Elsevier

European Economic Review

Volume 49, Issue 1, January 2005, Pages 227-250
European Economic Review

Tilting the supply schedule to enhance competition in uniform-price auctions

https://doi.org/10.1016/S0014-2921(02)00324-0Get rights and content

Abstract

Uniform-price auctions of a divisible good in fixed supply admit underpricing equilibria, where bidders submit high inframarginal bids to prevent competition on prices. The seller can obstruct this behavior by tilting her supply schedule and making the amount of divisible good on offer change endogenously with its (uniform) price. Precommitting to an increasing supply curve is a strategic instrument to reward aggressive bidding and enhance expected revenue. A fixed supply may not be optimal even when accounting for the cost to the seller of issuing a quantity different from her target supply.

Introduction

In the last few years, uniform-price auctions have become a popular mechanism to allocate divisible goods. For instance, since September 1998, the U.S. Department of Treasury has switched from a traditional discriminatory format to the uniform-price auction to issue all its securities.1 Similarly, uniform-price auctions are now commonly used to run on-line initial public offerings (IPOs) of unseasoned shares (open IPOs), as well as in electricity markets and in markets for emission permits.

In a uniform-price auction, bidders compete by simultaneously submitting their demand schedules for the divisible good on offer. The seller compares the aggregate demand with her aggregate supply and computes a clearing (stop-out) price. Demand above the stop-out price is awarded in full, while marginal demand at the stop-out price is prorated. Since all buyers pay the same price, the uniform-price auction is analogous to a Walrasian market, with the only difference that demand schedules are submitted strategically; see Nyborg (2002).

This difference makes uniform-price auctions susceptible to substantial underpricing, because bidders can submit high inframarginal demands that prevent competition on prices and support equilibria where the stop-out price is lower than its Walrasian equivalent. The possibility of underpricing equilibria was first proven in Wilson (1979), Maxwell (1983) and Back and Zender (1993). This result has been shown robust to different model specifications by Ausubel and Cramton (1998), Biais and Faugeron-Crouzet (2002), Engelbrecht-Wiggans and Kahn (1998), Noussair (1994) and Wang and Zender (2002).

A common assumption across these papers is that the supply of the auctioned good is fixed in advance. This seemingly innocuous assumption implies a strategic asymmetry between the bidders and the seller: The former can use their demand schedules to inhibit price competition, but the latter cannot alter her supply schedule to enhance it. It is plausible to expect that the introduction of an adjustable supply should prevent at least some underpricing equilibria. Intuitively, while the steepness of the competitors’ demand curves has a price effect which increases the marginal cost of a higher bid, an increasing supply function induces a quantity effect that raises its marginal revenue. Making the quantity effect greater than the price effect inhibits coordination on low prices.

Only a few papers have studied the equilibria of a uniform-price auction with a variable supply. Back and Zender (2001) shows that, if the seller reserves the right to decrease her supply after receiving the bids, underpricing – while still possible – is severely curtailed. McAdams (2001) derives a similar result and then shows that underpricing is eliminated if the seller reserves the right to increase or arbitrarily adjust her supply. Lengwiler (1999) assumes that the seller produces the good at a constant marginal cost which is private information to her and studies how the right to restrict supply affects the bidders’ demand schedules.

These papers share the assumption that the supply is adjustable after the seller has observed the bid schedules. However, there are situations where it may be necessary to precommit and declare the supply schedule before observing the bid schedules. For instance, declaring the supply schedule ex ante increases transparency in IPOs of unseasoned shares and thus should reduce the winner's curse. In electricity markets2 near peak capacity, there may simply be no time to allow for ex post adjustments.

This paper studies the existence of underpricing equilibria when the seller precommits to an increasing supply schedule, as suggested in Pavan (1996). We find that underpricing is still possible, although to a lesser extent than in the case of an ex post decreasable supply. Committing ex ante to an increasing supply attaches a positive quantity effect to price competition. This effect more than compensates the flexibility lost by giving up ex post reductions. On the other hand, note that precommitment entails the risk of losing control on the quantity sold. Therefore, we show also that a fixed supply is in general suboptimal even if the seller faces increasing costs for selling a quantity diverging from her supply target. The expected gain from reducing underpricing may offset the expected loss from selling a quantity potentially different from the target.

A variable supply is not the only means for the seller to obstruct underpricing in uniform-price auctions. Kremer (2001) and Nyborg (2002) suggest adopting different rationing rules. McAdams (2001) proposes to offer discounts to marginal bidders. Some fine-grained institutional details also hamper underpricing: Nyborg (2002) considers allowing only a finite number of bids, or imposing a tick size for price or quantities; Back and Zender (1993) considers the uncertainty about supply induced by the presence of noncompetitive bidders.

