Three friends, Arsenio, Berta and Vernica, are organizing a party. They do not agree on the number of people to invite. Each person i has a quasilineral utility function of the form mi + ui (x), where mi is the number of euros that i has to spend and x is the number of people invited to the party. Suppose that for each i,

u_i = a_ix - 1/2 x²

Everyone knows the functional form of the utility functions of others and knows what his own value of ai is but not the value of the others. Suppose that the effective values of ai are 20 for Arsenio, 40 for Berta, and 60 for Veronica.

a How many people should be invited to maximize the sum of the profits of the three people

b Suppose the three friends decide to use the VCG merchandise to decide the number of guests. If each one chooses their best strategy, how many people will they invite

c In the VCG mechanism, if the quantity supplied of the public good is x, Arsenio would receive a payment equal to the sum of the utility that x reports to Berta and Vernica. If Berta and Vernica choose their best strategies without colluding and if the amount of the public good is x, this payment will be _____. If everyone chooses their best strategy, this payout will be _____.

d The VCG mechanism, in addition to requiring payments, requires each person to pay an amount equal to the maximum possible sum of the other two people's profits. If Berta and Veronica choose their best strategies, this amount is _____. The net amount that Arsenio has to pay is the difference between this amount and the payment he receives. If everyone chooses their best strategy, what is the net amount that Arsenio has to pay?

e If everyone chooses their best strategy, what is the net amount that Berta has to pay What is the net amount that Vernica has to pay

f Suppose that the party is not organized by just three people but by a university residence in which 21 students live. All these residents have useful functions in the same way as those of Arsenio, Berta and Vernica. Seven have ai = 20, seven have ai = 40, and seven have ai = 60. To maximize the sum of residents' profits, how many people should be invited If there were only six people who had ai = 20, seven would If they saw ai = 40 and seven had ai = 60, how many people would have to be invited to maximize the sum of the s profits

g If everyone chooses their best strategy in the VCG game, once all the payments and taxes have been collected, how many net taxes will each of the people who have ai = 20 have to pay How many net taxes will they have to pay? pay each of the people who has ai = 40 How many net taxes will each of the people who have ai = 60 have to pay