# «Abstract From bidding data, we estimate the underlying value distribution for Forest Service timber. We nd that bidder values decrease $2=mbf ...»

Mergers, Cartels, Set-Asides, and Bidding

Preferences in Asymmetric Oral Auctions

Lance Brannman

National University of Kyiv-Mohyla Academy

Kyiv, Ukraine

leb@ukma.kiev.ua

Luke M. Froeb

Owen Graduate School of Management

Vanderbilt University

Nashville, TN 37203

luke.m.froeb@vanderbilt.edu

March 22, 1999

Abstract

From bidding data, we estimate the underlying value distribution

for Forest Service timber. We nd that bidder values decrease $2=mbf

thousand board feet with each mile from the tract and that small rms less than 500 employees have values that are $72=mbf lower than large rms. The empirical value distribution is used to simulate various hypothetical scenarios designed to inform public policy. The most anticompetitive mergers raise price by less than three percent, and a four-percent decline in marginal costs through greater merger e ciencies is enough to o set a one-percent anticompetitive price increase. Eliminating the SBA Set-Aside program would raise timber revenues by 15 percent. A policy of granting bidding preferences to small and more distant bidders would raise revenue by about one-tenth of one percent.

Steven Tschantz and Philip Crooke co-developed the theory, wrote much of the Mathematica code used in this paper, and provided valuable advice. Two referees, Patrick Bajari, Bruce Cooil, Vivek Ghosal, and Katherine Wolfram also provided valuable comments. Support for this project was provided by the Dean's fund for faculty research at the Owen Graduate School of Management.

Keywords: auctions, mergers, cartels, Forest Service, SBA Set-Asides.

JEL Classi cation: D44 auctions, L41 horizontal anticompetitive practices.

1 Introduction Competition policy typically requires answers to questions like will this merger raise price?" or how much does a set-aside program cost?" These questions implicitly compare two states of the world, e:g:, pre and post merger, or with and without the set-aside program, but only one of these states is observed. In some cases, it is possible to draw inference about the unobserved state of the world through the use of natural experiments," like backcasting from a competitive regime to a collusive one to determine the e ects of a conspiracy Froeb, Koyak, and Werden 13 . However, data permitting such comparisons are rare. An alternate approach, and the one taken here, is to estimate a structural oligopoly model and then use the estimated model parameters to simulate the unobserved state of the world.

Through simulation, we are able to draw inference about the e ects of various policy interventions.

This paper studies the price e ects of mergers or bidding coalitions, and the e ects of giving some bidders competitive advantages, through either the Small Business Administration SBA Set-Aside program or a system of bidding preferences. To do this, we specify a Vickrey 30 or second-price model of bidding competition for oral auctions of Forest Service timber. We use a within-auction estimator, based on the di erence between losing bids, to recover the bidders' value distribution. The estimated value distribution implies that bidder values decrease $2=mbf thousand board feet with each mile from the tract; and that small rms less than 500 employees have values that are $72=mbf lower than large rms.

Using the empirical value distribution, we simulate the e ects of various policy interventions. We nd that the price e ects of mergers or bidding coalitions are small, and that a one-percent anticompetitive price increase due to a merger is o set using a consumer-welfare standard by a fourpercent decline in marginal costs, occurring from merger e ciencies. Eliminating the Small Business Set-Aside program, where auction participation is restricted to rms with less than 500 employees, would raise timber revenues by 15 percent. Finally, a policy of granting bidding preferences to small and more distant bidders would raise revenue by less than one-tenth of one percent.

2 An Asymmetric Vickrey Auction Model Timber value to each potential bidder is speci ed as the sum of two independent components: a bidder-speci c component, and a common and commonly-observed component. Formally, Vi = Y + Xi; Xi = i + Ui : 2.1 Vi is the value drawn by the i-th bidder for a tract; Y is a common shock to all bidders' values; and Xi is the bidder-speci c value component.

