Subject: Metastable oscillators
From: "Stephen J. Turnbull" <stephen@xemacs.org>
Date: Thu, 27 Feb 2003 15:10:05 +0900

>>>>> "ben" == Benjamin J Tilly <" <ben_tilly@operamail.com>> writes:

    ben> Rich Bodo <rsb@ostel.com> wrote:

    >> Many robots and aircraft, like pencils on their tips, hold
    >> themselves in unstable positions during operation.  They remain
    >> unstable, but their inherent instability is handled by fast
    >> feedback systems.

    ben> Note that the feedback systems convert the overall system to
    ben> one which satisfies the description of Le Chatelier's
    ben> Principle.  Therefore it appears unstable, but is not.

That assumes that the the feedback system is capable of handling
extreme conditions.  It typically is not, in fact, unlike the
"natural" stable system, the controlled system typically responds
catastrophically (in both the mathematical and vernacular senses) to
conditions outside of its design range.[1]

Note how your "unicycle" example works.  There, the _goal_ is
stability.  It's not surprising that a rather stable control system
results from finding a solution.  However, economic stability of the
kind that is found in "traditional societies" is clearly not an
acceptable macroeconomic goal!  Who needs macroeconomics in such a
society?

BTW: Nash equilibria are fixed points of mappings, not steady states
of dynamical systems.  The same is true of market equilibria, with one
important difference in current models.  Market equilibria can fairly
naturally be embedded in dynamical systems for which the steady state
condition is equivalent to the equilibrium condition for the market.
This is not so for Nash equilibria.  There are several generally
accepted theories of stability for market equilibria, and they
coincide for the "canonical" case of upward-sloping supply and
downward-sloping demand (all equilibria of such markets are stable).
There is as yet not even one generally accepted dynamical model
embedding Nash equilibria.

    >> When he talks about feedback loops and the paradox of logical
    >> indeterminacy I interpret Soros as describing a sort of
    >> uncertainty principle that makes it difficult to view any
    >> economic system as stable.

The analogy is reasonably apt.  Fortunately, your (or Soros's) ability
to imagine a stable economic system doesn't have anything to do with
its possibility.  With a few exceptions, markets are analogous to the
realm of Newtonian, not quantum, mechanics.  The exceptions are
important, such as "tipping" in markets for products with significant
network externalities.  But the overall system is remarkably stable.
Remember, a 5% drop in production is considered disastrous.

Prices?  Yes, prices are volatile.  _But that's precisely because they
are the dual variables to the "real" quantities in the system, which
are stable._  Consider: the markets where we insist on controlling
prices (labor, housing, agricultural products) are precisely with ones
with rather unsatisfactory quantity outcomes.

Thus, price volatility, in some sense, is exactly the "quantum judo"
you refer to as "a wonderful thought."  You just don't notice it
because it's such an everyday miracle.

And, of course, because in situations like the labor market neither
price nor quantity volatility should be accepted without a fight.

    >> [...]  When everyone recognizes this, markets become inherently
    >> unstable.

This is, I suspect, confusing "instability" of static equilibrium with
"volatility" (periodic or even chaotic) of a dynamic process.  It is
true that in a system with multiple steady states, volatility can
drive the system from one basin of attraction to another (aka,
"economic depression").  However, it is important to recognize that
this is evidence _for_, _not_ against, stability of the system as a
whole.  And it is true that "socially optimal trajectories" are
typically not steady states[2], and thus attempting to force social
optimality turns the system into a controlled system fraught with
catastrophic potential if the controls fail.

And it is most definitely true that large fluctuations cause huge
amounts of pain that would be completely unnecessary if the economy
would only hew strictly to trend.  But the equation of economic pain
with instability is incorrect.

    >> He therefore argues for strong market authorities.  He wants
    >> control systems in place.  It's not a totally absurd theory.

    ben> I agree, and I have the same major qualm about
    ben> macroeconomics.  There are fundamental questions about what
    ben> solutions are stable, and my feeling is that well-meant
    ben> attempts to induce stability can itself be the cause of
    ben> future problems.

You've been reading Friedman and von Hayek again, I can tell.  :-)
(That's a joke, I realize you're coming from advanced diff eq.  But
their intuition maps quite precisely to the mathematical description
you give.)

    ben> Therefore I wonder whether the government should deliberately
    ben> attempt to oscillate interest rates - to try to create
    ben> periodic mild amounts of risk so that businesses avoid things
    ben> like the dot bomb excess.

I think maybe they should just let them alone, except for controlling
inflation.  Inflation does not affect the real economy to a first
approximation.  Therefore applying negative feedback to control
inflation can probably be decoupled from nasty side effects in the
real economy.  Not so if you actually try to control interest rates.

    ben> Unfortunately the social sciences generally fail of the most
    ben> basic requirement for scientific progress.  Which is the
    ben> availability of examinable systems where useful simplifying
    ben> approximations can be applied.

Actually, there's a yet more basic requirement, which is separation of
the observer from the observed.  Even in economics, which mostly deals
with quantities that can be measured in the most primitive way (one
dollar, two dollars, three dollars, ...), it is all too easy to feel
that (to take the example that kicked off this thread) the genuine
pain felt by the unemployed is something that must be addressed by
urgent action.

Unfortunately, the actions taken are all too often inexcusably
destabilizing, themselves.  But they are justified on the grounds that
they _might_ work as projected, and if they do, the immediate pain
will be dramatically reduced and the economy will be able to "return
to (putatively natural) trend".  Then, when they blow up (years or
even decades later), enormous effort (both by policy advisors and
academic researchers) goes into adjusting the remedies, when really
the whole line of attack should be scrapped.  It's hard enough to
"scrap a whole line of attack" when it's "just an operating system";
when you're talking about (eg) AFDC or the EU's CAP or rent control,
and the transition is going to inevitably destroy people's current
lifestyles, well, objectivity goes right out the window.[3]

And if your measurements are of "variables" like "educational
achievement" or "sexual harassment", well, it's just not within the
realm of human ability to treat it with mathematical rigor.



Footnotes: 
[1]  Of course natural systems also respond catastrophically to
extreme conditions.  Eg, an avalanche.  However, the usual goal of
control systems is to push things toward optimal solutions, which are
by definition extremes.

[2]  Note that by defining "growth rate" as the state variable, a
steady state need not imply stasis.  A steady state could include (eg)
a constant rate or constant growth rate of technological "capital".

[3]  And maybe it should.  I disagree, but that's more "religion" than
"science."

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