- 1 -

Accompanying Notes and Documentation 

on Development of DICE-2007 Model: 

Notes on DICE-2007.delta.v8 as of September 21, 2007

William Nordhaus 

Yale University 

October 5, 2007 

Preliminary 

I. Description of the Accompanying Notes ................................................................... - 3 -

II. Estimates of the backstop technology .................................................................. - 4 -

A. Description ............................................................................................................ - 4 -

B. Lab Notes .............................................................................................................. - 4 -

III. Estimates of the parameters of the abatement cost function ............................ - 7 -

a. General Description ............................................................................................. - 7 -

b. Lab notes ............................................................................................................... - 7 -

IV. Calibrating the preference function and discounting ........................................ - 9 -

a. General Description ............................................................................................. - 9 -

b. Lab Notes on the Calibration ............................................................................. - 9 -

c. Further notes on analysis of the preference function ................................... - 13 -

V. Lab notes for other GHG forcings ...................................................................... - 18 -

VI. Lab notes for the Climate and Carbon-Cycle Module ..................................... - 19 -

a. Basic philosophy ................................................................................................ - 19 -

b. MAGICC calibration: Methodology ............................................................... - 19 -

VII. Lab notes for Impacts and Damage Function ............................................... - 23 -

a. General background .......................................................................................... - 23 -

b. Lab notes ............................................................................................................. - 24 -

VIII. Lab notes for output and sigma estimates ..................................................... - 27 -

a. General philosophy ........................................................................................... - 27 -

b. Lab notes on second round estimates ............................................................. - 27 -

c. First round estimates ......................................................................................... - 31 -

IX. Lab notes on population ...................................................................................... - 32 -

a. Background ......................................................................................................... - 32 -

b. Lab notes ............................................................................................................. - 33 -

X. Lab notes specification of participation function ............................................. - 33 -

XI. Lab notes for calibration of length of no-control period ................................. - 34 -

XII. Lab notes for Importance Of Backstop Price Revision ...................................... - 35 -

XIII. Lab notes on the carbon cycle. .............................................................................. - 38 -
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XIV. Lab notes on adjustment of climate module.................................................. - 40 -

XV. Lab notes on changing time steps in climate module ........................................ - 51 -

XV. Miscellaneous Modeling Details ..................................................................... - 54 -

a. Initial Period Time Step .................................................................................... - 54 -

b. Kyoto Runs ......................................................................................................... - 54 -

XVI. Notes on uncertain parameter ......................................................................... - 55 -

a. The Monte Carlo estimates ............................................................................... - 55 -

b. Estimation of uncertain variables .................................................................... - 57 -
- 3 -

I. Description of the Accompanying Notes 

 These notes are intended to provide more detail on the derivation and 

calibration of the different components of the DICE-2007,delta.v8 model revision 

underlying the DICE model of September 5, 2007 underlying the DICE model. The 

current study, William Nordhaus, “The Challenge of Global Warming: Economic 

Models and Environmental Policy,”is dated September 11, 2007. These notes are 

necessarily incomplete because some of the steps involved detailed spreadsheets, 

but the major elements should be clear. 

 Where the entry is labeled “Lab notes,” these are drawn from 

contemporaneous notes on the procedures. They have sometimes been edited but 

not been rewritten for the present Accompanying Notes. These lab notes will help 

users understand the reasoning underlying the estimates. 

Cautionary note on the detailed estimates in these Accompanying Notes: The data 

in these notes are not published at the present stage because their quality is 

significantly less than the aggregated data in the published study. They are 

provided so that researchers can understand how the aggregate numbers were 

developed. The detailed estimates are often based on judgmental estimates, 

extrapolations from other estimates, or on less reliable source data. 
- 4 -

II. Estimates of the backstop technology

A. Description 

This was a major new element in DICE-2007. The mathematics of the 

backstop is straightforward. By solving the cost function for the derivative of cost 

w.r.t. emission, the marginal cost at miu = 1 is the backstop price. 

 The estimated level of the backstop technology was around $1170 per ton C in 

2005, declining by a logistic function at an initial rate of 0.5 percent per year to an 

asymptotic cost of $585 per ton C. These are the cost of 100 percent displacement of 

carbon emissions. 

B. Lab Notes 

For empirical estimates, we looked to three sources. First, the IPCC study of 

sequestration is shown in the following table. The results are as follows. The 

numbers are quite a bit lower than our backstop technologies. For electricity, 

including transportation and storage, the upper estimate is around $500 per ton C 

(for ocean storage). This is about half of our estimate. We can see the difference in 

the results from the following, which substitutes a backstop of $500 per ton C: 

CCS system components Cost of capture or injection (2002 $) Remarks

per ton CO2 (range) per ton C (range)

Capture from a coal- or gas-fired 15 75 55 275 Net costs of captured CO2, compared to the same plant

power plant without capture.

Capture from hydrogen and 5.0 55 18 201 Applies to high-purity sources requiring simple drying and

ammonia production or gas compression.

processing

Capture from other industrial sources 25 115 92 421 Range reflects use of a number of different technologies and

fuels.

Transportation 1.0 8.0 3.7 29 Per 250 km pipeline or shipping for mass flow rates of 5

(high end) to 40 (low end) MtCO2 yr-1.

Geological storage 0.5 8.0 1.8 29 Excluding potential revenues from EOR or ECBM.

Geological storage: monitoring and 0.1 0.3 0.4 1.1 This covers pre-injection, injection, and post-injection

verification monitoring, and depends on the regulatory requirements.

Ocean storage 5.0 30 18 110 Including offshore transportation of 100-500 km, excluding

monitoring and verification.

Mineral carbonation 50 100 183 366 Range for the best case

Source: IPCC Special Report: Carbon Dioxide Capture and Storage, Bert Metz, Ogunlade Davidson, Heleen de Coninck, Manuela Loos, and Leo Meyer, eds.,
- 5 -

 Second, we looked at estimates provided by Jae Edmonds (personal 

communication, shown next. For this, we took the Edmonds estimates of marginal 

cost at high miu (personal communication, January 9, 2007, included at the end). 

Estimating Jae’s function, we get the following. The estimated backstop for 2100 is 

also shown. The exponent is higher in this round (closer to DICE 1994). Based on 

this, we set exponent = 2.8 and 2100 backstop = 800. 

 We have adjusted the model with this introduction. We then define the base 

run as no controls for 150 years, then optimize after that. (Later, we adjusted that to 

250 years.) The carbon tax with Jae’s parameterization is 19.18 v. 19.86, this is 

slightly lower. This is probably because the initial cost is lower with this 

parameterization, and therefore the initial miu is lower. 

 Finally, we examined the preliminary compilation of estimates from the draft 

report for the FAR. A review of preliminary results from second draft FAR IPCC 

(SDFAR) indicates that the Edmonds estimates are relatively optimistic. The 

exponent is virtually identical the Edmonds runs (1.79+1), so the strong convexity 

seems to be realistic. The estimate of the cost of the backstop technology for 2050 is 

$1634 for the SDFAR and $621 for Edmonds for 2100. Note that few of the SDFAR 

scenarios go beyond 60 percent reduction, and the upper limit is $150 per t CO2 

($550 per t C). This further calibration suggests a great deal of uncertainty about the 

backstop technology cost and that the estimates here are consistent with the existing 

model estimates. The following graph shows the model and formulaic estimates for 

the two sources. MC is Edmonds, SDFAR is from the review. The results are in 

jae_calibration.wf1. 
- 6 -

0

400

800

1200

1600

2000

0.0 0.2 0.4 0.6 0.8 1.0

MIU

IPCCF_NEW

MCF_NEW

IPCC

MC

Model and functional MCemissions reductions (/tC)
- 7 -

III. Estimates of the parameters of the abatement cost function

a. General Description

 The estimates of the abatement cost, or miu, function takes the abatement cost 

estimates described above and estimates a log-linear function. Since the data are 

marginal cost data, we can integrate this to get the cost function. The important 

finding is the great convexity of the function, with a well-determined exponent near 

3 from all the data sets. 

b. Lab notes

In early runs, we had limited the increase in miu at .25 per decade. This gave 

poor results in many of the constrained runs, and it was therefore dropped in the 

d.8 version. The only major difference is in the highly constrained runs. 

 Given the problems with the base run, it seems best to have that a variant of 

the optimal with the miu at the Hotelling or at zero for the first few periods. 150 

years seems a reasonable guess. This also keeps cumulative C consumption well 

below the limit. 

obs MIU MC MCF MCFLN

1 0.88 635 602.55 662.50 

2 0.78 420 470.30 522.68 

3 0.52 221 204.47 235.59 

4 0.2 54 28.7185 36.028 

5 0.001 0.001 0.00053 0.0010 

6 1 783.508 851.71 

Dependent Variable: MC 

Method: Least Squares 

Date: 01/11/07 Time: 15:10 

Sample (adjusted): 1 5 

Included observations: 5 after adjustments 

Convergence achieved after 1 iteration 

MC=C(1)*MIU^C(2) 

 Coefficient Std. Error t-Statistic Prob. 
- 8 -

C(1) 783.5084 67.83342 11.55048 0.0014

C(2) 2.054282 0.344458 5.963815 0.0094

R-squared 0.983807 Mean dependent var 266.0002

Adjusted R-squared 0.978409 S.D. dependent var 263.4395

S.E. of regression 38.70945 Akaike info criterion 10.43922

Sum squared resid 4495.263 Schwarz criterion 10.28299

Log likelihood -24.09805 Durbin-Watson stat 2.675909

Dependent Variable: LOG(MC) 

Method: Least Squares 

Date: 01/11/07 Time: 15:10 

Sample (adjusted): 1 5 

Included observations: 5 after adjustments 

Variable Coefficient Std. Error t-Statistic Prob. 

