Lessons in Oil Reserves from an Ancient Greek

With gas prices rising, Americans are more than ever questioning our dependence on oil. Beside rising costs, the worry is that current demand is unsustainable, oil reserves will soon be depleted, and the American standard of living will be eroded for all future generations. I am happy to put those worries to rest: the depletion our supply of oil is inevitable so long as we continue to rely on it. And I have the wisdom of over 2,000 years to back me up.

The Greek philosopher Zeno (circa 450 B.C.) formulated a famous paradox who’s solution can shed light on the situation. Zeno’s paradox can be paraphrased as follows: Suppose I want to cross a room of length D. In order to get across, I must first travel half the distance. But before reaching half the distance, I must first travel half of that distance. And before getting there, I must travel half of that distance and so on until I can’t actually go anywhere since doing so would involve an infinite number of intermediate steps. This logic calls into the question the very possibility of motion.

Since we know motion is possible, we resolve Zeno’s paradox by noticing the following. Suppose I could walk half the original distance, which would be ½D. I could then walk half the remaining distance, ¼D, again half the remaining distance, 1/8D, and so on. Continuing in this manner, how far would I eventually walk? Adding up all my individual steps gives us ½D+ ¼D + 1/8D +… or [ ½ + ¼ + 1/8 +...]D. The term inside the brackets is the well-known geometric series, the sum of which is simply: G(r)=r+r²+r³+…=r/(1-r) for any fraction r. Letting r= ½, we have G(r)=1. This is remarkable: not only is the sum of an infinite number of increments a finite number, but by taking smaller and smaller steps we can eventually get to our destination having traveled the entire distance D. Any bewilderment should be relieved by realizing that any finite distance, broken up into an infinite number of small parts, must have the small parts add up to the original distance. Zeno’s paradox is resolved by understanding that while the number of intermediate steps is infinite, they must still add up to something finite. So motion is possible—good news for commuters.

So what does this have to do with oil reserves? Suppose D is the existing supply of some natural resource, say oil. And suppose we are given the job of determining a rate of extraction so that we will always have some oil left over for future generations. Assume that the resource is non-renewable, so once we extract D, there is none left. But to extract D, we must extract half of D. And to extract half of D, we must first extract half of that and so on. From Zeno’s paradox, we know that continuing in this manner, we will eventually extract all of our oil reserves and there will be no more. Fearing this, we may want to reduce our rate of extraction. If we consume say 1/3 of our original stock, then 1/3 of the remaining stock and so forth, we can insure that there is always some amount left over, right? Wrong. As it turns out, any manner in which we extract a constant fraction of our available reserves ultimately result in the complete depletion of the resource. To see this, suppose we decide to extract some fraction r of our existing reserve in every year. In year 1, we will extract rD, leaving D-rD=(1-r)D in reserve for year 2. In year 2, we extract fraction r of our existing reserve (1-r)D for a total extraction r(1-r)D, leaving (1-r)D-r(1-r)D =(1-r)²D for year 3. Continuing in this manner, we will extract r(1-r)^t-1D in any given year t. Summing our per-year extraction over all years, we will eventually extract [1+(1-r)+(1-r)²+(1-r)³+…]rD total. The term in brackets is 1+G(1-r)=1+(1-r)/r=1/r. Thus, our total resource extraction will be [1/r]rD=D, and we use up all of our resource regardless of the rate at which we extract it! The same principle that guarantees you will eventually make it to work in rush hour traffic guarantees that someone in the future will run out of gas and never make it.

With regard to the current discussion over oil consumption, the issue is not whether oil reserves will eventually be depleted but when. The when should be determined by balancing the benefits of current consumption with such benefits in the future while taking into account the possible development of alternative energy sources. Since consumption decisions are made on the level of the individual consumer and producer, any reduction in oil consumption would require intervention by a central authority possibly through the imposition of a tax. A tax would serve to raise the price of gas and of any product produced using oil, reducing consumption of these products and indirectly of oil itself, but in doing so lower our standard of living. The development of an alternative energy source that rendered oil useless would stop the process of depletion resulting in left-over reserves without the need to cut back consumption. Of course the remaining oil would have no value by virtue of having been rendered useless.

So I went to a lot of trouble to prove something obvious. At least the math was fun.