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Linkages between emissions trading systems (ETSs) are crucial for the cost-effective implementation of the Paris Agreement. Yet we know little about the determinants of economic gains in a multilaterally linked system, how they are shared among participating jurisdictions and less still about their magnitude. We characterize these gains for an arbitrary linkage group, decompose them into gains in the group's internal bilateral linkages and prove linkage is superadditive. Relative to autarky linkage reduces permit price volatility on average but not necessarily for individual linkage group members. In a quantitative application calibrated to five hypothetical ETSs covering the power sectors in Canada, continental Europe, South Korea, the UK and the USA, linking generates gains of up to $370 million (constant 2005US$) per year relative to autarky. Focusing on linkage groups with two and three members which are themselves not linked, we find that maximum aggregate gains decline by $43-178 million.
