Adjustable Supply in Uniform Price Auctions: Non-Commitment as a Strategic Tool

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1 Adjustable Supply in Uniform Price Auctions: Non-Commitment as a Strategic Tool David McAdams March 23, 2006 Abstract In some uniform price auctions, the auctioneer retains flexibility to adjust the total quantity sold after receiving the bids. Would such an auctioneer be better off committing to a fixed quantity and reserve price? Not necessarily. In the simplest possible model, this paper shows that auctioneer expected profit and social welfare are each strictly higher in all equilibria given adjustable supply than in all equilibria given any fixed quantity and reserve price. MIT Sloan School of Management. Post: E52-448, MIT Sloan, 50 Memorial Drive, Cambridge, MA I thank Susan Athey, Alessandro Pavan, Robert Wilson, participants at FERC, Maryland, Northwestern, and UC Energy Institute seminars, and especially Jeremy Bulow and Paul Milgrom. This research has been supported by the State Farm Companies Foundation and National Science Foundation grant SES

2 Following Wilson (1979), an extensive literature explores the fact that uniform price auctions have low-price equilibria. Even when all bidders are small relative to the market, there can be Nash equilibria of uniform price auctions in which prices remain far below the competitive price... The issue of [such] equilibria is plainly of great practical importance (Milgrom (2004) pp. 262, 264). More recently, several authors have explored ways for the seller to fight back by changing the rules, notably: by committing to a supply curve that is more elastic than true supply [LiCalzi and Pavan (2005)]; by restricting bids to a coarse set of quantities [Kremer and Nyborg (2004b)]; by committing to an allocation rule that does not necessarily assign objects to the highest bidders [Kremer and Nyborg (2004a), Damianov (2005) 1 ]; and by allowing the seller to adjust the quantity sold after receiving the bids, subject to a maximum constraint Q [Back and Zender (2001) 2 ] Many real-world uniform price auctions have the feature that the auctioneer has some latitude in deciding how much to sell after receiving the bids. For example, the Treasuries in Mexico, Italy, and Finland sometimes reduce the quantity of issued bonds after the bids have been received (Umlauf (1993), Scalia (1997), and Keloharju, Nyborg, and Rydqvist (2004)). IPOs in the United States incorporate the so-called Greenshoe Option allowing issuing firms to increase the amount of shares being offered by up to 15% after the bids. 3 1 Damianov (2005) allows for adjustable supply and shows that all equilibria lead to the Walrasian price, but the results are incomparable. Damianov (2005) s model is more general in that it allows for any elasticity of demand, but his results depend importantly on the allocation rule. 2 Back and Zender (2001) s constraint is natural when selling physical assets that must be produced before the auction. When selling financial assets or buying physical assets, however, the cost of production is typically incurred after production. For example, the cost of selling Treasury bonds is that they must be re-paid later. Similarly, the value from buying goods in a procurement auction is that they can be consumed later. In these situations, there is no upper bound on quantity at the time of auction. 3 Often only a portion of this option is exercised. For instance, when IT consulting firm Wincor 2

3 This paper examines the consequences of giving the seller total latitude to increase or decrease the quantity sold after receiving the bids ( adjustable supply ). 4 This would be the case if the seller were unable to commit to any rule constraining her ability to adjust quantity. In the simplest possible setting with complete information and perfectly elastic demand, all equilibria lead to the Walrasian price and quantity. Consequently, all equilibria given adjustable supply lead to strictly greater expected profit and welfare than all equilibria given any fixed quantity and reserve price. Model. The (female) seller has total cost C(Q) = Q MC(x)dx when producing Q > 0 x=0 units of a homogenous, perfectly divisible good. (The analysis can easily be extended to include fixed costs.) Marginal cost M C(Q) is strictly increasing, continuous, and unbounded above. Each of n bidders has constant marginal value MV (q) = v for all q. v is commonly known to the bidders but the seller only knows its distribution. v has interval support [0, v] for some v <. 5 Each bidder i simultaneously submits a non-increasing and left-continuous demand schedule (or bid ) d i specifying his total demand d i (p) at each price p 0. 6 Given a profile of bids (d 1,..., d n ), aggregate demand D(p) n i=1 d i(p). The seller sets total quantity so as to maximize her ex post profit given the bids ( adjustable supply ). The allocation among the bidders and the price paid is then Nixdorf went public in June 2004, its underwriter Goldman, Sachs, & Co. had the right to issue an additional 1.2 million shares. It chose to issue 852,131 additional shares and hence ultimately had the ability to either increase or decrease the number of shares. 4 This paper subsumes the complete information example in McAdams (1998), which analyzes adjustable supply in a richer model with incomplete information and downward-sloping demand. The first to study uniform price auctions with adjustable supply was Lengwiler (1999). 5 This simple information structure is common in the literature. See e.g. Wilson (1979), Back and Zender (1993), LiCalzi and Pavan (2005), and Kremer and Nyborg (2004a, 2004b). 6 Bolded items refer to functions whereas unbolded items refer to scalars. 3

