Preferred Suppliers and Vertical Integration in Auction Markets Roberto Burguet Institute for Economic Analysis (CSIC) Campus UAB Bellaterra, Barcelona, Spain 08193

Martin K. Perry* Department of Economics Rutgers University New Brunswick, New Jersey, USA 08901 September 2005 (First Draft: September 2003)

Abstract We consider a model of preference in an asymmetric procurement auction with multiple suppliers. The buyer can award the contract to a preferred supplier at the lowest bid of the competing suppliers. As such, the preferred supplier has a right-of-first-refusal. The preferred supplier may have paid for the preference or may be a subsidiary of the buyer. We first examine the case in which the suppliers are symmetric with general cost distributions. If the buyer is not compensated for the preference, the expected price in the symmetric preference auction will be higher than in an efficient auction. However, if the buyer can sell preference in a pre-auction, there exist natural conditions on the distribution of costs which ensure that the net expected price paid by the buyer (net of the payment for preference) will always be lower than an efficient auction. We then examine the case in which there are two asymmetric suppliers with different cost distributions. If the stronger supplier with the more favorable cost distribution has a greater willingness to pay for preference in the pre-auction, the net expected price paid by the buyer is also lower than in an efficient auction. With the family of power distributions, we find that that the stronger supplier outbids the weaker supplier to acquire preference in the pre-auction. Numerical computations for this power distribution show that the net expected price after selling preference is also lower than in a standard first-price auction. We also study the effect of selling preference on the expected profits of the suppliers. We find that the stronger preferred supplier can actually earn higher expected profits after acquiring preference, but that the weaker competing supplier will always have lower expected profits. Finally, we discuss the acquisition of one supplier by the buyer as a preferred subsidiary. If the buyer has some bargaining power relative to the suppliers, he will have a stronger incentive to acquire the stronger supplier than to auction preference. *

Martin would like to acknowledge the financial support of the Instituto de Analisis Economico (IAE, Institute for Economic Analysis) and the Institucio Catalana de Recerca i Estudis Avancats (ICREA, Catalan Institute for Research and Advanced Studies) in Barcelona Spain. Roberto would like to acknowledge financial support of FEDER and the Spanish Ministry of Science and Technology, project SEC 2002-02506 and CREA. In addition, we would like to acknowledge Maria Lauve, Martin’s research assistant at Rutgers University, for her help with various computations, tables, and figures.

2 1.

Introduction In this paper, we examine preference in a procurement auction. Governments and

corporations frequently have a preference for particular suppliers of various goods. Preference may arise in a variety of settings. First, preference for one supplier may arise from successful contractual relationships in the past between the buyer and this supplier. Second, preference for one supplier may also arise from the special capabilities possessed only by this supplier. Third, preference may arise from bribery of a buyer’s procurement officer by the supplier. In contrast to these explanations, the model of this paper will assume that preference is explicitly sold by the buyer to one of the suppliers. In particular, a buyer may auction preference to one supplier or may acquire one of the suppliers as a preferred internal subsidiary. Preference occurs when the buyer creates a right-of-first-refusal for the preferred supplier, allowing him to accept the contract at a price equal to the lowest bid by the competing suppliers. The preferred supplier will clearly accept the contract whenever his cost is below the lowest bid of the competing suppliers. This affects the bidding behavior of the competing suppliers, the allocation of the contract, and the expected price paid by the buyer. Bikhchandani, Lippman, and Ryan (2005) have recently examined the right-of-firstrefusal in a symmetric sealed-bid second-price auction with one seller and many buyers. For private-value, common-value, and affiliated-value settings, they focus on the gains and losses to a seller, the buyer with the right-of-first-refusal, and the other buyers. In the private-value setting, they find that the buyer with the right-of-first-refusal gains while the other buyers and the seller incur losses. With the second-price auction, the gain of the buyer with the right-of-first-refusal is equivalent to the loss of the seller. Thus, they conclude that there is no incentive for the seller to award a right-of-first-refusal to one of the buyers. In contrast, we investigate whether a buyer in a procurement auction with many suppliers could gain from selling the right-of-first-refusal to one of the suppliers. One main finding is that the buyer always benefits from selling a right-of-firstrefusal if the suppliers are symmetric with the same cost distribution. Thus, the inefficiency created by preference is more than offset by the reduction in expected profits of the competing suppliers caused by the purchase of the right-of-first-refusal by one supplier. When the suppliers are asymmetric, the comparisons are more difficult and complex. Thus, we focus on the case of two suppliers. We can show that if the stronger supplier wins

3 preference by outbidding the weaker supplier in a pre-auction, then the net expected price paid by the buyer is again lower when preference is auctioned. However, we cannot prove that the stronger supplier would always outbid the weaker supplier in the pre-auction for preference for arbitrary cost distributions. As a result, we examine the power family of cost distributions, and find that the stronger supplier does indeed outbid the weaker supplier for any specification of the asymmetry within this family. We can then show that the net expected price paid by the buyer is lower after selling preference than it would be with an efficient auction or a first-price auction without preference.1 In Section 2, we construct a procurement auction with preference for one of many potential suppliers. The competing suppliers bid for the contract, while the preferred supplier has a right-of-first-refusal to accept or reject the contract at the lowest bid of the competing suppliers. We first characterize the equilibrium bidding function for the competing suppliers. In Section 3, we analyze the symmetric case where all suppliers obtain their cost realizations from the same cost distribution. For this case, we show that the buyer can always benefit from selling preference. In Section 4, we consider the asymmetric case with two suppliers. For general cost distributions, we show that if the stronger supplier wins the pre-auction for preference, then the net expected price paid by the buyer is again lower when he sells preference rather than holding an efficient auction between the two suppliers. Using the power family of cost distributions, we show that the stronger supplier does indeed have a higher willingness to pay for preference. For this family of cost distributions, we also show that the net expected price for the buyer is also lower than the expected price in a first-price auction. In Section 5, we consider the effects of selling preference on the net expected profits of the suppliers. The supplier without preference always has lower expected profits. In the symmetric case, the preferred supplier also has lower expected profits. However, this need not be the case when the suppliers are sufficiently asymmetric. For the power family of cost 1

Lee (2004) examines the right-of-first-refusal with alternative family of asymmetric cost distributions and finds that the buyer can benefit by awarding preference to one of the suppliers at no charge. In particular, if one supplier is sufficiently weaker than the other supplier, the buyer can reduce the expected price by awarding preference at no charge to this weaker supplier. This is consistent with results in the auction and screening literature. Under the assumption that the suppliers know their costs before the auction, it is well known (Myerson (1981)) that the buyer can benefit by discriminating against the stronger supplier. However, if the buyer can contract with the suppliers prior to the time at which they realize their costs, the optimal screening mechanism would favor the stronger supplier in the ex post contracting in exchange for a higher up-front payment (Courty and Li (2000)).

