The Extraction of Exhaustible Resources

On December 28th, the Energy Bulletin included a link to a paper called Technology and Petroleum Exhaustion: Evidence from Two Mega-Oilfields (pdf) by two economists, John Malcolm Gowdy and Roxana Julia, from the Rensselaer Polytechnic Institute in New York. Here's the abstract.
In this paper we use results from the Hotelling model of non-renewable resources to examine the hypothesis that technology may increase petroleum reserves. We present empirical evidence from two well-documented mega-oilfields: the Forties in the North Sea and the Yates in West Texas. Patterns of depletion in these two fields suggest that when a resource is finite, technological improvements do increase supply temporarily. But in these two fields, the effect of new technology was to increase the rate of depletion without altering the fields' ultimate recovery - in line with Hotelling's predictions. Our results imply that temporary low prices may be misleading indicators of future resource scarcity and call into question the future ability of current mega-oilfields to meet a sharp increase in oil demand.
The paper is fairly standard fare for the peak oil community but what turns out to be of interest is the application of the work of Harold Hotelling regarding the Extraction of Exhaustible Resources and their discussion of the economic view of resource scarcity as regards oil. Examining the use of EOR technology in the historic production of Yates (West Texas) and the Forties (UK North Sea), Gowdy and Julia conclude that temporary incremental production gains are offset by later steeper decline rates in the tail end of production without increasing the overall URR. Their main conclusions are essentially that 1) oil is not being treated as a finite resource as the oil field analyses predict and 2) temporary production gains mask real scarcity and result in misleading low oil prices. Let's look at the work of Hotelling in the context of peak oil and see where that goes. This post runs a bit long so I hope you'll bear with me here.
The Hotelling Result

First, here is the Hotelling equation as regards a finite resource from Gowdy and Julia. [Editor's note--open this graphic in a new window, keep it available and continue to read along below. I will refer to this equation again near the end of this post.]

The main insight regards the rate of discount δ, which is essentially the same as the interest rate. The main idea, of course, is to maximize returns (profits) at all times. From the Extraction link cited above, we see that
...the society maximizes its profit over all times, not just the present time. For this, the value of future profits must be discounted at a rate that is more or less equivalent to the interest rate.

... this is the fundamental problem in all of exhaustible resource economics, and it was first posed and solved by Harold Hotelling in 1931....

This formulation is known as the Hotelling Rule, and predicts that prices increase at an exponential rate that is equal to the exhange rate. That means, in an economic sense, a mineral deposit in the ground has the same significance as a bond, and is in some sense interchangeable with such a financial instrument.

An overview of the hotelling price path is given in the figure below. As the resource is depleted, the price rises and as the price rises the demand, and hence the consumed quantity, falls. All the while the quantity remaining in the reservoir sinks.
Here is the Hotelling price path from that same paper.
As production of the finite resource decreases (lower left quadrant), the price path rises (upper right quadrant) and demand drops off (upper left quadrant). Note that the resource is never actually exhausted because there is a backstop price at which the "resource is said to be economically exhausted". However, the time to depletion is increased with use of EOR in mature oil fields. See Figures 1 and 6 from Gowdy and Julia. Presumably, substitutes are available at the point when production reaches the backstop price. The implications of all this for peak oil and current production behaviour are clearly explained by Francis de Winter at
The Harold Hotelling paper is often referenced because of its description of the "Free Market" mechanism by which our descendants are systematically deprived of any significant access to the finite natural resources which we currently have at our disposal. This mechanism works as follows.

If one "owns" an oil well, one has to decide whether to "produce" (i.e. extract) and sell the oil now or later. If one produces and sells the oil now, one can put the money in the bank, and it will grow because of the interest it will earn. If on the other hand one produces and sells the oil later, one has to discount the money one will get for the oil, because one is getting the money later, and not now. With either way of looking at this, there is only an incentive to produce and sell later (rather than now) if one has the certainty that the oil in the ground is appreciating as fast as the money in the bank (in annual percentage rate), or as fast as the annual discount rate used in the operation one runs. Many operations (companies, etc.) use an annual discount rate of 10% or even larger.

Consider an annual discount rate (or interest rate) of 10% and a 25 year-long generation. In 25 years, a compounded 10% annual interest rate will multiply an original investment by a factor of 10 (yes, ten), and an annual discount rate of 10% will decrease a later money receipt by a factor of 10. What does this mean? It means that an oil well owner using a discount rate of 10% has no incentive to leave any oil in the ground for our children 25 years from now unless there is a certainty of getting the children to pay 10 times as much for the oil as we are paying now, and our grandchildren 50 years from now would have to pay 100 times as much as we are paying now.
Finally, as Gowdy and Julia note, "theoretical [economic] models that address the issue of resource scarcity such as those of Hotelling and Ricardo assume 1) fully informed agents, 2) efficient allocation through time and 3) that present and future resource stocks are known". This quote is from section 1 of their paper, The Economic View of Resource Scarcity. I suggest you read that section--actually, I suggest you read the whole paper, it's quite good.

The Contentious Issues

Analysts like Michael Lynch, the IEA and others contend that Hotelling's rule does not apply to world oil production, principally because future resource stocks are not known [from Lynch].
There appear to be two primary errors in the design of these models. First, Hubbert-style forecasts take URR as a static variable when it is dynamic. This is a serious error. URR refers not to total resources, which is arguably a fixed amount, but to the proportion of the total which is recoverable....

