Optimal selling mechanisms under imperfect commitment: Extending to the multi-period case

This paper studies the optimal mechanism for a seller (she) that sells, in a sequence of periods, an indivisible object per period to the same buyer (he). Buyer's willingness to pay remains constant along time and is his private information. The seller can commit to the current period mechanism but not to future ones. Our main result is that a seller cannot do better than posting a price in every period. We give a complete characterization of the optimal mechanism and equilibrium payoffs for every prior. Also, we show that, when agents are arbitrarily patient, the seller does not learn about buyer's type except in extreme cases, posting a price equal to the minimum buyer's willingness to pay in every period. This result is a reminiscence of the Coase's conjecture, where a monopolist cannot exert her monopoly power due to the lack of long-term commitment.