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The B.E. Journal of EconomicAnalysis & Policy

TopicsVolume 9, Issue 1 2009 Article 35

Vertical Integration and Sabotage with aRegulated Bottleneck Monopoly

Alvaro E. Bustos∗ Alexander Galetovic†

∗Catholic University of Chile and Northwestern University, [email protected]†Universidad de los Andes, [email protected]

Recommended CitationAlvaro E. Bustos and Alexander Galetovic (2009) “Vertical Integration and Sabotage with a Reg-ulated Bottleneck Monopoly,” The B.E. Journal of Economic Analysis & Policy: Vol. 9: Iss. 1(Topics), Article 35.Available at: http://www.bepress.com/bejeap/vol9/iss1/art35

Copyright c©2009 The Berkeley Electronic Press. All rights reserved.

Vertical Integration and Sabotage with aRegulated Bottleneck Monopoly∗

Alvaro E. Bustos and Alexander Galetovic

Abstract

We study the vertical integration and sabotage decisions of a regulated bottleneck monopolythat sells “access” to independent firms and may own a subsidiary downstream. We extend theliterature in four directions by: (i) endogenizing vertical integration and linking it with the inten-sity of vertical economies or diseconomies a la Kaserman and Mayo (1991); (ii) systematicallystudying how vertical economies and diseconomies affect the intensity of sabotage; (iii) showingthat the intensity of sabotage is determined by either a standard Lerner condition augmented by thedirect cost of sabotage or a relation between the market share of the subsidiary and the elasticityof the derived demand for access; and (iv) systematically examining the welfare effect of verticalintegration.

KEYWORDS: bottleneck monopoly, sabotage, vertical integration, free entry, welfare

∗We are very grateful to Felipe Balmaceda, the editor, Ching-to Albert Ma and two anonymousreferees for helpful comments and Alvaro Stein for research assistance. Galetovic gratefully ac-knowledges the financial support of Fondecyt (project 1020808), Telefonica CTC Chile S.A., In-stituto Milenio P05-004F “Sistemas complejos de Ingenierıa,” the Tinker Foundation, and thehospitality of the Center for International Development and the Center for Latin American Studiesat Stanford University. All opinions are our own and do not represent those of Telefonica CTCChile S.A.

1. Introduction

Many network industries such as telecommunications, electricity, gas, portsor postal services, were unbundled, privatized and liberalized during the lasttwenty years. While the details of each restructuring vary between industriesand across countries, in most cases the typical vertical structure that emergesincludes some segments which are open to entry and competition and at leastone segment which remains a regulated �bottleneck�monopoly selling an es-sential input. Then the following regulatory tradeo¤ emerges: on the onehand, the bottleneck monopoly may realize vertical economies à la Kasermanand Mayo (1991) by owning downstream subsidiaries; on the other hand, priceregulation stimulates sabotage� degrading service quality to raise the costs ofcompetitors.1 The implications were hotly debated during the discussion ofthe 1996 Telecommunications Act and have attracted considerable legal andregulatory attention ever since.2

This paper studies how the equilibrium intensity of sabotage varies withobservable cost and demand parameters. We extend the literature in four di-rections. First, vertical integration is endogenous and linked with the intensityof vertical economies or diseconomies à la Kaserman and Mayo (1991). Sec-ond, we systematically study how vertical economies and diseconomies a¤ectthe intensity of sabotage. Third, we show that the equilibrium intensity ofsabotage is determined by two mutually exclusive conditions which depend onobservable market parameters. Last, we perform a systematic social welfareanalysis of vertical integration.

In our model a continuum of perfectly competitive independent �rms pro-duce a homogeneous good. A bottleneck monopoly sells access to independent�rms at a regulated and exogenous access charge, which is higher than mar-ginal cost. She can sabotage independent �rms and raise both their cost ofentry and their variable cost of production. Sabotage increases the pro�t of thebottleneck�s monopoly subsidiary which competes with downstream indepen-

1Sabotage is also known as non-price discrimination, which may increase entry or op-eration costs. An example of sabotage that raises entry cost is a transmission companywho claims technical di¢ culties and delays the interconnection to the grid of independentgenerators, e¤ectively increasing the construction lead time and the interest cost duringconstruction (a direct cost) and delaying the moment that the plant begins to produce rev-enues (an opportunity cost). An example of rising rivals�costs is to degrade the quality ofservice with protracted breakdowns of service to independent �rms. For evidence see Mini(2001), Rei¤en et al. (2000) and Rei¤en and Ward (2002).

2See, for example, Verizon Communications Inc. v. Law O¢ ces of Curtis V. Trinko LLP,540 U.S. 398 (2004) which has more than 2000 citing references in Westlaw.

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Bustos and Galetovic: Vertical Integration and Sabotage

Published by The Berkeley Electronic Press, 2009

dent �rms. We study how the incentives to sabotage vary with the subsidiary�ssize and vertical economies and make vertical integration endogenous.

Most of the literature assumes that the subsidiary and independent �rmscoexist but ignores the endogeneity of the vertical integration decision. En-dogenous vertical integration reveals that there are three possible downstreammarket outcomes: (i) the bottleneck monopoly sabotages to exclude indepen-dent �rms and extends her monopoly into the downstream market; (ii) thebottleneck monopoly�s subsidiary coexists with independent �rms; (iii) andthe bottleneck monopoly does not vertically integrate in the �rst place.

We show that when the subsidiary coexists in equilibrium, sabotage ismore intense with larger vertical economies. The reason is that vertical econo-mies increase the subsidiary�s market share and, ceteris paribus, the intensityof sabotage.

We also show that the decision whether to sabotage and its intensity de-pends on a simple and intuitive relationship between quanti�able and observ-able market parameters. Call � the subsidiary�s market share, " the elasticityof demand, � the access charge and p� the equilibrium downstream price.Then, for sabotage to emerge in equilibrium a necessary condition is that

��

p�" > 0; (1.1)

i.e. the subsidiary�s market share must be greater than the product of theelasticity of demand " and the share of the access charge in the downstreamprice, �

p�; moreover, equilibrium sabotage is increasing in this di¤erence. Thus,

sabotage is of little concern if the demand is very elastic and the share ofthe access charge in the downstream price is large. Nevertheless, in practicethe contrary often occurs, as demand is inelastic or the share of the accesscharge in the downstream price is not large. To see this, consider the followingexample taken from the Chilean electricity market, where the elasticity of theresidential demand for electricity is 0:3 and the share of distribution costsin the �nal price is 40%. Then if a distributor (the bottleneck monopoly)competes with independent retailers, she forbears sabotage only if her marketshare is less than 0:3� 40% � 12%.

Sabotage may also be used to exclude downstream competitors, but thisoccurs only if the subsidiary is large. Then the subsidiary sets a limit price andthe aim of sabotage is to bring the downstream price closer to the monopolist�sunconstrained optimum. With exclusion, optimal sabotage is determined bya standard Lerner condition augmented by the direct cost of sabotage. Moreimportant, the intensity of sabotage falls with the size of vertical economies,

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The B.E. Journal of Economic Analysis & Policy, Vol. 9 [2009], Iss. 1 (Topics), Art. 35

http://www.bepress.com/bejeap/vol9/iss1/art35

for the same reason that a textbook monopolist charges less when her costfalls. We also show that when vertical economies are very large, the subsidiarydoes not sabotage, optimally charges the unconstrained monopoly price andconsumers pay a lower price with vertical integration. Nevertheless, this is theonly case when consumer welfare increases with vertical integration, and heris unlikely. For example, if the average cost of independent �rms is 100, theelasticity of demand 2, and the access charge increases the cost of independent�rms by 20% to 120, then the subsidiary�s cost net of access would have to be60 (i.e. 40% lower) to increase consumer welfare. The cost must be even lowerwith smaller elasticities.

We derive the intuitive result that sabotage reduces consumer welfare, be-cause her raises downstream prices in the long run. The economics is simple:sabotage increases the long-run average cost of independent �rms and, when-ever the bottleneck monopoly sabotages, vertical integration reduces consumerwelfare.

We also show that a bottleneck monopoly who su¤ers vertical diseconomiesmay vertically integrate nonetheless, only because she can sabotage. At somepoint, however, vertical diseconomies become too large and the monopolistis better o¤ if vertically separated. Thus, endogenizing the vertical integra-tion decision suggests that a well known result in the literature, namely thatsabotage is less intense when the subsidiary is considerably less e¢ cient thanindependent �rms, is quite misleading. On the one hand, if the bottleneckmonopolist decides not to sabotage, then she should also prefer vertical sepa-ration, and no subsidiary would be observed in the �rst place! On the otherhand, an ine¢ cient subsidiary may exist precisely because she can sabotageand raise rival�s costs.

Sabotage has generated a fairly large literature.3 Nevertheless, it is fairto say that progress in establishing general conditions has been rather slow.Initial contributions by Weisman (1995) and Economides (1998) pointed outsome of the determinants of sabotage, but reached seemingly contradictoryconclusions: while Weisman�s results suggested that the bottleneck monop-olist may not have any incentives to discriminate against downstream rivals,Economides�s indicated that she may want to go all the way until excluding in-dependent �rms altogether.4 Research by Beard et al. (2001), Bergman (2000),Mandy (2000) and Rei¤en (1998) somewhat clari�ed this seeming contradic-tion: on the one hand, the incentives to sabotage are weak or even nonexistent

3For surveys see Mandy (2000) and Sappington (2005, 2006a).4In any case, Weisman (1995, p.257) indicated that the bottleneck monopolist does not

sabotage to lower the equilibrium downstream price to increase the demand for access.

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when the access charge is close to the bottleneck monopolist�s unregulated op-timum or the subsidiary is considerably less e¢ cient than independent �rms.On the other hand, the bottleneck monopoly sabotages when price regulationconstrains her upstream market power.

Further research has added determinants to the sabotage decision. Soby now it is well known that whether sabotage pays depends on: the costof sabotage;5 the shape of the downstream cost function;6 the size of theupstream margin granted by the regulated access charge;7 the subsidiary�smarket share in the downstreammarket;8 the intensity and type of downstreamcompetition;9 and the extent of downstream product di¤erentiation.10

Also, a central tradeo¤ has been identi�ed: sabotage increases pro�tsmade downstream, but reduces access charge revenue and may, as Mandy(2000) puts it, �kill the goose that may have laid the golden egg.�There alsoseems to be a regulatory tradeo¤: vertical integration may stimulate sabo-tage but generates vertical economies and has ambiguous e¤ects on social andconsumer welfare (Rei¤en, 1998; Crew et al., 2005; Sappington 2006a).11

This paper shows that most of these factors coalesce into the inequality(1.1), even if downstream competition is imperfect. In addition, while thee¤ects identi�ed by the literature assume coexistence, we make vertical inte-gration endogenous, and show that the bottleneck monopoly may also excludein equilibrium or, on the contrary, choose not to vertically integrate.

Last, our model also con�rms the by now well known fact that an unreg-ulated bottleneck monopolist does not sabotage� sabotage is caused by priceregulation (see Beard et al. 2001; Bergman, 2000; La¤ont and Tirole, 2000,section 4.5; Mandy, 2000; Sand; 2004). At the same time, we show that itwould not pay to deregulate to avoid sabotage, because for plausible values ofthe elasticity of demand the unregulated bottleneck monopolist�s markup isquite large.

5Mandy (2000), Rei¤en (1998), Weisman (1995, 1998).6Mandy (2000).7Beard et al. (2001), Bergman (2000), Engel et al. (2003), Kondaurova and Weisman

(2003), La¤ont and Tirole (2000), Mandy (2000), Rei¤en (1998), Sand (2004) and Weisman(2001).

