The Future of Oil Prices 3: Nonrenewable Resource Pricing

Oil prices have been climbing rapidly in the past few years, and especially in recent months. Some point to speculation, others suggest that the fundamentals justify high prices. In 2006 I wrote a couple of posts here about how you could predict the future of oil prices. Unfortunately, reality did not align with theory, and my predictions were not accurate. Here is another approach to the problem which aims to look at the fundamentals and determine what is the rational market price for oil.

The starting point is Harold Hotelling’s 1931 model of the economics of nonrenewable resources. Hotelling showed that, based on very simple and general assumptions, nonrenewable resources like oil should follow a predictable price path. The price should increase gradually, year after year, with the rate of increase equal to prevailing interest rates. This price increase is likely to suppress demand and lead to gradually reduced production levels. So we see prices go up, demand and production go down, until the nonrenewable resource is exhausted and completely consumed.

Here’s a figure stolen from a website which stole it from somewhere else:

This is a rather complex diagram. The upper right quadrant depicts price on the vertical axis rising exponentially as time moves to the right. Then the upper left quadrant is a simple demand model which shows demand level on the horizontal axis decreasing as price rices on the vertical axis. We then move to the lower left quadrant which shows production level (which equals demand level, in equilibrium) on the horizontal axis decreasing as time on the vertical axis moves downward.

A couple of interesting points are, first, that prices become much higher than the costs to produce the resource, the opposite situation from competitive markets in other kinds of commodities. Yet resource owners restrain production without the need for a cartel or any coordination. They are motivated solely by their own individual profits. And further, this production schedule actually maximizes the net value to society of the renewable resource. An enlightened dictator would impose a decreasing production schedule in exactly the same form that Hotelling showed happens automatically in a free market.

However one of the most surprising aspects is that despite its generality and seeming soundness, the Hotelling model has universally failed to accurately predict the behavior of markets in nonrenewable resources. Oil has been produced for over a century, and rather than a gradual and predictable increase in price, its price was generally stable for most of the 20th century, with various ups and downs. Then in the past few years and months its price has shot up far faster than the rate that Hotelling would predict, the rate of interest:

One possible explanation of this long-term failure is that in the past, oil was treated more like a renewable resource; new oil was reliably being discovered, as quickly as it was being consumed. However this has stopped being true for twenty years now, and we are finally seeing a realization that oil is very much a finite resource. It is possible that we may be witnessing a transition to an era where the Hotelling model dominates. In that case, the current rapid price increase can be interpreted as a transition from a renewable-resource price to a nonrenewable-resource price.

Where, then, will this price transition end up? Although the Hotelling prediction of the yearly change in prices is robust and simple, deriving an initial price level from his model is quite difficult and uncertain. The basic idea is to postulate a so-called “backstop” price for the commodity, the price at which there is an adequate substitute, where demand for the commodity will drop to zero. Then the initial price level should be such that as the price rises gradually, at the rate of interest, and demand drops due to the price increase, we reach the backstop price at exactly the same time that we run out of the commodity; in the case of oil, as the last economically recoverable drop of oil is coming out of the ground.

I’ll illustrate with a sample scenario, but it will be clear that many alternative assumptions are possible. Let us assume that oil uses can be replaced, in the fullness of time, by solar-powered electricity. A barrel of oil has 5,800,000 BTUs of energy, which is equivalent to 1700 kiloWatt-hours. Presently, solar electricity is available for about $0.30 per kWh in favorable locations such as the American southwestern deserts. However the effective price will be higher in less sunny locales; for example, in Germany the price is $0.50 per kWh. Depending on distribution scenarios, solar electricity may have to be produced relatively locally, meaning less favorable conditions, so we will adopt the higher figure as a starting point. Today, then, we could replace oil for 1700 * $0.50 = $850 per barrel using solar energy, if we had the infrastructure in place to run transportation on batteries and replace other uses of oil.

Solar electricity is likely to fall in price due to technological improvements, although demand increases may moderate that fall. Installed solar prices have not decreased for the past 5 years, for example, as demand increases have more than kept up with production improvements due to economies of scale and improved technology.

As far as the Hotelling price increase, the rate of interest in constant dollar terms is generally quoted as being in the 3-5% range. Let us use a value of 4%, which implies a doubling time of 18 years. We will see oil rice gradually in price, as solar electricity gradually falls, and when they meet, we will stop using oil. All that time we are converting our infrastructure to rely on electricity rather than oil. At the point where the prices meet, we should have used up our oil.

