One of the thorniest issues within global climate negotiations is how to most fairly share out what’s left of the carbon ‘pie’—the remaining few thousand gigatons or so of CO2 that humanity can safely release into the atmosphere without causing catastrophic climate change.

Even agreeing on how big that pie is to begin with is a sensitive topic. We have a certain sized pie, usually referred to as a ‘carbon budget’ or quota, if we want to limit average global warming to two degrees Celsius, but the pie is a bit bigger if we want to limit it to three degrees, and of course the pie is a bit smaller if we want to limit it to 1.5 degrees.

As a result, mere mention of a carbon budget has been seen as too sensitive to be mentioned in any international agreement. In October, UNFCCC executive secretary Christiana Figueres appeared to rule out the concept. “I don’t think it’s possible,” she told the Guardian newspaper last month. “Politically it would be very difficult. I don’t know who would hold the pen [in setting out allocations of future budgets].”

But now a group of climate scientists, earth systems researchers and quantitative policy analysts led by Michael Raupach of the Australian National University’s Climate Change Institute have come up with an elegant formula to consider how to share out this remaining carbon budget, whatever that may be, using a spectrum of sharing principles.

Until now, negotiators’ stances have largely relied on appeals either to equity or to pragmatism. Poorer nations tend to argue the former while wealthier nations tend to argue the latter. The problem here is that no matter how legitimate a particular claim might be, such appeals are fairly abstract. Whether I subscribe to “realism” or “climate justice”, how is either value supposed to be translated into a set of hard numbers? Is Europe’s pledged target of a CO2 emissions reduction of 40 percent on 1990 levels by 2030 more on the equitable side of things or is it more on the pragmatic side of things, and is it in line with the cuts necessary to avoid catastrophic warming?

Meanwhile, confounding all this, the job of assessing what the remaining global carbon quota is, and of sharing out the burden of emissions reductions, have been two distinct endeavours.

What should make the formula, published in Nature Climate Change in September, so useful to negotiators isn’t so much that it automatically calculates for them what the fairest way to share out the carbon pie is. The battle over fairness is still strictly a matter of politics.

Rather, it is that the formula attaches precise figures to these abstract values, regardless of one’s negotiating position and independent of how many degrees of warming one feels is acceptable.

The formula is a mathematical simplification of the principle of the longstanding and widely supported climate policy principle of ‘contraction and convergence’, in which overall global emissions contract to a safe level while all countries steadily converge on a common per capita emissions level. It takes the two extreme ends of a spectrum of sharing out the remaining emissions and blends them together. On the one extreme is a business-as-usual allocation that the researchers term “inertia”, in which quotas are portioned out according to current emission rates. This ‘pure’ inertia scenario would significantly disadvantage poorer countries’ ability to develop.

At the other end is what the researchers describe as the “equity” option. Under this scenario, the remaining carbon emissions allowances are shared out according to an emitter’s proportion of the global population. In other words, carbon rationing per person. A ‘pure’ equity option may appear maximally fair, but would require extremely high emissions reductions from wealthy countries. Indeed in such a situation, some countries will have already exceeded their quota, meaning that their allotment would actually be negative.

Image: Global Carbon Project/Sharing a quota on cumulative carbon emissions, Raupach et. al., Nature Climate Change 4, 873–879 (2014)

In essence, the formula employs a sharing index, or what could be described as a ‘variable for fairness’ to produce a sliding scale between these two scenario extremes as desired, and which can be used for all countries or regions in the world. The formula can also be easily modified to take into account other metrics such as historic emissions, a delay in initiating reductions, and chance of success.

In addition, the formula works independently of what any diplomatically agreed upon global total quota of greenhouse gas emissions might be. The researchers use the formula to show how carbon emission quotas could be allotted for three different levels of warming: 2° Celsius, 2.5°, and 3° at a 50 percent chance of success, equivalent to 1,400, 2,300 and 3,200 remaining available gigatons of CO2, but the formula could be used for any agreed level, such as 1.5° or 4°.

In 2011, Christiana Figueres, the executive secretary of the UNFCCC, said that the world should aim for a 1.5 target, rather than the currently agreed 2 target, because even at two degrees, we are locking in a ruinous impact in terms of sea-level rises that would drown a number of low-lying islands and coastal communities, and a measure of average temperature increases in sub-Saharan Africa that could cripple agriculture. “We can’t have as our goal something that we already know does not guarantee the survival of low-lying states and sub-Saharan Africa,” she said at the time.

However, the researchers did not investigate a 1.5° scenario because is almost certainly not possible without colossal changes. Meanwhile a six-degree scenario—with its possibility of a mass extinction event akin to what occurred at the end of the Permian period, the last time that temperatures rose by such an amount—was also not explored, because it would not require sharing emissions. Everyone could just keep growing as they pleased.

