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.