1

Buyer Power and Industry Structure

David E. Mills

Department of Economics

P.O. Box 400182

University of Virginia

Charlottesville, VA 22904-4182

434.924.3061 (phone)

434.924.7659 (fax)

mills@virginia.edu

February, 2010

Forthcoming in Review of Industrial Organization

Final publication available at www.springerlink.com

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Abstract

This paper investigates the exercise of market power by a large buyer who emerges via growth,

merger, or group purchasing. It explores the efficiency and redistributive effects of such an

event when a competitive fringe of small buyers remains in the market. Terms of trade,

including those for small buyers, depend on structural conditions on the supply side of the

market and the nature of interactions between the newly emerged dominant buyer and suppliers.

Predicted aggregate welfare effects have implications for antitrust.

Key Words

Antitrust, Buyer Power, Dominant Buyer, Waterbed Effect

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I. Introduction

Large, powerful buyers can be found in many markets. Examples include health plans, in

their dealings with physicians or hospitals; producers of food and beverage ingredients, in their

dealings with agricultural commodity growers; aircraft manufacturers, in their dealings with

aircraft engine manufacturers; and hospital group purchasing organizations, in their dealings with

the suppliers of health care products. When a large buyer exercises market power to influence

the terms of trade with its suppliers, the firm is said to possess “buyer power.” This term applies

to a variety of situations.1 A uniquely large buyer may emerge by growth or merger. Or the

large buyer may be a cooperative association of small buyers, such as a group purchasing

organization or even a buyer cartel.

The starting point for this paper’s analysis is an input or intermediate product market with

many small buyers who are not direct downstream competitors.2 Next a dominant buyer

emerges in the market because one of the buyers grows large in relation to the rest, or because a

significant number of buyers merge or form a group purchasing organization. The dominant

buyer exercises its market power to obtain favorable terms from sellers. But this conduct may

affect sellers’ terms to the remaining buyers. The dominant buyer’s effects on prices, quantities

and welfare depend on structural conditions on the supply side of the market and on the

interactions between the dominant buyer and suppliers.

If the good is produced by a perfectly competitive industry with infinitely elastic supply,

then a dominant buyer cannot influence the good’s price. But if industry supply is upward

1 Inderst and Shaffer (2008, p. 1612) define buyer power as “the ability of buyers to obtain advantageous terms of

trade from their suppliers”. Similarly, Noll (2005, p. 589) defines it as “the circumstance in which the demand side

of a market is sufficiently concentrated that buyers can exercise market power over sellers”. 2 The model also applies to retail distribution where buyers are retailers and where the dominant buyer does not

compete in the same geographic market as other retailers.

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sloping, the dominant buyer can induce a lower price by reducing its demands. In this scenario,

the low price that the dominant buyer forces on sellers becomes the market price, so the welfare

effect of the dominant buyer’s emergence is positive for all buyers. The welfare effect on

suppliers is negative, and the aggregate welfare effect is negative as well because unit sales

decrease. The dominant buyer’s emergence creates a dead weight loss.

Outcomes are different for buyers who purchase an input or intermediate product from a

monopolist or the supplier of a strongly differentiated good. This is because a dominant buyer

may negotiate terms of trade that do not apply to small buyers. In what follows, the dominant

buyer’s dealings with the seller are modeled as a Nash bargaining game. With this, the

emergence of a dominant buyer has no effect on small buyers if the seller has constant marginal

costs. But if the seller has increasing marginal costs, Nash bargaining creates a “waterbed

effect” on prices: The dominant buyer pays a lower price than if it were a passive price-taker and

the remaining price-taking buyers pay a higher price. In this scenario, the welfare effect of the

dominant buyer’s emergence is negative for non-dominant buyers, but aggregate welfare may

increase or decrease. The likelihood that a dominant buyer increases aggregate welfare is greater

where the supplier’s marginal costs do not increase rapidly and where the small buyer segment is

small.

The predicted effects on aggregate welfare have implications for antitrust. Buyers who

purchase inputs or intermediate products from suppliers in a competitive market should be

prohibited from merging or forming group purchasing organizations solely to create a dominant

buyer. If the input or intermediate product is sold by a monopolist, however, the formation of a

dominant buyer should not be prohibited in every instance. Leniency may be warranted if

market conditions portend a positive effect on aggregate welfare.

