Path:  physics insights > misc > A Trivial Oil Price Model

What Went Wrong? Farther down on this page, we present a model for the price of oil. That model was constructed about a third of the way through 2008. The price we projected for the end of 2008 was about $148/bbl. The actual price, as of the end of 2008, was about $50/bbl. We were off by almost a factor of three. What went wrong? In a sense the answer is obvious: The housing market in the United States crashed, and the repercussions rolled like thunder through the banking industry, and the world economy tanked as a result. But that begs the question -- in terms of the model presented here, what changed which caused the price to be so far off? One might at first guess that the magnitude of the elasticity of the price of oil increased radically. This seems plausible; during a recession people become more price-sensitive. So, elasticity increases, and that (somehow!) results in a price drop. It's plausible but it's wrong. If elasticity increases, people become more sensitive to price changes. But the price of oil plummeted. If elasticity had increased, and nothing else changed, surely that would have resulted in increased consumption -- but in fact consumption plummeted, and OPEC has cut production in an effort to avoid even steeper price declines. In fact the answer seems to be far simpler: Demand dropped, pure and simple. The year-on-year demand growth, instead of being small and positive, was large and negative. The model turns out to be extremely sensitive to a change in baseline demand: If we plug in a value of -2.5% for baseline growth in demand, in place of the default value of 1.79%, with no other change, the model's projected price drops to $50/bbl. So what does this all mean, anyway? In short, it means that the price of oil is very sensitive to world economic conditions. But more than that, it means that, if we dig into the new production and consumption figures (something I haven't done as yet), we'll very probably find that the actual rate of oil consumption hasn't changed all that much. A small drop in consumption can have a large impact on the price. But if consumption hasn't dropped all that much, then the longer term effects of peak oil -- and the consequences on the climate which stem from burning fossil fuels -- are not likely to have changed much, either. One other thing deserves mention. Both baseline demand and long-term elasticity can change, and over the past couple years we've been seeing signs that they were changing even before the crash. While the worldwide recession has chopped the price of oil, the fact that, for a while, the price went over $100/bbl and the price of gasoline at the pump in the United States shot to several dollars a gallon had a major effect on behavior. Prices have come down since, but past experience indicates that the changes in behavior are likely to persist. We're seeing electric cars coming to market, we're seeing more interest in alternative energy supplies -- in general people are switching to a less petroleum oriented operating mode. That, also, is not accounted for in the trivial model on this page, and it may have a major long term moderating effect on oil prices. For the time being, I'm not making any changes to the model, nor to the default assumptions. I'm leaving them as they are; it will be interesting to see if prices eventually rebound to anything like what the model predicted for the end of last year. And now, here's the model.

Whither goest the price of oil?

In today's world that's a fascinating question.  There's a huge amount of misinformation, disinformation, and just plain confusion on the subject, which prompted me to try to come up with an answer on my own.

The problem the world is facing is that demand for oil is continuing to rise, while supply is limited by the available extractable oil.  According to what is now conventional wisdom, the "peak" in oil production should be occurring ... now.  As you can find on some websites, the scenario which one then imagines is that demand, continuing to increase, "tears away" from supply; they no longer match, and prices head for the moon.

But assuming we really are at "peak oil", what actually should happen to prices?  What happens when the "natural" demand for an item is above the maximum which can be supplied?  The answer depends in the price elasticity of demand.  Demand (for most goods) depends on the price; there really is no single "natural" level of demand.  The elasticity is the percentage by which demand changes, for a given percentage increase in the price.  Elasticity is normally negative -- if the price rises, demand drops.

Conversely, if we know the elasticity of demand, and we postulate a change in the amount of a good for sale, we can determine what price people will be willing to pay for exactly that amount of the good.  That's the price at which the market "clears" and there's no unsatisfied demand, and no unsold supply.  The trivial model on this page uses that principle to determine the price of oil a year from now:

• We have an idea what the growth in demand for oil "naturally" would be, if the price were flat.
• We can hypothesize that, because we're at "peak oil", the supply will not rise to meet the demand, but will remain static.
• We have an idea what the price elasticity of demand for crude oil is.
• And from that we can determine what the price should be a year from now.

We now present the model (which is, indeed, trivial; the computation it does consists of one line of PHP code).  Later on the page we'll present the formulas used, discuss the default values in the form (which are my best guesses), and discuss vagaries which may affect the actual price of oil.

Starting price:  Price was around $95 in January 2008 Price elasticity of demand:  See discussion of elasticity, below Demand growth, year on year (percent):  Average growth for 1995-2005 was 1.79% Supply growth, year on year (percent):  At the peak, supply growth would be 0 Projected price after 1 year: 148.03

The Formulas Used in the Model

We use Q to refer to total demand for oil (in barrels per day) and P to refer to the price per barrel.