Indeed, URR estimates do seem to expand by time, with the average estimate from the 1950s and 1960s being 1.0 trillion barrels or less, while recently, numbers are 2.5-3.0 trillion. Examining individual authors--to correct for methodological differences--also finds the same pattern. The USGS has increased their estimates of URR from 1.7 trillion in 1984 to 3 trillion last year. Even Campbell has raised his estimates by 150 billion barrels from 1991 to 1997, an amount greater than consumption during that period....
This is not the view of the peak oil community. Roughly speaking, the current view is that world conventional oil supply (including condensates and NGLs) is roughly 2.3 trillion barrels (+/- 10%) and if, say, 43% of that is recoverable, that leaves us with about 1 trillion barrels left to exploit. According to the Lynch Cornucopian model, the oil in place (OIP) resources are not known at present and URR is expanding over time due mainly to advances in technology (like EOR techniques or super deepwater drilling) and assumed price rises. Required and sufficient investment is presumed to follow inevitably given the magical efficacy of technology and price. On the other hand, peak oil modellers base their analysis on Hubbert Linearizations, the (P/Q)/Q graph using past cumulative production, the well known new discoveries trend and some roughly calculated (but still vague) depletion rate for the existing world resource base. Total P5 and P50 reserves (the Qt) in individual cases (fields, countries) is predicted and not simply asserted. The world Qt is based on the big picture. Stuart's latest number is 2.35 trillion barrels. Importantly, peak oil theory rests on a posteriori extrapolations for fields like Ghawar or Burgan based on cases where good data is available (Texas, the UK/Norway North Sea, etc.) wherein it is assumed that the well-attested past also predicts the future. This seems like a reasonable assumption.

A Scarce Resource? Current Oil Production Practice

I think that the Gowdy & Julia result--namely, that EOR technology is masking real scarcity in the future, that technology does not create higher URR in existing fields (with the exception of some cases like Weyburn) and that prices are in fact too low due to temporary supply boosts--would not be disputed by many in the Peak Oil community. In fact, current production practices seem to follow de Winter's assertion; oil companies (IOCs or state-owned) are pumping it out like there's no tomorrow to meet ever growing demand. This practice in turn has kept oil prices within "reasonable bounds". These prices are still relatively low despite 30%/year rises over the last few years. Arguably, the price should be much higher but since URR is not regarded as fixed and it is assumed that daily flows (in mbpd) can be increased almost indefinitely (eg. by CERA), there is every incentive to keep pumping the oil. Regarding Hotelling, the OIP and what is ultimately recoverable is not perceived to be appreciating in value at anything even close to the discount rate, so there is every reason to produce it. No sharp drop-off in production (like that seen in the UK/Norway North Sea) is anticipated for world production. In fact, there are lingering fears of a sharp decline in oil prices among OPEC, the IOCs and other producers. This seems absurd on the face of it but there it is.

So, the problem for the peak oil community is to convince the people and the policy makers that oil is a finite resource that should be treated as precious. Furthermore, prices should reflect this. If that were the case, we would have a better chance of a flatter depletion curve later and could, at least partially, solve the "Hirsch Gap" problem. However, at this point, a couple trillion (or more if you believe the USGS) barrels of oil is never going to perceived as a finite, scarce resource in the current timeframe in any case.

Hotelling Revised--An Immodest Proposal

Well, if world OIP and URR are not regarded as finite and scarce, what would change that view? If you trust peak oil theory, you might also believe that perceived P5 and P50 reserves, which are always going up as they did mysteriously for the OPEC countries in the 1980's, simply don't matter. What matters is peak flows, the oil that can produced on any given day (mbpd). Peak oil theory implies that there is a number, say 88/mbpd, that will come at some point and never be exceeded. There may be a great deal of supply fluctuation below that number but it is never exceeded-- this is the infamous undulating plateau. It is argued here that it is this maximum flow number that indicates scarcity because the available supply at some given point at time--the time of the peak--is finite and known, even if it is only known in retrospect. None of us know when that time will come but many of us are confident that it will be within a few number of years (at most by 2015).

Perhaps the Hotelling model of the economics of resource scarcity should reflect the peak production flows, rather than some too-far-in-the-future abstract assumption of a total finite OIP and URR number (though those of us in the peak oil community do make this assumption). Let's change the Hotelling equation a bit. [Editor's note: have you got that window with the Gowdy & Julia formulation of the Hotelling equation available?]

In that equation, the term S is the remaining (recoverable) stock and q(t) is the time path for resource extraction that maximizes the present value of the stream of net benefits from extraction. Further,

S(t) = -q(t), S(t) ≥ 0, q(t) ≥ 0, S(0) = S0 (S at time 0)

Suppose we make the following revisions.

S(t) = -f(t), S(t) ≥ 0, f(t) ≥ 0, S(0) = S0 (S at time 0)

where f(t) = q(t,a(t))
and a(t) = the available flows (in mbpd)

What is this re-formulation intended to capture? The term f(t) is the time path for resource extraction that maximizes the present value of the stream of net benefits from extraction on the basis of current or anticipated available supply at every time t (the term a(t)), not on the basis of the whole couple trillion barrels Qt. Thus, this different calculation reflects the scarcity and finiteness of the resource in question based on the available incremental flows. Like any such model, it is an idealization. It depends particularly on much more transparency in the oil business which allows reasonable predictions, rational agents and is meant to promote efficient allocation in oil production. Even if TOD readers take issue with my simple re-formulation of Hotelling--and I have no doubt you will rake me over the coals on this one--please consider it a discussion point. This is all it is meant to be.

