Marx's limit cycles (fwd)
Steve.Keen at unsw.edu.au
Steve.Keen at unsw.edu.au
Thu Nov 30 13:47:38 MST 1995
Chris asks for a definition of "limit" in "limit cycle".
The term itself is a mathematical one, and refers to
a class of solutions which, rather than ending up in
a point, or a system which heads off infinitely in
one direction or another, ends up in an ever-repeating
cycle. The cycle is therefore the "limit of the
system as time goes to infinity", hence "limit
cycle".
There are many types of limit cycles, ranging from
rigid ones (like the one in Goodwin's 1967 paper) where
the system traces out exactly the same path time and
time again, to ones where if the system starts "inside"
the cycle then it gradually converges to it, ditto
outside, and finally ones like my 1995 extension to
Goodwin, where the system converges to the cycle from
both inside and outside it (it spirals around the
eventual cycle, moving ever closer to it).
They are one of the foundations of coupled systems,
where the rate of change of x depends upon the value
of y, and the rate of change of y depends upon value
of x. There is thus feedback from one to the other,
and to me they are the mathematical expression of
"dialectics", where the x/y system forms what
Marx would have called a unity.
The cycle described by Marx in Ch. 25 is clearly a
limit cycle, with the wages share of output being one
axis (the x) and the rate of employment being the
other (the y). The way Marx describes it implies that
he doesn't see it as a rigid cycle, but as one which can
take a range of values, though ultimately the cycle
will again occur: workers share will never rise to 100%,
nor will it ever fall to zero, while wages themselves
may rise or fall.
As to whether what the "capitalist system comes up against
repeatedly in its cycles, is a somewhat elastic sort of
ceiling of the total exchange value of the society",
the last time Westernc capitalism really approached that
barrier was in 1973. Since then, unemployment has cycled
about an increasing trend, while wages share has cycled
about a decreasing one, and the turning points have been
well before wages share ever began to challenge capitalist
profits, and in my opinion driven by issues of the
realisation of surplus (rather than its generation) and
finance.
Under the 1945-1973 system, then Goodwin's cycle was an
apt characterisation--hence the limiting feature you're
looking for, Chris, was the share of surplus. Since then,
I'd argue that it is crises in effective demand, and
finance, that have determined the cycle.
Cheers,
Steve
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