In a model inspired by the EU Emissions Trading Scheme, non-cooperative countries allocate their emissions to internationally trading and non-trading sectors. Each country is better off with trading than without, and aggregate welfare is maximized with all sectors in the trading scheme. We analyze the effects of extending the sectoral coverage of the trading scheme in a two-country model with quadratic abatement costs. If only the original trading sector is asymmetric between countries, the welfare change is always positive and the same in both countries. If the original and additional trading sectors are asymmetric, one country might lose, but there is an aggregate welfare gain. If the original trading sector and the non-trading sector are asymmetric, both countries always gain.