"A road not taken"

It has long been known that textbook economics, based both explicitly and implicitly on the neo-classical model, is completely wrong. Recent work in econophysics has driven the last nails into the coffin of neo-classical thought, but the inhabitants of 'the citadel' have not yet realized that their mathematized ideology is dead and about to be buried. As Alan Kirman quoted in one of his papers, "No amount of attention to the walls will prevent the citadal from being empty."

One can say that two main schools of thought have evolved as alternatives to neo-classical thinking: econophysics, based on empirical modelling, and computable economics, based on the requirement of computable models. The second school of thought is strongest in Italy. E.g., there are schools of alternative economic thought in Pisa,Trento, Firenze, Sienna, Ancona, and Salerno. Professor Velupillai, now in Galway, taught earlier in Trento and is a well-known advocate for the idea of comnputable economic modelling.

Velupillai's monograph begins with a sketch of the contributions of Alain Lewis (who did computable analysis, following Radner's key 1969 paper on equilibrium under uncertainty) and points out that there is no reason to prefer set theoretic models over computable ones. Leaving the question of stability aside, the next question is whether equilibrum is recursively defined, is algorithmically calculable. The latter is the main point emphasized and discussed by Velupillai. Leading neo-classical theorists like Arrow have mightily resisted the imposition of computability restrictions on their modelling. In other words, complexity was ruled out of consideration out of hand. Even if equilibrium would 'exist' mathematically in a given model and would be stable, neo-classical thinking takes for granted that the brain could perform the calculations that would be required in order for agents cooperatively to locate equilibrium. In general, such an assumption places impossible information acquisition and processing demands on agents, and indeed there exists no empirical evidence whatsoever for 'market clearing' in any known market!

Velupillai, who is apparently very well-read in the history and philosophy of economic thinking, attributes the first considerations of complexity in economic thought to Simon, Scarf, Menger (the author of the famous traveling salesman problem), and others. Hayek, who worried about 'socialist calculation' but did no quantatitive work, is also brought into the picture. Simon's program is outlined for the readership in detail.

Chapter 3, "Computable Rationality", begins with Witgenstein's quote: 'Turing's "Machines". These machines are humans who calculate.' Indeed, Turing's original idea (outlined for the reader in the Appendix) is based on the idea of typing symbols onto an arbitrarily long strand of paper. Turing began implicitly mechanically with what the brain can do, in contrast with neo-classical modelling.

In Chapter 4 we are introduced to the latest thinking about dynamics and computability through a model suggested by Cris Moore. As Velupillai points out in the text, Moore (in the early ninties) gave examples of simple one and two dimensional iterated maps that are equivalent to Turing machines. This was an enormous breakthrough in dynamical systems theory that goes far beyond Feigenbaum's discovery of the universal period doubling sequence to chaos, but is hardly known in physics and mathematics, let alone in economics. In my opinion, it is a great credit to Velupillai that he makes the reader aware of Moore's pioneering work on the origins of complexity in deterministic dynamics, especially the stark difference between chaos and complexity.

My favorite chapter is the ninth, where I learned for the first time about an integral equation with computable kernel and initial condition where the solution is not computable. The trick (shown in Aberth and Pour-El & Richards) is that the Lipshitz condition must be violated: nonuniqueness and branching of solutions are required for the noncomputability of the solution.

While in complete agreement with the criticism of standard economic theory given in the book, my main point of departure is that economic models should be firmly empirically based. I would expect that empirically based models will be computable, or at least decimally approximable. Because of this difference in emphasis, I disagree with some a few specific proposals made in the text. Multiple equilibria are discussed as alternative to single equilibria, but evidence for equilibria of any sort has not yet been found empirically (and there is enormous confusion over 'equilibrium' in economics and finance texts and papers). Velupillai argues that models should be nonlinear, but he assumes deterministic dynamics. Financial markets are instead stochastic, and are described very well empirically via linear stochastic models (leaving out complexity). It remains to be seen whether business cycle data are strong enough to permit the deduction of a reliable model, analogous to the success of modelling financial markets. In fact, success analogous to empirically-based finance market modelling should be the main aim of economic theory.

In closing, I strongly recommend this short, entertaining monograph to the mathematically prepared reader as strong antidote to the terribly misleading standard econ texts by 'leading authors' like Varian, Samuelson, Mankiw and Barro.

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