Global scenario selection

The 2 °C-scenario is RCP2.6 (Ref.51), the only of the four IPCC-AR5 Representative Concentration Pathways that offers a likely chance of limiting global warming to 2 °C (and results in a 1.7 °C median-warming at the end of the century45). The 1.5 °C-scenario in this study is the average of the 39 scenarios selected in Ref.8 to have both net-zero GHG emissions before 2100, including emissions from Land Use, Land-Use Change and Forestry (LULUCF) and international shipping and aviation, and results in a median-warming below 1.5 °C in 2100. Warmings are expressed in comparison with pre-industrial levels. Two scenarios are from the IPCC-AR5 database (hosted at the International Institute for Applied Systems Analysis and available at: https://tntcat.iiasa.ac.at/AR5DB) complemented by 37 scenarios of refs52,53,54. These scenarios from IAMs represent a commonly used framework to discuss global mitigation under various Shared Socioeconomic Pathways (SSP) that model possible futures with different equity settings55. However, many technologic assumptions used in these scenarios can adversely impact vulnerable populations, depending on their implementation (for example land-based mitigation to achieve negative emissions56). The global emissions scenarios used to derive the range of 2030-allocations under the hybrid approach are from the SSP database (85 emissions scenarios with temperature assessment, hosted at the International Institute for Applied Systems Analysis and available at: https://tntcat.iiasa.ac.at/SspDb). The 2100-warming median assessments of these SSP-scenarios range from 1.7 °C to 5.1 °C. These are complemented by lower emissions scenarios from ref54 (36 emissions scenarios) whose 2100-warming median assessment ranges from 1.2 °C to 1.5 °C.

The relationship between national 2030-emissions levels and the 2100-temperature responses presented in Fig. 4 is derived from a representative sub-selection of global emissions scenarios. We standardize the data across both dimensions (2030-emissions, excluding LULUCF and bunkers and 2100-warming) and derive the third-degree polynomial fit (Supplementary Fig. 7). Using a second-degree polynomial fit would result in a plateau where high global warming hardly depends on 2030-emissions levels. We then select a subset of scenarios with the least standardized distance to the fit, starting at the lowest 2100-warming and every 0.5 °C (nine scenarios, Supplementary Table 4, Supplementary Figs. 7, 8).

The USA’s policy projection for 2030 is 6.74 GtCO2eq without the Clean Power Plan taken from http://www.climateactiontracker.org/.

Global scenario preparation

We used and extended the Potsdam Real-time Integrated Model for the probabilistic Assessment of emission Paths19,57 (PRIMAP) to model allocations approaches. The database contains population, GDP and GHG emissions historical and projected data from composite sources as detailed in ref27.

The aggregation of Kyoto–GHG emissions follows the SAR GWP-100 (Global Warming Potential for a 100-year time horizon), consistently with the reporting under UNFCCC (http://unfccc.int/ghg_data/items/3825.php).

The national emissions allocations derived in this study do not cover the LULUCF sector. Emissions from the LULUCF and from international shipping and aviation are removed from the global scenarios before allocating their emissions across countries using the methods and data indicated in ref8.

For RCP2.6 and the 85 SSP scenarios, we subtracted CO 2 emissions from LULUCF. For the 36 scenarios of ref54. (including the 1.5 °C-scenario) where no specific LULUCF emissions were available, we subtracted the CO 2 emissions that do not come from fossil fuels combustion.

The historical emissions of Fig. 2b are from PRIMAP19,57 until 2010 and follow the growth rates of ref.58 until 2014.

Hybrid allocation

We name complete bottom-up allocation of a global scenario the allocation to each country of the least-stringent of the scenarios calculated under the CAP, EPC, CPC, GDR and CER (grandfathering) approaches. The modelling and parametrization of these five approaches follows that of ref.8,27 (and their Supplementary Information) with the same limitations regarding the data missing for 27 countries and territories. Similar modelling to the EPC is also named per-capita convergence11, equity12 or similar. The EPC dynamically shares the emissions of the global scenario across countries based on their projected population trajectories, and thus it does not result in equal cumulative per capita emissions (i.e. equal cumulative emissions over cumulative populations). Comparing two countries with equal given cumulative population, a country with increasing shares of the global population will have lower allocations under decreasing global emissions scenarios, and higher allocations under increasing global emissions scenarios, than a country with decreasing shares of the global population.

Under the hybrid approach, every country picks the least-stringent approach, in terms of cumulative emissions over 2010–2100 (Fig. 2) or 2030-emissions levels (Figs. 3 and 4), while staying below a warming threshold. The modelling of the hybrid allocation consists in iterative steps to derive a global aspirational pathway whose bottom-up allocation matches any chosen emissions scenarios from IAM.

