On the Local Conservation Laws in the Von Neumann Model

Abstract

P. Samuelson was the first economist to introduce the idea of the law of conservation in classical mechanics into economics. He has shown in his very important paper [7] that two types of conservation laws operate along the optimal path in a closed, consumptionless system of the von Neumann type. The famous theorem [4] on the conservation laws states that the Hamiltonian is conserved in cases where the Lagrangian does not involve time t explicitly. Samuelson applied this theorem to his own model and derived the following two results. Namely, 1) Ω₁=λY=const., that is, the product of the implicit price λ and the national income Y is conserved and 2) Ω₂=λW=const., that is, the product of the implicit price λ and the national wealth W is also conserved. These two laws are reducible to Ω₁/Ω₂=Y/W=const., that is, the aggregate output-capital (wealth) ratio is always constant.

The author is greatly indebted to Professor R. Sato, Brown University who has encouraged me to solve the problem treated herein for publication in the present form.