Knowledge that Transforms

The Leontief matrices of inter-industry transactions are large, a row for each industry in a nation. It would be desirable to invert such matrices of an order of from 100 to 200. The present paper suggests using the sum of a power series to approximate the inverse of a Leontief matrix with any desired degree of accuracy. This requires many more multiplications than do such direct methods as the Gauss-Doolittle process. But the method proposed in this paper is especially well adapted to automatic computation on the new electronic machines, in which case the large number of multiplications is not serious. The main advantage of the proposed method is that it provides an upper bound to the error of any element in the estimated inverse. A short cut method is also indicated for computing the approximation when the number of terms of the power series needed to obtain the desired degree of accuracy is large.

A new treatment is presented of a classical economic problem, one which occurs in many forms, as bargaining, bilateral monopoly, etc. It may also be regarded as a nonzero-sum two-person game. In this treatment a few general assumptions are made concerning the behavior of a single individual and of a group of two individuals in certain economic environments. From these, the solution (in the sense of this paper) of classical problem may be obtained. In the terms of game theory, values are found for the game. См. также: Two-person cooperative games, автор - Джо Нэш.