## Abstract

I introduce an empirical auction model where in addition to the private value that each bidder receives upon winning the auction, losing bidders incur a negative externality that depends on the identity of the rival winner. I show how the externalities and private value distributions can be identified and estimated in such a model. I then apply the model to U.S. Forest Service timber auctions. I find that mill bidders impose significant externalities on one another of between 10% and 22% of the heterogeneous portion of their valuation. This leads the timber tracts to be misallocated in 5.2% of the auctions in my sample.

## I. Introduction

IN many auction settings, competition among bidders outside the auction can affect bidding strategies and auction outcomes. For example, if bidders later compete in some downstream market, then a rival obtaining the auctioned object may lead to profit losses downstream. In this case, bidders are not only concerned about whether they win the auction, but also the identity of the winner if it is not them. This will affect their bid strategies as they bid not only to acquire the object but also to keep it away from certain rivals.

U.S. Forest Service (USFS) timber auctions are a recent example of a setting where outside competition is alleged to have affected bidding. In the 2007 U.S. Supreme Court case of Weyerhaeuser Company v. Ross-Simmons Hardwood Lumber Company, the plaintiff, Ross-Simmons, accused Weyerhaeuser of predatory bidding in timber auctions. The plaintiff's claim was that Weyerhaeuser was bidding high only to keep timber away from downstream competitors in the lumber market.

The goal of this paper is to look at how downstream competition in the lumber industry affected timber auction bidding and outcomes, as was alleged in this case. I do so by extending the method of Guerre, Perrigne, and Vuong (2000), which does not in general allow for outside competition among bidders to affect strategies and outcomes. The key feature of my model is that bidders may care about whom the object is allocated to even if it is not them. Thus, bidding strategies will depend not just on the number of competitors a bidder faces, but also their identities. I introduce an identity-dependent negative payoff that bidders incur if they lose the auction to a particular rival: the negative externality. I show that the negative externalities and distributions of valuations in such a model can be identified from observations on bids and bidder identities.

I then apply this model to USFS timber auctions to identify the degree to which timber auction bids are driven by bidders' own valuations for the tract, compared to their desire to keep the tract away from rivals. Distinguishing between the two motives is important for answering questions about market participation, object allocation, bidder subsidies, bidder collusion, and auction design. Identifying the externality effect is also useful in determining whether the incentives exist for timber auction bidders to engage in predatory bidding, as was alleged in the case against Weyerhaeuser.

Such an auction model also applies to other settings as well. Potential buyers in an M&A may care not only about the benefits from acquiring the target firm but also the potential losses if their rival instead acquires the target. This also applies to the exclusive sale of inputs to downstream competitors (such as a patent or the early exclusivity deal between Apple and AT&T) and the awarding of important projects such as government contracts in the aerospace industry. This model could also be used to explore the contracting of athletes in professional sports, where teams are known to “overpay” athletes in order to prevent their rival from signing the player.

The model I use is based on Jehiel, Moldovanu, and Stacchetti (1996, 1999). In addition to the private values that bidders receive on winning the object, losing bidders suffer negative externalities that depend on both their type and that of the winner. Extensions of this model have been studied previously in the theory literature by Jehiel and Moldovanu (1996, 2000) and Das Varma (2002). The model differs from other auction models estimated in the literature in that losing bidders are affected differently depending on the particular externality value between themselves and the auction winner. As a result, bidders care who wins the auction if they do not. The differing values of these externalities are ostensibly based on the degree of competition between the bidders outside the auction.

I show that both the externality parameters and the value distributions can be identified and estimated in this model from observations on auction bids and auction participant identities. The identification strategy requires observing enough exogenous variation in the set of participating bidders. As bidders of a given type face varying sets of competitors who confer differing levels of externalities on the bidder, this will shift their observed bid strategies. The externality parameters are then identified based on how bids fluctuate as the types of competitors a bidder faces vary.

Identification based on exogenous variation in bidder sets has been used in several prior papers on timber auctions. Aradillas-Lopez, Gandhi, and Qunit (2013) assume exogenous participation to get bounds on seller revenue in ascending correlated private values auctions. Aryal et al. (2016) use the assumption to identify value distributions when bidders face ambiguity. Coey et al. (2017) use exogenous bidder participation to identify bounds on optimal reserve prices, while Guerre, Perrigne, and Vuong (2009) use it to identify risk aversion in first-price auctions. Haile, Hong, and Shum (2003) make the assumption to test between common and private values, and Gillen (2010) uses it for identification in a level-k auction model.

This identification strategy is implemented here by first estimating bidder valuations as a function of the externality parameters. I then search for the parameter values that lead to bidder valuation distributions that are the same for bidders of the same type, across auctions with different sets of competing bidders. I introduce three estimators, which differ in which feature of the value distributions they match. I also show how this identification and estimation strategy can be extended to deal with observed and unobserved auction heterogeneity, particularly that which affects the exogeneity of bidder participation. In particular, identification of the externality parameter in the application comes from variation in an instrument—the set of potential bidders (defined by their geographic proximity to the auction and their recent activity).

I apply this estimation strategy to USFS timber auctions, for which there is a substantial prior literature. In addition to the studies already mentioned, Baldwin, Marshall, and Richard (1997) test timber auctions for collusion, Haile (2001) looks at the effects of resale on bidder valuations, and Lu and Perrigne (2008) use timber auctions to estimate bidder risk aversion. To my knowledge, no prior work has looked at the effect of downstream competition on timber auction outcomes.

I find that downstream competition between mills (bidders with manufacturing capacity) is sufficient enough that a mill bidder acquiring the timber tract will cost a rival mill between 10% and 22% of the heterogeneous portion of their valuation for the timber. This has important consequences in this application. It indicates that mill bids are generally higher than the bids of logger companies, not because of a stochastic dominance in their valuations for timber tracts but because rival mills compete more heavily with one another in the downstream lumber market. This affects the allocation of timber if the firms that win the auctions are not those that value the timber tract the most, but are instead firms that bid high in order to keep the tract away from rivals. I find that this is true for 5.2% of the auctions in my sample, where the presence of externalities leads the timber tract to be misallocated to a bidder that does not have the highest valuation for the tract.

The rest of the paper is organized as follows. Section II presents the model and equilibrium bidding strategies. Section III discusses the identification and estimation of the distributions of bidder valuations and the externality parameters. Section IV presents the results from a Monte Carlo experiment, and section V applies the estimator to USFS timber auctions. Section VI concludes.

## II. Model

The model is an auction consisting of $n≥2$ risk-neutral bidders competing for one indivisible object. The set of bidders is denoted by $B$. Bidders are partitioned into $K$ groups based on their type $k$ in the downstream market. There is no restriction on the number of types $K$. In the application, I separate bidders into three types: mills that have manufacturing capacity, loggers that do not, and small businesses. The set of bidders of type $k$ is denoted by $Bk$ (where $⋃k=1KBk=B$ and $⋂k=1KBk=∅$), and the number of bidders of type $k$ is denoted by $nk$ (where $n1+⋯+nK=n$). The auction mechanism is a first-price auction where each bidder $ik$ submits a bid $bik$, and the bidder with the largest bid gets the object and pays a price equal to the bid submitted.

Each bidder, $ik∈Bk$, has a valuation for the object, $vik$, drawn independently from the distribution $Fk(·)$. Valuation $vik$ is assumed to be private information to bidder $ik$. Each $Fk(·)$ has support $[vk̲,vk¯]$, and is common knowledge to all bidders. The distributions $Fk(·)$ may be the same for all $k$, in which case we say that bidders have symmetric valuations, or they may be different for bidders of different types, in which case we say that bidder valuations are asymmetric.

In addition to each bidder's payoff from winning the auction, losing bidders suffer a negative payoff that depends on their identity and the identity of the winning bidder. This negative payoff is referred to as a negative externality. It is denoted by $αik,jk'$, where $jk'$ is the winning bidder and $ik$ is the sufferer of the externality. Thus, the gross payoff for each bidder $ik∈Bk$ is
$vikifikwins-αik,jk'ifjk'≠ik∈Bk'wins.$

The externality value, $αik,jk'$, represents the lost profit to bidder $ik$ from bidder $jk'$ winning the auction object, ostensibly due to downstream competition between the two bidders.1 Because the size of the negative payoff depends on the identity of the winner, bidders will care who wins the object if they do not and will alter their bid to keep the object away from certain rivals. This is what separates this auction model with externalities from a standard auction model without them.

Accounting for these externalities is important in settings where auction outcomes have significant consequences on the downstream market. The antitrust case brought against Weyerhaeuser illustrates how USFS timber auctions can affect the downstream lumber market. Other examples include auctions for spectrum licenses, government contracts, and landing slots at an airport, as well as bidding in M&As. When analyzing bids in these settings, it is important to account for the effect downstream competition has on bidding behavior, yet this is mostly ignored in other auction models. In section A of the online appendix, I lay out three motivating examples of different models of downstream competition that lead to the above auction model with externalities.

For identification and estimation of the model, I assume that the value of the negative externality ($αik,jk'$) depends on both the type of the imposing bidder ($k'$) and the type of the receiving bidder ($k$) and is a parameter of the model, $αk,k'$. This is not as restrictive as it initially seems. A model that is closer to that of Jehiel et al. (1999) would have multidimensional bidder types. In that case, externalities are private information to their imposer and are drawn from a common-knowledge pair-type distribution, $Fαk,k'$, that depends on both the type of the imposer of the negative payoff and the receiver. Due to the uncertainty that a bidder then has about the negative payoff they receive if they lose to a particular rival, under the assumption that private valuations are uncorrelated with the negative externalities, the observed bid strategies from such a model would be equivalent to that of the model I estimate where externalities are type-dependent parameters.

This can be seen by considering the model with multidimensional bidder types. Bidders' strategy depends on their valuation for the object, the bidder set they face, and their expectation about their negative payoff if they lose to a particular rival $jk'$, which is given by $E[αik,jk']=αkk'$. If bidders are risk neutral, then bid strategies are going to depend on these means of the externality distributions. Therefore, nonparametrically identifying the externality distributions, or even identifying their variance in such a model, is not possible from bid data. Instead, all that can be recovered from observations on bids is the means of these distributions or the parameters, $αkk'$. These can equivalently be recovered using the model of this paper, where externalities are known parameters that depend on the types of both the imposer and receiver of the negative payoff.

