Dominant Firms, Barriers to Entry Capitaland Antitrust Policy�

Mika Katoyand Willi Semmlerz

November 26, 2006

AbstractThe main idea of our paper comes from earlier industrial organi-

zation literature that has shown that the threat of entry limits theprice setting power of dominant �rms and stimulates the incumbentsto undertake innovations. We provide a theoretical framework of thedynamics of competition where incumbents attempt to restrict or in-hibit competition through competition restricting investments such asadvertisement, political lobbying, protection of innovations throughpatents, and creating excess capacity, etc. Depending on how other�rms and the regulatory institutions respond to this type of invest-ment, complex dynamics may emerge. Since the e¤ectiveness of com-petition restricting investments depends on regulatory rules set andenforced by antitrust institutions, we show how an antitrust and com-petition policy can be designed that may prevent the build up of such acompetition restricting capital, strengthening incentives for price andinnovation competition.

1 Introduction

In the last decade the theory of competition has moved away from the statictheory, based on the perfect-imperfect of competition paradigm, to a dy-

�We want to thank Buz Brock, Herbert Dawid, and Ekkehard Ernst for helpful discus-sions, Armon Rezai for help with research.

yDepartment of Economics, Howard University, Washington, DCzDepartment of Economics, New School for Social Research, New York and CEM,

Bielefeld

1

namic theory. Competition in the traditional sense is price competition andthe deviation from perfectly competitive prices is shown to result in welfarelosses. Accordingly antitrust and competition laws in the U.S. and Europehad adhered to the static blueprint of the perfect competition paradigm.The recent research direction, however, moves away from the structure-

conduct-performance paradigm, a long time framework for industrial orga-nization studies and regulatory policy, and stresses that the dynamics ofcompetition does not necessarily depend on market structure. The new di-rection gives more relevance to the competitive behavior (for example rivalryin an oligopolistic setting). It views competition as price competition as wellas competition for product and process innovation. Accordingly, industrialorganization and antitrust literature have attempted to integrate more dy-namic and evolutionary view points into the studies. The major change ofthe paradigm came from both, �rst, the view that entry dynamics is alwaysan important source of potential competition and, second, the view thatstrategic behavior of incumbents may result in prices below monopoly prices(limiting pricing) and in a drive for new product and process innovation toprevent entry or to preempt the rivals�strategies. As the overall usefulness ofperfect competition framework has become more questionable as a guidelinefor antitrust regulation and competition policy it is still controversial whatthe features of a new antitrust rule and competition policy should be and howthey should be designed for the new paradigm of competition dynamics.1

One of the essential points of the new paradigm of competition is that,as �rms are exposed to the dynamics of competition, they are likely to at-tempt to restrict or inhibit competition through competition restricting in-vestments. Our paper which pursues this point is based on earlier work byBrock (1983) and Brock and Dechert (1985) that studied barriers to entrycapital to restrict competition. Brock (1983) argued that when dominant�rms face a threat of competitive fringe �rms in the industry they will havean incentive to prevent it. Investing into barriers to entry capital through en-gaging in production activities with increasing returns and high adjustmentcost of investment as well as through advertising, lobbying, creating excesscapacity and patent protection, the dominant �rm can create a thresholdabove which fringe �rms cannot induce price competition and stimulate in-novations.Yet, nowadays we know that �rms not only build up entry preventing

1For details on the new paradigm of competition dynamics, see Audretsch et al. (2001).

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capital to reduce market competition (through engaging in increasing returnsactivity, advertisement, political lobbying, protection of innovations throughpatents, creating excess capacity) but also can restrict competition throughinvestments that inhibit competitive behavior (for example, investment incoalition formation, lobbying and pressuring for anti-competitive regulatorymeasures, etc.). These are all examples of investments that restrict the dy-namics of competition (price competition as well as competition in welfareimproving product and process innovation). This paper is concerned withsuch type of investments.We also show that depending on how fringe �rms and the regulatory in-

stitutions respond to this type of investment, complex dynamics, multiplesteady states and a threshold, separating di¤erent domains of attraction,may emerge, where below the threshold there are forces that revive compe-tition while above the threshold incumbents achieve a high market share.The e¤ectiveness of competition restricting investments may indeed dependon the fringe �rms� reaction and on regulatory rules set and enforced byantitrust institutions, e.g., fringe �rms undermine the success of dominant�rms by accelerating the latter�s obsolescence of the applied technology andan antitrust and competition policy can be designed to prevent the build upof such a competition restricting capital, strengthening incentives for priceand innovation competition. In the paper these are represented by someparameters.2

