Bounding WTP distributions to reflect the ‘actual’ consideration set

The Journal of Choice Modelling ] (]]]]) ]]]–]]]

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The Journal of Choice Modelling journal homepage: www.elsevier.com/locate/jocm

Bounding WTP distributions to reflect the ‘actual’ consideration set Danny Campbell a,n, David A. Hensher b, Riccardo Scarpa c,d a

Economics Division, Stirling Management School, University of Stirling, Stirling FK9 4LA, Scotland Institute of Transport and Logistics Studies, The Business School, University of Sydney, Australia c Gibson Institute for Land, Food and Environment, School of Biological Sciences, Queen's University Belfast, Northern Ireland d Department of Economics, Waikato Management School, University of Waikato, Hamilton, New Zealand b

a r t i c l e i n f o

abstract

Article history: Received 5 August 2013 Received in revised form 12 December 2013 Accepted 4 February 2014

In this paper we extend the independent availability logit and combined latent class mixed logit models to accommodate respondents with different consideration sets due to their cost thresholds and cut-offs. Pertinent features of our model are that it is estimated in WTP-space and that the class-specific WTP distributions are truncated to be within the bounds deduced from the cost level(s) within each class-specific consideration set. Our analysis shows that our approach is well suited at uncovering the heterogeneity in the cost thresholds and cut-offs used by respondents. This is shown to help build a richer insight into respondent's behaviour as well as raise a number of concerns about the appropriateness of assuming deterministic choice sets. We discuss the implications of our results for welfare analysis. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Independent availability logit model Combined latent class mixed logit model Willingness to pay Consideration set Cost threshold Elimination-by-aspects

1. Introduction When analysing discrete choice data, especially stated preference data, it is typically assumed that respondents consider all offered alternatives and their choices are based on the alternatives that provide them with the highest utility. However, at least in principle, one may postulate the hypothesis that due to differences in personal budget constraints, as well as preferences, respondents may restrict their consideration set to those alternatives that do not exceed their cost (as well as other) thresholds and cut-offs (cf. Swait, 2001; Han et al., 2001; Cantillo et al., 2006; Cantillo and Ortúzar, 2006; Chou et al., 2008; Mørkbak et al., 2010; Campbell et al., 2011, 2012). If this is indeed the case, failing to account for this processing strategy is likely to be suboptimal, and perhaps lead to misguided inferences, as the model does not reflect actual choice behaviour. The alternatives taken into account by a respondent cannot be known with certainty. Indeed, recent studies ask whether an alternative is acceptable or not (e.g., see Hensher and Rose, 2012; Doherty et al., 2013). However, observed choice behaviour helps make probabilistic statements about the likelihood of competing consideration sets being the true choice set. Following Manski (1977), a probabilistic framework can be formulated to model this type of behaviour to help distinguish between the deterministic choice set, as generated by the experimental design, and the respondent's ‘actual’ consideration set—which we define as the subset of alternatives respondents actually considered when making their choice and is distinct from the universal set, which includes all alternatives presented to respondents. For this type of analysis we extend the independent availability logit (IAL) model (cf. Swait and Ben-Akiva, 1987; Swait, 2001) to accommodate

n

Corresponding author. E-mail address: [email protected] (D. Campbell).

http://dx.doi.org/10.1016/j.jocm.2014.02.004 1755-5345/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: Campbell, D., et al., Bounding WTP distributions to reflect the ‘actual’ consideration set. The Journal of Choice Modelling (2014), http://dx.doi.org/10.1016/j.jocm.2014.02.004i

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2

respondents with different consideration sets due to their cost thresholds and cut-offs. Since a respondent's true consideration set cannot be known with certainty, this model assumes that the choice sets are latent, and, in our case, the conditional choice model is mixed logit. Our work is motivated by the fact that greater behavioural insights into the alternatives that actually have an influence on respondent's choices should help at the analytical stage, particularly when it comes to deriving reliable welfare estimates. Indeed, an especially important feature of the insight into the cost level(s) of the restricted choice sets is that it gives an indication of the bounds of the aggregate marginal willingness to pay (WTP) distribution. If some respondents systematically restrict their actual choice set to only include alternatives that are below a given cost level, their maximum aggregate marginal WTP can be safely assumed to be less than this cost threshold. Similarly, a lower bound for their minimum aggregate marginal WTP can be identified, which is, clearly, at least the value of the highest cost level within their true consideration set. To accommodate this insight we further broaden the IAL model using a latent class consideration set approach. To accommodate within-class heterogeneity, we use continuous distributions of random parameters within classes. Albeit the latent classes in this case denote competing consideration sets as opposed to different taste structures.1 Salient features of our proposed model are that (i) it is estimated in WTP-space and (ii) that the WTP distributions for each class-specific consideration set are truncated to be within the bounds deduced from the cost level(s). Our analysis shows that our approach is well suited at uncovering the heterogeneity in the cost thresholds and cut-offs used by respondents. This is shown to help build a richer insight into respondent's behaviour as well as raise a number of concerns about the appropriateness of the conventional practice of assuming deterministic choice sets. The empirical application of our modelling approach shows that it has important implications for welfare analysis. The rest of the paper is organised as follows. Section 2 describes our econometric approach and Section 3 presents our empirical context. Section 4 reports the model estimation and post estimation results. Section 5 concludes.