4 distributions, we provide examples in which the expected profits of the preferred supplier, net of the payment for preference, are higher than what his expected profits would have been without preference. In Section 6, we examine vertical integration in which the buyer acquires one of the suppliers in a vertical merger. We construct a bargaining model that determines which supplier is acquired by the buyer at what acquisition price. With the power family of cost distributions, the buyer may still acquire the stronger supplier, but the acquisition price would depend on the bargaining power of the buyer. This solution illustrates that vertical integration is equivalent to selling preference in a pre-auction when the buyer has no bargaining power relative to the suppliers. Finally, a discussion of related literature in an appendix.

2.

The Model of a Preference Auction with Symmetric Suppliers

The buyer has a value v for a good with a fixed quantity and quality. There are (n+1) suppliers with independent and identical cost distributions for producing the good. The buyer could employ an efficient auction (EA), such as a second-price auction (SPA) or an open auction. If so, the contract would be awarded to the supplier with the lowest cost at a price equal to the second lowest cost. The buyer could also employ a sealed-bid first-price auction (FPA). If so, the contract would be awarded to the supplier with the lowest bid at a price equal to that bid. The winning supplier need not have the lowest cost, so this FPA need not be efficient. As an alternative to either an efficient auction or a first-price auction, we allow the buyer to employ a preference auction (PA) in which one supplier is the preferred supplier (PS) and the other n suppliers are the competing suppliers (CS). In the preference auction, the CS will bid for the contract, but the PS will then be offered the contract at a price equal to the lowest bid of the CS. The contract will be accepted by the PS if his cost is below this lowest bid of the CS, and rejected otherwise. Thus, preference means that the PS has a right-of-first-refusal at the lowest bid of the CS. The PS would not bid against the CS because under-bidding all CS would only lower the expected price he would receive.2 2

If the PS won, his bid must have been below the lowest bid of the CS. However, the lowest bid of the CS is the price at which the PS would be offered the right-of-first-refusal. Of course, this assumes that any payment for the right-of-first-refusal is independent of any bid that the PS might make. If so, the optimal bid of the PS is unity, the

5 We assume that each supplier draws his cost of production ci, where i = 1,2,…,(n+1) as independent realizations of the same distribution G(c) with support [0,1], and a positive, continuous density g(c) over this support. The cost ci is private information for each supplier. For simplicity, we also assume that the value of the buyer exceeds the highest possible cost realization (v > 1). Thus, we will examine preference in a symmetric independent private value auction.3 If preference is awarded, then we will denote variables related to the preferred supplier (PS) by p and those related to the competing suppliers (CS) by k. We now characterize a symmetric, monotone equilibrium bidding function for the CS, b(c). Assuming that the PS will not reject contracts at a price above his cost, the equilibrium bidding function has to satisfy: (1)

c = arg max Π k [b( z ); c ] = (b( z ) − c) ⋅ [1 − G (b( z ))][1 − G ( z )]n - 1 . z

The first-order condition for this problem can be written as (2)

d{b(c) ⋅ [ 1 − G (b(c))] ⋅ [ 1 − G (c) ]n − 1} d{[ 1 − G (b(c))] ⋅ [ 1 − G (c) ]n − 1} = c⋅ , dc dc

By integrating (2), we have that a solution to (1) defined by the following condition b (c ) =

(3a)

n − 1} 1 d{[ 1 − G (b( z ))] ⋅ [ 1 − G ( z ) ] dz ∫c z n −1 dz

1

[ 1 − G (b(c))] ⋅ [ 1 − G (c)]

= c +

.

n − 1 dz 1 ∫c [ 1 − G (b( z ))] ⋅ [ 1 − G ( z )] n − 1 [ 1 − G (b(c))] ⋅ [ 1 − G (c)] 1

This condition can be rewritten as the following differential equation (3b)

b − c=

[1 − G (b)] (n − 1) ⋅ (b − c) [1 − G (b)] g (c ) − ⋅ ⋅ g (b) b′ g (b) [1 − G (c)]

The optimal bidding function for the CS, b(c) , is implicitly defined by (3b). Note that b(1) = 1 , and b(0) > 0 . The problem (1) is dominance solvable when there are only two suppliers (n = 1).

highest possible cost realization. In effect, the PS submits the maximum possible bid so that he would never win the contract outright. See Burguet and Perry (2000). 3 We assume that the bid of the CS is verifiable in that the buyer can show the PS a signed document with the lowest bid of the CS.

6 In this case, the bidding function of the only CS is the best take-it-or-leave-it offer to a buyer who has the option of buying at the cost of the PS. We can now compute the expected profits of both CS and PS when they compete for the contract. We use an indirect approach. In a differentiable monotone equilibrium, incentive compatibility requires that the slope of the expected profit function Πk (c ) of a CS with cost c satisfies dΠ (c) k dc

(1 − G(c))`n − 1 (1 − G(b(c)))

=

.

By integrating this expression, we obtain (4)

Πk (c) = ∫c1 (1 − G ( z ) )`n−1 (1 − G (b( z )) ) dz .

Similarly, we obtain the following expression for the expected profits of the PS with cost c

(

) (1 − G(b n

(5a)

Π p (c) = ∫c1 1 − G (b −1 ( z )) dz , if c > b(0) , and

(5b)

Π p (c) = b(0) − c + ∫b1( 0)

−1

( z ))

)

n

dz , otherwise.

With these expressions for the expected profits, we can now analyze the allocation of preference.

3.

The Net Expected Price from Selling Preference with Multiple Symmetric Suppliers

After preference is assigned to one of the suppliers, and each supplier learns its cost, suppliers compete for the contract in the procurement auction. Independent on who wins the contract, the expected price paid by the buyer is equal to the lowest bid of the CS. Thus, the expected price from the preference auction is (6)

EPPA = ∫01 b(c)ng (c)(1 − G (c) )n −1 dc .

If no preference is assigned, all suppliers compete for the contract in some symmetric auction. Absent the possibility of not assigning the contract, the best auction for this symmetric setting is an efficient auction. In an efficient auction, the expected price paid by the buyer is (7)

EPEA = ∫01 c(n + 1)ng (c)G (c)(1-G (c) )n −1 dc .

7 As found by Bikhchandani, Lippmann, and Ryan, (2005), EPEA is lower than EPPA .4 That is, granting preference without a payment is not in the buyer’s interest. However, the buyer may gain if he can sell preference to one of the suppliers before they learn their costs. Preference, or the right-of-first-refusal, provides one of the suppliers with an advantage in the procurement auction and the other supplier with a corresponding disadvantage. As a result, each supplier has a willingness to pay for preference. Thus, the buyer can sell preference in a preauction to the supplier making the highest bid or negotiate with one of the suppliers over a price for preference. Since suppliers are symmetrically informed at this stage, they have the same willingness to pay for preference. Thus, as long as more than one supplier is present, the buyer will be able to extract this value from one of the suppliers irrespective of the format of the preference auction or negotiation. From the viewpoint of the suppliers, the willingness to pay for preference is the difference between the expected profits as the PS and expected profits as one of the CS, Π p − Π k . Using (4) and (5), and changing the order of integration, we obtain

(8a)

Πk = ∫01 ∫c1 (1 − G ( z ) )`n−1 (1 − G (b( z )) ) dz ⋅g (c)dc = ∫01 G (c)(1 − G (c) )`n−1 (1 − G (b( z )) ) dz ,

(8b)

1 b( 0 ) −1 Π p = ∫b( G (c)dc . 0 ) G (c) 1 − G (b (c )) dc + ∫0

(

)

n

Thus, the difference between these two expressions is the amount that any supplier is willing to pay for preference. Thus, when the buyer sells preference, the net expected price paid by the buyer is the difference between the expected price in the preference auction minus the payment for preference. In this symmetric setting, the revenue from selling preference is the willingness to pay for preference by any supplier, so that (9)

NEPPA = EPPA − [Π p − Π k ] .