In our context the antitrust and competition policy should be to stim-ulate, encourage, and if necessary, restore the dynamics of markets by pro-hibiting the restrictions of competition. We show that a competition policy,through some regulatory instruments, can increase the domains of attractionwhere competition takes place. Yet, when no competition policy is pursued,as for example, suggested by the theory of contestable markets, where evendominant �rms have to behave competitively since hit and run strategies ofentrants enforce competition,3 the dominant �rms may achieve a high mar-ket share as inferior domains of attraction. We use dynamic programming tocompute these domains of attraction.The remainder of the paper is organized as follows. Section 2 introduces

2One can also view the dominant �rms as playing a game against the fringe �rms andregulatory agencies, but the fringe �rms and the regulatory agencies set adverse conditions,as for example has been discussed in the robust control literature (see Zhang and Semmler,2005).

3See Baumol et al. (1982) and Brock (1983).

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the preliminary model, taking �rst, prices as constant. We present a numberof numerical examples. Section 3 introduces price reaction by employing adownward sloping demand function. Here we also compute the welfare lossesdue to restricted competition established through competition restricting in-vestment. Section 4 studies antitrust and competition policy as resultingfrom our theoretical and numerical study. Section 5 concludes the paper.The appendix gives a brief summary of the dynamic programming methodused to solve some of our model�s variants.

2 Model

This section introduces a preliminary version of our model where we assumethe product price constant. This assumption will be relaxed in Section 3 byintroducing a downward sloping demand curve where prices respond.

2.1 Industry environment

We presume a dominant �rm in an industry. We can also interpret thedominant �rm as a group of �rms whose activities are highly coordinated.Yet for short we will use the term dominant �rm. We presume that thedominant �rm and the competitive fringe � again these could be many �compete for a given market demand d. The dominant �rm may have anincentive to restrict the other �rms� behavior through investing in entry-deterring capital.The dominant �rm maximizes the discounted future net cash �ows:

maxx

Z 1

0

e�rt [q � C(q)� x� '(x)] dt (1)

where r is the discount rate, q is the output of the dominant �rm with theproduct price �xed at 1, C is the linear cost of production with C 0 = c > 0, xis the entry-deterring investment with the price of a unit of investment good1, and ' is the adjustment cost with '0 T 0 for x T 0 and '00 > 0.Entry-deterring capital accumulation is

_E = x� �EE (2)

where �E is the obsolescence rate of the entry-deterring capital.

4

Figure 1: Dominant �rm�s market share

Then the dominant �rm�s market share s depends on the level of entry-deterring capital

q = s(E)d (3)

where 0 < s(E) < 1 with properties s(0) = 0, s(+1) = 1, s0(E) � 0, ands0(0) = s0(+1) = 0. We assume economies of scale and then decreasingreturns as shown in Figure 1.Since the entry-deterring capital cannot be negative �E � 0,4 we impose

a non-negativity constraint to the equation of motion for E:

h = �E � 0) _h = � _E = �(x� �EE) � 0 whenever h = 0. (4)

Let the Lagrangian be written as

L = s(E)d� C(q)� x� '(x) + �(x� �EE)� � _h. (5)

Applying Pontryagin�s maximum principle, the �rst-order conditions are

Lx = �1� '0(x) + �+ � = 0, (6)

L� = � _h = x� �EE � 0 � � 0 �L� = 0, and (7)

4Since h is not allowed to exceed 0, then whenever h = 0, we must forbid h to increase.Thus, the problem has a state-space constraint.

5

�E � 0 �E = 0 (8)

where (8) is the complementary-slackness condition appended to (7) thatensures that (7) is valid only when the constraint is binding (E = 0).At points where � is di¤erentiable,

_� � 0 ( = 0 when � E < 0), (9)

_� = (r + �E)�� (1� c)s0(E)d+ ��E, (10)

and transversality conditions.

