Econ Om Ides Network Ex Tern Ali Ties EJPE 1996

8/8/2019 Econ Om Ides Network Ex Tern Ali Ties EJPE 1996

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European Journal of Political EconomyELSEVIER vol. 12 (1996) 21 1 2 3 3

European Journal ofPOLITICALECONOMY

Network externalities, complementarities, andinvitations to enter

Nicholas Economides.Sten1 .SC/IOO/4B U \ I I I ~ A. N P M , ork ( ! ~ l i ~ . ( , r ~ i r ~ .bre1t.Yot-k. N Y 10006, L'S.4

Accepled 1.5 March 1995

Abstract

We discuss the incentive of an exclusive holder of a technology to share it withcompetitors in a market with network externalities. We assume that high expected salesincrease the willingness to pay for the good. This is named the 'network effect'. At a stablefulfilled expectations equilibrium, where the actual sales are equal to the expected ones, it isshown that. if the network effect is sufficiently strong, a quantity leader has an incentive toinvite entry and license his technology without charge. If the quantity leader has theopportunity to use lump sum license fees, he will invite a larger number of competitors. Ifno lump sum fees are allowed, the leader will charge a decreasing fee in the intensity of thenetwork externality and will invite entry. In markets with very strong network externalities.the leader pays a subsidy to the invited followers. We also show that the results hold underuncertainty, and when the post-entry competition is Cournot.

K~jrr.orcl.s:Network extrrnalit~es:Monopoly: Quantity leader\hip; Entry: Licen~ing

I . Introduction

In the bat t le for the es tabl i shm ent of technical s tanda rds , sponsorship of a' s tandard ' i s of special s igni f icance. However , sponsorship is of ten insuff icient tol aunch a new p la t fo rm or ' s tandard ' in an industry wi th s igni f icant networkex terna l i ti es . For e xample , on Apr i l 2, 1987, IBM cam e o u t w i th a n ew an d mo r eef fi c ien t bus a rch it ec tu re fo r personal c omputer s ca l l ed the M icro Channel Arch i -0176-2680/96/$15.00 Copyright 0 996 Elsevier Sc~ence .V . All rights reserbedSSD l 01 76-2680(9.5)00014-3


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Itecture. or MCA . It was first announ ced as an exclusive technological advantag eof IBM PS/2 personal computers. In the next 18months, there were signs that theMCA 'standard' had significant difficulty in being accepted. Few other firmsproduced com ponents that were compa tible with MCA com puters. In 1989 IBMannounced that thereafter it would license the MCA bus without charge. In sodoing, IBM invited other firms to compete with it in a product line in which IBMcould have remained a monopolist. In another interesting example, some commen-tators have explain ed the dem ise and final withdrawal from the U.S. m arket of theBetamax VCR recording format by referring to the disregard of network effects byits sponsor, Sony. If Sony had freely licensed its Betamax technology to potentialcompetitors, it might have created a sufficiently strong network and survived. Inthis paper we develop a model that explains the decision by a monopolist to invitecompetition and entry.

W e focus on m arkets with significant network externalities. T his means that thewillingness to pay for a unit of a good sold in this comarket increases with thetotal number of units sold. ' We restrict expectations of sales to be equal to theactual sales, at equilibrium. We compare two regimes. In the first, there is only asingle producer. This single firm is an innovator, and only it has the technologyfor the production of this good. In the second regime, the innovator provides thetechnology to other firms, so that there are n active firms. The innovator is aquantity leader, and the other firms are quantity followers. It is shown that, when

I The internal bus defines the way in which informat~on is transferred among the internalcomponents of a computer, such as memory boards, video boards and microprocessors.' The positive consumption externalities, commonly called 'network externalities'. arise from theexistence and provision of complementary goods. Network externalities are a natural feature ofnetworks (see Rohlfs (1974)). They also arise In many non-network markets where complementarygoods are important, as in our two examples. In the MCA bus case. the complementary goods are thevariety of add-on boards that get attached to the motherboard and communicate with it at the MCAspecifications. These boards perform a variety of functions, such as video display support, terminalemulation. memory provision, etc., and users demand their availability at many quality levels. Theearly difficulty in the acceptance of the MCA bus by other-than-IBM personal computer manufacturerslead to inadequate provision of these complementary goods. This led to a further hesitation in theincorporation of the MCA bus in new PC 'IBM-clone' designs. In the Betamax case. the complemen-tary good was pre-recorded movies in that format. Although unanticipated by Sony, large numbers ofcustomers used VCRs to view pre-recorded commercially available material. Thus the existence andeasy availability of large number.; of titles in a particular recording format became crucial. Originallyaided by a longer duration recording capability and by a strong U.S. distribution network of manymanufacturers. the competing VHS format became dominant in market share in the recorders/playersmarket and in the number of titles of pre-recorded movies and their ava~lability.Eventually. Sonywithdrew from the U.S. market. See Economides and Salop (1992) for a discussion of pricing withcomplementaries, and Economides (1989). and Economides (1991) for the incentives of firms toproduce compatible components. See Economideh and White (1994) for a discussion of the economicsof networks in the context of antitrust.

The particular quantity-leader/follower structure is not crucial for the results, which also hold ingeneral terms for other oligopolistic models.


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the externalities are sufficiently strong, it pays for the monopolist to give away histechnology to other firms that become its competitors.