C 6.747257 0.152544 44.23158 0.0000

LOG(MIU) 1.965246 0.047851 41.07000 0.0000

R-squared 0.998225 Mean dependent var 2.994654

Adjusted R-squared 0.997633 S.D. dependent var 5.613898

S.E. of regression 0.273139 Akaike info criterion 0.531504

Sum squared resid 0.223815 Schwarz criterion 0.375279

Log likelihood 0.671241 F-statistic 1686.745

Durbin-Watson stat 2.274837 Prob(F-statistic) 0.000032

 
- 9 -

IV. Calibrating the preference function and discounting

a. General Description 

 Calibrating the utility function is one of the major new approaches in DICE2007. The major assumption is that the two taste parameters need to be set in a way 

that is consistent with observed market returns. This raises deep questions that are 

discussed in the literature.

b. Lab Notes on the Calibration 

 For this calibration, we adjust the elasticity of the MU of consumption to fit 

the path of interest rates conditional on the time discount rate. We begin with a time 

path of the real return on capital. The calibration was that the first four period give 

an average rate of return of around 6.5 percent. The following shows a table with 

results: 

prstp alpha r1 r2 r3 r4 av ctax

0.0% 2.93 6.80% 6.60% 6.30% 6.00% 6.43% 24.1651

1.0% 2.50 6.80% 6.80% 6.70% 6.40% 6.68%

1.0% 2.00 6.80% 6.20% 5.70% 5.40% 6.03%

1.5% 2.50 6.80% 7.20% 7.20% 7.10% 7.08%

1.5% 2.00 6.800% 6.600% 6.300% 6.000% 6.425% 21.2003

1.0% 2.31 6.800% 6.600% 6.300% 6.000% 6.425% 21.8687

0.5% 2.62 6.800% 6.600% 6.300% 6.000% 6.425% 22.7858

0.1% 2.87 6.800% 6.600% 6.300% 6.000% 6.425% 23.8328

0.0% 2.93 6.800% 6.600% 6.300% 6.000% 6.425% 24.1651

The columns are the time discount rate, consumption elasticity, first four period real 

returns, average return, and carbon tax. (These were a little different from the final 

runs). The last five rows show that the initial carbon tax is slightly affected by the 

calibration, with zero discount rates giving about 15 percent higher initial carbon 

tax than our base run. There is a slight increase in the current C tax with lower 

PRSTP with given interest rates, presumably because of higher decline in r in the 

future. However, to a first approximation, it is the near-term interest rate that 

determines the current policy, not the discount rate. We can see the impact of the 

PRSTP on the current carbon tax as follows: 
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21

21.5

22

22.5

23

23.5

24

24.5

0.0% 0.5% 1.0% 1.5% 2.0%

 The following shows how real returns differ with different parameters (these 

have slightly different other parameters, but the pattern is similar to the latest 

version): 

Return on capital

5.5%

5.7%

5.9%

6.1%

6.3%

6.5%

6.7%

6.9%

7.1%

7.3%

7.5%

2005 2015 2025 2035 2045 2055 2065

r=.03 a=1 dr=.0025

r=.015 a=1.7 dr=0.005

r=.015 a=1.8 dr=.01

r=.015 a=1.6 dr=0.005
- 11 -

Return on capital

0%

1%

2%

3%

4%

5%

6%

7%

8%

2005 2015 2025 2035 2045 2055 2065

r=.03,a=1,dr r=.015,a=1

r=.035,a=1 r=.001,a=1

r=.015,a=2 r=.01,a=2.8
- 12 -

Carbon tax 2005 2015 2025 2035 2045 2055

r=.03 a=1 dr=.0025 19.3 29.7 43.6 61.4 83.5 110.4 

r=.015 a=1.7 dr=0.005 20.2 31.8 47.6 68.7 95.9 129.9 

r=.015 a=1.6 dr=0.005 22.6 35.5 53.0 76.1 105.6 142.1 

r=.015 a=1.3 dr=.005 32.7 50.9 74.7 104.9 142.3 187.5 

r=.02 a=1.5 dr=.01 23.6 37.8 57.8 85.0 120.2 164.7 

r=.015 a=1.8 dr=.01 18.2 28.5 43.0 62.3 87.3 118.9 

r=.005 a=2.1 19.6 30.6 45.4 65.0 89.9 120.9 

r=.001 a=2.3 19.8 31.0 46.4 67.0 93.4 126.6 

Savings rate

r=.03 a=1 dr=.0025 22.2% 22.0% 21.8% 21.7% 21.7% 21.7%

r=.015 a=1.7 dr=0.005 21.8% 21.7% 21.8% 21.9% 22.0% 22.2%

r=.015 a=1.6 dr=0.005 22.4% 22.3% 22.4% 22.4% 22.6% 22.7%

r=.015 a=1.3 dr=.005 24.6% 24.4% 24.2% 24.2% 24.3% 24.4%

r=.02 a=1.5 dr=.01 21.8% 22.1% 22.3% 22.6% 22.9% 23.2%

r=.015 a=1.8 dr=.01 21.2% 21.2% 21.2% 21.3% 21.5% 21.7%

r=.005 a=2.1 21.9% 21.7% 21.7% 21.7% 21.8% 22.0%

r=.001 a=2.3 21.8% 21.6% 21.6% 21.7% 21.9% 22.1%

Return on capital

r=.03 a=1 dr=.0025 6.91% 6.99% 6.79% 6.52% 6.23% 5.94%

r=.015 a=1.7 dr=0.005 6.91% 7.10% 6.92% 6.59% 6.22% 5.84%

r=.015 a=1.6 dr=0.005 6.91% 6.90% 6.63% 6.28% 5.90% 5.54%

r=.015 a=1.3 dr=.005 6.91% 6.26% 5.76% 5.35% 4.99% 4.67%

r=.02 a=1.5 dr=.01 6.91% 7.09% 6.81% 6.38% 5.90% 5.44%

r=.015 a=1.8 dr=.01 6.91% 7.29% 7.20% 6.91% 6.53% 6.14%

r=.005 a=2.1 6.91% 7.06% 6.90% 6.60% 6.26% 5.92%

r=.001 a=2.3 6.91% 7.10% 6.95% 6.65% 6.28% 5.91%

Emissions control rate

r=.03 a=1 dr=.0025 10.2% 13.2% 16.5% 20.2% 24.3% 28.8%

r=.015 a=1.7 dr=0.005 10.5% 13.7% 17.4% 21.6% 26.3% 31.5%

r=.015 a=1.6 dr=0.005 11.2% 14.5% 18.4% 22.8% 27.7% 33.1%

r=.015 a=1.3 dr=.005 13.7% 17.8% 22.3% 27.3% 32.7% 38.7%

r=.02 a=1.5 dr=.01 11.4% 15.1% 19.3% 24.3% 29.8% 36.0%

r=.015 a=1.8 dr=.01 9.9% 12.9% 16.4% 20.4% 25.0% 30.0%

r=.005 a=2.1 10.3% 13.4% 16.9% 20.9% 25.4% 30.3%

r=.001 a=2.3 10.4% 13.5% 17.1% 21.3% 25.9% 31.1%

Discount rate

r=.03 a=1 dr=.0025 -2.96% -2.88% -2.81% -2.74% -2.67%

r=.015 a=1.7 dr=0.005 -1.49% -1.42% -1.35% -1.28% -1.22%

r=.015 a=1.6 dr=0.005 -1.49% -1.42% -1.35% -1.28% -1.22%

r=.015 a=1.3 dr=.005 -1.49% -1.42% -1.35% -1.28% -1.22%

r=.02 a=1.5 dr=.01 -1.98% -1.79% -1.62% -1.47% -1.33%

r=.015 a=1.8 dr=.01 -1.49% -1.42% -1.35% -1.28% -1.22%

r=.005 a=2.1 -0.50% -0.50% -0.50% -0.50% -0.50%

r=.001 a=2.3 -0.10% -0.10% -0.10% -0.10% -0.10%
- 13 -

c. Further notes on analysis of the preference function

Calibration of the utility function is a difficult issue that involves both the discount 

rate and the elasticity of the marginal utility of consumption. The background from 

the earlier model is the following. The original DICE model publication had an 

extensive discussion of discounting issues. In Managing the Global Commons, I 

summarized as follows: 

 Beginning with the fundamentals, a discount rate is a pure number per unit time that allows 

us to convert values in the future into values today. The most common form of discount rate is the 

nominal or money interest rate, which is applied to future dollar values so that they can be converted 

into present values. When the nominal interest rate is corrected for inflation, we obtain the real 

interest rate, which represents the rate used to convert future constant-dollar values into today's 

dollars. 

 To understand the economics of real interest rates, economists often use the optimal-growth 

framework that underlies this study, namely, the Ramsey model. The Ramsey model derives the real 

interest rate, or the discount rate on goods, from a combination of time discounting, the elasticity of 

marginal utility, and growth in consumption. We begin with the fact that society shows different levels 

of concern about real incomes of different generations. We call this phenomenon time discounting. 

For example, if, as between equally well-off generations, society is indifferent between an increment 

of real income today and (1+ρ)

100 increments in 100 years, we say that the "pure rate of time 

preference" is ρ per year. Most economists and political philosophers find it hard to defend a pure 

rate of time preference above zero on ethical grounds (just as it may be hard to defend the fact that 

the United States devotes only 1/20th of one percent of its national income to overseas development 

assistance). On the other hand, it would be unrealistic to make decisions based on the premise that 

there is in fact no time preference given that many social decisions are in fact tilted in favor of present 

generations. 

 A second and more justifiable reason for favoring present consumption, or in the context of 

this study postponing GHG control costs, comes from the fact that different generations enjoy 

different levels of consumption. Industrial countries have witnessed more or less continual growth in 

living standard for more than a century; thus the per capita real consumption in the U.S. has growth 

by a factor of four over the twentieth century. Society might well feel that it is appropriate for later, 

richer generations to pay a larger fraction of GHG control costs, just as high-income people pay a 

larger fraction of their income in income taxes. We might then discount future costs if average living 

standards were improving -- a phenomenon we call growth discounting. 

