Confidence intervals of willingness-to-pay for random coefficient logit models

Random coefficient logit (RCL) models containing random parameters are increasingly used for modelling travel choices. Willingness-to-pay (WTP) measures, such as the value of travel time savings (VTTS) are, in the case of RCL models estimated in preference space, ratios of random parameters. In this paper we apply the Delta method to compute the confidence intervals of such WTP measures, taking into account the variance–covariance matrix of the estimates of the distributional parameters. The same Delta method can be applied when the model is estimated in WTP space. Compared to simulation methods such as proposed by Krinsky and Robb, the Delta method is able to avoid most of the simulations by deriving partly analytical expressions for the standard errors. Examples of such computations are shown for different combinations of random distributions.     

Using nonparametrics to specify a model to measure the value of travel time

Using a range of nonparametric methods, the paper examines the specification of a model to evaluate the willingness-to-pay (WTP) for travel time changes from binomial choice data from a simple time–cost trading experiment. The analysis favours a model with random WTP as the only source of randomness over a model with fixed WTP which is linear in time and cost and has an additive random error term. Results further indicate that the distribution of log WTP can be described as a sum of a linear index fixing the location of the log WTP distribution and an independent random variable representing unobserved heterogeneity. This formulation is useful for parametric modelling. The index indicates that the WTP varies systematically with income and other individual characteristics. The WTP varies also with the time difference presented in the experiment which is in contradiction of standard utility theory.     

Willingness to pay for public transportation options for improving the quality of life of the rural elderly

Transportation for the rural elderly is an increasing concern as baby boomers age and young people continue to exit rural communities. When the elderly are no longer able to drive, they rely on alternative forms of transportation, including public transportation systems. Currently, such systems are usually not good substitutes for driving a private car, especially in rural areas. Because expanded rural transportation systems would likely be funded by taxpayers, an understanding of their preferences and willingness to pay (WTP) for non-medical transportation options is essential. To help understand WTP and preferences, a choice experiment survey was administered to taxpayers in three counties (Atascosa, Polk, and Parker) in Texas. Results indicate taxpayers’ value transportation services for the elderly and are willing to support them. They value more flexible options over base levels of the attributes presented, but they may not always prefer the most flexible options. Respondents’ WTP for the same transportation attribute was similar across counties, but differences in socio-demographic coefficients suggest that transportation systems may need to be customized to meet local needs. Furthermore, county residents’ WTP may not cover the cost of desired improvements to the transportation systems.     

Estimating the willingness to pay and value of risk reduction for car occupants in the road environment

In recent years there has been a re-focus on the valuation of a statistical life from the ex post or human capital method to an ex ante willingness to pay (WTP) approach. This is in part a recognition that we may have been undervaluing the cost of fatalities and injuries to society associated with crashes, but also a strong belief in the need to focus on establishing the amount, ex ante, that individuals are willing to pay to reduce the risk of exposure to circumstances that might lead to death or degree of injury on the road network. This study has developed a framework in which to identify the degree of preference heterogeneity in willingness to pay by individuals who are drivers or passengers in cars to avoid being killed or injured. A stated choice experiment approach is developed. The empirical setting is a choice of route for a particular trip that a sample of individuals periodically undertakes in Australia. The particular trip is described in enough detail to provide the respondent with a familiar market environment, providing all the relevant background information required for making a decision. Mixed logit models are estimated to obtain the marginal (dis)utilities associated with each influence on the choice amongst the attribute packages offered in the stated choice scenarios. These estimates are used to obtain the WTP distributions for fatality and injury avoidance, which are then aggregated to obtain estimates of the value of risk reduction (VRR), of which the fatality class is also known as the value of a statistical life (VSL).     

