Priorities in Coordinated Effects of Mergers
Working Paper

Akihiko Nakagawa
Professor of Law, Hokkaido University Law School(Email: aki@juris.hokudai.ac.jp)
March 2023

Abstract
This study addresses two questions regarding coordinated effects of horizontal mergers. First,
when should there be a major concern with coordinated effects of mergers? Through a numericalanalysis of the profit increase from collusion in triopoly markets, we argue that markets in which
the collusive profit increase is durable despite cost asymmetries deserve to be prioritized. The
second question relates to why mergers increase outsiders' incentives to collude. We argue thatthe identity of exiting firms can inform the remaining firms in the market that they are eligible
to engage in profitable collusion with tolerable cost asymmetries.
Keywords: Horizontal mergers; Coordinated effects; Collusion; Cost asymmetry11. Introduction
Currently, competition agencies assess the potential competitive effects of horizontal
mergers by referencing two theories of competitive harm: unilateral and coordinated effects.
Horizontal mergers give rise to coordinated effects when both the merged entity and othercompetitors in the market are likely to reduce competition and increase their prices after amerger. Coordinated effects are easily confused with parallel price increases, but competition
agencies must distinguish between them; Coordinated effects are profitable only whencompetitors follow suit, whereas parallel price increases can be the result of unilaterally
profitable price increases due to industry-wide cost shocks.
From a competition policy perspective, coordinated effects theories of harm leave two
fundamental questions unanswered. First, when should we focus on coordinated effects? The
extant economic literature has examined how mergers change the minimum (critical) discountfactor 𝛿 ∗ for collusion to be sustainable. One of the most supported propositions in theliterature is that symmetry facilitates collusion (Compte et al. 2002; Kuhn 2004; Vasconcelos2005). However, near-perfect symmetries are rare in reality; both large and small firms exist.
Thus, the practical question we face is as follows: how large are the asymmetries between firms
consistent with coordinated effects of mergers? What market structures are more concerning?
This issue entails the priority setting in the analysis of coordinated effects. Without knowledgeof typical cases concerning coordinated effects, lawyers cannot provide adequate ex-ante advice.
They are ineffective at their job if all they can say is that "Only the econometricians know theanswer." Section 2 discusses this question and argues that we should prioritize markets in which
the profit increase from collusion after a merger is both sufficient for each collusive member anddurable with cost asymmetries between them. Specifically, we use triopoly models with different
configurations (market shares) of firms and gain insights from numerical examinations on how
the profit increase from collusion varies depending on cost asymmetries. We argue that inhomogeneous goods quantity competition, markets where the top and second-largest firms havethe most similar costs should be prioritized. In differentiated goods price competition, markets
where the top large firm competes with two or more smaller firms should be prioritized.
The second fundamental question relates to why mergers increase outsiders' incentives tocollude. We use the word "outsiders" to mean all firms other than the merging parties. Comparedwith unilateral theories, coordinated effects theories of harm are more encumbered andintricate theories. That is, it is enough for unilateral theories to explain why the merged entity's
incentives to compete changes after the merger (for example, owing to enlarged capacity, a larger
stock of brands, or the elimination of the runner-up). However, coordinated theories must
explain why both the incentives of the merger entity and other competitors will change. Why do2outsiders find it profitable to collude with other competitors, knowing that the merging parties
will be larger? This question can be posed in a more legalistic way. Competition laws generally
prohibit conduct that restrains competition and harms other trading parties that would
otherwise benefit from competition in the market. As coordinated effects arise from the conductof both the merged entity and outsiders, the merger not only restrains competition between themerging parties but also between the merged entity and outsiders. Without explaining how themerger reduces outsiders' incentives to compete, it cannot be properly called a coordinatedtheory of harm. Based on the insights gained in Section 2, we argue in Section 3 that the identityof the exiting firm can inform outsiders about the possibility of profitable collusion.

