June 20th


It has been four days since my last diary entry? The question on the lips of thousands of rapt readers hanging on my every word is “What happened?” The answer is: I ran into a brick wall. Consider, for example, fluidized bed technology. This concept has been around since the 1960s: it was originally developed for certain types of industrial processes. Then somebody realized that the basic technology could be used in coal-burning power plants. You just mix pieces of limestone in with the crushed coal and the limestone absorbs the sulfur while the coal is burning. No sulfur emissions!

Of course, there was a catch: it costs a lot more to grind the coal down to little pellets and mix in tons of limestone before burning it all in a specialized combustion chamber with air being blown upward from underneath. It’s much cheaper to simply dump the coal down a tall furnace, burning on its way down, which is how it’s down nowadays. So yes, you can get lower sulfur emissions (and lower emissions of other baddies as well); but it costs more. So this is not a matter of technology, it’s a matter of economics. Right now, it’s too expensive to use, but the value of research is that it can developer cheaper and more efficient fluidized bed technologies. If we can get the price down, then at some point it becomes cost-effective. But what do we mean by “cost effective” in the case of fluidized bed technology? That depends on the price you set on sulfur emissions. If you require coal plants to pay a Pigovian tax (read all about Pigovian taxes here on Wikipedia), then you can set a price on sulfur emissions and make the installation of fluidized bed technology dependent upon the tax on sulfur. Set the tax high enough and everybody will use fluidized bed combusters.

But we don’t do it that way: we use state intervention instead of market mechanisms. The government simply declares “Thou shalt not emit more than this many million tons of sulfur dioxide per year” and the coal plant operators fiddle around to come in just underneath the ceiling. The results can be truly idiotic. So I’m just going to assume a market mechanism here and assign Pigovian taxes to emissions of various types.

But how do I factor new technology into my economic calculations of supply as a function of price and demand? Currently I use a relaxation algorithm that starts with the current price and the current demand, then tracks up the supply-as-a-function-of-price curve until the supply matches the demand. A Pigovian tax on sulfur emissions would push some portion of the coal supply out of economic reach. This means that we have two curves to consider:

First, the left graph is a bit misleading. The Y-axis represents the cumulative amount of coal with LESS THAN the sulfur content specified in the X-axis. In other words, if you accept humongous amounts of sulfur in your coal (“Would you like some coal with your sulfur, sir?”), then you can get almost 100% of the total supply of coal.

What we really need to do is prepare a 3D graph of the amount of coal available (on the Z-axis) as a function of price (on the X-axis) and sulfur content (on the Y-axis). But I’m too cheap to buy 3D graphing software (and too lazy to spend all the time learning how to use it), so you’ll just have to use your imagination. That 3D surface will show the basic cross-section above in both X and Y. Now, the true supply of coal under any given situation must fall somewhere on that surface. Let’s say we start off with zero Pigovian tax on sulfur emissionsk, which means that we can accept any amount of sulfur and we’re way out at the asymptote of the left curve. Since we already have a price for coal, we can set the supply point at any point on a line parallel to the Y-axis at an X-coordinate matching the price. Obviously, since there’s no Pigovian tax, we’ll use the asymptote for the sulfur content at the specified price. It’s way off at infinity on the Y-axis. So first, let’s put a small Pigovian tax on sulfur emissions just to bring our test point back from infinity. So now we have our test point at some high value of sulfur content on the price-line. Here’s where we get down to business: if we increase the Pigovian tax, then the point moves in BOTH the X and the Y dimension. It moves in the Y dimension (towards lower sulfur content) because the tax has changed the profitability calculation; coal plant operators must purchase lower-sulfur coal, which is more expensive. This reduces the supply of coal available at that price point — which forces the price of coal-generated electricity upwards. In other words, an increase in Pigovian tax always triggers a price increase — which makes perfect sense.

The problem is, how do I calculate all this? There is an analytic solution using partial differential equations, I’m sure, but I really don’t want to get into partial differential equations — I’m too rusty. Fortunately, I can use relaxation algorithms on the computer. But there’s still the matter of specifying the shape of the supply-versus-sulfur-content curve.

Now, the simplest way to approximate an asymptotic curve is with the formula I explained on June 2nd:

Supply(price) = Total Supply - Scaling Factor 1 / Price

I can use exactly the same form for the supply as a function of sulfur content:

Supply %(sulfur content) = Total Supply - Scaling Factor 2 / Sulfur Content

But that STILL doesn’t get me where I want to go, because my input is the Pigovian tax (T) instead of the sulfur content S. Here’s where the relaxation method applies. I start with the current price of coal P and the current value of the Pigovian tax T. I use P to calculate the supply available at that price and then I use T to calculate what percentage of that supply is actually available. Then I compare that supply with the demand and, if it’s not big enough, I increase P. I continue doing this until I have a supply large enough to meet the demand.

OK, I think I’ve got it figured out now. Time to write some code.