Mergers with Technological and Cultural Differences
Patrick de Lamirandea and Can Erutkub
January, 2009
Abstract
We examine mergers involving asymmetric firms with cultural or managerial
differences. The acquiring firm can either transfer its cost-efficient technology
to the target firm or shut it down. In this framework, we identify the privately
optimal merger, which always involves the most efficient firm. The transfer of
technology only takes place when cultural or managerial differences are relatively
small. Otherwise, the target firm is shut down often resulting in a decrease of
welfare post-merger.
JEL Classification Number: L120, L190, L420.
Keywords: Merger, Technology Transfer, Cultural or Managerial Differences.
a
Department of Financial & Information Management, Cape Breton University, P.O. Box 5300, 1250
Grand Lake Road, Sydney (NS), B1P 6L2, Canada, email: Patrick deLamirande@cbu.ca.
b
Economics Department, York University, Glendon College, 2275 Bayview Street, Toronto (ON), M4N
3M6, Canada, email: cerutku@glendon.yorku.ca. (Corresponding author)
1
Introduction
In an organization, sets of workable principles and routines can be constructed to enhance
the effectiveness of its members. They create shared expectations as to what constitute right
and wrong behaviors, for example. This, in turn, can help managers in making decisions and
in solving disputes. Those principles and routines, however, are often based on a common
culture and on actions that have worked well in the past. As such, they work best in
typical circumstances. But when they are applied to changing conditions, such as a merger,
and imposed to newcomers (with a different background than the members of the original
organization), principles and routines can translate into a clash of culture.
In addition, the coordination of activities between individuals within a firm or between
firms within a larger organization can sometimes be best handle by management. The issue of
technology transfer post-merger is one example. Decisions about the use of new equipments,
the reliance on a new supply-chain model, or the necessity to put in place training programs
are often made at a higher level in the firm. But this centralization is not without costs.
Information that rest at the individual levels need to be collected and analyzed. And when
a decision is made, it must be communicated downward and implemented in such a way
that individuals understand what needs to be done and what objectives are being pursued.
These managerial tasks are costly to execute and can create tensions.
Clashes of culture and tensions between merging firms have often been observed. In a
merger brief discussing the Citicorp-Travelers merger,1 The Economist noted that incompatible computer systems, power struggle at the top management, a lack of coordination to
1
“First Among Equal,” published August 24, 2000.
2
cross-sell financial products, and the absence of a clear Internet strategy all led to a costly
and delayed integration of the two companies, which remained distinct by name.
Another example is the failed DailmerChrysler merger, which was not based on a strategy
of establishing combined dealerships. Aside from the geographic distance between headquarters and factories, there was a need to overcome different national and corporate culture. In
another merger brief,2 The Economist highlighted the potential clash between “process-led
German engineers with Chryslers hunch-inspired, risk-taking bosses” and the necessity to
join Dailmer’s bureaucratic and formal business approach to America’s more spontaneous
behavior. As a result, “[e]fforts to link product development at Chrysler and MercedesBenz always met with fierce resistance” and failed to produce the anticipated workable and
cost-efficient ways of doing. These issues were compounded with large pay disparities as
American managers were earning much higher salaries than their German counterparts.
These examples illustrate what we have in mind, which is to analyze mergers involving firms with distinct technologies and cultures that may want to integrate into a multidivisional structure. Kamien and Zang (1990) introduce the idea that firms participating in a
merger may not be consolidated, preventing the business-stealing (or free-riding) effect that
leads Salant et al. (1983) to find that a merger becomes profitable only when the newly created firm captures 80% of the market. It may nonetheless be possible to observe post-merger
shutdowns, especially in industries with overcapacity problems (Fauli-Oller, 2002).
A paper related to our work is Tombak (2002) who looks at a sequence of horizontal mergers between firms having different marginal production costs. In his analysis, an acquiring
firm can decide to either consolidate its operations with those of the acquired firm, forming
2
“The DailmerChrysler Emulsion,” published July 27, 2000.
