By Malinvaud E & Bacharach M
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Additional info for Activity Analysis in the Theory of Growth and Planning: Proceedings of a Conference held by the International Economic Association
Our reversal of the order in which these two properties are established is favoured by the simpler mathematical tool we use. We shall use the following: Separation theorem for convex sets. 1 If A and B are convex sets in three-dimensional space which have no point in common, then there exists a plane P which separates A and B, in the sense that every point of A is either in P or on one side of P, whereas every point of B is either in P or on the other side of P. The application we shall make of this theorem will at the same time serve as an illustration of its meaning and use.
On the one hand, a comparison path is constructed which is a path of proportional expansion, and which starts from y such that p (y) > 0. Of course, this means that such a y must be attainable, and it is simpler to bypass the problem of reaching a y with p (y) > 0 by assuming a starting vector y 0 > 0. Then y can be reached by free disposal. The comparison path establishes a floor beneath the value of p (yt), which is valid for every period, and which, loosely speaking, grows at some proportional rate characteristic of a von Neumann equilibrium.
XV. A CATENARY PROPERTY OF MAXIMAL GROWTH PATHS Proposition 4 is silent on the question as to where in a maximal growth path the exceptional activities more than S removed from a unique von Neumann activity z* may occur. The considerations that led Dorfman, Samuelson, and Solow to formulate the turnpike conjecture, however, suggested from the beginning that the exceptional steps can occur only near the beginning and/or near the end. Speaking very roughly, the notion is that any sufficiently long maximal growth path has somewhat the shape of a chain suspended from the points x1 , yT in a field of gravitational attraction towards the ray of the von Neumann path.