Consumer behavior involves two types of decisions. On one hand, people decide which goods to purchase or not purchase; on the other, people decide what quantity to purchase of the commodities they have chosen to acquire. Researchers would like to explain the two decisions using a unified utility model. This model should also account for other characteristics of consumer behavior. First, consumer demand is affected by other attributes of the goods besides prices and income, for instance, brand selection depends on the objective and subjective attributes of the different brands (Hendell, 1999; Dubé, 2004). Second, individual consumer choice behavior is characterized by the prevalence of corner solutions, wherein consumers are observed not to purchase any quantity of certain commodities.
Since the work by Hanemann (1978, 1984), Wales and Woodland (1983) the approach used to estimate demand models with a general corner solution, has been based on the Kuhn-Tucker (KT) conditions for the solution to the consumer's utility maximization problem. The KT approach starts with the formulation of a utility function whose arguments include the level of consumption of each commodity and also the quality or other attributes of the commodities. The maximization of this utility function subject to the budget and nonnegativity constraints generates the first order conditions (KT) governing whether a positive or zero quantity of each good is consumed and, if the former, how large a quantity. Adding a random term to the utility function makes it possible to generate probability statements for the consumption bundle observed vector which serve as the building blocks for maximum likelihood estimation.
An additional complication arises from the fact that one often wants to use the estimated utility model to predict commodity demands under different scenarios, and to calculate welfare measures for changes in prices or attributes. In both of these cases, one needs to predict the new general corner solution that will be chosen under the new scenarios. This is complex because, if there are M commodities, there are 2^{M-1} alternative possible solutions to the consumer's utility maximization.
In the last few years there has been renewed interest in the KT model triggered by three major developments. First, it has proved feasible to apply the KT approach to demand functions with a significant number of commodities based on a clever use of the extreme value distribution which provides a closed form representation of the likelihood function (Von Haefen et al., 2004; Bhat, 2007). Second, the development of simulation techniques allows researchers to avoid the problems associated with numerical evaluation when integrals in the likelihood function lack a closed form representation. Furthermore, simulation introduces greater flexibility in the random structure of the model and facilitates the calculation of welfare measures. Third, the use of an additively separable utility function has provided a simple way to calculate welfare measures without requiring an explicit comparison of all possible solutions to the utility maximization problem (Von Haefen and Phaneuf, 2004; Von Haefen et al., 2004), increasing significantly the number of alternatives that can be handled with this framework.
An additively separable utility function is unfortunately a restrictive functional form since it reduces the flexibility of the utility function in terms of substitution patterns.
In this paper we present a utility model that serves as the basis for modeling discrete/continuous consumer choices with a general corner solution. The model involves a more flexible representation of preferences than what has been used in the previous literature and, unlike most of this literature, it is not additively separable. This functional form can handle richer substitution patterns such as complementarity as well as substitution among goods. We focus in part on the Quadratic Box-Cox utility function and examine its properties from both theoretical and empirical perspectives. We identify the significance of the various parameters of the utility function, and demonstrate an estimation strategy that can be applied to demand systems involving both a small and large number of commodities.