Another Iteration on the Speculation Model

Here's another iteration on the model of oil markets we've been developing. [Update: Paul Krugman comments on the model and provides a simple, continuous time version.]

(I doubt very many of you will have much interest in digging into the details, it's tedious, so the results are summarized at the end).

Here's where it stands. After posting the graphical version of the speculation model, and incorporating Steve Waldman's comments on convenience yields, Paul Krugman noted:

Steve Waldman is right. But we should add that the convenience yield isn’t a fixed number. Actually — I should have realized this earlier — the demand for inventories is like the demand for money. People hold some money for convenience even though money normally yields less than bonds. The amount of money they hold, however, depends on the yield differential. Similarly, the demand for oil inventories should depend negatively on the interest rate minus the contango.

I’ll reformulate my little model to reflect this when I get a chance. ...

And Tyler Cowen remarked:

Mark Thoma has an exhaustive post on convenience yield.  The models used are too piecemeal and they allow "inventories," "convenience yield," and "speculation," to serve as free-floating, not necessarily attached concepts.  The discussion here pays insufficient attention to Holbrook Working, who knew that convenience yield was front and center of the entire analysis, just as "the demand for money" is the centerpiece of the quantity theory. ...

I don't think I will deal sufficiently with Workings discussion for Tyler's taste on this iteration, maybe later, but let me try to incorporate some of the other comments and make convenience yield more "front and center".

The previous version mostly illustrated how the model worked, and used Krugman's model as a baseline for adding in Steve Waldman's suggestion that:

...the future price of a storable commodity is determined by the spot price plus the total cost of storage, defined as foregone interest, plus storage costs, minus ... the convenience yield...

Thus, all that was required was to subtract the convenience yield from the costs specified in Krugman's model, and show how that changed the results.

This model will be quite a bit different than the previous iteration, and, as you can guess from the introduction, I view this as a work in progress. Also, this is a general model of integrating stocks and flows, there may be particulars of oil markets that still need to be incorporated or that I've missed. But it hopefully captures the essence of what we are after.

The model assumes there is a flow demand and flow supply for oil each period. A period could be a day, a week, a month, something like that, I'll use a day just to fix ideas. For example, we might be at a flow equilibrium where we use 100 units of oil each day. Thus, at equilibrium there is a constant flow of oil - we bring in 100 units each day and burn it in our cars, use it to heat our homes, etc. The market is in equilibrium so both the flow demand and flow supplies are 100 at the current spot price.

There are also stock supply and stock demand curves in the model (stock meaning a quantity, not the financial asset sold on Wall Street). For example, suppose a speculator enters the market and wants to purchase 10 units of fuel to store and sell at a later date. The extra demand for fuel that day will increase the spot price. As the spot price goes up, the flow quantity demanded falls, the flow quantity supplied increases, say to 95 and 105 for an excess supply of 10. This 10 units can then be used to augment the speculators stock, as desired.

At this price, then, both markets clear. In the flow market, the price is above equilibrium and there is an excess supply of 10 units, but the extra 10 units are used to increase the existing stock of oil.

Here are the pieces of the model in more detail:

Stock Demand

The stock demand curve has two components, a speculative component, and a service or convenience yield component. The speculative demand, E, depends upon the expected net return from storing oil which, following the lead of others, is assumed to be

[(expected yield per unit) - (cost per unit)] = [((f-p)/p) - (i + c)],

where f is the expected future price, p is the spot price, i is the interest rate, and c is the storage cost. The first term, (f-p)/p, is the expected yield from holding oil and selling it at a future date. As the expected appreciation increases, demand increases. The term i is the opportunity cost of holding the oil, and as i goes up the demand goes down. When the storage costs, c, are higher, demand is lower. Therefore, speculative demand is

E = E[(f-p)/p - (i + c)],

and E is increasing in f, and decreasing in p, i,  and c.

The other term is the convenience yield. Here, I'll take Krugman's suggestion above and model this as depending upon the difference between i and contango

C = C[i-(f-p)/p, c],

that is, it is assumed that the convenience yield depends negatively on the interest rate differential, and also negatively on the storage cost.

Note that all arguments have the same effect on demand as they did in the speculative demand case. That is, as p increases, the expected appreciation falls causing the interest rate differential to increase and this causes C, and hence overall demand for stock balances, to fall. As f increases, the appreciation term increases, the differential falls, and C goes up. Finally, as i and c increase, C falls.

The bottom line, then, is that total stock demand, the sum of speculative demand plus the convenience yield demand, is

D = E + C = D(p, f, i, c) ,

with signs -, +, -, -. [I need to think more about the specification of the arguments of the E and C functions above, but the bottom line assumption about the general form of D in terms of its arguments and their signs seems correct.]

Stock Supply

Stock supply at time t is assumed to be the stock supply at time t-1 plus any additions or subtractions from the stock due to imbalances in the flow market. That is,

S_t = S_t-1 + (s-d),

where s-d is the excess supply in the flow market. For example, if the flow supply is s=100, but the flow demand is only d=90, then the stock today, S_t, will rise above the stock yesterday, S_t-1, by (s-d) = 10.

The connection between the stock supply curve and excess supply or excess demand in the flow market shown in the equation will be important below, so here's how it fits together graphically:

Start at the long-run flow equilibrium shown by Q₀ and p*. At this price, the flow supply and flow demand curves are in balance.

