Mathematical models of climate justice

1. A Climate Justice Based
Model for Carbon
Budget Allocation
Malcolm Fabiyi, PhD
OptimaBiome

2. Agenda
• Climate Justice
• Attribution and Allocation Models
• Climate Justice Model – method,
cases & application
• Next Steps | Future Work
Climate Justice Climate Justice Based Model for Carbon Budget Allocation 2

3. Overview: The Challenges of Climate Action
• >1,500 GtCO2 emitted since 1750
• Richest, most powerful nations have
highest contributions
• Poorest nations have lowest emissions
• Challenge: Attribution cannot be
enforced
3
• Highest per capita emitters have largest
economies
• ~2 tCO2 per capita per year to avoid
reaching 1.5oC tipping point over 30 years*
• Challenge: Allocation of emissions that is
both Safe and Fair requires hard choices
* Based on ~500 GtCO2 remaining carbon budget
A
B

4. Challenge: Attribution and
Allocation
• How do we fairly allocate remaining
carbon budget?
• How do we fairly accommodate
current economic output?
• How do we fairly consider historical
responsibility?
• How do we justly support growth of
developing economies
4

5. “Safety and fairness cannot
be achieved simultaneously
for strict definitions of
both”
Socolow & Co Workers, 2011
Tavoni, Massimo and Chakravarty, Shoibal and Socolow, Robert, Safe vs. Fair: A Formidable Trade-Off in Tackling Climate Change (August 29,
2011). FEEM Working Paper No. 61.2011, Available at SSRN: https://ssrn.com/abstract=1918764 or http://dx.doi.org/10.2139/ssrn.1918764 5

6. Integrated Assessment Models: The Limitations
IAMs: Integrated Assessment Models (IAMs) combine complex interplays of socio-
economic, technological, and biogeochemical variables in their conceptual frameworks
• Do not sufficiently address fairness of adaptation and mitigation strategies (Jafino et
al, 2021; Rivadeneira & Carton, 2022).
• Reliance on aggregated perspectives of benefits and costs which does not permit a
clear outlook on burdens and benefits for different actors across time and space
(Stanton et al, 2009).
• Limited representations of inequality and distributional impacts, reproduce types of
economy that are not sustainable and can be unrealistic in their representation of Earth
and Human systems and the relation between the two (Jafino et al, 2021; Asefi-
Najafabady et al, 2021; Keen, 2020).
6

7. Fairness and Justice Models
FJMs: There has been an increasing focus on the development of more robust
distribution models that incorporate fairness and justice into emissions attribution and
carbon budget allocations (Hohne et al, 2014; Hickel, 2020; Williges et al, 2022).
• Fair shares attribution: An attribution modeling approach which establishes
differentiated national responsibility for climate breakdown consistent with a 350-ppm
atmospheric CO2 planetary boundary and equal per capita access to atmospheric
commons (Hickel, 2020)
• Integrated PCC and EPC: Williges et al (2022) evaluated two commonly utilized allocation
mechanisms (per capita convergence, PCC and equal per capita, EPC) and developed a
qualified equal per capita allocation approach that incorporates grandfathering to
address fairness concerns.
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8. Modeling of CO2 Budget Allocations
Assure Safety & Fairness
• Ensures global targets for carbon
mitigation are met
• Requires incorporating factors
such as historical emissions,
differences in current levels of
emissions, and differing
capacities of societies to cope
with reductions in emissions
rates (Baskin, 2009; Williges et
al, 2022).
Incorporate Justice
• Should not deny rights to viable
development pathways for
societies that have made minimal
contributions to the climate
problem
• Enable possibility that just
allocations might lead to an
increase in emissions levels for
some nations, relative to the status
quo.
Addresses Geopolitics
• Must support dramatic reductions
in GHGs
• Should have methodological tools
for not increasing inequality
• Should not increase potential for
international conflict
• Provide actionable policy outcomes
The lack of alignment around climate action is due to the absence of robust justice principles in climate
policy formulation (Baskin, 2009)
8

9. Proposing a Novel Climate Justice Model
Integrated Attribution and Allocation Model: Fundamentally incorporates
historical responsibility, current emissions, rights to viable future growth
1. Individual emissions are the elemental basis of national carbon emissions i.e., aggregate
emissions in regions are determined by summing up emissions at an individual level
2. Responsibility for historical and current emissions are to be borne by the current
inhabitants in the region
3. Poor nations have rights to attain threshold economic and human development outcomes
consistent with widely accepted global development milestones
4. Prescribed transition pathways for mitigating climate action should not erode competitive
or comparative advantage
5. Convergence of per capita emissions across geographical regions is a necessary outcome
of ensuring an equitable level of access to the global atmospheric commons
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10. Outline of the Fairness and Justice Model
10
𝐸𝐺,𝑅 = ෍ 𝑓 𝐸𝐶,𝑅 + ෍ 𝑓 𝐸𝐹,𝑅 − ෍ 𝑓 𝐸𝐻,𝑅
Allocated
emissions per
capita
Historical emissions per
capita (𝐸𝐻,𝑅),
Current per capita
emissions (𝐸𝐶,𝑅)
Per capita emissions
due to climate
justice
considerations
(𝐸𝐹,𝑅).

