Chukyo University Institute of Economics
Discussion Paper Series
December 2021

No. 2103
A Note on Leading Mergers with Free Entry

Yoshitaka Ishino and Yoshihiro Tomaru

A Note on Leading Mergers with Free Entry
Yoshitaka Ishino
Graduate School of Economics, Chukyo University
Yoshihiro Tomaru †School of Economics, Chukyo University

Abstract
This article analyzes the leading mergers that give insiders the strategic advantage ofbecoming a Stackelberg leader in free-entry markets. It is shown that mergers enhance welfare
if and only if they are profitable. It is also shown that the welfare-deteriorating leading mergers
do not appear in equilibrium if the entry costs are sufficiently large.

JEL Classification: L13; L22
Keywords: Horizontal mergers; Stackelberg competition; Free entry†Corresponding author’s address: School of Economics, Chukyo University, 101-2 Yagoto-Honmachi,Showa, Nagoya 466-8666, Japan; E-mail: yoshihiro.tomaru@gmail.com; Tel: +81 5 2835 7111
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1. Introduction
How is merger profitability affected by market structure? This issue has been extensively
discussed in the growing literature on horizontal mergers. Salant et al. (1983) showed thatmergers are likely to be harmful to insiders under Cournot competition unless they involve avast majority of industry participants. However, this result depends crucially on competition
structures (or the order of firms’ moves). Using a Stackelberg model, Huck et al. (2004) showedthat the merger between a leader and a follower is profitable regardless of the number ofoutsiders. Heywood and McGinty (2008) also proved the robustness of their results after
considering convex costs.
As Davidson and Mukherjee (2007) point out, competitiveness (or the number of firms) isalso important in merger profitability. They demonstrated that horizontal mergers amongincumbent firms are always profitable in free-entry markets when both incumbent firms andentrants incur fixed costs of production. The result might differ if entry costs are incurred.
Davidson and Mukherjee (2007) have also shown that merger profitability relies on the degreeof cost synergy that arises from mergers. In particular, mergers with dramatic cost synergy arebeneficial for insiders.
As shown in the studies above, competition structures and competitiveness have aninfluence on merger profitability. However, to the best of our knowledge, no studies have everexamined the influence of both these factors at the same time. The main purpose of this study
is to address how the endogenous market structure, which is induced by horizontal mergers,
influences merger profitability. Our study is most similar to that of Liu and Wang (2015). In
their model, the merger, or the leading merger in their terminology, allowed insiders to enjoy
the strategic advantage of becoming a Stackelberg leader. They found that the leading merger
was always profitable when the number of firms is exogenous. We built off their study byintruding free entry and endogenizing the order of firms’ moves and the number of firms toexamine how competition structures and competitiveness affect merger profitability. Our
2results show that the leading merger does not benefit if the entry costs are sufficiently large,
which is in sharp contrast to the findings of Liu and Wang (2015). Furthermore, our studies
also show that profitable leading mergers are always welfare-enhancing.

