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We show that kidney exchange markets suffer from traditional market failures that can be fixed to increase transplants by 25%-55%. First, we document that the market is fragmented and inefficient: most transplants are arranged by hospitals instead of national platforms. Second, we propose a model to show two sources of inefficiency: hospitals do not internalize their patients’ benefits from exchange, and current mechanisms sub-optimally reward hospitals for submitting patients and donors. Third, we estimate a production function and show that individual hospitals operate below efficient scale. Eliminating this inefficiency requires a combined approach using new mechanisms and solving agency problems.