Developing the Integrated Attribution Allocation (IAA) model framework
The global carbon budget provides the maximum value for a range of nottobeexceeded emission levels needed to maintain a trajectory towards < 1.5oC average global temperature rise (Friedlingstein et al, 2021; Matthews et al, 2020). We define the global carbon budget thus:
$$\begin{array}{c}{C}_{G,T}=\sum _{t=1}^{t={T}_{d}}{C}_{G,t} \#1\end{array}$$
Where \({C}_{G,T}\) (GtCO2) is the global carbon budget over a drawdown period, \({T}_{d}\) and \({C}_{G,t}\) (GtCO2per year) is the annual global carbon budget.
For any given year, within the drawdown period, we posit that:
$$\begin{array}{c}{C}_{G,t}=\sum {C}_{G,R} \#2\end{array}$$
\({C}_{G,R}\) (GtCO2 per year) is the equitable allocation of carbon emission rights for individual regions. At the global level, the regions of interest are nation states.
The carbon budget in any region is obtained by aggregating per capita emissions (tCO2 per capita) for the entire population domiciled within that region. Hence, we can define the relationship:
$$\begin{array}{c}{C}_{G,R}={N}_{R}{E}_{G,R } \#3\end{array}$$
Where \({N}_{R}\) is the population living within the region and \({E}_{G,R}\) (tCO2 per capita) is the average allocated per capita emission for the region.
We further expand the definition of \({E}_{G,R}\) to include three per capita emissions categories that should be incorporated into an equitable and justicebased carbon budget, i.e., historical per capita emissions (\({E}_{H,R}\)), current per capita emissions (\({E}_{C,R}\)) and projected future per capita emissions (\({E}_{F,R}\)). Hence, we have that:
$$\begin{array}{c}{E}_{G,R}=\sum f\left({E}_{C,R}\right)+\sum f\left({E}_{F,R}\right)\sum f\left({E}_{H,R}\right) \#4\end{array}$$
We can rewrite the formulation in Eq. 4 as:
$$\begin{array}{c}{C}_{G,R}=\sum \theta {N}_{R}{E}_{C,R}+\sum \phi {{N}_{R}E}_{F,R}\sum {\beta }_{R}{{N}_{R}E}_{H,R} \#5\end{array}$$
Where \(\beta\), \(\theta\) and \(\phi\) are weighting factors whose values reflect the extent to which the historical, current and future economic developmental pathways influence carbon budget allocations.
Equation 5 can also be expressed as:
$$\begin{array}{c}{C}_{G,R}={N}_{R}\left(\sum \theta {E}_{C,R}+\sum \phi {E}_{F,R}\sum {\beta }_{R}{E}_{H,R}\right) \#6\end{array}$$
The economic underpinnings of\({E}_{F}\)
The future emissions allocation, \({E}_{F}\) is best aligned with climate justice. It addresses the requirement for climate action to allow poor and developing countries to grow and industrialize without being forced down developmental pathways that erode their competitive advantage and denies them access to energy options that nations that are now at higher levels of economic and human development utilized for attaining their own growth (Baskin, 2009; Davidson, 2021). Such an allowance for the possibility for higher levels of emissions to support development must necessarily be bounded (Fuhr, 2021). We assume that the bounding variable is a per capita level of emissions, \({E}_{T}\) that is functionally correlated with the threshold Gross Domestic Product (GDP) per capita value corresponding to the transition to a high development economy status, \(G{DP}_{T}\).
We can redefine the justice based, future emissions allocation as:
$$\begin{array}{c}{E}_{F,R}=f\left\{{E}_{T}{E}_{R}\right\} \#7\end{array}$$
Where \({E}_{R}\) is the current per capita CO2 emissions (tCO2 per capita) for the region and \({E}_{T}\) is the CO2 emissions per capita level associated with the GDP per capita threshold that marks the transition to developed economy status.
Given the well documented positive relationship between economic growth, energy consumption and CO2 emissions (Onofrei et al, 2022) we assume that for any region and within a given period, a functional relationship can be established between emissions during that period, \(E\) (tCO2) and GDP (USD) of the form:
$$\begin{array}{c}GDP=kE \#8\end{array}$$
Where \(k\) is a transitory constant that establishes the relationship between CO2 emissions and economic output in the specified region, for the specified period.