 One must take into account both time and growth discounting to understand the 

phenomenon of goods discounting. Goods discounting refers to the real interest rate defined above 

and concerns the relative valuation of units of goods or consumption at different points in time; the 

real interest rate combines goods discounting and time discounting.

If we follow the approach set out in the Ramsey model developed in Chapter 2, we can derive the 

condition for an optimal path of investment and consumption as: 

(6.2) M{u'[c(t)]}/Mt = u'[c(t)] {MY(t)/MK(t) - δK - ρ} 

which states that the time rate of change of the marginal utility of consumption equals the marginal 

utility of consumption times the net marginal product of capital {MY(t)/MK(t) - δK} minus the pure rate of 
- 14 -

social time preference (ρ). Assuming riskless competitive markets, the net marginal product of capital 

will equal the instantaneous real interest rate [r(t)], so (6.2) reduces to: 

(6.3) r(t) = M{u'[c(t)]}/Mt / u'[c(t)] + ρ = αg(t) + ρ 

where g(t) is the growth rate of per capita consumption. In steady state, with stable population and a 

constant rate of growth of per capita consumption, (6.3) becomes: 

(6.4) r* = αg* + ρ 

where asterisks represent steady-state values. In equation (6.4), if ρ = .03 , g = .03, and α = 1 -- 

parameters which might reflect today's conditions -- the real interest rate on goods would be 6 

percent. However, if all economic growth ceases and the growth discount evaporates, the real 

interest rate would fall to 3 percent. It is crucial to understand that a high real interest rate can be 

generated either by a high rate of time preference or by a high elasticity of marginal utility in a society 

where living standards are improving.... 

 The methodology of this study is to anchor the parameters in actual observations wherever 

possible, and evidence on rates of return are reviewed in the next section. We have constructed the 

DICE model so that the parameters on the right-hand side of equation (6.3) are constrained to be 

consistent with observed market returns as represented by the left-hand side of (6.3). The base 

parameters in the model determine a real interest rate of around 6 percent per annum in the first few 

periods, with the rate declining slowly in the future as population and economic growth slows. In the 

next section, we discuss the appropriateness of this approach. (p. 122-25) 

In other words, the calibration relied primarily on an estimate of the rate of 

return on capital. Additionally, however, there were difficulties in using an 

alternative elasticity, as I explained in the review of the Stern Review: 

When the DICE model was constructed fifteen years ago, I assumed logarithmic utility for 

computational reasons – alternative utility functions would not converge numerically. This calibration 

led to a social discount rate of 3 percent per year, which was calibrated to match the growth of 

consumption, savings rates, and market rates of return on capital. Because of improvements in 

computers and software, we can now easily calibrate alterative utility functions. Experiments with the 

DICE-2006 model indicate that a social discount rate of 0.1 percent per year is consistent with a 

utility curvature parameter of 2.25. However, the Review’s social discount rate of 0.1 percent per year 

is inconsistent with its utility curvature assumption of 1. The Review’s calibration gives too low a rate 

of return and too high a savings rate compared to macroeconomic data, but the alternative calibration 

proposed here fits the macroeconomic data underlying the DICE model. (Stern Review, pp. 16-17) 

We must not make a virtue out of necessity. Sound policy should not rely 

upon assumptions that are unnecessarily philosophically justifiable. In this round of 

modeling, I therefore continue to rely on the fundamental empirical assumption that 

the rate of return must be calibrated to observed market levels. However, from the 

point of view of welfare economics, it seems preferable to change the assumption 

about the pure rate of social time preference to a level that is much closer to zero or 

generationally neutral. Many paradoxes and puzzles arise with discount rates near 

zero. We take a pure rate of time preference of 1 percent per year to be a reasonable 
- 15 -

assumption for the current round. We also compare with time preference rates of 

0.5 and 0.1 percent. 

 Lowering the PRSTP requires, however, that the elasticity of the marginal 

utility of consumption (EMUC) be changed. For this purpose, I select a EMUC that, 

along with the assumed PRSTP, yields a rate of return on capital that is stable in the 

first three periods indicating long-run growth equilibrium. This yields a rate of 

return of approximately 7 percent per year in the first three decades with the 

current data. This rate is consistent with safe returns on capital in advanced 

economies, although it may be lower than estimated returns in high-growth 

economies such as China (see particularly the rates of return on human capital 

estimated by Heckman, 

http://www.journals.uchicago.edu/EDCC/journal/issues/v51n4/510401/510401.

html for a very big example). 

 The following shows the combination of parameters and values of key 

variables in early periods. There is not much operational difference among the 

calibrations for rho of 0.1, 0.5 and 1.0 percent per year for the first half century. The 

first-period carbon taxes are all between $19 and $20 per ton C, which is virtually 

identical to the base period. So it is mainly a matter of aesthetics. The long-run 

temperature profiles are shown here. There is some difference in the profiles after a 

century or so because the savings rates are slightly higher for the low discount rate 

cases as growth slows. 

 Taking all this information, unless the economic data are revised 

significantly, the best parameterization appears to be a PRSTP of 1 percent along 

with a EMUC of 1.85. 
- 16 -

Mean temperature

0%

50%

100%

150%

200%

250%

300%

350%

2005

2025

2045

20652085

2105

2125

2145

21652185

r=.03,a=1,dr r=.03,a=1

r=.01,a=2 r=.001,a=1

r=.005,a=2 r=.01,a=1.9
- 17 -

Carbon tax 2005 2015 2025 2035 2045 2055

r=.03,a=1,dr 19.3 29.7 43.6 61.4 83.5 110.4 

r=.03,a=1 16.9 25.6 36.6 50.3 66.8 86.3 

r=.01,a=2 16.8 26.0 38.4 54.8 75.6 101.3 

r=.001,a=1 117.1 169.2 225.0 285.5 351.3 422.1 

r=.005,a=2 21.8 34.0 50.3 71.7 98.6 131.8 

r=.01,a=1.9 19.5 30.1 44.4 62.9 86.2 114.6 

r=.005,a=2.1 19.6 30.6 45.4 65.0 89.9 120.9 

r=.001,a=2.3 19.8 31.0 46.4 67.0 93.4 126.6 

Savings rate

r=.03,a=1,dr 22.2% 22.0% 21.8% 21.7% 21.7% 21.7%

r=.03,a=1 22.1% 21.7% 21.3% 21.0% 20.8% 20.6%

r=.01,a=2 21.2% 21.0% 20.9% 20.9% 21.0% 21.1%

r=.001,a=1 33.0% 31.7% 31.0% 30.5% 30.2% 30.0%

r=.005,a=2 22.5% 22.3% 22.3% 22.3% 22.4% 22.5%

r=.01,a=1.9 22.1% 21.9% 21.8% 21.7% 21.8% 21.8%

r=.005,a=2.1 21.9% 21.7% 21.7% 21.7% 21.8% 22.0%

r=.001,a=2.3 21.8% 21.6% 21.6% 21.7% 21.9% 22.1%

Return on capital

r=.03,a=1,dr 6.91% 6.99% 6.79% 6.52% 6.23% 5.94%

r=.03,a=1 6.91% 7.01% 6.91% 6.73% 6.55% 6.37%

r=.01,a=2 6.91% 7.28% 7.24% 7.01% 6.70% 6.38%

r=.001,a=1 6.89% 4.34% 3.36% 2.89% 2.61% 2.41%

r=.005,a=2 6.91% 6.87% 6.63% 6.30% 5.96% 5.62%

r=.01,a=1.9 6.91% 7.00% 6.83% 6.55% 6.23% 5.92%

r=.005,a=2.1 6.91% 7.06% 6.90% 6.60% 6.26% 5.92%

r=.001,a=2.3 6.91% 7.10% 6.95% 6.65% 6.28% 5.91%

Emissions control rate

r=.03,a=1,dr 10.2% 13.2% 16.5% 20.2% 24.3% 28.8%

r=.03,a=1 9.5% 12.1% 15.0% 18.1% 21.5% 25.1%

r=.01,a=2 9.5% 12.2% 15.4% 19.0% 23.0% 27.5%

r=.001,a=1 27.8% 34.6% 41.1% 47.6% 54.1% 60.6%

r=.005,a=2 10.9% 14.2% 17.9% 22.1% 26.7% 31.8%

r=.01,a=1.9 10.3% 13.3% 16.7% 20.5% 24.8% 29.4%

r=.005,a=2.1 10.3% 13.4% 16.9% 20.9% 25.4% 30.3%

r=.001,a=2.3 10.4% 13.5% 17.1% 21.3% 25.9% 31.1%
- 18 -

V. Lab notes for other GHG forcings

 This is clearly an area of major scientific uncertainty. We adopted the other 

forcings from the FAR: 

 The total non-CO2 forcings are way down from the TAR, possibly because of 

cloud albedo effects. We lower the overall level and keep the GISS change in the 

total over the 2000-2100 period the same. Note that CO2 forcings are more than 10 

times other forcings over the period. 
- 19 -

VI. Lab notes for the Climate and Carbon-Cycle Module

a. Basic philosophy 

 We decided to keep the same basic structure for the climate model, but we 

used the MAGICC program to tune the parameters. For this purpose, we did the 

tuning in two stages. First, we set the parameters of the carbon cycle to match the 

projections for concentrations in the MAGICC model. We did this for a number of 

runs, but concentrated on the A1F1 run. 

 We next took the forcings from the CO2 and exogenous forcings and 

compared the DICE climate model with the various MAGICC models. We then 

tuned the DICE parameters to align with the MAGICC model projections. More 

details are in the next section. 

b. MAGICC calibration: Methodology 

The method of the new recalibration is the following: 

1. We based the runs primarily off the A1F1 run in MAGICC. Using this, we 

looked at the carbon cycle, the forcings, and the temperature predictions for 

2.6 and 4.5 degrees for 2x CO2. We could not get the user option with 3 oC to 

run.

