Joint Venture for Product Innovation and Cartel Stability under ...amsacta.unibo.it/689/1/385.pdfJoint Venture for Product Innovation and Cartel Stability under Vertical Di¤erentiation

Joint Venture for Product Innovationand Cartel Stability under Vertical

Di¤erentiation

Cristina Iori# - Luca Lambertinix

Department of Economics, University of BolognaStrada Maggiore 45, 40125 Bologna, Italy

fax +39-051-2092664# e-mail: [email protected]

§ e-mail: [email protected]

July 18, 2000

Abstract

We describe a vertically di¤erentiated market where …rms choosebetween activating either independent ventures leading to distinctproduct qualities, or a joint venture for a single quality. Then, …rmseither repeat the one-shot Nash equilibrium forever, or behave col-lusively, according to discount factors. We prove that there exists aparameter region where the joint venture makes it more di¢cult for…rms to sustain collusive behaviour, as compared to independent ven-tures. Therefore, public policies towards R&D behaviour should bedesigned so as not to become inconsistent with the pro-competitiveattitude characterising the current legislation on marketing practices.

J.E.L. classi…cation: C72, D43, L13Keywords: product quality, R&D investment, implicit collusion,

joint venture, independent ventures

1

1 IntroductionOligopoly theory has produced a relevant literature on repeated market inter-action. The relative e¢ciency of Bertrand and Cournot competition in stabi-lizing cartels composed by …rms whose products are imperfect substitutes hasbeen analysed by Deneckere (1983), Rothschild (1992) and Albæk and Lam-bertini (1998), showing that when substitutability between products is high,collusion is better supported in price-setting games than in quantity-settinggames, while the reverse is true in case of low substitutability.1 Majerus(1988) has proved that this result is not con…rmed as the number of …rmsincreases. These contributions compare Cournot and Bertrand supergamesto conclude that a quantity-setting cartel should almost always be preferredto a price-setting cartel on stability grounds.2 Finally, the in‡uence of en-dogenous product di¤erentiation on the stability of collusion in prices hasbeen investigated by Chang (1991, 1992), Ross (1992) and Häckner (1994,1995, 1996). The main …nding reached by these contributions is that, undervertical di¤erentiation, collusion is more easily sustained, the more similarthe products are, while the opposite applies under horizontal di¤erentiation.

The consequences of collusion on the extent of optimal di¤erentiation inthe horizontal di¤erentiation model have also received attention. Friedmanand Thisse (1993) have considered a repeated price game in the horizontalframework and found out that minimum di¤erentiation obtains if …rms col-lude in the market stage. In most of these models, although di¤erentiationcan be endogenously determined by …rms through strategic interaction, theissue of cartel stability is studied by making the degree of di¤erentiationvary symmetrically around the ideal midpoint of the interval of technolog-ically feasible or socially preferred varieties, leading to the conclusion thatproducers may prefer to choose the characteristics of their respective goodsdi¤erently from what pro…t maximization would suggest, if this helps themminimize the incentive to deviate from the implicit cartel agreement.

1The same question is addressed in Lambertini (1996), where the evaluation of cartelstability under Bertrand and Cournot behaviour is carried out in terms of the concav-ity/convexity of the market demand function.

2This approach cannot grasp any strategic interaction behind the choice of the marketvariable. Using the same demand structure as in Deneckere (1983) and analysing asymmet-ric cartels where one …rm is a Bertrand agent while the other is a Cournot one, Lambertini(1997) proves that the choice of the market variable in order to stabilize implicit collusionproduces a Prisoner’s Dilemma.

2

To our knowledge, little attention has been paid so far to the interplaybetween …rms’ technological decisions and their ability to build up and main-tain collusive agreements over time. This is a relevant issue, in that publicauthorities prosecute collusive market behaviour, while they seldom discour-age cooperation in R&D activities. Indeed, there exist many examples ofpolicy measures designed so as to stimulate the formation of research jointventures.3 However, encouraging cooperative R&D and discouraging marketcollusion can be mutually inconsistent moves, if R&D cooperation tends tofacilitate collusion in the product market.