4 determined by standard uniform price auction rules. (This formulation follows LiCalzi and Pavan (2005).) Every bidder pays the stop-out price p (d 1,..., d n ) for all quantity that he receives, defined as p (d 1,..., d n ) = sup{p D(p) SR(d 1,..., d n )} when {p D(p) SR(d 1,..., d n )} ; otherwise define p (d 1,..., d n ). (Negative prices play no part in the analysis, but allowing for them eliminates some technical complications.) When it can not cause confusion, I will simply refer to the stop-out price as p. E(p) D(p) SR(d 1,..., d n ) 0 is the excess demand. When E(p) = 0, each bidder is allocated quantity d i (p) = d i (p). Otherwise excess demand is rationed pro rata at the margin. Let d i (p) d i (p) lim p p d i ( p) and D(p) n i=1 d i(p). Then bidder i receives d i (p) = d i (p) d i(p) D(p) E(p). Theorem 1. In all equilibria of the uniform price auction given adjustable supply, Pr(p = v) = 1 and Pr(Q = MC 1 (v)) = 1. Proof. For any v, a bidding equilibrium exists: all bidders submit their true, perfectly elastic demand at price v and the seller supplies the efficient quantity. Given that others announced demand is perfectly elastic at v, there is no way for bidder i to make positive profits whatever he bids. Thus, he is willing to announce perfectly elastic demand at v as well. To complete the proof I need to show that, for all v, all bidding equilibria lead to price p = v and total quantity Q = MC 1 (v). Suppose that (d A 1,..., d A n ) is an equilibrium that leads to price p A, total quantity Q A, and allocation (q A 1,..., q A n ). 7 (Recall that v is fixed and common knowledge among the bidders but unknown to the seller.) 7 This proof also rules out mixed-strategy equilibria in which price is less than v with positive probability. In particular, let p A be the infimum of all prices that are realized in equilibrium. When suitably modified, the arguments developed here apply to this lowest realized price and hence prove that p A = v. 4

5 Part I: Price equals v. Clearly price can never exceed v in equilibrium. Suppose that there is an equilibrium in which p A < v. Several steps will lead to a contradiction. First step. The seller does not withhold any quantity, i.e. Q A = MC 1 (p A ). Suppose otherwise, that each bidder i gets q A i the deviation d 1 for bidder 1 from his equilibrium bid d A 1 : units with Q A = n i=1 qa i < MC 1 (p A ). Consider d 1 (p) = d A 1 (p) for all p > p A (1) = for all p p A When bidder 1 deviates in this way, the seller will supply MC 1 (p A ) at the same price p A and bidder 1 will get an additional MC 1 (p A ) Q A units. Since p A < v, this deviation makes bidder 1 strictly better off. (See Figure 1. The unlabelled curve is the seller s ex post marginal revenue curve before the deviation.) Given others bids, define bidder i s residual supply RS i (p) MC 1 (p) j i da j (p). The previous step implies that q A i = RS i (p A ) for each bidder i. Some of bidder i s deviations may induce a price and quantity not on his residual supply curve, but residual supply specifies the maximum quantity that he can possibly get at any given price. A definition is useful as shorthand for the next step. A non-decreasing function f is perfectly elastic to the right of x if either there is a right-discontinuity, lim ε 0 f(x+ε) > f(x), or an infinite right-derivative, lim ε 0 f(x+ε) f(x) ε =. Second step. Note that the seller withholds no quantity only if announced aggregate demand is perfectly elastic to the right of p A. Otherwise the seller can raise profits by raising the price and reducing the quantity sold. This in turn requires that some bidder i must announce demand that is perfectly elastic to the right of p A. Without loss, suppose that i 1. Thus, bidder 1 s residual supply must be perfectly elastic to the right of p A as in Figure 2. 5

