An Increase in Worldwide Demand for Oil

Another round on the oil market model, this time to show what happens when there is an increase in the world demand for oil due to some factor such as increased demand from developing economies. The point is to show that, in this simple model, the increase in demand would increase in the long-run equilibrium price, but it would not change the level of inventories in the long-run. There are also other results to note, e.g. the possibility of overshooting the new long-run equilibrium and mimicking a bubble.

Case 1: An Increase in the Expected Future Price of Oil

First, the continuous time version of an increase in the expected price. I did a discrete time-version of this yesterday, but the continuous time version of this case Paul Krugman did yesterday is much simpler, so let's use that. Here's a quick review of that case:

In this model, the initial equilibrium is at point a. Then, there is an increase in the expected future price or a drop in the interest rate that increases the stock demand, N^d, and the equilibrium moves to point b. At this point, the spot price is above the equilibrium value in the flow market shown on diagram on the right, and there is excess supply as indicated by the red line. This excess supply increases the stock so, as shown by the arrow, the stock supply curve begins shifting out. Eventually, the economy settles at the new equilibrium shown at point c.

Summarizing the results for an expected increase in the future price:

1. The spot price, p, rises in the short-run, but is unchanged in the long-run.
2. The stock of inventories, N, rises.
3. The long-run flow equilibrium is unaltered.
4. The change in inventories depends upon the horizontal shift in the stock demand curve. This can be related to the slope of the stock demand curve, but it is not necessarily the case that a steeper demand curve leads to a smaller horizontal shift.
5. Steeper flow supply and demand curves affect the size of the change in the spot price during the transition (and the slopes can affect the transition path for inventories), but this does not change the ultimate size of the inventory change since this depends only upon the size of the shift in the stock demand curve.

Case 2: An Increase in Worldwide Demand for Oil

Moving to the next case, what happens if there is an increase in the world demand for oil due to worldwide economic growth. This shifts the flow demand curve outward:

Starting at the equilibrium a, as the flow demand curve shifts out, this causes an excess demand for oil as shown by the red line on the diagram. This excess demand is met by reducing stocks, so the stock supply curve begins shifting left and the economy moves to point b. Thus, so far there is an increase in price, and a decline in inventories.

But this isn't the end of the story. Because the increase in flow demand is permanent, the increase in price is permanent, and this will increase the expected future price. The increase in the expected future price will shift the demand curve out as shown in the next diagram:

As the demand curve shifts out to reflect the higher expected future price, the price moves up to point c. At point c, the flow market has excess supply as shown by the orange line segment, and this pushes the stock supply curve outward as the excess flow supply is absorbed as new stocks. Eventually, the economy reaches point d which, compared to point a, reflects a higher price but no change at all in inventories. Notice that the spot price overshoots its long-run value as it moves from b to c, then back down to b.

Why does the demand curve go through the same point for inventories as before? Recall from Krugman's post that  N^d = N(i-(p^e-p)/p), where i is the interest rate, p is the spot price, and p^e is the expected future price. At the initial long-run equilibrium, it must be that p^e=p, otherwise there would be a tendency for something to change (and hence it wouldn't be a long-run equilibrium [Update: I should add that there is no mechanism in this model to force p^e=p, but adding this in is a simple fix, e.g. just add an equation that says dp^e/dt = f(p^e-p), f'<0, or go to a more complicated rational expectations set-up. In this model, the requirement that p^e=p arises from making the model internally consistent with the definition of a LR equilibrium]). Thus, at the long-run equilibrium, the stock demand is just N(i). That means that the long-run equilibrium for stocks is independent of the spot price and its expected future value. Thus, the inventory level will be the same as its initial value after the offsetting changes in p and p^e.

Summarizing the results for an increase in flow demand:

1. The long-run spot price rises.
2. In the short-run, the spot price can overshoot in the new long-run equilibrium. The model doesn't predict overshooting, the a-b-c-d progression shown in the diagram is just for exposition, the actual path can be different (e.g. there's no reason for the expected spot price to increase only after the economy reaches point b). But the model is consistent with overshooting, and therefore the change in the spot price can look like a bubble that is inflating, then deflating even though the change is driven purely by fundamentals, i.e. by a shift in world demand.
3. The level of inventories can change in the short-run, but is unchanged in the long-run. (However, in a more general model, there might be a change in, say, the interest rate or convenience yield and this would cause an additional shift in the stock demand curve and change inventory levels. But it's still possible for the variation in inventories to be small.)

Finally, this is just a "vintage" Branson-style exchange rate model applied to commodities, so if you are familiar with those models and the bells and whistles that can be added to them, or with alternative models, for the most part the results and intuition ought to carry through to this case.

Update: Paul Krugman in response to Tyler Cowen.