The intuition for this 'paradox' is simple . Entry of competitors reduces pricesand profits ceteris paribus. But the addition of their production to the size of thenetwork shifts the inverse demand functions facing all network members, includ-ing the innovator (former monopolist). This allows the innovator to sell higherquantities and c harge a higher price. Thus , if the externality is strong, the networkeffect overshadows the standard cnrnpetitizle eject of entry. As a result. theinnovator is better off a s one of n oligopolistic firms rather than as a mon opolist. "Why couldn't the monopolist produce a high quantity and create a largenetwork effect without inviting competition? Our model is based on consumersexpectations of network-wide sales that are fulfilled at equilibrium. Given a levelof expected m arket-wide sales, firms choose their outputs in oligopolistic competi-tion with each other. We require that the expected market output be realized atequilibrium. For any level of expected sales, a profit maximizing monopolist willchoose a smaller output than total output in an industry with a larger number offirms. This implies that higher expectations of sales can only be fulfilled atequilibrium in a more com petitive market with a larger num ber of participants. Tobe able to realize the benefits of a larger network effect, the innovator has toinduce a higher market-wide output. Acting as a monopolist that uses onlyquantity as a strategic variable, the innovator is not able to commit credibly toproduce a larger output (that would induce a higher willingness to pay through alarger netwo rk effect). Even if the mon opolist claim ed that he would produce alarge amoun t of output (to support at equilibrium a large network effect) he wouldnot be believed because as a monopolist he has an incentive to reduce output forany given level of consumers expectations. Thus. the innovator creates thedesirable network effect by inviting entry. The invitation to free entry supports thehigh expectations of sales, because the consumers know that a more competitiveindustry will have higher sales. By inviting entry and freely licensing, theinnovator commits to a larger industry output for any level of initial expectations.and therefore to a larger equilibrium output at fulfilled expectations.In an alternative reinterpretation of our model, abundance of varieties of a

' This i \ shown for the quanti ty lea dc r\h ~p game. and I \ extended to a simultaneous-actingquantity-setting gam e a-1;-Cournot in Section 6 2 .

This possibility was first raised in a s i m ~ l a rmodel by Katz and S hapiro (1985. p. 431). Theywrlte. "It IS interesting to note that the monopolists's profits may be lower than the profits of aduopolist in the 2-active-firms sym metr ic equ~ libr iu m. n other words, a monopoly may benefit f romentry. This unusual result fol lows from the fulf il led expectation condit ion: a m onopolist wil l ex pl o~ t isposit ion with high pr lces and consumers know t h ~ s . hus. consumers expect a smaller network and arewill ing to pay less for the goo d. If the monopoli \ t co mm it h ~m sel f o higher sale$, he would be betteroff. but this commitment is not credible \o long as he is the sole producer".

In thi \ Gngle-period ga me there i \ no p oss~bil i ty for the m onopo list ' s pr lce to be dr iven dv wnaccording to the Coase conjecture.


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3. Quantity leadership equilibrium with given expectationsSuppose that a market is described by inverse demand function ''

P ( Q ; O ) = A - Q , ( 2 )so that, with the network externality. the inverse market demand is

P ( Q : S ) = A - Q + f ( S ) . (3)In a quantity-setting game, let there be a leader and n - 1 followers, so that

. , I - I IQ = q , + A , = I qr5 ( 4 )where q, and q+' are the quantities chosen by the leader and the ith follower.I IAssume no costs. The profit functions for the leader and followers are

Maximizing a follower's profits while keeping the production levels of all otherfirms constant. and then setting equal the production levels of all followers, resultsin

Substituting in the profit function of the leader and maximizing i t results in theequilibrium production level for the leader:4 , ( S ) = [ A + f ' ( ~ ) 1 / 2 . ( 7 )

Substituting back in the follower(s) best reply results in41-(S ) = [ A + f ( ) 1 / ( 2 f z ) . (8)

Actual market-wide sales, market price. and realized profits areQ ( S , = [ A + f ( ~ ) l ( 2 n 1 ) / ( 2 n ) . P ( S ) = [ A + f ' ( ~ ) l / ( 2 n ) , (9)

Eq. (7) - ( 10) sum marize the quantity leadership equilibrium for a given expecta-tion of market size S. As expected. quantities, prices. and profits increase in salesexpectation s. Given any expectation S, prices and profits decrease in the numb erof active firms. In the next section, we restrict expectations to be fulfilled at therealized equilibrium.

I 0 N o r m a l i ~ i n g h e slre units, we set without lors of generality the coefficient of t) o I" The rehults are identical if constant marginal costs are ahsumed. Then 'prices' in the text are

interpreted as differences between price and marginal cost. The general flavor of the results will not belort ~f general cost functions ar e assumed.


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4. Fulfilled expectations equilibriumAt the overall equilibrium , the expec tations have to be fulfilled. Thu s, the levelof the expected sales is the realized one. This defines the equilibrium level ofexpected (and realized) sales S * as the solution of

where Q(S) was substituted from Eq. (9). See Fig. I . Q(S) can be thought of as amapping of sales expectations into actual sales. Fulfilled expectations then define afixed point S ' of function Q (S).

Lrninia I . Thrrr c~.~i.stst least one ,fidfillrd e.upec.tations rquilihriurn lrrbel oj"snles.Proof: By property (iv) of ,f (S) , eventually, for large S > S , , f l ( S )< 1; i.e.. themarginal contribution of increased expectations of sales on the network externalityis smaller than the increase in the size of the network. Th e slope of Q (S ) isd Q / d S = , f1 ( S )( 2 n- 1)/( 2n ) < f1 (S ), and for these large S > S , , d Q / d S < (2n- 1) / (2n) < 1. Q(S) starts (for S = 0) at Q(0) =A(2n - ) / ( 2 n ) > 0, i.e., abovethe 45" line. Since eventually, for large S, the slope of Q (S ) is less than 1 .eventually. for large S, Q(S) < S. Therefore. since Q(S) is continuous. it crossesthe 45" line at least once. QED.

