In this paper we analyze pricing strategies of firms that compete for the demand of an assortment of goods. In particular, we model the competition between two symmetric firms in a market that consists of two types of consumers, each of which may buy one unit of both goods sold by the two firms. We show that price dispersion for each good and across stores arises as the unique Nash Equilibrium and provides a rationale for the existence of price dispersion across an assortment of goods where one firm charges a higher price for one good while the other charges a higher price for the other good and each firm sells both goods. We also show that under certain conditions these Nash equlibrium prices are the same as those that maximizes joint profits. In this sense, the firms are able to completely price discriminate across consumers and derive the maximum consumer surplus.