2.2 Entry-deterring investment rules

Whenever entry-deterring capital is positive, E > 0 (constraint not binding),from (8) and (9), � = _� = 0. Thus, from (6), the dominant �rm�s optimalentry-deterring investment rule is8<:

x > 0 � > 1x = 0 for � = 1x < 0 � < 1

when E > 0, (11)

where � is the value of future marginal net cash �ows by increasing a unit ofentry-deterring capital5. The rule (11) suggests that if � is greater (smaller)than 1 (the price of a unit of investment good), the dominant �rm invests(disinvests) until � decreases (increases) to 1.When the nonnegativity constraint is binding for some time period, it fol-

lows that E = _E = 0. Thus, from (2), the optimal entry-deterring investmentrule is

x = 0 when E = 0. (12)

5From (10), _� � (r + �E)� = �s0(E; �; �)d: Thus solving this �rst-order di¤erentialequation gives�t = d

R1ts0(E; �; �)e�(r+�E)�d� .

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Figure 2: Dynamics of x and E

Out-of-steady-state dynamics

From (6) and (10), the equation of motion for x is

_x =1

'00(x)[(r + �E)(1 + '

0(x))� (1� c)s0(E)d� �r + _�]. (13)

Note that � = _� = 0 for E > 0. The phase diagram is shown in Figure 2.In the industry, there are possibly three steady states, two at x�; E� > 0

and one at x� = E� = 0 (when the nonnegativity constraint is binding.)Let�s call these steady states E�1 , E

�2 , and E

�3 in order of size of E

� where E�1suggests a competitive environment where the market share of the dominant�rm is zero (negligibly small) while E�3 suggests a higher market share bythe dominant �rm. An industry tends toward a higher concentration ofpower when the dominant �rm accumulates entry-deterring capital beyonda certain level that is called a "threshold". The threshold separates twodomains of attraction where below the threshold there are forces that revivecompetition while above the threshold the dominant �rm achieves a highmarket share. If this is not the case, we can have a sole attractor at zeroThis is likely to happen when the obsolescence of the entry-deterring capitalis high or/and when the regulatory agency imposes a strong regulation. Bothcases discourage the dominant �rm to accumulate and hold entry-deterringcapital.

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The local stability about each steady state is studied by checking theJacobian matrix J :

J =

�@ _E@E

@ _E@x

@ _x@E

@ _x@x

�x�;E�>0

=

���E 1� r + �E

�. (14)

where

� � �1'00(x)

(1� c)s00(E)d. (15)

At E�2 , the phase diagram indicates@ _x@E< 0 in the vicinity and thus � < 0.

Therefore,

J =

�� +� +

�, (16)

Det J = ��E(r + �E) + 1'00 (1 � c)s

00d, and Tr J = r > 0. Det J > 0 holdsfor relatively small r, �E, c and a large d where multiple steady states aremore likely to arise. Those suggest that the local dynamic property in thevicinity of E�2 is a source, i.e. E

�2 is a repellor. It can be a spiral source if the

discriminant � = (Tr J)2 � 4Det J < 0 or a node source if � > 0.

At E�1 and E�3 , the phase diagram indicates @ _x

@E> 0 in the vicinity and

thus � > 0. Therefore,

J =

�� ++ +

�(17)

and Det J < 0. This means that the local dynamic property in the vicinityof E�3 is a saddle.

Yet, overall we want to remark here that the local analysis of computingthe number of steady states and the stability does not necessarily imply thatthose are actually reached. We may detect one of the following four scenarios:

1. Dominance of high market share: There are three steady states. Yet,for any E (0), the dominant �rm chooses to increase the entry-deterringcapital to E�3 through competition restricting investment. Therefore,in the long run, this will entail a high concentration of the industry.

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2. Threshold dynamics: There are three steady states, but there existsa threshold level of E separating di¤erent domains of attraction. Thedominant �rm, for any E (0) above the threshold, has an incentive tobuild up entry-deterring capital to E�3 and therefore high concentrationof the industry is realized in the long run. The dominant �rm, on theother hand, may let the fringe �rms enter the industry rather thanblocking them by building up E. This happens for any E (0) below thethreshold since the marginal costs of building up E is larger than thebene�ts from it. The dominant �rm reduces E to E�1 and the industryrestore a competitive state in the long run.