8Kondaurova and Weisman (2003), Sibley and Weisman (1998a,b), and Weisman (1995).9Mandy (2000, 2001).10Kondaurova and Weisman (2003), Mandy and Sappington (2007), Rei¤en (1998), Sibley

and Weisman (2005) and Weisman (1995).11Beard et al. (2001) show that this regulatory tradeo¤ can be relaxed if entry into the

upstream segment can be successfully promoted.

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2. The model

2.1. Description

Production A bottleneck monopoly produces access at zero marginal costand sells it at a regulated access charge � to a continuum of n identical andperfectly competitive �rms in the downstream market.12

Let q be the quantity produced, k the �xed investment cost, c(q) the totalvariable cost of operation, with cq; cqq > 0, and s � 0 the intensity of sabotage.Each independent �rm uses access in �xed proportions to produce with cost

(1 + s)C(q) + �q � (1 + s)[k + c(q)] + �q:

If the bottleneck monopolist vertically integrates into the downstreammarket she operates a subsidiary who owns a measure m > 0 of plants, eachwith cost function (1��)C(q), where � 2 (�1; 1). When � > 0 there are ver-tical economies: plants owned by the subsidiary have lower costs for any givenproduction q. When � < 0 there are vertical diseconomies. In addition, weassume that m is exogenous. Given this, the subsidiary is fully characterizedby a pair (m; �). Hence, in what follows we refer to �subsidiary (m; �).�

QM is the total quantity produced by subsidiary (m; �), which minimizescosts by producing qM = QM=m in each plant. Her total cost function ism(1� �)C(qM).

Sabotage Sabotage proportionally increases both the cost of setting up aplant (the entry cost) and the variable cost� sabotage s increases costs by(100 � s)%.13 Also, like most of the literature, we assume that sabotage doesnot increase the subsidiary�s cost� its e¤ect is asymmetric. As Beard et al.

12Does it make sense to assume perfect competition? The premise behind the wave ofnetwork industry restructuring is that the cost function of some segments is not subadditivein the relevant production scale. Consequently, these segments can be opened to compe-tition and functionally separated from bottleneck monopolies. For example, in electricitygeneration the e¢ cient scale of operation is between 300 and 500 MW, which is far smallerthan many electrical systems (for example, installed capacity in the main system of a smallcountry like Chile is around 8,000 MW). Thus, generation can be liberalized, but high volt-age transmission, a natural monopoly, must be regulated. Sea shipping is quite competitive,but economies of scale in ports are signi�cant and some, especially in small countries, arebottleneck monopolies. Similarly, telecomm service providers like ISPs or long-distance car-riers do not seem to enjoy signi�cant economies of scale, but density economies in the localloop are important.13The main results of the paper hold with a non-linear disentangled relationship between

sabotage and variable and �xed costs. See Lemma A.4 in Appendix A.

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(2001) point out, these characteristics di¤er somewhat from the traditionalstrategies to raise rivals�costs described by Krattenmaker and Salop (1986),because they involve non-price conditions of supply and provide neither di-rect nor indirect control of the independent �rms�outputs. Also, Salop andSche¤man (1983, 1987) assume that cost-raising strategies increase the aver-age and marginal costs of rivals, as well as the average cost of the saboteur,although they have in mind horizontal cost-raising strategies, not a bottleneckmonopoly.

Sabotage intensity s costs the bottleneck monopoly (s), with (0) = 0, 0 � 0 and 00 � 0. This function allows for free and convex sabotage.Sabotage costs may refer to the direct costs of sabotage as well as �nancial�nes imposed by regulators or image costs if caught. As Mandy (2000) pointsout, the direct cost of some sabotage activities like issuing standards that arecostly for rival �rms may be very small, but the cost of in�uence activitiesaimed to obtain regulations that damage competitors or, on the other hand,of regulatory and antitrust backlash if sabotage is convincingly revealed, maybe substantial.

Demand D is the demand for the good and Q = D(p) is the quantitydemanded at price p. Also, " � �D0p

Dis the elasticity of demand. The following

property of the demand functions ensures the regularity of the solution:

Property 1 (downward-sloping marginal revenue): Let P � D�1. Forall Q � 0, 2P 0(Q) +QP 00(Q) < 0.

Timing and competition The timing is as follows. First, the bottleneckmonopoly decides whether to establish a subsidiary and the intensity of sabo-tage s. Next the following actions take place simultaneously: the independent�rms decide whether to enter the downstream market by investing k(1 + s)and both independent �rms and the subsidiary choose their output levels.

REMARK (Alternative sequences of moves, a competitive fringe) A sequentialgame where the bottleneck monopoly moves �rst might be more intuitive. InAppendix A.2 (Proposition A.3) we show that results do not change if weassume that the subsidiary confronts a competitive fringe and chooses QM ,which is taken as given by the independent �rms when they decide whether toenter to the market and how much to produce. Essentially, with free entry theprice is still determined by the minimum average cost of independent �rms.This result is standard in the literature (e.g. see Carlton and Perlo¤, 2005,pp. 116-119).

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2.2. Sabotage and equilibrium in the downstream market

Assume that the bottleneck monopoly sabotages s > 0 in equilibrium. Thenfor all � and s, q� (s) = q0, where q0 is the minimum e¢ cient scale, with� = s = 0 (see Lemma A.2 in Appendix A.1). Hence in equilibrium

p� (s) =(1 + s) [c(q0) + k] + �q0

q0= (1 + s)cq(q0) + � : (2.1)

Note that p� (s) is determined by independent �rms and does not depend onthe subsidiary�s production. Hence the bottleneck monopolist cannot extendher market power just by vertically integrating, no matter her size.

How much does the subsidiary produce? Assume that the subsidiary co-exists with independent �rms. Given �, m and s she equates marginal cost top� (s). Thus, in equilibrium qM(s) is such that

p� (s) � (1� �)cq (qM(s)) + � ; (2.2)

and QM(s) = mqM(s).14 It follows that that qM > q0 if s > ��.15Note that the access charge � is part of the subsidiary�s marginal cost.

Why? With perfect competition the subsidiary�s substitutes the output ofindependent �rms one by one. Hence, each unit has an opportunity cost, thelost access charge.

It also follows from condition (2.2) that the subsidiary�s production andmarket share increases with s. This is shown in Figure 1, which depicts thedemand curve and the marginal cost function of a given subsidiary (m; �).

With s = 0 the equilibrium price is p� (0) and the subsidiary optimallyproduces QM(0). Sabotage o¤ers the possibility of raising p� . As the intensityof sabotage increases, the subsidiary increases output and captures an increas-ing share of the market. Call es the minimum intensity of sabotage needed toexclude all independent �rms. Then as long as p� (s) < p� (es) the subsidiarycoexists with independent �rms, hence QM < D.

Now when the bottleneck monopoly sabotages es and the price reachesp� (es), QM = D� the subsidiary grabs all sales. From then on, for any s > es,the subsidiary sets a limit price and qM = D=m.

Note that with free sabotage the bottleneck monopoly would choose s toincrease the price all the way up to the monopoly price pM . But if

0 > 0,raising rivals�costs is costly and sabotage varies with �, m, � and .

14We write QM (s) for QM (p� (s)) and qM (s) for qM (p� (s)).15See Lemma B.1 in Appendix B.

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)0(τp

)~(sQM ))0(( τpD

Demand

Figure 1: The figure shows how the subsidiary’s sales depend on the intensity of sabotage. As the intensity of sabotage rises so does the equilibrium price, the subsidiary moves up her marginal cost curve and her market share increases. When the intensity of sabotage reaches s~ , the subsidiary grabs the whole market. Note that the subsidiary’s marginal cost includes the access charge τ −when her sales increase, access sales to independent firms fall.

τη +− qc)1(

)~(spτ

Quantity

Price

QM(0)

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3. Vertical integration and the incentive to sabotage

3.1. A road map

Whether sabotage pays depends on the values of m, �, � and function . Thissection is a systematic exploration of this dependence. Nevertheless, thereare several cases and the formal analysis is easier to follow if we begin bydescribing outcomes in Figures 2 and 3, which show the optimal decision ofeach subsidiary in the (m; �) space. The formal derivation is in appendices Cand D.

As a benchmark consider vertical integration when sabotage is not anoption and � > 0 (Figure 2).16 It can be seen that there are four regions:17

� E¢ cient exclusion (EE) In Region I both m and � are large and thesubsidiary e¢ ciently excludes �rms: the unconstrained monopoly price,call it pM , is less than p� (0), the equilibrium price when only independent�rms sell in the downstreammarket. Note that along locus �EE(m), pM =p� (0).

� Limit pricing: In Region II subsidiaries still optimally exclude but theylimit price charging p� (0), since pM > p� (0).

� Coexistence In Region III m is small and vertical economies are weak.The bottleneck monopoly vertically integrates but the equilibrium priceis still p� (0) and part of D(p� (0)) is produced by independent �rms.

� Vertical separation Last, in Region IV � < 0 the subsidiary su¤ers verticaldiseconomies and the bottleneck monopoly does not vertically integrate.

Basically, vertical integration is pro�table when the subsidiary is more e¢ cientthan independent �rms; in Appendix C we prove that this occurs wheneverthere are some vertical economies, however small. On the contrary, if there arevertical diseconomies the bottleneck monopoly does not vertically integrate.

Figure 3 is the analogue of Figure 2, but now the bottleneck monopolycan sabotage. As can be seen, there are two types of (m; �) combinations suchthat there is no sabotage in equilibrium:

16Let �M be the access charge that maximizes the pro�t of a vertically-separated bottle-neck monopoly. Figure 2 also assumes that � < �M = cq(q0)=("� 1).17The frontiers between regions are not necessarily straight lines, but they are always

downward sloping. Nevertheless, let n� be the number of independent �rms that wouldenter the market when the access charge is � and the monopolist does not establish asubsidiary. Then it is always true that �LP(n� ) = 0.

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Figure 2: Vertical integration with no sabotage The figure shows the incentives to vertically integrate as a function of economies of scope (η ) and the size of the subsidiary (m). The monopolist integrates if there are some economies of scope ( 10 ≤<η ), which occurs in regions I, II and III; she does not integrate with diseconomies of scope 0<η , as in Region IV. A very efficient subsidiary excludes all independent firms by setting the unconstrained monopoly price; this is Region I, where economies of scope are large. In Region II the subsidiary sets a limit price. All sales are made by the subsidiary, but the price is set by the minimum average cost of a potential entrant. Finally, in Region III the subsidiary coexists with independent firms, which set the price (see Figure 1). Note that all lines end when τnm = , the number of independent firms that enters the market with no subsidiary. This is the maximum number of independent firms that ever enters.

ηLP(m)

ηEE(m)

II: Limit pricing

I: Efficient exclusion 1

III: Coexistence

nτ m

η

IV: Vertical separation

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A[i]: Efficient exclusion

1

0

D: Coexistence & no sabotage

A[ii]: Limit pricing & no sabotage

B[ii] Limit pricing & sabotage

C[ii] Coexistence & sabotage

)(mAη

)(mCη

η

nτ

)(EE mη

B[i]: Limit pricing & sabotage

C[i]:Coexistence & sabotage

m

E[i]: Vertical separation

)(mBη

m*mC

E[ ii]

Figure 3: Vertical integration with sabotage The figure shows the incentives to vertically integrate and sabotage. When the subsidiary is very efficient (region A) or small (region D) the bottleneck monopoly does not sabotage. The bottleneck monopoly does sabotage when the subsidiary is efficient and has a large enough market share (regions B[i] and C[i]). Also, when an inefficient subsidiaries are observed, they always sabotage (regions B[ii] and C[ii]), but when the subsidiary is very inefficient, the bottleneck monopolist does not integrate (region E).