3
a centralized firm, or can transfer its technology to the acquired firm and operate both firms
as separate entities in the product market, creating a decentralized firm.3 In comparison
to Tombak (2002), however, we focus on the profitability and, therefore, the likelihood of
technology transfer post-merger rather than on the possibility of monopolization through
sequential mergers. Indeed, Tombak (2002) leaves the cost of transferring technology undefined and simply assumes that the acquiring and low-cost firm can profitably transfer its
efficient technology to the acquired high-cost firm. Our goal is to determine how the profitability of technology transfer is influenced by various factors such as the pre-merger size
of the acquired firm, the marginal production cost difference between merging firms, and
cultural or managerial (that is, non-technological) differences. The added structure to the
technology transfer cost function is important as it reverses some of the results found by
Tombak (2002). It is also useful as it can help antitrust authorities to identify circumstances
under which alleged post-merger efficiencies are more or less likely.
Another relevant paper is Banal-Estanol et al. (2008) who also consider mergers that can
result in efficiency gains. The size of those efficiencies gains are related to internal conflict
within the merged firm. These conflicts are linked to the magnitude of relation-specific
investments made by managers to lower the marginal production cost of the newly created
firm. In making their investment decisions, managers can either cooperate or pursue their
own interests. In comparison to our work, however, Banal-Estanol et al. (2008) do not
consider the cost of technological transfer as firms are identical pre-merger. As a result,
while they can make predictions as to whether the industry will move toward a monopoly,
3
Car manufacturers are often cited as examples of firms consolidating production and using the most
efficient technology available, but decentralizing marketing between multiple divisions.
4
duopoly, or triopoly, they cannot predict the identity of the merging firms. Moreover, the
dichotomous decision of managers is somewhat embedded in our model. While Banal-Estanol
et al. (2008) make the decision to cooperate or not endogenous, we have an exogenous but
continuous parameter that can capture managerial cooperation or conflict.
With both market power and efficiency effects included in our analysis, we find that a
decentralized firm emerges post-merger when cultural or managerial differences are small and
when the technological difference between the merging firms is either small or large. The
transfer of technology from the efficient to the inefficient firm increases welfare (measured
by total surplus). When a centralized merger takes place, there is a shift of production from
the inefficient firm to the efficient one. The increase in post-merger market power is offset
by the increase in efficiency only if the centralized merger involves the most and the least
efficient firms and when the marginal production cost differential is not too low.
The paper is organized as follows. Section 2 describes the model and section 3 presents
the equilibrium analysis as well as the results. Section 4 provides concluding remarks.
2
The model
Suppose an industry with three firms, 1, 2, and 3, offering a homogeneous good and facing
the inverse demand function
p = a − q1 − q2 − q 3 .
(1)
In (1), a denotes the absolute size of the market and qi refers to firm i’s output with i = 1, 2, 3.
Firms are asymmetric and can be ranked according to their constant marginal production
cost. Firm 1 has the lowest marginal production cost with c1 = c − , where c < a and
5
∈ [0, c). Firms 2 and 3’s marginal production costs are respectively given by c2 = c and
c3 = c + . We impose, in section 3, a condition on those parameters such that no firm can
be forced out of the market because it is too inefficient compared to its rivals.4 There are
no fixed costs of production.5
Interactions between firms are described by a two-stage game. In the first stage, an acquiring firm decides whether or not to merge with a target firm. We make the assumption
that the merging decision is always undertaken by the low-cost firm. The latter can also
decide to shut down the target firm, and to operate in a centralized manner, or to become
a decentralized firm by operating both firms post-merger and transferring its efficient technology to the target firm.6 We assume that the merger that takes place is profitable and is
the one generating the greatest private surplus. A merger is profitable if the merged firm’s
post-merger profit is greater than the combined profits of the acquiring and target firms
pre-merger.7 A profitable merger generates the greatest private surplus when the difference
between the merged firm’s profit and the combined profits of the acquiring and target firms
pre-merger is greater than this difference for any other merger. Because we assume that the
cost of acquiring the target firm equals the latter’s pre-merger profit, decentralized mergers
without technology transfer are not analyzed. They would lead to the equality between the
post-merger profit of the merged firm and the sum of the pre-merger profits of the partic4
This assumption is not as restrictive as it might appear. Indeed, according to the Federal Trade Commission and US Department of Justice’s Horizontal Merger Guidelines “...efficiencies are most likely to make
a difference in merger analysis when the likely adverse competition effects, absent the efficiencies are not
great. Efficiencies almost never justify a merger to monopoly or near-monopoly.” (p. 32)
5
We assume the presence of sunk costs of entry to stay away from entry considerations. Horizontal
mergers with free entry have been analyzed in Spector (2003), for example.
