Carbon Sensitivity and Externalities

OK, let’s not drive science off the page altogether!


It’s widely agreed that a crucial quantity is what is called the sensitivity of the climate system.

This word is used in a technical sense. It is rather like an efficiency or even a commodity cost. A sensitivity is a ratio of an output to an input. It is meaningful only in conditions where the relationship of the output to the input is roughly linear and doesn’t wander about too much.

So if limes are four for a dollar, they are eight for two dollars: that is a linear relationship between cost and benefit. But if little keychains with a 3-ball on them are 5 dollars a piece or 50 dollars for a hundred, the cost per item is not constant and a linear estimate fails. In order for sensitivity to be a useful quantity, it must be roughly constant over the conditions of interest. No bulk deals are allowed.

Among the human forcings of the atmosphere, we are most concerned with carbon dioxide because, unlike the others, it is both large and approximately cumulative. So we would like to know how much change a unit of CO2 emission effects, and express this as a sensitivity.


Now basically, change is trouble when it comes to climate. It’s important to point this out.

(One of the redder herrings often proposed to duck out of responsibility for the problem is to say that it is arrogant to claim that there is an ideal temperature for the planet. Leaving aside that we don’t want to freeze or boil the oceans, perhaps not. But we have built our civilization during a time of stability. We expect rain on our farmlands, and a certain sea level at our ports. We expect frosts in some places, and not in others. If these things change, we find ourselves maladapted. We may be able to adapt over a broader range than pine forests do, but these changes are costly nonetheless).

For various reasons, some good and some not, we have chosen to focus on global average temperature as the measure of change. I will be coming back to this point later on: global temperature is not the only measure of climate change that we could use and in some ways it is misleading. It certainly is overstressed in discussions of climate change, though there is no question that it alone would lead directly to various disruptions. Indeed, despite the efforts of people who prefer precision in language, the whole issue is often and apparently irrevocably called “global warming”.

Note that it is conceivable that we could have a great deal of anthropogenic climate change and very little temperature change. As a thought experiment, consider the possibility that as the result of some human activity, some places get much colder while others get much warmer. This would be bad for everybody, but would amount to a zero change in average temperature. In the end, the sensitivity we are looking for is the ratio of units of utility lost per unit of CO2 emitted; the temperature change is something of a crude proxy for that utility.

For the rest of this essay, we’ll accept that, though it’s something worth revisiting. For present purposes, we accept that carbon in the atmosphere is the cause and global warming is the effect.

What’s more, as Mark Lynas examines in his book Six Degrees, the utility is not proportional to the temperature change, but increases sharply. This is obvious: a small change is harmless, but an increase of no more than 85° C would be required to effectively destroy the world altogether since the oceans would boil off. So the idea that there is a specific dollar impact to each unit of emission is a very bad one: it depends crucially on what the total concentration peak is. (That is, the commonly held idea that the sensible purpose of a carbon tax is to capture the externality is wrong!)

Now it turns out that for very large changes in atmospheric CO2, the relationship between concentration and temperature is not linear at all, but in fact is best approximated in different ways at different concentrations. But for the range concentrations and conditions that we expect to find ourselves in, the relationship is logarithmic. In a rare moment of successful clarity, someone long ago found a way to express this so that it is easily understood by people who don’t often think about logarithms. We express the sensitivity in degrees of temperature per doubling of CO2 concentrations. There are several streams of evidence that have converged on this quantity being in the range between 2.5° C and 3° C per doubling.

And much of the argument, from those who would like to put policy on hold until “the science is in”, centers around this number. “Lukewarmers” have convinced themselves that this number is much too high, that it is perhaps closer to 0.5 or 0.75.

That is, they ask us to bet the entire future of the planet on all of the following

  • What is at best a minority view of the competent science
  • The belief that warming is the only significant issue in anthropogenic climate forcing
  • The belief that the utility cost of a temperature change remains small for some significant range of emissions


I think it’s clear that uncertainty counts against the delayers’ arguments. Not only are they asking us to believe that the sensitivity is a quarter of what we think, and that the other things break our way, but to believe that we are sure of all of that. Because, even if you take the minority view as the central estimate, it’s not that far from the consensus. If the uncertainty is fourfold, that brings the risk right back into the IPCC range.

So when the same person tells you the following two things (as many of them do)

  • the uncertainty in sensitivity is large
  • policy can be delayed and may not be necessary

they are not holding a consistent position; you can be sure that they are not people who have thought through the situation carefully. You simply cannot accept all three propositions in a coherent way, except if you simply don’t care whether our descendants have a very nasty time of it.

That’s not all, but it will do for today. I will follow up soon with an explanation of why even a low sensitivity with low uncertainty in itself is not sufficient for reassurance, even on the global temperature front. There’s at least one more key risk that often gets short shrift. Tune in for our next episode.


