Bargaining, Bundling, and Clout: The Portfolio Effects of

Horizontal Mergers

Daniel P. O’Brien1

and

Greg Shaffer2

December 2003

Abstract

This paper examines the output and profit effects of horizontal mergers between

upstream firms in intermediate-goods markets. We consider market settings in which

the upstream firms sell differentiated products to, and negotiate nonlinear supply contracts with, a downstream retail monopolist. If the merging firms can bundle their

products, transfer pricing is efficient before and after the merger. Absent cost savings,

consumer and total welfare do not change, but the merging firms extract more surplus.

If the merging firms cannot bundle their products, the effects of the merger depend on

the merged firm’s bargaining power. If the merged firm’s bargaining power is low, the

welfare effects are the same as with bundling; if its bargaining power is high, and there

are no offsetting cost savings, the merger typically reduces welfare. We evaluate the

profit effects of mergers on rival firms and the retailer for the case of two-part tariff

contracts. In this setting, a merger that harms rival firms and the retailer may still

reduce final-goods prices.

1Economist, U.S. Federal Trade Commission, 601 New Jersey Avenue, N.W., Washington, D.C., 20580, USA. The

views expressed herein are my own and do not purport to represent the views of the Federal Trade Commission or

any Commissioner. O’Brien can be reached at dobrien@ftc.gov or danobrien@cox.net.

2Professor of Economics and Management, William E. Simon Graduate School of Business, University of Rochester,

Rochester, N.Y. 14627, USA. Shaffer can be reached at shaffer@simon.rochester.edu.
I. Introduction

Merger policy in the industrialized countries is largely motivated by classical theories of oligopoly

whose roots trace to the theories developed by Cournot (1838) and Bertrand (1883).1 The Merger

Guidelines in the U.S., for example, rely heavily on modern variants of these theories, which predict

that a merger between competitors with market power can raise prices significantly unless the

merger generates offsetting efficiencies or attracts sufficient post-merger entry.2 Unfortunately,

however, the classical theories and their progeny generally make no distinction between final good

and intermediate-goods markets. The theories assume that firms set take-it or leave-it prices

that apply to all buyers, which is a reasonable assumption for most final-goods markets and some

intermediate-goods markets, but it is not descriptive of pricing in many intermediate-goods markets.

A common feature of pricing in manufacturing sectors is that contracts are negotiated with

individual downstream firms. For example, manufacturers of products sold through retail outlets

like supermarkets, convenience stores, and mass merchants often negotiate different contracts with

each distributor. Moreover, these contracts are a far cry from the simple, linear price set unilaterally

by firms in the classical theories. The payment schedules one observes in reality are often highly

nonlinear, with features like slotting fees, minimum quantity thresholds, and quantity discounts.

They may also involve variants of bundling, such as aggregate rebates and full-line forcing.

In this paper we incorporate nonlinear supply contracts, bargaining and bundling (defined as

inter-dependent price schedules) into a model of upstream competition to examine the effects of

horizontal mergers. We focus on the simplest market setting in which these factors are present: N ≥

2 upstream manufacturers and a single downstream retailer, with no uncertainty, no asymmetric

information, and no moral hazard. We find that the effects of horizontal mergers in this simple

environment differ substantially from the effects predicted by the classical theories of oligopoly.

1See the 1992 Horizontal Merger Guidelines, at http://www.ftc.gov/bc/docs/horizmer.htm. The unilateral effects

section of the Guidelines discusses markets in which the firms are distinguished by their capacities, and markets in

which they are distinguished by product differentiation. The former is motivated by Cournot’s model of oligopoly

among producers of homogeneous products, while the latter is motivated by Bertrand competition among producers

of differentiated products. The coordinated effects section of the Guidelines follows closely the ideas in Stigler’s (1964)

theory of oligopoly, which is understood today as a repeated game among Cournot or Bertrand competitors.

2See Deneckere and Davidson (1985), Farrell and Shapiro (1990), and the discussion in Willig (1991).

1
One key difference between an environment in which firms distribute their products through

a common retailer and an environment in which firms sell their products directly to consumers

is that, in the former environment, all pricing externalities can be internalized. As Bernheim

and Whinston (1985; 1998) and O’Brien and Shaffer (1997) have shown, when two single-product

firms make take-it-or-leave-it offers to a common retailer, the supply contracts yield the verticallyintegrated outcome (maximize overall joint profit). In this paper we show that this result also holds

under simultaneous Nash bargaining for the case of N ≥ 2 single-product firms, and for the case

of N firms selling single or multiple products if the multi-product firms can bundle their products.

This has surprising implications for merger policy. If bundling is feasible, it means that a merger

between two single-product firms that sell differentiated products to a common retailer need not

have any effect on input choices, output choices, wholesale prices, or final-goods prices. Thus, in

contrast to conventional wisdom, a merger between upstream competitors with significant market

power need not lead to higher prices even if there are no offsetting cost efficiencies or post-merger

entry. In this environment, the U.S. Merger Guidelines fail to offer much in the way of guidance.

Nor is it be possible to deduce appropriate public policy based on whether the buyer (retailer) feels

it would benefit or be harmed. The merging firms would have an incentive to merge even absent

cost savings because the combined entity would be able to extract more surplus from the retailer.

Although the retailer would oppose the merger, consumer and total welfare would be unchanged.

Things are more complicated if bundling is not feasible (or is prohibited). In that case, the

effect of a merger on final-goods prices, and hence on consumer and total welfare, depends on the

relative bargaining powers of the merged firm and the retailer, as measured by their bargaining

weights in an asymmetric Nash bargaining solution. If the merged firm’s bargaining power is low

enough, post-merger contracts are efficient and the welfare effects are the same as in the case with

bundling: welfare is higher if there are cost-efficiencies related to the merger, and otherwise there

is no change. But if the merged firm’s bargaining power is sufficiently high, the merged firm no

longer has an incentive to negotiate an efficient contract. This contrasts with what one might

expect from models of common agency. However, it has a simple, intuitive interpretation. When

2
bundling is not feasible, the distortion arises because the retailer can credibly threaten to drop one

of the merged firm’s products if the merged firm attempts to extract more than the incremental

surplus generated by each product. In this case, the merged firm does better negotiating higher

than efficient marginal transfer prices. Although this allows the firm to extract more surplus than it

would otherwise obtain (because it reduces the retailer’s profit from dropping one of its products),

it also tends to reduce total welfare. Unless there are offsetting efficiencies, the merger would

unambiguously lead to higher prices. Thus, when post-merger bundling is not feasible and the

merged firm’s bargaining power is sufficiently high, the qualitative tradeoffs of mergers predicted

by the classical theories of oligopoly re-emerge.

In addition to its implications for mergers, our model has implications for the on-going policy

debate over the use of “bundled discounts” by multi-product firms. Several high-profile cases in

Europe involving vitamins, tires, and soft drinks have hinged on the alleged use of these discounts

by larger firms to exclude smaller rivals.3 To satisfy the EC Commission, for example, Cola Cola

Export was forced to remove from its contracts with large distributors any rebates that were

conditioned on the buyer’s purchases of other beverages in addition to its purchases of Coca Cola.

Bundled discounts are sometimes viewed skeptically by antitrust authorities because they may raise

barriers to entry, allowing the larger firm to charge higher prices.4 However, in our setting bundled

discounts can lead to lower transfer prices if the bundling firm’s bargaining power with respect to

the retailer is high enough. Thus, a policy of prohibiting bundled discounts can lead to higher

final-goods prices and lower welfare if it fails to induce offsetting entry. Prohibiting such discounts

might also make mergers less profitable, preventing efficient mergers from taking place. A complete

assessment of the effects of bundled discounts requires balancing the potential static benefits we

identify against the potential dynamic effects on entry and exit.

The literature on mergers in intermediate-goods markets is surprisingly sparse. The seminal

3See Hoffman-La Roche v EC Commission [1979] European Commmission Report 461, at p. 547 (para. 111),

Michelin v EC Commission [1983] European Commission Report 3461, at pp. 3520-3522 (para. 92-99), and Coca

Cola Export Corporation, Filiale Italiana, Commission press release IP (88), 615 of 13 October 1988.

4Similar concerns have been expressed in the U.S. by Tom, Balto, and Averitt (2000). There are also private

litigation cases. Recently, a major supplier of hospital beds and burial caskets got sued for bundling discounts on

standard hospital beds to purchases of its specialty hospital beds. The parties agreed to an out-of-court settlement.

3
work in this area is Horn and Wolinsky (1988), who use the Nash bargaining solution to analyze

incentives for mergers in markets where competing downstream firms acquire inputs from independent suppliers, and in which they acquire inputs from a monopoly supplier. Horn and Wolinsky

differ from us in that their upstream suppliers do not compete, bargaining takes place over linear

prices only, and the inputs are assumed to be homogeneous. von Ungern-Sternberg (1996) and Dobson and Waterson (1997) also use the Nash bargaining solution to analyze the effects of mergers on

input prices. They too, however, restrict attention to linear prices and do not consider bundling.

The market structure they consider consists of a single upstream firm. Other papers in this area

look at different market structures, do not allow for bargaining, and do not consider bundling.5

Much of the literature on multiproduct pricing focuses on the use of bundling to extract surplus

from heterogeneous buyers (Adams and Yellen, 1976; McAfee, et al. 1989; Mathewson and Winter,

1997) or to leverage monopoly power across markets (Whinston, 1990; Choi and Stefanadis, 2001;

and Carlton and Waldman, 2002). In contrast, bundling is profitable in our model even when there

is a single buyer (the downstream retail monopolist) and no opportunity to leverage across markets.

