Three roommates, Jim, Saleem, and Ritesh, are thinking about buying a new speaker system for their apartment. The speaker system would be a public good if they buy them, and the total cost would be $300 which would be shared equally among the three. Jim values it at $80, Saleem values it at $140, and Ritesh values it at $70. If the speakers are purchased each person gets a payoff equal to their net valuation (valuation minus cost share); if they are not purchased each gets zero payoff. a)Is it socially efficient that they buy the speaker system? With reference to the definition of a public good, why or why not? The roommates decide that they will write down their net valuations, and if these reports sum to more than zero, they will buy the TV (sharing the cost equally), or else they will not. b) If all were to report their true valuations, what payoff would each agent earn? c) Show that if Jim and Ritesh submit their true net valuation, then Saleem does better by submitting a false report than his true net valuation. Now say that there will be a Clarke tax. That is, pivotal agents will have to pay a tax that is equal to the absolute value of the sum of the other agents’ reported net valuations (whether the good is provided or not). d) If all were to report their true valuations, what payoff would each agent earn now? e) If Jim and Ritesh submit their true net valuation, show that Saleem now prefers to submit a true report rather than the false report from c). f) Explain why in e) the Clarke tax aligns Saleem’s private interest with the social interest when he is deciding on his report

Please solve part d,e and f!!!

Three roommates, Jim, Saleem, and Ritesh, are thinking about buying a new speaker system for their apartment. The speaker system would be a public good if they buy them, and the total cost would be $300 which would be shared equally among the three. Jim values it at $80, Saleem values it at $140, and Ritesh values it at $70. If the speakers are purchased each person gets a payoff equal to their net valuation (valuation minus cost share); if they are not purchased each gets zero payoff.

a)Is it socially efficient that they buy the speaker system? With reference to the definition of a public good, why or why not?

The roommates decide that they will write down their net valuations, and if these reports sum to more than zero, they will buy the TV (sharing the cost equally), or else they will not.

b) If all were to report their true valuations, what payoff would each agent earn?

c) Show that if Jim and Ritesh submit their true net valuation, then Saleem does better by submitting a false report than his true net valuation.

Now say that there will be a Clarke tax. That is, pivotal agents will have to pay a tax that is equal to the absolute value of the sum of the other agents’ reported net valuations (whether the good is provided or not).

d) If all were to report their true valuations, what payoff would each agent earn now?

e) If Jim and Ritesh submit their true net valuation, show that Saleem now prefers to submit a true report rather than the false report from c).

f) Explain why in e) the Clarke tax aligns Saleem’s private interest with the social interest when he is deciding on his report.

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