11. Fairness and Justice Models
11
𝐸𝐺,𝑅 = ෍ 𝑓 𝐸𝐶,𝑅 + ෍ 𝑓 𝐸𝐹,𝑅 − ෍ 𝑓 𝐸𝐻,𝑅
𝐶𝐺,𝑅 = ෍ 𝜃𝑁𝑅𝐸𝐶,𝑅 + ෍ 𝜑𝑁𝑅𝐸𝐹,𝑅 − ෍ 𝛽𝑅𝑁𝑅𝐸𝐻,𝑅
𝐶𝐺,𝑅 = 𝑓(𝑁𝑅𝐸𝐺,𝑅 )
EQN 1
EQN 2
EQN 3
𝐶𝐺,𝑅 = Carbon budget (tCO2/yr)
𝑁𝑅 = Population in region, R
𝜃, 𝜑, 𝛽 = Weighting factors
𝐸𝐶,𝑅 = Current per capita emissions
𝐸𝐹,𝑅= Future (justice) per capita emissions
𝐸H,𝑅 = Historical per capita emissions

12. Economic Underpinnings of EF,R
*We assume 𝑘𝐺 to be the inverse of the average global carbon intensity value of GDP (IEA, 2022). The estimate for the average global CO2 emissions intensity of GDP for the
period 2019 to 2021 is 0.26 tCO2 per $1,000 i.e., 𝑘𝐺 =$3,846 per tCO2 (IEA, 2022).
12
𝐸𝐹,𝑅 = 𝑓 𝐸𝑇 − 𝐸𝑅
𝐺𝐷𝑃 = 𝑘𝐸
𝐸𝐹,𝑅 = 𝑓 𝐸𝑇 − 𝐸𝑅 =
𝐺𝐷𝑃𝑇 − 𝐺𝐷𝑃𝑅
𝑘𝐺
• Assumes a relationship between economic output
and carbon emissions
• 𝐺𝐷𝑃𝑇 is the threshold GDP per capita level beyond
which a region is considered to have transitioned to
a high development status,
• 𝐺𝐷𝑃𝑅 is the current per capita GDP in the region
• 𝑘𝐺 is a “representative” carbon intensity level used
for translating the gap in economic output between
𝐺𝐷𝑃𝑇 and 𝐺𝐷𝑃𝑅 into an emissions equivalent*
EQN 4
EQN 5
EQN 6
𝐸𝑇= Emissions associated with a threshold
level of economic output
𝐸𝑅= Emissions in the region 𝐺𝐷𝑃  Measure of economic output

13. Economic Underpinnings of EC,R
13
• For fairness, the level of permissible net
current emissions per capita, 𝐸𝐶 should reflect
the difference between current emissions from
that region, 𝐸𝑅 and a fairness based budgetary
allocation based on the available global carbon
budget, 𝐸𝐵,𝑅.
• 𝐸𝐵,𝑅 is the gross annual per capita carbon
allocation is determined by proportionally
assigning every global citizen equal emissions
allocation from the global carbon budget over
a defined drawdown period.
𝐸𝐶,𝑅 = 𝐸𝐵,𝑅 − 𝐸𝑅
𝑁𝑅
𝑁𝑇
𝐶𝐺,𝑡
𝑁𝑅
= 𝐸𝐵,𝑅
EQN 7
EQN 8
𝑁𝑅 = Population in region, R 𝑁𝑇 = Global population

14. Fairness and Justice Models
14
Global carbon
budget
Consideration for
historical emissions
(𝐸𝐻,𝑅),
Contribution from
Current per capita
emissions (𝐸𝐶,𝑅)
Justice based
contributions
considering
allowable future
per capita
emissions (𝐸𝐹,𝑅).
𝐶𝐺,𝑡 = ෍ 𝜃𝑁𝑅𝑓 𝐸𝐵,𝑅 − 𝐸𝑅 + ෍ 𝜑𝑅𝑁𝑅
𝐺𝐷𝑃𝑇 − 𝐺𝐷𝑃𝑅
𝑘𝐺
− ෍ 𝛽𝑁𝑅𝐸𝐻,𝑅

