FS IV 94 - 7

discussion papers

FS IV 94 - 7

The Desirability of Compatibility in the Presence of More than One Strategic Variable

Anette Boom

Juni 1994

ISSN Nr. 0 7 2 2 - 6748

Forschungsschwerpunkt Marktprozeß und Unter­

nehmensentwicklung (IIMV) Research Unit

Market Processes and

Zitierweise/C itation:

Anette Boom, The Desirability of Compatibility in the Presence of More than One Strategic Variable, Discussion Paper FS IV 94 - 7, Wissenschaftszentrum Berlin, 1994.

Wissenschaftszentrum Berlin fur Sozialforschung gGmbH,

ABSTRACT

The Desirability of Compatibility in the Presence of More than One Strategic Variable

This paper analyzes the incentive o f duopolists who compete in prices and in production technologies to offer compatible components o f a system good which is exogenously horizontally differentiated in two dimensions. The firms interact in a noncooperative two stage game where they first decide on their marginal costs and then choose their prices adequately. With incompatibility, despite the symmetry o f the two firms, two asymmetric multistage perfect equilibria in pure strategies can occur. The firms and society prefer costless compatibility to incompatibility which is preferred by the consumers. With com­

patibility costs excess standardization can occur whereas inefficient incompatibility is impossible.

ZUSAMMENFASSUNG

Die Attraktivität der Kompatibilität bei mehr als einer strategischen Variable

In dieser Arbeit wird der Anreiz von Duopolisten untersucht, kompatible Komponenten eines Systemgutes anzubieten, das exogen horizontal in zwei Dimensionen differenziert ist. Die Unternehmen konkurrieren miteinander sowohl in ihren Preisen als auch durch die Wahl ihrer Produktionstechnologie. Sie spielen ein zweistufiges nichtkooperatives Spiel, bei dem sie zunächst die Höhe ihrer Grenzkosten bestimmen und dann ihre Preis­

entscheidung entsprechend treffen. Bei inkompatiblen Komponenten treten trotz ange­

nommener symmetrischer Unternehmen zwei asymmetrische teilspielperfekte Gleichge­

wichte in reinen Strategien auf. Die Unternehmen und die Gesellschaft präferieren aller­

dings Kompatibilität, wenn sie nichts kostet, während die Konsumenten das Angebot in­

kompatibler Komponenten bevorzugen würden. Berücksichtigt man Kompatibilitätsko­

sten, kann eine suboptimale Standardisierung eintreten, während eine ineffiziente Inkom­

patibilität nicht möglich ist.

1 I n tr o d u c tio n

The desirability or the disadvantage of com patibility has been a ra th e r pop­

ular subject in recent publications in the field of industrial economics. On the one hand, this has been discussed in the context of netw ork externalities where standardization of products enlarges th e entity which produces posi­

tive externalities in consum ption.1 The most well-known exam ple for such a product is th e telephone. On th e oth er hand, th e question w hether firms will decide in favour of standardization and w hether social welfare is favoured by their decision is raised concerning horizontally differentiated system goods.* 2 Here one can think of com puters and software.

This paper focuses on system goods which can be divided into com plem ents thus creating th e opportunity for th e producing firms to m arket them sepa­

rately and for the consumers to combine them to an individual system as long as they are standardized.3 Standardization can be achieved if com ponents be­

come identical or if they become com patible with unchanged characteristics, thus increasing th e variety of th e system goods available. However, w ith sep­

arated com ponent suppliers instead of integrated system suppliers, w ith free en try and scale economies in one com ponent m arket, th e variety of th e system goods m ight even increase by a reduction of th e variety in one com ponent m arket where th e num ber of suppliers is exogenous.4 We will concentrate here on suppliers who produce com ponents of the com plete system and can unilaterally achieve com patibility w ith com peting com ponents by an adaptor technology.

One rath er robust result in the literatu re concerning this ty p e of stan d ard ­ ization is th a t integrated system suppliers prefer com patibility of th eir com­

ponents, if it can be realized w ithout costs and if the dem and for th e system good of a single supplier and for th e hybrid products which consist of compo­

nents produced by different suppliers is sym m etric.5 However, th e consumers

^ e e e.g. Katz, Shapiro (1985) and Farrell, Saloner (1992).

2See e.g. Maiules, Regibeau (1988), Economides (1989a), and as an exception Einhorn (1992), who asks the same question for a vertically differentiated system good market. See also Chou, Shy (1990), and Church, Gandal (1992) where system goods create indirect network externalities and standardization can be the outcome of the market process but is not a decision variable of the involved firms.

3See for the definition of a system good Economides (1989a), p. 1165.

4See Chou, Shy (1990), and for a similar setup Church, Gandal (1992) as well as Church, Gandal (1993).

5See Matutes, Regibeau (1988), p. 225 f. and p.229, Economides (1989a), p. 1177, and Economides (1991), p. 47. Even the results for vertically differentiated system goods point in the same direction. See Einhorn (1992), p. 542 ff.

are worse off w ith com patiblity th a n w ithout if th e m arket is covered.6 7 In addition, prices w ith standardized com ponents are higher th a n w ith incom ­ patible ones, and a higher social welfare is realized as long as th e num ber of firms is exogenous.' Following this literatu re, incom patible com ponents will only be supplied by integrated system good suppliers if th e ad ap to r technol­

ogy is either prohibitively costly or does not exist or needs th e consensus of the involved firm s.8

However, one m ain feature of the above cited literatu re is th a t th e firms have only one strategic variable in which they com pete, and this is th e price of their com ponents.9 T he m ain interest of this paper is how these ra th e r robust results m ight change if th e firms com pete in an additional strategic variable.

Here two firms are considered which supply two com ponents of a system good which is exogenously differentiated in two characteristics. T hey decide on th e production technology in a first stage preceding th e pricing gam e by sim ultaneously choosing th e constant m arginal cost of production from the sam e m enu. In order to be able to produce w ith lower m arginal costs th e firm has to incur a higher fixed and sunk cost. In the second stage th e two firms decide sim ultaneously about their prices. T he m ulti-stage perfect equilibrium of this two stage gam e in the case of com patible com ponents is com pared to th e possible equilibria w ith incom patible com ponents. It tu rn s out th a t w ith incom patible com ponents two assym m etric m ulti-stage perfect equilibria in pure strategies exist despite the assum ed sym m etry of the firms. T he above described result still holds.

The concrete assum ptions of the model are presented in the n ext section.