So how much oil is there? This is also an area of controversy, largely due to assumptions about how much more oil will become economically recoverable because of higher prices, and also the possibility of future oil discoveries in relatively unexplored areas such as the arctic sea floor. My personal opinion is that the more pessimistic estimates are more likely to come true, for two reasons. In the first place, the rate of new discoveries over the past 10-15 years seems lower than would have been predicted. Second, many of the estimates of economically recoverable oil make an obvious mistake, in that they do not take into consideration the increase in expenses due to higher energy costs, when estimating what price would allow producing certain resources. Therefore, for this rough estimate I will adopt a more pessimistic scenario, corresponding to the USGS 95%-likelihood low bound.

This graph from the US EIA illustrates a number of possibilities:

The blue lines represent the most pessimistic scenario, an ultimate recovery of 2.2 trilliion barrels. The green lines reflect various demand scenarios. These are uninformed by the Hotelling model, so I would use the horizontal green line to approximate demand which is suppressed by constantly rising prices, until we are almost out of oil, at which time a blue line falls from this horizontal green line to zero. This zero point is only approached asymptotically in this graph, but a ballpark figure for substantially reduced production levels would be the decade of the 2060s, a bit more than 50 years in the future.

Going back to our oil price doubling time of 18 years, this is about 3 doublings, implying a price change from today to the endpoint by a factor of 8. With today’s backstop solar price of $850 to replace the energy in a barrel of oil, this corresponds to $106 per barrel as an estimate of what today’s oil price should be. To the extent that solar electricity falls in price over the next 50 years, the backstop price will be lower. If it falls by a factor of 2, the Hotelling oil price today would be only about $50.

These considerations suggest that the recent oil price of about $130/barrel is unsustainable in the medium term (over the next few years). If we maintained this price, oil demand would be over-suppressed, too much effort would be devoted to conversion to solar and other alternatives, and we would find ourselves in a position where we could stop using oil long before we needed to. This would hurt oil producers in the long run, as they would find demand dropping for their product while they still had plenty of easily recoverable oil available. Worse, our resources in this critical time would be mis-allocated and the net wealth available to future generations would be reduced.

However it’s also worth noting that we were at an arguably reasonable price level only a few months ago. The recent price spike may turn out to be a short-term bubble and we might well see a return to prices in the $50-100 range within the next year or two. That would put us at a sustainable price level that would allow for steady price increases, doubling every 18 years or so, as alternatives ramp up and are ready to fully replace oil in about 50 years, just as we run out. It will be interesting to see how closely actual events approximate this theoretical scenario.

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  • Thomas Ryan

    Seeing this posted here is getting me excited. Oil use has been a concern within my family for a very very long time. My father, especially, put much of his efforts into funding research projects. Yeah… much less time actually developing.

    It does seem that the problem is that people treat oil as a renewable substance. The prices seem dependent on people not understanding the implications of using a non-renewable substance as their main source of energy. Oil should be much more expensive.

    I often question my efforts in the field of alternative energy research because I rarely meet people concerned about the problem. So, I wonder sometimes if I’m being rational with the amount of energy I put into this problem. I’m certainly inclined towards it due to my family’s history.

    Hopefully I can finally get my my concerns with rationality resolved: is it rational to prepare for the worst?

  • Ben

    That model assumes the only use for oil is burning it for energy. There are many other uses for it, especially as a feedstock for many other industrial processes, some of which have no alternatives within that ‘factor of 8’ price increase (*citation required). That would make the optimal pricing of oil dependant on it’s intended use, with one price for transport, one for energy, one for feedstock. I suspect it’s more optimal to rapidly approach the backstop price over a period of perhaps only a couple of years, so the net benfit to society of the “cheaper industrial feedstock” can provide us all with more long lasting products (eg, producing cheap plastic buckets for the world which get reused thousands of times should have a greater net benefit to society for that oil than burning it up wholesale in a “price it to burn every last drop” scenario.

    There is also the environmental costs not calculated into this. To use “price” as the optimisation variable, and then exclude things from varying price basically guarantees you’re going to get the wrong decisions made. This is the primary reason artificial controls such as “carbon trading” schemes make sense, as it starts wrapping up the real long term costs of the system as a whole into todays price.