2. Method We derive marginal WTP estimates using the framework of discrete choice models based on random utility maximisation (RUM). We start by introducing the required notation so as to provide a base model against which comparisons can be made. Then, we illustrate our model to facilitate the fact that some respondents may have rationally and systematically excluded some of the proposed alternatives from their consideration set at the moment of choice. We further extend this model by making provision for the fact that the marginal WTP estimates reflect the bounds implied by the restricted choice sets. We go on to explain how we concurrently address the heterogeneity in respondent's marginal WTP. Finally, we round-up this section by detailing our approach to model estimation.

2.1. Background notation We use the conventional approach to analyse discrete choice experiment data, based on RUM, where individuals are assumed to select the choice alternative that yields the greatest expected utility to them. In particular, where respondents are indexed by n, choice sets by t, the price and non-price attributes are represented by p and x respectively, the utility of the chosen alternative i can be written as U nit ¼  αpnit þβxnit þεnit ;

ð1aÞ

where α and β are parameters to be estimated for the price and non-price attributes respectively, and ε is an iid type I extreme value (EV1) distributed error term, with constant variance of π 2 =6. The specification in Eq. (1a) parameterises utility in ‘preference-space’. Given the focus of this paper is to uncover respondent's marginal WTP, we prefer to work in ‘WTPspace’, as promoted by Train and Weeks (2005) and Scarpa et al. (2008). The implied marginal WTP for an attribute is the ratio of the attribute's coefficient to the price coefficient: w ¼ β=α. Using this definition, utility can be rewritten as U nit ¼  αpnit þðαwÞxnit þ εnit ;

ð1bÞ

which is called utility in WTP-space because w represents a vector of marginal WTPs for the respective attributes. Expressing utility in this manner has the advantage that the estimates of marginal WTP are reported directly in the model ^ Moreover, the coefficient estimates obtained for w ^ are independent from those output by the vector of estimates w. ^ meaning that the instability associated with marginal WTP estimates derived from the obtained for the price coefficient α, ratio of random variables in preference-space is reduced (Balcombe et al., 2010; Hensher and Greene, 2011).2 1 We note, in passing, that this framework used here only for the cost attribute can easily be generalised to account for thresholds in more than one attribute. In fact, the approach lends itself to exploring a range of ‘elimination-by-aspects’ behaviour (i.e., where the choice procedure is a process of eliminating alternatives that do not fulfil certain aspects). 2 For further discussion on WTP-space models see Train and Weeks (2005) and Scarpa et al. (2008).

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Given the assumptions outlined above, the probability of the sequence of choices made by individual n can be represented by the MNL model: expð  αpnit þ ðαwÞxnit Þ

Tn

Prðyn jpn ; xn Þ ¼ ∏

J t ¼ 1 ∑j ¼ 1

expð  αpnjt þ ðαwÞxnjt Þ

;

ð2Þ

where yn gives the sequence of choices over the Tn choice sets for respondent n, i.e., yn ¼ ½in1 ; in2 ; …; inT n .3

2.2. Accounting for endogenous WTP bounds While the MNL choice probability expressed in Eq. (2) directly uncovers estimates of marginal WTP for various attributes, when it is estimated from typical stated preference data it does so in a manner that assumes all respondents consider all offered alternatives, including those that are priced above their maximum marginal WTP. However, respondents may rationally restrict their consideration set to include only those alternatives that do not exceed their maximum marginal WTP. So, the standard consideration set assumption may be inappropriate. Instead, this paper considers a semicompensatory choice process whereby if a respondent's maximum marginal WTP is surpassed the alternative is rejected and the remaining options constitute the consideration set. The respondent then makes their choice among the alternatives in their consideration set following a utility maximisation compensatory rule. Following Manski (1977), a probabilistic model can be formulated to account for this type of choice behaviour to help distinguish between deterministic choice sets, as generated by the experimental design, from the respondent's actual consideration set. In order to achieve this we take as a point of departure the independent availability logit (IAL) model (cf. Swait and Ben-Akiva, 1987; Swait, 2001; Frejinger et al., 2009; Kaplan et al., 2012; Richardson, 1982; Ben-Akiva and Boccara, 1995; Louviere et al., 2000, for some examples). The choice probability of the IAL model is given by S

Tn

s¼1

t¼1

Prðyn jpn ; xn Þ ¼ ∑ π s ∏ Prðyn jC s ; pn ; xn Þ;

ð3aÞ

where Prðyn jC s ; pn ; xn Þ is the conditional probability of the sequence of choices given the choice set is C s DS, where S is the set of all subsets, πs is the probability that Cs is the ‘true’ choice set. Specifically, S is the set of all non-empty subsets of Cs (i.e., all the potential choice subsets, which we describe below in the context of our study). Since a respondent's true consideration set cannot be known with certainty, the model assumes that choice sets are latent and vary across the S classes, while conditional on the consideration set (and hence the class) the choice probability is MNL: expð αpnit þ ðαwÞxnit Þ : ∑ j A C s expð  αpnjt þ ðαwÞxnjt Þ t¼1 Tn

Prðyn jC s ; pn ; xn Þ ¼ ∏

ð3bÞ

Typically in an IAL model, the number of classes, S, is determined as a function of the number of alternatives (e.g., for a universal set with J alternatives, there are 2J  1 possible choice sets). In this paper, however, we are interested in exploring whether respondents restrict their choice set on the basis of the cost attribute. For example, in the situation where there are four cost levels, p ¼ fd0; d1; d2; d3g, four types of behaviour can be identified: Class Class Class Class

1 2 3 4

(C s1 ): (C s2 ): (C s3 ): (C s4 ):

a a a a

subset subset subset subset

whose actual choice set is the same as the set offered in the choice experiment; who restrict their actual choice set to include alternatives that cost less than d3; who restrict their actual choice set to include alternatives that cost less than d2; and who restrict their actual choice set to include alternatives that cost less than d1.4

