Please solve all parts

02 (30 points) Rocia lives in a house with 9 other people. All the housemates want to purchase a coffee machine for everybody to share. If x tens of dollars are spent for the coffee machine, Rocia's gross utility is 113(27) = %]n :1: whereas each of the other housemates' utility is u(a:) = gar x? (a) Find the Pareto efcient amount of money to spend in the coee machine. Call it 3'. If the equilibrium has 2 solutions, take the positive one. (5 points) (b) Individually, how much does each want to be spent on the coffee machine if the cost is split evenly? Call the amounts at}; for Rocia and 2:,- for each other housemate. {5 points) (c) Does Rocia prefer 3' or 17,; to be spent under the even split rule? If a vote between the housemates is cast about spending 23' or 17g, which amount will be chosen? (4 points) (d) Rocia has heard about Lindhal, who proposed personalized contributions. Let T]; and 1',- , 1' = 2, . . . , 10 be the individual share of the expense each has to pay. Those shares are determined by equating the marginal utility (with respect to 3:) of each with their respective share. Under this rule, nd the individual shares and the spending amount 3L (don't forget shares add up to 1). Compare your answer with 13'. (10 points) (e) In practice, the rule proposed by Lindhal requires each agent to report their preferences (i.e. their utility function), so that the contributions can be computed for each agent. What is the problem with the implementation of that rule? If the housemates have to vote on whether to split the cost evenly or use Lindhal' rule, which rule will win? Explain. (6 points)