Note in Eq. (9) that Q (S ) is a linear function of , f( S) . Since f( S ) is relativelyunrestricted. de pending on the shap e of ,f( S) , there can be many intersections of

- -SA SB s, sF IE 1 Fulfilled expectation\ r q u l l ~ b r ~ d


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21 8 ,V. E~orlomidc.\ Euro[)rcrn Journul o f Polific~crlC . o n om y 12 ( 19961 211-233

Q(S) with the 45" line, each defining a fulfilled expectations equilibrium. In thenetworks externalities literature there has been wide use of concave and linearnetwork externality functions. For such functions, the slope of ,f(S) (and thereforeof Q ( S ) > is a weakly decreasing function of S. Thus, there is no possibility of asecond crossing of Q(S) with the 45" line and the fulfilled expectations equilib-rium is unique.C o r o l l a n 1. If the netwrork exterrzulity ,functio n is weakly c onca c3e , f U ( S ) 0,rherz the ,fulfilled expect ations equilibrium is unique .

The equilibrium is locally stable irz exprctatior~s f and only if in the neighbor-hood of equilibrium S ' the slope of Q (S ) is less than 1,d Q ( S ' ) / d S < I , i .e ., f " (S x ) < 2 n / ( 2 1 1- I ) . ( 1 2 )In Fig. 1. equilibria A and C fulfill this condition, but equilibrium B does not.Starting with expectations in the neighborhood of an unstable equilibrium but notexactly at the equilibrium value, there will be a tendency to move away from it.Given an unstable equilibrium. such as B, with Q1(S)> 1 , there always existsanother stable equilibrium. such as C at a higher level of sales, S , > S,. This isbecause. as shown in the proof of Lemma 1 , for large S, Q1(S)< 1, and eventuallythere will be a crossing of Q(S) and the 45" line with Qf(S)< 1 . Thus, it may notbe unreasonable to expect that an unstable equilibrium will be avoided in favor ofa stable equilibrium at a higher S.Leinriza 2. A fuljilled expecta tions ey~lilibriurir .s loca lly stuble if and only ! f thernar-gincrl rzrtvvork externnlity is 17ot too I L I ~ ~ P ,"(S ' ) < 2 n / ( 2 n - ).Coroll~ct?. . For bveakly coizcu1.e t~c~hvorkuter izal itv unctions, , f1 ' (S)5 0, the~ in iy u e yuilibriunz is globally .stable.

Intuitively. we expect that an increaje in market production for any given levelof consumers expectations S should support higher fulfilled expectations and

Fig. 2 . Upward \hif t \ in Q ( S )


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therefore higher equilibrium production. This intuition is confirmed for stableequilibria. An increase in the number of firms n increases the quantity producedfor any consumers expectations S. That is, an increase in n shifts up the Q(S)function. As a result of the shift. Q ( S ) intersects the 4.5" line at a larger S - if theslope of Q(S) is less than 1 (as in panel a of Fig. 2); conversely. the upward shiftof & (S ) results in a smaller S * if the slope of Q (S ) is larger than 1 (as in panel bof Fig. 2). Thus, increases in n lead to increases in S ' if and only if the fulfilledexpectations equilibrium is locally stable.

Formally, from total differentiation of the fixed point condition ( I I ) we derived S Z / d n : ''d S " / d n > 0 is equivalent to the stability condition of the equilibrium in expecta-tions S. I ?P r p o s i t i o r ~ . Irzcrease.s in the tzumber oJ:firms n result in incre~ised ales at thelocal ,fiilJilled e,xpec,tations equilibrium if und only the network e x te rn a li ~ t thernurgin is not too high. This is rq ui ~~ al en to the equilibrium being locally stable in

I4e.upectations.Higher market-wide sales (induced by entry) imply higher sales for the leader.

But prices could fall as a result of entry. It is a priori unclear if the leader shouldinvite entry. This is examined in the next section.

5. The leader's incentive to invite entryWe are interested in the effects of increases in the number of competitors on the

leader's profits. If we can show that the leader's profits increase in the number of

" d S ~ / d ~ ~ = - [ i i ( ~ ( . ~ ~ ) - . / z ] / [ . ) - ~ ' ) / i i S l = [ A - , f ~ ( S ' ) l / { r ~ [ ? r ~ - ( ? r ~ -I ) , f ' ( .S ' ) I ) = 2 s ' /{(2r1 - )[21r ( 2 r 1- ) f ' ( S , )I). where we subs t~ tu tedA + f l S x ) = ZnS ' j ( 2 r 1-I ) f rom Eq. (11) .

I ' Note that . in c a w of rnul tiple eq uil ibr ~a. he upward shift of Q ( S ) esulting from an increaw inthe num ber of compe titors can alho eliminate the low sa les equilibria such as A .

I 4 It may \ee m perverbe. that in situatio n\ with high network externalities at the margin (which ~ m p l yQ ' ( S ) > 1 ). increases in the nurnher of firms result in decreases In the size of S ' The clue lies in thefact that S' i \ a /zilfillrtl a.rp;l,rt~tcrtior~.squilihrlurn Starting at equilibrium point B (Fig. Zb) whereexpected bales equal actual sale\ . Q(S,) = S,. a n u p a a r d s h ~ f t f Q ( S ) ( b ec a us e of a n i n c r e a ~ en 11)creates a pap between re al i~ ed ales and expected \ales (point B"). Increasing S abobe S, would onlylend to fu rther di se qu il~ br ium ecaube fo r ebery Llnlt of increase in S. Q (S ) increases more. The onlyway to reach equi l~hr iu rnocally is to reduce S by going to point B'. A market sire S b . vnal ler that S,is the only level of fulfil led expectat ion \ ( ~ nhe neighbo rhood of B ) that is consi\tent with the shiftedup Q ( S ) unctron.