2. We then went step by step through the various steps using a Excel version of 

the geophysics module. It is somewhat messy, but the files are 

modelDICE_recalib_v4.4_a1f1.xls and modelDICE_recalib_v4.4_b1min.xls. 

3. First, we looked at the carbon cycle from Magicc. We reoptimized the 

coefficients, assuming that the system was in equilibrium in 1750 (this 

imposes linearity). This yielded the following comparison for the history and 

projection. These are imperfect, and particularly the historical is not a good 

match, perhaps because of our emissions numbers.
- 20 -

580

600

620

640

660

680

700

720

1760

1780

1800

1820

1840

1860

1880

1900

1920

1940

1960

dice

magicc

700

900

1100

1300

1500

1700

1900

2100

1980

2000

2020

2040

2060

2080

2100

dice

magicc
- 21 -

4. We then checked the forcings, which were reasonable: 

Forcings

0.000

1.000

2.000

3.000

4.000

5.000

6.000

7.000

8.000

9.000

10.000

1 2 3 4 5 6 7 8 9 10

DICE

magic 2.6

Magicc 4.5

5. We then checked the temperature projections in magicc and dice and 

recalibrated. This ended up with the following. DICE is a little low, but this 

reflects the slightly lower forcings just shown. This calibration seems 

reasonable for now. 
- 22 -

Temp 4.5

0.000

1.000

2.000

3.000

4.000

5.000

6.000

7.000

1 2 3 4 5 6 7 8 9 10

MAgicc

DICE

Temp 2.6

0.000

0.500

1.000

1.500

2.000

2.500

3.000

3.500

4.000

4.500

5.000

1 2 3 4 5 6 7 8 9 10

Magicc

DICE

6. We also looked at the B1min emissions scenario. These also were a close 

match for the carbon cycle, as shown below: 
- 23 -

700

800

900

1000

1100

1200

1300

1400

1500

1980

2000

2020

2040

2060

2080

2100

dice

magicc

 

VII. Lab notes for Impacts and Damage Function 

a. General background 

The impacts analysis follows the earlier framework in DICE/RICE-2000. We 

recalibrated the damage estimates. No major changes in the accounting format were 

introduced. However, the earlier version was somewhat mechanical. We 

particularly were concerned that the negative damages for some regions were 

continued at high T increases, and that the catastrophic was not carefully calibrated. 

 The changes were as follows: (1) We removed the risk aversion from the 

catastrophic because we would later introduce risk analysis. (2) We introduced 

Cline’s agricultural studies. These were generally more positive than our earlier 

estimates, and we took a judgmental average. (3) We recalibrated using the new 

estimates for GDP and population. (4) We took the output weighted as that is the 

measurement convention. 

 The major summary table is the following: 
- 24 -

Notes to 2007 revisions:

1. All revisions are in blue.

2. The basic sectoral breakdowns are the same.

3. There have been substantial changes in the regional definitions. These are indicated on page "Regions."

However, we have not carefully redefined regions to match new regions. Problems are primarily low and middle income.

These were poorly estimated in last round, so the estimates are probably about the same quality.

4. All numbers have been updated to use 2005 GDPs in PPP and 2005 US dollars per the data on the spreadsheet macro_011807.xls.

5. The major changes are in the catastrophic estimates. See that sheet for discussion.

6. We have parameterized by assuming that the damage function is to the 2th power. It is then calibrated to 2.5 degrees C. 

The estimated elasticity is very close to quadratic, so this is a reasonable specification.

7. Last round had negative damages at 0. This is doubtful given all results and has been constrained so that 

new linear term is zero. Specification is therefore simply quadratic. 

8. We use the output weights.

The new summary table is shown below (linked to the source).

Region Other vul- market Total non-catastrophic Catastrophic impact TOTAL TOTAL

Agriculture nerable mkt Coastal Health time use Settlements

[2.5 

degree]

[6 

degrees]

[2.5 

degree]

[6 

degrees]

[2.5 

degree] [6 degrees]

US 0.03% 0.00% 0.10% 0.02% -0.28% 0.10% -0.03% 1.34% 0.94% 4.00% 0.70% 5.34%

WE/Euro 0.03% 0.00% 0.46% 0.02% -0.43% 0.25% 0.33% 4.26% 1.09% 4.80% 3.65% 9.06%

OHI -0.05% -0.32% 0.09% 0.02% -0.35% 0.10% -0.50% 4.19% 1.11% 4.80% -2.79% 8.99%

Russia -0.82% -0.80% 0.05% 0.02% -0.75% 0.05% -2.25% 3.63% 1.12% 4.80% -16.36% 8.43%

EE/FSU 0.03% 0.00% 0.01% 0.02% -0.36% 0.10% -0.21% 0.76% 0.94% 4.00% -0.66% 4.76%

Japan 0.02% 0.00% 0.27% 0.02% -0.31% 0.25% 0.24% 4.04% 1.07% 4.80% 2.96% 8.84%

China 0.02% 0.32% 0.08% 0.09% -0.26% 0.05% 0.30% 3.92% 1.04% 4.00% 3.33% 7.92%

India 0.32% 0.29% 0.09% 0.40% 0.30% 0.10% 1.51% 6.94% 1.57% 6.00% 13.29% 12.94%

MidEast 0.35% 0.20% 0.04% 0.23% 0.24% 0.05% 1.12% 4.41% 0.96% 4.00% 9.61% 8.41%

SSA 0.67% 0.32% 0.02% 1.00% 0.25% 0.10% 2.35% 9.55% 1.78% 7.00% 20.00% 16.55%

LA 0.42% 0.28% 0.10% 0.32% -0.04% 0.10% 1.18% 5.16% 1.30% 5.20% 10.47% 10.36%

OthAsia 0.52% 0.21% 0.09% 0.32% -0.04% 0.10% 1.20% 5.00% 1.23% 5.00% 10.55% 10.00%

Global: 2105 weights: 2.5 oC

 Output weighted 0.61% 3.51% 1.16% 4.72% 1.77% 8.23%

 Population weighted 0.97% 5.60% 1.26% 5.05% 2.24% 10.65%

b. Lab notes 

 The major issue at this stage is that the database for impact studies continues 

to be relatively small. 

One new finding that informed the revision was the Gecon results, W 

Nordhaus, Proc Nat Acad Sci (US), Feb. 2006, which indicated that the damage 

function at the current climate was negative rather than positive. 

 Calibration is the following: 
- 25 -

6. To calibrate the new damage function, we set a power function to these estimates. (See below for calibration.) This yields the following:

Damage estimate Damage estimate at T*

D/Y= 0.0028388 *T^ 2.00000 1.77%

Calibrated at T = 2.5

D/Y= 0.0010262 *T^ 2.50000 9.05%

Calibrated at T = 6

D/Y= 0.0031112 *T^ 1.90000 1.77%

Calibrated at T = 2.5

Damage functions Orig New Alt1 Alt2 Alt3

Calibrated at 2.5 2.5 2.5 6

a1 -0.0045 0 0 0 0

a2 0.0035 0.0028388 0.0031112 0.0010262 

a3 2 2 1.9 2.5000000 

T Orig New Alt1 Alt2 Old

0 0.00% 0.00% 0.00% 0.00% 0.00%

1 -0.10% 0.28% 0.31% 0.10% 0.32%

2 0.50% 1.14% 1.16% 0.58% 1.27%

2.5 1.06% 1.77% 1.77% 1.01% 1.98%

3 1.80% 2.55% 2.51% 1.60% 2.85%

4 3.80% 4.54% 4.33% 3.28% 5.07%

5 6.50% 7.10% 6.62% 5.74% 7.93%

6 9.90% 10.22% 9.36% 9.05% 11.41%
- 26 -

Just for information, the following shows the damage functions at a function of T, 

which suggested IPCC TAR and FAR range: 

 

 July 24, 2007. There was a small correction in the tabulation in the earlier 

version, but this did not affect the results.

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

10%

11%

0 1 2 2.5 3 4 5 6

Mean T increase

Damage/Output

Orig New

Alt1 Alt2

IPCC
- 27 -

VIII. Lab notes for output and sigma estimates

a. General philosophy 

The general approach was to start with a database on major countries and 

then to aggregate up to regions. We then further aggregated to the DICE global 

total. The current data base is 71 major countries totaling about 96 percent of 

emissions, output, and slightly less of population. The database and calculations are 

contained in DATA SET 5.16.2007_WNadj_v5.xls. 

b. Lab notes on second round estimates 

 We have the problem of adjusting our RICE country totals for the world total. 

We decided to keep the global emissions total and the population total. The 

question of how to treat the remainder is an issue because the missing countries are 

low income. We have decided to calibrate to the world totals, as shown below: 

2004 2004 2004 2005 2005 2005

RICE Total ratio RICE Total ratio

Population 5456.27 6312.75 0.864325

GDP PPP 53928.25 57,262.56 0.941771

Emissions

7158.667 7375.654 0.970581 7457.55 7683.594 0.970581

26,247.97 27,043.57 0.970581

GDP/capita 9.883717 9.070933 1.089603

co2-gdp 0.138286 0.134182 1.03059
- 28 -

We adjusted China’s PPP GDP as it was unrealistically high and distorts the overall 

numbers. The following shows the calculations. We have rescaled the level to 60 

percent of the IMF PPP numbers. 