Then th e m ulti-stage perfect equilibria, given com patible com ponents and given incom patible com ponents, are derived in th e following two sections of th e paper. In section five the results of a com parison of th e m ulti-stage p er­

fect equilibria are sum m arized and conclusions on the above raised question are drawn. The final section discusses the im plications of th e analysis for eco­

nomic policy m easures aim ed at th e stan d ard izatio n activities of firms w ith and w ithout costs of com patibility.

6See Maiutes, Regibeau (1988), p. 226, and Economides (1989a), p. 1179.

7See Maiutes, Regibeau (1988), p. 225 f., and for the ambiguous social welfare result in the case of free market entry Economides (1989a), p. 1179 f.

8See Maiutes, Regibeau (1988), p. 229.

9In related contributions an intermediate decision stage has been introduced where the firms decide whether to give a discount on the purchase of their whole system compared to the single components’ prices which does not change the structure of the compatibility decision. See,Maiutes, Regibeau (1992), p. 44, and p. 47, as well as Andersen, Leruth (1993), p. 56 f.

2 T h e m o d e l

We assume th a t there are N consumers w ith N = 1 for th e sake of simplicity.10 11 Every consumer buys one unit of the horizontally differenti­

ated system good. If she chooses a system good, one of th e two com ponents contains th e pro p erty i, th e other th e property j and the surplus she realizes from th e system good ( i , j ) is characterized by:11

y = (1)

where c is a positive param eter representing th e consum er’s surplus if she obtains her m ost prefered variant of th e system good for free instead of paying p for th e whole system . Each consum er has a single m ost prefered variant described by (?*, J*) which is uniformly distributed over a unit square defined by i 6 [0,1], j E [0,1]. The consumer buys th e system good (z ,j) which ensures th e highest surplus.

Each of th e two producers supplies a whole system consisting of two com­

ponents i and j . One firm, further indexed by 0, produces th e system (0,0),- the other, indexed by 1, the system (1 ,1 ).12 T he production technology of producer k, k E {1,0} is characterized by the following cost function:

C k = CikXik + CjkXjk + (c,jk - + (cjk ~ Z j)2 (2) w ith Zi, Zj > 0,

where c,*, (c,fc) is firm k's constant m arginal cost of producing Xik units of com ponent i (xjk units of com ponent j respectively). The term (c,;. — Zi)2 represents firm k's fixed and sunk cost of producing com ponent i and (cyjt ~ Zj)2 is th e analogous for com ponent j . The considered cost function ensures th a t low m arginal costs of com ponent production can only be achieved by high fixed and sunk costs. In.addition, th e function is additive in th e two com ponents in order to be able to com pare th e com patibility case w ith th e incom patibility case.

10The demand side of the market is equivalent to the case presented by Economides (1989a), p. 1178.

11The utility is assumed to be quadratic in the Euclidean distance which is a widely used assumption in spatial product differentiation models. See e. g. d ’Aspremont, Jaskold Gab- szewicz, Thisse (1979), p. 1148, and Economides (1989b), p. 181. Alternatively Matutes, Regibeau (1988), p. 224, assume a utility function which is block metric in the distance which is intensively discussed in Eaton, Lipsey (1980), whereas Fetter (1924), as well as Economides (1986), p. 434 f. consider preferences which are linear in the Euclidean distance.

12The product space, the distribution of consumers in the product space, and the loca­

tion of the supplied system goods is modelled analogously to Matutes Regibeau (1988), p.

223 f., and Matutes, Regibeau (1992), p. 38 f.

1. They choose th eir production technology sim ultaneously by selecting (ca-,Cjfc) from the intervals c.^ G [0,2;], and Cjk G [0, Zj] in th e first stage.

2. In th e second stage they decide sim ultaneously on th e prices of th eir com ponents or th e price of th eir com plete system depending on w hether com patibility is assum ed or not.

At any decision stage they have com plete b u t im perfect inform ation about the dem and side of th e m arket, th eir own and th e pay-off function of th eir rival. Since they can thus anticipate the consequences of th eir decision in the first stage for the possible outcom e in th e second stage, the adequate solution concept for th e whole decision problem is th e m ulti-stage perfect equilibrium .13 14 T he equilibrium is derived by working backw ards through th e decision stages and looking for th e B ertrand-N ash-E quilibrium first which is then taken into account for th e derivation of the N ash-Equilibrium in m arginal costs.

The two firms maximize th eir profits and decide on two different issues:13

3 T h e S u b g a m e P e rfe c t E q u ilib riu m w ith C o m p a tib ility

3.1 The Bertrand-Nash-Equilibrium

If th e supplied com ponents of th e two firms are com patible th e consum ers can choose am ong four system goods (z ,j): (0,0), (0 ,1 ), (1,0), (1,1).' From th e consum er’s surplus function (1) and the conditions of indifference between buying, e.g., on th e one hand (0,0) or (0 ,1 ), and on th e other han d (1,0) or (1,1) it is easy to calculate x^k th e quan tity of com ponent z and Xjk th e quan tity of com ponent j dem anded from firm k, k G {0,1}, which are given by:

%ik —

0 for p ik > P ik' + 1

1 -P ,k + P ,k> £o r p .ki _ £ < pkQ < p .ki + £

1 for pik < Pik> - I-

(3)

13The assumed order of decisions can be justified by the fact that the production technol­

ogy contrary to product prices has to be chosen before production and cannot be changed on a short term basis.

14The concept originates in Selten (1965), p. 306 ff.

1

1 - P io + P .i 2

1-PjO+Pjl

2

(1,0) (1,1)

(0,0) (0,1)

1-P jO + PjI 1 j

Figure 1: M arket Areas for th e System Goods with C om patibility 0

i - p j k + P j k ' 2

1

for pjk > pjf!> + 1

for p j k ' ~ 1 < P jk < P jk ' + 1 for p j k < P jk ’ ~ 1.

(4)

with k, k' e {0,1}, k k'

T he dem ands coincide w ith the m arket areas of the respective system goods in th e product space which consist of th e com ponent of th e considered firm .15 Assum ing at this m om ent th a t none of th e two firms is forced out of one com ponent m arket, yields for th e profit functions of firm k, k € { 0 ,1}:16

fk ( P ik ,P jk ;P ik ',P jk > ) / \ f P ik T P ik' f \

= (Pifc - C ik )--- --- + \P jk - C jk)

~ (c ik - Ziy - (cjk - 2 jy with k, k' € {0,1}, k k'

1 “ P jk + P jk '

2

(5)

By solving sim ultaneously th e four first order conditions of profit m axim iza­

tion for pik, Pjk, k £ {0,1} the B ertrand-N ash-E quilibrium can be derived:17

P*k = 1 + 2 c , + cik>

3 (6)

15Every consumers whose most prefered product specification of component i given by i* fulfills, e.g., i* < 1~P,gd'P'1 decides in favour of firm 0’s component i whereas all the others prefer the one of firm 1. See figure 1 for the shape of the market areas for the different system goods.