Does the preference payment compensate for the higher expected price in the procurement auction? The answer is affirmative if the CS bid more aggressively after preference is assigned to the PS. More aggressive bidding may seem natural, but Arozamena and Weinschelbaum (2004) and Porter and Shoham (2005) have shown that this occurs only if the inverse hazard rate is decreasing and convex. Thus, we can state the following proposition. 4

This is a direct corollary of the optimal auction analysis of Myerson (1981). Indeed, when a sale takes place with probability one, any efficient procedure is an optimal selling or buying procedure.

8

If the inverse hazard rate, [1 − G (c)] / g (c) , is decreasing and convex, then the

Proposition 1:

net expected price when the buyer sells preference is lower than the expected price in a standard, efficient auction.

Proof of Proposition 1: The net expected price when selling preference is NEPPA = EPPA − [Π p − Π k ].

Note that

Πk = ∫01 G (c)(1 − G (c) )`n −1 (1 − G (b(c)) )dc = ∫01 G (c)(1 − G (c) )`n dc − ∫01 G (c)(1 − G (c) )`n −1 [(1 − G (c) ) − (1 − G (b(c)) )] dc. Integrating the first term in the last line by parts [with v(c)=c so that v’(c)=dc, and u(c)= G (c)(1-G (c)) n ], we obtain

Πk = ∫01 (n + 1)cg (c)G (c)(1 − G (c) )`n−1 dc − ∫01 cg (c)(1 − G (c) )`n−1 dc − A , where A = ∫01 G (c)(1 − G (c) )`n −1 (G (b(c)) − G (c) )dc > 0. The first term in Π k equals

EPEA . Thus, taking into account the value of NEPPA , n `n −1

NEPPA − EPEA = ∫01 nb(c) g (c)(1 − G (c) )

dc − Π p −

[

n −1 n

`n −1

EPEA + 1n ∫01 ncg (c)(1 − G (c) )

]

dc − A.

Let Y(i; m) denote the ith order statistic (from low to high) of m realizations of costs. Note that EPEA equals the expected value of Y( 2; n +1) , whereas

[

]

`n −1

E Y(1;n ) = ∫01 ncg (c)(1 − G (c) )

dc .

Thus, `n −1

NEPPA − EPEA = ∫01 n(b(c) − c) g (c)(1 − G (c) )

where B =

n −1 n

dc − Π p − A − B ,

(E[Y(2;n+1) ] − E[Y(1;n) ] )

Note that B > 0 for n > 1 and that B = 0 for n = 1. Now, we can write n −1 1 Π p = ∫01 ∫max {b −1(c),0}(b( x) − c )ng ( x)(1 − G ( x) ) dx⋅ g (c)dc

9 where we use max{b − 1 (c) ,0} to indicate that the lower limit is 0 whenever b −1(c) does not exist, that is, whenever c < b(0). Also, n-1 n-1 1 1 1 ∫0 (b(c)-c )ng (c)(1-G (c) ) dc = ∫0 ∫0 (b(c) − c )ng (c)(1-G (c) ) dc ⋅ g ( x)dx

Thus, changing the name of the variables of integration, `n −1

∫0 (b(c) − c)ng (c)(1 − G (c) ) 1

dc − Π p

1 = ∫01 ∫01 (b( x) − x )ng ( x)(1-G ( x) ) dx ⋅ g (c)dc − ∫01 ∫max {b −1 ( c ), 0}(b( x) − c )ng ( x)(1-G ( x) ) dx ⋅g (c)dc n-1

n-1

[

]

= ∫01 ∫01 (c − x )ng ( x)(1-G ( x) ) dx ⋅ g (c)dc − C = E [c ] − E Y(1;n ) − C , n-1

where C = − ∫b1(0) ∫0b

−1

(c)

(b( x) − c )ng ( x)(1-G( x))n-1dx ⋅ g (c)dc > 0.

Thus, (10)

[

]

NEPPA − EPEA = E [c ] − E Y(1;n ) − A − B − C.

Note that for n = 1, NEPPA − EPEA = − A − B − C < 0 . With a change of variable, z = b −1 (c),

and then a change in the order of integration, note that −1

(b( x) − c )ng ( x)(1-G( x))n-1dx ⋅g (c)dc > 0 n-1 = − ∫01 ∫0z (b( x) − b( z ) )ng ( x)(1-G ( x) ) g (b( z ))b ' ( z )dxdz n-1 = − ∫01 (∫x1 (b( x) − b( z ) )g (b( z ))b ' ( z )dz ) ng ( x)(1-G ( x) ) dx.

C = − ∫b1(0) ∫0b

(c)

Now, 1 1 ' ∫x (b( x) − b( z ) )g (b( z ))b ( z )dz = ∫x (b( x) − b( z ) )dG (b( z ))

= b( x) − 1 + ∫1x G (b( z ))b ' ( z )dz = b( x) − 1 + ∫b1( x ) G (c)dc < b( x) − 1 + ∫1x G (c)dc , where the second equality is obtained from integrating by parts and the third equality is obtained by a change of variable, c = b(z). Thus,

(

)

C > − ∫01 b( x) − 1 + ∫1x G (c)dc ng ( x)(1-G ( x) )n-1 dx

[ = − E [b(Y = − E [b(Y

] )] + 1 − ∫ ∫ ng ( x)(1-G ( x) ) dxG (c)dc )] + 1 − ∫ G (c)dc + ∫ G (c)(1-G (c) ) dc.

= − E b(Y(1;n ) ) + 1 − ∫01 ∫1x G (c)dc ng ( x)(1-G ( x) )n-1 dx (1;n ) (1;n )

n-1

1 c 0 0 1 0

1 0

n

The second equality is obtained by changing the order of integration. Integrating by parts, we see that ∫01 G(c)dc = 1 − E [c ] . Thus, we find that

10

[

] [

]

NEPPA − EPEA < E b(Y(1;n ) ) − E Y(1;n ) − A − B − ∫01 G (c)(1-G (c) ) dc. n

Note that the last term equals the expected profits of a supplier competing in a standard auction with n other suppliers. Thus, letting bEA ( x) represent the equilibrium bidding function in a symmetric FPA or SPA with n+1 suppliers, we have NEPPA − EPEA < ∫01 (b(c) − bEA (c) )d (1-G (c) )n .

Thus, the result follows from Proposition 1 in Arozamena and Weinschelbaum (2004). QED.

