BLS WORKING PAPERS

U.S. DEPARTMENT OF LABOR

Bureau of Labor Statistics

 

OFFICE OF PRICES AND LIVING

CONDITIONS

The Effects of Mergers in Open Auction Markets

Keith Waehrer

and Martin K. Perry

Working Paper 322

June 1999

 

Any opinions expressed in this paper are our own and do not constitute policy of the U.S. Bureau of Labor Statistics. This

paper benefited from the comments of Roberto Burguet, Robert Bradford, Serdar Dalkir, Luke Froeb, Ian Gale, Patrick Greenlee, Preston McAfee, Rich McLean, Mike Rothkopf, Jozsef Sakovics, Guofu Tan, and Charles Thomas.
The E§ects of Mergers in Open Auction Markets

Abstract. A buyer solicits bids from suppliers with di§erent cost distributions.

The cost distribution of a supplier is deÖned by its capacity. The expected market

share of each supplier is the ratio of its capacity to the industry capacity. If the buyerís

reserve price is Öxed, mergers increase industry concentration, increase the expected

price, and reduce the buyerís welfare. Moreover, suppliers have an incentive to merge.

If the buyer can optimally lower the reserve price, he can partially or fully o§set the

e§ects of a merger. However, a merger still reduces the buyerís welfare because he must

forego some gains from trade when he lowers the reserve price. The optimal reserve price

can undermine the incentive for larger suppliers to merge and result in stable industry

structures for which no further mergers would be proÖtable.

1
The E§ects of Mergers in Open Auction Markets

The e§ects of mergers have been examined by a number of authors using the traditional

Nash-Cournot model with homogeneous products. Several of these papers consider models in

which the Örms have di§erent capacities and thus di§ering levels of output in equilibrium.1

These asymmetric models allow a more realistic assessment of the competitive e§ects of

mergers because one can examine mergers between Örms of varying sizes. In this paper

we investigate the e§ects of mergers in an auction market where the Örms are asymmetric

because they have di§erent distributions for their costs. In the context of this model, we

examine the traditional merger questions. Does a merger increase the price and reduce the

welfare of buyers? Do Örms have an incentive to merge? Do mergers reduce total welfare?

The answers to these questions in the context of this auction model will have some similarities

and some di§erences from those found using the asymmetric Nash-Cournot models.

We consider mergers in an asymmetric auction model of procurement in which the Örms

are interpreted as suppliers bidding to win contracts to supply some input that the buyer

needs for the production of a Önal good. This is typical of an industrial product setting in

which auctions are a common method for the purchase of inputs.2 The model employs a

1Perry and Porter (1985) consider the incentive to merge in a Nash-Cournot model with a homogeneous

product. In their model, Örms have di§erent capacities in that the marginal cost of production for one Örm is

linearly increasing and shifts horizontally outward with more capacity. Using a linear demand function, they

solve for an asymmetric Nash-Cournot equilibrium with two types of Örms, one type with twice the capacity

of the other type. The merger of two small Örms into one large Örm increases the price and reduces consumer

welfare. There is typically an incentive to merge when the two Örms already have large market shares, but

there need not be an incentive to merge when the two Örms have smaller market shares.

Farrell and Shapiro (1990a and 1990b) examine a general version of the capacity model used by Perry and

Porter (1985), and focus on the welfare e§ects of mergers. They Önd that large Örms operate at a lower

output per unit capacity than small Örms. This results in an ine¢cient allocation of a given industry output

across the Örms. Mergers between large and small Örms eliminate some of this ine¢ciency and thus can

increase welfare despite the resulting higher prices. Farrell and Shapiro then provide some general conditions

for identifying mergers that would increase welfare. With the linear version of this capacity model, McAfee

and Williams (1992) characterize the proÖtable mergers between two Örms that would increase welfare. Their

computational results suggest a limited set of circumstances in which mergers between large and small Örms

are both proÖtable and increase welfare. Werden (1991) also reexamines the linear model in order to illustrate

the relevance of the HerÖndahl-Hirschman index (HHI) in assessing the price e§ects of mergers. 2This auction model is unlikely to be an appropriate representation of competition in consumer product

markets. In those markets, the more appropriate model would be either the Nash-Cournot model if products

are homogeneous or the Nash-Bertrand model if the products are di§erentiated. In recent years, various

economists have proposed and the antitrust agencies have employed empirical methodologies to estimate or

simulate the price e§ects of mergers in di§erentiated consumer product industries. For an overview of these

methodologies and their applications, see the Spring 1997 issue of Antitrust published by the American Bar

Association. This empirical work has its theoretical foundations in a paper by Deneckere and Davidson

(1985). Deneckere and Davidson investigate the incentive to merge in a model with heterogeneous products.

Their model does not have a capacity variable on the cost side. However, the merged Örm continues to sell

both products. In this sense, a merged Örm is larger than the other Örms. They Önd an incentive to merge

2
second-price auction mechanism. In the second-price auction, suppliers submit sealed bids

to a buyer and the input is sold to the lowest bidder at a price equal to the second lowest

bid. If the costs of the suppliers are private information and stochastically independent,

second-price auctions are strategically equivalent to open auctions, also called descending

price oral auctions.

In an open auction, the auction begins at some high price, declines with progressively

lower bids, and the contract is awarded to the last bidder active in the auction.3 The

dominant strategy for each supplier in such auctions is to submit a bid equal to its cost.

Thus, the supplier with the lowest cost will win the contract and receive a contract price

equal to the cost of the supplier with the second lowest cost. We only consider second-price

and open auctions since analytical results for the e§ects of mergers in Örst-price auction

markets are di¢cult to obtain. The equilibrium in Örst-price auctions is much less tractable

when the cost distributions are not identically distributed.4

In this paper we also allow for the possibility that the buyer in this auction can set a

reserve price to maximize his expected proÖt from the employment of the input. A binding

reserve price would require that the buyer commit to forego purchasing at a price above the

reserve price even when such a price is below the buyerís value for the input.

Several recent papers examine mergers in auction markets with asymmetric Örms.5

Waehrer (1997) examines mergers in both Örst-price and second-price auction markets. Using the same structure on cost distributions employed in this paper and assuming a Öxed

reserve price, Waehrer Önds that the proÖt per market share for larger Örms is higher than

for smaller Örms in second-price auction markets, while the opposite is true in Örst-price

auction markets. It follows that non-merging Örms beneÖt from a merger in Örst-price auction markets, while the non-merging Örms are una§ected by a merger in second-price auction

markets. This suggests that the incentive to merge may be stronger in second-price auction

markets. We return to this point later in the paper.

Using simulations to derive equilibrium bidding strategies for Örst-price auctions, Dalkir,

Logan, and Masson (1998) show that failure to consider the asymmetries generated by mergfrom the ability to coordinate in the pricing of the two products. A merger results in higher prices for both

the products of the merged Örm and the products of the nonmerging Örms. 3The second-price auction model might be an appropriate representation of the market even though the

buyers do not employ a strict descending price oral auction. For example, buyers might invite an initial sealed

bid from the suppliers, suggesting a Örst-price auction. However, if the buyers subsequently use the low bids

to ìwhipsawî other suppliers into lowering their bids, a second-price auction model seems more appropriate. 4General treatments of asymmetric Örst-price auctions can be found in Maskin and Riley (1992), Lebrun

(1995), and Waehrer (1997). 5There is also a related literature on collusion in auction markets. For example, see DeBrock and Smith

(1983), Graham and Marshall (1987), Von Ungern-Sternberg (1988), Mailath and Zemsky (1991), McAfee

(1994), and McAfee and McMillan (1992).