6
For example, if the merger involves firms 2 and 3, we assume that firm 2 has initiated the acquisition
and that the decisions to shut down firm 3 or transfer the low-cost technology to firm 3 rest with firm 2.
7
We only look at mergers solving the merger paradox.
6
ipating firms, and the existence of a small acquisition cost, such as legal fees, would make
them unprofitable. In the case of a decentralized merger, the technological transfer comes
at a (fixed) cost of
θ(cH − cL )qH .
(2)
In (2), cH (cL ) denotes the marginal production cost of the high (low)-cost firm participating
in the merger. Also, qH corresponds to the output of the target firm pre-merger. This
function captures the fact that the transfer of technology can be costly if either the difference
in the marginal production cost between the acquiring firm and the target firm is important or
if the latter is a large firm pre-merger.8 In the first case, transferring the low-cost technology
is costly because the inefficient firm’s equipment needs to be replaced (e.g., technologies may
not be compatible, IT systems may need to be upgraded) and/or employees require extensive
training. In the second case, transferring the low-cost technology is costly simply because
of the size of the target firm. The larger the target firm, the more equipment needs to
be replaced and/or the more employees require training even if the technological difference
between firms is not great. The parameter θ ∈ [1, +∞) captures the non-technological
difficulties in transferring the technology from one firm to the other. Differences in culture
or managerial conflicts can make a merger less likely to realize anticipated efficiencies. For
instance, it is too costly to transfer the technology and we should expect a merger to be
centralized when θ → ∞. If θ = 1, however, the cost of transferring technology is solely
explained by marginal production cost differences and the size of the target firm pre-merger.
While the technological transfer cost differs across mergers, we restrict the parameter θ to
8
Based on the work of Kreps and Scheinkman (1983), it is easy to justify that the size of the high-cost
firm is positively correlated to its pre-merger equilibrium output in a setting of quantity competition with
homogeneous products.
7
be common to all mergers to preserve some symmetry.
In the second stage, firms set their output simultaneously. In the case of centralization,
the merged firm is and acts as one entity. Because quantities are strategic substitutes, the
outsider has an incentive to increase its output post-merger. In the case of decentralization,
the acquiring and target firms continue to operate as two separate entities. This can be
seen as a commitment to produce a high combined output post-merger and to prevent the
outsider from engaging in business-stealing post-merger.
We are looking for a subgame perfect equilibrium (SPE) of this game.
3
Equilibria in the Marketplace
Before solving the game, we need to introduce some notation. Each outcome is denoted by
(m, n, t) where





∅









{1, 2}
m=




{1, 3}









{2, 3}
n=
if there is no merger
if firms 1 and 2 merge
if firms 1 and 3 merge
if firms 2 and 3 merge




2 if the high cost firm involved in the merger is shut down post-merger



3 if there is no merger or if the merger is decentralized
t=




0 if there is no technology transfer



1 if there is a technology transfer between merging firms.
8
This allows us to denote firm i’s equilibrium output and profit by qi (m, n, t) and πi (m, n, t),
respectively. Also, let us define K = (a − c)/.9 For given values of a and c, a large (small)
K implies a small (large) , that is a small (large) difference in the marginal production cost
between firms. We assume throughout that K > 5 as this ensures that no firm outside the
merger is forced out of the market post-merger as it earns a positive profit after any merger.
3.1
Downstream Competition
At the second stage of the game, firms simultaneously set their quantities knowing the structure of the industry. Because many cases must be considered, we start with the benchmark
scenario of no merger or pre-merger case. Second, we look at centralized mergers where the
target firm is shut down. Third, we analyze the situation of decentralized mergers, which
involve a technology transfer and where the target firm remains active.
3.1.1
No Merger
The firms’ Cournot equilibrium quantities pre-merger are
(K + 4) 4
K
q2 (∅, 3, 0) =
4
(K − 4) q3 (∅, 3, 0) =
.
4
q1 (∅, 3, 0) =
Firm i’s profit is πi (∅, 3, 0) = [qi (∅, 3, 0)]2 with i = 1, 2, 3.
9
This notation is commonly used in the licensing literature and avoids making unnecessary assumptions
about the parameters a and c.