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competitors, then clearly it is in the interests of the leader to license its technologywithout charge and to invite entry.

Let the fulfilled expec tations equilibrium profits l 5 of the leader be denoted byn; ,As the num ber of firms increases, there are two opposite effects on the lead er's

profits. First, because the number of competitors increases, profits of the leaderfall. This is the competitille effect. Second , as the numb er of comp etitors increases,the market can support larger expected sales as a fulfilled expectations equilibriumS ' . Increases in expected sales increase the leader's profits because they push upthe industry demand through the expansion of the network. This is the networkeffect. These effects can be identified on nJ as follows:

IhConsider ny. as a continuous function of n and S ' where S " = S ' n )depend s on n through the fulfilled exp ectations condition (1 1). I' Then the changein profits because of an increase in the number of market participants can bedecomposed into the competitive effect and the network effect:

As expected. the direct effect of an increase in the number of firms is negative:

Increases in expected sales increase profits:

and from the discussion of the fulfilled expectations equilibrium in Section 4 wehave

After substitution of all terms in Eq. (15) and simplification, the total effect ofincreases in 11 on the leader's profits is

Thus. the sign of the profits change in n depends only on the slope f ' ( S ' ) of the

I' From (I I ). 211s (211 - ) w as s u b s t ~ t u t ed b r A + /(S' .I n W e uhe n as a continuous variable and then evaluate the functions for integer n. Clearly. if

d I l i / d n > 0. it follows that profits for a leader Increase ah the integer number of market participantsincreases.

I ' Note that the formula for 11, is v a l ~ d ven for monopoly wi th 11 = I and no fol lowers .


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network externality function. An increase in the number of active firms nincreases the leader's profits, drI;/dn > 0 , if and only if

2 n / ( 2 n + I ) < f f ( S * < 2 n / ( 2 n - I ) . (19)For slopes f 1 ( S ) below 2 t z / ( 2 n + 11, the network effect is not sufficientlystrong to overcome the competitive effect of an increase in n . S l o p e s , f r ( S ' )larger than 2 n / ( 2 n - 1) imply an unstable equilibrium and a tendency to over-shoot S * . Since eventually (for large S ) f ' ( S ) < 1 , for every unstable equilibriumS,, there will exist a stable equilibrium with larger S, S,. > S,. It makes sense inthis model for the leader to pick the equilibrium with the largest S " , which will bestable and therefore will fulfill the R H S of Eq. (19).Proposition 2. An exclusir~eholder of a technology in a market with strongnehtvrk externalities at the margin. , f ' ( S x ) > 2 n / ( 2 11 + 1 ) , has an incentir.e toinr,ite competitors to enter the indust? and compete with him.

Note that the crucial parameter of the network externality function in these andthe earlier results is f 1 ( S ) , the marginal increase in the aggregate willingness topay created by the expansion of the expected size of the network by one more unit.This is as it should be, since firms consider changes in strategic variables thataffect total sales at the margin, which at equilibrium coincide with their expectedlevel. Because of the importance of , fl (S ). we consider next a linear example with, f l ( S )= h. a constant.For example. let the network externality function be linear, ,f(S) = bS, b < 1 .From Corollaries 1 an d 3 , the fulfilled expectations equilibrium is unique andstable. The leader's profits at equilibrium increase in the number of firms for

h > 2 1 1 / ( 2 n+ 1 ) . (2 0 )Therefore, given h , the leader should invite entry as long as condition ( 2 0 ) holds.Solving ( 2 0 ) with equality gives t ~ '= h / [ 2 ( 1 - h ) ] . rI; is maxim ized at n' .Because this number is not in general an integer. the leader should compare hisprofits at I [ n X and I [ n y+ I] (where I [ . ] enotes the integer part function). andpick the number that corresponds to higher profits. Thus in an industry with verylow fixed costs. the leader would like to invite entry but also restrict the number offirms that have free access to his technology.

6. Extensionsi ';6.1. Effects o f uncertainty und dlflbsc~preferet1c.e.c

We discuss next the market equilibrium and incentives to invite entry underunr~ertainty.We assume that the expectation of sales is stochastic with mean ? and

I X I thank Angelos Ant /ou ld to\ to r h ~ \ncouragement dnd he lp In de ve lo p~n g h ~ \ectlon


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222 N. Gonomide. s/E~rop eun ournul ofPoliricu1 Economy 12 (1996)21 1-233

variance a'. Fulfilled expectations are defined by a rational e.upectations eguilib-riurn where the expected mean sales are actualized, S = ~ ( 3 ) .n principle, onecan imp ose fulfilled expe ctations on any num ber of m ome nts of the distribution ofexpected sales. We follow the macroeconomic tradition of imposing fulfilledexpectations only on the first moment of the distribution. 19

W e find that the invitation to entry result holds under uncertainty. W e also findthat increases in the variance of expectations decrease the fulfilled equilibriumsales, and invite entry. The reduction of the equilibrium sales is purely an adverseeffect of uncertainty. Starting with a lower sales equilibrium. the incentive toinvite entry is stronger.

