EJ wrote:Efficiency by definition is "achieveing a goal as cheaply as possible" (Colander, G4) There is no reason to suppose that perfect efficiency is even possible or that the model makes this assumption. I agree the actual working models will be difference equations but ODE's are the continuous analogs of difference equations, I am making no logical err by starting here.
That was not a critisism of your post, it was explaining my understanding of it. I did not say or think that model assumes perfect efficiency, in fact quite the opposite. If that is not your meaning then we aren't going to get very far. So I'll be a bit more explicit. Try not to be too critical of the math from a non mathy:
D(t) = the "demand" at time t given current prices P(t).
S(t) = the supply at time t given current prices.
D(t) does not equal S(t) at any given t.
However P(t+dt) = P(t) + k(D(t)-S(t))
I was saying that in an ideal "efficient" market where information was processed instantaneously and where there is no lag in response then k approaches infinity and (D(t)-S(t)) approaches 0.
So you are assuming that k does not approach infinity which is a GOOD thing. But it does make life complicated because unless you're not interested in testing your hypothesis now you are going to have to find a way to measure (D(t)-S(t)).
Yes we have to get an estimate of D[0] and S[0], since we are interested in the long term behavior of the systems and most trajectories will be captured by the trend we can feel ok with being off by possibly as much as 15%. This is a calibration issue separate from the notion of process realism.
By D[0] and S[0] you mean you nead the initial values at time t =0? What I was meaning is that you need D(t) and S(t) for any point in time. These are MEASURED values aren't they? Or do you propose to model D and S as well. I don't know how model D and S might be modelled, but then I don't really know how you can measure D and S either so I'll let that go. We really do seem to be on different wave lengths.
How small do you think is to samll to account for with mathematics? This point does not make sense nero. I can keep a math model that tracks to 30 digits of precision in both Mathematica and Maple certainly that is good enough to stay wihtin a percentage error of any real trend. Unless of course the trend we are looking at is chaos.
No too small to be measured by experiment, due to limitations in the experiment.
I think your conclusion is totally invalid the study of far from equlibbrium systems is gainaing ground in mathematical ecology as we speak. You've done nothing to show that the model is invalid. The argument you are using is often called "The Strawman Fallacy." It means you constructed an argument different than my position without really understanding my position. Then you defeated your own imagined foe and declared victory. You'll have to do better than that my friend. Try this link for starters
http://www.datanation.com/fallacies/ . He does a pretty good job with it. Of course any good math program is going to teach formal logic and set theory (which is the foundation of everything logical).
I didn't set out to show that the model was invalid I was attempting to explain why I thought that the model wasn't useful.
Don't go throwing around "strawman fallacy" unless you're pretty darned sure it isn't simply a matter of misinterpretation. Them's fighting words.