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FEDERAL COMMUNICATIONS COMMISSION
MEDIA BUREAU STAFF RESEARCH PAPER
Asymmetric Bargaining
Power and Pivotal Buyers
By NodirAdilovandPeterJ. Alexander
September 2002
2002 - 13
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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 bargain-ing
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 Commu-nications 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. 1 Chipty, 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.
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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 divi-sion
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 in-creasing
in firm size. First, a merger may augment the set of useful infor-mation
regarding prices and other contractual terms the previously non-merged
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 asymmet-ric
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 bar-gaining
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 asym-metric
bargaining power is by example. We preface the example by in-troducing
a bargaining power parameter that can vary across firms, and
denote the i th buyer’s bargaining power by ®i 2 (0; 1), where a higher
2 We thank Alex Raskovich for his discussion relating to these reasons.
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value of ® 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
®A = :8, ®B = :4, and ®C = :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 ¡ ®A) ¢ vA =
(0:2 ¢ 80) = 16 and that TC = (1 ¡ ®C) ¢ vC = (0:7 ¢ 40) = 28. It is
immediately clear that TA + TC = 44 < 50 = F. Further, we note that
TB = (1 ¡ ®B) ¢ (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 equilib-rium
still exists. Next, we show that the introduction of asymmetric bar-gaining
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 mis-leading
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 asym-metric
bargaining power. We begin by constructing the transfer prices
faced by pivotal and non- pivotal buyers and then show that an equilib-rium
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 = P n 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) +
n
X i= 1 Ti ¸ 0 (1) 3 For Raskovich (2000), ®1 = ®2 = ®n = 1 2 . In fact, Raskovich’s pivotal result will hold for any constant value ® = ®1 = ®2 = : : : ®n where ® 2 (0; 1) : Note that ®i represents the share of surplus kept by buyer i.
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We also note that:
q¤ i = arg x max[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
6 j=i
Tj < 0 (3)
and
x max[vi( x; q ¡ i) + V (Q ¡ i + x)] + X j
6 =i
Tj ¸ 0 (4)
where vi( 0; q ¡ i) = 0. 5
The transfer price (incorporating asymmetric bargaining power and
using ® notation) becomes, for a non- pivotal buyer, Ti = (vi+( V ¡ V ¡ i))( 1 ¡
®i) ¡ (V ¡ V ¡ i) which can be written as:
Ti = vi( 1 ¡ ®i) ¡ ®i( V ¡ V ¡ i) (5)
Next, noting that P j 6 =i Tj + V ¡ i < 0, we see that the transfer price for a
pivotal buyer can be written as Ti = [vi + ( P 6 j=i Tj + V )]( 1 ¡ ®i) ¡ V ¡
P 6 j=i Tj, or as: Ti = vi( 1 ¡ ®i) ¡ ®i( X 6 j=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 simul-taneously
for all i:
q¤ i = arg x max(vi( x; q¤ ¡ i ) + V ( X
j 6 =i
q¤ j + x)) (7)
Ti = vi( x; q¤ ¡ i )( 1 ¡ ®i) ¡ ®i( V (Q¤) ¡ V (Q¤ ¡ q¤ i )) (8)
if P 6 j=i Tj + V (Q¤ ¡ q¤ i ) ¸ 0
Ti = vi( x; q¤ ¡ i )( 1 ¡ ®i) ¡ ®i( X
j 6 =i
Tj + V (Q¤)) (9)
if 6 j=i Tj + V (Q¤ ¡ q¤ i ) < 0
4 We assume that the surplus from trade Pis positive at the optimal quantity for any buyer. This implies that
vi + V ¡ V ¡ i > 0 for all i. 5
Raskovich has the restriction that V ¡ 1 · V ¡ 2 · ::: · V ¡ n · V , while we allow V ¡ i to vary across buyers.
