The Conun Drum

From a post on The Oil Drum, Rapier pondered the mysteries of reserve growth.

I've hinted at this before, but I wanted to nail a simple idea centered around reserve growth. As a premise, let me create a hypothetical situation. Say starting from right now, i.e. Time=0, we find a growth in reserves that goes like 1/(Time+k), where "k" is some small number to keep the starting number finite. Let's say that this reserve growth falls in the provable category to indicate that we can extract it.

Three interesting results spring from this premise.

1. The amount of reserve left from now until eternity sums to an infinite number. This derives from a property of integrating a hyperbola (1/Time) over all of time. In other words, we get a URR of infinity.

2. If production follows a rate proportional to the current reserves (the classic "greed is good" assumption which explains man's and the free market's capitalistic instincts), the position of peak won't change too much. This has everything to do with rate considerations; the rate of reserve growth cannot match consumption rates, and new discoveries clearly continue to dwindle.

And most importantly, the one thing that explains Rapier's puzzlement.

3. The draw-down from reserves can become vanishingly small in this scenario. Taking finite production from an infinite pool leads to the conundrum that we will continue to extract an infinitesimal fraction of that eventually available.

I consider the argument quite subtle, so that if interpreted incorrectly, it gives ammunition to the cornucopians, who can assert that huge reserves lay in wait. However, in reality, since oil depletion occurs proportionally to current reserves, we end up seeing the classic effect of "diminishing returns". Of course this has real ramifications for a continuously growing energy-based global GDP economy, but the cornucupians will not spin it that way. They instead point to a continuously finite reserve that doesn't get drawn down by as much as one's expectations can intuit.

The above figure shows an extraction term corresponding to an exponential and a reserve growth indicated by a 1/(T+k) function. The convolution of the two -- shown below¹ -- roughly gives an idea of the overall extraction. (A variation of this forms the basis of the oil shock model)

Having to face and account for this argument, a cornucopian would have to propose a reserve growth rate that will keep pushing the peak into the future. Unfortunately, this would result in a growth even more aggressive that the 1/Time variant, which already has an infinite URR ! Fortunately, anyone actually proposing such a growth rate puts themselves in a situation of endless mockability.

We have to continue to question the numbers because otherwise we fall into the logical conundrum B.S. traps that politicians and corporations and scam artists have historically used to try to separate us from our money. Simple thought experiments like I have shown here remain one of the few options that we have to eliminate the rhetorical arguments from the public discourse.

As a recent and complementary example of where people have gotten hoodwinked in this fashion, google the "infinite horizon" argument to escalating Social Security costs. Bush's people have actually suggested huge future costs of S.S. based solely on a hidden assumption of an "infinite horizon". It takes time for the economists to dig this stuff out of the rhetorical arguments, but by that time the spinners have inflicted the damage and people get a completely misleading impression of the issue. I remember hearing Al Franken debunk this argument quite effectively by saying that, "yes we may have a huge SS deficit, but will have infinite time to pay it off, so it looks like our current funding is no problem". Touche.

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¹ Spreadsheets don't have a convolution function as far as I have found, but as a trick that Khebab would appreciate, use the Fourier transform on each function, do a complex multiplication, and then do an inverse transform on the result. This does the convolution effectively, albeit in a round-about way.