In this respect, Martin (1995) analyses the strategic e¤ects of a researchjoint venture (JV henceforth) designed to achieve a process innovation foran existing product. Then, the product is marketed by …rms engaging inrepeated Cournot behaviour over an in…nite time horizon. Martin showsthat cooperation in process innovation enhances implicit collusion, which canjeopardise the welfare advantage of eliminating e¤ort duplication through theJV . This result has potential implications for the case of product innovationas well.4

We reassess Martin’s framework, by considering a vertically di¤erentiatedmarket where …rms are given the possibility of choosing between activatingeither independent ventures leading to distinct product qualities, or a jointventure for a single quality, aimed at reducing the initial R&D expenditurevis à vis independent ventures. Then, …rms market the product(s) over an in-…nite horizon. In doing so, they either repeat the one-shot Nash equilibriumforever, or behave collusively, according to their intertemporal discounting.In such a setting, we prove that there exists a parameter region where Mar-tin’s conclusion is reversed, i.e., the JV makes it more di¢cult for …rms tosustain collusive behaviour in the market supergame, as compared to inde-pendent ventures. This holds independently of whether …rms set prices orquantities during the supergame. Our result entails that public policies to-wards the R&D behaviour of …rms should be tailored case by case, so as notto become inconsistent with the pro-competitive attitude characterising the

3See the National Cooperative Research Act in the US ; EC Commission (1990) ; and,for Japan, Goto and Wakasugi (1988).

4Lambertini, Poddar and Sasaki (1998) adopt the same view as in Martin (1995), al-though they consider the relationship between standardization and the stability of implicitcartel agreeements. See also Lambertini, Poddar and Sasaki (2000). Cabral (1996), in asomewhat dissimilar vein, proves the possibility that competitive pricing is needed tosustain more e¢cient R&D agreements.

3

current legislation on marketing practices.The remainder of the paper is structured as follows. The basic model

of vertical di¤erentiation is described in section 2. Section 3 describes thecase of collusion along the frontier of monopoly pro…ts. Section 4 deals withpartial collusion under either Cournot or Bertrand behaviour. Finally, section6 provides concluding remarks.

2 The vertical di¤erentiation modelWe adopt a well known model of duopoly under vertical di¤erentiation (seeGabszewicz and Thisse, 1979, 1980; Motta, 1993; Aoki and Prusa, 1997;Lehmann-Grube, 1997; Lambertini, 1999, inter alia).5 Two single-product…rms, labelled as H and L, produce goods of (di¤erent) qualities qH andqL 2 [0;1), with qH ¸ qL; through the same technology, C(qi) = cq2i ; withc > 0: This can be interpreted as …xed cost due to the R&D e¤ort needed toproduce a certain quality, while variable production costs are assumed away.Products are o¤ered on a market where consumers have unit demands, andbuy if and only if the net surplus from consumption vµ(qi; pi) = µqi ¡ pi ¸ 0;where pi is the unit price of the good of quality qi, purchased by a genericconsumer whose marginal willingness to pay is µ 2 [0; µ]: We assume that µis uniformly distributed with density one over such interval, so that the totalmass of consumer is µ.

Firms interact over t 2 [0;1); as follows:² At t = 0; they conduct R&D towards the development of product

quality, through either a joint venture (JV henceforth) or independentventures (IV henceforth). If …rms undertake a joint venture, thenqi = qj = q and each …rm bears half the development cost, cq2=2.Otherwise, …rms market di¤erentiated products, each of them bearingthe full development cost of their respective varieties, cq2i .

6

² Over t 2 [1;1); …rms market the product(s) resulted from previousR&D activity, either à la Cournot or à la Bertrand.

5A di¤erent model is used in Shaked and Sutton (1982, 1983), where …xed costs areexogenous.

6The R&D e¤orts of …rms operating in vertically di¤erentited markets are investigatedin Beath et al. (1987), Motta (1992), Dutta et al. (1995), Rosenkranz (1995, 1997), vanDijk (1996). In particular, Motta (1992) and Rosenkranz (1997) describe the incentivestowards cooperative R&D.