Consumers’ willingness to pay for alternative fuel vehicles: A comparative discrete choice analysis between the US and Japan

This paper conducts a comparative discrete choice analysis to estimate consumers’ willingness to pay (WTP) for electric vehicles (EVs) and plug-in hybrid electric vehicles (PHEVs) on the basis of the same stated preference survey carried out in the US and Japan in 2012. We also carry out a comparative analysis across four US states. We find that on average US consumers are more sensitive to fuel cost reductions and alternative fuel station availability than are Japanese consumers. With regard to the comparative analysis across the four US states, consumers’ WTP for a fuel cost reduction in California is considerably greater than in the other three states. We use the estimates obtained in the discrete choice analysis to examine the EV/PHEV market shares under several scenarios. In a base case scenario with relatively realistic attribute levels, conventional gasoline vehicles still dominate both in the US and Japan. However, in an innovation scenario with a significant purchase price reduction, we observe a high penetration of alternative fuel vehicles both in the US and Japan. We illustrate the potential use of a discrete choice analysis for forward-looking policy analysis, with the future opportunity to compare its predictions against actual revealed choices. In this case, increased purchase price subsidies are likely to have a significant impact on the market shares of alternative fuel vehicles.     

Recovery of inter- and intra-personal heterogeneity using mixed logit models

Most applications of discrete choice models in transportation now utilise a random coefficient specification, such as mixed logit, to represent taste heterogeneity. However, little is known about the ability of these models to capture the heterogeneity in finite samples (as opposed to asymptotically). Also, due to the computational intensity of the standard estimation procedures, several alternative, less demanding methods have been proposed, and yet the relative accuracy of these methods has not been investigated. This is especially true in the context of work looking at joint inter-respondent and intra-respondent variation. This paper presents an overview of the various different estimators, gives insights into some of the theoretical properties, and analyses their performance in a large scale study on simulated data. In particular, we specify 31 different forms of heterogeneity, with multiple versions of each dataset, and with results from over 16,000 mixed logit estimation runs. The findings suggest that variation in tastes over consumers is captured by all the methods, including the simpler versions, at least when sample size is sufficiently large. When tastes vary over choice situations for each consumer, as well as over consumers, the ability of the methods to capture and differentiate the two sources of heterogeneity becomes more tenuous. Only the most computationally intensive approach is able to capture adequately the two sources of variation, but at the cost of very high run times. Our results highlight the difficulty of retrieving taste heterogeneity with only cross-sectional data, providing further evidence of the benefits of repeated choice data. Our findings also suggest that the data requirements of random coefficients models may be more substantial than is commonly assumed, further reinforcing concerns about small sample issues.     

Valuing reductions in environmental pollution in a residential location context

A stated preference ranking experiment is designed to estimate the willingness-to-pay (WTP) for reducing the amount of atmospheric pollution in a group-based residential location context. Important issues are the proper definition of the context and the variable metric for the environmental attribute. Sample members were asked to rank 10 options arising from variations in the attributes travel time to work and to study, rent of the house and an environmental attribute (the number of days of Alert, in terms of concentration of PM₁₀, at a dwelling’s location). Multinomial logit models based on a consistent microeconomic framework were estimated for various stratifications of the data (income, pollution sensitivity, and type of dwelling currently inhabited). From these subjective values of time and WTP were derived for reductions in the number of days of alert and hence the amount of pollutant concentration at a given location. The WTP came out at about 1% of the family income for reducing one contingence day per year; this is approximately 60% higher than an estimate reported for the city of Edmonton, Canada, but the average PM₁₀ concentration in Santiago is about six times higher.     

Embedding risk attitude and decision weights in non-linear logit to accommodate time variability in the value of expected travel time savings

In recent years we have seen important extensions of logit models in behavioural research such as incorporation of preference and scale heterogeneity, attribute processing heuristics, and estimation of willingness to pay (WTP) in WTP space. With rare exception, however, a non-linear treatment of the parameter set to allow for behavioural reality, such as embedded risk attitude and perceptual conditioning of occurrence probabilities attached to specific attributes, is absent. This is especially relevant to the recent focus in travel behaviour research on identifying the willingness to pay for reduced travel time variability, which is the source of estimates of the value of trip reliability that has been shown to take on an increasingly important role in project appraisal. This paper incorporates, in a generalised non-linear (in parameters) logit model, alternative functional forms for perceptual conditioning (known as probability weighting) and risk attitude in the utility function to account for travel time variability, and then derives an empirical estimate of the willingness to pay for trip time variability-embedded travel time savings as an alternative to separate estimates of time savings and trip time reliability. We illustrate the richness of the approach using a stated choice data set for commuter choice between unlabelled attribute packages. Statistically significant risk attitude parameters and parameters underlying decision weights are estimated for multinomial logit and mixed multinomial logit models, along with values of expected travel time savings.     