Let S be stochastic, distributed with density h ( S ) , mean S and variance a'. Asbefore. P ( Q ; S ) = A + ( S ) - Q . Firms maximize expected profits, L',=E ( q, P ( Q ; S ) ) , II,' = E (q ; P ( Q : S ) ) , where E ( . ) denotes the expectation function.Approximating n, in a Taylor expansion we have

S~milar ly .II, = q ; [ A- Q + f ( S ) + " ( ~ ) u ' / ? ] . ( 2 2 )

Follower i choosesq ; = [ A + ~ ( s ) f " ( ~ ) n ' / Z - q , ] / n .

and the leader choosesq , = [ A+ ( ( S ) + f . . ( S ) U 2 / 2 ] / 2 . ( 2 3 )

xo that at market equilibrium,q; = [ A f ( S ) + f r f ( S ) n ' / 2 ] / ( ~ n ) . ( 2 4 )Q ( S ) = [ A+ ( S ) + f " ( S ) u 2 / 2 ] ( 2 n- I ) / ( 2 n ) .

P = [ A + f ( S ) + f " ( S ) 0 2 / 2 ] / ( 2 n ) . ( 2 5 )

I1 To impose fulfilled expectations up to the Mth moment. we would approx~~nate,. up to orderM + I in the Taylor expansion. and then equate the expected and actual valuea of the first M moments.


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A I-urionul expec~tation s equilibriunl is defined by actual equal to expectedmean sales.

Define S * = S as the solution of this equation, i.e.,

This equilibrium is meaningful when S ' > 0 . Assuming f " < 0, a positive equilib-rium size results when the variance is not too high, (T ' > 2 [ A +f ( S x >I/[-f"(S ' I]. Intuitively, when there is extreme uncertainty in the expecta-tions of the consumers, both types of firms will not provide the good because theyexpect that positive sales will not be realized at equilibrium.

A concave network externality function implies decreasing q, and q , in u'.Therefore, with f " < 0, the rational expectations equilibrium S * is decreasing invariance (T? . It follows that equilibrium sales and profits are smaller underuncertainty than under certainty.Proposition -3. For c,onc,ur3e eht,ork e.rtrrntrlities,func.fiorz.v,he rutiorzul esp ectu -tioizs equili hriu ~n roduction decreas es in the i.uriunc.e of tlze expectutiorz.~.

We now consider the effect of inviting entry. i.e., of increasing 11. Theequilibrium profits at the rational expectations equilibrium are

+ , f1 " ( s - f f 2 / ? < 2 1 1 / ( 2 1 ?- ) . (31 )We expect that the criterion fl(S' 1 +,f""(Sx ) a 2 / 2 will increase with the

degree of uncertainty as measured by the variance u'. An increase in u' causesS ' to be lower. as we have seen. This increases j"(S * ) because of the concavityof ,f. Also many network externality functions, such as f(S) = log(S), have apositive third derivative. Then the positive effects of increases in c~' on thecriterion function are accentuated. Formally. d(f' + ' " i r 2 / 2 ) / d u 2 =f / 2 + (f'"+ f " b/2)(dS ' d c~ '). which is positive if ,f "< 0, ,f l"2 0, ,f""1 . Thereforethe incentive to invite entry increases with uncertainty. and a monopolist facing amarket with diffuse expectations has stronger incentives to invite entry. Intuitively.


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in the presence of uncertainty, the total market output is more precisely predictablein the presence of n firms than with only the monopolist present. Thus, underuncertainty, the presence of entrants has a law-of-large -numb ers stabilizing effecton expected output, and this is seen favorably by the market.Proposition 4. The incentice to inr3ite etztn increases with the r.ariance o f th eexpectcltions, pror>ided that the ,first four der irutic es of the network externality

70function alternate in sign: f' > 0. f " < 0, f" ' 2 0, f""I. -6.2. Courn ot oligopoly

To stress that the results hold not only for the particular market structureemployed this far (quantity leadership), we briefly discuss the symmetric Cournotmarket structure, and confirm that, with sufficiently strong network externality atthe margin, a monopolist will invite entrants.

Here we discuss the decision to invite entry by a monopolist if after entry theresulting competition will be Cournot oligopoly. Starting with industry demandfunction P ( Q ;S) = A + , f( S ) - Q. firm i maximizes nl= q jP ( Q ;S) by choosingy,, where Q = q,+2,'; ,q, . The first-order condition of firm i is

The implied symmetric market equilibrium is.Y , = ( A + f ( S ) ) / ( n + l ) , Q = l ~ ( A + f ( S ) ) / ( r z + I ) , ( 3 3 )P = ( A + f ( S ) ) / ( n + I ) . H , = ( A + f ( S ) ) ' / ( n + I ) ' . ( 3 4 )

At fulfilled expectations.

An increase in the number of competitors increases market production iff

This condition is equivalent to local stability of the fulfilled expectations equilib-rium.

Th e equilibrium profits of a firm at an 17-firm fulfilled expe ctatio ns equilibriumare

i l These condition\ are fulfilled by many network externality functions, including the logarithm of\ales. f (S) = log(S) .


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N. Eronomidr , / Eurol~ ean ournal oj'Poli t irr~ l l.onotn! 12 f 19961 21 1-233 22 5

Profits of the innovator increase as a result of the introduction of competitors ifd n , * / d n = &',*/an + ( a n ; / a s * ) ( d s * / d n ) > 0. (38 )By substitution and simplification we findd n , " / d t z = ~ * ' [ n f ' ( ~ * ) n - l ) ] / [ n 3 ( n + 1 - f ( ~ * ) ) ] .

