from the New York Times
June 4, 2001
By EVAR D. NERING
SCOTTSDALE, Ariz. — When I discussed the exponential function in the
first-semester calculus classes that I taught, I invariably used consumption
of a nonrenewable natural resource as an example. Since we are now engaged
in a national debate about energy policy, it may be useful to talk about the
mathematics involved in making a rational decision about resource use.
In my classes, I described the following hypothetical situation. We have a
100-year supply of a resource, say oil — that is, the oil would last 100
years if it were consumed at its current rate. But the oil is consumed at a
rate that grows by 5 percent each year. How long would it last under these
circumstances? This is an easy calculation; the answer is about 36 years.
Oh, but let's say we underestimated the supply, and we actually have a
1,000-year supply. At the same annual 5 percent growth rate in use, how long
will this last? The answer is about 79 years.
Then let us say we make a striking discovery of more oil yet — a bonanza —
and we now have a 10,000-year supply. At our same rate of growing use, how
long would it last? Answer: 125 years.
Estimates vary for how long currently known oil reserves will last, though
they are usually considerably less than 100 years. But the point of this
analysis is that it really doesn't matter what the estimates are. There is
no way that a supply-side attack on America's energy problem can work.
The exponential function describes the behavior of any quantity whose rate
of change is proportional to its size. Compound interest is the most
commonly encountered example — it would produce exponential growth if the
interest were calculated at a continuing rate. I have heard public
statements that use "exponential" as though it describes a large or sudden
increase. But exponential growth does not have to be large, and it is never
sudden. Rather, it is inexorable.
Calculations also show that if consumption of an energy resource is allowed
to grow at a steady 5 percent annual rate, a full doubling of the available
supply will not be as effective as reducing that growth rate by half — to
2.5 percent. Doubling the size of the oil reserve will add at most 14 years
to the life expectancy of the resource if we continue to use it at the
currently increasing rate, no matter how large it is currently. On the other
hand, halving the growth of consumption will almost double the life
expectancy of the supply, no matter what it is.
This mathematical reality seems to have escaped the politicians pushing to
solve our energy problem by simply increasing supply. Building more power
plants and drilling for more oil is exactly the wrong thing to do, because
it will encourage more use. If we want to avoid dire consequences, we need
to find the political will to reduce the growth in energy consumption to
zero — or even begin to consume less.
I must emphasize that reducing the growth rate is not what most people are
talking about now when they advocate conservation; the steps they recommend
are just Band-Aids. If we increase the gas mileage of our automobiles and
then drive more miles, for example, that will not reduce the growth rate.
Reducing the growth of consumption means living closer to where we work or
play. It means telecommuting. It means controlling population growth. It
means shifting to renewable energy sources.
It is not, perhaps, necessary to cut our use of oil, but it is essential
that we cut the rate of increase at which we consume it. To do otherwise is
to leave our descendants in an impoverished world.
Evar D. Nering is professor emeritus of mathematics at Arizona State
University.