Hence, we have that:
$$\begin{array}{c}{E}_{F,R}=f\left\{{E}_{T}{E}_{R}\right\}=\frac{G{DP}_{T}G{DP}_{R}}{{k}_{G}} \#9\end{array}$$
Where \(G{DP}_{T}\) is the threshold GDP per capita level beyond which a region is considered to have transitioned to a high development status, \(G{DP}_{R}\) is the current per capita GDP in the region and \({ k}_{G}\) is a “representative” carbon intensity level used for translating the gap in economic output between \(G{DP}_{T}\) and \(G{DP}_{R}\) into an emissions equivalent. We assume \({k}_{G}\) 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., \({k}_{G}=\) $3,846 per tCO2 (IEA, 2022).
Establishing \({GDP}_{T}\)
A critical question to be resolved for practical applications of this framework is the question of how \({GDP}_{T}\) gets established and what constitutes an appropriate level of development to set as the high development GDP per capita threshold. There are at least three possible options that can be considered, and these include i) the United Nations definition for high income nations which sets a GNI of >$12,500 as a transitional threshold (Serajuddin & Hamadeh, 2020) ii) setting the threshold as the average GDP per capita for the OECD of >$45,000 (OECD, 2022) and iii) using the average output of >$30,000 GDP per capita observed for most developed economies as the threshold level for economic development (UNCTAD, 2019).
Estimating \({E}_{C}\)
For fairness, the level of net current emissions per capita, \({E}_{C}\) should reflect the difference between current emissions from that region, \({E}_{R}\) and a fairness based budgetary allocation based on the available global carbon budget, \({E}_{B,R}\).
Hence, we define the relationship:
$$\begin{array}{c}{{E}_{C,R}=E}_{B,R}{E}_{R} \#10\end{array}$$
\({E}_{B,R}\) , which 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. Hence, we have that:
$$\begin{array}{c}\left(\frac{{N}_{R}}{{N}_{T}}\right)\frac{{C}_{G,t}}{{N}_{R}}={E}_{B,R} \#11\end{array}$$
Where \({N}_{T}\) is the global population, \({N}_{R}\) is the regional population, \({C}_{G,t}\) is the annual global carbon budget, and \({E}_{B,R}\) is the gross annual per capita emissions allocation.
We can therefore rearrange the global carbon allocation budget, \({C}_{G,t}\) as:
$$\begin{array}{c}{C}_{G,t}=\sum \theta {N}_{R}f\left\{{E}_{B,R}{E}_{R}\right\}+\sum {{\phi }_{R}N}_{R}\left(\frac{G{DP}_{T}G{DP}_{R}}{{k}_{G}}\right)\sum {{\beta N}_{R}E}_{H,R} \#12\end{array}$$
Specifying Time Horizons
There are twotime horizons of interest. The first is the drawdown period, \({T}_{d}\), which is the time over which the available carbon budget is to be applied. The second is the historical emissions coverage horizon, which is the time span for which historical emissions are to be considered and requires the definition of a start time, \({T}_{s}\), and a baseline time, \({T}_{b}\).
Estimating \({E}_{H}\)
In this conceptual framework, the responsibility for historical carbon emissions generated in any region is assumed to be borne by the extant populations domiciled in those regions. This is consistent with principles of reparative justice and is supported by the reality that current inhabitants of those regions that have contributed the most to the addition of carbon to the atmosphere experience relatively higher standards of living and economic attainment that have been made possible by developmental pathways that are directly linked to the historical release of higherthanaverage CO2 emissions into the atmosphere from those regions (Chapman and Ahmed, 2021).
Defining \({C}_{H,R}\) as the cumulative value of atmospheric CO2 emissions in a region from a starting period, \({T}_{s}\) through to a defined baseline period \({T}_{b}\), then the per capita historical emissions in that region, \({E}_{H,R}\) for a specified drawdown period, \({T}_{d}\) can be defined as:
$$\begin{array}{c}\frac{{C}_{H,R}}{{N}_{R}}\left(\frac{1}{{T}_{d}}\right)={E}_{H,R} \#13\end{array}$$
\({N}_{R}\) is as previously defined.
Reviewing model conditionalities and constraints

