Unobserved Di¤erentiation in Discrete Choice Models

Discrete choice models used in statistical applications typically interpret an unobservable term as the interaction of unobservable horizontal di¤erentiation and idiosyncratic consumer preferences. An implicit assumption in most such models is that all choices are equally horizontally di¤erentiated from each other. This assumption is problematic in a number of recent studies that use discrete choice frameworks to evaluate the welfare e¤ects from di¤erent numbers of goods (e.g. Berry and Waldfogel, 1999; Rysman, 2000). Researchers might think that it is possible for product space to “...ll up” and that ignoring this issue might lead to an overestimate of welfare as the number of new products increases. This paper proposes a solution whereby the researcher estimates the decrease in value that agents receive from higher numbers of products as a result of the decreasing importance of horizontal di¤erentiation. The paper reviews previous results on how a linear random utility model (LRUM) can be mapped into an address (Hotelling) model. The paper shows how realistic assumptions on di¤erentiation in an address setting can be mapped into an LRUM. LRUM models imply that all choices are strong gross substitutes. In order to preserve that condition in an address model, n choices must be di¤erentiated along at least n ¡ 1 dimensions. This paper proposes that utility drawn from di¤erent dimensions be weighted di¤erently. Mapping this feature into an LRUM requires weighting the utility from each choice based upon the dimension along which it is di¤erentiated from others. As researchers will typically be unwilling to make assumptions about which dimension products di¤er on, the paper discusses integrating over the di¤erent possibilities in a computationally inexpensive way that still allows the researcher to relax the assumption of symmetric di¤erentiation. ¤I would like to thank Dan Ackerberg and Phil Haile for helpful discussion.