For each of these three cases that were explored, the formula demonstrated how the fairness variable (which in the equation is assigned the letter ‘w’) influenced the share of this remaining global carbon quota that would be allotted to 14 representative countries from ten key regions: Europe, North America, wealthy Pacific countries (Japan, South Korea, Australia and New Zealand), former Soviet countries, the Middle East, China, India, the rest of Asia, Africa and Latin America. The last four regions are both the poorest per capita and have the per capita emissions below the global average.

If we look at the scale of the emissions mitigation challenge that different regions face, and assuming the internationally agreed target of 2°, we find things do not seem as diplomatically bleak as we might think. A compromise on emissions reduction burden sharing where everybody walks away relatively happy should be an easier compromise to achieve than it has been.

The two degree target means that globally, we have 1,400 gigatons of CO2 left that we can pump into the atmosphere. To achieve this, the world as a whole once emissions have peaked has to reduce its emissions by 5.5 percent per year over the next 45 years—independent of the fairness variable.

Who gets what share of this global 5.5 percent—that is, the scale of their mitigation challenge—changes as the fairness variable slides between pure equity and pure inertia.

The amount of time a country or region has to achieve its target is also highly sensitive to the fairness variable. At pure inertia, all countries have the same 45 years. Both Europe and North America would be required to cut by 2 percent a year over that time, for example, while Africa would have to cut by 5 percent and India 7 percent.

At pure equity however, the mitigation rates and time to perform this task changes dramatically. North America now has to cut its emissions by over 15 percent a year, and has just ten years to do so, while India would have to cut its emissions by just three percent a year over the course of about 165. Africa’s target is 1.4 percent a year, but for almost two centuries. Africa, India, Latin America and the poor countries of Asia get both lower emissions reductions targets and over a longer period, while the wealthy countries get higher emissions targets over a shorter period. China however, even at pure equity, would still have to reduce its emissions by 11 percent. At pure inertia, the Middle Kingdom’s target is 7 percent.

Interestingly, Europe’s mitigation rate at pure equity (its least favourable position), at just over 5.5 percent, changes relatively little from pure inertia (its most favourable position), of 2 percent, compared to the U.S..  Similarly for China, the difference between pure equity and pure inertia is not large compared to the differences for the US on the one hand and Africa on the other, almost 11 percent over the next 30 years compared to 7.5 percent over 45 years. Although China would have very steep emissions reduction targets, however the carbon allotments are shared out, and, curiously, this analysis does put China more on the side of the likes of the US, Europe and Japan than its traditional climate allies in Africa, India and Latin America, in that less global fairness gives Beijing an easier target to achieve.

However, at the midpoint between inertia and equity, at which the fairness variable is precisely half way between the two extremes, something curious happens. It appears that negotiators might not be as far apart as their current rhetoric suggests. Once you crunch the numbers, a compromise appears much more achievable, even if the task for the world as a whole remains gargantuan. Poor countries still get much more forgiving reduction targets and over longer periods than they would under pure inertia, but wealthy countries now do not have to commit to anywhere near the same high reduction rates.

This sweet spot,  or “blended sharing principle” as the researchers call it, offers substantial development benefits to the Global South, while being significantly less demanding on the Global North.

In addition, if the formula is adapted to include historic emissions, the change to how the remaining carbon quota is shared out for wealthy countries, unsurprisingly, is substantial, because historic emissions come close to or even exceed what is supposed to be left over for future emissions. Adapting the formula in this way, the researchers compared how things change if the baseline for attribution of historic emissions were set to 1750—the start of the industrial revolution; to 1972—the date of the UN Scientific Conference in Stockholm where climate change was first recognized internationally as a concern; and to 2012.

Yet while the burden on wealthy countries is dramatic, it turns out that the benefit to poor countries of the varying scenarios is surprisingly small, regardless of what date is chosen. This remarkable finding could offer climate negotiators a way around the impasse over historic responsibility for global warming—one of the most intractable diplomatic disputes amongst states.

There is some bad news however, very bad news. If we hike up the threshold for probability of success from a 50 percent chance of avoiding 2°C of global warming to a 66 percent chance, then the necessary global mitigation rate jumps from 5.5 percent to 7 percent per year.

And 5.5 percent is already a stunning scale of reduction, as the researchers know. “Reaching 2°C is extremely difficult. I think there is a lack of appreciation for the challenges,” Glen Peters, one of the authors of the paper and a researcher with Norway’s Centre for International Climate and Environmental Research and the Global Carbon Project, which tracks emissions and mitigation pledges, told Road to Paris. “Particularly given the current bottom up approach to climate policy, 2°C seems highly unlikely.”

Conversely, if we give ourselves a bit more leeway, going for a 3°C limit instead, with its much greater risk of dangerous climate change, this would still require substantial annual emissions reductions—globally about 3 percent a year, which translates to 2 percent annually for the US and 6 percent for China.

But we already knew that the global challenge was immense. Now at least, instead of arguing blindly over numbers seemingly picked out of the air, negotiators, campaigners, scientists, journalists and the wider public have a simple tool to see how fair everyone is being and how one state’s pledges affect everyone else.