When someone signed up, according to an early internal estimate, he or she was worth forty or fifty dollars a month in gross revenues and eight to ten dollars in gross profit—for the foreseeable future. “This is the equivalent of a perpetual-motion machine and cannot go on forever, but it means that rider spending is accelerating at a rate that is bigger than our churn rate,” Kalanick wrote to his fellow investors in an e-mail that year.