3
ers can result in an overstatement of their price e§ects.6 Tschantz, Crooke, and Froeb (1997)

simulate equilibria for Örst-price auctions assuming costs follow the extreme value distribution. They Önd evidence that the anti-competitive e§ects of mergers are smaller for Örst-price

auctions than for second-price auctions. Thomas (1998) derives analytical solutions for an

asymmetric Örst-price auction by assuming that costs are binomially distributed. Thomas

obtains the surprising result that a merger between two weaker Örms may in fact beneÖt the

buyer even when there are no e¢ciencies generated by the combination. However, in such

situations the Örms have no incentive to merge.

The asymmetric second-price auction is more tractable than the Örst-price auction because the dominant strategy for a supplier in second-price auctions is to submit a bid equal

to its cost. Thus, the expected equilibrium price is simply the expected value of the secondorder statistic of the distributions on the costs for all suppliers. Brannman and Froeb (1997)

and Froeb, Tschantz, and Crooke (1998) study asymmetric second-price auctions when costs

follow the extreme value distribution. They examine the price e§ects of mergers assuming a

Öxed reserve price. For a common-value second-price auction, Krishna and Morgan (1997)

show that joint bidding can actually beneÖt the bid-taker.

In this paper, we characterize a family of distributions for the costs of the suppliers that

are parameterized by variables which can be interpreted as the sizes, scales, or capacities of

the suppliers. The family of cost distributions is assumed to have properties which eliminate

both (1) positive or negative externalities across suppliers and (2) economies or diseconomies

of scale. Thus, mergers do not a§ect the cost distribution of non-merging suppliers and

generate constant returns to scale for merged suppliers. We then consider a private-value

second-price auction with a reserve price. We Önd that mergers always reduce the welfare

of the buyer, even if the buyer optimally adjusts the reserve price in response to the merger.

When the reserve price is Öxed, mergers result in a higher expected price. However, when

the buyer adjusts the reserve price in order to maximize its expected proÖt, the expected

price can fall as a result of a merger.

The results also provide a systematic way of ranking di§erent mergers by the magnitude

of their e§ects on the buyers. The incentive to merge and the overall welfare e§ects depend

on whether the reserve price is Öxed or set optimally by the buyer. When the reserve price

remains the same before and after a merger, total welfare is una§ected by a merger and

suppliers always have an incentive to merge. However, when the reserve price is adjusted

optimally by the buyer, total welfare decreases as a result of the merger because the buyerís

optimal response is to reduce the reserve price. In this case suppliers may or may not have

an incentive to merge and the expected price may increase or decrease. Numerical examples

6Also using simulations, Marshall, et.al. (1994) Önd evidence that the bid-takerís payo§ is higher in

Örst-price auctions than in second-price auctions when cost distributions are independent and asymmetric.

4
suggest that smaller suppliers are more likely to have an incentive to merge than larger

suppliers.

The paper is organized as follows. In Section I, we deÖne the second-price auction with

asymmetric capacities. In Section II, we then solve for the expected probability of winning

each contract by the suppliers and the expected price paid by the buyer. In Section III, we

deÖne industry concentration and explain its relation to the HHI. In Section IV, we examine

the welfare questions of whether mergers increase the expected price and reduce the expected

proÖt of the buyer. Finally, in Section V, we examine the incentive to merge and the e§ects

of a merger on total welfare.

1. Model

We assume that a buyer requests bids from suppliers who can provide an input necessary

for production of a Önal good. The buyer employs an open auction to select the winning

supplier. More generally, the market would be composed of a series of such auctions by a

number of buyers.

The buyer has a value v>c for the input where v is known to the buyer and to all of the

suppliers. The buyer would clearly reject a bid price for the input above v. However, we will

also consider cases in which the buyer can reject bid prices below his value v by committing

to a single reserve price r<v. With a reserve price r<v, the buyer is committing not to

purchase the input from any supplier at a bid p where r<p, either during the auction or

after an auction in which no supplier o§ered a bid less than r.

7 When the reserve price is

set below v and below the upper support on the cost distributions of the suppliers, an ex

post ine¢ciency could arise from the lack of trade at some price less than the buyerís value

but greater than some supplierís cost. Note that if the buyer cannot commit to a reserve

price less than its value, we will simply assume that r = v.

Let N = f1;:::;ng denote the set of potential suppliers of the input. We assume that

each supplier i has a capacity parameter ti and draws his cost ci of producing the input from

the distribution G(¢jti) with a support of [c; c1] common to all suppliers. Denote the proÖle

of capacities for the suppliers as t = (t1;:::;tn) and deÖne t

^ = Pn

i=1 ti as the industry

capacity. We assume that the suppliersí costs are independently distributed.8 Furthermore,

each supplier obtains his cost of production prior to submitting a bid to the buyer and need

7We only consider the case where the buyer uses a single reserve price that applies equally to all of the

suppliers regardless of size. When the suppliers have di§erent capacities, the buyer may be able to gain by

setting di§erent reserve prices for the di§erent suppliers. 8This independence assumption would be violated if the costs of suppliers depended on the uncertain

future prices of raw materials and if each supplier received a correlated signal about those future prices. Our

model assumes that no such correlation exists. Thus, if raw material prices are relevant, we assume that

these costs are known and common to the bids of all suppliers. Any remaining uncertainty about the costs

of each supplier would depend only on characteristics unique to that supplier.

5
not incur this cost unless he wins the auction to supply the input.

We assume that the suppliers are asymmetric in that they have di§erent capacity parameters which deÖne di§erent distributions for their cost of producing the input. With di§erent

distributions on costs, the equilibrium will necessarily have the property that suppliers will

have di§erent expected probabilities of winning the auction, and thus, have di§erent market

shares over a large number of auctions. When two suppliers merge, they have a larger combined capacity, and thus, the cost of producing the input is drawn from a new distribution

that places a higher probability on low costs. This is the sense in which the merger results

in one new larger supplier who will have a higher probability of winning the auction.

In deÖning the notion of size, scale, or capacity which generates the di§erences in the cost

distributions for suppliers, we want to impose three properties. First, we assume that there

are no positive or negative externalities across suppliers. In particular the cost distribution

of each supplier is independent of the capacities of the other suppliers. As such, we eliminate

any real or pecuniary externalities from mergers which a§ect the costs of the other suppliers.

Second, we assume that capacity is a homogeneous parameter. In particular, any suppliers with the same capacity have the same cost distribution. As a result, the equilibrium will

depend only on the current capacity of each of the suppliers, and not the history of mergers.

Third, we assume that there are no economies or diseconomies of scale in these cost

distributions. In order to accomplish this, we assume that the probability distribution of the

minimum cost draw of all the suppliers does not depend on how capacity is distributed across

suppliers. That is, as long as total capacity is constant, the distribution of the minimum

cost is the same whether there are only a few large Örms or many small Örms. This property

is equivalent to assuming that the cost distribution of a merged supplier is the same as the

distribution of the minimum cost of the two suppliers before the merger. By eliminating

economies or diseconomies, we can focus on the price and welfare e§ects that result solely

from increasing concentration after a merger.9

The three properties of the cost distributions can be formally expressed as follows.

Property 1 (No Externalities): The distribution of each supplierís cost depends only on

its own capacity and not on the capacities of other suppliers.

Property 2 (Homogeneity): If two suppliers have the same capacity, then they also have

the same cost distribution.

