[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


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


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


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

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 

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

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
s.keen@uws.edu.au 61 2 4620-3016 Fax 61 2 4626-6683
Home 02 9558-8018 Mobile 0409 716 088
Home Page: http://bus.macarthur.uws.edu.au/steve-keen/

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