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II. Competitive Sellers

Consider a homogeneous good that is produced and sold in a perfectly competitive

industry. Let the industry’s increasing and continuously differentiable supply function be

y S( p ) for p p . There are many buyers, each of whose demand for the good is decreasing

and continuously differentiable on the interval (0, p ) , where p p . Aggregate demand

y D( p ) is decreasing and continuously differentiable on the same interval, so the competitive

price and output level c c( p , y )are uniquely determined by c c cy D( p ) S( p ) , with cy 0 and

cp ( p, p ) .

Now suppose that a group of small, independent buyers merge to form a single dominant

buyer whose demand x f ( p ) is decreasing and continuously differentiable on the interval

(0, p ) . In the alternative, the dominant buyer may be a group purchasing organization

consisting of independent buyers whose combined demand is f ( p ) .3 In either case, the

remaining independent buyers’ aggregate demand is y g( p ) , where g( p ) D( p ) f ( p ) .

These assumptions imply that cf ( p ) 0 and cg( p ) 0 .

If the dominant buyer is large enough in relation to the rest, it is plausible that the firm’s

size and prominence give it some influence over prices in the market. Indeed acquiring this

influence may be the raison d’être of the dominant buyer. A buyer such as this is unlikely to

exhibit the passive, price-taking behavior of atomistic buyers in a competitive market.

To focus on the immediate welfare effects of a dominant buyer, assume that this firm

does not compete directly with other buyers in a downstream market. (Whether non-dominant

3 For convenience, the dominant buyer will be referred to as a “firm” even if it is a group purchasing organization.

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buyers compete with each other downstream is immaterial.) This assumption applies to several

scenarios. The good may be an intermediate product where buyers are producers of unrelated or

strongly differentiated final goods. Or the good may be a locally produced intermediate product

that is purchased by local producers of a final good that is sold in a global market in which even

the dominant buyer has no market power. Finally, non-dominant buyers may be dealers or

retailers who distribute the good in different marketing channels or at different locations than

does the dominant buyer. In none of these scenarios is a small buyer’s demand for the good

dependent on the price paid by the dominant buyer.

This model is an exact demand-side analogy of the familiar “dominant firm” model that

has a uniquely large seller in a market with many small sellers. In that model, the dominant

seller chooses a price to maximize its own profit on the presumption that small, competing

sellers are passive price-takers (e.g., Stigler, 1965). This exercise of market power sustains a

price that is higher than the competitive price. In the dominant buyer case, a large buyer uses its

price-setting ability to impose a lower than competitive price (Blair and Harrison, 1993). The

dominant buyer’s ability to impose such a price stems from the firm’s ability to reduce

significantly the total quantity demanded at any price.4

To see how a dominant buyer affects outcomes in the industry, note that the residual

supply of the good available to the dominant buyer is x S( p ) g( p ) . This quantity is positive

4 This paper emphasizes the effects of group purchasing organizations that are formed to exercise monopsony

power. Other papers have emphasized the ability of group purchasing organizations to exploit nonlinear pricing or

vertical restraints. Marvel and Yang (2008, p. 1091) show that a buying group may form to elicit discounts from

competing suppliers and “extract attractive prices from suppliers not through enhanced bargaining power, but rather

through their ability to obtain [nonlinear] tariffs that pit suppliers against one another effectively”. In a model that is

an extension of O’Brien and Shaffer’s (1997) analysis of interactions between a monopsonist and a duopoly, Dana

(2006) shows that buying groups may organize in order to win discounts from duopolists by committing to purchase

exclusively from the supplier who offers the lower price.

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for any price greater than p , where cp ( p, p ) is defined by S( p ) g( p ) .5 For any price less

than or equal to p , the residual supply is zero because producers would not even supply enough

output to meet the demands of the small buyers. The dominant buyer’s residual supply function

is increasing and continuously differentiable on the interval ( p, p ) . By the inverse function

theorem, this function may be inverted to give a function p ( x ) that is increasing and

continuously differentiable on the interval (0,S( p )) . Accordingly, the dominant buyer may

purchase any quantity x (0,S( p )) of the good for a total cost of ( x ) x .