    Price needs to be simple. It needs to include everything, or it can no longer be used as a variable to optimise for to get optimal outcomes. Look at mobile phone contracts, with cap plans. Effectivly re-defining the value of a dollar. Creating your own currency, or undervaluing or overvaluing a dollar should be illegal in a competitive market, as it effectivly kills our ability to compare products/services fairly on price. I shouldn’t have to spend the time and effort to re-normalise the dollar before I can compare products. As soon as you remove this easy ability to compare, such that a large portion of the market gives up comparing because it’s too much effort, you wind up killing competition, and so should be outlawed under some anticompetitive behaviour bill. (yes, I’m implying not including the future costs of carbon dioxide emmissions in the price of oil has a similar impact)

  • There are three important points left out of your analysis of the Hotelling model.

    1. The model assumes secure property rights. If I am the Shah of Iran c. 1975, effective owner of a lot of underground oil, reducing production today in order to be able to produce more fifty years from now doesn’t look like a very attractive gamble, since I, or my heir, may well have lost control long before that. The same argument applies to a U.S. oil company which suspects that it will be hit with a windfall profits tax if the price of oil gets too high.

    2. The model predicts price net of extraction cost, not price. If we are imagining, as we should, a world where new reserves can be discovered at a cost, the predicted price is net of extraction plus search costs.

    3. The model assumes that the decision makers are fully informed. One possible explanation of past price movements is that people kept revising their estimates of future supply and demand due to discoveries, technological change, and the like.

    Interested readers will find a fairly detailed discussion of some of these issues in the chapter on time of my webbed Price Theory, at:

  • Shalmanese

    Solar power is not a substitute for oil. One is an energy, the other is a fuel. Currently, we have plenty of energy, coal alone is forecast to last 300 years and nuclear can substitute after that. But we can’t yet put coal or uranium into our cars and planes. What we don’t currently have is a good fuel substitute but Oil Shale, Biodiesel, Electric & Hydrogen seem to be the top contenders.

  • Only in economics can someone begin an article by saying: “Unfortunately… my predictions were not accurate. Here is another approach to the problem.”

    What on earth could persuade anyone to believe that your next round of predictions will be any more accurate than the previous one; or, more pertinently, than a coin toss.

  • Saifedean – I suggest starting at the beginning of Overcoming Bias and working upwards. Your question would be answered.

    But here are some links to help:

    As for admitting to errors:

    “The Importance of Saying Oops”

    “Tsuyoku Naritai”

    I’m sure there are others.

    And for the coin toss:

    Some evidence has been presented in this post. Weigh it appropriately and update. It isn’t particularly rigorous, but it provides some direction for future individual inquiry. While the evidence may not be overwhelming, it’s better than a coin toss. πŸ™‚

  • Douglas Knight

    But we can’t yet put coal or uranium into our cars and planes

    The Nazis used coal liquefaction 70 years ago. South Africa does it today, also having started under an oil embargo. So a backstop price of of $850 is way, way off.

  • Mike K

    It’s interesting but I think you’re inflating the equivalent cost of solar power to favour oil use. Firstly I’d content that if you were going to build on a large scale you would not build in Germany, but in a desert, where there is a lot of sunshine – I would contend that a more accurate price for “barrel of electricity” is around $500.

    Also it’s not a like-for-like comparison, because burning petrol in a car only has a mechanical efficiency of 20-25%, whereas electric motors are close to 90% efficient or higher. On that measure you’d get parity with solar power at around $100-$150 per barrel (that’s before refining costs etc). Even power stations are only around 33% efficient (though that is improving) which would equate to $166 per barrel if you just burned the oil for electricity.

    Even burning oil for domestic heating isn’t a good use of oil either: A good heat pump can provide 3-5 kWh heating power for each kWh put in (depending on outside temperature), so again the break-even point for domestic heating is $166 at most. There are applications where oils cannot so easily be replaced

    What I see a lot is the blinkered “let’s wait until it all runs out before we figure out what to do” attitude, and of course the “oh, no these prices are killing us, someone has got to do something about it” from free-market non-interventionists, some very head-in-the-sand thinking.

    The only certain thing is that oil will run out one day, and if we have a duty to the next generation, it is to provide them with the infrastructure to live an oil-free lifestyle, and do as much research as possible into this while we still can. One sure way of destroying the future generation’s wealth is to put a lot of CO2 into the atmosphere and watch extremes of hot and cold weather make the planet less fertile and usable. I don’t think they’d be thanking us for that.

