A choice model for packaged goods: dealing with discrete quantities and quantity discounts

Utility maximizing solutions to economic models of choice for goods with either discrete quantities or non-linear prices cannot always be obtained using standard first-order conditions such as Kuhn–Tucker and Roy's identity. When quantities are discrete, there is no guarantee that derivatives of the utility function are equal to derivatives of the budget constraint. Moreover, when prices are nonlinear, as in the case of quantity discounts, first-order conditions can be associated with the minimum rather than the maximum value of utility. In these cases, the utility function must be directly evaluated to determine its maximum. This evaluation can be computationally challenging when there exist many offerings and when stochastic elements are introduced into the utility function. In this paper, we provide an economic model of demand for substitute brands that is flexible, parsimonious, and easy to implement. The methodology is demonstrated with a scanner panel data set of light-beer purchases. The model is used to explore the defects of price promotions on primary and secondary demand, and the utility of product assortment.