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� In Region A both m and � are large, the subsidiary is very e¢ cient andthe bottleneck monopoly vertically integrates. Subsidiaries such thatpM < p� (0) e¢ ciently exclude by setting their unconstrained monopolyprice� Region A[i], which is the same as Region I in Figure 2. Sub-sidiaries (m; �) such that pM > p� (0) limit price but do not sabotage�Region A[ii].

� In Region D the bottleneck monopolist does not sabotage when eitherm or � are small because the subsidiary�s sales are too small to bene�tfrom sabotage. In Region E, by contrast, subsidiaries su¤er verticaldiseconomies which are not compensated by size� there is no sabotagebecause there is no vertical integration in the �rst place.

Consider now subsidiaries (m; �) that prompt the bottleneck monopoly tovertically integrate and sabotage; these are in regions B and C.

� In Region B sabotage increases the equilibrium price to p� (s) > p� (0)and independent �rms are limit-priced.

� In Region C the subsidiary coexists with independent �rms.

Interestingly, in regions B[ii] and C[ii] subsidiaries who su¤er vertical disec-onomies vertically integrate because they can sabotage. As we will see, raisingthe cost of independent �rms through sabotage compensates vertical disec-onomies, and size makes sabotage pro�table.

Last, the arrows in Region B indicate that the intensity of sabotage fallswith � and m with limit pricing. By contrast, in Region C, where indepen-dent �rms and the subsidiary coexist, the intensity of sabotage increases withvertical economies and the subsidiary�s size.

3.2. The basic economics of sabotage

3.2.1. Sabotage always increases p� and reduces consumer welfare

Result 3.1. dp�ds= cq(q0) > 0 and

d2p�ds2

= 0.

Proof. Because p� = (1 + s)cq(q0) + � , p� is linear and increasing in s.

One might think that there is a tradeo¤between sabotage and vertical economies.Result 3.1 shows that, from the point of view of consumers, there is none.With sabotage they pay more price regardless of the size and sign of verti-cal economies. Indeed, next we show that sabotage intensi�es with verticaleconomies when independent �rms and the subsidiary coexist.

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3.2.2. The sabotage decision

Sabotage with coexistence: the tradeo¤ Consider �rst Region C inFigure 3. Such subsidiaries coexist and the optimal intensity of sabotage, so,maximizes

�(s;m; �; �) ��p� (s)QM(s)�m(1� �)C(qM(s))+

� [D(p� (s))�QM(s)]� (s)

�: (3.1)

Note that QM is an implicit function of s, because the subsidiary chooses QMoptimally according to equation (2.2) for a given intensity of sabotage. Hencethe �rst order condition that determines so is

d�

ds(so;m; �; �) = QM

dp�ds

+ �D0dp�ds

� 0 = 0: (3.2)

Condition (3.2) shows the basic tradeo¤. The �rst term, QMdp�ds, is the increase

of the subsidiary�s revenues. Against this bene�t, two costs are traded o¤: thedirect cost of sabotage, 0(so), and the opportunity cost of lost access sales,�D0 dp�

ds. Further, one can rearrange (3.2) as

d�

ds(so;m; �; �) = D

��

p�"

�dp�ds

� 0 = 0; (3.3)

where � � QM=D is the subsidiary�s market share and �p�" is the elasticity

of the derived demand for access in the case of �xed proportions (see Brofen-brenner, 1961). The tradeo¤ is summarized by the term ��

p�".18 On the one

hand, a larger market share stimulates sabotage, because a given increase inp� implies more revenue for the subsidiary. On the other hand, the intensityof sabotage falls with a more elastic derived demand for access because accesssales fall.

Result 3.2. If the subsidiary coexists, sabotage increases with her marketshare.

Result 3.3. If the subsidiary coexists, sabotage falls the more elastic thederived demand for access, i.e. the more elastic the demand for the �nal goodand the higher the share of the access charge in the downstream price.

18Note that

��

p�" =

0

dp�ds D

;

where the RHS is an increasing function of s.

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REMARK (Coexistence and the cost of sabotage) Note that if sabotage is free( (s) = 0(s) = 0 for all s), d�

ds> 0 for a large enough market share and

condition (3.3) does not hold. Hence, the bottleneck monopolist sabotagesuntil excluding rivals. It follows that whenever coexistence with large enoughmarket shares is observed, the unobserved cost of sabotage must be positiveand convex in s.

REMARK (Sabotage and imperfect competition) Simple manipulation of Sibleyand Weisman�s (1998) condition (6), or Sand�s (2004) condition (7), whichcome from a Cournot model, yield the same parenthesis as in (3.3). Thisis important, for it says that the determinants of sabotage are similar withimperfect competition.

Now ultimately the subsidiary�s market share depends on her size ande¢ ciency. Thus how does the intensity of sabotage vary with m and �? Astraightforward application of the implicit function theorem shows that

@so@m

= �qM�

dp�ds

> 0

and@so@�

= �cq �m

(1��)cqq�

dp�ds

> 0;

with the second order condition

� =d2�

ds2=

�dQMdp�

+ �D00��

dp�ds

�2� 00 < 0: (3.4)

Result 3.4. With coexistence sabotage increases with size and vertical economies.

It may not be surprising to �nd that size and vertical integration reduce con-sumer welfare. But it is probably somewhat surprising that vertical economies,far from increasing consumer welfare, actually reduces it. Why? The reasonis that a more e¢ cient subsidiary has a larger market share ceteris paribus,which makes sabotage more pro�table at the margin.

Note, last, that the second order condition (3.4) is informative. BecausedQMdp�

> 0, either a convex cost of sabotage or a very concave demand arenecessary for coexistence. Otherwise, the FOC identi�es a minimum and thebottleneck monopoly would sabotage to exclude independent �rms. Thus, themere fact that we observe coexistence suggests that bottleneck monopoliescannot sabotage at will (for technical details see the Remark in Appendix D).

14



Sabotage with exclusion and limit pricing Consider next Region B,where the subsidiary sets a limit price and the bottleneck monopoly sabotagesin equilibrium. Now the problem is to maximize

�(s;m; �; �) � p� (s)D(p� (s))�m(1� �)C(qM(s))� (s):

This time the �rst order condition is

d�

ds(so;m; �; �) = D

�1� p� � (1� �)cq

p�"

�dp�ds

� 0 = 0: (3.5)

All the economics is in the term in brackets, which is proportional to

1

"� p� � (1� �)cq

p�� 0

and resembles the Lerner condition. With limit pricing the subsidiary�s marketshare equals 100% and sabotage has no opportunity cost. Instead, it is used topush the downstream price closer to pM , the monopoly price. Indeed, that�sthe price the subsidiary charges if 0 = 0 for all s. With costly sabotage, theinequality is strict and the optimal price p� (so) is lower than pM .

What is now the relation between size and e¢ ciency on the one hand, andsabotage on the other? Another application of the implicit function theoremyields

@so@m

= �1��m2 D

0cqq

�� dp�ds

< 0

and@so@�

= �cqD0

�� dp�ds

< 0;

with

� =d2�

ds2=n2D0 � 1��

mcqq (D

0)2+ [p� � (1� �)cq]D

00o�dp�

ds

�2� 00 < 0;

the second order condition, which holds with a downward-sloping marginalrevenue curve (Property 1) and convex c; see Proposition B.2 in AppendixB.2. Hence:

Result 3.5. With limit pricing sabotage falls with size and vertical economies.

Essentially, p� (so) is closer to pM the larger and more e¢ cient the subsidiary.Hence, less is gained by sabotaging a bit more.

15



The maximum intensity of sabotage Thus, we have seen that the in-tensity of sabotage increases with m and � if the subsidiary coexists; on thecontrary, it falls with m and � if the subsidiary sets a limit price. It followsthat there exists a maximum intensity of sabotage, call it smaxo , which is cho-sen with subsidiaries that are just indi¤erent between coexistence and limitpricing.

Indi¤erence indicates where to look to characterize smaxo . On the one hand,with coexistence price is determined by the independent �rms�marginal cost:

(1 + smaxo )cq (q0) = p� (smaxo )� �

= (1� �)cq

�QM (s

maxo )

m

�:

At the same time, with limit pricing QM(smaxo ) = D(p� (smaxo )). It follows that

smaxo is implicitly de�ned by

(1 + smaxo )cq (q0) = (1� �)cq

�D((1+smaxo )cq(q0)+�)

m

�:

Lemmas B.4 and B.5 in Appendix A show that smaxo exists, is unique and itis the highest equilibrium sabotage intensity� for all (m; �), smaxo � so(m; �).Moreover, smaxo de�nes locus �B(m) in Figure 3.19

Ine¢ cient vertical integration Figure 3 also indicates that the monopolymay vertically integrate despite vertical diseconomies. To see why note thatshe gains by vertically integrating and sabotaging even when � = 0, providedthat the subsidiary is large enough. Continuity implies there must be sub-sidiaries which su¤er vertical diseconomies but can still pro�t from sabotage.

To show it formally, think of a subsidiary with � = 0. When s = 0 the bot-tleneck monopolist is indi¤erent between integration and separation because�(0;m; 0) = �D(p� (0)).20 But if the subsidiary is large enough, so(m; 0) > 0and �(so(m; 0);m; 0) > �D(p� (0)) because we are inside Region C.

19Note that for subsidiaries (m; �) such that so(m; �) = smaxo , es(m; �) = so(m; �).20If vertically separated, monopolist (m; 0) earns �D [p� (0)]; if integrated it earns

p� (0)QM �mC(qM ) + � [D(p� (0))�QM ]

which, (using (2.1)) is equal to

�D(p� (0)) +mq0 � [cq(0)� C(q0)=q0] :

But the second term equals zero because of free entry, hence total pro�ts are �D(p� (0)).

16



Now consider subsidiary (m; ��) who su¤ers small vertical diseconomies�� < 0. If the bottleneck monopoly decides to vertically integrate, sabotagesso(m; 0) and mimics subsidiary (m; 0) then total pro�ts are

�(so(m; 0);m; ��) > �D(p� (0)) > �(0;m; ��):

The result follows after we notice that monopolist (m; �")maximizes her pro�tsselecting sabotage intensity so(m; ��) and not so(m; 0).

Sabotage makes vertical integration pro�table because it reduces the rela-tive cost disadvantage of a subsidiary with � < 0 and increases the equilibriumprice. Thus, the subsidiary can sell at a higher price and pro�t by verticallyintegrating.

Vertical integration and the subsidiary�s size So far we have assumedthatm is �xed� that is, the bottleneck monopolist does not choose the numberof plants when integrating. This is equivalent to assuming that the monopolistcan choose the size of her subsidiary but regulators limit it to a maximum ofm plants (this limit need not be explicit, as regulators can use precedents orlawsuits to educate potential entrants of what is or not acceptable).

To see the equivalence note that the derivative of the pro�t function withrespect to m equals (1� �) [q� (so)cq(q� (so))� c(q� (so))], which is greater thanzero for all values of q� (so) > 0. Consequently, if the monopolist had thefreedom to choose how many plants to integrate it would choose exactly m�and the rest of the analysis of the paper follows as before.

3.3. When sabotage isn�t a concern

An unregulated bottleneck monopoly does not sabotage Assume fora moment that the bottleneck monopolist is free to set the access charge � .Then the following result follows:

Result 3.6. An unregulated bottleneck monopoly does not sabotage.

Proof. The derivative of (3.1) with respect to � implies that an unregulatedbottleneck monopoly sets � = �M = p� (s)=" with coexistence. Hence

d�

ds(s;m; �; �) = �D (1� �)

dp�ds

� 0 � 0

for all s � 0 and s = 0 is optimal.

