Consider a hypothetical environmental externality: produced as the byproduct of industrial activity, each unit of emission of a pollutant that is emitted once, today, causes $10 of external harm to society each and every year, starting immediately, forever.

Lets say that policymakers apply a discount factor of (0, 1) to future gains and losses, in an analog of the discounted utility model except for cash payoffs rather than utilities. What would the socially efficient Pigouvian tax on the emission of this pollutant be if we applied a discount factor of (i) = 0.5, (ii) = 0.9, and (iii) = 0.99?

Give an intuitive explanation of how to interpret the parameter in this context, and explain precisely but in simple terms what the goal and effect of the Pigouvian tax would be.