Suppose that in the market for paper, demand is p = 100 - Q. The private marginal cost is MCp = 10 + Q. Pollution generated during the production process creates external marginal harm equal to MCe = Q

What specific tax would result in a competitive market producing the socially optimal quantity of paper?

The socially optimal quantity of paper is found by setting MCp + MCe = p or
10 + Q + Q = 100 - Q. Rearranging yields Q = 30. At this level of output society incurs an external cost of 30. This is the specific tax that would yield the socially optimal quantity. To check, set
MCp + tax = 100 - Q. This yields Q = 30.