2. How large cost asymmetries are consistent with coordinated effects ofmergers?
2.1 Practical guidance is still needed
Most pertinent to this study is the extant literature on the stability of collusion andasymmetry. Compte et al. (2002) examine how mergers affect firms' incentives to collude in thecontext of Bertrand-Edgeworth competition with different capacities but with equal marginal
costs. As larger capacities increase firms' abilities to punish deviations and their incentives todeviate, capacity increases facilitate collusion when they enhance the retaliation power of small
firms. This study is the most articulate about symmetry and collusion. They also criticize thatthe Herfindahl-Hirschman Index tells the opposite relationship between coordinated effects andsymmetry. Kuhn (2004) analyzes mergers under symmetrically differentiated Bertrand
competition with equal marginal costs. He examines not only firms' incentives to deviate fromthe collusive price but also their incentives to deviate from the punishment price. Mergers
increase the merged entity's varieties (brands) and reduce their incentives to compete becauseof externalities or cannibalization between brands. He argues that mergers that reduce
asymmetries in varieties increase collusive prices. 1 Vasconceles (2005) analyzes firms'
incentives to deviate from collusive output and their incentives to deviate from the punishment
phase in Cournot competition with different quadratic costs. He argues that mergers increase
the scope of collusion when they increase or decrease the size of the smallest or largest firms,
respectively.
Although the abovementioned studies use different models, they are strikingly similar in thatsymmetry facilitates collusion. Thus Fabra & Motta (2018) state that "there is a robust resultthat says that firms' asymmetries hinder collusion." That being said, line drawing between
1

Davis (2006) reaches a similar conclusion regarding critical discount factors.
3symmetric and asymmetric mergers is quite difficult in practice. Even the most specific guidance,
given by Fabra & Motta (2018), is also ambiguous, stating "in general tacit collusion is unlikely
to arise unless after the merger there will be two or three firms with a very important share of
the market (say, more than 70 percent), and there will be considerable symmetry among them."
To clarify the range of "considerable symmetry," we fix the number of firms in the market tobe three, and numerically analyze when the profit increase from collusion is sufficient anddurable for each collusive member notwithstanding cost asymmetries
The literature most relevant to our analysis is Kovacic et al. (2007, 2009). They calculate
payoffs from explicit collusion for all subsets of firms in the post-merger market and comparethem with pre-merger non-cooperative profits. They prioritize merger enforcement when theincrease in payoffs from collusion after the merger is largest.2 Davis & Huse (2010) employ asimilar approach but also analyze incentive compatibility constraints in a repeated gameframework.3 These works consider differentiated Bertrand models because unilateral effects
analysis is straightforwardly applied to the context of coordinated effects; collusion after amerger is the same as another merger between the colluding members. 4 Our focus in thissection is to identify markets where collusion is mutually profitable for collusive members
despite cost asymmetries. In other words, we focus on the profitability of collusion after a merger,
and not on the sustainability of collusion. We do not focus on the minimum discount factors
because condensing various market configurations into a single index is contrary to the purposeof this study. We analyze collusions in both Cournot competition (Section 2.2) and differentiatedBertrand competition (Section 2.3).

