2. National Defense (40 points). There are 11 countries in Europe who get utility from general...

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2. National Defense (40 points). There are 11 countries in Europe who get utility from general consumption c, and from European national defense G. The utility of a generic country i is u(ci, G) = c.G0.1 11 where G j=1 9j, and 9j denotes the contribution to national defense by country j. Prices of general consumption and of national defense are both equal to 1. Countries are identical and each has income of $100. 1 (a) Find the optimal level of national defense G* when each country independently chooses its own national defense contribution gi. (Hint: this is equivalent to find the private optimum, with the difference that individuals are replaced by countries.) (10 points) (b) What is the social optimum quantity G? That is, what is the quantity chosen by a utilitarian social planner who maximizes the sum of the 11 countries' utilities? (10 points) (c) Is G* lower than G? Why? Briefly Explain. (5 points) (d) Now suppose that the United States provides a $10 dollar overall amount of national defense at no cost for the European countries. Solve the problem as in part (a) and show that the US contribution crowds out European purchase of national defense. (10 points) (e) How would the answer to part (c) change if all the 11 countries were altruistic? Briefly explain (You can use math if you want, but it is not required). (5 points) 2. National Defense (40 points). There are 11 countries in Europe who get utility from general consumption c, and from European national defense G. The utility of a generic country i is u(ci, G) = c.G0.1 11 where G j=1 9j, and 9j denotes the contribution to national defense by country j. Prices of general consumption and of national defense are both equal to 1. Countries are identical and each has income of $100. 1 (a) Find the optimal level of national defense G* when each country independently chooses its own national defense contribution gi. (Hint: this is equivalent to find the private optimum, with the difference that individuals are replaced by countries.) (10 points) (b) What is the social optimum quantity G? That is, what is the quantity chosen by a utilitarian social planner who maximizes the sum of the 11 countries' utilities? (10 points) (c) Is G* lower than G? Why? Briefly Explain. (5 points) (d) Now suppose that the United States provides a $10 dollar overall amount of national defense at no cost for the European countries. Solve the problem as in part (a) and show that the US contribution crowds out European purchase of national defense. (10 points) (e) How would the answer to part (c) change if all the 11 countries were altruistic? Briefly explain (You can use math if you want, but it is not required). (5 points)

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a To find the optimal level of national defense G when each country independently chooses its own national defense contribution g we need to maximize ... View the full answer