Auctions with Entry

A robust prediction of auction theory is that reserve prices will be used in auctions. The optimal selling mechanism for the owner of a unique, indivisible item is to sell to the bidder with the highest valuation unless this valuation is below a cut-off level; and this cut-off valuation is strictly above the seller’s own valuation. Thus the monopolist imposes a distortion: with positive probability he will refuse to sell the item even though there is a bidder who values it more than he does. ’ In practice, reserve prices are often, perhaps usually, not used in auctions [Cassady (1967, pp. 226-227)]. Thus, there appears to be a discrepancy between theory and practice. In this paper, we relax the assumption, universally used in auction theory, that there is a fixed number of bidders. Instead, we make the assumption, arguably more economically reasonable, that any bidder can enter the bidding upon paying an entry cost (which may be interpreted as the cost of learning what the item is worth or the cost of preparing a bid). We find that the seller should never reject a bidder with a valuation higher than his own. In addition we find that, if the seller uses a first-price sealed-bid auction, the optimal number of bidders will enter. Thus, the optimal auction in this case is the first-price sealed-bid auction with a reserve price equal to the seller’s own valuation. The seller designs an efficient selling mechanism. If we add the non-restrictive assumption that any bidder’s valuation exceeds the seller’s, then, consistent with observation, no reserve price is used. * In the usual auction analysis, with the number of bidders given exogenously, the seller uses the reserve price to extract rents from the winning bidder. But with entry of bidders, the winner’s rent is fixed, regardless of the seller’s policy. Thus there is no gain from introducing a distortion by using a reserve price.