To meet the targets requires decreases in cumulative emissions to 2100, but the NDCs are expressed in terms of decreases in annual emissions. We now describe a method for calculating the increases in the NDCs needed to meet the targets, given the required cumulative emissions to 2100.

In the absence of additional efforts, the median forecast turns out to be for total global annual emissions to remain roughly constant from 2016 to 2100, at around A = 3108/85 = 36.6 Gt CO2 per year (see Figure 1). Suppose that to achieve a given climate target with a given probability would require keeping cumulative emissions in 2016–2100 to at most X Gt CO2. Let a be the rate of decline in annual global emissions needed to achieve this, such that if Et is global annual emissions in year 2015 + t, then Et = E0e−at. Then 100 × a is approximately the percent annual decline in global emissions.

To find a, note that cumulative emissions from 2016 to year (2015 + T) from a starting point of 1 Gt CO2 per year in 2016, is

Then a is the solution of the nonlinear equation C(a, T)A = X, and this can be found using a numerical univariate root-finding method.

Let aP be the value of a needed for meeting the Paris Agreement NDCs, assuming that for countries for which the NDCs refer to years before 2030 (such as the USA), the declines in annual emissions continue beyond the NDC target date to 2030 at the same average annual rate. In that case, X = 2083, and solving the equation C(a, T)A = X yields aP = 0.0101, or an average annual rate of decline in emissions of just over 1%.

Each target and probability corresponds to a different value of X, and the corresponding value of a can be calculated in the same way as for the NDCs. For example, to have a 50% chance of staying below 2°C in 2100 requires X = 1579, which corresponds to a = 0.0182, or an average annual rate of decline about 80% higher than needed to meet the Paris NDCs.

We can calculate the corresponding needed increase in NDCs as follows, taking Germany as an example. The NDC for Germany is to reduce carbon emissions by 40% from 1990 to 2030. Germany’s carbon emissions were 1,052 Mt CO2 in 1990, and 795 Mt CO2 in 2015. Thus the NDC for Germany corresponds to a target of 1052 × 0.6 = 630 Mt CO2 in 2030. This requires an annual rate of decline of 1(630795)115=0.0154 from 2015 to 2030. To stay below 2°C in 2100 requires a rate of decline that is 0.0182/0.0101 = 1.802 times higher, or 0.0278. This leads to a revised target for 2030 of 795 × (1 − 0.0278)15 = 521. This is a reduction of 50% over the 1990 level, which is 25% more than the NDC level of a 40% reduction. Thus we say that to stay below 2°C in 2100, Germany would need to increase its NDC by 25%.

The calculation is slightly different for countries whose NDCs are expressed in terms of carbon intensity rather than carbon emissions. We assume that GDP is measured in current values in local currency, and we use the numbers reported by the World Bank. We will take China as an example. China’s NDC is to reduce carbon intensity by 60% from 2005 to 2030. China’s carbon emissions in Mt CO2 were 5771 in 2005 and 9717 in 2015. Its GDP was 18.73T yuan in 2005 and 68.60T yuan in 2015. Thus its carbon intensity was 308.1 in 2005 and 141.7 in 2015. We then do the same calculation as we did for Germany, but for carbon intensity instead of carbon emissions. The result is that China’s NDC should become a reduction of 64.2% instead of 60%, an increase of 7% in the promised reduction.

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