In this section we analyze the sensitivity of the results with respect to some critical changes in the assumptions. A summary table (Table S1) can be found in the SM.

**4.1 Nordhaus’ damage cost and discounting parameters**

We first present an alternative case with the damage function and SDR determinants that Nordhaus uses (Nordhaus 2018) (see SM).

As shown in Figure 4, both SCC and SCM are significantly smaller with Nordhaus’ parametrization due both to the lower damage cost and the higher discount rate. The SCC is seven times lower in 2020 (27 USD/tCO2) and three times lower in 2100 (210 USD/tCO2) while the SCM is 4.5 times lower in 2020 (826 USD/tCH4) and two times lower in 2100 (9,550 USD/ton CH4). The difference in the SCM is smaller than the one for SCC because SCM estimates are less affected by variations in the SDR due to the short lifetime of methane, while the SCC is highly affected by these variations.

It is interesting to observe that the ratio of SCM to SCC is larger under Nordhaus’ parametrization. The fundamental reason is Nordhaus’ use of higher discount rate parameters. With a high discount rate one values the future less, so both SCC and SCM will drop. But the SCM will not drop as much because methane disappears more quickly from the atmosphere and hence the way one values the far future matters less, while a unit emission pulse of CO2 stays much longer (significant amounts remains for centuries), so the SCC will drop much more (when the future is valued less). Therefore, the ratio of SCM to SCC is higher with Nordhaus’ parametrization (higher discount rate) than in our base case.

**4.2 Social cost estimates under a business-as-usual scenario**

The main results presented in this study correspond to a case where the social cost of carbon and methane are evaluated along the optimal temperature path, so the social cost estimates and their ratio reflect a case where the optimal emission trajectories are followed. However, it is far from certain that the world will succeed in reducing GHG emissions. It is therefore also of interest to study scenarios that do not follow the optimal emissions trajectory. To analyze these, we run our model under a business-as-usual scenario (BaU) in which no emissions reductions take place at all.

Figure 5 shows that both the SCC and SCM are higher under the BaU scenario, but the impact on these two metrics is not symmetrical: the SCC increases much more than the SCM as we move from the optimal to the BaU scenario. Hence, the ratio of SCM to SCC is lower under the BaU scenario.

Both SCC and SCM increase because marginal damages are proportional to the temperature increase, so higher temperature means higher marginal damages and therefore higher social cost estimates. Consider an emission pulse in the year 2020. For methane, the social cost is mostly determined by what happens during the next few decades when the business-as-usual temperature and the optimal temperature are rather similar. However, for carbon dioxide, where the temperature impact of a pulse emission lingers for centuries, the difference between the BaU temperature and the optimal temperature towards the end of the century and beyond will matter (in particular with lower discount rates). Hence, under these conditions the SCC will increase more than SCM under the BaU scenario, which explains why the ratio SCM/SCC drops.

**4.3 No NETs**

Negative Emission Technologies (NETs) are a key element of most IPCC emission pathways that stay below 2ºC or 1.5 °C (Edenhofer et al. 2014; Masson-Delmotte et al. 2018), although they are also criticized for being uncertain (Minx et al. 2018; Fuss et al. 2018). Therefore, we also explore results if no NETs were available.

As observed in Figure 6, the SCC is significantly higher when no NETs are considered compared to our base case, 323 USD/tCO2 versus 194 USD/tCO2 in 2020, a finding consistent with (Hänsel et al 2020). On the other hand, the SCM is approximately the same in both cases until 2055 and then it becomes higher in the no NETs case. Consequently, the ratio of SCM to SCC is much lower than in our base case.

This is explained by the difference in temperature increase from 2095 onwards. As seen in 6d, the temperature increase during the 21st century is relatively similar, but it starts to deviate towards the end of the century and during the 22nd century it drops a lot in the base case but stays roughly constant at approximately 1.6 ºC in the case without NETs. Due to its long lifetime, the temperature impact of a pulse emission of CO2 lingers for centuries, so the SCC in 2020 is affected by the difference in temperature in the 22nd century between these two cases. However, the SCM is not affected by the difference in temperature until the end of the 21st century because of the relatively short lifetime of methane.

**4.4 Variations in the damage function**

The quantification of damages caused by a specific temperature increase is highly uncertain. In order to explore the impact of different values on SCM and SCC we carry out a sensitivity analysis on the proportionality coefficient φ2 so that damages caused by a 2ºC temperature increase are varied in the range 1% to 5% of global GDP (3% corresponds to the proportionality coefficient that we use in our base case).

As observed in Figure 7, the ratio of SCM to SCC is larger the larger the damage coefficient. A larger damage coefficient would increase equally both SCM and SCC if the optimal temperature path were not affected, but this is not the case in an optimisation model with endogenous abatement. A higher damage coefficient means that emissions will be abated more and hence the optimal temperature increase will drop after a few initial decades in which the temperature increase will be about the same. This second order effect will cause both SCC and SCM to drop in comparison to a case where the temperature paths were unaffected by changes in the damage coefficient. This drop in the optimal temperature will affect the SCC more than the SCM due to the much longer lifetime of CO2 than CH4. Hence, the ratio of SCM to SCC increases with an increasing damage cost coefficient.

**4.5 Variations in the SDR determinants**

To test how sensitive our results are to social discount rate (SDR) values, we try different values for δ while holding η constant at 1. The δ values tested here are 0.5, 1, 1.5 and 2%/year.

Figure 8 shows that the SCC is higher the lower the SDR is. The same holds for the SCM but only until around 2080 (given that the optimal paths for each is followed). If we stretch beyond 2100 also, at some point in the future, lower values of SDR will also give lower values of SCC. This happens because there are two countervailing mechanisms acting here: On the one hand, the SCC and SCM are reduced with a higher discount rate because the value of future damages matters less. On the other hand, a high discount rate will also imply that emissions will be abated less, and therefore the temperature will become higher in the future. Given that marginal damages are proportional to the temperature increase when the damage function is quadratic, higher temperature in the future will imply higher damages and therefore higher social cost estimates.

One can also observe in Figure 8 that the SCC is relatively more sensitive to variations in the SDR than the SCM. This is due to the much longer lifetime of CO2 compared to CH4 (see section 4.1 for a more detailed explanation).