OBSERVATIONS ON THE TIME PROFILE FOR THE SOCIAL COST OF CARBON.
John Rhys. April 2008.
The arguments in this piece have been updated in the more recent April 2016 posting
The ideas in this piece were subsequently developed for an OIES Working Paper and an OIES Energy Comment on UK Treasury Guidance, later discussed with DECC and HM Treasury officials.
In December 2007 DEFRA published recommendations on the social cost (SCC) and shadow price (SPC) of carbon dioxide (CO2) emissions to inform policy and investment appraisals across government. The importance of the subject, in a policy setting, is that it provides at least a starting point, and some necessary if not sufficient conditions, for “joined up” government that embraces climate change policy. The purpose of the exercise, whether in investment or policy appraisal, is to enable comparisons to be made of streams of CO2 emissions in future years as between project or policy alternatives, and to estimate their net benefits or costs.
The subject of this note is the time profile of carbon. Quite apart from its role in the technical requirements of net present value calculations for appraisal purposes, the presentation of that profile, and in particular the answer to the question of whether current emissions do more or less damage than future emissions, conveys an important message for the urgency of policy on climate change.
It is generally assumed, correctly, that CO2 emissions are essentially cumulative or have such a long life in the atmosphere that they can be regarded as very nearly so for most practical purposes. Logically, this implies a time profile for social costs, measured in terms of their current net present value (ie as at 2008), in which significantly higher values should attach to reductions in current emissions than to reductions in emissions in (say) ten years time. This simply reflects the fact that this year’s emissions are still contributing an increment to CO2 concentration in ten years time, but have had an additional ten years of impact. The cumulative effect of CO2, without re-absorption, implies higher social costs should attach to current emissions.
A first reading of the DEFRA paper might, however, suggest that the opposite is true. The paper proposes a time profile for the social cost of carbon which rises over time, by 2.0% per annum, and seeks to explain this as follows:
• As time goes on, the damage comes closer, and is discounted less heavily; so its present value rises, increasing the SCC.
• The concentration of carbon in the atmosphere is rising towards its long-run stabilisation level, and expected climate-change damages accelerate with higher concentrations. An extra unit of carbon will do more damage at the margin the later it is emitted because, even with a plausible concentration goal, it will be in the atmosphere while concentrations are higher and higher concentrations mean larger climate-change impacts at the margin (as damage is a function of the cumulated stock); this too increases the SCC. Additionally, as incomes grow, so the monetary value of damage is likely to grow, owing to an associated higher willingness to pay to avoid warming damage.
The first explanation is clear. DEFRA is presenting a time profile for the SCC in which damage of emissions in each year is presented as a net present value (NPV) of all future damages discounted to the year of the emission, so that the comparison of damage, as between emissions now and in the future, cannot be deduced directly from the profile. Given that DEFRA appears to use a 3.5% per annum discount rate in this context, one would expect this factor alone to result in a 3.5% per annum rate of increase in the SCC, and so this more than explains the profile growth, taken on a year of emission basis, of 2.0 % per annum. If the DEFRA series were discounted back at 3.5% per annum to a common base, it would show, as we should expect, more damage from earlier than from later CO2 emissions.
However the second explanation makes an assertion about the physical nature of CO2 concentration which is at odds with the hypothesis outlined above, that the essential link for climate change is to cumulative concentrations. It would imply that emissions now cause less damage than those in the future. One example of a perverse conclusion for policy, that would arise from acceptance of this argument, is the following:
Question. Suppose we have a large store containing thousands of tonnes of CO2, held under pressure in large corroding metal vessels. Technical experts have advised me that there is no means of permanently sealing the vessels, but that I can at some expense treat the seals of the vessels in a way that will prolong their expected life from 5 years to 20 years. What should I do, given an objective of minimising adverse climate impact?Answer. According to an analysis of social and environmental costs which tells us that later emissions are more damaging, the answer is obvious. We should be prepared to spend money not on reinforcing the vessels, but on breaking them open immediately, since the social cost will be significantly higher in 5 years time and even more so in 20 years time.
This is clearly absurd if we regard CO2 emissions as purely cumulative.