Bundling also takes place with substitute goods in our model, in contrast to the well-studied cases

of bundling with independent or complementary goods. The closest paper to ours in the literature

on multiproduct pricing is Shaffer (1991), who considers bundling in a bilateral monopoly setting

with take-it-or-leave-it offers. However, his model does not allow for upstream rivalry, mergers, or

bargaining, nor does he consider the welfare implications of a policy prohibiting bundling.

The remainder of the paper is organized as follows. Section II presents the model and solves for

the pre-merger equilibrium. Section III solves for the post-merger equilibrium and discusses output

and welfare effects with and without bundling. Section IV examines the profit effects of mergers

and explores the effects of cost savings. Section V discusses extensions and concludes the paper.

5Ziss (1995) considers mergers-to-monopoly in a market setting with two manufacturer-retailer pairs. Colangelo

(1995) looks at pre-emptive mergers. Neither considers bundling because each firm sells only one product.

4
II. The Model and Pre-Merger Equilibria

N ≥ 2 upstream firms (manufacturers) each distribute a single differentiated product through a

downstream monopolist.6 Manufacturer i’s production cost is Ci(qi) ≥ 0, where Ci(0) = 0 and

qi ≥ 0 is the quantity it produces. The downstream firm (retailer) resells the manufacturers’

products to final consumers. Its net revenue from selling q ≡ (q1, q2, ..., qN ) units, after subtracting

all costs other than what it pays the manufacturers, is R(q). We assume that R(q) −

P

i Ci(qi) is

concave and differentiable for all qi > 0, and that it has a unique maximum at q

I ≡ (qI

1

, qI

2

, ..., qI

N ).

We refer to the quantity vector q

I and associated joint profit as “the fully-integrated outcome.”

Competitive interactions are modelled as a two-stage game. In stage one, the manufacturers

simultaneously negotiate contracts with the retailer over the price of their products. In stage two,

with all contracts in place, the retailer decides how much of each product to buy, and hence how

much to resell to final consumers. As in Horn and Wolinsky (1988), we assume the outcomes

of bargaining are determined by the set of simultaneous, asymmetric Nash bargaining solutions

between the retailer and each manufacturer. In the event a negotiation breaks down, each firm

in the negotiation earns its disagreement payoff. For the manufacturer, who has only the retailer

as a trading partner, we normalize this payoff to zero. In the case of the retailer, we assume its

disagreement payoff with each manufacturer is the profit it could earn without that manufacturer.7

Manufacturer i’s contract with the retailer is a function Ti(·), which specifies the retailer’s payment for any quantity qi purchased. We assume Ti(0) = 0 and Ti(qi) ≥ Ci(qi) for all qi

. Additional

restrictions on contracts are discussed as needed. This formulation permits most contractual forms

observed in practice, including linear wholesale prices, two-part tariffs, and quantity forcing. It does

not permit payments to depend on final-goods prices (resale price maintenance), or on the retailer’s

purchases of other manufacturers’ products (e.g., we do not allow exclusive dealing provisions).

6Many of our results go through with M > 2 retailers if the retailers do not observe each others’ contracts and

have passive beliefs (O’Brien and Shaffer, 1992; McAfee and Schwartz, 1994). This is discussed further in section V.

7The simultaneous Nash bargaining solution also arises in the literature on labor market negotiations, which

are analogous to negotiations between upstream and downstream firms. For example, Davidson (1988) considers

bargaining between a union and two employers, and Jun (1989) examines bargaining between two unions and one or

two employers. These papers show that the simultaneous Nash bargaining solution is equivalent to perfect equilibria

of natural alternating offer bargaining games in the limit as the time between offers goes to zero.

5
We now characterize the bargaining equilibrium. Given the vector of contracts T ≡

(T1(·), T2(·), ..., TN (·)), the retailer chooses quantities to maximize profits. Let Ω(T) be the set

of quantity vectors that maximize the retailer’s profit given the contract vector T. That is,

Ω(T) ≡ arg max

q

R(q) −

X

j

Tj (qj ).

In the first stage, the retailer and each manufacturer negotiate their contract recognizing that

the retailer will subsequently choose quantities from the set Ω(T). Let T−i denote the vector of

contracts of firm i’s rivals, e.g., T−1 ≡ (T2(·), T3(·), ..., TN (·)). Then, given T−i, we can define the

feasible set of quantity-contract combinations available to manufacturer i and the retailer as

Ai(T−i) ≡ {(qi, Ti(·)) | q ∈ Ω(T), Ti(0) = 0, Ti(qi) ≥ Ci(qi)}.

Thus, the Nash bargaining solution between manufacturer i and the retailer solves

max

(qi,Ti(·))∈Ai(T−i)

(πi − di)

αi

(πr − dri)

1−αi

(1)

where πi = Ti(qi) − Ci(qi) is manufacturer i’s profit; πr = R(q) −

P

j Tj (qj ) is the retailer’s

profit; di and driare the disagreement profits of manufacturer i and the retailer, respectively; and

αi ∈ [0, 1] is manufacturer i’s bargaining weight. As discussed previously, the disagreement profit

for manufacturer i is di = 0. The disagreement profit of the retailer with manufacturer i is

dri = max

q−i

R(0, q−i) −

X

j6=i

Tj (qj ),

where, for ease of exposition, we use R(0, q−i) in place of R(q1, ..., qi−1, 0, qi+1, ..., qN ).

A bargaining equilibrium is a set of quantities and contracts that solve (1) for each product i. We

refer to an equilibrium in which K products are sold as“K-product” equilibria, K ∈ {1, 2, ..., N}.

Characterization of equilibrium quantities and payoffs

Unfortunately, the maximization problem in (1) is not easy to work with because it involves the

choice of a function Ti(·) and hence it is not amenable to calculus. However, as we show below, we

can nevertheless characterize equilibrium quantities and payoffs by solving an equivalent problem

6
in which manufacturer i and the retailer choose a quantity-forcing contract with two parameters,

T

F

i

(qi) =







0 if qi = 0

Fiif qi = q

0

i

∞ otherwise

,

and quantity qi, from the feasible set of quantity-contract combinations

A

F

i

(T−i) ≡ {(qi, Fi, q0

i

) | q ∈ arg max

q

R(q) − T

F

i

(qi) −

X

j6=i

Tj (qj ), Fi ≥ Ci(q

0

i

)}.

With this restriction to quantity-forcing contracts, the maximization problem in (1) becomes

max

(qi,Fi,q0

i

)∈AF

i

(T−i)



T

F

i

(qi) − Ci(qi)

αi



R(q) − T

F

i

(qi) −

X

j6=i

Tj (qj ) − dri





1−αi

. (2)

We will henceforth assume the Nash product in (2) has a unique solution. Then Lemma 1 below

implies that the solution to (2) yields the same quantities and payoffs as any solution to (1).

Lemma 1 Suppose (ˆqi, Fˆ

i

, qˆ

0

i

) ∈ AF

i

(T∗

−i

) is the unique solution to the maximization problem in

(2) given the vector of rival contracts T∗

−i

. Suppose (q

∗

i

, T∗

i

(·)) ∈ Ai(T∗

−i

) is a solution to the

maximization problem in (1) given the vector of rival contracts T∗

−i

. Then qˆi = q

∗

i

and Fˆ

i = T

∗

i

(q

∗

i

).

Proof: Suppose (q

∗

i

, T∗

i

(·)) ∈ Ai(T∗

−i

) is a solution to (1) given the vector of contracts T∗

−i

, and let

q

∗ ∈ Ω(T∗

). Note that AF

i

(T−i) ⊂ Ai(T−i), and that the quantities q

∗ and payments T∗

j

(q

∗

j

) can be

obtained when quantity-contract combinations are restricted to the set AF

i

(T∗

−i

) by setting qi = q

∗

i

,

Fi = T

∗

i

(q

∗

i

), and q

0

i = q

∗

i

. Thus, (q

∗

i

, T∗

i

(q

∗

i

), q∗

i

) ∈ AF

i

(T∗

−i

) solves the maximization problem in (2)

for manufacturer i and the retailer, and the two maximization problems yield the same quantities

and payoffs for manufacturer i and the retailer, conditional on contracts T∗

−i

. Q.E.D.

Suppose (q

∗

, T∗) is a vector of quantities and contracts that form a bargaining equilibrium,

where q

∗ ≡ (q∗

1

, ..., q∗

N ) and T∗ ≡ (T

∗

1

, ..., T∗

N ). Then Lemma 1 implies that we can characterize the

equilibrium quantity and payoffs for manufacturer i and the retailer by solving the problem:8

max

(qi,Fi,q0

i

)∈AF

i

(T∗

−i

)

(Fi − Ci(qi))αi



R(q) − Fi −

X

j6=i

T

∗

j

(qj ) − dri





(1−αi)

= max

qi,Fi,q−i

(Fi − Ci(qi))αi



R(q) − Fi −

X

j6=i

T

∗

j

(qj ) − dri





(1−αi)

(3)

8Note that this does not imply that the forcing contracts that solve the restricted problem and the contracts T∗

−i

form a bargaining equilibrium because the best responses of rivals to the forcing contract T

F

i may differ from T

∗

−i

.

The appeal to forcing contracts is only a means to characterize the bargaining equilibrium quantities and profits.

7
such that

Fi ≥ Ci(qi), (4)

R(q) − Fi −

X

j6=i

T

∗

j

(qj ) ≥ dri, (5)

where constraints (4) and (5) ensure that manufacturer i and the retailer earn at least their disagreement profits. The equality in (3) follows because the constraint (qi

, Fi, q0

i

) ∈ AF

i

(T∗

−i

) requires

that q−i maximize R(q) −

P

j6=i T

∗

j

(qj ). Since Fi − Ci(qi) is independent of q−i, and Fi and driare

fixed when the retailer chooses q−i, this amounts to choosing q−i to maximize the Nash product.

