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PONTTFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
where A, B, and C are matrices A,, B,, and C, evaluated at 
pu=p; (i=0, 1, ..., n). In view of (2), (9) and 3 g;=1 we find 
i=1 
that one of the characteristic numbers {4,, …, |, is I+r. We 
can also show that the eigen-vector % associated with 1 +7 is 
non-negative. It is clear that we have 
(16) x B(1+—d)+(1+n)x([—+A—B—C)=o. 
Thus (14) has a particular solution (1 + 7)! which is referred to 
as a balanced growth solution. We also refer to a state fulfil- 
ling (8) (9), and (16) as a state of balanced growth. 
2. So far we have treated the rate of interest as a given 
constant and have shown that to any assigned value of it there 
corresponds a state of balanced growth. It is impossible, how- 
ever, for the rate of growth of outputs to exceed the rate of 
growth of the working population for a long time, because the 
scarcity of labour will sooner or later emerge. In the contrary 
case where the labour force is increasing at a rate higher than 
the rate of growth of outputs, the ratio of the number of unem- 
ployed to the number of employed workers continues to rise. 
In the following, therefore, we are concerned with finding a 
rate of balanced growth at which the growth of outputs is in 
harmony with that of the labour force. 
We begin with examining the effects of a change in the 
rate of interest on the long-run equilibrium prices. Differentiat- 
ing (10) with respect to », we get 
a log : 
ax — (us — @-— 0)" 
ar 
y| Morishima - pag. 8