»
- 2 -
UK +L*
The major oscillation, with a) in the range (O, ir) for O < h < 1 or/(0, n/2)
for h > 1, appears if and only if (o) > a + b, that is iff
1
(6) (1 + r ) < bh > 1+h > a + b
n
Condition (6) is a constraint on the parameters a, b and h. This condition is
satisfied if and only if
(7) a < 1
(8) b JL < b < b 2
where b^, b^ are two non-negative roots of the equation <J> (D) = a + b. It may
also be noted that (0) - (a + b) has a maximum/for b = 1/h.
N wiR -fe b
From the definition of n (m) it follows that the major oscillation, if
exists, explodes when
(9)
sin a) /h
o
sin w
> 1/b
o
where o) o is given by the equation <> (u3 q ) = a + b.
Thus it is sufficient that b > 1/h for the oscillation to explode.
Case h = 1 (as in your letter).
Condition (6) implies 2 /b > a + b. Hence b^ ^ = 2 - a + 2 A-a. Since
b < 1 for a < 1, the major oscillation exists ' and is damped iff
b < b <1. The oscillation exists and explodes iff 1 < b < b^. The same
conditions hold for the oscillations of the higher order. There is no periodic
oscillation for b outside the range (b , b^)•
I was unable to get any general result of the kind that given 1 > a > a > O
there is a lower bound h depending on a such that the major oscillation explodes
for all b > O and 0 > h > h.
But in view of the proven possibility of all the oscillations being damped,
I must and have withdrawn my paper on Kalecki from the OEP.
I do not know how to proceed now. It is clear that my criticism of Kalecki * 1 * * * s
trend equation (see his Eq. 35', footnote 2, p. 146): b ^
X t+1 - 5 *t + " Vh t " x t T ' i ' i
remains valid for some values of n* notably for frl > J-/h, but i£ is not valid
generally. This relativeness of the results makes me think that perhaps we
should try to estimate n and h from actual data?
Anyway thank you again for your letter. You ^ v-vooP
VAjsA^J /V t/V V^h/ ,
Yours sincerely,
Stanislaw Gomulka