An original point I have made here is this: there cannot be a proper Pigouvian tax on carbon.


A Pigovian tax (also spelled Pigouvian tax) is a tax levied on a market activity that generates negative externalities. The tax is intended to correct the market outcome. In the presence of negative externalities, the social cost of a market activity is not covered by the private cost of the activity. In such a case, the market outcome is not efficient and may lead to over-consumption of the product. A Pigovian tax equal to the negative externality is thought to correct the market outcome back to efficiency.

But the externality of carbon fuels is weird, because the impact per unit of carbon goes up the more other units are consumed. You cannot actually set the Pigouvian tax high enough to cover the externalities because there is no limit on the externalities. Sadly, it is possible to imagine emissions literally destroying everything humans value anywhere.

The tax on fossil-carbon-based emissions must be high enough that use of fossil-carbon-based fuels is economically foolish and thus very rare. That is not the same as balancing the externality, because impact goes up rapidly with future emissions.

Image: Oil wells in Kilgore, Goliad Co. Gregg Co., TX, 1943. John Vachon, 1914-1975, believed public domain via


  1. "But the externality of carbon fuels is weird, because the impact per unit of carbon goes up the more other units are consumed."

    How does that differ from any other pollution tax? If demand for widgets goes up, and production is increased accordingly, widget-based pollution goes up. There's also nothing to stop you from changing the tax rate based on circumstances - so, for instance, in the case of carbon, finding some way to tie it to an overall carbon output level. Not saying that's easy, but it's not ruled out, AFAIK.

    Interestingly, just been having exactly this kind of argument about uncertainty. There's that claim: it's all too uncertain, but it's certain that the costs of dealing with it are low.

  2. As a commendation of the proofreading services of "Every Word Proper", here's the proof of the article as published yesterday. I stubbornly refuse to put punctuation inside of quotation marks, which may cause EWP to abandon me at some point.

    For now, let me suggest that if you see grammatical errors on this site, it's most likely because they haven't been proofed yet, but if you see commas and periods outside quotation marks, it is because I am stubborn.


    strike Now | add At this time,
    strike comma after dioxide | add comma after because | add comma after others | add comma after So
    How about subscripting 2 in CO2?

    is to say that it is arrogant to claim that there is an
    Leaving aside that we don't want to
    strike commas after farmlands and after places
    but these changes are costly nonetheless).

    would amount to zero change in average temperature.

    for is the unit of

    essay, we’ll | purposes, we

    italicize Six Degrees | add degree sign ° after 85 | boil off. | flat wrong or is, flatly, wrong

    be rid of Now | substitute ° for degrees

    close space between argument and comma | in," | that it is

    In other words, they ask | competent science, | forcing, and

    delayers' | strike See, cap Not | unitalicize that

    reassurance, even


    there is no limit on the externalities | Sadly, it is possible to imagine

    fossil-carbon-based (2x)

  3. "But the externality of carbon fuels is weird, because the impact per unit of carbon goes up the more other units are consumed"

    It seems like we need to raise the price of carbon as the impact rises. Ross McKitrick has suggested a mechanism for this - tying the price of carbon to the mean temperature.

    This was discussed on Only In It For The Gold: The In It crowd was generally not in favour of the plan, although the following changes made it somewhat more palatable:

    1) Since equatorial mean temperature is not a good proxy for warming due to CO2 we should instead use arctic mean temp which has the advantage of being very sensitive to CO2.

    2) We should use the GISS base period 1951-1980 as our starting point instead of today's temperature. This would mean that today's price would already account for the warming since the 70's.

    Some additional concerns were:
    1) Temperature is not the sum total of the problem, it is merely a proxy. (but it is one that is easy to measure
    and is already widely associated with the impact of CO2)

    2) The instantaneous temperature is less than the equilibrium temperature. (The only downside here is that temperature will continue to rise even after we stop releasing CO2.)

    3) There are fluctuations that can mask the trend. (Using arctic mean temperature largely eliminates the fluctuations. Adding a 10 year mean would reduce this further still)

    Michael's statement that the impact per unit of carbon increases for each unit spent is another point in favour of McKitrick's plan. Perhaps it is time to reevaluate it?

  4. The problem with the temperature link is more fundamental: you want a carbon pricing mechanism that limits the amount of carbon being put into the atmosphere to an agreed-upon figure that we know stands a chance of keeping temperatures within safe limits. That figure needs to have been worked out. Any current temperature tells you / tells the market price next-to-nothing about future impacts.

    It's equivalent to pressing the brake pedal harder the closer you get to crashing into the back of a truck: it might work, but you're better off keeping beyond the known stopping distance for your speed, because equally - if your goal is `don't crash into the truck' - it may well not work.

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.