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Data is nonrival: a person’s location history, medical records, and driving data can be used by any number of firms simultaneously without being depleted. Nonrivalry leads to increasing returns and implies an important role for market structure and property rights. Who should own data? What restrictions should apply to the use of data? We show that in equilibrium, firms may not adequately respect the privacy of consumers. But nonrivalry leads to other consequences that are less obvious. Because of nonrivalry, there may be large social gains to sharing data across firms, even in the presence of privacy considerations. Fearing creative destruction, firms may choose to hoard data they own, leading to the inefficient use of nonrival data. Instead, giving the data property rights to consumers can generate allocations that are close to optimal. Consumers appropriately balance their concerns for privacy against the economic gains that come from selling data to all interested parties.