9Brannman and Froeb (1997), Waehrer (1997), Dalkir, Logan, and Masson (1998), and Froeb, Tschantz,

and Crooke (1998) also model mergers in this way. This type of merger is equivalent to e¢cient collusion as

discussed by Mailath and Zemsky (1991) and McAfee and McMillan (1992).

The quadratic cost model in Perry and Porter (1985) possesses this property of constant returns. DeÖne

x(c; s) as the quantity of output that can be produced at or below a marginal cost of c, given the capacity s.

The quadratic cost function implies that x(c; si + sj ) = x(c; si) + x(c; sj ):

6
Property 3 (Constant Returns): The probability distribution of the lowest cost draw

out of all of the suppliers depends only on total capacity. In particular, it does not

depend on the number of suppliers or how total industry capacity is distributed across

suppliers. That is, for any two capacity proÖles t1;:::;tn and s1;:::;sm such that

Pn

i=1 ti = Pmi=1 si and for any c 2 [c; c],

1 °Yn

i=1

[1 ° G(cjti)] = 1 °Ym

i=1

[1 ° G(cjsi)] : (1)

For mergers, we assume that if two suppliers with capacities ti and tj merge, then the

capacity of the resulting supplier is tm = ti + tj. Thus, Property 3 implies that the cost

distribution of a merged supplier is the minimum of the two cost draws from the original

suppliers. That is,

G(cjti + tj)=1 ° [1 ° G(cjti)] [1 ° G(cjtj )] :

The following result characterizes Properties 1, 2, and 3 in terms of the form the distribution must take.

Theorem 1. Properties 1, 2, and 3 hold if and only if there exists a distribution function

F with a support of [c; c] such that for c 2 [c; c),

G(cjti)=1 ° [1 ° F(c)]ti :

Proof in the Appendix.

According to Theorem 1, any cost structure that satisÖes Properties 1, 2, and 3 must take

the form described in the theorem. In other words, there is no cost structure more general

than this form that satisÖes Properties 1, 2, and 3. Alternatively, for any distribution F,

when the cost distribution G takes the form described, then Properties 1, 2, and 3 must

be satisÖed. Fortunately, this cost structure simpliÖes the analysis. The remainder of our

analysis will be conducted assuming that Properties 1, 2, and 3 hold and thus, making use

of the functional form provided by Theorem 1.10

If f is the density function associated with the distribution F, then the resulting density

function for G(¢jti) can be written as g(cjti) = ti [1 ° F(c)]ti°1 f(c). The distribution F

1 0Waehrer (1997) makes use of the same family of distributions. The structure here generalizes the cost

structure assumed by Marshall, et.al. (1994) and Dalkir, Logan, and Masson (1998) in that those analyses

assume that F is the uniform distribution. Brannman and Froeb (1997) and Froeb, Tschantz, and Crooke

(1998) employ the extreme value distribution for which a parameter similar to t can be deÖned. However,

the analysis is limited to this family of distributions and the convenience of this family requires that positive

probability be assigned to any value or cost in (°1;1).

7
deÖnes a family of cost distributions G for suppliers with di§erent capacity parameters. If a

supplier has a capacity of one, then its cost distribution is simply G(¢j1) = F. For a supplier

with a capacity parameter greater than one, the supplier would have a more favorable cost

distribution in that there would be a higher probability of obtaining a cost below c. That

is, for any c 2 (c; c) and ti > 1, G(cjti) > G(cj1).

The capacity parameter ti can be interpreted as the number of draws from a cost distribution F, where the supplier uses the lowest draw for its cost of producing the input. As

such, G(¢jti) is the distribution function of the order statistic for the lowest cost from among

ti independent draws from F. Similarly, the capacity parameter ti can be interpreted as

the number of plants from which the supplier could produce the input, each plant having

a cost distribution of F. This is the sense in which the parameter ti can be interpreted

as measuring the size, scale, or capacity of a supplier.11 Even though ti does not measure

capacity in the sense of an absolute physical limit on production, we will refer to ti as the

capacity of the ith supplier since it is a measure of size.

2. Market Equilibria and Expected Price

In a second-price auction, the supplier with the lowest bid wins and sells at a price equal

to the lower of the reserve price or the second lowest bid. When setting the reserve price,

the buyer knows the capacities of the suppliers but does not know their actual cost draws.

In equilibrium, each supplier has a dominant strategy to submit a bid equal to its cost. If

the seller submits a bid below its cost, then there is a possibility that the supplier will be

required to sell at a price below its cost. On the other hand, if a supplier submits a bid

above its cost, then there is a possibility that the supplier will forego an opportunity to sell

the input at a price above its cost. Conditional on winning, a supplierís bid has no e§ect on

the price it receives. Therefore, there is no gain to any supplier from raising its bid above

its cost. Thus, in equilibrium, each supplier submits a bid equal to its cost.

For any given contract, there would be at most one winning supplier at a given price.

However, in order to assess the industry performance, we examine the expected market

share. The expected market share of a supplier is its probability of winning a given contract

conditional on some supplier winning the contract. We can now state the following lemma.

Lemma 1. Suppose the capacity proÖle is t. (a) The probability distribution for the

lowest cost in the industry is 1 ° [1 ° F(c)]t

^

. (b) The ex ante expected market share of each

supplier is the ratio of its capacity and the total industry capacity, ti=t

^.

1 1Piccione and Tan (1996) obtain the same functional form from a property which they call ìcomplete

stochastic orderingî. In Piccione and Tan (1996) and an earlier paper, Tan (1992), the exponent is interpreted

as research and development activity. Both papers examine the symmetric equilibrium in research and

development expenditures by Örms competing in an auction for a production contract.

8
Proof in the Appendix.

We can now derive expressions for the suppliersí proÖts, the buyerís proÖt, and the

expected price paid by the buyer conditional on the contract being awarded. Let oi

(ci; rjt)

denote supplier iís expected proÖt in equilibrium when i has costs ci, the reserve price is

r, and the capacity proÖle is t. The distribution function of the lowest cost of supplier iís

rivals is 1 ° [1 ° F(ci)]t

^°ti . Thus, we have

oi

(ci; rjt) = ·Z r

ci

[a ° ci](t

^° ti)[1 ° F(a)]t

^°ti°1f(a)da + (r ° ci)[1 ° F(r)]t^°ti

 ̧

¢ 1fci6rg

=

·Z r

ci

[1 ° F(a)]t

^°tida ̧

¢ 1fci6rg (by integration by parts).

The indicator function, 1fci6rg, is equal to one when the condition within the braces is true

and zero otherwise. Supplier iís ex ante expected proÖt is ¶i

(rjt) = E[oi(ci; rjt)] where the

expectation is taken over supplier iís cost.

¶i

(rjt) = Z r

c

Z r

ci

[1 ° F(a)]t

^°ti ti[1 ° F(ci)]ti°1f(ci)dadci

=

Z r

c

n

[1 ° F(ci)]t

^°ti ° [1 ° F(ci)]^t

o

dci (by integration by parts). (2)

Holding the reserve price constant, a supplierís expected proÖt (both oi

(ci; rjt) and ¶i(rjt))

depends only on its capacity ti and the industry capacity t

^. Expected proÖt does not depend

on how the capacity t

^° ti is distributed among the rival suppliers. Thus, a merger does

not alter the expected proÖts of the non-merging suppliers when the buyerís reserve price

remains unchanged. Similarly, a merger would not increase the expected proÖts of potential

entrants since potential entrants can be thought of as potential non-merging suppliers.12 It

is also easy to see that for r 2 (c; c), a supplierís expected proÖt is increasing in the reserve

price. A lower reserve price does not alter a supplierís market share, but it does reduce the

probability of purchase by the buyer.

