Julio Huato wrote:
Les,
Consider a much simpler example, from anybody's world: A can with some
liquid. Let the state of the system at $t$ be $x_t$. Now shake the
can. At $t+1$, after the shaking, the system is $x_{t+1}$. Fixed
point theorems say that there'll be at least one point along the
mapping from $x_t$ to $x_{t+1}$ that will be fixed.
yea yea yea .. and there is always someplace on the earth's windy
surface where there is no (horizontal) wind:
http://en.wikipedia.org/wiki/Hairy_ball_theorem
But i already said, Julio, there are a thousand examples in the physical
sciences, and now you have stated #1001. though this one is more
esoteric, in my opinion, than electronic circuits, or fluid flow
(pump:supply, pipe:demand), or equilibrium vapor pressures
(evaporation:supply, condensation:demand), or ..., or ..., or ..., or ...
on fixed points and equilibria: think about modern thermodynamics and
the definition of the second law in terms of stable equilibrium states.
its a deep abstract principle. but it is itself built on hundreds of
years of connection to mundane and practical things: steam, engines,
thermometers, pumps, and so forth.
Now, watch this, magic: instead of your can, take the earth's
gravitational field. instead of the foam, take the atmosphere bound to
the planet's surface. instead of shaking the can, stir up the air by
shining a strong light on one side of the global. do or don't rotate the
globe, that's up to you.
what have we concocted? of course: a model for the earth's weather and
climate. and the beautiful fixed point theorem says that no matter what
weather the sun shakes up in the atmosphere, there is at least one point
on the earth's surface where the weather is relatively calm.
but how useful is this fixed point for understanding earth's weather???
practically useless.
instead, one has to probe more in depth the air currents that circulate
due to uneven heating of the surface between the equator and the poles.
and that's just for starters.
so, coming back to economics, what is the status of a theory like the
law of supply and demand? see my questions in previous post.
Les
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