844. J. M. Keynes to Harrod , 26 September 1938 [a]
[Replies to 840 , answered by 879 ]
Tilton, Firle, Lewes. #
26 September 1938
As a result of your last letter, I have, at last, seen in a flash what it is all about. My intuition told me that your conclusion could not be true in general, but only subject to specified conditions; and I have been trying, confusedly, to discover what assumptions your conclusion required without the clue, until I got your latest letter, of knowing what argument it was that you found convincing. Where I went wrong appears in what follows.
Your argument as now expounded shows, quite correctly, that if there is a warranted rate of growth, an increase in excess of this rate will lead to the results you indicate. But this assumes that there is a warranted rate. That is the basic assumption I have been allowing you to get away with. In general there is no warranted rate, and special conditions are required for a warranted rate to be possible. The conditions, which I have been trying to establish as being necessary for a rate in excess of those warranted to satisfy your requirements, are in fact those which are necessary for there to be a warranted rate at all.
Take the example which I gave in my last letter.  That example did not depend (as you suppose) on a variability in s.  Nor were my assumptions in any way unrealistic;  for a community without durable goods my quantitative assumptions are quite plausible. What I ought to have pointed out is that with the assumptions in question no rate of growth is warranted. With every rate of growth saving will be excessive.
Indeed, once one's mind is directed to the point, it becomes obvious. Three variables are concerned--(i) the amount of additional capital normally required to provide a given increment of daily income, (ii) the proportion of daily income saved, (iii) the length of the interval at which the geometrical rate of growth is compounded.
(i) If C is the amount of capital and Y the amount of daily income, let DC = tDY where DC is the capital appropriate to an increase DY in daily income. I stress that it is which is relevant, not .
(ii) Let be the amount of daily saving corresponding to an income Y. I agree with you that the average, not the marginal, propensity is relevant;--it is not worth bothering about a possible change in s, corresponding to the change from Y to Y + DY.
(iii) Let the interval elapsing between new sets of entrepreneurs' decisions be m days; i.e. we suppose that a planning committee or Board of Directors meets every m days and that during those m days income and output are at an unchanged level. I am not quite clear what determines the relevant value of m in practice. But it seems to me to depend on the period of production.
Now if there is no warranted rate (i)
if there is one determinate warranted rate, namely (ii) 
You are dealing exclusively with case (ii), and is, as I said before, the assumption necessary to validate your conclusions.
So much for the formal argument, which is what has been troubling me hitherto. As soon as you explain that this is your basic assumption, all goes swimmingly; except that for reasons given below, you require in practice that t should be much greater than .
There remains the question whether actual experience can be safely assumed to conform to case (ii); or whether case (i) also deserves exploring. At first I was ready to concede that actual experience conforms to your case (ii) and I was ready to neglect case (i) in practice; but on reflection I feel that both cases have to be considered if one's mind is to be clear, and that approximations to case (i) are not impossible.
There is, first of all, the question of the value of m. The larger m is, the greater the risk of case (i). If you concede that m is a year, you will, I think, be giving enough away and weighting the argument with fully sufficient fairness in favour of case (i). A year should give industry long enough to appreciate the results of its existing scale of output and to revise them if necessary. Having made this assumption, we can get rid of m from our equations by defining t in terms of annual, instead of daily, income. But it is dangerous to drop out m in the first instance, because it confuses the dimensions to drop the time dimension out of sight by defining it as unity. (This is a point about which a mathematician is very particular.)
Now if t were to be equal to where our unit of time is a year, I agree that anything approximating to case (i) would be extremely unlikely in the modern world. For is of the order of magnitude of 4, whilst s is more like to . But with the position is not so clear. If growth represents a growth of population and not of standard of life, there is, indeed, no reason for much difference between and , and you will be on strong ground. But let us suppose a stationary population in an old country, the rate of growth relating solely to the standard of life; or, even worse, a declining population. Let us suppose that the assumed rate of interest has been in force for a long time, so that all the capital which can be advantageously employed at that rate of interest with the existing level of income is already employed. Let us suppose that the transport system and public utilities and housing are all on a satisfactory standard, and no foreign investment. Let us suppose disarmament, sinking funds rampant, distribution of profits conservative and belated, incomes unequally divided, fairly good employment so that there is no large dissaving on this head. In such circumstances we might have . Is it certain that t will be much greater than s?