4

² In the in…nitely long marketing phase, …rms may collude if their respec-tive time discounting allows them to do so. Otherwise, they always playà la Nash. De…ne as ±i the discount factor of …rm i; and ±Ii (K) the crit-ical threshold for the stability of collusion, with superscript I = B;Cstanding for Bertrand and Cournot, and K = IV; JV; indicating theorganizational design chosen for the R&D phase.

As a …rst step, observe that the locations of indi¤erent consumers along[0; µ] are:

µH =pH ¡ pLqH ¡ qL

; µL =pLqL

(1)

where µH is the marginal willingness to pay of the consumer who is indi¤erentbetween qH and qL; and µL is the marginal willingness to pay of the consumerwho is indi¤erent between qL and not buying at all. Then, market demandsare

xH = µ ¡ µH ; xL = µH ¡ µL : (2)Notice that (2) can be inverted to yield the relevant demand functions forthe Cournot case:

pH = qH³µ ¡ xH

´¡ qLxL ; pL = qL

³µ ¡ xH ¡ xL

´: (3)

At any t ¸ 1; …rm i obtains revenues RIi = pixi; I = B;C: The discounted‡ow of pro…ts over the whole game is then:

¼Ii =

8>>><>>>:

±i1¡ ±i

¢RIi (qi; qj)¡ cq2i under IV±i

1¡ ±i¢RIi (q)¡

cq2

2under JV

(4)

To model collusion in marketing, we adopt the Perfect Folk Theorem (PFThenceforth; see Friedman, 1971), where the in…nite reversion to the one-shotNash equilibrium is used as a punishment following any deviation from theprescribed collusive path.7 The collusive path can instruct …rms to colludeeither fully (i.e., at the Pareto frontier of monopoly pro…ts) or partially, at

7There exist other (less grim) penal codes (see Abreu, 1986; 1988; Abreu, Pearce andStacchetti, 1986; Fudenberg and Maskin, 1986), using symmetric optimal punishments.However, the asymmetry of our model prevents us from adopting optimal punishments.For the application of optimal punishments in a symmetric duopoly model with productdi¤erentiation, see Lambertini and Sasaki (1999, 2000).

5

any pair of prices or quantities such that per-period individual revenues areat least as large as the Nash equilibrium revenues.

De…ne:

[1] The instantaneous best reply of …rm i as ®¤i :

[2] The collusive action as ®coll 2³min

n®N ; ®M

o;max

n®N ; ®M

oi; ® = p; x:

[3] The collusive revenues to …rm i as RIcolli (¢) ; (¢) = f(q); (qi; qj)g :

[4] The one-shot Nash revenues to …rm i as RINi (¢):

[5] The one-shot deviation revenues to …rm i as RIDi (¢):

The rules of the PFT establish what follows:

² At t = 0; …rms play ®coll:

² At t ¸ 1; …rms play ®coll i¤ ®i = ®coll at t¡ 1 for all i ;…rms play ®¤i otherwise.

De…nitions [3-5] and the rules of PFT yields that implicit collusion at®coll is sustainable i¤

±i ¸ ±Ii (K) =RIDi (¢)¡RIcolli (¢)RIDi (¢)¡RINi (¢)

for all i : (5)

In the next section, we quickly deal with the case of full collusion, where®coll = ®M :

3 Full collusionFirst, notice that when …rms operate along the frontier of monopoly pro…ts,they are indi¤erent between settting prices or output levels. Therefore, wecon…ne our attention to the Bertrand case.

Suppose …rms choose independent ventures at t = 0: Then, over t 2[1;1), they should market di¤erent products. We are going to show thatthis cannot be an equilibrium. At any t 2 [1;1), the cartel aims at

maxpH ; pL

RM = RBH(qH ; qL) +RBL (qL; qH) : (6)

6

Monopoly prices are:

pMH =µqH2; pML =

µqL2; (7)

at which xMH =µ

2, while xML = 0: Therefore,

¼BH =±H

1¡ ±H¢ µ

2qH4

¡ cq2H ; ¼BL = ¡cq2L : (8)

On the basis of the above result, independent ventures imply that, for allqL 2 (0; qH); the low-quality …rm would exit , getting thus zero pro…ts.Alternatively, …rm L may produce qL = qH : This immediately entails that±Bi = 1=2 for all i; as …rms o¤er homogeneous goods.

