In a share economy, individuals don't owe a sum. They have exchanged a share of their income for something else, a lump sum, a car, a home. Their firm has payments coming in. For example, each time a customer pays for a nights stay, each employee and each investor gets a share, a micropayment. As a computer professor, my University would have exchanged a share of each individuals future income for their "tuition." Each time, their employer gets a payment, they get a little share, and the University gets a little share and then myself as a computer science professor when the individual was a student get a share. And then my landlord gets a share and my "credit card" gets a little share, and the investors in the landlord and to the credit card company gets a share....
This means that all accounts, incomes and payments would be on a central system. Computers would have lots of divisions to create the little sums, each of which would be divided and subdivided as per the share. It would have to be centralized to prevent evasion of one's share. The computers in the system would be doing lots of divisions, and then additions to other accounts. Fortunately, computers do divisions and additions very fast!
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