17



Consider now a bottleneck monopolist who e¢ ciently excludes. In thiscase, for each s the monopolist chooses � such that p� (s)�(1��)cq

p� (s)= 1=". Hence

d�

ds(s;m; �; �) = � 0 � 0

for all s and so = 0.

Result 3.6 is well known. Beard et al. (2001) explain the intuition with coex-istence. A higher access charge or an increase in the intensity of sabotage hasthe same e¤ect on the subsidiary�s revenue. Nevertheless, while a higher � in-creases the margin earned by the bottleneck monopoly on sales to independent�rms, sabotage does not, and a higher access charge is always better than moresabotage. Exactly the same economics explains why a slightly higher accesscharge always reduces the intensity of sabotage in equilibrium:

Result 3.7. Let so > 0 for � < �M . Then sabotage falls with � .

Proof. Straightforward di¤erentiation of (3.2) or (3.5) as the case may beyields

@so@�

= �D0

�

dp�ds

< 0;

with � = d2�ds2, the second order condition.

Results 3.6 and 3.7 show that the bottleneck monopoly sabotages be-cause her market power is successfully curtailed. Hence, one might think thatsometimes price regulation should be abandoned to avoid sabotage, but thisis unlikely. For example, if the elasticity of demand is 1:1, �M is 10 timesthe downstream marginal cost cq(q0); if " = 3, �M still is substantial, equalto one-half the downstream marginal cost cq(q0).21 At the same time, if theaccess charge is equal to marginal cost, the monopolist sabotages to achieveher desired margin. Thus, unless sabotage is veri�able, marginal cost pricingis not optimal; see Sand (2004).

No sabotage if the subsidiary is �small� Consider now subsidiaries whocoexist. A straightforward implication of monotonicity in m, is that at somepoint the subsidiary becomes too small to warrant sabotage; this is Region

21To obtain this note that �Mcq(q0)

= 1"�1 at the monopolist�s optimum.

18



D in Figure 3. In particular, Lemma B.6 in Appendix B shows that for any� > 0 there exists �(�) such that for all � in [0; �(�)]

D��(�)� �

p� (0)"� dp�ds

� 0(0) � 0: (3.6)

Hence (3.6) tells that the bottleneck monopoly does not sabotage if

� � �

p�"; (3.7)

i.e. the subsidiary�s market share is smaller than the elasticity of the deriveddemand for access �

p�" (if sabotage is free then (3.7) is su¢ cient and necessary).

This no-sabotage condition is useful because it relates incentives with observ-able market parameters. Now for a given ratio �

p�and elasticity of demand ",

Table 1 shows the maximum market share of the subsidiary such that condi-tion (3.7) holds. For example, if �

p�= 0:2 (i.e. the access charge is equivalent

to 20% of the downstream price) and " = 0:7, then the su¢ cient no-sabotagecondition (3.7) holds for subsidiaries with market share of 14% or less.

Note that when the access small compared with the �nal price (say 0:15or less) and demand is inelastic, the su¢ cient condition holds only for smallmarket shares. Moreover, if the access charge is equal or less than marginalcost, the bottleneck monopolist sabotages however small her market share.

Result 3.8. If �p�is small, demand is inelastic, and 0(0) its not too large,

then only very small subsidiaries do not sabotage.

Application (A bottleneck monopoly with small �p�) First consider two examples

of Chilean services with small ratios �p�. In Chile�s Central Interconnected

System, high-voltage transmission costs are about 6% of the average wholesaleelectricity price22 and current estimates indicate that the price elasticity of theresidential demand for electricity in Chile is about 0:3.23 Hence, a market shareof more than 0:3� 6% = 1:8% is enough for condition (3.7) not to hold.

Consider now long distance. In Chile local �xed-line companies receivea per-minute access charge for originating and delivering long-distance calls,which is paid by independent carriers and currently equalsCh$5 :07 per minute.24

22See Galetovic and Muñoz (2006).23The elasticity of demand and the share of distribution charges in the Chilean residential

electricity price is taken from Galetovic et al. (2004).24We thank Patricio Cáceres of Telefónica CTC Chile for sharing data on access charges.

In June 2009 $1 = Ch$540.

19



Table 1: Largest market share such that the bottleneck monopoly doesn’t sabotage (in percentage)

↓ε →τ

τp

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.0 0.1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 0.2 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0 0.3 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 16.5 18.0 19.5 21.0 22.5 24.0 25.5 27.0 28.5 30.0 0.4 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 38.0 40.0 0.5 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0 42.5 45.0 47.5 50.0 0.6 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0 33.0 36.0 39.0 42.0 45.0 48.0 51.0 54.0 57.0 60.0 0.7 3.5 7.0 10.5 14.0 17.5 21.0 24.5 28.0 31.5 35.0 38.5 42.0 45.5 49.0 52.5 56.0 59.5 63.0 66.5 70.0 0.8 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 40.0 44.0 48.0 52.0 56.0 60.0 64.0 68.0 72.0 76.0 80.0 0.9 4.5 9.0 13.5 18.0 22.5 27.0 31.5 36.0 40.5 45.0 49.5 54.0 58.5 63.0 67.5 72.0 76.5 81.0 85.5 90.0 1.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0 75.0 80.0 85.0 90.0 95.0 100 1.5 7.5 15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 75.0 82.5 90.0 97.5 100 100 100 100 100 100 100 2.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100 100 100 100 100 100 100 100 100 100 100 3.0 15.0 30.0 45.0 60.0 75.0 90.0 100 100 100 100 100 100 100 100 100 100 100 100 100 100 5.0 25.0 50.0 75.0 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100

Note: For a given ratio ττ p/ and demand elasticity ε , the table shows the maximum market share μ of the subsidiary such that the bottleneck monopoly has no incentive to sabotage. For example, if 2.0/ =ττ p and 7.0=ε , then the sufficient no-sabotage condition (3.6) will hold only for subsidiaries with a market share equal or less than 14.0%.

20



In turn, carriers charge on average Ch$171:10 per minute for internationalcalls, and Ch$41:90 per minute for national calls. Hence �

p�= 5:07

171:10� 0:03 for

international calls and �p�= 5:07

41:90� 0:12 for national calls. Unfortunately, we

don�t have estimates for the elasticity of the demand for long distance calls.If equal to the 0:7 estimate by Taylor (1994) for the United States, then onlysubsidiaries with market share of 2:1% or less meet the no-sabotage condition.

Application (Downstream liberalization and sabotage) Depending on initial con-ditions a �liberalization�may mean allowing independent �rms to competewith a vertically integrated incumbent; or allowing the upstream provider toenter the downstream market (as when the 1996 Telecommunications Act al-lowed RBOCs to enter the InterLATA market in the United States).

Incumbents often retain large market shares after liberalization and thedemand for such services is likely to be inelastic. The North-East quadrant ofTable 1 suggests, thus, that sabotage should be a concern when independent�rms are allowed to enter to compete with a dominant vertically integratedincumbent. Consider, for example, electricity distribution, which can be un-bundled from retailing. As said before, current estimates indicate that theprice elasticity of the residential demand for electricity in Chile is about 0:3and �

p�is about 0:4.25 Thus the no-sabotage condition holds only if the incum-

bents�shares falls from 100% today to 12% or less. More generally, if " = 0:5or less, any incumbent which retains more than 20% of the market sabotages.

Sibley and Weisman (1998a) did a similar calculation to estimate sabo-tage after allowing RBOCs to enter the interLATA market, and their parame-trization yielded a threshold market share of 26%. Yet they concluded thatsabotage would probably not be a concern because RBOCs were new to themarket and had to gain market share from incumbents.

No sabotage if the subsidiary is very e¢ cient The bottleneck monop-olist does not sabotage if her subsidiary is very e¢ cient. Such a subsidiaryoptimally sets pM < p� (0) < p� (s) for any s > 0. Hence

�(s; �;m) = pMD(pM)�m(1� �)C�D(pM )m

�� (s);

which falls with s and:

Result 3.9. There is no sabotage with e¢ cient exclusion.

25The elasticity of demand is taken from Benavente et al. (2005). The share of distributioncharges in the Chilean residential energy price is taken from Galetovic et al. (2004).

21



Furthermore, Figure 3 also shows that at some point vertical economies becometoo strong for sabotage to be pro�table, regardless of the subsidiary�s size m.Then even subsidiaries who set a limit price optimally choose not to sabotage.

To see why �x m arbitrarily at m. Note that for subsidiaries (m; �) wholimit price when sabotage is not feasible

d�

ds(0;m; �) = D

�1� p� � (1� �)cq

p�"

�dp�ds

;

and p� � pM(m; �). As � and vertical economies increase, the term

p� � (1� �)cqp�

increases and d�=ds (0;m; �) falls. For � = �0 such that pM(m; �0) = p� (0),this fraction equals 1=" and d�=ds(0;m; �0) = 0. Thus, when 0(0) > 0,there exists some continuous interval such that d�=ds(0;m; �0) � 0(0) < 0.Since d�

ds(s;m; �) is decreasing in s, the bottleneck monopoly never sabotages.

Essentially, with strong vertical economies p� is close to pM and sabotage isnot worth its cost.

Application (E¢ cient exclusion is unlikely) In antitrust cases it is sometimesclaimed that vertical integration increases consumer welfare, because verticaleconomies lower prices. Our analysis shows that this claim is suspect becauseprices do not fall with vertical integration unless the subsidiary (m; �) is �con-siderably�more e¢ cient than independent �rms. To see this, we parametrizecost di¤erences between the subsidiary and independent �rms necessary fore¢ cient exclusion.

To proceed, note that the frontier between Regions A[i] and A[ii] in Figure3, points such that pM = p� , is the implicit function �EE(m) obtained from

p� � (1� �EE(m))cq�Dm

�� 1

"p� : (3.8)

De�ne �� =cq(q0), the mark-up above marginal cost imposed by the accesscharge. Because p� = cq(q0) + � , one can manipulate (3.8) to yield

(1� �)cq�Dm

�cq(q0)

= (1 + �� )(1� 1="): (3.9)

The ratio on the LHS is a lower bound such that e¢ cient exclusion is anequilibrium (for points inside Region A[i] the cost di¤erence is even higher

22



and, since qM � q0, average cost is below marginal cost at qM). Because theratio cannot be negative, " > 1 a fortiori at the observed equilibrium price.

Table 2 shows the minimum cost advantage that the subsidiary must enjoyfor e¢ cient exclusion to be pro�table. If the observed elasticity of demand isclose to 1, or � is small relative to the independent �rm�s marginal cost, thesubsidiary�s cost advantage has to be very large. For example, if " = 1:1 and�� = 0:1, the subsidiary�s marginal cost has to be one-tenth, or 10% of anindependent�s �rm marginal cost, for e¢ cient exclusion to occur. And in anycase, for plausible values of " and �� the monopolist�s cost advantage needs tobe substantial for e¢ cient exclusion to be pro�table. For example, if " = 2,even if � increases �rms�costs by 50% (�� = 0:5), the subsidiary�s cost wouldneed to be 75% of the cost of an independent �rm. Hence:

Result 3.10. Unless demand is very elastic, vertical integration increases con-sumer welfare only if the subsidiary is substantially more e¢ cient.

No vertical integration if the subsidiary is very ine¢ cient Manyauthors have noted that sabotage becomes unattractive when the subsidiaryis considerably less e¢ cient than independent �rms.26 This, of course, alsoappears in our model: as can be seen in Figure 3, for each m there is nosabotage if vertical diseconomies are strong enough. The economics is wellknown: the ine¢ cient subsidiary substitutes production of independent �rms,but her margin is smaller than the access charge. Thus, substitution reducespro�ts when the subsidiary�s costs are high.

