Can someone help me solve this? Thank you!

2. There are two consumers, Jim and TJ, and six bottles of a single good, beer. Both Jim and TJ like to consume beer but dislike when the other drinks beer, since then they have to put up with the other being drunk. Their utility functions are as follows, where 1;; is the number of bottles of beer allocated to Jim and Mn the number allocated to TJ: UJ = $J ($702 (1) UT = SET ($Jl2 (2) a) If Jim could dictate the allocation of beer, what would he choose? b) What number would a minimax social welfare function attach to the allocation in a)? c) Is an equal division of beer a Pareto improvement over the allocation in a)? Explain why or why not. (1) Is an equal division of beer welfare-improving under the minimax SWF? Ed is observing the situation and has decided to intervene. Jim and TJ agree to accept whatever allocation Ed decides. Ed believes that a utilitarian social welfare function is the best metric to judge allocations. e) What number would a utilitarian SWF attach to the allocation in a)? f) Ed is considering which allocation to implement. Assume first that he will not let any bottles of beer go to waste, so that aT = 6 -xJ. Solve Ed's welfare-maximization problem to find the best allocation under the utilitarian SWF when no bottles of beer are wasted. g) Ed thinks he can do better. Show that if Ed throws some of the beer away he can increase social welfare under the utilitarian SWF. h) Assuming the allocation cannot include fractions of a bottle (so that Jim and TJ must both end up with a whole number of bottles), at least how many bottles will Ed throw away? i) Is an equal division allocation of 3 bottles each Pareto efficient? Why or why not