Note that the total expected proÖt to the suppliers can be written as

Xn

i=1

¶i(rjt)=(p(rjt)°E [z1jz1 6 r]) Prfz1 6 rg (3)

where p(rjt) is the expected price paid by the buyer conditional on a sale taking place and

1 2As Waehrer (1997) points out, this would not be true in a Örst-price auction. A merger in a Örst-price

auction would increase the expected proÖt of both non-merging suppliers and entrants. Similarly, this is not

true in the asymmetric Nash-Cournot model. A merger in that model increases the proÖts of the non-merging

Örms because the higher price after the merger induces them to expand output.

9
where z1 = minfc1;:::;cng. Making use of Lemma 1, it is straightforward to see that

Prfz1 6 rg = 1 ° [1 ° F(r)]t

^ and E [z1jz1 6 r] =

R r

c c t

^[1 ° F(c)]t

^°1f(c) dc

1 ° [1 ° F(r)]t

^ :

Hence, the right-hand side of (3) is the expected price net of the expected cost, times the

probability of purchase. Rearranging (3) yields an expression for expected price

p(rjt) =

Pn

i=1 ¶i

(rjt)

Pr fz1 6 rg + E [z1jz1 6 r] : (4)

An expression for the proÖt of the buyer U(rjt) can be written as the total gains from

trade minus the expected proÖt to the suppliers. That is,

U(rjt) = 31 ° [1 ° F(r)]t

^

¥

(v ° E [z1jz1 6 r]) °Xn

i=1

¶i(rjt) (5)

The total gains are the probability of purchase times the di§erence between the value of the

input to the buyer and the expected minimum cost draw conditional on it being less than

the reserve price.

If the reserve price is Öxed, then the capacity proÖle enters p(rjt) and U(rjt) through

its e§ect on the total expected proÖt of the suppliers. We model mergers as changes in how

capacity is distributed across suppliers. The e§ect of a merger on the expected price and

the expected proÖt of the buyer will follow directly from its e§ect on the suppliersí total

expected proÖts. If the reserve price is optimally set by the buyer, then the e§ects of a

merger on the expected price and proÖts of the buyer are more complicated. Section IV

discusses the e§ect of concentration and mergers on the buyer. Our analysis is based on the

deÖnitions of concentration deÖned in the following section.

3. Changes in the Concentration of Market Capacity

In this section, we deÖne a concentration ordering which allows us to compare industry

performance for di§erent capacity proÖles. This ordering facilitates our examination of the

welfare e§ects of mergers and the incentive for mergers.13 Consider two capacity proÖles t

and s. The proÖles t and s are equivalent if s is simply a rearrangement of the elements of

t. For parts of the analysis, it is useful to order the capacities in a proÖle by size. Thus,

we deÖne the index notation t(i) such that t(1) > ¢¢¢ > t(n). When the suppliers f1;:::;ng

follow their equilibrium strategies, the proÖles (t1;:::;tn) and (t1;:::;tn; 0) will result in the

same outcomes. This follows from the fact that a supplier with zero capacity has no chance

of being the lowest cost supplier and winning the auction. Hence, it is always possible to add

1 3Encaoua and Jacquemin (1980) discuss the application of concentration measures to entry and mergers.

10
suppliers with zero capacity to a market without a§ecting the equilibrium outcomes. This

is convenient because our deÖnitions of concentration will require that the proÖles being

compared have the same number of suppliers.

An equalizing transfer between two suppliers is a transfer of capacity from one supplier to

another such that the absolute di§erence in capacities between the two suppliers is reduced.

Let t be the pre-transfer capacity proÖle and s be the post-transfer capacity proÖle. More

formally, an equalizing transfer of capacity from t to s for Örms j; k 2 N requires that (i)

j tj ° tkj > jsj ° skj, (ii) tj +tk = sj +sk, and (iii) ti = si, for all i 2 Nnfj; kg.

14 In applying

this deÖnition, it is important to note that any proÖle can be arbitrarily reindexed without

materially changing its properties.

Definition 1. Capacity proÖle t is more concentrated than s by the transfer principle

(t ¬T s) if and only if s can be constructed from t by applying a Önite series of equalizing

transfers.

It is not di¢cult to see that if t1 = ¢¢¢ = tm, then there is no proÖle s such that t ¬T s.

Thus, the symmetric capacity proÖle is the least concentrated proÖle for any given total

industry capacity. The concentration ordering implied by the transfer principle is irreáexive,

transitive, and incomplete. Irreáexivity follows from the fact that an equalizing transfer is

deÖned with a strict inequality. Transitivity follows from the fact that the order is deÖned

in terms of a series of equalizing transfers. The transfer principle is incomplete because not

all proÖles can be ranked. This fact is illustrated in Example 4 below.

While our results on concentration will make use of the transfer principle, at times it is

useful to employ the following equivalent deÖnition.

Definition 2. Capacity proÖle t is more concentrated than s by second-order dominance

(t ¬SD s) if and only if for all m = 1;:::;n,

Pn

i=m t(i) 6 Pni=m s(i) with the inequality strict

for at least one m.

In words, t is more concentrated than s by second-order dominance if for all m = 1;:::;n,

the sum of the capacities of the (n°m)th smallest suppliers is not greater for t than for s.

15

In order to apply this ordering to capacity proÖles with a positive capacity for a di§erent

total number of suppliers, suppliers with zero capacity can be added to the shortest proÖle

1 4The Pigou-Dalton condition holds that inequality should increase when income is transferred from a

poorer individual to a richer individual. While stated di§erently, the Pigou-Dalton transfer condition implies

a concentration ordering that is equivalent to the transfer principle deÖned here. 1 5Shorrocks and Foster (1987) and Foster and Sen (1997) describe a number of concentration or inequality

concepts such as generalized Lorenz dominance that are equivalent to second-order dominance (also referred

to as second-order stochastic dominance).

11
so that the total number of suppliers in the two proÖles is equal. Note that second-order

dominance can order two proÖles even if the total capacity of each proÖle is di§erent. If t >^ s^,

then the capacity proÖle t cannot be more concentrated than s by second-order dominance.

However, s can be more concentrated than t by second-order dominance.

In practice, it is common to calculate and compare the shares of the largest Örms.16

Second-order dominance allows this alternative approach when the proÖles being compared

have the same total capacity. If t

^= ^s, then it is clear that for all m = 2;:::n,

Pn

i=m t(i) 6

Pn

i=m s(i) if and only if Pm°1i=1 t(i) > Pm°1i=1 s(i). Therefore, when t

^= ^s, t ¬SD s is equivalent

to the criteria that, for all m, the sum of the capacities of the m largest Örms for t is not

less than for s.

Finally, we deÖne the HerÖndahl-Hirschman Index (HHI), which the Justice Department

and the Federal Trade Commission use in the Merger Guidelines.17

Definition 3. Capacity proÖle t is more concentrated than s by the HerÖndahl-Hirschman

Index (t ¬H s) if and only if Pn

i=1 t

2

i > Pn

i=1 s2

i and t

^= ^s.

Note that a merger clearly results in a more concentrated industry under all three deÖnitions. The following result relates the three deÖnitions of concentration for arbitrary

capacity proÖles.

Proposition 1. For capacity proÖles t and s such that t

^= ^s,

(a) t ¬T s if and only if t ¬SD s

(b) If t ¬T s, then t ¬H s:

These results are known from the literatures on mergers and income inequality.18 Part (a)

of Proposition 1 is an equivalence result for second-order dominance and the transfer principle

when comparing market structures with equal total capacities. Part (b) of Proposition

1 states that whenever the transfer principle provides a ranking of capacity proÖles by

concentration, the HHI will provide the same ranking. However, the reverse implication

is not true. A capacity proÖle which is more concentrated by the HHI need not be more

concentrated by the transfer principle. Consider the following example.