The first-order conditions for Fi and qi at an interior solution of (3) are

αiπ

(αi−1)

i

(πr − dri)

(1−αi) − (1 − αi)π

αi

i

(πr − dri)

−αi = 0. (6)

−C

0

i

(qi)αiπ

(αi−1)

i

(πr − dri)

(1−αi) +

∂R(q)

∂qi

(1 − αi)π

αi

i

(πr − dri)

−αi = 0. (7)

Substituting (6) into (7) and simplifying yields

∂R(q)

∂qi

− C

0

i

(qi) = 0, (8)

which implies that q

∗

i maximizes the joint profit of manufacturer i and the retailer given T∗

−i

. Since

this must be true for all i, the bargaining equilibrium quantities must maximize overall joint profits,

i.e., q

∗

i = q

I

i

, provided (3) has an interior solution for each i. This proves the following proposition.

Proposition 1 All N-product bargaining equilibria replicate the fully-integrated outcome.

Proposition 1 extends to bargaining with N upstream firms the well-known result in the agency

literature that a common retailer internalizes all pricing externalities when manufacturers make

take-it-or-leave-it offers, and thus that marginal transfer pricing is efficient (Bernheim and Whinston, 1985). This has an intuitive interpretation. Fix the contracts of all manufacturers other than

i, and consider negotiations between the retailer and manufacturer i. Since nonlinear contracts

are feasible, they will choose their quantity to maximize their bilateral profits and divide the surplus with a non-distortional transfer. Note that choosing qi to maximize their bilateral profits,

R(q) −

P

j6=i T

∗

j

(qj ) − Ci(qi), is the same as choosing qi to maximize overall joint profit.

8
III. Post-merger Equilibria and Output Effects

Suppose manufacturers 1 and 2 merge. This alters negotiations in potentially three ways. First,

it affects the retailer’s disagreement profit with the merged firm. After the merger, the retailer’s

disagreement profit is the profit it would earn if it did not sell products 1 and 2. Second, it may

affect the retailer’s bargaining power. After the merger, the retailer’s bargaining weight in the Nash

bargaining solution with respect to the newly merged firm, αm ∈ [0, 1], may differ from what it was

with respect to each firm separately. Third, it affects the contracts the merged firm may be able

to negotiate. After the merger, contracts in which the payments for q1 and q2 are interdependent

may be feasible. We say that the merged firm engages in bundling if the payment for one of its

products depends on the amount purchased of the other. Formally, let Tm(q1, q2) be the merged

firm’s contract with the retailer, where q1 is the quantity the retailer purchases of product 1 and q2

is the quantity the retailer purchases of product 2. Then Tm(q1, q2) exhibits bundling if and only

if there does not exist T1(q1) and T2(q2) such that Tm(q1, q2) = T1(q1) + T2(q2) for all q1, q2 ≥ 0.

A. Mergers with Bundling

Let Cm(q1, q2) be the post-merger cost of producing q1 units of product 1 and q2 units of product 2.

We say that the merger generates cost efficiencies at quantities (q1, q2) if and only if Cm(q1, q2) <

C1(q1) + C2(q2). If there are no cost savings at (q1, q2), then Cm(q1, q2) = C1(q1) + C2(q2).

Let πm denote the merged firm’s post-merger profit. Then πm = Tm(q1, q2) − Cm(q1, q2). As

before, the retailer will choose quantities from the set Ω(T) in stage two, where T now equals

(Tm(·, ·), T3(·), ..., TN (·)). Thus, given the rivals’ contracts, T−1,2 ≡ (T3(·), ..., TN (·)), we can define

the feasible set of quantity-contract combinations available to the merged firm and retailer as

Am(T−1,2) ≡ {(q1, q2, Tm(·, ·)) | q ∈ Ω(T), Tm(0, 0) = 0, Tm(q1, q2) ≥ Cm(q1, q2)} .

The feasible set of quantity-contract combinations available to rival firm j is still Aj (T−j).

Suppose q

B ≡ (qB

1

, ..., qB

N ) and TB ≡ (T

B

m , T B

3

, ..., T B

N ) form a bargaining equilibrium when

bundling is feasible. Then the Nash bargaining solution between the merged firm and retailer

9
solves

max

(q1,q2,Tm(·,·))∈Am(TB

−1,2

)

(πm − dm)

αm (πr − drm)(1−αm)

, (9)

where dm and drm are the disagreement profits of the merged firm and retailer. The disagreement

profit of the merged firm is dm = 0. The disagreement profit of the retailer with the merged firm is

drm = max

q−1,2

R(0, 0, q−1,2) −

X

j6=1,2

Tj (qj ).

Characterization of equilibrium quantities and payoffs

We can use the same method that we used in the previous section to characterize equilibrium

quantities and payoffs. In particular, let T

F

m(·, ·) be a quantity-forcing contract with T

F

m(0, 0) = 0,

T

F

m(q

0

1

, q0

2

) = Fm, and T

F

m(q1, q2) = ∞ otherwise. Then, as we show in the appendix, we can

characterize the equilibrium quantities and payoffs for the merged firm and retailer by solving

max

q1,q2,Fm,q−1,2

(Fm − Cm(q1, q2))αm



R(q) − Fm −

X

j6=1,2

T

B

j

(qj ) − drm





(1−αm)

(10)

such that

Fm ≥ Cm(q1, q2), (11)

R(q) − Fm −

X

j6=1,2

T

B

j

(qj ) ≥ drm. (12)

where constraints (11) and (12) ensure that the merged firm and retailer earn at least their disagreement profits when q1, q2 > 0. The constraints that the retailer would rather choose q1, q2 > 0

than q1 = 0, q2 > 0, or q1 > 0, q2 = 0, are not binding because T

F

m(q1, q2) in these cases equals ∞.

The first-order conditions for Fm and qi at an interior solution of (10) are

αmπ

(αm−1)

m (πr − drm)

(1−αm) − (1 − αm)π

αm

m (πr − drm)

−αm = 0. (13)

−

∂Cm(q1, q2)

∂qi

αmπ

(αm−1)

m (πr − drm)

(1−αm) +

∂R(q)

∂qi

(1 − αm)π

αm

m (πr − drm)

−αm = 0. (14)

Substituting (13) into (14) and simplifying yields

∂R(q)

∂qi

−

∂Cm(q1, q2)

∂qi

= 0, (15)

10
which implies that q

B

i maximizes the joint profit of the merged firm and retailer given TB

−1,2

. Since

this must be true for i = 1, 2, and since rival manufacturers solve the same problem as before (prior

to the merger), it must be that the bargaining equilibrium quantities maximize overall joint profit,

i.e., q

B

j = q

I

j

, provided (3) and (10) have interior solutions. This proves the following proposition.

Proposition 2 Suppose manufacturers 1 and 2 merge. Then, if the merged firm can bundle its

products, the fully-integrated outcome is realized in all N-product bargaining equilibria.

Proposition 2 extends the common-agency result to the case of N firms selling one or more

products to a common retailer if the multi-product firms can bundle their products. It implies,

among other things, that a merger between upstream competitors with market power need not lead

to higher prices for consumers even if there are no offsetting cost efficiencies. In the absence of cost

efficiencies, we see from Propositions 1 and 2 that a merger between two upstream rivals will have

no effect on input choices, output choices, wholesale prices, or final-goods prices. These results are

surprising because they differ substantially from the effects of mergers that arise in the classical

theories of oligopoly. Perhaps the biggest difference is the diminished role of concentration indices

and cross elasticities in our model as predictors of the competitive effects of a merger. For example,

if bundling is feasible, a merger without cost savings is benign regardless of the degree of market

power in the upstream market or the degree of substitution between the merging firms’ products.

In these instances, the U.S. Merger guidelines fail to offer much in the way of guidance.

B. Mergers without Bundling

Suppose bundling is not feasible. Then the merged firm and retailer must negotiate a contract that is

additively separable in q1 and q2, i.e., Tm(q1, q2) = T1(q1)+T2(q2). Recall that T

F

i

(0) = 0, T

F

i

(q

0

i

) =

Fi, and T

F

i

(qi) = ∞ otherwise. Suppose q

NB ≡ (qNB

1

, ..., qNB

N ) and TNB ≡ (T

NB

1

, ..., T NB

N ) form

a bargaining equilibrium when bundling is infeasible. Then, as we show in the appendix, we can

characterize the equilibrium quantities and payoffs for the merged firm and retailer by solving

max

q1,q2,F1,F2,q−1,2

(F1 + F2 − Cm(q1, q2))αm



R(q) − F1 − F2 −

X

j6=1,2

T

NB

j

(qj ) − drm





(1−αm)

(16)

11
such that

F1 + F2 ≥ Cm(q1, q2), (17)

R(q) − F1 − F2 −

X

j6=1,2

T

NB

j

(qj ) ≥ drm, (18)

R(q) − F1 − F2 −

X

j6=1,2

T

NB

j

(qj ) ≥ max

q2,q−1,2

R(0, q2, q−1,2) − T

F

2

(q2) −

X

j6=1,2

T

NB

j

(qj ), (19)

R(q) − F1 − F2 −

X

j6=1,2

T

NB

j

(qj ) ≥ max

q1,q−1,2

R(q1, 0, q−1,2) − T

F

1

(q1) −

X

j6=1,2

T

NB

j

(qj ), (20)

where constraints (17) and (18) ensure that the merged firm and retailer earn at least their disagreement profits when q1, q2 > 0. Constraints (19) and (20) are individual rationality constraints

that ensure that the retailer earns weakly higher profit by choosing q1, q2 > 0 than by dropping

product 1 (constraint 19) or product 2 (constraint 20). The right-hand sides of (19) and (20) are

weakly larger than the right-hand side of (18). With bundling, these constraints are always satisfied

because in these cases T

F

m(·, ·) = ∞. Without bundling, however, these constraints may bind.