Nevertheless, making vertical integration endogenous shows that it iswrong to conclude that ine¢ cient subsidiaries do not sabotage. On the onehand, if the bottleneck monopolist decides not to sabotage, then she prefersvertical separation, and no subsidiary is observed in the �rst place! On theother hand, the bottleneck monopoly may create an ine¢ cient subsidiary be-cause she can sabotage. Thus, if an ine¢ cient subsidiary is observed, socialwelfare falls a fortiori and society is better o¤ prohibiting vertical integration.

3.4. Social welfare

Sabotage and social welfare How does social welfare vary with sabotage?Because downstream �rms are perfectly competitive and make zero pro�ts,social welfare equals the sum of the monopolist�s pro�t and consumer surplus,both evaluated at s0. Hence, with exclusion social welfare equals

26See, for example, Mandy (2000), Sand (2004) and Weisman (1999).

23



Table 2: Subsidiary’s marginal cost such that efficient exclusion is profitable (as percentage of an independent’s firm marginal cost)

↓ε →≡)( 0qcq

τλτ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

)( 0qcq

Mτ

1.1 9.1 10.0 10.9 11.8 12.7 13.6 14.5 15.5 16.4 17.3 18.2 10.0

1.2 16.7 18.3 20.0 21.7 23.3 25.0 26.7 28.3 30.0 31.7 33.3 5.00

1.3 23.1 25.4 27.7 30.0 32.3 34.6 36.9 39.2 41.5 43.8 46.2 3.33

1.4 28.6 31.4 34.3 37.1 40.0 42.9 45.7 48.6 51.4 54.3 57.1 2.50

1.5 33.3 36.7 40.0 43.3 46.7 50.0 53.3 56.7 60.0 63.3 66.7 2.00

2 50.0 55.0 60.0 65.0 70.0 75.0 80.0 85.0 90.0 95.0 100 1.00

3 66.7 73.3 80.0 86.7 93.3 100 - - - - - 0.50

4 75.0 82.5 90.0 97.5 - - - - - - - 0.33

5 80.0 88.0 96.0 - - - - - - - - 0.25

10 90.0 99.0 - - - - - - - - - 0.11

Note: For a given ratio )(/ 0qcqτ and demand elasticity ε the table shows the subsidiary’s marginal cost, as a percentage of an independent firm’s marginal cost, such that )0(τppM = . For example, if

2.0=τλ and 2.1=ε , then for the subsidiary to set )0(τppM = , its marginal cost would have to be 20.0% of an independent’s firm marginal cost. The last column shows the maximum τλ that could possibly be observed for the respective elasticity─when Mττ = and the access charge maximizes the bottleneck monopolist’s profits.

24



Z D(p� (so))

0

[P (x)� p� (so)] dx+ p� (so)D(p� (s0))

�m(1� �)C (qM(so))� (so):

On the other hand, with coexistence, social welfare equalsZ D(p� (so))

0

[P (x)� p� (so)] dx+ p� (so)QM(so)

�m(1� �)C (qM(so)) + � [D(p� (s0))�QM(so)]� (so):

Now note that so is a function of � andm. If the bottleneck monopoly excludes,then sabotage is more intense the smaller the subsidiary or the less intense arevertical economies. Because smaller and less e¢ cient subsidiaries make lowerpro�ts and sabotage more, the following follows:

Result 3.11. With exclusion social welfare falls with sabotage.

Things are di¤erent when the subsidiary coexists. Here the bottleneckmonopoly sabotages more when the subsidiary is more e¢ cient. Hence:

Result 3.12. With coexistence social welfare may increase or fall with sabo-tage.

Vertical integration and social welfare A related question is how verticalintegration a¤ects social welfare. The tradeo¤ is simple: integration createsvertical economies but consumers pay more and independent �rms have highercosts.

The change in social welfare when the bottleneck monopolist verticallyintegrates and sabotages, which is depicted in Figure 4, is

�Z D(p� (0))

D(p� (so))

[P (x)� cq(q0)] dx (3.10)

�s0cq(q0) [D(p� (so)�QM ]

+QM

�cq(q0)�

(1� �)C (qM(so))

qM(so)

�� (s0):

Three of these four terms indicate a welfare loss. The �rst term in (3.10), whichis the shaded area in the �gure, is the loss due to the fall of consumption�with vertical integration price increases, consumption falls and access charge

25



cq(q0)

Figure 4: Welfare and vertical integration The figure shows the change in welfare wrought by vertical integration when the subsidiary and independent firms coexist. The shaded area is the welfare loss due to the fall of consumption. The gridded area is the welfare loss due to the increase in the independent firms’ cost caused by sabotage. Last, the checkered area is the welfare gain produced by the subsidiary’s economies of scope. Note that the direct cost of sabotage incurred by the bottleneck monopolist is not shown.

Price

τ + (1 + s0)cq(q0)

τ + cq(q0)

M

Mq

qC )()1( η−

))0(( τpD ))(( 0spD τ )( 0sQM

Demand

Quantity

26



revenues are lost. The second term, which is the gridded area in the �gure,is the higher cost that independent �rms have to bear. And sabotage has adirect cost, (s0). But consider the third term. We know that qM(so) � q0,so that C(qM (so))

qM (so)� cq(q0). Thus, if � is small, zero or negative, social welfare

must fall:

Result 3.13. Social welfare falls with integration unless vertical economiesare large.

Now the checkered area in Figure 4 is positive whenever the subsidiary canrealize vertical economies. With su¢ ciently large vertical economies, thus,social welfare could increase with vertical integration. We now show this witha simulation.

A simulation with �large� sabotage costs To explore the tradeo¤ be-tween vertical economies and welfare we consider a simple example. Assumethat the demand for the �nal good is 250� 6p; the total cost of production ofa one-plant independent �rm is 100 + q2; the access charge is �xed at � = $2;and the cost of sabotage is 12; 500 � s2. Then 11:8 independent �rms enter themarket with vertical separation, Q = 118, p = $22, consumer welfare equals$1; 160 and social welfare equals $1; 396. Note that in this example sabotagecosts can be called �large�: to increase the cost of independent �rms by 1%the bottleneck monopolist must spend 12; 500 � 0:01 = $125, or about 5% oftotal downstream production costs.

Table 3a shows the change of social welfare, and Table 3b the change ofconsumer welfare, with vertical separation = 100 in both. Pairs (�;m) suchthat there is e¢ cient exclusion (Region A) are shown in italics; pairs (�;m)such that the bottleneck monopolist sets a limit price (Region B) are shownwith continuous shading; pairs (�;m) such that the subsidiary coexists andsabotages (Region C) are shown in normal script; last, pairs (�;m) such thatthe subsidiary coexists but does not sabotage (Region D) are shown withdiscontinuous shading.

Table 3a indicates that social welfare increases when vertical economiesyield cost savings of 7% or more. On the other hand, if vertical economies are6% or less, social welfare falls for most (�;m) pairs, unless the subsidiary issmall. Social welfare can fall up to 7:6%, which occurs when the subsidiarycoexists with � = 0 and m = 10. Note that welfare uniformly increases withvertical economies for a givenm, despite that sabotage intensi�es with verticaleconomies in Region C.

27



Table 3a: Aggregate welfare and sabotage (vertical separation = 100) ↓η →m 1 2 3 4 5 6 7 8 9 10 11

1 279,8 279,8 279,8 279,8 279,8 279,8 279,8 279,8 279,8 279,8 279,8 0,90 163,3 213,2 230,5 239,1 244,0 247,1 249,1 250,4 251,3 251,9 252,9 0,80 130,0 161,1 192,8 208,9 218,2 224,1 227,9 230,5 232,2 233,3 233,9 0,70 119,1 137,6 157,6 180,0 193,3 201,7 207,2 210,9 213,3 214,9 215,8 0,60 113,4 125,7 138,7 152,6 169,2 179,9 187,0 191,7 194,8 196,7 197,8 0,50 109,7 118,1 126,9 136,3 146,1 158,7 167,1 172,8 176,5 178,8 180,1 0,40 107,0 112,6 118,5 124,7 131,3 138,2 147,7 154,2 158,4 161,1 162,5 0,30 104,9 108,3 112,0 115,8 119,9 124,3 128,9 135,9 140,6 143,5 145,1 0,20 103,2 104,8 106,5 108,5 110,6 112,9 115,5 118,2 123,1 126,2 127,9 0,10 101,6 101,7 101,9 102,2 102,6 103,3 104,0 104,9 106,0 109,1 110,8 0,09 101,5 101,4 101,4 101,6 101,9 102,4 103,0 103,7 104,6 107,4 109,1 0,08 101,4 101,1 101,0 101,0 101,2 101,5 101,9 102,5 103,2 105,7 107,4 0,07 101,2 100,8 100,6 100,4 100,5 100,6 100,9 101,3 101,9 104,0 105,7 0,06 101,1 100,6 100,2 99,9 99,7 99,7 99,8 100,1 100,5 102,3 104,0 0,05 100,9 100,3 99,7 99,3 99,0 98,9 98,8 98,9 99,2 100,6 102,3 0,04 100,8 100,0 99,3 98,8 98,3 98,0 97,8 97,8 97,9 98,9 100,6 0,03 100,7 99,7 98,9 98,2 97,6 97,2 96,8 96,6 96,6 97,2 98,9 0,02 100,5 99,5 98,5 97,7 96,9 96,3 95,9 95,5 95,3 95,5 97,2 0,01 100,4 99,2 98,1 97,1 96,3 95,5 94,9 94,4 94,0 93,8 95,6 0 100 98,9 97,7 96,6 95,6 94,7 93,9 93,3 92,8 92,4 93,9

Table 3b: Consumer welfare and sabotage: (vertical separation = 100)

↓η →m 1 2 3 4 5 6 7 8 9 10 11

1 112.2 112.2 112.2 112.2 112.2 112.2 112.2 112.2 112.2 112.2 112.2 0,90 85.0 92.7 95.7 97.2 98.1 98.7 99.2 99.5 99.8 100.0 101.0 0,80 93.7 85.0 90.0 92.7 94.5 95.7 96.5 97.2 97.7 98.1 98.4 0,70 96.5 90.8 85.0 88.6 91.1 92.7 94.0 94.9 95.7 96.3 96.8 0,60 97.9 93.7 89.4 85.0 87.8 90.0 91.5 92.7 93.7 94.5 95.1 0,50 98.7 95.3 92.0 88.5 85.0 87.3 89.2 90.6 91.8 92.7 93.5 0,40 99.2 96.5 93.7 90.8 88.0 85.0 86.9 88.6 90.0 91.1 92.0 0,30 99.6 97.3 94.9 92.5 90.0 87.5 85.0 86.6 88.2 89.4 90.5 0,20 99.9 97.9 95.8 93.7 91.5 89.4 87.2 85.0 86.4 87.8 89.0 0,10 100 98.4 96.5 94.7 92.8 91.0 89.1 87.2 85.2 86.3 87.5 0,09 100 98.4 96.5 94.7 92.8 91.0 89.1 87.2 85.2 86.1 87.4 0,08 100 98.4 96.6 94.8 92.9 91.1 89.2 87.3 85.4 86.0 87.2 0,07 100 98.4 96.6 94.8 93.0 91.2 89.3 87.5 85.6 85.8 87.1 0,06 100 98.5 96.7 94.9 93.1 91.3 89.5 87.6 85.8 85.7 87.0 0,05 100 98.5 96.8 95.0 93.2 91.4 89.6 87.8 86.0 85.5 86.8 0,04 100 98.5 96.8 95.1 93.3 91.5 89.8 88.0 86.1 85.3 86.7 0,03 100 98.6 96.9 95.1 93.4 91.6 89.9 88.1 86.3 85.2 86.5 0,02 100 98.6 96.9 95.2 93.5 91.8 90.0 88.2 86.5 85.0 86.4 0,01 100 98.6 97.0 95.3 93.6 91.9 90.1 88.4 86.6 84.9 86.3 0 100 98.7 97.0 95.3 93.7 92.0 90.3 88.5 86.8 85.0 86.1

28



Variations in the intensity of sabotage can be assessed looking at consumerwelfare in Table 3b. We know that it must fall in regions B and C and reachits nadir along the frontier that separates both region. Then in this exampleconsumer welfare can fall up to 15% with vertical integration. Interestingly,there seem to be two types of pairs (�;m) that reduce consumer welfare themost. On the one hand, when vertical economies are very large, but thesubsidiary is small (this, however, seems unlikely, for welfare falls a lot onlyif vertical economies are very large). On the other hand, a large subsidiary(m � 7, say) sabotages more.

