Is von Neumann square? (Q1082236)

Considering two semi-positive rectangular matrices A and B whose rows are interpreted as the respective input and output vectors of production processes, the maximal rate of growth of the economy is the solution G to max\(\{\) g; there exits \(q\geq 0\) such that \((1+g)qA\leq qB\}\); the positive components of q define the activity levels of efficient processes. Von Neumann showed that the problem can be solved by duality: an adequate price vector exists, for which the efficient processes are those which yield the maximal profit rate R; moreover \(R=G\) and goods whose growth rate is greater than G have a zero price. We show that the submatrices (Ā,\=B) made of the efficient processes and the positively-priced goods are generically square and satisfy the inequality \((\bar B-(1+r)\bar A)^{-1}>0\) for \(r=R-\epsilon\). This property is interpreted in terms of a connexion between von Neumann's systems and ''good'' (all-engaging) Sraffa systems at high rates of profit - leading to a single-production-like behaviour - and in terms of a non- substitution property in joint production at high rates of growth.