Therefore,d I l , * / d n > 0 a n - ) / n < f l ( S * ) < ( n + l ) / n . (39)

Propositiot1 5. At1 exclusice holder c$. a t e c l z t z o l o ~has an incenti~se o inr'itee tz tp if there a re .strong network externalities, ,f l( S' > ( n - ) /TI, and theuf ter -entg n~ ark et t ructure is C o u r t ~ n t ligopoly.

7. An alternative interpretation of the basic modelThis far. our basic model was assuming the existence of a network externality

without a detailed analysis of the source of the externality. Next we present adifferent interpretation of the basic model in the context of two industries thatproduce com plem entary goods. This interpretation a lso helps justify the existenceof the externalities.

Let there be two industries that produce complementary goods that are com-bined in I : 1 ratio in consu mption . Suppose that one firm is the exclusive holder ofthe rights to the technology in industry I . but there is free entry in industry 2. Themonopolist in industry 1 may invite n , - 1 2 0 competitors. When productionlevels in industries 1 and 2 are Q, and Q,, the willingness to pay for product 1 is

P ( Q l ; Q ? ) = A - Q , + . f (Q , ) . (40)In this context, Q , plays the role of Q, and Q2 plays the role of S of our previousdiscussion. Why is the willingness to pay for product 1 increasing in theproduction level of product 2. i.e.. where is the source of the network externality'?Higher production Qz implies a larger number of varieties of product 2 and alower price for them. Thus, with higher Q, , the surplus realized by consumers ofproduct 7 is higher. Since products I and 7 are complementary, higher surplusgenerates a higher willingness to pay for product I. This is captured by f(Q ,).Thus, the network externality arises out of mutual positive feedbacks in a pair ofmarkets for complementary goods.

T o understand the structure of the mutually com pleme ntary markets, we presenta simple example. Suppose that there are ri , symmetrically located differentiatedproducts on a circumference as in Salop (1979). Let consumers be distributeduniformly with density p. according to their most preferred variety, have adisutility of distance equal to 1 , and a reservation price of R which is sufficiently


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Fig 3 . Cornplsmcntary marker5 w ~ t heedback5

large so that all consumers buy a differentiated good. " The symmetric equilib-rium price is p (12 1= 1/11?. and profits are I I ( n,) = p / n l - F. With freee n t r y , n ( n 2 )= 0; therefore there will be 11; = d ( p / F ) active firms. The averagebenefit of a consumer from the consumption of one unit of product 2 is then

Interpreting as Q l (since every consum er on the circumfe rence buys a differen-tiated good), the average benefit to a consum er when there are Q, sales in market2 is , f (Q,) = R - 5/ 4) J(F /Q l) . Since products 1 and 2 are consumed in 1 :1ratio. this average benefit should be added to the willingness to pay of consumersfor good 1 . Th e network externality ,f(Q ,) is increasing in Q2 because a high levelof production in industry 2 implies a larger number of varieties n , and a higherdegree of competition.

Firms play an oligopoly game in market 1. taking Q 2 as given. Let the resultingequilibrium output be Q;( Q,; n , , This is a direct reinterpretation of Q(S; n ) . SeeFig. 3. Firms in market 2 take Q , as given. Let equilibrium ou tput in market 2 beQ, (Q1) . In the expectations model. we used S ^ Q ) = Q, i.e., Q,' ( Q , ) = Q , , a ndthis applies well in a model where the two types of products are consumed in 1 : 1ratio. as in our circumference example. In general, when goods 1 and 2 are notconsumed in a I : 1 ratio, Q; (Q , ) will not be the identity function; it is shown inFig. 3 as an upward-sloping curve.

Equilibrium across markets defines Q ; ' , Q l as the intersection ofQ ; ( 9 , ; n l and Q,' ( 0 , ) . Increasing the number of firms n , shifts Q; ( Q ? ;n , ) othe right. The effect of the increase of 11 , on equilibrium output in industry 1 is

- ' We could a150 allon the leservatlon price K in indu5try 2 ro vary in Q , ~v ith ou t hang ing there5ult.


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d Q ,- * / d n , = (dQ, ' /d t z , ) / [ l - dQ r/dQ ,)(dQ ;/dQ ,)], which is positive if theequilibrium is stable (1 > (dQ ; /d Q , )(dQ; /d Q ,) ) provided that an increase ofthe number of active firms in industry I results in an increase of its equilibriumoutput, i .e . . dQ;/dn, > 0. See Fig. 3.

The effect of increases in n, on the profits of the original monopolist dependson the particulars of the oligopolistic interaction in markets 1 and 2, as well as onthe degree of complementarity between the two markets. As we have seen, ifgoods 1 and 2 are consumed in 1:1 ratio, this model of interaction acrosscomplementary markets is an exact reinterpretation of the expectations model. Inthat case, all propositions of the expectations model can be directly reinterpretedfor complementary goods model. In more general settings of variable degrees ofcom plem entarity between markets, one still expects results of the sam e flavor, i.e..strong network externalities on the margin leading to invitations to enter.

8. Licensing8. . Lump .sum t e e s

We have shown that it is beneficial to the leader to invite entry while charginga zero licensing fee. but that he may want to restrict the number of firms to whichhe freely gives his innovation. One way to achieve the latter without creatingincentives for firms to cut output is to put licensing fees per firm. Suppose theinnovator imposes a marginal fee of k per unit of output of an invited firm, and alump sum fee of A per invited firm. A follower's profits.

will be set to zero as the licensing fee will be set so as to absorb all profits offollowers. Then the leader's profits are ''

Note that, because all profits are absorbed through the lump sum fee. the marginalfee is immaterial.