Notice that the sufficient condition of Proposition 1 is far from being necessary. In fact, the following corollary follows immediately from equation (10) in the proof of Proposition 1:

Corollary: For n small enough, the net expected price when the buyer sells preference is lower

than the expected price in a standard efficient auction.

For larger n, we have not been able to eliminate the possibility that selling preference would induce bidding by the CS which is sufficiently less aggressive that the buyer is better off using an efficient auction.

4.

The Net Expected Price from Selling Preference with Two Asymmetric Suppliers

When suppliers have different distributions for their costs, they will also have different willingnesses to pay for preference. When suppliers are asymmetric, there is a new question about what procedure the buyer should use to sell preference. If the buyer has bargaining power with respect to the suppliers, he could negotiate with either supplier for the price of preference. Alternatively, the buyer could use a pre-auction for preference. Moreover, when suppliers are asymmetric, the standard auction and bargaining models generate different benchmarks for comparing any procedure for selling preference. For example, the first-price auction (FPA) and the second-price auction (SPA) result in different revenues for the buyer. In addition, neither the FPA nor the SPA are optimal selling procedures, despite the fact that the SPA (or oral ascending auction) is efficient. Thus, asymmetries open several new questions. In this section, we first examine whether the stronger supplier or the weaker supplier would win the pre-auction for

11 preference. Second, we can identify whether asymmetries provide additional reasons for the buyer to sell preference rather than use either a FPA or SPA. In order to examine these questions, we examine the case of two suppliers (n = 1). Assume that each of the two suppliers draws his cost of production ci, where i = 1,2 from a distribution Gi(c) with a common support [0,1], and a positive, continuous density gi(c) over this support. We assume that these distributions can be (weakly) ordered in terms of stochastic dominance. Thus, we assume that G1(c) ≥ G2(c), so that supplier 1 has a (weakly) more favorable cost distribution than supplier 2. Either of the two suppliers may be the CS or the PS, so we will use subscripts to indicate which supplier is the PS and which is the CS. With two suppliers, the bidding function of the CS from (3b) reduces to

b − c= i

(11)

  



1 − G (b) j  , g (b) j

where i,j = {1,2}. We begin by considering the case in which the buyer has limited bargaining power relative to the suppliers. In particular, he cannot negotiate with the suppliers, but simply holds a pre-auction to sell preference. The stronger supplier 1 would have a higher willingness to pay for preference, and then win the pre-auction for preference, if [Π p =1 − Π k =1 ] − [Π p =2 − Π k =2 ] > 0 . Rearranging this condition, we see that supplier 1 will win the pre-auction for preference if the combined profits of both suppliers is larger when supplier 1 has preference rather than supplier 2, that is, if Π p =1 + Π k = 2 is larger than Π p = 2 + Π k =1 . Using (4) and (5), we can write Π p =1 + Π k =2 = ∫01 [G2 (c) + G1 (c)] dc − ∫01 G2 (c)G1 (b2 (c))dc − ∫b12 ( 0) G1 (c)G2 (b2−1 (c))dc ,

where bi (c) is the bidding function for the case of i=k. After changing the variable in the third integral above, the combined profits with the stronger supplier 1 as the preferred supplier can be expressed as Π p =1 + Π k =2 = ∫01 [G2 (c) + G1 (c)] dc − ∫01 [1 + b' 2 (c)] G2 (c)G1 (b2 (c))dc.

Since the first term is the same irrespective of which supplier is preferred, the combined expected profits are greater when supplier 1 is preferred if and only if ∫0 [1 + b'1 (c)] G1 (c)G2 (b1 (c))dc − ∫0 [1 + b' 2 (c)] G2 (c)G1 (b2 (c))dc > 0 . 1

1

12 Thus, this condition determines whether supplier 1 is willing to pay more for preference than supplier 2. It is unclear whether this condition is satisfied for general distributions. The mild condition that

G1 (c) is decreasing implies that G1 (c)G2 (b1 (c)) > G2 (c)G1 (b2 (c)) . However, G 2 (c )

this condition also implies that b'1 (c) < b' 2 (c) . Thus, there is a potential ambiguity, and we have found no simpler general condition on the cost distributions which would determine who wins preference in the pre-auction. Despite this fact, we can prove, under a mild strengthening of the dominance conditions, that if the stronger supplier 1 wins the pre-auction for preference, the buyer will pay a lower net expected price than if he had conducted an efficient auction without preference. We can prove this result with the following assumptions: A1.

1 − G ( c ) 1 − G (c ) 1 2 for all c in [0,1]. ≤ g (c ) g (c) 1 2

A2. g1 (c) − g 2 (c) is decreasing in c.

Proposition 2: Assuming A1 and A2, if the stronger supplier 1 wins the pre-auction for

preference, the net expected price when the buyer sells preference to supplier 1 is lower than the expected price in a efficient standard auction.

Proof of Proposition 2: Let EPPA(p=1) be the expected price in the preference auction when supplier 1 is the preferred supplier. The net expected price when selling preference is

[

]

NEPPA = EPPA ( p = 1) − Π p =2 − Πk = 2 = ∫01 b2 (c) g 2 (c) dc - Π p =2 + Πk =2 .

The term in brackets is the winning bid of the pre-auction and is equal to the willingness to pay for preference by supplier 2. As in Proposition 1, note that Πk =2 = ∫01 [1 − G1 (b2 (c))] G2 (c) dc < ∫01 [1 − G1 (c)] G2 (c) dc

= ∫01 c ⋅ [g 2 (c)G1 (c) + G2 (c) g1 (c)] dc − ∫01 c ⋅ g 2 (c) dc . The first term on the right-hand side of the second expression is simply EPEA . Thus, NEPPA − EPEA < ∫01 (b2 (c)-c )g 2 (c) dc - Π p =2 .

Now, we can write Π p =2 = ∫0b1( 0 ) ∫01 (b1 ( x) − c )g1 ( x)dx⋅ g 2 (c) dc + ∫b11( 0 ) ∫b11−1(c) (b1 ( x) − c )g1 ( x)dx ⋅ g 2 (c) dc,

13 and ∫0 (b2 (c)-c )g 2 (c) dc = ∫0 ∫0 (b2 (c) − c )g1 ( x)dx⋅ g 2 (c) dc 1

1 1

so that NEPPA − EPEA < ∫01 ∫01 [b2 (c)-b1 ( x)]g1 ( x)dx ⋅ g 2 (c)dc + ∫b11( 0 ) ∫0b1

−1

(c)

= ∫01 b2 (c) g 2 (c)dc − ∫01 b1 (c) g1 (c)dc + ∫b11( 0 ) ∫0b1

(b1 ( x) − c )g1 ( x)dx g 2 (c)dc −1

(c)

(b1 ( x) − c )g1 ( x)dx g 2 (c)dc

.

Using the expression for the willingness to pay for preference of each supplier, we see that NEPPA − EPEA
0 is a capacity parameter which can differ for the two 5

This has been used since Marshall et al. (1994), and is one of the two families for which the program BidComp obtains computations.