Some of these factors may go towards explaining why, in spite of their theoretical ubiquity, the degree in which underpricing equilibria occur is still controversial. The empirical literature has concentrated mostly on the question whether more revenue is raised by a discriminatory or by a uniform-price auction; see Binmore and Swierzbinski (2001) for a critical review. However, the experimental evidence reported in Goswami et al. (1996) shows that bidders manage to coordinate on underpricing, at least in environments where nonbinding preplay communication is possible. Evidence of underpricing is reported by Tenorio (1997) for foreign currency auctions in Zambia, by Kandel et al. (1999) for IPO auctions in Israel, and by Bjonnes (2001) for Treasury auctions in Norway. Keloharju et al. (2002) confirms the underpricing in Treasury auctions in Finland, but argues that it is not due to strategic manipulation.

The rest of the paper is organized as follows. Section 2 describes the model, which is a straightforward variation on the setup in Back and Zender (1993). Section 3 characterizes a large class of symmetric equilibria under fixed supply, which contains as special cases all the symmetric underpricing equilibria studied in the literature. Section 4 studies the effects of an increasing supply schedule and generalizes the equilibria of Section 3 to the case of an increasing and concave supply schedule. Section 5 analyzes the symmetric underpricing equilibria under a linear supply schedule. Section 6 studies the seller's ex ante choice of a linear supply schedule that maximizes her expected profit and provides an example with an explicit derivation. Finally, Section 7 rounds up the paper with a few comments. All proofs are in Appendix A.

Section snippets

The model

A single (female) seller wishes to auction a homogenous and perfectly divisible good using a uniform-price format. She can offer a fixed supply Q or, more generally, she can post a (weakly) increasing3 and right-continuous (aggregate) supply schedule S(p). She can also set a reserve price pL⩾0, under which no sale occurs.

There are n⩾2 (male) risk-neutral bidders. The per unit value of the good to

Underpricing equilibria under fixed supply

Throughout this section, we assume that the divisible good is in fixed supply at a level Q. Except for her early choice of the uniform-price format and the reserve price pL, the seller plays no strategic role and we restrict attention to the (sub)game among the n bidders engaged in the auction. The payoff to bidder i is πi=(v−P)d̂i(P), where P now depends only on the bidders’ choice of their demand schedules. If v<pL, participating in the auction is not profitable. We focus on the case where vp

Underpricing equilibria under increasing supply

Under a fixed supply Q of the divisible good, Proposition 1 shows that the n bidders can sustain an underpricing equilibrium at a stop-out price p in [pL,v) and split symmetrically the quantity Q by posting the profile of demand schedules {dj(p)}j=1n as in Proposition 1.

Suppose from now on that the seller commits ex ante to an increasing supply schedule S(p). Given p, assume that S(p)=Q so that coordination on p is still feasible. The next proposition establishes that, if S(p) is

Underpricing equilibria under linear supply

Proposition 2 suggests how the seller might be able to induce more aggressive bidding by posting an increasing supply schedule. In essence, what she has to accomplish is making the quantity effect sufficiently high to compensate for the highest possible price effect that bidders’ strategies can achieve. On the other hand, bidders submit their demand schedules only after the supply curve has been announced. Thus, it seems reasonable to assume that they can try to contrast the quantity effect

The choice of a supply schedule

In this section we let the seller explicitly use her supply schedule as a strategic variable. We consider a two-stage game where the seller first publicly commits to an increasing linear supply curve and then bidders compete simultaneously on demand schedules within a uniform-price auction.

We assume that the seller's payoff πs is the difference between the revenue she collects when selling a quantity Q at a uniform price of P and a cost function C(Q); that is, πs=P·QC(Q). In the standard case

Concluding remarks

We close the paper with some remarks on the implications of our analysis for the two prominent examples of Treasury auctions and initial public offerings.

The market for Treasury securities is by far the most relevant example of a widespread use of uniform-price auctions for divisible goods. This paper suggests that, for uniform-price auctions, the practice to combine a fixed supply with a reserve price below market values can be suboptimal for the Treasury. The adoption of an elastic supply

Acknowledgements

We are greatly indebted to Mario Gilli, who participated in the early stages of this project. We are grateful for their helpful comments to the editor, Xavier Vives, two referees, and participants at the 11th Italian Conference in Game Theory and Applications, at the 52nd European Meeting of the Econometric Society, at the 9th Hebrew University Summer School in Economic Theory and at the 2002 EuroConference on Auctions and Market Design. We benefited from discussions with Bruno Biais, Bengt

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