Lemma 2 There exists αm ∈ (0, 1) such that for all αm > αm constraints (19) and (20) bind.

Proof: See the appendix.

Lemma 2 says that if the manufacturer’s bargaining weight is sufficiently high, (19) and (20)

must bind in any N-product bargaining equilibrium. To see this intuitively, suppose the merged

firm had all the bargaining power (αm = 1). If constraints (19) or (20) did not bind, the merged

firm would raise one of the fixed fees to the point where the retailer earns its disagreement profit

drm = maxq−1,2 R(0, 0, q−1,2) −

P

j6=1,2 T

NB

j

(qj ). Since drm is weakly smaller than the right-hand

sides of (19) and (20), this contradicts the assumption that one of the constraints does not bind.

When the constraints do not bind (αm < αm), the problem in (16) is equivalent to the problem

in (10) with Fm = F1 + F2. In this case, bargaining without bundling yields the fully-integrated

outcome. When the constraints bind (αm > αm), the equilibrium quantities for the merged firm

and retailer can be found by substituting the binding constraints into (16). For convenience, define

v1(q1) = max

q−1,2

R(q1, 0, q−1,2) − F1 −

X

j6=1,2

T

NB

j

(qj ), (21)

12
v2(q2) = max

q−1,2

R(0, q2, q−1,2) − F2 −

X

j6=1,2

T

NB

j

(qj ). (22)

The function vi(qi) is the profit of the retailer if it purchases product i but drops product j.

Substituting these definitions into constraints (19) and (20), and then substituting the constraints

into the objective in (16), the merged firm and retailer’s maximization problem becomes

max

q1,q2



R(q) −

v1(q1) + v2(q2)

2

−

X

j6=1,2

T

NB

j

(qj ) − Cm(q1, q2)





αm 

v1(q1) + v2(q2)

2

− drm

(1−αm)

.

(23)

To gain insight into the solution, assume that T

NB

j

(qj ) is continuously differentiable (i.e., v1(q1)

and v2(q2) are differentiable). After some algebra, the first-order condition for q1 can be written as

∂R(q)

∂q1

−

∂Cm(q1, q2)

∂q1

=

1

2



1 −

(1 − αm)πm

αm(πr − drm)



∂v1(q1)

∂q1

. (24)

A symmetric condition holds for q2. The term on the left-hand side of (24) is the derivative of overall

joint profit with respect to q1. Since profits are concave and single-peaked, this derivative equals

zero at the fully-integrated outcome, exceeds zero if q

NB

1 < qI1

, and is less than zero if q

NB

1 > qI1

.

The term in square brackets on the right-hand side of (24) is positive. This is true because when

the constraints (19) and (20) bind, the merged firm receives less than its “fair share” of the profits

from Nash bargaining, i.e., (1 − αm)πm < αm(πr − drm). Therefore, the sign of ∂R(q)

∂q1

−

∂Cm(q1,q2)

∂q1

is the same as the sign of ∂v1(q1)

∂q1

. Using the envelope theorem, the derivative of v1(q1) is

∂v1(q1)

∂q1

=

∂R(q1, 0,  ̃q−1,2(q1))

∂q1

> 0. (25)

where  ̃q−1,2(q1) ≡ arg maxq−1,2 R(q1, 0, q−1,2)−

P

j6=1,2 T

NB(qj ). Therefore, at the bargaining equilibrium, ∂R(q)

∂q1

−

∂Cm(q1,q2)

∂q1

> 0, which implies that q

NB

1 < qI1

. An analogous argument implies that

q

NB

2 < qI2

. This proves Proposition 3 for the case in which T

NB

j

(qj ) is continuously differentiable.

Proposition 3 Suppose manufacturers 1 and 2 merge. If the merged firm cannot bundle its products, then whether or not the fully-integrated outcome is realized depends on αm: (i) if αm < αm,

the fully-integrated outcome is obtained in all N-product bargaining equilibria; (ii) if αm > αm, the

merged firm distorts its quantities downward and the fully-integrated outcome is not obtained.

13
Proof: See the appendix for the case in which T

NB

j

(qj ) is not continuously differentiable.

Proposition 3 contains the main result of the paper. It says that if the merged firm’s bargaining

weight vis a vis the retailer is sufficiently low, then the constraints (19) and (20) do not bind and the

incentives of the two firms are to maximize bilateral joint profit. However, if the merged firm has

a lot of bargaining power (αm > αm), then maximizing bilateral joint profit is not optimal because

the negotiated F1 and F2 will be constrained by the ability of the retailer to drop one or both of

the products. For example, if the merged firm attempts to extract ‘too much’ surplus by raising

F1, then the retailer can drop product 1 (constraint (19) is violated), and similarly, product 2 will

be dropped if the merged firm attempts to extract ‘too much’ surplus by raising F2 (constraint

(20) is violated). To relax these constraints, it is optimal for the merged firm to induce an upward

distortion in its input pricing (decrease its quantities) in order to decrease the retailer’s payoff. By

reducing the retailer’s quantity of product 2, for example, the retailer is harmed in the event it

sells products 1 and 2, but it would be harmed even more if it were to drop product 1 (because

products are substitutes). The former is a second-order effect while the latter is a first-order effect.

This result is surprising because it contrasts with the common intuition that overall joint profits

tend to be maximized in situations of common agency and complete information. We have shown

that this intuition does not necessarily extend to a negotiations setting in which the upstream firm

has sufficiently high bargaining power. In that case, the merged firm (or any multiproduct firm)

will find it optimal to knowingly reduce the overall profit pie because in doing so it can capture a

larger share for itself. With a larger share of a smaller pie, the manufacturer can gain.

Our results have implications for the output and welfare effects of mergers. They imply that

a merger without bundling either does not affect output (αm < αm) or causes the merged firms’

outputs to fall (αm > αm). In the former case the post-merger contracts are efficient and the

welfare effects are the same as in the case with bundling: welfare is higher if there are efficiencies

related to the merger, and otherwise there is no change. In the latter case, the merged firm no

longer has an incentive to negotiate an efficient contract, and welfare would typically fall. Because

the goods are substitutes, rival firms would respond by increasing their quantities, but typically

14
not by enough to offset the negative welfare effect of the reduction in the merged firms’ quantities.

Our results also have implications for policy toward bundled discounts. If the bargaining power

of the merged firm is high enough, prohibiting bundling leads to higher marginal transfer prices

for the merged firm’s products. Any attempt by authorities to prevent a multi-product firm from

increasing its “clout” through bundling may therefore result in higher prices for final consumers.

This finding suggests that antitrust concerns with bundling by dominant, multiproduct firms may

be misguided unless there is reason to believe that bundling has foreclosed, or is likely to foreclose

rivals. In our model bundling arises not to foreclose rivals but to extract rent from the retailer.

IV. Profit Effects

Expressions for equilibrium profits can be derived for each case by solving the restricted (quantityforcing) negotiations of each firm for its optimal fixed fee and then substituting back into the

expressions for profits. The resulting equilibrium profit expressions for the pre-merger case are

π

∗

i = αi



R(q

I

) − Ci(q

I

i

) −

X

j6=i

T

∗

(q

I

j

) − d

∗

ri



 , i = 1, ..., N (26)

π

∗

r = R(q

I

) −

X

i

Ci(q

I

i

) −

X

i

π

∗

i

(27)

d

∗

ri = max

q−i

R(0, q−i) −

X

j6=i

T

∗

j

(qj )

where π

∗

i

is manufacturer i’s equilibrium profit, π

∗

r

is the retailer’s equilibrium profit, and d

∗

ri

is the

retailer’s disagreement profit with manufacturer i under the equilibrium contracts T

∗

j

(·), ∀j 6= i.

Notice that manufacturer i’s profits are expressed in terms of rival firms’ equilibrium contracts

and not just in terms of the revenue and cost primitives. This is because the contracts are not

uniquely determined in equilibrium. The reason for this is that there are many different contracts

for product j that induce the retailer to select a given quantity q

∗

j

, and the disagreement profit of

the retailer in negotiations with firm i depend on the contract with firm j at quantities other than

q

∗

j

. Thus, firm i’s equilibrium profits depend on the type of equilibrium contracts employed by rival

firms j 6= i. The non-uniqueness of profits means that it is not possible to compare pre-merger

15
and post-merger profits at this level of generality. Further restrictions are needed to make this

comparison. In the remainder of this section we restrict attention to two-part tariff contracts, and

we assume that the manufacturers have constant marginal costs, i.e., C

00

i

(qi) = 0, i = 1, ..., N.

Before proceeding we need some more notation. Let w

I

i = C

0

i

(qi), i = 1, ..., N, be the constant

per-unit prices (wholesale prices) that yield the vertically-integrated outcome. Define

Π ≡ R(q

I

) −

X

i

w

I

i

q

I

i

, (28)

Π−i ≡ max

q−i

R(0, q−i) −

X

j6=i

w

I

j

qj , (29)

Π−1,2 ≡ max

q−1,2

R(0, 0, q−1,2) −

X

k6=1,2

w

I

k

qk. (30)

These are the retailer’s maximized profits (net of fixed fees) if it sells all N products, all but product

i, and all but products 1 and 2, respectively, when its marginal costs are given by w

I

i

, i = 1, ..., N.

