Saturday, May 3, 2008

Mathematics of Growth

OK, three end-of-semester snarky posts ought to do for awhile. Lets get down to business.

A week or so ago I posted a "homework problem" in the comments on a blog about whether there is really 200 years of coal left. I was quite impressed that someone (Bartlettman) got it right, albeit on the second try.

Because answering a question like "How long will 847 Gt of coal last if the current consumption rate of 6 Gt/yr increases at 5% per year?" is extremely important for a variety of reasons (ranging from national and international energy policies to local decisions about encouraging growth of a community), I thought I should post the solution here.

If you don't want to see the mathematics, you can skip to the section with the final formula and the answer to this problem. Evaluating it for any problem of interest (such as "when will my city have the same population as (name) if it grows at X% per year") is then a simple matter with any basic scientific calculator.


Use R(t) for the Rate of consumption of coal at a particular time t, measured in years from now. The initial rate is called R0 = R(0).

Use C(T) for the total consumption of coal between now [t = 0] and some future time T. This is calculated as the integral of R(t) from 0 to T. We use this to calculate how long some specific amount of coal, C, will last.

The same analysis applies to something like the population of a city if we replace R(t) with P(t) everywhere and simply ignore the discussion of the consumption C(t).

(A) Rate does not change

The rate of change is zero, dR/dt = 0.

Answer for the rate at any future time: R(t) = R0 is constant.

C(T) = integral (from 0 to T) of R(T) = R0*T.

Answer for how long an amount of coal C will last: T = C/R0.

(B) Rate increases at 5%

dR/dt = 0.05*R. Notice that the change in R depends on R. As R gets bigger, it increases by a larger amount each year. This is called a "differential equation". This particular one is rather easy to solve by a method called separation of variables, where we cross multiply (saying this makes a real mathematician react like someone scraped fingernails on the board) to get dR/R = 0.05*dt so we can easily integrate both sides of the equation. The result is ln(R) = 0.05*t + R0.

Answer: R(t) = R0 exp(0.05*t), where I use exp to denote the exponential function ex.

The consumption is just the integral of R(t). The nice thing about the integral of the exponential function is that any child can do it: the answer is equal to the itself - the exponential function. Well, almost any child, because you have to be careful about that 0.05 in there. The result is [R0/0/05] * exp(0.05*T) evaluated from 0 to T, which gives:

C(T) = [R0 / 0/05] * [exp(0.05*T) - 1].

Solving this for the T when the consumption is C is, however, not child's play. We have to unwrap the exponential function and use the natural logarithm to get rid of it. Skipping some details, the key intermediate step is to get

C * 0.05 / R0 = exp(0.05*T) - 1
1 + C*0.05/R0 = exp(0.05*T)
ln(1 + C*0.05/R0) = 0.05*T

Answer for how long an amount of coal C will last: T = ln(1 + C*0.05/R0)/0.05

If you have a scientific calculator, there will be an ln(x) or ln key, with the ex function right above it.


This calculation is for a rate of 5% (0.05 in the formulas). You can do it for any other rate of increase by changing 0.05 everywhere to whatever you want. You would use 0.03 for a 3% rate, etc.

How long will the coal last?

We have C = 847 Gt and R0 = 6 Gt/year, with T in years.

Evaluate ln(1 + C*0.05/R0) / 0.05 = 41.7 years.

Compare C / R0 = 141 years.

We should really round these answers to 42 years and 140 years given the lack of precision in the numbers being used.

"Bartlettman" got 42.8, which is probably due to a different assumption about when the rate was 6 (start of the year or end of the year), or was based on an amortization type calculation where the rate is constant during the year. Like I said, good enough for government work, since we are talking about a 1 year difference compared to a 100 year effect!


It is really easy to calculate the projected life of the resource if someone gives you a number calculated under the assumption of no growth, such as the 141 years, because that is equal to C/R0 in the other formula. So if we want to know the answer for 5% growth if the resource could actually last 200 years with no growth, we simply evaluate

ln(1+0.05*200) / 0.05 = 47.96 years = 48 years

Instead of lasting 60 more years, it only lasts 6 more years! This is due to the fact that the consumption rate is increasing exponentially, because its increase gets bigger as R gets bigger.

DOUBLING TIME: the rule of 70

A population with double in 70/(% rate of growth) years.

If you read this far, you are probably interested enough to want to know a simple formula that does not require the use of a calculator. This is particularly useful for questions about the rate of consumption itself, or related questions about population growth, where we want to solve the equation (again for 5% growth)

P = P0 * exp(0.05*T)

for the time T when we reach a certain population. The answer is

T = ln(P/P0) / 0.05

In the special case where the population doubles, P = 2*P0, we get

T = 0.693/0.05 = 13.9 years.

Al Bartlett (see below) and others (investment counselors) often talk about the rule of 70. This comes from noticing that 69.3% (the natural log of 2) is really close to 70%, and it is easier to divide a number into 70 than 69.3 and nothing we are doing is accurate enough to worry about that third digit anyway.

I used this to estimate that the coal consumption rate will double in 70%/5% per year = 14 years and double again in 14 more years. That is how I was able to write in another comment that we would be burning 24 Gt of coal per year in 28 years if our consumption rate grew at 5% per year ... without using a calculator. Just divide 70 by 5 to get 14, then double 6 twice to get the rate in 28 years.


The assumptions behind these calculations are false.

An economy based on constant growth is based on false assumptions.

A constant percentage rate of increase is not sustainable to the end.

The assumption is, basically, that it is possible to use 48 Gt/yr (17% of the entire earth's coal resource) in the last year !! before we run out. That is not how "extraction" industries operate. [But it is how population growth operates; by the time you realize you are running out of space, you have already run out of space.] The last half will be harder and more expensive to get out, for the simple reason that you will dig up the easiest half first.

That is the essential idea behind "peak oil", something we are seeing play out right now. (Just as peak production by the US was correctly predicted, it would seem that peak production by the Saudis was also predicted pretty accurately. It is highly likely that OPEC is not refusing to increase production because it wants to keep the price high. It is quite likely that some major producers can no longer increase production.) When exponential growth runs up against a finite resource, a peak will result somewhere around the time you have consumed more than half of the resource.

Better to ask when we will use up half of all of the coal out there. (The answer would be 30 years at 5% growth rates.) After that, the growth rate would become negative and consumption would tail off. It might take a century or more to get at the last bits of coal, but it is only in that sense that coal will last centuries.

One reference and closing comment:

Since (as I explained in that comment) I first learned about this subject from a beautiful talk given by Prof Al Bartlett of the Univ of Colorado, I thought I should link to a transcript of a more recent version (circa 2004) of that talk. In scanning it for what might have changed from the 1970s (not much except for verification of his predictions), I was intrigued to see a reference to a Newsweek magazine article that was extremely optimistic about the amount of coal left in the US. There was a similar article in Time that prompted me to write them a letter documenting that the coal reserves they described would not last 600 years because we would burn coal equal to the mass of the earth in 600 years if we kept increasing its use at the rates discussed in the article. Needless to say, they did not publish it. Yet here we see that, primarily due to increased global consumption, there are now saying that there are less than 200 years of coal left. Surprising? No. The lifetime of the resource has dropped because the rate of using it has tripled (about right for 4% growth over 30 years) since that previous estimate.

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