Commuters’ willingness-to-pay for improvement of transfer facilities in and around metro stations – A case study in Kolkata

Although several cities in India are developing the metro system, there are lacunas associated with transfer facilities in and around metro stations. The present work aims to investigate the perception of commuters of Kolkata city, India in terms of their willingness-to-pay (WTP) for improvement of transfer facilities. A stated preference survey instrument was designed to collect choice responses from metro commuters and the database was analysed by developing random parameter logit (RPL) models. The decomposition effects of various socioeconomic and trip characteristics on mean estimates were also investigated in random parameter logit models with heterogeneity. The work indicates significantly high WTP of metro commuters as compared to the average metro fare for improvement of various qualitative attributes of transfer facility such as ‘facility for level change’, ‘visual communication’, ‘pedestrian crossing’, and ‘pedestrian environment’. The WTP values are also found to vary across different groups of commuter formed on the basis of ‘trip purpose’, ‘monthly household income’, ‘station type’ and ‘metro fare’. ‘Work trip’ commuters are found to have higher WTP for improvement of access time, pedestrian environment and use of an escalator over the elevator. On the other hand, ‘high-income group’ commuters have shown higher WTP for improvement of access time, pedestrian crossing, and pedestrian environment. While ‘high fare group’ commuters have higher WTP for access time and pedestrian environment, heterogeneity is also observed in WTP for facility for level change, pedestrian crossing, and pedestrian environment across commuters using different ‘station type’ (underground, at-grade, and elevated). The findings from the study provide a basis for formulating policies for the improvement of transfer facilities in and around metro stations giving due attention to the preference of commuters having different socioeconomic and trip characteristics.     

Parameter transfer of common-metric attributes in choice analysis: implications for willingness to pay

There is a growing literature that promotes the presence of a mix of compensatory and semi-compensatory processing strategies in the way that individuals evaluate packages of attributes in real or hypothetical markets, and make choices. This paper proposes a specification for the utility form in a choice model to test if, given a pair of attributes with a common-metric (e.g., components of travel time or cost), the attribute with the dominating level defines the marginal (dis)utility that is assigned to both attributes. We refer to this processing strategy as a parameter transfer rule. We use a stated choice data set, in the context of car driving individuals choosing between tolled and non-tolled routes, to estimate a mixed logit model which incorporates the presence of the parameter transfer rule and the conventional fully compensatory rule, both existing up to a probability. We find that if this parameter transfer heuristic is part of the mix, the WTP is more than 30% higher, on average, than when only a fully compensatory rule is imposed. We also contrast the parameter transfer rule with other semi-compensatory heuristics which have been investigated in other papers, and show that the finding adds further support to the accumulating evidence that a semi-compensatory attribute processing rules tend to result in higher mean WTP estimates compared to the fully compensatory attribute processing rule.     

Random vehicle dispatching with options and optimal fleet size

A dispatching problem with random availability of vehicles and options to send rented vehicles is considered. We assume passenger arrivals to be described by a pure-birth process. Such a problem is analytically attractive and is shown to have practical applications in vehicle dispatching models. An average cost criterion is used to determine firm's fleet size and option (renting) strategy.     