The definition of the elasticity is the percent change in demand, divided by the percent change in price:

(1)    

The percent change is the total change divided by the value, times 100.  Since we're dividing one percentage by another, the "times 100" factors cancel out.  In fact, unless E is constant, this definition of elasticity is only really correct in the limit for very small changes in the price and demand.  For an infinitesimal change we can rewrite the formula in terms of derivatives, as:

(2)   

Rearranging, we can separate P and Q:

(3)    

Now, by making the entirely unjustified (but none the less standard) assumption that E is independent of the price, we can easily integrate both sides:

(4)    

This formula describes a situation in which the price changes continuously from P₁ to P₂, and demand changes smoothly along with it, with the elasticity remaining constant every step of the way.  Alternatively, it describes a situation in which supply changes from Q₁ to Q₂, and price follows along smoothly, which is the situation we're interested in.  Either way, it's not completely realistic but it's close enough to be useful.

We can now solve for P₂:

(5)    

To apply this to oil prices, we must make some assumptions.
We first assume that the rate of oil consumption is allowed to rise in line with rising demand for a year, while prices remain fixed.  Of course, this won't happen, but we pretend it does, as it simplifies things for us.  (In fact, since (5) is a pure exponential, and Q₁ and Q₂ are also exponential in the time which passes (they compound), we get the same final result this way as we would by having prices rise smoothly throughout the year.)
At the end of the year, the rate of oil consumption is reduced (or increased) to match the actual available supply of oil.  We assume the elasticity remains constant while the supply is reduced; in consequence, the price changes when that happens, and the new price will be given by formula (5).  We also assume the demand at the beginning of the year is for 100 units (this value is arbitrary).

Thus, the values we plug into formula (5) to find the new price are:

(6)    

The Starting Price

The default starting price is currently set to the price of a 30 day oil future contract in January 2008, which was around $95.  That leads to a final price of almost $150/bbl, which is what we might expect to see in January 2009.
If we set the starting price to what we were seeing early in April, 2008, which was about $110/bbl, we find that the projected price for April, 2009 is about $170/bbl.
In January 2007, the price was around $60/bbl.  If we plug that value in, we get a projected price for January 2008 of about $93/bbl -- which is very close to the actual value!  It would be interesting to see how the world oil consumption rate changed from January 2007 to January 2008, but unfortunately I don't yet have those numbers.

Elasticity of Demand for Oil

The biggest unknown here is the price elasticity.  If one searches the Internet for anything having to do with elasticity of oil prices, one finds several things.

• The elasticity of demand for gasoline was, historically, around -0.2.  However, I ran across a paper claiming that, in recent years, its magnitude has dropped to a value roughly 1/10 that size.  This suggests that historical data may not always be very useful in making projections about petroleum use!  It also suggests that something strange and rather disturbing is going on -- people are becoming less sensitive to the price of gasoline, perhaps due to increasing prosperity.
• The elasticity of demand for crude oil depends on the grade.  Demand for light, sweet crude is less elastic than demand for sour crude, perhaps because many refineries require the better stuff and can't function without it. 
• Overall, the demand for crude oil is far less elastic than the demand for gasoline:  Values I found ran from -0.01 to -0.04.  These are incredibly small magnitudes; they imply that when the supply changes, the price moves between 25 and 100 times more than the degree to which the supply changed.
• Long-run and short-run elasticity of oil demand are very different.  We've been talking about the short-run elasticity.   In the short run, it's very hard to reduce oil use; there is no product which can be substituted for it without substantial changes to many systems.  However, over the longer term it is practical to move away from oil use for many things, and the long-term elasticity reflects this:  Its magnitude is believed to be far larger than the short-run elasticity.  Long-run elasticity may be close to -1.
  Unfortunately I didn't find a good definition of the "short" versus "long" run, and the distinction seems a little vague at best.  In any case this suggests that using the short-run elasticity to project oil prices probably won't work well for periods of time longer than a few months or a year.  For longer periods, one should use a larger magnitude for elasticity.
• Elasticity isn't constant in any case.  As prices rise, the magnitude of the elasticity also usually rises.
  The classic example of a "totally inelastic" good is a heart.  If you need a heart transplant, you'll pay any price for the new heart.  On the other hand, if you don't, then you won't pay anything for a new heart.  Consequently the demand is claimed to be completely independent of the price.  But ... it's really not.  If the price of a new heart rises to $20,000,000, the demand will drop off -- almost nobody will be able to afford one.  So, if the price rises high enough, the magnitude of the elasticity also rises to some nonzero value.  The same is true of food -- at high enough prices, many people cannot afford the food, and starve; at very high prices the elasticity is nonzero, even for necessities.
  Similarly, if the price of oil rises high enough, consumption will drop off radically.  In projecting a year or more out, we're considering price changes which may be on the order of a factor of 2 or more.  That may very well be a large enough change to affect the elasticity of demand, so our assumption of constant E may not be valid.  If so, that would suggest that prices may not go as high as predicted by the model.

