# [OPE-L:5132] Re: Re: Re: comparative statics

From: Steve Keen (s.keen@uws.edu.au)
Date: Fri Mar 09 2001 - 00:48:32 EST

```I promised to answer Jerry's question some time ago, and then got caught
up. So I'll try a fast answer now.

Dynamic systems are ones in which the model is specified in terms of rates
of change, rather than as a set of simultaneous equations. These rates of
change can be either in continuous (differential) or discrete (difference)
form (mixed difference/differential versions are also feasible, but that's
real rocket science stuff). The general form is thus

dX/dt=F(X)
X[t+1]=F(X[t],[X[t-1],...)

where X signifies a vector of variables, and F a vector of functions. The
latter can be linear, but the really interesting stuff is where those
functions are nonlinear.

Boring dynamic models have a dependence of the form

dx/dt=f(y)

ie, the dependence is of one variable on the values of another; interesting
ones have the form

dx/dt=f(x,[X]),

where the rate of change of variable x depends on itself and on other
variables.

Solutions to these models can be closed form for low dimensional models,
but the norm is for high ( >2 ) dimensions, in which case for nonlinear
models, closed form solutions do not exist. Instead, the time paths of the
models can only be found by simulation and exploration of what's known as
the phase space.

There is no presumption that dynamic models will converge to an equilibrium
solution. In this case, comparative static models can be seen as a subset
of dynamic models in which convergence to equilibrium is assumed.

Chaotic models occur in continuous time models of dimension 3 or above;
they cannot occur for lower dimensional models. The essential feature which
allows you to add the moniker 'chaotic' is that the overall dynamics of the
system are such that points which are very close together initially lead to
highly divergent time paths. A > 2 dimensional model can have this
characteristic, but it needn't necessarily.

From what I have seen of TSS, the models there are not fully specified
dynamic models: they are rather numeric examples of what could be the
outcome of dynamic models. Dynamic modelling is much more difficult than
comparative statics, precisely because the techniques of linear algebra
can't be used to derive closed form solutions--and because continuous time
problems of dimension >2 don't have solutions.

Some rules of consistency from comparative statics carry over to dynamics.
While it is possible to generate dynamic models which have no equilibrium
for some parameter values (see my paper in "Commerce, complexity and
evolution", CUP 2000), normally dynamic models will have equilibria. It is
possible that a poorly specified model will have no equilibria, not because
one does not exist, but because the model is either over or
underdetermined. This is my expectation of TSS, though it would take some
serious work to put the argument in a form where that expectation could be
tested.

That's a very quick and dirty pastiche. I'll try to provide something more
detailed when I have time.

Steve
At 12:17 AM 3/9/01 -0500, you wrote:
>Re [5128] and [5130]:
>
>Comparative statics or what?  It seems to me that
>there is a lot more talk about dynamic (and chaotic)
>theories and models than actual dynamic (and
>chaotic) models: it is easy to say that one needs
>dynamic analysis, it is harder to do it.
>
>I asked a related question in [OPE-L:4960]
>on "dynamic and chaotic systems": namely, I asked
>anyone to specify the *formal characteristics* of
>dynamic systems and chaotic systems.  Since
>nobody answered that question it was hard to move
>on to what would have been my next question:
>which (if any) Marxist theories and models could be
>said to be truly dynamic models and which could
>be said to be chaotic models?
>
>Let me be clear here. I am not asking whether a
>theory is consistent with the *possibility* of dynamic
>and chaotic modeling. I think that begs the question.
>I am asking whether a theory is actually *in a formal sense*  dynamic, etc.
>Until one can answer that, then
>all the talk against comparative statics is just talk, imo.
>
>
>In solidarity, Jerry

Dr. Steve Keen
Senior Lecturer
Economics & Finance
Campbelltown, Building 11 Room 30,
School of Economics and Finance
UNIVERSITY WESTERN SYDNEY
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s.keen@uws.edu.au 61 2 4620-3016 Fax 61 2 4626-6683
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