One may also wish to relax the independence between valuations and externalities. In section VI, I show how the estimation strategy of this paper can be extended to that case when I impose a particular form of the correlation between a bidder's private value and the negative payoff the bidder inflicts. In this case, externalities are identity dependent and private information to the imposer.

### A. Equilibrium

The expected utility of bidder $ik∈Bk$ with valuation $vik$, given that the person submits bid $b$, is given by
$uk(vik,b)=(vik-b)Pr(b≥bl,∀l∈B-ik)-∑k'αkk'∑jk'≠ik∈Bk'Pr(bjk'≥bl,∀l∈B-jk'|b),$
(1)

where $αkk'$ is the negative payoff a bidder of type $k'$ imposes on a bidder of type $k$ when winning the auction. At the Bayesian Nash equilibrium, each bidder chooses the bid that maximizes equation (1). I restrict attention to symmetric equilibria where each bidder of type $k$ follows the same strategy.

It can be shown that the above auction model satisfies the conditions of McAdams (2003) for the existence of an equilibrium with monotone pure-strategy bidding functions.2 Thus, I focus on equilibrium strategies that are differentiable and strictly increasing over a range of valuations. This range includes any valuation over a threshold that depends on both the bidder's type and the bidder's auction competition. One can think of bids above this threshold as equivalent to the notion of “serious bids” in Reny and Zamir (2004). The reason I have only monotone bidding strategies above this threshold is that with asymmetric bidders and externalities, bidders with valuations below the threshold are indifferent between a continuum of lower bids that provide the same expected utility.

Establishing a unique equilibrium is difficult in general and is further complicated by the presence of externalities. With externalities, an auction participant's bid depends on his or her belief about who will get the object if he or she does not. This belief is endogenously determined in equilibrium. One can imagine that several sets of consistent beliefs could be constructed that are consistent with an equilibrium.3 In the absence of a uniqueness result, I make the additional assumption in the estimation below that the observed bids are rationalized by a single, symmetric equilibrium.

I am interested in defining the monotone bidding function for all valuations above the indifference threshold. Let $βk(·)$ be the strictly increasing equilibrium bid strategy for a type-$k$ bidder, with an inverse denoted by $βk-1(·)$. A bidder $ik∈Bk$ solves
$maxb{(vik-b)∏k'Fk'(βk'-1(b))(nk'-1{k'=k})-∑k'[αkk'nk'-1{k'=k}∫βk'-1(b)vk'¯×∏k''Fk''(βk''-1(βk'(x)))(nk''-1{k''=k'}-1{k''=k})fk'(x)dx]},$
(2)
where $1{·}$ is the indicator function. Differentiating equation (2) with respect to $b$, for all $k$, leads to a system of $K$ first-order differential equations:
$∑k'[(vik-b+αkk')×(nk'-1{k'=k})fk'(βk'-1(b))βk'-1'(b)Fk'(βk'-1(b))]=1.$
(3)

This system of equations, along with the boundary conditions at the indifference thresholds, defines the equilibrium strategies $βk(·)$. Generally, solving this system of differential equations for each $βk$ is difficult, even with the use of numerical methods. Thus, I follow Guerre et al. (2000) and avoid directly solving for the equilibrium bid strategies.

Let $Hk(b|B)$ be the probability that a particular bidder $ik$ of type $k$ wins the auction with a bid of $b$, given that the set of bidders is $B$ (i.e., $Hk(b|B)=Prmaxj∈B-ikbj≤b|B$). Then using the strict monotonicity of the bidding functions, I can write the expected utility of a type $k$ bidder as
$uk(vik,b;B)=(vik-b)Hk(b|B)-∑k'[αkk'nk'-1{k'=k}×∫bk'(B)̲bk'(B)¯Hk'(x|bik=b,B)gk'(x|B)dx],$
(4)
where $Hk'(x|bik=b,B)$ is the probability that a specific bidder of type $k'$ wins the auction with a bid of $x$ given that $ik$ submits a bid of $b$, $gk'(x|B)$ is the bid density function for a bidder of type $k'$ given bidder set $B$, and $bk'(B)¯$ and $bk'(B)̲$ are the upper and lower bound, respectively, of the bid distribution for a bidder of type $k'$ given set of bidders $B$. The first-order condition of this expected utility with respect to $b$ is
$(vik-b)Hk'(b|B)=Hk(b|B)+∑k'[αkk'nk'-1{k'=k}∂∂b×∫bk'(B)̲bk'(B)¯Hk'(x|bik=b,B)gk'(x|B)dx].$
(5)
The key to getting a tractable expression for the last term in equation (5) is to look at how $ik$'s bid $b$ enters the integral. The integral gives the probability that a specific bidder of type $k'$, $jk'$, wins the auction given that $ik$ bids $b$. This probability is only affected by $b$ in that for any $jk'$ bid of $x, that probability of winning is 0. For all rival bids of $x>b$, the probability of $jk'$ winning is the probability that all other bidders, excluding $jk'$ and $ik$ (since $ik$ bids $b$), bid below $x$. This latter probability is independent of $ik$'s bid of $b$. Thus, I can simplify the above expression to an integral where $b$ only enters the limit of integration (derived in section B.1 of the online appendix):
$∫bk'(B)̲bk'(B)¯Hk'(x|bik=b,B)gk'(x|B)dx=∫bbk'(B)¯Prmaxl≠ik,jk'∈Bbl≤x|Bgk'(x|B)dx.$
(6)
Taking the derivative of equation (6) with respect to $b$, and substituting it back into equation (5) results in
$vik=b+Hk(b|B)Hk'(b|B)-∑k'[αkk'nk'-1{k'=k}×Prmaxl≠ik,jk'∈Bbl≤b|Bgk'(b|B)Hk'(b|B)].$
(7)

Equation (7) is a necessary condition for $b$ to be an optimal bid for a bidder of type $k$ with valuation $vik$. The first portion of this equation is similar to what is typically found in the literature. The second portion of the right-hand side of equation (7) represents the increase in the bid over the standard equilibrium bid due to the presence of rivals that can exert externalities on the bidder. The final term in equation (7) is interpreted as the probability that for bidder $ik$, one rival, $jk'$, bids $b$ and all other players in the auction have bid below $b$, so that the only way that $ik$ can prevent rival $jk'$ from obtaining the good is to bid $b$ or marginally better. The parameter $αkk'$ is the cost to $ik$ of $jk'$ getting the object, and so together, the last term in equation (7) is the increase in expected utility $ik$ receives from preventing rival $jk'$ from getting the object by making a bid of $b$.

## III. Identification and Estimation Strategy

### A. Identification

The goal of identification is to identify the set of externality parameters, ${αkk'}k,k'$, and the distributions of valuations, ${Fk}k$, from observations on bids and bidder identities from a sample of auctions. I assume that a series of T independent auctions are observed for the same object. For each auction, one observes the set of bidders, $Bt$, and the joint distribution of bids, $G(b1t,⋯,bnt|Bt)$. The $L$ externality parameters to be estimated are given by an $L×1$ vector, $α$, that belongs to the set $A⊂RL$. With no symmetry restrictions, $L=K2$. Imposing some symmetry lowers $L$ and eases the requirements for identification.4 I also allow for asymmetries between value distributions for bidders of different types. I say that $α$ is identified if for any $α*$, $α˜$$∈A$, and any $Fv*(·)$, $Fv˜(·)$$∈ℑ$, where $ℑ$ is the set of strictly increasing and continuous distributions, if $G·;α*,Fv*(·)B)=G·;α˜,Fv˜(·)|B$ for all observed bidder sets $B$, then $α*=α˜$ and $Fv*(·)=Fv˜(·)$.

The strategy behind identification is to use exogenous variation in the set of participating bidders, so that equation (7) identifies the externality parameters. Exogenous variation in the set of bidders occurs if there is some cost to participating in the auction that is uncorrelated with bidder valuations (Athey & Haile, 2002; Athey, Levin, & Seira, 2011), or if there are some exogenous participation restrictions imposed by the auction seller, as is sometimes the case in government auctions. In section IIIC, I show how the identification strategy can be adapted to allow for endogenous entry due to observed and unobserved auction characteristics. There, I allow for both observed and unobserved auction heterogeneity to affect participation decisions and bidder valuations, leading to potential correlation between bidder sets and value distributions. I deal with this by using geographic variation in the location of bidders as an instrument for bidder participation.

The technique in this paper could also be adapted to several other models of endogenous participation from the prior literature.5 Adapting those models to the current setup with externalities does require the additional assumption that bidders cannot just avoid the negative externality by not participating in the auction.6 Thus, any model of endogenous participation implicitly assumes that downstream market participants who are not observed to bid in the auction must have such a low combination of valuation draw and externality parameter that they do not meet the entry threshold.

Variation in bidder sets identifies the externality parameters since bidders will bid differently depending on the number and identities of their opponents. Observing how a bidder of a particular type changes his or her bidding strategy as the types of opponents change allows the econometrician to infer what the value of the externality parameter must be to cause the observed differences in bidding strategies. Thus, identification requires observing sufficient variation in bidder sets.

Formally, the first step for identification is to identify bidder valuations as a function of observed bids and the bidder set, and the unknown externality parameters. Equation (7) does this since each distribution in it is known from the observed joint distribution of bids, $G(·|B)$. Thus, bidder valuations are identified as a linear function of the externality parameters.

Under the assumption that bidder participation is exogenous, bidder value distributions do not depend on $B$. Therefore, value distributions for a given bidder type will be equal across all auctions with different sets of bidders. Exogenous variation in the observed sets will result in a series of equalities between distributions of valuations that are functions of the externality parameters.

