From: Paul Cockshott (wpc@DCS.GLA.AC.UK)
Date: Tue Apr 17 2007 - 18:14:14 EDT
"Can one develop through the analysis of the logical construct of simple reproduction an explanation of why capitalist dynamics will likely be characterized by constant non equilibrium, with equilibrium which is the rule presupposed by political economy only at best a momentary transitory point?" (Rakesh) I think the answer to that question is no, for exactly the same reason that supply-demand curves prove very little. It all depends on what numbers you feed into the equations and what assumptions you make. There exists no logical proof that capitalism must always tend towards equilibrium, or towards disequilibrium. There exists only the empirical evidence of a succession of booms and slumps. Metaphysicians want to deduce logically from first principles that capitalism must break down, or spontaneously balance itself, but scientific people want to explain the observables.(Jurrian) It may also be worth while examining the 'in principle' computability of the models of equilibrium that are presetented. Arguments about computability can themselves reveal more about the axiomatic foundations of economic theories than they do about the operation of real world economies. Arrow, for example, supposedly established the existence of equilibria for competitive economies. Let us term such an equilibrium 'classical mechanical' following Mirowski who showed that the conceptual apparattus used to define it is equivalent to that used for posing energy minimisation problems in classical mechanics. Vellapuli showed, Arrow's proof rested on theorems that are only valid in non-constructive mathematics. Arrow's use of non-constructive mathematics is critical because only constructive mathematics has an algorithmic implementation and is guaranteed to be effectively computable. But even if a) a classical mechanical economic equilibrium can be proven to exist, b) it can be shown that there is an effective procedure by which this can be determined : i.e., the equilibrium is in principle computable, there is still the question of its computational tractability, that is of determining the complexity order governing the computation process that arrives at the solution. An equilibrium might exist, but all algorithms to search for it might be NP-hard. Deng says that subject to Leontief utility functions, the problem of finding a market equilibrium is NP hard. Their result might at first seem to support the Austrian school's objections to Lange, since he relied on similar equilibrium concepts. Whilst NP hardness may show that the neo-classical problem of economic equilibrium was intractable for economic planners, even with large scale computers, it need not. Recent work shows NP problems have phase transtion regions within which they are hard to solve and have other, less constrained regions, where solutions are easy to find. It might be the case that in practice, the problem of finding a social welfare maximising equilibrium, falls into a non-critical region of the constraint space. If, on the other hand, we assume that real economies fall into the phase transition region of the problem space, then neither central planners, nor a collection of millions of individuals interacting via the market could solve the social welfare maximisation problem. This implies that a market economy could never have sufficient computational resources to find its own equilibrium. Clearly we cannot conclude from this that market economies are impossible, as we have empirical evidence that they exist. It would follow that the problem of finding the neo-classical equilibrium is a mirage: no planning system could discover it, but nor could the market. If we dispense with the notion of classical mechanical equilibrium and replace it with Farjoun's idea of statistical mechanical equilibrium we arrive at a problem that is much more tractable. Ian has shown that a market economy can rapidly converge on this sort of equilibrium. It should be noted that the notion of a statistical mechanical equilibrium, whilst quite alien to neo-classical economics, has something in common with the presumptions of the Austrian school who emphasise more the chaotic, non-equilibrium nature of capitalist economies.
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