The presumption here is that pumping all the oil you can all the time to meet rising demand is not an efficient allocation of the world's remaining oil supplies. Price is too low given the impending peak in available oil flows. We would not steal from future generations the resources they will need to live on (at least not as quickly). For example,
  • If EOR has temporary recovery rate benefits but entails steep declines later, then maybe it's better in maximizing benefits in the longer run to not bother and use stripper wells instead.
  • Rather than ramping up new fields (like Kashagan) up to their maximum daily production as quickly as possible, perhaps it maximizes benefits to get to and maintain lower production levels to maximize benefits over the longer term.
Calculations to maximize profit might consider that mature field X with reserves r is depleting at n%. Or field Y has just come online with an initial production p now which is expected to rise (on reasonable assumptions) to p+n next year. Or that new project Z has a 80% chance of coming online in timeframe t with an initial output of n. Etc.

In other words, we would live in a more rational world. Prices will be higher as they should be. Demand would have to adjust. Alternatives could be more efficiently developed based on real pricing. Perhaps, I repeat, perhaps, in a world like this, a precious scarce and finite resource like oil would not be wasted as it is now as we drive toward a Thelma And Louise cliff that most people refuse to recognize.

A very interesting post, as usual.  Some thoughts.

I don't see how substituting flow for reserves changes anything at this time.  It isn't certain at this time what flow will do in the future.  Many are predicting over 100 mbpd in 15 years.  In my opinion if the flow tops out, there will be a quick revision on the reserves to show they aren't as large as predicted.  Either measure will show we are at peak, but neither are accepted now by non peakers.

A second point is that even if people (oil insiders) know oil is a scarce resource there doesn't seem to be any incentive to stop producing if price isn't rising very quickly.  They want their money NOW.  Even if oil peaks the money can be used in some other technology at better return.  

I liken oil extraction to logging of old growth forests, trees larger than 36" in diameter.  This industry was clearly finite in the U.S. and was near peak in the 1980's.  There were not going to be anymore trees to feed into the mills in a few years at the rate they were being cut.  Theoretically, a very low harvest rate would have sustained some of the industry indefinately if replacement of trees equalled harvest.  But this is on a scale of hundreds of years and many would be out of business.  

So clearly the price of U.S. virgin forest wood should have gone through the roof, but it didn't.  Essentially all the available trees have now been cut. A few remain in protected areas but it is not enough to sustain an industry.  The mills that could only handle big trees went out of business or had to retool for small plantation trees or invested in Brazillian wood.  People did not behave rationally, with respect to price, when faced with a finite resource.  Even when the trees could be surveyed from the air there was no economic incentive to save them.  Many people who work with wood say that the old growth wood can't be replaced with tropical wood or plantation wood for many uses.  That wood was incredibly valuable for large beams and structures but much was turned into common 2x4's.

I forsee the same approach in oil.  There won't really be a substitute for high grade oil.  Most will recognize its uniqueness only after it is gone.  Like the Passenger Pigeon they won't value it while it is still here.

Forest equilibrium

This is done in Austria since empress Maria Theresa, the mother of Marie Antoinette, who was beheaded in the french revolution.
If you travel to Italy, you still can see the old border between the Austro-Hungarian Empire (sustainable forestry) and the Republic of Venice (Peak wood due to unsustainable demand levels ).

To play Devil's Advocate, though, we now make large beams by laminating layers of wood and glue into glulams and microlams that are stronger than old growth fir or pine, though not as aesthetic.  

What is the likelihood of making "better" gasoline from other sources?

Pretty much zero.

You can define the physical strength of beams in physical terms and then discover that there is no shortage of tensile strength or stiffness. With oil, one of the key values is energy density. Nothing short of nuclear fuel has the energy density of oil.

Furthermore, energy source, unlike wood beams, have the characteristic that they lose value in proportion to the amount they are modified.

You can take a pile of wood laths, add glue and energy and technique and make a beam which has more value than the starting ingredients. With energy sources, the more work you do to modify them the less net energy you wind up with. Hence, ethanol from corn, oil shales and tar sands are not going to be replacements for oil.

This is the miracle of oil. It has a very high energy density, you get it by sticking a pipe in the ground, you carry it in a bucket, you use it by lighting a match to it. Any other energy source we know of suffers by comparison in at least one of those characteristics.

This is the crunch the world is in. We've been enjoying the benefits of this fantastic source of energy for over a hundred years, treating it as if it will last forever, and now we are finally facing up to the fact that it won't.

Both zinc and aluminum have energy densities greater than petroleum products. Zinc in particular can be oxydised in a fuel cell and the byproducts can be recovered and recycled into fresh zinc and electrolyte. The byproducts of gasoline oxydation are CO2 and steam which are lost to the atmosphere thereby creating climate change.
Our evolutionary impulses to value the present more than the future carry over into the way markets value oil and gas stocks. Wall st firms care about current production and proven reserves. If a company decides not to use water injection to get all the oil out this year that they can (and in effect deferring production to a later time), that stock will underperform one that goes all out in production.