Let $G$ be the observed distribution of bids when the bidder set is $B$ and $Ga$ be that distribution when the bidder set is some alternative $Ba$. Then the identifying equalities can be written out as
$Fξk(ξk(b,G;α,B)|B)=Fξk(ξk(b,Ga;α,Ba)|Ba)∀B,Ba.$
(8)

Given that the observed bidder set is independent of bidder values and that when evaluated at the true externality parameters, the inverse bid function $ξk$ is a true description of bidding behavior, the above equality must hold for all observed bidder sets, $B,Ba$, at the true parameter value, $α*$.

In addition to holding at the true parameter value, identification requires that equation (8) not hold for at least one pair of observed bidder sets, $B,Ba$, for any alternative parameter values, $α˜≠α*$. Thus, point identification entails observing enough exogenous variation in the set of participating bidders to generate a sufficient number of exclusion restrictions like equation (8), to restrict the permissible $α$ parameters to a singleton. An easily verifiable sufficient condition that satisfies this identification requirement is formulated by looking at some percentile of the value distribution, such as the median. For the distribution of $ξk$ under $B$ to be the same as the distribution of $ξk$ under the alternative $Ba$, they must have the same median values (i.e., $ξk(bk,Bmed,G;α˜,B)=ξk(bk,Bamed,Ga;α˜,Ba)$). Thus, point identification is achieved if there is only one set of parameters ($α*$) for which the median values are equal for all pairs of observed bidder sets $B$ and $Ba$.

The condition that $ξk(bk,Bmed,G;α˜},B)-ξk(bk,Bamed,Ga;α˜},Ba)=0$, is linear in the parameters $α˜$. Observing more variation in bidder sets increases the number of these identifying equalities, leading to a system of equations that are linear in the parameters. Let $Sk$ be the number of observed bidder sets that contain a bidder of type $k$. This system can be represented in matrix form as $Cα˜=CLC$, where $C$ is a $∑k=1K(Sk-1)×L$ matrix and $CLC$ is a $∑k=1K(Sk-1)×1$ vector.

Then the problem of identification can be reposed as one of finding a unique solution to the above system. The uniqueness of the solution depends on the rank of the matrix $C$, which in turn depends on the observed variation in bidder sets. If $rank(C)≥L$, then there is at most one solution to this system of linear equations. Since the true parameter values, $α*$, must be a solution, then if $rank(C)$ is large enough, the only way alternative parameter values, $α˜$, will satisfy all of the observed equalities is if $α˜=α*$. So the externality parameters are identified.

Proposition 1.

Assume exogenous variation in bidder participation (i.e., $Fk(·;B)=Fk(·;Ba),∀B,Ba$) and that inverse bid function, $ξk(b,G;α,B)$, is strictly increasing in $b$ for $b∈βk(vk̲),βk(vk¯)$. Let $C$ be the matrix described above, constructed by stacking equalities of the form $ξk(bk,Bmed,G;α˜},B)=ξk(bk,Bamed,Ga;α˜},Ba)$, for all bidder types $k$ and all pairs of observed bidder sets $B$ and $Ba$. Then if enough variation in bidder sets is observed so that $rank(C)≥L$, the externality parameters $α$ are identified.

The proof is in section B.2 of the online appendix.

Making additional assumptions on bidder distributions can ease the identification requirement. For example, assuming that the median bidder valuation is the same for bidders of all types or further restricting value distributions to be symmetric increases $rank(C)$ and improves identification. Imposing symmetry on the externality parameters also eases identification by lowering $L$.

Once the externality parameters have been identified, the pseudo-valuations in equation (7) are also identified (since they are functions of the parameter values). I can then identify the distributions of valuations as in Guerre et al. (2000).

### B. Estimation

Estimation follows the identification argument from the previous section. I assume that I observe $T$ independent auctions labeled $t=1,⋯,T$. For each auction, I observe each bidder's bid, as well as each participating bidder's type. Each auction $t$ has $nkt$ bidders of type $k$ for $k=1,⋯,K$. I denote this set of bidders for each auction as $Bt$. I let $B∪$ be the nonrepeating set of bidder sets that are observed and $Bk∪⊂B∪$ be those sets for which $nk≥1$. I denote the cardinality of these sets as $S=|B∪|$ and $Sk=|Bk∪|$. Thus, $S$ is the number of different bidder sets observed by the econometrician, and $Sk$ is the number of those observed bidder sets that contain a bidder of type $k$. A bidder of type $k$ in auction $t$ with bidder set $B$ will have valuation denoted by $vk,Bit$ and bid denoted by $bk,Bit$. Finally, I let $p(i)$ be a function that returns the type of bidder $i$.

The idea behind estimation is similar to that of identification. The strategy is to first use the observed bids to compute the distributions and densities in equation (7). Then I use these estimates and the observed bids to compute an externality-dependent estimate for the distribution of valuations for each bidder type and bidder set. I use that for a given bidder type; the distribution of pseudo-values, or the median or mean pseudo-value, should be the same for each bidder set that bidder type is a part of. Variation in bidder sets is used to generate equalities of this type, which are then solved to get estimates for the externality parameters.

#### K-S estimator for $α$⁠.

The first of the three estimators identifies the externality parameters by equating the estimated pseudo-value distributions. This estimator is based on the Kolmogorov-Smirnov test statistic, which tests the equality of two distributions. Here, the two distributions that should be equal are the distributions of pseudo-values for a bidder of a specific type for any two distinct bidder sets. Since the pseudo-values in equation (7) are functions of the unknown parameters, the parameter values that equate the two distributions should be a good estimate of the true parameters.

The first step is to construct empirical counterparts for the distributions in equation (7): $Hk^(b|B)$, $Hk'^(b|B)$, $Pr^maxl≠ik,jk'∈Bbl≤b|B$, and $gk^(b|B)$. A discussion of how these are estimated is in section C of the online appendix. Then, for a given guess at the value of the externality parameters, $α'$, I compute the pseudo-values corresponding to each observed bid:
$vk,Bit^(α')=bk,Bit+Hk^(bk,Bit|B)Hk'^(bk,Bit|B)-∑k'αkk''nk'-1{k'=k}Pr^maxl≠ik,jk'∈Bbl≤bk,Bit|Bgk'^(bk,Bit|B)Hk'^(bk,Bit|B)×Pr^maxl≠ik,jk'∈Bbl≤bk,Bit|Bgk'^(bk,Bit|B)Hk'^(bk,Bit|B).$
(9)
For each $B$ and for each type $k$ in $B$, I estimate the distribution of valuations given guess $α'$:
$Fk,B^(v;α')=1TB∑t=1T1nkt∑i=1nt1vk,Bit^(α')≤v×1{Bt=B,p(i)=k}.$
(10)
I next create an objective function that is the sum of the maximum distances between successive estimates of the distributions. I search for the $α'$ that minimizes this objective function. Thus, if I order the bidder sets in $Bk∪$ (${Bk1,⋯,BkSk}$), my estimate for the externality parameters is
$α^KS=argminα'∈A{∑k=1K∑s=1Sk-1maxv∈[ξk̲(α')^,ξk¯(α')^]|Fk,Bk(s+1)^(v;α')-Fk,Bks^(v;α')|}.$
(11)
Proposition 2.

Assume $A$ is a compact subset of $RL$. Assume that $ξk(b,G;α,B)$ is the equilibrium inverse bid function for a bidder of type $k$ that is strictly increasing in $b$ and continuous in $α$. Also assume that the identification conditions from the previous section hold. Then the estimator $α^KS$ defined above is a consistent estimate for the true parameter value $α*$.

The proof is in section B.3 of the online appendix.

#### Median estimator for $α$⁠.

An alternative estimator for $α$ is to equate the median pseudo-values (or any percentile) for bidders of the same type in auctions with different bidder sets. The advantage of this estimator is that it is just the solution to a set of linear equations and requires no minimization procedure. The disadvantage is that it is dependent on the median bid and thus is susceptible to small sample bias.

For each type $k$ and each bidder set $B$ that $k$ is in, I calculate the pseudo-value for the median bidder of type $k$ in an auction with bidder set $B$. I first find the empirical median bid of a type $k$ bidder in auctions with bidder set $B$. I use the monotonicity of $ξ$ to calculate the corresponding pseudo-value for each median bid, as a function of the externality parameters, as in equation (9). Then for each bidder type, I set the pseudo-values for median bidders from auctions with different bidder sets, equal to each other. For each bidder type $k$, this gives me $Sk-1$ equations, where again $Sk$ is the number of observed bidder sets that include bidder type $k$. These equalities are linear functions of the externality parameters, and so they form a system of linear equations in the desired parameters.

For each $k$, I define the matrix $Ck^$ of size $(Sk-1)×L$ and the vector $CkLC^$ of size $(Sk-1)×1$, which define the system of equalities between the median pseudo-values of bidders of type $k$ in different bidder sets $B$. I combine the equations for bidders of different types with the $∑k(Sk-1)×L$ matrix $C^$, constructed by stacking each $Ck^$ for all $k$, and the vector $CLC^=C1LC^C2LC^⋯CKLC^'$. The system I then wish to solve for $α$ is given by $C^α=CLLC^$.

When there is enough variation in the bidder sets for identification, a consistent estimate for the externality parameter is $α^med=C^'C^-1C^'CLC^$. Consistency requires the assumptions made for identification, such as monotonicity of the equilibrium bid function, exogenous bidder participation, and that equation (7) holds in equilibrium. Then consistency of the estimator $α^med$ follows straightforwardly from the consistency of the estimated distributions and densities used to construct the pseudo-values and the consistency of the sample median bid:
$α^med=C^'C^-1C^'CLC^→pC'C-1C'CLC=α.$
(12)
Proposition 3.

Assume the conditions for identification hold, and the estimated distributions, densities, and median bid are consistent estimates of their true counterparts. Then the estimator $α^med$, as defined above, is a consistent estimate for the true parameter value $α*$.

#### Mean estimate for $α$⁠.

The parameter vector $α$ can also be estimated by finding the parameter values that equate the mean pseudo-valuations. To construct this estimator, I first take the mean of each component of equation (7) over the observed sample bids (see section C.1 of the online appendix). The estimator then uses the restriction that mean valuations for certain bidder types should be the same across auctions with different bidder sets (i.e., $vk,Bμ=vk,Baμ$, $∀B,Ba$). This assumption is used to construct similar matrices as before, $Cμ^$ and $CLC,μ^$. They have the same form as the previous matrices but with the mean estimates for the components of equation (7) replacing the median estimates. Once again, consistency can be shown for the estimator $α^μ=Cμ^'Cμ^-1Cμ^'CLC,μ^$.

