We clarify the relationship between the “double dividend” (associated with a marginal change from the existing tax system) and the situation in which the optimal environmental tax rate is greater than the Pigouvian tax rate. These two situations are generated by rather similar combinations of parameters. We also show that, if environmental taxes are increased beyond their second-best optimal level, there is a rapid increase in the non-environmental costs of tax distortions. Our calculations suggest that, for typical parameter combinations, there will be overall welfare losses when the tax rate is increased to somewhat more than twice the optimal level.