3. Restoration of competitive market: There are three steady states. Yet,for any E (0), building up E is too costly comparing to the bene�tsfrom it. Therefore, the dominant �rm reduces E to E�1 and graduallyloses the dominance of the market share. A competitive state of theindustry will be restored in the long run.

4. Competitive region as sole attractor: There is a unique steady stateE�1 . For any E (0), a competitive state of the industry will be restoredin the long run.

2.3 Numerical examples

We study numerical examples for those scenarios. Let us use the followingspeci�c functions:

C(q) = cq, (18)

'(x) = �x2, (19)

s(E) =E�

�� + E�< 1 (20)

where the functional form (20) can produce a s-shape as in Figure 1 for a pa-rameter � > 1. Let � measure the deterrence e¢ ciency by the dominant �rm.Smaller � suggests that the same amount entry-deterring capital contributesmore to an increase in the share of the dominant �rm.For convenience, we set up default parameters:

Default A r = :02; �E = :15; � = 5; � = 10; d = 10; c = :001; � = :5.

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It turns out that the default case has three steady states. The steadystate values of E, x, and s are summarized in Table 1.6 It shows that themarket share by the dominant �rm is, if E�3 is achieved, 82.8% while if E�1is achieved, the dominant �rm�s share is 0. We interpret the latter case as acompetitive state of the industry, i.e., a negligibly small market share by thedominant �rm.

E�1 E�2 E�3Entry-deterring capital E 0.00 5.0205 13.6949

Investment level x 0.00 0.7531 2.0542Market share s 0.00 0.0309 0.8281

Table 1: Default A

Deterrence e¢ ciency �

When the dominant �rm has a higher deterrence e¢ ciency, it can acquirea higher market share with the same amount entry-deterring capital. Table2, comparing to Default A (� = 10), shows that with higher deterrencee¢ ciency (� = 5), the dominant �rm realizes a higher market share (93%)with less entry-deterring capital (8.4) when the upper steady state E�3 isachieved. Table 3, on the other hand shows that with lower e¢ ciency (� =20), only a smaller market share (42%) is realized even with more entry-deterring capital (18.8) atE�3 . When the e¢ ciency is further lowered (� = 30)as Table 4 shows, a sole attractor emerges about E�1 . This implies that a highmarket share by the dominant �rm is transitory and a competitive marketwill be restored in the long run.

E�1 E�2 E�3E 0.00 1.9329 8.4019x 0.00 0.2899 1.2603s 0.00 0.0086 0.9305

Table 2: �=5Higher e¢ ciency

E�1 E�2 E�3E 0.00 15.9242 18.7755x 0.00 2.3886 2.8163s 0.00 0.2424 0.4217

Table 3: �=20Lower e¢ ciency

E�1E 0.00x 0.00s 0.00

Table 4: �=30Lower e¢ ciency

Obsolescence rate �E

The parameter �E represents a obsolescence rate of the entry-deterring cap-ital. The e¤ects of past patents, advertising and lobbying decline as time

6The steady states are computed by Mathematica 5.2.

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passes. One can also view �E as a policy parameter as the regulatory agencycan counteract the dominant �rm�s entry-deterring e¤orts, e.g. shorten thelifetime of patents, raise the patent renewal fees, restrict lobbying activities,invalidate some antitrust activities by stricter law enforcement, etc..Comparing to Default A (�E = :15), Table 5 shows that when the ob-

solescence rate is smaller (�E = :05) the dominant �rm�s market share risesto 95% at E�3 . This occurs because the entry barrier lasts longer and thedominant �rm is less exposed to competition. When the obsolescence rate islarger (�E = :25) as in Table 6, the opposite occurs and the dominant �rm�smarket share falls to 64% at E�3 . For the further greater obsolescence rate(�E = :35), a sole attractor emerges around E�1 (Table 7) and a competitivemarket will be restored in the long run.

E�1 E�2 E�3E 0.00 3.5967 17.9879x 0.00 0.1798 0.8994s 0.00 0.0060 0.9496

Table 5: �E=::05Lower obsolescence

E�1 E�2 E�3E 0.00 6.4724 11.2141x 0.00 1.6181 2.8035s 0.00 0.1020 0.6394

Table 6: �E=:25Higher obsolescence

E�1E 0.00x 0.00s 0.00

Table 7: �E=:35Higher obsolescence

Discount rate r

The discount rate re�ects uncertainty of the future market. When a lowerdiscount rate applies (r = :005), comparing to Default A (r = :02), thedominant �rm has a greater incentive for competition-restricting investmentsas it values the future market share. The resulting market share at E�3 isthus higher (Table 8). On the other hand, a higher discount rate (r = :05)discourages the dominant �rm to make such investments and the marketshare drops (Table 9). When uncertainty is even higher (r = :45), there willbe little e¤orts to make reservation of future market share. Thus a uniquecompetitive attractor emerges (Table 10).