#### Estimate for distribution of valuations.

Once I have estimates of the externality parameters, I follow the existing literature to construct estimates of the distributions of valuations. Given any estimator of the parameters, $α^$, I can use equation (9) to compute the corresponding pseudo-values $vit^(α^)$ for each observed bid $bit$. Then an estimate for the bidder-type specific distribution of valuations is constructed similar to equation (10):
$Fk^(v)=1Tk∑t=1T1nkt∑i=1nt1{vit^(α^)≤v}1{p(i)=k},$
(13)
where $Tk$ is the number of observed auctions that contain bidders of type $k$. This estimated distribution of valuations does not depend on the bidder set.

### C. Extension to Observed and Unobserved Auction Heterogeneity

The above identification and estimation strategy assumes that we observe $T$ independent auctions of an identical object. In many cases (including the application of this paper), considerable heterogeneity exists among the objects being auctioned off. I will first show how the above estimation strategy can be extended to allow for observable auction heterogeneity and then also unobserved heterogeneity. Both extensions draw heavily on Haile et al. (2003).

#### Observable auction heterogeneity.

Assume that for each auction $t$, there is a set of characteristics, $xt$, that are observable to bidders before the auction—and to the econometrician. One approach that maintains the nonparametric nature of the above procedure, is to estimate the distributions in equation (9), conditioning on both bidder sets and $x$. The rest of the estimation proceeds the same as in section IIIB, resulting in a nonparametric estimate for the valuation distribution that is conditioned on $x$. However, with a large number of characteristics, this is often infeasible to estimate without a larger data set.

An alternative is to use an index approach. This reduces the dimension of the observable characteristics to 1 by regressing the average auction bid on the auction characteristics. The predicted value from this regression is then used as an index to condition on. The distributions in equation (9) are estimated as before, except that instead of conditioning on the entire vector $x$, one only conditions on this single-dimensional index. This approach has the advantage of mitigating the curse of dimensionality of the first technique while still keeping the distribution estimates nonparametric.

In the application, I use a variation of this index approach that additionally imposes structure on how $x$ affects valuation distributions. This approach has been used widely in the previous literature (Haile et al., 2003; Krasnokutskaya, 2011; Luo & Wan, 2018). I assume that bidder valuations are multiplicatively separable in $x$ (a similar approach applies if values are instead additively separable in $x$):
$vit=νit*Γ˜(xt),$
(14)
$ln(vit)=ln(νit)+Γ(xt),$
(15)
where $νit$ is distributed independent of $xt$. Haile et al. (2003) show that separability in valuations implies separability in bids. Thus, equilibrium bids can be broken down as
$ln(bit)=υ(Bt)+Γ(xt)+εit,$
(16)
where $Γ(x)$ is often a linear function of $x$.
I then estimate $Γ(x)$ by regressing log bids on the observable characteristics and indicators for the number of auction participants of each type. Homogenized bids are then constructed as
$ln(bhit)=ln(bit)-Γ^(xt).$
(17)
This is the bid that $i$ would have submitted if auction $t$ was a “generic” auction with bidder set $B$ and auction characteristics $x0$ such that $Γ^(x0)=0$. I then apply the same estimation procedure as before, but use the log homogenized bids as the observed equilibrium bids. If $Γ(x)$ is specified as a linear function of $x$, then $Γ^(x)$ converges at a relatively faster rate than the distribution estimates, so using homogenized bids for the above estimators does not affect their asymptotic properties.

#### Unobservable auction heterogeneity.

There may also be unobserved auction heterogeneity that affects bidder valuations. This poses two problems for my estimation procedure. The first is the typical problem of unobserved heterogeneity affecting the identifiability of the valuation distributions. The second problem arises if auction participation is related to the unobserved heterogeneity. In this case, bidder participation is no longer exogenous, which is crucial to identifying the externality parameters. This problem has been recognized in prior auction work that relies on exogenous variation in bidder sets (Guerre et al., 2009; Campo et al., 2011; Gillen, 2010; Compiani, Haile, & Sant'Anna, 2017). In this paper, if the effect the unobserved characteristics have on auction participation and bidder valuations is the same across bidders of different types, then they do not interfere with the identification of the externality parameters. This is because externalities are identified from how bidders of a given type respond differently to increased competition from bidders of different types. However, if the effect of the unobserved heterogeneity differs across bidder types, then the externalities estimated using the above procedure may be biased.

To deal with endogenous participation due to unobserved heterogeneity, I again use the approach of Haile et al. (2003). I assume that the unobservable heterogeneity is represented by a $K×1$ vector, $ut$, that is independent of $xt$ but may be correlated with bidder valuations. Each element of $ut$ is a scalar representation of the effect the unobserved characteristics of auction $t$ have on a particular bidder type $k$ (which, as stated above, may differ across types). I also assume the existence of a vector of instruments, $zt$, that is also independent of $ut$. An instrument should be an observable that is correlated with the number of bidders of each type but is not related to valuations, conditional on $xt$. For the application, I use the set of potential bidders of each type.

I then model the number of participating bidders of each type as a function of the observable auction characteristics, the instrument vector, and the scalar unobservable for bidders of that type:7
$nkt=ηkxt,zt+ukt.$
(18)
For $zt$ to be a valid instrument, each $ηk(·)$ is required to be nonconstant in $zt$. Data on ${nkt,xt,zt}$ are then used to get consistent estimates ($ηk^$) of the predicted number of bidders of each type. For the application below, I specify each $ηk(·)$ as a linear function of the observable auction characteristics and the instruments, and estimate it using a simple linear regression.
To estimate the full auction model, I then adapt equations (15) and (16) for the presence of unobserved heterogeneity. In addition to assuming separability of valuations in $xt$, I also assume separability in $ut$ as well. Then valuations and equilibrium bids can be written as
$ln(vit)=ln(νit)+Γ(xt,ut),$
(19)
$ln(bit)=υ(Bt)+Γ(xt,ut)+εit.$
(20)
I then control for $ut$ using the consistent estimates for $ηk^$ from before. Following equation (18), consistent estimates of the unobservable scalars are given by
$u^kt=nkt-ηk^xt,zt.$
(21)
Stacking the estimates for the different bidder types results in the $K×1$ vector, $u^t$, for each auction. If each $ηk(·)$ is specified as a linear function of $x$ and $z$, these estimates have a sufficient rate of convergence to not affect the rest of the estimation procedure.
I then follow the same approach as above by specifying $Γ(xt,u^t)$ as a linear function of $xt$ and separable in $u^t$ (i.e., $Γ(xt,u^t)=γxt+Λ(u^t)$). I estimate $Γ(·)$ by regressing log bids on the observable auction characteristics, and indicators for the number of auction participants of each type and$u^t(nt,xt,zt)$. Homogenized bids are then given by
$ln(bhit)=ln(bit)-Γ^(xt,u^t).$
(22)
The resulting homogenized bids are free of the auction heterogeneity that is related to endogenous entry. Thus, variation in these homogenized bids due to variation in the bidder sets is due to the externalities rather than any observed or unobserved auction heterogeneity. Therefore, I can use the homogenized bids in the estimation procedure of section IIIB and get unbiased and consistent estimates of the externality parameters and the valuation distributions.

## IV. Monte Carlo Experiments

To assess the performance of the externality parameter estimators, I ran several Monte Carlo experiments. In the experiments, bidders are one of $K=2$ types denoted by $M$ and $L$. I make three assumptions on their value distributions. The first is that the distributions are asymmetric, with $FM(·)∼U[0,1]$ and $FL(·)∼U[0,2]$. The second is that the distributions are asymmetric but have the same median and mean ($FM(·)∼U[0.25,1.25]$ and $FL(·)∼U[0,1.5]$). The econometrician is assumed to know that the distributions have the same median and mean but not know what they are. The final assumption I make is symmetry between the two distributions ($FM(·)∼U[0,1]$ and $FL(·)∼U[0,1]$). Here, the econometrician knows that the distributions are the same but does not know what that distribution is. For each case, I set the externality parameters between bidders to $αMM=0.3$, $αLL=0.2$, and $αML=αLM=0.1$.

Under each assumption, I create auctions with four bidder sets: auctions with two mills, three mills, two loggers, and one mill and one logger. The bidder sets that I can simulate auctions for is restricted by the difficulty in calculating the bid functions. Even with uniform valuations and the smallest possible number of participants, I cannot get an analytic solution to equilibrium equation (3) for auctions with bidders of different types. Therefore, to get bid functions for the auctions with one mill and one logger, I have to solve equation (3) numerically. This may add error to the procedure.

For each Monte Carlo run, I simulate a sample of 100 auctions—25 for each of the four different bidder sets. For each sample, I calculate the K-S, median, and mean estimates for the parameters. I run the experiments 100 different times for each of the three assumptions on value distributions. (A discussion of the choice of kernel, bandwidth, and the trimming procedure is in section D.1 of the online appendix.) The K-S estimator is the most computationally demanding of the three since it involves an optimization routine. The median estimator takes the shortest time to compute since I only need to evaluate the distribution functions for one bid value (the median bid) for each bidder type and set of participants. This may come at the expense of efficiency, as the other two estimators make use of the entire distribution of pseudo-valuations.

### A. Results

The results of the Monte Carlo experiments for each estimator are in table 1. The mean and median parameter estimates, along with the 10th and 90th percentiles, are presented for each of the three different assumption made on bidder value distributions. The results indicate that all three approaches perform well in estimating the externality parameters. In all cases, the 10th percentile and 90th percentile estimates bound the true value of the parameter. In most cases, the mean and median estimates are also close to the true $α$ value. As would be expected, the estimates perform better as more restrictions are put on the distribution functions. This affects the mean estimator the most. With no symmetry imposed between the value distributions, the mean estimator has a difficult time pinpointing the parameter values, and there is a lot of variance in the estimates. In this case, there are only four restrictions on the parameters and three parameters to estimate, so it is not surprising that the estimator does not perform as well. As more restrictions are put on the distributions, the mean estimator's performance improves.

