In the standard competing auctions model, many sellers compete to sell a single good by offering auctions to buyers. In the first stage of the game sellers post auctions. In the second stage buyers choose one particular auction, place their bid and the good goes to the higest bidder paying the second highest bid. Both the resources available to buyers and the quantity of the good at each auction are exogenously given. In this paper we endogenize both. We first allow buyers to choose the amount of money they bring to an auction, trading off the cost of holding money with the expected surplus from participating in an auction. Second, we allow sellers to choose how much of their production good they want to put on auction, trading off the production cost of the advertised quantity against the expected number of potential buyers. Finally, we allow sellers to charge each buyer with a fee for participating to their auction. This fee, which can be positive or negative, trades off the additional revenue (or cost) from the fee with the number of buyers taking part into their auction. We use our model to study how monetary policy affects the equilibrium allocation of a competing auctions economy and derive recommendations for optimal monetary policy in this environment.
To conduct this exercise we embed the competing auctions framework into the Lagos and Wright (2005) model of monetary exchange with two&sided divisibility. This model is in the tradition of Kiyotaki and Wrightns (1991, 1993) environment in which a role for fiat money is determined endogenously from the frictions of the trading environment, i.e. money is essential for trade (Kocherlakota, 1998; Wallace, 2001). In terms of equilibrium we build on the limit equilibrium concept developed by Peters and Severinov (1997), and extends it to the context of a monetary economy. We build a competing auction environment, start with a finite number of buyers and sellers, characterize the posted contracts, payoffs and money holdings, and then take the limit of these expressions in the infinite game. This limit equilibrium enables to exploit the convergence properties of a competitive matching economy, especially that the deviation by one seller will not affect the payoff buyers can get by visiting him. This corresponds to the market utility property (Peters, 2000) by which the buyerns utility in competitive matching economies is determined by the market and is taken as given by sellers. Finally we assume rational expectations so that sellers believe that their payoff functions satisfy the market utility property.
We consider two variants of the model, which imply different equilibrium outcomes. In the first, each seller posts a quantity for sale and a fee, and allow the dollar price of goods to be determined, ex post, by an auction. In the second, sellers post a dollar price and a fee, and allow the quantity sold to be determined through the auction. In the first version we assume that prices are formed via second&price auctions. We prove that existence of a monetary equilibrium requires that money growth not be too high. Marginal increments in money growth decrease both the equilibrium posted quantity and buyers participation. Sellers charge a positive fee when inflation is low but subsidize buyers when inflation is high. Symmetric efficiency is attained at the Friedman rule where sellers post the efficient quantity, charge no fee to buyers and where entry is efficient.
In a second model we assume sellers post a dollar price and a fee, and allow the quantity sold to be determined through the auction. This protocol shares similarities with industrial procurement auctions. The difference is that in procurement auctions the bidders are sellers and the bid&taker is a buyer. In this case, we prove that existence requires that money growth is not too high and that the distribution of money holdings is degenerate and equal to the posted price. Marginal increments in money growth decrease the posted price and the quantities traded. Sellers subsidize buyers when inflation is low but charge a positive fee when inflation is high. Symmetric efficiency is attained at the Friedman rule where sellers subsidize buyers, agents trade an inefficiently low quantity in multilateral matches and an inefficiently high quantity in pairwise matches [how about entry?]. This second model is similar in spirit to monetary models with divisible goods but indivisible money (Shi (1995), Trejos and Wright (1995), Kultti and Riipinen (2003) and Julien, Kennes, and King (2008)). A key difference is that the fixed price posted by sellers is endogenous here while it is exogenously set to 1qthe indivisible unit of money in those models.
Auctions combined with monetary exchange have already been studied. Kultti and Riipinen (2003) and Julien, Kennes, and King (2008) introduce competing auctions in the so&called second generation of monetary search models (Shi (1995), Trejos and Wright (1995)). Since money is indivisible in these models buyers can compete only through adjustments in quantity. Conversly, Galenianos and Kircher (2008) consider second&price auctions with divisible money but goods are indivisible. By way of contrast, here, both money and goods are fully divisible. Also search is not directed in their paper. Here we allow sellers to post any quantity (or price), and allow buyers to decide which seller to approach so that the matching function is endogenous as in any directed search model. Other papers in the competing auctions literature are Julien (1997), Burguet and Sákovics (1999), Schmitz (2003) and Hernando Veciana (2005) which consider environments with finite numbers of buyers and sellers. Moldovanu, Sela and Shi (2008) have recently constructed a model in which two competing auctioneers can choose the supply of their good as we do here. Their focus, however, is on oligopolistic competition and on the coexistence of two competing auction sites. Also their model is non monetary. Finally this paper contributes to the literature on the micro foundations of money by examining another pricing mechanism, namely competing auction. In contrast to the bargaining, price taking, and competitive search pricing mechanisms examined in Rocheteau and Wright (2005), auctions generate terms of trade dispersion. Combined with the divisibility of goods and the fee charged by sellers, this produces interesting trade offs for both sellers and buyers that have not been previously studied.
