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.. Tht London School of Economics and Folitical Science is incorporated in England as a company limited by guarantee Linder the Companies Acts (Reg. No. 70527) Registered Office as above ﻿» - 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) 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