Nonlinear Pricing with Resale
2008
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Abstract
We consider the problem of a monopolist-choosing an optimal nonlinear pricing scheme-facing two consumers who can resell some or all of the goods to each other in a secondary market. We suppose that the valuations of the consumers are drawn independently from a continuous distribution. We …nd conditions for the optimum direct mechanism and show that the monopolist can be better o¤ or worse o¤ as compared to the without resale case, depending on the speci…cs of the cost function of the monopolist and the utility functions of the consumers.



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In this paper, we investigate the model of speculative resale in auctions. Resale (or secondary trade) is allowed for each bidder, but in equilibrium we show that there is only speculative resale. We consider a standard …rst-price auction in the …rst stage with symmetric independent private values (IPV) among N bidders and several speculators. In the second stage there is resale among the bidders. The winner in the …rst stage auction uses an optimal mechanism to sell the object to the losing bidders. We establish a supermodularity property without assuming monotonicity or symmetry in bidding. We show the equilibrium bidding function of the regular bidders to be symmetric and increasing. We give a simple computable equilibrium solution and prove it to be unique. The revenue formula is extended to our model with speculative resale. Revenue monotonicity holds, and the Bulow-Klemperer (1996) argument favoring participation rather than the choice of optimal reservation price is also valid here with even greater force. Speculators'active participation enhances the seller revenue, but for optimal revenue, the seller should prevent speculative resale by setting a su¢ ciently high reservation price. Thus …rst-price auction provides an implementation of optimal auctions with a unique equilibrium, while allowing speculative resale.

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