PPP MER US data US-C tradeRatio PPP MER Ratio

GDP, 2005 8817.00 2224.00 3.96 4497.82 2224.00 2.02

Exports G&S

0.368 818.43 762.00

Exports to US

0.47 384.66 244.00 243.00

GDP-exports 7998.57 1405.57 5.69 4497.82 1405.57 3.20

Ratio 0.51013

In earlier versions of PWT we have made estimates for the PRC based upon Bold is recalculated. 

quasi-benchmark estimates of individual researchers including Irving Kravis, Key assumption is the 3.2 PPP 

who, in the early 1980s, made PPP estimates for the year, 1975. Because of its ratio for non-exports.

large population, much interest has attached to real product estimates for China.

The present range of estimates for the PRC would put it as the 2nd or 3rd largest

economy in the world. The wide range of PPP estimates for China and the large

size of their difference from the exchange rate suggest that substantial

uncertainty is associated with these numbers. China participated in an ESCAP

benchmark study for 1993 limited to comparisons between Shanghai and Tokyo

and Hong Kong and Guangdong. In the end only the Hong Kong-Guangdong

comparison was completed. Currently there is a study of prices in 10 cities within

China and discussions have taken place about a possible link to the OECD

comparisons for 1996; however that has not yet been realized.

Draft: December, 2001

Alan Heston

Treatment of China in PWT 6.

Redid the emissions and output data. Retuned the population to the new IIASA, 

both by region and by total. The IIASA has more of a hump, which we have not 

tried to match, but the basic trend is the same. 

Then we redid the output per capita and emissions per output using the new data. 

Both have major uncertainties, which cannot be overcome with given data and 
- 29 -

methods. We have matched reasonably closely the new aggregates (see DATA SET 

5.16.2007_WNadj_v3.xls as well as the above file). Output is very slightly lower 

($284 tr in 2100 v. $300 in 2100). However, we did overestimate the sigma 

significantly, and the new 2095 sigma is 0.0681 v. 0.0877. This came about because 

we did not take into account the long run trend. 

The effect on the social cost of carbon in uncontrolled run is marginally changed, 

from 29.28 to 30.07. This paradoxical results arises because of consumption 

elasticity. The runs are pretty stable at this point, even though the trajectories 

change. 

Here is the comparison of old and new data and DICE: 

sigma

0.0600

0.0700

0.0800

0.0900

0.1000

0.1100

0.1200

0.1300

0.1400

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

DICE

New data

Old data

ajdusted new

ajdusted new

CO2

0

5

10

15

20

25

30

2005

2010

2015

2020

2025

2030

2035

2040

2045

2050

2055

2060

2065

2070

2075

2080

2085

2090

2095

2100

2105

CO2 (GtC)

DICE New data

Old data
- 30 -

Pop

6,000

6,500

7,000

7,500

8,000

8,500

9,000

9,500

2005

2015

2025

2035

2045

2055

2065

2075

2085

2095

DICE

new rescaled

Old data

IIASA

GWP

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

20052010201520202025203020352040204520502055206020652070207520802085209020952100

Output

DICE

New data

Old data

CO2-GWP

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

20052010201520202025203020352040204520502055206020652070207520802085209020952100

CO2-GDP

DICE

New data

Old data
- 31 -

c. First round estimates 

 We have revisited the output and emissions projections. These are contained 

in macro_011807.xls. The output data are taken from the IMF, World Development 

Report, although we use PPP aggregation. The TFP growth is adapted from 

Corderi’s estimates. The basic assumptions are the following with respect to the 

growth of per capita GDP: 

Asymptotic output ratio for US 10

Asymptotic ratio of region to US:

US 1.000

WE 1.000

OHI 1.000

Russia 0.500

EE/FSU 0.700

Japan 1.000

China 0.400

India 0.400

MidEast 0.400

SSA 0.333

LA 0.600

OthAsia 0.400

Rate of approach to long-run level (annual rate):

US 1.000%

WE 0.700%

OHI 0.800%

Russia 1.200%

EE/FSU 1.200%

Japan 0.500%

China 3.000%

India 2.500%

MidEast 1.000%

SSA 2.000%

LA 2.000%

OthAsia 1.500%
- 32 -

The estimates for emissions are taken from the EIA. We have then projected 

future emissions growth by region using the following logistics estimates: 

Ratio to 

US

Asymptotic 

ratio

Ratio 

asym/initi

al Initial growth

 0.03 1.00 0.196 -0.90% US

 0.03 1.00 0.191 -0.90% WE/Euro

 0.01 0.25 0.171 -0.90% OHI

 0.08 3.00 0.256 -1.00% Russia

 0.03 1.00 0.294 -1.00% EE/FSU

 0.01 0.50 0.151 -0.50% Japan

 0.04 1.50 0.312 -0.50% China

 0.04 1.30 0.379 -0.60% India

 0.08 3.00 0.361 -0.30% MidEast

 0.03 1.20 0.176 -0.50% SSA

 0.03 1.00 0.294 -0.50% LA

 0.03 1.00 0.158 -0.50% OthAsia

We then ran the model out to 2200 using the Hotelling estimates. We 

calibrated the sigmas and output levels of the new DICE model with the estimates 

from the aggregated regional RICE. This produced the following calibration, shown 

in sheet “calibrateDICE2005” of the above Excel sheet: 

 DICE MODEL RICE regional aggregated Ratio RICE to DICE 

year 2005 2105 2205 2005 2105 2205

output 61.063 291.445 587.23 61.036 246.571 496.974 1.000 0.846 0.846

pccon 7.432 27.531

savrate 0.22 0.22

indem 7.576 19.998 22.49743 7.619 20.282 26.709 1.006 1.014 1.187

sigma 0.125 0.072 0.0456 0.125 0.082 0.072 1.000 1.146 1.583

The emissions are reasonably close, with the sigma and output errors offsetting. 

Note that the Hotelling rents are important by 2200, so those estimates will not 

match. 

IX. Lab notes on population

a. Background

We got the new population estimates from Wolfgang Lutz. The totals 

are not all that different, but the distribution is very different from the earlier. 

We tuned the totals and the uncertainty estimates to the new IIASA 

projections. These are a little higher than our earlier. The main problem is the 

shape over the 21st century (see above).
- 33 -

b. Lab notes

 The projections are in pop_compare.xls 

***** 

X. Lab notes specification of participation function

We next add a model for participation. The modeling equations are the following: 

** Participation 

 PARTFRACTper1 Fraction of emissions under control regime 2005 /.1/ 

 PARTFRACTper21 Fraction of emissions under control regime 2205 /.5/ 

PARTFRACT(T) = PARTFRACTper1+ .05*(PARTFRACTper21-PARTFRACTper1)*(ORD(T)-1)$(ORD(T) LT 

22)+ (PARTFRACTper21-PARTFRACTper1)$(ORD(T) GE 22); 

ABATEEQ(T).. ABATECOST(T) =E= (PARTFRACT(T)**(1-

expcost2))*YGROSS(T)*(cost1(t)*(MIU(T)**EXPcost2)); 

*ABATEEQ(T).. ABATECOST(T) =E= YGROSS(T)*(cost1(t)*(MIU(T)**EXPcost2)); 

YY(T).. Y(T) =E= YGROSS(T)*((1-(PARTFRACT(T)**(1-

expcost2))*cost1(t)*(MIU(T)**EXPcost2)))/(1+a1*TATM(T)+ a2*TATM(T)**2); 

 The general theory is the following: We assume that only a fraction of 

countries participates in the climate protocols. In this example, the fraction begins at 

1 percent and rises to 50 percent in equal increments. 

 The impact is the following: Assume that the parameter π represents the 

participation rate. With this participation, if the control fraction of the participants 

in miu = 1, then the overall miu = π. The cost is the cost of that fraction, given by 

C = a*miu^b 

And the total cost is then the following fraction of total reductions: 

C(miu) = π a (miu/ π)^b = π^(1-b) a*miu^b 

So there is an inefficiency given by the parameter π^(1-b). Because our estimate of b 

is relatively large (2.8), this means that nonparticipation is relatively costly. 
- 34 -

XI. Lab notes for calibration of length of no-control period

Because including carbon constraints is a problem, another approach is to have 

miu=0 for a limited period, after which the carbon constraints kick in. This is not as 

efficient as Hotellling pricing, but will get most of the impact. The following shows 

the impact, with objective function and periods of zero miu. This is very close to the 

solution where have Hoteling miu for 20 periods and then optimal (for that). 

0

1

2

3

4

5

6

0 1 2 5 10 15 20 25

Case

Obj 

function

Diff from 

opt (trill)

Diff from 

opt (bill)

opt 1332.656 0 - 

1 period 1332.626 0.03 30 

2 periods 1332.578 0.078 78 

5 periods 1332.363 0.293 293 

10 periods 1331.498 1.158 1,158 

15 periods 1330.295 2.361 2,361 

20 periods 1329.12 3.536 3,536 

1327.677 4.979 4,979 

Optimal after 20 period 1329.167 3.489 3,489 
- 35 -

XII. Lab notes for Importance Of Backstop Price Revision

One of the major differences in the new DICE is the estimate of the backstop 

technology. This was not a central issue in the earlier DICE model as it was not a 

variable which affected the optimal policy (which is the central design strategy of 

the modeling). 

In the DICE version of the last round, the backstop (miu=1) rose rapidly over 

time. The following shows a comparison of estimates of the backstop in the current 

and the old version (these are approximate): 

. 

MC carbon reduction

miu=.5 Old New

2050 323 $ 300 $ 

2100 1,100 $ 275 $ 

2200 1,480 $ 230 $ 

miu=1

2050 730 $ 915 $ 

2100 1,250 $ 833 $ 

2200 4,000 $ 740 $ 

The current version is much more optimistic about future carbon technologies. It 

reflects both a modeling error in the last round and much technological work on the 

issue. 
- 36 -

 To test the importance of the backstop issue, we ran the new model with a 

rising rather than falling backstop price. The alternative model was 

D_011207d_opt.GMS, while the new model was D_011207c_opt.GMS. Some results 

are shown in test_base and opt_011207.xls. 

