Seller supplies only one good to the consumer. Let x be 1 if trade takes place and 0 if not. Let tQ be the amount of money that the seller receives and tB the amount that the buyer pays. Let v be the buyer’s valuation and s be the seller’s valuation of the object and assume preferences are quasi-linear. We can then normalize utilities so that the seller’s utility is tQ – cx2 and the buyer’s utility is vx2 – tB. Assume the preference parameters v and c are independently drawn from a uniform distribution on [0,1]. The buyer knows v and the seller knows c.
a. What is the efficient rule for trade?
b. Let mS and mB be the seller’s respectively the buyer’s reported preference for the object. Determine the set of direct mechanisms (expressed as a function of the reports) that admit truth telling as a dominant strategy and implement efficient trade
c. Show that there is a unique Groves mechanism that has the property that whenever trade does not occur, the transfer payments are set equal to zero (tB = tS = 0). Is this mechanism feasible?
d. Show that there is no Groves mechanism for which the budget breaks even for all reported preferences.
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