The Rocky Mashed Potato Factory produces output at costs \(C=Q^{2}\) (marginal costs \(2 Q\) ), where \(Q\) is the quantity of mashed potatoes produced, in tons. In addition, 2 units of emissions are produced for each ton of mashed potatoes \((E=2 Q)\). Pollution damage is \(\$ 2\) for each unit of emissions, which leads the government to charge \$2 per unit of emissions as a Pigovian fee. The firm's output sells competitively for \(\$ 10\) per ton.

a. How many tons of mashed potatoes will the Rocky Mashed Potato Factory produce? How much does it pay in emission fees? What are its profits?

b. A device is invented that would reduce the firm's emissions to one unit for each ton of output \((E=Q)\). How much would the firm be willing to pay for such a device?

c. How would your answer to part (b) change if there were no government regulation of pollution emissions? What does this lead you to say about the relationship between government regulation and the market for pollution abatement equipment?