546 PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 
28 
where v;=g;/v,, and the i-summation takes place over all goods 
t with g>0 and the j-summation over all goods j with gi =o. 
It follows from (35) and (36) that v is bounded and hence 
zr=Nx* - x° is also bounded. 
When T becomes large with x, and x* remaining unchang- 
ed, it is seen from (33) that those œ, which are associated with 
|A;|>I and those B, with |\,,;|>I, as well as «, are shown to 
become small, because the absolute values of SE Ar. and 
I+D+ ... +D" are shown to become very large, but zr is 
bounded. All the other a; and 8; are not very small, but those 
A; and A, which are associated with them have absolute values 
less than unity. It can now be seen that when the programming 
period T is sufficiently long, 2’, well approximates f; £,,; most 
of the period T. From the definitions of E,,, and (x°) it can 
be shown that they are proportionate to each other, i.e. 
Eur1=0 (x°). By virtue of (27) we finally find that in the very 
long-run programme outputs of industries almost always grow 
like x, 2 (1+7°) (B,0+1)x°. The turnpike property is thus 
established. 
5. TECHNICAL CHANGE 
I. In this section we concentrate our attention on the state 
of the voN NEUMANN balanced growth, the convergence to which 
nas been discussed in the preceding section. We shall be con- 
cerned with comparative statical analysis of effects on the von 
NEUMANN equilibrium of technological changes (or changes in 
the parameters of the production functions) which may be 
arranged in the following classes: (1) an increase in F;, all 
other parameters remaining unchanged, (2) a substitution of b,; 
for a, accompanying an increase in F; and (3) a converse 
substitution between b,; and a,; associated with an increase 
in F,. If there were no changes in prices and the wage rate, 
9] Morishima - pag. 18