 The following shows a rough simulation where we use a rising backstop in 

the spirit of DICE-1998. The rising backstop means that carbon policies are much 

more expensive, so that constraining policies are more advantageous in the present 

circumstance. 

OLD

Run Objective functionDifference from base/150: Objective function

Base

150 year delay 1327.345 -2.465 0

50 year delay -3.757 1.932 1,329.28

Optimal 1330.16 -3.893 2.815

Limit to 2 degree C 1322.167 -9.877 -5.178

Limit to 2.5 degree C 1327.451 -6.521 0.106

Limit to 2X CO2 1329.504 -4.549 2.159

NEW

Run

Objective 

function

Difference 

from 

base/150: 

Objective 

function

Base

150 year delay 1329.810 0.000

50 year delay 1333.034 3.224

Optimal 1334.053 4.243

Limit to 2 degree C 1332.044 2.234

Limit to 2.5 degree C 1333.972 4.162

Limit to 2X CO2 1334.053 4.243

Note that the initial carbon taxes are not as heavily affected, however, indicating 

that the long-term outlook is different but current optimal policy is relatively little 

affected: 
- 37 -

NEW 

Carbon tax comparison 2005.00 2055.00 2105.00 2205.00

Base

150 year delay 0.03 0.36 2.73 407.92

50 year delay

Optimal 19.18 112.67 329.53 742.31

Limit to 2 degree C 28.75 271.68 677.79 745.95 

Limit to 2.5 degree C 19.88 124.45 449.92 743.52

Limit to 2X CO2 19.18 112.67 329.53 742.31

Hotelling 0.03 0.36 2.73 64.53

OLD 

Carbon tax comparison 2005.00 2055.00 2105.00 2205.00

Base

150 year delay 0.00 0.00 0.00 0.00

50 year delay

Optimal 19.79 120.2 380.2 1893.69

Limit to 2 degree C 36.94 405.81 1,578.71 3,779.72

Limit to 2.5 degree C 23.65 185.47 1048.94 3483.8

Limit to 2X CO2 20.32 130.6 506.4 2949.67

The new model has slightly lower C taxes in the short run (because the long run is 

more optimistic). There is little short-run difference, but an enormous long-run 

difference in the optimal policy because emissions reductions in the long-run are 

economical. The long-run are unrealistically high and would only be applicable in a 

world where technology is worse that stagnant. 

Note primarily that policy is now much more important because the long-run cost 

of reductions are much less expensive. 
- 38 -

XIII. Lab notes on the carbon cycle.

 We have recalibrated the carbon cycle to recent data on sources and sinks. We 

have used GISS data on atmospheric carbon concentrations. We assume that 110 

GtC of deforestation has occurred from 1850 to 2005, based on Marland. (IPCC SAR 

WGII Tables B.3 and B.4 in Sect B.3.3.1 and Gregg Marland, Oak Ridge National 

Laboratory.) The earlier carbon cycle model overpredicts atmospheric 

concentrations by a substantial margin given the revised data. The time pattern of 

land-use emissions is from Joos et al. 

 We then optimized the coefficients to minimize the squared errors over the 

last 20 years. The change for the 10-year model is the following for a baseline run 

with unlimited carbon resources.. The climate trajectory is lower (because the old 

model overpredicted atmospheric concentrations). The marginal retention rate is in 

line with other models. The impact on the short-term carbon tax is lower by 4 

percent. 

Coefficient Old New 

b11 0.66616 0.6608035

b12 0.33384 0.3391965

b21 0.27607 0.2764262

b22 0.60897 0.5968861

b23 0.11496 0.1266877

b32 0.00422 0.0038996

b33 0.99578 0.9961004

 
- 39 -

NEW

year 2005 2105 2205

temp 0.71 2.76 4.65 

conc 787.0 1,417.6 2,358.9 

ctax 16.9 289.9 1,019.0 

indem 75.2 221.9 349.8 

Cumulative emission 1,630.6 75.2 4,577.5 

Average atmos ret ratio 38.7% 34.3%

OLD 2005 2105 2205

temp 0.71 2.88 4.824

conc 787.00 1,481.30 2,485.61 

ctax 17.6 292.4 1,007.5 

indem 75.2 221.8 349.2 

Cumulative emission 1,630.3 75.2 4,573.4 

Average atmos ret ratio 42.6% 37.1%

Change (%) 2005 2105 2205

temp 0.00% 4.42% 3.65%

conc 0.00% 4.49% 5.37%

ctax 3.96% 0.88% -1.13%

indem 0.00% -0.05% -0.18%

Cumulative emission 0.00% -0.02% -0.09%

Average atmos ret ratio 10.12% 8.16%
- 40 -

XIV. Lab notes on adjustment of climate module

The following are the lab notes on the adjustment of the climate module: 

 

The basic structure of the climate model has not been changed since the last 

round. However, there has been much careful analysis of GCMs, particularly in the 

IPCC TAR, and we have incorporated that in the latest model. 

 One first change is to incorporate the latest view on forcings. For this, we took 

the total forcings from GISS. We then examined the model using both GISS and 

Hadley global temperature record. 

 We first looked at the average temperature (using the average of GISS and 

Hadley). We have estimated the model using both GISS and Hadley data. Using 

GISS total forcings, we get the following results. Constraining the adjustment 

parameter to .15, the key parameter is c(1), which is t2xco2. This is lower than the 

estimate from GCMs or our parameterization, which is 2.9. The parameter 

reasonably well determined in this tight specification. Estimates using well-mixed 

GHGs are not significant. 

 The continuing puzzle is why the time series estimate of the temperaturesensitivity coefficient is lower than the models. The estimate is not significantly 

below 3. Time will tell. 

Dependent Variable: TAV 

Method: Least Squares 

Date: 01/11/07 Time: 19:50 

Sample (adjusted): 1891 2006 

Included observations: 116 after adjustments 

Convergence achieved after 4 iterations 

TAV=TAV(-10)+.15*( TOTFORC_GISS-(4.1/C(1))*TAV(-10)-.44 

 *(TAV(-10)-TLOW(-10)) ) 

 Coefficient Std. Error t-Statistic Prob. 

C(1) 2.268675 0.445698 5.090166 0.0000
- 41 -

R-squared 0.712229 Mean dependent var -0.064793

Adjusted R-squared 0.712229 S.D. dependent var 0.243178

S.E. of regression 0.130451 Akaike info criterion -1.227048

Sum squared resid 1.957022 Schwarz criterion -1.203310

Log likelihood 72.16878 Durbin-Watson stat 0.906687

We also estimate with t2xco2=3 degrees, yielding: 

Dependent Variable: TAV 

Method: Least Squares 

Date: 01/11/07 Time: 19:47 

Sample (adjusted): 1891 2006 

Included observations: 116 after adjustments 

TAV=TAV(-10)+C(2)*( TOTFORC_GISS-1.366*TAV(-10)-.44*(TAV( 

 -10)-TLOW(-10)) ) 

 Coefficient Std. Error t-Statistic Prob. 

C(2) 0.155163 0.017754 8.739675 0.0000

R-squared 0.708578 Mean dependent var -0.064793

Adjusted R-squared 0.708578 S.D. dependent var 0.243178

S.E. of regression 0.131276 Akaike info criterion -1.214443

Sum squared resid 1.981846 Schwarz criterion -1.190705

Log likelihood 71.43771 Durbin-Watson stat 0.943899

Freeing both coefficients yields: 

Dependent Variable: TAV 

Method: Least Squares 

Date: 01/11/07 Time: 19:55 

Sample (adjusted): 1891 2006 

Included observations: 116 after adjustments 

Convergence achieved after 4 iterations 

TAV=TAV(-10)+C(2)*( TOTFORC_GISS-(4.1/C(1))*TAV(-10)-.44 
- 42 -

 *(TAV(-10)-TLOW(-10)) ) 

 Coefficient Std. Error t-Statistic Prob. 

C(2) 0.150269 0.018181 8.265170 0.0000

C(1) 2.271184 0.478918 4.742319 0.0000

R-squared 0.712229 Mean dependent var -0.064793

Adjusted R-squared 0.709705 S.D. dependent var 0.243178

S.E. of regression 0.131022 Akaike info criterion -1.209808

Sum squared resid 1.957018 Schwarz criterion -1.162333

Log likelihood 72.16889 Durbin-Watson stat 0.906993

ESTIMATING ALL THREE COEFFICIENTS 

Dependent Variable: TAV 

Method: Least Squares 

Date: 01/11/07 Time: 19:57 

Sample (adjusted): 1891 2006 

Included observations: 116 after adjustments 

Convergence achieved after 4 iterations 

TAV=TAV(-10)+C(2)*( TOTFORC_GISS-(4.1/C(1))*TAV(-10)-C(3) 

 *(TAV(-10)-TLOW(-10)) ) 

 Coefficient Std. Error t-Statistic Prob. 

C(2) 0.172415 0.022721 7.588394 0.0000

C(1) 2.721348 0.647524 4.202697 0.0001

C(3) 1.037278 0.330157 3.141768 0.0021

R-squared 0.718649 Mean dependent var -0.064793

Adjusted R-squared 0.713670 S.D. dependent var 0.243178

S.E. of regression 0.130124 Akaike info criterion -1.215130

Sum squared resid 1.913357 Schwarz criterion -1.143916

Log likelihood 73.47752 Durbin-Watson stat 0.913028
- 43 -

ONE YEAR ADJUSTMENT: 

Using one-year adjustment, we get the following, with even lower temp sensitivity: 

Dependent Variable: TAV 

Method: Least Squares 

Date: 01/11/07 Time: 20:06 

Sample (adjusted): 1882 2006 

Included observations: 125 after adjustments 

Convergence achieved after 6 iterations 

TAV=TAV(-1)+C(2)*( TOTFORC_GISS-(4.1/C(1))*TAV(-1)-.044 

 *(TAV(-1)-TLOW(-1)) ) 

 Coefficient Std. Error t-Statistic Prob. 

