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First steps toward a broad climate agreement, such as the Kyoto Protocol, have focused on less than global geographic coverage. We consider instead a policy that is less comprehensive in term of greenhouse gases (GHGs), including only the non-CO2 GHGs, but is geographically comprehensive. Abating non-CO2 GHGs may be seen as less of a threat to economic development and therefore it may be possible to involve developing countries in such a policy even though they have resisted limits on CO2 emissions. The policy we consider involves a GHG price of about $15 per ton carbon-equivalent (tce) levied only on the non-CO2 GHGs and held at that level through the century. We estimate that such a policy would reduce the global mean surface temperature in 2100 by about 0.55° C; if only methane is covered that alone would achieve a reduction of 0.3° to 0.4° C. We estimate the Kyoto Protocol in its current form would achieve a 0.25° C reduction in 2100 if Parties to it maintained it as is through the century. Furthermore, we estimate the costs of the non-CO2 policies to be a small fraction of the Kyoto policy. Whether as a next step to expand the Kyoto Protocol, or as a separate initiative running parallel to it, the world could well make substantial progress on limiting climate change by pursuing an agreement to abate the low cost non-CO2 GHGs. The results suggest that it would be useful to proceed on global abatement of non-CO2 GHGs so that lack of progress on negotiations to limit CO2 does not allow these abatement opportunities to slip away.