A. Profit Effects with No Cost Savings

Using (26) and the definitions (28)-(30), manufacturer i’s pre-merger profit under two-part tariffs

is

π

∗

i = αi



R(q

I

) − Ci(q

I

i

) −

X

j6=i

w

I

j

q

I

j − Π−i



 . (31)

The profit of the merged firm when it bundles its products can be found by solving the first-order

conditions for (10) and then substituting them into the expression for profit. This gives

π

B

m = αm



R(q

I

) − Cm(q

I

1

, qI

2

) −

X

j6=1,2

w

I

j

q

I

j − Π−1,2



 . (32)

Suppose the merger does not increase the manufacturers’ bargaining power or affect their costs

(Cm(q1, q2) = C1(q1) + C2(q2)). Then the benefit to manufacturers 1 and 2 from merging is

∆π

B

m = π

B

m − π

∗

1 − π

∗

2

= αm ([Π − Π−1,2] − [Π − Π−1] − [Π − Π−2]). (33)

The term Π − Π−1,2 is the cost to the retailer (net of fixed fees) of failing to reach an agreement

with the merged firm. The term Π − Π−i, i ∈ {1, 2}, is the cost to the retailer of failing to reach

16
an agreement with manufacturer i prior to the merger. Equation (33) indicates that the merger

will be profitable if the expression in parenthesis is positive, i.e., if the retailer’s cost of failing to

reach an agreement with the merged firm is greater than the sum of the costs of failing to reach

agreement with of each of the merging firms prior to the merger. This is intuitive. A manufacturer’s

bargaining strength comes in part from its ability to inflict a loss on the retailer by refusing an

agreement. If the loss imposed by the merged firm exceeds the sum of the losses imposed by the

merging firms prior to the merger, then the merged firm will extract greater rents from the retailer.

In general, the concavity of joint profits ensures that this will be the case. Since the products are

substitutes, the loss imposed by the merged firm will indeed exceed the sum of the losses imposed

by the merging firms prior to the merger (see the proof of Proposition 4 below).9 Thus, we have

that ∆π

B

m > 0, implying that mergers are profitable for the merging firms when bundling is feasible.

Next we consider the profitability of a merger when bundling is infeasible. If the merged

firm’s bargaining weight is less than αm, then the constraints (19) and (20) do not bind and the

maximization problem in (16) is the same as the maximization problem in (10) with Fm = F1 +F2.

In this case, the merger is profitable and the profit of the merged firm is the same with or without

bundling. However, if αm > αm, then at the integrated quantities the merged firm is constrained

from capturing its share of the incremental profits from its products. That is, an unconstrained

Nash bargaining solution would require (1 − αm)πm = αm(πr − drm), but constraints (19) and (20)

force (1 − αm)πm < αm(πr − drm). This establishes an upper bound on πm. Since the wholesale

price of each non-merging firm is unchanged whether or not bundling is feasible, it follows that the

merged firm is worse off when αm > αm and bundling is infeasible than when bundling is feasible.

To determine whether the merger itself is profitable when bundling is infeasible and αm > αm,

let π

NB

m denote the profit of the merged firm in this case. Then, using the fact that the constraints

(19) and (20) will bind in any bargaining equilibrium, and that when αm > αm the merged firm

prefers to introduce a distortion by inducing the retailer to choose the vector of quantities q

NB in

9

Inderst and Wey (2003;14) also obtain this result, in a different context, using the Shapley value as their solution

concept to multilateral bargaining. As they note, “Broadly speaking, a merger shifts bargaining away from the margin.

If the created net surplus is smaller at the margin ... the respective market side prefers to become integrated.”

17
equilibrium rather than q

I

, it can be shown that (see the proof of Proposition 4 below)

π

NB

m >

X

i=1,2



R(q

I

) −

X

j6=i

w

I

j

q

I

j − Π−i



 − Cm(q

I

1

, qI

2

). (34)

Assuming, as before, that the merger does not affect relative bargaining weights or the merged

firm’s costs (Cm(q1, q2) = C1(q1) + C2(q2)), the benefit to manufacturers 1 and 2 from merging is

∆π

NB

m = π

NB

m − π

∗

1 − π

∗

2

>

X

i=1,2

(1 − αi)



R(q

I

) − Ci(q

I

i

) −

X

j6=i

w

I

j

q

I

j − Π−i



 , (35)

which is positive if pre-merger profits are positive and αi < 1. Intuitively, the merger is profitable

even when the manufacturer is constrained for two reasons. First, the merged firm’s fixed fees rise

to the point where constraints (19) and (20) bind, whereas they do not bind prior to the merger

unless αi = 1. Second, the merged firm earns additional profit by reducing its output of each

product (raising its wholesale price) in order to capture more profit from selling the other product.

We summarize these results for the bundling and no-bundling cases in the following proposition.

Proposition 4 A merger between manufacturers 1 and 2 is profitable whether or not bundling

is feasible, even if there are no cost-savings from the merger and no increase in their collective

bargaining weight. If αm < αm, then the merged firm’s profits are the same with and without

bundling. If αm > αm, then the merged firm’s profit is higher with bundling than without bundling.

Proof: See the appendix.

The result that mergers are always profitable in our model even if there are no cost savings

contrasts with the results in the standard models of horizontal mergers in final-goods markets where

the profitability of a merger often turns on whether the firms’ strategies are strategic substitutes or

strategic complements. In the latter case, we know from Deneckere and Davidson (1985) and others

that mergers of any size are profitable because, in addition to the usual gains from coordination,

they induce less aggressive pricing by the non-merging firms. In the former case, however, we know

from Salant et al. (1983) and others that, in the absence of cost savings, mergers may not be

18
profitable because they induce rival firms to respond by increasing their outputs. In our model,

mergers are profitable even without cost savings because (a) they allow the merging firms to impose

losses on the retailer by jointly withholding their products, and (b) the non-merging firms’ wholesale

prices do not change pre and post merger (that is, the rival firms do not respond aggessively).

Effects on the Non-merging Firms’ Profits

Next, consider the effects of a merger on the non-merging firms’ profits. The nature of the bargaining

problem between these firms and the retailer does not change after the merger. A non-merging

firm’s profit in any regime has the same structure as (31) but is evaluated at the wholesale prices

and quantities corresponding to the particular regime. In all the efficient regimes—pre-merger,

post-merger with bundling, and post-merger with no bundling and αm < αm—the wholesale prices

and quantities are the same. A non-merging firm’s profits are the same across these regimes.

In the post-merger regime with no bundling and αm > αm, the merged firm reduces its outputs

by raising its wholesale prices above the fully-integrated prices. To see the effect of this on rival

firms, let q

e

(w1, w2) ≡ (q

e

1

(w1, w2), ..., qe

N (w1, w2)) denote the vector of bargaining equilibrium

quantities, where q

e

j

(w1, w2) is the bargaining equilibrium quantity of product j when the merged

firm has wholesale prices w1 and w2. Then we can write a non-merging firm i’s profit as

πi = αi(R(q

e

(w1, w2)) − Ci(q

e

i

(w1, w2))

− w1q

e

1

(w1, w2) − w2q

e

2

(w1, w2) −

X

j6=1,2,i

w

I

j

q

e

j

(w1, w2)

− max

q−i

[R(0, q−i) − w1q1 − w2q2 −

X

j6=1,2,i

w

I

j

qj ]), (36)

where we have substituted w

I

j

for wj , j 6= 1, 2, i, because the wholesale prices of each non-merging

firm equals marginal cost in any bargaining equilibrium. Let  ̃qj (w1, w2), for all j 6= i, solve the

maximization problem in the third line of (36). Then, using the envelope theorem, we have

∂πi

∂w1

= αi( ̃q1(w1, w2) − q

e

1

(w1, w2)). (37)

Since  ̃qj (w1, w2) − q

e

j

(w1, w2) > 0 for all j when the products are substitutes, it follows that a nonmerging firm i’s profit will be increasing in w1. A symmetric argument implies that a non-merging

19
firm i’s profit will be increasing in w2. Since w1 and w2 both rise after the merger if αm > αm and

bundling is infeasible, the non-merging firms will benefit from the merger in this case.

Proposition 5 A merger between manufacturers 1 and 2 that generates no cost savings has no

effect on the non-merging firms’ profits if bundling is feasible or if the merged firm’s bargaining

power is sufficiently low (αm < αm). However, the same merger raises the non-merging firms’

profits if bundling is infeasible and the merged firm’s bargaining power is high (αm > αm).

Proposition 5 implies that the non-merging firms will oppose bundling by a merged firm when

the merged firm has a lot of ‘clout’ vis a vis the retailer. However, we know from Proposition 3

that prohibiting bundling in this case leads to higher wholesale prices (lower outputs) and thus

such opposition does not necessarily coincide with the interests of social welfare. When bundling

is allowed, Proposition 5 implies that the non-merging firms will be indifferent to the merger.

Effects on the Retailer’s Profit

Lastly, consider the effects of the merger on the retailer when there are no cost savings. Proposition

4 implies that the merging firms always benefit from the merger, and Proposition 5 implies that

the non-merging firms either benefit or are not affected. Since overall joint profits are maximized

pre-merger, it follows that the retailer is harmed by the merger if there are no cost savings.