The dominant firm’s inverse demand function 1p f ( x ) is decreasing and continuously

differentiable on the interval (0, f (0 )) . The firm’s surplus from acquiring x units is

x

1

0f ( z )dz ( x ) x . To maximize this surplus, the dominant buyer purchases x* units, where

x* uniquely satisfies the first-order condition:

1( x*) '( x*) x* f ( x*) . (1)

The firm pays the price p* ( x*) for these units. At this price, the remaining buyers purchase

y* g( p*) units.

The main effects of the dominant buyer’s emergence are summarized in:6

Proposition 1: c c c cp* < p , x* < f(p ), y* > g(p ) and x* +y* < y .

The dominant buyer causes a price decrease by withholding purchases to support the lower price.

This means that the dominant buyer purchases fewer units than if the firm acted as a passive

price-taking buyer. The remaining buyers are passive beneficiaries of the dominant buyer’s

5 The existence of a unique p is assured by previous assumptions about and S( p ) g( p ) .

6 Proofs of the propositions are in the Appendix.

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market power because they buy more units of the good at a lower price than before the dominant

buyer emerged.

The welfare effect of an emerging dominant buyer is positive for every buyer in the

market. For producers, the welfare effect is negative because total output falls; they sell fewer

units at a lower price. The aggregate welfare effect is negative because the producers’ loss

exceeds the buyers’ gain. With perfectly competitive suppliers, the emergence of a dominant

buyer creates dead-weight loss in the market.

The predictions in Proposition 1 hinge on the assumption that the industry supply

function is increasing. The price effects are smaller the more elastic is S( p ) . If the industry

supply function is infinitely elastic, a dominant buyer would have no impact on market outcomes

because it could not force a lower price on sellers by purchasing fewer units. There would be no

“power buyer” incentive for buyers to merge or form group purchasing organizations if supply is

infinitely elastic. Profitably exercising market power on the buyers’ side of the market requires a

less-than-infinitely elastic industry supply.

III. A Single Seller

Now consider a good produced and sold by a monopolist. In the alternative, the good

may be produced and sold by a single firm in an industry with several firms whose products are

strongly differentiated. The seller’s cost function C( y ) is increasing and continuously

differentiable with for all C''( y ) 0 y 0 . Aggregate demand y D( p ) is the same as before,7

and may be inverted to give for all y1p D ( y ) (0,D(0 )) . The seller’s profit function is

7 If the good is a differentiated product, D( p ) depends on the prices of competing “brands,” although cross price

elasticities are assumed to be small. This dependence is suppressed in the notation.

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1( y ) D ( y ) y C( y ) . The firm’s profit-maximizing output levelmy is given by m'( y ) 0 ,

and its price by 1

m mp D ( y ) . Call the seller’s resulting profit m m( y ) .

As before, suppose that a group of buyers merge to form a single dominant buyer or

organize a group purchasing organization with demand x f ( p ) . The combined demand of the

remaining buyers is y g( p ) . An example of a dominant buyer and a single seller might be a

health plan (the buyer) and a regional hospital (the seller) where the health plan purchases

hospital services for a large fraction of the patients who reside in the region served exclusively

by the hospital. Another example might be an airline (the buyer) and an airport (the seller)

where there are no other nearby airports and where the airline accounts for a significant fraction

of the airport’s fees and lease payments for flight-related services. The emergence of a dominant

buyer in this situation has different effects on market outcomes than with competitive suppliers.

In particular, seller competition no longer prevents an outcome in which the dominant buyer

pays a different price than others.

The relationship between the dominant buyer and the seller resembles a bilateral

monopoly. Since this arrangement can be vulnerable to the double marginalization distortion,

the firms have strong incentives to reach mutually beneficial terms of trade to extract as much

surplus as possible from their transaction.8 Assume that the dominant buyer and the seller

negotiate a contract in a bilateral bargaining game instead of an arrangement where the buyer

chooses a quantity after the seller sets the price. In market structures where there is both a buyer

and a seller with market power, and where the firms contract in isolation under conditions of

8 Inderst and Shaffer (2008) distinguish market structures where firms interact via a “market interface” versus a

“bargaining interface.” In the former, impersonal market forces set prices. In the latter, prices are set in bilateral

bargaining between individual buyers and sellers.

