Extending the Multiple Discrete Continuous (MDC) modelling framework to consider complementarity, substitution, and an unobserved budget

Many decisions can be represented as interrelated discrete and continuous choices, i.e. what and how much to choose from a set of finite alternatives (incidence and quantity of consumption). In the last twenty years, several models of Karush–Kuhn–Tucker demand systems have been developed and used to study these kinds of decisions. While strongly grounded in economic theory, most of these models have two limitations: they require specifying a budget, and usually omit any complementarity effects. In this paper, we propose two extensions to the Multiple Discrete Continuous (MDC) modelling framework: (i) an MDC model including explicit complementarity and substitution effects, and (ii) an MDC model with complementarity, substitution that requires no budget definition. Model (ii) relies on the hypothesis that total expenditure on the alternatives under consideration is small compared to the overall budget. This allows using a linear utility function for the numeraire good, leading to a likelihood function without the budget or numeraire good in it. The lack of a budget is specially useful when forecasting, as it avoids cascading errors due to an inaccurate budget specifications. The inclusion of complementarity and substitution effects enriches the interpretability of the models, while the resulting functional form avoids theoretical issues present in previous formulations. Alongside the derivation of the models, we discuss their main properties and propose an efficient forecasting algorithm for (ii). We also report four applications to datasets about time use, household expenditure, supermarket scanner data, and trip generation. Free estimation code for both models is made available online.