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Payment card networks, such as Visa, require merchants' banks to pay substantial “interchange” fees to cardholders' banks on a per transaction basis. This paper shows that a network's profit-maximizing fee induces an inefficient price structure, oversubsidizing card usage and overtaxing merchants. We show that this distortion is systematic and arises from the fact that consumers make two distinct decisions (membership and usage), whereas merchants make only one (membership). In general, we contribute to the theory of two-sided markets by introducing a model that distinguishes between extensive and intensive margins, thereby explaining why two-part tariffs are useful pricing tools for platforms. (JEL D42, D85, G21, L12)