J M Keynes’s Mathematical Style: Very Concise, Precise , and Exact

Abstract

Keynes’s mathematical style, starting with his First Fellowship Dissertation in 1907 for Cambridge University, England, through his formulation of a linear, first order difference equation that incorporated the interaction of the Multiplier and Accelerator( called the Relation) for Harrod’s use in his August,1938 correspondence with Harrod, and ending with his exchanges over probability and statistics with J. Tinbergen, an advocate of the Limiting Frequency Interpretation of Probability in 1939-40,was always very concise, precise and exact. Specifically, Keynes always provided the first steps in a mathematical analysis and the last step. However, he would rarely put in the intermediate steps. Keynes’s view was that he always provided a clear, literary, prose explanation of his analysis that would allow any reader of his work to grasp the same basic,fundamental points that were being made in the mathematical analysis. A reader concentrating on Keynes’s supplementary mathematical analysis would also grasp the basic fundamental point being made. The intermediate mathematical steps in a Keynesian analysis need to be formulated by working backwards from the final step by the reader. This is precisely what economists have failed to do. They have been unable to generate the intermediate steps that connect the first and last steps. For instance, the results that Keynes presented in the General Theory regarding his IS-LM(LP) and D-Z models were never correctly grasped by any mainstream economist in the 20th and 21st centuries because they were not able to reconstruct the mathematical analysis in chapters 20 and 21 of the General Theory. This paper will concentrate on illustrating Keynes’s style by analyzing the mathematical connection between Keynes’s mathematical results in chapter 26 on page 315 in footnote one of the A Treatise on Probability and the identical type of result that appears on page 183 of Richard Kahn’s June, 1931 article in the Economic Journal. It will be demonstrated that, once the differences in mathematical notation are compared and taken into account, the results presented by Kahn are identical to those presented by Keynes in 1921 in the A Treatise on Probability and reflect a style of presentation that is identical to Keynes’s mathematical style. We will also examine the belief among modern economists that Keynes’s General Theory does not use math except for defining or conceptualizing relationships, in the form of y =f(x) or z=g(q) type notation, like the D=ф(N) and Z=θ(N) specifications that appear in chapter 3 of the General Theory. Such economists will be completely lost when they reach chapters 20 and 21 of the General Theory, where Keynes derives and presents his actual mathematical models.