Why the global carbon price should probably be around $50/tonne (nerdy/wonkish, but not too difficult, I hope)

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[1]. 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.[2] 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.

48 thoughts on “Why the global carbon price should probably be around $50/tonne (nerdy/wonkish, but not too difficult, I hope)

  1. John its a nice back of the envelope calculation.

    You didn’t want to discuss discounting but can I point out that if you set a targeted emissions cut in the future and use a low discount rate the current price will be high but there will be low future growth in carbon prices. If you use a Nordhaus type rate then the current price is low but you get high growth rate in the future. I assume politicians would prefer the procrastinating latter option but those concerned with climate change might be skeptical of the willingness of future politicians to grow prices strongly when they won’t set a moderately high price now. I’d be happy with $50.

  2. I have nothing substantive to add, but I thought if I could help fix some typos this (IMO excellent) post it might help others who read it in the future.

    Forgot function at the end of this sentence:

    “Costs increase slowly at first, then much more rapidly, giving rise to a convex damage function.”

    Errant $ sign:

    “if you repeat the exercise I’ve just done, using a 5 per cent damage estimate, and plug in a target of $550 ppm”

    Missing is:

    “The equilibrium issomething like a 500 ppm target”

  3. @Tapen Sinha
    I don’t know if there is a paper, but the story is pretty well known. All national goverments expected meeting the targets to be really hard, so they pushed for generous allocations, most of which were then given away free to polluting industries. As it turned out, meeting the targets was so easy that, towards the end of the first market period, there were more permits available than people who wanted them (they couldn’t be banked for the next period, so there wasn’t much value in increasing emissions for a few months).

  4. So this would be a price that would last forever? We’d be paying $50/tonne emitted right through the century?

  5. Nitpick: “So, we have the pair D=0.1, C=650” should be “So, we have the pair C=0.1, X=650”.

    What price human life? We are already, at 390 ppm CO2 (not CO2-e, John[1]), seeing increasing tightness in world food stocks, due to increasing incidence and severity of regional droughts and floods in the Americas, Eurasia, and East Asia. Simple extrapolation and climate models both tell us to expect greater volatility in food production, with the risk all on the down side. Some of the posited 10% reduction in Gross World Product is likely to come from much larger reductions in food production.

    Removing the demand from many of the 800 million Indians who live on less than USD 0.50 per day [2], and the hundreds of millions of others in a similar situation elsewhere, wouldn’t affect GWP greatly — it’s unlikely that the effect in the data could be distinguished from random noise. But from an ethical point of view, it’s unconscionable. And unfortunately, linearly proportional measures like a carbon tax won’t prevent it.

    FN1. http://blog.greens.org.nz/2007/10/10/flannery-says-ghgs-hit-455ppm-co2e-in-mid-2005/

    FN2. http://www.bostonreview.net/BR36.3/siddhartha_deb_india_food_crisis.php “The Indian government’s own data show that 800 million Indians live on about twenty rupees (about $0.50) a day.”

  6. @wilful
    In fact, $50/tonne rising at 4 per cent (real) a year in this analysis. But the idea is to get net emissions down to zero by mid-century, so the point that they would still be taxable after that is a bit hypothetical.

  7. @Greg
    Your numbers on poverty in India are way off the mark. Here’s some World Bank figures, showing that the proportion of the Indian population on less than $1.00 a day (2005 prices) is now about 24 per cent, down from 42 per cent in 1980. That’s way below the 800 million you cite for $0.50. Of course, that’s still a huge problem, but if you start your analysis with wrong information, and get the trend wrong as well, you’re going to come up with bad answers.

    http://www.sajaforum.org/2008/08/poverty-world-b.html

    The Flannery number is also wrong

    http://www.realclimate.org/index.php/archives/2007/10/co2-equivalents/

    I’ll check my sources and make sure I get the right current number right

  8. Of course Quiggin’s model is based on a parabola, which is fair enough for illustration. However the real curve is steeper. This implies a greater cost per additional greenhouse emission.

    Lets not get too fixated on a illustrative model, unless the function itself is validated. I would expect this to be better represented by an exponential function, particularly if population growth, deforestation and industrial development, effects all pile up on each other.

    The SEARCH foundation wants $70.

