5. Two power plants produce greenhouse gas (GHG) emissions. Currently, plant 1 produces E1 = 50 tons of GHGs annually, while plant 2 produces E2 = 25 tons. The EPA has estimated that each ton of emitted GHG causes $10 of external damage to society. Each plant can take costly action to lower its emissions. The total abatement cost functions for each plant are Cz(a2) = baz + 70; If plant 1 chooses abatement level aj, it emits e, = E - a, tons of GHGs. When plant 1 chooses the most cost-effective means of reducing emissions by a1, its profits are reduced by Ci(a, ) relative to emitting E, tons. (a) We'll first determine the socially optimal level of emissions for each plant. i. Derive the marginal abatement cost function for each plant. ii. What are the socially optimal abatement and emissions levels for each plant? Denote your solution (ap , a;) for abatements and (er , e?) for emissions. ili. What is the total abatement cost paid by each plant at the social optimum? (b) Suppose the EPA mandates that each plant reduce its emissions by 15 tons (a] = 02 = 15). i. What are the total abatement costs for the two plants? How do total emissions damages and abatement costs compare to the social optimum? ii. Explain in words why requiring equal emissions reductions by the two plants is suboptimal. (c) Suppose instead that each plant must reduce emissions by 40% relative to the status quo, i.e. a1 = 20 and as = 10. What are total abatement costs and emissions damages for the two plants, and how do they compare to the social optimum? (d) Now suppose that the EPA implements a "cap-and-trade" system which allows plants to buy and sell pollution rights. Specifically, the EPA gives the two plants credits which allow them to emit ef and e; tons of GHGs (this is equivalent to requiring abatement levels of = E1 - ef and a? = Ex - e?). Then, plant 1 may sell plant 2 a credits at any per-unit price p that plant 2 will accept. This means plant 2 pays p .a to plant 1 and may emit ef + a tons, while plant 1 may emit at most ef - a tons. Of course, firm 2 may instead sell some of its right to pollute to firm 1. i. Suppose the EPA initially assigns credits (ei, e;) = (40, 5). In other words, it assigns plant 1 a 10-ton reduction, and plant 2 a 20-ton reduction. What is the equilibrium emissions price and the emissions level of each plant? ii. Suppose instead the initial credit assignment is (ei, e;) = (20, 25). How does your answer to (i) change? iii. Compare the profits of each plant under (i) and (ii). iv. Explain in words how your answer to parts (i)-(iii) is related to the Coase Theo- rem