On supply-constrained equilibria

The paper gives sufficient conditions for the existence of an equilibrium with price rigidities and quantity rationing where: (i) demand is never rationed; (ii) net trades of an a priori chosen numeraire are never rationed; and (iii) supply is rationed only when relative prices are downward rigid.     

Compensating for the winner's curse: Experimental evidence

We conduct experiments on common value auctions with rationing. In each auction, the good is randomly allocated to one of the k highest bidders, at the (k+1)st highest price. When k>1, bidders are rationed. As the degree of rationing increases, the equilibrium bid function increases. Consistent with prior literature, we find that bidders suffer from the winner's curse and lose money on average. However, bids in the experiments do adjust in the appropriate direction as the degree of rationing changes, providing support for the comparative statics implications of the theory. Our results are consistent with subjects having an intuitive understanding of the winner's curse, but being unable to compute the equilibrium bid levels.     

Demand reduction and preemptive bidding in multi-unit license auctions

Multi-unit ascending auctions allow for equilibria in which bidders strategically reduce their demand and split the market at low prices. At the same time, they allow for preemptive bidding by incumbent bidders in a coordinated attempt to exclude entrants from the market. We consider an environment where both demand reduction and preemptive bidding are supported as equilibrium phenomena of the ascending auction. In a series of experiments, we compare its performance to that of the discriminatory auction. Strategic demand reduction is quite prevalent in the ascending auction even when entry imposes a (large) negative externality on incumbents. As a result, the ascending auction performs worse than the discriminatory auction both in terms of revenue and efficiency, while entrants’ chances are similar across the two formats.     

Reserve price signaling

In a general auction model in which bidders’ signals are affiliated, we characterize the unique separating equilibrium in which the seller can use reserve prices to credibly signal her private information. When the buyers’ signals are independent, the optimal reserve price is shown to be increasing in the number of bidders under certain conditions. We also demonstrate that the probability that the item is sold at the reserve price can increase as the number of bidders increases, which indicates a more central role for reserve prices than perceived in the standard auction models.     

Budget-constrained sequential auctions with incomplete information

I study a budget-constrained, private-valuation, sealed-bid sequential auction with two incompletely-informed, risk-neutral bidders in which the valuations and income may be non-monotonic functions of a bidder's type. Multiple equilibrium symmetric bidding functions may exist that differ in allocation, efficiency and revenue. The sequence of sale affects the competition for a good and therefore also affects revenue and the prices of each good in a systematic way that depends on the relationship among the valuations and incomes of bidders. The sequence of sale may affect prices and revenue even when the number of bidders is large relative to the number of goods. If a particular good, say α, is allocated to a strong bidder independent of the sequence of sale, then auction revenue and the price of good α are higher when good α is sold first.     

Commitment in sequential auctioning: advance listings and threshold prices

This paper models sequential auctioning of two perfect substitutes by a strategic seller, who learns about demand from the first-auction price. The seller holds the second auction only when the remaining demand is strong enough to cover her opportunity cost. Bidding in anticipation of such a contingent future auction is characterized, including a sufficient condition for existence of an invertible (increasing symmetric pure-strategy) bidding equilibrium that facilitates the seller’s learning. A unique invertible bidding equilibrium exists for the Dutch auction format, but only when the second auction is sufficiently discounted by the bidders. In the equilibrium, high-valuation bidders shade their bids down as if the second auction were guaranteed. To counter such strategic bidding, the seller would value ex-ante commitment to hold the second auction less often. Three forms of such commitment are analyzed: commitment to list future auctions in advance, commitment to not hold the second auction unless the first price exceeds a publicly announced threshold, and commitment to a reserve-price in the second auction. I would like to thank Georgios Katsenos, Thomas Jeitschko, Miguel Villas-Boas, George Deltas, and an anonymous referee for thorough and insightful feedback.     

Multiunit Pay‐Your‐Bid Auction with One‐Dimensional Multiunit Demands*

An arbitrary number of units of a good is sold to two bidders through a discriminatory auction. The bidders are homogeneous ex ante and their demand functions are two‐step functions that depend on a single parameter. We characterize the symmetric Bayesian equilibrium and prove its existence and uniqueness. We compare this equilibrium with the equilibrium of the multiunit Vickrey auction and with the equilibria of the single‐unit first price and second price auctions. We examine the consequences of bundling all units into one package. We study the impacts that variations of the “relative” supply have on the equilibrium, on the bidders' average payoffs per unit, and on the efficiency of the equilibrium allocation.     

