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Article 17 of the Kyoto Protocol allowed emissions trading; Article 12 defined a Clean Development Mechanism (CDM). In a simple theoretical model we adapt standard gains from trade results to demonstrate that, relative to any status quo with specified worldwide aggregate emissions: first, emissions trading with a suitable international distribution of permit rights generally allows each nation more consumption while aggregate emissions remain constant; second, the CDM generally adds to these efficiency gains by reducing further the total cost of achieving any target worldwide emissions level. The provisions of the 2015 Paris Agreement that grant credit for carbon removals allow even further potential gains. Contrary to some claims, the Kyoto Protocol is not "defunct"; instead, retaining its key provisions of emissions trade and the CDM while including credit for carbon removals could make emissions reductions much more affordable. Yet an essential part of the institutional background required to make international trade in emissions permits function well has been destroyed by the Paris Agreement, since it lacks mandatory limits of the kind that the Kyoto Protocol imposed.