9
3.1.2
Centralized Mergers
After the merger, the acquiring (and efficient) firm shuts down the target (and inefficient)
firm, which implies a reallocation of production to more efficient firms. This beneficial
effect, however, can potentially be offset by the creation of more market power because of
the reduction in the number of firms. Three mergers need to be analyzed.
First, consider the merger between firms 1 and 2, which implies that only firms 1 and 3
remain active post-merger. The resulting equilibrium Cournot quantities are
(K + 3) 3
(K − 3) q3 ({1, 2}, 2, 0) =
3
q1 ({1, 2}, 2, 0) =
with firm i’s profit being πi ({1, 2}, 2, 0) = [qi ({1, 2}, 2, 0)]2 for i = 1, 3.
Second, when firms 1 and 3 merge, the equilibrium Cournot quantities for firms 1 and 2
are respectively
(K + 2) 3
(K − 1) q2 ({1, 3}, 2, 0) =
.
3
q1 ({1, 3}, 2, 0) =
Firm i’s profit is πi ({1, 3}, 2, 0) = [qi ({1, 3}, 2, 0)]2 for i = 1, 2.
Finally, the ({2, 3}, 2, 0)-merger leads to the following quantities
(K + 2) 3
(K − 1) q2 ({2, 3}, 2, 0) =
3
q1 ({2, 3}, 2, 0) =
and to a profit of πi ({2, 3}, 2, 0) = [qi ({2, 3}, 2, 0)]2 for firm i, i = 1, 2.
10
3.1.3
Decentralized Mergers
Here, the acquiring firm continues to operate the target firm post-merger after transferring
its low-cost technology to the high-cost firm. Once again, three mergers need to be examined.
When firms 1 and 2 merge, the latter’s marginal production cost becomes c − and the
equilibrium Cournot quantities are
(K + 3) 4
(K − 5) q3 ({1, 2}, 3, 1) =
.
4
q1 ({1, 2}, 3, 1) = q2 ({1, 2}, 3, 1) =
Recall that we assume K > 5. The profits of the merged firm, gross of the technology
transfer cost, and of the outsider are 2 [q1 ({1, 2}, 3, 1)]2 and [q3 ({1, 2}, 3, 1)]2 , respectively.
When the merger involves firms 1 and 3, the marginal production cost of each of those
firms is c − . The resulting equilibrium Cournot quantities are
(K + 2) 4
(K − 2) q2 ({1, 3}, 3, 1) =
.
4
q1 ({1, 3}, 3, 1) = q3 ({1, 3}, 3, 1) =
The merged firm’s profit, gross of the technology transfer cost, is 2 [q1 ({1, 3}, 3, 1)]2 , while
the outsider obtains a profit of [q3 ({1, 3}, 3, 1)]2 .
When the acquiring and the target firms are 2 and 3, respectively, they both end up with
a marginal production cost of c. As a result, the equilibrium Cournot quantities are
(K + 3) 4
(K − 1) q2 ({2, 3}, 3, 1) = q3 ({2, 3}, 3, 1) =
.
4
q1 ({2, 3}, 3, 1) =
The merged firm earns a profit of 2 [q2 ({2, 3}, 3, 1)]2 gross of the technology transfer cost.
As for firm 1, its profit equals [q1 ({2, 3}, 3, 1)]2 .
11
3.2
Privately Optimal Merger
In the first stage of the game, firms decide whether or not to merge as well as the structure
taken by the merged firm. If the post-merger structure is centralized, then the (low-cost)
acquiring firm shuts down the (high-cost) target firm. However, if a decentralized structure
is adopted after the merger, a technology transfer from the acquiring firm to the target firm
takes place at a cost given by (2). Recall that we assume that the merger that takes place
is profitable and is the one generating the greatest private surplus.
Lemma 1. For K ∈ (5, 28/5] the ({1, 2}, 2, 0)-merger is more profitable than the ({1, 3}, 2, 0)merger. For K ≥ 28/5, the ({1, 3}, 2, 0)-merger is more profitable than the ({1, 2}, 2, 0)merger, but it becomes unprofitable for K ≥ 28. The ({2, 3}, 2, 0)-merger is never the most
profitable centralized merger for K > 5.
Proof. See the appendix.