16Whether not to force one’s rival out of one component market is the result of a multi­

stage perfect equilibrium is tested in the next subsection.

17See Appendix A for the first and second order conditions of profit maximization in prices.

. _ , 2cjfc + Cj k , (V) - 1 + 3 >

w ith k, k' E {0,1}, k 0 k1

From the dem and functions given in (3), and (4) and th e B ertrand-N ash- Equilibrium in prices it is obvious th a t both firms are selling both com ponents only if in equilibrium th e following conditions are fulfilled for th e m arginal costs:

cik' - 3 < cik < dk> + 3 (8)

Cjk' 3 Cjk Cjk1 4” 3 (9)

w ith k, k1 E {0,1}, k k'

W hether this is tru e will be shown in th e following subsection where the N ash-Equilibrium in m arginal costs is derived.

3.2 The Nash-Equilibrium in Marginal Costs

F irst it is assum ed th a t th e conditions (8) and (9) are fulfilled. A fter calcu­

lating for th e resulting N ash-Equilibrium in m arginal costs it is proven th a t firms do not w ant to deviate from these levels in order to force th eir rival out of one com ponent m arket.

In th e first stage of th e gam e th e two firms choose th eir m arginal costs c^., Cjk, k E {0,1}, in order to maximize th eir profits and take into account th e consequences of th eir cost decisions for th e prices in th e second stage.

S ubstitu tin g the B ertrand-N ash-E quilibrium given in (6), and (7) into the profit functions (5) yields:

f k ( Cik > C jk , C ik', C jk ') (3 - cik + Cjfc>)2 (3 - Cjk + Cjk')2

ro 18

- ( c , t - 2i)2 - (ci t - ^ ) 2 (10) w ith k, k' E {0,1}, k / k'

By solving sim ultaneously th e four first order conditions of profit m axim iza­

tion w ith respect to d o , Cj0 , c n , and c}1 we ob tain :is

c*o = c*i = Z i~ ^ = Zi ~ 0,16 ( U )

(12) 18See Appendix B for the first and second order conditions of profit maximization in marginal costs.

If our assum ption th a t none of th e two firms is forced out of th e m arket were correct, the N ash-Equilibrium in m arginal costs would be characterized by (11) and (12). In this case m arginal costs would rise w ith Zi or respec­

tively, which represent the main determ inants of the costs of m arginal cost reductions. By inserting (11) and (12) into th e price equations (6), and (7) the resulting prices can be calculated:

Pio = P*i = zi + (13)

Pjo = Pji = zi + (14)

which means for th e system goods price:

= j + « ( + zy = 1.6 + -fPy, w ith k e {0,1}. (15) Prices also positively depend on Zi and Zj if the m arginal costs in (11) and (12) describe th e m ulti-stage perfect equilibrium . S ubstitu tin g th em into the profit function (10) yields the profit of th e two firms:

» = - = 0.94,17 (16)

which is a constant and does not depend on th e param eters Zi and Zj of the cost function.

However, do th e m arginal costs given in (11), and (12) form a Nash- Equilibrium of the first stage of the gam e and therefore also a subgam e perfect equilibrium ? If c*^ and c*-k are positive which is equivalent to:

Zi,Zj - g = ° -16 (17)

this is the case. Since (11) and (12) is th e N ash-Equilibrium given th a t both firms com pete in the m arket, unilateral deviation in the range where still both firms supply b o th com ponents does not pay. In addition, if only firm k deviates in such a way th a t the lower levels of (8) and (9) are no longer fulfilled and its rival is pushed out of the m arket it can only realize negative profits.19 Therefore the m arginal costs given in (11) and (12) are th e Nash Equilibrium of the second stage of th e gam e if they are feasible which means th a t (17) holds.

19See Appendix C for the proof.

T he consum er surplus C S can be calculated by sum m ing up th e individual surpluses of the consumers defined in (1) from th e four system goods which is equivalent to:

2 / 2

C S = [ f c - ( r ) 2 - ( / ) 2 - p * 0 - p * od j * d r ( is )

Jo Jo

+ j a 1 C - (i*)2 - (1 - / ) 2 - p"0 - p 'od / d i -

, . ■-■’W

+ A - - , » »

Z 1

c - ( 1 - n 2 - ( j ' ) 2 - j> : > .- p JW d > '' + c “ (1 “ ’" )2 - (1 - ’ * - p>“d j ' di If we assum e th a t th e m arginal costs given in (11) and (12) are feasible because (17) holds, th e consum er surplus can be calculated by su b stitu tin g th e equilibrium prices given in (13) and (14) into (18) which yields:

CS* = c — Zi — Zj — ^ - = c — Zi — zj — 1.83 (19) T he consum er surplus rises if the difference between th e m axim um utility th e consum ers can realize by the free consum ption of th eir m ost prefered variant of th e system good and the param eters of the cost function increases.

The social welfare can be derived by adding up th e profits of the two firms and th e consum er surplus and is, in this case where (17) holds, represented by:

IT = c — Zi — Zj + — = c — Zi — Zj + 0.05 (20) 1 b

T he higher the difference between the m axim um possible u tility and th e m axim um possible m arginal costs (or fixed costs) th e higher is th e social welfare.

4 T h e S u b g a m e P e rfe c t E q u ilib riu m w ith In c o m p a tib ility

A fter having derived th e subgam e perfect equilibrium for com patible com­

ponents we will concentrate here on the subgam e perfect equilibrium for incom patible com ponents of the two firms. The problem is solved by back­

ward induction and therefore we sta rt with the last stage of th e gam e where the prices are determ ined.

