Should a carbon tax be differentiated across sectors?

If some, but not all, countries are cooperating to reduce CO2 emissions, one could make the following argument: a high carbon tax for carbon intensive tradeable sectors in the cooperating countries will reduce the production of goods from these sectors, and therefore CO2 emissions, in the cooperating countries. However, this will to a large extent be counteracted by increased production of such goods in the countries that have no climate policy. And since it is only total CO2 emissions from all countries that are relevant for the climate, there is no point in a policy that simply relocates CO2 emissions from the cooperating countries to the countries that have no climate policy. According to this line of reasoning, carbon intensive tradeable sectors should thus face a lower carbon tax than other sectors of the economy. This paper shows that a carbon tax should not be differentiated across sectors in the economy, provided one can use import and export tariffs on all traded goods. It is also shown that such a differentiation of carbon taxes is optimal for the cooperating countries if they are prevented from using tariffs on the traded goods. However, informational or political factors constraining the use of tariffs are likely also to constrain the possibility of differentiating carbon taxes between sectors.