In this paper we consider a class of discrete choice models in which consumers care about a finite set of product characteristics. These models have been used extensively in the theoretical literature on product differentiation, but have not as yet been translated into a form that is useful for empirical work. Most recent econometric applications of discrete choice models implicitly let the dimension of the characteristic space increase with the number of products. The models in this paper have very different theoretical properties. After developing those properties and comparing them to the properties of models where there is a taste for the product per se, we provide an algorithm for an estimator for the parameters of our model. In this version of the paper, we particularly consider how the modeling choices discussed in the paper affect the calculation of ideal consumer price indices, especially in the presence of new goods.