These four patterns ðS ¼ fC s1 ; C s2 ; C s3 ; C s4 gÞ can be dealt with using an IAL model with four latent classes, where each class describes a unique consideration set. As noted above, the alternatives taken into account by a respondent cannot be known with certainty. However, their observed choice behaviour helps make probabilistic statements about the likelihood of competing consideration sets being their true choice set. These class membership probabilities can be derived using a further constant only MNL model: πs ¼

expðθs Þ ; ∑Ss ¼ 1 expðθs Þ

ð3cÞ

3 We note that accounting for the panel effect is immaterial in the MNL model due to the independence of choice probabilities. We, nevertheless, present the MNL model in this manner to introduce the necessary terms as early on as possible so that differences in models are clearer as we progress through this section. 4 We recognise that further classes could be defined if we also consider lower thresholds (e.g., see Campbell et al., 2012).

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where θs denotes the constant corresponding to the class with consideration set Cs, and where, for identification purposes, one constant (say θS) is set to zero (or alternatively ∑Ss ¼ 1 θs ¼ 0).5 An especially important insight from the cost level(s) of the restricted choice sets is that it gives an indication of the bounds of the aggregate marginal WTP values for each latent class. That is, if some respondents systematically restrict their actual choice set to only include alternatives that are below a given cost threshold, their maximum aggregate marginal WTP can be safely assumed to be less than this upper cost threshold, pus . Similarly, a lower bound, pls , for their minimum aggregate marginal WTP can be identified, which is, clearly, at least as high as the highest cost level within their inferred consideration set. To accommodate this insight we further broaden the IAL model. That is, we specify the IAL model with class-specific marginal WTP values as follows: S

expð  αpnit þðαws Þxnit Þ ; expð  αpnjt þðαws Þxnjt Þ

Tn

Prðyn jpn ; xn Þ ¼ ∑ π s ∏ s¼1

t ¼ 1 ∑j A C s

ð4Þ

where pls r ∑Kk ¼ 1 wnks opus restrictions are imposed to ensure that the aggregated marginal WTP estimates for the K noncost attributes are within the bounds implied by the consideration set. We remark that for the latent class associated with the deterministic choice it is only necessary to impose the restriction ∑Kk ¼ 1 wnks ¼ 1 Z pls ¼ 1 . We also highlight that it is not possible to compute marginal WTP estimates for the class that restricted their choice sets to only zero priced alternatives. We are mindful of the potential confounding between the type of behaviour we are trying to capture and the recovery of preference heterogeneity for the cost attribute, a point to which we return later. 2.3. Accounting for the heterogeneity in marginal WTP While the assumption of homogeneity in the values that respondents are willing to pay may hold in some cases, for a variety of reasons the values are likely to be heterogeneous across respondents. Consequently, we are also interested in capturing the heterogeneity in respondents' marginal WTP within consideration set classes. For this reason, we treat α~ n and ~ n as continuously distributed random terms entering the WTP-space utility function. w ~ n , as well as C sn , were known with certainty for each respondent n, then the probability of If the values of α~ n and w respondent n's sequence of choices would be given by expð αn pnit þ ðαn wsn Þxnit Þ : ∑ j A C sn expð  αn pnjt þðαn wsn Þxnjt Þ t¼1 Tn

Prðyn jC sn ; αn ; wn ; pn ; xn Þ ¼ ∏

ð5Þ

However, it is clearly not possible to know αn, wn or C sn with certainty for each respondent n. For this reason, in estimation, we accommodate heterogeneity across respondents by allowing for random variation. Denote the joint density of ½αn ; wn1s ; wn2s ; …; wnK s  by f ðΘns jΩs Þ, where Θns represents the vector comprised of the random parameters and Ωs denotes the class-specific parameters of these distributions (e.g., the mean and variance). The unconditional choice probability is the ~ n weighted by the class probabilities to also integrate out the integral of the logit formula over all possible values of α~ n and w different C sn : Z Tn S    ~ sn Þxnit Þ expð  α~ n pnit þ ðα~ n w f Θns jΩs Þd Θns : ∏ Prðyn jwn ; pn ; xn ; ΩÞ ¼ ∑ π s ð6Þ ~ sn Þxnjt Þ ~ n pnjt þðα~ n w t ¼ 1 ∑j A C sn expð  α s¼1 In this combined latent class random parameters logit (LC-RPL) specification (e.g., Bujosa et al., 2010; Greene and Hensher, 2013; Campbell et al., 2014, for applications focussing on taste heterogeneity and Hess et al., 2012; Hensher et al., 2012, for applications focusing on processing strategies), parameters of the distributions (i.e., Ωs) and probabilities associated with each consideration set based on cost thresholds are obtained. Again, restrictions can be placed on the marginal WTP distributions to ensure that they are within the bounds characterised by each class-specific consideration set. 2.4. Model estimation The MNL model (Eq. (2)) and our IAL model to facilitate marginal WTP bounds (Eq. (4)) are estimated using maximum likelihood estimation. However, the choice probabilities in the LC-RPL model (Eq. (6)) cannot be calculated exactly because the integrals do not have a closed form. So, these are approximated by simulating the log-likelihood with 350 quasi-random draws via Halton draws. In the case of the models that retrieve class probabilities, we are also mindful of their vulnerability to local maxima of the simulated sample-likelihood function. Thus, in an attempt to reduce the possibility of reaching a local rather than a global maximum, we started the estimation iterations from a variety of random starting points. Specifically, we do this by estimating these models many times, but each time using a different vector of starting values, which are chosen randomly. 5 We note that socio-demographic variables could, of course, be included as covariates to help establish profiles of respondents with different consideration sets. However, the inclusion of such individual characteristics can lead to potential endogeneity bias. While it could be avoided under a latent variable framework, this would further complicate our analysis and draw attention away from our main focus. On this basis we felt that the inclusion of such variables was beyond the aims and scope of the paper.

