Standard discrete choice models such as logit, nested logit, and random coefficients models place very strong restrictions on how unobservable product space increases with the number of products. We argue (and show with Monte Carlo experiments) that these restrictions can lead to biased conclusions regarding price elasticities and welfare consequences from additional products. In addition, these restrictions can identify parameters which are not intuitively identified given the data at hand. We suggest two alternative models that relax these restrictions, both motivated by structural interpretations. Monte-Carlo experiments and an application to data show that these alternative models perform well in practice.