E�1 E�2 E�3E 0.00 4.8813 13.9492x 0.00 0.7322 2.0924s 0.00 0.0270 0.8408

Table 8: r=:005Lower discount

E�1 E�2 E�3E 0.00 5.2809 13.2413x 0.00 0.7921 1.9862s 0.00 0.0395 0.8028

Table 9: r=:05Higher discount

E�1E 0.00x 0.00s 0.00

Table 10: r=:45Higher discount

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Initial entry-deterring capital E (0)

The initial entry-deterring capital size E (0) is important when thresholddynamics (Scenario 3) arise. Some industries (for example in utilities, likeGas, Electricity, etc.) causes by nature a high initial barrier to entry. Theincumbent �rms in such industries thus may have an above-threshold E(0)and tend to achieve a high market share in the long run.

3 Restricted Competition and Loss of Bene�t

In the previous section, the dominant �rm simply maximizes its market sharefor a given market demand. The maximization of the market-share, however,makes more sense with an inelastic market demand curve. In this section, weintroduce a downward sloping demand. This section also studies the possibleloss of economic bene�ts arising from the competition-restricting activitiesby the dominant �rm.

3.1 Model

The dominant �rm�s objective is to maximize the discounted future net cash�ows:

maxx

Z 1

0

e�rt [pq � C(q)� x� '(x)] dt (21)

subject to (2). By presuming price setting power by the dominant �rm,we conveniently assume that the product price is a monotonically increasingfunction of the market share of the dominant �rm, p(s) > 0 for 0 � s � 1with p(0) = pc, p(1) = pm where pc is the competitive price and pm is themonopolistic price, as depicted in Figure 3.With a downward sloping market demand, the dominant �rm�s output is

q = s(E)d(p), (22)

and the revenue is R(s) = p(s)sd(p). Note that most empirical studies showsthat there is some positive correlation of market share and rates of return,R0(s) > 0.7

7See for example Weiss (1963). For an extensive survey of earlier literature, see Semmler(1984).

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Figure 3: Market share and price

The Lagrangian is written as

L = p(s)s(E)d(p)� C(q)� x� '(x) + �(x� �EE)� � _h. (23)

We have the common �rst order conditions (6)-(9) from the previous sectionand the equation of motion for � is modi�ed as

_� = (r + �E)�� p0(s)s0(E)s(E)d(p) (24)

�(p(s)� c)fs0(E)d(p) + s(E)d0(p)p0(s)s0(E)g+ ��E.

Out-of-steady-state dynamics

The equ. (24) modi�es the economic system as follows:

_x =1

'00(x)[(r + �E)(1 + '

0(x))� p0(s)s0(E)s(E)d(p) (25)

�(p(s)� c)fs0(E)d(p) + s(E)d0(p)p0(s)s0(E)g � �r + _�]

and (2). The system again shows the possibility of both two attractorsor a sole attractor.

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3.2 Numerical examples

We use simple linear functions for the market price and the demand:

p(s) = pc + (pm � pc)s for 0 � s � 1 (26)

d = b� ap. (27)

where parameters pc, pm, b and a are chosen so that R0(s) > 0 as empiricalevidence shows.We set a default parameter set as follows:

Default B r = :02; �E = :15; � = 5; � = 30; c = :001; � = :5; pm = 8;pc = 2; b = 10; a = :5

The default case has three steady states (Table 11). The dominant �rmwill have a high (81%) market share if the upper steady state E�3 is achievedand the corresponding price level is 6.6.