2. ModelConsider a homogeneous-good market with infinite potential entrants, which can enter themarket by incurring a fixed cost 𝑓𝑓 > 0. In the market, there are two groups of firms (groups Iand O), the members of which are indexed by 𝑆𝑆𝐼𝐼 = {1,2, ⋯ , 𝑛𝑛𝐼𝐼 } ( 𝑛𝑛𝐼𝐼 ≥ 2 ) and 𝑆𝑆𝑂𝑂 ={𝑛𝑛𝐼𝐼 + 1, 𝑛𝑛𝐼𝐼 + 2, ⋯ , 𝑛𝑛𝐼𝐼 + 𝑛𝑛𝑂𝑂 }, and are producing at a constant marginal cost 𝑐𝑐 ≥ 0. Group I isa set of incumbent firms that are weighing the option to merge, whereas group O is a set ofentrants in the market. It is assumed that no entrants merge. Following previous studies, werefer to the members of group I as insiders and those of group O as outsiders.
Let the output of firm 𝑖𝑖 ∈ 𝑆𝑆𝐼𝐼 ∪ 𝑆𝑆𝑂𝑂 be denoted by 𝑞𝑞𝑖𝑖 . Likewise, the total output of group𝑘𝑘 ∈ {𝐼𝐼, 𝑂𝑂} is denoted by 𝑞𝑞𝑘𝑘 , that is, 𝑞𝑞𝑘𝑘 = ∑𝑖𝑖∈𝑆𝑆𝑘𝑘 𝑞𝑞𝑖𝑖 . The inverse demand function for the goodis given by 𝑃𝑃(𝑞𝑞) , where 𝑞𝑞 = 𝑞𝑞𝐼𝐼 + 𝑞𝑞𝑂𝑂 is the industry output. The following assumption ismade:Assumption 1. There exists a positive real number 𝑞𝑞� such that 𝑃𝑃(𝑞𝑞) = 0 if 𝑞𝑞 ≥ 𝑞𝑞� and𝑃𝑃(𝑞𝑞) > 0 otherwise. Furthermore, for any 𝑞𝑞 ∈ [0, 𝑞𝑞�) , 𝑃𝑃(𝑞𝑞) is three-times continuously
differentiable with 𝑃𝑃′(𝑞𝑞) < 0.

The profit of firm 𝑖𝑖 is given by[𝑃𝑃(𝑞𝑞) − 𝑐𝑐]𝑞𝑞𝑖𝑖
Π 𝑖𝑖 (𝑞𝑞𝑖𝑖 , 𝑞𝑞−𝑖𝑖 , 𝑓𝑓) = �[𝑃𝑃(𝑞𝑞) − 𝑐𝑐]𝑞𝑞𝑖𝑖 − 𝑓𝑓𝑖𝑖𝑖𝑖 𝑖𝑖 ∈ 𝑆𝑆𝐼𝐼 ,𝑖𝑖𝑖𝑖 𝑖𝑖 ∈ 𝑆𝑆𝑂𝑂 ,where 𝑞𝑞−𝑖𝑖 = ∑𝑗𝑗≠𝑖𝑖 𝑞𝑞𝑗𝑗 . The profits of groups I and O are Π 𝐼𝐼 (𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 ) ≡ ∑𝑖𝑖∈𝑆𝑆𝐼𝐼 Π 𝑖𝑖 (𝑞𝑞𝑖𝑖 , 𝑞𝑞−𝑖𝑖 , 𝑓𝑓) andΠ 𝑂𝑂 (𝑞𝑞𝑂𝑂 , 𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 , 𝑓𝑓) ≡ ∑𝑖𝑖∈𝑆𝑆𝑂𝑂 Π 𝑖𝑖 (𝑞𝑞𝑖𝑖 , 𝑞𝑞−𝑖𝑖 , 𝑓𝑓), respectively. The welfare is𝑞𝑞

