618. [D. H. Robertson's note on Foreign Trade Multiplier] [a] , 
[Attached to a letter not found,  answered by 619 ]
Let Y = income
V = home investment
X = exports
M = imports
p = proportion of income spent on home-produced consumption goods.
q = proportion of income spent on [b] imports.
Then (i) (Harrod's equation.--Trade Cycle, p. 149).
(ii) (Keynes' equation 1 ).
Since M = qY, the two equations are equivalent; but form (ii) suggests that (X - M) and q have a causal significance which form (i) makes it plain that they have not.
Thus let us start with the values in pencil,  and consider two cases:
(1) X rises to 35 owing to enhanced foreign demand, p and V remaining unchanged. Equation (1) shows at once that income becomes 150. This is compatible with
(2) p is reduced by "liberalising" measures from 7/10 to 6/10, X and V remaining unchanged. Equation (i) shows at once that income becomes 75. This is compatible with
2. This note was first sent to Keynes on 28 January 1937, as part of an exchange regarding the consequences at home of the different ways in which the part of income disbursed at home is expended. In the accompanying letter, Robertson described the point "on which Harrod has always insisted" as follows: if, "for the sake of argument", one accepts the multiplier method and its implication, "the level of income does not depend on what happens to the balance of payments on economic account, but on what happens to (a) the credit side of that balance (`exports'), (b) the proportion of income spent on home-produced goods." "It does not make any difference to home activity whether Englishmen spend on bonds or on bananas that part of their income which they disburse abroad." (Robertson to Keynes, in JMK CO/3/44). In his reply, Keynes asked "Where's Harrod's equation to be found?", and confessed that he did not understand it. Since he interpreted p and q to be proportions of consumption and not of income, so that p+q=1, it would follow that M =
= = V+X, "from which it follows that home investment is always equal to the adverse balance of trade" (Keynes to Robertson, undated, in JMK CO/3/46). Robertson replied that he always took p and q as proportions of income, and referred to his pencilled arithmetical illustration (see note 4 to this letter). He provided the reference to Harrod's Trade Cycle ( 1936:8 ), pp. 145 and following, in particular pp. 149 and 155-56. Finally, Robertson announced that he intended to ask Harrod to check whether his exposition was correct (Robertson to Keynes, undated, in JMK CO/3/47).
3. J. M. Keynes, The Means to Prosperity, 1933 (pp. 10-36 correspond to pp. 339-65 of the version printed in Keynes's Collected Writings, vol. IX, save for the alterations there indicated).
4. The following values were indicated in pencil, written above the variables in equations (i) and (ii): Y=100; V=10; M=20; p=7/10; X=20; q=6/30.
- a. AD, one page, in DHR 3/2 11 and JMK CO/3/45 (enclosed with a letter to Keynes dated 28 January 1939).
b. Ms: five words dittoed.
1. Inferred from "The Means to Prosperity", pp. 10 & 36  . [Robertson's footnote].
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