I fear that we are at the point in the curve where we are passed/passing what normally would have been the peak, but technology is making it look like the curve is still rising.
Robbing Peter to pay Paul. And Paul has alot and Peter is going to need it.

In my corner of the Oil Patch in West Central Texas, I have in fact been arguing for lower production rates and larger spacing in new discoveries.  We are in a very mature area, basically looking for missed fields.  

However, the fields can be quite profitable, including one that I found with recoverable reserves of 5 million BOE.  Of course, we are nothing but a rounding error in the global energy scene.  

I'm not in rush to get the money for two reasons:  (1)  I think oil prices are going to skyrocket and (2)  I'm not sure that we can profitably invest the money anyway, given the mature area that we are in.

I suspect that a lot of energy exporters are going to gradually reach the same conclusion.  Why should they rush to sell depleting energy assets when they can sell less energy for more money in future years?  

It has also been my contention for some time that the only true capital is measured in BTU's.  The Fed can inflate the money supply from here to the moon, but it won't create one BTU of energy.

I suspect that a lot of energy exporters are going to gradually reach the same conclusion. Why should they rush to sell depleting energy assets when they can sell less energy for more money in future years?

Some of the decisions will be driven by the "real" discount rate that they expect. And in at least some cases, the discount rate is odd indeed. The telecom industry experienced the results of that in the late 1990s. Executive compensation in the form of limited-term options created a situation for the decision-makers that had a discount rate that went to infinity when the options expired. This led to an outcome that should have been expected: the executives did whatever it took to increase share prices in the near term so they could cash in before that infinite discount kicked in.

The situation in energy is different, of course, but the possibility of so-called hyper-discounting remains. Decisions are made by individuals, and the discount rate over time for those individuals may not match the simple uniform discount rate used in the Hotelling and other models.

I see several problems with this analysis but I will just mention a couple. A minor point is that the equation you wrote as S(t) = -q(t) is actually supposed to be S-dot, not S. If you look at you see there is a dot above the S. This is dS/dt, the rate of change of S. It makes sense that S, the remaining supply, is decreasing at the rate of q, which is the flow rate.

Another problem is that you quote Lynch supposedly criticizing the Hotelling model when in fact he is criticizing Hubbard peak theory. These are not the same! In fact I read a paper recently comparing both the classical economic model of Hotelling and the peak oil model of Hubbard against American historical experience. Neither one worked well, but they are certainly not the same. Here's the paper:

A few other points: as others have pointed out, your reformulation is meaningless, changing q to f. q is simply the rate at which oil is being extracted. There are no assumptions about its functional form or what it depends on in the bare Hotelling model.

Also you quote Stuart's estimate of 2.35 trillion barrels but that comes from Hubbard linearization which should mean that it is recoverable reserves. But in that same paragraph you suggest that world recoverable reserves are only 1 trillion barrels and quote other figures that are also inconsistent.

This is a very complicated area and the bottom line is that there are no good models for predicting the future course of oil production. There is too much uncertainty politically, technologically and economically to come up with a credible opinion. That doesn't stop people from trying, of course, but I can't help noticing that everyone gets a different answer.

Of course I meant Hubbert not Hubbard. No idea where that came from.

Oops, sorry, I've got to go check the cupboard, my dog's hungry...

As regards "S dot", that's how I understood it but left the "dot" out of my reformulation.

With respect to Lynch criticizing Hubbert Linearizations --since Lynch does not effectively believe that fossil fuels are finite in extent, then he does not believe that any economics of finite resources (like that of Hotelling) applies to the problem.

Regarding the 2.35 trillion barrels and what is recoverable, that was my mistake. I don't know what I was thinking there. The main point, that there is a finite URR, remains untouched.

My reformulation was supposed to be a "straw man". If URR is finite but very large and its exhaustion is beyond the time window that our economic behaviour allows--which for the futures markets appears to be at best a few years--then my thinking was that available flows are finite ie. have an upper bound. Since that is the case, as Peak Oil theory predicts, then efficient allocation of the resource might be based on those instead of the entire estimated Qt.

Perhaps an economist like yourself or someone else might step in here, rewrite macroeconomics a bit and show us a light that leads us out of the darkness. Because it is certain in my view that in using the current economics of extraction we are stealing the scarce, precious resources from future generations. To make that less abstract, that would be our kids and their kids.

In talking about the economics of this, I knew I was setting myself up but I was hoping and still am that something positive might come from the discussion. After you've fed the dog, I'll be looking forward to any proposed solution to the mess we're in that you care to make.
I'm actually a cryptographer.
Great post, Dave. Economists take a beating on TOD, often undeservedly so. This is evidence that there is good economic thought being produced on peak oil.

"In this paper we ... support the counter-argument that technology, rather than "increasing" the lifetime of present reserves, masks increasing absolute scarcity, in accordance with Hotelling's theoretical predictions. Technology does have the effect of increasing extraction initially, but it does so at the cost of increasing the rate of depletion - and the lifetime of the resource. Temporary low prices may be then misleading indicators of the ability of the economy to adjust to resource exhaustion."