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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 ¡ ®i) ¸ (vk +( V ¡ V ¡ k))( 1 ¡ ®k). 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 ¡ ®i) vi ¡
®i( V + P 6 j=i Tj) and that Th = (1 ¡ ®h) vh ¡ ®h( V ¡ V ¡ h). Then, Th ¡ Ti =
(1 ¡ ®h) vh ¡ ®h( V ¡ V ¡ h) ¡ (1 ¡ ®i) vi +®i( V + P 6 j=i Tj)=( 1 ¡ ®h) vh +( 1 ¡
®h)( V ¡ V ¡ h) ¡ (V ¡ V ¡ h) ¡ (1 ¡ ®i) vi ¡ (1 ¡ ®i)( V ¡ V ¡ i)+( 1 ¡ ®i)( V ¡ V ¡ i)+
®i( V + P 6 j=i Tj). Let bi = (vi +V ¡ Vi)( 1 ¡ ®i). Next, by substitution, we
re- write this expression as Th ¡ Ti = bh ¡ bk+V ¡ h ¡ V ¡ i +®iVi+®i( P 6 j=h Tj+
Th ¡ Ti) or Th ¡ Ti = 1 1 ¡ ®i (bn ¡ bk) + (V ¡ h ¡ V ¡ i) + ®i 1 ¡ ®i ( 6 j=h Tj + V ¡ h).
Noting that 1 1 ¡ ®i (bh ¡ bk) ¸ 0 and that ®i 1 ¡ ®i ( j 6 =h Tj +V ¡P h) ¸ 0, we write
Th ¡ Ti ¸ V ¡ h ¡ V ¡ i and thus, V ¡ i ¡ Ti ¸P V ¡ h ¡ Th. Adding P j Tj to
both sides we get Vi + P j 6 =i Tj ¸ V ¡ h + P j 6 =h Tj ¸ 0. This implies that
Vi + P 6 j=i Tj ¸ 0, which is a contradiction. Q. E. D. 6
Lemma 2: If production is efficient, P n j= 1 vj + V ¸ 0, then the out-come
in which all buyers are pivotal satisfies the supplier’s participation
constraint.
Proof of Lemma 2: P n j= 1 Tj+V = ::: = 1 1+ P j · p ®i 1 ¡ ®j ( P n j= 1 vj+V ) ¸ 0
Now, denote by Ti( p) the transfer price for buyer i when first p buyers
are pivotal.
6 Consider a possible equilibrium with p pivotal buyers. Lemma 1 implies that (5) holds for i > p
and that (6) holds for (i · p). Next, we note that (6) can be written as Ti = vi( 1 ¡ ®i) ¡ ®i( V +
P allj Tj) + ®iTi or as Ti = vi ¡ ®i 1 ¡ ®i (V + P j Tj). Summing across the i’s we see that P j Tj = 1 1+ P j · p ® j 1 ¡ ® j ( P j · p vj + P j>P (1 ¡ ®j)vj ¡ P j · p ®j 1 ¡ ®j V + P j>p ®j(V ¡ j ¡ V )) which we can write as: Ti = vi ¡ ®i 1 ¡ ®i V ¡ ® i 1 ¡ ® i 1+
P j · p ® j 1 ¡ ® j ( P j · p vj + P j>P (1 ¡ ®j)vj ¡ P j · p ®j 1 ¡ ®j V + P j>p ®j(V ¡ j ¡ V )) (11)
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Lemma 3: If P i 6 =p Ti( p) + V ¸ 0 then P i 6 =p Ti( p ¡ 1) + V ¸ 0
Proof of Lemma 3: By contradiction assume that P i 6 =p Ti( p)+ V ¸ 0
and P i 6 =p Ti( p ¡ 1)+ V < 0. Then, ( P i 6 =p Ti( p)+ V ) ¡ ( P i 6 =p Ti( p ¡ 1)+ V ) =
P i 6 =p Ti( p) ¡ ( P i 6 =p Ti( p ¡ 1) = P i 6 =p ¡ 1 ( ®i 1 ¡ ®i [ P n j= 1 Tj (p ¡ 1) ¡ P n j= 1 Tj (p)]). Next, we see that Tp( p ¡ 1) ¡ Tp( p) = (1 + P i · p ¡ 1 ®i 1 ¡ ®i )( P n j= 1 Tj (p ¡ 1) ¡ P n 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 i 6 =p Ti( p) ¡ P i 6 =p Ti( p ¡ 1) < 0, which is a contradiction. Q. E. D. Proposition 1: If production is efficient, then there exists an equilib-rium 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 ¡ V S ¡ A )®A, and the net surplus for buyer B before a merger is