It needs no proof to show that the same holds in the case of a jointventure, as this would yield product homogeneity as a result of technologicaldecisions taken at t = 0: We have thus proved the following:

Lemma 1 Under full collusion, the low-quality product enjoys zero demand.As a consequence, …rms will only supply homogeneous goods, with JV Â IVdue to the cost-saving e¤ect.

Corollary 1 Under full collusion in prices, ±Bi = 1=2 for all i; independentlyof …rms’ venture decisions.

As to the Cournot case, notice that, as long as …rms provide di¤erentqualities, we have

xMH =µ

2; xML = 0 (9)

which again entails that the low-quality …rm survives only if qL = qH ; ei-ther because …rms activate a JV , or because …rms develop the same qualityindependently of each other. As a result, we can state the following:

Lemma 2 Under full collusion in quantities, ±Ci = 9=17 for all i; indepen-dently of …rms’ venture decisions.

In summary, independently of the market variable chosen for the su-pergame over t 2 [1;1), the …rms’ venture decisions at t = 0 have nobearings on the stability of collusion, as setting either monopoly prices orquantities induces …rms to play a supergame with homogeneous goods.

7

4 Partial collusionHere, we investigate the bearings of technological choices on cartel stability,under the assumption that …rms may activate partial collusion, i.e., they maycollude at any ®coll 2

³min

n®N ; ®M

o;max

n®N ; ®M

oi; ® = p; x:

4.1 Cournot behaviour

Consider partial collusion at xcoll 2³xM ; xN

´, for a generic quality pair

fqH ; qLg : In the limit, as qL ! qH ; we obtain the description of the JV case.We de…ne the partially collusive output of …rm i as:

xcolli = axNi + (1¡ a)xMi ; a 2 (0; 1) ; (10)

where xMi = xM=2 = µ=4 and8

xCNH =µ (2qH ¡ qL)4qH ¡ qL

; xCNL =µqH

4qH ¡ qL: (11)

The associated Nash equilibrium revenues are:

RCNH =µ2qH (2qH ¡ qL)2(4qH ¡ qL)2

; RCNL =µ2q2HqL

(4qH ¡ qL)2: (12)

Substituting (11) into (10) and rearranging, we have:

xcollH =µ [4qH(1 + a)¡ qL(1 + 3a)]

4 (4qH ¡ qL); xcollL =

µ [4qH ¡ qL(1¡ a)]4 (4qH ¡ qL)

(13)

which allow to calculate RCcolli :

RCcollH =µ2[4q2H(3¡ a)¡ qHqL(7¡ 3a) + q2L(1¡ a)] [4qH(1¡ a)¡ qL(1 + 3a)]

16 (4qH ¡ qL)2

RCcollL =µ2qL [2qH (2¡ a)¡ qL(1 + a)] [4qH ¡ qL(1¡ a)]

8 (4qH ¡ qL)2(14)

8We omit the explicit derivation of the Nash equilibrium quantities, as it is well knownfrom previous literature (see Motta, 1993).

8

The deviation from xcolli remains to be described. The best reply of …rmj to xcolli is given by:

9

xDCH =µ

h(4qH ¡ qL)2 ¡ aq2L

i

8qH (4qH ¡ qL); xDCH =

4µqH(3¡ a)¡ 3µqL(1¡ a)8 (4qH ¡ qL)

(15)

yielding deviation revenues:

RDCH =µ2

h(4qH ¡ qL)2 ¡ aq2L

i2

64qH (4qH ¡ qL)2; RDCL =

µ2qL

h4qH(3¡ a)¡ 3µqL(1¡ a)

i2

64 (4qH ¡ qL)2:

(16)We are now able to write the expressions for the critical threshold of thediscount factors:

±CH =(1¡ a) (2qH ¡ qL)2 (4qH ¡ qL)2q2L [32q

2H ¡ 16qHqL + q2L(1¡ a)]

; (17)

±CL =(1¡ a) (4qH ¡ qL)2

(4qH ¡ 3qL) [4qH(5¡ a) ¡ 3qL(1¡ a)]: (18)

Notice that the above critical thresholds are independent of µ; and can beplotted over the space fa; qLg ; after setting qH = 1.10 This is done in …gures1 and 2.

