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We introduce a simple adverse selection problem arising in credit markets into a standard textbook continuous-time real business cycle. We show that such adverse selection generates multiple steady states and both local and global indeterminacy, and can give rise to equilibria with probabilistic jumps in credit, consumption, investment and employment driven by Markov sunspots under calibrated parameterizations and fully rational expectations. Introducing reputational effects eliminates defaults and results in a unique but still indeterminate steady state. Finally we generalize the model to firms with heterogeneous and stochastic productivity, and show that indeterminacies and sunspots persist.