We study optimal insurance contracts for an agent with Markovian private information. Our main results characterize the implications of constrained efficiency for long-run welfare and inequality. Under minimal technical conditions, there is Absolute Immiseration: in the long run, the agent’s consumption and utility converge to their lower bounds. When types are persistent and utility is unbounded below, there is Relative Immiseration: low-type agents are immiserated at a faster rate than high-type agents, and “pathwise welfare inequality” grows without bound. These results extend and substantially generalize the hallmark findings from the classic literature with iid types, suggesting that the underlying forces are robust to a broad class of private information processes. The proofs rely on novel recursive techniques and martingale arguments. When the agent has CARA utility, we also analytically and numerically characterize the short-run properties of the optimal contract. Persistence gives rise to qualitat- ively novel short-run dynamics and allocative distortions (or “wedges”) and, quantitatively, induces less efficient risk-sharing. We compare properties of the wedges to their counterparts in the dynamic taxation literature.