  • The difficult thing is that you also have to model the expected rate of return on every other asset, and not ignore price changes due to what an asset isn’t as opposed to only what it is. This sort of Hegelian trick is very familiar to currency traders, and the energy as currency meme is seductive.

  • PaulHunt

    @ Thomas Ryan:

    I think that the majority of people know that oil isn’t a renewable resource and that we do need to find a renewable one to replace it with but I think the mind set is “I’m on a budget and until its affordable I can’t do anything about it.” And I somewhat agree that oil should be higher for a few reasons but it is really hurting at the same time. Its kind of like a blessing, because its forcing us to find alternatives, in a really ugly disguise. And here is some more bad news this article I was reading called $225 Per Barrel – The New Gov’t Standard for Oil talks about what the government is expecting to pay for oil eventually and how they’re planning around it. Its very interesting and reminds me to plan carefully as I watch the gas pump price rise.

  • I am surprised you didn’t talk about the Hubbert’s peak model in this context. The exploration-exploitation phenomenology of oil has a couple of other key features that need to be modeled. One is that oil isn’t finite in the sense of a big pool sitting somewhere — it is distributed in increasingly harder-to-pump locations, which begin to be economically exploitable as demand begins to sufficiently overtake supply. This moderates the shock dynamics somewhat. Also, substitution effects mess really badly with basic models, and the rudimentary discussion of substitution in your model is not enough. I like the Saudi Oil minister’s quote, which goes “the stone age did not end because te world ran out of stone. The oil age will not end because the world runs out of oil.”

    i.e. the abstract recognition of resource finiteness plays into how energy markets behave, but the proximal drivers that ultimately change it will be other things.

  • will chris

    Hal, thanks for the analysis, it was thought provoking for me.

    I’m skeptical that future solar power cost will go down or even remain constant. You note that many estimates of future oil production oddly “do not take into consideration the increase in expenses due to higher energy costs, when estimating what price would allow producing certain resources.” In a similar manner, much of the cost of creating solar power is tied up in the up-front energy needed to make the solar panels and other equipment. The cheapest current source of this energy is oil, or at least carbon-based power. As oil rises through, say, $400/barrel this will likely drive the overall cost of solar up.

  • Why is Hotelling accepted as a model for price movements? Has it ever correctly predicted the price movements for any commodity? Is oil the first resource that the human race has ever been on the verge of using up? I guess we have to exclude carrier pigeons and sabertooth tigers because they were unpriced.

    I’ll admit that it’s a nice clean model, but without some corroboration, it seems like an extreme move to base much planning on an argument that this forecast is reliable. It’s possible, but so are lots of other outcomes.

  • LeBleu

    I predict this prediction will not be accurate. πŸ™‚

    Your backstop price is unrealistically high, and as other posters have commented, doesn’t account for non-energy uses of oil. Look at other sources such as which shows a way to produce gasoline by re-capturing atmospheric CO2 using nuclear energy to power the endothermic reaction, which is supposed to be economical using current technology when gas hits $4.60/gallon at the pump.

    Also, you’re assuming rational economics. Humans, as we well know, are not rational. Confirmation bias encouraged treating oil as a renewable resource. Anchoring bias encourages purchasers and speculators to see a price near the current one as reasonable.

    Oil prices are (currently) denominated in dollars, however the dollar itself has a value determined by supply and demand.

  • Silas

    nuclear can substitute after that. But we can’t yet put coal or uranium into our cars and planes.

    Actually, nuclear-powered octane may very well become viable soon. It’s the method of using nuclear power to store atmospheric CO2 in gasoline (basically, the chemical reverse of combustion), effectively making all cars carbon neutral, and thus killing 2+ birds with one stone.

    From the link:

    “This plan has a minor hurdle, too; the electricity for driving the chemical processes, according to a white paper describing the overarching concept, would come from nuclear power. The proposal says it’d be worth it to have a payoff of steady, secure streams of methanol and gasoline with no carbon added to the atmosphere (and a price for gasoline at the pump of perhaps $4.60 a gallon β€” comparable to petroleum-based fuels as oil becomes harder to find).”

