1. Lecture 4 on Auctions Multiunit Auctions We begin this lecture by comparing auctions with monopolies. We then discuss different pricing schemes for selling multiple units, the choice of how many units to sell, and the joint determination of price and quantity.

2. Are auctions just like monopolies? Monopoly is defined by the phrase “single seller”, but that would seem to characterize an auctioneer too. Is there a difference, or can we apply everything we know about a monopolist to an auctioneer, and vice versa? How does a multiunit auction differ from a single unit auction? What can we learn about market behavior from multiunit auctions?

3. Two main differences between most auction and monopoly models The two main differences distinguishing models of monopoly from a auction models are related to the quantity of the good sold: 1.Monopolists typically sell multiple units, but most auction models analyze the sale of a single unit. In practice, though, auctioneers often sell multiple units of the same item. 2.Monopolists choose the quantity to supply, but most models of auctions focus on the sale of a fixed number of units. But in reality the use of reservation prices in auctions endogenously determines the number the units sold.

4. Other differences between most auction and monopoly models 1.Monopolists price discriminate through market segmentation, but auction rules do not make the winner’s payment depend on his type. However holding auctions with multiple rounds (for example restricting entry to qualified bidders in certain auctions) segments the market and thus enables price discrimination. 2.A firm with a monopoly in two or more markets can sometimes increase its value by bundling goods together rather than selling each one individually. While auction models do not typically explore these effects, auctioneers also bundle goods together into lots to be sold as indivisible units.

5. Auctioning multiple units to single unit demanders Suppose there are exactly Q identical units of a good up for auction, all of which must be sold. As before we shall suppose there are N bidders or potential demanders of the product and that N > Q. Also following previous notation, denote their valuations by v 1 through v N. We begin by considering situations where each buyer wishes to purchase at most one unit of the good.

6. Open auctions for selling identical units Descending Dutch auction: As the price falls, the first Q bidders to submit market orders purchase a unit of the good at the price the auctioneer offered to them. Ascending Japanese auction: The auctioneer holds an ascending auction and awards the objects to the Q highest bidders at the price the N - Q highest bidder drops out.

7. Multiunit sealed bid auctions Sealed bid auctions for multiple units can be conducted by inviting bidders to submit limit order offers, and allocating the available units to the highest bidders. In discriminatory auctions the winning bidders pay different prices. For example they might pay at the respective prices they posted. In a uniform price auction the winners pay the same price, such as a k th price auction (where k could range from 1 to N.)

8. Revenue equivalence revisited Suppose each bidder: - knows her own valuation - only want one of the identical items up for auction - is risk neutral Consider two auctions which both award the auctioned items to the highest valuation bidders in equilibrium. Then the revenue equivalence theorem applies, implying that the mechanism chosen for trading is immaterial (unless the auctioneer is concerned about entry deterrence or collusive behavior).

9. Prices follow a random walk In repeated auctions that satisfy the revenue equivalence theorem, we can show that the price of successive units follows a random walk. Intuitively, each bidder is estimating the bid he must make to beat the demander with (Q+1) st highest valuation, that is conditional on his own valuation being one of the Q highest. If the expected price from the q s+1 item exceeds that of the q s item before either is auctioned, then we would expect this to cause more (less) aggressive bidding for q s item (q s+1 item) to get a better deal, thus driving up (down) its price.

10. Multiunit Dutch auction To conduct a Dutch auction the auctioneer successively posts limit orders, reducing the limit order price of the good until all the units have been bought by bidders making market orders. Note that in a descending auction, objects for sale might not be identical. The bidder willing to pay the highest price chooses the object he ranks most highly, and the price continues to fall until all the objects are sold.

11. Clusters of trades As the price falls in a Dutch auction for Q units, no one adjusts her reservation bid, until it reaches the highest bid. At that point the chance of winning one of the remaining units falls. Players left in the auction reduce the amount of surplus they would obtain in the event of a win, and increase their reservation bids. Consequently the remaining successful bids are clustered (and trading is brisk) relative to the empirical probability distribution of the valuations themselves. Hence the Nash equilibrium solution to this auction creates the impression of a frenzied grab for the asset, as herd like instincts prevail.

12. Why the Dutch auction does not satisfy the conditions for revenue equivalence We found that the revenue equivalence theorem applies to multiunit auctions if each bidder only wants one item, providing the mechanism ensures the items are sold to the bidders who have the highest valuations. In contrast to a single unit auction, the multiunit Dutch auction does not meet the conditions for revenue equivalence, because of the possibility of “rational herding”. If there is herding we cannot guarantee the highest valuation bidders will be auction winners.

13. Summary We began this lecture by comparing auctions with monopoly, and establishing some close connections. We found the revenue equivalence theorem applies to multiunit auctions if each bidder only wants one item. Intermediaries exploit their monopolistic position, by creating a wedge between their buy and sell prices. Although fixed price monopolies create inefficiencies, by restricting supply, perfect price discriminators produce where the lowest value consumer only pays the marginal production cost, an efficient outcome.