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spoons at spoons at
Tue Dec 20 02:00:41 MST 1994

From: Steve.Keen at
Subject: Re: Causality

A quick reply to some of Lulu's comments.

Re chaotic limit cycles multi-pole strange attractors, and other
terminology. The jargon is still pretty fluid, as is always the case
in a new area of research. I have developed a simple model of Minsky's
"financial instability hypothesis", and in checking it out against
one of the best books on the subject [HW Lorenz's _Nonlinear Dynamical
Economics and Chaotic Motion_, Springer-Verlag, 1993], it appears that
it was classifiable as a "chaotic limit cycle" (see p. 39 ff of above);
but I am more comfortable with the term you used, of single pole vs
multi-pole strange attractors.

For those who are not familiar with them, think of a plot showing the
level of employment on one graph, and workers' share of output on the
other. Goodwin developed a model with workers' wage demands depending
on the level of employment, and capitalists investing all their profits.
The product was a model in which wage share and employment would
cycle endlessly: high employment led to high wage demands, resulting
in low investment, which slowed employment growth, leading to
unemployment, a fall in wage demands, a consequent rise in employment,
and so on.

This is a strict "limit cycle", in that (a) the system always follows
exactly the same path (the set of initial values repeats periodically)
and (b) unless you begin with equilibrium values for employment and
wages share, the system will never get to equilibrium: thus cycles
are endemic.

A "chaotic limit cycle", or to use Lulu's term, a single-pole attractor,
has at least one extra component to the model: bankers as well as
capitalists and workers, for example. Such a system can settle down
to an equilibrium (constant workers' share, employment, capitalist
share and bankers share of output), it can collapse (an economic
breakdown, with--for example--capitalists' debt resulting in bankers
taking all output, investment ceasing, and unemployment becoming total),
or it can cycle. But the difference with the above simple cycle is that
the values never exactly repeat themselves.

A multui-pole attractor has two or more interlinked "orbits". To extend
the economic example, a model with prices might show that there are
several "poles": high employment with low inflation (the 50s&60s);
low employment with high inflation (the 70s); high employment and
high inflation (Weimar Germany); low employment and low inflation
(Great Depression and now?). The system could seem to settle down
around one of these (as in the 50s); but then behaviours alter as
the apparent stability seems to be the norm, and suddenly you
fly off to a different orbit.

The best known such model is Lorenz's model of the weather. It has
two poles, and a plot of the system gives a beautiful "butterfly"
pattern, no matter what set of initial values you start with.
However, no two patterns are the same (in fact, any set of the
three values which occurs in one simulation cannot occur in any
other simulation with different initial values). It is
instructive--if you can do it--to see two plots of Lorenz's system,
not as X-Y plots, where the symmetry makes them appear identical,
but as plots of the variables against time (which is the way that
Lorenz himself first saw them). With infinitesmal differences in
initial conditions, the plots at first appear identical. Then they
start to diverge, and suddenly, they look entirely different.

What's in this for marxists? If capitalism were a linear system,
then it could settle down to the kind of equilibrium which
conventional economists fervently believe in. If it is nonlinear
and chaotic, then such an outcome is impossible--it is necessarily
unstable (though it does not have to necessarily breakdown: stumbling
from one crisis to another is more likely). With the exception that
marxists have tended to focus on final breakdown analysis rather than
cyclical, this mindset fits in very well with a marxian perspective.
And this time the tools of mathematical analysis support a marxian
perspective, rather than undermining it (as in the Sraffian critique).

Steve Keen


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