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Supplementary Information:
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On our rapidly shrinking capacity to comply with the planetary
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boundaries on climate change
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Jean-Denis Mathias1,1, John M. Anderies2,3,4, Marco A. Janssen2,4
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1
IRSTEA, UR LISC, 9 avenue des landais, 63170 Aubiere, France. 2School of Sustainability, Arizona
State University, United States. 3School of Human Evolution and Social Change, Arizona State
University, United States. 4Center for Behavior, Institutions and the Environment, Arizona State
University, United States. Correspondence and requests for materials should be addressed to JDM (email: [email protected])
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2010 scenario
2025 scenario
2035 scenario
Supplementary Figure S1: Impact of delayed policies on relative GWP Qd according to the GWP of the
baseline scenario, CO2 concentration and emission control rate in 2035 (blue points), 2055 (green
points), 2075 (cyan points) and 2100 (red points) from the 2010 initial states and for the 2010- 2025- and
2035- scenarios. The GWP is impacted in the first years in main cases but then the absence of climate
damages will benefit to GWP until 2100. The slope between the relative difference of GWP and the CO 2
concentration shows the trade-off between climate damages and abatement costs: the first years,
implementing emission reduction policies has a negative impact on the GWP until 2075
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Relative difference of net ouput Qd (%)
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2100
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2010 states: (381 ppm, 0%)
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0
2075

-2
max

=1
2055
-4

max

=3
-6

350
360
370
2035
max
=2

380
390
400
410
420
430
440
CO concentration (ppm)
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Supplementary Figure S2: The impact of an increase of the acting capacity  
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  =  max , t (optimistic scenario). Having  max = 2 (or  max = 3 ) instead of 1 (we double or triple
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the acting capacity) enables a decrease of 20ppm in 2100. Indeed, even with
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against 420ppm in 2025 for
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max
. We consider
 max = 3 , the climate lag
makes difficult the decrease of CO2 concentration in the atmosphere below 350ppm. The maximum
max
peak of CO2 is as important as the emission control rate: the peak reaches 440ppm in 2035 for   = 1
 max = 2 . Besides, as expected, the GWP is more impacted in the first years
than in the case of staying below 550ppm and the main economic benefits of climate mitigation are
delayed after 2075
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DICE model (2013R)
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Main equations of the DICE (2013) model are recalled here. E (t ) is the CO2 emissions of carbon per
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year, composed of industrial emissions Eind (t ) and emissions from land-use changes Eland (t ) :
E (t ) = Eind (t )  Eland (t )
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Emissions from land-use changes Eland (t ) writes:
Eland (t ) = Eland (t  1)(1   E
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land
Eind (t ) =  (t )(1   (t )) A(t ) K (t ) L(t )1
 (t ) =  (t  1)[1  g (t )]
g (t ) =
g (t  1)
1  
(7)
(see next section "adaptive policy"). A(t ) constitutes the total factor productivity:
A(t ) = A(t 1)[1  g A (t )]
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(6)
 (t ) corresponds to the emissions-reduction rate. The latter will constitute one of our controls
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(5)
with:
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(4)
 (t ) is the estimate of the baseline carbon intensity:
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)
Industrial emissions Eind (t ) writes:
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(3)
(8)
with:
g A (t ) =
g A (t  1)
1 A
(9)
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The function L (t ) represents the population as well as the labor inputs. We take the trajectory of
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L (t ) with the UN medium trajectory (11 213 millions in 2100). The capital stock writes as:
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K (t ) = I (t  1)Q(t  1)  (1   k ) K (t  1)
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(10)
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I (t  1) represent the reinvestment rate from the net economic output Q(t  1) . I (t  1) represents a
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control (see section "‘adaptive policy"’). The global net output writes:
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Q(t ) =
2
(t ) = 1TAT  2TAT
2
(t ) = 1 (t ) (t )
(13)
The abatment cost function coefficient 1 (t ) writes:
B p (t ) (t )
1 (t ) =
1000 2
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(12)
TAT represents the mean surface temperature. The function (t ) represents the abatement costs:
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(11)
The function (t ) represents climate damages:
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1  (t )
A(t ) K (t ) L(t )1
1  (t )
(14)
with B p (t ) the backstop price (1000$ per tons of CO2):
B p (t ) = B p (t  1)(1   B )
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(15)
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The atmospheric temperature TAT (t ) (degree celsius above 1900) and the deep temperature of oceans
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TLO (t ) interact as follows:
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TAT (t ) = TAT (t  1)  1[ F (t )   2TAT (t  1)  3 (TAT (t  1)  TLO (t  1))]
(16)
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TLO (t ) = TLO (t  1)   4 (TAT (t  1)  TLO (t  1))
(17)
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Variations in temperature depend on the change in total radiative forcings F (t ) of greenhouse gases
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since 1750:
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F (t ) = log 2 (
M AT (t )
)  Fext (t )
M AT (1750)
Fext (t ) represents exogenous forcings:
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(18)
Fext (t ) = f 2000  0.01( f 2100  f 2000 )(t  2000)
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(19)
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where f 2000 are 2000 forcings, non-CO2 GHG and influences and f 2100 are expected 2100 forcings,
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non-CO2 GHG and influences. Finally, the variables M AT (t ), M UP (t ) and M LO (t ) represent carbon
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in the atmosphere, carbon in a quickly mixing reservoir in the upper oceans and the biosphere, and
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carbon in the deep oceans. Carbon flows in both directions between adjacent reservoirs:
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M AT (t ) = E (t )  11M AT (t  1)  21M UP (t  1)
(20)
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M UP (t ) = 12 M AT (t  1)  22 M UP (t  1)  32 M LO (t  1)
(21)
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M LO (t ) = 23M UP (t  1)  33M LO (t  1)
(22)
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Values of the parameters are described in Table S1. We consider M LO (t ) as a forcing equation:
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simulations show that whatever the policy, it does not influence the carbon stock in deeper oceans at
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time horizon 2100.
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Sobol indices
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Sobol indices were calculated from the range analysis defined above. Considering a model Y=f(X),
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first-order Si of variable X i writes as follows:
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Si =
Var E Y X i 
(26)
Var (Y )
and the second-order indices Si , j of variables X i and X j are:
S ij =



