# Keynes, and Only J. M. Keynes, Was Responsible for the Logical and Mathematical Development of the Multiplier Concept in 1921(1908) in His a Treatise on Probability (1921) That He Later Used in the General Theory (1936)

16 Pages Posted: 5 Dec 2018 Last revised: 11 May 2019

Date Written: November 10, 2018

### Abstract

J M Keynes had already developed the theory of the Multiplier concept mathematically, logically, and technically in his "A Treatise on Probability" (1921). The same analysis can be found in his second Cambridge Fellowship Dissertation of 1908.

Samuelson explicitly covered the material, Keynes’s risk (R) formula, presented by Keynes on page 315 of the A Treatise on Probability, in his 1977 article in the Journal of Economic Literature. However, Samuelson overlooked the footnote, footnote 1 on page 315 of the A Treatise on Probability, in which Keynes applies an explicit multiplier analysis to a geometric, declining, infinite series because Keynes left out the intermediate steps of taking the limit of the series as n, the number of terms in the series, approached infinity. Keynes simply gave the final answer one will obtain after he has taken the limit. Samuelson also overlooked Keynes’s generalized risk analysis on page 353 of the A Treatise on Probability ,which extended Keynes’s Risk analysis ,which used Chebyshev’s Inequality to derive a lower bound .This would be an imprecise probability that would become more accurate as more observations were obtained, leading eventually to a precise estimate of probability from the normal distribution if the decision maker could afford to delay action for the period of time needed.

R. Kent’s 2007 article in the History of Political Economy leaves completely unresolved the issue of where Richard Kahn got the idea for the use of the multiplier from. Kahn, in 1936, stated, in a note on a paper of Neisser’s that appeared in the Review of Economic Statistics, that “…my own ideas were largely derived from Mr. Keynes.” (Kahn,1936, p.144).

Kahn’s contribution originated in private conversations with Keynes, where Keynes showed Kahn his chapter 26 analysis contained in the A treatise on probability on page 315 in footnote 1. Keynes, and no one else in history, was the person who had already originated the mathematical theory of the multiplier, which Keynes in the General Theory called the logical theory of the multiplier. Keynes's discussion of the logical theory of the Multiplier on pp.122-123 of the General Theory is simply a literary description of the mathematical analysis presented by Keynes on page 315 of the A Treatise on Probability. Kent's belief that Keynes had presented a multiplier analysis in 1929 is correct. The issue of whether Keynes made an arithmetic error in adding up the terms of the series is completely irrelevant to the main issue, which is “who is the person who created the theory of the multiplier?”

**Keywords:** Keynes Multiplier Kahn a Treatise on Probability General Theory Kent Declining, Infinite, Geometric Series

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

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