On the foundations of mathematical economics

II. Problems for Economic Theory

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II. Problems for Economic Theory

A. Explosive Phenomena

Velupillai draws on the work of Zambelli ([77], [78]) to consider the problem of the possible uncomputability of economic processes. The economic process Zambelli focuses on is that of endogenous growth, and the function he identifies as a metaphor for its possible uncomputability is the busy beaver function of Radó [49 ]. Velupillai ([74], Section 3.2) proposes a method of minimimalization due to Greenleaf [29] to “tame” the otherwise uncomputable busy beaver function, although he states after making his proposal ([74], p. 39) that he is “not sure” that his “suggestion” is a “way out of, and beyond, this renewed perplexity” (of the busy beaver function). What is going on here?

The busy beaver function is defined for n-state Turing machines as the maximum number of 1’s (or “productivity”) for any member of the n-state that it writes in a square on a blank, two-way infinite tape and halts, and is labeled Σ (n), with this productivity equaling zero if it fails to halt. This function is the largest of a finite set of non- integers. Dewdney ([23], pp. 10-11) explains the non-computability of this well-defined function as arising because it “grows too fast.” Any formula used to compute it will end up generating more 1’s for an n-state busy beaver than the formula specifies.

Velupillai sees the problem as being of a naming or semantical nature along lines similar to Chaitin ([15], [16]) and a case of the Berry Paradox, first identified by Bertrand Russell [57]. His classic example ([57], p. 222) takes the form of “the least integer not nameable in less than nineteen syllables.” That refers to a particular number, 111,777, but the phrase in the quotation marks itself only contains eighteen syllables, thus establishing the paradox.4

The solution advocated by Velupillai is to use Greenleaf’s [29] method to properly restrict the range and domain of the busy beaver function. This method was used to guarantee the primitive recursiveness of the Akerman function, although as noted, Velupillai is uncertain that this method is fully satisfactory.

What then is the economic significance of this problem? For Zambelli it is a metaphor for the development of technological ideas in endogenous growth theory, a possible paradox that can arise from the “production of ideas by means of ideas.” The problem is in some sense that the ability to create new ideas can outrace the ability to name what is being created. On the other hand, the arbitrary nature of the halting that occurs can become a metaphor for the limits of the production of knowledge as well, which he shows in his analysis using the busy beaver function [78].

It occurs to me that another area of economics where the questionable computability of the busy beaver function might arise might be is with the problem of explosive hyperinflation, where the very act of agents attempting adjust their expectations to the ever-accelerating rate of inflation itself triggers a further acceleration that pushes it ever beyond the ability of their expectational formulations to compute. An aspect of this that is rarely commented upon has a certain connection with the famous remark by Keynes ([38], Chap. 12) regarding the beauty contest regarding how agents may begin to think in terms of higher order expectations, not merely guessing the average guess of the other contestants, but the average guess of the others about the average guess of the others, and higher. However, in this case it may be a matter of focusing on higher order derivatives without limit. As the hyperinflation accelerates agents may cease forecasting the rate of inflation but instead focusing on the rate of change of the rate of inflation and then the rate of change of the rate of change, and so forth, with these rising expectational dynamics themselves driving the system to accelerate ever ahead of these evolving expectations.

Yet another possible example is that implied in the markomata argument of Mirowski [43]. While he limits his discussion to the four level hierarchy of Chomsky [17], in principle there is no reason that there might not be more levels, with no clear upper bound. This is a model in which simpler markets generate higher order markets that embed the lower level ones, much as a futures market may embed a spot market, and an options market may then embed a futures market. We may have seen the outcome of such a process in the financial collapses 2007 and 2008 as ever higher order derivatives were created out of lower order ones in a way that kept the system from achieving a computable general equilibrium solution, although this must be admitted to be a rather speculative possible application of these ideas, with such a process not clearly related to the busy beaver or any other clearly uncomputable function.

Of course we should keep in mind that in the applications by Zambelli the ultimate process being modeled is not explosive, even if the function that is being used to model them is.

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