There are two countries, A and B. Their rms emit a chemical that needs to be controlled. However reducing emissions is costly, and because of technological differences, costs vary both across rms in a given country and across the two countries. Call 2' an individual rm in country A, emitting emissions er, and call 3' an individual rm in country B, with emissions ej.The costs of reducing emissions is given by the cost functions CA1 = (Z (16202 '2 03; = (j ejlz There are N ,4 firms in country A, and N 3 rms in country B. Hence 7; = 1,2, ...,NA and j = 1,2,...,NB. Suppose rst that each country wants to enforce its own ceiling on emissions, E A in country A and BB in country B. The two countries independently decide to do so via separate cap-andtrade systems. Each license is the right to emit one unit of the chemical compound. You can assume that in both countries the market for permits will be competitive. 1. Consider country A. (a) (5 points) Accounting for the cap-and-trade system, what level of emissions does rm 2' choose (as a function of pp, and parameters i and a)? (b) (5 points) If IDA: the price for permits in country A, is strictly positive (13,; > 0}, what is pA as a function of EAand NA, for given or? To save some time, note that 2:213. 2 NA (NA + 1)/2. (c) (3 points) What condition on E A must be satised for 19;; to be strictly positive? Suppose o: = 1/2 and NA = 10. (d) (3 points) What does your answer to (c) above become? Suppose now EA 2 100. As above, and for the remainder of the exam, a 21/2 and NA = 10. (e) (3 points) What is pA? (f) (3 points) What is e15? (g) (5 points) The rms in country A differ in their abatement costs. Since they are all facing the same price of cutting emissions, does this mean that some rms will be able to choose the same level of emissions they would have chosen in the absence of any regulations, and others will not? (h) (5 points) What is 13A and what are the emissions of rm '1' if EA 2 110? 2. Consider now country B. (a) (4 points) Accounting for the cap-and-trade system, what level of emissions does rm 3' choose (as a function of p5 and parameters 3' and B)? (b) (4 points) If p 3, the price for permits in country B, is strictly positive (PB > 0), what is 1);; as a function of EB and NB, for given ,8? (c) (3 points) What condition on E3 must be satised for p3 to be strictly positive? Suppose 6 = 1/4 and N3 = 20. (d) (3 points) What does your answer to (c) above become? Suppose now E B = 800. As above, and for the remainder of the exam, ,8 = 1/4 and N3 = 20. (e) (3 points) What is p3? (f) (2 points) What is 8,? (g) (4 points) Suppose E3 = 1000. What is p3 and what are the emis- sions of rm 3' now