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Specification of random taste variation models requires the assumption of adequate distribution functions for each of the random parameters in the utility (Hensher and Greene, 2003; Hess et al., 2005). Random parameters can take a number of predefined functional forms. Given the theoretical expectations of disutility for increasing cost, the fact that it is not possible to separately identify the price and scale parameters, as well as the established practice6 in WTP-space models (e.g., Balcombe et al., 2010; Scarpa et al., 2008; Thiene and Scarpa, 2009), we specify α~ as having a log-Normal distribution to ensure strictly negative values for the price/scale coefficient as follows: α ¼  expðμ þsυÞ, where υ is a standard Normal deviate and μ and s are the parameters to be estimated. We note that to facilitate identification, we do not estimate class specific values for μ and s. A key aspect of our approach is that the class-specific marginal WTP distributions, wks , respect the bounds implied by the consideration set. This means that limits for lower (aks ) and upper (bks ) values of the interval of variation are necessary (i.e., ∑aks Z pls and ∑bks o pus ). After evaluating the results from various specifications and distributional assumptions, for the distributions of wks we opt for Uniform distributions. Specifically, the distribution for each wks is given as follows: wks ¼ aks þðbks  aks Þυks , where υ is a draw from a standard Uniform distribution and where bks  aks gives the range of wks . We further note that the class-specific marginal WTP distributions are assumed to be independent. While this may be somewhat restrictive, it enables us to explore the key issue subject to investigation, namely the bounding of marginal WTP distributions to reflect the alternatives actually included by respondents in their consideration sets. Importantly, this also avoids the proliferation of parameters that would be needed to represent class-specific diagonal and off-diagonal terms of the Cholesky matrix (cf. Campbell et al., 2014), making the model more parsimonious. While the Cholesky matrix can be used to approximate covariance in Uniform distributions, the approximation is poor.7 As a result, under our specification the modeller is limited to class-specific univariate distributions. We acknowledge that this is an important limitation. We do note, however, that since our LC-RPL model retrieves class-specific distributions of marginal WTP, correlation in the marginal WTP distributions is an inherent characteristic. This should go some way to alleviating this limitation. All models were coded and estimated in Ox version 6.2 (see Doornik, 2009, for further details).

3. Case-study: demand for assured, safe and traceable food This section starts with a general description of the empirical case-study. Then we discuss the experimental design procedure used to generate the choice sets, which we follow with an outline of the survey implementation and data collection.

3.1. Stated preference choice experiment for value-added food products Increasingly, food safety legislation is focusing on ensuring safety and traceability of food from ‘farm-to-fork’. Connected with this, the use of various labelling policies as a method of ensuring integrity in the food chain and enhanced consumer confidence is becoming more widespread. In addition to the obvious health benefits of ensuring food safety, there are numerous economic benefits that can be achieved. One such economic benefit is the value-added to food products through additional features, such as country of origin labelling and safety assurance labels (cf. Hu et al., 2012; Martínez et al., 2011, for recent stated preference studies looking at food labelling). In this paper, we apply our methodology to the data used in Campbell and Doherty (2013) and Doherty and Campbell (2014). Specifically, the case-study explored the WTP for value-added services to chicken meat, specifically, two uncooked chicken breasts. To identify the relevant food safety attributes and levels associated with this chicken product, the study design was informed by expert opinion from food scientists involved in developing methods to verify the safety and authenticity of food. Although these discussions helped establish the attributes of interest, we gathered information from food stores and undertook a series of focus group discussions with members of the general public and pilot surveying to further ensure that the attributes and levels used to describe the product alternatives in the experiment were understandable and relevant to the general public. Following this, three main food safety attributes were identified: (i) food testing standards; (ii) traceability standards; and (iii) animal health/welfare standards. All three of these value-added services were defined as having two standards: (i) an enhanced standard and (ii) a current standard. For food testing, the enhanced standard represented the use of additional testing to ensure safer food. For traceability, the enhanced standard consisted of the use of technology to verify the exact origins of the meat so that labelling fraud could not occur. For the animal health/welfare attribute, respondents were informed that the enhanced standard tested the animals for the presence of any drugs or diseases, whilst the current standard only tested for the presence of drugs. A region of origin attribute was also included to decipher preferences for chicken produced within the British Isles versus chicken products that came from outside this area. Price was the final attribute included to explore sensitivity to income loss for the purchase. The price attribute, which was reflective of supermarket prices, varied over six levels, ranging between d2.00 and d4.50 in d0.50 increments. 6 7

Although, see Doherty et al. (2013) for application using a mixture of two continuous distributions. We are grateful to an anonymous referee for this insight.

