to the men DCE. Consequently, the aparameter for women and the aparameter for men are on the same
scale. Furthermore, normalizing the a, by fixing it to one, for the women DCE has the affect of setting the scale of the afor
the men DCE. Due to the scale issue care must be taken still in contrasting the parameters from the different DCEs. However,
the primary result for the influence model is the partitioning of variance in the joint DCE. It is a proportion, and it is
scale free.
Variance
The model examines the preference variance between respondents of the same sex in the sample. For any one attribute it
establishes the extent to which its preference variance between men and between women can account for its preference
variance between the joint. To illustrate, we start by considering just one attribute, OD. In the utility equations for the
women, men and joint the three OD covariate, x
OD,w
,x
OD,m
and x
OD,j
each had random coefficients where the sources of
randomness are the expressions: for women s
OD;w
~
β
OD;w
, for men s
OD;w
~
β
OD;w
and for joint s
OD;j
~
β
OD;j
.
Let there be weights
c
OD;w
¼s
OD;j
s
OD;w
and c
OD;m
¼a
OD;m
s
OD;j
s
OD;m
ð12Þ
The values are reported in Table 9. Adapting (11) gives that the random component of the joint systematic utility is a
weighted linear sum of the primitive random components with weights c
OD;w
and c
OD;m
. Following (3).
V
OD;j
¼β
OD;j
þs
OD;j
~
β
OD;j
no
x
OD;j
¼β
OD;j
þc
OD;w
s
OD;w
~
β
OD;w
þc
OD;m
s
OD;m
~
β
OD;m
no
x
OD;j
ð13Þ
Table 9
Result estimates of the weights, c, for the influence model (S). c
w
¼a
w
s
j
/s
w
and c
m
¼a
m
s
j
/s
m
.
Attribute Women Men SUM
c
w
|tValue|c
m
|tValue|c
w
þc
m
Odour 0.71 2.75 0.66 2.47 1.37
Taste 1.30 3.20 0.71 1.70 2.01
Turbidity 0.97 3.04 0.77 3.41 1.74
Stain 0.92 3.27 0.58 3.19 1.50
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and
Var s
OD;j
~
β
OD;j
¼c
OD;w
2
Var s
OD;w
~
β
OD;w
þc
OD;m
2
Var s
OD;m
~
β
OD;m
ð14Þ
This provides a useful benchmark. Consider the hypothetical case where the variances are the same for the primitive and
joint and Var s
OD;w
~
β
OD;w
¼Var s
OD;m
~
β
OD;m
¼Var s
OD;j
~
β
OD;j
. Then 1 ¼c
OD;w
2
þc
OD;m
2
. Under such a constraint it is
easily shown that 1rc
OD;w
þc
OD;m
rffiffiffi
2
pwith the minimum occurring when one weight is zero and maximum when the
weights are equal (just as the hypotenuse of a right angle triangle is less than the simple sum of the other two sides). Unlike
the Power model where the expectation is that the sum of the weights c
OD;w
þc
OD;m
will be 1 there is now, with this
argument reflecting the variance of random coefficients, an expectation that the sum will be slightly greater. But this is
based on the assumption that the variances for the primitives, and joint are equal. If the variance of the joint is less than the
primitive then the Power model is more likely supported. However, the discussion of joint preference formation above
suggests that if anything the joint will involve polarization of preferences and so will have greater variation, with the sum of
the weights c
OD;w
þc
OD;m
increasing to a maximum of approximately 2. The empirical results are in Table 9 and fall within
the anticipated range. The lowest value of c
OD;w
þc
OD;m
is 1.4 for odour, and the highest is 2 for taste. This indicates that
taste was the attribute in which the collaboration involved in joint preferences formation lead to the most polarization,
synergy and strengthening of preference.
Joint preference formation
The model developed here reflects the discussion above on joint preference formation. Data for all three discrete choice
experiments—i.e. for the man and woman primitive choices and for the joint choices—were analysed using the logit kernel
specification. The joint systematic utilities have been specified as a weighted linear combination of the primitive systematic
utilities. In the joint preference formation the model has estimated separate weights for the women and men for each
attribute. As proposed in the discussion of joint preference formation, the weights are all positive and generally their sum is
more than 1 and less than 2. Furthermore, as proposed, for each attribute the sum of the weights is typically not 1 as
required by the Power model. The results generally support the arguments given above regarding joint preference
formation.
Conclusions
The study of preferences underlying group decisions can be conducted by adequately developed surveys and the data of
which are consistently analyzed by employing specifically developed choice models. While previous work has mostly
employed power function approaches at the individual indirect utility level (Dosman and Adamowicz, 2006) or at the single
attribute level (Beharry et al., 2009), we offer an “influence model”based on a special structure of the idiosyncratic
components of the joint choice. This is a special case of a broader approach to choice modeling developed by Rungie et al.,
(2011,2012) and Coote et al. (2011) called structural choice modeling. As a proof of concept, we explore this approach in a
small sample but high quality set of discrete choice experiments conducted in households and investigating preferences for
tap water. Tap water is a multi-attribute good that is appreciated differently by each member of a household. Yet, one single
contract provides this utility at the household level. Household preferences should hence be based on joint decisions by
members of the household. In an identical choice experiment conducted first individually and separately by husband and
wife, and then jointly, we find that a structural model of choice greatly improves model fit.
We stop short of deriving estimates of welfare measures for specific policies because we favor the uncovering of structure
in the heterogeneity of joint decisions. Overall we find the preliminary results worth of attention and the modeling
approach informative. Further research should focus on other plausible specifications of influence across individual and
joint choice as well as on deriving welfare estimates for specific policy proposals. Future work should also explore the
predictive power of the model for joint group decisions in observations held out of sample during estimation, but based on
the individual preferences of the group.
Acknowledgments
Senior authorship is equally shared by the authors. We wish to thank Jordan Louviere for promoting and encouraging this
research partnership. The responsibility of all remaining mistakes rest solely with the authors.
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