FEDERAL COMMUNICATIONS COMMISSION

MEDIA BUREAU STAFF RESEARCH PAPER

Asymmetric Bargaining 

Power and Pivotal Buyers

By Nodir Adilov and Peter J. Alexander

September 2002

2002 -

13
Asymmetric Bargaining Power and Pivotal

Buyers

Nodir Adilov and Peter J. Alexander

September 25, 2002

ABSTRACT

Raskovich (2000) suggests that becoming pivotal through merger

worsens the merging buyers bargaining position. We show that these

results hold in the case where buyer bargaining power is equal across

buyers, but not in the case where bargaining power is asymmetric.

We demonstrate it is possible when there are asymmetries in bargaining power that larger buyers, including pivotal buyers, can extract

greater gains from trade than smaller buyers. We show that this

result holds even if the supplier’s value function is convex. These

results imply that horizontal merger might be used as a strategy to

enhance bargaining position.

Introduction

In this paper, we extend the work of Raskovich (2000) and explore the

case of asymmetric bargaining power. Building on the work of Chipty

and Snyder (1999), Raskovich demonstrated that, under the assumption

of constant bargaining power across firm size, ‘pivotal’ (i.e., large) buyers

would be systematically disadvantaged in negotiations with sellers.1 We

show that if bargaining power increases with the size of the buying firm,

Raskovich’s results do not necessarily hold. On the contrary, large firms

may be systematically advantaged in negotiations with sellers.

Chipty and Snyder (1999) and Raskovich (2000) explore simultaneous

bilateral bargaining models in which there is a single seller and more than

∗Adilov: Department of Economics, Cornell University, email: na47@cornell.edu; Alexander: Federal Communications Commission, email: palexand@fcc.gov. We are indebted to David Sappington for his many thoughtful

and useful comments, and ongoing support. Any errors are our own. The views expressed in this paper are those

of the authors, and do not necessarily represent the views of the Federal Communications Commission, any of its

Commissioners, or other staff.

1Chipty, Tasneem and Christopher Snyder, ”The Role of Firm Size in Bilateral Bargaining: A Study of

the Cable Television Industry,” The Review of Economics and Statistics, May, 1999, 81(2), 326-340; Raskovich,

Alexander, ”Pivotal Buyers and Bargaining Position,” Economic Analysis Group Discussion Paper 00-9, U.S.

Department of Justice, Anti-Trust Division, October, 2001.

1
several buyers. Both assume that the gains from trade are divided equally

(i.e., 50-50), irrespective of firm size. Chipty and Snyder suggest that the

effect on bargaining position of a merger by two (or more) buyers can

be determined by the curvature of the supplier’s value function, and they

demonstrate that if the supplier’s value function is concave, the merger will

enhance the buyer’s bargaining position; if the value function is convex,

the merger will worsen the the buyer’s bargaining position. Raskovich

generalizes Chipty and Snyder’s model by introducing a pivotal buyer;

that is, a buyer so large that only the buyer can completely cover the

supplier’s costs. Thus, the large firm is “on the hook” for the supplier’s

costs. The result is that merger worsens a buyer’s bargaining position.

In what follows, we generalize the approach of Chipty and Snyder

(1999) and Raskovich (2000) by relaxing the assumption of equal division of the gains from trade. We demonstrate that an equilibrium exists

when the division of the surplus varies across firms, and we analyze the

case where bargaining power is assumed to increase in firm size.

We offer several plausible reasons why bargaining power might be increasing in firm size. First, a merger may augment the set of useful information regarding prices and other contractual terms the previously nonmerged firms’ possessed. Second, if there are differences in bargaining

skills between the merging firms, the merger may result in the retention

of the more-skilled bargaining team. Third, the merged firm may have

a lower risk aversion coefficient. Fourth, the merged firm may be more

patient, i.e., it may not discount the future as much as the previously

non-merged firms may have.2 Regardless, our goal in this paper is simply

to explore the outcome of the bilateral bargaining model as if bargaining

power is asymmetric, an assumption we see as no more or less heroic than

any other.