Please cite this article as: Campbell, D., et al., Bounding WTP distributions to reflect the ‘actual’ consideration set. The Journal of Choice Modelling (2014), http://dx.doi.org/10.1016/j.jocm.2014.02.004i

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3.2. Experimental design Having established the attributes and their levels, in an attempt to maximise sampling efficiency and account for the uncertainty with regard to the assumed parameter values, a Bayesian efficient experimental design was generated using the Ngene software, based on the minimisation of the Db-error criterion (for a general overview of efficient experimental design literature, see e.g., Scarpa and Rose, 2008, and references cited therein). Our prior parameter estimates were informed on the basis of initial estimations produced from a MNL model on our pilot study of 400 choice observations (gathered from 50 respondents). We also used the feedback from the expert opinions, focus group discussions as well as evidence from the literature to help define the random priors and to establish the appropriate number of tasks that should be used in the final design. The final experimental design comprised 16 choice sets, which were blocked into two smaller designs, such that each respondent completed a panel of eight choice sets and a complete design was obtained every two respondents. For each task, respondents were asked to choose between two experimentally designed alternatives and a ‘buy none’ option. When making their choices, respondents were asked to consider only the information presented in the choice set and to treat each task separately.

3.3. Survey implementation and data collection The choice data was collected during September 2010 via an on-line survey. In total, we recruited 622 respondents residing in Great Britain resulting in 4976 choice observations used in model estimation. The sample consisted of approximately equal numbers of male (49 per cent) and female (51 per cent) respondents, which is comparable to the national population statistics. In accordance with the regional breakdown, 86, 9 and 5 per cent of respondents resided in England, Scotland and Wales, respectively. Also in line with the breakdown of the adult population of Great Britain, the majority of respondents were younger than 45 years (69 per cent) and worked, either on a full- or part-time basis (63 per cent). Of the 51 per cent of respondents who disclosed their income, the average annual gross income was almost d28,000, which is also comparable with current national statistics.

4. Results We begin this section with the results obtained from the discrete choice models outlined in Section 2. Following this, we compare the marginal WTP distributions obtained under the different model assumptions.

4.1. Estimation results Estimation results are presented in Table 1. As a point of reference our analysis starts with the MNL model, with a parameter for price/scale (denoted by α); marginal WTP parameters for the value-added services8; and a marginal WTP for either chicken product compared to the ‘buy none’ option, whose coefficient can be interpreted as the marginal WTP for the ‘standard’ or ‘baseline’ chicken product. In line with our expectations, we find that the marginal WTP coefficients for the three food safety attributes are all significant and estimated as having positive signs—implying that respondents are willing to pay a price premium for these value-added services. Comparing the magnitudes of these coefficients suggests that respondents place the highest value on chicken that has undergone enhanced food testing to ensure food safety (d1.15) and that the chicken was produced under enhanced animal health/welfare standards (d1.07). The ability to fully trace the chicken is predicted as having a considerably lower price premium (d0.67). Also, in accordance with prior expectations, the marginal WTP coefficient for the locally produced value-added service is found to be positive, and significant—revealing that respondents are more likely to purchase chicken breasts that are produced in the British Isles, compared to chicken produced elsewhere. Nevertheless, we find that the price premium respondents are willing to pay for locally produced chicken breasts (d0.28) is considerably lower than all of the food safety value-added services. As anticipated, respondents dislike the situation of not having any chicken breasts and are willing to pay almost d3.50 to obtain a ‘baseline’ chicken product (i.e., a chicken product with no value-added services and that has not been reared within the British Isles) compared to having none. When aggregated, the marginal WTP estimate we arrive at is d6.63. This, therefore, represents the maximum marginal WTP for the ‘premium’ chicken product (i.e., a chicken product with all value-added services and that has been reared within the British Isles) versus buy none. On the face of it, this would imply that all respondents have a maximum marginal WTP that is above the highest price level of d4.50, and—by model specification—that all alternatives (irrespective of their price level) were considered by all respondents. 8 The value-added service attributes are entered as dummy variables, with the value of one representing the enhanced standard for the three food safety attributes and chicken that has been produced within the British Isles in the case of the region of origin attribute.

Please cite this article as: Campbell, D., et al., Bounding WTP distributions to reflect the ‘actual’ consideration set. The Journal of Choice Modelling (2014), http://dx.doi.org/10.1016/j.jocm.2014.02.004i

MNL

RPLa

LC-IAL

est.

jtrat:j

est.

jtrat:j

LC-RPL-IALa

est.

jtrat:j

est.

jtrat:j

μ s

 0.668

24.70

 0.677

23.90

0.128 0.837

2.15 10.11

 0.030 1.085

0.47 11.24

Food testing

aC s ¼ 1 bC s ¼ 1 aC s ¼ 2 bC s ¼ 2 aC s ¼ 3 bC s ¼ 3 aC s ¼ 4 bC s ¼ 4 aC s ¼ 5 bC s ¼ 5 aC s ¼ 6 bC s ¼ 6

1.146

19.81

1.130

19.04

o 0:001 2.531

o 0.01 19.09

2.634

5.56

o0.001

o 0.01

0.453

o 0.01

o0.001

o 0.01

0.663

0.33

o 0.001 2.899 1.812 1.812 o 0.001 o 0.001 o 0.001 o 0.001 0.073 0.073 1.069 1.071

o 0.01 14.42 1.63 o 0.01 o 0.01 o 0.01 o 0.01 o 0.01 0.05 o 0.01 0.94 o 0.01

aC s ¼ 1 bC s ¼ 1 aC s ¼ 2 bC s ¼ 2 aC s ¼ 3 bC s ¼ 3 aC s ¼ 4 bC s ¼ 4 aC s ¼ 5 bC s ¼ 5 aC s ¼ 6 bC s ¼ 6