𝑊𝑊(𝑞𝑞, 𝑛𝑛𝑂𝑂 , 𝑓𝑓) = � 𝑃𝑃(𝑧𝑧)𝑑𝑑𝑑𝑑 − 𝑐𝑐𝑐𝑐 − 𝑛𝑛𝑂𝑂 𝑓𝑓.
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We analyze leading mergers using the following game. In stage 1, insiders decide onwhether to merge. Once they agree to the merger, all insiders are integrated into one firm with
one plant. In stage 2, potential entrants decide on whether to enter the market. As in Liu andWang (2015), we assume that the merger gives insiders a strategic advantage in this game. To
materialize it, we assume that stage 2 is followed by different subsequent subgames inaccordance with the insiders’ decisions in stage 1. If the merger is agreed upon, the merged
firm commits to its output as a Stackelberg leader, while the outsiders choose their outputs asfollowers after observing the decision by the merged firm. If the merger falls through, both
insiders and outsiders engage in Cournot competition. For convenience, we refer to a subgame
consisting of both stage 2 and its subsequent subgame with Cournot (res. Stackelberg)
competition as the Cournot (res. Stackelberg) game.
To solve for the subgame perfect Nash equilibrium (SPNE), the usual method begins bycharacterizing the equilibrium of the output-setting stage before proceeding to the entry stage.
However, we deal with both stages simultaneously in this study. For the preparation, weprovided some definitions and assumptions.
Definition. For 𝑘𝑘, 𝑘𝑘 ′ ∈ {𝐼𝐼, 𝑂𝑂} (𝑘𝑘 ≠ 𝑘𝑘 ′ ), 1(a) 𝑞𝑞𝑘𝑘 = 𝑅𝑅𝑘𝑘 (𝑞𝑞𝑘𝑘 ′ , 𝑛𝑛𝑘𝑘 ) such that𝑛𝑛𝑘𝑘 𝑃𝑃(𝑅𝑅𝑘𝑘 (𝑞𝑞𝑘𝑘 ′ , 𝑛𝑛𝑘𝑘 ) + 𝑞𝑞𝑘𝑘 ′ ) + 𝑃𝑃′ (𝑅𝑅𝑘𝑘 (𝑞𝑞𝑘𝑘 ′ , 𝑛𝑛𝑘𝑘 ) + 𝑞𝑞𝑘𝑘 ′ )𝑅𝑅𝑘𝑘 (𝑞𝑞𝑘𝑘 ′ , 𝑛𝑛𝑘𝑘 ) − 𝑛𝑛𝑘𝑘 𝑐𝑐 = 0,

� 𝐼𝐼 (𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 ) ≡ Π 𝐼𝐼 �𝑞𝑞𝐼𝐼 , 𝑅𝑅𝑂𝑂 (𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 )�,(b) Π(c) 𝐺𝐺(𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 , 𝑛𝑛𝑂𝑂 ) ≡ Π1𝐼𝐼 (𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 ) + Π2𝐼𝐼 (𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 )𝑅𝑅1𝑂𝑂 (𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 ), and� 𝐼𝐼 (𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 ).
(d) 𝑄𝑄�𝐼𝐼 (𝑛𝑛𝑂𝑂 ) = argmax𝑞𝑞𝐼𝐼 ∈[0,𝑞𝑞�) Π𝑖𝑖 (𝑞𝑞′′′Assumption 2. Π12�) and 𝑞𝑞𝑖𝑖 ∈ [0, 𝑞𝑞).
𝑖𝑖 , 𝑞𝑞−𝑖𝑖 , 𝑓𝑓) = 𝑃𝑃 (𝑞𝑞) + 𝑃𝑃 (𝑞𝑞)𝑞𝑞𝑖𝑖 for any 𝑞𝑞 ∈ (0, 𝑞𝑞Assumption 3. 𝐺𝐺 satisfies the following:
1We use subscripts to indicate the partial derivatives of multivariable functions. For instance,𝑖𝑖 (𝑞𝑞2𝑖𝑖
Π12𝑖𝑖 , 𝑞𝑞−𝑖𝑖 , 𝑓𝑓) = (𝜕𝜕 ⁄𝜕𝜕 𝑞𝑞−𝑖𝑖 𝜕𝜕𝑞𝑞𝑖𝑖 )Π (𝑞𝑞𝑖𝑖 , 𝑞𝑞−𝑖𝑖 , 𝑓𝑓).
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Π3𝑖𝑖 (𝑞𝑞𝑖𝑖 , 𝑞𝑞−𝑖𝑖 , 𝑓𝑓) = (𝜕𝜕⁄𝜕𝜕 𝑓𝑓)Π 𝑖𝑖 (𝑞𝑞𝑖𝑖 , 𝑞𝑞−𝑖𝑖 , 𝑓𝑓) and(a) 𝐺𝐺1 (𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 , 𝑛𝑛𝑂𝑂 ) < 0, 𝐺𝐺2 (𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 , 𝑛𝑛𝑂𝑂 ) < 0, and |𝐺𝐺1 (𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 , 𝑛𝑛𝑂𝑂 )| > |𝐺𝐺2 (𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 , 𝑛𝑛𝑂𝑂 )|, and(b) 𝐺𝐺2 (𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 , 𝑛𝑛𝑂𝑂 )𝑅𝑅2𝑂𝑂 (𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 ) + 𝐺𝐺3 (𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 , 𝑛𝑛𝑂𝑂 ) < 0 and 𝐺𝐺3 (𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 , 𝑛𝑛𝑂𝑂 ) ≥ 0.