The value I chose for the default, -0.04, is the high end of the magnitudes I found mentioned on the Web for crude oil.  My primary reason for picking this value rather than, say, -0.02 (the middle of the range) or -0.01 (the bottom of the range) is my gut reaction to values that small:  I just can't believe it's really that inelastic!  I also had another, more rational reason, which is that we're considering prices in the course of a full year, not just a few months.  This is a long enough period that we could very well be seeing the effect of the long-run elasticity of demand for oil creep in.  Since the long-run elasticity has much larger magnitude than the short-run elasticity, that would have the effect of raising the magnitude of the effective elasticity.  Thus it might be that we should actually be using an even larger value for elasticity here, which would in turn result in a lower final price.

Historic Oil Consumption Growth Rates

I located a table, here, showing world oil consumption dating back to 1960.  The annualized compound growth rate for all years in the table is 3.08%.  However, the growth rate from 1995 to 2005 was smaller: it was just 1.79%.
One might wonder if recent slower consumption rise rates might be due to rising prices.   Perhaps we're already looking at the consequences of the nonzero elasticity in petroleum demand.
I located  table of historic oil prices here,  and looked for periods when prices weren't rising.  It turns out that from 1988 through 2003, prices were fairly level (though they jiggled up and down a bit).  From 1991 through 2001, inflation-adjusted prices were nearly level, and from 1992 to 1999 prices were relatively flat in nominal dollars.
From 1991 to 2001 demand grew at 1.42% per year.  From 1988 to 2003 it grew at 1.36% per year.  And from 1992 to 1999, it grew at 1.65% per year.
From this, we might conclude that the "natural" -- constant-price -- demand growth rate was about 1.5% or 1.6% during the 1990's.  However, since the data from 1995 to 2005 spans a more recent set of years than that, and demand grew faster during that period than it did in the earlier years despite increasing oil prices, it seems more realistic to use the value from that period.  So, that's what I've used for the default demand growth rate.

Income Elasticity of Demand;  China and India:  Wild Cards

The concept of elasticity can be applied to more than just prices.  In particular, one can consider the income elasticity of demand.  When income increases, the capability to buy goods increases.  When world income increases, world oil demand increases as well.  The ratio of the percent increase in demand resulting from a particular percent increase in income is the income elasticity of demand.
The income elasticity of crude oil appears to be substantially higher than the price elasticity.  Consequently, when world income increases significantly, demand for oil increases significantly as well.  If consumption is restricted by supply lagging the new demand, the result is a large increase in prices.
This issue is particularly important with regard to the developing world, and in particular it is significant with regard to India and China.  While demand for oil in Europe and the Americas has been increasing at a relatively even rate for many years, and may even have dropped slightly in the past year, demand from India and China is moving in tandem with the rapidly -- and recently -- increasing incomes in those countries.  This could push the constant-price increase in demand up well above the "historic" rate of increase, which was dominated by demand from the United States and Europe.

To put it simply, "new" demand from China and India may result in world demand growing by a lot more than the 1.79% I used for the default value.

The Oil Supply

As I mentioned earlier, it's widely believed that oil production is "peaking".  This was originally based on sophisticated models of the petroleum industry, and I'm not in a position to second-guess that modeling.  However, we can simply look at recent news to get an idea of whether this claim looks reasonable.

• Saudi Arabia, Russia, and Mexico are having a hard time maintaining their production levels.
• Saudi Arabia is finally opening up their next-to-last large field, the Khurais field, which will require water injection from day 1 to get the oil out.  This will require an enormous engineering effort on the part of Aramco.  They have put off tapping this field for quite a long time.  They are probably looking at declining output from their existing fields, which is one major reason for bringing this field on stream now; they are also looking for ways to meet world demand, which continues to increase.  Without opening up new oil fields, they can't turn up the flow rate beyond where it currently is.
• Brazil has just made a much-ballyhooed find:  Perhaps 40 billion barrels of oil have been discovered.  But it's under a mile of seawater, and it's not going to hit the market any time soon.  Furthermore, it's not going to be easy to get the oil out, and it's quite likely that no more than half of it will actually be recoverable (and I've read claims that as little as a third may ultimately be recovered).  So, we're looking at maybe 20 billion barrels of recoverable oil, which won't be available for years (it takes a while to open up a deep ocean field).  World oil consumption is about 30 billion barrels per year.  So, the new field in Brazil could supply world oil demand for about eight months ... and the last reported find of that magnitude, anywhere on Earth, was decades ago.

None of this is conclusive, of course, but it seems to fit with the claim that oil production is at a peak, and it can only go down in the coming years as the huge oil fields we've been drawing on gradually run down.

So, What Can We Do About It?

As a purely passive thing, I suppose you should avoid making long-term investments in airlines or oil tanker operators.  The former will suffer if oil hits $170/bbl, as the default values for the model suggest; the latter may find there's less and less oil being shipped around the world if supplies are indeed running out.

Something you can do which is more active, and which also helps with global warming, is conserve.  And the best way for you as an individual to do that is to switch to a plant based diet.  Raising animals for food consumes an enormous fraction of the world's petroleum supply; see the FAO report, Livestock's Long Shadow.  It's a rather lengthy report, but there's a short summary page on it here.




Page created on 4/25/2008.  Added "What Went Wrong?" box, 1/25/2009