2.2 Cournot Competition
We consider an industry comprising three firms (Firm1, 2 and 3) that produce homogeneous
products. Firms make output decisions simultaneously.5 The inverse demand curve is given by:
𝑃 = 𝑎 − 𝑞1 − 𝑞2 − 𝑞3 ,where 0 < 𝑎.
Kovacic et al. (2009) add calculations of the increase in profit by cheating and the decrease inprofit from Nash reversion. They argue that comparing these figures "provides a measure of
the stability of a post-merger cartel."3 See also Sabbatini (2006) for a different collusive equilibrium.
4 Other models have been applied as well. Kovacic et al. (2007) analyze the increase in profit
from collusion under the auction model. Igami & Sugaya (2019) estimate the net value ofcollusion, that is, the present value of collusion profits minus the defection profits and thepresent value of Nash equilibrium profits, in the dynamic Cournot model.
5 We do not examine a leader-follower game, as it is hard to find the mutually profitablecollusive output. We do not assume side payments either, because such an assumption is too
speculative in the merger review context.
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There are only two types of firms in the market: large (low cost) and small (high cost). The costfunction of a large firm is given by
𝐶𝐿 = 𝑐𝑞𝐿 ,and the cost function of a small firm is given by𝑑
𝐶𝑆 = 𝑐𝑞𝑆 + 𝑞𝑆2 ,2
where 0 < 𝑐 < 𝑎 and 0 < 𝑑.
Here, d represents the inefficiency parameter. The simplified cost functions make calculations ofmarket shares quite simple because the large firm's output in Nash equilibrium (NE) is (1 + 𝑑)times larger than that of the small firm: 𝑞𝐿𝑁𝐸 = (1 + 𝑑)𝑞𝑆𝑁𝐸 .
We calculate and compare Nash equilibrium profits and collusive profits of each firm. We
assume that collusion is industry-wide and that all three firms participate. As our focus is to
identify the markets where collusion is profitable despite cost asymmetries, we do not need tocompare profits before and after a merger. In other words, we treat the triopoly market as themarket structure after a merger. We compare two configurations of markets. One market
comprises two large firms and one small firm (we call this an “LLS market”). The other market
comprises one large firm and two small firms (an “LSS market”).
Before presenting the numerical results, we will explain some of the assumptions of ourmodel. First, we do not assume quadratic cost functions with capital cost 𝑘𝑖 , given by 𝐶𝑖 = 𝑐𝑞𝑖 +𝑞𝑖22𝑘𝑖, which was assumed by Vasconcelos (2005). The reason is, first, that such cost functions

presume that mergers automatically decrease a merged entity's marginal cost. However, in themerger review practice of competition agencies, such efficiencies are not presumed, but they
have to be proved by the merging parties. Second, the technical reason is that Cournot Nashequilibria in a triopoly does not always result in output proportional to capital (proportionalSPE), which is quite different from his analysis. Therefore, we selected different cost functions.
Second, when choosing the collusive output that maximizes joint profits, we assume it to be
the monopoly output when the most efficient firm (the low-cost firm) monopolizes the market.
The allocation of collusive output is assumed to reflect the cost of each firm. For example, in amarket with one large firm and two small firms (an LSS market), the collusive output isallocated at a ratio of (1 + 𝑑): 1: 1. More specifically, 𝑞1𝑐 =(𝑎−𝑐)(1+𝑑)
6+2𝑑

𝑎−𝑐

, 𝑞2𝑐 = 𝑞3𝑐 = 6+2𝑑. In ourmodel, a merger is assumed to reduce costs only when a small firm and a large firm merge andshut down high-cost plants (rationalization). Therefore, if two small firms merge, their costs
1

remain the same after the merger and their allocated collusive share in duopoly is 2+𝑑. One of
the reasons for selecting such an allocation rule is its simplicity. Another possible allocation5

rule is that the merged entity is allocated a share based on the sum of the merging parties’ premerger market share. However, such an allocation is not mutually profitable and we haveexcluded such a rule.6 The assumption about the allocation of collusive output suggests that if
a merger increases efficiency, outsiders do not know the merged entity's new costs, which
frustrates the formation of collusion. Therefore, a merger between small firms with increased
efficiency is procompetitive not pro-collusive. This implication is contrary to that found in theextant literature.
In Table 1, we show the calculated profit increases for the LLS and LSS markets according to
the specific parameter values of d. Profit increases are collusive profits minus Nash equilibrium
profits. For simplicity, we assume (𝑎 − 𝑐)2 = 1, which does not affect our analysis.
Table 1. Profit increase from collusion compared to Cournot NE
LLS

LSS

𝑑 = 1/8

0.0204(L)𝑑 = 1/8

0.0200(L)(35,35,30)

0.0199(S)(36,32,32)

0.0196(S)𝑑=1

0.0184(L)𝑑=1

0.0139(L)(40,40,20)

0.0144(S)(50,25,25)

0.0130(S)𝑑=20.0171(L)𝑑=20.0094(L)(43,43,14)0.0106(S)(60,20,20)

