[jimk]

User westexas at TheOilDrum has been pointing out that oil producing countries are actually consuming more and more of the oil they produce. When production is flat, this increasing consumption means that exports will be declining. There was some discussion of how to model this, so I thought I would take a swing.

My thought is that the main difference between consumers in a producing nation versus consumers in a non-producing nation is that the money spent on oil in a producing nation stays in the nation, or most of it anyway. So an increasing price doesn't hurt consumers in the producing nations as much as it hurts in non-producing nations. I.e. the difference is in the elasticity of the demand curve.

So here are the equations I propose to model all this. I present them as a collection of multiple producers and multiple consumers.



The producers are each governed by something like a logistic equation, but with a term added for price sensitivity. I need the price to change to make evident the different price sensitivities of the various consumers. The price will get set by the balancing of supply and demand in the marketplace.

I want to capture somehow the idea that countries grow or shrink. The amount consumed by a nation is not just a function of the price, but also of how "big" the country is. If a country consumes a lot, then the country can grow quickly. But if the country isn't able to consume enough, it will shrink.

I wrote a little simulator in excel, for a system with just one producer and two consumers. The consumers have different elasticities, i.e. different sensitivities to the price. The lower sensitivity consumer is intended to be that of the producing nation.

One tricky bit that I don't know how to put in equations is - when the price gets high enough, my simple linear demand curve ends up in negative consumption territory, which is obviously nonsense. So my simulator just knocks consumers out of the market place, i.e. assigns 0 consumption, when the price gets that high.

So here is the evolution of the amounts consumed by the two consumers. I set the initial conditions so the producing nation starts out much smaller than the non-consuming nation. But the curves cross eventually


Here is the total consumption on the same graph as the price. The price goes up slowly, but then when the price sensitive consumer is knocked out, only the price insensitive consumer is left, and the price really takes off.

An interesting exercise!

[MD]

...
The price goes up slowly, but then when the price sensitive consumer is knocked out, only the price insensitive consumer is left, and the price really takes off....

Thanks Jim!

Next in line after the price-sensitive consumer will be the non-productive consumer.

[aldente]

It is somewhat difficult to follow your point, but then again we all are overwhelmed.

[basil_hayden]

Albente is as lost with a model of reality as we are with his random pic posts! I love it! Whodathunkit....

Jim - thanks for the illustration that accompanies the land export model, it's easy to see where things are headed.

[jimk]

Sometimes there seems to be a conflict between economics and reality. An economic perspective might seem to imply that if the price goes high enough, we can just keep discovering oil. An interesting feature of my tweaked logistic equation is that it is still governed by a fixed amount of recoverable resource, but economics can shift around when we extract that resource. The more we extract early in the game, the less we'll have to extract later.

So here are five plots, superimposed, of total consumption versus time, for the same system of one producer and two consumers. The only difference across the plots is the price sensitivity of the producer.

[jimk]

One thing that makes this simulation hard to understand is that I took a big leap from the usual logistic - I added price sensitivity to the production, and I created two competing consumers with different price sensitivities.

It seems useful to take a step back and consider the single consumer case:



What this shows ought to be obvious... the downside of the slope will be steeper than the upslope. On the downslope we have more people more dependent on petroleum. So this shifts the demand curve. If the production is at all price sensitive, the logistic will surely be distorted to a more rapid decline.

[jimk]

A curious feature of some solutions of this model - the price can go negative. This means that production is actually less that the straight logistic equation would predict. Thinking about this, I realized - this really happens! It's like what OPEC does, or the Texas Railroad Commission. The whole point of these cartels is to supress production.

[jimk]

I am starting to learn C#. So that gives me a bigger hammer than Excel to let me crunch equations!

The linear supply/demand curves I had above were clearly too simplistic - the possibility of them going negative being the most obvious trouble. So here are some fixed-up equations:



This shifts the meaning of the price - it becomes more realistic, for starters. Now a price of 1 means a producer is following the pure logistic equation. As the price goes higher, the producer will pump more crude - of course, this just depletes the fixed-size reservoir more quickly! And when the price is less than 1, the producer will pump less that the pure logistic would predict. Similarly, consumer demand is price sensitive.

My C# simulator puts out a text file which I feed into Excel to get graphs like:

This is a system with two producers and three consumers. The two producers have all parameters equal except the price sensitivity exponent - one is 1.5 and the other 1.7 - that small difference makes their peaks look quite different!

The three consumers differ in starting size as well as price sensitivity exponent. That way I can get the curves to cross in various ways.

Anyway I think these new power-law supply & demand curves are a lot more realistic or at least less temperamental numerically - the wierd discontinuities have disappeared from the simulations.

[jimk]

Since oil prices are in the news these days, I thought I should post the variation in price generated by the simulation that produces the production/consumption graph above:

[aldente]

All jibberish to me..


There is always the option to drop out, as Timothy Leary suggested back in the day, and some even before him..