The marginal effect of an increase in 1 1 on the leader's profits (comparable toEq. (15)) is

7 7- - W e find the la\ t cxprr\ \ ion of profit\ in Eq. (4.3) from Eq. ( I l ) evaluated at S "


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Therefore, with lump sum licensing fees the leader prefers an even larger numberof competitors tzL * > n * . The licensing fee A" can be increased abruptly at n L 'the optimal number of competitors from the leader's point of view, to thwartfurther entry.Proposition 6 . A n e x c l u s i ~ ~ eolder of a technology in a market with strongnetcvork externalities at the margin will incite a larger number qf competitors toenter i f he can charge lum p sum l icm sing fees.8.2. Marg inul licensing ,fees

Often lump sum fees are unfeasible. Thus, we consider licensing with fees perunit sold for linear network externality functions, f ( S ) = bS. Let the leader collecta licensing fee of k per unit of outpu t of the follow ers. The profit func tions for theleader and the followers are now

Given S an d k , the equilibrium in the leader-follower gam e isq / ( S ) = [ A + b S ] / 2 , q , ( S ) = [ A + b S - 2 k ] / ( 2 n ) , ( 4 6 )Q ( S ) = [ ( A+ b S ) ( 2 n - ) - 3 k ( n - 1 ) ] / ( 2 n ) ,

P ( S ) = [ A+ b S + 2 k ( n - 1 ) ] / ( 2 n ) . (47 )

Imposing self-fulfilled sales expectations, i.e., S * = Q ( S * ), determines thefulfilled equilibrium sales:

This equilibrium exists and is unique and globally stable provided thatd ~ / d ~I , - I < 2 n / ( 2 n - I ) . ( 5 0 )

The i nd~ vid ual uan tit~ es, rlce and profit$ at the equillbnum areq , ( S * )= [ n A b k ( n - 1 ) ] / [ 2 n b ( 2 r z I ) ] ,

y , ( S * )= [ A - ( 2 - b ) k ] / [ 2 n b ( 2 n - l ) ] . ( 5 lP ( S * )= [ A+ 2 k ( n - ) ( l - b ) / [ 2 1 1- h ( 2 n - I ) ] . ( 5 2 )r I , ( s X ) = q / ( S h ) P ( S * ) X ( n ) y , ( S X ) ,n , ( ~ * )q t ( ~ * ) [ ~ ( ~ 7 )k ] . ( 5 3 )


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Equilibrium production decreases in the size of the licensing fee

Equilibrium price increases in the size of the licensing fee if the network effect isweak, but decreases in the size of the licensing fee if the network effect is strong:

The leader chooses the marginal license fee to maximize II,(S' ). He solvesd n , ( S ' ) / d k = 2 ( n - 1 ) [ 2 ~ n ( l h ) - ( h 2 + 4 n ( 1- h ) ]

and chooses the optimal license fee

Because of second -order conditions. '' the licensing problem is well-definedonly the denominator of k * is positive, i.e., h' - 4 h n + 4 n > 0. The optimal feedecreases 'I n b and is positive for 0 I < 1 and negative for 1 < h < h , = 2 ( n- t d n - I)]'"'}, the smaller of the roots of the deno mi nato r of k ' Note that thesigns of k * and d P ( S ' /d k coincide. Thus. for weak externalities it is optimalfor the leader to charge a positive fee, and this increases the market price. Forstrong network externalities, the leader gives a subsidy, and this increases theequilibrium price above its level with no subsidy. The licensing fee increases inthe number of competitors n fo r b < 1 (while the fee is positive). For b > 1 theoptimal subsidy increases in n .Proposition 7. The leader- charge s N positii.e licerzsing,fee in a ma rke t with weaknetwork externalities. Conr,erse ly, in a ma rket with strong network externalities,the leader is willing to give a subsidy to his c.otnperitors to encourage higherproduction. 117 the case when the licensing t e e chosen by the inn ol.ator is positice,the ,fee rilso increases rlith the num ber of com petitors. W hen the leade r cho ose s to.suh.ridize the ,follower.s, his subsid! increases with the n um ber of his competitors.

These results are rather intuitive. In a market with small network externalities,the result is the same as in a market with no externalities. i.e.. the leader charges apositive licensing fee. Since in this case the benefit from the externality is small.the leader increases his profit by restricting at the margin the level of output of his

> 1 d 2 I l , ( ~ ' ) / d k ' = - 2 ( r ~ - 1 ) ( h ' + 3 r ~ ( l - b ) ] / [ 2 i r - h ( l r r - 1 ) ] ' < O t t h ' + J n ( l - h ) t OThe root\ ot thl\ are h , = 2 ( n + [ n o 7 - ) ] ' '), h , = 2 (n - [ r ~ ( r i- ) ] ' ') or (h' +11z(I - h ) > O thehindlng condltlon Ir h < h 2'' d X ' / d h = 2 n h ~ ( h - 2 ) / ( b ' - i l h t r + 4 n ) ~ < O" d k * / d n = A ~ ' ( I h ) / ( h 2 4 h n + 4 r 1 ) '


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competitors through a positive licensing fee. This fee is higher if there are morecompetitors, to compensate for the higher output. Conversely. when the networkexternalities are strong, the leader gives a subsidy to his compe titors to encourageincreased production and greater network effects from which he will benefit. Inthis case the optimal subsidy increases in the number of competitors to create thestrongest externality.

The profits of the leader who invites 11 competitors and uses the optimalmarginal licensing fee are

They are an increasing function of 11. '' We have already shown that the optimallicensing fee is also increasing in 11. Thus, for small network externalities, togenerate the same externality (from the same amount of industry output) it is moreprofitable for the monopolist to invite many competitors and collect high royaltiesfrom them rather than invite few and collect low royalties. This is because themonopolist collects higher total royalties in the former case.