0.666

0.654

11.42

0.075 0.938

0.25 1.31

1.222

3.29

o0.001

o 0.01

0.709

0.01

0.081

0.07

0.292

0.10

0.571 0.571 2.188 2.188 0.025 0.025 0.003 0.003 o 0.001 0.004 o 0.001 o 0.001

3.92 o 0.01 4.56 o 0.01 0.02 o 0.01 0.01 o 0.01 o 0.01 0.01 o 0.01 o 0.01

aC s ¼ 1 bC s ¼ 1 aC s ¼ 2 bC s ¼ 2 aC s ¼ 3 bC s ¼ 3 aC s ¼ 4 bC s ¼ 4 aC s ¼ 5 bC s ¼ 5 aC s ¼ 6 bC s ¼ 6

1.069

1.078

17.38

o0.001 2.107

o 0.01 13.98

o0.001

o 0.01

3.408

4.31

0.503

0.03

o0.001

o 0.01

1.105

0.38

o 0.001 2.407 o 0.001 0.500 1.986 1.986 o 0.001 o 0.001 o 0.001 o 0.001 1.025 1.025

o 0.01 10.55 o 0.01 0.94 1.37 o 0.01 o 0.01 o 0.01 o 0.01 o 0.01 0.47 o 0.01

aC s ¼ 1 bC s ¼ 1 aC s ¼ 2 bC s ¼ 2

0.282

0.256

3.56

o0.001 0.374

o 0.01 3.66

0.181

0.44

0.001 0.513 o 0.001 o 0.001

0.01 1.23 o 0.01 o 0.01

Traceability

Animal health/welfare

Region of origin

11.98

17.74

4.06

7

Price/scale

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Please cite this article as: Campbell, D., et al., Bounding WTP distributions to reflect the ‘actual’ consideration set. The Journal of Choice Modelling (2014), http://dx.doi.org/10.1016/j.jocm.2014.02.004i

Table 1 Estimation results.

8

MNL jtrat:j

est. aC s ¼ 3 bC s ¼ 3 aC s ¼ 4 bC s ¼ 4 aC s ¼ 5 bC s ¼ 5 aC s ¼ 6 bC s ¼ 6 Baseline chicken

Class probabilitiesb

LL K ρ2 AIC BIC a b

aC s ¼ 1 bC s ¼ 1 aC s ¼ 2 bC s ¼ 2 aC s ¼ 3 bC s ¼ 3 aC s ¼ 4 bC s ¼ 4 aC s ¼ 5 bC s ¼ 5 aC s ¼ 6 bC s ¼ 6

3.467

πCs ¼ 1 πCs ¼ 2 πCs ¼ 3 πCs ¼ 4 πCs ¼ 5 πCs ¼ 6 πCs ¼ 7

1.000

RPLa

LC-IAL

30.68

Fixed

 4168.532 6 0.236 8349.230 8388.043

jtrat:j

est. 0.554

0.68

0.791

0.01

o0.001

o 0.01

o0.001

o 0.01

4.865

26.90

o0.001

o 0.01

o0.001

o 0.01

0.781

0.01

2.890

1.23

o0.001

o 0.01

0.888 0.053 0.019 o0.001 0.009 0.009 0.020

22.86 4.48 2.66 o 0.01 1.82 1.23 2.63  3762.318 37 0.305 7598.850 7839.688

LC-RPL-IALa jtrat:j

est.

o0.001 12.067

o 0.01 14.57

1.000

Fixed

 3571.084 12 0.345 7166.168 7244.317

To facilitate the interpretation of heterogeneity, the t-ratios for bC s test H0 : bC s ¼ aC s . To ease interpretation, rather than reporting θs, we report πs. We also used the Delta method to obtain the t-ratio for the baseline probability.

est.

jtrat:j

0.679 0.679 o 0.001 o 0.001 o 0.001 o 0.001 o 0.001 o 0.001

0.90 o 0.01 o 0.01 o 0.01 o 0.01 o 0.01 o 0.01 o 0.01

4.950 4.950 o 0.001 o 0.001 1.311 1.311 3.497 3.497 2.442 2.442 o 0.001 o 0.001

9.27 o 0.01 o 0.01 o 0.01 1.91 o 0.01 5.00 o 0.01 1.74 o 0.01 o 0.01 o 0.01

0.835 0.033 0.014 0.082 0.007 0.012 0.018

18.27 2.01 2.83 3.02 1.50 2.82 3.38  3534.428 68 0.341 7204.856 7647.697

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Please cite this article as: Campbell, D., et al., Bounding WTP distributions to reflect the ‘actual’ consideration set. The Journal of Choice Modelling (2014), http://dx.doi.org/10.1016/j.jocm.2014.02.004i

Table 1 (continued )

D. Campbell et al. / Journal of Choice Modelling ] (]]]]) ]]]–]]]

9

We now turn to the LC-IAL model, which accounted for cost thresholds and bounded marginal WTP estimates. The six different price levels (i.e., values between d2.00 and d4.50 in d0.50 increments) used to describe chicken products and the zero-priced ‘buy none’ alternative leads to a seven-class model: Class 1: actual choice set included alternatives priced up to and including d4.50 (i.e., is the same as the set all respondents were presented with in the choice experiment); Class 2: consideration set only included alternatives priced up to and including d4.00; Class 3: consideration set only included alternatives priced up to and including d3.50; Class 4: consideration set only included alternatives priced up to and including d3.00; Class 5: consideration set only included alternatives priced up to and including d2.50; Class 6: consideration set only included alternatives priced up to and including d2.00; and Class 7: consideration set is confined to the zero-priced ‘buy none’ alternative.

