Hayek’s focus [in Economics and Logic, 1937] is on the assumptions that economists make in their models about the knowledge that agents are assumed to possess, and the implications that these have for equilibrium analysis. He asserts that the notion of equilibrium has a clear meaning as applied to an individual: the agent is assumed to have a plan, and to know her own tastes and preferences and constraints, and if these are all known and do not change, the choice made is simply a matter of logic.
When one discusses equilibrium for a society, however, one enters into a different sphere. For society to be in equilibrium, the plans of all the agents must be compatible with one another. Plans are based on expectations about future states of the world, and that includes actions of other agents. The usual assumption made about knowledge in equilibrium theory – namely, that all agents have access to the same, correct information – automatically brings about such compatibility, in which all expectations are met. But if agents have access to different bits of information – if knowledge is “divided,” as Hayek puts it – then how does such a compatibility, how does equilibrium, ever come about? This is the essence of the knowledge problem.
The assumptions made in Axiomatic Theory of Economics about the knowledge that agents are assumed to possess applies only to individuals. The agent is assumed to have a plan and to know his or her own tastes and preferences and constraints, and if these are all known and do not change, the choice made is simply a matter of logic. Observe:
Value changes each day by a proportion (called 1+j, with j denoting the day), of the previous day's value.
ii)
In the long run, the j's may be considered random as they are not directly related to each other nor are they uniquely a function of value.
iii)
The j's are taken from an unspecified distribution with a finite mean and a non-zero, finite variance.
Thus it is that, in Hayek’s 1937 essay, Economics and Knowledge, Hayek anticipated Axiomatic Theory of Economics. But, lacking the mathematical training necessary to produce an axiomatic theory of economics, economists had to wait until 1999 for such a theory. So it is that, in 1945 or so, Hayek turned instead to such fields as psychology and philosophy of mind, intellectual history, political philosophy, and methodology of the social science.
Hayek does not solve the problem in the paper [Economics and Logic, 1937]. Instead, he speculates that its solution will require further analysis of what kinds of knowledge are relevant for agents to possess, and the process by which they come to possess that knowledge.
So we see that Bruce Caldwell acknowledges that Hayek did not produce an axiomatic theory of economics, but that he understood the need for one like Axiomatic Theory of Economics, where each agent is assumed only to have a plan and to know his or her own tastes and preferences and constraints, unlike that of Debreu where the plans of all the agents must be compatible with one another and where they are assumed to all have access to the same, correct information.
In a way all the economic calculus is concerned with is the classification of goods according to their economically relevant characteristics; not concerned with their physical characteristics but with position in the means-end order.
Observe that the theory Hayek yearned for is identical to Axiomatic Theory of Economics (p. 104-105):
The location parameter, μ (mean), quantifies the importance of a phenomenon relative to money and the scale parameter, σ (standard deviation), quantifies the difficulty of substituting other phenomena for the one in question…. Both μ and σ must be positive. With u(s), μ and σ describe all phenomena. Thus, every phenomenon is associated with a point in u(s), μ, σ space where u(s) is a negative monotonic probability density function on Ɍ+ [positive real numbers] and μ and σ are both from Ɍ+. For the purpose of economics, nothing else distinguishes one phenomenon from another.
It is important that from the beginning we look at competition not as a state of affairs in which everybody knows everything but as a process by which knowledge is dispersed and acquired – how effectively this happens under different conditions we shall gradually see (emphasis in original).
Again, observe that the theory Hayek yearned for is identical to Axiomatic Economics:
Suppose that some of the market participants come to the realization that their position is foolish and they do something about it: They move leftward on the graph. So the shape of the demand distribution changes, becoming fatter on the left and thinner on the right. This is like if one has a plastic water bottle and suddenly gives the bottom a squeeze, splashing water up and leaving a new bottle shape, one that is fatter on top and thinner on the bottom.
An axiomatic economist is not a visionary. I cannot predict that these people will suddenly have a forehead-slapping moment and decide to change their position. What I can predict, during the resulting turmoil, is where the price and stock will settle, provided that I can estimate the changes in the parameters that have occurred. In the same way, I can predict the new water level in the squeezed bottle if I can estimate the changes to its shape.
Hayek would continue to develop that which was most new in the Virginia lectures, his ideas about complex orders, in future work. He would also frequently point out that order could be represented by mathematical structures, always with the caveat that care had to be used in interpreting them. But he would never return to the economic calculus as another exemplar of such a structure. Why not?
One reason, certainly, was that the economic calculus basically refers to the sort of exercises that one presents in an introductory microeconomics class…. Hayek did not have the mathematical preparation to go beyond that level of mathematics to try to update what he was saying using more recent developments…. [Hayek] lacked the mathematical background to formalize his ideas, a fact that he recognized and which doubtless frustrated him, as it kept him from publishing his ideas in the form in which he originally developed them.
So we see that, as late as 1961, Hayek was still yearning for someone of sufficient mathematical preparation to create an axiomatic theory of economics. Unfortunately, I would not be born for another five years.
In conclusion, I want to thank Bruce Caldwell for his painstaking historical research. Thanks to Bruce Caldwell, we now know that, as early as 1937, Hayek understood that there were seven serious problems with his economic theory and, lacking an answer, deferred the solution to someone of sufficient mathematical preparation. Had Hayek lived to 1999, he would have applauded the publication of Axiomatic Theory of Economics. But Hayek did not live that long. Instead, Hayek’s good name was hijacked by sycophants who would form a cult-like religion that worships his 1931 lectures, which he had clearly already moved past six years later in anticipation of an axiomatic theory of economics.
It was actually very brave of Bruce Caldwell to defy all the other Hayek worshipers and come out in support of Axiomatic Theory of Economics. Three cheers for Bruce Caldwell! Now if only the rest of the Hayek worshipers will follow Bruce Caldwell’s example, renounce their cultish religion, and take up real science.
REFERENCES
Aguilar, Victor. 1999. Axiomatic Theory of Economics. Hauppauge, NY: Nova Science Publishers, Inc.
q and q 
j, with j denoting the day), of the previous day's value.