Thanks. So far I’ve managed to replicate “Bank Accounts in a Growing Economy” in your Dijon05 (after finally noticing the log scale).

The world is nonlinear. Most models are based on linearity because it is more intuitive, less surprising and analytical solutions exist (i.e. for the convenience of the modeller) and it often does a good job.

My polynomial example is relatively simple to express as a differential equation, but difficult to mathematically solve without resorting to numerical approximations. At no place does my example lead precisely to exponential growth, but for finite regions it does come arbitrarily close to exponential growth (and exponential decay in other regions). We can quickly determine this behaviour because the regions near the zero crossings are closely approximated by straight lines.

This is the process by which models are constructed — work in a local region where linear approximations are good enough for the model to give sensible results, then carefully explore the nonlinear boundaries of that region.

The excitement of our world comes from the nonlinear bits, and the transitions from one operating mode to another. Do we expect harsh discontinuities like a strut snapping within a bridge truss or do we expect gentle smooth transitions such as my example? Either case is possible from a mathematical perspective, so only by close study of real-world measurements can we decide which case applies to the situation at hand.

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Steve, with respect to nomenclature, I’ve always used the following:

* When a cleanly delimited discrete event is followed later by another discrete event, I call it a time delay.

* When an exponential curve (or approximation thereof) is in evidence I call it either an exponential time constant to be proper or just a time constant as shorthand.

* When a cyclic signal is in evidence and one feature of the cycle comes a bit later than some other feature of the cycle I call it a phase angle

* I tend to use lag and lead as just general indicators of which came first (e.g. A lags B in this situation).

That’s just the way I’ve seen these things used.

I have a copy of “Modern Control Engineering” by Ogata who uses lag to describe any elements of a system that would cause the output the lag behind the input for whatever reason, and lead has similar usage in the reverse direction. Thus, for example a lag network is another name for a low pass filter, a lag compensator is some device that deliberately adds lag to the system in order to compensate for some other system behaviour and so on.

Ogata brings up the concept of transportation lag to describe the situation where you make a change in one place but you can’t see the effect of that until something works its way through. In a finance context, if a cheque has a fixed clearing time of three days, that would be a transportation lag.

Ogata defines a specific term delay time to be the time it takes for a system to react to a step function input such that the system output gets halfway to its final resting state. This is in the context of terms such as rise time, and settling time.

The term time constant is used to describe most any time value that turns up in an equation, also Ogata fluidly switches between Bode analysis, complex pole/zero analysis and time domain so time constants often become corner frequencies or just points on the complex plane with minimal explanation.

Ogata is a reasonably well respected book in control engineering so if you borrow it from a library and scan through for the basic concepts you will at least be in the ballpark w.r.t. usage.

I was about to ask about the “time lag” term. Shouldn’t it be called the “time constant”?

http://en.wikipedia.org/wiki/Time_constant
http://en.wikipedia.org/wiki/RC_time_constant

I know what you mean however time lag may be confused with the latency (or delay) by someone with electronics/networking background:

http://en.wikipedia.org/wiki/Lag

In your case I think we are talking about the parameter inversely related to velocity of money
http://en.wikipedia.org/wiki/Velocity_of_money
which is the average time of the cycle of getting and spending a certain amount of money like wage.

Does it make sense or is it even more confusing?

When I have time I’ll look at your paper.

Thanks for that. I’ve got it pretty much running but my curves aren’t much like yours yet. Maybe it’s units. When you write r=5%, you mean r=0.05 yr^(-1)? My background in this kind of thing is in kinetics, so I prefer to write k where you have 1/w. So when you write tau=1/26, I write k=26 yr^(-1)? Meaning that the corresponding variable “turns over” (roughly speaking; there’s a problem with terminology here) 26 times per year?

Again, any comments you have would be appreciated.

The equation you wrote indeed has 3 equilibria (I didn’t have time to verify whether they are stable or not).

But the point BS1 was making was slightly different. In most of the cases real systems which embed both exponential growth and limits end up in oscillations. Technically you are right – it is possible to create a highly non-linear model which behaves exponentially for a certain amount of time and later gets into saturation. However people who claim that real systems (like for example our housing assets) which sustained a period of exponential growth will later saturate rather than collapse are quite often wrong. It is because very special conditions have to be met to prevent instability – that is equilibria must be stable.
http://en.wikipedia.org/wiki/Equilibrium_point

We will see whether claims made by Ric Batellino about ever rising and them moderating house prices come true. Has he evaluated his Jacobian matrix?

Thank you for giving us enough rope. I will play with the intermediate model (in scilab) for sure – most likely over the weekend.

yes, sorry that my documentation is a bit tardy. It would help if I derived my equations in the same engine that I publish them in, but I use mathcad for the latter and various programs (Word, Scientific Workplace) for the latter. I’m thinking of moving to just one platform (Publicon/Mathematica or Scientific Workplace) to get around this hassle.

In the meantime, I set initial price to 1, and population to such a level that, given a value for L, the employment rate (L/N) was 95%.

I’ve uploaded a PDF of two simulations for a forthcoming conference here and here. They necessarily have the initial conditions embedded in them at the Given … Odesolve() block.