821. J. M. Keynes to Harrod , [29 August 1938] [a]
[Replies to 819 , answered by 827 ]
[29 August 1938] 
Interim minority report
Y income or output during unit of time assumed;
s proportion of income saved during any period;
DS increment of saving for unit of time assumed
so that .
= increment of capital, corresponding to increment of income at a rate DY per unit of time, required by "normal" technological considerations.
Then the warranted proportional growth during the unit of time assumed , where t is a function of the unit of time adopted but s is not.
Suppose, however, that ex-ante growth of output exceeds the warranted growth in the assumed unit of time by an amount X, then (I say) you are assuming that
t(DY w + X) > s(Y + X)
where t and s have the same values as before, i.e. that t > s (for that unit of time).  The longer the unit of time, the less likely is this to be true. The relevant unit of time is presumably the interval which has to elapse before previous decisions can be effectively revised in the light of current facts. In any likely actual circumstances, however, there should be plenty of margin to ensure that in fact t > s for the appropriate time unit. Thus with the basic assumption that t > s I agree with your conclusion as to instability. But I still insist on the necessity of this basic assumption.
But there is a lot else to take account of. t(DY w + X) must equal s(Y + X) at the values of t and s actually prevailing. Thus if growth is unwarranted, t and s are forced from their previous "normal" values. The extent of the changes in t and s depends on the marginal values of t and s for short-period (unexpected) changes in output. Haven't you, therefore, to make some assumption as to the readiness of t and s to respond in such circumstances?
This is leading into deep and (probably unnecessarily) complicated water. My present point is merely that you cannot assume absolute rigidity of s and t and a departure from warranted growth. You have to make some assumption as to the changes in s and t in unwarranted conditions. Moreover in this context it is the marginal s and t that matter. The actual course of the cycle will be largely influenced by the short-period factors.
At the same time the point is important. For the actual values of t and s must always be such that
t(DY w + X) > s(Y + X)
which means that ex-post the marginal value of t corresponding to the increment of output X must equal the corresponding marginal value of s. Now if (as in your numerical example  ) the average value of t is 50 times s, this means that the distortion of the marginal from the average values of the quantities is very violent in the event of unwarranted output. The fact is, of course, that there are large fluctuations in unused capacity (or in capacity employed below its maximum) or in inventories or in both, and (especially for very short periods) in the marginal propensity to consume. You cannot assume all these complications away and yet allow unwarranted output to occur. But you are justified in pointing out that t will only fall below its normal if the prices of investment goods rise and s will only rise above its normal if the prices of consumption goods rise or incomes are redistributed in favour of profit. [b]
Consider such a case as this. If there is all-round surplus capacity (both plant and inventories), the position may be stable in the sense that an unwarranted increase in output will be followed by a relapse; for it will increase DS without involving any increased demand for investment. In fact there is probably no moment in the trade cycle at which t has its "normal" value in the sense that the actual values of C and Y are so related that C = t 0 Y where t 0 is the technologically "normal" value of t. Thus you are assuming that there is no surplus capacity. [c]
In general, you must prove your point from stated assumptions, and not merely assert it as an intuition from unstated ones--especially when the assumptions are (avowedly) unrealistic and therefore not easily supplied by the reader. The above points are not intended as fundamental--they are directed to elucidating the nature of the assumptions which you are making in your schematism. [d]
J. M. K.
2. Keynes's statement is inaccurate, as s and t are not homogeneous quantities. An explicit reference to the time interval would have clarified Keynes's thought, as the instability condition would have been
, where J stands for time: see Keynes's letter 832 of 9 September and, for a comment, D. Besomi, "From The Trade Cycle to the `Essay in Dynamic Theory': The Harrod-Keynes Correspondence, 1937-38", History of Political Economy 27:2, 1995, in particular pp. 328-29n.
3. Letter 814 , [jump to page] .
- a. TDI, with autograph corrections, formulas and additions, three pages on three leaves, numbered from the second, in HP II-87. CcI, corrected, in JMK EJ/1/5/313-15. At the top of page 1 there is the indication «R. F. H.» in type. Printed in Keynes, Collected Writings, vol. XIV, pp. 333-35. Reproduced by kind permission of the Provost and Scholars, King's College, Cambridge.
b. This sentence, autographed, was added as an afterthought.
c. This sentence, autographed, was added as an afterthought.
d. This sentence, autographed, was added as an afterthought.
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