Quick and Easy Math: Doubling Time and the Rule of 72

[Dogmudgeon]

You may have noticed in a number of my posts that I cite doubling times, and throw around numbers without giving them much thought. Although part of it can be attributed to my poor math skills, others come from a trick I learned called The Rule of 72. It allows you to calculate doubling time quickly, and as often as not, in your head.

Here's the Rule of 72:
T_D = 72 / R% --- or --- Doubling Time = 72 / Rate in Percent

One of the rules of thumb economists use is that our economy requires an energy growth rate of 2.5 to 3% per year, to power the industries that prevent economic collapse. An even 4% is considered the "normal" rate to keep the economy our of recession. Short term declines are usually not destructive, but no one know how even a five-year-long decrease in energy demand would affect our economic system, since we've never experienced one as a high-tech society.

A 2.5% rate means a doubling time of 28.8 years, because 72 / 2.5 = 28.8 --- and those 3% and 4% rates yield 24 and 18 years, respectively, after 72 / 3 and 72 / 4.

For instance, if a renewable technology development white paper reports that the installed technology investment is increasing at 18% per year, its doubling time is 72 / 18 or 4 years. From that point, you can do the math on the number of doublings required to make the investment growth large enough to prevent catastrophe, problems, or inconvenience.

By the same token, in an era where agriculture is facing ruin, the climate is unpredictable, and the resources to keep people fed and warm are too expensive for the world's peasantry, a 5% loss in population per year will give a reduction to half in 72/5 or 14.4 years. If this die-off situation starts in 2020, it means that the population of the world will fall from 7.5 billion to 3.75 billion by early 2035.

Here's another frightening situation: Instead of maintaining a 3% increase in energy demand, once we've entered the post-Peak Oil era, the declines are likely to exceed 5% per year. That's an 8% shortfall per year, leading to a 50% dropoff in the energy we need in nine years -- or less.

The interaction of those rates is not too difficult to calculate, either.

Albert Einstein is said to have liked the simple beauty of the Rule of 72, and is quoted as saying "It is the greatest mathematical discovery of all time". (That's an apocryphal quotation, of course.) And as famed American philosopher Barbie once said, "Math is hard!"

--p!

[NNadir]

1. Here is a simple derivation of the rule.

http://www.pli.edu/public/newsletters/newsletter.asp?stid=2147483621&ID=EN00000000029612

None of this makes much sense, though, in the context supposed. There is no reason to suppose that the rate will, in fact, remain constant.

In any case the model, the exponential curve, is very simplistic.

Many kinds of events could invalidate the model, including the onset of sudden catastrophe, such as may be represented by some kind of feedback loop.

If the model were valid over some short range, the rate would begin to change in any case as the carrying capacity of the earth were approached. One thing is for sure, the ultimate carrying capacity of the earth will be lower than what it was at the beginning of the industrial revolution, just as the carrying capacity of Easter Island is much lower than it was when the first humans arrived there.

I think (off the top of my head) that the earth's real sustainable carrying capacity for humanity may have once been about a billion people, but it may not be that high now. We don't know. It could be zero in the near future.

[depakid]

3. Two overshoot models

are discussed in the 30 update to Limits to Growth. One of these involves collapse, which of course can and does happen to given populations- or it occurs in the context of larger systems, such as complex societies or ecosystems -or in the case of certain, more pernicious feeback loops, on a continental or planetary scale.

From the systems perspective, overshoot and collapse results when signals or responses to problems are delayed and limits are erodable (irreverable degraded when exceeded). That's basically the Easter Island case- population pressure, slow regeneration rates and lack of trade. Other Polynesian societies with faster regenaration rates could sustain their historic populations and cultures on smaller islands.

The other model is overshoot and oscillation (which is what we'd like to hope for). Obviously, both signals and especially responses have been delayed- almost suicidally: think Reagan tearing down the solar panels on the White House. However, the limits (fish stocks, topsoil and other ecological services) can, with wise management- recover relatively quickly from erosion- or, depletion and degradation.

There you get the classic exponential growth curve, with smaller collapses and recoveries until a more or less stable state is reached.

It should be obvious to anyone looking at the cumulative evidence that we've overshot. The questions are: by how much, what places fare worst, where will it level out again and how hard will the landing be?

Personally, I'm not an adherent of Oldavai theory- but there's one thing for certain, fossil fuels (and ultimately, fissionable materials) are finite. Once they're gone, we're back to basically a solar (and small geothermal) budget, based on photosynthesis, passive and perhaps modest (or more likely small) scale direct solar power capture, wind and hydro-electric. Small potatoes based on our current throughput.

[bananas]

4. Under a billion? What do you base that on?

[NNadir]

5. Oh, a few things I guess.

Some things are irreversible.

If arable land is desertified, and the top soil blows away and falls in the ocean, many thousands of years may be required to restore it for instance.

If you remove mountain ranges to mine coal, and the acid runoff and heavy metal pollution kills the river's fish, those fish cannot sustain a population.

If you pump an aquifier like the Ogalalla dry to grow corn, it may take hundreds of thousands of years to restore it.

If you plow rainforests under, they don't spontaneously restore themselves, since the climate that depended on them as much as it created them, may settle into another self sustaining state.

My guess is that the sustainable world population will be less than it was at the outset of the Industrial Revolution, when it was about 1 billion, because the industrial age had greatly damaged the earth in many ways that are irreversible. Therefore I reason that the carrying capacity has been lowered.


http://www.prb.org/Content/NavigationMenu/PRB/Educators/Human_Population/Population_Growth/Population_Growth.htm

[bananas]

6. Well in that case there's no need for more nuclear power plants

the existing ones should be more than enough for under a billion people.

[NNadir]

7. Only if you kill off 5 billion people immediately.

That, I think, is my constant point. I always make that clear, that billions of existing lives are at state.

Sometimes I suspect that the anti-nuclear crowd is proposing exactly that: Population reduction by catastrophe. I have never concealed the fact that I consider this grossly immoral.

Unless we build nuclear reactors we must kill off 5 billion people immediately. There is no other workable option.

I have always favored another approach, which is ethical population control, control using attrition. As I often state, ethical population control involves the following liberal agenda: Respect for the rights of women; reduced infant mortality; equitable distribution of wealth; decent living standards; an educated population; respect for diversity; respect for science; the promotion of cooperative secular values...

The liberal agenda is most closely realized in countries where there is wealth. It is not realized in countries with low per capita incomes or energy demand.

We can have Finland, where the population is slowly decreasing in an orderly way, or we can have Rwanda and Cambodia where the population is decreasing through anarchy and violence too horrible to contemplate.

I note that I see that the nuclear option is being exercised, that the world has by consensus more or less rejected anti-nuclear mysticism. However I strongly suspect, with tremendous sadness, that it probably already far too little too late.

[Dead_Parrot]

2. Dammit

I wish someone had mentioned that to me 30 years ago...