When redistribution makes personalized pricing of externalities useless

We consider a standard optimal taxation framework in which consumers' preferences are separable in consumption and labor and identical over consumption, but are affected by consumption externalities. For every nonlinear, income-dependent pricing of goods there is a linear pricing scheme, combined with an adjusted income tax schedule, that leaves all consumers equally well-off and weakly increases the government's budget. The result depends on whether a linear pricing scheme exists that keeps the aggregate amount of consumption at its initial level observed under nonlinear pricing. We provide sufficient conditions for the assumption to hold. If adjusting the income tax rate is not available, personalized prices for an externality can enhance social welfare if they are redistributive, that is, favor consumers with a larger marginal social value of income.