After extending the model of Raskovich (2000) to incorporate asymmetric bargaining power, we then show that: (1) the results of the bargaining

solution employed by Chipty and Snyder and Raskovich are robust to any

constant division of the trade surplus (e.g., 80-20, 60-40, etc.) and not

simply 50-50; (2) the curvature of the value function may no longer be

a reliable rule-of-thumb method for evaluating the change in bargaining

position and hence the effect of mergers on sellers; (3) the post-merger bargaining position of the merged firm may improve even though the merged

firm becomes pivotal; and (4) a merger may decrease the merged firms’

transfer payments and decrease the seller’s transfer revenues.

Perhaps the simplest way to demonstrate the potential effects of asymmetric bargaining power is by example. We preface the example by introducing a bargaining power parameter that can vary across firms, and

denote the i

th buyer’s bargaining power by αi ∈ (0, 1), where a higher

2We thank Alex Raskovich for his discussion relating to these reasons.

2
value of α means greater bargaining power.3

Now, assume that we have three buyers, each with different valuations

of the seller’s product, and each with different levels of bargaining power.

For example, assume that vA = 80, vB = 56, and vC = 40, and that

αA = .8, αB = .4, and αC = .3. Ti denotes the transfer price for the i

th

buyer. The level of seller costs, F, is 50. It is easy to demonstrate that

under these conditions, buyer B is pivotal, whereas buyers A (with the

highest valuation of the seller’s product) and C (with the lowest valuation

of the seller’s product) are not pivotal. Note that for Raskovich (2000),

buyers A and B would be pivotal. We see that TA = (1 − αA) · vA =

(0.2 · 80) = 16 and that TC = (1 − αC) · vC = (0.7 · 40) = 28. It is

immediately clear that TA + TC = 44 < 50 = F. Further, we note that

TB = (1 − αB)·(vB − F + TA + TC) + (F − TA − TC) = (0.6 · 50 + 6) = 36.

Observing that TA + TB = 16 + 36 = 52 > 50 and TB + TC = 64 > 50,

it is clear that buyer A and buyer C are not pivotal, and that buyer B

is pivotal. In fact, as we see from the example, TB > TC > TA, i.e., the

buyer with the highest valuation pays the least. Thus, in a framework

with asymmetric bargaining power, pivotal buyers can derive significant

benefits.

The rest of the paper is organized as follows. First, we extend Raskovich’s

(2000) model and show that under more general assumptions an equilibrium still exists. Next, we show that the introduction of asymmetric bargaining power can improve the buying firm’s bargaining position (even if

the firm is pivotal). We also show that in the presence of asymmetric

bargaining power the ‘curvature test’ of the value function can be a misleading indicator of the effects of merger on bargaining position, i.e., that

the bargaining position of the merged firm can improve even if the the

value function is convex. Finally, we make some concluding remarks.

Nash Equilibrium with Bargaining Power

In this section, we extend Raskovich’s (2000) model to accommodate asymmetric bargaining power. We begin by constructing the transfer prices

faced by pivotal and non-pivotal buyers and then show that an equilibrium exists under conditions more general than Raskovich’s.

Following Raskovich (2000), we assume the i

th buyer’s surplus is given

by vi = (qi, q−i), while the supplier’s gross surplus equals V (Q), where

Q =

Pn

i=1 qi

. Specifically, V (Q) = A(Q) − C(Q), where A(Q) ≡ ancillary

revenue, and C(Q) ≡ total cost. The supplier will produce iff:

V (Q) +Xn

i=1

Ti ≥ 0 (1)

3For Raskovich (2000), α1 = α2 = αn =

1

2

. In fact, Raskovich’s pivotal result will hold for any constant value

α = α1 = α2 = . . . αn where α ∈ (0, 1). Note that αi represents the share of surplus kept by buyer i.

