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The first part of this lecture put auctions in a more general context, by highlighting the similarities and differences between auctions and monopolies. In this spirit we investigate the sale of multiple units by auction, to see when the selling mechanism affects the outcome, and how.
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The first part of this lecture put auctions in a more general context, by highlighting the similarities and differences between auctions and monopolies. In this spirit we investigate the sale of multiple units by auction, to see when the selling mechanism affects the outcome, and how. The latter parts of the lecture analyze other aspects of monopolistic practices. We discuss mechanisms for setting prices and quantities, the role of commitment, market segmentation, and product bundling. Lecture 4Multiunit Auctions and Monopoly
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? • We now begin to make the transition between auctions and markets by noting the similarities and differences.
The two main differences distinguishing models of monopoly from auction models are related to the quantity of the good sold: • 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. • 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. Two main differences between most auction and monopoly models
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. • 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. Other differences between most auction and monopoly models
An agenda for the first part of the lecture • We will focus on three issues: • How does a multiunit auction differ from a single unit auction? • What can we learn about market behavior from multiunit auctions? • How does a uniform pricing monopolist set price and quantity?
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 v1 through vN. We will begin by considering situations where each buyer wishes to purchase at most one unit of the good.
Decisions for the seller to makein multiunit auctions • The seller must decide whether to sell the objects separately in multiple auctions or jointly in a single auction. • The seller must choose among different auction formats.
Open auctions for selling identical units • Descending Dutch auction: • Suppose the auctioneer has five units for sale. As the price falls, the first five 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 five highest bidders at the price the sixth bidder drop out.
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.
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.
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, either at the respective prices they posted, or at some common price that all the buyers have indicated they are willing to pay. It could invite sealed bid offers from customers in a kth price auction (where k could range from 1 to N.)
Prices follow a random walk • In the repeated Dutch and English auctions, 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.
Revenue equivalence revisited • Suppose that each bidder: - knows her own valuation, or alternatively has an independent signal about her valuation drawn from the same probability distribution • is risk neutral • Consider two auctions that have the same allocation mechanism (the mapping from the valuations to the winner(s) of the auction. • 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).
Multiunit demanders By a multiunit demander we mean that each bidder might desire (and bid on) all Q units for himself. We now drop the assumption that N > Q. Relaxing the assumption that each bidder demands one unit at most seriously compromises the applicability of the Revenue Equivalence theorem. Typically auctions will not yield the same resource allocation even if the usual conditions are met (private valuations, risk neutrality, lowest feasible expects no rent from participation).
Example: Two unit demanders in a third price sealed bid auction Consider a third price sealed bid auction for two units where there are two bidders, each of whom wants two units. Thus N = Q = 2. Each bidder submits two prices. We suppose the first bidder has a valuation of v11 for his first unit and v12 for for his second, where v11 > v12 say. Similarly the valuations of the second bidder are v21 and v22 respectively, where v21 > v22.
Example continued • The arguments given for single unit second price sealed bid auctions apply to the highest price of each bidder. One of his prices is highest valuation. • There is some probability that each bidder will win one unit, and in this case the price paid by one of the bidders will be determined by his second highest bid. Recognizing this in advance, he shades his valuation on his second highest bid.
Vickery auctions defined A Vickery auction is a sealed bid auction, and units are assigned according to the highest bids (as usual). To calculate how much each bidder pays for the unit(s) he has won, we define the losing bids he displaced. The losing bids he displaced would have been included within the winning set of bids if the bidder had not participated in the auction, and everybody else had submitted the same bids. In a single unit auction this corresponds to the second highest bidder. The total price a bidder pays in a Vickery auction for all the units he has won is the sum of the bids on the units he displaced.
Vickery auctions are efficient A Vickery auction is the multiunit analogue to a second price auction, in that the unique solution (derived from weak dominance) is for each bidder to truthfully report his valuations. This implies that a Vickery auction allocates units efficiently, in contrast to many multiunit auction mechanisms.
Choosing quantity • When analyzing monopoly, an important issue is the quantity the monopolist chooses to supply and sell. • Regulators argue that compared to a competitively organized industry where there are many firms supplying the product, a monopolist restricts the supply of the good and charges higher prices to high valuation demanders in order to make rents out of his position of sole source. • Is this true in practice?