We can also rank the retailer’s preferences over bundling versus no bundling in the post-merger

regime. The two cases can be nested by writing (19) and (20) for the no-bundling regime as

πr ≥ max

q2,q−1,2

R(0, q2, q−1,2) − T

F

2

(q1) −

X

j6=1,2

T

NB

j

(qj ) − b, (38)

πr ≥ max

q1,q−1,2

R(q1, 0, q−1,2) − T

F

1

(q1) −

X

j6=1,2

T

NB

j

(qj ) − b. (39)

In constraints (38) and (39), b is a parameter that represents the tightness of the constraints. The

no-bundling case occurs when b = 0. The bundling case occurs when b is large enough that the

constraints (38) and (39) do not bind. The effects of prohibiting bundling on the retailer can be

found by evaluating the derivative of πr with respect to b starting from the value of b at which the

20
constraints just begin to bind.10 As we show in the appendix, we find that the derivative is negative

at this point, implying that the retailer is typically worse off under bundling than it is with no

bundling. Mathematically, tightening the no-bundling constraint (decreasing b) has a first-order

positive effect on the retailer’s profits, as shown in (38) and (39). It also has a second-order effect

that comes through equilibrium adjustments in wholesale prices and quantities as the bundling

constraint is tightened. However, the second-order effects are outweighed by the first-order effects.

Proposition 6 A merger between manufacturers 1 and 2 reduces the retailer’s profit if there are

no cost savings from the merger. If αm < αm, then the retailer’s profit is the same with and without

bundling. If αm > αm, then the retailer’s profit is lower with bundling than without bundling.

Proof: See the appendix.

As in the previous case with the non-merging firms, we would expect the retailer to oppose

bundling by a merged firm when the latter has a lot of ‘clout’ vis a vis the retailer. However, as

we have seen, such opposition does not necessarily coincide with the interests of social welfare.

Whether or not bundling is allowed, Proposition 6 implies that we should expect the retailer to

oppose any merger that does not generate cost savings. But this opposition is driven solely by the

retailer’s desire to preserve its profit; such a merger need not have any social welfare consequences.

B. Profit Effects with Cost Savings

To investigate the effects of cost savings from a merger between manufacturers 1 and 2, we rewrite

the merged firm’s costs as Cm(q1, q2; θ), where θ is a cost-shift parameter on the merged firm’s

products. Assume ∂Cm(q1,q2;θ)

∂θ > 0, ∂

2Cm(q1,q2;θ)

∂qi∂θ ≥ 0, and ∂

2Cm(q1,q2;θ)

∂q2

i

= 0. An increase in θ

increases the total cost and may increase the marginal cost of producing product i.

In the regimes where joint profits are maximized, the merged firm’s profit can be written as

πm = αm



R(q

I

) − Cm(q

I

1

, qI

2

; θ) −

X

j6=1,2

w

I

j

q

I

j − max

q−1,2

[R(0, 0, q−1,2) −

X

j6=1,2

w

I

j

qj ]



 (40)

10Note that either both constraints bind or neither constraint binds. If only one constraint was binding, the merged

firm could adjust F1 and F2 to raise F1 + F2 and at the same time relax the binding constraint.

21
Differentiating the expression in (40) with respect to θ and using the envelope theorem gives

∂πm

∂θ = −αm

∂Cm(q

I

1

, qI

2

; θ)

∂θ (41)

Since ∂Cm(q1,q2;θ)

∂θ > 0 by assumption, condition (41) implies that ∂πm∂θ < 0. Thus the merged firm

benefits from cost savings, whether the cost savings are fixed or marginal. The retailer will also

benefit in this case because Nash bargaining will allow it to share in the cost savings.

The effects of the merged firm’s cost savings on the non-merging firms’ profits depend on whether

the cost savings reduce fixed or marginal costs. Marginal-cost savings will have the same effect on

profits as a reduction in w1, as expressed in (37), and therefore will reduce the non-merging firms’

profits. Fixed cost savings, on the other hand, do not affect the non-merging firms’ profits.

It can be shown that cost savings in the no-bundling regime with αm > αm have the same

qualitative effects. The retailer benefits from fixed and marginal cost savings; non-merging firms

are harmed when the savings reduce the merged firm’s marginal costs and are not affected otherwise.

Proposition 7 Suppose the merger between manufacturers 1 and 2 reduces its costs. Fixed-cost reductions benefit the merged firm and the retailer and do not affect the non-merging firms. Marginalcost reductions benefit the merged firm and the retailer and harm the non-merging firms.

In this model, the merged firm’s outputs increase if and only if its marginal costs decrease. It

follows that the merger harms the non-merging firms if and only if it reduces the merging firms’

marginal costs. Since the merged firm extracts greater rents from the retailer, the merger will

still harm the retailer if the cost savings are small. Thus, in this model the preferences of the

non-merging firms are an indicator of the output effects of a merger, while the preferences of the

retailer may be misleading.

V. Conclusion

A phrase often used by business people in explaining the reasons for a merger is that it will increase

the merged firm’s ‘clout’ in negotiations with buyers or suppliers. As such, antitrust investigations

22
of upstream mergers in intermediate-goods markets often focus on the effects of the merger on the

combined entity’s bargaining strength vis a vis the customer, and whether the customer will be

harmed as a result.11 In this paper, additional clout may come from three sources: bargaining

power, as measured by a firm’s bargaining weight in its asymmetric Nash bargaining solution; the

ability to negotiate contracts on products jointly rather than separately; and the ability to bundle

products via interdependent price schedules, for example, by offering discounts and rebates that

are applied ‘across-the-board.’

We find that the merging firms may benefit from an increase in their clout even if it does not

affect their marginal transfer prices. Absent cost savings, for the reasons noted above, mergers

with bundling increase the merging firms’ profits, decrease the retailer’s profit, and leave rivals’

profits unchanged. The profit effects are purely rent transfers; there is no effect on consumer or

total welfare. In contrast, with cost savings, mergers with bundling increase the merging firms’

profits and decrease rival firms’ profits. If the cost savings are small, the retailer’s share of the cost

savings will be less than the rent transfer to the merged firm and the retailer’s profit will decrease.

Otherwise, if the cost savings are large enough, the retailer will gain. Thus, it is possible for the

merger to increase welfare while harming both rival firms and the retailer, or just rival firms. Basing

policy on the retailer’s perceived benefit or harm is uninspired and uninformative in this case.

When bundling is not feasible, the increase in the merging firms’ profits can be attributed to the

benefits of negotiating terms on products jointly rather than separately. Because the products are

substitutes, the loss imposed on the retailer by the merged firm if it withholds both products exceeds

the sum of the losses that can be imposed by each merging firm prior to the merger. However, if the

merged firm’s bargaining power is high enough to induce it to raise wholesale prices, the benefits of

merging are smaller than they are when bundling is allowed. Rival firms benefit if transfer prices

increase, but the retailer is always worse off than with no merger, and worse off with bundling than

without it after the merger if the merged firm’s bargaining power is sufficiently high.

We view this paper as a first step toward understanding the effects of mergers in intermediate11See the European Commission Decision in Case No COMP/M.2220–General Electric/Honeywell, 2001, and In

the Matter of Pepsi, Inc./Quaker Oats Company, Federal Trade Commission File No 0110059, August 1, 2001.

23
goods markets when contracts are negotiated. The model is too simple at this point to be definitive

for policy conclusions. However, it is rich enough to show that the effects of mergers in this

environment can be substantially different than the effects predicted by classical oligopoly models.

A simplification in this paper is the restriction to a single downstream firm. Under this assumption, equilibrium contracts are efficient (in the sense of replicating the fully-integrated outcome)

before and after the merger except when bundling is prohibited and the merging firm’s bargaining

power is sufficiently high. This result has strong implications for the effects of mergers. If bundling

is allowed, so that contracts are efficient before and after the merger, the merger increases the

merged firm’s output if and only if it reduces marginal costs. This result is independent of the

degree of market power in the upstream market and the degree of substitution among upstream

products. The merger also increases the merging firms’ clout in negotiations with the retailer by

increasing the combined loss the merging firms can impose by refusing to sell. An implication is

that a merger with small cost savings enhances welfare even though it reduces the profits of rival

firms and the retailer.

The obvious next step is to extend the model to an environment with downstream oligopoly.

Once there is downstream competition, the rents to be split by a manufacturer and retailer will

depend inter-alia on the amount of competition the retailer faces from rival retailers who sell the

same product. In this case, contracts generally will not lead to the vertically-integrated outcome.

An additional complication is that the nature of the equilibrium will depend on whether downstream

firms can observe each others’ contracts. If contracts are not observable, it can be shown that perunit transfer prices will still equal marginal cost in a bargaining equilibrium. In this case, many of

the results in this paper carry through. However, the implication that per-unit transfer prices equal

marginal cost does not appear to be consistent with pricing in many intermediate-goods markets in

which non-linear contracts are negotiated. If contracts are observable, then firms have incentives

to negotiate contracts that dampen competition so as to increase the size of the total surplus to be

split.12 This generally leads to per-unit transfer prices that exceed marginal cost. The analysis of

12One factor that tends to make contracts more observable is the Robinson-Patman Act, which constrains the

24
mergers when contracts are observable among rivals is more complicated and awaits further work.

Another extension would be to allow for non-contractible investments by upstream or downstream firms. The need for ongoing, non-contractible investments in marketing or quality is another

reason for upstream firms to earn positive economic margins, as one often observes in practice.

ability of manufacturers to price discriminate. See O’Brien and Shaffer (1994).

25
Appendix

Characterization of equilibrium quantities and payoffs with bundling

To characterize equilibrium quantities and payoffs, we solve an equivalent problem to the one in

(9). In the equivalent problem, the merged firm and retailer choose a quantity-forcing contract

T

F

m(q1, q2) =







0 if q1 = q2 = 0

Fm if q1 = q

0

1

and q2 = q

0

2

∞ otherwise

,

and quantities q1 and q2, from the feasible set of quantity-contract combinations

A

F

m(T−1,2) ≡ {(q1, q2, Fm, q01

, q0

2

) | q ∈ arg max

q

R(q) − T

F

m(q1, q2) −

X

j6=1,2

Tj (qj ), Fm ≥ Cm(q

0

1

, q0

2

)}.