There is an important point here. Everybody basically agrees that today's flows of oil are larger than they would be without enhanced recovery, but this information is interpreted in two diametrically opposed ways. The CERA/Lynch argument is that better technology actually increases available reserves, and total recovery. Under this scenario, higher current flows (and the resulting lower prices) are accurate signals that the resource is large, or even increasing.

The Gowdy and Julia argument is that higher flows and lower prices are a short-term aberration: higher current flows mask faster depletion, and are therefore inaccurate signals. We are lulled into believing that the resource is larger than it is, we use it too quickly, and we don't plan until it is too late.

There is a central issue here. Everybody sees the same data: strong current oil flows, and somewhat moderated prices. CERA sees the data as correct indicators in an efficient market, which implies that future supply is acceptable. Gowdy and Julia argue that the exact same indicators are spurious. It's like climate change research: one set of data, two very different interpretations.

One argument to resolve this impasse: economic theory assumes that oil is traded in completely free markets with accurate information equally available to all parties. We know this isn't so. Since we violate assumptions, we have a market failure. Therefore we can't work backwards and say that price signals tell us anything accurate about the long-term situation.

Should we live in a more rational, fairly-priced oil world? Absolutely. But who has the nerve to impose short-term pain for longer-run benefits?

Excellent points.

It struck me that the Gowdy/Julia view, shared by many on this site (that technology is just masking what would normally be a peak), is yet another form of leverage in our system.

So, as an American, many of my fellow citizens live beyond their means on credit cards, our government certainly lives beyond its means by running a huge federal devicit and balance of payments deficit, and on top of that the entire global infrastructure is being supported by borrowing from future oil production (another form of leverage).

We are triply on margin.

Also, Westexas point is a good one - there will come an inflection point where the owners of valuable oil assets will choose to sit on them instead of produce them - this will be the sea change that will cause the permanent spike in prices. But as public companies with shareholders, how would that work at annual meetings: "Dear shareholder, we'd like to begin our important meeting today by discussing the ideas of a man named Marion King Hubbert..."

If I understand the Hotelling formulation correctly the price/production elasticity is regarded as static. Even if you accept the economists faith that if you knock on God's door with enough money he will put more oil in the ground for you, you have to accept that He gives pretty slow service. For many  processes the response to increased prices is fast with respect to the rate of change in demand and a static price/ production elasticity is a reasonable assumption. With oil exploration and development the response to a step change in demand has a time constant of about 10 years if you include thinking time by the oil companies and this is increasing as we are driven more extreme places to find the oil. This is long with respect to changes in demand and is also a time span over which there is appreciable depletion of resources.

I suspect a strongly time dependant elasticity function will give a very different optimised production rate.

All of this assumes that we buy into the sod posterity assumption of economics. An assumption you clearly are not accepting Dave, from the evidence of all the work you are putting into these posts. Thanks very much.

There seems to be some confusion about the discount rate here. I'm not an economist, but from what I've read I believe the rate that applies in such a situation depends critically on the level of uncertainty about the other parameters in the model. I.e. If you're very uncertain how much of the resource remains or what future benefits or costs will be, a high discount rate applies; if you have an  accurate view of the remaining resources, the rate would be lower; if you knew every parameter precisely (including time-dependence of your costs and benefits) the discount rate could even be zero (at least inflation-adjusted - i.e. less than that for bonds, which have their own uncertainties).

Transparency of international oil resources helps, but only with the resource component of the uncertainty here. Another major component of the uncertainty is the presence of alternatives. If alternatives are actually viable, then future benefits from oil extraction are going to be lower (and the price will be too) even if the resource is severely depleted. Hence the increased benefits of extracting now rather than later.

Nevertheless, the plateau in the production curve (one of the other posts here recently) suggests that despite that incentive to extract now, we've hit some limits. I guess we'll see how that continues...

I don't think that the discount rate used in this model has anything to do with the resource itself or with any other parameters in the model. It's actually driven by the performance of the rest of the economy and simpy tells you how much your money in the bank will appreciate with time. That's what creates the problem, since it's some sort of artificial index that has little connection with real life, just like money flows become detached from flows of actual goods and services.

That's just part of the problem. If we were operating within some fair economic system driven by only market forces, we should already be seeing some signals and reactions to them. Indeed if the oil prices are growing at 30% while the discount rate is 8-10% max, somebody should already start realizing that keeping more in the ground, is a better deal than pumping it out, converting into $$ and reinvesting. Unfortunately it's not just economic decisions (no matter how bad they can be) that regulate the system. All the markets are distorted by subsidies, politics, private and corporate interests, etc., so the prices as they are are not realistic, and economic models have very limited applicability.

What's nice about Gowdy's paper is that it explains some pretty obvious dynamics (if you have a limited resource and use it faster, then you get less left and it will hit you harder and sooner when you run out of it, duh?) using models and terminology that economists can finally understand. So there is hope.

Well, I'm not sure what the intention is in the Hotelling case, but generally the discount rate used to, say, evaluate an investment, contains a "risk premium" that depends on the uncertainty of expected benefits/paybacks. For example, see the wikipedia definition of "discount rate":

"Typically, the discount rate is arrived at by beginning with the appropriate interest rate for the length of time in question, then adding an additional sum to account for risk. For example, some companies add 15% to term-specific risk-free rates."

Those who "own" oil in the ground face a peculiar position.  Each has to look at all the other owners and make a guess at when is the best time to offer his resource to the market place.  

