As with any good economic model/principle, let's start with a few assumptions to a) motivate the problem, b) set the stage, and c) simplify the explanation to cut off the fat and get to the meat.

Assumptions:

- Suppose a private owner owns the complete stock of a natural resource.
- The complete stock of the resource is fully known and there is no more.
- Once some of the stock is withdrawn, the resource withdrawn is used completely with no waste and nothing left over for reuse.
- The stock can never regenerate itself.
- The cost of withdrawing a unit of the resource is always the same (to make things really simple, we will assume the cost of extraction is $0).
- There are no alternatives to the resource.

For those who like to picture what these assumptions mean, think of yourself as being the sole owner of a giant bowl of all of the existing M&M's in the world (and the recipe for M&M's and all other candy has been lost so no more M&M's or other candy exist or will ever be made)*.

Given the 6 assumptions, how would you manage your stock of M&M's? And what affect would that have on the price of M&M's. Read on...

Being the greedy business owner that you are, you have one simple goal--maximize the amount of money you have to spend on other things (because, really, do you want to live on M&M's alone?).

So how do you maximize your profits from a resource that will eventually run out?

If you sell the entire stock of M&M's today (extract everything and sell), you could take that money and put it in the bank. That would give you the money you get from the sale, plus all of the interest you earn on the bank principal over time, minus anything you spend on other stuff.

On the other hand, if you leave all of the M&M's in the bowl, you don't have any money today, but you have the option of selling the M&M's for some other price in the future. In order to make it worth leaving the M&M's in the bowl, the future price would have to be high enough to make up for the fact that you could have extracted the M&M's and invested the money elsewhere. So we would expect the value of the stock in the ground to increase over time.

How fast will the value of the M&M's in the bowl increase? Well, it depends on the interest rate you could earn if you extract and sell.

Let's suppose (to keep the math simplish) that the interest rate you can earn on money in the bank is 5%. If you sell everything in the bowl today for $1,000 and put the money in the bank, one year from now you would have $1,050. You will have made $50 by selling your M&M's. So, if you leave the M&M's in the bowl for a year, you need the value of the M&M's to increase by 5% to make up for the lost interest you would have earned if you had sold everything. Even though the amount of M&M's in the bowl hasn't changed, and never will, the value of the stock has to rise as fast as the interest rate to make it worth keeping the M&M's in the bowl and not selling everything. Otherwise you would just pull everything out as fast as possible and sell.

So what happens if you only take out some of the M&M's today and leave some in the bowl for the future?

The result is similar.

Let's say I take 10% of the M&M's out of the bowl today and sell them for $100. If I put that $100 in the bank, I would have $105 plus the value of the 90% of the M&M's in the bowl. We've already figured out that the value of and M&M's in the bowl has to increase at the interest rate, but it should also now be clear that the price of the M&M's that you take out of the bowl has to increase at least 5% each year too. Otherwise, you would leave the M&M's in the bowl and let their value increase just sitting there.

So, to make the most money, the price of the M&M's has to increase at the interest rate.

That's Hotelling Rule in its simplest form. For a non-renewable, exhaustible resource with completely known stock, no discoveries possible, no alternatives, no recycling, private ownership and constant costs of extraction, the price of the resource will increase at the interest rate over time.

What does that look like?

Here's a picture:

Now that I'm sure I have convinced you of the sensibility of Hotelling, here's a graph of the inflation adusted index prices of the five metals from 1950-2010 from the infamous Ehrlich-Simon bet (I found this here on the internets because I didn't have the time to recreate the graph from USGS data):

The graph of real prices doesn't look a whole lot like the predicted price path by Hotelling, does it? I'm sure you're wondering why...

That's going to have to wait for Hotelling's Rule Part 2. When I get around to it.

(hint, assumptions make an ass of u and mptions--or something like that).

*For those of you who don't like silly candy examples, substitute the words 'Oil' for 'M&M's' and 'ground' for 'bowl' and it's a little more realistic. But where's the fun in that?

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