Reservation prices for auctions • One reason for an auctioneer to set a reservation price is because of the value of the auctioned item to him if it is not sold. This value represents the opportunity cost of auctioning the item. For example he might sell it at another auction at some later time, and maybe use the item in the meantime. • Should the auctioneer set a reservation above its opportunity cost?
Auction Revenue • What is the optimal reservation price in a private value, second price sealed bid auction, where bidders are risk neutral and their valuations are drawn from the same probability distribution function? • Let r denote the reservation price, let v0 denote the opportunity cost, let F(v) denote the distribution of private values and N the number of bidders. Then the revenue from the auction is: K N N 1 N 2 ? X ? F Ý r Þ v NF Ý r Þ ß 1 ? F Ý r Þ à N v Ý N ? 1 Þ F Ý v Þ F v Ý v Þ dv + + 0 r
Solving for the optimal reservation price • Differentiating with respect to r, we obtain the first order condition for optimality below, where r0 denotes the optimal reservation price. • Note that the optimal reservation price does not depend on N. • Intuitively the marginal cost of the top valuation falling below r, so that the auction only nets v0 instead of r0, equals the marginal benefit from extracting a little more from the top bidder when he is the only one bidder to beat the reservation price.
The uniform distribution • When the valuations are distributed uniformly with: • then: o r Ý v v Þ / 2 = + 0
Designing a monopoly game with a quantity choice • In the game below, the valuations of buyers are uniformly distributed between $10 and $20. • Each buyer is endowed with $20. • The monopolist’s production capacity is 100 units of the good. The marginal cost of producing each unit up to capacity is constant at $10. • What is the equilibrium quantity bought and sold?
A static approach The traditional argument can be framed as follows. Let: c denote the cost per unit produced, consumers demand quantity q(p) when the price is p and the function Assume q(p) is differentiable and declining in p, and write p(q) as its inverse function. That is q(p(q)) = q. The monopolist chooses q to maximize (p(q)– c) q
Marginal revenue equals marginal cost Let qm denote the profit maximizing quantity supplied by the monopolist. Then qm satisfies the first order condition for the optimization problem, which is: p(qm) + p’(qm) qm = c The two terms on the left side of the equation comprise the marginal revenue from increasing the quantity sold. When an additional unit is sold it fetches p(q) if we ignore any downward pressure on prices. The traditional argument is that the monopolist will only produce sell an extra unit if the marginal revenue from doing so exceeds the marginal cost.
Demand schedule In this example the marginal cost is $10. 18 16 14 12 10 8 6 4 2 0
Eleven buyers and one seller 20 19 18 17 16 15 14 13 12 11 10 - - - - - - - - - - - MC=10 | | | | | | | | | | | q 1 2 3 4 5 6 7 8 9 10 11
Solution to game • There are two outputs that yield the maximum profit, which is $30. • If the monopolist offers 6 units for sale, the market will clear at a price of $15. • If the monopolist offers 5 units for sale, the market will clear at a price of $16.
The solution illustrated Price in dollars 20 inverse demand curve Uniform price solution unit cost 10 marginal revenue curve quantity 0 Uniform quantity solution
Intermediaries with market power • We typically think of monopolies owning the property rights to a unique resource. Yet the institutional arrangements for trade may also be the source of the monopoly power. Consider the NYSE, in which a dealer for each stock sets the spread, and traders only have the opportunity to make market buy and sell orders. • In auction terminology, each dealer is playing a multiunit Dutch auction on both sides of the market, simultaneously determining the units to be traded. • How should a dealer set the spread on the stock he manages? A small spread encourages greater trading volume, but a larger spread nets him a higher profit per transaction.
A Summary This first part of this lecture has emphasized the links between auctions and monopoly, and thus established the close connections between auctions and markets. We discussed pricing multiunit sales, setting quantities, segmenting the market to achieve discrimination, the role of commitment in multiunit settings, and product bundling which induce consumers to self select product packages. As we noted in our application to the dealer markets, the lessons form auction theory carry over to more complicated settings than single unit auctions, but nevertheless they are harder to apply.