15. Model Constraints
1. 𝛽, 𝜃 and 𝜑 can range from 0 𝑡𝑜 ∞. They are expected to vary as economic, development and
environmental conditions change
2. 𝐸𝑇, 𝑘𝐺, and 𝐺𝐷𝑃𝑇 have values ≥ 0, that can change over time
3. Given that 𝐸𝐹,𝑅 =
𝐺𝐷𝑃𝑇−𝐺𝐷𝑃𝑅
𝑘𝐺
then as 𝑘𝐺 → ∞,
𝐺𝐷𝑃𝑇−𝐺𝐷𝑃𝑅
𝑘𝐺
→ 0; and 𝐸𝐹,𝑅 ≈ 0. This represents conditions
in which the region has a high rate of economic output per ton of CO2 emitted i.e., a high energy
productivity economy
4. Solution sets that reduce inequality are preferred. The coefficient of variation for the allocated per capita
emissions, 𝐸𝐺,𝑅 represents a suitable equality criterion (Bendel et al, 1989).
5. We assume a historical emissions coverage horizon that spans from a starting period, 𝑇𝑠 = 1750 through to
a baseline, or current period, 𝑇𝑏
6. Convergence values of 2 tCO2 per capita are set for all instances where 𝐸𝐵,𝑅 < 2 and a positive carbon
budget is to be allocated.
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16. Case Studies: Outline of Scenarios
16
Cases
GDPT per
capita
Model
Constraints
Drawdown
(Td, yrs.)
Ts Tb
CG,T
GtCO2
Case 1 $12,500 None 10 1750 2019 500
Case 2 $30,000 None 10 1750 2019 500
Case 3 $45,000 None 10 1750 2019 500
Case 4 $12,500 Min CoV 10 1750 2019 500
Case 5 $30,000 Min CoV 10 1750 2019 500
Case 6 $45,000 Min CoV 10 1750 2019 500
Case 7 $12,500 Min CoV 30 1750 2019 500
Case 8 $30,000 Min CoV 30 1750 2019 500
Case 9 $45,000 Min CoV 30 1750 2019 500
Case 10 $30,000 Min CoV 100 1750 2019 500
Case 11 $30,000 Min CoV 30 1850 2019 500
Case 12 $30,000 Min CoV 30 1950 2019 500
Case 13 $30,000 Min CoV 30 1990 2019 500
Case 14 $30,000 None 10 1750 2019 -500
Case 15 $30,000 None 30 1750 2019 -500
• Cases 4 to 13 have equality constraints
• Cases 14 & 15 represent negative
carbon allocation scenarios i.e., a
period where the remaining global
carbon budget has been exhausted

17. Case Studies: Results Summary
17
Cases Mean EG,R SD EG,R CoV EG,R q b j
GDPT per
capita
Model
Constraints
Drawdown
(Td, yrs.)
Ts Tb
CG,T
GtCO2
Case 1 6.61 5.21 0.79 1.73 0.68 1.38 $12,500 None 10 1750 2019 500
Case 2 6.36 5.11 0.80 1.07 0.96 1.10 $30,000 None 10 1750 2019 500
Case 3 6.37 5.11 0.80 0.91 1.06 0.81 $45,000 None 10 1750 2019 500
Case 4 7.31 3.92 0.54 1.69 0.00 0.43 $12,500 Min CoV 10 1750 2019 500
Case 5 5.71 2.60 0.46 0.00 0.00 1.12 $30,000 Min CoV 10 1750 2019 500
Case 6 5.82 2.27 0.39 0.00 0.00 0.69 $45,000 Min CoV 10 1750 2019 500
Case 7 2.15 0.24 0.11 0.00 0.01 0.85 $12,500 Min CoV 30 1750 2019 500
Case 8 2.13 0.17 0.08 0.00 0.00 0.32 $30,000 Min CoV 30 1750 2019 500
Case 9 2.14 0.15 0.07 0.00 0.00 0.21 $45,000 Min CoV 30 1750 2019 500
Case 10 0.55 0.34 0.63 0.00 0.00 0.12 $30,000 Min CoV 100 1750 2019 500
Case 11 2.13 0.17 0.08 0.00 0.00 0.32 $30,000 Min CoV 30 1850 2019 500
Case 12 2.13 0.17 0.08 0.00 0.00 0.32 $30,000 Min CoV 30 1950 2019 500
Case 13 2.13 0.17 0.08 0.00 0.00 0.32 $30,000 Min CoV 30 1990 2019 500
Case 14 -7.90 14.93 -1.89 0.63 0.30 1.19 $30,000 None 10 1750 2019 -500
Case 15 -3.39 12.69 -3.74 0.69 0.67 1.26 $30,000 None 30 1750 2019 -500
𝐶𝐺,𝑡 = ෍ 𝜃𝑁𝑅𝑓 𝐸𝐵,𝑅 − 𝐸𝑅 + ෍ 𝜑𝑅𝑁𝑅
𝐺𝐷𝑃𝑇 − 𝐺𝐷𝑃𝑅
𝑘𝐺
− ෍ 𝛽𝑁𝑅𝐸𝐻,𝑅

