# On Keynes’s Formal Development of the Logical Theory of the Multiplier in the A Treatise on Probability in 1921: It Was Keynes Who Helped Kahn, Not Kahn Who Helped Keynes

19 Pages Posted: 11 Dec 2018

Date Written: November 18, 2018

### Abstract

Keynes had completely developed the Logical Theory of the Multiplier in his A Treatise on Probability in 1921 in chapter 26 on page 315 and in footnote 1 on page 315. This same analysis appears in his second, 1908, Fellowship Dissertation at Cambridge University. Keynes, however, had no interest in actually publishing a worked out application of the Logical Theory of the Multiplier where one would use arithmetic to actually calculate a worked out example of the process. As pointed out by Kent, Keynes did work out all of the arithmetic steps involved in the multiplier process in 1929, but also made an arithmetic error which he was not concerned with since he already knew that the Multiplier process was mathematically and logically airtight.

Keynes left the actual presentation of the arithmetic of a completely worked out example of a Multiplier problem, which would be based on the Logical Theory of the Multiplier that he had already worked out in 1908 and 1921, to Kahn. The mathematics of the Investment multiplier, presented on pp. 114-115 of chapter 10 of the General Theory in 1936, is identical to the mathematics used by Keynes in both 1908 and 1921 with respect to his example involving a series of reinsurances aimed at shifting the risk.

The main problem in the vast literature of the Keynesian Multiplier concept is that no economist was familiar with Keynes’s Risk model in chapter 26 of the A Treatise on Probability. The only economist in the 20th Century to take Keynes’s risk model seriously was Paul Samuelson in a paper published in 1977 in the Journal of Economic Literature. Unfortunately, Samuelson overlooked the footnote that contained Keynes’s worked out analysis in which he took the limit of a geometrical, infinite, declining series to arrive at a finite single number answer.

**Keywords:** risk,multiplier, Kahn Keynes, A treatise on probability, geometric declining, infinite series with a finite limit

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

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