Bayesian Estimation of Willingness‐to‐pay Where Respondents Mis‐report Their Preferences*

AuthorA. Chalak,A. Bailey,I. Fraser,K. Balcombe
Publication Date01 June 2007
Date01 June 2007
Bayesian Estimation of Willingness-to-pay Where
Respondents Mis-report Their Preferences*
K. Balcombe,A. Baileyà,A. Chalakàand I. Fraserà
Department of Agricultural and Food Economics, University of Reading, Reading, UK
àAEBM Group, Kent Business School, University of Kent, Wye, Kent, UK
We introduce a modified conditional logit model that takes account of uncertainty
associated with mis-reporting in revealed preference experiments estimating will-
ingness-to-pay (WTP). Like Hausman et al.[Journal of Econometrics (1988) Vol.
87, pp. 239–269], our model captures the extent and direction of uncertainty by
respondents. Using a Bayesian methodology, we apply our model to a choice
modelling (CM) data set examining UK consumer preferences for non-pesticide
food. We compare the results of our model with the Hausman model. WTP estimates
are produced for different groups of consumers and we find that modified estimates
of WTP, that take account of mis-reporting, are substantially revised downwards. We
find a significant proportion of respondents mis-reporting in favour of the non-
pesticide option. Finally, with this data set, Bayes factors suggest that our model is
preferred to the Hausman model.
I. Introduction
Stated preferences are now commonly used in conjunction with the choice modelling
(CM) framework in environmental valuation (e.g. Bennett and Blamey, 2001;
Hanley, Mourato and Wright, 2001). Naturally, an assumption underpinning such
studies is that agents are able to answer in a way that reflects their preferences, as
would be revealed by a ‘true’ choice. Part of the rationale for preferring CM to
contingent valuation (CV) is that by setting out a number of attributes, and varying
*We are grateful for the comments from seminar participants at Deakin University and the University
Newcastle, Australia. This research was partly funded by UK DEFRA grant number: PS2302.
JEL Classification numbers: C25, C11, Q51.
doi: 10.1111/j.1468-0084.2006.00198.x
Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2006. Published by Blackwell Publishing Ltd,
9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
those attributes, the outcomes of a well-designed experiment should closely reflect
the preferences of respondents (Hanley et al., 2001). Yet, arguably, even in a well-
designed experiment, many respondents may be uncertain that their stated
(hypothetical) choices would be in accordance with their actual choices, and there
is also the possibility that some respondents deliberately state preferences they know
to be false, or probably false.
Unlike some other studies in the environmental valuation literature (e.g. Welsh
and Poe, 1998; Evans, Flores and Boyle, 2004; Vossler et al., 2003), which have
addressed this problem, this paper introduces an approach that does not require
explicit acknowledgement by respondents about their uncertainty. In common with
other CM studies, the essential idea behind our approach is that individuals are
assigned a linear utility function, with differences in choices being due to a random
residual. In addition, each respondent is assigned a probability that they will respond
in a manner not consistent with this utility function (which we term mis-reporting).
Moreover, each option within the choice set is assigned a probability that,
should respondents mis-report their preferences, is in favour of that option.
These probabilities are then estimated along with the usual coefficients of the logit
The model we develop is similar to that introduced in Hausman, Abrevaya and
Scott-Morton (1998) (henceforth referred to as the Hausman model or Hausman
logit). As in the Hausman model, our approach assumes that respondents potentially
give inaccurate responses, deliberately or otherwise, and the propensity for them to
do so is explicitly estimated within the model (by generalizing the likelihood
function). However, whereas the Hausman model specifies conditional probabilities
for mis-reporting within the likelihood that depend on the nature of the ‘true’ choice,
we specify an unconditional probability that there is mis-reporting. Consequently,
henceforth we refer to our model as the unconditional probability of mis-reporting
(UPMR) logit to distinguish it from the Hausman model which we refer to as the
conditional probability of mis-reporting (CPMR) logit. The UPMR and CPMR
approaches to the treatment of mis-reporting produce models that are non-nested.
However, the UPMR generalizes more easily into the multinomial logit, and is more
parsimonious than the CPMR model except in the binomial case when the number of
parameters in each model are equal.
Within the environmental valuation literature, the approach proposed in this paper
is indirectly related to Welsh and Poe (1998), Alberini, Boyle and Walsh (2003),
Evans et al. (2004), Vossler et al. (2003) and Vossler and Poe (2004), among others.
In this literature, researchers have explicitly allowed survey participants as part of the
survey elicitation format to indicate qualitative and/or quantitative levels of
uncertainty associated with particular choices in non-market CV studies. This richer
survey design has led to a number of modifications in how WTP is estimated such as
the dual-uncertainty decision estimator (DUDE) of Evans et al. (2004). In addition,
it has allowed calibration of responses to ‘deflate’ WTP estimates (e.g. Vossler et al.,
414 Bulletin
Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2006

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