    So, it doesn’t require oil to get that much more expensive before this method is profitable. By my rough gas->oil reckoning, it only needs oil to hit $160/barrel. In much of the world, if you exempted such gasoline (as would be reasonable) from GW-related regulations, it’s probably already profitable.

    I wouldn’t worry.

  • Silas

    Neat coincidence there, LeBleu πŸ˜‰

  • nick

    Hal describes the surprising results of the Hotelling model of nonrenewable commodities: prices become much higher than the costs to produce the resource, the opposite situation from competitive markets in other kinds of commodities. Yet resource owners restrain production without the need for a cartel or any coordination.

    This is because, under the Hotelling assumptions that the commodity is not renewable and cannot be substituted for (and of secure property rights, as David points out) the commodity producer can choose between producing today or keeping it in the ground to produce tommorrow, with no fear of permanently losing sales to some competitor. Any sales lost to a competitor today draws down the competitor’s fixed stores and can be recouped at any time in the future.

    What the Hotelling model does not properly account for is that these very characteristics also make these goods great substitutes or hedges for money. Hotelling looks at a “fixed” or “prevailing” interest rate, but we should really be looking at expectations of future interest rates, taking into account expectations of future inflation. And we should account for changes in these expectations. If we do so, I think we will find the model to have high explanatory power (albeit not high predictive power, since we can’t outguess the market about future monetary conditions — but the search for predictive as opposed to explanatory power in economic models is largely futile anyway).

    Imagine a world with two currencies C1 and C2 and two nonrenewable minerals, M1 and M2. C1 money supply outstrips money demand by 10%/year, and for C2 by 5%/year (in other words, C1 “inflates” or “falls” by 10%/year and C2 by 5%/year). If the price has not properly increased to account for future expected inflation, people will prefer holding M1 and M2 to holding either of the currencies. They will play “hot potato” with the currencies: as soon as they obtain a currency in trade, they will try to purchase M1 or M2 with it. All holders of M1 or M2 will demand a stiff premium to exchange their minerals for falling currencies. What premium will they charge? In theory, the inflation premium is infinite: it is a “net present value” calculation of the depreciation of the currency into the infinite future. In practice, (as with the St. Petersburg paradox, and because no commodity is perfectly immune to substitution over the long term), the premium will just be very high: not only far higher than the cost of production, but also far higher than the Hotelling model under the assumption of no inflation. Inflation expectations will dominate the prices of G1 and G2 in C1 and C2.

    Now imagine that inflation expectations change. If expectations of inflation in C2 to infinity go down from 5% to zero, we have in theory a change of net present value of revenue streams from G1 and G2 in currency C2 from infinity to zero, and in practice just a very high drop. It takes only a small change in inflation expectations to send the prices of G1 and G2 in C1 and C2 soaring or plummeting.

    Since the standard Hotelling model predicts no abrubt price changes, and the monetary model does, the monetary model is a better choice for explaining the actual dramatic price movements in relatively nonrenewable commodities such as oil that we have observed.

    Under imperfect information, since we have far better information about geology and technology than we do about future monetary conditions, and since uncertainty in monetary conditionsn produces far greater price changes than uncertainty about geology and technology, we can again conclude that abrupt changes such as we’ve seen are due almost entirely to changes in monetary expectations rather than in “fundamentals”.

    Furthermore, when we see dramatic changes across a wide variety of commodities, being led by the less renewable commodities like oil, Occam’s Razor (or equivalently, basic probability) tells us that there are not 100 different explanations for why 100 different commodities have all gone up dramatically. Rather, we need only two quite related explanations: changes in inflation and inflation expectations for the currencies they are priced in.

    Two commodities that come close to being Hotelling nonrenwable commodities are gold and oil. For gold, the above-ground stockpiles are far greater than the annual production. For oil, the below-ground stockpiles are far greater than the annual production. The price of gold has always historically been dominated by its role as money, monetary substitute, or hedge, or equivalent, and the price of oil is also coming to be so in an age of floating currencies and in the current decade of rising inflation expectations.

  • zeropointzero

    nice to read the arrogant comments of a certain mr. perry metzger to a certain mr. david matthews from 2 years ago. i’d like to read again 2 years after, what he thinks of them.

  • mitchell porter

    I don’t know what Perry thinks about current oil prices, but here he is a month ago arguing for the reality of anthropogenic global warming. Maybe he’s a bit more complex than you think.