[The_Virginian]

Albente,

The graphs are clear, I'm surprised you don't understand.

My interpretation is that as peak oil hits because everyone is using more of their own dwindling supplies, and exporting less, we can expect more assaults on person and property.




Therefore what our NY friend is saying is that EVERYONE be trained to use sub-machineguns NOW:


Grandma with a MP-40

[aldente]

Your reply is correct, Virginian. It was just one of those obnoxious comments that sometimes I can't help to make. Some people drive 'under the influence', others post while under the influence.

[The_Virginian]

Albente,

I think we know each other enough to be able to say:

"DON'T GET HIGH ON YOUR OWN SUPPLY"

Wine is too precious to waste it on PO.com

If you read this, then you can see my own post above is very tung in cheek as well....

[aldente]

[The_Virginian]

OK LET"S MAKE SENSE of this FOR THE ARTSY-FARTSY CROWD
[web]http://www.youtube.com/watch?v=MQlkwJvH2J0[/web]

[TheDude]

Death to embedded video!

Granted the Onion is a scream.

The ELM is an interesting notion but there are a flock of parameters that could affect how things turn out for a producer on the downslope - for one thing the poster children for the phenomenon, the UK and Indonesia, have massive populations; will a ceiling on demand be reached for other countries? What would be the reaction among non-OPEC to all of this de facto hoarding - an embargo on auto shipments? Or massive tarrifs? Or demanding that the ME nations purchase CNG vehicles? What if there's so much demand destruction the auto industry is severely impacted? Etc. etc.

[jimk]

The ELM is an interesting notion but there are a flock of parameters that could affect how things turn out for a producer on the downslope

There are two ways to work with models like this - the logistic equation of King Hubbert or more elaborate versions like the ones I am playing with. This paper:

http://www.ifi.uib.no/sd/workingpapers/ ... oversy.pdf

helped me see these two approaches more clearly. One can use the model to get a feel for the basic shape of the likely trajectories a system is liable to traverse. Or, one can be looking for specific predictions. Probably this just comes down to long term versus short term, since long term predictions are impossible anyway.

Perhaps it's like climate modeling versus weather prediction. Climate modeling is useful for trying to get a handle on large scale phenomena like el nino etc. Weather prediction is more about whether it will rain on my picnic tomorrow.

Here are a couple aspects of my model that I don't like.

1) I treat money just as a way to facilitate trading, to find the market balance so that the total amount consumed is the total amount produced. Somebody on TheOilDrum commented that I don't take profits into account. Another way to look at that is that I don't take borrowing and lending into account. Someday the Saudis will stop buying T-bills and start selling them instead. What is that going to look like?

2) I am not so happy about treating each producer as its own little pseudo-logistic. What a logistic does is plot a transition between an exponentially increasing production rate at the beginning of history, to an exponentially decreasing rate at the end. It seems clear enough why the production rate decreases as the resource becomes depleted.

But why does the production rate increase gradually at the beginning? If we start with scads of petroleum in the ground, why doesn't the production rate just start at its maximum? Somehow this has to do with our extraction technology. There are two aspects to extraction technology: quantitative and qualitative. We use energy to make drilling rigs, and we use drilling rigs to produce energy. That's the quantitative aspect. But we also learn how to make better drilling rigs - that's the qualitative aspect. Or we learn what sorts of seismic signatures indicate likely pools of crude underground.

What bugs me is that a lot of the technological growth should not be producer-specific. The energy, materials, and skills that one producer uses to build drilling rigs - those rigs can be used by a different producer. Perhaps a different producer will have to borrow some money to buy those rigs, though.... which begs the question: what is a producer, anyway? Is Exxon a producer, or is Texas a producer, or what?

Oftentimes it turns out that adding lots of model details like these won't really change the structure of the equations - the details all just get combined into the various constants. But sometimes an extra term can become crucial. The interesting thing is to look at such models in terms of regimes, the way the nonlinear systems or chaos theory folks do. A system will often stay in one pattern for a while, then flip into a different pattern. Each single pattern can be modeled simply, with something like maybe a logistic. But more terms can be needed to see how the system can shift from one mode to another.

[jimk]

Here's a tweak on the growth and collapse of national appetite. Really, demand can only grow at some limited rate, but it can collapse overnight. So how about:


Some sample simulation output... the total volume produced & consumed doesn't look too wierd:


But national appetites do collapse precipitously:

[sparky]

.
An interesting concept , keep going ,
I have some problem with your "appetite"
is it demand increase or gratuitous increase beyond production growth

Also for the supply /demand effect on price it's some log function
with a added fixed number for cost .
that's not just the technical cost , but also the producing country taxes
This number ( the extraction cost ) is a variable for each supplier
the cost is the price of the ultimate barrel to bring supply and demand in balance
if 100 barrels are needed and 90 of those cost 5$ while 10 barrels cost 50$ , then the price is 50$