For we need not be so extreme, for practical purposes, as to suppose t to be no greater than s. Let us admit that t > s, so that a warranted rate is theoretically possible. Nevertheless, unless t is much greater than s, the warranted rate will have to be so great as not to be practicably feasible, since the risks of acting on a sufficiently drastic scale will be too great for the entrepreneurs. For example if t is ten times s, the warranted rate of growth will have to be 11 per cent per annum. (For 10DY = Y + DY). If you imagine a prospect of this persisting year after year with a stationary population, the change in the direction of new demand and of technique will be so great and so unpredictable as to baffle the entrepreneurs. The rate of general change will be too great to allow investment in durable instruments to be safe. Yet it is precisely this that you require. With stationary population, no abnormal government expenditure, and an unequal distribution of incomes, I doubt if the warranted rate can be attained in practice unless it approaches 40 times the value of s (i.e. a warranted rate not exceeding per cent per annum). Yet it is quite likely that may be considerably less than 40.
One may mention in passing that, if we start from a position where there is unused capacity, t may be temporarily very small; and it may be difficult to surmount the position where temporarily s > t.
In actual conditions, therefore, I suspect the difficulty is, not that a rate in excess of the warranted is unstable, but that the warranted rate itself is so high that with private risk-taking no one dares to attain it, except momentarily by [b] accident. In prosperous conditions in U.S.A. to-day it may be that t is nearer ten times s than forty times; yet an all round continuing expansion at the rate of 11 per cent per annum is simply not practicable.
In such a case the remedy or mitigation must take the form of either
(i) a burst of new inventions of the right kind, or a fall in the rate of interest sufficiently violent to introduce drastic changes of technique, so as substantially to increase t, temporarily at least; or
(ii) abnormal government loan expenditure on unproductive purposes or a drastic change in the distribution of incomes so as to reduce s.
Otherwise t will not sufficiently exceed s, except for brief periods when the arrears of a sufficiently prolonged previous slump are being overtaken. I doubt if, in fact, the warranted rate--let alone an unstable excess beyond the warranted--has ever been reached in U.S.A. and U.K. since the war, except perhaps in 1920 in U.K. and 1928 in U.S.A. With a stationary population, peace and unequal incomes, the warranted rate sets a pace which a private risk-taking economy cannot normally reach and can never maintain.
Well, you see that I find your analysis very interesting, clarifying and thought-provoking. And there is no reason why in this article you need consider in detail case (i) or cases sufficiently approximating to case (i) to raise practical difficulties. Case (ii), theoretically at least, is well worth studying for its own sake. But you make it, I think, impossibly difficult for your readers if you don't distinguish the two cases and explain the assumptions necessary for case (ii), before concentrating on your analysis of the characteristics of this case. On the other hand, once you have made the distinction and the necessary assumption, you can get the intuition of your readers working on the right lines and they will see what it is all about.
J M Keynes
I have found it rather comforting in these days to muddle one's head with this sort of stuff.
R. F. Harrod Esq., Christ Church, Oxford.
2. Letter 840 , [jump to page] .
3. Letter 840 , [jump to page] .
4. See note 2 to letter 821 .
- a. TLS with autograph corrections and addition, nine pages on nine leaves, in HP II-89. Corrected CcI in JMK EJ/1/5/337-45. Printed in Keynes, Collected Writings, vol. XIV, pp. 345-50. Reproduced by kind permission of the Provost and Scholars, King's College, Cambridge.
b. Ts: «momentarilyby».
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