The price increases behind the welfare changes shown in Table 3 are signif-icant but not very large� at most 7% . One might thus be tempted to concludethat important but yet plausible vertical economies might be enough to com-pensate whatever welfare losses consumers bear with vertical integration. Yetthis conclusion is probably unwarranted, because we have assumed that thedirect cost of sabotage is quite large. A smaller cost of sabotage signi�cantlyenlarges the set of pairs (�;m) such that social welfare falls and then consumerwelfare losses are compensated only if vertical economies are much larger.

Sabotage, vertical divestitures and social welfare Some recent pa-pers have begun to analyze the costs and bene�ts of vertical divestitures (seeCrew et al., 2005 and Sappington, 2006b). In our model, Result 3.1 impliesthat a vertical divestiture increases consumer welfare whenever the bottleneckmonopoly sabotages� p� (0) < p� (s) regardless of vertical economies. More-over, because the subsidiary is a price taker, p� (0) is the minimum price thatconsumers pay when independent �rms coexist with the subsidiary. Hence:

Result 3.14. With coexistence divestitures increase consumer welfare.

On the other hand a vertical divestiture that prevents e¢ cient exclusionreduces consumer welfare. But, as we saw, e¢ cient exclusion is unlikely� inour simulation, it occurs only if the subsidiary�s cost is close to zero.

Our results are similar to Sappington�s (2006b), who studies vertical di-vestitures in a model where an incumbent �rm (the equivalent of the sub-sidiary) competes à la Bertrand with rivals of varying e¢ ciency. He �nds thatvertical economies do not a¤ect the price that consumers pay when either theincumbent is the lowest-cost producer or the costs of rivals are su¢ cientlysimilar. In both cases, the equilibrium price is determined by the costs of in-dependent rivals, which are not a¤ected by vertical economies. Consequently,a vertical divestiture increases consumer welfare because the price falls whensabotage ceases, just as in our model.

29



How does a vertical divestiture a¤ect aggregate welfare? Crew et al.(2005) tackled this question with a Cournot duopoly, and found that wel-fare rises with a divestiture, unless vertical economies are substantial.27 In oursimulation (see Table 3b) a divestiture increases aggregate welfare if, roughlyspeaking, vertical economies are larger in magnitude than the price increaseprompted by sabotage.

4. Conclusion

This paper links sabotage with equilibrium market structure and observablemarket parameters. We obtain two simple conditions which tell us when sabo-tage pays. Moreover, we make vertical integration endogenous and characterizeequilibrium market structure and sabotage as a function of vertical economiesand diseconomies. Let us brie�y put together what we found.

First, vertical integration increases consumer welfare only if the subsidiarydoes not sabotage, enjoys large and unlikely vertical economies, and e¢ cientlyexcludes downstream competitors. But whenever the bottleneck monopolysabotages, downstream prices rise, regardless of the size of vertical economies.This is the most likely outcome and vertical divestitures seldom reduce con-sumer welfare.

Second, the decision to vertically integrate is endogenous. Bottleneckmonopolies that enjoy vertical economies always integrate, and most sabotage.And bottleneck monopolies that su¤er moderate vertical diseconomies mayintegrate if the subsidiary is large because they can sabotage competitors. Onthe other hand, with large vertical diseconomies, integration is not observed.

Third, because the decision to vertically integrate is endogenous, observedmarket structure and parameters inform about sabotage. For example, withcoexistence only subsidiaries with small market shares forbear sabotage be-cause they loose too much revenue from access sales (the so-called �killing thegoose�e¤ect). Thus, unless the subsidiary�s market share is small, coexistencesuggests that there is sabotage. Moreover, sabotage intensi�es with verticaleconomies and subsidiary size, because both increase the subsidiary�s marketshare. Last, coexistence implies that something restrains sabotage and sub-sidiary expansion, perhaps increasing and convex sabotage costs or constraintsto subsidiary growth.

27In their simulations, and for their speci�c parameter values, they �nd that marginalcosts have to be at least 28:61% higher for a divestiture to decrease welfare. See also theirTable 3.

30



How general are our conclusions? Let us emphasize that assuming freeentry, while not common in the literature, is not restrictive. On the contrary,when writing regulations (either rules or statutes), a central question is: does itmake sense to allow bottleneck monopolies to own subsidiaries, or is it better toprohibit vertical integration? This is, essentially, a question about the long-runperformance of alternative regulatory regimes which are inevitably a¤ected byentry. Moreover, such long-run analysis is useful because ex post regulation ofsabotage is frequently ine¤ective. For example, American courts have seldomupheld claims that sabotage raises rival�s cost and violates Section 2 of theSherman Antitrust Act. As mentioned by Goetz and McChesney (2006): �Forthe most part, causes of action under §2 based essentially on allegations ofraising rivals� costs have not fared well.� In such a case the plainti¤ (theFTC or the DOJ, say) not only has to prove that the defendant has engagedin activities with the intention to monopolize the market and has injuredthe plainti¤, but also has to prove that competition has been harmed. The�rst requirement is daunting by itself, as the plainti¤ has to present materialevidence that the defendant raised his costs. Expert testimony is usually notenough because, as City of Tuscaloosa, 877 F. Supp. 1504 (N.D. ALA. 1995)suggests, courts are reluctant to base their decisions on economic theoriesunless these are completely developed and accepted by the profession. Thesecond requirement is no less demanding, for if the plainti¤ proves personalinjury, then he must show that the social losses wrought by sabotage outweighthe productive e¢ ciencies wrought by vertical integration. In the few availablecases (e.g. Oahu Gas Serv., 838 F.2d 360, 368 (9th. Cir. 1988) and ViacomInternational, 785 F. Supp 371, 376 n.12 (S.D.N.Y. 1992)) plainti¤s failed toachieve that goal, suggesting that ex ante regulation of industry structure, along-run decision, may be the only e¤ective remedy against sabotage.

The main limitation of our paper is that we assumed perfect competitiondownstream. So it seems natural to systematically explore how imperfect com-petition a¤ects the incentives to sabotage and to vertically integrate. Clearly,double marginalization is an additional incentive to vertical integration and,with imperfect competition, an additional boost to e¢ ciency ceteris paribus,as it reduces double marginalization (see Galetovic and Sanhueza, 2009). Butthe similarity of condition (3.3) with Sibley and Weisman�s (1998a) condition(6), or Sand�s (2004) condition (7), which come from a Cournot model, showthat the simple result derived in this paper extends to imperfectly competi-tive markets. Of course, one would like results and conditions that are validfor all market structures between perfect competition and monopoly, not onlyCournot. Mandy (2001), who models downstream competition with a conjec-tural variations model, suggests a promising way to proceed.

31



Appendix

A. Equilibrium in the downstream market

A.1. A price-taking subsidiary

We begin by restating the de�nition of a competitive equilibrium.

De�nition A.1. An equilibrium of the downstream market with free entryand vertical integration with input access charge � and sabotage intensity s isa price p� (s); a combination of independent �rms and integrated �rms outputsq� (s) and qM(s) respectively; and a number n� (s) of independent �rms suchthat:

(i) each independent �rm i maximizes pro�ts:

q� (s) = argmax fp� (s)qi � [(1 + s)c(qi) + �qi]g : (A.1)

(ii) each independent �rm makes zero pro�ts:

p� (s)q� (s)� (1 + s)c(q� (s))� �q� (s) = (1 + s)k: (A.2)

(iii) the subsidiary maximizes pro�ts:

qM(s) = argmax fm[p� (s)qM � (1� �)c(qM)] + �(D(p� (s))�mqM)g : (A.3)

(iv) the market clears:

D(p� (s)) = n� (s)q� (s) +mqM(s): (A.4)

We denote this equilibrium with the tuple (p� (s); q� (s); qM(s); n� (s); � ; s). Thenext lemma characterizes the tuple (p� (s); q� (s); qM(s); n� (s); � ; s).

Lemma A.2. Consider an equilibrium with coexistence. Then the symmetricequilibrium of the downstream market with free entry and vertical integrationwith input access charge � and sabotage intensity s is

p� (s) = (1 + s)cq(q0) + � ;

q� (s) = q0;

qM(s) = c�1q

�p� (s)� �

1� �

�;

32



n� (s) =D(p� (s))�m � qM(s)

q0;

with q0 implicitly de�ned by

q0 � cq(q0)� c(q0) � k:

Proof. In a competitive equilibrium two conditions must hold. First, indepen-dent �rms equalize marginal cost with price: p� (s) = (1+ s)cq(q� (s))+ � . Sec-ond, independent �rms make zero pro�ts: p� (s)q� (s)�(1+s)c(q� (s))��q� (s) =(1 + s)k. Simple substitution and simpli�cation yield that in equilibrium

q� (s)cq(q� (s))� c(q� (s)) = k; (A.5)

which must hold for all values of q� (s). To show that q� (s) = q0 it is enough toshow that (A.5) de�nes a unique q� (s)� which consequently must be q0. Butthat is the case because q� (s)cq(q� (s))�c(q� (s)) is a strictly increasing functionin q� (s) as @ [q� (s)cq(q� (s))� c(q� (s))] =@q� = q� (s)cqq > 0 and q� (s)cq(q� (s))�c(q� (s)) = 0 only when q� (s) = q0.

To show that all plants of the subsidiary produce the same, note that forany QM , cq(qj) = cq(q

0j) minimizas costs, since the cost function is convex.

Next, note that in equilibrium the subsidiary�s �rst order condition is

p� (s)� (1� �)cq(qM(s))� � = 0:

Straightforward manipulation of (A.3) yields qM(s). Last, n� (s) is obtaineddirectly from the market clearing condition.

A.2. The subsidiary confronts a competitive fringe

How does the equilibrium in the downstream market look like when the sub-sidiary is a dominant �rm and independent �rms are a competitive fringe? Wenow show that is the same as when the subsidiary is a price taker.

Proposition A.3. Suppose that the subsidiary commits QM before indepen-dent �rms enter and decide how much to produce. Then the equilibriumoutcome of the dynamic game is characterized by the same tuple as in LemmaA.2.

Proof. Consider the subsidiary�s problem. In order to calculate qM we solve(A.3), but now we must allow for QM to potentially a¤ect p(s) in the last stage

33



of the game. Hence, the subsidiary sets qM(s) so that

m [p� � (1� �)cq(qM)� � ] + [qM + �D0(p� )]@p�@qM

= 0 (A.6)

(where we have omitted the dependence on s to ease notation). The �rst termin brackets on the left hand side is the standard �rst order condition of a�rm in a competitive market. The second term in brackets typically appearswhen a �rm has some market power. As long as @p�

@qM< 0, it would imply

that the subsidiary would be prompted to produce less than in a competitiveequilibrium.