Table 1.
Monte Carlo Results
Experiment 1: Asymmetric DistributionsExperiment 2: Same Median/MeanExperiment 3: Symmetric Distributions
$αMM=0.3$Mean50%10%90%Mean50%10%90%Mean50%10%90%
$α^KS$ 0.310 0.316 0.232 0.392 0.276 0.290 0.215 0.316 0.324 0.328 0.279 0.360
$α^med$ 0.323 0.304 0.246 0.465 0.309 0.320 0.275 0.332 0.320 0.315 0.290 0.364
$α^μ$ 0.397 0.341 0.280 0.568 0.289 0.292 0.281 0.302 0.297 0.298 0.289 0.308
$αLL=0.2$ Mean 50% 10% 90% Mean 50% 10% 90% Mean 50% 10% 90%
$α^KS$ 0.208 0.207 0.161 0.243 0.216 0.205 0.199 0.256 0.205 0.198 0.184 0.234
$α^med$ 0.200 0.200 0.061 0.282 0.194 0.185 0.1406 0.245 0.185 0.189 0.138 0.225
$α^μ$ 0.070 0.141 −0.128 0.255 0.208 0.209 0.187 0.227 0.204 0.203 0.197 0.210
$αML=0.1$ Mean 50% 10% 90% Mean 50% 10% 90% Mean 50% 10% 90%
$α^KS$ 0.089 0.092 0.073 0.104 0.102 0.103 0.091 0.109 0.094 0.102 0.089 0.104
$α^med$ 0.086 0.075 0.044 0.108 0.098 0.107 0.053 0.136 0.095 0.099 0.064 0.127
$α^μ$ 0.134 0.102 0.068 0.197 0.103 0.103 0.086 0.117 0.099 0.103 0.083 0.108
Experiment 1: Asymmetric DistributionsExperiment 2: Same Median/MeanExperiment 3: Symmetric Distributions
$αMM=0.3$Mean50%10%90%Mean50%10%90%Mean50%10%90%
$α^KS$ 0.310 0.316 0.232 0.392 0.276 0.290 0.215 0.316 0.324 0.328 0.279 0.360
$α^med$ 0.323 0.304 0.246 0.465 0.309 0.320 0.275 0.332 0.320 0.315 0.290 0.364
$α^μ$ 0.397 0.341 0.280 0.568 0.289 0.292 0.281 0.302 0.297 0.298 0.289 0.308
$αLL=0.2$ Mean 50% 10% 90% Mean 50% 10% 90% Mean 50% 10% 90%
$α^KS$ 0.208 0.207 0.161 0.243 0.216 0.205 0.199 0.256 0.205 0.198 0.184 0.234
$α^med$ 0.200 0.200 0.061 0.282 0.194 0.185 0.1406 0.245 0.185 0.189 0.138 0.225
$α^μ$ 0.070 0.141 −0.128 0.255 0.208 0.209 0.187 0.227 0.204 0.203 0.197 0.210
$αML=0.1$ Mean 50% 10% 90% Mean 50% 10% 90% Mean 50% 10% 90%
$α^KS$ 0.089 0.092 0.073 0.104 0.102 0.103 0.091 0.109 0.094 0.102 0.089 0.104
$α^med$ 0.086 0.075 0.044 0.108 0.098 0.107 0.053 0.136 0.095 0.099 0.064 0.127
$α^μ$ 0.134 0.102 0.068 0.197 0.103 0.103 0.086 0.117 0.099 0.103 0.083 0.108

The left-most columns show the distribution of estimates when the value distributions are assumed to be completely asymmetric ($FM(·)∼U(0,1)$ and $FL(·)∼U(0,2)$). The middle columns show the distribution of estimates when the value distributions are assumed to have the same median and mean ($FM(·)∼U(.25,1.25)$ and $FL(·)∼U(0,1.5)$). The right-most columns show the distribution of estimates when the value distributions are assumed to be symmetric ($FM(·)∼U(0,1)$ and $FL(·)∼U(0,1)$).

## V. Application to USFS Timber Auctions

I now apply the strategy to timber auctions held by the U.S. Forest Service (USFS). This is an interesting setting to look at externalities in as there have recently been a series of antitrust cases brought against a lumber mill, accusing the mill of predatory bidding.

### A. Industry Background

In 2007, the U.S. Supreme Court heard the case of Weyerhaeuser Company v. Ross-Simmons Hardwood Lumber Company, Inc. Weyerhaeuser Company was accused of placing higher bids in timber auctions in order to achieve monopsony power in the market for timber. According to Rausser and Foote (2007), testimony indicated that “Weyerhaeuser sometimes bought more sawlogs than it needed with the purpose of keeping them from competitors.” This is consistent with a model of auction externalities, where certain firms are induced to bid more in order to keep the object away from specific competitors. The Weyerhaeuser case was not the only example of smaller sawmills alleging that larger mills were hoarding timber to push them out of the market. In 2015, the Small Business Administration published an Advance Notice of Proposed Rulemaking (ANPRM), requesting public comments on proposed rule changes to the Forest Service's small business set-aside program. According to the Federal Register (Small Business Set-Aside Program, 2016), “Several commenters noted that large multinational companies have begun to aggressively pursue both timber program full-and-open and stewardship sales in an attempt to drive small businesses from the playing field.”

The identification and estimation strategy of this paper can be used to investigate whether mill bidders did indeed exhibit bidding behavior consistent with trying to keep timber away from downstream competitors. By applying the model to this setting, I can better understand how outside competition affected bidding behavior in timber auctions and the degree to which it actually affected auction outcomes, as was alleged in the instances above. To my knowledge, no prior work has looked at the effect downstream competition has had on timber auction outcomes.

Instead of focusing on specific cases of alleged predatory bidding, I broadly look at USFS timber auctions held in all regions between 1982 and 1990. This provides a more general understanding of how downstream competition affected timber auction markets. These auctions are conducted by the USFS in order to allocate the right to remove all timber from a given tract. The Forest Service initially identifies a tract to be sold off and conducts a “cruise” of the tract to get appraisal estimates of the total value of the tract. The appraisal is used to determine the reserve price set in the auction, which is generally viewed as nonbinding.8 The characteristics of the tract and the estimated appraisal values are made public to the bidding firms prior to the auction.

The Forest Service uses a mix of open and sealed bid auction formats, but I focus here on sealed bid auctions.9 Bids are made on a per unit (thousand board-feet of timber) basis. Most of the literature has used a model of private values for these “scaled sale” auctions. This is because bidders are generally specialized firms that differ in their inventories of both uncut timber and the end-product lumber, and differ in terms of production costs.10 Once the auction is completed, the winning bidder has a set period of time (set in the contract) to harvest the timber on the tract.

The bidders in these auctions range from large, vertically integrated conglomerates to small logging companies. Based on Athey et al. (2011), I classify bidders into three groups: mills that have manufacturing capacity, loggers that do not have manufacturing capacity, and small businesses with fewer than 25 employees. Loggers only harvest the timber on each tract, while mills manufacture the timber into lumber. Therefore, downstream competition in the lumber market involves only mill firms. I do not further classify bidders based on their geographic location because while mill assembly costs and transportation costs restrict the timber supply for each mill to a local geographic zone, the downstream market for lumber is generally considered to be global in nature. Thus, externalities based on geographic location appear to be less important in this setting.

### B. Data

I look at sealed bid auctions in all nine USFS regions between 1982 and 1990. I restrict the sample to after 1981 since policy changes in 1981 reduced the significance of resale affecting bidder valuations, an important issue discussed in Haile (2001). I further restrict the sample to auctions where more than one bid was actually received and the item was sold. In a small portion of auctions, entry is restricted to small businesses. I do not include those auctions in my sample.

The data are from the USFS (via Phil Haile's website). For each auction, I know the date and location of the auction, the auction format used, the contract length, the bids and bidder identities, and the cruise estimates. The bid data contain the per unit bid price placed by each firm for each species. Using the data on the volume of each species, I combine the separate bids for each species to get a total bid for each bidder, which in practice is what is used to determine the auction winner. The cruise estimates include the volume and density of the tract and estimates of the selling value, manufacturing costs, road construction costs, logging costs, and advertised rate (which is set as the reserve price). I also compute the Herfindahl index for the concentration of species across a tract, since specialized mills may place a higher value on tracts where the timber is concentrated in only a few species.

Table 2 provides summary statistics of the tract characteristics and auction outcomes. Mills win 57% of auctions, loggers win 42%, and small businesses win less than 1% of auctions. On average, auctions contain 2.05 mill participants and 1.67 loggers. The tracts also exhibit considerable heterogeneity in the observable characteristics, which I control for in the estimation below.