The implications on willingness to pay of respondents ignoring specific attributes

Individuals processing the information in a stated choice experiment are typically assumed to evaluate each and every attribute offered within and between alternatives, and to choose their most preferred alternative. However, it has always been thought that some attributes are ignored in this process for many reasons, including a coping strategy to handle ones perception of the complexity of the choice task. Nonetheless, analysts typically proceed to estimate discrete choice models as if all attributes have influenced the outcome to some degree. The cognitive processes used to evaluate trade-offs are complex with boundaries often placed on the task to assist the respondent. These boundaries can include prioritising attributes and ignoring specific attributes. In this paper we investigate the implications of bounding the information processing task by attribute elimination through ignoring one or more attributes. Using a sample of car commuters in Sydney we estimate mixed logit models that assume all attributes are candidate contributors, and models that assume certain attributes are ignored, the latter based on supplementary information provided by respondents. We compare the value of travel time savings under the alternative attribute processing regimes. Assuming that all attributes are not ignored and duly processed, leads to estimates of parameters which produce significantly different willingness to pay (WTP) to that obtained when the exclusion rule is invoked.     

Valuing of attributes influencing the attractiveness of suburban train service in Mumbai city: A stated preference approach

The paper presents valuing of qualitative and quantitative travel attributes influencing the attractiveness of suburban train service in Mumbai city, India. A stated preference experiment is designed to capture the data of sub-urban train mode choice behavior. The behavioral data are analyzed using different modeling techniques such as multinomial logit (MNL) and mixed logit (ML). In ML model, the random parameters are assumed to follow constrained triangular distribution, where mean equals its spread. The decomposition of preference heterogeneity around the mean estimate of random parameter is also investigated using ML model. The study shows the influence of headway time and train ride time associated with a particular crowding level (expressed in density of standing passengers/m²) in choosing the sub-urban train mode by calculating their willingness-to-pay (WTP) values and highlights the importance of WTP for addressing policy issues in the reduction of in-vehicle crowding level. The present study documents new findings of the effect of crowding level on train ride time in the context of a developing country and suggests some important directions for future suburban train transport crowding valuation research.     

The Signs of the Times: Imposing a Globally Signed Condition on Willingness to Pay Distributions

A feature of recent developments in choice models that enable estimation of the distribution of willingness to pay (WTP) is that the sign of the distribution can change over the range. Behaviourally this often makes little sense for attributes such as travel time on non-discretionary travel, despite a growing recognition of positive utility over some travel time ranges. This can in part be attributed to the analytical distribution that is selected (except the cumbersome lognormal), many of which are unconstrained over the full range. Although a number of analysts have imposed constraints on various distributions for random parameters that can satisfy the single-sign condition, these restrictions are, with rare exception, only satisfied for the mean and the standard deviation estimates of a random parameter. When heterogeneity around the mean and/or heteroscedasticity around the standard deviation is allowed for, however, the constraint condition is often not satisfied. Given the popularity of distributions other than the lognormal, in order to satisfy the sign condition under the most general form of parameterisation, we need to impose a global sign condition. In this paper we show how this might be achieved in the context of the valuation of travel time savings for car commuters choosing amongst an offered set of route-specific travel times and costs. We illustrate the impact of the constraint under a globally constrained Rayleigh distribution for total travel time parameterisation, contrasting the evidence with a multinomial logit model and a range of other distributional assumptions within the mixed logit framework. Discussions with Bill Greene, John Rose, Ken Train and especially Juan de Dios Ortuzar have been invaluable as have the comments of referees.     

Making use of respondent reported processing information to understand attribute importance: a latent variable scaling approach

In recent years we have seen an explosion of research seeking to understand the role that rules and heuristics might play in improving the predictive capability of discrete choice models, as well as delivering willingness to pay estimates for specific attributes that may (and often do) differ significantly from estimates based on a model specification that assumes all attributes are relevant. This paper adds to that literature in one important way—it explicitly recognises the endogeneity issues raised by typical attribute non-attendance treatments and conditions attribute parameters on underlying unobserved attribute importance ratings. We develop a hybrid model system involving attribute processing and outcome choice models in which latent variables are introduced as explanatory variables in both parts of the model, explaining the answers to attribute processing questions and explaining heterogeneity in marginal sensitivities in the choice model. The resulting empirical model explains how lower latent attribute importance leads to a higher probability of indicating that an attribute was ignored or that it was ranked as less important, as well as increasing the probability of a reduced value for the associated marginal utility coefficient in the choice model. The model does so by treating the answers to information processing questions as dependent rather than explanatory variables, hence avoiding potential risk of endogeneity bias and measurement error.     

