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This paper examines the effects of combining an international cap-and-trade scheme with national carbon taxes. We consider a two-country stochastic partial equilibrium model with log-normally distributed uncertainty. The situation is analogous to the situation where European countries impose national carbon taxes in addition to the EU emissions trading. The allowance price in the joint cap-and-trade scheme depends on the tax rate, the relative size of countries and abatement options, the magnitude of uncertainty, and correlation of abatement costs. In most cases, the additional tax will not lead to additional production of the public good beyond the fixed targets. The additional tax results in higher costs of abatement to the country introducing the additional tax, and higher costs overall.