Table 2.
Timber Auction Data Summary Statistics
VariableMeanMedianSD
Winning bid ($) 101,942 32,827 194,408 Per unit winning bid ($/mbf) 96.54 82.00 74.71
Number of bidders 3.76 3.00 1.85
Number of mills 2.05 2.00 1.87
Number of loggers 1.67 1.00 1.87
Number of small businesses 0.04 0.00 0.41
Mill wins auction 0.57
Logger wins auction 0.42
Timber volume (mbf) 1,032 390 10,786
Density (mbf/acre) 4.65 1.97 9.20
Advertised rate ($) 61,233 15,636 130,406 Selling value ($) 578,924 167,110 1,313,678
Manufacturing costs ($) 338,089 108,663 870,118 Logging costs ($) 225,522 76,373 484,346
Road construction costs ($) 6,600 394 123,308 Per unit advertised rate ($/mbf) 64.39 51.43 52.47
Per unit selling value ($/mbf) 344.60 361.23 125.72 Per unit manufactruing costs ($/mbf) 178.38 178.18 45.63
Per unit logging costs ($/mbf) 130.83 127.05 45.97 Per unit road construction costs ($/mbf) 15.30 2.59 231.88
Contract length (years) 1.96 1.81 1.48
Salvage sale 0.13 0.00 0.34
Species HHI 0.60 0.55 0.22
VariableMeanMedianSD
Winning bid ($) 101,942 32,827 194,408 Per unit winning bid ($/mbf) 96.54 82.00 74.71
Number of bidders 3.76 3.00 1.85
Number of mills 2.05 2.00 1.87
Number of loggers 1.67 1.00 1.87
Number of small businesses 0.04 0.00 0.41
Mill wins auction 0.57
Logger wins auction 0.42
Timber volume (mbf) 1,032 390 10,786
Density (mbf/acre) 4.65 1.97 9.20
Advertised rate ($) 61,233 15,636 130,406 Selling value ($) 578,924 167,110 1,313,678
Manufacturing costs ($) 338,089 108,663 870,118 Logging costs ($) 225,522 76,373 484,346
Road construction costs ($) 6,600 394 123,308 Per unit advertised rate ($/mbf) 64.39 51.43 52.47
Per unit selling value ($/mbf) 344.60 361.23 125.72 Per unit manufactruing costs ($/mbf) 178.38 178.18 45.63
Per unit logging costs ($/mbf) 130.83 127.05 45.97 Per unit road construction costs ($/mbf) 15.30 2.59 231.88
Contract length (years) 1.96 1.81 1.48
Salvage sale 0.13 0.00 0.34
Species HHI 0.60 0.55 0.22

mbf: 1,000 board feet.

### C. Preliminary Evidence

If externalities play an important role in timber auctions, then bidders will be observed to be adjusting their bids based on the types of competitors they are facing in the auction. The key for identification of the externalities is to separate the adjustment due to externalities from standard competition effects. For example, a bidder may increase its bid when facing more mill competitors due to either externalities with mills or because mills have higher valuations for timber.

To distinguish between the two cases, figure 1 graphs the bid distributions for mills and loggers when facing different competitors. The distributions are formed by first regressing each bid on the observable auction characteristics described in section VB, as well as dummies for the year and region of the auction. I then take the bid residuals from this regression and separate them based on the type of bidder (mill or logger) and the number and types of competitors in the auction.

Figure 1.

Distributions of Bid Residuals for Mill and Logger Bidders Facing Different Competition Sets

Figure 1.

Distributions of Bid Residuals for Mill and Logger Bidders Facing Different Competition Sets

Figure 1a shows the residual bid distributions for mills and loggers in auctions where they are facing three logger competitors and no mill competitors. If the main externality in timber auctions is coming from mill bidders (the bidder type that competes downstream), then externalities should play no role in these auctions. However, if mill bidders have higher valuations than loggers, then even when facing competition only from other loggers, mill bidders should bid higher than loggers. Looking at figure 1a, the residual bid distributions for the two types of bidders are instead very similar, indicating that mill bidders do not necessarily have higher valuations than loggers. A Kolmogorov-Smirnov test that the two samples of bid residuals are drawn from the same distribution fails to reject the null at a 10% significance level (the $p$-value is 44.6%), confirming their visual similarity.

Figure 1b then compares the mill distribution in figure 1a, with the distribution of mill bid residuals when one of the logger competitors is replaced with a mill competitor. It shows that facing an additional mill bidder rather than a logger leads to more aggressive bidding by mills. Figure 1c does the same for loggers, comparing their bid residual distribution when facing three loggers and zero mills, with that when facing two loggers and one mill. Contrary to figure 1b, there is no discernible difference between the distributions in figure 1c. A Kolmogorov-Smirnov test of the two samples of mill bids rejects the null that the two samples come from the same distribution at a 1% level (the $p$-value of the test is 0.089%), while the same test on the two samples of logger bids fails to reject the null that they come from the same distribution at a 10% significance level (the $p$-value of that test is 22.7%). Together, figures 1b and 1c imply that mills bid more aggressively when competing against mills rather than loggers, but that loggers do not adjust their bids based on the types of the competitors they are facing. This is evidence of the presence of externalities between mill bidders.

Figures 1d to 1f are similar to figures 1a to 1c, but for auctions with five total bidders rather than four. Again, figure 1d shows that when facing competition only from loggers, mills and loggers bid similarly, indicating that mills do not have systematically higher valuations than loggers. Figures 1e and 1f then compare the mill (logger) residual bid distribution when facing no mills, with that when facing three loggers and one mill, and when facing two loggers and two mills. Figure 1e shows that mills increase their bids as competing loggers are replaced with competing mills. The mill residual bid distribution when facing three loggers and one mill stochastically dominates the mill distribution when facing four loggers and no mills. These are then each stochastically dominated by the mill bid distribution when facing two loggers and two mills. If this was due to mills having higher valuations, then I would expect loggers to react similarly to the change in the composition of competitors from loggers to mills. However, figure 1f shows that the bid distributions are relatively similar for loggers when facing the three different bidder sets, reinforcing that externalities between mill bidders are what drive the changing bid behavior in figure 1e.

For further evidence, I run a regression of the log of each bid on the total number of competitors, a mill dummy, the number of mill competitors if the bidder is a mill, and the number of mill competitors if the bidder is a logger. I also include the observed auction characteristics and dummies for the year and region of the auction. In running this regression, I am concerned about the endogeneity of bidder participation. In particular, some auctions may be more attractive to bidders (and to bidders of particular types) for unobservable reasons, and these auctions are also likely to elicit more aggressive bidding. To deal with this, I use the number of potential bidders of each type as an instrument for the actual number of participants of each type. I define potential bidders for each auction as the number of firms of each type that participated in an auction in the same county as the auction of interest, within the previous year. This is a valid instrument as long as the number of potential bidders is uncorrelated with the unobserved auction heterogeneity.

The main regression results are in table 3. Column 1 presents the results of a simple OLS regression. Column 2 contains the results of the IV regression (the first-stage results are in table A1 of the online appendix). The results in column 2 show that mill bidders do not bid higher, holding their competition constant. The coefficient on the mill dummy is small (−0.0002) and not statistically significant. This indicates that mill bidders do not have higher valuations than loggers for timber. The coefficient on the number of mills if a bidder is a mill is 0.0749 and is statistically significant. This coefficient is interpreted as saying that mills will increase their bid by roughly 7.5% for each additional mill competitor, holding constant the total number of bidders in the auction. Thus, mills are increasing their bids by more when facing additional competition from other mill bidders than if those additional competitors are loggers. In contrast, the coefficient on the number of mill bidders if the bidder is a logger is small (0.0081) and not statistically significant. Thus, loggers are not responding to additional competition from mills in the same way.

Table 3.
Preliminary Regression of $ln(bid)$ on Observed Auction Characteristics
OLS (1)IV (2)Only the Advertised Rate (3)
Mill Dummy 0.0358* −0.0002 −0.0002
(0.0102) (0.0104) (0.0104)
Number of Bidders 0.0413* 0.0660* 0.0692*
(0.0016) (0.0116) (0.0103)
Number of mills if mill 0.0283* 0.0749* 0.0537*
(0.0023) (0.0093) (0.0087)
Number of mills if logger 0.0236* 0.0081 −0.0134
(0.0033) (0.0097) (0.0091)
ln(Advertised Rate) 0.5719* 0.5666* 0.5685*
(0.0030) (0.0035) (0.0034)
Density −0.0020* −0.0027* −0.0023*
(0.0003) (0.0004) (0.0004)
ln(Volume) 0.3146* 0.3078* 0.3107*
(0.0036) (0.0046) (0.0044)
ln(Selling Value) −0.0076* −0.0049023*
(0.0023) (0.0025)
ln(Manufacturing Costs) 0.0115* 0.0055
(0.0038) (0.0041)
ln(Logging Costs) −0.0026 0.0058
(0.0037) (0.0038)
ln(Road Costs) −0.0039* −0.0050*
(0.0010) (0.0011)
Salvage scale dummy −0.0004 0.0303* 0.0262*
(0.0086) (0.0097) (0.0095)
Species HHI −0.2646* −0.1756* −0.1998*
(0.0133) (0.0201) (0.0191)
Contract length 0.1417* 0.1222* 0.1227*
(0.0030) (0.0030) (0.0030)
Constant 2.6621* 2.6951* 2.6951*
(0.0281) (0.0404) (0.0404)
Region dummies
Year dummies
Number of observations 33,239 33,239 33,239
$R2$ 0.8913 0.8858 0.8856
OLS (1)IV (2)Only the Advertised Rate (3)
Mill Dummy 0.0358* −0.0002 −0.0002
(0.0102) (0.0104) (0.0104)
Number of Bidders 0.0413* 0.0660* 0.0692*
(0.0016) (0.0116) (0.0103)
Number of mills if mill 0.0283* 0.0749* 0.0537*
(0.0023) (0.0093) (0.0087)
Number of mills if logger 0.0236* 0.0081 −0.0134
(0.0033) (0.0097) (0.0091)
ln(Advertised Rate) 0.5719* 0.5666* 0.5685*
(0.0030) (0.0035) (0.0034)
Density −0.0020* −0.0027* −0.0023*
(0.0003) (0.0004) (0.0004)
ln(Volume) 0.3146* 0.3078* 0.3107*
(0.0036) (0.0046) (0.0044)
ln(Selling Value) −0.0076* −0.0049023*
(0.0023) (0.0025)
ln(Manufacturing Costs) 0.0115* 0.0055
(0.0038) (0.0041)
ln(Logging Costs) −0.0026 0.0058
(0.0037) (0.0038)
ln(Road Costs) −0.0039* −0.0050*
(0.0010) (0.0011)
Salvage scale dummy −0.0004 0.0303* 0.0262*
(0.0086) (0.0097) (0.0095)
Species HHI −0.2646* −0.1756* −0.1998*
(0.0133) (0.0201) (0.0191)
Contract length 0.1417* 0.1222* 0.1227*
(0.0030) (0.0030) (0.0030)
Constant 2.6621* 2.6951* 2.6951*
(0.0281) (0.0404) (0.0404)
Region dummies
Year dummies
Number of observations 33,239 33,239 33,239
$R2$ 0.8913 0.8858 0.8856

Column 1 contains the results of the simple OLS regression of $ln(bid)$ on observed auction participants and characteristics. Column 2 instruments for the number of participants of each type, with the number of potential bidders of each type. Column 3 excludes the auction characteristics that are used in determining the advertised rate and includes only the advertised rate. Coefficient estimates market with an * are statistically significant at the 5% level.