14 suppliers.6 The corresponding density function is g(c;t) = t ⋅ [1 − c] t −1 . With this family of distributions, assumptions A1, A2, are satisfied if t1 ≥ t2. The equilibrium bidding function of the CS from (3b) has the following linear form: bk (c) =

(12)

tp 1 + ⋅ c, 1+ t p 1+ t p

where t p is the parameter that defines the distribution of the PS. The expressions for the expected profits from (4) and (5) can be written as (13)

  tp tk  1 Π p t p , tk = ⋅ + ⋅ 1 + t p 1 + t k 1 + t p ⋅ 1 + t p + t k  1 + t p 

(14)

 tp tk Π k (t p , t k ) = ⋅ (1 + t p ) ⋅ (1 + t p + t k )  1 + t p

(

)

tp

(

)(

)

   

   

tp 

  ,  

tp

.

We can now state the following proposition about the willingness to pay for preference.

Proposition 3:

For the family of cost distributions G(c;t) = 1 − [1 − c] t , the stronger supplier

has a higher willingness to pay for preference than the weaker supplier.

The proof of Proposition 3 is contained in Appendix 2. For this family of cost distributions, an immediate corollary of Propositions 2 and 3 is that the buyer benefits from holding a pre-auction to sell preference to the stronger supplier.

Corollary: For the family of cost distributions G(c;t) = 1 − [1 − c] t , the buyer pays a lower net

expected price after selling preference to the stronger supplier than he would pay using an efficient auction such as a second-price auction or oral ascending auction.

6

Waehrer and Perry (2003) show that this distribution function is not as restrictive as it might seem. Distributions of this form follow directly from natural properties, particularly a property corresponding to constant returns to scale. Note that power distributions of the form cs are not members of this family, except for s=1. Instead, these distributions would be members of a larger family of power distributions: G(c;s,t) = 1 – [1-cs ]t .

15 In order to compare preference with the FPA, numerical calculations of the expected price in the FPA are required. Although the SPA is efficient, the expected price using a SPA is often higher than using a FPA. As we will see, this clearly occurs for some combinations of the capacities (t1,t2) in the power family of distributions. Thus, we examine whether the net expected price paid by the buyer after selling preference is also lower than the expected price using a FPA. Using Bidcomp2 to compute the expected price for the FPA, we can state the following finding.

Finding: For the family of cost distributions G(c;t) = 1 − [1 − c] t , and for a fine grid of supplier

capacity parameters (t1,t2) where t1 > t2, the net expected price paid by the buyer when the stronger supplier acquires preference in a pre-auction is lower than the expected price with a first-price auction.

Table 1 provides examples of expected prices for these three auctions: preference auction, NEPPA(t1,t2), first-price auction, EPFPA(t1,t2), and efficient auction, EPEA(t1,t2). These expected

prices are calculated for different industry capacities T = t1 + t2, and different degrees of asymmetry, t1 = λ⋅T where λ ≥ 0.5. The expected prices NEP(t1,t2) and EPEA(t1,t2) have closedform solutions, but EPFPA(t1,t2) is computed using Bidcomp2.7 In Table 1, we see that the net expected price from the preference auction is substantially less than the expected price using a FPA, which in turn is less than the expected price using an EA. The reductions in the net expected price can be significant. For example, when T = 4, the price reduction relative to the FPA is 24.4% when λ = 0.5, 21.5% when λ = 0.6, and 17.5% when λ = 0.7.

4.

The Effect of Preference on the Expected Profits of the Suppliers

Another question that we briefly explore is the effect of preference on the expected profits of suppliers. From an ex ante viewpoint (before the suppliers learn their costs), the answer is trivial in the symmetric case. In particular, the expected profits of each supplier are lower when preference is sold. First, total expected surplus is lower because preference introduces an

16 inefficiency in the allocation of the contracts that is not present in either a symmetric FPA or SPA. Second, since the net expected price paid by the buyer is lower with preference, the buyer is extracting more rents from the suppliers. Finally, in the symmetric case, the suppliers are indifferent between winning preference and not, irrespective of which supplier obtains preference. When the suppliers are asymmetric, it is also clear that the expected profits of the CS are lower. The optimal bidding function of the CS is bk (c) when the other supplier has acquired preference from the buyer. If the CS used the same bidding function in a FPA without preference for the other supplier, the expected profits would be the same in each case that he wins the procurement auction. Moreover, he would win the auction more often because the other supplier without preference would be bidding above his cost bk (c) , rather than being offered the contract at his cost. As a result, by bidding optimally when no preference is awarded to the other supplier, the CS can do no worse than bidding bk (c) in a FPA without preference. Thus, the optimal bidding function of either supplier in the FPA without preference for the other supplier must generate at least as much expected profit as bk (c) in the preference auction. The expected profits of the CS are also lower than what he would earn in an efficient auction such as a SPA without preference. In an efficient auction the expected profits of supplier i with cost c are (15)

[

]

Π iEA [c ] = ∫c1 1 − G j ( z ) dz .

Comparing this expression with (4) for n = 1, the fact that bk ( z ) > z for z < 1 implies that the expected profits of the CS are lower than with an efficient auction for any cost c. The behavior of expected profits for the PS is less clear. The PS benefits from preference in the preference auction, but the PS must pay the price for preference to the buyer. The expected profits of the PS, Π p = i [c] , are clearly higher than the expected profits from an efficient auction Π iEA [c] . This follows from the fact that b p−1 ( z ) < z for z < 1. However, the PS would have to pay to the buyer the willingness to pay for preference of the CS, Π p = j - Π k = j . For symmetric suppliers, we already know that the net expected profits of the PS are lower than his expected profits in an efficient auction. By continuity, this would also be true if the PS was somewhat 7

The Bidcomp2 program to calculate the expected price in a FPA only allows for powers greater than unity. We

17 stronger than the weaker CS. However, using the power family of cost distributions, we can find cases when the opposite is true. For instance, when t2 = 1, the stronger PS who wins preference would earn higher expected profits when t1 ≥ 2.8 In these cases, the gains from having preference in the preference auction exceed the price of purchasing preference. In sum, when the stronger supplier is sufficiently stronger than the weaker supplier, both the buyer and the stronger supplier may mutually benefit from the auction of preference.

5.

Vertical Merger and Bargaining

Our analysis of the preference auction also applies when the preferred supplier is interpreted as an internal subsidiary of the buyer. With this interpretation, the buyer would optimally choose to award its subsidiary a right-of-first-refusal to supply the good. In particular, the cost-minimizing strategy of the buyer is to purchase from an external supplier only if the price offered is lower than the cost of the internal subsidiary. The external supplier would bid against the internal subsidiary just as a CS in the preference auction bids against the PS. As such, the procurement auction would remain unchanged. The difference arises in how the preference is created. With a PS, preference is awarded to the highest bidder in a pre-auction, but with an internal subsidiary, we need to examine the incentives of the buyer to acquire one of the suppliers. In particular, we characterize a bargaining model that determines which supplier is acquired by the buyer and at what price. Assume that the buyer announces his intention to acquire one of the two suppliers. If this announcement is credible, each supplier can only guarantee itself the expected profits of a CS in the preference auction after the other supplier is acquired. In any bargaining between one supplier and the buyer, this expected profit is the supplier’s threat point. The threat point for the buyer is more complex because the buyer can negotiate to acquire either supplier. Thus, there are two possible agreements, a vertical merger with supplier 1 or with supplier 2. Bargaining with

know of no reason why the comparisons in Table 1 might differ for powers less than unity. 8 The FPA typically generates a lower expected price for the buyer by distorting the allocation in favor of the weaker supplier, thereby reducing the informational rents of the stronger supplier. For this reason, the stronger supplier earns a lower expected profit in a FPA than he would in an EA. Thus, the preference auction would also generate higher expected profits for the stronger PS relative to the expected profits in an FPA when t1 is sufficiently larger than t2.