Note that in the last stage independent �rms take QM as given. But, eachindependent �rm i still maximizes pro�ts choosing

q� (s) = argmax fp� (s)qi � [(1 + s)c(qi) + �qi]g ;

and in equilibrium makes zero pro�ts:

p� (s)q� (s)� (1 + s)c(q� (s))� �q� (s) = (1 + s)k:

Hence, the equilibrium price is p� (s) = (1 + s)cq(q0) + � , which is the same aswith perfect competition and independent of qM . Thus, @p�=@qM = 0 becausep� (s) is only a function of the costs of the independent �rms and the levelof sabotage! Thus, the bottleneck monopolist cannot a¤ect p� through qMand it follows from A.6 that the subsidiary�s �rst order condition is p� � (1��)cq(qM)� � = 0, the same as with perfect competition.

A.3. The general relationship between sabotage and the cost struc-ture

We show that the �qualitative�results of the paper are preserved with a non-linear disentangled relationship between sabotage and variable and �xed costs.

Lemma A.4. Assume that the cost of an independent �rms is

f(s)c(q) + g(s)k + �q;

where f and g are increasing in s and f(0) = g(0) = 1. Then @p� (s)@s

> 0 andequations (3.3) and (3.5) hold.

34



Proof. The competitive equilibrium characterized by Lemma A.2 changesbecause q� (s) is not equal to q0 anymore. Instead, it is implicitly de�ned by

[q� (s)cq(q� (s))� c(q� (s))]�f(s)

g(s)� k:

Additionally, the equilibrium price is

p� (s) = f(s)cq(q� (s)) + � :

It follows that

@p� (s)

@s= f 0(s)cq(q� (s)) + f(s)cqq(q� (s))

@q� (s)

@s:

But from the de�nition of q� (s)

@q� (s)

@s= � @

@s

�f(s)

g(s)

�[q� (s)cq(q� (s))� c(q� (s))]

q� (s)f(s)cqq(q� (s))g(s);

which, after substitutions and some algebra, implies

@p� (s)

@s=f 0(s)c(q� (s))

q� (s)+f(s)g0(s)

g(s)

[q� (s)cq(q� (s))� c(q� (s))]

q� (s)> 0:

That is, sabotage still unambiguously increases the equilibrium price. More-over, because the bottleneck monopolist�s problems with and without exclusiondo not directly depend on f or g� those functions only appear in the problemfaced by the independent �rms� it follows that equations (3.3) and (3.5) arepreserved and, with them, the main results of the paper.

B. Proofs of lemmas

B.1. The subsidiary�s scale of production

Lemma B.1 (Producing beyond the MES). Let q0 be a plant�s minimume¢ cient scale MES when s = 0, that is where average cost equals marginalcost and s > ��. Then QM

m� q0.

35



Proof. We know that p� (s) = (1+s)cq(q0)+� and p� (s)�(1��)cq(qM)�� = 0in equilibrium. Thus

0 = (1 + s)cq(q0) + � � (1� �)cq(qM)� �

= (1 + s)cq(q0)� (1� �)cq(qM):

But because s > ��, this can hold i¤ qM > q0, as cqq > 0.

B.2. An implication of Property 1 and convex c

We claim in section 3.2.2 that Property 1 (a downward-sloping marginal rev-enue curve) and convex c is su¢ cient for the second-order condition to holdwhen the bottleneck monopoly limit-prices. Now we prove it.

Proposition B.2. If P +P 0Q is decreasing in Q for all Q, (Property 1), withP (Q) � D�1(Q), and (1� �)c(q) is convex then

2D0 � 1��mcqq (D

0)2+ [p� (1� �)cq]D

00 < 0 (B.1)

for all p 2 [0; p], with p > pM .

To prove the Proposition, the following lemma is useful:

Lemma B.3. Let P (Q) � D�1(Q). Then (i)D0(P (Q)) = 1P 0 ; (ii)D

00(P (Q)) =

� P 00

(P 0)3.

Proof. P (D(p)) � p. Hence

P 0(D(p))D0 � 1; (B.2)

from which (i) follows after straightforward substitutions. Next, totally di¤er-entiating (B.2),

P 00(D(p))(D0)2 + P 0(D(p))D00 � 0;from which (ii) follows.

Proof of Proposition B.2 The proposition is clearly true if D00 � 0. Thusassume D0 > 0 but Property 1 holds and c is convex. Use (i) in Lemma B.3to substitute 1

P 0 for D0 in (B.1) and (ii) to substitute P 00

(P 0)3for D00, and obtain

2

P 0� 1� �

m

cqq

(P 0)2� [p� (1� �)cq]

P 00

(P 0)3: (B.3)

36



Now note that for all p � pM ,

p� (1� �)cq �p

"� � p

D0(p) pD(p)

= �QP 0;

where the second equality follows from Lemma B.3 (i). Hence

[p� (1� �)cq]P 00

(P 0)3� �QP 0 P

00

(P 0)3:

But then

2

P 0� 1� �

m

cqq

(P 00)2� [p� (1� �)cq]

P 00

(P 0)3

� 2

P 0� 1� �

m

cqq

(P 00)2+QP 0

P 00

(P 0)3

=1

(P 0)2�2P 0 +QP 00 � 1��

mcqq�< 0;

where the last inequality follows directly from Property 1 and the convexityof c. Last, existence of p > pM follows directly from continuity.

B.3. Properties of smaxo

Lemma B.4 (Existence and uniqueness). For all values ofm there existsa unique smaxo that satis�es

(1 + smaxo )cq (q0) = (1� �B(m))cq


m

�(B.4)

Proof. The LHS is strictly increasing in smaxo converging to in�nity as smaxo

tends to in�nity, and the RHS is strictly decreasing in smaxo . Hence showingthat ,

cq (q0) < (1� �B(m))cq

�D(p� (0))

m

�;

when (B.4) evaluated at s = 0is su¢ cient to show that both functions intersectonly once.28 But that is direct from

cq (q0) < (1 + smaxo )cq (q0)

= (1� �B(m))cq


m

�< (1� �B(m))cq

�D(p� (0))

m

�:

28Note that given m; �B(m) is the minimum value of � that satis�es (3.2) when QM = D:

37



Lemma B.5 (Maximum sabotage). For all (m; �), smaxo � so(m; �).

Proof. By de�nition when the monopolist sabotages smaxo then QM = D andall independent �rms are limit priced. But then inside Region C in Figure 3(sabotage with coexistence) so(m; �) is always smaller than smaxo because @so

@m>

0 and @so@�

> 0. We also know that inside Region B in Figure 3 (sabotage withlimit-pricing) so(m; �) is always smaller than smaxo this time because @so

@m< 0

and @so@�

< 0.

B.4. A monopolist who owns a small subsidiary does not sabotage

Lemma B.6. (Minimummarket share) For all � > 0 there exists a marketshare �(�) such that for all � � �(�)

D(p� (0))��

p� (0)"� dp�ds

� 0(0) � 0

Proof. The proof is in two parts. First, for any � > 0 we �nd �(�) such thatoptimal sabotage is 0: Second, we show that a bottleneck monopolist with asubsidiary with market share less than �(�) does not sabotage.

We know that 0 > �D(p� (0)) �p� (0)

"dp�ds� 0(0) and that

D(p� (0))�1� �

p� (0)"� dp�ds

� 0(0) � D(p� (so))�1� �

p� (so)"� dp�ds

� 0(so) = 0:

Hence, by continuity there exists �(�) 2 [0; 1] such that

D(p� (0))��(�)� �

p� (0)"� dp�ds

� 0(0) = 0: (B.5)

From (3.3) this implies that a monopolist with subsidiary (m; �) such that

qM(s) = c�1q

�p� (s)��1��

�= D(p� (0))

�(�)

m

does not sabotage. The second part of the proof is direct: clearly, for any� < �(�), the LHS of (B.5) is strictly negative.

C. Vertical integration without sabotage

In this appendix we formally derive Figure 2. To analyze vertical integrationwe study the monopolist�s decision, which the next proposition characterizes.

38



Basically, the result says that, depending on � and m, the monopolist�s sub-sidiary may exclude independent �rms, limit price or coexist acting as a pricetaker.

Proposition C.1. Subsidiary (m; �) with � � 0:(i) sets pM < p� and e¢ ciently excludes independent �rms if

� > �EE(m); (C.1)

(ii) limit-prices at p� if

�LP (m) � � � �EE(m); (C.2)

(iii) coexists and sets qM such that

p� � (1� �)cq(qM) = � (C.3)

if0 � � < �LP (m): (C.4)

�EE(m) = 1�(p� (1� 1=")) =cq�D(p� )m

�and �LP (m) = 1�(p� � �) =cq

�D(p� )m

�.

(iv) If � < 0 the monopolist remains vertically separated.

Proof. The proof consists in comparing pro�ts under di¤erent alternatives.(a) With vertical separation the monopolist makes pro�ts equal to �D(p� ).(b) With vertical integration, pro�ts are

� =

�p�QM �m(1� �)C(qM) + � [D(p� )�QM ] if QM < D(p� )

p�D(p� )�m(1� �)C(qM) if QM � D(p� )(C.5)

The kink in � occurs exactly where the subsidiary�s market share is 100%.The FOC for maximizing (C.5) are

p� � (1� �)cq(qM)� � = 0 if QM < D(p� )

p� � (1� �)cq

�D(p� )m

�� 0 if QM � D(p� )

Proof of part (i) We can rewrite (C.1) as follows

(1� �)cq

�D(p� )m

�(1� 1=") < p�

39



which means that if (C.1) holds then, pro�t maximization implies that themonopolist wants to set a price pM = (1 � �)cq[D(p� )=m]=(1 � 1=") which issmaller than p� . Consequently, the subsidiary e¢ ciently excludes independent�rms. We only have to show that the monopolist is better-o¤ integrating, thatis

pMD(pM)�m(1� �)C(qM) > �D(p� ):

To see that that this is indeed the case note that

hpM � (1� �)C(qM )

qM

iD(pM) >

24p� � (1� �)C�D(p� )m

�D(p� )m

35D(p� ) (C.6)> p� � (1� �)cq

�D(p� )m

�D(p� );

where the �rst inequality follows from pro�t-maximization and the secondfrom cq

�D(p� )m

�> C(D(p� )=m)

D(p� )=m. In addition, p� � (1� �)cq

�D(p� )m

�� , because

for all � 2 [0; �M ] there exists m = n� = D(p� )=q0 and �� 0 such thatp��(1�� )cq(q0) = p�=", i.e. pM = p� . Since p� = p0+� = cq(q0)+� , it followsthat p� � (1�� )cq(q0) = ��p� + � � � . Hence, p�=" = p� � (1�� )cq(q0) � � .This establishes the result.

Proof of part (ii) We can rewrite (C.2) as

� � p� � (1� �)cq

�D(p� )m

�� 1

"p� :

Then, we have that pM > p� which means that the monopolist cannot e¢ -ciently exclude and must take price p� . But the subsidiary grabs the wholemarket because the necessary FOC condition to maximize (C.5) holds atQM = D(p� ). To see that vertical integration is pro�table, note that theFOC implies

p� � (1� �)cq

�D(p� )m

�� ;

but because cq�D(p� )m

�= C

�D(p� )m

�=D(p� )

m,

p� � (1��)cqD(p� )m

� �

() p�D(p� )� (1� �) �m � C�D(p� )m

�� D(p� )

which completes the proof.

40



Proof of part (iii) We can rewrite (C.3) as p� � (1 � �)cq(D(p� )=m) < � ,which implies pM > p� . In addition, p� � (1 � �)cq(qM) = � satis�es theFOC, hence the subsidiary coexists with independent �rms and QM < D(p� ).Now note that a subsidiary with � = 0 and m < n� sets qM = q0, and earnsp� � cq(q0) = � per unit sold. Hence, the monopolist is indi¤erent betweenintegrating and not. For any subsidiary with � > 0, the price�cost margin is �for the last unit sold, and greater than � for the inframarginal units. Hence,on average the monopolist earns higher pro�ts with vertical integration.