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complete information, Whinston’s (2006, p. 139) “bilateral contracting principle” predicts that

the firms “will reach an agreement that maximizes their joint payoff”.9

In accordance with this principle, the dominant buyer and the seller negotiate a contract

that maximizes the sum of the dominant buyer’s surplus and the seller’s profit. The market

structure assumed here is distinguished from a bilateral monopoly because there is a competitive

fringe of small buyers. Although they are passive, the latent surplus of these buyers affects

bilateral negotiations between the dominant buyer and the seller. It is natural to assume that the

firms’ contract reflects endogenous effects on the seller’s profit from sales to its remaining

buyers. Let the terms of the contract between the firms be those predicted by the Nash

bargaining solution.

The dominant buyer and the seller negotiate a contract for x units of the good and a total

payment of T. This contract anticipates sales of y units to small buyers at the price 1g ( y ) . With

Nash bargaining, the output levels ˆ ˆ( x, y )are those that maximize:

x

1 1

0M( x, y ) f ( z )dz g ( y ) y C( x y ) . (2)

The division of ˆ ˆ( , )M x y between the dominant buyer and the seller is achieved by the dominant

buyer’s payment T . With Nash bargaining, T depends on each firm’s bargaining power and

their outside options: the firms’ payoffs should an agreement not be reached.

9 The insight that buyers and sellers with market power can achieve the vertically integrated solution is invoked by

Bernheim and Whinston (1998), Chipty and Snyder (1999), Mathewson and Winter (1987), Campbell (2007), and

Tirole (1988), among many others. Blair and Harrison (1993) discuss the idea’s antecedents in early economic

theory. Baker, Farrell, and Shapiro (2008) are less sanguine about the generality of the vertically integrated solution

because of obstacles raised by asymmetric information and incomplete contracts.

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Assume that if the firms failed to reach an agreement, the seller would charge the price

mp to all buyers and the dominant buyer would purchase mf ( p )units. With this, the seller’s

profit would bem and the dominant buyer’s surplus would be:

mx

1

m m m0

v f ( x )dx p x . (3)

Based on these outside options, the firms’ joint gain from bilateral bargaining is:

ˆ ˆ( , ) m mM x y v . (4)

Let (0,1) designate the bargaining power of the dominant buyer and1 the

bargaining power of the seller. The Nash bargaining solution predicts that the contract between

the firms fixes T so that the dominant buyer retains the fraction of expression (4) and the seller

gets the rest. This division implies that the seller’s profit , the dominant buyer’s surplus v , and

the payment T are jointly determined by:

y

x1

0

m m

ˆˆ ˆ ˆ ˆ ˆp y T C( x y )

ˆv f ( x )dx T

ˆ ˆ ˆT (1 ) ( v v )

. (5)

Under the contract ˆˆ( x,T ) , the dominant buyer pays the seller an effective price of

xˆˆ ˆp T x . The main effects of the dominant buyer’s emergence are:

Proposition 2: If

=C'' 0

>, then ˆ ˆ ˆ ˆ ˆ ˆ

x m y m m m

= =p < p p , x > f(p ), y g(p ) and x + y > y

< <.

The first result to notice is that the contract between the seller and the dominant buyer

calls for a lower price and more output than if the buyer had instead acted as a passive price-

taker. The second result to notice is that the dominant buyer’s emergence increases total unit

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sales in the market. The final result to notice is that the dominant buyer’s impact on small

buyers depends on the seller’s costs.

Small buyers are unaffected by the dominant buyer’s emergence if the seller has constant

marginal costs. But if the seller’s marginal costs are increasing, the dominant buyer creates a

waterbed effect on prices. That is, the price paid by small buyers increases while the dominant

buyer’s price decreases. By negotiating superior terms with the seller, the dominant buyer

secures a lower effective price thanmp . But this precipitates a higher price than

mp for small

buyers. In effect, the dominant buyer wrests a “discount” from the seller. But this discount

triggers a price increase for remaining small buyers because supplying more output to the

dominant buyer increases the incremental cost of supplying the rest.

The source of the waterbed effect in this model is the seller’s increasing marginal costs.

This pricing phenomenon arises for different reasons in some other models. Inderst and Valletti

(2006) identify a waterbed effect that stems from the fixed costs that buyers may bear if they

search for or switch to a backup source for the seller’s good, or if they integrate upstream to

produce the good themselves. In that model, a seller must offer a lower price to a large buyer

than to small buyers to induce the large buyer to forgo alternatives. Majumdar (2005)

demonstrates a waterbed effect in a model with two suppliers who have decreasing marginal

costs. In that model, a large buyer elicits a low price by inviting bids to become its sole supplier.

