8. Bob, Carol, and Ted are residents of a tiny commune in darkest Peru. Bob currently has a utility level ( U b ) of 55 utils, Carol’s utility ( U c ) is 35 utils, and Ted’s utility ( U t ) is 10 utils. Alice, the benevolent ruler of the commune, has discovered a policy that will allow her to redistribute utility between any two people she chooses in a util-to-util transfer.

a. If Alice believes the social welfare function is given by W = min( U b , U c , U t ), i. Recommend a transfer that will improve social welfare, if any such transfers are possible.

ii. What is the highest level of welfare that the commune can achieve, and how must utility be divided among Bob, Carol, and Ted?

b. If Alice believes the social welfare function is given by W = U b + U c + U t , i. Recommend a transfer that will improve social welfare, if any such transfers are possible.

ii. What is the highest level of welfare that the commune can achieve, and how must utility be divided among Bob, Carol, and Ted?

c. If Alice believes the social welfare function is given by W = U b × U c × U t , i. Recommend a transfer that will improve social welfare, if any such transfers are possible.

ii. What is the highest level of welfare that the commune can achieve, and how must utility be divided among Bob, Carol, and Ted?