The iterative process starts by calculating the difference D(1) between the chosen IAM scenario (IAMscenario) and the bottom-up allocation of that chosen IAM scenario BU(IAMscenario). We then build a first aspirational emissions scenario A(1) that is IAMscenario discounted by half the calculated difference D(1)/2.

$${\mathrm{A}}(1) = {\mathrm{IAMscenario}} - \left( {{\mathrm{BU}}\left( {{\mathrm{IAMscenario}}} \right)-{\mathrm{IAMscenario}}} \right){\mathrm{/}}2.$$ (1)

The following step consists in calculating the difference D(2) between IAMscenario and the bottom-up allocation of the new aspirational pathway A(1). We then build a new aspirational emissions pathways A(2) that is A(1) discounted by the difference D(2)/2:

$${\mathrm{A}}( 2 ) = {\mathrm{A}}( 1 )-( {{\mathrm{BU}}( {{\mathrm{A}}( 1 )} )-{\mathrm{IAMscenario}}}){\mathrm{/}}2$$ (2)

These steps are repeated iteratively until BU(A(n)) = IAMscenario or until A(n + 1) = A(n):

$${\mathrm{A}}( {{{n}} + 1} ) = {\mathrm{A}}( {{n}} )-\left( {{\mathrm{BU}}( {{\mathrm{A}}( {{n}} )} )-{\mathrm{IAMscenario}}} \right){\mathrm{/}}2$$ (3)

Note that A(0) = IAMscenario. Supplementary Fig. 2 shows the national emissions allocations of the aspirational 2 °C-scenario under the five selected effort-sharing allocation approaches (Supplementary Figs. 2a–e) and under its complete bottom-up allocation which is also the complete hybrid allocation of the original 2 °C-scenario (Supplementary Fig. 2f). The most favourable approach for a country may differ when considering a different global scenario, and therefore may also change over the iterative process (Supplementary Fig. 3).

The allocations of country-groups presented in this study (EU28, G8 + China or the rest of the world) are calculated based on the least-stringent approach of each of their members individually, rather than the least-stringent approach for the country group.

We update the CPC modelling approach8 when applied to global emissions scenarios with positive emissions in 2100 to avoid national positive emissions after a period of negative emissions. The CPC approach then derives national ratios of the global emissions scenarios that are positive in 2100. These national ratios are a linear interpolation between 2010-emissions ratios and the 2100-ratios that result in equal cumulative per capita emissions. The impact of high historical per capita emissions has therefore a lower impact on 2030-emissions than under the CPC modelling applied to global scenarios with negative 2100-emissions. The lesser equity stringency on 2030-emissions aligns with the lesser global stringency. For example, the influence and importance of equitable allocation is lower when applied to business-as-usual scenarios. The accounting of historical emissions since 1990 and the autonomous energy efficiency improvement index are similar in both CPC setups.

Under the CBDR-RC hybrid setup used in Fig. 4, the hybrid approach is based on countries’ least-stringent of three equity approaches only (CAP, EPC and CPC), following their 2030 allocations. Using this methodology, the least-stringent approaches applied to the 2 °C-scenario (RCP3PD), corresponding to a CBDR-RC bottom-up situation, are shown in Supplementary Fig. 3a. The least-stringent approaches of the aspirational 2 °C-scenario, corresponding to a CBDR-RC hybrid approach, are shown in Supplementary Fig. 3b. Only few countries have different least-stringent approaches under the CBDR-RC bottom-up and CBDR-RC hybrid cases.

The comparison between a complete hybrid (attributing the least-stringent of all five effort-sharing approaches over the 2010–2100 period, as used in Fig. 2) and a CBDR-RC hybrid (attributing the least-stringent of only the CAP, EPC and CPC equity approaches in 2030), is shown in Supplementary Fig. 4e. The current NDCs of China and Russia imply higher 2030-emissions than even the most favourable effort-sharing approach applied to the 2 °C-scenario, under a complete bottom-up approach (Supplementary Fig. 4a). The national allocations of the complete hybrid approach happen to yield similar results to the complete average of the five effort-sharing approaches’ allocations8 for most G8 countries (including the 28 EU countries), and China (Supplementary Fig. 4b). The variability across national results is greater than under the CBDR-RC hybrid (Fig. 3b). Reaching the complete hybrid allocation, including the status-quo grandfathering approach, implies raising the NDC’s ambition by around 30 percentage-points of 2010-emissions for the USA and the EU28, and 77 percentage-points for China (Supplementary Fig. 4c). Aiming at 1.5 °C, rather than 2 °C, under the complete hybrid approach requires five additional percentage-points from the G8 countries and China, 20 for the other economies altogether, and more for LDCs (Supplementary Fig. 4d).