0.0088(S)The profit increases for each value of d are calculated for large firms (first row with (L)) andsmall firms (second row with (S)). The numbers in parenthesis under the values of d indicate themarket shares of the three firms.
There are three insights evident from Table 1. First, in both LLS and LSS markets, an increasein d reduces the profitability of collusion for both large and small firms.7 In other words, greater
symmetry makes collusion profitable mutually. Second, by comparing the increase in profits
between the two markets for the same value of d, we observe that the increase in profits is larger
in the LLS market than in the LSS market for each collusive member. This is one reason forprioritizing the LLS market over the LSS market. Aggregating all collusive members’ profits does
not seem to be the right criterion for priority setting because the sum of collusive profits is larger
with a large number of firms. Third, in the LLS market, profitability of collusion is less immune
to an increase in d. In the LSS market, in contrast, the small firm’s collusive profits shrink more

It is easily shown that the outsider’s collusive profit is lower than its Nash equilibrium profit
after the merger.
7 A similar result was obtained in Schmalensee (1987), where one low-cost firm faces
competition from N identical high-cost firms.
66quickly as cost asymmetries widen. When 𝑑 = 2, the profit increase of a small firm decreases to
0.0088. In this respect, the collusion in the LLS market is more durable despite cost
asymmetries. 8 The profitability of collusion, even with a large d, signifies that collusion is
profitable with wider margins of error in estimating competitors' costs. All these observations
allow us to conclude that the priority target markets for coordinated effects in quantity
competition are those markets where the costs of the top firm and the second largest firm are
the closest. Caveat is that we do not mean to say that the LSS market is not subject to coordinatedeffects. When the cost differences are small, both the LSS and LLS markets are equally concerningwith respect to coordinated effects. Caution should be taken not to dismiss coordinated effects
theory of harm in LLS markets when there are sufficient cost asymmetries.9

2.3 Bertrand Competition
Next, we consider a symmetrically differentiated triopoly. The inverse demand curves are
given by, 𝑝1 = 𝑎 − 𝑞1 − 𝑑𝑞2 − 𝑑𝑞3 , 𝑝2 = 𝑎 − 𝑞2 − 𝑑𝑞3 − 𝑑𝑞1 , 𝑝3 = 𝑎 − 𝑞3 − 𝑑𝑞1 − 𝑑𝑞2 , where
0 < 𝑎, 0 < 𝑑 < 1. The parameter d stands for differentiation. A lower d means that the goods ofthe three firms are more differentiated and a higher d means that they are more homogeneous.
We assume that marginal costs are constant. As in the Cournot competition model, we assume
that there are only two types of firms: large (low cost) firms with marginal cost 𝑐1 , and small(high cost) firms with marginal cost 𝑐2 , where 0≤ 𝑐1 < 𝑐2 .
There are two points to note regarding this model: Firstly, we confine our model to asymmetrically differentiated market because competition is fiercer compared to one which isasymmetrically differentiated. When firms are highly differentiated, they benefit from highprices even with Nash equilibrium, and the profit increase from collusion is minimal.10 Secondly,
we assume that each firm sells only one product. In other words, we analyze a triopoly as amarket after a merger and do not incorporate the unilateral effects of a merger that brings two
or more brands together into the merged entity. Nevertheless, our simple model is justified if we
consider the case in which, after a merger, a remote competitor becomes a closer competitor to
top firms by obtaining a new brand.
In duopoly, cost asymmetries have a larger effect of destroying the profitability of collusion.
For example, in a market with one large firm and one small firm, the profit increase fromcollusion for a large firm decreases to 0.0067 when 𝑑 = 1. This finding admonishes us that
duopoly might not always be a pro-collusive market structure.
9 We report the minimum discount factors for collusion to be sustainable with grim strategies(Nash reversion) here (when 𝑑 = 1/8). For the LLS market, it is 𝛿 ∗ = 𝛿𝐿 = 0.5669. For the LSS
market, it is 𝛿 ∗ = 𝛿𝐿 = 0.5623. The large firm’s incentive constraint is binding.
10 It is easily shown numerically that collusion is barely profitable when products are moredifferentiated. For example, when 𝑎 = 1, 𝑑 = 0.1, 𝑐1 = 0.1, 𝑐2 = 0.5, profit increase for alarge firm is 0.0021.
8

