For the purpose of these exercises, let the firm's profits be given by P(x) = 20x - x^2, and let damages be given by D(x) = 2x^2. 

You are asked to describe equilibrium outcomes for this economy under several alternative regulatory and/or legal regimes. For each of the situations described below, compute the following quantities:

• Level of output x
• Producer surplus, which equals profits, plus the net effects of any taxes, subsidies, or side-payments agreed to under the terms of a contract with the consumers; and
• Net damages to the consumer, likewise net of any payments made to or received from the polluting firm or the government.

Compute these three quantities under each of the following regulatory/legal regimes.

3. A regulator imposes a unit Pigovian tax on pollution at the level t=2 per unit.

4. Same as (3), but the government returns to tax revenues collected to the consumers.

5. A regulator imposes a unit Pigovian tax on pollution, at a level designed to maximize total economic surplus net of damages, i.e., P(x)-D(x).

6. Same as (5), except tax revenues are returned, with one-half share going to the firm, and the other half going to the consumers.