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We consider a principal repeatedly allocating a single resource in each period to one of multiple agents, whose values are private, without relying on monetary payments over an infinite horizon with discounting. We design a dynamic mechanism that induces agents to report their values truthfully in each period via promises/threats of future favorable/unfavorable allocations. We show that our mechanism asymptotically achieves the first-best efficient allocation (the welfare-maximizing allocation as if values are public) as agents become more patient and provide sharp characterizations of convergence rates to first best as a function of the discount factor. In particular, in the case of two agents we prove that the convergence rate of our mechanism is optimal—that is, no other mechanism can converge faster to first best.