We now need to compute the merged firm’s equilibrium profit for each possible decentralized merger. If the merger involves firms 1 and 2, the profit of the merged firm is
(K + 3) π1 ({1, 2}, 3, 1) + π2 ({1, 2}, 3, 1) − θq2 (∅, 3, 0) = 2
4
2
θK2
−
.
4
When firms 1 and 3 merge, the merged firm obtains
(K + 2) π1 ({1, 3}, 3, 1) + π3 ({1, 3}, 3, 1) − 2θq3 (∅, 3, 0) = 2
4
2
2
− 2θ
K −4
4
.
In the case of the ({2, 3}, 3, 1)-merger, the merged firm’s profit is
(K − 1) π2 ({2, 3}, 3, 1) + π3 ({2, 3}, 3, 1) − θq3 (∅, 3, 0) = 2
4
12
2
− θ
2
K −4
4
.
Lemma 2. If θ = 1, then the ({1, 3}, 3, 1)-merger is the most profitable merger with technology transfer for K > 5. If θ ∈ (1, 2], then the ({1, 3}, 3, 1)-merger is the most profitable
merger with technology transfer for K ≤ K ({1, 3}, 3, 1) = (4θ − 3)/(θ − 1). If θ ∈ (1, 2]
and K > K ({1, 3}, 3, 1) or if θ ≥ 2, then there is no profitable decentralized merger with
technology transfer.
Proof. See the appendix.
Using Lemmas 1 and 2 we obtain the following proposition.
Proposition 1. The privately optimal merger is the
1. ({1, 2}, 2, 0)-merger for θ ≥ 1 when K ∈ (5, 28/5];
2. ({1, 3}, 3, 1)-merger for K ∈ [28/5, K(θ)] when θ ∈ [1, 361/360] with K(θ) = (18θ − 2)−
p
6 3θ (3θ − 2) and for K ∈ K̄(θ), (4θ − 3)/(θ − 1) when θ ∈ [1, 25/24] with K̄(θ) =
(18θ − 2) + 6
p
3θ (3θ − 2);
3. ({1, 3}, 2, 0)-merger for K ∈ max{K(θ), 28/5}, min{K̄(θ), 28} .
Proof. See the appendix.
Figure 1 depicts the results presented in Proposition 1, which can be interpreted as
follows. First, the most profitable merger is not a decentralized one if cultural or managerial
differences are not small, i.e., if θ > 25/24. For instance, suppose that the merging firms’
top managers cannot agree among each other about the type of equipments to be installed in
the target firm or the nature of the training required for the target firm’s employees, or that
the target firm’s employees do not accept the procedures imposed by the new management.
Then, it becomes too costly to transfer the low-cost technology to the high-cost firm even if
13
it is small or if the marginal production cost differential is small. Consequently, it is more
profitable to merge with a rival and shut it down. Antitrust authorities should, therefore,
be suspicious at claims about potential efficiencies resulting from a merger in these cases.
K
4" # 3
" #1
K(" )
!
!
28
2(8 + 3 3)
!
({1,3},2,0) " merger
({1,3},2,0) " merger
!
2(8 "!
3 3)
28/5
!
({1,2},2,0) " merger
K(" )
5
1
361
360
25
24
!
!
"
2
!
!
!
Figure 1: Privately Optimal Mergers
Second, suppose that θ is relatively small. Then a decentralized merger is more profitable
than a centralized merger if the target firm is small pre-merger, i.e., is sufficiently large.
Based on the work of Kreps and Scheinkman (1983), a small output pre-merger translates in
14
a small production capacity and/or a relatively small workforce. This, in turn, implies that it
should not be too costly to change the target firm’s equipment or to start a training program
for its employees. Tombak (2002)’s Lemma 1, in comparison, states that a technology transfer
does not take place when the merging firms are small in output.
Third, Proposition 1 also suggests that when the cost differential between merging firms
is small, i.e., is small, a decentralized merger is more profitable than a centralized merger
if θ is relatively small. It should not be costly to adapt the target firm’s technology to the
one of the acquiring firm when they both use initially very similar production processes. A
similar result is found in Tombak (2002)’s Lemma 1.
Fourth, according to Proposition 1 the merging firms may find it profitable to centralize
their operations if is large. Tombak (2002) finds in his Lemma 3, however, that centralization emerges at an equilibrium post-merger only when the industry becomes a monopolist.