Example 1. Suppose that t = (0:77; 0:18; 0:05) and s = (0:75; 0:25; 0:0). Since Pn

i=1 t

2

i =

0:6278 and Pn

i=1 s2

i = 0:625, t ¬H s. However, it is not true that t ¬T s. By Proposition 1,

1 6 It is typically easier to identify the large Örms in an industry and estimate their capacity or market share. 1 7See U.S. Department of Justice/Federal Trade Commission, Horizontal Merger Guidelines, April 2, 1992

(Revised: April 8, 1997). 1 8See for example Encaoua and Jacquemin (1980) and Shorrocks and Foster (1987).

12
t ¬T s only if for each m = f1; 2; 3g,

Pn

i=m ti 6 Pni=m si. However, for m = 3;

Pn

i=m ti =

0:05 > 0 = Pn

i=m si.

Although second-order dominance and the transfer principle deÖne the same ordering

when the total industry capacities are equal, each generates a di§erent computational method

for comparing two capacity proÖles. With the transfer principle, one would look for an

appropriate set of transfers of capacity. On the other hand, with second-order dominance,

one would compare the sums of capacity for subsets of the suppliers from the supplier with

the smallest capacity on up or the largest capacity on down. Depending on the two capacity

proÖles being compared, one method or the other may be easier to apply.

It is important to note that neither the transfer principle nor second-order dominance

can compare every pair of capacity proÖles even when the total industry capacity is Öxed.

However, we show in the following section that when two proÖles can be ordered, that

ordering unambiguously predicts how the buyerís expected proÖt and other variables of

interest will compare for the two proÖles.

4. The Effects of Concentration and Mergers on Buyers

In this section, we examine the e§ect of industry concentration on the expected price and

the welfare of buyers. The following lemma states that the aggregate expected proÖts of the

suppliers is higher for capacity proÖles that are more concentrated by the transfer principle.

This lemma makes the proof for many of the subsequent results follow quite easily.

Lemma 2. If t ¬T s, then for any r>c,

Pn

i=1 ¶i

(rjt) > Pn

i=1 ¶i

(rjs).

Proof in the Appendix.

The next proposition is the key result. It states that increases in concentration by the

transfer principle decrease the expected proÖt of the buyer and increase the expected price

when the reserve price is held constant. It also states that an increase in concentration

lowers the optimal reserve price.

Proposition 2. Suppose t ¬T s. Then for any r>c, (a) U(rjt) < U(rjs), (b) p(rjt) >

p(rjs), and (c) U(rjt) ° U(rjs) is decreasing in r. Further suppose that r§

t and r§

s are the

proÖt-maximizing reserve prices for the buyer under capacity proÖles t and s. Then (d)

U(r§

t jt) < U(r§

s js) and (e) r§

t 6 r§

s .

Proof. (a) Making use of (5), for any given r>c, we have

U(rjt)°U(rjs) = Xn

i=1

¶i(rjs) °Xn

i=1

¶i(rjt) < 0.

13
The inequality follows from Lemma 2.

(b) Similarly, making use of (4), for any given r>c, we have

p(rjt) ° p(rjs) =

Pn

i=1 ¶i

(rjt) ° Pn

i=1 ¶i

(rjs)

Prfz1 6 rg > 0:

(c) Note that,

@ [U(rjt) ° U(rjs)]

@r = @

£Pn

i=1 ¶i

(rjs) ° Pn

i=1 ¶i

(rjt)

§

@r

= °

Z Æ

0

@2¢(rja)

@r@a da

= °

Z Æ

0

n

[1 ° F(r)]t

^°tk°a ° [1 ° F(r)]t^°tj+a

o

ln[1 ° F(r)]da < 0:

(d) When the reserve price is adjusted to maximize the expected proÖt of the buyer,

U(r§

s js) > U(r§

t js) > U(r§t jt). The Örst inequality follows from the fact that r§

s is the

proÖt-maximizing reserve price when the buyer is faced with capacity proÖle s. The second

inequality follows from part (a) of the proposition.

(e) The result follows from part (c) of this proposition. Q.E.D.

Proposition 2 is a general result about the e§ect of concentration on the expected price

and the expected proÖts of the buyer. An immediate implication is that for a given number

of suppliers, a given total industry capacity, and a given reserve price, the expected price

paid by the buyer is minimized when all of the Örms have the same capacity.19

In the remainder of this section, we examine various merger scenarios and consider their

impact on concentration, the expected price, the expected proÖts of the buyer, and the

optimal reserve price.

Proposition 3. A merger of two or more suppliers results in a more concentrated market

by the transfer principle. Thus, when the reserve price is Öxed, a merger results in a higher

expected price and a lower expected proÖt to the buyer. When the reserve price is set to

maximize the expected proÖt of the buyer, a merger results in a lower reserve price20 and a

lower expected proÖt to the buyer.

Proof. Suppose that suppliers j and k merge. For an initial proÖle s, the resulting

capacity proÖle is can be deÖned as tj = sj + sk, tk = 0, and ti = si, for all i 2 Nnfj; kg.

1 9This result is also true in the asymmetric Nash-Cournot model. See Farrell and Shapiro (1990a) and

Werden (1991). 2 0Graham and Marshall (1987) and Mailath and Zemsky (1991) prove a similar result relating the optimal

reserve price. They consider e¢cient collusion among bidders of di§erent sizes. Both show that as the number

of colluding bidders rises the bid-taker sets a more aggressive reserve price.

14
It is straightforward to see that t ¬T s. The proof is completed by applying Proposition 2.

Q.E.D.

With a Öxed reserve price, buyers will obtain a lower expected proÖt because of the

higher expected price of the input even though there is no change in the probability of

purchase. The higher expected price arises because for cases in which the two subsidiaries

of the merged supplier have the two lowest costs among all the suppliers, the price then

becomes the third highest cost. By adjusting the reserve price optimally, the buyer may be

able to reduce the adverse e§ects of the merger, but the envelope theorem implies that the

merger reduces the expected proÖt of the buyer after the buyer adjusts the reserve price

optimally.21

When the buyer is allowed to set the reserve price optimally, the buyer possesses some

bargaining power as a monopsonist. A decrease in the reserve price is analogous to a monopsonist reducing his consumption to lower the price. When the reserve price is set optimally

by the buyer, the e§ect of a merger on the expected price is ambiguous. Higher levels of

concentration have a positive e§ect on the expected price, while the lower reserve price has

a negative e§ect on the expected price. The following two examples demonstrate that when

the reserve price is set optimally, expected price can rise or fall following a merger.

Example 2. Suppose that s = ( 1

2 ; 12 ) and F is the uniform distribution over the unit

interval. Now consider the merger of suppliers 1 and 2. Thus, t = (1; 0). When v = 1; the

optimal reserve prices are r§

s = 5=9 and r§t = 1=2, and the expected prices are p(r§s js) = 47=90

and p(r§

t jt)=1=2. Thus, expected price falls after the merger.

Example 3. Assume the same setup as in Example 2 except that v = 1:8. The optimal

reserve prices are r§

s = 1 and r§t = 9=10. Under capacity proÖle s, it is optimal to set

the reserve price greater than or equal to the upper bound on the support of the cost

distributions. The expected prices are p(r§

s js)=5=6 and p(r§t jt)=9=10. Thus, expected

price rises after the merger.