1

1 Po-Pl

x 2

(0.0)

(1,1)

1 po-p.1, 1

1 2 j

Figure 2: M arket Areas w ith Incom patibility and p0 > p x

4.1 The Bertrand-Nash-Equilibrium

If th e com ponents supplied by the two firms are incom patible the consumers can only select from two system goods (0,0), and (1,1), supplied as a whole by one seller. Therefore, the dem and for each firm ’s com ponent is identical w ith the dem and for the whole system which can form ally be described by: r.-j-. = Xjk = Xk, k € {0,1}. Thus, com ponent prices need not be considered. The system price of firm k is represented by pk, k 6 {0,1}. The dem and for the system A’, k £ {0,1}, Xk is derived sim ilarly to th e com patibility case by solving (1) for the (z*,j*) com binations of those consum ers who are indifferent between buying th e system of seller k or th e system of seller k1 with k, k' € {0,1}, and k k' and yields:

0 for pk > pk< + 2

< 2 _ for p k> + 2 > p k > p k’

1 - | ( l - EitZPi.)2 for p k, > pk > pk, - 2 1 for Pk < Pk’ ~ 2

(21)

T he dem and for firm k's system , Xk, coincides w ith th e m arket area of firm k in th e product space which differs depending on w hether Pk < Pk1 or pk > pk’

holds. The two situations are depicted in figure 2 and 3. F irst we assum e again th a t Xk > 0 holds for both firms k £ {0, l} .20 In addition we assum e,

20Later on it will be shown th at not to push the rival out of the market is a subgame perfect equilibrium.

1

Figure 3: M arket Areas with Incom patibility and p0 < p±

w ithout loss of generality, th a t pk > pk> for A:, k' € {0,1}, k k'. Taking this and th e dem and function (21) into account, th e profit functions of th e two firms read as follows:

- (cifc - ^ ) 2 - (cjfc - z j)2 (22) fk(Pk',Pk‘) = (P k-C ik-C jk) } - ( l Pk Pk ' ~ ' 2

f k '( P k '', P k ) = (P k ' ~ Cik' — C jk') 1 _ I f 1 _ Pk ~ Pk‘

2 \ 2 (23)

-(C iv - Zi)2 - (cjk. - Zj)2, w ith k, k' € {0,1}, k k '.

Solving sim ultaneously th e first order conditions of profit m axim ization w ith respect to p* and pk>, yields th e B ertrand-N ash-E quilibrium for pk > pk1'21

Pk* — Ö {2 + + Cj k ) + c ik' + Cjk' + [36 + ^{( c ik} + Cjfc)' (24) T(o«fc' T Cjk1} 4(ct’fc + Cjk Cik' Cjk') 2(ct’fc T Cjfe)(cjfc» 4- Ojfc')] >

Pj.* — ö { — 10 4- 5(c,-fc + C jk) 4- 3(c,-fc/ 4- Cjk') 4- 3 [36 4- (c,fc + c^.)2 (25) 4"(Oifc' + Cjk') 4:(Cik T Cjk Cik' Cjk') 2(cjk T Cjk)(Cik' T fijA:')] • 21See Appendix D for the first and second order conditions of profit maximization with respect to the system goods prices.

It is obvious th a t the firms k and k' ju st reverse roles in th e B ertrand-N ash- Equilibrium for pk < Pk1 which can be derived com pletely analogously to (24) and (25), which is given by:22

p** = - { — 10 + 5(c,-fc< 4- cjfc») + 3(c,a- + cjjt) + 3 [36 + (c,*/ + c^./)2 (26) .

—* b

+(c,-fc + c j k ) 2 - 4 ( c ifc, + Cjk' - Cik - c j k ) - 2 ( c ik ' + Cjk ' ) ( c ik + C jfc)]2 } ,

Pk* — q + 7(cjt» + Cjk>) + d k + Cjk + [36 + {cik< + C jk ') 2 (27) b

+ (c !jk + C jk ) 2 - 4 \ c ik ' + Cjk' - Cik - C jk) - 2 ( c ik ' + C j k ') ( c ik + Cjjfc)]2 } .

Substituting (24) and (25) into th e basic assum ption pk > p k> and (26) and (27) into p k < Pk' as well as into the condition for a positive dem and for each single firm from (21) yields for th e validity range of the two B ertrand-N ash- Equilibria in th e m arginal cost space:

p** + 2 > p*f* > p** c ik' + Cjk> + 2 > Cik + Cjk > Cik< + Cjk>, (28) p £ - 2 < Pk P ^ Cik' + ci k' ~ 2 < Cik + c>k Cik' + c>k'- (29) As in th e com patibility case it is possible to prove for the m arginal costs in equilibrium th a t deviating in such a way th a t th e above conditions are no longer fulfilled and th e rival is pushed out of the m arket results in negative profits.23 *

4.2 The Nash-Equilibrium in Marginal Costs

In order to find a N ash-Equilibrium in m arginal costs it is assum ed th a t condition (28) holds and then it is checked w hether one of th e two firms has an incentive to deviate w ith its m arginal costs in such a way th a t instead of (28) the condition (29) is fulfilled.

We reform ulate th e problem of th e two firms in order to simplify th e presen­

tatio n of th e problem and th e necessary calculations. Choosing th e m arginal cost of each single com ponent sim ultaneously w ith one’s rival, is logically equivalent to selecting sim ultaneously th e m arginal cost of th e whole system

22See figure 2 and 3.

23The proof is analogous to the one in Appendix C for the equilibrium values of the marginal costs derived in the next subsection.

good Ck and th e share of m arginal cost of com ponent i in th e m arginal cost of th e whole system of firm k, k € {0,1}- Therefore we define:

Q = cik + Cjk w ith ck G [0, Zi + Zj], (30)

Cik = s kck w ith s k G (31)

Substituting the B ertrand-N ash-E quilibrium for th e considered casep£* > pk*

given in (24), and (25) into the profit functions (22) and (23) and applying th e definitions given in (30) and (31), yields for the profit of th e two firms:

fk(ck,Sk',ck'') = (2 — ck + ck! + >/32 + (2 — Ck + Q ,)2) (32)

" ( W - Zi)2 - ((1 - SkYk ~ Zj)2,

fk'{ck',Sk'\Ck) = - —10 + 5ck — 5ck< + 3 y/32 + (2 — cfe 4- c v )2j (33) 2 — ca- + Ck> + a/ 3 2 + (2 — Ck + Ck')2

128

(■^A.'^A’' ) ((1 ^fc')cfc' Z j) ,

with k, k 1 G {0,1}, k k', and ck > ck>.

Solving th e system of the four first order conditions of profit m axim ization with respect to q, s k, ck>, and yields for th e m arginal costs and th e share of th e m arginal cost of com ponent z:24

*

k = c ^ Z i + Zj - 0.226159, (34)

^ - 0 . 1 1 3 0 8 2, + ^ - 0.226159’ (35)

'k< — c ~ Z{ Zj — 0.448908, (36)

„ Zi - 0.224454 fc' ’ zi + zj — 0.448908’ (37)

w ith A, k' G {0,1}, k k’, and ck > ck>.

