After formulating a general model involving two populations of microorganisms competing for two nonreproducing, growth-limiting resources in a chemostat, we focus on perfectly substitutable resources. León and Tumpson considered a model of perfectly substitutable resources in which the amount of each resource consumed is assumed to be independent of the concentration of the other resource. We extend their analysis and then consider a new model involving a class of response functions that takes into consideration the effects that the concentration of each resource has on the amount of the other resource consumed. This new model includes, as a special case, the model studied by Waltman, Hubbell, and Hsu in which Michaelis-Menten functional response for a single resource is generalized to two perfectly substitutable resources. Analytical methods are used to obtain information about the qualitative behavior of the models. The range of possible dynamics of model I of León and Tumpson and our new model is then compared. One surprising difference is that our model predicts that for certain parameter ranges it is possible that one of the species is unable to survive in the absence of a competitor even though there is a locally asymptotically stable coexistence equilibrium when a competitor is present. The dynamics of these models for perfectly substitutable resources are also compared with the dynamics of the classical growth and two-species competition models as well as models involving two perfectly complementary resources.