Three friends, Archie, Betty, and Veronica, are planning a party. They disagree about how many people to invite. Each person I has a quasilinear utility function of the form mi + ui(x) where mi is the NAME 463 number of dollars that i has to spend and x is the number of guests invited to the party. Suppose that for each i,
ui = aix – 1/2 x2
Everyone knows the functional form of the others’ utility functions and knows his own value of ai but does not know anyone else’s value. Let us suppose that the actual values of ai are 20 for Archie, 40 for Betty, and 60 for Veronica.
(a) How many guests should be invited to maximize the sum of the three persons’ utilities?
(b) Suppose that the three friends decide to use the VCG mechanism to determine the number of guests. If each plays his or her best strategy, how many guests will be invited?
(c) In the VCG mechanism, if the amount of public good supplied is x, Archie would receives a side payment equal to the sum of Betty’s and Veronica’s utility for x. If Betty and Veronica play their best strategies (without colluding) and if the amount of public good is x, this side payment will be __________. If everybody plays their best strategy, the amount of this side payment in dollars is _________
(d) In addition to receiving side payments, the VCG mechanism requires that each person must pay an amount equal to the maximum possible sum of the other two persons’ utilities. If Betty and Veronica play their best strategies this amount is ___________ On net, Archie has to pay the difference between this amount and the side payment that he receives. If everybody plays their best strategy, what is the net amount that Archie must pay? _________
(e) If everybody plays their best strategy, what is the net amount that Betty has to pay? ________ What is the net amount that Veronica has to pay? __________
(f) Suppose that the party is organized not by just three people, but by a dormitory with 21 residents. All of these residents have utility functions of the same form as Archie, Betty, and Veronica. Seven of them have ai = 20, seven have ai = 40, and seven have ai = 60. In order to maximize the sum of the residents’ utilities, how many guests should be invited? ____________ If there were only six persons with ai = 20, seven with ai = 40 and seven with ai = 60, how many guests would have to be invited in order to maximize the sum of utilities? _________
(g) If everybody plays their best strategy in the VCG game, then after all side payments and taxes are collected, how much net tax will each of the people with ai = 20 have to pay? ______ How much net tax will each of the people with ai = 40 have to pay? __________ How much net tax will each of the people with ai = 60 have to pay? ___________