Last modified: 28 March 2017

#### Abstract

The multiple discrete continuous extreme value (MDCEV) model as introduced by Bhat (2008) in the transportation literature has received significant attention in recent years across different sectors from both scientific and applied perspectives (e.g. van Nostrand et al. 2013, Lu et al. 2016). The MDCEV model describes a decision framework where consumers maximise their utility by choosing a bundle of goods with different quantities of each good being ‘consumed’ whilst allowing for corner solutions (i.e. zero consumption of certain goods). The model is closely related to the Kuhn-Tucker models discussed in the environmental economics literature on recreation demand (e.g. von Haefen et al. 2004).

The purpose of this paper is to revisit the origins of the MDCEV utility function to better understand its foundations in economic theory and improve our understanding of the model parameters. Most importantly, we identify an implicit trade-off between the empirical confounding of key MDCEV model parameters and satisfying weak complementarity. We provide a solution by contrasting the MDCEV model against alternative utility functions satisfying weak complementarity whilst allowing for positive minimum consumption. Additionally, we evaluate efficient algorithms to derive welfare measures for MDCEV models.

The origins of the MDCEV utility function can be traced back to the Deaton and Muellbauer (1980) additively separable utility functions where goods are perfect substitutes. We illustrate how the MDCEV model specification arises after i) introducing a set of traditional translation parameters (the γ parameters), which allow for either minimum consumption or corner solutions by shifting the asymptote of the indifference curves; and ii) subsequently applying a Box-Cox transformation only to a part of the translated utility function. More specifically, by using the γ parameters both inside and outside of the Box-Cox transformation, confounding (see Bhat, 2008) between the α (Box-Cox) and γ (translation) parameters arises as both parameters now account for satiation effects.

The partial Box-Cox transformation was driven by the need of the MDCEV direct utility function to satisfy weak complementarity (von Haefen 2007), which implies that quality improvements in non-consumed goods do not affect a consumer’s utility. The empirical confounding between the α and γ parameters is the first price paid by this utility specification. The second price paid is that the translation parameters can no longer be interpreted as such and they are therefore no longer the appropriate tool to introduce minimum consumption. Non-zero minimum consumption may, however, well be a reasonable proposition for some ‘goods’. Work arounds are now being developed (e.g. van Nostrand et al. 2013) whilst the necessary tools are already present in the existing model specification.

In the empirical part of the paper, we contrast the MDCEV model with two alternative model specifications in which the translation parameters are no longer confounded with the Box-Cox parameters. The first model sacrifices weak complementarity and applies the Box-Cox transformation to the entire translated utility function, whereas the second model takes the form of von Haefen et al. (2004) which treats the way in which quality attributes enter the utility function in an alternative way in order to satisfy weak complementarity. We conduct these analyses on time use data from an activity-based research study in Denmark (Vuk et al. 2015), where there is clear potential for non-zero constraints for some ‘goods’ such as working.

The paper presents the derivation of the Marshallian demand function for the MDCEV model in the absence of outside goods thereby extending the work of Pinjari and Bhat (2011). The latter paper also made an important contribution in the context of efficiently forecasting future consumption patterns. While analysts are accustomed to standard choice models replicating the observed market shares in predictions, we highlight that, due to the nature of the model specification, it is nearly impossible to replicate the consumption patterns in the estimation sample when using unconditional forecasting routines (von Haefen 2003). We further extend the work of Pinjari and Bhat (2011) by applying their efficient forecasting routine to the welfare analysis of price and quality changes. We specifically contrast its empirical performance against an alternative efficient method proposed by von Haefen (2007) in the same empirical context of time use in Denmark.

**References:**

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.

Deaton, A. and Muellbauer, J. (1980). An almost ideal demand system. *The American Economic Review*, 70(3), 312-326.

Lu, H., Hess, S., Daly, A. and Rohr, C. (2016) Measuring the impact of alcohol multi-buy promotions on consumers’ purchase behaviour. *Journal of Choice Modelling, *in press.

Pinjari, A.R. and Bhat, C.R. (2011) Computationally Efficient Forecasting Procedures for Kuhn-Tucker Consumer Demand Model Systems: Application to Residential Energy Consumption Analysis. *Department of Civil and Environmental Engineering, The University of South Florida*.

van Nostrand, C., Sivaraman, V. and Pinjari, A.R. (2013) Analysis of long-distance vacation travel demand in the United States: a multiple discrete–continuous choice framework. *Transportation*, 40, 151-171.

von Haefen, R. H. (2003). Incorporating observed choice into the construction of welfare measures from random utility models. *Journal of Environmental Economics and Management*, 45(2),145-165.

von Haefen, R. H. (2007) Empirical strategies for incorporating weak complementarity into consumer demand models. *Journal of Environmental Economics and Management*, 54,15-31.

von Haefen, R. H., Phaneuf, D. J., and Parsons, G. R. (2004). Estimation and welfare analysis with large demand systems. *Journal of Business & Economic Statistics*, 305 22(2):194-205.

Vuk, G., Bowman, J.L., Daly, A. and Hess, S. (2015) Impact of family in-home quality time on personal travel demand. *Transportation*, 43(4), 705-724.