7

As in the previous section, we compare collusive profits to Nash equilibrium profits acrossthe LLS and LSS markets and examine how the profit increase from collusion changes dependingon the cost asymmetries of the three firms. Collusive profits are obtained by maximizing the joint
profits of all three firms. The collusive price is 𝑝𝐿𝑐 =𝑎+𝑐1
2, for large firms and 𝑝𝑆𝑐 =𝑎+𝑐2
2, for small

firms.
Table 2 shows the calculated profit increase for specific parameter values, where 𝑎 = 1, 𝑑 =0.9, 𝑐1 = 0.1.
Table 2. Profit increase from collusion compared to Bertrand NE
LSS

LLS

𝑐2 ≤ 0.19

𝑐2 ≤ 0.1473

𝑐2 = 0.11

0.0707(L)𝑐2 = 0.11

0.0647(L)(36,32,32)

0.0512(S)(35,35,30)

0.0451(S)𝑐2 = 0.14

0.1061(L)𝑐2 = 0.14

0.0827(L)(43,28,28)

0.0297(S)(38,38,23)

0.0053(S)For small firms to supply a positive output in the LSS and LLS markets, they must satisfy theupper limits on 𝑐2 , which are described in the first row of the table. As the LLS market has alower limit for small firms’ costs, it is more difficult for them to survive in the LLS market thanthe LSS market.
If we apply the same approach as in the previous section, the LSS market should be a priority
market we are more concerned with. First, when comparing the increase in profits between thetwo markets for the same parameter of 𝑐2 , we observe that the increase in profits is larger inthe LSS market than the LLS market for each collusive member. Second, in the LSS market,
profitability of collusion is less immune to an increase in 𝑐2 . In the LLS market, in contrast, thesmall firm’s collusive profits shrink more quickly as cost asymmetries widen. The same caveat
applies to this model as well. This is not indicative of a lack of concern for the LLS market, butmore caution should be taken to consider coordinated effects theory of harm in the LSS market
when there are sufficient cost asymmetries.
Based on the above analysis, we conclude that we should be more concerned withcoordinated effects in differentiated Bertrand markets when the products are less differentiatedand when the cost asymmetries between the second largest and the lower ranked firms are thesmallest.11

3. why do mergers increase outsiders' incentives to collude?
We report the minimum discount factors for collusion to be sustainable in both markets
here (when 𝑎 = 1, 𝑑 = 0.9, 𝑐1 = 0.1, 𝑐2 = 0.11). For the LSS market, it is 𝛿 ∗ = 𝛿𝑆 = 0.8562.
For the LLS market, it is 𝛿 ∗ = 𝛿𝑆 = 0.8724.
11

8

The analysis in Section 2 assumes that, after a merger, firms in the market have incentives tocollude. This section examines the reasons justifying this assumption. The simplest answer isthat horizontal mergers reduce the numbers of competitors (at least one). Although this is an
important reason, it does not specify the most concerning cases of coordinated effects nor does
it draw a line between competitively benign and harmful mergers. Another possible explanationis that a merger with a maverick firm provides outsiders incentives to collude. As Baker (2002,2019) exposits, a maverick is "any firm that is nearly indifferent between coordination andcheating” and that prevents “coordination from becoming more effective.” Although themaverick theory is the most compelling explanation of why mergers facilitate collusion, it does
not identify who among the outsiders would have incentives to collude after a merger. Strictly
speaking, the maverick theory sidesteps specifying the would-be collusive members and isunbale to provide an answer to the second question.
In Section 2, we have concluded that merger enforcement should prioritize markets where
collusion is mutually profitable despite tolerable cost asymmetries. If we accept this criterion,
we can answer the second question by relating mergers to outsiders’ costs and profitability. A
merger can send two messages; that the market is not worthwhile for the exiting firm to keep
investing, or that the exiting firm has high costs and cannot survive in the market. Outsiders withequal to or higher than the exiting firm’s costs will know that their prospective businesses arealso unprofitable in spite of their competitive efforts. Such outsiders can reasonably bepresumed to have incentives to collude with, or at least to follow price increases. When theexiting firm is efficient, especially when the top firm exits,12 such a message is the strongest. Aseven the top firm does not see high stakes in investing in the market, a merger incentivizes
outsiders to live and let live. Conversely, if the top firm keeps investing substantially in themarket, coordinated effects are highly unlikely. When an inefficient firm exits through merger, itmeans that it cannot survive in the market. The remaining competitors that are more efficientthan the exiting firm understand that they can engage in profitable collusion, similar to a small
firm in the LSS market in the differentiated Bertrand model. As the number of significant
competitors or relatively close competitors decreases, outsiders are more likely to know thatthey are within tolerable cost asymmetries for profitable collusion.

