Recent studies have highlighted divergent change as a more common outcome of language contact than previously thought. While convergent change is often attributed to bilingual cognitive pressures, divergent change has usually been explained by appealing to sociocultural factors. We argue that the effects of social pressures on linguistic systems must nevertheless be realized in how language is processed in the individual bilingual speaker and, therefore, that divergent change is also ultimately rooted in bilingual cognition. Since lexical forms are most susceptible to contact-induced divergent change we focus on their production. We begin by developing a cognitive model that combines Grosjean's language mode with a later output-monitoring stage. The parameters to the model are then fit to the results of an experiment in which bilinguals are seen to avoid shared lexical items. These best-fit parameters form the basis of a series of multi-agent simulations that show rapid divergence in the lexica of languages with large proportions of bilinguals. We consider the implications of these findings for the psycholinguistic study of bilingual lexical selection, the construction of phylogenies, and the reconstruction of language family histories.