Thinking Quantitatively about CO2 Uncertainty

Anyone who claims to know what the social cost of carbon is, is a liar or a fool.[Infallible Preacher]

If we don't know what the Social Cost of Carbon (SCC) is, how can we make any decisions relating to CO2 emissions? If you are a full on Bayesian, the answer is easy: you have a subjective probability density on the SCC; use it.

To illustrate how this works, I will use myself as an example. A full on Bayesian can determine his probability density on just about anything by considering a string of lottery choices. For example, to determine the median of my SCC density, I ask myself to choose between:

1) a flip of a fair coin in which my win is a million dollars with probability 0.5 (the median level) and my win is zero dollars with probability 1.0 - 0.5,

2) a gamble in which my win is a million dollars if the SCC of carbon is less than x $/ton CO2 and my win is zero if it's more than x.

Figure 1 sketches this choice.

Figure 1. Decision Tree to Determine My Subjective Median SCC. The x at which I am indifferent between the two gambles is my median.

I start out with a low x of zero $/ton. I pick the coin toss. But if x is $1000/ton, I pick the SCC lottery. I start narrowing in. $20/ton -> coin. $500/ton ->SCC. And so on. When I get to around an SCC of $100/ton, I'm indifferent between the two gambles. $100/ton is the SCC at which my SCC Cumulative Density has a value of 0.5.

A full-on Bayesian can repeat this process replacing the coin flip probability 0.5 with every probability between 0.0 and 1.0, after which he has his complete density on the SCC. I consider myself close to a full-on Bayesian; but I have never actually generated a subjective density on anything this way. Rather I pick a family of densities and then try to determine which member of that family best represents my state of uncertainty.

In this case, I wanted a density with a fat upper tail reflecting the fact that the SCC could be extremely large. To do this I chose to use a lognormal density. The way I think about the lognormal, is that in a "normal" (pun intended) density a 2 sigma event is 2 sigmas away from the most likely outcome, a 3 sigma event is 3 sigmas away, and an n-sigma event is n sigmas away. But in the lognormal an n-sigma event is e^n sigmas away from the peak. A 1 sigma event is 2.72 sigmas away from the peak, a 2 sigma event is 7.4 sigmas away, a 3 sigma event is 20 sigmas away. The probabilities fall off far, far slower than for a normal density, as you move to larger and larger SCC's.

A lognormal density has three parameters, so I need to specify three constraints on my SCC density to determine which lognormal I should use. I've already got my median. I think there's a slight probability that the SCC could be negative. I'll call that probability 0.01. The key decision is the upper tail. So I ask myself to think about the choices sketched in Figure 2. I can go with a lottery that has a probability p of a million dollars or I can have a bet that pays me a million dollars if the SCC is greater than a $1000/ton. At what p would I be indifferent between those two gambles?

Figure 2. Decision Tree to Determine My Probability the SCC is greater than a $1000/ton

Of course, there is no way I can be precise about such questions, but I can close in on this p. At a p of 0.05, I clearly prefer the lottery. At a p of 0.005, I clearly prefer the SCC gamble. I decide to compromise on 0.02. Figure 3 shows the lognormal which is consistent with these three numbers.

Figure 3. Jack's Social Cost of Carbon Probability Density. The area under the density curve between any two numbers is the probability that the SCC is between those two numbers. The green area is my probability the SCC is between 100 and 150 $/ton.

Now when I'm crowned king of the world, I'm in business. Being the benevolent guy I am, I choose to minimize the expected value of the social cost (sum of grid cost and CO2 cost) of providing the electricity, that is, I weight each SCC by my probability of that SCC in determining the optimal mix of power sources.

This will require a lot of computation. Once I'm made king, that will not be an issue. Until then, I will have to live with a sub-optimal approximation.

1) I compute the optimal grid for each of a full range of SCC's. Conveniently, we have already done this for Germany. These grids are efficient (aka not stupid). You can't have both less CO2 and less grid cost. With stupid grids, you can.

2) I compute my expected social cost, for each of these unstupid grids. The GKG Grid model tells me the grid cost and the amount of CO2 produced by each of these grids, so this is a trivial amount of extra work.

3) I pick the grid that has the lowest expected social cost. This may not be my true optimum, but usually it will be pretty close.

I've done this for our German trade-off mesh, Figure 4. I was a little surprised that for every nuke Capex, this calc picked the $200 per ton CO2 grid. My curve and the green dashed line are the same.

Figure 4. Jack's Optimal Choices for Germany as a Function of Nuke CAPEX.

The reason I was surprised is my SCC density peaks at $25/ton CO2, far below $200. But on reflection, I'm willing to live with my choices. $200 is not that far from my mean SCC. I'm willing to accept up to a doubling in grid cost, if I had to, but not more. It seems I have more money than empathy for the poor. They had better hope nuke is near its should-cost when I become king.

The correspondence between my effective SCC, the slope of the trade off curve at my optimal point, and the mean of my SCC density should not have come as a surprise. For a Bayesian, they are equal. You can find a proof at the PDF of this piece. The $200 dollar per ton points were the closest to my mean of $186/ton on Figure 4 that had actually been computed.

Figure 5 shows some of the grids on my curve. At a nuke Capex of $4000/kW or less, I go with a nuke plus gas peaking grid. Since all my capacity is dispatchable, the total installed capacity, adjusted for availability, is very close to the peak hourly demand The CO2 intensity is around 20 g/kWh. Some where above $8000/kW, I give up on nuclear, and go with a wind/solar/gas grid. My total installed capacity more than doubles. I'm paying for two grids. My CO2 intensity is a mediocre 200 g/kWh.

Figure 5. Jack's Grid for 6 Different Nuke Capexes

But what is the point of all this navel gazing? Nobody is going to make me king of the world. Nobody cares what my SCC density is. I submit this egotistical exercise makes a couple of points.

1) Our uncertainty about the impact of CO2 need not paralyze us. We can think about that uncertainty and attempt to quantify it.

2) Nor should it force us to assume a worst case and base our decisions as if this worst case is what will happen, no matter how unlikely it is.

3) Different people will have very different SCC densities. But perhaps by focusing on differences in densities rather than framing the issue in binary, black and white terms, we can make some progress in resolving those differences. Admitting the other guy could be right, is a big first step toward a constructive discussion. I've often been called a climate denier by some of my alarmist friends. How far away from the green dashed line are they? How far away are you?

Alice the Alarmist

Consider my friend Alice. Alice is a bit of an alarmist and with good reason. Her SCC density, Figure 6, has a mean of about $750/ton and a monster standard deviation.

Figure 7. Alice's Social Cost of Carbon Density.

It turns out her CAPEX curve is the $800/ton CO2 curve in Figure 4. These were the closest points to her mean SCC that had actually been computed.. That's pretty far away from mine. But look at her grids, Figure 7.

Figure 7. Alice's Grid for 6 Different Nuke Capexes

Alice's top row is very similar to mine. Neither of use use any wind/solar. Alice buys a bit more nuke capacity than I do and uses a bit less OCGT, but the differences are minor. Even at $8000/kW nuclear, the differences are not worth shouting at each other. Alice does stick with nuclear much longer than I do, when nuclear get preposterously expensive. She is willing to install an enormous amount of wind to push the fossil fuel usage down when even she can no longer stomach nuclear's cost. But she never stops using some gas.

Overall I find it hard to get excited about the differences between Figures 5 and 7. What both Alice and I should be exercised about is the differences between the top row and the bottom row in both figures, the difference between affordable, low CO2 power, and crushingly expensive, mediocre CO2 power.