840. Harrod to J. M. Keynes , 22 September 1938 [a]
[Replies to 838 , answered by 844 ]
22 September 1938
I apologize for worrying you again. But I am worried by your reaction. You seem to have some substantial objection to my argument which I am quite unable to see. I cannot feel that my mind will move in the coming months since I cannot comprehend the force of your criticism.
It may be that I have expressed myself very badly, tho' as I read my thing it appears to me lucid enough. On the other hand I build my hopes on the idea that you are yourself so much interested in kindred problems that you have been psychologically inhibited from reading what I have actually written. This is suggested by your arithmetical example.
You complain of lack of proof of my instability propositions. The proof is given subject to the proviso that the propensity to save does not change. In all economic arguments we have to assume some variables constant pro tem. But of course you will object that it is begging all questions to assume propensity to save (s) constant.
I might make the formal point that it is not to be supposed that s changes in response to changes in the rate of growth, but only to changes in the level of income. A changed rate of growth will have to endure some time before an appreciable consequential change in the level of income occurs.
But I will not take that point, and consider the reaction through time straight away. What I now say is that my proposition regarding instability, rigidly proved for s constant, also holds with s variable save in unlikely circumstances. That seems to me to be a footnote matter. It is desirable for logical reasons that the failure of my proof in certain abnormal conditions should be recognized, but it is not of much consequence from a practical point of view.
Now in one breath you say that my assumptions are reasonable, but elsewhere you suggest that failure to take account of the other possibility vitiates the article generally. This is what I dont understand.
What appears to me is that you have not grasped the force of the proof I have given, the force of the point, namely, that in the warranted position the saving from the whole of income just suffices to supply the extra capital required in association with an increment of turnover. If that is so, how unlikely it is that if you make a trial extra increment of output, the saving from that increment of income suffices to "support" the increment of turnover. The marginal propensity to save must be so immensely much greater than the average propensity. Just how much greater may be stated precisely. I forget what I said exactly in former letters, but the proposition is this. For your condition of neutral equilibrium to apply it is necessary that the ratio of marginal propensity to save to average propensity should be as great as the ratio of total income to increment of income.  For the condition of stable equilibrium, the marginal propensity to save must be greater still. I think this may be grasped intuitively, but a formal proof is appended at the end. Now the reason why this seems to me a footnote point is that this condition is not likely to be realized.
Take figures. Suppose warranted rate of 3%. This means that if your condition of neutral equilibrium is to be realized, even taking marginal propensity to save as 100% (which is absurd, if we remember wage-earners) the average propensity to save can only be 3%, which is unduly [b] low. If marginal s is 50%, average s must be %! Is this a likely state of affairs? Does the neglect of it vitiate my whole article? I cannot but feel that you havent given what is vouchsafed by the per cent warranted equation adequate consideration. And this is my main point.
You might object (tho' you do not) that I am giving too much emphasis to the acceleration principle. Let us bring in long range capital outlay (my K).  If K accounts for half all savings in the warranted position, you still have to assume for neutral equilibrium that with marginal s at 50% average s is only 3%. You may want to assume a still larger K, e.g. absorption in this way of ths of saving, which I am sure is too large for the real world, considering what happens to capital outlay in a slump. But even if we can swallow that, other absurdities arise. This assumption does indeed raise the average propensity to save required for your neutral equilibrium, assuming 50% marginal s, to %, but since only % is used to support current output this means that with warranted growth of 3% that the capital required for current output is only 6 months' [c] income. And this is surely far too low. If we take less exaggerated figures for marginal s and K, the capital deemed to be required is reduced in proportion. These curious combinations may at times occur, but if they are very improbable, I think my case for instability is good enough.
What particularly distresses me is your example. For it is the sort of example which a man might give who had never read my arguments at all. On the face of it, it is absurd to suppose extra capital required only of annual output, when the capital required in association with the pre-existent level of incomes in England to day is 4 or 5 times annual output. But apart from that it assumes a fantastic warranted position, which you do not define; it assumes namely if average s = a warranted increase of 100% per annum, if average s = a warranted increase of 50% etc. It is no good saying "get away from your formula"; what you are really asking me to do is to get away from my major premise (tho' you do not criticise that).
You are bound to get these fantastic results if the point of departure assumed is a warranted position, that is if the saving from the whole of income just suffices to "support" the increment of turnover. Of course if you take for your point of departure some other position, one, say, in which saving from the whole of income exceeds that required to "support" the increment, it is possible applying the marginal method to get results like yours. But such results would not show neutral or stable equilibrium since the point of departure would not be an equilibrium position at all. A point that is not an equilibrium position cannot be one of stable or neutral equilibrium. In the example given in this paragraph, the point of departure would be one in which strong depressive forces were operating. Thus, I fear, your criticism does not move me. I am sure that I have something important to say; I think my account of dynamic instability much more systematic than anything I have seen. But whether I have said it is another matter. There you make me feel un-comfortable!
To show how large marginal saving must be by comparison with average saving to get neutral equilibrium. 
is average propensity to save in initial warranted position.
is average propensity to save in trial position, assuming neutral equilibrium.
s m is marginal propensity to save required for neutral equilibrium.
c is capital coefficient, i.e. amount of capital required per unit increment of output measured in those units.
x is warranted increment of output, measured in absolute units.
y is total income.
1. (desired investment = saving) (in warranted position).
2. [d] (derived from above).
3. If equilibrium is neutral, [e] (i.e. desired investment = saving in trial position).
4. (combining 2 and 3) \
(av[erage]. saving of new income = av[erage]. saving of old income + marginal saving of increment).
\ (combining 4 and 5)
i.e. the ratio of the marginal propensity to save to average propensity to save must be as high as the ratio of total income to warranted increment of income.
2. Harrod, "Essay in Dynamic Theory" ( 1939:7 ), pp. 26-27; preliminary draft, essay 19 , section 10, [jump to page] .
3. Harrod added two new sections (10 and 13) to the "Essay in Dynamic Theory" to argue this point: 1939:7 , pp. 24-26 and 27-28 (see letter 879 , [jump to page] ).
- a. ALS, five pages on five leaves, in JMK EJ/1/5/332-36. Printed in Keynes, Collected Writings, vol. XIV, pp. 342-45.
b. Ms: «unduely».
c. Ms: «months».
d. Ms: Harrod omitted the brackets on the left hand side of the equation.
e. Ms: Harrod omitted the brackets on both sides of the equation.
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