Thursday, 20 November 2008

A new sort of Chaos?

I've been thinking. I've also been reading this book, which is about some of the great equations of modern science. It covers a range of fields; so far it has covered quantum theory, exobiology, nuclear energy, and chaos. Or, for the scientists, E=hf, the Drake equation, E=mc2, and x(n+1)=ax(1-x). The latter two desperately need some subscripts and superscripts, but (sadly) Blogger is frustrating me on that score.

The latter equation is fascinating (Stay with me on this - I'll return to a familiar theme soon). It was developed to model animal populations. The constant 'a' represents a growth rate - roughly representing a number of offspring. So if 'x' is a number between 0 and 1 representing the fraction which the current population represents of the total possible population that can be sustained in the locality concerned, 'ax' is the number after a generation. Of course, some will die; generally, the higher the population, the more disease will spread, the more attractive the colony will be to predators, and so on. So the (1-x) factor limits the population in this way; as the population grows, (1-x) shrinks and this the next generation is reduced by this factor.

The result is weird. If a is small (less than 3), then the population grows steadily to a stable point and then stays there. For values of a between 3 and 3.57, the population oscillates between two (or more) stable values. Above 3.57, the population varies widly between generations and becomes wholly unpredictable - i.e. chaos.

This is usually shown in a graph known as the "Logistic Map", in which the eventual stable point is shown (on the y axis) varying with the chosen value of a (on the x axis):

So, in other words, at low growth rates, things are stable. Push the growth rate, and things start to oscillate, alternately growing and crashing. Push harder still, and the whole thing goes haywire.

Does this sound familiar?

When I read that, I wondered if it could be applied to economics. Go for gentle growth,and things can be stable albeit disappointing. Push harder, and the good times will be interspersed with bad times. Push harder still, and all bets are off; anything might happen.

Maybe, when our Glorious Leader And Saviour Of The World's Economies said he had put an end to boom and bust, he really had. If so, his mistake was to push the economy even harder; by taking more out of the productive parts of the economy to fund the unproductive public sector, he forces the private sector to work correspondingly harder in order to maintain its position. And the result has indeed begun to look like the right hand side of the graph, I have to say.

Guess I'm just a frustrated economist, really....


  1. "Guess I'm just a frustrated economist, really...."

    No kidding!

  2. No you are far better than an economist.

    I am not into stability, I am into development. Does anyone suppose that we would have had the industrial revolution if we had been as risk-averse as the present regime? I can just imagine some technocrat saying "those railways don't look up to much, why would anyone want to take stuff around more quickly than by canal?"

  3. ps should anyone say "BE you are evil, you would let people be destroyed by the economic gyrations" I say to them: people would have to take responsibility for saving up while they had good jobs in order to tide themselves over during a downturn. Isn't that what used to happen?

  4. I think you're right, BE. Stability is very tempting, but the difference between that and stagnation is too slim for comfort. We have to push for growth, but at the same time be aware of the risk and be prepared for it.

    And there is the great failure of Brownism; by pronouncing an end to boom and bust, he gave himself permission to ignore the downside risk. But he was not willing to accept the lower returns of mere stability, so allowed the economy to steam on. The policy is and always was self-contradictory.

    Where we all (as a nation) failed was in not noticing in 2001 or 2005 that Brown actually believed his own rhetoric. I don't think anyone else genuinely believed that we really had seen an end to boom and bust; the problem was that we didn't spot that he did, and was acting on it.

  5. LFAT - there is a very thin line between "brilliant polymath" and "raving lunatic". :-)

  6. Did you used to have that "Fox and Rabbit" program on your early 286 VGA
    The one where you put in any number for each and the program went by until equilibrium had set in, or both sets were wiped out.

  7. No, I had it on my ZX Spectrum ;-)