Can rational individuals keep money under their mattresses forever?

Summary Conditions are given for an infinite horizon consumption-savings model under which savings are bounded away from 0 with probability 1 even in the long-run. That is, with probability 1 there exists a timeT and a minimum level of savingsS such fort > T savings will always be greater thanS. I would like to thank Graziella Bertocchi, Oded Galor, Basilis Gidas, Gil Skillman, David Weil and seminar participants at Brown University. I am also very grateful to an anonymous referee for many useful suggestions.