With this restriction to quantity-forcing contracts, the maximization problem in (9) becomes

max

(q1,q2,Fm,q0

1

,q0

2

)∈AFm(TB

−1,2

)



T

F

m(q1, q2) − Cm(q1, q2)

αi



R(q) − T

F

m(q1, q2) −

X

j6=1,2

Tj (qj ) − drm





1−αi

.

(A.1)

We will henceforth assume that the Nash product in (A.1) has a unique solution. Then, since

AF

m(T−1,2) ⊂ Am(T−1,2), and the choices q

0

1 = q

B

1

, q

0

2 = q

B

2

, and Fm = T

B

m (q

B

1

, qB

2

) are feasible

when the merged firm and retailer choose a quantity-forcing contract, it follows that the solution to

(A.1) yields the same quantities and payoffs for the merged firm and retailer as the solution to the

problem in (9), conditional on contracts TB

−1,2

. This means that we can characterize the equilibrium

quantities and payoffs for the merged firm and retailer by solving the restricted problem:

max

(q1,q2,Fm,q0

1

,q0

2

)∈AFm(TB

−1,2

)

(Fm − Cm(q1, q2))αm



R(q) − Fm −

X

j6=1,2

T

B

j

(qj ) − drm





(1−αm)

= max

q1,q2,Fm,q−1,2

(Fm − Cm(q1, q2))αm



R(q) − Fm −

X

j6=1,2

T

B

j

(qj ) − drm





(1−αm)

(A.2)

such that

Fm ≥ Cm(q1, q2),

R(q) − Fm −

X

j6=1,2

T

B

j

(qj ) ≥ drm,

26
which correspond to (10)–(12), respectively. The rest follows from the discussion in the text.

Characterization of equilibrium quantities and payoffs without bundling

When bundling is not feasible, the merged firm and retailer must negotiate a contract that is

additively separable in q1 and q2: Tm(q1, q2) = T1(q1) + T2(q2). In this case, we define the feasible

set of quantity-contract combinations available to the merged firm and retailer as

Aˆm(T−1,2) ≡ {(q1, q2, T1(·), T2(·)) | q ∈ Ω(T), T1(0) + T2(0) = 0, T1(q1) + T2(q2) ≥ Cm(q1, q2)} .

The feasible set of quantity-contract combinations available to rival firm j is still Aj (T−j).

Suppose (q

NB, TNB) form a bargaining equilibrium when bundling is infeasible. Then the Nash

bargaining solution between the merged firm and retailer solves

max

(q1,q2,T1(·),T2(·))∈Aˆm(TNB

−1,2

)

(πm − dm)

αm(πr − drm)(1−αm)

. (A.3)

Following the technique used to characterize the equilibrium in the bundling case, suppose that

the merged firm and retailer restrict attention to a pair of quantity-forcing contracts in the set

AˆF

m(T−1,2) ≡ {(q1, q2, F1, F2, q01

, q0

2

) | q ∈ arg max

q

R(q) −

X

i=1,2

T

F

i

(qi) −

X

j6=1,2

Tj (qj ),

F1 + F2 ≥ Cm(q

0

1

, q0

2

)}.

With this restriction to quantity-forcing contracts, the maximization problem in (A.3) becomes

max

(q1,q2,F1,F2,q0

1

,q0

2

)∈AˆFm(TNB

−1,2

)





X

i=1,2

T

F

i

(qi) − Cm(q1, q2)





αi 

R(q) −

X

i=1,2

T

F

i

(qi) −

X

j6=1,2

Tj (qj ) − drm





1−αi

.

(A.4)

We will henceforth assume that the Nash product in (A.4) has a unique solution. Then, since

AˆF

m(T−1,2) ⊂ Aˆm(T−1,2), and the choices q

0

1 = q

NB

1

, q

0

2 = q

NB

2

, F1 = T1(q

NB

1

) and F2 = T2(q

NB

2

)

are feasible when the merged firm and retailer choose quantity-forcing contracts, it follows that

the solution to (A.4) yields the same quantities and payoffs for the merged firm and retailer as

the solution to (A.3), conditional on contracts TNB

−1,2

. This means that we can characterize the

27
equilibrium quantities and payoffs for the merged firm and retailer by solving the restricted problem:

max

(q1,q2,F1,F2,q0

1

,q0

2

)∈AˆFm(TNB

−1,2

)

(F1 + F2 − Cm(q1, q2))αm



R(q) − F1 − F2 −

X

j6=1,2

T

NB

j

(qj ) − drm





(1−αm)

= max

q1,q2,F1,F2,q−1,2

(F1 + F2 − Cm(q1, q2))αm



R(q) − F1 − F2 −

X

j6=1,2

T

NB

j

(qj ) − drm





(1−αm)

(A.5)

such that

F1 + F2 ≥ Cm(q1, q2),

R(q) − F1 − F2 −

X

j6=1,2

T

NB

j

(qj ) ≥ drm,

R(q) − F1 − F2 −

X

j6=1,2

T

NB

j

(qj ) ≥ max

(q2,q−1,2)

R(0, q2, q−1,2) − T

F

2

(q2) −

X

j6=1,2

T

NB

j

(qj ),

R(q) − F1 − F2 −

X

j6=1,2

T

NB

j

(qj ) ≥ max

(q1,q−1,2)

R(q1, 0, q−1,2) − T

F

1

(q1) −

X

j6=1,2

T

NB

j

(qj ),

which correspond to (16)–(20), respectively. The rest follows from the discussion in the text.

Proof of Lemma 2: Suppose (19) or (20) does not bind. Without loss of generality, let (19) be

the non-binding constraint. Then the merged firm and retailer will negotiate F1 to maximize the

objective in (16). After some algebra, the first-order condition for F1 can be written as

F1 + F2 = αm(R(q) −

X

j6=1,2

T

NB

j

(qj ) − drm) + (1 − αm)Cm(q1, q2). (A.6)

Substituting (A.6) into the expression for the retailer’s profit gives

πr = R(q) − F1 − F2 −

X

j6=1,2

T

NB

j

(qj )

= (1 − αm)(R(q) − Cm(q1, q2) −

X

j6=1,2

T

NB

j

(qj )) + αmdrm. (A.7)

Note that

lim αm→1πr = drm (A.8)

Since (19) does not bind by assumption, condition (A.8) implies that for sufficiently large αm,

drm = max

q−1,2

R(0, 0, q−1,2) −

X

j6=1,2

T

NB(qj )

28
> max

q2,q−1,2

R(0, q2, q−1,2) − T

F

2

(q2) −

X

j6=1,2

T

NB

j

(qj )

≥ max

q−1,2

R(0, 0, q−1,2) − T

F

2

(0) −

X

j6=1,2

T

NB(qj )

= max

q−1,2

R(0, 0, q−1,2) −

X

j6=1,2

T

NB(qj ),

which is a contradiction. Q.E.D.

Proof of Proposition 3: Assume that T

NB

j

(qj ) is differentiable almost everywhere. For convenience, we repeat conditions (21) and (22):

v1(q1) = max

q−1,2

R(q1, 0, q−1,2) − F1 −

X

j6=1,2

T

NB(qj ), (A.9)

v2(q2) = max

q−1,2

R(0, q2, q−1,2) − F2 −

X

j6=1,2

T

NB(qj ). (A.10)

Assume that the solutions to the maximization problems in (A.9) and (A.10) are unique.

The strategy of the proof is to show that even if the non-merging firms’ contracts are not

differentiable, the functions vi(·), i ∈ {1, 2}, are differentiable at the equilibrium quantities and, in

particular, condition (25) holds. The analysis presented in the text then establishes Proposition 3.

Let q

1

−1,2

(q1) solve the maximization problem in (A.9) and q

2

−1,2

(q2) solve the maximization

problem in (A.10). Define

r

1

j

(q1, q−1,j) ≡ arg max

qj

R(q1, 0, qj , q−1,2,j) − F1 −

X

j6=1,2

T

NB(qj ), j ∈ {3, 4, ..., N}. (A.11)

The solution to the maximization problem in (A.9) can be characterized as the simultaneous solution

to the N − 2 “sub-maximization” problems in (A.11). Define r

2

j

(q2, q−2,j), j 6= 1, 2, symmetrically

as the quantities that solve the analogous sub-maximization problems that correspond to (A.10).

Step 1. If T

NB

j

(q

NB

j

) is discontinuous in either direction from q

NB

j

, it must jump upward; otherwise

the retailer could increase its profits by choosing a different quantity.

Step 2. Suppose T

NB

j

(q

NB

j

) jumps up to both the right and the left of q

NB

j

. Consider an arbitrarily small change in q

NB

1

to q

NB

1 + x. This will have an arbitrarily small effect on the

29
marginal revenue of product j, so the solution to firm j’s maximization problem in (A.11) will

not change. That is, r

1

j

(q

NB

1

, q−1,j) = r

1

j

(q

NB

1 + x, q−1,j) provided that x is small. Similarly,

r

2

j

(q

NB

1

, q−2,j) = r

2

j

(q

NB

1 + x, q−2,j) for small x.

