Last modified: 28 March 2017

#### Abstract

Many real-life situations involve making discrete and continuous choices simultaneously: which activities to perform during a day and how much time to allocate to each of them; or how many types of a product to purchase and how much money to spend on each. The family of Multiple Discrete-Continuous Extreme Value (MDCEV) models, first proposed by Bhat (2005) and later extended in different directions (Bhat 2008, Pinjari & Bhat 2010, Bhat et al. 2015) represents the state of the art in modelling multiple discrete-continuous choices. The model framework is based on a direct utility specification which is assumed to be maximised by consumers subject to a budget constraint setting the upper limit on total consumption.

The assumption of an overall budget on the entire choice set is useful in constraining total consumption across all goods. This ensures that a consumer cannot choose a combination of amounts of different goods where the total implied ‘cost’ (in whatever unit is used) exceeds the available budget. In its simplest form, this however directly allows for the possibility of a given consumer spending all her/his budget on a single good, and this may be unrealistic in many contexts that are governed by upper limits on consumption of specific goods and/or subsets of goods. Such upper limits might arise due to at least the following three reasons. First, in addition to the total consumption budget on the entire choice set, subsets of choice alternatives might be governed by additional budget constraints. For example, consider weekly activity participation and time allocation behaviour: in addition to an overall weekly time budget, activities conducted on weekend days are governed by an additional budget constraint of 48 hours. Ignoring such additional budget constraints can potentially lead to unrealistic forecasts such as weekend activity durations exceeding 48 hours. Second, there might be externally imposed, product-specific maximum consumption allowances. For example, parents might impose a maximum daily limit on children’s time allocation to video gaming. In another example, supermarket promotions on consumer products might be associated with a fine print that only a certain maximum number of units can be purchased by the same shopper. Third, the consumer’s preference structure might be such that she/he might not derive additional utility from additional consumption beyond a certain maximum level of consumption of a product.

The standard MDCEV model cannot accommodate upper limits on consumptions due to any of the above reasons. In this paper, we explore alternative approaches to recognize upper limits on consumption due to each of the above three reasons. To incorporate additional budget constraints on subsets of goods, we propose to explore an extension of the MDCEV model allowing for multiple budgets proposed by Castro et al. (2012). While the Castro et al. work proposed this approach in the context where each good has a cost along more than one dimension (say a money cost and a time cost), we exploit it with the specific view of imposing upper limits on consumption for subsets of goods. While each good still incurs a cost in relation to the overall budget (say a weekly time budget of 168 hours across all activities), there are additional budgets that apply to individual goods (say a separate 48 hour budget shared across all weekend activities, with only those activities having a non-zero ‘cost’ in relation to that budget). We apply both the standard version of the Castro et al. (2012) framework, which leads to consumption probabilities becoming multivariate integrals of the dimension equal to the number of constraints as well as variants of it (as in Satomura et al., 2011) that make simplifying assumptions on the stochastic terms of the utility functions for obtaining closed form probability expressions. Finally, we also test the impact of specifying utility functions in such a way that the marginal utility becomes zero beyond a certain consumption level.

The work we describe has widespread applications. For our empirical example, we rely on time-use data from Concepción, Chile, namely a two-day activity diary including a weekday and a weekend day. Respondents could freely describe their activities, which we grouped into 12 macro-categories: basic needs, drop-off & pick-up, family, household obligations, in-home recreation, out-of-home recreation, social, services, shopping, study, travel and work. While most of the activities can be performed on both weekday and weekend days, some of them are more likely to be performed on one or the other. We accommodate this by defining subcategories for the activities (weekday vs weekend) and specifying additional constraints as described above.

Estimation results will be compared with the classic MDCEV model to draw conclusions on the advantages provided by the proposed methods. We also will be able to provide insights into time use and activity choice in the context of the study.

**References**

Bhat, C. R. (2005). A multiple discrete–continuous extreme value model: formulation and application to discretionary time-use decisions. Transportation Research Part B: Methodological, 39(8), 679-707.

Bhat, C. R. (2008). The multiple discrete-continuous extreme value (MDCEV) model: role of utility function parameters, identification considerations, and model extensions. *Transportation Research Part B: Methodological*, *42*(3), 274-303.

Bhat, C.R., M. Castro, and A.R. Pinjari (2015). Allowing for Complementarity and Rich Substitution Patterns in Multiple Discrete-Continuous Models. *Transportation Research Part B*, Vol. 81, No. 1, pp. 59-77.

Castro, M., Bhat, C. R., Pendyala, R. M., & Jara-Díaz, S. R. (2012). Accommodating multiple constraints in the multiple discrete–continuous extreme value (MDCEV) choice model. *Transportation Research Part B: Methodological*, *46*(6), 729-743.

Pinjari, A. R., & Bhat, C. (2010). A Multiple Discrete–Continuous Nested Extreme Value (MDCNEV) model: formulation and application to non-worker activity time-use and timing behavior on weekdays. *Transportation Research Part B: Methodological*, 44(4), 562-583

Satomura, T., Kim, J., & Allenby, G. M. (2011). Multiple-constraint choice models with corner and interior solutions. *Marketing Science*, *30*(3), 481-490.