Proof of part (iv) Last, note that with vertical diseconomies (� < 0), p� �(1 � �)cq(qM) = � if and only if qM < q0. Hence the monopolist is better o¤by not integrating into the downstream market.

Before moving on we discuss the economics of Proposition C.1. FigureC1 shows the demand curve confronted by the subsidiary for a given � . Withfree entry �rms produce any quantity demanded at price p� . Thus the demandcurve is kinked with the traditional discontinuity of the marginal revenue curve,and the four types of equilibria emerge.

To begin, assume that the subsidiary is very e¢ cient and her marginalcost curve is, say, MC1. Then the subsidiary ignores independent �rms andsets p = pM < p� . Consequently, Figure 2 indicates that for (m; �) in RegionI independent �rms are e¢ ciently excluded.

Clearly, the access charge � is irrelevant inside Region I. On the otherhand, � is relevant for a subsidiary with marginal cost curve MC2 + � , wholimit-prices �rms. In this case the subsidiary enjoys of vertical economies andcollects a unit margin higher than � which increases with � . Subsidiaries (m; �)who limit-price are in Region II of Figure 2.

Next, as the proposition indicates, any monopolist with � > 0 at leastcoexists with �rms. Consider a monopolist with marginal cost curve MC3 +� . Initially � yields a cost advantage over �rms, and it always pays to takeadvantage of this margin until diminishing returns set in. After marginal costsreach p� , the monopolist earns more by selling access.

Last, in Region IV, where � < 0, the bottleneck monopoly does not in-tegrate because she would obtain a smaller margin than � at any scale ofproduction, even when � = 0. This again highlights the importance of entry.If �rms were price takers but their number �xed, then an ine¢ cient monopolistgains setting up a subsidiary and restricting production. But, as we have seen,entry transforms the subsidiary into a price taker in the downstream marketand erodes the rents that could be appropriated by vertically integrating andrestricting output.

41



Marginal revenue

pτ

MC3 + τ

MC2 + τ

MC1

pM

Price

Quantity D(pM) D(pτ)

Demand

Figure C1: Vertical integration with no sabotage The figure shows how the vertical integration decision depends on the subsidiary’s cost. A low-cost subsidiary with marginal cost curve MC1 will set a monopoly price below pτ; this is efficient exclusion, Region I en Figure 2. A subsidiary with a marginal cost curve like MC2 will limit price independent firms by charging pτ; this is region 2 in Figure 2. Subsidiaries with still higher costs, like MC3 will, coexist; this is Region III in Figure 2.

42



D. Vertical integration with sabotage

We now show that if � is low enough, sabotage is characterized by Figure 3.We proceed as follows. Lemma D.1 shows that there exist downward-slopingfunctions �A, �B and �C . Then Lemma D.2 shows that they are ordered suchas in the �gure. Last, a corollary completes the characterization. In whatfollows de�ne �, the pro�t function gross of direct sabotage costs:

�(s;m; �) � �(s;m; �) + (s):

Lemma D.1. The following continuous downward-sloping functions exist on(0; n� ]:

(i) �A de�ned byd�

ds(0;m; �A(m))� 0(0) � 0 (D.1)

with d�ds(s;m; �) = D(p� (s))

h1� p� (s)�(1��)cq(qM )

p� (s)"idp�ds. This characterizes

limit pricers who set so = 0 but satisfy the FOC;

(ii) �B de�ned by

p� (smaxo )� � = [1� �B(m)]cq

�D(p� (smaxo ))

m

�; (D.2)

(iii) �C de�ned by

�[so;m; �C(m)]� (so) =

��(0;m; �C(m)) if m � mC ;�D(p� (0)) if m � mC ;

(D.3)

with mC de�ned by

�(so;mC ; 0)� (so) = �(0;m

C ; 0) = �D(p� (0))

and

�(s;m; �)) =

�p� (s)QM(p� (s))�m(1� �)C (qM(p� (s)))+

� [D(p� (s))�QM(p� (s))]

�:

REMARK Note that two di¤erent cases can occur for monopolists who coex-ist and are indi¤erent between sabotaging and not. First, the marginal costfunction 0 may intersect the marginal bene�t function d�

dsonly once. In that

case equation (D.3) does not de�ne an implicit function when � > 0 because itis not only satis�ed for those who are just indi¤erent, but for all monopolists

43



with subsidiaries that coexist without sabotage. Additionally, the implicitfunction is obtained from d�

ds(0;m; �C(m)) � 0(0) = 0, a case treated like

(i) changing the marginal utility function. Given that, we only consider thecase where 0 and d�

dsintersect either twice or don�t intersect at all. Then

�(so;m; �)� (so) = �(0;m; �) does indeed de�ne an implicit function.

Proof. The proof is an application of the Implicit Function Theorem. Ineach case we have an expression of the form F (m; �(m)) = 0 which de�nesimplicitly � as a function of m. The slope of this function is d�

dm= �Fm

F�(when

F� 6= 0) with Fm and F� the respective partial derivatives.Proof of part (i) F (m; �) � d�

ds(0;m; �)� 0(0). Then

Fm =@

@m

�d�

ds(0;m; �)

�< 0;

F� =@

@�

�d�

ds(0;m; �)

�< 0

because, for subsidiaries (m; �) who limit price, marginal pro�t falls at 0 asthe subsidiary becomes larger or more e¢ cient. Hence �Fm

F�< 0.

Proof of part (ii) F (m; �) � p� (smaxo )� � � [1� �]cq

�D(p� (smaxo ))

m

�. Then

Fm = (1� �)cqqDm2 ;

F� = cq:

Because cq and cqq are positive.

d�B

dm= �(1� �)D

m2

cqqcq

< 0:

Proof of part (iii)Whenm � mC , F (m; �) � �(so;m; �)� (so)��(0;m; �).Then

Fm =

�d�

ds� d

ds

�@s

@m+@�

@m(so)�

@�

@m(0)

=@�

@m(so)�

@�

@m(0)

44



because so satis�es the �rst order condition d�ds(so)� 0(so) = 0. Similarly

F� =@�

@�(so)�

@�

@�(0):

Let qM � QM [p� (s);m; �]=m. The partial derivatives @�@m(s) and @�

@�(s) are

@�

@m(s) = (1� �) [qM � cq � C] ;

@�

@�(s) = m � C;

Both derivatives are increasing in s because

d

ds

�@�

@m

�= (1� �) (cq + qMcqq � cq)

dqMds

= (1� �) � qM � cqqdqMds

> 0;

d

ds

�@�

@�

�= m � cq dqMds > 0

because dqMds

is positive for all subsidiaries who optimally coexist. Thus Fmand F� have the same sign and

d�A

dm= �Fm

F�< 0:

When m � mC , �(0;m; �) is replaced by �D[p� (0)], which neither depends onm nor �. Thus

d�A

dm= �

@�@m(so)

@�@�(so)

;

which is negative because � increases with size and vertical economies.

Lemma D.2. (i) limm!0 �A = limm!0 �

B = limm!0 �C = 1; (ii) �A(n� ) > 0;

(iii) �B(n� ) < �C(n� ) < 0; (iv) for all m 2 (0; n� ], �A(m) > �B(m); (v)�B(m) > �C(m) i¤ m 2 (0;m�) and �B(m) < �C(m) i¤ m 2 (m�; n� ] withm� > mC .

Proof of part (i)We show that for � arbitrarily close to 1 there always existsan m such that �A(m) = �, that is, �A gets as close as possible to 1 and the

45



pair (m; �) satis�es equation (D.1). First rewrite equation (D.1) as

(1� �)cq

�D(p� (0))

m

�= p� (0)

"1� 1

"+

0(0)

"D(p� (0))dp�ds

#(D.4)

Here only (1� �)cq

�D(p� (0))

m

�depends on � and m. Now rewrite (D.4) as

�A(m) = 1� p� (0)

cq

�D(p� (0))

m

� "1� 1"+

0(0)

"D(p� (0))dp�ds

#:

Let H be a bounded constant and note that limm!0 cq(D (p� (0)) =m = 1.Hence:

limm!0

�A(m) = 1� limm!0

H

cq

�D(p� (0))

m

� = 1Next, consider the equation satis�ed by (m; �B(m));

p� (smaxo )� � = (1� �)cq

�D(p� (smaxo ))

m

�;

and rewrite it as

�B(m) = 1� p� (smaxo )� �

cq

�D(p� (smaxo ))

m

� ;which, as before, implies limm!0 �

B(m) = 1. Last, consider the equationsatis�ed by (m; �C(m)) when m � mC ,(

p� (so)QM(p� (so))�m(1� �C(m))C�QM (p� (so))

m

�+

� [D(p� (so))�QM(p� (so))]� (so)

)= p� (0)QM(p� (0))�m(1� �C(m))C

�QM (p� (0))

m

�+ � [D(p� (0))�QM(p� (0))]

which implies

�C(m) = 1�

�� [D(p� (so))�D(p� (0)] +QM(p� (0))�QM(p� (so))+p� (so)QM(p� (so))� p� (0)QM(p� (0))� (so)

�mhC�QM (p� (0))

m

�� C

�QM (p� (so))

m

�iand limm!0 �

C(m) = 1 as limm!0 so(m) = 0:

46



Proof of part (ii) All subsidiaries (m; �A(m)) limit price with so = 0 and thebottleneck monopoly always prefers to integrate. Then all such subsidiariesmust have � > 0, otherwise the bottleneck monopoly would remain verticallyseparated.

Proof of part (iii) With subsidiary (n� ; �B(n� )) the bottleneck monopolyloses money with vertical integration and does not satisfy the participationconstraint. Then �B(n� ) must lie below �C(n� ), which de�nes that participa-tion constraint.

Proof of part (iv) Fix m and consider subsidiary (m; �B(m)). Then thebottleneck monopolist sabotages to set a limit price, but is indi¤erent betweensabotaging and not. Hence �A(m) > �B(m), otherwise the subsidiary onlycoexists.

Proof of part (v) First, whenm < mC we apply a similar argument as in (iv).Fixm and consider subsidiary (m; �B(m)). This subsidiary must coexist with amarket share strictly less than one and must be more e¢ cient than subsidiary(m; �C(m)). Now, when m > mC , �C(m) < 0; �B(mC) > �C(mC) = 0;and �B(n� ) < �C(n� ) < 0. By continuity they must intersect at some pointm�, and the intersection is unique because on �B(m) pro�ts monotonicallyfall as m increases and along �C(m) pro�ts are zero by de�nition. Obviously�B(m�) = �C(m�) < 0. This completes the proof.

Corollary D.3. Functions �A(m), �B(m) and �C(m) split the (m; �) space inopen sets A, B, C, D and E, where

(i) A is the open set of all (m; �) with � > �A(m) such that the bottleneckmonopoly strictly prefers to integrate, does not sabotage and has � = 1.

(ii) B is the open set of all (m; �) with �A(m) � � > �B(m) for all m < m�

and �A(m) � � > �C(m) for all m � m� such that the bottleneck monopolystrictly prefers to integrate, sabotage and has � = 1.

(iii) C is the open set of all (m; �) with �B(m) � � > �C(m) such that thebottleneck monopolist strictly prefers to integrate, sabotage and has � < 1.

(iv)D is the open set of all (m; �) with �C(m) � � > 0 such that the bottleneckmonopolist strictly prefers to integrate, does not sabotage and has � < 1.

(v) E is the open set of all (m; �) with min(0; �C(m)) � � such that thebottleneck monopolist strictly prefers to remain vertically separated.

Proof. The Corollary follows directly from the former analysis.

47



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vertical integration - northwestern university

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