Var E Y X i , X j  Var EY X i   Var E Y X j
Var (Y )

(27)
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In our case, the variables X i are represented by the 6 state variables (carbon stocks and temperatures
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in the ocean and the atmosphere as well as the capital stock and the emission control rate) and the
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variable Y represents the viability of the system. The most important variables are the emission
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control rate  (t ) , the carbon stocks in the atmosphere M AT and in the upper oceans M UP . Then, then
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the viable sets of Figure 2 have been plotted through these variables and the values of other
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dimensions correspond to the ones found in the baseline scenario.
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Variable
E (t )
Eland (t )
Type
Output
Name
Total CO2 emissions
Unit
Giga tons of CO2 per year
Value
E (2010) = 31.4
Forcing equation
CO2 emissions due to land-use
changes
Decrease of CO2 emissions due to
land-use changes
Industrial CO2 emissions
Giga tons of CO2 per year
Eland (2010) = 1.54
per 5 year
0.2
Giga tons of CO2 per year
E (2010) = 29.860
tons of CO2 per 1000$
0.3
 (2010) = 0.489
% of change per year
g (2015) = -1
 Eland (t )
Parameter
Eind (t )
Output
 (t )
Parameter
Forcing equation
g (t )
Forcing equation
Elasticity
Estimate of baseline carbon
intensity
rate change of carbon intensity

 (t )
  (t )
Parameter
parameter of g
% of change per 5 years
-0.1
State variable
Emissions-reduction rate
% of change per 5 years
 (2010) = 3.9
Control
Variation in emissions-reduction
rate  (t )
-
A(t )
Forcing equation
Total factor productivity
-
A(2010) = 3.8
g A (t )
Forcing equation
rate change of total factor
productivity
% of change per 5 year
g A (2015) = 7.9
A
Parameter
% of change per 5 years
0.6
L (t )
K (t )
Forcing equation
World population size
million
State variable
World capital stock
trillion of 2005 $
L(2010) = 6838
K (2010) = 135
k
Parameter
% of change per year
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Q(t )
Output
trillion of 2005 $
-
I (t )
(t )
1
Control
Depreciation rate of the world
capital stock
Output net of damages and
abatement
Reinvestment rate
-
[0.2366-0.2592]
Output
Climate damages
trillion of 2005 $
-
Parameter
-
0
2
Parameter
-
0.0027
C (t )
(t )
2
Output
Parameter of the climate
damages function
Parameter of the climate
damages function
Cost of climate damages
trillion of 2005 $
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Output
Abatement costs
trillion of 2005 $
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Parameter
-
2.8
1 (t )
Forcing equation
-
1 (2010) = 0.060
B p (t )
Forcing equation
Parameter of the abatement
costs
Abatement cost function
coefficient
Backstop price
1000 $ per ton of C02 $
B p (2010) = 344
C P (t )
Output
Carbon price
per ton of C02 $
C p (2010) = 1
B
Parameter
per year
0.025
F (t )
Output
Depreciation of the backstop
price
Change in total radiative forcings
Watts per square meter
F (2010) = 1.824