Once the large buyer selects a sole supplier, competition between the suppliers for sales to the

remaining buyers supports a higher price than the large buyer pays. Mathewson and Winter

(1996) show that if a group of independent buyers in a monopolistically competitive market form

a group purchasing organization to negotiate exclusive contracts with a subset of the suppliers,

they will pay lower prices than do outside buyers. Scale economies of one kind or another play a

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role in each of these models. But in the present analysis, the waterbed effect is caused by

diseconomies of scale.

The only price or quantity in Proposition 2 that depends on the relative bargaining power

of the seller and the dominant buyer is ˆxp . The quantities and ˆ ˆx y and the small buyers’ price yp

are not affected by a change in . The dominant buyer’s payment T varies inversely with , so

the effective price ˆxp paid by that buyer decreases as its bargaining power increases.

10 The

payment T is bounded above and below as follows: First,

m0

ˆ ˆlimT x p

, (6)

because v would be less than mv if the dominant buyer paid an effective price of mp for

mx f ( p ) units of the good. This would be unacceptable to the dominant buyer. Also,

1

1

ˆ ˆ ˆlimT x f ( x ),

(7)

because would be less than m if the dominant buyer paid an effective price of 1 ˆf ( x ) for x

units of the good. An effective price this low would award the entire incremental surplus created

by increasing x to the dominant buyer while reducing the seller’s profit from sales to the small

buyers. This would be unacceptable to the seller. These bounds on T mean that the dominant

buyer’s effective price is never as high as mp and never as low as 1 ˆf ( x ) .

While the welfare effect of a dominant buyer on small buyers is positive when there are

many sellers, the same does not hold when there is a single seller with market power. With a

single seller, the welfare effect on the seller and on those buyers who grow, merge, or organize is

positive; but the welfare effect on the remaining buyers is non-positive. This is because small

10

Inderst and Shaffer (2008) provide an extensive discussion of factors that affect the value of firms’ outside options

and comparative bargaining power in bilateral negotiations between buyers and sellers with market power.

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buyers cannot free ride on the advantageous terms negotiated by the dominant buyer when there

is a single seller. If the seller’s marginal costs are constant, the dominant buyer has no effect on

small buyers. In this instance the aggregate welfare effect of the dominant buyer is positive

because the contract between the dominant buyer and the seller reduces dead weight loss. But if

the seller’s marginal costs are increasing, the welfare effect on small buyers is negative because

of the waterbed effect on prices. Even if small buyers are losers when the dominant buyer

emerges, the aggregate welfare effect of the dominant buyer still may be positive. This occurs if

the small buyer segment is small relative to the dominant buyer,11

or if C''( y ) is sufficiently

small. Either condition would make the loss incurred by small buyers less than the gain jointly

captured by the dominant buyer and the seller.

Proposition 2 applies where the dominant buyer and the seller negotiate an efficient

contract. But most of the price and quantity relationships predicted in the Proposition apply

more generally. Suppose that there are obstacles to efficient contracting between the firms,

which causes them to strike a bargain ( x,T ) that is suboptimal in the sense that their joint payoff

is less than ˆ ˆ( , )M x y .12

Any such contract would be mutually beneficial to the dominant buyer

and the seller.

A mutually beneficial contract ( x,T )between the dominant buyer and the seller is one

that has:

and m m( x ) v( x ) v , with at least one strict inequality, (8)

where

11

At the limit, if there were no small buyers, this would be an instance of bilateral monopoly/monopsony, and any

negotiated outcome between the two parties must improve the welfare of both parties and thus improve aggregate

welfare. 12

See Baker, Farrell, and Shapiro (2008) for a discussion that challenges the feasibility of efficient contracting in

these kinds of commercial relationships.

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y

h( x ) arg max[ M( x, y )] , (9)

and where and ( x ),v( x ), T( x ) jointly satisfy:

1

x1

0

m m

( x ) g ( h( x ))h( x ) T( x ) C( x h( x ))

v( x ) f ( z )dz T( x )

T( x ) (1 ) ( ( x ) v( x ) v )

. (10)

Under such a contract, the dominant buyer pays an effective price xp T( x ) / x , and the

monopolist charges small buyers the price 1

yp g ( h( x )) .

