Quantifying the cost of excess market thickness in timber sale auctions

Abstract

In auctions with endogenous entry, theory predicts that too many potential bidders, or the excess market thickness, may actually decrease the seller's expected revenue and the social welfare generated by the auction. This paper proposes a computationally easy method for estimating the optimal number of potential bidders in timber sale auctions with endogenous entry and an uncertain number of active bidders and then quantifies the cost of excess market thickness. It is found that the welfare loss due to the excess market thickness is moderate in this market.

Introduction

This paper uses the structural approach to analyze and quantify the cost, both to the seller and the society, due to excess market thickness in timber sale auctions. Every year, both the federal and state governments in the United States sell many timber harvesting rights to logging companies using auctions. The total revenue from such auctions is large. For example, in the state of Michigan alone, where the data used in the empirical part of this paper come from, every year, $20 to $30 million worth of timber are sold through auctions. Because of the large amount of money involved, a central research question, which is of interest to both academic researchers as well as government policy makers, is which auction mechanism to use such that the seller's revenue and/or the welfare of the society is maximized.

From an auction mechanism design point of view, an auction is characterized by three instruments that are under the seller's control. These instruments determine the likely outcomes of the auctions. First, the seller decides which auction format to use.¹ Second, the seller sets the reserve price and decides whether to adopt a public reserve price strategy or a secret reserve price strategy. Finally, the seller decides on how many bidders to invite for the auction, that is, the number of potential bidders.

The theoretical relationships between the three auction instruments and the likely outcomes of the auction are well understood in the literature. First, in the standard auction model where bidders' private values are independent (IPV framework), it is well known that the four auction formats mentioned above yield the same expected revenue for the seller (Vickrey, 1961, Riley and Samuelson, 1981, Myerson, 1981).² Regarding the reserve price, in the IPV framework, both Laffont and Maskin (1980) and Riley and Samuelson (1981) have shown that the optimal reserve price is above the seller's own value of the auctioned object and depends on the density of bidders' private value distribution. Recently, Levin and Smith (1994) find that in auctions with endogenous entry, the optimal reserve price equals the seller's own value of the auctioned good. Finally, for the relationship between the number of potential bidders and the outcome of the auction, in the IPV framework, the seller's expected revenue is a monotone increasing function of the number of potential bidders and therefore, the optimal strategy for the seller is to invite as many potential bidders as possible. However, when the auction environment changes to the common value (CV) framework or to auctions with endogenous entry, this result no longer holds. For example, Levin and Smith (1994) find that there is a non-monotone relationship between the number of potential bidders and the seller's revenue or the social welfare for auctions with entry and a certain number of active bidders. Therefore, inviting as many potential bidders as possible is not an optimal strategy. Li and Zheng (2008) extend their result to auctions with entry and an uncertain number of active bidders.

On the other hand, since many auction data are available from federal and state government agencies, it provides a unique opportunity for empirical researchers to evaluate the usefulness of theoretical models using field data. Indeed, starting with Paarsch (1992), various econometric methods have been proposed to estimate and test the implications of various auction models. Examples include, Guerre et al. (2000), Li et al. (2000), Haile et al. (2002) and Athey and Haile (2002), to mention just a few. In this literature, some studies focus upon the welfare implications of the first two instruments of auction mechanism design, that is, auction format and reserve price. Regarding the auction format, for example, Athey et al. (2004) find that sealed bid auctions attract more small bidders, shift the allocation towards these bidders, and can also generate higher revenue than open ascending auctions in U.S. Forest Service timber auctions. For another example, Kang and Puller (2007) find that discriminatory auctions yield a larger expected revenue as well as better efficiency than uniform-price auctions in the multi-unit Korean treasury auctions. Regarding the optimal use of reserve price, both Paarsch (1997) and Li and Perrigne (2003) find that switching to the optimal reserve price strategy, the seller's revenue can increase substantially in timber sale auctions.

However, to date, there exists no empirical study examining the welfare implications of the optimal use of the third instrument, that is, number of potential bidders. As discussed above, in the standard IPV auction, this is not an issue as the seller's revenue is monotone increasing in the number of potential bidders. But for auctions with endogenous entry as for the timber sale auctions studied in this paper, it matters. According to the theory, limiting the number of potential bidders may benefit the seller, and/or the society. How much can the seller gain by adopting an optimal strategy regarding the number of potential bidders? Furthermore, in order to adopt such a strategy, policy makers need to know the optimal number of potential bidders in the first place. This paper aims to answer these questions and fill the gap between the theoretical and empirical auction studies with respect to the optimal use of the number of potential bidders and empirically quantify the cost of allowing a sub-optimal number of potential bidders in timber sale auctions.

In order to achieve these two goals, a computationally easy approach is proposed to estimate a structural auction model with endogenous entry and uncertain number of active bidders for timber sale auctions.³ Using the estimated model primitives, the optimal number of potential bidders is calculated for each auction in the dataset. Then new auction outcomes under the optimal number of potential bidders are simulated. It is found that by switching to auction mechanisms with optimal number of potential bidders, the median winning bid increases $45.46 and the median social welfare increases $1006.29, which account for approximately 0.15% and 3.39% of the median winning bid observed in data, respectively. Therefore, the welfare loss due to excess market thickness is moderate in this market.

This paper makes several contributions to the literature. First, this is the first paper that quantifies the welfare implications of allowing a sub-optimal number of potential bidders into the auction. It demonstrates the empirical importance of the number of potential bidders as a tool for the seller in the process of designing optimal auction mechanism. As mentioned above, most of the previous empirical studies in this literature focus on other aspects of the design like auction format and reserve price. Second, theoretically, in an auction model with entry and uncertain number of active bidders, it is proved that from a seller's point of view, the optimal number of potential bidders is the maximum number of potential bidders that still induce all the potential bidders to enter into the auction and become active bidders. This generalizes the result in Levin and Smith (1994) to more general auction models with entry. Finally, this paper offers a computationally easy approach to estimate the optimal number of potential bidders in auctions with entry and uncertain number of active bidders, thus providing an easy way for policy makers to assess the optimality of their auction designs.

The rest of the paper is organized as follows. Section 2 describes the data and discusses results from a reduced-form analysis of the data. The auction model and theoretical results are presented in Section 3. Section 4 outlines the estimation method and reports the estimation results. Counterfactual analyses are conducted in Section 5 to quantify the cost of excess market thickness. Finally, Section 6 concludes. Technical proofs are collected in Appendix A.