Parameter of
gA
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[0-  
max
]

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Fext (t )
Parameter
Forcing equation
f1
Parameter
f2
Parameter
TAT (t )
TLO (t )
of greenhouse gases since 1750
Forcings at CO2 doubling
Exogenous forcings
Watts per square meter
Watts per square meter
3.8
F (2010) = 0.008
Watts per square meter
-0.06
Watts per square meter
0.62
State variable
2000 forcings, non-CO2 GHG and
influences
2100 forcings, non-CO2 GHG and
influences
Atmospheric temperature
degree Celsius (above 1900)
State variable
Deep oceans temperature
degree Celsius (above 1900)
TAT (2010) = 0.83
TLO (2010) = 0.0068
1
Parameter
-
0.104
2
Parameter
-
1.1875
3
Parameter
-
0.088
4
Parameter
-
0.025
M AT (t )
M UP (t )
M LO (t )
State variable
Parameter of the change in
atmospheric temperature
Parameter of the change in
atmospheric temperature
Parameter of the change in
atmospheric temperature
Parameter of the change in deep
oceans temperature
Atmospheric carbon stock
Giga tons of carbon
State variable
Upper ocean carbon stock
Giga tons of carbon
State variable
Lower ocean carbon stock
Giga tons of carbon
M AT (2010) = 818.985
M AT (2010) = 1527
M AT (2010) = 10010
11
Parameter
per 5 years
0.912
21
Parameter
per 5 years
0.03833
12
Parameter
per 5 years
0.088
22
Parameter
per 5 years
0.95917
32
Parameter
per 5 years
0.00034
23
Parameter
per 5 years
0.0025
33
Parameter
Parameter of the atmospheric
carbon stock
Parameter of the atmospheric
carbon stock
Parameter of the upper ocean
carbon stock
Parameter of the upper ocean
carbon stock
Parameter of the upper ocean
carbon stock
Parameter of the deeper ocean
carbon stock
Parameter of the deeper ocean
carbon stock
per 5 years
0.99966
Table S1. Description of the variables used in the DICE model (2013R). A state variable is a variable that
is monitoring and used for designing policy through controls. The model parameters are the intrinsic variables
of the DICE model. Forcing equations are exogenous dynamics that we cannot control. Outputs are functions
needed for calculations of the state variables.
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Variable
Type
Range of
analysis
[0.039-1]
 (t ) State variable
TAT (t ) State variable [0.8-4.2]
Range of viability
calculation
[0.039-1]
[0.8-4.2]
Comments
Ranges are the same (  (t ) > 0 and
 (t )  1 )
e
TAT
q (atmosphere / oceans temperature) is between 1.3 and 3.2 (also
matches the IPCC results)
[0-1.8]
TLO (0) > 0 . Maximum of TLO (t ) =1.8 with the maximum values of the
K (t )
State variable [135-1500]
[135-1640]
K (135) > 0 whatever the values of the parameters (within the analysis
range). K = 1640 is the maximum calculated value of K in the analysis
M AT (t )
State variable [819-1181]
[723-1181]
M UP (t ) State variable [1527-2300]
[1527-2500]
TLO (t )
State variable
[0-0.9]
analysis range.
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range (case of high population, high reinvestment and high capital stock).
M AT (t ) reaches 723 in the extreme case of no CO2 emission, during 90
years. The maximum value is the planetary boundary (constraint).
M UP (t ) > 0 on the analysis range. M UP (t ) =2500 in the case of
M UP (t ) and M AT (t ) equal the maximum value of the analysis range.
Table S2. Description of the variables used in the viability analysis. The minimum and maximum
values of the analysis range correspond to the minimum and maximum values from the baseline and
the temperature-limited scenarios. The calculation range corresponds to the set of states necessary for
calculate the viability of the states within the analysis range.
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