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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