Any mutually advantageous contract between the seller and an emergent dominant buyer

has the following effects:

Proposition 3: If

=C'' 0

>, then

x m y m m

= =p < p p , x > f(p ) and y g(p )

< <for any contract

(x,T(x)) that satisfies (8).

This Proposition indicates that if the seller has constant marginal costs, any mutually beneficial

bargain that a seller strikes with a dominant buyer leaves small buyers untouched. The aggregate

welfare effect of the dominant buyer’s emergence in this instance is positive because the contract

reduces dead weight loss. If the seller’s marginal costs are increasing, the Proposition indicates

that any mutually beneficial contract between the two firms brings on a higher price and fewer

unit sales for small buyers. The size of the waterbed effect, and the sign of the effect on

aggregate welfare, depends on the terms of the contract, the relative size of the small buyer

segment, and the magnitude of C''(y).

IV. Policy Implications

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In recent years, several Federal Trade Commission conferences and reports have studied

potential antitrust problems related to the purchasing practices of dominant firms in industries as

diverse as e-commerce, health care, and petroleum.13

There is general recognition that the

exercise of market power on the demand side of a market may create allocative inefficiency that

is similar to market power on the supply side. Because the effects of monopsony and monopoly

“are basically the same,” Noll (2005) argues that antitrust policy “should be symmetrical” (p.

623). The relevance of antitrust for monopsony was underlined recently by the Supreme Court’s

decision in Weyerhaeuser Co. v. Ross-Simmons Hardwood Lumber Co., Inc., 549 U.S. 312

(2007).14

The analysis in this paper suggests that the emergence of dominant buyers is problematic

in some circumstances but not all.15

Insofar as antitrust is guided by the goal of promoting

economic efficiency, input and intermediate product buyers should be prohibited from merging

or forming group purchasing organizations to achieve dominance when purchasing from

suppliers in a competitive market. But the correct prescription is less clear-cut when the

dominant buyer’s trading partner is a seller with market power. If the seller has constant

marginal costs, the emergence of a dominant buyer is socially beneficial and does not harm small

buyers. An event like this does not call for antitrust intervention. However, if the seller has

13

Noll (2005) summarizes these studies and reviews the implications of “buyer power” for antitrust policy. 14

In Weyerhaeuser the plaintiff (a small buyer) claimed that the defendant (a large buyer) bid prices up so as to

impose losses on the plaintiff; i.e., the behavior was allegedly predatory. In the present analysis, the dominant

buyer’s conduct reduces its own price but may raise the price that small buyers pay because of the seller’s

diseconomies of scale. 15

The notion that the “countervailing power” of a dominant buyer can subdue the market power of a large seller and

improve market performance has a long history, beginning with Galbraith (1952, 1954). Galbraith’s main

hypothesis was that the “countervailing power” of a dominant retailer lowers retail prices. Chen (2003), Dobson and

Waterson (1997), and von Ungen-Sternberg (1996) each have found specialized structural conditions and other

features of retail markets that support this hypothesis. The mechanisms and qualifications that they identify are

different, but all of them rely on vigorous competition among retailers to transfer benefits downstream to consumers.

The analysis in this paper indicates that countervailing power may pit the interests of one class of buyers against

another, whether or not it improves market performance.

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increasing marginal costs, the dominant buyer’s emergence may or may not be socially

beneficial. This is because the dominant buyer’s emergence creates a waterbed effect that

reduces the welfare of small buyers. If the effect on the small buyer segment is sufficiently

large, intervention to prohibit a merger or the formation of a group purchasing organization to

achieve dominance would be warranted. The two factors that raise a red flag for antitrust under

this scenario are if the seller has significant diseconomies of scale and if the size of the buyer

segment left to absorb the waterbed effect is relatively large.

These conclusions must be qualified by two considerations. The first qualification is that

some dominant buyers may not be created by growth or by the merger or group purchasing of a

large number of small buyers. Some may be created when two oligopsonists merge who already

are large enough to exert some influence over their terms of sale with the seller. With this kind

of merger, some of the gains unlocked by mutually advantageous contracts between a dominant

buyer and the seller may already have been captured. This reduces the incremental welfare

gains, if any, that would arise if the buyers merged.