Imagine, if you will, a card game wherein each player has one card (his marketable resource.)  He is allowed to surreptitiously falsify or withhold the information on his card in order to trick the others into playing their hands first.  Some players have enormous resources; some have few.  Some are driven by immediate needs; others are not.  The strategy of individual play runs the gamut.

How best to play your own hand is the question for each.

Mathematical models assume a clarity of "play" in this game that is simply not there, i.e., that total resources are known, or that that knowledge is willing to be shared.  In fact, secrecy will be rule.  Profits are the name of the game.  

For those of us on the outside watching, we can only guess the future.  What we can know is what has happened to date.  In short, the peak will pass us by well before we know it.

Don't oil companies have bills to be paid now and can't sit on a recoverable resource for long without going belly up?
As I am not an economist, perhaps these comments may be simplistic or totally off base, but at the risk of being very wrong I will make them anyway.

The decision on whether to exract a finite resource sooner rather than later is based on many things, not the least of which is the understandable desire to maximize one's return on the capital investment required to enable you to extract that resource in the first place.

 Fundamental to such decisions is the concept of the time value of money. It is taken as almost a law of nature that a dollar in your hand 10 years from now is worth less than a dollar in your hand today, and that the difference between the two values is determined by that application of some agreed-upon discount rate. In many ways, the 'law' of the time value of money is essentially nothing more than a formalized expression of the fundamental human tendency to value the present over the future. Nothing wrong with that, as long as one knows the inplications.

Now, here's where I see a possible flaw in the way the time value of money is applied. My present is now what used to be my grandfather's future; and one day my grandson's present will be what was is now my future. If one extrapolates the time value of money to such an intergenerational relationship, it automatically calls for my grandfather to maximize his monetary benefits at my expense, and for me to maximize my monetary benefits at my grandson's expense.  This game can be summed up as: Screw the next generation.

So, a equivalent barrel of  useable oil products is more valuable to me now than to me when my grandson becomes an adult. But that barrel of oil products is just as valuable to him at that future time as it is to me in my present time.

On a personal level, people often do set aside wealth and real assets for the next generation, but on a group or public level, particularly when large amounts of capital are involved, it's back to the game of screw the next generation.  I can't put my finger on it but, I think some flaw has creeped into the way we apply the concept of the time value of money to things for which it might not be entirely appropriate.  And the longer the time frame involved, the more this possible distortion is likely.

I am not offering these comments as my dogmatic version of 'the truth' but just as something to consider when we deal with the present/future problem or resource allocation.  

What you are forgetting is this: that making more capital available to earlier generations actually enriches later ones. Much of the fantastic technological and economic progress of the last century was fueled by oil. If all those years people had spent conserving oil and living meaner, poorer lives, there would not have been the excess wealth necessary for the great improvements that we saw. We would have ended the century in much the same position that we began it. Instead, the world of 2000 is so advanced over that of 1900 that a person of that older time would consider true tales of 2000 as the ravings of a madman.
So, if spending more finite resources makes more capital available, then would it not follow that the more oil and gas we burn the more capital we will 'create'? Are you actually saying that driving your Lincoln Navigator those extra miles actually creates more capital for future generations?  Pardon my ignorance, but I just don't get it.

See, this is the exact sort of thing I was trying to get at in my previous post: How do we relate present and future wealth, capital, or whatever you want to call it, in a way that makes sense from the standpoint of the long-range survivability of the human race?

Not all capital is equal.  Some of it is used for self-destructive purposes, while some of it is used for things that will improve rather than impoverish the future. It's a matter of choice and values.  

So, it should not be assumed that just because  one creates captial that one is automatically doing something useful, for very often the exact opposite is the case. Some capital formation is little more than the transfer of wealth from person A to person B, while other capital formation is the genuine creation of something useful.

I think we need to look at some of these things less in terms of financial language and more in terms or what is actually physically taking place.

The purchase of a $10,000 domestic solar cell system and  $10,000 in legal fees from the settlement of a lawsuit theoretically increase the GDP by the same amount; but in the former case something useful has resulted whereas in the latter nothing useful at all has transpired.

You seem to be trying to formulate an unwarranted law. Utilization of oil resources in the past has enriched us today, but are we still not enriching ourselves through the same utilization of resources? I don't think we can draw any strong conclusions about the wealth of the future from this.

OK, this is all a bit (a lot, perhaps) out of my area.  I printed out the paper, and will read it later.  I did skim it, and it looked interesting - thanks for bringing this up.
A picayune point here - when using abbreviations in a technical post such as this, it would be helpful to define the abbreviation (such as EOR) the first time it is used, even if that is tedious.  Then I could forward it to people who are capable of following the concepts, but not familiar with the terminology.  As it is, I hesitate because I know they'll get lost in the acronyms!  Otherwise, an interesting post.
Good suggestion, I have made a mental note to make sure I follow that practice.
Dave, I don't know much about math, but I know what I like. Fascinating stuff, congrats. The logic is impeccable and the facts are irrefutable -but  my caveat is that the underlying moral principle is somewhat dubious. You write:

The presumption here is that pumping all the oil you can all the time to meet rising demand is not an efficient allocation of the world's remaining oil supplies. Price is too low given the impending peak in available oil flows. We would not steal from future generations the resources they will need to live on (at least not as quickly).

