We demonstrate that multiple stationary rational-expectationsequilibria exist in a version of Lucas's island economy. Theexistence of these equilibria follows from the fact that there is an indeterminate set of monetary equilibria in the two-period overlapping-generations model. We show how to construct stationaryrational-expectations equilibria by randomizing over the set ofnonstationary monetary equilibria.In some of our equilibria, apositively sloped Phillips curve exists even though our economycontains no signal-extraction problem as in the original Lucas paper.Our equilibria are indexed by beliefs and are examples of theexistence of sunspot equilibria in which allocations may differacross states of nature for which preferences, technology, andendowments are identical.Our technique for constructing stationarysunspot equilibria should prove useful in a wide class of models inwhich an indeterminate stationary equilibrium exists.