# Mathematical Interpretations of Keynes’s General Theory

Introduction
Many changes had occurred in the last century in every science. Economics was not the exception and a transformation into the theoretical approach took place in the 30s. In 1936 Keynes published “The General Theory of Employment, Interest and Money”. Its main ideas set the foundations of macroeconomics and had influenced economics since.
Unfortunately, Keynes had (and has) a bad reputation as a writer, because of this there has been constant analysis and interpretations of what he meant. Economists affirmed that Keynes’s innovating work, as any new approach, had some “inconsistencies”. Further, the book had small use of mathematics and, thus, had small proof of what it stated. Therefore, these ideas need interpretation and testing into the “real world”.

Keynes Main Ideas
Kriesler and Nevile (2000) define the book’s main points as follows: “…in a capitalist economy employment, and hence unemployment, is determined by effective demand […] decisions about production and investment are made on the basis of expectations”, and “monetary variables influence real variables such as output and employment and real variables, in turn, influence monetary ones.”
One of the most daunting tasks was to set up a mathematical foundation of the cited Keynes’s works. The most important economists who performed this job were, in a “crossed fertilization process” as Heller (2000) points it, Roy Harrod, James Meade and mainly John Hicks. In fact, in 1937 Hicks published an influential article, “Mr. Keynes and the Classics: A suggested interpretation” that rapidly became the standard of Keynesian economics theory.
Mathematical Foundations
The Keynes’s ideas stated in the paragraphs above turned “into simple mathematical models of the macro-economy; [and] the most long-lived and flexible, the “IS-LM model”, came from John R. Hicks” (Morgan 2001). Despite that, some authors suggest that Harrod and Meade had inspired the IS-LM model proposed by Hicks (Young, qt. in Heller, 2002)[1]. Others suggest that Champernowne and Reddaway also had some contribution into the model’s formalization (Barens 1998, qt in Heller, 2002)
Besides the multiple interpretations of Keynes’s work, Hicks’ one remains as one of the most important.
As pointed by Heller (2000), the models proposed as explanations had similar expressions which are systems of simultaneous equations. And the mathematical formulation’s success of Keynes’s General Theory is because of the “mathematical elegance” and the exact nature of systems of simultaneous equations, which many consider alike; and particularly to Hicks who was the only one in representing the theory through diagrams. (Heller, 2002)
As pointed, Harrod was another economist who performed a mathematical foundation for Keynes’s work. According to him, Keynes system corrects the traditional theory; allowing the Income level to be not given, the price level does not depend on the money, and money demand is divided in two. Due to all of this Harrod states that Keynes system is better than the traditional one. Keynes had a very positive view of Harrod’s development of his own work, which Harrods presented in the same conference where Hicks did. Hence, Keynes thought that Harrod works interpreted correctly his own ideas.
But these authors’ success in interpreting Keynes’s ideas is not free of detractors. Kriesler and Nevile (2000) made a clear stand of this: economists reject the “IS-LM framework as being neither a valid simplification of the arguments in the General Theory nor a reliable model for analyzing macroeconomic issues.” Economists think the IS-LM model ignores expectations and it is not useful to analyze a particular economy beside the static equilibrium[2]. They also affirm that Hicks took Keynesian macroeconomics to another direction from the one intended by the work’s author.  The same authors assert that Keynes’s own vision on Hicks model “did have the faults that post Keynesians typically ascribe to IS-LM.”[3]
Conclusion
Keynes’s work and the posterior mathematical development gave, to governments and to economists, answers that can be easily explained and understood by everyone only using analytical tools as diagrams or simple mathematics. What is more, those ideas were beyond the economic “common sense” of the time: governments can spend during depressions. Affirming if these formalizations of the model are what Keynes’s work pointed might be intimidating. Regardless of this and the different views, Hicks simplification and formalization of the model is useful, and catches the spirit of the work. As any first interpretation improvements were (and still are) made but the first task was accomplished.
References
Heller, Claudia, The General Theory of Employment, Interest and Money According to Brian Reddaway Economia em Revista, Vol. 10, pp. 15-32, 2002
Heller, Claudia, The ‘General Theory’ Synthesis According to Roy Harrod in ‘Mr. Keynes and Traditional Theory’, Revista de Economia (Curitiba), Vol. 23, pp. 27-49, 2000
Keynes, John Maynard, The General Theory of Employment, Interest, and Money (New York: Prometheus Books, 1997).
Morgan, Mary, The formation of “Modern” Economics: Engineering and Ideology, Department of Economic History, London School of Economics, May 2001, Available at ;http://www.lse.ac.uk/economicHistory/home.aspx;
[1] Young proposed that to call it the “IS-LM Harrod-Meade” model.
[2] It is important to make clear that many have pointed rational expectations as one of the inconsistencies in Keynes’s work.
[3] Post Keynesian economists main critic to the IS-LM model is its static equilibrium nature, thus has no means to deal economy’s path of adjustment.

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