The fact that the expected price can fall after the merger should not be interpreted as an

increase in the buyerís welfare. Proposition 3 demonstrates that the expected proÖts of the

buyer decline after a merger even though the buyer optimally lowers the reserve price. The

buyer is using the reserve price to moderate his welfare reduction after the merger. Thus,

in auction markets with a reserve price, one cannot simply examine the price e§ect in order

to assess the welfare e§ects of mergers on buyers.22

2 1Thomas (1998) Önds some support for the conjecture that allowing two small Örms to merge may beneÖt

the buyer since they will be better able to compete with the larger Örm. Since Proposition 2 does not

distinguish between small and large Örms, this model clearly does not support such a result. 2 2 In the asymmetric Nash-Cournot model, mergers unambiguously increase the equilibrium price and reduce

consumer welfare.

15
We can also use the transfer principle deÖnition of market concentration to compare the

e§ects of di§erent mergers. The following proposition allows us to compare a variety of

mergers between di§ering suppliers in the industry.

Proposition 4. For any initial capacity proÖle t, if tj > tk and tg > th, then a merger

between suppliers j and g results in a more concentrated industry by the transfer principle

than a merger between suppliers k and h: Furthermore, a merger between Örms j and g

results in a higher expected price (when the reserve price is Öxed), a lower expected proÖt

to the buyer, and a lower optimal reserve price than a merger between suppliers k and h.

Proof. We Örst show that for any j; k; g 2 N such that tj > tk, a merger between suppliers

j and g results in a more concentrated market than a merger between suppliers k and g.

Let s1 denote the capacity proÖle after the merger of j and g, and let s2 denote the capacity

proÖle after the merger of k and g. SpeciÖcally, s1

g = s2g = 0, s1

j = tg+tj , s2

k = tg+tk, s1k = tk,

s2

j = tj, and for all i 2 Nnfj; kg, s1i = s2i = ti. Note that Pn

i=1 s1

i = Pn

i=1 s2

i . Therefore,

s1 ¬T s2 if

Ø

Ø

Ø

s1

j ° s1

k

Ø

Ø

Ø >

Ø

Ø

Ø

s2

j ° s2

k

Ø

Ø

Ø. This inequality is equivalent to jtg + tj ° tkj > jtg + tk ° tjj

which follows from tj > tk.

Now consider the capacity proÖles that result from the mergers of suppliers j and g and

suppliers k and h. For the case where tg = th, the proof is complete. Now suppose tg > th.

With an initial proÖle t, let s3 denote the capacity proÖle after the merger of Örms k and

h. Note that s2 ¬T s3 follows from the result established in the Örst part of the proof.

Therefore, by transitivity s1 ¬T s3. The proof is completed by applying Proposition 2.

Q.E.D.

Proposition 4 provides one straightforward conclusion for industries in which some suppliers are interested in acquiring other suppliers. First, if a supplier is shopping for another

supplier to acquire, the acquisition of a larger supplier will result in a more concentrated

industry than the acquisition of a smaller supplier. Conversely, if a supplier is for sale, the

acquisition of this supplier will result in a less concentrated industry if it is acquired by a

smaller rather than larger supplier.

In order to illustrate the implications of Proposition 4 consider four suppliers from a

capacity proÖle t such that t1 > t2 > t3 > t4. There are three possible merger comparisons

between two distinct pairs of suppliers: (a) a merger of suppliers 1 and 2 versus a merger

of suppliers 3 and 4, (b) a merger of suppliers 1 and 3 versus a merger of suppliers 2 and

4, and (c) a merger of suppliers 1 and 4 versus a merger of suppliers 2 and 3. According to

Proposition 4, we easily can rank the mergers in (a) and (b). That is, a merger of suppliers

1 and 2 results in a more concentrated market than a merger between suppliers 3 and 4, and

a merger of suppliers 1 and 3 results in a more concentrated market than a merger between

suppliers 2 and 4.

16
The comparison of the merger between suppliers 1 and 4 versus the merger between

suppliers 2 and 3 cannot be deduced from Proposition 4. However, with the additional

condition that t2 +t3 > t1 +t4, we can conclude that a merger of suppliers 2 and 3 results in

a more concentrated market than a merger of suppliers 1 and 4. These two mergers require

a comparison of the following two capacity proÖles (t2 + t3; t1; t4) and (t1 + t4; t2; t3). The

former is more concentrated than the latter under second-order dominance because t3 > t4

and t2 + t3 > t1 + t4.

If, on the other hand, t1 + t4 > t2 + t3, then neither merger results in a market that is

more concentrated than the other and either merger may result in a higher price than the

other. Consider the following example.

Example 4. Suppose that t = (9; 5; 3; 1). Now compare the merger of suppliers 1 and 4

with the merger of suppliers 2 and 3. Note that t1+t4 > t2 +t3. Hence, s1 = (10; 5; 3; 0) and

s2 = (9; 8; 1; 0). Neither s1 ¬T s2 nor s2 ¬T s1. This ambiguity follows from an application

of second-order dominance. To see this, note that s1

1 > s21 while s11 + s12 < s21 + s22.

Even though the merger of suppliers 1 and 4 creates a larger Örm than the merger

suppliers 2 and 3, it need not result in a lower expected proÖt for the buyer.23

5. The Incentive to Merge and Total Welfare

The incentive to merge and the e§ects of a merger on total welfare depend on how the reserve

price is a§ected. If the buyer sets the reserve price optimally given the capacity proÖle for

the industry, then a merger decision will presumably take into account this reaction by the

buyer. When the reserve price is Öxed, the incentive to merge is always present. However,

when the buyer sets the reserve price optimally, the negative e§ect on supplier proÖts from

the lower reserve price implies that it is possible for suppliers to have no incentive to merge.24

Two suppliers have an incentive to merge if their combined proÖts after the merger are

higher than the sum of their individual proÖts before the merger. For the Öxed reserve

price case, Lemma 2 implies that the total proÖts of all suppliers increases after a merger.

Inspection of (2) indicates that non-merging suppliers are una§ected by a merger because

2 3For cases where the transfer principle fails to order capacity proÖles, it may be tempting to use the

HHI. However, in such cases, the HHI cannot be trusted to predict correctly which proÖle is associated with

lower expected proÖts for the buyer. It is straightforward to construct examples where the HHI indicates

higher concentration but the buyerís expected proÖt increases. Using the proÖles in Example 1, t ¬H s,

even through a straightforward calculation shows that U(1jt) > U(1js), when v = 1 and F is the uniform

distribution on [0; 1]. 2 4Our presumption is that when the buyer is able to set the reserve price optimally, he sets the reserve

price to whatever level is optimal for the capacity proÖle that he faces. However, suppose that the buyer

announces to the market that any merger will be met with an adjustment of the reserve price to c. If the

suppliers believe that such a reaction would occur, no merger would take place because any merger would

result in zero proÖts for all suppliers.

17
neither their capacity nor total industry capacity is a§ected. Therefore, when the reserve

price is Öxed, the expected proÖts of merging suppliers must increase after the merger.

Total welfare W(rjt) can be written as

W(rjt) =31 ° [1 ° F(r)]t

^

¥

(v ° E [z1jz1 6 r]):

Total welfare does not depend directly on the allocation of market capacity among Örms, but

only on the total market capacity t

^and the reserve price r. This is a direct implication of our

assumption of constant economies of scale. Therefore, when the reserve price is constant, a

merger has no e§ect on total welfare. Hence, for the case of a constant reserve price, the

gains from a merger are simply a transfer of expected proÖts from the buyer to the merging

suppliers. The non-merging suppliers are una§ected.