C(2) 0.050109 0.013030 3.845539 0.0002

C(1) 1.239215 0.314128 3.944936 0.0001

R-squared 0.824945 Mean dependent var -0.079852

Adjusted R-squared 0.823522 S.D. dependent var 0.241038

S.E. of regression 0.101258 Akaike info criterion -1.726415

Sum squared resid 1.261148 Schwarz criterion -1.681162

Log likelihood 109.9009 Durbin-Watson stat 2.356257

Dependent Variable: TAV-.07*.044*TLOW(-1) 

Method: Least Squares 

Date: 01/11/07 Time: 20:25 

Sample (adjusted): 1882 2006 

Included observations: 125 after adjustments 

Variable Coefficient Std. Error t-Statistic Prob. 

(1-.07*1.366-.07*0.044)*TAV(-1) 0.923321 0.046935 19.67229 0.0000

.07*TOTFORC_GISS 0.711085 0.184331 3.857664 0.0002

R-squared 0.825068 Mean dependent var -0.078965
- 44 -

Adjusted R-squared 0.823645 S.D. dependent var 0.241000

S.E. of regression 0.101207 Akaike info criterion -1.727423

Sum squared resid 1.259877 Schwarz criterion -1.682170

Log likelihood 109.9639 Durbin-Watson stat 2.359090

 
- 45 -

There are nasty AR and MA problems in the regressions however. For example: 

Dependent Variable: TAV-.07*.044*TLOW(-1) 

Method: Least Squares 

Date: 01/11/07 Time: 20:27 

Sample (adjusted): 1882 2006 

Included observations: 125 after adjustments 

Convergence achieved after 13 iterations 

Backcast: 1881 

Variable Coefficient Std. Error t-Statistic Prob. 

(1-.07*1.366-.07*0.044)*TAV(-1) 1.061581 0.018635 56.96689 0.0000

.07*TOTFORC_GISS 0.365049 0.077425 4.714897 0.0000

MA(1) -0.665235 0.073908 -9.000895 0.0000

R-squared 0.848090 Mean dependent var -0.078965

Adjusted R-squared 0.845600 S.D. dependent var 0.241000

S.E. of regression 0.094698 Akaike info criterion -1.852535

Sum squared resid 1.094066 Schwarz criterion -1.784656

Log likelihood 118.7835 Durbin-Watson stat 1.684536

Inverted MA Roots .67 

These might be introduced by the data smoothing, but it is not clear. This makes it difficult to rely 

upon the regression analysis for parameterization. (I’m not sure if this has turned up in the 

statistical analysis.) 
- 46 -

 We tried an alternative specifcation and adjusted it only to 0.15 as a compromise 

between models and data. The “NEWNEW” is with the adjusted and the “NEW” is 

with the carbon cycle adjustment only. 

NEW

year 2005 2105 2205

temp 0.71 2.758 4.654

conc 787 1417.594 2358.884

ctax 16.9 289.86 1019.02

indem 75.1822 221.9427 349.8124

Cum emis 75.1822 1630.577 4577.455

average ret ratio 0.386731 0.343397

NEWNEW

temp 0.71 2.616 4.592

conc 787 1481.758 2488.904

ctax 13.49 239.69 871.04

indem 75.1822 222.0999 350.0969

temp 1 0.948513 0.986678

conc 1 1.045263 1.055119

ctax 0.798225 0.826916 0.854782

indem 1 1.000708 1.000813

 As additional runs, we tested the model against the IPCC TAR model 

simulations. Using the GISS forcings, we estimate the following patterns of model 

temperature and forcings: 

1905 1915 1925 1935 1945 1955 1965 1975 1985 1995 2005 2000

Atmospheric temperature (degrees Celsius above 

preindustrial) 0.000 0.010 0.042 0.087 0.143 0.191 0.247 0.337 0.485 0.693 0.920 0.806

Total increase in radiative forcing since preindustrial 

(Watts per square meter) 0.042 0.160 0.278 0.408 0.476 0.603 0.855 1.277 1.816 2.289 2.539 2.414

Lower ocean temperature (degrees Celsius above 

preindustrial) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

This matches very closely Figure 8.15 in TAR. 
- 47 -

 In addition, the 1% co2 growth scenarios matches the TAR, but it has a oneperiod lag. That is, it gets to the 1.8 degree one period late because of the integration 

period. 
- 48 -
- 49 -
- 50 -

Figure 9.5: (a) The time evolution of the globally averaged temperature change relative to 

the years (1961 to 1990) of the DDC simulations (IS92a). G: greenhouse gas only (top), GS: 

greenhouse gas and sulphate aerosols (bottom). The observed temperature change (Jones, 

1994) is indicated by the black line. (Unit: °C). See Table 9.1 for more information on the 

individual models used here. (b) The time evolution of the globally averaged precipitation 

change relative to the years (1961 to 1990) of the DDC simulations. GHG: greenhouse gas 

only (top), GS: greenhouse gas and sulphate aerosols (bottom). (Unit: %). See Table 9.1 for 

more information on the individual models used here. 

http://www.grida.no/climate/ipcc_tar/wg1/fig9-5.htm
- 51 -

XV. Lab notes on changing time steps in climate module

 One of the early problems was the timing of the climate model. The lags in a 

10-year time step mean that we cannot reproduce the results of climate models 

accurately. We therefore changed the atmospheric equation to include 

contemporaneous rather than one-period lag. 

 The tests seemed to have no technical problems. Temperature was higher as 

follows: 

year 2005 2055 2105 2205

OLD OPT 

temp 0.71 1.671 2.534 2.703

ctax 16.4 93.24 277.93 631.61

 

NEW OPT 

temp 0.71 1.827 2.614 2.38

ctax 19.59 114.71 330.87 634.24

 This seems analytically superior. It needs, however, to have the climate 

model parameters adjusted. 

We begin with Twentieth-century temperature and precipitation trends in ensemble 

climate simulations including natural and anthropogenic forcing, Anthony J. 

Broccoli,1 Keith W. Dixon, Thomas L. Delworth, Thomas R. Knutson, and Ronald J. 

Stouffer, NOAA/Geophysical Fluid Dynamics Laboratory, Princeton University, 

Princeton, New Jersey, USA 

Fanrong Zeng, RSIS, Princeton, New Jersey, USA. Their report is for basically the 

same forcings as described: 

The time series of radiative forcing (relative to the preindustrial atmosphere) for the G ensemble (Figure 1a, 

red curve) features a slow rise from a value of _0.4 W m_2 in 1865 to _1.2 W m_2 in 1960. A more rapid 

rise follows thereafter, with the radiative forcing approaching 3 W m_2 by 1997.
- 52 -

In comparing with the Broccoli et al model, that model with the forcings leads to an 

increase of of about 1.05 oC over the period. This is consistent with the earlier 

calibration. 

We therefore recalibrated as follows: 

1. We changed the equilibrium T* to 3o C. 

2. We then changed the atmos/ocean coefficient to calibrate with both the 

doubling and the CFDL scenarios. The parameter of .215 yielded a good 

match, a little low for the CO2 doubling and a little high for the GFDL 

scenario: 

Difference from 1880-1920

1880-1920 2000.000

0.245 1.296 1.051

CO2 doubling from IPCC

1.784

Speed of adjustment

parameter for 

atmospheric temperature 0.215

Comparing to the results pre-adjustment, we get the following. The adjustments 

tend to make the temp path a little higher (because simultaneous), increasing the 

carbon tax by about 10 percent to $20. 
- 53 -

OPTIMAL

year 2005 2055 2105 2205

Carbon tax

 Pre carbon and climate adjusted 17.08 96.27 284.28 529.24

 Latest 19.86 116.52 337.44 633.93

Temp

 Pre carbon and climate adjusted 0.710 1.723 2.636 2.724

 Latest 0.710 1.834 2.643 2.420

Concentrations

 Pre carbon and climate adjusted 787.0 1055.2 1291.2 1094.1

 Latest 787.0 1020.2 1218.9 990.3

BASE

Temp

 Pre carbon and climate adjusted 0.710 1.762 2.891 4.777

 Latest 0.710 1.903 3.013 4.824

Concentrations

 Pre carbon and climate adjusted 787.0 1086.7 1486.9 2406.1

 Latest 787.0 1055.1 1422.9 2279.1
- 54 -

XV. Miscellaneous Modeling Details 

a. Initial Period Time Step 

 The initial time step is a complication because it include largely the historical 

period. However, we still want to include it. We assume that the period includes the 

first two years of the KP, which we estimate to have a global reduction rate of 3 

percent for the last two years, which is a decadal reduction rate of 0.3 percent. 

(macro_kyoto_032107.xls, sheet “Kyoto”, lines AZ:BA72). We raise this to 0.5 

percent to capture anticipatory effects and non-participants. Note that this is the 

same in all runs. It means, however, that first-period prices are non-informative, 

and second-period ones are the useful ones. 

b. Kyoto Runs 

The Kyoto runs have a computational problem because the control rates are 

so low. The Hotelling rents and carbon taxes cross relatively early, so the Kyoto 

“def” files need to ensure that the Hotelling rent takes over at the crossover point. 

The following shows an example for the Kyoto without US (from sheet 

“Economics”): 

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1 2 3 4 5 6 7 8 9 10 11 12 13 14

base25

kyoto

kyoto w US

strong kyoto

c.
- 55 -

XVI. Notes on uncertain parameter 

a. The Monte Carlo estimates 

 We have undertaken a preliminary Monte Carlo estimate to determine if a 

larger sample would produce higher variances or evidence of fat tails. For this 

purpose, we took the random 100 runs and estimated a “response surface” as a 

function of the random variables. The estimated equation was the following for a 

simple and for a polynomial function: 

Dependent Variable: MARGY 

Method: Least Squares 

Date: 07/18/07 Time: 16:41 

Sample (adjusted): 1 100 

Included observations: 100 after adjustments 

Variable Coefficient Std. Error t-Statistic Prob. 