    But it must be a realistic price. Big business wants it at a laughable $10 per tonne, Prime Minister Gillard appears to want it set at less than $30 per tonne. This is also too low to drive investment in renewable wind and solar energy. It will only enable a shift from coal to gas, promoting destructive coal seam gas extraction, and locking in high carbon emissions for another 30 years. The starting price must be at least $70 per tonne to help the shift to renewables. But first we have to defeat the climate deniers led by Tony Abbott, Barnaby Joyce, Nick Minchin and the energy corporations.

    [ http://www.search.org.au/archives/2525 ]

  9. @Chris Warren
    That’s right. The point about the quadratic is that you can easily do the whole thing in a blog post. And I’m not claiming precision to within $10 or $20, just giving a ballpark number.

  10. @John Quiggin

    The discussion seems to be only about damage costs.

    Is it right to assume that abatement costs will be lower or similar in the short to medium term?

    Up to $20 / tonne CO2-e this might be a reasonable assumption. But, above this figure, most marginal abatement cost curves for carbon emissions generally depict; unproven new technology, scaling up of renewables in a manner that has yet to be proven, energy efficiency measures which ignore Jevon’s paradox, and/or reserves (and EROEI) of natural gas being very optimistic.

    I’m assuming some combination of all these abatement measures may prove viable at $50 / tonne. But, I’m not optimistic, or really convinced.

    In the short term, a $50 / tonne price will shift emissions to countries that don’t have the price.

    In the mid term, most polluters will just pay the bare minimum tax they can wiggle down to and keep on polluting (especially in the absence of quick and viable abatement opportunities).

  11. In the short term, a $50 / tonne price will shift emissions to countries that don’t have the price.

    Interesting point but I assume that this would be visible. All policies cause such games.

    In time countries with a carbon price should not have to accept carbon-content imports from countries without a carbon price. There will then be little benefit shifting emissions as a form of gaming the new policy.

  12. @Chris Warren

    The most practical example is integrated steel manufacturing. Above $20 / tonne there are no big marginal abatement opportunities available in the short – mid term for this industry.

    You are basically hoping for a miracle to happen overnight with blast furnace technology.

    A $50 / tonne price signal will create no further meaningful abatement opportunities for this industry.

  13. iain, there’s an obvious abatement opportunity – make less steel. Which we will, because the price will be higher and demand curves tend to slope downwards. It’s not as though there are no substitutes these days.

  14. For primary metals there is always improved recycling rates. I helped deliver a ute load of steel to a scrap yard yesterday. The going rate I believe was $205/t for those with an ABN. There was a queue of trailers at the weighbridge with wrecked cars, tyres removed. Every time we quaff a can of softdrink (outside SA) we really should get a deposit refunded. Recycling is one industry that will go well with carbon tax. It should have been done years ago.

  15. @iain

    It is not clear what you are saying. I assume you are concerned that a carbon price will impact on steel smelting, so that metal products from Australia cannot compete with metal products from overseas where there is no carbon price?

    In this case – if the steel industry vacates Australia, a vacuum arises for substitutes eg recycling. But also high-steel comprised cars become more expensive which is useful.

    You lessen the tendency for an industry to move offshore, if all OECD countries block unfair trade from states with no or weak carbon pricing.

    If your concerns are purely about the fate of the steel industry in Australia, then this is the whole point of carbon pricing. Those sources of emissions which “no big marginal abatement opportunities” will contract. Due to the seriousness of the climate problem this is the road to the future.

    There may be other ways to produce steel other than using coke, but I do know. This is not the key point. Climate first – steel second.

  16. iain, charcoal can be substituted for coke and this could be cheaper than paying the carbon price. It can be mixed in with coke, or new blast furnaces can be built to use all charcoal. Since I think a blast furnace is only good for about 15 years, it won’t take long to turn over the existing stock of blast furnaces.

  17. The big end of town must be listening
    http://news.smh.com.au/breaking-news-business/rio-tinto-boss-defends-carbon-record-20110603-1fkhp.html
    Iron ore could be reduced with hydrogen but I think we’re looking at $2-$5 at kg for hydrogen vs. maybe 20c a kg for coke.

    I think there may be an element of bluff in threats by smelters to relocate to China since that country’s coal output has hit a flat spot of 3,200 Mt a year and may steeply decline from now on. Our paltry coal exports of 260 Mt to all countries will help little.