Multi-objective bilevel construction material transportation scheduling in large-scale construction projects under a fuzzy random environment

Abstract

This paper investigates a transportation scheduling problem in large-scale construction projects under a fuzzy random environment. The problem is formulated as a fuzzy, random multi-objective bilevel optimization model where the construction company decides the transportation quantities from every source to every destination according to the criterion of minimizing total transportation cost and transportation time on the upper level, while the transportation agencies choose their transportation routes such that the total travel cost is minimized on the lower level. Specifically, we model both travel time and travel cost as triangular fuzzy random variables. Then the multi-objective bilevel adaptive particle swarm optimization algorithm is proposed to solve the model. Finally, a case study of transportation scheduling for the Shuibuya Hydropower Project in China is used as a real world example to demonstrate the practicality and efficiency of the optimization model and algorithm.     

A class of RUM choice models that includes the model in which the utility has logistic distributed errors

A class of random utility maximization (RUM) models is introduced. For these RUM models the utility errors are the sum of two independent random variables, where one of them follows a Gumbel distribution. For this class of RUM models an integral representation of the choice probability generating function has been derived which is substantially different from the usual integral representation arising from the RUM theory. Four types of models belonging to the class are presented. Thanks to the new integral representation, a closed-form expression for the choice probability generating function for these four models may be easily obtained. The resulting choice probabilities are fairly manageable and this fact makes the proposed models an interesting alternative to the logit model. The proposed models have been applied to two samples of interurban trips in Japan and some of them yield a better fit than the logit model. Finally, the concavity of the log-likelihood of the proposed models with respect to the utility coefficients is also analyzed.     

The role of socio-economic,built environment and psychological factors in parental mode choice for their children in an Iranian setting

This research investigated the role of parental psychological and socio-economic factors as well as built environment for the choice of their children’s (primary school pupils, aged 7–9 years) travel mode to school in Rasht, Iran. A total of 1078 questionnaires were distributed (return rate of 80 percent) among pupils in nine primary schools in January 2014. A mixed logit (ML) model was employed due to its ability to test heterogeneity among parents and also to determine its possible sources. Results of random coefficient ML modelling showed that several psychological, socio-economic and built environment characteristics were significant factors in parental mode choice. Only walking time perception to school had a significant random normal distribution coefficient and no other psychological and socio-economic variable had a random effect. Further investigation by random coefficient analysis showed that the possible source of household preference heterogeneity could be to own two or more cars. Regarding psychological variables, strong parental worry about their children walking alone to school had a negative impact on allowing them to walk to school. Parents who evaluated poor contextual and design preconditions for walking tended to choose school service more than private car and walking. Parents with stronger environmental personal norms were more willing to allow their children to walk. The findings suggest that infrastructural measures, such as sidewalk facilities, neighborhood security and safety, encourage parents to allow children to walk to school. Information campaigns targeting environmental norms may increase walking among pupils in an Iranian setting.     

Can scale and coefficient heterogeneity be separated in random coefficients models?

There is growing interest in the notion that a significant component of the heterogeneity retrieved in random coefficients models may actually relate to variations in absolute sensitivities, a phenomenon referred to as scale heterogeneity. As a result, a number of authors have tried to explicitly model such scale heterogeneity, which is shared across coefficients, and separate it from heterogeneity in individual coefficients. This direction of work has in part motivated the development of specialised modelling tools such as the G-MNL model. While not disagreeing with the notion that scale heterogeneity across respondents exists, this paper argues that attempts in the literature to disentangle scale heterogeneity from heterogeneity in individual coefficients in discrete choice models are misguided. In particular, we show how the various model specifications can in fact simply be seen as different parameterisations, and that any gains in fit obtained in random scale models are the result of using more flexible distributions, rather than an ability to capture scale heterogeneity. We illustrate our arguments through an empirical example and show how the conclusions from past work are based on misinterpretations of model results.