3
We also note that:

q

∗

i = arg max

x

[vi(x, q−i) + V (Q−i + x)] (2)

where we assume there exists a q

∗

t

that maximizes joint surplus.4 Buyer i

is pivotal iff:

V (Q−i) +X

j6=i

Tj < 0 (3)

and

max

x

[vi(x, q−i) + V (Q−i + x)] +X

j6=i

Tj ≥ 0 (4)

where vi(0, q−i) = 0.5

The transfer price (incorporating asymmetric bargaining power and

using α notation) becomes, for a non-pivotal buyer, Ti = (vi+(V −V−i))(1−

αi) − (V − V−i) which can be written as:

Ti = vi(1 − αi) − αi(V − V−i) (5)

Next, noting that P

j6=i

Tj + V−i < 0, we see that the transfer price for a

pivotal buyer can be written as Ti = [vi + (P

j6=i

P

Tj + V )](1 − αi) − V −

j6=i

Tj, or as:

Ti = vi(1 − αi) − αi(

X

j6=i

Tj + V ) (6)

Definition 1: A Nash Equilibrium in purchased quantities (q

∗

1

, q∗

2

, ..., q∗

n

)

and transfer prices (T1, ..., Tn) is that for which the following hold simultaneously for all i:

q

∗

i = arg max

x

(vi(x, q∗

−i

) + V (

X

j6=i

q

∗

j + x)) (7)

Ti = vi(x, q∗

−i

)(1 − αi) − αi(V (Q

∗

) − V (Q

∗ − q∗

i

)) (8)

if P

j6=i

Tj + V (Q∗ − q

∗

i

) ≥ 0

Ti = vi(x, q∗

−i

)(1 − αi) − αi(

X

j6=i

Tj + V (Q

∗

)) (9)

if P

j6=i

Tj + V (Q∗ − q

∗

i

) < 0

4We assume that the surplus from trade is positive at the optimal quantity for any buyer. This implies that

vi + V − V−i > 0 for all i.

5Raskovich has the restriction that V−1 ≤ V−2 ≤ ... ≤ V−n ≤ V , while we allow V−i to vary across buyers.

4
X

j=1,...,n

Tj + V (Q

∗

) ≥ 0 (10)

In what follows, we rank order the i < k buyers such that (vi + (V −

V−i))(1−αi) ≥ (vk + (V −V−k))(1−αk). This implies that the buyer with

the highest valuation is not necessarily the buyer with the highest transfer

price.

Lemma 1: If buyer i satisfies the conditions for being pivotal, then

buyer h, such that h < i, also satisfies the condition for being pivotal.

Proof of Lemma 1: The proof is by contradiction. Suppose that i is

pivotal and that h, h < i, is not pivotal. We note that Ti = (1 − αi)vi −

αi(V +

P

j6=i

Tj ) and that Th = (1−αh)vh −αh(V −V−h). Then, Th −Ti =

(1−αh)vh −αh(V −V−h)−(1−αi)vi +αi(V +

P

j6=i

Tj )=(1−αh)vh + (1−

αh)(V −V−h)−(V −V−h)−(1−αi)vi−(1−αi)(V −V−i)+(1−αi)(V −V−i)+

αi(V +

P

j6=i

Tj ). Let bi = (vi + V − Vi)(1 − αi). Next, by substitution, we

re-write this expression as Th−Ti = bh−bk+V−h−V−i+αiVi+αi(

P

j6=h

Tj+

Th − Ti) or Th − Ti =

1

1−αi

(bn − bk) + (V−h − V−i) + αi

1−αi

(

P

j6=h

Tj + V−h).

Noting that 1

1−αi

(bh − bk) ≥ 0 and that αi

1−αi

(

P

j6=h

Tj + V−h) ≥ 0, we write

Th − Ti ≥ V−h − V−i and thus, V−i − Ti ≥ V−h − Th. Adding P

j

Tj to

both sides we get Vi +

P

j6=i

Tj ≥ V−h +

P

j6=h

Tj ≥ 0. This implies that

Vi +

P

j6=i

Tj ≥ 0, which is a contradiction. Q.E.D.6

Lemma 2: If production is efficient, Pn

j=1 vj + V ≥ 0, then the outcome in which all buyers are pivotal satisfies the supplier’s participation

constraint.