given by (vB + V S ¡ V S ¡ B )®B. The net surplus after a merger is (vAB +
V M ¡ V M ¡ AB )®AB, 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 ¡ V M ¡ AB )®AB > (vA +V S ¡ V S ¡ A )®A +( vB +V S ¡ V S ¡ B )®B (12)
We can write (12) as vAB +V M ¡ V M ¡ AB > (vA +V S ¡ V S ¡ A ) ®A ®AB +( vB +
V S ¡ V S ¡ B ) ®B ®AB , letting DE = vAB ¡ vA ¡ vB where DE is downstream
efficiency, UE = (V M ¡ V M ¡ AB ) ¡ (V S ¡ V S ¡ AB ) where UE is upstream
efficiency, and:
BP = (vA + V S ¡ V S ¡ A ) ®AB ¡ ®A ®AB + (vB + V S ¡ V S ¡ B ) ®AB ¡ ®B ®AB
+( V S ¡ A + V S ¡ B ¡ V S ¡ V S ¡ AB ) (13)
7 Raskovich notes that the equilibrium may not be unique.
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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 ¡ V S ¡ A and vB +
V S ¡ V S ¡ B are positive. Therefore, if ®AB > ®A and ®AB > ®B, then
(vA + V S ¡ V S ¡ A ) ®AB ¡ ®A ®AB + (vB + V S ¡ V S ¡ B ) ®AB ¡ ®B ®AB > 0. Noting that for
Chipty and Snyder (1999), BP CS = V S ¡ A + V S ¡ B ¡ V S ¡ V S ¡ 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 under-estimates
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:
®ABvAB +®AB( X
j 6 =AB
(T M j +V M )) > ®A( vA+ V S ¡ V S ¡ A )+ ®B( vB +V S ¡ V S ¡ B )
which we note is equivalent to vAB + P j 6 =AB (T M j + V M ) > (vA + V S ¡
V S ¡ A ) ®A ®AB +( vB +V S ¡ V S ¡ B ) ®B ®AB . We decompose this expression into three
parts: DE = vAB ¡ vA ¡ vB, UE = (V M ¡ V M ¡ AB ) ¡ (V S ¡ V S ¡ AB ), and
BP = (vA + V S ¡ V S ¡ A ) ®AB ¡ ®A ®AB + (vB + V S ¡ V S ¡ B ) ®AB ¡ ®B ®AB
+( V S ¡ A + V S ¡ B ¡ V S ¡ V S ¡ AB ) + µ( P j 6 =AB T M j + V M ¡ AB ) (15)
where µ = 1 if AB is pivotal, and µ = 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 ¡ V S ¡ A ) ®AB ¡ ®A ®AB + (vB + V S ¡ V S ¡ B ) ®AB ¡ ®B ®AB + BP R
Clearly, BP > BP R . According to Raskovich, if the merged buyer be-comes
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 posi-tive
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
8 Under Chipty and Snyder, concavity (convexity) of the value function implies the bargaining position of the
merged firm improves (worsens).
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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 posi-tion
of the, now pivotal, merged firm.
Conclusion
Raskovich (2000) suggested that becoming pivotal through merger wors-ens
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 possi-bility
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 posi-tion
of the, now pivotal, merged firm.
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