18 multiple parties has been analyzed in Burguet, Caminal, and Matutes (2002). The solution proposed in that paper considers both potential agreements simultaneously and endogenously identifies the threat point for the buyer in the bargaining with one supplier as the payoff that the buyer obtains in an agreement with the other supplier. Denote the endogenous threat points of the buyer relative to each supplier as r1 and r2 , and denote the exogenous bargaining power of the buyer relative to each supplier as δ1 and δ 2 . The surplus in the negotiation with supplier i is v − EPPA ( p = i ) + Π p =i . Assuming that the buyer’s threat points with respect to each supplier are both above the payoff for the buyer from running an efficient auction without preference, the threat points can be defined as

[ ] [ ]} { r2 = min{(1 − δ 1 )r1 + δ 1 [v − EPPA ( p = 1) + Π p =1 − Π k =1 ] , [v − EPPA ( p = 1) + Π p =1 − Π k =1 ]}. r1 = min (1 − δ 2 )r2 + δ 2 v − EPPA ( p = 2) + Π p = 2 − Π k =2 , v − EPPA ( p = 2) + Π p = 2 − Π k = 2

The buyer’s threat point with respect to one supplier is the bargaining solution with the other supplier. Thus, the threat points are endogenous. If the buyer has no bargaining power relative to supplier 2 (δ2 = 0), then his threat point with respect to supplier 1 is equivalent to running the pre-auction for preference, r1 = v – NEPPA . If the buyer also has no bargaining power relative to supplier 1 (δ1 = 0), then the buyer will receive exactly r1, the expected profits from running the pre-auction for preference. Thus, if the buyer has any bargaining power with respect to either supplier, he can negotiate a merger with one of the suppliers and receive more than he would from running the pre-auction for preference. We first apply this bargaining solution to the symmetric case in which the two suppliers have the same cost distribution. If suppliers are also symmetric with respect to their bargaining power with the buyer (δ = δ1 = δ2), the common solution for the threat point is (16)

r = v − EPPA + Π p − Π k = v − NEPPA .

For this fully symmetric case, the threat points of the buyer and each supplier are located on the Pareto frontier for each agreement. As such, the threat points are the bargaining solutions for the vertical mergers. Moreover, the net expected price paid by the buyer is the same as running a pre-

19 auction for preference. In other words, with symmetric suppliers, the pre-auction for preference allows the buyer to exercise his full bargaining power. The asymmetric case is more complex so consider again the power family of cost distributions. This family generates a special result that the expected price in the preference auction

is

the

same

irrespective

EPPA = EPPA ( p = 1) = EPPA ( p = 2) .

From

of

which

Proposition

supplier 3,

we

also

is

preferred: know

that

Π p =1 − Π k =1 > Π p = 2 − Π k = 2 for this family. Thus, for symmetric bargaining power (δ = δ1 = δ2), r2 > r1 = v − EPP = 2 + Π p = 2 − Π k = 2 for any δ > 0 . The buyer’s threat point r1 with respect to supplier 1 is lower than the threat point r2 with respect to supplier 2 because supplier 2 generates a lower total surplus. Moreover, the buyer’s threat point with respect to supplier 1 is interior to the Pareto frontier for an agreement. Thus, the payoff u1 of the buyer in any such agreement with supplier 1 is (17)

[

] ) + δ (Π

u1 = (1 − δ )r1 + δ v − EPPA + Π p =1 − Π k =1

(

= v − EPPA + (1 − δ ) Π p =2 − Π k = 2

p =1

)

− Π k =1 .

The payoff of the buyer coincides with the payoff from auctioning preference only when the buyer has no bargaining power (δ = 0). Whenever δ > 0 , u1 is larger than v − NEPPA ( p = 1) , the payoff of the buyer from auctioning preference. Thus, auctioning preference to the stronger supplier is equivalent to a vertical merger when the buyer has no bargaining power. However, if the buyer has some bargaining power, then negotiating to acquire one of the suppliers will generate a higher payoff. This result should not be surprising. As we mentioned above, the allocation of the contract is the same whether the buyer merges with the stronger supplier or whether the stronger supplier acquires preference in the pre-auction. The only possible difference is how the buyer and the stronger supplier share in the surplus. A pre-auction for preference is the smallest exercise of bargaining power by the buyer and ensures him the lowest possible payoff. The only threat of the buyer in a pre-auction for preference is to sell preference to the weaker supplier at his willingness to pay. When suppliers are symmetric, each supplier has the same willingness to pay for preference, and a pre-auction for preference is all that the buyer needs to capture the maximum rents from either supplier for preference. However, when suppliers are asymmetric, a pre-auction

20 for preference cannot extract all the rents of the stronger supplier from preference. The buyer can extract only some of the rents of the stronger supplier, in particular, those rents corresponding to the willingness to pay of the weaker supplier for preference. Apart from possible differences in ex post price EPPA which depends on which supplier is preferred, the pre-auction for preference limits the buyer to his reservation value. As such, the pre-auction for preference is equivalent to the buyer having no bargaining power relative to the suppliers. Selling preference in a pre-auction requires very little bargaining power for the buyer. Alternatively, if the buyer had all the bargaining power, an optimal mechanism for the buyer would be to run an ex post efficient auction (after the suppliers learn their costs) but then charge both suppliers an ex ante entry fee (before the suppliers learn their costs) equal to their expected profits from the efficient auction.9 Selling preference to the stronger supplier appears to be contradictory to the results in optimal mechanism design that a buyer would favor the weaker of two asymmetric suppliers. However, there is no contradiction, and this can be illustrated using the findings of Courty and Li (2000). This paper examines a buyer who faces a supplier that knows some information about of his costs, but not the true cost realization. Assuming that the supplier learns his true cost at a later point in time, the authors show that the optimal screening mechanism for the buyer distorts the allocation more for an ex ante stronger supplier. Our results follow the same logic in a twosupplier case. The key to this result is the existence of a first stage in which the buyer can extract rents prior to a second stage in which the agents learn their private information.

6.

Conclusions and Policy Implications

The sale of preference clearly empowers the buyer relative to running a standard auction at the procurement stage. In particular, the buyer can extract rents from the suppliers because part of the willingness to pay for preference arises from the desire to avoid the reduction in expected profits if the other supplier is preferred instead. With a power family of cost distributions, the buyer will always pay a lower net expected price for the good after the sale of preference to the

9

The optimal mechanism with entry fees requires that the buyer allow each supplier to bid for the contract in the procurement auction, and prevent any other new supplier, such as a future entrant, from bidding.