\(\beta\) , \(\theta\) and \(\phi\) can range from \(0 to \infty\). They are expected to vary as economic, development and environmental conditions change

\({E}_{T}\) , \({k}_{G}\), and \({GDP}_{T}\) have values ≥ 0, that can change over time

Given that \({E}_{F,R}=\frac{G{DP}_{T}G{DP}_{R}}{{k}_{G}}\)then we have that as \({k}_{G}\) \(\to \infty\), \(\frac{G{DP}_{T}G{DP}_{R}}{{k}_{G}}\to 0\); and \({E}_{F,R}\approx 0\). 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

Solution sets that reduce inequality are preferred. The coefficient of variation for the allocated per capita emissions, \({E}_{G,R}\) represents a suitable equality criterion (Bendel et al, 1989).

We assume a historical emissions coverage horizon that spans, \({T}_{s}=1750\) and \({T}_{b}=2019\), except stated otherwise

Convergence values of 2 tCO2 per capita are set for all instances where \({E}_{B,R}<2\) and a positive carbon budget is to be allocated. For instances where the carbon budget is negative, no convergence values are specified.
Case Studies
The following section will demonstrate the use of the model using fifteen (15) case studies. Historical carbon emissions data is sourced from Ritchie et al (2020). All other data is sourced from the World Bank (World Bank, 2022). The case studies address one or more of these conditions:
1. For twelve cases (Cases 1 through 13) a carbon budget of 500 GtCO2 is specified. The carbon budget represents the total amount of permissible CO2 emissions to limit global warming within ≤ 1.5oC (Xu and Ramanathan, 2017; Matthews et al, 2020).
2. For two cases (cases 14 and 15) a carbon reduction target of 500 GtCO2. These scenarios are intended to demonstrate the model’s capacity for carbon mitigation allocation which could be relevant in future periods where humanity has exhausted the carbon budget and a net reduction in atmospheric carbon is required to mitigate global warming
3. Setting constraints on the maximum emissions allocation in each region, \({{E}}_{{G},{R}}\) (see Table 1)
4. Setting the drawdown period, \({T}_{d}\) (see Table 1)
5. Setting the historical emissions coverage horizon by varying \({T}_{s}\) while maintaining \({T}_{b}=2019\) (see Table 1).
6. Specifying the threshold GDP per capita level, \({GDP}_{T}\) to which developing nations can aspire using unfettered development pathways (see Table 1)
Table 1 contains a summary of all the conditions used in each of the 15 case scenarios. For each case, the generalized reduced gradient (GRG) nonlinear optimization routine is used to solve for a set of \(\beta\), \(\theta\), \(\phi\) values that satisfy the requirements which in all cases is allocating a specified annual global carbon budget on an equitable basis across all regions, subject to specific constraints. The GRG routine evaluates the gradient of the objective function as the input values change and reaches a local optimum solution when the partial derivative equals zero (Lasdon et al, 1978; Fylstra et al, 1998). We note that the use of the GRG approach does not preclude the existence of other values of \(\beta\), \(\theta\), \(\phi\) that satisfy the model’s objective function within the specified constraints since it returns local optima. For the GRG approach, the starting values for \(\beta\), \(\theta\), \(\phi\) can have an impact on the local optima that meet the model specifications. We utilize starting values for the solver routine of \(\beta =1\), \(\theta =1\), \(and \phi =1\) in all cases.
Table 1
Summary table of key variables in all the evaluated case examples
Cases

\({G}{{D}{P}}_{{T}}\)
per capita

Constraints

Drawdown (Td, yrs.)

\({{T}}_{{s}}\)

\({{T}}_{{b}}\)

\({{C}}_{{G},{T}}\)
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

Minimize Coef. of Var

10

1750

2019

500

Case 5

$30,000

Minimize Coef. of Var

10

1750

2019

500

Case 6

$45,000

Minimize Coef. of Var

10

1750

2019

500

Case 7

$12,500

Minimize Coef. of Var

30

1750

2019

500

Case 8

$30,000

Minimize Coef. of Var

30

1750

2019

500

Case 9

$45,000

Minimize Coef. of Var

30

1750

2019

500

Case 10

$30,000

Minimize Coef. of Var

100

1750

2019

500

Case 11

$30,000

Minimize Coef. of Var

30

1850

2019

500

Case 12

$30,000

Minimize Coef. of Var

30

1950

2019

500

Case 13

$30,000

Minimize Coef. of Var

30

1990

2019

500

Case 14

$30,000

None

10

1750

2019

500

Case 15

$30,000

None

30

1750

2019

500

The determination of a preferred state between two states that satisfy the model will require the setting of additional constraints such as the selection of the state that has the minimum value for the coefficient of variation for the objective function, \({E}_{G,R}\) (Bendel et al, 1989). Generally, a lower coefficient of variation suggests that less “inequality” exists within the sample set.