Step 3. Suppose T

NB

j

(qj ) jumps up to the right of q

NB

j

, but is continuous to the left. At the

solution to (A.11), it must be true that

"

∂R(q

NB

1

, 0, r1

j

, q

NB

−1,2,j

)

∂qj

−

∂T NB

j

(r

1

j

)

∂qj

#

−

≥ 0 (A.12)

where the notation [ ]− indicates the left-hand derivative. Suppose the inequality in (A.12) is

strict. Consider an arbitrarily small change in q1 to q

NB

1 + x. Since the marginal revenue

function is continuous, the inequality in (A.12) will still hold at r

1

j

(q

NB

1 + x, q

NB

−1,j

). Therefore,

r

1

j

(q

NB

1

, q

NB

−1,j

) = r

1

j

(q

NB

1 + x, q

NB

−1,j

). Suppose that (A.12) holds with equality. This means that

the first-order condition holds for movements of qj in the leftward direction. Movements in the

rightward direction will not occur given small changes in marginal revenue because T

NB

j

jumps

upward in that direction. Analogous conditions hold for r

2

j

(q

NB

1 + x, q

NB

−2,j

).

Step 4. Suppose T

NB

j

(qj ) jumps up to the left of q

NB

j

, but is continuous to the right. At the

solution to (A.9), it must be true that

"

∂R(q

NB

1

, 0, r1

j

, q

NB

−1,2,j

)

∂qj

−

∂T NB

j

(r

1

j

)

∂q1

j

#

+

≤ 0 (A.13)

where [ ]+ denotes the right hand derivative. Suppose the inequality in (A.13) is strict. By the same

argument as in the preceding paragraph, a small change in q1 to q

NB

1 + x will leave r

1

j unchanged,

i.e., r

1

j

(q

NB

1

, q

NB

−1,j

) = r

1

j

(q

NB

1 + x, q

NB

−1,j

). Suppose that (A.13) holds with equality. This means

that the first-order condition holds for movements of qj in the rightward direction. Movements in

the leftward direction will not occur given small changes in marginal revenue because T

NB

j

jumps

upward in that direction. Analogous conditions hold for r

2

j

(q

NB

1 + x, q

NB

−2,j

).

30
Step 5. Steps 1-4 establish how the solution to each product’s sub-maximization problem

changes in response to small changes in q1 starting at the equilibrium quantity q

NB

1

. In particular,

product j’s quantity either does not change or it changes to satisfy its first order condition. We

now establish that this is true for the solutions to the maximization problems in (A.9) and (A.10).

The solution to the problem in (A.9) is given by the simultaneous solution to the N − 2 submaximization problems in (A.11). For a given change in q1 to q

NB

1 + x, let S be the subset of

products for which the solution to the product’s sub-maximization problem changes according to

its first-order condition. By the implicit-function theorem, the simultaneous solution to the submaximization problems for products in S (holding constant the quantities of products not in S) are

continuous functions of q1 on the interval (q

NB

1

, qNB

1 +x). This means that a small change x results

in a small change in these quantities, and hence a small change in the marginal revenues of the

other products whose sub-maximization solutions do not change in response to changes in q1. Since

the change in marginal revenue from all the adjustments for products in S is small, the quantities

of the products not in S will not change in response to a small change in q1 and the associated

adjustments in quantities for products in S. Therefore, in the solution to (A.9), the quantity qj

either does not respond to a small change in q1, or it responds according to its first-order condition.

Now differentiate (23) in the text, and recognize that ∂v1(qNB

1

)

∂q1

=

∂R(qNB

1

,0, ̃q−1,2(qNB

1

))

∂q1

regardless

of whether T

NB

j

(qj ) is smooth or continuous at q

NB

j

. Proposition 3 follows as explained in the text.

Q.E.D.

Proof of Proposition 4: We know from Propositions 1, 2, and 3 that the fully-integrated outcome

is achieved pre and post merger if bundling is feasible or αm < αm. In these cases, when there are

no cost savings from the merger, the benefit to manufacturers 1 and 2 from merging is given by

∆π

B

m = π

B

m − π

∗

1 − π

∗

2

= αm ((Π − Π−1,2) − (Π − Π−1) − (Π − Π−2)). (A.14)

31
To see that the right-hand side of (A.14) is positive, define

M(q1, q2) ≡ max

q−1,2

R(q1, q2, q−1,2) − w

I

1

q1 − w

I

2

q2 −

X

j6=1,2

w

I

j

qj . (A.15)

Since the objective in (A.15) is concave in (q1, q2, q−1,2), it follows that M is concave in (q1, q2).

Let  ̃q1 ≡ arg maxq1 M(q1, 0) and  ̃q2 ≡ arg maxq2 M(0, q2). Using these definitions along with the

definitions of Π, Π−1,2, Π−1, and Π−2 in the text, we have

Π − Π−1,2 = M(q

I

1

, qI

2

) − M(0, 0) (by definition)

> [M(q

I

1

, qI

2

) − M(0, qI

2

)] + [M(q

I

1

, qI

2

) − M(q

I

1

, 0)] (by concavity and uniqueness)

≥ [M(q

I

1

, qI

2

) − M(0, q ̃2)] + [M(q

I

1

, qI

2

) − M( ̃q1, 0)] (by the definition of  ̃q1 and  ̃q2)

= (Π − Π−1) + (Π − Π−2) (by definition),

which implies that the merger is profitable when bundling is feasible or αm < αm.

If bundling is infeasible and αm > αm, then Lemma 1 implies that constraints (19) and (20) will

bind in any bargaining equilibrium. Suppose q

NB ≡ (qNB

1

, ..., qNB

N ) and TNB ≡ (T

NB

1

, ..., T NB

N )

form a bargaining equilibrium. Then, after some algebra, we can rearrange constraint (19) as

F1 = R(q

NB) −

X

j6=1,2

w

I

j

q

NB

j −



max

q−1,2

(R(0, qNB

2

, q−1,2) −

X

j6=1,2

w

I

j

qj )





= R(q

NB) −

X

j6=1

w

I

j

q

NB

j −



max

q−1,2

(R(0, qNB

2

, q−1,2) − w

I

2

q

NB

2 −

X

j6=1,2

w

I

j

qj )





> R(q

NB) −

X

j6=1

w

I

j

q

NB

j −



max

q−1,2

(R(0, q2, q−1,2) − w

I

2

q2 −

X

j6=1,2

w

I

j

qj )





= R(q

NB) −

X

j6=1

w

I

j

q

NB

j − Π−1, (A.16)

where we have used the fact that the non-merging firms offer their products at marginal cost to

the retailer whether or not bundling is feasible. Similarly, we can rearrange constraint (20) as

F2 > R(q

NB) −

X

j6=2

w

I

j

q

NB

j − Π−2. (A.17)

It follows that the profit of merged firm when bundling is infeasible and αm > αm is

π

NB

m = F1 + F2 − Cm(q

NB

1

, qNB

2

)

32
>

X

i=1,2



R(q

NB) −

X

j6=i

w

I

j

q

NB

j − Π−i



 − Cm(q

NB

1

, qNB

2

)

>

X

i=1,2



R(q

I

) −

X

j6=i

w

I

j

q

I

j − Π−i



 − Cm(q

I

1

, qI

2

). (A.18)

The first inequality follows from (A.16) and (A.17). The second inequality follows from the observation that the merged firm’s profit increases when it induces the retailer to choose quantities

q

NB rather than qI

. Note that (A.18) corresponds to (34) in the text, as was to be proved. Q.E.D.

Proof of Proposition 6: Let w

e

1

(b) and w

e

2

(b) be the bargaining equilibrium wholesale prices

for firms 1 and 2, respectively, and let q

e

i

(b), for all i, be the bargaining equilibrium quantities.

Rearranging (38) and (39), the upstream profits for products 1 and 2 can be written as

π1 = F1 + w1q1 − C1(q1)

= R(q

e

(b)) − C1(q

e

i

(b)) − w

e

2

(b)q

e

2

(b) −

X

j6=1,2

w

I

j

q

e

j

(b)

− max

q−1



R(0, q−1) − w

e

2

(b)q2 −

X

j6=1,2

w

I

j

qj



 + b, (A.19)

π2 = F2 + w2q2 − C2(q2)

= R(q

e

(b)) − C2(q

e

i

(b)) − w

e

1

(b)q

e

1

(b) −

X

j6=1,2

w

I

j

q

e

j

(b)

− max

q−2



R(0, q−2) − w

e

1

(b)q1 −

X

j6=1,2

w

I

j

qj



 + b. (A.20)

Using (36), the profit of a rival firm i can be written as

πi = αi[R(q

e

(b)) − Ci(q

e

i

(b)) − w

e

1

(b)q

e

1

(b) − w

e

2

(b)q

e

2

(b) −

X

j6=1,2

w

I

j

q

e

j

(b)

− max

q−1



R(0, q−1) − w

e

1

(b)q1 − w

e

2

(b)q2 −

X

j6=1,2,i

w

I

j

qj



], ∀i 6= 1, 2. (A.21)

Total profits can be written as

π = R(q

e

(b)) −

X

i

Ci(q

e

i

(b)). (A.22)

33
The retailer’s profits are given by

πr = π − π1 − π2 −

X

i6=1,2

πi. (A.23)

Let q

j

i maximize the retailer’s profits when the retailer drops product j. For example, q

1

2

is the

quantity of product 2 that solves the maximization term in equation (A.19). Substituting (A.19)-

(A.22) into (A.23), differentiating πr with respect to b, and using the envelope theorem gives

∂πr

∂b = −2 −







X

i6=1

αi(q

i

1 − q

e

1

)

∂we

1

(b)

∂b +

X

i6=2

αi(q

i

2 − q

e

1

)

∂we

2

(b)

∂b







.

The terms involving q

i

j − q

e

j

for all i, j 6= i, are positive because the products are substitutes and

an increase in b induces increases in w1 and w2. It follows that ∂πr

∂b < 0. Q.E.D.

34
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