The bidder-speci c component has observed, i, and unobserved components, Ui. The unobserved component is assumed to follow an extremevalue distribution with location and spread parameters 0; , implying that the bidder-speci c value component of each bidder, Xi, has an extremevalue distribution with parameters i; . The mean and variance of this extreme-value component are 2 EXi = i + ; V arXi = 62 2.2 where is Euler's constant :57721.

This is essentially an independent private-values model with a common shock Y to account for large observed price variation across auctions. The common shock includes commonly-observed information about tracts, and about future lumber prices, driven largely by variation in housing starts.

A private-values model for the unobserved components, rather than a common-values speci cation, is chosen based on interviews with mill owners in the context of a timber-mill merger investigation. In addition to haul distance and SBA classi cation, the most-frequently mentioned determinant

**of value was the match" between the size and species mix of the forest e:g:,**

**large-diameter pine stems and the specialization" of the timber mill e:g:,**

small-diameter r stems. This factor is a private rather than a common value. Note however that the Forest Service has an o cial estimate or cruise" of the size and number of trees by species e:g:, r, pine and bidders can game" the auction by bidding more on the overestimated species, and less on the underestimated ones. While this does send a privately-observed signal about the common tract value to each bidder who conducts a cruise, not every bidder cruises the tract, implying that this signal is relatively weak.

The private-values speci cation follows a long literature of others who have used private-values to model similar procurement auctions e.g., Bajari 3 and Forest Service auctions in particular e.g., Hansen 17, Paarsch 24 , although this speci cation decision is driven partially by the di culty of modeling common-value equilibria Baldwin, Marshall, and Richard 4 . A formal test of the private versus common values framework comes from Paarsch 23 who rejects a private-values speci cation in favor of a commonvalue one, but he uses a symmetric model, a speci cation that is strongly rejected by our data.

The usefulness of the extreme-value distribution for modeling auctions is derived from its closure under the maximum function. If bidders are drawing from independent extreme-value distributions with the same variance, but di erent means, then the maximum of their values has an extreme-value distribution with the same variance, but a higher mean. The maximum function is used to compute the winning probabilities the probability that a bidder will have a value higher than the maximum of rivals' values and prices the maximum of rivals' values, and to compute the e ects of a merger the merged rm has a value equal to the maximum of coalition member values. The following proposition states a well-known property of independent extreme-value distributed variates see e.g. Ben-Akiva and Lerman 5 .

Proposition 1.

**If X1; X2; : : : ; Xn are distributed as independent extremevalue variates with parameters i,, then Xmax = maxfXi : 1 i ng is also distributed as an extreme-value variate with parameters max, where "n 1 log X exp :**

max = 2.3 k k=1 The next proposition is a standard result for models using the extreme-value distribution. It provides a formula for the probability that bidder i has the highest value for a tract and thus wins the auction.

Proposition 2. Let Xi = maxfXk : 1 k n; k 6= ig, and let pi be the winning probability of bidder i; i:e:, the probability that Xi Xi. Then

3 Estimation of Value Distribution In this section, we describe the estimator used to recover the value distribution; i:e:, the parameters fig and , from oral-auction data. The estimation requires data on bidder identities, bidder characteristics including losing bidder characteristics, and bids across a sample of auctions. We treat each auction as an independent event, but recognize that this assumption may not be appropriate in the presence of collusion, as in a bid-rotation scheme, or with bidder capacity constraints.

Auctions are typically used to sell or purchase unique items. This imparts a considerable degree of heterogeneity to auction data. We have modeled this hetereogeneity by adding a commonly-observed component, Y, to each bidder's value. This common component masks variation in winning bids due to the idiosyncratic component and implies that estimators based only on the winning bids will have a di cult time distinguishing within from between-auction variance. To solve this problem, Froeb, Tschantz and Crooke 16 suggest eliminating the common shock by di erencing the losing bids.

Proposition 4. Let Bi be the second-highest value of the Xs after Xi. Let Bi;j be the third-highest value of the Xs behind Xi and Xj, in that order.