  18. Looking at coke and charcoal prices, there doesn’t seem to be much difference between the two. Of course, for a blast furnace near a coal mine, the marginal cost of using coke might be quite low. None-the-less, it looks like it might only take a fairly modest price on carbon to cause a swich to using charcoal for steel production.

  19. Ronald,
    Not that it’s important, but I think your 15 years relates to a blast furnace lining, not the furnace itself which will be designed to last much longer. More importantly, you can’t build a big charcoal blast furnace (it’s all driven by the compressive strength of the coke). So I think we’ll be stuck with the big dinosaur coke BFs for a while. Rio Tinto tried to develop a coke-free ironmaking process (HIsmelt) and it’s currently being developed further in Europe. Maybe its time has finally come….though it is a much smaller capacity. Alternatively, there are proven gas-based direct-reduced iron (DRI) plants eg Midrex.
    For what it is worth, the BF is a very efficient use of coal because the coke oven gas and the blast furnace gas are mixed and used as fuel through the plant.

  20. Ah, right. Thanks Andrew. It makes sense that it is just the lining. I thought they looked a bit robust to only last 15 years. I don’t know how big a charcoal blast furnace can be, but Brazil produced a lot of steel using charcoal. I have no idea how big their blast furnaces were.

  21. I really dont know why the obession with costs and prices. I wonder if, last century, in this country whether our leaders decided on the basis of immediate costs and immediate prices whether it wwas worth rolling out rail to everywhere in the state of NSW, whether they costed in such minute detail whether it was worth rolling out telgraph and phone lines, whether they costed in such detail whether it was worth doing kerbing and guttering or the sewerage system or whether they had a “private market for it”.
    We need to do something about dirty energy, we could have vision if it wasnt for the short term focus on markets and costs and profits. We could just say “solar is the future and we need a grid capable of handling it and that needs minds in government not focused on short term costs and prices in tiddling little private markets.

    Vision isnt about costs. Vision is about investment and being entrepreneurial and willing to take risks.

    Its about building a railway when yoyu find out later some lines are uneconomival. Big deal. Close them down later (which governments did) but dont delay building or taking risks because you are obessed with current costs.
    Sounds like just a delay to me.

  22. At the rate we are going Australia looks likely to be one of the nations facing international sanctions/imposts on goods produced without a carbon price. When it does hit the fan all those effected nations looking for someone else (besides their own foot-dragging, scamming of mitigation and sequestration schemes etc) will see a hypocritical nation that’s been lining it’s pockets at the climate related expense of others – a perfect scapegoat.

    I’d like to think we are near a tipping point in acceptance of the existence and seriousness of the problem but the fossil fuel industry is very cashed up and has barely begun to shift from back door influence to openly fighting – with big and sudden economic hits just one of the ploys yet to be pulled out and used. And the electricity industry continues it’s ‘too hard, can’t be done, we can’t be taken in by mad green schemes’ approach to the problem. Whether they will continue to get behind closed doors assurances that reliability and growth of supply is far more important than a path to zero emissions and don’t worry, you’ll get exemptions or even grants to keep fiddling is a question but I can’t see the Coalition gov’t’s of NSW or Victoria making emissions reductions a priority. And I can’t see any Australian Gov’t taking action that would reduce the growth of fossil fuel exports.

  23. I looked up Brazillian steel making and it appears they use about 6 million tonnes of charcoal a year for steel making in furnaces about a sixth the size of a big blast furnace. They pay about one third as much for charcoal as they do for coke, so I guess that they feel that losing out on any economy of scale that may come with a larger blast furnace is worth it. They mostly use eucalyptus, which I guess makes sense because it can make charcoal that’s hard as a rock (but not as hard as a hard rock). Interestingly, they don’t pyrolyse it on site, which is wasteful of energy and results in it containing water by the time it reaches the furnace. (Brazil can be really humid.) I also found out that some types of charcoal can be as hard as coke, but I have no idea how easy they would be to produce on a large scale.

    Maybe with a carbon price it would be worthwhile to build smaller charcoal blast furnaces in Australia. Of course, that would require there be an actual demand for steel. If we are at peak steel demand at the moment, it’s possible not many new blast furnaces will be built in the future.

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