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Task switching has been used as an experimental index of limitations on human cognitive flexibility. When switching from one task to another, participants exhibit increased reaction time, even when given ample time to prepare for the switch. Classical explanations of this “residual switch cost” have assumed that it is a fixed limit of human cognition. More recently however, research has shown that aspects of this limitation may in fact be optional, driven by effort minimization. Here we fully explore this effort minimization idea by providing a formal rational analysis of the task switching paradigm. This leads to a precise account of residual switch costs that makes testable predictions. We then conduct two new experiments to test the model against two alternative formulations. The experiments support our model. Overall, this work formally demonstrates that the tension between mental effort and performance demands will indeed lead to residual switch costs. This validates recent ideas that residual switch costs are “optional” rather than an irreducible limit on human cognition, and provides a principled and rigorous framework for testing additional implications of this idea in future work. This work is joint with Michael Todd, Matthew Botvinick, Jonathan Cohen, and Peter Dayan.