Fortunately the second DEFRA explanation above is incorrect, and is contradicted by DEFRA’s own research. The question of how to compare the options of emissions now and emissions in the future, on a comparable basis, clearly matters, not least for the urgency that should be attached to early action. It is worth checking first, that current understanding of the climate science does indeed support the notion that CO2 emissions are essentially cumulative; and second, whether the modelling of economic costs confirms, as logically it should, the hypothesis of higher costs associated with current emissions.
Interpreting the Science on Re-absorption
The re-absorption rate for carbon in the atmosphere is a crucial measure in determining whether emissions are cumulative in their effect, a little less than wholly cumulative, or “more than cumulative” with positive feedback. To be precise, it is not the average rate of re-absorption of the stock of CO2 concentrated in the atmosphere that is most relevant in this context, but the incremental re-absorption rate for the additional units of CO2 emission. This requires interpretation of the available science.
Climate science is too complex, and many of its individual parameters too uncertain, to allow an unqualified statement of any simple mathematical relationship between CO2 re-absorption rates and concentration levels. Actual re-absorption depends on a wide range of factors, and will change over time with the state of other climate and climate system variables. For example one possible feature of terrestrial and oceanic carbon sinks might be that they become saturated, but their cumulative absorption limit is likely to depend on a variety of climatic and other factors, and will not necessarily be driven directly by CO2 concentration levels.
Pursuit of a fully estimated mathematical function, remaining broadly unchanged over time, and differentiable with respect to concentration levels, may therefore be neither realistic nor meaningful. However it is possible to make sensible inferences, in context, from a number of sources, including Stern, other studies referenced in Stern, and IPCC publications, about current best estimates of re-absorption.
Stern, in Chapter 8 of his review, describes a current stock in the atmosphere of around 3000 GtCO2, and annual man made emissions of 35 GtCO2, of which about half, or about 17 GtCO2 per annum are currently removed. This indicates an annual re-absorption rate, expressed as an average in relation to the stock, of about 0.56 % per annum. This is presented as a guide to the future, but only with the proviso that there are no feedbacks into the carbon cycle, such as those that might be associated with a maximum level of cumulative re-absorption. Other sources, such as the IPCC, indicate similar estimates of the “historic” rate of re-absorption.
Stern, drawing on the climate literature, warns very clearly that carbon feedback can have a dramatic effect, quoting a recent study showing that, if feedbacks between the climate and carbon cycle are included in a climate model, the resulting weakening of natural carbon absorption means that the cumulative emissions at stabilisation are dramatically reduced. The “with feedback” calculation allows emissions of 1600 GtCO2, of which 1050 GtCO2 are removed in the course of a century; this equates to about 10.5 GtCO2 or 0.3 % per annum.
The inclusion of the effect of feedbacks to the carbon cycle should be expected to have an even more pronounced effect on the measurement of the incremental impact. Climate model projections incorporating carbon cycle feedbacks imply that net absorption is falling in absolute terms. This makes it much harder to assert that an incremental tonne of carbon this year results in less than an additional tonne in five years time. Indeed it suggests that the process may well be purely cumulative, possibly as carbon sinks approach capacity limits, or that there may be a positive feedback.
Overall this reading of current scientific understanding on the subject suggests that it is hard, even on the most optimistic reading of the data, to support re-absorption rate estimates much higher than 0.5 % per annum, and more likely figures are much lower at around 0.2 %. The decline in re-absorption associated with feedback into the carbon cycle suggests that, incrementally, re-absorption rates might even be zero or negative. Re-absorption is therefore likely to provide no more than a minor adjustment to the assumption of a cumulative effect. It is easy to show mathematically that the rate of growth in SCC cannot exceed the rate of “decay” of CO2 in the atmosphere.
Results from modelling of economic impacts
Economic models of the impact of climate change may be subject to qualification, but the work commissioned by DEFRA , and which provides the basis for their cost estimates, serves to provide further confirmation. This has the time profile of SCC discounted to a common base year 2000, which shows a pattern consistent with our hypothesis, of a falling time profile of around 1% per annum. This confirms that any re-absorption effects, as currently understood, do not have a significant effect on the time profile. Describing the SCC in the year of emission shows annual increases of about 2% per annum, the difference being entirely attributable to the 3.5% per annum discount factor.
