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Utility and Probability

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I will review this book by concentrating on those essays that discuss Ramsey,Keynes,De Finetti or the subjectivist Bayesian approach to probability .
The subjectivist, Bayesian approach regards probability as another name for the purely mathematical laws of the probability calculus that require 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) ,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 assumption of additivity and linearity. Keynes's demonstration ,taken from chapter 26 of his A Treatise on Probability(1921;TP),of the special case nature of any expected value approach ,based on the purely mathematical laws of the probability calculus,is given below .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 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. 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 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 probabiity calculus.

Keynes specifies his conventional coefficient of risk and weight,c, model in chapter 26 of the TP on p.314 and fotnote 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 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.