# On Keynes's a Treatise on Probability and General Theory: Why the Basic, Essential Foundations for Both Works Were Not Understood by Economists and Philosophers

27 Pages Posted: 5 Feb 2018

See all articles by Michael Emmett Brady

California State University, Dominguez Hills

Date Written: January 26, 2018

### Abstract

The failure of economists and philosophers to grasp and understand the basic, essential, logical, mathematical and ethical building blocks that were used by Keynes in his construction of the A Treatise on Probability and General Theory explain why Keynes’s work in ethics, decision theory and macroeconomics is caricatured by the question, “What did Keynes really, really mean?”

The foundation for both the A Treatise on Probability and General Theory is Keynes’s interval valued approach to probability, based on G. Boole’s 1854 The Laws of Thought, combined with his related concept of the weight of the evidence (not the weight of the argument), which underlies Keynes’s concept of uncertainty in the General Theory.

There are two basic kinds of probabilities that Keynes used to operationalize his logical theory of probability, indeterminate, non additive, non linear interval valued probabilities and imprecise, interval valued probabilities based on Boole’s interval valued upper and lower probabilities. Part II of the A Treatise on Probability provided the first axiomatic foundation for additive, linear, mathematical probability and an explicit discussion of non additivity in chapters 10-14. Chapters 15-17 provided Keynes’s own version of Boole’s technical apparatus. Chapter 26 of the A Treatise on Probability summarizes Keynes’s interval valued approach and provides a mathematical decision rule that Keynes called a conventional coefficient of weight and risk, c. Keynes’s c is the first decision rule in history that provides a generalization of the linear and additive probability calculus that extends the theory of decision making to include both non linear and non additive decision weights. In Chapter 29 of the A Treatise on Probability, Keynes provided the first “safety first” approach using lower bounds to establish imprecise probabilities.

Keynes’s General Theory is based on Keynes’s own unique, certainty equivalent approach in order to incorporate his interval valued and weight concepts from the A Treatise on Probability. There are two different models in the General Theory-the microeconomic foundations in the theory of the firm, production function, and labor market are provided by the expectational D (aggregate demand function)-Z (aggregate supply function) model that allowed Keynes to construct his Aggregate Supply curve (ASC), a locus of all possible D-Z intersections that are optimal for the entrepreneur. Only one of these intersections will be actually realized. The realized value is called Y. This is used to provide a specific Y, actual or realized, value that is then used in Keynes’s second model-the IS-LP(LM) model. This model is composed of two parts. The first part, Y=C I=C S, leading to the I=S, or IS, equation, incorporates the actual consumption function, marginal propensity to consume, investment function, and investment multiplier. The second part incorporates the L=M, or LM, equation given by Keynes on page 199. It specifies the demand and supply of money, or Liquidity Preference Function. The IS-LM model is not independent from the D-Z-ASC analysis. Keynes divides everything through by w, the money wage, so that questions of money wage flexibility or stickiness will have no bearing on the set of possible full employment or involuntary unemployment results. Changes in w simply shift the ASC up or down. They have absolutely no effect on any of the set of D-Z equilibriums.

Keynes brings the IS-LP(LM) and D-Z-ASC models together in chapter 21 of the General Theory in sections IV to VI. One needs to have worked through chapter 20 to understand chapter 21.

There is no theory of ordinal probability developed in the A Treatise on Probability. Of course, Keynes’s theory easily allows applications of ordinal probability to be applied. However, Keynes’s theory is mainly a development of Boole’s. There is no ordinal theory of probability developed in Boole’s 1854 The Laws of Thought.

There is no conflict between the IS-LP (LM) model and the D-Z model because Keynes is the original creator of both IS-LP(LM) and the D-Z model. Each model plays an important role.

Keywords: IS-LM, IS-LP(LM), Reddaway, Champernowne, Keynes, chapter 21, chapter 15, Keynes's views of math

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

Suggested Citation

Brady, Michael Emmett, On Keynes's a Treatise on Probability and General Theory: Why the Basic, Essential Foundations for Both Works Were Not Understood by Economists and Philosophers (January 26, 2018). Available at SSRN: https://ssrn.com/abstract=3111061 or http://dx.doi.org/10.2139/ssrn.3111061

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