But how quickly is quickly? How many future generations should we not `steal' from? Ten? A hundred? A thousand? In other words, what ethical principle should we apply?  No doubt one could actually calculate the planet's entire amount of non-renewable fuel resources and, by dividing that by the number of  Btus consumed by a `standard middle-class human being' in the course of a `standard middle-class lifetime', arrive at the number of generations we should factor in for a given world population. Or should we set the standard of living lower, say at that of Indian peasants, or medieval monks, or hermits in the Arabian desert? What's the reasonable man's acceptable energy footprint?

And say the `future generations' end up being benighted Muslims, or born-again Christians awaiting the Rapture, which may be the way things are going. What do I care about the resources they would have had, if I hadn't got there first and extracted most of the honey from the pot? And why SHOULD I care? For example, why should I live a life of voluntary simplicity and, as a result, not even be in a position to procreate part of the `future generation' because no chick will have me if I ride a bicycle rather than drive the proverbial SUV? Girls don't fall into the arms of the holy fools of the peak oil community.

There is something almost `future-counterfactual' about even adumbrating the possibility  that human beings will restrict jam today so that somebody else will have jam in a thousand years' time - or even a hundred years' time. If there is anything in life that is certain, it is that human animals will continue to behave as human animals and go for broke, hoping that the die-off won't occur in THEIR lifetime. Try proposing to your wife and kids that your family should move into a log cabin so that the end of the human race won't be as nigh as it will be if you stay put in your MacMansion! I know I am almost caricaturizing your position - but we most follow the argument wherever it goes, and that is what I am doing.

But all is forgiven - your contribution is certainly great food for thought!

Re: "But how quickly is quickly?"

Your comment actually speaks to the so-called "Hirsch gap". Here is a good summary of the Hirsch report.

The basic insight is in this graph taken from that article.

If we pump less and consume less oil starting now based on what I would consider its real price, we have more time to take mitigation measures of all kinds (efficiency, alternative fuels, etc.) that bequeath to future generations a sound basis on which to operate. When you say
If there is anything in life that is certain, it is that human animals will continue to behave as human animals and go for broke, hoping that the die-off won't occur in THEIR lifetime.
you may be right but, as in so many other things, all we can do is suggest the right thing and hope for the best. As Rick so eloquently said here
Should we live in a more rational, fairly-priced oil world? Absolutely. But who has the nerve to impose short-term pain for longer-run benefits?
I'm not suggesting that anyone leave their "McMansion" and move into a log cabin--although current economic practice suggests that will almost certainly happen if we maintain business as usual. But one would hope that everybody cares about the quality of the lives their kids and grandkids will lead, right?
Thanks for the Hirsh reference -- but mitigation has already started and if the conventional peak doesn't hit us until 2020 the Hirsh 20-year scenario will have been a reality in retrospect. Given that oil futures are now approx. US$54 for December 2012 (approx. $47 in real terms assuming 2% inflation), any investment costing less than $47 per barrel extracted will be worth the candle: profit guaranteed since futures are present prices from the business angle. Including the conversion of oil shale into shale oil (and a windfall for Estonia with a second peak in production), coal-to-liquids, gas-to-liquids etc .etc. Mitigation is another word for going for broke the second time round. So I reckon our grandchildren won't be at risk -- though THEIRS almost certainly will be.

Unfortunately, all business is 'business as usual'.

Recommended reading on prices and their impact on non-conventional extraction at: "Oil price Assumptions for Energy Outlook" (Samuelson and Tsylor) -- sorry, for some reason I can't cut and paste the web address of this PDF document. But googling in the title will get you there.

You shouldn't apply a deflator to the 2012 futures. If you want to buy one today, the seller will insist on being paid with 54 2005 dollars, not 54 2012 dollars (or 47 2005 dollars).
Err, that would be 2006 dollars, yeah, that's it. I knew what year it is really...
I have to strongly disagree with those who state that "there will come an inflection point where the owners of valuable oil assets will choose to sit on them instead of produce them."  It's an interesting economic theory that has no basis, IMO, in reality.  If owners chose to sat on their resources, the price of oil would skyrocket so high that economies would be crushed, demand would drastically drop and the price of oil would plummet.  Hmm, so much for that valuable asset.  But this scenario would never happen because private oil companies wouldn't dare do this... oil demand is so vital to society that the public outcry would force governments to seize fields so that they could be produced.  Look at the outcry that high prices have generated.  Things may get trickier with government owned assets, but what county is going to squeeze the tap shut with the US casting its shadow over the world?  Maybe Russia.  Certainly not any member of OPEC, barring maybe Mr. Chavez.  Just my opinionated $0.02.
I am talking about marginal decreases in production and an accompanying structural adjustment of our economies based on a realistic price for a scarce and finite resource. I am not suggesting that everyone stop pumping out oil. If I were going to make that argument, I would base it on a realistic assessment of climate change. But that's another story.
This inflection point has happened many times already in Saudia Arabia, as they have in effect been the swing producer. What do you think Russia did this past weekend, of a sort?
We are conditioned to believe the future will be pretty much the same as today - I agree with you thatpre-peak and during the plateau sharp price increases will cause demand destruction, but looking 5-10 years past peak, no amount of supply will come close to satisying demand. If oil prices are doubling every year then, would it make sense if you owned 100 units at a 30% depletion rate to maybe tighten the spigot and sell 10% a year rather than 30% of your 100? Would you do that? Its only my opinion of course, but the rules completely change in that environment - throw neo-classical economics out the window. Energy will be the primary store of value (and through it water, food etc)