Choosing c from (34) is the optim al decision for firm k given th a t firm k' selects ck' = c from (36) and given th a t it is restricted w ith its variable ck

24See Appendix E for the derivation of (34), (36), (35), and (37). They satisfy the first and second order conditions of profit maximization and are found by numerical calculations.

to th e interval [q>, Zi + zj], Analogously can be argued for cJ7 = c from (36).

However, since firm k has no incentive to reduce Ck below c, the m arginal cost of firm k', and firm k 1 cannot improve its profit by exceeding th e m arginal cost level c of firm k, th e m arginal cost levels and th e shares of the m arginal costs for com ponent i given in (34), (36), (35) and (37) are a N ash-Equilibrium as long as they are feasible.25

The m arginal cost levels and shares of com ponent i given in (34),(36), (35) and (37) represent two N ash-Eqilibria in pure strategies because eith er firm 0 is th e high cost supplier (k = 0) and firm one is th e low cost supplier (£' = 1) in equilibrium or th e other way round. In addition it should be noticed th a t each single N ash-Equilibrium is asym m etric although th e whole set-up of th e model is com pletely symmetric.

The two N ash-Equilibria described above are feasible if they fulfill th e con­

ditions (30) and (31), which is equivalent to dem anding:

zx + z, > 0.44S90S. (38)

Assuming th a t (38) holds, it is easy to calculate from (32) and (33) th e profits, from (30) and (31) the m arginal costs of each firm ’s com ponent and from (24) and (25) each firm ’s system price in th e two equilibria by sub stitu tin g (34), (35), (36), and (37). By this procedure we obtain for the profits:

/ ( c , s , ( c ) ; c ) « 0.421428 (39) / (^c,5^a-(^c); c ) « 0.495672 • (40) The high cost provider realizes a lower profit th a n th e low cost provider. How­

ever, both profits are not influenced by th e param eters of the cost function z,- and Zj. T he m arginal costs of each firm ’s com ponents are given by:

ct(c) = c, ~ Zi — 0.11308 c?(c) = Cj ~ zj — 0.11308 (41) c,(c) = c- sa zx - 0.224454 C j ( c ) = c ; sa zj - 0.224454 (42) The resulting prices of th e firm s’ system s in th e two possible equilibria are derived from su bstituting (41) and (42) into (24) and (25). They can be characterized by:

p(c;c) « 0.737181 + z,-+ Zj (43) p(c; c) « 0.663862 + z, + Zj (44) 25See Appendix F for the proof.

1 0

c c

c 0,474426

0,474426

0,421428

0,495672

c 0,495672

0,421428

0,399241

0,399241

Table 1: T h e Profits Contingent on Playing c or c

T he consum er surplus is derived by sum m ing up individual surpluses of th e consumers of th e two system goods and yields:26 27

CS** = c - z i - Zj - 1.03253 (45) By sum m ing up th e consum er surplus and th e profits of th e two firms, we obtain th e social welfare, given by:

IP** = C- Z i - z j - 0.115428 (46) However, even if the strategies of the N ash-Equilibria are feasible and (38) holds, it is contingent on the assum ption th a t th e two firms can solve th e coordination problem . If this coordination problem cannot be solved by th e two firms they can instead play mixed strategies eith er over th e whole range of th eir decision variable Cfc, ck> G [0, Zi + Zj] with k, k' e {0,1}, k / k' or over th e two pure strategies c, c respectively.

As an exam ple, we will consider here th e N ash-Equilibrium in mixed s tra te ­ gies if th e two firms random ize concerning th e roles they play in th e game.

The profits from playing either cjt = c or q = c are given in tab le l . 2” T he prices which result if th e two firms by accident choose th e sam e roles are derived from inserting ck = ck- = c or ck = ck> = c into (24) or (25) and are characterized by:

p(c, c) « 0.773841 + Zi + Zj (47)

p(c,c) ft; 0.551092 + (48)

The expected pay-off of firm A’, k G {0,1} playing c w ith probability qk and c w ith probability (1 — qk) is given by:

E [f(c, c)] = qk [qk'.f(c- c) + (1 - </fc»)/(c; c)] (49) +(1 ~ f c '/ ( c ; c) + (1 - <Jp)/(c;c)]

w ith k' k.

26The derivation is analogously to the consumer surplus in the case with compatibility which is given in (18).

27They are calculated by substituting the respective marginal cost level from (34) and (36) in (32) and (33).

X CS

W

c, c c - Z i - Zj - 1.03253 c - Z i - Zj - 0.115428 c, c c — Zi — Zj — 1.10717 c — Zi — Zj — 0.158322 c, c c — Zi — Zj — 0.884426

c —

Zi — Zj — 0.0859438 E M c — Zi — Zj — 1.01657 c — Zi — Zj — 0.119566

Table 2: The Possible IF and C S from Random izing about c o r e T he best response qC(qk') can be derived from th e first order condition of maxim izing the expected profit (49) w ith respect to qk- Taking into ac­

count th a t th e pay-offs of both firms are completely sym m etric th e Nash- Equilibrium in mixed strategies can be calculated:

*’ = * * = * * = ________ / ( c ; c ) - / ( c ; c ) ________

& 0.51083 (50)

The probability q** of playing c is slightly higher th an th e probability (1 — 9**) of playing c. S ubstituting th e probability q** from (50) and th e profit levels from table (1) into (49) yields for th e expected profits of the two firms:

£ [ / ( c ,c ) ] « 0.448501 (51)

T he results concerning th e consumers surplus, th e social welfare and th e respective expected values for th e possible outcomes of th e gam e if the firms random ize over their roles are given in table 2.

5 T h e C o m p a riso n o f th e S u b g a m e P e rfe c t E q u ilib ria w ith a n d w ith o u t C o m p a tib il­

ity

F irst of all we restrict our com parison of the subgam e perfect equilibria w ith and w ithout com patibility to those cases where th e param eters of th e cost function z,- and Zj fulfill th e feasibility constraints of both considered cases

given in (17) and (38).28 From com paring the resulting m arginal costs of th e two com ponents given in (11), (12), (41), and (42), w ith and w ithout com patibility th e following proposition can be concluded:

P r o p o s i t i o n 1: If the com ponents are com patible, b o th firms choose m arginal costs c* and c* which are in between the two m arginal cost lev­

els (c,-,c,) and (c j,c j) possibly selected w ithout com patible com ponents because th e following holds:

Ci_< c* < Ci and Cj < c* < c3

Given th a t the two firms can solve the coordination problem which evolves from the two possible N ash-Equilibria in pure strategies, w ith incom patibility, Proposition 1 m eans th a t th e m arginal costs of th e high-cost provider exceeds the levels chosen w ith com patible com ponents, whereas the low-cost provider produces w ith lower m arginal costs. D espite the am biguous relation between the m arginal costs in th e considered two cases a ra th e r strong proposition concerning the prices of th e system s can be derived from (15), (43), (44), (47), and (48):

P r o p o s i t i o n 2: If the com ponents of th e rival suppliers are incom patible th e price a consum er has to pay for a com plete system is in any case lower than th e system price dem anded in th e case of com patible com ponents because the following holds:

p(c; c) < p(c; c) < p(c; c) < p(c- c) < p*

Thus, it does not m a tte r w hether the two firms play one of th e two Nash- Equilibria in pure strategies or w hether they random ize over th e equilibrium cost levels in th e incom patibility case. A com plete system is cheaper for every consum er in the product space com pared to the situation w ith com patible com ponents.29 Concerning the profits of th e th e two firms we can also derive a clear statem en t from (16), (39), (40), table 1 and (51):

P r o p o s i t i o n 3: The two firms realize in any case lower profits if th eir com ­ ponents are incom patible, more so th a n w ith com patible com ponents because the profits can be ordered in th e following way:

> / (^c,^sa-(^c);^c) > E [f(c ,c )} > f ( c , s k(c);c) 28This is essentially done in order to shorten the presentation. The consequences and conclusions of the various corner solution cases do not essentially differ from the inner solutions of the profit maximization problem of the two firms considered here.

29Those authors who restrict their analysis to the effects of compatibility on the price competition draw the same conclusion concerning the equilibrium prices. See Economides (1989a), p. 1178, and Malates, Rcgibe.au (1988), p. 225 f.

A consequence of Proposition

3

is th a t th e two firms would always prefer com patibility independent of the N ash-Equilibrium really played in th e in­

com patibility case if it can be achieved costlessly.30 However, a com parison of the consumer surplus from (19) with (45) and table 2 reveals th a t this is not in the interest of the consumers because th e following proposition holds:31

Proposition 4:

Regardless of th e kind of N ash-E quilibrium really played in th e incom patibility case, the consumers will always realize a higher surplus th a n in th e com patibility case because th e following relationship holds:

CS* < CS** < £ [C S ]

Nevertheless, from (20), (46) and table 2 it is obvious th a t com patibility is still prefered from a social welfare point of view because th e following statem ent is true:

Proposition 5:

T h e social welfare w ith com patibility is always higher th an w ithout com patibility because th e following relationship holds:

IE* > IE** > £[IE]

The source of th e welfare improvement in the com patibility case is th e fact th a t th e system goods in the m arket are b e tte r adjusted to at least some of the consum ers’ m ost prefered variant. However, Propositions 1 to 5 reveal th a t th e consumers cannot realize this advantage, because th e firms m anage to acquire an even greater piece of their surplus presum ably because of th e less tough price com petition with com patibility.32 The additional com petition in m arginal costs on which this paper focuses creates a cap on th e piece of the social surplus th e firms can acquire in b o th considered cases because the profit becomes a constant. In each case th e social surplus rises w ith a decrease in th e costs Zi and Zj of th e production technology w ithout influencing th e realized profits. However, it does not change th e ranking of com patibility versus incom patibility in the eyes of producers, consumers, and society.

6 C o n clu sio n s

The desirability of com patibility is not essentially changed by considering an additional strategic variable, as long as it can be achieved unilaterally by one

30The endogeneity of the marginal costs does not change this result of the existing literature. See Economides (1989a), p. 1178, and Matutes, Regibeau (1988), p. 225 f.

31See Matutes, Regibeau (1988), p. 226, who find the same result for a covered market concerning the consumer surplus and the social welfare if firms compete only in prices.

32See Matutes, Regibeau (1988), p. 226 f., for this interpretation.

of th e two firms w ithout costs. In this case it is desired by th e firms and by society b u t paradoxically not by th e consumers. Economic policy measures aim ing at influencing th e standardization decision of th e two firms can thus only be justified by distributional considerations bu t not by efficiency reasons.

However, costless achievement of com patibility is not a very realistic as­

sum ption. If we alternatively assum e th a t th e firm which decides in favour of com patible com ponents w ith its rival has to bear a constant cost it will do so as long as its additional profit exceeds th e com patibility costs. Com paring th e possible differences in profits w ith th e differences in social welfare under com patibility versus incom patibility it tu rn s out th a t th e increm ental profit from standardization is always much higher th an th e increm ental social wel­

fare. This means th a t w ith com patibility costs th e firms m ay decide in favour of com patibility in situations where th e com patibility costs already exceed the increm ental welfare gain. In these situations governm ent could improve efficiency by m aking the achievement of com patibility by a firm even m ore expensive. C ontrary to the result of M atutes, Regibeau who consider only price com petition excess incom patibility can never occur, m ainly because our analysis is restricted to an inelastic overall m arket dem and.33 Therefore a prom otion of com patibility by th e governm ent cannot be justified from the result of th e m odel presented here.

A p p e n d ix A

D iffe re n tia tin g th e pro fit fu n c tio n s re p re se n te d by (5 ) w ith re sp e c t to th e firm ’s ow n c o m p o n e n t prices yields th e follow ing se t of first o rd e r co n d itio n s:

d f k , , 1 + Pik' + cik

= --- 2--- K t = 0 d f k , x 1 + Pjk' + qi-

= --- J---

» = o w ith k ,k ' € { 0 ,1 } , a n d k ± k!

(52)

(53)

T h e second o rd e r co n d itio n s o f p ro fit m a x im iz a tio n a re also fulfilled for th e p rofit fu n c tio n o f firm I’, k € { 0 ,1 } b ec au se th e follow ing holds:

-x-2-(?hfc;/ht-') = - 1 < 0 d 2 A- z x , . n ä t

(

i w w = - i < o

(54)

(55)

33See Matvtes, Regibeau (1989), p. 230 f.

