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A flexible multiple discrete–continuous probit (MDCP) model: application to analysis of expenditure patterns of domestic tourists in India

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Abstract

Traditional multiple discrete–continuous (MDC) choice models impose tight linkages between consumers’ discrete choice and the continuous consumption decisions due to the use of a single utility parameter driving both the decision to choose and the extent of choice. Recently, Bhat (Trans Res Part B Methodol 110:261–279, 2018) proposed a flexible MDCEV model that employs a utility function with separate parameters to determine the discrete choice and continuous consumption values. However, the flexible MDCEV model assumes an independent and identically distributed (IID) error structure across the discrete and continuous baseline utilities. In this paper, we formulate a flexible non-IID multiple discrete–continuous probit (MDCP) model that employs a multivariate normal stochastic distribution to allow for a more general variance–covariance structure. In doing so, we revisit Bhat (Trans Res Part B: Methodol 109: 238-256, 2018) flexible utility functional form and highlight that the stochastic conditions he used to derive the likelihood function are not always consistent with utility maximization. We offer an alternate interpretation of the model as representing a two-step decision-making process, where the consumers first decide which goods to choose and then decide the extent of allocation to each good. We demonstrate an application of the proposed flexible MDCP model to analyze households’ expenditure patterns on their domestic tourism trips in India. Our results indicate that, if the analyst is willing to compromise on the strict utility-maximizing aspect of behavior, while also enriching the behavioral dimension through the relaxation of the tie between the discrete and continuous consumption decisions, the preferred model would be the flexible non-IID MDCP model. On the other hand, if the analyst wants the model to be strictly grounded on utility-maximizing behavior (which may also have benefits by way of welfare measure computations), and is willing to assume a very tight tie between the discrete and continuous consumption decision processes, the preferred model would be the non-IID traditional MDCP model.

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Notes

  1. The baseline preference parameter of a good is the marginal utility at zero consumption of that good.

  2. The discussion here pertains to the \(\gamma - {\text{profile}}\) utility form since it has been generally the case that the \(\gamma - {\text{profile}}\) comes out to be superior to the \(\alpha - {\text{profile}}\) function (see Bhat 2018b; Jian et al. 2017). However, the discussion in the rest of the paper, including the model formulation, is applicable for the \(\alpha - {\text{profile}}\) utility function as well.

  3. \(\begin{gathered} {\text{At}}\,x_{k} = 0,\,\,\left. { \, \frac{{\partial U({\varvec{x}})}}{{\partial x_{k} }}} \right|_{{x_{k} = 0}} = \mathop {\lim }\limits_{{h \to 0^{ + } }} \,\left. {\frac{{U({\varvec{x}} + h) - U({\varvec{x}})}}{h}} \right|_{{x_{k} = 0}} \, = \,\mathop {\lim }\limits_{{h \to 0^{ + } }} \,\frac{{\left. {U({\varvec{x}} + h)} \right|_{{_{{x_{k} = 0}} }} - \left. {U({\varvec{x}})} \right|_{{_{{x_{k} = 0}} }} }}{h} = \mathop {\lim }\limits_{{h \to 0^{ + } }} \,\frac{{\left. {U({\varvec{x}} + h)} \right|_{{_{{x_{k} = 0}} }} - \left. {U({\varvec{x}})} \right|_{{_{{x_{k} = 0}} }} }}{h} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{{h \to 0^{ + } }} \,\frac{{\psi_{kc} \gamma_{k} \ln \left( {\frac{h}{{\gamma_{k} }} + 1} \right) - 0}}{h} = \mathop {\lim }\limits_{{h \to 0^{ + } }} \,\frac{{\frac{\partial }{\partial h}\psi_{kc} \gamma_{k} \ln \left( {\frac{h}{{\gamma_{k} }} + 1} \right)}}{{\frac{\partial }{\partial h}h}} = \psi_{kc} .{\text{ Note:}}U({\varvec{x}} + h) = U(x_{1} ,x_{2} ,...,(x_{k} + h),...,x_{K} ). \hfill \\ {\text{At}}\,x_{k} > 0,\, \, \left. { \, \frac{{\partial U({\varvec{x}})}}{{\partial x_{k} }}} \right|_{{x_{k} = 0^{ + } }} = \mathop {\lim }\limits_{{x \to 0^{ + } }} \,\,\psi_{kc} \left( {\frac{{x_{k} }}{{\gamma_{k} }} + 1} \right)^{ - 1} = \psi_{kc} . \hfill \\ \end{gathered}\) Given the above, the \(\psi_{k\,d}\) parameters do not enter the marginal utility functions of Bhat’s (2018b) utility form, which as we discuss later, has implications for whether the formulation is always consistent with utility maximization.