E�1 E�2 E�3Entry-deterring capital E 0.00 19.6056 39.8492

Investment level x 0.00 2.9408 5.9774Market share s 0.00 0.1065 0.8053Price Level p 2.00 2.6391 6.8316

Market Demand d 9.00 8.6805 6.5842Table 11: Default B

3.3 Loss of bene�t

As a consequence of the dominant �rm�s competition-restricting activities,the price level diverges from the competitive price. Using basic microeco-nomic theory, we can �nd the resulting welfare loss (Figure 4).The economic surplus in the competitive market ESc is the sum of pro-

ducer�s surplus and consumer�s surplus:

ESc = (pc � c)d(pc) +Z 1

pcd(p)dp =

Z 1

pcd(p)dp (28)

where c = C 0(q) is the constant marginal cost and pc = c in the competitivemarket while the economic surplus ES� where the dominant �rm has s�

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Figure 4: Welfare loss

market share is

ES� = (p(s�)� c)d(p(s�)) +Z 1

p(s�)

d(p)dp. (29)

Therefore, the welfare loss l can be computed as

l � ESc � ES� (30)

=

Z p(s�)

c

d(p)dp� (p(s�)� c)d(p(s�)) > 0.

The welfare loss for Default B is

E�1 E�2 E�3Welfare Loss l 0.00 1.74 11.664

Table 12: Default B �Loss

Table 12 implies that competition-restricting activities can cause a welfareloss when no competition policy is pursued by the regulatory agency.

3.4 Global analysis

As we stated in Subsection 2.2, when multiple steady states emerge, tradi-tional approaches such as the study undertaken by Pontryagin�s maximum

15

principle and the subsequent local stability analysis are not enough to obtaina complete picture of the global dynamics. To study the global dynamics,dynamic programming is helpful. Through this approach, we can �rst com-pute the local value function associated with each candidate path and thenobtain the global value function. This additional information allows us toknow which steady state is actually reached. We here use the dynamic pro-gramming algorithm applied in Grüne and Semmler (2004).8 It enables us tosolve the dynamic problem numerically, compute the global value functionand the corresponding policy function, and detect a threshold(s) if it exists.The phase diagram for Default B is depicted in Figure 5. This reveals

the equilibrium candidates of the respective equations. Figure 6 shows thecorresponding value function and the policy function computed with theGrüne-Semmler algorithm. The direction of the arrows on the horizontal axissuggests the motion of E that fetches the dominant �rm the largest pro�ts. Itis obvious that Default B has Scenario 1 (dominance of high market share).Although we know from the local analysis that E�1 is also a stable steadystate, it will never be actually reached. The dominant �rm instead choosesto increase the entry-deterring capital to E�3 for any E(0) and achieves 81%of the market share in the long run that incurs 11.7 welfare loss.

4 Antitrust Policy

Based on the previous discussion, the regulatory agency�s intervention tothe market may be justi�ed under the following circumstances; the marketis in Scenario 1 (dominance of high market share) like Default B or themarket is in Scenario 2 (threshold dynamics) where the dominant �rm ownsentry-deterring capital beyond the threshold level. In this section, we studywhether the regulatory agency can reduce or eliminate the possibility thatthe dominant �rm under these circumstances may achieve a high marketshare. We consider �E, the obsolescence rate of the entry-deterring capital,to play the role of a policy instrument for the reason as stated in Subsection2.3.

8For a short description, see the Appendix.

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Figure 5: Dominance of high market share (Default B)

Figure 6: Global value function (Default B)

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Regulatory environment �E

The global analysis in Subsection 3.4 detects Scenario 1 for Default B. In thisscenario, when no competition policy is pursued, the dominant �rm increasesthe entry-deterring activities to reach E�3 , and this causes welfare loss of 11.7in the long run. The following numerical examples in Table 13 and 14, how-ever, demonstrate that the regulatory policy might e¤ectively work to alterthe scenario to the preferable scenario, Scenario 3 or 4, where the competitivestate of the industry is restored in the long run. The obsolescence rate ofthe entry-deterring capital �E can be viewed as a policy instrument since theregulatory agency can counteract the dominant �rm�s entry-deterring e¤ortssuch that it accelerates the obsolescence of entry-deterring capital, (e.g., byimposing a shorter lifetime of patents, higher patent renewal fees, lower limitsin political donations, more severe restrictions on political lobbying, stricterlaw enforcement to antitrust activities, etc.). Table 13 shows that in a tighterregulatory environment (�E = :24), Scenario 3 (Restoration of competitivemarket) arises while there are still three steady states. The dominant �rmin this scenario gradually reduces the entry-deterring activities for any givenE(0) and the industry restores a competitive state in the long run. Figures 7and 8 show the phase diagram and the corresponding global value function.When a further higher depreciation rate (�E = :26) is set (Table 14), Scenario4 can arise. This suggests a possibility that the regulatory agency can evenmake the competitive state E�1 a sole attractor. Figures 9 and 10 show thephase diagram and the global policy function for this case.

