Brief von Stanisław Gomułka an Josef Steindl
Stanisław
Gomułka
1538565550087
' The London School of Economics and Political Science
(University of London)
Houghton Street,
London, WC2A 2AE
Telephone: 01-405 7686
Department of Economics
SG/JM 28th June, 1976.
Professor Steindl,
A-llo3,
"ien Postfach 91,
AUSTRIA.
Dear Professor Steindl,
My reaction to your letter of 13th June was, it appears,too quick and
ill-founded. Over the weekend I was trying to reproduce my original discussion
of the equation:
I
(1) +
where a, b and h are all non-negative, and a < 1. Unfortunately, I could
not find my notes with the discussion (of ten years ago!), so that I have had
to do it again. However, the results are different than I thought they were.
Let 1^ = E e^. On substituting to (1) we have
= a + b |jL-e^J
a + i a)
(2) e
Let X = a + i a). Since e
(3)
a
= e (cos a) + i sin a)) , we have
a , . , -ha ,
e cos a) = a + b- be cos h a)
t a\ a , -ha
(4) e sin a) = b e
a i+h
From Eq.4 e = (b sin h w/sin ai) S n (w) . Svibstituting into (3) gives
—h
(5) <f> (to) = n (u) cos a) + b n (w) cos h ai = a + b
Continued..
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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) bJL < b < b2
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 <> (u3q) = 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 ^
Xt+1 - 5 *t+ " Vh t " xt 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