= 1 * 0 < « >

w ith k1 G { 0 ,1 } , a n d k' / k

Since th e p rofit fu n ctio n s o f th e tw o firm s a re th u s q u asi-concave a n d co n tin u o u s in c o m p o n e n t prices a B e rtra n d -N a sh -E q u ilib riu m in P u re S tra te g ie s ex ists w hich is given in (6 ), a n d (7 ).34

A p p e n d ix B

D ifferen tiatin g th e p rofit fu n c tio n s re p re se n te d by (10) w ith re sp e c t to th e firm ’s ow n m a rg in a l costs yields th e follow ing set o f first o rd e r con d itio n s:

d h

dcik(cik; cik>) - 3 - 17c,-fc - cik> + 18g,

9 (57)

dfk

dcjk(cjk; Cjk') ^{3 17cjfc C jk i} + 18^j _

9 (58)

w ith k ,k ' G { 0 ,1 } , a n d k k'

T h e second o rd e r co n d itio n s of p ro fit m a x im iz a tio n a re also fulfilled for th e p ro fit fu n c tio n o f firm fc, k G { 0 ,1 } b ec au se th e follow ing holds:

# 2A , x 17

dc^ c lk,c tk') - 9 < 0 (59)

d 2 f k( x 17 . .

QcA C3k,C]kl} - g < 0 (60)

d 2f k , , 92f k , , ( d 2f k ( /

Qc2k (ctk, ctk>) (cjk ,pjk.) [ d c.kdc.k (c‘*’c^-')) 2 = <17’2 > o

} 81 - (61)

w ith k1 G { 0 ,1 } , a n d k' 0 k.

A p p e n d ix C

If firm k w a n ts to force firm k' w ith c*fc, = Zi — 1 /6 o u t o f th e m a rk e t o f co m p o n e n t i, k, k' G { 0 ,1 } , k / k', it h a s to re d u ce its m a rg in a l co sts to :

cik = zi - 1 9 /6 , (62)

34For the conditions which ensure that a Na-sh-Equilibrium in Pure Strategies exists see, e. g. Fudenberg, Tirole (1991), p. 34.

in o rd e r to en su re t h a t (8 ) is n o t fulfilled. Since c,jt £ [0,2;] h a s to be fulfilled th is is only po ssib le if 2; > {ß holds. T h e re su ltin g p rice for firm k w ould be:

Pik = ( 63 )

T h e p ro fit o f firm k from p ro d u c in g co m p o n en t i is eq u iv ale n t to :

z x2 n / 1 9 \ 2 289

Pifc - - (cifc - 2,)2 = 2 - ( — 1 = . (64)

Is it possible to im p ro v e th is p ro fit by red u cin g c,fc from (62) by d, d > 0, a n d stick in g to th e p rice level o f (63)? P ro fit w ould ch an g e to:

w hich is sm aller th a n (6 4 ) for any d > 0. T h e sam e re aso n in g is tr u e for co m p o n e n t J .

A p p e n d ix D

F ro m d iffe re n tia tin g (22) w ith re sp ect to pj.. a n d (23) w ith re sp e c t to pf.i th e first o rd e r co n d itio n s can b e derived:

dpk^{(Pk\ Pk')}

Pk - Ph'

2 ~ Pk + Cik + Cjfc = 0, (65)

i _ i A _ Pk ~ Pk'

2 V 2 (6 6 )

Z A1 A P k - P k '

-(.P k1 ~ cik' ~ cj k ' ) ^ I 1 --- --- ^{= 0}

w ith k .k ' € { 0 , 1 } , k / k'.

F or th e second p a r tia l d e riv a tio n o f (22) w ith re sp e c t to p^ a n d (23) w ith re sp e c t to pk/ th e follow ing c o n d itio n m u st hold in o rd e r to e n su re th e co n c av ity o f th e p ro fit fu n c tio n in th e prices of th e sy ste m goods:

d 2fk

dp2k (pk;Pk') P k - Pk>

A

2

J

Pk - Pk'

2 < 0 ,

(67)

— Pk + Cik + cjl-

d 2Ä (68) d Pk>

- ^ 1 “ 2 _ 4 ^ ' _ Cik> ~ Cjk'^ ~ °

w ith k, k' € { 0 ,1 ) , k k'.

G iven th e a ssu m p tio n s o f th e considered case, t h a t is pk> + 2 > pr- > p^.», (67) is fulfilled as long as (6 5 ) h a s a so lu tio n , an d (68) holds as long as th e so lu tio n of (65) yields a p rice abo v e m a rg in a l costs.

A p p e n d ix E

D ifferen tiatin g th e p ro fit fu n c tio n s (32) w ith re sp e c t to q- a n d s k a n d (3 3 ) w ith re sp ect to ck> a n d sk> yields for th e n ecessary co n d itio n s o f p ro fit m a x im iz a tio n :

s k- ck,) = - ^ 2 - e t + ck> + y /3 2 + (2 - ck + cfc/)2) - 2 s k(skck - zfi - 2(1 - s fc) ( ( l - sk)ck - Zj)

= 0

Ck + Zi ~ Z j

(69)

d ft Sk' Ck') ~ 2c4 ( l _ 2sk)ck + Zi - zj) = 0 <=> sk = ⁽⁷⁰⁾

d fk>, x ö — \Ck',Sk<'-, Ck)

dck, ^{- 5 +}

3(2 - ck + ck>) y/32 + (2 - ck + cfc,) 2_

(2 — ck + ck> + ^ /3 2 + (2 — ck + cki)2^

(71)

128

+ 10 - 5ck + 5ck, - 3y/32 + (2 -- ck + Q .,)2 5 1 2 7 3 2 + (2 - Cfc + ck, y

• (2 — ck + cki + y/32 + (2 — ck + Q -')2)

- 2 s k>(sk>ck> - Zi) - 2(1 - JSfc,)((l - sk>)ck> - Zj)

= 0

d fki , „ ,, „ „ ck> + z; — Zj

■7r^-(ck>sk>;ck) - 2cfc»((l - 2sk>)ck> + Zi - Zj) & sk> - --- ---

cfski 2ck! ⁽⁷²⁾

In o rd e r to solve th e e q u a tio n sy stem (69), (7 0 ), (7 1 ), a n d (72) sk a n d s ki from (70) a n d (72) a re s u b s titu te d in to (69) an d (71), (71) is s u b tr a c te d fro m (6 9 ), a n d th e follow ing d efin itio n for th e difference b etw een th e m a rg in a l co sts o f th e tw o riv als’ sy stem s is ap p lied to th is difference:

& = ck - ck, > 0 (73)

T h e w hole p ro c e d u re re su lts in:

d fk, f x , d fk ', , x 1

^ ( a , « ( « ) ; c t .) - = _ _ ^ = = =

• [3A 3 - IS A 2 + 140A - 232 - (3 A 2 + 116A - 6 0 )> /3 2 + (2 - A ) 2]

= 0

O 3 A 5 + 3 7 A 4 - 136A 3 + 1964A 2 - 1760A + 296 = 0 (74) N u m erically solving (74) reveals t h a t th e re is only o n e p o sitiv e a n d re a l so lu tio n for A :

