Hi Anders,
> Very interesting discussion. Jurriaan has already replied, making
> some very important points. But I think the discussion could be more
> to the point if one introduces the question of reaction speed.
> Because if the equilibrating forces never do get us fast enough to
> eq. before the eq. point (or attractor) changes again, then you have
> to have a theory of the "moving target", i.e. the moving eq./attractor.
Yes you do. In fact, I think such a theory is necessary.
> So far I have not seen any formal modelling of such an eternally
> moving eq. point. Mathematically it is possible, there is a (too!)
> developed formalism of hitting moving targets. The military have tons
> of missiles that chase other missiles using control theory (Kalman
> filters etc.). But in economics this type of models are underdeveloped.
Yes agreed. Allow me to widen the discussion a little. We have been
discussing "equilibrium" and "equilibrium point" but of course these
are natural language echoes of a highly developed mathematical theory
of deterministic dynamical systems (systems of ordinary differential
equations) in which "equilibrium points" play a small part. Some of
this theory reappears in economics. In other fields -- such as physics
-- there is a much longer and more intimate connection between the
domain of application and the mathematics.
It's commonplace in physical models and control theory to "stitch
together" a model that keeps some input constant with another model
that then drives that constant and makes it variable. In fact, many
stability theorems exploit our ability to analytically separate the
dynamics of the complete system in this manner. For example,
dissipativity theory, a generalization of Lyapunov theory to systems
which are "open" to inputs, works this way. We can prove stability
properties for the base system assuming the inputs are zero. Then we
can take another system that is also "open" to inputs and prove the
same. Then we can join those systems together via their input/output
ports. We then use the stability proofs for the separate systems to
construct a stability proof for the joined systems.
My point is this -- the "counterfactual" approach of assuming some
things constant (when factually they are not) is an *essential* tool
for understanding the causality of complex dynamic systems. The
mathematicians exploit this analytical procedure. It is really after
all simply "divide and conquer". I feel that critics of this
methodological approach must either be ignorant or naive.
> Ian:
>>This is semantic. I take "natural prices" to refer to those prices
>>that obtain when there is no profit incentive to reallocate capital in
>>circumstances of constant technique and final demand.
>
> The problem that Jurriaan points to here is that the same forces that
> drives you towards eq. - the forces of competition also continuously
> changes the technique and the preferences.
Yes of course they do. I want to understand these kinds of dynamics.
My approach is to take one step at a time. You can't enjoy the full
scientific meal at one sitting, that is all.
> The problem here is that the value of the thermostat is continuously
> changing. The thermostat might change in an orderly way so that you
> can have a theory of the moving attractor. Given that we can estimate
> the *reaction speed* we could get - if not an equilibrium theory, so
> at least a theory of "non-crisis" path of development - like the
> golden decades after the 2nd WW.
I agree with your general point. But my example was constructed to
show that -- even if the reference temperature of the thermostat is
constant -- it is not the case that the whole system will reach its
equilibrium point; nonetheless the attractor is real and causally
efficacious. In other words, it is wrong-headed to deny the importance
of theorizing equilibrium points and their stability properties simply
because empirical reality seems not to exhibit such properties.
> You are right - capitalism still exist - it reproduces itself - but
> on a ever expanding diseq. path with periodic crisis (not regular,
> not repetitive) - so the meaning of non-constant technology
> equilibrium and corresponding prices is not still clear to me.
It is an analytical abstraction to hold some things constant, just as
a physical experiment is designed to isolate a particular mechanism
from the interference of other causes.
I cannot expand on this now, but I am deeply influenced by early Roy
Bhaskar (i.e., critical realism). So I view the empirical stream of
events as jointly determined by multiple "hidden" causal mechanisms in
"open systems". Part of the work of science is to discover the laws
that describe how those mechanisms operate in isolation or "closed
systems". How those mechanisms work in isolation may be very different
from how they manifest in open systems. Once you've done that, put the
mechanisms back together again and you'll have a good explanation of
the empirical. The abstraction of natural price is a "moment" in this
scientific project.
I also think this is essentially Marx's method, for what it's worth.
>>Having said that, I do think there are important sources of
>>instability in capitalist economies.
>
> Problem is not that there are "sources" of instability - but that
> they are the same as the equilibrating forces, i.e. the un-cordinated
> - not mutually consistent from the outset - actions of individual firms.
I am not sure they are the same. For instance, I think that price and
quantity adjustment by firms is connected to but quite different from
sources of technical change.
> I agree that to hold things still might be useful, but we have been
> waiting for the full story for more than hundred years - so one might
> ask if not economic modelling should start with modelling real
> competition, real change of technology - because it has shown itself
> extremely difficult to modify these models to make them dynamic afterwards.
I can't speak to the poor progress of economic science, although of
course the obsession with static, Walrasian equilibrium models is
largely to blame.
But I can say that "modeling everything at once" is easy to do --
especially with computational agent-based models -- but this approach
has the danger of creating a proxy object that is almost as complex as
the real world. You don't get insight.
> The problem between Cap. vol 1 and 3 - the transformation problem has
> shown itself very resistant - although Farjoun and Machower,
> your work, the TSSI - points to another and hopefully more fruitful
> research agenda - but I think Jurriaan is correct in stressing that
> using static models to model "moving attractor systems" with slow
> reaction speeds for the eqilibr. processes compared to technological
> development is a real problem.
My model is not a static model. There's lots of interesting dynamics
in there. I do (unapologetically) assume a constant technique, because
my current (short-term) aim is to formally analyze some outstanding
conceptual problems in value theory (rather than "model the economy").
>From my reading of the classical authors they too assumed a constant
technique; that's part of the meaning of "natural price".
>From a mathematical perspective, I also know that it is easy to add
technical change to this kind of cross-dual model. I have already done
this in simulation. But developing a good understanding of the
resultant system requires a lot of time and effort. Again, as a
practical "working scientist", it pays dividends to analyze simpler
models first.
Best wishes,
-Ian.
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Received on Fri Feb 4 15:19:13 2011
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