Proposition 8. Wh en the leader uscJs marg inul licensing ,fees, profits increase inthe nunzher of(, oinp etitor s. Thus, the leclder bus an incentirle to incite comp etitors.

Note that it is to the benefit of the leader to invite competitors, both when thelicense fee is positive and again when it is negative. The intuitive reasons aredifferent in each case. When network externalities are strong, the leader invitescompetitors, and provides them with a subsidy to enjoy the strong network effects.When the network externalities are relatively weak, the first objective of the leaderis to collect the licensing fees; cultivating the network effects is secondary. Ofcourse, in both cases, optimal marginal fee licensing is superior to free licensing.so that.

It may be infeasible to subsidize direct competitors at significant levels that arerequired in the case of strong network externalities. Subsidization of direct(horizontal) competitors may raise eyebrows even in today's liberal antitrustclimate. Thus. the upper limit in the amount of the subsidy imposed by the legalenvironment may implicitly determine the number of the firms that the leader willinvite.


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9. Concluding remarks

We have found that in a market with strong network externalities, if there areno other means of commitment to high production (such as binding contractualcommitments or vertical integration), an innovator quantity leader has incentivesto license his technology freely to competitors. This seemitzg parudox occursbecause the leader benefits from the increase in the size of the network that comeswith the introduction of competitors and the increase in competition. The expan-sion of output required for the creation of a large network cannot be done in theabsence of competitors. The innovator-monopolist cannot credibly com mit him-self to create a large network (and reap its benefits) because given any level ofconsumers' expectations of sales. the monopolist has an incentive to produce arelatively low output. Nevertheless. the innovator can use the fact that a morecompetitive market will result in a higher output (for any given expectations). Byinviting competition. the innovator commits to an expanded amount of marketoutput for any given expe ctatio ns. Th us. the innovator credibly sustains theexpectation of a high production by inviting competition, and thereby creates thedesired large network effect. These results also hold in the presence of uncertaintyand under different cond itions of oligopolistic com petition.

We have also shown that the expectations model is formally equivalent to amodel of strategic interaction between two complementary markets. In thisframework. the size of sales in industry 2 affects positively the surplus realized inindustry 2. This in turn affects positively the willingness to pay for the comple-mentary good 1. Because of the formal equivalence. all results can be reinterpretedin the framework of two complementary markets.

The innovator acting as a quantity leader does even better and invites morecompetitors if he can charge lump sum licensing fees. If the leader can charge onlymarginal licensing fees, his optimal licensing fee will be positive for markets withweak network externalities. and negative (i.e.. a subsidy) when the externalities arestrong. For both weak or strong network externalities. the leader invites entry aswell. and has higher profits than when licensing was free.

There are a number of dimensions in which this research can be extended. First.i t can be used as a basis for the construction of a model of competing networksthat also compete in acquiring members. In such an endeavor, a discussion ofnon-cooperative coalition formation and stability as in Economides (1988) andmore recently in Yi and Shin (1992) is essential. '' Second. incorporating bothvertically and horizontally differentiated products in the industry exhibiting theexternality is desirable. There are many industries (e.g.. VCR players) where theexternality brought to a network (firms that adhere to the sam e technical standard)by a consumer of a low quality good could be as large as the externality created by

7 -- ' Se e ~ l l s o o n s ~ m o n ~t '11. (1086)


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23 2 N . E c o ~ w m ~ d e rEuropean Jounlul of Polit~c.ulEconomy 12 ( 1996) 2 11-23.3

a consumer of a high quality good. For example, it could be that the propnsity torent pre-recorded movies is equal for consumers that buy high quality VCRs as forthose who buy low quality VCRs. In such cases we expect to observe subsidiza-tion of the low quality buyers and producers by the high quality producers. Forexample, subsidization of low quality producers can occur through the use ofdifferential licensing fees. Seen in this context, the cross subsidization commonlyobserved in networks may not be undesirable.

AcknowledgementsI thank Angelos Antzoulatos, Bob Dansby, Rob Masson, Dennis Mueller, Joe

Farrell, Frank Fisher, Bill Greene, Charlie Himmelberg, Manfred Holler, PaulJoskow, B runo Jullien, Barbara Katz. M ichael Katz, D avid Salant, Jacques Thisse,Richard Schmalensee, Ralf Winkler, Glenn Woroch, an anonymous referee. theparticipants at seminars at C.O.R.E, E.N.S.A.E, M.I.T., the University of Mary-land, the Pricate Networks and Public Objecticrs Conference of the ColumbiaInstitute for Tele-information, and the Eastern Economic Association Meetings foruseful suggestions.

ReferencesCrampes. Claude and Abraham Hollander. 1993, Umbrella pricing to attract early entry. Economics 60.

465-474.Donsimoni, Marie-Paule. Nicholas Economides. and Heraclis Polemarchakis. 1986. Stable cartels.

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of Economics, Columbia University, New York).Econom ides. Nicholas, 1989 , Desirability of compatibility in the absence of network ex ternalities.

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Guerrin-Calven and Steven Wildman. eds.. Electronic services networks: A business and publicpolicy challenge (Praeger Publishing Inc.. New York).Econom ides, Nicholas and Charles H immelberg , 1995. C r~ tic al mass and network ev olution intelecommunications, In: Gerard Brock, ed., Toward a competitive telecommunications industry:Selected papers from the 1994 Telecommunications Policy Research Conference.

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