4. Conclusion
This study has addressed two questions regarding coordinated effects of mergers from anovel perspective than that endorsed in the extant economic literature First, we have identified
12

The famous example of this type is Nestle/Perrier merger. See Compte et al. (2002).
9

the markets in which mutually profitable collusion with cost asymmetries is predicted. These
are the primary targets about which we should be more concerned with. We have also found thatthe relationship between the number of firms and collusion may not be monotonic (see footnote
8). Second, based on the analysis in Section 2, we obtained an answer to why mergers incentivize
outsiders to collude. Mergers can inform competitors that they are within the boundary of
profitable collusion.
The questions discussed in this study may resonate more with lawyers than economists. The
method used in this study is quite simple. However, showing changes in indices without
incorporating the various aspects of individual markets is not a practical guide for lawyers. We
hope to see more research that is nuanced and compelling for lawyers in the near future. We
have not analyzed all possibilities of collusive output (price) or all possible allocations ofcollusive profits. There may be more subtle and effective ways of collusion, but we submit these
explorations for future studies.

Acknowledgments
This work was supported by the Japan Society for the Promotion of Science KAKENHI under
grant number JP17K03401.

10References
Baker, J. (2002), "Mavericks, Mergers, and Exclusion: Proving Coordinated Competitive Effects
under the Antitrust Laws." New York University Law Review, Vol.77(1), 135-203.
Baker, J. (2019), The Antitrust Paradigm: Restoring a Competitive Economy. Cambridge, MA:
Harvard University Press.
Compte, O., F. Jenny, and P. Rey (2002), "Capacity constraints, mergers and collusion." European
Economic Review, Vol. 46, 1-29.
Davis, P. (2006), "Coordinated effects merger simulation with linear demands"
Davis, P. and C. Huse (2010), "Estimating the 'coordinated effects' of mergers."
Fabra, N. and M. Motta (2018), "Assessing coordinated effects in merger cases." In Corchón L. andMarini M. (Eds.), Handbook of Game Theory and Industrial Organization, Vol. 2, 91-122.
Cheltenham, UK: Edward Elgar Publishing.
Igami, M. and T. Sugaya (2019), "Measuring the Incentive to Collude: The Vitamin Cartels, 19901999."
Kovacic, W., R. Marshall, L. Marx, and S. Schulenberg (2007), "Coordinated effects in merger
review: quantifying the payoffs from collusion." In Hawk B. (Ed.), International Antitrust
Law and Policy: Fordham Law 2006, 271-285.
Kovacic, W., R. Marshall, L. Marx, and S. Schulenberg (2009), "Quantitative analysis of
coordinated effects." Antitrust Law Journal, Vol. 76, 397-430.
Kuhn, K.-U. (2004), "The Coordinated Effects of Mergers in Differentiated Products Markets."
Sabbatini, P. (2006), "How to simulate the coordinated effect of a merger."
Schmalensee, R. (1987), "Competitive Advantage and collusive optima." International Journal ofIndustrial Organization, Vol. 5, 351-367.
Vasconcelos, H. (2005), "Tacit collusion, cost asymmetries, and mergers." RAND Journal of
Economics, Vol. 36(1), 39-62.

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