Proof of Lemma 2:

Pn

j=1 Tj+V = ... =

1

1+P

j≤p

αi

1−αj

(

Pn

j=1 vj+V ) ≥ 0

Now, denote by Ti(p) the transfer price for buyer i when first p buyers

are pivotal.

6Consider a possible equilibrium with p pivotal buyers. Lemma 1 implies that (5) holds for i > p

P

and that (6) holds for (i ≤ p). Next, we note that (6) can be written as Ti = vi(1 − αi) − αi(V +

allj Tj ) + αiTi or as Ti = vi −

αi

1−αi

(V +

P

j

Tj ). Summing across the i’s we see that P

j

Tj =

1

1+P

j≤p

αj

1−αj

(

P

j≤p

vj +

P

j>P (1 − αj )vj −

P

j≤p

αj

1−αj

V +

P

j>p αj (V−j − V )) which we can write as:

Ti = vi −

αi

1−αi

V −

αi

1−αi

1+P

j≤p

αj

1−αj

(

P

j≤p

vj +

P

j>P (1 − αj )vj−

P

j≤p

αj

1−αj

V +

P

j>p αj (V−j − V ))

(11)

5
Lemma 3: If P

i6=p

Ti(p) + V ≥ 0 then P

i6=p

Ti(p − 1) + V ≥ 0

Proof of Lemma 3: By contradiction assume that P

i6=p

Ti(p)+V ≥ 0

and P

i6=p

Ti(p−1)+V < 0. Then, (P

i6=p

Ti(p)+V )−(

P

i6=p

P

Ti(p−1)+V ) =

i6=p

Ti(p)−(

P

i6=p

Ti(p−1) = P

i6=p−1

(

αi

1−αi

[

Pn

j=1 Tj (p−1)−

Pn

j=1 Tj (p)]).

Next, we see that Tp(p − 1) − Tp(p) = (1 + Pi ≤ p − 1

αi

1−αi

)(Pn

j=1 Tj (p −

1) −

Pn

j=1 Tj (p)). Since Tp(p − 1) − Tp(p) < 0, i.e., the pivotal payment is

always greater than the non-pivotal, we get P

i6=p

Ti(p)−

P

i6=p

Ti(p−1) < 0,

which is a contradiction. Q.E.D.

Proposition 1: If production is efficient, then there exists an equilibrium where only the first p buyers are pivotal.

Proof of Proposition 1: See Raskovich(2000).7

Merger Effects

Using the results from the previous section, we explore the potential effects

of merger on bargaining power, and compare these results with Chipty and

Snyder (1999) and Raskovich (2000). As we demonstrate, once potential

asymmetries are introduced into the bargaining solution, the results of

Chipty and Snyder and Raskovich may not hold. In fact, the introduction

of even a modest amount of bargaining power can have significant effects

on bargaining position.

We begin by assuming there are two non-pivotal merging firms, A and

B, and then show the conditions under which a merger between the firms

increases their bargaining position.

Note that the net surplus for buyer A before a merger is given by

(vA + V

S − VS

−A)αA, and the net surplus for buyer B before a merger is

given by (vB + V

S − VS

−B)αB. The net surplus after a merger is (vAB +

V

M −VM

−AB)αAB, assuming, that AB is non-pivotal as in Chipty and Snyder

(1999). We note that A and B have the incentive to merge iff:

(vAB +V

M −VM

−AB)αAB > (vA +V

S −VS

−A)αA + (vB +V

S −VS

−B)αB (12)

We can write (12) as vAB +V

M −VM

−AB > (vA +V

S −VS

−A)

αA

αAB

+ (vB +

V

S − VS

−B)

αB

αAB

, letting DE = vAB − vA − vB where DE is downstream

efficiency, UE = (V

M − VM

−AB) − (V

S − VS

−AB) where UE is upstream

efficiency, and:

BP = (vA + V

S − VS

−A)

αAB−αA

αAB

+ (vB + V

S − VS

−B)

αAB−αB

αAB

+(V

S

−A + V

S

−B − V

S − VS

−AB)

(13)

7Raskovich notes that the equilibrium may not be unique.