A** = c £ - « 0.222748, (75)

w hich is th e o p tim a l difference b etw een q a n d c y for q- > c/.-/ if th e sufficient co n d itio n s for a p ro fit m ax im u m a re fulfilled a t th e re su ltin g c/., c^, Sk a n d s/../.35 T h e second o rd e r co n d itio n s w ith re sp e c t to q an d q, can also b e p re se n te d in te rm s of A fro m (73):

d 2f k

ö c 2 (Q ,5fc; Q-')

A « A ’ * O

3(2 - A + 7 3 2 + (2 - A ) 2)2 3 2 7 3 2 + (2 - A ) 2

1 2 - A + 7 3 2 + ( 2 - A ) 2 '

’ 7 3 2 + ( 2 - A ) 2 + 16

- 2 [ ( 1 -s, ) 2 + s2] < 0

d2fk ~ -1 -8 9 7 0 1 + 4 ^ .(1 - s fc) < 0, (76)

A « A** #

d2fk dcl

-(C k'yS k'^C k) 2800 - 672A + 2 1 6 A 2 - 2 4 A 3 + 3 A 4 128 ( 7 3 2 + ( 2 - A ) 2) 3

3A + 250 128

+ 4sk' — 4 s 2 < 0 d 2f k>

d c l (ck',Sk'-,Ck) -1 .8 5 8 6 3 + 4 ^ ( 1 - M < 0, (77) a n d (76) a n d (7 7 ) a re fulfilled for an y Sk a n d $k>. T h e second o rd e r c o n d itio n s w ith re sp e c t to Sk a n d are given by:

^ t (q,^ -;q-') = - 4 c * < 0 , (7 8 )

-Q ^ ~ ( ck ^ s k'',Ck) = ~^Ck> < 0, (79)

35The resulting c* is calculated by substituting the definition (73) and A* into (69) which yields (34). Applying (34) and (75) to (73) yields Ck> from (36). The resulting sj and Sk1 are calculated by inserting (34) into (70) and (36) into (37).

a n d h o ld for an y q.,q, > 0. T h e sam e is tr u e fo r th e second o rd e r co n d itio n s concerning th e cross p a r tia l d erivatives. A p p ly in g (7 6 ), (7 8 ), (70), (7 7 ), (7 9 ), a n d (72) yields:

02h , 'P h , . ( a2h , .

3.58803c(. > 0,

d2f k

^ c k,,sk,-,ck) ^ - ( c k,,Skl-,ck) - ^ ± ^ ( c k,,ske,ck})

(81)

dck'

3.58803c3/ > 0, w hich h av e th e d e m a n d e d signs.

A p p e n d ix F

If firm k red u ces its m a rg in a l cost below c « zi + z j — 0.448908, th e m a rg in a l cost o f firm k', th e roles o f k a n d k' a re re v ersed a n d ta k in g th is in to ac co u n t its p ro fit fu n c tio n is given by (33). T h e first o rd e r co n d itio n o f p ro fit m a x im iz a tio n w ith re sp e c t to ck a n d sk coincides w ith (71) a n d (72) w ith rev ersed indices k a n d k'.

S u b s titu tin g (72) in to (71) a n d ta k in g in to a c c o u n t t h a t ck = c — & w ith A > 0, c « zi + z j — 0.448908, yields for th e first o rd e r co n d itio n of p ro fit m a x im iz a tio n w ith re sp e c t to ck in th is case:

d f k , . , . 296 - 172A + 18A - 3 A 3

= ■ 2 5 6 ^ 2 + ( 2 -A )*

| 3 ( A 2 - 13.6932) + 244A

(82)

T h e e q u a tio n (82) h a s no so lu tio n for a p o sitiv e A w hich m ean s t h a t th e p ro fit fu n c tio n of firm k is e ith e r m o n o to n ic ally in cre asin g o r d ec rea sin g in ck for ck < c.

W h ich case ap p lies can b e d e te rm in e d by co n sid erin g th e m a rg in a l p ro fit fu n c tio n given in (82) fo r th e lim itin g case ck —> c:

Cfc—£lim (cjt,Sfe(cfc) ;c ) 5_

12 « 0.0322413 > 0. (83) -i + ^{Z j} - c -

T h is im plies t h a t th e p ro fit fu n c tio n o f firm k in cre ases in th e w hole a r e a ck € [0, c]

w hich m ean s t h a t it w ould p re fer to choose th e lim itin g value c if it w ould b e re s tric te d to th is area.. H ow ever, as one can easily reco g n ize from ta b le 1 th e p ro fit w ith ck — c is low er th a n w ith ck = c w hich is th e even m o re p refered choice.

In a d d itio n firm k' h as no in cen tiv e to in cre ase its m a rg in a l co sts ab o v e th e level of firm k w hich is c « zi + Zj - 0.226159. If it w ould choose ck> > c its p ro fit

w ould b e re p re se n te d by (32) w ith reversed roles. T h e first o rd e r c o n d itio n o f p ro fit m a x im iz a tio n coincides w ith (69) a n d (70) w ith in te rc h a n g e d k a n d k'. S u b s titu tin g (70) in to (6 9 ), ta k in g in to ac c o u n t t h a t ck> = c + A w ith A > 0 a n d c m z, + Zj — 0.226159, yields for th e first o rd e r co n d itio n o f p ro fit m a x im iz a tio n o f firm k:

d f k>

dck,(ck>,sk'(ck,); c) = 0.226159 - A + 3 ( - 2 + A - y z32 + ( 2 - A ) 2)3

1 0 2 4 ^ /3 2 + (2 - A ) 2 (84) T h e re is no p o sitiv e A w hich solves (84) an d th e m a rg in a l p ro fit fu n c tio n o f firm k' is e ith e r m o n o to n ic ally in cre asin g o r d ec rea sin g fo r cki > c C o n sid e rin g th e m a rg in a l p ro fit fu n c tio n fo r th e lim itin g case c ck>:

= Zi + Zj - c — i w -0 .1 9 4 6 4 9 < 0, (85) (cfc',<sfc'(cfc');c )

lim

reveals t h a t th e p ro fit of firm k1 is m o n o to n ic ally d ec rea sin g for cki > c. T h e re fo re it w ould choose th e lim itin g value cki = c if it w ould be re s tric te d to th e in te rv a l ck> G [c,Zi + Zj], H ow ever, as ta b le 1 show s, its p ro fit is even h ig h er if it stick s to Cfc' = £ w hich is th e re fo re p refered .

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