An auction with positiveexternalities. Suppose that there is a single object to be allocated among 3 individuals. Each individual i has a willingness to pay vi for the object. Individuals 1 and 2 have a zero value when she does not getthe object.
Individual 3 on the other hand would have a positive value e ifindividual 1 were allocated the object and a zero value if individual 2 wereallocated the object.

(a) Determine the utilitarian outcome as a function of v1; v2; v3and e:

(b) Just as in the case of a negative externality, the VCG mechanism reduces to a modi ed Vickrey auction for this problem. Calculate the VCGpayments for all three individuals in the case that v1 +e > v2 > v1 > v3:Explain these payments in terms of the \bene t/harm to society" intuition developed inclass.