Some of the coefficients on the appraisal variables in column 2 make less sense (e.g., the negative coefficient on selling value). That most likely has to do with those values already being incorporated in the advertised rate. Thus, as a robustness check I run the same regression as in column 2, but without the observed auction characteristics that are incorporated in the advertised rate. Column 3 contains the results. The coefficient on the number of mill competitors for a mill bidder is still positive (0.0537) and statistically significant, while neither the mill dummy nor the number of mill competitors for a logger have statistically significant coefficients.

### D. Estimation Procedure and Dealing with Endogenous Participation

Given the preliminary evidence of externalities between mill bidders, I apply the estimation technique of this paper to timber auctions to identify the externality parameters and value distributions. I assume that bidders of the same type have the same value distribution but allow for asymmetries between different bidder types. There is considerable observed heterogeneity between the timber tracts, and so I homogenize the bids (discussed in section IIIC) using the auction characteristics available in the data. Most of this observed heterogeneity is captured by the advertised rate set by the Forest Service, but $xt$ also includes the species HHI, the contract length, and dummies for the year and region of the auction. I specify $Γ(x)=γx$ and estimate the $γ$ parameters.

I am also concerned about unobserved auction heterogeneity. The estimation procedure relies on exogenous variation in bidder participation. Unobserved auction characteristics that increase bidder valuations and auction participation among bidders of certain types preclude participant exogeneity and may be misinterpreted as externalities. I deal with this using the approach outlined in section IIIC. The basic idea is similar to the instrumental variables approach of the previous section, where I use exogenous variation in the set of potential bidders for identification. In particular, the potential number of bidders and the observed auction characteristics are used to predict the number of bidders of each type for a given auction. The difference between the actual number of bidders and the predicted number of bidders is then assumed to be due to unobserved heterogeneity, which can then be controlled for in the subsequent estimation.

I define potential bidders as before. I assume that the observed auction characteristics and the number of potential bidders of each type, are independent of the unobserved heterogeneity. I then specify the bidder participation functions—($ηk(xt,zt)$ in equation (18))—as linear functions of the observable auction characteristics and the number of potential bidders of each type. I get consistent estimates for each $ηk$ by regressing $nkt$ on the observed auction characteristics ($xt$) and the number of potential bidders of each type ($zt$). This is done for each bidder type and is the same as the first-stage regression from the previous section (results are in table A1 of the online appendix). These estimates are then used to construct the predicted number of auction participants of each type for each auction. Since this prediction is generally not an integer value, I pool the predictions into bands centered around integer values. I then get consistent estimates of the unobserved heterogeneity (for each type) from equation (21), by subtracting the predicted number of auction participants of each type from the actual number of participants of that type. I then use these estimates for the unobserved heterogeneity ($u^kt$) in the bid homogenization procedure.

To homogenize the bids, I first regress log bids on the auction characteristics and indicators for both the number of bidders of each type and for the estimates of the unobserved heterogeneity for each type:
$ln(bit)=γxt+∑k=13∑j=1Jk¯λk,j1nkt=j+∑k=13∑l=Lk̲Lk¯φk,l1u^kt=l,$
(23)
where $Jk¯$ is the maximum number of bidder participants of type $k$ and $Lk¯$ ($Lk̲$) is the maximum (minimum) possible value of $u^kt$. The homogenized bids are then constructed as
$ln(bhit)=ln(bit)-γ^xt-∑k=13∑l=Lk̲Lk¯φ^k,l1u^kt=l.$
(24)
These are the bids that would have been submitted by bidders facing bidder set $B$ if each auction were instead a generic auction with observed and unobserved characteristics $x0$ and $u0$, respectively, such that $Γ(x0,u0)=0$. Thus, these bids tell us about the distribution of the idiosyncratic portion of bidder valuations—$ln(νit)$ in equation (19). Auction participation, which may have previously been endogenous if it was correlated with $xt$ or $ut$, which also affects bidder valuations through $Γ(·)$ in equation (19), is independent of $ln(νit)$, and so valuation distributions estimated from the homogenized bids no longer depend on the bidder set $B$. Therefore, I can again use variation in the observed bidder sets to identify the externality parameters as was done in section III. Estimation proceeds as before, but on the homogenized bids. It is important to note that because I am using log homogenized bids for estimation, the estimated externality parameters are in the same units as $ln(ν)$ (rather than that of $v$). I also compute the valuation distributions as the distributions of $ln(ν)$. This is done simply to ease the interpretation of these estimates, and it would not have been difficult to de-log the estimates or add back in the estimated heterogeneity.

I estimate the externalities using each of the three approaches outlined above: the median estimator, the mean estimator, and the K-S estimator. I use the kernel, bandwidths, and trimming procedure outlined in section D.1 of the online appendix. To get standard errors, I use a bootstrap method based on 100 subsamples of the observed bids.

### E. Results

Initially, I only allow for externalities to exist between two mill bidders and between two loggers. The estimation results are in table 4. Across the different methods, the externality parameter between mill bidders remains considerably large. The estimates indicate that a mill bidder winning the auction will cost a rival mill by between 10.4% and 21.8% of the idiosyncratic portion of their valuation for the timber.11 Ostensibly this is due to the competition between mill bidders in the downstream lumber market. The estimated externality between loggers is not statistically significant for any of the estimation methods. This is not unexpected given that logging firms do not produce lumber for the downstream market and thus do not compete outside the auction in the same way that mills do.

Table 4.
Externality Parameter Results: Mills and Loggers
MedianMeanKS
Ignore Unobserved HeterogeneityAccount for Unobserved HeterogeneityIgnore Unobserved HeterogeneityAccount for Unobserved HeterogeneityIgnore Unobserved HeterogeneityAccount for Unobserved Heterogeneity
Mill-mill 0.1822 0.2184 0.1595 0.1696 0.1041 0.1058
(0.0232) (0.0218) (0.0375) (0.0164) (0.0459) (0.0209)
Log-log 0.0779 0.0074 0.0570 −0.0120 −0.0246 0.0083
(0.0465) (0.0539) (0.0462) (0.0328) (0.0280) (0.0465)
MedianMeanKS
Ignore Unobserved HeterogeneityAccount for Unobserved HeterogeneityIgnore Unobserved HeterogeneityAccount for Unobserved HeterogeneityIgnore Unobserved HeterogeneityAccount for Unobserved Heterogeneity
Mill-mill 0.1822 0.2184 0.1595 0.1696 0.1041 0.1058
(0.0232) (0.0218) (0.0375) (0.0164) (0.0459) (0.0209)
Log-log 0.0779 0.0074 0.0570 −0.0120 −0.0246 0.0083
(0.0465) (0.0539) (0.0462) (0.0328) (0.0280) (0.0465)

Standard errors in parentheses are found using a bootstrap method based on 100 subsamples of the observed bids.

Mill bid functions constructed using the median estimator that accounts for unobserved heterogeneity are illustrated in figure 2. The figure shows the bid functions for a mill bidder facing two logger opponents, facing three loggers, and facing one mill and two loggers. In going from facing two loggers to facing three loggers, the mill bidder sees an increase in competition for the timber but no increase in the probability of incurring an externality from losing the auction. Thus, the shift up in the bid function there is driven purely by competition, not by externalities. If the additional bidder is instead a mill, then the bid function shifts up even higher to the function for auctions with two mills and two loggers. This additional shift up in the bid function is due the mill wanting to keep the timber away from the rival mill because of the externality.

Figure 2.

Mill Bid Function for Different Bidder Sets

Estimated mill bid functions (using the median estimates accounting for unobserved heterogeneity) when the mill bidder is facing two loggers, facing three loggers, and facing one mill and two loggers.

Figure 2.

Mill Bid Function for Different Bidder Sets

Estimated mill bid functions (using the median estimates accounting for unobserved heterogeneity) when the mill bidder is facing two loggers, facing three loggers, and facing one mill and two loggers.

The externality effect could affect the allocation of timber if the highest bids are not coming from firms with the highest valuation for the timber, but instead from mills that are bidding high to keep the tract away from rival mills. Using the median estimator that accounts for unobserved heterogeneity, I find that for 5.20% of the auctions in my sample, the presence of externalities made it so that the timber tract was not won by and allocated to the highest-value bidder. In the majority of those cases, the highest-value bidder was a logger, but the highest bidder was a mill that bid up the price to keep the tract away from another mill (see section E of the online appendix).

Figure 3 illustrates how not accounting for externalities can affect estimates for the valuation distributions. The graph on the left shows the estimated distributions of valuations for mills and loggers when externalities are taken into account, while the graph on the right shows the estimated distributions when externalities are ignored. In the graph on the right, the mill bidders' valuation distribution stochastically dominates the logger bidders' valuation distribution. But in the graph on the left, there is no stochastic dominance. This shows that if externalities are not accounted for when estimating the value distributions, it will appear as if mill bidders have higher valuations for timber tracts than their logger counterparts do. My estimation results instead say that mill bidders are bidding higher in certain auctions to keep the timber away from other mills that they compete with in the downstream lumber market. I separate this from the asymmetric distributions explanation by comparing mill and logger bids in auctions where they face more competition from rival mills with mill and logger bids where they face more competition from rival loggers. I find that mills, on average, bid higher than loggers only when facing more mills and not when facing more loggers. This is evidence that the higher bids placed by mills are due to externalities from outside competition between mills rather than asymmetries in the valuation distributions between mills and loggers.

Figure 3.

Estimated Distribution of Logarithm of Valuation Residuals

(Left) The estimated CDF for both mill and logger valuations using the estimation method of this paper based on the auction model with externalities. (Right) The estimated CDF for both mill and logger valuations using a typical auction estimation procedure that ignores externalities.

Figure 3.

Estimated Distribution of Logarithm of Valuation Residuals

(Left) The estimated CDF for both mill and logger valuations using the estimation method of this paper based on the auction model with externalities. (Right) The estimated CDF for both mill and logger valuations using a typical auction estimation procedure that ignores externalities.