21 stronger supplier. If the buyer is competing in some subsequent final good market, this could benefit the ultimate consumers if the lower input costs are passed on to these consumers through competition in the final market. Despite this benefit to the buyer and his consumers, the creation and sale of preference raises several concerns for efficiency and future competition. The first concern that the reduction in costs for the buyer occurs from rent extraction rather than new efficiencies. The standard justification for exclusive dealing is that the interests of the buyer and the supplier are more closely aligned, and that various externalities in their production and distribution decisions are internalized. On the contrary, preference arises even though it actually creates an allocative distortion in the award of the procurement contract. The allocative distortion in favor of the preferred supplier creates an artificial asymmetry in the procurement auction which generates the willingness to pay for preference by both suppliers. The second concern is that the pre-auction for preference distorts the procurement auction in favor of the stronger supplier. With the power family of cost distributions, the stronger supplier always outbids the weaker supplier for preference. Moreover, if the stronger supplier is sufficiently stronger, he may actually earn higher expected profits even after paying the buyer for preference. Thus, preference accentuates the underlying asymmetry from the differing capacities of the suppliers. In an extended model with investments by multiple suppliers and multiple buyers, there could be other competitive effects on consumers in the final market. Consider the question of investments by suppliers. In our related paper Burguet and Perry (2000), we examined the incentives of two suppliers to make investments when they can bribe the auctioneer for favoritism, which corresponds to preference in this model.

Although

favoritism enhanced the investment incentives of the favored supplier, it severely undermined the investment incentives of the other supplier. Moreover, aggregate investment in capacity was lower with favoritism and was less than the social optimum. Consider the question of competition among multiple buyers of the input in a final market, and more suppliers than the buyers. In the current model with one buyer, preference to one supplier does not withdraw that supplier’s capacity from the input market. However, with multiple buyers, the capacity available to bid on contracts by one buyer may diminish as each buyer auctions preference to another one of the suppliers. This would clearly occur if preference

22 for one supplier also requires exclusivity between the buyer and that supplier. This will alter competition among the suppliers in the input market and among the buyers in the final market.

23

References Aghion, Philippe and Patrick Bolton, “Contracts as a Barrier to Entry,” American Economic Review 77 (1987), 388 –401. Arozamena, Leandro and Estelle Cantillon, “Investment Incentives in Procurement Auctions,” Review of Economic Studies 71(1) (2004), 1-18. Arozamena, Leandro and Federico Weinschelbaum, "The Effect of Corruption on Bidding Behavior in First-Price Auctions", working paper, U. Torcuato di Tella and U. San Andrés, 2004 Bernheim, Douglas B. and Michael D. Whinston, “Common Agency,” Econometrica 54 (1986), 923-942. Bernheim, Douglas B. and Michael D. Whinston, “Exclusive Dealing,” Journal of Political Economy 106 (1998), 64-103. Besanko, David and Martin K. Perry, “Exclusive Dealing in a Spatial Model of Retail Competition,” International Journal of Industrial Organization 12 (1994), 297-329. Bikhchandani, Sushil, Steven Lippmann, and Reade Ryan, “On the Right-of-FirstRefusal”, working paper, University of California at Los Angeles, 2003. Burguet, Roberto, Ramon Caminal and Carmen Matutes, "Golden Cages for Showy Birds; Endogenous Switching Cost in Labor Contracts", European Economic Review, (46)7 (2002) pp. 1153-1185. Burguet, Roberto and Martin K. Perry, “Bribery and Favoritism by Auctioneers in SealedBid Auctions,” working paper, Institut d’Analisi Economica (CSIC, Barcelona, Spain) and Rutgers University, 2000. Cantillon, Estelle, “The Effect of Bidders’ Asymmetries on Expected Revenue in Auctions,” working paper, Harvard Business School, 2003. Courty, Pascal and Hao Li, "Sequential Screening", Review of Economic Studies 67( 4) (2000), 697-717 Lebrun, Bernard, "First Price Auctions in the Asymmetric N Bidder Case," International Economic Review 40 (1999), 125-142.

24 Lee, Joon-Suk, "Favoritism in Asymmetric Procurement Auctions", working paper, University of North Carolina, Chapel Hill, 2004. Li, Huagang and John G. Riley, “Auction Choice,” working paper, University of California at Los Angeles, 1999. Marshall, Robert C., Michael J. Meurer, Jean-Francois Richard, and Walter Stromquist, "Numerical Analysis of Asymmetric First Price Auctions," Games and Economic Behavior 7 (1994), 193-220. Martimort, David, “Exclusive Dealing, Common Agency, and Multiprincipals Incentive Theory,” The Rand Journal of Economics 27 (1996), 1-31. Mathewson, G. Frank and Ralph A. Winter, “The Competitive Effects of Vertical Agreements: Comment.” American Economic Review 77 (1987), 1057-1062. McAfee, R. Preston and John McMillan, "Government Procurement and International Trade," Journal of International Economics 26 (May 1989), 291-308. Myerson, Roger B., "Optimal Auction Design," Mathematics of Operations Research 6 (1981), 58-73. O’Brien, Daniel P. and Greg Shaffer, “Non-Linear Contracts, Foreclosure, and Exclusive Dealing,” Journal of Economics and Management Strategy 6 (1997), 757-785. Porter and Shoham (2005). Segal, Ilya R. and Michael D. Whinston, “Exclusive Contracts and Protection of Investments,” The RAND Journal of Economics 31(4) (2000), 603-633. Waehrer, Keith, “Asymmetric Private Values Auctions with Application to Joint Bidding and Mergers”, International Journal of Industrial Organization 17(3) (1999), 437-452. Waehrer, Keith and Martin K. Perry, “The Effects of Mergers in Open Auction Markets”, The RAND Journal of Economics 34(2) (2003), 287-304. .

25

Appendix 1: Related Literature This paper builds on the model of Burguet and Perry (2000). That earlier paper examines various models of bribery in a model with one auctioneer and two suppliers competing to provide an input for the ultimate buyer. The auctioneer represents the buyer in selecting the supplier to receive the contract, but the auctioneer can be bribed by one of the suppliers. In the basic model of bribery, the auctioneer gives one supplier a right-of-first-refusal at the price bid by the other supplier in return for a bribe equal to a share of the profits on the contract. The paper then examines the implications of bribery for the expected price, the allocative distortion, the expected profits of the suppliers, and the ex ante investments by the suppliers. This paper uses the same auction model between the suppliers, but eliminates the auctioneer and allows the buyer to hold a pre-auction for the right-of-first-refusal. One related paper is Aghion and Bolton (1987). An incumbent seller enters into a contract with a monopoly buyer in which there is a payment for the good when an entrant seller with a lower cost of producing the good does not arise. However, if a lower cost entrant arises, the buyer makes a lower payment to seller to breach the contract and then purchase the good from the entrant.

The incumbent seller and buyer can mutually benefit from this contract by

extracting rents from the entrant seller.