The idea explaining our result is easy to understand. When the marginal production cost
differential is large, the inefficient firm, in this case firm 3, cannot successfully engage in
business-stealing. The merged firm, which consists of firms 1 and 2, can therefore exercise
its market power relatively unchecked.
Finally, Tombak (2002)’s Proposition 2 states that “In the absence of consideration of any
subsequent rounds of acquisitions, the incentive to acquire is greatest for the most efficient
(largest) firm, and that firm would prefer to acquire the most efficient (largest) rival firms...”
(p. 529) We also find that the most efficient and largest firm participates in all possible
equilibrium mergers. But the low-cost firm can either merge with the next most efficient
firm or with the least efficient firm depending on the values taken by the different parameters.
15
3.3
Welfare Impacts
We now discuss the impact of the various mergers on welfare and, in particular, the tradeoffs
between the post-merger efficiencies and the increased market power.
Proposition 2. When the privately optimal merger is the
1. ({1, 2}, 2, 0)-merger, it always decreases total surplus;
2. ({1, 3}, 2, 0)-merger, it increases total surplus for K ∈ [28/5, 100/7] and it decreases
total surplus for K ∈ 100/7, min{K̄(θ), 28} ;
3. ({1, 3}, 3, 1)-merger, it always increases total surplus.
Proof. See the appendix.
Few remarks are in order. First, when a merger leads to a decrease in total surplus it
also results in a decrease in consumer surplus.
Second, when the privately optimal merger results in the shut down of the inefficient
firm, there is a transfer of production from the inefficient firm to the efficient one. This shift
in production increases total surplus only when the merger is between the most and the least
efficient firms and when the marginal production cost differential (measured by the parameter
) is not too low. It is only in this case, when the inefficient firm is sufficiently inefficient,
that the increase in post-merger market power is offset by the increase in efficiency.
Third, consumer surplus and total surplus both increase when the privately optimal
merger involves a technology transfer from the most efficient firm to the least efficient one.
In these circumstances, when all firms remain in the market, assets divestitures imposed by
16
antitrust authorities to alleviate anticompetitive concerns could have the opposite effects as
they could prevent the realization of efficiencies.
4
Conclusion
Merging firms are unlikely to be identical. They can differ in terms of cultural or managerial
background or of cost-efficiency and size. Our model encompasses these dimensions. Furthermore, it allows the transfer of technology between merging firms or the closure of one of
the merged firm’s divisions.
We find that the privately optimal merger, that is the profitable merger generating the
greatest private surplus, always involves the most efficient firm. The latter will merge with
and transfer its technology to the least efficient firm only when cultural or managerial differences are relatively small. Otherwise, the most efficient firm will merge with and shut down
the second most efficient firm or the least efficient one depending on the size of the target
firm and of the technological difference.
When the merger results in the transfer of technology, welfare, measured by total surplus,
increases compared to the pre-merger situation. In such a case, antitrust remedies, if imposed, can prevent an increase in welfare. When the target firm is shut down post-merger,
the increase in market power coming from the reduction in the number of firms must be
offset by the reallocation of production from the inefficient (and closed) firm to the efficient
one for welfare to increase. This scenario arises only when the merger involves the most
and least efficient firms and when the difference between the marginal production costs is
sufficiently large.
17
Future research could focus on situations where managerial or cultural differences vary
across mergers. Indeed, we have assumed that the parameter capturing such considerations
is the same for any merger. Introducing a merger specific parameter could be corollated,
for example, to some measure of spatial differentiation. Managerial or cultural differences
could be smaller (larger) for mergers involving firms located close to (far from) each other or
producing highly substitutable (differentiated) goods. However, the results we have found
would be modified only if the parameter associated with the merger between firms 1 and 3
would become sufficiently high compare to the parameters associated with the other mergers.
Another possibility for future work would be to estimate empirically the parameter θ,
which captures cultural and managerial differences between merging firms. For instance, the
distance between or location of headquarters or factories (e.g., Europe v. US), differences in
languages (e.g., French v. English), or in managerial pay (e.g., in terms of size or incentives)
could all play a role in facilitating or hindering the ease of merging. Understanding the key
variables of success would facilitate the work of antitrust authorities in identifying mergers
that are more likely to deliver the promised efficiencies.