A_CARBCYC 3298.726 285.3171 11.56161 0.0000

A_DAMCOEF 12678.03 5763.093 2.199865 0.0303

A_DECARB -1044.854 382.2601 -2.733360 0.0075

A_FOSSIL 0.008743 0.006374 1.371721 0.1735

A_PBACK 31.11352 16.26451 1.912970 0.0589

A_POP 0.045886 0.003963 11.57901 0.0000

A_T2XCO2 9.297585 6.819875 1.363307 0.1761

A_TFPG -796.7031 191.9248 -4.151120 0.0001

R-squared 0.043463 Mean dependent var 1177.421

Adjusted R-squared -0.029317 S.D. dependent var 74.31845

S.E. of regression 75.39998 Akaike info criterion 11.56011

Sum squared resid 523034.4 Schwarz criterion 11.76852

Log likelihood -570.0055 Durbin-Watson stat 2.028405

Dependent Variable: MARGY 

Method: Least Squares 

Date: 07/18/07 Time: 16:40 

Sample (adjusted): 1 100 

Included observations: 100 after adjustments 

Variable Coefficient Std. Error t-Statistic Prob. 

A_CARBCYC 9336.260 372.5297 25.06179 0.0000
- 56 -

A_DAMCOEF -1873.304 3373.255 -0.555340 0.5802

A_DECARB -566.9492 328.9890 -1.723308 0.0886

A_FOSSIL 0.020930 0.008258 2.534523 0.0132

A_PBACK 9.118749 11.91313 0.765437 0.4462

A_POP 0.041152 0.004502 9.140391 0.0000

A_T2XCO2 4.457665 5.388065 0.827322 0.4105

A_TFPG -1288.599 260.3073 -4.950298 0.0000

A_CARBCYC^2 -24526.64 1040.204 -23.57867 0.0000

A_DAMCOEF^2 127666.9 563876.8 0.226409 0.8214

A_DECARB^2 -2896.838 2211.598 -1.309840 0.1939

A_FOSSIL^2 -1.71E-06 6.72E-07 -2.552498 0.0125

A_PBACK^2 -3.611497 4.924094 -0.733434 0.4654

A_POP^2 -5.81E-07 2.62E-07 -2.216236 0.0294

A_T2XCO2^2 0.124151 0.882171 0.140733 0.8884

A_TFPG^2 2867.141 3049.376 0.940239 0.3499

A_TFPG^3 -4023.028 10442.88 -0.385241 0.7011

A_T2XCO2*A_TFPG -65.35669 29.68070 -2.201993 0.0305

R-squared 0.976919 Mean dependent var 1177.421

Adjusted R-squared 0.972134 S.D. dependent var 74.31845

S.E. of regression 12.40599 Akaike info criterion 8.035784

Sum squared resid 12620.50 Schwarz criterion 8.504715

Log likelihood -383.7892 Durbin-Watson stat 2.104314

 We then predicted the major variables using a sample of 10,000 realizations of the 

random variables. The following shows the major statistics for the SCC, temperature at 

2205, and the marginal utility of consumption: 

 MARGYF 

MARGYF_QU

AD SCC1F 

SCC1F_QUA

D TEMP21F 

TEMP21F_QU

AD 

 Mean 1178.760 1179.265 26.70280 26.57500 5.033002 4.967514 

 Median 1179.226 1180.559 26.71182 27.84185 5.043746 5.004213 

 Maximum 1447.814 1449.938 85.37925 85.83153 12.45189 11.38508 

 Minimum 899.5732 874.3436 -43.92759 -59.45535 -2.033592 -2.724209 

 Std. Dev. 73.51238 74.66149 17.13403 17.42335 1.823906 1.876188 

 Skewness -0.014920 -0.049511 -0.016256 -0.488811 -0.017054 -0.163506 

 Kurtosis 2.970143 3.067010 3.005680 3.633053 3.069014 3.160268 

 

 Jarque-Bera 0.742441 5.956634 0.453865 565.2080 2.469309 55.25962 

 Probability 0.689892 0.050878 0.796974 0.000000 0.290935 0.000000 

 Sum 11787603 11792648 267028.0 265750.0 50330.02 49675.14 

 Sum Sq. Dev. 54035302 55737812 2935455. 3035428. 33263.01 35197.29 

 Observations 10000 10000 10000 10000 10000 10000 
- 57 -

 Most of the specifications are close to normal, although two show high JarqueBera statistics, indicating significant kurtosis. There is no indication of an increasing 

variance as the sample size increases. The standard deviations are virtually identical 

for a sample of 100, 1000, and 10000. A test of the coefficient of the Pareto 

distribution for the largest 1000 values of SCC gives a coefficient of -9, and similar 

results are found for the other variables. The tail has finite variance for a coefficient 

> -2. 

b. Estimation of uncertain variables 

September 4, 2007 

 We recalculated the parameter calculation for the temperature-sensitivity 

coefficient and refined the description, adding the new description in the text. 

July 18, 2007 

 We have incorporated the revisions of delta.8 runs in the uncertainty analysis. 

We first created the new file (in C:\Major\Resclim\DICE\DICE_delta8_unc\Base_newparams). We then verified that it gave the 

same results as the last runs. Note that we widened the upper and lower limits in 

the new runs. 

 The next step was to perform the “sigma tests.” These examine the impact of 

single variables on the outcome, with the variable going from -6sigma to +6sigma of 

its value. 

June 7, 2007 

 The estimates of the uncertain parameters were drawn from the existing 

literature as well as on the author’s judgment. The starting point was the earlier 

uncertainty analysis in the study, Managing the Global Commons, Chapter 6. The top 

uncertain variables in terms of contributing to overall uncertainty were the 

following: 

• decline of rate of growth of population (asymptotic population) 

• decline of rate of growth of productivity (growth in total factor productivity) 

• pure rate of social time preference (same) 

• rate of decline in the CO2-output ratio (same) 

• intercept of the damage function (same) 

• climate-GHG sensitivity coefficient (same) 
- 58 -

• intercept of the mitigation-cost function (price of backstop technology) 

• rate of atmospheric retention of CO2 (transfer coefficient) 

 We have added in parentheses the corresponding parameter in DICE-2007 

model or “same” where the concept is the same. For the current runs, we omit the 

pure rate of time preference because that is a preference rather than a technological 

parameter. We have added an additional parameter, the resources of fossil fuels, 

which was not in the original DICE model. 

 There are multiple sources for the estimates of the uncertainty of the 

parameters. One comparison is from DICE 1994, and the following shows the 

coefficients of variation of the parameters which are easily compared. Additionally, 

a number of alternative estimates have been used for comparison. We have found 

no useful comparative studies of the uncertainty of damages. In general, the 

calibration is to the coefficient of variation in the different sources. 

 It should be emphasized that the uncertainty estimates are themselves very 

uncertain. Few of them have been calibrated to actual historical data, and some, like 

future productivity growth, are inherently unknowable. 

Variable Definition Units Mean

Standard 

deviation

Coefficient 

of variation, 

DICE-2007

Coefficient 

of 

variation, 

DICE-1994

Alternative

estimates

ga0

Rate of growth of total 

factor productivity Per year 0.9200 0.4000 0.435 0.680 0.19-0.54 (b, g)

gsigma Rate of decarbonization Per year 0.7000 0.2000 0.286 0.650 0.250 (b)

t2xco2

Equilibrium temperature 

sensitivity coefficient

oC per CO2 

doubling 3.0000 1.1100 0.370 0.354 0.372 (i)

a2

Damage parameter 

(intercept of damage 

equation)

Fraction of 

global 

output 0.00278 0.00130 0.467 0.846

pback

Price of backstop 

technology

$ per ton C 

replaced 468 1,170 0.400 0.551 0.300 (b)

popasym

Asymtotic global 

population Millions 1,892 8,600 0.220 0.224 (d)

b12

Transfer coefficient in 

carbon cycle Per decade 0.017 0.189 0.090 0.090 (e, f)

fosslim

Total resources of fossil 

fuels

Billions tons 

carbon 1,200 6,000 0.200 0.445 (g)
- 59 -

Source 

(a) Uncertainty Analysis of Climate Change and Policy Response, M. Webster, C. Forest, J. Reilly, M. Babiker, D. 

Kicklighter, M. Mayer, R. Prinn, M. Sarofim, A. Sokolov, P. Stone and C. Wang, Report No. 95, December 2002, 

(b) Mort D. Webster, Uncertainty in Future Carbon Emissions: A Preliminary Exploration, Massachusetts 

Institute of Technology,Joint Program on the Science and Policy of Global Change, Report #30, November 1997

(d) IIASA stochastic population projections, personal communication, Wolfgang Lutz, May 2007.

(e) The uncertainty in DICE-2007 is set on the basis of the uncertainty generated by the MAGICC simulations, 

which have a standard deviation of approximately 40 ppm in 2100 for the A1F1 scenario.

(f) From the cross-sectional uncertainties of the model comparison in Riahi, Keywan, Arnulf Gruebler, and 

Nebojsa Nakicenovic [2007]. “Scenarios of long-term socio-economic and environmental development under 

climate stabilization,” Technological Forecasting & Social Change, forthcoming. 

(g) Nordhaus and Yohe, "Future Paths of Energy and Carbon Dioxide Emissions," NRC, 1982.

(h) H. Rogner, Ann. Rev. Energy and Env. 22, p. 249. But this is the range with and without speculative resources,

which is interpreted as 4 sigma.

(i) IPCC, Fourth Assessment Report, Science, Chapter 10. For this estimate, we take the average of results for the 

18 AOGCMs and the estimates constrained by climatology, which yield a 5-95 percent range of 3.63 oC. This is