In our model, the buyer auctions the right-of-first-

refusal to one of the suppliers, and purchases from the other supplier only if his bid is below the cost of the PS. In a similar manner, the buyer and possibly the PS can increase their expected profits at the expense of the competing supplier. The paper by Aghion and Bolton (1987) spawned a subsequent literature that is less related to our paper. Awarding a right-of-first-refusal to one supplier is similar to exclusive dealing with the buyer or vertical integration between one supplier and the buyer. A number of authors have examined models in which two manufacturers compete to foreclose each other by executing an exclusive dealing contract with a downstream retailer who has a monopoly in some local market. These papers employ an explicit or implicit model of consumer demand with differentiated products, and focus on complete foreclosure. This literature includes papers by Bernheim and Whinston (1986), Mathewson and Winter (1987), Besanko and Perry (1994), Martimort (1996), O’Brien and Shaffer (1997), and Bernheim and Whinston (1998). The latter two papers examine

26 a similar reduced form model and obtain the same key result. So we discuss the relationship between these two papers and our paper. In O’Brien and Shaffer (1997) and Bernheim and Whinston (1998), two manufacturers, A and B, offer exclusive and non-exclusive contracts to a single monopoly retailer. The retailer must choose between the contracts. The most interesting case occurs when an exclusive contract with either manufacturer would result in a lower total surplus. Let ΠC be the maximized total surplus when the retailer is a common dealer for both manufacturers, and let ΠA and ΠB be the maximized total surplus when the retailer has an exclusive contract with one manufacturer, respectively. The interesting case arises when ΠC > ΠA > ΠB, but ΠC < ΠA + ΠB. The first inequalities imply that exclusivity reduces total surplus and that the product of manufacturer A is more preferable than B to consumers. The second inequality implies that the products are substitutes and not complements. With this framework, both papers demonstrate that there exists a Pareto undominated equilibrium for the manufacturers in which efficient common distribution occurs and each manufacturer receives a payment equal to its marginal contribution to the maximized total surplus. Thus, manufacturer A receives ΠC - ΠB ; manufacturer B receives ΠC - ΠA ; and the retailer retains the remainder of the surplus equal to ΠA + ΠB - ΠC . However, this equilibrium is undominated only for the manufacturers. If the retailer could auction the exclusive contract to one of the manufacturers, it could guarantee itself a payment of ΠB. Manufacturer A would outbid manufacturer B for the exclusive contract, ensuring that the bid would extract all the surplus of the less popular product of manufacturer B. If so, the retailer would clearly prefer to auction an exclusive contract to manufacturer A rather than to accept the undominated equilibrium payoff for common distribution. The assumption ΠC > ΠA implies that ΠB > ΠA + ΠB - ΠC . This point is most clearly made in Proposition 5 of the paper by O’Brien and Shaffer (1997), and the subsequent discussion. By auctioning an exclusive contract, the retailer and manufacturer A can increase their profits at the expense of manufacturer B, even though the total surplus in reduced by the exclusive contract. Manufacturer B is completely foreclosed and earns zero profits. This finding can also apply to the other dominated equilibrium of common distribution by the manufacturers. Let PA and PB be the payments to the manufacturers in such an equilibrium.

27 In order for the retailer to prefer auctioning an exclusive contract, it must be true that ΠB > ΠC – PA – PB . Depending on the equilibrium payments to the manufacturers, this condition can be satisfied even though ΠC > ΠA. In particular, this can occur whenever the sum of the equilibrium payments to the manufacturers is such that PA + PB > ΠC - ΠB > ΠC - ΠA > 0 . The nice feature about both papers is that the results are independent of specific functional forms for demand. However, the resulting disadvantage is that one does not know which equilibria for common distribution would invite the retailer to auction an exclusive contract. Our paper focuses on one standard auction model to capture the equilibrium without preference. Thus, we are able to make clear comparisons after the pre-auction for preference. The primary difference is that these models assume the retailer must either accept or reject the offers of exclusive or non-exclusive contracts from the manufacturers. The retailer is more passive in that it cannot auction an exclusive contract. If it could auction an exclusive contract, it would place the manufacturers in a more competitive situation. Although it is natural to examine the equilibrium between the manufacturers, these models seem to impart less bargaining power to the retailer than it deserves based on its monopoly position in the final market. Our model imparts more power to the retailer because he can hold a procurement auction without preference or pre-auction preference and then hold the procurement auction. The residual bargaining power of the suppliers then derives from their private information and their relative status as the preferred or competing supplier. That said, the results would be very similar if the retailer could auction an exclusive contract. The retailer could extract all the profits of the weaker manufacturer B, and thereby split the combined profits under the exclusive contract with the stronger manufacturer A. There are likely to be many cases in which the retailer would increase its profits, even though the total surplus declined with the exclusive contract. Indeed, there are also likely to be cases in which manufacturer A receives higher profits even though it must bid aggressively to obtain the exclusive contract. The secondary difference is that these models result in complete foreclosure of the weaker manufacturer B when the retailer auctions an exclusive contract. This arises because there is no private information. In our model, the buyer must auction preference before holding the procurement auction and the suppliers have private information about their costs of

28 producing the good. The weaker supplier is not completely foreclosed by the sale of preference to the stronger supplier because he may still have sufficiently low enough costs to underbid the cost of the stronger supplier in some cases. The more recent models of exclusive dealing have focused on issues of investment by buyers and sellers. See Segal and Whinston (2000) for a model and other references. Our model does not include investment by the suppliers or the buyer. It is clear that the return on capacity is higher for the preferred stronger supplier. If investment were incorporated in the model after preference is sold, it seems that the PS would invest in more capacity prior to the procurement auction. However, if investment were incorporated before the sale of preference, the investments would then affect the pre-auction for preference.

Appendix 2: Proof of Proposition 3 Proof of Proposition 3: The difference in the willingness to pay of the stronger and weaker

suppliers can be rearranged as follows:  t1t 2 (1 + 2t 2 ) 1  ⋅  − (1 + t1 + t 2 )  t 2 (1 + t 2 ) t ⋅ (1 + t ) 2 2 2 

 t2     1 + t2 

t2 

 (1 + 2t1 ) 1   −   t (1 + t ) − 1 t1 ⋅ (1 + t1 ) 2   1

t   t1  1     .   1 + t1   

The term in braces is the difference between two expressions having the form:  1 (1 + 2t )  −  t (1 + t ) t ⋅ (1 + t ) 2 

 t  ⋅  1 + t 

t

  .  

Thus, the term in braces is positive if this expression is decreasing in t. Taking the derivative of this expression, we find that: t t d ( −) (1 + 2t ) 1 (1 + 2t + 2t 2 )  t    t  t  = − ⋅  + ln ⋅   +   < 0 . dt t (1 + t )(1 + 2t )  1 + t   t (1 + t ) 2  t  1 + t  1 + t 

The first and third terms are obviously positive, but the second term is negative. However, the first term dominates the second term because 1/t + ln[t/(1+t)] > 0. This follows from the fact that [1/t + ln(t/(1+t))] approaches zero as t → ∞ and is decreasing for all t. Since this derivative is negative, V1 > V2 whenever t1 > t2. Q.E.D.

29