Let i;j = Bi, Bi;j be the di erence between the second and third-highest values. Let Xij = maxfXm : 1 m n; m 6= i; jg and let pij = PrXij maxfXi ; Xj g. The cumulative-distribution function of i;j is p +p ij j Fi;j t = 1, p + p expt p + p + p expt 3.1 ij ij j i j Expressing the distribution in terms of the pi s instead of the i s suggests a two-step limited-information maximum-likelihood LIML estimator. First, by assuming that the highest-value bidder from the set of all potential bidders wins each auction, we can use rm-speci c characteristics to estimate the logit probabilities of winning e:g:, Train 28 . Second, use the tted probabilities to construct the likelihood for the di erence between the second and third-highest bids based on 3.1. The parameter can then be estimated by maximizing the constructed likelihood.

The way that the Forest Service conducts oral auctions allows recovery of losing oral bids. Bidders must qualify" by submitting a written bid at the reserve price or appraisal. Oral bidding proceeds with quali ed bidders taking turns. If a bidder fails to bid at his turn, that bidder drops out of the auction and the bidder's last bid is recorded. It is a dominant strategy for losing bidders to bid up to their values. Hence, if the second and thirdhighest bidders are present and bidding competitively, the di erence between the second and third-highest bids is equal to the di erence between the second and third-highest values. In contrast, the di erence between the twohighest bids carries no information about the value distribution because the winner tries only to outbid the second-highest-value bidder. In our data, the average di erence between the two-highest bids is less than one-percent of the reserve price while the mean di erence between the second and thirdhighest bids is about twenty percent of the reserve price.

To estimate the logit coe cients and the spread parameter, we use a full-information maximum-likelihood FIML estimator using starting values recovered from the two-step LIML estimator. To aid convergence, we simplify the analytic complexity of the logit likelihood by setting equal to one. We then rescale" the logit coe cients, mulitplying by the factor 1=. A secant optimization methodology is used instead of Newton's method Crooke, Froeb, and Tschantz 10 . The secant methodology has the advantage of not requiring computation of the analytic derivatives of the likelihood function.

The assumption of competitive behavior by bidders is maintained and is necessary to recover the value distribution from the bidding data. However, some authors e.g., Mead 22 suggest that the oral Forest Service auction environment facilitates collusion. To assess how a mistaken assumption of competition would a ect the estimates, note that an e cient cartel would allocate the timber to the highest-valued bidder. The logit estimates of the probabilities of winning are therefore robust with respect to e cient collusion. However, the second part of the likelihood assumes that the di erence between the second and third-highest observed bids is equal to the di erence between the second and third-highest values. In constructing the price likelihood, we exclude 17 of the 51 auctions where the second and third-highest bids are not at least ten percent higher than the reserve price. In these auctions, we suspect that the losing bidders are not bidding competitively.

Screening on low bids eliminates certain kinds of non-competitive behavior, but it does not address the informal quid-pro-quo bidding that would occur between bidders who refrain from bidding against one another. If this were the case, then the estimated variance, based on a the mistaken assumption that the second and third-highest bidders are present when it could really be the third and sixth-highest bidders, would be biased upward.

This would impart an upward bias to the simulated merger e ects, to the simulated costs of the SBA Set-Aside program, and to the simulated bene ts of bidding preferences.

4 Data and Results The joint value distribution is estimated from bidding data covering 51 oral auctions held in Oregon's Lane and Douglas counties during 1977. The data mainly come from the Forest Service 2400-17 Report of Timber Sale" forms.

Ten of the auctions were SBA Set-Aside sales. All of the auctions were on a at-rate" pricing basis, where the nominal winning bid remains constant and the winner makes payments to the government on an as-harvested basis.

Bids are measured in dollars per thousand board feet mbf of timber.

Any bidder who participated in any of the 51 auctions is considered a potential bidder in each auction, subject to the constraints of the SBA SetAside program. The use of an uncensored sample of all potential bidders eliminates selection bias associated with the auction participation decision.