Also, the analysis and conclusions in this paper do not account for distortions, if any, that

may arise if the dominant buyer competes downstream with the small buyers. Where this

happens, small competitors incur a cost disadvantage because of the waterbed effect. If there is

downstream competition (e.g., buyers are competing retailers), this disadvantage might create a

vicious cycle where upstream market power confers a downstream cost advantage that bolsters

downstream market power. This, in turn, amplifies the firm’s upstream market power, and so on

(Dobson and Inderst, 2008). In extreme cases, it is possible that this chain of events might

culminate in the exclusion of downstream rivals, or deterrence of downstream entrants.

Appendix

18

Proof of Proposition 1:

Recall that 1f ( x ) is decreasing on (0, f (0 )) . To show that * ( ) cx f p , it is sufficient to

show that the dominant firm’s incremental surplus would be negative if the firm purchased

cf ( p )units. When purchasing x units, the dominant firm’s incremental surplus is:

x1 1

0

df ( z )dz ( x ) x f ( x ) ( x ) '( x ) x

dx. (11)

Evaluating the r.h.s. of this expression at cx f ( p )gives:

c c c cp ( f ( p )) '( f ( p )) f ( p ) . (12)

Next, inverting both sides of c c cf ( p ) S( p ) g( p )gives:

c c( f ( p )) p . (13)

Substituting (13) into (12) simplifies (12) to c c'( f ( p )) f ( p ) , which is negative. This

establishes that * ( ) cx f p .

It follows from * ( ) cx f p that ( *) ( ( )) cx f p and hence that cp* p . Further,

because cp* p and g'( p ) 0 , we have cy* g( p*) g( p ) . Finally,

cS( p*) S( p ) , (14)

because cp* p and S'( p ) 0 . Inequality (14) implies that cx* y* y .■

19

Proof of Proposition 2:

Without a dominant buyer, those buyers who otherwise merge would purchase mf ( p )

units of the good, and the remaining buyers would purchase mg( p )units. These quantities

satisfy:

11 m

m m m m

11 m

m m m m

df ( f ( p ))f ( f ( p )) f ( p ) C'( f ( p ) g( p ))

dx

dg ( g( p ))g ( g( p )) g( p ) C'( f ( p ) g( p ))

dy

. (15)

The first order conditions for maximizing M( x, y )are:

and 1

1 1 ˆdg ( y )ˆ ˆ ˆ ˆ ˆ ˆ ˆf ( x ) C'( x y ) g ( y ) y C'( x y )

dy

. (16)

Suppose mˆ ˆx y y . Because C''( ) 0 , this relationship would imply:

mˆ ˆC'( x y ) C'( y ) . (17)

Because and 1 1f ( x ) g ( y ) are decreasing, and because m m my f ( p ) g( p ) , (15) - (17) imply

that mx f ( p ) and my g( p ) . This establishes a contradiction and shows that:

mˆ ˆx y y . (18)

Next, (18) implies that:

m m

=ˆ ˆC'( x y ) C'( f ( p ) g( p ))

>

if

=C'' 0

>

. (19)

Equations (15) and (16) together with (19) imply that:

m

=y g( p )

<

if =

C'' 0>

. (20)

Because g( ) is decreasing, (20) implies that:

20

ˆy m

=p p

>

if =

C'' 0>

. (21)

Together with (18), (20) also establishes that mx f ( p ) . Finally ˆx mp p because the dominant

buyer would preferm m( p , f ( p )) to ˆ( p,x )for any mp p .■

Proof of Proposition 3:

For (8) to hold, it is straightforward that mx f ( p ) . This inequality implies that x mp p

because otherwise mv( x ) v , which contradicts (8). For any x , (9) requires that y h( x ) must

satisfy:

1

1 dg ( y )g ( y ) y C'( x y )

dy

. (22)

With mx f ( p ) , equation (22) implies that m y m

= = =p p and y g(p ) if C'' 0

< < >

because

g( p ) is decreasing.■

21

Acknowledgements

The author thanks the General Editor, two referees, Simon Anderson, Federico Ciliberto,

Kenneth Elzinga, Maxim Engers, and Nathan Larson for helpful comments, and the Bankard

Fund for Political Economy at the University of Virginia for financial support.

22

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