In the end though we are biological animals - there is no precedent for any successful species that defers consumption what they could consume at the moment. Most animals are limited by the size of their bellies (squirrels, lions etc), we are limited only by our imagination, and dollars, and their interaction with the leverage, which is called oil. Though it hasnt been mentioned in a while, 1 barrel of oil has the amount of BTUS generated by 25,000 man hours of labor, which is 12 years... (10.5 using american workweeks)

No different than speculators who buy property and only slowly develop it, holding chunks aside, knowing that as time goes by the value will increase dramatically as buildable land grows scarcer. The has been done for years in CA, and only requires knowing that ultimate supply is limited while demand will continue to grow.

This is most easily done by private, independent firms. Large shareholder owned companies have to try to maximize short-term profit and look good in sales growth to get their managers the bonuses they seem to need, and please stockholders.

One has to remember that private oil companies have a very little share of total production and reserves, this data is probably a bit old, but shouldn't have changed a lot:

Ranking Company Reserves (million barrels)
1 Saudi Arabian Oil Co 259.300
2 Iraq National Oil Co 112.500
3 National Iranian Oil Co 99.060
4 Kuwait Petroleum Corp. 96.500
5 Abu Dhabi National Oil Co 92.000
6 Petroleos de Venezuela SA 77.783
7 National Oil Corp. (Libia) 29.500
8 Petroleos Mexicanos 25.425
9 Nigerian National Petroleum 24.000
10 Qatar General Petroleum Corp. 15.207

Ranking Company Production (liquids, in million barrels)
1 Saudi Arabian Oil Co 3.055,70
2 National Iranian Oil Co 1.384,80
3 Petroleos Mexicanos 1.299,00
4 Petroleos de Venezuela S.A. 1.192,50
5 ExxonMobil Corp. 889,00
6 Royal Dutch/Shell 810,00
7 Nigerian National Petr. 766,50
8 PetroChina Co. Ltd 763,50
9 Kuwait Petroleum Corp. 745,30
10 Iraq National Oil Co 715,40

I'm not sure this model hits the nail on the head in explaining resistance to price rises. A major factor has to be negative feedback on incomes without the need to use time or discount rates. Oil producers are co-dependent on oil users having enough purchasing power. Therefore price rises should be moderate and not destabilising. It follows that there is a price, say $US300 a barrel, at which the system collapses and no-one can continue to generate sufficient income to afford that high price. The objective becomes staying in business, not maximising returns.
Yes, and that I suppose is what I was alluding to in my earlier post re the problem of valuating present/future resources.   I think we can (or should) believe that the fundamental objective of the human race is to 'stay in business'.  As such, one  generation gorging out on all the available resources works to put the human race out of business.  I would like to think that we have more foresight than that and will adopt at least some measure of planning ahead.

I used to have a huge pet iguana.  We fed him a salad once a day.  Sometimes when we'd go away for the weekend, we'd leave him two bowls, one for now and one for later on. However, our beloved iguana was so present-oriented that after he finished Bowl No. 1, he'd immediate start in on Bowl No. 2 till there was nothing left. Then he'd sit under his heat lamp and have a long pleasant digestion. So much for forward thinking.  I'd like to think that we have a bit more sense than an iguana, but sometimes I am not so sure.

My wife likes to say,"A person is smart but people are stupid." The larger the group of people looking at the future the dumber the plans they come up with. A case in point; The US Congress.
Nice work! I was not aware of Hotelling's work. I was wondering why the price increase is found to be exponential. Is it because the supply decreases as an exponential as it seems to be the case in the lower left quadrant.
Much is said here about a) how wise it would be to hold out for a higher price, and b) how we should save some for our children. We can all give thanks to OPEC for doing exactly that, to the best of their (fallible) ability, balancing maximum price/minimum supply against the need to keep prices low enough to not destroy western economies and their continued consumption. Note that their withholding production has come during times when they (saudi) were running large fiscal deficits.  OPEC contrasts very favorably, from a peak oiler's viewpoint, with western oil companies that have always pumped every barrel possible, absolutely including those times when the price was less than overall production costs, behaving quite like US cod fishermen who depleted the Georges banks, once thought to be 'inexhaustible'. (These fishermen, currently on the dole, did not even save some for their own generation, much less the next - or, maybe their dad's didn't save enough for them.)

OPEC and peak oiler views are nevertheless not necessarily correct, and it would be much better if, in fact, the human race develops something to effectively take oil's place, before we either run out or boil the planet. Higher prices will no doubt reduce consumption, possibly drastically, but most PO's think we will really run out in, say, 100-1000 years.  What then? Breeders might go for a long time, but for the HR to survive indefinitely in the manner to which we have become accustomed, we will have to go to solar, either 'indirect' or with fusion, fabricating liquid fuels (if we still want them) from basic constituents. Who can say now whether a bucket of natural oil would then be worth a king's ransom, or not very much?

THere's simply too little information to predict the HR demise on account of running out of oil, just as there was insufficient info to predict WWIII would do it, even though that verdict is still out.