� 𝐼𝐼 and 𝑄𝑄�𝐼𝐼 are the merged
𝑅𝑅𝑘𝑘 is the aggregate reaction function of group 𝑘𝑘 ∈ {𝐼𝐼, 𝑂𝑂} . Πfirm’s objective function and its profit-maximizing output respectively. Note that�1𝐼𝐼 (𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 ) . This equation allows us to express the first-order
𝐺𝐺(𝑞𝑞𝐼𝐼 , 𝑅𝑅𝑂𝑂 (𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 ), 𝑛𝑛𝑂𝑂 ) = Πcondition of the merged firm by 𝐺𝐺(𝑞𝑞𝐼𝐼 , 𝑅𝑅𝑂𝑂 (𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 ), 𝑛𝑛𝑂𝑂 ) = 0. Assumption 2 ensures the second-𝑖𝑖 (𝑞𝑞𝑘𝑘order condition Π11𝑖𝑖 , 𝑞𝑞−𝑖𝑖 , 𝑓𝑓) < 0 and the strategic substitution 𝑅𝑅1 (𝑞𝑞𝑘𝑘 ′ , 𝑛𝑛𝑘𝑘 ) ∈ (−1,0)when outsiders exist in the market (i.e., 𝑛𝑛𝑂𝑂 ≥ 1). Assumption 3 (a), which is borrowed fromGal-Or (1985), ensures that the second-order condition of the Stackelberg leader holds and that

the Stackelberg equilibrium is stable. Assumption 3 (b) is a natural assumption that themarginal profit will decrease as the number of entrants increase.
The equilibrium of the Cournot game is characterized by the optimality conditions and thezero-profit

condition,

thatis,𝑞𝑞𝑘𝑘 = 𝑅𝑅𝑘𝑘 (𝑞𝑞𝑘𝑘 ′ , 𝑛𝑛𝑘𝑘 )( 𝑘𝑘, 𝑘𝑘 ′ ∈ {𝐼𝐼, 𝑂𝑂}, 𝑘𝑘 ≠ 𝑘𝑘 ′ )andΠ 𝑂𝑂 (𝑞𝑞𝑂𝑂 , 𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 , 𝑓𝑓) = 0. Likewise, the equilibrium of the Stackelberg game is characterized by𝐺𝐺(𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 , 𝑛𝑛𝑂𝑂 ) = 0, 𝑞𝑞𝑂𝑂 = 𝑅𝑅𝑂𝑂 (𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 ), and Π 𝑂𝑂 (𝑞𝑞𝑂𝑂 , 𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 , 𝑓𝑓) = 0. Let the former (res. latter)equilibrium

bedenoted

by(𝑞𝑞𝐼𝐼 , 𝑞𝑞𝑂𝑂 , 𝑛𝑛𝑂𝑂 ) = �𝑄𝑄𝐼𝐼∗ (𝑓𝑓), 𝑄𝑄𝑂𝑂∗ (𝑓𝑓), 𝑁𝑁 𝑂𝑂∗ (𝑓𝑓)�

(res.