6
where BP is the firm’s bargaining position. Combining these conditions

yields:

DE + UE + BP > 0 (14)

Recall that by assumption (see footnote 4) vA + V

S − VS

−A and vB +

V

S − VS

−B are positive. Therefore, if αAB > αA and αAB > αB, then

(vA + V

S − VS

−A)

αAB−αA

αAB

+ (vB + V

S − VS

−B)

αAB−αB

αAB

> 0. Noting that for

Chipty and Snyder (1999), BP CS = V

S

−A + V

S

−B − V

S − VS

−AB, and given

our formulation in (13), clearly, BP > BP CS. Thus, in the presence of

asymmetric bargaining power, Chipty and Snyder’s (1999) result underestimates the positive effect of bargaining power on post-merger bargaining

position, since bargaining position in the context of asymmetric bargaining

power can be positive even if BP CS < 0. Thus, bargaining position can

increase even if V

00(Q) > 0, i.e., even if V is convex.8

Next, following Raskovich (2000), assume that buyers A and B merge

and become pivotal. The merger is profitable iff:

αABvAB +αAB(

X

j6=AB

(T

M

j +V

M)) > αA(vA +VS−VS

−A)+αB(vB +V

S−VS

−B)

which we note is equivalent to vAB +

P

j6=AB(T

M

j + V

M) > (vA + VS −

V

S

−A)

αA

αAB

+ (vB + V

S − VS

−B)

αB

αAB

. We decompose this expression into three

parts: DE = vAB − vA − vB, UE = (V

M − VM

−AB) − (V

S − VS

−AB), and

BP = (vA + V

S − VS

−A)

αAB−αA

αAB

+ (vB + V

S − VS

−B)

αAB−αB

αAB

+(V

S

−A + V

S

−B − V

S − VS

−AB) + θ(

P

j6=AB T

M

j + V

M

−AB)

(15)

where θ = 1 if AB is pivotal, and θ = 0 if AB is not pivotal. It is

immediately clear that (15) is the general case of (13). Thus, (15) can be

written as

BP = (vA + V

S − VS

−A)

αAB − αA

αAB

+ (vB + V

S − VS

−B)

αAB − αB

αAB

+ BP R

Clearly, BP > BP R. According to Raskovich, if the merged buyer becomes pivotal, its bargaining position worsens, since the last term in (15)

is negative. However, this worsening of bargaining position can be offset

by an increase in bargaining power that increases the first two terms of

(15).

The measures of Chipty and Snyder (1999) and Raskovich (2000) may

under-estimate bargaining position because they abstract from any positive effects of bargaining power for the merging firm. Once this effect is

accounted for, the curvature of the value function is no longer a reliable

8Under Chipty and Snyder, concavity (convexity) of the value function implies the bargaining position of the

merged firm improves (worsens).

7
rule-of-thumb method for evaluating the change in bargaining position and

hence the effects of the merger on sellers. Moreover, despite Raskovich’s

prediction that pivotal buyers would be disadvantaged by merger, we have

shown that increasing bargaining power can improve the bargaining position of the, now pivotal, merged firm.

Conclusion

Raskovich (2000) suggested that becoming pivotal through merger worsens the merging buyers’ bargaining position. We have shown that these

results hold in the case where buyer bargaining power is constant, but not

necessarily in the case where bargaining power increases with firm size.

We demonstrated that larger buyers, including pivotal buyers, can extract

greater gains from trade than smaller buyers when there are asymmetries

in bargaining power. Chipty and Snyder (1999) and Raskovich (2000)may

under-estimate bargaining position because they abstract from the possibility that bargaining power may increase with firm size. Once this effect

is accounted for, the curvature of the value function is no longer a reliable

rule-of-thumb method for evaluating the change in bargaining position and

hence the effects of the merger on sellers. Moreover, despite Raskovich’s

prediction that pivotal buyers would be disadvantaged by merger, we have

shown that increasing bargaining power can improve the bargaining position of the, now pivotal, merged firm.

8