For r < minfc; v 1 g, total welfare is increasing in the reserve price. Therefore, for the

case where the buyer can set the reserve price optimally, any increase in concentration that

reduces the reserve price, also reduces total welfare. The preceding discussion is summarized

by the following proposition.

Proposition 5. If the reserve price is una§ected by a merger, then there is always

an incentive to merge, but total welfare and the proÖts of the non-merging suppliers are

una§ected. However, if the buyer sets the reserve price optimally given the capacity proÖle

and if the optimal reserve price is initially below c1 or falls below c1 after the merger, then

mergers reduce total welfare and the proÖts of the non-merging suppliers.

The implication of Proposition 5 for the non-merging suppliers is interesting. If the

reserve price is constant, the non-merging suppliers are una§ected by the merger. However,

if the buyer sets the reserve price optimally, then the buyer reacts to the merger by lowering

the reserve price making the non-merging suppliers worse o§. They have the same expected

market share but lower expected proÖts because the probability of purchase by the buyer is

lower. The buyer lowers the reserve price to mitigate the increased concentration and this

harms the non-merging suppliers.25

When the buyer optimally sets the reserve price below c, there may or may not be an

incentive for suppliers to merge. Consider the following two examples. In Example 5, there

is an incentive to merge, while in Example 6 there is no incentive to merge.

2 5This result is di§erent from the asymmetric Nash-Cournot model. In that model, non-merging suppliers

obtain higher proÖts by expanding production in response to the higher market price after the merger. This

insight has been used in Antitrust cases to argue that non-merging suppliers cannot successfully challenge a

merger because they would have no damages. This auction model provides an explanation of why non-merging

suppliers, in addition to buyers, would be harmed by a merger.

18
Example 5. Suppose that s = ( 1

2 ; 14 ; 14 ) and F is the uniform distribution over the

unit interval. Now consider the merger of suppliers 2 and 3. Thus, t = ( 1

2 ; 12 ; 0). When

v = 1, the optimal reserve prices are r§

s = (25 ° 3

p5)=32 and r§

t = 5=9. The expected

proÖts for the merging suppliers, pre- and post-merger are ¶2(r§

s js)+¶3(r§s js) t 0:0671 and

¶2(r§

t jt) = 11=162 t 0:0679. Thus, suppliers 2 and 3 have an incentive to merge.

Example 6. Using the same scenarios as in Example 2, suppose that s = ( 1

2 ; 12 ) and F is

the uniform distribution over the unit interval. Now consider the merger of suppliers 1 and 2.

Thus, t = (1; 0). When v = 1, the optimal reserve prices are r§

s = 5=9 and r§t = 1=2, and the

expected proÖts of merging suppliers are ¶1(r§

s js)+¶2(r§s js) = 11=81 and ¶1(r§t jt)=1=8.

Thus, suppliers 1 and 2 have no incentive to merge.

These two examples provide some intuition about the incentive to merge. In Example

6, the negative e§ects of the lower reserve price fall completely on the merged supplier since

it is the only supplier in the market after the merger. On the other hand, in Example 5,

the negative e§ect of the lower reserve price on the suppliers in the market is shared by the

merged supplier and the non-merging supplier.

The surprising result of Example 6 is that merger to monopoly is not proÖtable for

a symmetric duopoly.26 In this auction model the buyerís optimal reserve price facing

a monopoly completely undermines the incentive of the duopolists to merge. Although

Examples 5 and 6 are calculated using the uniform distribution for F, they suggest that the

optimal reserve price of the buyer can undermine a merger wave.27 As such, the buyerís

ability to commit to a reserve price r cannot only mitigate his loss in expected proÖts for a

given industry structure, it can also maintain a ìstableî industry structure in which there

would be no incentive for mergers that would further concentrate the industry.

However, it is not always the case that merger to monopoly is unproÖtable. Consider

the following example.

Example 7. Maintain the same assumptions as in Example 6 except let v = 1:8. That

is, we consider the same scenario as in Example 3. The optimal reserve prices are r§

s = 1

2 6Using a models that are mathematically Thomas (1998) and McAfee (1994) also Önd that in some cases

there is no incentive for merger to monopoly. These results from the auction models are clearly at variance

with the symmetric or asymmetric Nash-Cournot models. The symmetric model of Salant, Switzer, and

Reynolds (1983) Önds an incentive for symmetric duopolists to merge even though there is no incentive for

oligopolists to merge. In Perry and Porter (1985), larger Örms are more likely to have an incentive to merge,

but asymmetric duopolists would always have an incentive to merge. All the gains from a merger to monopoly

by duopolists in these models would be internalized. 2 7Perry and Porter (1985) Önd cases such that if a merger between two Örms with the same capacity is

proÖtable, all subsequent mergers between Örms of the same size will also be proÖtable. Examples 4 and 5

illustrate that the opposite conclusion may arise in this auction model.

19
and r§

t = 9=10, and the expected proÖts of merging suppliers are ¶1(r§s js)+¶2(r§s js)=1=3

and ¶1(r§

t jt) = 81=200. Thus, suppliers 1 and 2 have an incentive to merge.

With a higher value for the input, the reserve price adjustment is not large enough to

eliminate the incentive to merge. In Example 6, we showed that under certain assumptions

the industry proÖle of ( 1

2 ; 12 ) is stable. In the following example we consider what other

industry proÖles are stable.

Example 8. As in Examples 5 and 6, let F be the uniform distribution over the unit

interval and v = 1. Starting from a symmetric industry structure, we search for the industry

structures that can result after all proÖtable merger opportunities have been exhausted.28

The following table gives the stable proÖles that can be reached from an initial symmetric

industry with n Örms each with a capacity of 1=n, considering all possible merger paths.

n Stable ProÖles

3 (1=3; 1=3; 1=3)

4 (1=2; 1=2)

5 (2=5; 2=5; 1=5)

6 (1=2; 1=2) and (1=3; 1=3; 1=3)

7 (4=7; 3=7)

8 (1=2; 1=2) and (3=8; 3=8; 1=4)

9 (5=9; 4=9) and (1=3; 1=3; 1=3)

10 (1=2; 1=2), (2=5; 2=5; 1=5), and (2=5; 3=10; 3=10)

The numerical calculations reported in Example 8 provide some interesting insights.

First, small suppliers have an incentive to merge but large Örms may not. Second, di§erent

stable capacity proÖles can arise from the same initial proÖle through a di§erent sequence

of mergers. Third, in the particular example considered, the buyerís optimal reserve price

eliminates the incentive for suppliers to merge once they approach half the capacity of the

industry. As v becomes larger than c, the buyer is more reluctant to set a reserve price

below c and reduce the probability of purchase. If v is su¢ciently large, the buyerís ability

to set the reserve price optimally does not even prevent merger to monopoly.

6. Conclusion

Using this auction model, we have found both similarities and di§erences from the traditional asymmetric Nash-Cournot model of mergers. Consider the simplest case in which the

2 8An e§ective but ine¢cient program written in Mathematica checks all possible merger paths reporting

the resulting stable proÖle when all proÖtable mergers have been exhausted along a particular path. Our

program is available upon request.

20
reserve price is constant. As in the traditional model, we Önd that the welfare of buyers

declines because mergers increase the expected price paid by the buyers. However, unlike the

traditional model, total welfare is unchanged because there is no ine¢ciency in production

and no change in the probability of purchase. The lowest cost supplier wins the contract

before and after the merger, but the merger results in a higher price to the buyer when the

merging suppliers are the two lowest cost suppliers. As a result, there is always an incentive for any merger. The non-merging suppliers are una§ected by the merger because they

receive the same expected price when they have the lowest cost.