E�1 E�2 E�3E 0.00 26.3148 30.7209x 0.00 6.3156 7.3730s 0.00 0.3418 0.5296p 2.00 4.0507 5.1779d 9.00 7.9746 7.4111l 0.00 4.1001 6.7000

Table 13: �E=:24Tighter regulation

E�1E 0.00x 0.00s 0.00p 2.00d 9.00l 0.00

Table 14: �E=:26Tighter regulation

Deterrence e¢ ciency �

It should be noted, however, that the regulatory policy should be industry-speci�c. When the dominant �rm has a lower deterrence e¢ ciency, the reg-

18

Figure 7: Restoration of competitive market (�E = :24)

Figure 8: Global value functioin (�E = :24)

19

Figure 9: Competitive region as sole attractor (�E = :26)

Figure 10: Global value function (�E = :26)

20

ulatory agency may not need to use the regulatory instruments as much asbefore to revive competition. In Table 15, the dominant �rm, comparing toDefault B, has a lower deterrence e¢ ciency (� = 40). Figure 11 depicts thephase diagram and Figure 12 shows the corresponding global value functionand the policy function. Those suggest that it leads to Scenario 2 (thresh-old dynamics). Hence, the high market share is reached by the dominant�rm only when the dominant �rm�s entry-deterring capital is larger than thethreshold level. This is more likely to occur, e.g., when it acquires a criticalpatent or when it forms a coalition or a merger, etc.. Thus, with no regula-tory intervention to the market, there is some possibility that the dominant�rm achieves a considerable market share in the long run. The regulatoryagency, given Scenario 2, can alter the market environment to Scenario 3or Scenario 4 where competition takes place with less use of the regulatoryinstruments. Similarly, when the dominant �rm has even lower deterrencee¢ ciency (� = 50, Table 16), Scenario 4 (a sole attractor) emerges.9 As Fig-ures 13 and 14 suggests, a competitive state of the industry will be restoredfor any given E(0) in the long run. In this case, the regulatory interventionis not necessary even when a high market share of the dominant �rm becauseit should be temporary.

E�1 E�2 E�3E 0.00 30.5258 46.5924x 0.00 4.5789 6.9889s 0.00 0.2056 0.6820p 2.00 3.2337 6.0918d 9.00 8.3831 6.9541l 0.00 2.6126 9.2743

Table 15: �=40Lower e¢ ciency

E�1E 0.00x 0.00s 0.00p 2.00d 9.00l 0.00

Table 16: �=50Lower e¢ ciency

5 Conclusion

This paper provides a theoretical framework of the dynamics of competi-tion where incumbents attempt to restrict or inhibit competition through

9We want to note that there is likely to be a scenario, similarly to the �rst scenario,where, however, the lower equilibrium becomes the sole attractor even though there arethree equilibrium candidates.

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Figure 11: Threshold dynamics (� = 40)

Figure 12: Global value function (� = 40)

22

Figure 13: Competitive region as sole attractor (� = 50)

Figure 14: Global value function (� = 50)

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competition restricting investments such as advertisement, political lobby-ing, protection of innovations through patents, and excess capacity, etc. Wealso studied of how an antitrust and competition policy can be designed toprevent the build up of such a competition restricting capital, strengtheningincentives for price and innovation competition.We commence our study by introducing a preliminary version of our

model, assuming the product price is �xed, where the dominant �rm andthe competitive fringe compete for a given market demand. In this simplesetting, the dominant �rm simply maximizes its market share for a givenmarket demand. We �rst show that the model possibly gives rise to multi-ple steady states, a competitive state and a highly concentrated state. If themarket tends to competitive or to be largely dominated by the dominant �rmin the long run depends on the deterrence e¢ ciency of the dominant �rm,the obsolescence rate of the entry-deterring capital, the regulatory policy bythe regulatory agency, and the discount rate. Secondly, we indicate four pos-sible scenarios. Which of the scenarios emerges depends on how a regulatoryinstitution responds to this type of investment and other market conditions.We present a number of numerical examples to illustrate di¤erent outcomesof our morel.Thereafter we endogenize the product price by assuming price setting

