## 683. Harrod to J. Tinbergen

, 1 July 1937[a]

I am very grateful to you for sending me an offprint of your review of my book and also for the very generous things you said about it. [1]

May I refer to your main criticism. I fear that my mathematics are rather rudimentary and that any single-handed attempt to give a rigid mathematical formulation to my theory would not be successful. But I am c ontemplating a quasi-mathematical article elaborating my central thesis.

I am aware that my fundamental propositions do not yield a sine curve of the kind that your soul delights in. I do not think it follows that they necessarily fail to demonstrate the inevitability of the cycle. On the look out for a certain type of equation you have, I think, done less than justice to my argument at this point.

I take your formulation of my fundamental proposition [2]

I add two further conditions:--

(i) There is a given finite magnitude which x cannot exceed (full employment). 2

(ii) For any value of x below a certain value, y = 0. 3 (This specific value may rise with time in accordance with a secular trend).

These equations and conditions give you this kind of curve:--[fig. 1]

The empirical trade cycle is not of course quite like this.

My idea is that my curve gives the basic fact of the cycle on which various lags must be superimposed. 4 In particular of course we do not get the absolute discontinuity indicated for the slump. I made various suggestions for the operation of certain lags at this point. (see especially p. 99). But I am strongly of the opinion that there is a strong element of discontinuity in the slump, [4] and of a-symmetry as between slump and boom, and that conditions remain for a while in the slump, <more> decidedly out of short period equilibrium even than they do in the upward swing.

I must confess also that my treatment of the recovery, as Mrs Robinson pointed out in her review in the Economic Journal, [5] was rather too brief. Why does not the system run on at the bottom of the slump on the basis of investment = 0? I think there is a straightforward explanation of why it does not do so, connected with the replacement question, but I grant that this requires more elaborate treatment than I gave it. 5

I very much hope therefore that you will pause a little further to consider the significance for the cycle of my multiplier/relation propositions, and not dismiss them because the solution is not so neat as <those> you get by certain lag assumptions. My own intuition, for what it is worth, is that you will not get at the vera causa 6 of the cycle by looking at lags only. I have no doubt they play some part in the whole thing, but I believe it will be found to be a relatively minor part.

I see you convict me of bringing lags into my argument at various points. [6] Of course I do. But I do not think that the assumption of a lag is present in the fundamental part of my argument formulated in the equations above.

Again with many thanks for your very handsome review.

With regard to my curve, no doubt it will be plain to you why the breaks occur. When x reaches its maximum value, must be zero. But cannot be zero for high values of x. \ there must be a sharp discontinuous drop in the value of x.

- 1. J.
Tinbergen, "Harrod, R. F., The Trade Cycle. An Essay",
Weltwirtschaftliches Archiv XLV, 1937, pp. 89*-91*.
2. Tinbergen's formula was obtained as follows (p. 90*):

- Let x stand for the production of consumption goods and y for net investment. Both variables are taken as deviation from their equilibrium values. Let us assume further that the price level and replacements are both constant. These hypotheses are only meant to simplify our argument, but one could assume more general conditions without altering the conclusion. Under these assumptions, the "relation" can be expressed as

, where . The "multiplier" can be couched in the form . Taken together, these equations enable us to formulate the differential equation for x (and also for y): . [My translation--Ed.]

3. Harrod's formula is incorrect, as the units of measurement are not homogeneous: x

*t*represent production per time unit, whilerepresents the increase of production per time unit squared. The formula should therefore be completed with a reference to the interval t

*n+1*- t*n*. Omission of this term later caused confusion in the correspondence with Keynes on Harrod's "Essay in Dynamic Theory" ( 1939:7 ): see in particular letter 832 .4. The discontinuity at the turning point was the topic of a "brisk little correspondence" between Harrod and Robertson about Tinbergen's review, which unfortunately does not survive: Robertson reported that

- I now believe that at heart I am on his [Harrod's] side! i.e. that though his world is too catastrophic, a scheme which starts from the assumption of sharp discontinuities is more likely to resemble the real world of cyclical fluctuations than a smooth system of sine curves. (Letter to Tinbergen, 1 July 1937, in LoN 10B/26666/12653)

Previously Robertson had held instead that Tinbergen had "put very neatly in 6 lines all my fumbling intuitions that H[arrod]'s whole apparatus makes nonsense unless a time element is explicitly introduced!" (Robertson to Tinbergen, 20 June 1937, in LoN 10B/26666/12653). Robertson was therefore looking forward to discussing the matter à trois with Harrod and Tinbergen at Geneva, where they all were to attend in September a meeting to discuss a draft of the first part of Tinbergen's Statistical Testing of Business Cycle Theories, 1939 (see letter 686 ).

5. J. V. Robinson, "The Trade Cycle by R.F. Harrod" (1936), p. 692.

6. After remarking that in a theory of the kind examined lags are essential for the possibility of explaining fluctuations, Tinbergen wrote (p. 90*):

- If one follows the author's procedure, it is apparent that he also is working with lags. In fact, in the crucial passages of his exposition (e.g. p. 94) he considers a period of time, which for convenience he calls a half day [in reality, Harrod is considering "a short period, say a day", in which "orders are homogeneous": p. 88] admitting, however, at a later stage that the period is probably longer (p. 95). He also expresses the view that one cannot neglect the length of the production process, especially in the case of investment goods (p. 88). Finally, he himself offers a successful elaboration of the multiplier proposition in that he makes a distinction between "short run" and "a slightly longer period" (p. 74). This indicates that he uses lags for his argument. [My translation--Ed.]