  4. Note also that the equality condition for the chosen goods, which is based on the C-preference parameters, automatically implies an inequality that \(\psi_{kc} - \lambda p_{k} > 0\). Such an inequality based on the C-preference parameters is not explicitly stated in the above conditions since it would be redundant.

  5. In the special case that M = 0 (that is, none of the inside goods is consumed), Eq. (13) collapses to the following:

    \(P\left( { \, {\varvec{x}}_{{}}^{*} } \right)\,\, = P(0,0,0,...0) = \,\,\,\,\displaystyle\int\limits_{{{\varvec{\eta}}_{D2} = - {\mathbf{\infty }}}}^{{{\varvec{\eta}}_{D2} = \tilde{\boldsymbol{V}}_{D2} }} {} \,{\varvec{f}}_{K - 1 + M} ({\varvec{\eta}}_{D2} ;\,{\mathbf{0}}_{K - 1 + M} ,{{\varvec{\Theta}}})\,d{\varvec{\eta}}_{D2} ,\)

    which takes the form of a simple multivariate cumulative normal distribution function.

  6. The symmetric nature of the multivariate normal density function is a distinct advantage over the asymmetric multivariate logistic density function used in Bhat (2018b). While the multivariate logistic has a closed form expression for the cumulative distribution function, computing the integral of the multivariate logistic density function with a combination of upper and lower limits cannot be collapsed to the evaluation of a single cumulative distribution function. However, while the multivariate normal cumulative distribution (MVNCD) function is not available in closed-form, the integral of the multivariate normal density function with a combination of upper and lower limits can be collapsed to the evaluation of a single MVNCD function. This is a particularly useful result for the proposed flexible MDCP model.

  7. A small nuance is in order here. Bhat’s (2018b) flexible MDCEV uses the same variance for \(\tilde{\lambda }_{kk}\) and \(\omega_{kk}\), which leads to a closed-form likelihood expression. However, other than the distribution form, the IID D-C MDCP model is similar to the MDCEV in Bhat’s paper, because the scale does not matter in the discrete part of the IID D-C MDCP, and we can as well normalize \(\tilde{\lambda }_{kk} \,\) to \(0.5\mu^{2}\) (k = 1,2,…K). However, just to be consistent with the model development in Sect. "Revisiting Bhat’s (2018a, b) flexible MDC model structure", we will keep to the assumption that \(\tilde{\lambda }_{kk} \, = 0.5\), because it more clearly shows that the IID D-C MDCP model is a restricted version of the full flexible MDCP model proposed in Sect. "Revisiting Bhat’s (2018a, b) flexible MDC model structure".

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Acknowledgements

This research was undertaken as part of a SPARC project funded by the Indian Ministry of Education for encouraging international collaborations. Empirical data used in this study were collected by the National Sample Survey Office of the Indian Ministry of Statistics and Program Implementation. The authors are grateful for the support provided by both ministries. The second author would like to thank Tarun Rambha for helpful discussions on non-linear optimization.

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Authors and Affiliations

  1. Department of Civil Engineering, Indian Institute of Science, Bengaluru, Karnataka, 560012, India

    Shobhit Saxena & Abdul Rawoof Pinjari

  2. Centre for infrastructure, Sustainable Transportation, and Urban Planning (CiSTUP), Indian Institute of Science (IISc), Bengaluru, 560012, India

    Abdul Rawoof Pinjari

  3. Department of Civil, Architectural and Environmental Engineering, The University of Texas at Austin, Austin, TX, 78712, USA

    Chandra R. Bhat & Aupal Mondal

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  1. Shobhit Saxena
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Author contributions to the paper are as follows: study conception and design: CB, SS, AP; analysis and interpretation of results: SS, AP, CB, AM; draft manuscript preparation: SS, AP, CB, AM. All authors reviewed the results and approved the final manuscript.

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Correspondence to Abdul Rawoof Pinjari.

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Saxena, S., Pinjari, A.R., Bhat, C.R. et al. A flexible multiple discrete–continuous probit (MDCP) model: application to analysis of expenditure patterns of domestic tourists in India. Transportation (2023). https://doi.org/10.1007/s11116-022-10364-y

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