Controlling economic? chaos

Steve.Keen at Steve.Keen at
Mon Mar 20 17:03:18 MST 1995

Chris Burford recently cited the following from a work on chemistry:

"It is [now] possible to stabilise otherwise unstable states in chaotic
systems by applying feedback methods that are analogous to the clown's
balancing act. Unstable oscillatory behaviour can be stabilised with small,
controlled perturbations to the operating conditions of the system,
transforming irregular chaotic responses into regular periodic oscillations"."
"Although control methods were developed some time ago, early methods
required an accurate model of the system so the governing equations could
be suitably modified to produce the desired stabilisation. The new feedback
methods, however, require only a means to monitor the system and access to
an operating condition that can be appropriately perturbed."

and asks whether anyone can take this further.

The mechanism described above turns up as a model of the effect of
"automatic stabilisers" on an unstable economy in my paper "Finance and
Economic Breakdown: Modelling Minsky's Financial Instability Hypothesis",
in the Journal of Post Keynesian Economics, Vol. 17, No. 4 (forthcoming).
While I wasn't aware of the effect--"Unstable oscillatory behaviour can
be stabilised with small, controlled perturbations"--that is what
happened when I added a "government sector" to my model there.

Briefly, the paper starts with Goodwin's "predator-prey" model of the
trade cycle, which he describes, accurately, as Marxian in spirit. In
fact, I'll let Marx "summarise" it -:) :

"accumulation slackens in consequence of the rise in the price of labor,
because the stimulus of gain is blunted. The rate of accumulation lessens;
but with its lessening, the primary cause of that lessening vanishes, i.e.
the disproportion between capital and exploitable labor power. The
mechanism of the process of capitalist production removes the very
obstacles that it temporarily creates. The price of labor falls again to
a level corresponding with the needs of the self-expansion of capital,
whether the level be below, the same as, or above the one which was normal
before the rise in wages took place... To put it mathematically, the rate
of accumulation is the independent, not the dependent variable; the rate
of wages the dependent, not the independent variable." [Capital I,
Progress Press edition, pp. 580-581; Ch. 25, about 7 paras in]

What that gives you is a limit cycle, where the same pattern of wages
share of output and employment is repeated cyclically, for all time.

Then I add in the existence of banks, who lend money to capitalists at
interest to finance investment. Capitalists invest more during booms
than during slumps (more than the rate of profit at the peak of booms),
leading to an accumulation of debt, which is only partly repaid in a
slump (because the reduced rate of profit diminishes the ability of
capitalists to discharge debt). Eventually, falling wages share on
account of increased unemployment lets profit share recover sufficiently
to set of another boom, and so on; but with a different level of debt
than initially.

This model has two outcomes: for a low rate of interest, it eventually
stabilises to a single "point" where the wages share of output, bankers'
share, capitalists' share, and the rate of employment, are all
constant (while the economy continues to grow). However, for higher
rates of interest, the point is a "strange attractor": while the system
tends in that direction, the debt being accumulated in booms continues
to mount, leading to the stage where debt starts to "blow out". The
system continues to be cyclical, but debt grows with every cycle,
leading eventually to a collapse in workers share and capitalists share,
with the bankers taking the lot, but with the economy collapsing.

The introduction of a government sector, with only two rules of
behaviour, dramatically changes the outcome. The government sector
increases the rate of change of subsidy payments to capitalists on the
basis of the level of unemployment; and it increases the rate of change
of taxes imposed on capitalists, on the basis of the rate of profit.
(I realise this isn't how governments behave now; but it is a parody
of how they behaved in the early post-WWII period [remember progressive
tax scales? -:)], and it is how Keynes argued they should behave).

The inputs are only slight, but the effect is to generate a cyclical,
but stable, system: given almost any set of initial conditions and any
rate of interest, the system eventually settles down to a cycle rather
like the original limit cycle, where the location and shape of that
cycle is determined almost completely by the small government intervention.

So the effect mentioned in the chemistry paper is general--because systems
analysis is general.

Steve Keen

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