18
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19
A
Proof of Lemma 1
Pre-merger, firm i’s profit is πi (∅, 3, 0) = [qi (∅, 3, 0)]2 with i = 1, 2, 3. If firm 1 merges with
and shuts down firm 2, then πi ({1, 2}, 2, 0) = [qi ({1, 2}, 2, 0)]2 for i = 1, 3. The ({1, 2}, 2, 0)merger is profitable if ∆ ({1, 2}, 2, 0) = π1 ({1, 2}, 2, 0) − [π1 (∅, 3, 0) + π2 (∅, 3, 0)] > 0,
which is the case when K ∈ (5, 12]. If firm 1 merges with and shuts down firm 3, then
πi ({1, 3}, 2, 0) = [qi ({1, 3}, 2, 0)]2 for i = 1, 2. The ({1, 3}, 2, 0)-merger is profitable if
∆ ({1, 3}, 2, 0) = π1 ({1, 3}, 2, 0) − [π1 (∅, 3, 0) + π2 (∅, 3, 0)] > 0, which is the case when K ∈
(5, 28]. If firm 2 merges with and shuts down firm 3, then πi ({2, 3}, 2, 0) = [qi ({2, 3}, 2, 0)]2
for i = 1, 2. The ({2, 3}, 2, 0)-merger is profitable if ∆ ({2, 3}, 2, 0) = π2 ({2, 3}, 2, 0) −
[π2 (∅, 3, 0) + π2 (∅, 3, 0)] > 0, which is the case when K ∈ (5, 16]. Because ∆ ({1, 3}, 2, 0) >
∆ ({2, 3}, 2, 0) for K ∈ (5, 28], the ({1, 3}, 2, 0)-merger is profitable and more profitable than
the ({1, 3}, 2, 0)-merger for K ∈ (5, 28]. Because ∆ ({1, 2}, 2, 0) > ∆ ({2, 3}, 2, 0) for K ∈
(5, 8], the ({1, 2}, 2, 0)-merger is profitable and more profitable than the ({2, 3}, 2, 0)-merger
for K ∈ (5, 8]. For K ∈ [8, 16], the ({2, 3}, 2, 0)-merger is profitable and more profitable
than the ({1, 2}, 2, 0)-merger. Because ∆ ({1, 3}, 2, 0) > ∆ ({1, 2}, 2, 0) for K ∈ [28/5, 28],
the ({1, 3}, 2, 0)-merger is profitable and more profitable than the ({1, 2}, 2, 0)-merger for
K ∈ [28/5, 28]. For K ∈ (5, 28/5], the ({1, 2}, 2, 0)-merger is profitable and more profitable
than the ({1, 3}, 2, 0)-merger. Hence, Lemma 1.
B
Proof of Lemma 2
A decentralized merger between the low-cost firm i and the high-cost firm j is profitable if
[πi ({i, j}, 3, 1) + πj ({i, j}, 3, 1) − θ(cj − ci )qj (∅, 3, 0)]−[πi (∅, 3, 0) + πj (∅, 3, 0)] ≥ 0. We refer to this difference as ∆ ({i, j}, 3, 1). The ({1, 2}, 3, 1)-merger is profitable if ∆ ({1, 2}, 3, 1) ≥
0, which is the case when K ≤ 1/[2(θ−1)] = K ({1, 2}, 3, 1). The ({1, 3}, 3, 1)-merger is profitable if ∆ ({1, 3}, 3, 1) ≥ 0, which is the case when K ≤ (4θ−3)/(θ−1) = K ({1, 3}, 3, 1) with
K ({1, 3}, 3, 1) ≥ 5 for θ ∈ [1, 2]. The ({2, 3}, 3, 1)-merger is profitable if ∆ ({2, 3}, 3, 1) ≥ 0,
which is the case when K ≤ (8θ − 7)/2[(θ − 1)] = K ({2, 3}, 3, 1). Because ∆ ({2, 3}, 3, 1) ≥
∆ ({1, 2}, 3, 1) for θ ≥ 1, the ({2, 3}, 3, 1)-merger is more profitable than the ({1, 2}, 3, 1)merger for θ ≥ 1. Because ∆ ({1, 3}, 3, 1) ≥ ∆ ({1, 2}, 3, 1) for K ≤ (16θ − 13)/[2(θ − 1)],
the ({1, 3}, 3, 1)-merger is more profitable than the ({1, 2}, 3, 1)-merger for K ≤ (16θ −
13)/[2(θ − 1)]. Since (16θ − 13)/[2(θ − 1)] > K ({1, 3}, 3, 1) > K ({1, 2}, 3, 1) for θ ≥ 1, the
({1, 3}, 3, 1)-merger is more profitable than the ({1, 2}, 3, 1)-merger when they are both profitable. Because ∆ ({1, 3}, 3, 1) ≥ ∆ ({2, 3}, 3, 1) for K ≤ (8θ − 5)/[2(θ − 1)], the ({1, 3}, 3, 1)merger is more profitable than the ({2, 3}, 3, 1)-merger for K ≤ (8θ − 5)/[2(θ − 1)]. Since
(8θ − 5)/[2(θ − 1)] > K ({1, 3}, 3, 1) > K ({2, 3}, 3, 1) for θ ≥ 1, the ({1, 3}, 3, 1)-merger
is more profitable than the ({2, 3}, 3, 1)-merger when they are both profitable. This leaves
the ({1, 3}, 3, 1)-merger as the most profitable merger. However, it becomes unprofitable for
θ ≥ 2 and K ≥ 5. Hence, Lemma 2.