Now let the price rise above p* to p₁.  At p₁, there is excess flow supply as shown by the orange line, and this increases the stock above Q₀ as shown by the corresponding orange line in the stock diagram. If the price increases again to p₂, excess flow supply will go up as shown by the red line, and the stock supply rises to match, again as shown by the corresponding red line in the stock diagram.

For a decrease in the price to p₃, there is excess demand in the flow market as shown by the green line, and the excess demand is met by reducing the stock as indicated by the green line in the stock diagram showing the stock falling below its initial value of  Q₀.

One further point on the mechanics of the model. Suppose the price rises to p₁ as in the diagram increasing the stock supply by the amount indicated by the orange line. Since supply today is supply yesterday plus any additional stock added today from the flow market, i.e. since

S₂ = S₁ + (s-d),

the stock at the beginning of the second period will be higher than in the first period by the amount of the orange line.

This means that at the start of the next period the supply curve will shift out to reflect the higher initial stock:

After the shift, the starting point at the beginning of the second period is p* and Q₁, and the supply curve is S₂.  Any changes in the stock above or below Q₁ must come from excess supply or demand in the flow market, as before. That is, the beginning second period looks just like the diagram above for the first period except that the initial stock is labeled  Q₁, not  Q₀, and the supply curve is S₂ rather than S₁.

Flow Demand

Nothing fancy here, will simply assume

d = d(p), d'<0

Flow Supply

Again, simply

s=s(p), s'>0

We can add more to these functions later as needed. For example, one key question to ask the model is what happens if d, the daily demand for fuel, increases permanently due to higher world demand, and this can be modeled by adding a variable to represent growth in world demand (e.g. world GDP).

We also want to ask the model about speculation, so let's do the graphs for that case next.

Here's a picture of the initial equilibrium:

At the spot price of p*, both the stock market and flow markets are in equilibrium.

Now let f, the expected future price, go up for some reason. The increase in f will increase both the convenience yield and the expected appreciation, and this will shift the stock demand curve out:

After the shift in demand due to the increase in f, and with price still at p*, there is excess stock demand. This excess demand begins to bid the spot price, p, up above p* and as it does excess supply begins appearing in the flow market. As the price gets higher, the excess flow supply, s-d, increases until at some point, p₁ in the diagram, enough excess supply has been created to satisfy the increased stock demand.

Now lets move to the next period. As explained above, because the stock increased from Q₀ to Q₁, the stock supply curve shifts out to start the next period. The shift sets a new round in motion:

The increase in the stock supply causes an excess of stock supply over stock demand at the temporary price of p₁, that drives the spot price down to p₂, and at p_2, the new equilibrium, the excess flow supply just meets the amount needed to clear the stock market.

Again though, since the stock supply went up, there will be a new baseline stock for the next period. One more iteration to fix ideas:

And finally, after enough time, we reach a new long-run equilibrium:

At the new equilibrium, the spot price is the same as its initial value, the amount stored has gone up, and the flow equilibrium is unchanged.

Summarizing, an increase in the expected future price causes

1. a temporary increase in the spot price, but not a permanent increase. When the spot price goes up, some of the daily flow is diverted into storage, and this happens each day that the spot price is above p*. However, after enough time periods have passed, the stock of fuel in storage will be as high as desired, and there's no need or desire to divert any more of the flow into storage. At this point, the spot price returns to where it started, and the flow market will be in balance once again.

2. the amount stored to increase.

In addition,

3. the increase in the expected future price that shifts the stock demand curve isn't driven by a change in the fundamentals in the model, i.e. the change did not shift the S, D, s, or d curves. Thus, if the expected future price returns to its long-run value of p* over time, as it should since nothing fundamental changed, this would all reverse itself. The whole process described above would run backwards.

4. a signature of speculation of the type modeled here is changes in stocks. When the expected future price goes up storage increases, when it goes down, storage decreases.

5. an increase in the spot price over long periods of time is not likely to be a signature of speculation. Speculation can and does drive the price in the short-run, but not the long-run.

Interesting that an increase in speculative demand for oil causes a temporary price spike, but not a permanent one. I missed that before. I guess that's why you build models (and this is a work in progress - more iterations may be needed...).

There are lots of questions we can ask this model, and I probably haven't asked the model the questions Arnold Kling or Tyler Cowen might want to ask of it, but next up: Use the model to show what happens in the short-run and long-run when there is an increase in d, the flow demand, due to an increase in the world demand for oil [Update: see here for details]. I think I can show this is consistent with very little change in inventories, i.e. with little change in stocks, but we shall see. If so, then an increase in flow demand, d, from growth in world demand would be consistent with rising prices and stagnant (or even falling) inventories, while an increase in speculation would show little change in the price and higher inventories. Thus, if we were to observe rising prices and stagnant inventories, that would be consistent with a world demand growth story, but inconsistent with a story that involves an increase in speculative activity to a higher level (though spot prices would rise in the short-run, and a continual infusion of new speculative activity could keep the spot price rising until the increase in speculative activity leveled off).

[In the link in the update at the beginning of the post, Paul Krugman recasts this model in continuous time rather than discrete time.

For example, as he notes, at a point in time with inv=inventories and p = the spot price, let the reduced form equations be p=f(inv, expected future p) and dinv/dt = S(p)-D(p). In this model, as in the model above, the equilibrium price is determined the stock equilibrium and the flows from the commodity market change the stocks over time. It's "basically Bill Branson's 1970-something portfolio balance exchange rate model adapted to commodities." The continuous time formulation is a much simpler framework, both mathematically and graphically.]