�𝑄𝑄𝐼𝐼∗∗ (𝑓𝑓), 𝑄𝑄𝑂𝑂∗∗ (𝑓𝑓), 𝑁𝑁 𝑂𝑂∗∗ (𝑓𝑓)�). As observed, the Cournot and Stackelberg games share the same

conditions 𝐹𝐹 𝑂𝑂 (𝑞𝑞𝑂𝑂 , 𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 ) = Π 𝑂𝑂 (𝑞𝑞𝑂𝑂 , 𝑞𝑞𝐼𝐼 , 𝑛𝑛𝑂𝑂 , 𝑓𝑓) = 0. Shedding light on this, let the solution tothe equation system be denoted by (𝑞𝑞𝑂𝑂 , 𝑛𝑛𝑂𝑂 ) = �𝐻𝐻 𝑞𝑞 (𝑞𝑞𝐼𝐼 , 𝑓𝑓), 𝐻𝐻 𝑛𝑛 (𝑞𝑞𝐼𝐼 , 𝑓𝑓)�.
𝑞𝑞Lemma 1. From the implicit function theorem, it follows that 𝐻𝐻1 (𝑞𝑞𝐼𝐼 , 𝑓𝑓) = −1, 𝐻𝐻1𝑛𝑛 (𝑞𝑞𝐼𝐼 , 𝑓𝑓) <𝑞𝑞0, 𝐻𝐻2 (𝑞𝑞𝐼𝐼 , 𝑓𝑓) < 0, and 𝐻𝐻2𝑛𝑛 (𝑞𝑞𝐼𝐼 , 𝑓𝑓) < 0.
𝑞𝑞

The inequalities 𝐻𝐻1𝑛𝑛 < 0 , 𝐻𝐻2 < 0 , and 𝐻𝐻2𝑛𝑛 < 0 are so intuitive that it requires no𝑞𝑞explanation. More interestingly, 𝐻𝐻1 = −1. This equality implies that the industry output isinvariable regardless of whether the insiders become a Stackelberg leader through a leadingmerger. This prominent feature is a key factor in the main results.

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3. Results
After the preliminaries, we now present our main results in order.
Proposition 1. When there is free entry by outsiders, the following are equivalent:(a) The leading merger is profitable.
(b) 𝑄𝑄𝐼𝐼∗ (𝑓𝑓) < 𝑄𝑄𝐼𝐼∗∗ (𝑓𝑓).

(c) 𝑁𝑁 𝑂𝑂∗ (𝑓𝑓) > 𝑁𝑁 𝑂𝑂∗∗ (𝑓𝑓).

(d) 𝑊𝑊(𝑄𝑄𝐼𝐼∗ (𝑓𝑓) + 𝑄𝑄𝑂𝑂∗ (𝑓𝑓), 𝑁𝑁 𝑂𝑂∗ (𝑓𝑓), 𝑓𝑓) < 𝑊𝑊(𝑄𝑄𝐼𝐼∗∗ (𝑓𝑓) + 𝑄𝑄𝑂𝑂∗∗ (𝑓𝑓), 𝑁𝑁 𝑂𝑂∗∗ (𝑓𝑓), 𝑓𝑓).

Proof: Invoking Lemma 1 (a), let 𝑄𝑄(𝑓𝑓) be the equilibrium industry output of both the Cournotand Stackelberg games. It is apparent that for Π 𝐼𝐼∗ (𝑓𝑓) ≡ Π 𝐼𝐼 �𝑄𝑄𝐼𝐼∗ (𝑓𝑓), 𝑄𝑄𝑂𝑂∗ (𝑓𝑓)� and Π 𝐼𝐼∗∗ (𝑓𝑓) ≡Π 𝐼𝐼 �𝑄𝑄𝐼𝐼∗∗ (𝑓𝑓), 𝑄𝑄𝑂𝑂∗∗ (𝑓𝑓)�, there holdsΠ 𝐼𝐼∗∗ (𝑓𝑓) > Π 𝐼𝐼∗ (𝑓𝑓) ⇔ �𝑃𝑃�𝑄𝑄(𝑓𝑓)� − 𝑐𝑐�𝑄𝑄𝐼𝐼∗∗ (𝑓𝑓) > �𝑃𝑃�𝑄𝑄(𝑓𝑓)� − 𝑐𝑐�𝑄𝑄𝐼𝐼∗ (𝑓𝑓),⇔ 𝑄𝑄𝐼𝐼∗∗ (𝑓𝑓) > 𝑄𝑄𝐼𝐼∗ (𝑓𝑓).