Retailers and consumers in sequential auctions of collectibles

Abstract.  We analyse an independent private-value model, where heterogeneous bidders compete for objects sold in sequential second-price auctions. In this heterogeneous game, bidders may have differently distributed valuations, and some have multi-unit demand with decreasing marginal values (retailers); others have a specific single-unit demand (consumers). By examining equilibrium bidding strategies and price sequences, we show that the presence of consumers leads to more aggressive bidding from the retailers on average and heterogeneous bidders is a plausible explanation of the price decline effect. The study of the expected revenue of the seller confirms the interest of auctioneers in inviting different types of bidders.     

All equilibria of the multi-unit Vickrey auction

This paper completely characterizes the set of equilibria of the Vickrey auction for multiple identical units when buyers have non-increasing marginal valuations and there are at least three potential buyers. There are two types of equilibria: In the first class of equilibria there are positive bids below the maximum valuation. In this class, above a threshold value all bidders bid truthfully on all units. One of the bidders bids at the threshold for any unit for which his valuation is below the threshold; the other bidders bid zero in this range. In the second class of equilibria there are as many bids at or above the maximum valuation as there are units. The allocation of these bids is arbitrary across bidders. All the remaining bids equal zero. With any positive reserve price equilibrium becomes unique: Bidders bid truthfully on all units for which their valuation exceeds the reserve price.     

Indeterminacy and directed search

The directed search approach assumes each seller posts a fixed price and, ex post, randomly allocates the good should more than one buyer desire the good. This paper assumes sellers can post prices which are contingent on ex post realized demand; e.g. an advertisement might state the Bertrand price should there be more than one buyer, which corresponds to an auction outcome. Competition in fixed prices and ex post rationing describes equilibrium behavior. There is also real market indeterminacy: a continuum of equilibria exists which are not payoff equivalent. Sellers prefer the equilibrium in auctions.     

Ascending price Vickrey auctions

We show that an ascending price auction for multiple units of a homogeneous object proposed by Ausubel (i) raises prices for packages until they reach those nonlinear and non-anonymous market clearing prices at which bidders get their marginal products and (ii) the auction is a primal–dual algorithm applied to an appropriate linear programming formulation in which the dual solution yields those same market clearing prices. We emphasize the similarities with efficient incentive compatible ascending price auctions to implement Vickrey payments when there is a single object or when objects are heterogeneous but each buyer does not desire more than one unit. A potential benefit of these common threads is that it helps to establish the principles upon which Vickrey payments may be implemented through decentralized, incentive compatible procedures.     

Auctions in which losers set the price

We study auctions of a single asset among symmetric bidders with affiliated values. We show that the second-price auction minimizes revenue among all efficient auction mechanisms in which only the winner pays, and the price only depends on the losers' bids. In particular, we show that the kth price auction generates higher revenue than the second-price auction, for all k>2. If rationing is allowed, with shares of the asset rationed among the t highest bidders, then the (t+1)st price auction yields the lowest revenue among all auctions with rationing in which only the winners pay and the unit price only depends on the losers' bids. Finally, we compute bidding functions and revenue of the kth price auction, with and without rationing, for an illustrative example much used in the experimental literature to study first-price, second-price and English auctions.     

Multi-unit auctions with uniform prices

Summary. Auctions in which individuals can purchase more than one unit of the good being sold differ in striking ways from multi-unit auctions in which individuals may purchase only one unit. The uniform price auction in particular frequently yields Nash equilibria in which bidders underbid for their second unit and therefore pay very low prices for the good. This paper characterizes equilibria for the uniform price auction. Received: July 31, 1995; revised version: May 28, 1997     

Multiple unit auctions with strategic price-quantity decisions

Summary. I study a multiple unit auction where symmetric risk-neutral bidders choose prices and quantities endogenously. In the model, bidders (a) may place non-linear valuations on the auctioned units, and (b) bid for several units at the same price (“lumpy” bids). I characterize quantity-symmetric and strictly monotone-increasing price equilibria for discriminatory and competitive auctions, and show that (i) if quantity strategy profiles are equal across auctions revenue- equivalence holds, (ii) expected revenue is higher if bidders bid for the entire supply rather than for shares of it, and (iii) equilibrium allocations may fail to be Pareto-optimal. Received: April 14, 1995; revised version: September 3, 1997     

