This paper introduces the rest of this issue of Economic Systems Research, which is dedicated to the contributions of Sir Richard Stone, Michael Bacharach, and Philip Israilevich. It starts out with a brief history of biproportional techniques and related matrix balancing algorithms. We then...

Biproportional methods project a matrix <Emphasis Type="Bold">A to give it the column and row sums of another matrix; the result is <Emphasis Type="Bold">R A S, where <Emphasis Type="Bold">R and <Emphasis Type="Bold">S are diagonal matrices. As <Emphasis Type="Bold">R and <Emphasis Type="Bold">S are not identified, one must normalize them, even after computing, that is, ex post. This article starts from the idea developed in...</emphasis></emphasis></emphasis></emphasis></emphasis></emphasis>

Analysts often are interested in learning how much an exchange system has changed over time or how two different exchange systems differ. Identifying structural difference in exchange matrices can be performed using either 'directed' or 'undirected' methods. Directed methods are based on the...