Discussion on the monotony and uncertainty

The bijectivity between NDCs ambition and their temperature assessment relies on the strict monotony of the relationship between global scenario’s 2030-emissions and 2100-warming. We selected nine global scenarios every 0.5 °C to achieve such strict monotony at the global level. The 2030 emissions levels dependency to the NDC temperature assessment of Fig. 4 is a second-degree polynomial fit based on the allocations derived from the selected nine global scenarios. A second-degree polynomial fit smoothens the variability while preserving the greater sensitivity of national 2030-emissions allocations at lower 2100-warmings.

For 36 countries, the relationship between 2030-emissions and 2100-warming is non-strictly monotonous. In each case, the local maxima are at 4.8 °C or more, higher than their NDC assessments of the corresponding countries. The national emissions allocations at high temperature are only indicative for these countries. In the absence of interpolation, local maxima are found for 14 countries and are at lower temperature than the NDCs of only: Iraq, Trinidad and Tobago, Jordan, Brunei Darussalam and the Maldives.

The allocation of high global emissions pathways using effort-sharing approaches reflects equitable contributions to high global warming. However, the allocation of global BaU scenarios results in national scenarios that are, de facto, no longer business-as-usual. The range of 2030-emissions levels from global BaU scenarios reflects a range of modelling assumptions rather than a range of ambitions. The high-warming allocations derived in this study can indirectly be used to assess NDCs. Clearly, though, the equity allocations of BaU scenarios, taking into account the impacts the world is facing at global warming of 3.9 °C or more (Supplementary Table 4), cannot represent an equitable outcome28.

The uncertainty resulting from the various NDC quantifications is shown in Fig. 4 by the height of the NDC ranges. The high assessment reflect the most optimistic end of the assessment range, including the application of conditional targets44. The low NDC assessment takes the least optimistic assumptions of unconditional targets. The maps of global warming responses under the CBDR-RC hybrid approach associated with high and low NDC assessments are shown in Supplementary Figs. 9 and 10.

Choice of equity approaches

The IPCC-AR5 presented quantifications of emissions reduction following five effort-sharing categories (Chapter 6, Figure 6.28 ref. 35), which reflect combinations of three underlying equity principles: responsibility, equality and capability (Chapter 6, Table 6.5 ref. 35). In addition, the IPCC-AR5 presented a category (but did not present a quantification) based on responsibility only, as proposed by Brazil in 1997, that derives emissions goals without allocation (IPCC-AR5 Chapter 6, table 6.5, refs. 20,35). The historical responsibility of countries for their past emissions is modelled here with the equal cumulative per capita approach (CPC) that uses only historical emissions and population data to calculate countries’ emissions allocations. The GDR approach also uses historical emissions and accounts for countries historical responsibility. Other approaches of distributive justice (e.g., based on sufficiency59, or using the Human Development Index) and other metrics (e.g., accounting for consumption-based emissions or exported emissions), not currently used in the IPCC report or under the UNFCCC, are not modelled here but would bring useful perspectives that could be integrated in the hybrid approach.

The sensitivity of effort-sharing approaches to their input data and parameters indirectly affects the results of their hybrid combination and thus of the NDCs' warming assessments presented in Fig. 4. The sensitivity of the effort-sharing framework used in this study8 was studied under a range of parameters consistently across the five effort-sharing approaches48,49. The sensitivity analysis was performed by quantifying 3 to 81 combinations of parameters, depending on the approach. Because of their flexible parameterization, the most sensitive approaches are the CPC (sensitive to the period covering historical emissions) and GDR approaches (also sensitive to business-as-usual emissions projections and internal parameters, such as wealth threshold and responsibility-capability ratio16). The GDR approach is included in the complete hybrid quantifications, but not in the NDC assessment of Fig. 4. In addition to the allocation approaches uncertainty, various exogenous assumptions have equity implications. For example, an earlier (later) convergence date for the EPC and CAP approaches, and earlier (later) starting date to account for historical emissions is expected to favour—result in a lower NDC warming assessment—developing (developed) countries compared with the current NDCs.

The selection of effort-sharing approaches to derive countries’ least-stringent allocations directly influences the hybrid allocations of all countries. Removing the least-stringent approach of a country group would penalize these countries that would have to follow a more stringent approach and would consequently favour all other countries.

The complete hybrid allocation, which uses the five effort-sharing approaches, results in lower temperature assessments of developed countries’ NDCs and consequently in higher temperature assessments of other countries’ NDCs (Supplementary Fig. 5). The assessment of China’s NDC is still higher than the temperature scale range.