Equivalence to (b) and (c) comes from Propositions 1 (a) and 1 (a). ∎Proposition 1 suggests that a profitable merger is always welfare-improving. Thiscounterintuitive result is explained by the excess entry. Given that the industry output isinvariant, merger profitability is determined only by market share. If the merger gives a higher
market share to the insiders, the outsiders should reduce their outputs due to strategic
substitution, thus confronting higher average costs. This encourages outsiders to exit the market.
Accordingly, by alleviating excess entry, the leading merger improves welfare.
Proposition 1 indicates that socially undesirable mergers do not occur because such mergers
are not profitable. This does not mean that insiders necessarily disagree with the merger in theSPNE. Unfortunately, our general model prevents us from deriving the necessary and sufficientconditions for insiders to disagree with the leading merger. Nevertheless, we can propose asufficient condition.
6

Proposition 2. Suppose that in the Stackelberg equilibrium with one leader and one follower,
the follower earns a positive profit. The insiders never merge in the SPNE if 𝑓𝑓 is sufficientlylarge.

Proof: Consider 𝑓𝑓̃ such that 𝑁𝑁 𝑂𝑂∗ �𝑓𝑓̃� = 1. Since Π 𝑂𝑂 �𝑅𝑅𝑂𝑂 (𝑞𝑞𝐼𝐼 , 1), 𝑞𝑞𝐼𝐼 , 1, 𝑓𝑓̃� is decreasing in 𝑞𝑞𝐼𝐼 ,the inequality,

Π 𝑂𝑂 �𝑅𝑅𝑂𝑂 �𝑄𝑄�𝐼𝐼 (1), 1�, 𝑄𝑄�𝐼𝐼 (1), 1, 𝑓𝑓̃� > Π 𝑂𝑂 �𝑅𝑅𝑂𝑂 �𝑄𝑄𝐼𝐼∗ �𝑓𝑓̃�, 1�, 𝑄𝑄𝐼𝐼∗ �𝑓𝑓̃�, 1, 𝑓𝑓̃� = 0,implies that 𝑄𝑄�𝐼𝐼 (1) < 𝑄𝑄𝐼𝐼∗ �𝑓𝑓̃� . Next, we prove that 𝑄𝑄� 𝐼𝐼 (1) > 𝑄𝑄𝐼𝐼∗∗ �𝑓𝑓̃� . Define Φ(𝑞𝑞𝐼𝐼 ) ≡𝐻𝐻 𝑞𝑞 �𝑞𝑞𝐼𝐼 , 𝑓𝑓̃� − 𝑅𝑅𝑂𝑂 �𝑞𝑞𝐼𝐼 , 𝑄𝑄�𝐼𝐼−1 (𝑞𝑞𝐼𝐼 )� , where 𝑄𝑄�𝐼𝐼−1 is the inverse function of 𝑄𝑄�𝐼𝐼 . Using thisdefinition, we obtain Φ �𝑄𝑄𝐼𝐼∗∗ �𝑓𝑓̃�� = 0 . It follows from the definition of 𝐻𝐻 𝑞𝑞 and thepresumption in Proposition 2 that 0 = 𝑃𝑃 �𝑄𝑄� 𝐼𝐼 (1) + 𝐻𝐻 𝑞𝑞 �𝑄𝑄�𝐼𝐼 (1), 𝑓𝑓̃�� − 𝑐𝑐 < 𝑃𝑃 �𝑄𝑄� 𝐼𝐼 (1) +𝑅𝑅𝑂𝑂 �𝑄𝑄�𝐼𝐼 (1), 1�� − 𝑐𝑐. Accordingly, we obtained Φ �𝑄𝑄�𝐼𝐼 (1)� > 0. Moreover,Φ′ (𝑞𝑞𝐼𝐼 ) = −1 +𝐺𝐺1 𝑅𝑅2𝑂𝑂 − 𝐺𝐺3 𝑅𝑅1𝑂𝑂
𝐺𝐺2 𝑅𝑅2𝑂𝑂 + 𝐺𝐺3