9Both xDH and xDL are admissible for all a 2 (0; 1] and qL 2 (0; qH ]: As usual, deviation

against a collusive output never drives the cheated …rm out of business, and never makesthe deviator a monopolist.

10Note that this normalisation involves no loss of generality, since the same plots wouldobtain by rewriting ±Ci in terms of the quality ratio qL=qH 2 (0; 1]:

9

0

0.2

0.4

0.6

0.8

1

a

0

0.2

0.4

0.6

0.8

1

qL

0

0.2

0.4 dH

0

0.2

0.4

0.6

0.8

1

a

Figure 1. Plot of ±CH over fa; qLg , with a 2 [0; 1] and qL 2 [0; 1] .

0

0.2

0.4

0.6

0.8

1

a

0

0.2

0.4

0.6

0.8

1

qL

0

0.2

0.4

dL

0

0.2

0.4

0.6

0.8

1

a

Figure 2. Plot of ±CL over fa; qLg , with a 2 [0; 1] and qL 2 [0; 1] .Observe …gure 1. The range of ±CH is truncated at 9/17 to put into evi-

dence the parameter region wherein independent ventures make it easier for

10

the high-quality …rm to sustain quantity collusion, as compared to a jointventure. The equation of the border at which ±CH = 9=17 is:

ba =2q4L ¡ 15q3L + 149q2L ¡ 408qL + 2722q4L ¡ 51q3L + 221q2L ¡ 408qL + 272

: (19)

All combinations of fa; qLg de…ning a point along the downward slopingsurface in …gure 1, de…ne levels of partial collusion and low quality such thatindependent ventures favour collusion as compared to a joint venture. Theopposite holds for any point such that

a 2Ã0 ;

2q4L ¡ 15q3L + 149q2L ¡ 408qL + 2722q4L ¡ 51q3L + 221q2L ¡ 408qL + 272

!: (20)

Consider now …gure 2. For any combination of a and qL in the admissiblerange, ±CL · 9=17; holding as an equality at fa = 0; qL = qHg :11

The foregoing analysis allows us to state the following:

Proposition 1 For all a 2 (ba; 1] ; implicit collusion is more easily sustainedunder independent ventures than under a joint venture. For all a 2 [0; ba) ;the opposite holds.

This means that, given a generic quality ratio qL=qH ; independent ven-tures are preferable to a joint venture in terms of cartel stability, if …rms col-lude not too far above the disagreement point given by the one-shot Cournotequilibrium. The shape of ba shows that, as far as cartel stability is con-cerned, IV tends to become more and more advantageous compared to JVas product di¤erentiation decreases. In the limit, as qL=qH ! 1; IV ensures±Bi < 1=2 for all a 2 (0; 1] :

Alternatively, the above result can be reformulated as follows. As a in-creases (that is, as the level of collusion weakens towards the Cournot-Nashoutput), the range of qL=qH wherein IV ensures ±

Bi < 1=2 increases. The

intuition is that, if collusion is only slightly above the Nash equilibrium prof-its, than deviation is scarcely pro…table and this drastically contributes tostabilise implicit collusion.

11Notice that, in both plots, ±Ci becomes negative if a is su¢ciently large and qL=qH issu¢ciently low, due to the fact that deviation pro…ts become lower than collusive pro…ts.In such a case, it can be assumed ±Ci = 0; so that any ±i ¸ 0 ensures that the low-quality…rm does not cheat. Clearly, this has no particular bearings on our analysis.

