By Malinvaud E & Bacharach M

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Example text

Hence, in any maximal growth path now, the number of steps outside a Z 8 which closely hugs S* is limited. ty2 x, FIG. 11. CASE OF MANY VON NEUMANN ACTIVITIES Further complications arise if P and Z intersect in a line segment S not contained in S(t-t*). In that case, S and S(t-t*) have only the unique von Neumann activity z* in common. A case of this kind is not in any way a result of coincidence. In fact it is likely to arise except for coincidences in any technology with a finite basis. In McKenzie's work (1963b, c, d), cases of this kind are analysed in detail for n-commodity models.

For any other point z of Z, since the original profit is negative in z, there exists an interest-rate reduction 3 small enough to leave the modified profit negative in z. It follows that the slices Z 0 , nested one inside the other as 3 becomes smaller, shrink down to the point z* as 3 tends to zero. This mathematical fact is responsible for the turnpike theorem in its present, simplest, version. It also suggests that it will be convenient to use 3 as a measure of the distance, in the three-dimensional space of normalized activities, between z* and any other maximal 2 point z of Z, instead of Radner's measure of angular distance between points of the (two-dimensional) commodity space.

1 2 K• Studies on Growth Theory We shall now follow Radner in a calculation showing that only a limited number of the factors f:-tt can fall substantially below the upper bound JL""· To this end we consider a comparison path (xt, y 1), t = 1, ... , T, constructed as follows. The initial stock X1 = x 1 is the same as before. The first two steps are used to arrive at the highest multiple ji 2 = vx*, v > 0, of the von Neumann point x* of Assumption A that can be attained in two steps. The next T- 4 steps proceed along the von Neumann path at maximum growth jit=~:-t*xt, t=3, ...