One of the big frustrations with trying to follow the debate on climate change is that most of the key questions are best answered with large, complicated models. Learning enough to assess these models in one subject area, even in general terms, is a huge task, and learning the details of any particular model is a full-time occupation. But, if we are going to make any real progress, we need numbers we can understand. It seems hopeless, but it isn’t entirely so. One thing I learned very early on about modelling is that, for almost any large complicated model, there’s a small simple model that gives much the same answers to the key questions of interest, if you use it correctly, and choose input parameters consistent with those in the big model. The big model (if it’s a good one) imposes consistency conditions you might miss in a simple model, and also gives detailed answers to lots of more specific questions, but a lot of the time, you can do without that. I’m writing a paper at the moment, trying to answer some of the important in a way anyone can check without spending years mastering a big model.
The biggest question of the moment is: what is the right price for carbon? I’m going to look at this question for the world as a whole, disregarding national differences and so on. If you’ve read the title of the post, you’ll know what answer I reach.
I’m going to start by talking about the damages caused by uncontrolled climate change. The pre-industrial concentration of CO2 (+ equivalents) was about 280, and we are now around 430. The most common candidate for a ‘safe level’, where the risk of damaging climate change is small, is 350 ppm. I’m going to assume that level is associated with zero net damage. A plausible, somewhat optimistic, Business as Usual path might take us to 650 ppm, which implies eventual warming somewhere between 2 and 8 degrees C, with a median value around 4.5 degrees C. Even the low end of this range would entail significant economic and environmental costs, while the high end would almost certainly be catastrophic. Finally, the target in most international discussions is a path that ends up with CO2-equivalent concentrations stabilised at 450 ppm. I’m going to assume (more posts to come on this) that this target could be reached with a global price starting at $50/tonne of CO2 and rising gradually over time.
What’s crucial here is that the expected damage associated with a given emissions trajectory isn’t a linear function of the final atmospheric concentration. Costs increase slowly at first, then much more rapidly, giving rise to a convex damage function. The most convenient kind of convex damage function is the quadratic, which some will remember from high school math, in which the function value increases with the square of its argument. Given our assumption that damages are zero at 350 ppm, we want a function of the form.
(1) C = k (X – 350)^2
where C is cost (expressed as a proportion of global income), X is CO2-e concentration and k is a constant. But what is k? We can work it out if we have an estimate for the damage associated with the Business As Usual scenario. To illustrate the point, I’ll use a (conservative) estimate that the expected cost is the same as a 10 per cent permanent reduction in global income. So, we have the pair D=0.1, C=650, and high school algebra gives us k = 1.1*10^-6. I’ll simplify this to k=1*10^-6
Now we need the tiniest bit of calculus (take it on faith if you have to, this is as hard as the math is going to get). Given the cost formula (1), the marginal damage associated with an additional PPM of final concentration is given by
(2) MC = 2k (X-350)
To finish up, we need to convert the marginal cost on the left-hand side into dollars/tonne of CO2, so we need a conversion factor for ppm/tonne of CO2. The conversion part of this is reasonably simple – after taking account of sinks, emitting around 10 Gigatonnes of CO2 raises the atmospheric concentration by 1 ppm.
For the left-hand side, we need to deal with another controversial question, that of discounting. I’m going to assume that income grows at 2 per cent a year, and use a real discount rate of 4 per cent [fn3]. Current world income is around $US50 trillion, so the present value of the income stream is $US50 trillion/(0.04-0.02) = $US2.5quadrillion. We’ve got some huge numbers on both sides here, but we can cancel them out, to get a marginal cost in $/tonne of 2.5*2*(X-350)/10, or more simply, 0.5*(X-350).
Now plug in the most popular target of X=450 ppm, and voila!, the answer is (drumroll ….) $50/tonne. And, as I said above, it seems pretty reasonable to expect that if the world moved reasonably rapidly to a carbon price of $50/tonne (which should rise at the discount rate of 4 per cent a year), that we could achieve climate stabilization at or below 450 ppm.
Obviously, I’ve picked numbers that give that particular answer. What if the cost of BAU is estimated at 5 per cent, or 20 per cent instead of the 10 per cent I’ve chosen. Won’t that make a big difference. Surprisingly, perhaps it doesn’t. If you repeat the exercise I’ve just done, using a 5 per cent damage estimate, and plug in a target of 550 ppm, you get, once again, a carbon price of $50/tonne. That’s because, with a quadratic cost function, doubling the difference (X-350) doubles the marginal cost, and exactly offsets the halving of the damage estimate. But that doesn’t mean the change in costs has no effect. Since a carbon price of $50/tonne is more than we need to hold the equilibrium concentration to 550 ppm, this pair can’t actually be realised.The equilibrium is something like a 500 ppm target with a price a little under $40/tonne. Doing a similar exercise with 20 per cent damages, a likely target would be around 425 ppm with a price of $75/tonne. Changing the discount rate can make a somewhat bigger difference, but not that much bigger. The logic of the problem forces us to an optimal carbon price somewhere close to $50/tonne.
Currently, of course, no country is anywhere near that. That’s not surprising. The nature of a collective action problem like this is that everyone tries to free ride, sharing in the benefits but not contributing enough to the costs. The best we can hope for in the short run is a collection of national policies that mostly give an effective price around $25/tonne, about half of what we ought to have.
fn1. These numbers are based on sensitivities from climate models. I’m not going to go over them in detail in this post, but will make the point that, for many purposes, you can sum up the output of a gigantic climate model by looking at the range of values it gives for sensitivity, that is, the equilibrium temperature change from a doubling of CO2.
fn2. Again, there are some fairly simple ways of getting a handle on this number, which is largely determined by the risk of catastrophic climate change. I’ll try to cover these later.
fn3. This is substantially higher than Stern used, and closer to the rates used by some of his critics. I don’t want to get hung up on this particular issue for the moment, so I’ll ask that we leave it to one side in the discussion.