𝐺𝐺2 𝑅𝑅2𝑂𝑂 + 𝐺𝐺3> −1 +𝐺𝐺2 𝑅𝑅2𝑂𝑂 + 𝐺𝐺3by Assumption 3=0Thus, by the intermediate value theorem, we have 𝑄𝑄�𝐼𝐼 (1) > 𝑄𝑄𝐼𝐼∗∗ �𝑓𝑓̃�, which yields 𝑄𝑄𝐼𝐼∗ �𝑓𝑓̃� >𝑄𝑄𝐼𝐼∗∗ �𝑓𝑓̃�. This inequality holds for large entry costs close to 𝑓𝑓̃ by continuity of 𝑄𝑄𝐼𝐼∗ �𝑓𝑓̃� and𝑄𝑄𝐼𝐼∗∗ �𝑓𝑓̃�. Therefore, using Proposition 1, we can conclude that the merger is not profitable withlarge entry costs. ∎The intuition behind Proposition 2 is as follows. The large entry costs discourage outsidersfrom entering the market under Cournot competition. Suppose then that the number of outsiders
remains the same even after the leading merger forms. By the presumption of Proposition 2,
positive profits induce outsiders to enter the market. This works to substitute the output of themerged firm with the output of the entrants. Consequently, the merged firm loses its market
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share, and thus its profit is reduced. Again, this comes from the fact that industry output isinvariant before and after the leading merger.
We conclude this article by making two remarks on Proposition 2.
Remark 1. As suggested in Proposition 2, it is possible that the leading merger is not profitable.
This is in sharp contrast with Liu and Wang’s (2015) result that the merger is always profitable.
This contrast may be emphasized in the linear demand model with 𝑃𝑃(𝑞𝑞) = 𝑎𝑎 − 𝑞𝑞 .

Straightforward computation shows that Π 𝐼𝐼∗ (𝑓𝑓) = 𝑛𝑛𝐼𝐼 𝑓𝑓 and Π 𝐼𝐼∗∗ (𝑓𝑓) = (𝑎𝑎 − 𝑐𝑐)�𝑓𝑓⁄2, which
results inΠ 𝐼𝐼∗ (𝑓𝑓) ⋛ Π 𝐼𝐼∗∗ (𝑓𝑓) ⇔ 𝑓𝑓 ⋛ 𝑓𝑓 ̅ ≡(𝑎𝑎 − 𝑐𝑐)2.
4𝑛𝑛𝐼𝐼2This implies that there is a unique entry cost 𝑓𝑓 ̅ such that the leading merger is profitable if
𝑓𝑓 < 𝑓𝑓,̅ but not profitable otherwise.

Remark 2. Davison and Mukherjee (2007) considered horizontal mergers in the presence of
free entry. They found that mergers with dramatic cost synergy would reduce the number offirms. On the other hand, Remark 1 combined with Proposition 1 suggests that profitable
leading mergers can decrease the number of firms, even if there are no cost synergy effects.

Acknowledgements
Tomaru gratefully acknowledges the financial supports of Grant-in-Aid from the Japan Society
for the Promotion of Science (No. 18K01592).

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Heywood, S.J., McGinty, M., 2008. Leading and merging: Convex costs, Stackelberg, and the8merger paradox. South. Econ. J. 74(3), 879–893.
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