We study a sequential investment model and offer a theory of the sunk cost fallacy as an optimal response to limited memory. As new information arrives, a decision-maker may not remember all the reasons he began a project. The initial sunk cost gives additional information about future net profits and should inform subsequent decisions. We show that in different environments, this can generate two forms of sunk cost bias. The Concorde effect makes the investor more eager to complete projects when sunk costs are high and the pro-rata effect makes the investor less eager. The relative magnitude of these effects determines the overall direction of the sunk cost bias. In a controlled experiment we had subjects play a simple version of the model. In a baseline treatment with no memory constraints subjects exhibit the pro-rata bias. When we induce memory constraints the effect reverses and the subjects exhibit the Concorde bias.