11

4.2 Bertrand behaviour

Turn now to the case where …rms are price-setters and try to collude atpcoll 2

³pN ; pM

´, for a generic quality pair fqH ; qLg : Again, in the limit, as

qL ! qH ; we obtain the picture of the JV case.De…ne the partially collusive price of …rm i as:

pcolli = apNi + (1¡ a)pMi ; a 2 (0; 1) ; (21)

where pMi = µqi=2 and12

pNH =2µqH (qH ¡ qL)4qH ¡ qL

; pNL =µqL (qH ¡ qL)4qH ¡ qL

: (22)

The associated Nash equilibrium revenues are:

RBNH =µ2qH (2qH ¡ qL)2(4qH ¡ qL)2

; RBNL =µ2q2HqL

(4qH ¡ qL)2: (23)

Substituting (22) into (21) and rearranging, we have:

pcollH =µqH [4qH ¡ qL(1 + 3a)]

2 (4qH ¡ qL); pcollL =

µqL [2qH (2¡ a)¡ qL(1 + a)]2 (4qH ¡ qL)

(24)

which allow to calculate RBcolli :

RBcollH =µ2qH [4qH ¡ qL(1¡ a)] [4qH ¡ qL(1 + 3a)]

4 (4qH ¡ qL)2

RBcollL =µ2aqHqL [2qH (2¡ a)¡ qL(1 + a)]

2 (4qH ¡ qL)2(25)

Now consider the deviation from pcolli . The best reply of …rm i againstthe collusive price pcollj is:

pBDH =µ [8q2H ¡ 2qHqL (3 + a) + q2L(1¡ a)]

4 (4qH ¡ qL)pBDL =

µqL [4qH ¡ qL(1 + 3a)]4 (4qH ¡ qL)

(26)

12Again, the explicit derivation of the Nash equilibrium prices is omitted for the sake ofbrevity (see Choi and Shin, 1992; Motta, 1993).

12

The corresponding output levels for the cheating …rm are:

xBDH =µ [8q2H ¡ 2qHqL (3 + a) + q2L(1¡ a)]

4 (4q2H ¡ 5qHqL + q2L)xBDL =

µqH [4qH ¡ qL(1 + 3a)]4 (4q2H ¡ 5qHqL + q2L)

(27)

Notice that deviation outputs (27) are admissible for all values of fa; qH ; qLgsuch that xBDi · µ; which entails the following restrictions, for all positiveµ :

xBDH · µ for allqLqH

2"0;7¡ a¡

pa2 ¡ 22a+ 253 + a

#; (28)

xBDL · µ for allqLqH

2"0;19¡ 3a¡

p9a2 ¡ 114a+ 1698

#: (29)

The admissible range for the quality ratio in (29) is larger than in (28), i.e.,

19¡ 3a¡p9a2 ¡ 114a+ 1698

¸ 7¡ a¡pa2 ¡ 22a+ 253 + a

8 a 2 [0; 1] :(30)

The above inequality entails that, as intuition would suggest, it is easier forthe high-quality than for the low-quality …rm to become a monopolist.

If (29) and (29) are met, then deviation revenues are:

RBDH =µ2[8q2H ¡ 2qHqL (3 + a) + q2L(1¡ a)]2

16 (qH ¡ qL) (4qH ¡ qL)2

RBDL =µ2qHqL [4qH ¡ qL (1 + 3a)]216 (qH ¡ qL) (4qH ¡ qL)2

(31)

Otherwise, the deviator becomes a monopolist. For the moment, we writethe critical threshold of the discount factors by using (31):

±BH =(1¡ a)qL (4qH ¡ qL)2

(2qH ¡ qL) [16q2H ¡ 2qHqL(7 + a) + q2L(1¡ a)]; (32)

±BL =(1¡ a) (4qH ¡ qL)23qL [8qH ¡ qL(5 + 3a)]

: (33)

Again, the above thresholds are independent of µ; and can be plotted overthe space fa; qLg ; after setting qH = 1. This is done in …gures 3 and 4, where

13

the range of both plots is bounded above at 1/2, corresponding to the criticallevel of discounting associated with a joint venture.13

0

0.2

0.4

0.6

0.8

1

a

0

0.2

0.4

0.6

0.8

1

ql

0

0.1

0.2

0.3

0.4

0.5

deltaH

0

0.2

0.4

0.6

0.8

1

a

Figure 3. Plot of ±BH over fa; qLg , with a 2 [0; 1] and qL 2 [0; 1] .13As in the Cournot case, whenever ±Bi < 0 because deviation is unpro…table, the

relevant threshold becomes ±Bi = 0:

14

0

0.2

0.4

0.6

0.8

1

a

0

0.2

0.4

0.6

0.8

1

ql

0

0.1

0.2

0.3

0.4

0.5

deltaL

0

0.2

0.4

0.6

0.8

1

a

Figure 4. Plot of ±BL over fa; qLg , with a 2 [0; 1] and qL 2 [0; 1] .