Ignoring the externalities also affects estimated markups. Table 5 compares estimates for bidder profit margins that account for externalities with those that do not. These margins are calculated with the auction heterogeneity added back into both bids and bidder valuations. The table shows that the median profit margin is 21.43% if externalities are accounted for but 27.77% without externalities. The rise comes almost entirely from mill bidders. The estimated median profit margin across all mills is 13.32% when externalities are accounted for, and it is 28.41% when they are ignored. This is because ignoring externalities implies that mills have higher valuations for the timber, meaning that the profit margins on their bids are higher. With externalities, mills instead sacrifice direct profit from the timber in order to keep it away from rivals. Loggers do not have the same incentive to forfeit profit on the timber to avoid losses in the downstream market. Thus, loggers see a much higher median profit margin of 26.49% with externalities (which is nearly equivalent to the median profit margin of 26.94% for loggers in the model without externalities).

Table 5.
Median Estimated Profit Margins
Model with ExternalitiesModel without Externalities
All bidders 21.43% 27.77%
Mills 13.32% 28.41%
Loggers 26.49% 26.94%
Model with ExternalitiesModel without Externalities
All bidders 21.43% 27.77%
Mills 13.32% 28.41%
Loggers 26.49% 26.94%

The first column presents the median bidder profit margins ($=Bid-ValueValue$) estimated using the model of this paper that accounts for externalities. The second column presents estimates for the median profit margins when externalities are ignored. These estimates are calculated using bids and estimated valuations that have the observed and unobserved auction heterogeneity added back in.

To further investigate the source of the externalities, I divide the sample by region and perform the estimation routine separately for different regions. Regions differ in the fraction of forest land that is publicly and privately owned. Forest land in the West is primarily publicly owned, while the South has the largest concentration of privately owned forest land (see figure A2 in the online appendix). This suggests that National Forest Service (NFS) auctions are the primary source of timber in the West. However, in the South, it is more likely for a large mill to own land that supplies its timber or to have negotiated a deal with a private landowner to supply its timber. This makes it harder for mills to keep timber away from rivals through the NFS auctions, as these rivals are more likely to have alternative sources of timber in the South. Thus, I expect the externalities to be stronger in the western regions.

The first row of table 6 shows the results of estimating the externalities model on data from only the six western regions (North, Rocky Mountain, Southwest, Intermountain, Pacific Southwest, and Pacific Northwest). The estimated externalities between mill bidders in the West are significantly higher than the estimates in table 4, when all regions were included. The second row contains the results of estimating the model only for the southern region. There, the estimated externalities are much smaller, which is expected if mills in this region have other sources of timber.

Table 6.
Externality Parameter Results by Region and Mill Type
MedianMeanKS
A. Western and southern regions
Mill-mill 0.2992 0.2555 0.2274
Western regions (0.1353) (0.1441) (0.1389)
Mill-mill 0.1824 0.1192 0.1710
Southern region (0.0602) (0.0593) (0.0978)
B. Small mills and large mills
Small mill–Small mill 0.1204 0.2802 0.0228
(0.0381) (0.2230) (0.1156)
Small mill–Large mill 0.3358 0.3135 0.2195
(0.0611) (0.2560) (0.0975)
Large mill–Small mill −0.0092 −0.0766 −0.0012
(0.0873) (0.1739) (0.2987)
Large mill–Large mill 0.0602 0.0670 0.1238
(0.1799) (0.1945) (0.2786)
MedianMeanKS
A. Western and southern regions
Mill-mill 0.2992 0.2555 0.2274
Western regions (0.1353) (0.1441) (0.1389)
Mill-mill 0.1824 0.1192 0.1710
Southern region (0.0602) (0.0593) (0.0978)
B. Small mills and large mills
Small mill–Small mill 0.1204 0.2802 0.0228
(0.0381) (0.2230) (0.1156)
Small mill–Large mill 0.3358 0.3135 0.2195
(0.0611) (0.2560) (0.0975)
Large mill–Small mill −0.0092 −0.0766 −0.0012
(0.0873) (0.1739) (0.2987)
Large mill–Large mill 0.0602 0.0670 0.1238
(0.1799) (0.1945) (0.2786)

Finally, I break down mills into small and large mills based on Small Business Administration size standards. I estimate the same model as before but look for externalities between the different mill types and ignore externalities between loggers. The results are in panel B of table 6. Standard errors are larger than in the base case, and the different estimators are less consistent with one another. Still, one can infer from the estimates that the most affected firms are small mills and that they are most affected by large mills. On the other hand, large mills do not appear to be affected by outside competition. Thus, the externality estimated in the base case appears to be driven mainly by the effect downstream competition has on small firms. This is likely a result of timber auctions being local in nature, so that bidding is most affected by local downstream competition. Small mills are more concerned with local competition since large mills garner most of their revenue from the global lumber market.

## VI. Conclusion and Extensions

In this paper, I study how downstream competition between bidders affects bidding strategies and auction outcomes. I introduce a simple auction model with externalities and establish identification and estimation of the model. I present three estimators for the externalities and provide Monte Carlo results showing that they perform well in a simple setting with two bidder types.

I then apply the identification and estimation strategy to USFS timber auctions. This is a setting where recent antitrust allegations have accused a mill bidder of raising its bid in order to keep timber away from competitors and affect the downstream lumber market. I find that between 1982 and 1990, downstream competition between mills was strong enough that a rival mill acquiring a timber tract cost a mill bidder by between 10% and 22% of the heterogeneous portion of its valuation of the tract. There was no such effect for logger bidders. This indicates that mills bid higher in timber auctions in order to keep the timber away from other mills. This helps to explain why mills generally bid higher than loggers do. This was previously explained by mills having valuations that stochastically dominated that of loggers, but I find that not to be true here. Furthermore, I find that these externality effects lead the auctioned-off timber tract to be misallocated to a bidder who does not have the highest valuation for the tract in 5.2% of the auctions in my sample.

The model of this paper can be extended in a variety of ways. One is to allow the externality that a bidder imposes on a rival to be correlated with their private value for the object. In that case the externality is bidder-pair-specific ($αikjk'$), and private information to its imposer ($jk'$). Jehiel et al. (1996) provide some arguments on why it makes sense in a variety of examples for $αikjk'$ to be private information and to be correlated with $vjk'$. To extend the estimation strategy of this paper to that case, I assume a specific form for this correlation—particularly, that $αikjk'=αkk'vjk'$ for some types-specific parameter $αkk'$. As far as I know, the estimation strategy extends only to this form of correlation. The imposed relationship is a result of applying Cournot competition to the downstream market, where the auction is for some good that lowers the winning firm's marginal cost. The parameter $αkk'$ is then a combination of the parameters from the Cornout model. This is shown in section A.3 of the online appendix for the case of two bidders.

Estimation of the parameters then follows the same strategy as before, except that the equation relating bidder valuations to bids is different. Instead of equation (7), I get its counterpart:
$vik=Hk(b|B)+bHk'(b|B)Hk'(b|B)+∑k'∑j∈Bk'αkk'Prmaxl≠i,j∈Bbl≤b|Bgj(b|B).$
(25)

I can then use this equation in the same way that I used equation (7): to identify and estimate $αkk'$.

Another extension is to have the externality depend on a set of covariates. In this case, one would let $αkk'=β1'Xk+β2'Xk'$, where $Xk$ and $Xk'$ are vectors of variables measuring characteristics of the type $k$ and $k'$ firms, respectively. Then ${β1,β2}$ are the parameters to be estimated. This would allow one to measure the effect certain characteristics have on the size of the externality.

## Notes

1

The focus of this paper is on negative externalities ($αik,jk'≥0$). Positive externalities would imply a different set of issues, such as whether firms free-ride by not participating in the auction.

2

McAdams (2003) derives conditions for the existence of a monotone pure strategy equilibrium for multiunit auctions. That setup can be applied here to establish existence.

3

See Jehiel and Moldovanu (2006) for more on externalities and multiple equilibria.

4

Some asymmetry in the externality parameters must remain, as the externalities we are looking to identify in this paper are intended to be type dependent. A common externality effect on all bidders cannot be separately identified from an upward shift in bidder valuation distributions. The purpose of this paper is instead to uncover how asymmetries in downstream competition between bidders of different types affect bidding strategies.

5

Haile et al. (2003) propose some alternative approaches to dealing with endogenous entry caused by information acquisition costs, binding reserve prices, or entry costs.

6

In the timber application, this assumes that even bidders who are not participating in the auction are affected by how that auction outcome affects downstream competition.

7

Haile et al. (2003) show how this bidder participation equation results from a more general model where $nkt=κkxt,zt,ukt$, and $κk(·)$ is assumed to be strictly increasing in $ukt$. They show that strict monotonicity and the discrete nature of $nkt$, lead to the representation in equation (18).

8

Haile (2001) discusses why reserve prices in timber auctions are generally nonbinding.

9

Athey et al. (2011) compare the open and sealed bid formats for timber auctions.

10

Haile et al. (2003) found for scale sales that only one out of ten tests rejected the null hypothesis of private values at the 10% level. Prior work that assumed private values for timber auctions includes Baldwin et al. (1997), Cummins (1994), Haile (2001), Haile and Tamer (2003), and Lu and Perrigne (2008).

11

By “idiosyncratic portion of their valuation,” I mean the bidder's valuation after the observed and unobserved auction-specific heterogeneity has been removed: $ln(νit)$ in the breakdown of bidder valuations in equation (19).

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## Author notes

I am grateful to the editor, Shachar Kariv, and two anonymous referees for their valuable comments and suggestions. I thank Connan Snider, Hugo Hopenhayn, Rosa Matzkin, Masanori Tsuruoka, Robert Zeithammer, and seminar audiences at UC Santa Cruz and the 2016 IIOC, for useful comments and suggestions. I also thank Phil Haile for providing the data used in this paper on his website. All errors are my own.

A supplemental appendix is available online at http://www.mitpressjournals.org/doi/suppl/10.1162/rest_a_00810.