6 Now, consider the deviation d ε 1 for bidder 1, also illustrated in Figure 2: d ε 1(p) = d A 1 (p) for all p > p A + ε (2) = for all p p A + ε After this deviation, the seller will supply S = MC 1 (p A + ε) and q 1 = MC 1 (p A + ε) j 1 da j (p A + ε). Consequently, bidder 1 must gets Z 1 (ε) ( MC 1 (p A + ε) MC 1 (p A ) ) ( + qj A ) d A j (p A + ε) j 1 j 1 additional quantity at price p A + ε and pays an additional ε on his equilibrium quantity q A 1. Yet, by the previous discussion, j 1 da j is perfectly elastic to the right at price p A. By presumption, however, MV (q A 1 ) > p A so we conclude (v p A )Z 1 (ε) lim ε 0 εq1 A = (3) Thus, this deviation is profitable for bidder 1 for all small enough ε > 0. Part II: The outcome is efficient, i.e. Q A = MC 1 (v). Suppose that p A = v but that Q A < MC 1 (v). Consider the deviation for bidder 1, as illustrated in Figure 3: d ε 1(p) = d A 1 (p) for all p > v ε (4) = for all p v ε For small enough ε, the seller s best response is to supply MC 1 (v ε) units at price v ε so that bidder 1 gets MC 1 (v ε) D(v ε) > 0 additional units. Furthermore, bidder 1 pays a lower price than before on the rest of his quantity. Thus, this is a profitable deviation for bidder 1. Finally, p A = v and Q A > MC 1 (v) is impossible since the seller must prefer to sell less. We have proven the desired result: in all equilibria, the outcome is always efficient and the seller always extracts all bidder surplus. 6

7 References Back, K., and J. Zender (2001): Auctions of Divisible Goods with Endogenous Supply, Economics Letters, 73(1), Damianov, D. (2005): The uniform price auction with endogenous supply, Economic Letters, 89(1), Keloharju, M., K. Nyborg, and K. Rydqvist (2004): Strategic Behavior and Underpricing in Uniform Price Auctions, Journal of Finance, 60(4), Kremer, I., and N. Nyborg (2004a): Divisible-Good Auctions: The Role of Allocation Rules, RAND Journal of Economics, 35(1), (2004b): Underpricing and Market Power in Uniform Price Auctions, Review of Financial Studies, 17(3), Lengwiler, Y. (1999): The Multiple Unit Auction with Variable Supply, Economic Theory, 14, LiCalzi, M., and A. Pavan (2005): Tilting the Supply Schedule to Enhance Competition in Uniform-Price Auctions, European Economic Review, 49(1), McAdams, D. (1998): Adjustable Supply and Collusive-Seeming Equilibria in the Uniform Price Auction, mimeo (Stanford). Milgrom, P. (2004): Putting Auction Theory to Work. Cambridge University Press. Scalia, A. (1997): Bidder profitability under uniform price auctions and systematic reopenings: the case of Italian Treasury bonds, Working paper (Bank of Italy). 7

8 Umlauf, S. (1993): An Empirical Study of the mexican Treasury Bill Auction, Journal of Financial Economics, 33, Wilson, R. (1979): Auctions of Shares, Quarterly Journal of Economics, 94,

9 Extra quantity price at no extra price v MC(Q) p A D A (p) Q A MC 1 (p A ) quantity Figure 1: If p A < v and Q A < MC 1 (v), then all bidders have a profitable deviation. price Extra quantity at ε extra price v p A + ε p A Bidder 1 s residual supply d A 1 (p) q A 1 q 1 1 s quantity Figure 2: If p A < v and Q A = MC 1 (v), then some bidder has a profitable deviation. 9

10 price Extra quantity at a lower price MC(Q) v v ε D A (p) Q A MC 1 (v ε) quantity Figure 3: If p A = v and Q A < MC 1 (v), then all bidders have a profitable deviation. 10

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