power by the dominant �rm, yet the price responds to a downward slopingmarket demand curve. We study the e¤ects of the dominant �rm�s com-petition restricting activities on the market price and the loss of economicbene�ts arising hereby.Furthermore, we attempt to answer the question whether any policy pa-

rameter can be used to reduce the possibility of the dominant �rm achievinga high concentration equilibrium in an industry and avoid the possible lossof economic bene�ts. We consider �E, the obsolescence rate of the entry-deterring capital, as a policy parameter as the regulatory agency can coun-teract the dominant �rm�s entry-deterring e¤orts, e.g. shorten the lifetime ofa patent, restrict political lobbying, invalidate antitrust activities by stricterenforcement of the law, etc.. Our numerical study shows that, by increasing�E, the domains of attraction where competition takes place enlarges. Thismeans that, once the regulatory agency is successful to change the competi-tive environment of the industry, it does not persistently have to intervene.Finally, we mention that the use of dynamic programming approach is

helpful to study the global dynamics and to detect the superior or inferiordomains of attraction.

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Appendix: Numerical Solution Method

We here brie�y describe the dynamic programming algorithm as applied inGrüne and Semmler (2004) that enables us to numerically solve the dynamicmodel as proposed in Section 4. The feature of the dynamic programmingalgorithm is an adaptive discretization of the state space which leads to highnumerical accuracy with moderate use of memory.Such algorithm is applied to discounted in�nite horizon optimal control

problems of the type introduced in Section 4. In our model variants we haveto numerically compute V (x) for

V (x) = maxu

Z 1

0

e�rf(x; u)dt

s.t. _x = g(x; u)

where u represents the control variable and x a vector of state variables.In the �rst step, the continuous time optimal control problem has to be

replaced by a �rst order discrete time approximation given by

Vh(x) = maxjJh(x; u); Jh(x; u) = h

1Xi=0

(1� �h)Uf(xh(i); ui) (A1)

where xu is de�ned by the discrete dynamics

xh(0) = x; xh(i+ 1) = xh(i) + hg(xi; ui) (A2)

and h > 0 is the discretization time step. Note that j = (ji)i2N0 heredenotes a discrete control sequence.The optimal value function is the unique solution of a discrete Hamilton-

Jacobi-Bellman equation such as

Vh(x) = maxjfhf(x; uo) + (1 + �h)Vh(xh(1))g (A3)

where xh(1) denotes the discrete solution corresponding to the controland initial value x after one time step h. Abbreviating

Th(Vh)(x) = maxjfhf(x; uo) + (1� �h)Vh(xh(1))g (A4)

25

the second step of the algorithm now approximates the solution on grid� covering a compact subset of the state space, i.e. a compact interval [0; K]in our setup. Denoting the nodes of � by xi; i = 1; :::; P , we are now lookingfor an approximation V �h satisfying

V �h (Xi) = Th(V

�h )(X

i) (A5)

for each node xi of the grid, where the value of V �h for points x which arenot grid points (these are needed for the evaluation of Th) is determined bylinear interpolation. We refer to the paper cited above for the description ofiterative methods for the solution of (A5). Note that an approximately opti-mal control law (in feedback form for the discrete dynamics) can be obtainedfrom this approximation by taking the value j�(x) = j for j realizing themaximum in (A3), where Vh is replaced by V �h . This procedure in particularallows the numerical computation of approximately optimal trajectories.In order the distribute the nodes of the grid e¢ ciently, we make use of a

posteriori error estimation. For each cell Cl of the grid � we compute

�l := maxk2cl

j Th(V �h )(k)� V �h (k) j

More precisely we approximate this value by evaluating the right handside in a number of test points. It can be shown that the error estimators �lgive upper and lower bounds for the real error (i.e., the di¤erence betweenVj and V �h ) and hence serve as an indicator for a possible local re�nementof the grid �. It should be noted that this adaptive re�nement of the grid isvery e¤ective for computing steep value functions and models with multipleequilibria, see Grüne and Semmler (2004).

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