# J M Keynes Had Already Mastered the Mathematical and Logical Theory Required to Develop the Concept of the Multiplier in His a Treatise on Probability in 1921 (and in His 1908 Second Fellowship Dissertation Submitted at Cambridge, England) Long Before Kahn’s 1931 Article on the Employment Multiplier

20 Pages Posted: 9 Dec 2018

Date Written: November 15, 2018

### Abstract

The idea that Keynes learned about the concept of the Multiplier from Richard Kahn is demonstrated to be incorrect. Keynes had already completely worked out the specification of such a problem that involved taking the limit of a series that was declining, geometrical, infinite and showing that the answer was a finite number. Thus, Keynes was the first to develop the theory of the multiplier. However, Keynes may not have been the first to actually apply the theory of the multiplier. It is extremely probable that Keynes showed Richard Kahn how to take the limit of a geometrical, declining, infinite series of observations in order to arrive at a finite limiting value. Keynes then left it to Kahn to finish the job of doing the arithmetic.

The reason why Keynes’s mastery of the method of applying the multiplier is not known by economists was that economists failed to understand Keynes’s mathematical exposition as presented in the A Treatise on Probability in 1921. It is in chapter 26, on page 315, in footnote one, that Keynes applies the method of the multiplier to a problem involving insurance and re-insurance. The answer obtained by Keynes is mathematically identical in form to the investment multiplier derived and presented on page 115 of the General Theory once notational questions are satisfied. However, Keynes did not append a footnote dealing with how to take the limit of a geometrical, declining, infinite series of observations in order to arrive at a finite limiting value in the General Theory. He expects a reader of the General Theory to have covered this in a calculus course taken before reading the General Theory. Keynes was already aware in 1936, as was Samuelson, that economists needed to be able to apply a higher level of mathematics in economics that went beyond the mathematics of functions dealing with one independent variable and one dependent variable.

Keynes already knew in 1908 and in 1921 what the logical theory of the multiplier involved and entailed. He already knew what Kahn’s father explained to Richard Kahn in 1913. However, Keynes was not interested in having to take the time in order to have to actually work out the arithmetic of a process that he knew was correct for certain .As Kent (2007) has pointed out, Keynes had finally gotten around to doing the arithmetic for his speech for the Liberal party in March,1929. Keynes informed Kahn of this technique sometime between March, 1929 and August,1930, which is when Kahn states that he started work on his multiplier paper. Keynes was more than willing to help Kahn publish an important article that only indirectly referred to Keynes’s contribution.

Keynes studied under the most illustrious mathematicians of the first quarter of the 20th century, while simultaneously earning his BA in mathematics, taking 12th Wrangler, and beginning work on his TP in 1904.The truly silly belief, that Keynes was taught by Kahn to apply a concept taught to Keynes in his lower division work at Cambridge when he was preparing to finish his bachelor degree work, is truly mystifying.

**Keywords:** Keynes, Samuelson, Multiplier, Kahn, Infinite Series, Limit, Taking the Limit, Geometrical Series, Infinite Series, Declining Series

**JEL Classification:** B10, B12, B14, B16, B20, B22

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