From: Paul Cockshott (wpc@DCS.GLA.AC.UK)
Date: Mon Jul 31 2006 - 04:43:22 EDT
Implications of a new research result
One of the Hayekian objections to the notion of economic planning
Has been to object that the sheer scale of the economic problem is
such that although conceivable in principle, such computations would
be unrealisable in practice {Hayek:1955 p. 43), and note 37 on pp.
212--213, of of The Counter-Revolution of Science. In the note, Hayek
appeals to the judgment of Pareto and Cournot, that the solution of a
system of equations representing the conditions of general equilibrium
would be practically infeasible. This is perhaps worth emphasizing in
view of the
tendency of Hayek's modern supporters to play down the computational
issue.
However neo-classical economists and the Austrian school have a very
particular concept of equilibrium. One possible concept of equilibrium
would be Statistical equilibrium: not a point in phase space, but a
region defined by certain macroscopic variables, such that there is a
large set of microscopic conditions that are compatible with it. This is
the concept
advanced by Machover.
The concept of equilibrium with which Hayek was familiar was, in
contrast, that of a mechanical equilibrium, a unique position in at
which all forces
acting on the economy come into balance.
Arrow supposedly established the existence of this sort of equilibria
for competitive economies, but as Velupillai{2003} showed, his proof
rests on theorems that are only valid in non-constructive mathematics.
Why does it matter whether Arrow used constructive or non-constructive
mathematics?
Because only constructive mathematics has an algorithmic implementation
and is guaranteed to be effectively computable. But even if
a) a 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 computation tractability. What
complexity
order governs the computation process that arrives at the solution?
Suppose that an equilibrium exists, but that all algorithms to search
for it are NP-hard, that is, the algorithms may have a running time that
is exponential in the size of the problem. This is just what has been
shown by Deng and Huang, (Information Processing Letters vol 97, 2006).
Their result might at first seem to support Hayek's contention that the
problem of rational economic planning is computationally intractable. In
Hayek's day, the notion of NP-hardness had not been invented, but he
would seem to have been retrospectively vindicated. Problems with
a computational cost that grows as exp(n) soon become astronomically
difficult to solve.
I mean astronomical in a literal sense. One can readily specify an
NP-hard
problem that involves searching more possibilities than there are atoms
in the universe before arriving at a definite answer. Such problems,
although in principle finite, are beyond any practical solution.
But this knife cuts with two edges. On the one hand it shows that
no planning computer could solve the neo-classical problem of economic
equilibrium. On the other it shows that no collection of millions of
individuals interacting via the market could solve it either. In
neo-classical economics, the number of constraints on the equilibrium
will be proportional, among other things, to the number of economic
actors n.
The computational resource constituted by the actors
will be proportional to $n$ but the cost of the computation will grow as
en. Computational
resources grow linearly, computational costs grow exponentially. This
means that
a market economy could never have sufficient computational resources
to find its own mechanical equilibrium.
It follows that the problem of finding the neo-classical equilibrium is
a mirage.
No planning system could discover it, but nor could the market. The
neo-classical problem of equilibrium misrepresents what capitalist
economies actually do and also sets an impossible goal for socialist
planning.
If you dispense with the notion of mechanical equilibrium and replace it
with statistical equilibrium one arrives at a problem that is much more
tractable. The simulations of Ian Wright show that a market economy can
rapidly converge on this sort of equilibrium. But this is because
regulation by the law of
value is computationally tractable. This same tractability can be
exploited in a socialist planning system. Economic planning does not
have to solve the impossible problem of neo-classical equilibrium, it
merely has to
apply the law of value more efficiently.
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