18. Case Studies: Emissions Allocations
18
• 𝐺𝐷𝑃𝑇 = $30,000
• 𝑇𝑑 = 10 years
• Carbon budget = 500 GtCO2
• 𝐶𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡 = 𝑁𝑜𝑛𝑒
• 𝐺𝐷𝑃𝑇 = $30,000
• 𝑇𝑑 = 30 years
• Carbon budget = 500 GtCO2
• C𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡 = 𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝐶𝑜𝑉 𝐸𝐺,𝑅
• 𝐺𝐷𝑃𝑇 = $30,000
• 𝑇𝑑 = 30 years
• Carbon budget = -500 GtCO2
• 𝐶𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡 = 𝑁𝑜𝑛𝑒
Case 5 Case 8 Case 15
• No obvious Global North vs Global South splits
• Equality constraints decrease per capita emissions variance across nations
• Drawdown scenarios will likely require more action from Developed economies

19. Case Studies (1 of 3)
19
Figure 1: Case 5 summary showing distribution of emissions allocations globally
Mexico = 5.76; Nigeria = 8.06; South Africa = 6.76; Peru = 6.68; Poland = 4.15;
Indonesia = 7.49; Iran = 7.75
• Developing nations have
higher carbon allocations – that
they cannot immediately use
• Developed nations are required
to draw down – which is
difficult, perhaps impossible
• Policy considerations: escrow
allocations, allocations trading

20. Case Studies (2 of 3)
20
Figure 2: Case 8 summary showing distribution of emissions allocations
globally for a scenario where equality is a model constraint i.e., minimization
of coefficient of variance for EG,R
Mexico = 2; Nigeria = 2.29; South Africa = 2; Peru = 2; Poland = 2; Indonesia =
2.13; Iran = 2.2
• Equality constraints reduce
emissions variance across nations
• Policy considerations: Establish
realistic constraint terms that make
action more likely

21. Case Studies (3 of 3)
21
Figure 3: Case 15 allocation of emissions. This represents a drawdown
scenario with a net negative carbon budget
Mexico = -0.99; Nigeria = 6.83; South Africa = -6.99; Peru = 3.59; Poland = -18.39;
Indonesia = 4.27; Iran = -2.41
• Drawdown scenarios will likely
bring more variance in allocations
– within and across regions
• Wealthiest, most powerful nations
will be required to do more
• Geopolitical tensions around
climate action could increase
• Policy considerations: Urgent
action should be occurring now,
under positive carbon budget
scenarios

22. Discussion: Applications
1. Supports creation of marketplace for emissions: This model can define the emissions volumes
that can be traded in a global carbon market and provide a fair means for determining which
nations come to the market with surplus or deficit emissions allocations.
2. Mechanism for funding global climate goals: Provides market-based mechanism for generating
funds to finance global climate change mitigation programs.
3. Can incorporate measures of fairness: Model allows incorporation of quantitative, statistical
fairness constraints. e.g., coefficient of variance of 𝐸𝐺,𝑅 22
A B

23. Summary
• Climate justice can be robustly integrated into CO2
emissions attribution and allocation models
• Demonstrates the lack of need for a priori categorizations
such as “west vs east”, “Global South vs Global North”,
“developed vs developing” of regions as the basis of climate
policy
• Provides tools for objective and quantitative climate policy
negotiations e.g., debates could be about how to specify
weights (𝛽, 𝜃 , 𝜑), 𝐸𝑇, 𝑘𝑇, and 𝐺𝐷𝑃𝑇 values that should be
utilized
• Future work will evaluate the extent to which additional
measures of inequality are relevant to consider as metrics
for evaluating equality and fairness.
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24. Thank you
24
Dr Malcolm Fabiyi
mfabiyi@optimabiome.com
www.optimabiome.com