Independence, Additivity, Uncertainty (Studies in Economic Theory)

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Volume 14 in the Studies in Economic Theory �series deals with the important economic problem of uncertainty. It contains all the classical results, but also� new results that give a solution to how uncertainty can be formalized.

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Independence, Additivity, Uncertainty (Studies in Economic Theory) Review

The authors recognize that the standard Subjective Expected Utility (SEU) approach based on the subjectivist Bayesian approach erected by F P Ramsey,B de Finetti and Leonard Savage is a special case of a far more general theory.Unfortunately,the approach used in this book ,based on the recognition that,instead of there being one unique probability measure (distribution), there is a set of probability measures( p.244),overlooks the fact that this is equivalent to the specification of probability as an interval valued measure.The book completely overlooks the fact that Boole and Keynes had devised a formal approach to dealing with uncertainty as distinct from risk.Keynes worked out a total of 17 problems in his 1921 A Treatise on Probability (TP) that explicitly deal with a formal approach to dealing with interval estimates.This work is identical to the work presented in his 1908 Cambridge University Fellowship Thesis.These problems can be found on pp.160-163 and pp.186-194 of the TP.
Second,the authors overlook Keynes's formal specification in the first paragraph of chapter 6 of the TP of a precise index to measure the degree of uncertainty.This is formally presented in chapter 26 on p.315 and footnote 2 on p.315 as w,which measures the weight of the relevant evidence supporting the estimate of the probability.w is defined on the unit interval [0,1].Keynes integrates w into a decision rule ,his conventional coefficient of risk and weight ,c,which represents the first decision weight approach in history.I will present Keynes's analysis after this review.
The authors are certainly correct to point out that " Keynes stressed the importance of a concept of uncertainty,which could not be expressed by probabilities..."(p.243).Unfortunately,errors are introduced when they claim that " Both Keynes and Knight insisted that "their " uncertainty could not be formalized "(p.196).The authors are correct only with regards to Knight.Knight did not have the technical and mathematical tools needed to engage in such an endeavor .However,they are incorect with regards to Keynes.This error appears on the very first page of their book : " The most important contribution of the book to economic theory is to formalize a concept of uncertainty.Ideas about uncertainty and this concepts importance for economics go back at least to Keynes and possibly Knight.The idea has been rejected with the argument that the concept could not be made precise and therefore not included in serious economic theory" (p.1).

Keynes presented a very precise analysis demonstrating that an analysis of uncertainty introduced non additivity and non linearity into the formal representation of decision making. The subjectivist, Bayesian approach regards probability as another name for the purely mathematical laws of the probability calculus that requires additivity and linearity. The Subjectivist approach makes the crucial error of conflating probability theory with decision theory.Keynes realized that ,due to the impact of the weight of the evidence (confidence)on decision makers ,as well as the optimism-pessimism of the decision maker,decision theory would have to be able to take into account the importance of non linearity and non additivity. The concept of expected value or utility is crucial to the Ramsey-De Finetti approach.Keynes demonstrated that expected value or expected utility can ,at best,be a special case only of a much more general theory .
The Ramsey-De Finetti approach is the mathematical translation of Jeremy Benthem's Benthamite Utilitarian approach.Bentham's approach was that the whole can not be anything more than the sum of the individual ,atomic parts. However, this requires the assumptions of additivity and linearity.Bentham assumed also that all decision makers can calculate the odds.Keynes showed that this was not the case. Keynes's demonstration ,taken from chapter 26 of his A Treatise on Probability(1921;TP),of the special case nature of any expected value(utility) approach ,based on the purely mathematical laws of the probability calculus,shows this to be a very special case when w=1.Bentham claimed that all individuals have the capability to calculate the odds and outcomes and act on the expected utility (the probability times the utility of the outcome) in a rational way.This can be expressed by the following ,where p is the probability of success,q is the probability of failure, and A is the outcome:

Maximize pA.

The modern version of this is to Maximize pU(A),where p is a subjective probability that is additive,linear,precise,and exact and U(A) is a Von Neumann-Morgenstern Utility function. The goal is to

Maximize pU(A).

The modern name for Benthamite Utilitarianism in neoclassical economics is SEU theory(Subjective Expected Utility). Therefore,a microeconomic foundation based on Utility Maximization is just Benthamite Utilitarianism updated with modern mathematical probability techniques.Modern macroeconomics is all disguised SEU theory.

Keynes rejected Benthamite Utilitarianism as a very special case that would only hold under the special assumptions of the subjectivist, Bayesian model-that all probabilities were additive,linear,precise,single number answers that obeyed the purely mathematical laws of the probability calculus.

Keynes specifies his conventional coefficient of risk and weight,c, model in chapter 26 of the TP on p.314 and footnote 2 on p.314,as a counter weight to the Benthamite Utilitarian approach of Ramsey.

Essentially, Keynes's generalized model is given by

c=2pw/(1+q)(1+w),

where w is Keynes's weight of the evidence variable that measures the completeness of the relevant, available evidence upon which the probabilities p and q are calculated.(Benthamite Utilitarians always assume that the value of w is always 1.)w is an index defined on the unit interval between 0 and 1,p is the probability of success,and q is the probability of failure.p+q sum to 1 if they are additive.This requires that w=1.Keynes's c coefficient can be rewritten as

c=p [1/(1+q)][2w/(1+w)].

Now multiply the above by A or U(A).One obtains

cA =p[1/(1+q) ][2w/(1+w)] A or

cU(A)= p[1/(1+q)][2w/(1+w)]U(A).

The goal is to maximuze cA or cU(A).The weight 1/(1+q) deals with non linearity of probability preferences.The weight 2w/(1+w) deals with non additivity.Modern Macroeconomics amounts to nothing more than the claim that c=p or cA [cU(A])= pA [pU(A)] .

It is now straightforward to see that the neoclassical microfoundations of macroeconomics assumes that all probabilities are additive and linear.This is nothing but a special case of Keynes's generalized decision rule to maximize cA,or cU(A),as opposed to the Benthamite Utilitarian rule to maximize pA or pU(A). Economists today have only a very vague,hazy,cloudy understanding of Keynes 's distinction between risk and uncertainty. It is this distinction that has to be grasped first before any economist can have any hope of understanding what Keynes meant in the GT.

The conclusion is very straightforward. SEU theorists use the rule to Maximize pU(A).Keynes used the rule to maximize cU(A).Keynes's rule is of the same kind or type of rule used by the overwhelmingly ambiguity averse decision makers that populated the real world in the past as well as in the present.Keynes's analysis of uncertainty is clearly related to non additivity and non linearity.It is only an anomaly in a neoclassical world of linearity and additivity that exists for Benthamite Utilitarian, SEU theorists.

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