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Unformatted text preview: Optimal Auctions We wish to analyze the decision of a seller who sets a reserve price when auctioning off an item to a group of n bidders. Consider a seller who chooses an optimal reserve for a secondprice auction with one bidder. Clearly the seller who faces a single bidder should set a positive reserve, otherwise the sale price will be zero. In fact, the optimal reserve is equal to the monopoly price. It can be obtained by solving max r (1 r 100 ) r ⇔ max r r r 2 100 The firstordercondition is 1 r 50 = 0 So, the optimal reserve (or monopoly price) is r = 50 . Rod Garratt ECON 177: Auction Theory With Experiments Note that if the seller had her own use value for the item, v s , then her optimization problem becomes max r (1 r 100 ) r + r 100 v s ⇔ max r r r 2 100 + r 100 v s The firstordercondition is 1 r 50 + v s 100 = 0 So, the optimal reserve (or monopoly price) is r = 50 + v s 2 . Rod Garratt ECON 177: Auction Theory With Experiments Question Consider two scenarios. In scenario 1 the seller sells an item using a firstprice auction with no reserve to 2 bidders. In scenario 2 the seller sells an item to a single bidder, but sets the optimal reserve. Assume the bidder(s)’ value is drawn from the uniform distribution on [0 , 100] . Compute the expected seller revenue in each case. Answer: In scenario 1, the expected revenue to the seller is 33.33. In scenario 2, the optimal reserve is 50, and seller’s expected revenue is (since there is only one bidder) . 5 * 0 + . 5(50) = 25 ....
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This note was uploaded on 12/26/2011 for the course ECON 177 taught by Professor Garratt during the Fall '09 term at UCSB.
 Fall '09
 GARRATT

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