INTERNATIONAL ENVIRONMENTAL AGREEMENTS: POLITICS, LAW AND ECONOMICS
RESEARCH ARTICLE
Bargaining and climate change
Supplementary online appendix: Damage function methodology
The simplified methodology used here for calculating damages from greenhouse
gas emissions (treated as CO2-equivalents in warming potentials) is based on the model
in Fankhauser and Kverndokk (1996). Although highly stylized and simplified, several
steps are nonetheless required to represent the mechanisms that translate emissions into
damages in this model.
A damage function is assumed in this form:
Dit ( It ) = ki ( 1 + hi ) t * ( It / Λ )
(1)
Annual damage is assumed to grow proportionally with income, where the rate of
economic growth in country i is denoted by hi ; ki denotes the damage for country I
caused by a hypothetical temperature increase of Λ˚C in period 0.1 It is the temperature
rise between period 0 and the current period.
1
See Fankhauser and Kverndokk (1996: 93). The analysis here differs most importantly from that of
Fankhauser and Kverndokk in that we are interested in the marginal damage resulting from the
temperature increase caused by each year's emissions after 2015 (the assumed baseline year for a
conjectural international agreement), rather than the total damage caused by the cumulative rise in
Since ki is a point estimate of damage at Λ˚C, the latter temperature level is
essentially the baseline for assessing potential damages from climate change. For many
studies, including that of Fankhauser and Kverndokk, this baseline is 2.5˚C above preindustrial levels, corresponding to a doubling of CO2 concentrations in the atmosphere.
For the purposes of the present study, two of the estimated k-values from Fankhauser and
Kverndokk are employed: kUS = 104.94, and kChina = 36.34. These two values correspond
to the low- and high-damage scenarios for each country, respectively, and have been
chosen in order to reflect better the higher-than-expected degree of economic
convergence that has occurred between the two countries since 1993.
Next, we assume a rate of economic growth (in present value terms, after
discounting, if any), hi, that is 1% for the US and 3% for China, which is consistent with
the consensus projections in the literature.2 Finally, for calculating It, which we derive
from Tt, or the cumulative industrial-era rise in temperature, we use a range of estimates
in accordance with the system of equations used in Fankhauser and Kverndokk(1996,
89ff.) to represent a stylized climate system.
First, the stock constraint relates previous-period emissions, Et-1, to the
concentration of CO2 in the atmosphere:
temperatures (and greenhouse gas concentrations) above pre-industrial levels – in other words, the
variable of interest here is It, instead of Tt. The other minor modification that has been made to the
damage function here is its simplification to exclude convexity, which does not meaningfully affect the
results.
2
See Manne and Richels (1991); Fankhauser and Kverndokk (1996: 94); Blanford, Richels, and Rutherford
(2009, Supplementary Data, "pessimistic scenario": 2).
Qt = λ Eit-1 +( 1 – σ ) Qt-1
(2)
Qt is the atmospheric greenhouse gas concentration at time t (in CO2-equivalents). The
parameter λ translates emissions units (GtC) into the concentration units (ppm), and is
estimated here to be 0.47. σ represents the constant rate at which is assumed to dissipate,
through absorption into the oceans and other natural processes; the assumed value here is
0.005, corresponding with an atmospheric lifetime of 200 years for CO2 .
Next, the temperature constraint represents how temperatures adjust over time in
response to a change in atmospheric CO2 concentrations:
Tt = α ∙ω ln ( Qt / QP ) + ( 1 - α ) Tt-1
(3)
QP is the pre-industrial atmospheric concentration of CO2. α is the delay parameter which
determines the speed of adjustment, 0 ≤α ≤ 1; here it is assumed that α = 0.10,
corresponding to a lag of 50 years. ω is a climate sensitivity parameter, assumed here to
have a value of 3.61.3
Finally, to obtain values for temperature increases attributable only to post-2015
emissions, or It, the rise in temperature until period 0 must be backed out:
It = α ∙ω ( ln ( Qt / QP ) - ln ( Q0 / QP ) ) + ( 1 - α ) Tt-1
(4)
3
The relationship between atmospheric concentrations and the equilibrium change in global mean
temperature (i.e. the change which will occur after full adjustment) is usually approximated by a simpler
logarithmic function, Tt * = ω ln ( Qt / QP ); equation (3) therefore represents a partial equilibrium
during a process of gradual adjustment characterized by thermal inertia.
Q0 for the purposes of this analysis is assumed to be 385 parts per million (ppm), while
QP is assumed to be 280 ppm. Lastly, the values for Eit and hence Qt are taken from the
projected estimates of the International Energy Agency's World Energy Outlook, 2009
and 2010.