# J M Keynes's Differential Calculus, D-Z Model of Chapters 3, 20, 21 and the Appendix to Chapter 19 of the General Theory Is a Full Blown Mathematical Model Like that in Pigou's 1933 The Theory of Unemployment and Is Not a Framework for Verbal Discussion

19 Pages Posted: 17 Apr 2017 Last revised: 26 Sep 2017

Date Written: April 15, 2017

### Abstract

J M Keynes's mathematical approach in all three of his major works, A Treatise on Probability, A Treatise on Money, and The General Theory of Employment, Interest and Money, was to provide his readers with the major, mathematical results of his formal, technical analysis. However, he expected his readers to have had the necessary training and knowledge of the required mathematical techniques, which he was using in order to be able to support his conclusions to fill in the missing initial and intermediate mathematical steps that he would not provide. This, of course, means that readers of these books, who are mathematically confused, illiterate, inept or innumerate, will not be able to follow what Keynes is doing technically in the General Theory or his other works. These readers will conclude that Keynes is only providing very general functional relationships that are not technically developed. This sort of reader of the General Theory usually adds on the following two claims. The first claim is that Keynes was a strict Marshallian, who would only provide a verbal summary of his privately, worked out, technical analysis in his published works. The second claim is to interpret Keynes's critique of pseudo mathematics on pp. 297-298 and p. 305 of the GT as an attack on the use of any mathematical and formal analysis whatsoever.

The latest author to take this approach is Roger Backhouse (2010), who argues that Keynes used the word "curve" only to describe the classical and neoclassical special, mathematical analysis that was being used to support their claims while using the word "function" only to describe his own general analysis, which could not be formally developed. It is a straightforward matter to demonstrate that there is no support for any of Backhouse's claims, who demonstrates repeatedly that he does not know the difference between algebra and differential calculus, which makes use of algebra. Keynes used the terms "curve" and "function" to describe his own work in the GT. He also used these terms to describe the 'classical' school which he demonstrated was a special case of his own GT. Keynes's framework actually reaches back, like the work of Adam Smith, to Aristotle's general theory of money, ethics, economics, politics, and civics.

Keynes did not use the word "function" to only describe his own analysis in the General Theory. Keynes did not use the word "curve" to only describe the classical analysis. The words "curve" and "function" have been completely misapplied, misconstrued, misused, and misinterpreted by Backhouse, Carabelli and Cedrini, and by Wray.

Keynes's Aggregate Supply Function, is a linear, straight line curve. Keynes's Aggregate Demand Function, D, is a concave curve. Keynes's Aggregate Supply Curve, a locus of the set of all possible Z and D intersections, is a convex curve by design. Keynes's Liquidity Preference Demand for Money function, L, when combined with M, the supply of money, is a curve that is identical to the Figure 2 curve in Hicks's 1937 article in Econometrica.

There is nothing original, new, novel, innovative, or creative in Hick's 1937 article on IS-LM except for one thing. He actually drew the IS-LM curves as exactly described by Keynes in the GT on p. 207 of the General Theory. Hicks's exposition is inferior to Champernowneâ€™s model and exposition.

It is quite disconcerting to realize that Keynes's GT still has not been read carefully and methodically some 81 years after its publication.

**Keywords:** curve, function, schedule first derivative, second derivative, Keynes, Hicks, Champernowne

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

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