It is interesting that the same result does not hold for all greenhouse gases. Methane for example has a much shorter life and is not therefore cumulative. Later emissions may indeed be more damaging in the case of methane, and this is confirmed by model results.
The presentation of a time profile for the social cost of carbon deserves care, and the DEFRA explanation of its own time profile, in terms which contradict the cumulative CO2 hypothesis, indicates the scope for confusion. The true position can be stated unequivocally: emissions now are more damaging than those in ten years time. The apparent paradox is that the social cost of emissions is falling at the same time as the perceived damages are rising; the paradox is however apparent, but not real.
This should reinforce the hand of all who argue for a post-Stern presumption in favour of early action. It remains true (as DEFRA suggest) that climate problems are likely to grow in severity over time, together with public perception of their economic and environmental impact. This may well be reflected in continuing willingness to increase the attention we pay to CO2, and to attach higher social costs to it. However that re-assessment would necessarily include, implicitly at least, a retrospective increase in the cost of emissions in past years and hence a higher “regret” for the opportunities foregone.
A higher value to early emission reduction, at least for CO2, and whatever values we might attach to climate impacts in the future, enhances the case for acting with urgency to reduce the economic impact of climate change. This is further emphasised by the fact that many of the possible short-term opportunities for reducing emissions do not depend on changes in capital stock, and represent comparatively low cost abatement opportunities, or “low hanging” fruit.
 Economics Group, DEFRA. The Social Cost of Carbon and the Shadow Price of Carbon. What They Are And How To Use Them In Economic Appraisal In The UK. December 2007.
 Nicholas Stern. The Economics of Climate Change. The Stern Review. Cabinet Office - HM Treasury 2007
 The calculation is performed with respect to the total stock of carbon.
 http://www.ipcc.ch/pdf/assessment-report/ar4/wg1/ar4-wg1-faqs.pdf; FAQ 10.3, for example; and numerous other IPCC sources
 Jones, C.D., P.M. Cox and C. Huntingford (2006): 'Impact of climate-carbon feedbacks on
emissions scenarios to achieve stabilisation', in Avoiding Dangerous Climate Change,
Schellnhuber et al. (eds.), Cambridge: Cambridge University Press.
 A brief mathematical demonstration of a point that may be fairly obvious to the reader intuitively:
Let Dn be the economic damage over all future years attributed to an incremental unit that exists in the atmosphere in year n only, discounted back to year 0. Let Zn be total impact over all years, ie an infinite series, of a one-off emission in year n of volume K, discounted back to year 0. [NB In this formulation impacts are always discounted back to the base year.]
Let the reduction factor V, assumed to be constant, be the proportion of incremental CO2 not re-absorbed after a year, so that the proportion remaining after n years is Vn . [NB V =1 if no re-absorption.]
Now let us compare Z0 and Z1 , the total effects of the same amount of emission K but a year apart.
Z0 = K x D0 + V x K x D1 + V2 x K x D2 + … + Vn x K x Dn + …
Z1 = K x D1 + V x K x D2 …. + Vn-1 x K x Dn + ……
So Z0 = K x D0 + V x [K x D1 + V x K x D2 …. + Vn-1 x K x Dn + ..... ]
= [K x D0 ] + [V x Z1]
Hence unless D0 is zero or negative, impact of emission in year 0 is greater than the impact of emission in year 1 times the reduction factor; ie if reduction is 1% pa, then SCC cannot increase by more than 1% pa. Setting V=1 equates to no re-absorption, and the simple form of the cumulative hypothesis on SCC, ie a decreasing profile.
 Appendix 3. Research on behalf of DEFRA carried out by AEA Technology et al (2005). Authors Watkiss et al. Available at http://www.defra.gov.uk/environment/climatechange/research/carboncost/pdf/aeat-scc-report.pdf
 These orders of magnitude suggest that the shape of the time profile for the social cost of carbon may have as much influence over the outcome of an appraisal as the more familiar debate over the appropriate choice of discount rate.