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C
Proof of Proposition 1
From Lemmas 1 and 2, we only need to compare the ({1, 3}, 3, 1)-merger with the ({1, 2}, 2, 0)merger and the ({1, 3}, 2, 0)-merger in the relevant range for K and θ. For K ∈ (5, 28/5],
the most profitable centralized merger is the ({1, 2}, 2, 0)-merger. The difference between
∆ ({1, 3}, 3, 1) and ∆ ({1, 2}, 2, 0) simplifies as (K − 6), which is less than zero for K ∈
(5, 28/5] when θ = 1. When θ increases, ∆ ({1, 3}, 3, 1) decreases and ∆ ({1, 2}, 2, 0) does not
change. For K ∈ [28/5, 28], the most profitable centralized merger is the ({1, 3}, 2, 0)-merger.
The difference between ∆ ({1, 3}, 3, 1) and ∆ ({1, 3}, 2, 0) simplifies as (K + 2)2 − 36θ(K −
with K(θ) =
4). This difference
is positive for K ∈ [28/5,
p
K(θ)] when θ ∈ [1, 361/360]
(18θ − 2) − 6 3θ (3θ − 2) and for all K ∈ K̄(θ), (4θ − 3)/(θ − 1) when θ ∈ [1, 25/24]
p
with K̄(θ) = (18θ − 2) + 6 3θ (3θ − 2). K(θ) is decreasing in θ and equals 28/5 when
θ = 361/360. K̄(θ) is increasing in θ, equals 28 at θ = 25/24, and equals K ({1, 3}, 3, 1) at
θ = 25/24. Hence, Proposition 1.
D
Proof of Proposition 2
Pre-merger total surplus can be expressed as T S (∅, 3, 0) = (64 + 15K 2 )/32. Total surplus
after the ({1, 2}, 2, 0)-merger can be written as T S ({1, 2}, 2, 0) = (18 + 4K 2 )/9, which is
less than or equal to T S (∅, 3, 0) for K ∈ (5, 28/5]. Total surplus after the ({1, 3}, 2, 0)merger can be written as T S ({1, 3}, 2, 0) = (11 + 8K + 8K 2 )/18, which is greater than or
equal to T S (∅, 3, 0) for K ∈ [28/5, 100/7] but less than or equal to T S (∅, 3, 0) for K ∈
[100/7, 28]. Total surplus after the ({1, 3}, 3, 1)-merger can be written as T S ({1, 3}, 3, 1) =
(28 + 64θ − 16θK + 20K + 15K 2 )/32. It is greater than or equal to T S (∅, 3, 0) if K ≤
(16θ − 9)/(4θ − 5) = K̃(θ). However, K̃(θ) > 0 only when θ > 5/4. In such a case, K̃(θ)
is greater than K ({1, 3}, 3, 1), which implies that T S ({1, 3}, 3, 1) > T S (∅, 3, 0) when the
privately optimal merger is the ({1, 3}, 3, 1)-merger as long as θ > 5/4. When θ ≤ 5/4,
T S ({1, 3}, 3, 1) > T S (∅, 3, 0) for K > 5. Hence, Proposition 2.
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