The price normalization problem in imperfect competition and the objective of the firm

Summary. General equilibrium models of oligopolistic competition give rise to relative prices only without determining the price level. It is well known that the choice of a numéraire or, more generally, of a normalization rule converting relative prices into absolute prices entails drastic consequences for the resulting set of Nash equilibria when firms are assumed to maximize profits. This is due to the fact that changing the price normalization amounts to altering the objective functions of the firms. Clearly, the objective of a firm must not be based on price normalization rules void of any economic content. In this paper we propose a definition of the objective of a firm, called maximization of shareholders' real wealth, which takes shareholders' demand explicitly into account. This objective depends on relative prices only. Real wealth maxima are shown to exist under certain conditions. Moreover, we consider an oligopolistic market and prove the existence of a Nash equilibrium in which each firm maximizes the real wealth of its shareholders. Received: July 10, 1997; revised version: July 27, 1998     

The multiple unit auction with variable supply

Summary. The theory of multiple unit auctions traditionally assumes that the offered quantity is fixed. I argue that this assumption is not appropriate for many applications because the seller may be able and willing to adjust the supply as a function of the bidding. In this paper I address this shortcoming by analyzing a multi-unit auction game between a monopolistic seller who can produce arbitrary quantities at constant unit cost, and oligopolistic bidders. I establish the existence of a subgame-perfect equilibrium for price discriminating and for uniform price auctions. I also show that bidders have an incentive to misreport their true demand in both auction formats, but they do that in different ways and for different reasons. Furthermore, both auction formats are inefficient, but there is no unambiguous ordering among them. Finally, the more competitive the bidders are, the more likely the seller is to prefer uniform pricing over price discrimination, yet increased competition among bidders may or may not enhance efficiency. Received: June 18, 1998; revised version: January 13, 1999     

Commitment and price competition in a dynamic differentiated-product duopoly

This paper characterizes linear Markov-perfect equilibrium in a duopolistic environment where firms engage in dynamic price competition. Firms have constant (but potentially different) marginal costs and produce differentiated products. We show that, for the case of linear demand, dynamically stable Markov-perfect equilibrium prices are strictly higher than one-shot Nash equilibrium prices, but lower than fully collusive (monopoly) prices. We provide closed-form solutions for the Markov-perfect equilibrium prices which, in principle, can be estimated given data on firm demand and costs. Our results suggest that static two-stage models of price commitment are on reasonably solid ground in that they might be viewed as a reduced form for more complicated dynamic models.     

Feedback effects between credit ratings and financial markets

Credit rating agencies often make sharp adjustments in their pronouncements during times of stress in financial markets. These adjustments typically happen with a delay relative to shocks in market prices. Since prices convey information about what market participants are doing and thinking, it is likely that rating agencies take into account market prices when issuing their pronouncements.In order to understand the relationship between credit ratings and financial prices, we develop a model of debt roll-over in which rating agencies incorporate information publicly available in financial markets. We find that (1) rating agencies respond to market prices, i.e. nonfundamental price volatility can shift financing conditions from a low risk spread and high credit rating equilibrium to an equilibrium with high spread and low rating, and (2) rating agencies can anchor expectations about the equilibrium in financial markets, thus serving as an antidote to nonfundamental price volatility.     

Reserve Prices without Commitment

When potential bidders in an auction have to incur a cost to prepare their bids and thus to learn their valuations, imposing a reserve price and announcing that in case no bid is submitted there will be another auction without a reserve price is both revenue and welfare improving. Reserve prices that induce less than maximum entry in the first auction may be optimal. Also, entry fees are not necessarily better instruments than reserve prices.Journal of Economic LiteratureClassification Number: D44.     

Monopoly Price Dispersion Under Demand Uncertainty

When a monopolist sets its price before its demand is known, then it may set more than one price and limit the availability of its output at lower prices. This article adds demand uncertainty and price rigidities to the standard model of monopoly pricing. When there are two states of demand and the ex post monopoly price is greater when demand is high then the monopolist's optimal ex ante pricing strategy is to set two prices and limit purchases at the lower price.