The di§erences from the traditional asymmetric Nash-Cournot model are more accentuated when the buyer can optimally commit to a reserve price. As in the traditional model,

we Önd that the welfare of buyers declines with a merger. In addition, total welfare declines

because the buyer reduces the reserve price after the merger, thereby decreasing the probability of purchase. However, unlike the traditional model, the expected price paid by the

buyers need not increase. The buyerís optimal reserve price may o§set the price e§ect of

the merging suppliers. The non-merging suppliers are harmed by a merger. Even though

the non-merging suppliers have the same market share as before the merger, the probability

of purchase is now lower because the reserve price has fallen. In addition, the lower reserve

price may reduce the price received when a non-merging supplier wins the contract.

This auction model need not always result in an incentive to merge when the buyer

optimally reduces the reserve price. In the traditional model, an incentive to merge need

not arise because the non-merging suppliers expand production in response to the higher

price. Thus, an incentive to merge is most likely to exist for two large suppliers rather

than two small suppliers. In this auction model, the buyer dulls the incentive to merge by

reducing the reserve price. The suppliers with larger capacities may have no incentive to

merge because they bear most of the burden of the reduction in the reserve price. On the

other hand, suppliers with smaller capacities typically have an incentive to merge even when

larger Örms do not. The examples suggest that the optimal use of the reserve price can

create a stable oligopolistic industry structure for which no further mergers are proÖtable.

Unfortunately these stable industry structures may be relatively concentrated.

21
Appendix

Proof of Theorem 1. (If): It is easy to verify that the distribution function as deÖned in

the theorem satisÖes P1, P2, and P3.

(Only if): For arbitrary t1; t2 > 0, deÖne the proÖles t = (t1; t2) and s = (t1 +t2). Note s

contains only one supplier with a capacity of t1 +t2. Making use of these proÖles in equality

(1), Properties 1, 2, and 3 imply that for all t1; t2 > 0 and all c 2 [c; c],

G(cjt1 + t2)=1 ° [1 ° G(cjt1)][1 ° G(cjt2)] (6)

First we prove that for any c 2 [c; c], G(cj¢) is nondecreasing in its second argument. For

ti > tj > 0,

G(cjti) ° G(cjtj )=1 ° [1 ° G(cjti ° tj)][1 ° G(cjtj)] ° G(cjtj )

= G(cjti ° tj )[1 ° G(cjtj )] > 0:

The Örst equality in the expression above follows from (6) since G(cjti)=1 ° [1 ° G(cjti °

tj )][1 ° G(cjtj)].

In order to show that G(cjti) is di§erentiable in its second argument for all ti > 0, choose

an arbitrary (c; ti) 2 [c; c]£R++. There exists a tk > ti such that G(cj¢) is di§erentiable in its

second argument at tk. Such a tk exists since being monotonic, G(cj¢) is di§erentiable almost

everywhere. DeÖne tj > 0 such that tk = ti + tj. By choice of tk, G(cj¢) is di§erentiable in

its second argument at tk, and thus, G(cjti + tj) is di§erentiable in ti. Since equality (6)

with ti and tj replacing t1 and t2 holds for all ti; tj > 0, the fact that the left-hand side is

di§erentiable in ti implies that the right-hand side and, thus, G(cjti) is di§erentiable in ti.

Replacing t1 with ti and t2 with tj and di§erentiating (6), noting that @G(cjti+tj )=@ti =

@G(cjti + tj )=@tj, we have, for any c 2 [c; c),

@G(cjti)=@ti

[1 ° G(cjti)] = @G(cjtj )=@tj[1 ° G(cjtj )] :

By Property 1, the left-hand side of the equality does not depend on tj. Therefore, since the

equality holds for all ti; tj > 0, the ratio [@G(cjti)=@ti] = [1 ° G(cjti)] is constant with respect

to ti. Therefore, G(cj¢) must satisfy a di§erential equation of the form @G(cjti)=@ti =

[1 ° G(cjti)] k(c) where k(c) can depend on c but not on ti. This is a Örst-order linear

di§erential equation which has a unique solution. In fact, the solution can be written as

G(cjti)=1 ° æ(c) [1 ° F(c)]ti

where F(c)=1 ° e°k(c) and æ(c) is a constant term with respect to ti.

22
The fact that G(cj0) = 0 for all c 2 [c; c) follows immediately from expression (6).

Therefore, æ(c)=1. Since G is a distribution function with a support of [c; c], F must also

be a distribution function with a support of [c; c]. Q.E.D.

Proof of Lemma 1. (a) a straightforward calculation of the probability yield the result.

(b) The probability that all of supplier iís rivals have costs higher than ci is Q

j6=i[1 °

F(ci)]tj = [1 ° F(ci)]t

^°ti . Since each supplier submits a bid equal to its cost and the

lowest bidder wins, supplier iís ex ante probability of winning the auction is equal to the

probability that all of the other suppliers have a higher cost and its cost is below the reserve

price. Supplier iís ex ante probability of winning is equal to

Z r

c

[1 ° F(ci)]t

^°ti ti[1 ° F(ci)]ti°1f(ci)dci = ti

Z r

c

[1 ° F(ci)]t

^°1f(ci)dci

= ti

t

^

3

1 ° [1 ° F(r)]t

^

¥

.

Hence, supplier iís ex ante probability of winning conditional on some bidder winning is

ti

t

^

3

1 ° [1 ° F(r)]t

^

¥ 3Pn

j=1

tj

t

^

3

1 ° [1 ° F(r)]t

^

¥¥°1

= ti

t

^ . Q.E.D.

Proof of Lemma 2. Consider two capacity proÖles t and s such that for some j; k 2 N,

tj > sj > sk > tk, tj + tk = sj + sk, and ti = si, for all i 2 Nnfj; kg. Let Æ = tj ° sj . Recall

that capacity proÖles can be reindexed without materially a§ecting the market outcomes.

Notice that t ¬T s because s can be constructed from t by taking Æ > 0 capacity from

supplier j and giving it to supplier k. It is su¢cient to prove the lemma for t and s, because

any two proÖles that can be ordered by the transfer principle di§er by a Önite series of

transfers.

DeÖne ¢ such that

¢(rjÆ) = Xn

i=1

¶i(rjt) °Xn

i=1

¶i(rjs)

=

Z r

c

3

[1 ° F(c)]t

^°tj ° [1 ° F(c)]t^°tj+Æ + [1 ° F(c)]t^°tk ° [1 ° F(c)]t^°tk°Æ

¥

dc:

The second equality follows after making use of (2) and canceling terms. Note that ¢(rj0) =

0. Therefore, ¢(rjÆ) > 0 and, hence, the conclusion of the lemma follows if @¢(rja)=@a > 0,

for a 2 (0; Æ). Di§erentiating the expression for ¢ yields

@¢(rja)

@a =

Z r

c

n

[1 ° F(c)]t

^°tk°a ° [1 ° F(c)]t^°tj+a

o

ln[1 ° F(c)]dc.

The desired inequality follows as long as for a 2 (0; Æ), t

^° tk ° a > t

^°tj +a or equivalently,

23
tj ° tk ° 2a > 0. Notice that for a 2 (0; Æ),

tj ° tk ° 2a>tj ° tk ° 2Æ = sj ° sk > 0:

The Örst equality follows from the fact that Æ = tj ° sj = sk ° tk. Q.E.D.

24
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26