Although homogeneity within each WTP bound was not permitted, we acknowledge that the confounding between heterogeneity and processing makes it difficult to identify whether the latent classes reflect threshold behaviour in consideration set formation or preference heterogeneity. We, nevertheless, feel this is logically consistent, since it would seem illogical not to bound marginal WTP variation where it is believed that respondents only considered alternatives that lie within their cost threshold. This issue aside, the model results do suggest that the majority of respondents (89 per cent) considered all proposed alternatives and have a total marginal WTP at least as high as the highest price level of d4.50.9 Given the large class size it is not surprising to find that marginal WTP estimates associated with this class are not dissimilar to those obtained from the MNL model. The only exception to this is the marginal WTP obtained for a baseline chicken product which is d1.40 higher than the estimate with the conventional MNL model. Aggregating the marginal WTP estimates for this class, who did not eliminate any alternatives from the proposed set, gives d7.98. Of the remaining respondents, almost half (a 5 per cent share) are suggested to have considered all alternatives apart from those with highest price d4.50. Interestingly, it would appear that this class would only consider buying chicken products that have undergone enhanced food safety testing, are traceable, and that have been reared within the British Isles. The aggregated marginal WTP estimate for this class is calculated to be d4.04, which is between the highest price level (d4.50) and the second highest price level (d4.00). For the remaining latent classes, the very small class sizes do not facilitate any meaningful interpretation. Notwithstanding this, we do draw attention to the fact that only 2 per cent of the respondents are predicted as having only considered the buy none alternative. This implies that there are very few serial non-participators in this sample (i.e., respondents who always select the buy none alternative, cf. von Haefen et al., 2005; Burton and Rigby, 2009) and, importantly, it suggests that 98 per cent of respondents are willing to pay at least the lowest price level (d2.00) for the chicken product. Comparing model fit reveals that the LC-IAL model is associated with an improvement of over 400 log-likelihood units compared to the MNL model. Although we do acknowledge that this improvement is in part due to the fact that the panel nature of the data is being accounted for in the LC-IAL model. Notwithstanding this, we note that the information criteria indicate that this improvement is supported, even after accounting for the large number of additional parameters. Furthermore, we do recognise that the values of these model fit diagnostics would have been even more convincing if we were to merge several of the smaller classes, and remove the many variables that were not statistically significant. We, nevertheless, decided to retain all possible consideration sets and to leave in insignificant variables, because we feel that it leads to a richer insight into the choice sets that are actually considered by respondents and their marginal WTP. Besides, we are interested in segmenting respondents according to their decision-making process and marginal WTP bounds, which means that the usual information criteria used in latent class analysis is less relevant. While the results of the LC-IAL model indirectly suggested heterogeneity of marginal WTP among the sample of respondents, the RPL model is intended to explicitly uncover this heterogeneity. In all cases, the lower bound of the Uniform marginal WTP distributions is essentially zero. The fact that the upper bounds of these distributions (with the sole exception of traceability) are all found to be significantly higher than the lower bound confirms the presence of heterogeneous marginal WTP. We remark the relatively high upper bound of d12.07 for the marginal WTP for a baseline chicken product. This means that the upper bound of the aggregate marginal WTP (which is bounded between d0.07 and d18.02) is also quite high, as are the measures of central tendency (d9.05). Despite the increase in fit to the data is almost 200 log-likelihood units, the high marginal WTP values implied by this model cast doubt on its reliability. Our final model, the LC-RPL-IAL, attempts to account for cost thresholds and accommodates random variation in marginal WTP within the bounds implied by the respondent's restricted choice set. We again find strong evidence that the majority (83 per cent) of respondents considered all alternatives when making their decision. Within this class, we highlight the identification of heterogeneous marginal WTP only for the food testing and animal health/welfare value added services. The maximum marginal WTP for the ‘premium’ chicken product for this class is bounded between d5.54 and d11.96, which, 9 While this class dominance suggests that most respondents found the upper price level acceptable, we cannot rule out that it may be an artefact of ‘yeah-saying’ behaviour or some sort of strategic behaviour. Given this finding and with prefect hindsight it would have been helpful to have used a larger price range, so that further comparisons could be made. The latter option though might have detracted from the realism of the choice context.

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10

Table 2 Conditional marginal WTP distributions per pack of two chicken breasts (d). 5%ile