Consider …rst ±BL (…gure 4). We have:

±BL =1

2(34)

ifqLqH=4

h5¡ 2a§ 3

p2a2 ¡ 1

i

17 + 7a: (35)

The above solutions coincide at a = 1=p2 ' 0:707; where qL ' 0:653:

Then, observe the behaviour of ±BH (…gure 3). The border along which±BH = 1=2 is everywhere to the north-west of the border (35).

Moreover, the curve xBDH = µ is also to the north-west of the border(35).14

14Indeed, the equation

4£5 ¡ 2a + 3

p2a2 ¡ 1

¤

17 + 7a=

7 ¡ a ¡p

a2 ¡ 22a + 253 + a

has no real root for a 2 [0; 1] ; with the r.h.s. being always larger than the l.h.s. over theunit interval.

15

The cases where deviation gives rise to a monopoly remain to be inves-tigated. This would entail recalculating ±Bi anew, taking into account theadditional information conveyed by the complements to (28) and (29). Yet,to the aims of the present paper, the following argument will su¢ce.

First, observe that, in general:

@±Ii@RIDi

=RIcolli ¡RINi(RIDi ¡RINi )2

> 0 : (36)

At the boundary where xBDH = µ; critical discount factors are given by (32)and (33). When

qLqH

2Ã7¡ a¡

pa2 ¡ 22a+ 253 + a

;19¡ 3a¡

p9a2 ¡ 114a+ 1698

!; (37)

the critical discount factor for …rm L is still given by (33), while that associ-ated to …rm H is:

b±B

H =RM ¡RBcollHRM ¡RBNH

> ±BH =RBDH ¡RBcollHRBDH ¡RBNH

>1

2: (38)

Finally, whenqLqH>19¡ 3a¡

p9a2 ¡ 114a+ 1698

; (39)

we haveb±B

L =RM ¡RBcollLRM ¡RBNL

> ±BL =RBDL ¡RBcollLRBDL ¡RBNL

>1

2; (40)

along with (38).The above discussion su¢ces to establish the following result:

Proposition 2 Implicit collusion in prices is more easily sustained underindependent ventures than under a joint venture, for all

qLqH

20@4

h5¡ 2a¡ 3

p2a2 ¡ 1

i

17 + 7a;4

h5¡ 2a+ 3

p2a2 ¡ 1

i

17 + 7a

1A

Outside the above range, the opposite holds.

As the intensity of collusion decreases towards the Bertrand-Nash equi-librium pro…ts, i.e., as a grows larger, the range of product di¤erentiationwherein collusion is easier under IV than under JV increases. The intuitiveexplanation behind this conclusion is the same as in the Cournot case.

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5 Concluding remarksWe have reassessed an issue previously raised by Martin (1995), under a newperspective, where …rms’ initial R&D e¤orts are aimed at product ratherthan process innovation. We have analysed the relationship between theorganizational design of R&D for product innovation and the stability ofimplicit collusion either in quantities or in prices, keeping unaltered the rulesgoverning the market supergame, i.e., using the Perfect Folk Theorem.

The main conclusion emerging from this setting is that a JV may or maynot facilitate collusion in the market supergame, depending upon (i) thedegree of di¤erentiation produced by …rms activating independent ventures;and (ii) the intensity of price or quantity collusion.

Independently of the market variable being set by …rms, we have foundthat, the lower is the level of collusion, the lower is the pro…tability of de-viation for any given degree of product di¤erentiation resulting from inde-pendent ventures. This drastically contributes to stabilise implicit collusion,in that a reduction of deviation pro…ts goes along with a reduction in thecritical threshold of the discount factor.

Therefore, public policies towards R&D behaviour should be designed soas not to become inconsistent with the pro-competitive attitude characteris-ing the current legislation on marketing practices.

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