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ECON 372 - Assignment #1

1. Consider a setting in which two potential contributors to a public good, measured as total quantity , decide on their own how much to contribute. The public good is, for example, clean up of a lake that is commonly shared by the two individuals. The benefits and costs are as follows where :

Individual 1: The utility function for individual 1 is:

Individual 2: The utility function for individual 2 is:

where  is individual ’s consumption of a private good, , and  is each individual’s income. Assume that . Note that for each player:.

A. How much would each player contribute to the public good in the symmetric Nash equilibrium? (2 points)

B. If a hypothetical government wishes to maximise , what is the optimal quantity of public good that should be provided? (2 points)

C. Assume that both players discuss the possibility of implementing an agreement where each player would contribute half of the socially optimal amount (which you calculated in item B above). Suppose that player 2 agrees and blindly supports the agreement; that is, player 2 provides half of the socially optimal quantity. What is player 1’s optimal response? (Assume that this player is individually rational and wishes to maximise his/her utility). (2 points)

D. Compute the payoffs for two strategies that each player can play:

N = no cooperation (i.e., the opportunistic payoff where a player maximizes his/her utility)

C = cooperation (i.e., the payoff a player obtains from playing the socially desirable option) (2 points)

E. Show that the game played by the two players, where the strategies are N or C, described in question D above imply that the dominant strategy for each player is N while the socially optimal choice would be C. This is the essence of a Prisoners’ dilemma. (2 points)

2. Consider a global economy containing three nations, which differ from each other only in terms of their income levels. The income levels are , , . The nations abate their carbon emissions. Let  denote the abatement level produced by nation ,  The nations’ payoffs are as follows:

,

where  is individual ’s consumption of a private good, .

A. What are the abatement levels produced by the nations in the symmetric Nash equilibrium? (2 points)

B. What are the payoff levels in the symmetric Nash equilibrium? (2 points)

C. Let  denote the payoff level obtained by nation , , in the symmetric Nash equilibrium which you calculated in part B above. Let  denote the level of global welfare in the symmetric Nash equilibrium. Consider now a universal International Environmental Agreement (IEA) in which nation  chooses  to maximize , where , , , . Calculate each nation’s abatement contribution and payoff in the symmetric Nash equilibrium for the universal IEA. (2 points)

D. Consider a partial IEA in which nations 1 and 2 participate but nation 3 does not. Nation  chooses  to maximize   where , . Nation 3 chooses  to maximize . Calculate each nation’s abatement contribution and payoff in this partial IEA setting. (2 points)

E. List the nations’ payoffs in the symmetric Nash equilibrium without an IEA (part B), in the symmetric Nash equilibrium in the universal IEA (part C) and in the Nash equilibrium for the partial IEA you calculated in part D. Evaluate nation 3’s decision of participating or not participating in the universal IEA when the nation is short sighted and when the nation is far sighted. (2 points)