10%ile

25%ile

Median

Mean

75%ile

90%ile

95%ile

Food testing

LC-IAL RPL LC-RPL-IAL

0.65 0.48 0.25

1.13 0.58 0.40

1.13 0.78 0.85

1.13 1.09 1.35

1.15 1.18 1.35

1.13 1.58 1.94

1.18 1.92 2.30

1.65 2.08 2.48

Traceability

LC-IAL RPL LC-RPL-IAL

0.30 0.40 0.06

0.65 0.42 0.33

0.65 0.46 0.63

0.65 0.50 0.66

0.65 0.51 0.58

0.65 0.55 0.66

0.67 0.61 0.66

0.85 0.61 0.66

Animal health/welfare

LC-IAL RPL LC-RPL-IAL

0.24 0.48 0.33

0.89 0.52 0.39

1.07 0.71 0.80

1.08 0.97 1.15

1.03 1.00 1.13

1.08 1.26 1.58

1.08 1.51 1.79

1.09 1.63 2.03

Region of origin

LC-IAL RPL LC-RPL-IAL

0.18 0.17 0.14

0.24 0.17 0.17

0.26 0.18 0.30

0.26 0.19 0.32

0.25 0.19 0.32

0.26 0.20 0.35

0.26 0.20 0.38

0.26 0.21 0.40

Baseline chicken

LC-IAL RPL LC-RPL-IAL

0.04 0.75 0.50

2.36 1.78 3.43

4.82 4.25 4.72

4.86 7.24 4.88

4.35 6.04 4.39

4.87 7.64 4.88

4.87 7.87 4.88

4.87 7.96 4.88

Optimum chicken

LC-IAL RPL LC-RPL-IAL

4.00 3.56 4.03

5.67 4.61 4.99

7.95 7.22 7.58

7.98 10.12 8.46

7.43 8.92 7.78

7.98 10.55 9.04

7.98 11.02 9.17

7.98 11.23 9.71

compared to that implied under the RPL model, is considerably much more in-line with prior expectations. We also remark that the measures of central tendency associated with this class (d8.75) are also much lower compared to that predicted under the RPL model. Again, while the small class sizes do not permit a rigorous investigation of the parameters retrieved for the remaining classes, we draw attention to the fact that in these classes there is not strong evidence of class-specific heterogeneous marginal WTP. We, once more, recognise that this may be an artefact of confounding between heterogeneity and processing. Nevertheless, the fact that the model fit obtained under the LC-RPL-IAL model is over 200 log-likelihood units higher compared to that associated with the LC-IAL model justifies the move to accommodating within class heterogeneity. Furthermore, this improvement in model fit is confirmed by the information criteria. The LC-RPL-IAL model also attains a superior fit compared to the RPL model. However, we acknowledge that this gain comes at a very high parametric cost and this is reflected in the increases in the information criteria.10 On the other hand, we cannot ignore the increase in almost 40 log-likelihood units nor the unexpected high marginal WTP estimates implied under the RPL model. While this means we are faced with a trade-off between model parsimony and plausibility of marginal WTP estimates, the fact that the LC-RPL-IAL also gives an insight into different segments of consumers leads us to favour it over the RPL model. 4.2. Comparison of WTP For a more detailed examination and sensitivity of the estimated marginal WTP distributions, we compute the mean of the conditional marginal WTP distributions for each respondent, based on their pattern of observed choices for the LC-IAL, RPL and LC-RPL-IAL models. While disregarding the fact that these conditional parameters themselves follow a random distribution around this mean, this approach nevertheless gives us some information about the most likely position of a respondent on the distribution. In Table 2 we report summary statistics of the means of these conditional distributions. The central tendency statistics for the value-added and region of origin attributes are quite similar for the three models and are in line with those obtained using the MNL model. This aside, the most striking differences are the associated marginal WTP distributions for the baseline chicken product. Both the LC-IAL and LC-RPL-IAL models identify the realistic situation of a share of consumers who would not be willing to pay a high value for the baseline chicken product. The central tendency statistics of marginal WTP attained under the RPL model for the baseline chicken product are found to be of a higher magnitude, which confirms the repercussions of not accounting for diversity of consideration sets based on WTP thresholds. Related to this finding, we remark that for the baseline chicken the upper tail of the marginal WTP distribution becomes less extreme as we move from the RPL model to the LC-RPL-IAL model. A similar pattern emerges for the aggregated marginal WTP for a ‘premium’ chicken product (i.e., a chicken product with all three value-added services and has been reared within the British Isles). Methodological issues aside, we draw attention to the fact that the attributes only contribute a fraction of the value respondents assign to the ‘premium’ chicken product. A substantial proportion of the total value respondents place on this product is reflected by their marginal WTP for the baseline chicken product. Nonetheless, the value-added and region of origin attributes do contribute approximately 40 per cent of the maximum value respondents are willing to pay the chicken 10 We again note that the differences in the information criteria could be reduced by merging several of the smaller classes and the removal of the many insignificant parameters.

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11

product. But a large share of this value is associated with food testing and animal health or animal welfare value-added services. 5. Conclusions In this paper, we explore the behavioural proposition that respondents may exhibit specific patterns in their use of thresholds and cut-offs to define their consideration set. Put simply, they would disregard those alternatives that fall outside their maximum WTP. To empirically address this issue we expand the independent availability logit model (similar in form to a latent class model, but where we define each class to describe a specific consideration set and endogenous WTP bound) to account for the complete range of cost thresholds that may be held by respondents. Additionally, we account for within class heterogeneity and use a WTP-space representation. A further feature is that the class-specific WTP distributions are truncated to be within the bounds deduced from the cost level(s) within each class-specific consideration set. Results, based on a stated preference choice experiment exploring attributes of chicken products, provide corroboration that a relatively small share of respondents do not attend to all alternatives, and, in particular, restrict their ‘actual’ consideration set to only those alternatives offered at a cost that does not exceed their maximum WTP. Although, this is found to apply to only approximately 15 per cent of respondents, the repercussions on the estimated marginal WTP distributions are substantive—the marginal WTP distribution for a ‘premium’ chicken product is, on average, up to 20 per cent lower when actual consideration sets and WTP thresholds are accounted for. While specific to this dataset, this result highlights the problem of ignoring the fact that even a small proportion of respondents eliminate alternatives that exceed their maximum WTP. Our findings provide, what we feel is preliminary, but quite compelling, evidence for further research in this area. While we recognise the potential confounding issues inherent in the approach, we feel that it does shed additional light on the manner in which respondents reach their final decision and provide a practical empirical solution for it. This represents an important challenge for those who are keen to improve the realism of stated preference choice data analysis in future studies.

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Please cite this article as: Campbell, D., et al., Bounding WTP distributions to reflect the ‘actual’ consideration set. The Journal of Choice Modelling (2014), http://dx.doi.org/10.1016/j.jocm.2014.02.004i