The choice of the five effort-sharing approaches includes the grandfathering approach (CER) that is only implicitly supported by some countries through their pledges. The grandfathering approach represents a status-quo in terms of equity where all countries conserve their share of global emissions and mitigate at a common rate, that of the global scenario. The GDR approach, categorized as a responsibility-capability-need35, preserves a right to development through the allocation of mitigation requirements, although the link between the objective right to development and the selected implementation criteria is subjective11,16,60. Excluding only the grandfathering approach, and including the GDR, represents four equity approaches representative of the four key equity principles dimensions: responsibility, capacity, equality and the right to sustainable development (IPCC-AR5 WG3 Chapter 4, Section 4.6.2, ref. 24).

The modelling of the GDR approach relies on business-as-usual (BaU) emissions that countries do not mutually recognize. The BaU emissions used here are downscaled from RCP8.5 resulting in allocations substantially higher than other allocations for Eastern European countries and Australia. Compared with the CBDR-RC hybrid setup presented in Fig. 4, the inclusion of the GDR approach in the hybrid setup results in a lower temperature assessment in favour of the NDCs of Eastern European countries, Australia and South Africa, and higher temperature assessment disfavouring India, Brazil, Mexico and Indonesia (Supplementary Fig. 6). The GDR approach was designed to allocate mitigation efforts to individuals with income above a certain threshold. The share of a country’s population above the income threshold is derived using the Gini index of inequality16. While the GDR is complex method accounting for more indicators than most other approaches in the literature, its reliance on hypothetical BaU emissions and Gini projections that are not commonly agreed indicators results in an important sensibility that cannot be easily resolved. The notions of responsibility and capacity that the GDR is based on are conveyed by the equal cumulative per capita (CPC) and capability (CAP) approaches, respectively.

Removing both the GDR and the grandfathering approaches from the hybrid allocation (Fig. 4) leaves equity approaches that rely on measurable population and GDP data that can be updated over time.

Warming assessment of the global scenarios

The evaluation of the warming resulting from the bottom-up scenarios requires the calculation of their GHG compositions (Fig. 2). The Equal Quantile Walk (EQW) method61 is used to derive a multi-gas scenario that is needed by the simple climate model MAGICC62,63 for the evaluation of the temperature response. Land-use CO 2 is taken directly from the target 2 °C-scenario and 1.5 °C-scenario. The probabilistic temperature projections of the complete bottom-up multi-gas scenario is constrained by historical global mean temperature observations and a priori estimates of uncertain model parameters, such as climate sensitivity64. We use MAGICC version 6.8 and climate sensitivity distribution from ref 40.

The aspirational scenarios are of purely numerical nature and have no underlying economic assumptions. These scenarios, which include bunkers but exclude land-use emissions, show a steep decline in the first half of the century with minima in 2070 (aspirational 2 °C-scenario, Supplementary Fig. 11a) and 2060 (aspirational 1.5 °C-scenario, Supplementary Fig. 11b) and very low emissions throughout the second half of the century. The EQW is based on older scenarios that did not have negative CO 2 emissions. The EQW thus cannot model negative fossil CO 2 emissions. However, these negative emissions are necessary to reach the low emissions levels in the second half of the century. We calculate multi-gas emissions scenarios consistent with global aspirational scenarios given in CO 2 -equivalent units as the aggregation of all Kyoto–GHG.

A dataset of full-gas 1.5 °C scenarios54 is harmonized to 2010-emissions19. Harmonization is carried out for the global aggregate Kyoto–GHG value excluding land-use emissions but including bunkers emissions. Bunkers emissions are included because for most IAM scenarios they are not available as independent time series but included in the world total and regional time series. The harmonization factor linearly converges to unity in 2040, and scenarios are unchanged onwards. The Kyoto–GHG harmonization factor is used for all substances.

From the harmonized database, we select the ten scenarios with the least mean-square distance to the aspirational scenario over the 2010–2100 period. Absolute distances are used because relative distances are not meaningful to compare positive and negative values, which is inevitable when using low emissions scenarios with negative emissions. The average across the ten selected scenarios is taken for each gas individually. In order to match the aspirational scenario, only the CO 2 emissions levels are reduced, which corresponds to additional mitigation implemented in the fossil CO 2 sector. Indeed, fossil CO 2 emissions can be mitigated more profoundly than other GHG (e.g., methane from agriculture). Carbon dioxide is the only gas where prototypes for large-scale negative emissions technologies exist (even though costs, side-effects and acceptability of large-scale projects are uncertain). Conversely, additional fossil CO 2 emissions are assumed where the aspirational scenario is higher than the average of the selected scenarios. Land-use CO 2 is taken directly from the target 2 °C-scenarios and 1.5 °C-scenarios.

The resulting multi-gas aspirational scenarios are then used for probabilistic temperature assessment as described above for the bottom-up scenarios.

Code availability

The code developed for this study is based on the PRIMAP modelling environment, developed at the Potsdam Institute for Climate Impact Research (PIK) and is not publicly shareable. Requests for code will be jointly considered by the authors and the PRIMAP research team.