What were the benefits of the new system of double-entry bookkeeping?
In the late 13th and early 14th centuries, a process took place – primarily in Italy but partly also in the Empire – that historians have termed a 'commercial revolution'. Among the innovations of the period were new forms of partnership and the issuing of insurance, both of which contributed to reducing the risk of commercial ventures; the bill of exchange and other forms of credit that circumvented the canonical laws for gentiles against usury, and eliminated the dangers of carrying bullion; and new forms of accounting, in particular double-entry bookkeeping, which allowed for better oversight and accuracy.
Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices. The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics. This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities, complementary slackness, and saddlepoint duality. In the proceedings of a conference on von Neumann's growth model, Paul Samuelson said that many mathematicians had developed methods useful to economists, but that von Neumann was unique in having made significant contributions to economic theory itself.