Research joint ventures, product differentiation, and price collusion

International Journal of Industrial Organization 20 (2002) 829–854 www.elsevier.com / locate / econbase

Research joint ventures, product differentiation, and price collusion Luca Lambertini a , *, Sougata Poddar b , Dan Sasaki c a

Department of Economics, University of Bologna, Strada Maggiore 45, I-40125 Bologna, Italy b Department of Economics, National University of Singapore, 10 Kent Ridge Street, Singapore 119260, Singapore c Department of Economics, University of Exeter, Exeter, Devon EX4 4 PU, UK Received 8 July 1997; received in revised form 1 August 1998; accepted 29 December 1999

Abstract We investigate the interplay between firms’ decisions in product development, either joint or independent, and their ensuing repeated pricing, either collusive or Bertrand–Nash. A joint venture develops a single product, whereas independent ventures develop their respective products which can be differentiated. We prove that joint product development and the resulting lack of horizontal product differentiation may destabilise collusion, whilst firms’ R&D decisions have no bearings on collusive sustainability if differentiation is vertical. We also discover the non-monotone dependence of firms’ venture decisions at the development stage upon their time preferences, as well as upon consumers’ willingness to pay.  2002 Elsevier Science B.V. All rights reserved. JEL classification: D43; L13; O31 Keywords: R&D; Product innovation; Collusive sustainability; Time discount factor; Optimal punishment

1. Introduction Whilst public authorities explicitly prohibit collusive market behaviour, they * Corresponding author. Tel.: 139-051-2092-623; fax: 139-051-2092-664. E-mail address: [email protected] (L. Lambertini). 0167-7187 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 00 )00082-5

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seldom discourage cooperation in R&D activities. On the contrary, there are examples of policy measures meant to stimulate the formation of research joint ventures (RJVs henceforth).1 However, encouraging cooperative R&D and discouraging market collusion can be inconsistent if cooperation in R&D tends to induce collusion in the product market. There is a large literature on the effects of product differentiation on implicit collusion (Deneckere, 1983; Chang, 1991, 1992; Rothschild, 1992; Ross, 1992; ¨ Friedman and Thisse, 1993; Hackner, 1994, 1995, 1996; Lambertini, 1997; Albaek and Lambertini, 1998; Lambertini and Sasaki, 1999; inter alia). There also have been studies dealing with R&D in differentiated markets. Some of them, such as Motta (1992), consider cooperation in the development phase. On the other hand, the effectiveness of RJVs in eliminating effort duplication has been well noted in a large number of contributions (Katz, 1986; d’Aspremont and Jacquemin, 1988, 1990; Kamien et al., 1992; Suzumura, 1992; inter alia). So far, however, few serious attempts have been made to consolidate these two streams of research. Among these few pioneering studies are Martin (1995), and Cabral (1996). Martin analyses the strategic effects of an RJV aimed at achieving a process innovation for an existing product, when the product is marketed by firms engaging in Cournot behaviour. He shows that cooperation in process innovation enhances implicit collusion, which can jeopardise the welfare advantage of eliminating effort duplication through the RJV. His finding has potential implications in the case of product innovation as well. Cabral, on the other hand, proves the possibility that competitive pricing is needed to sustain more efficient R&D agreements. Our effort in this paper broadly follows Martin, except that we take into account the possible effects of product differentiation resulting from the presence or the absence of cooperation in product development, as opposed to Martin’s analysis of process development. In particular, unlike most of the existing literature on repeated games under product differentiation, we explicitly model the effort-saving effects of RJVs, which affect firms’ incentives as well as social welfare. Namely, in an RJV, firms share the costs of product development by jointly developing a single product. An RJV thereby eliminates effort duplication, whilst it offers no product differentiation: all participant firms will have to market the same product. Independent ventures do the opposite: each firm bears the full costs of innovating its product, in return for the possibility of product differentiation. In brief, we investigate the effects of product innovation decisions, i.e. whether to form an RJV or independent ventures, on firms’ ability to sustain implicit price collusion. To pursue this theoretical analysis, there are two modelling alternatives, reflecting two distinct ways to conceptualise product differentiation. One is to parameterise the substitutability (or complementarity) between products. A semi1

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

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nal model due to Bowley (1924) has been used by Singh and Vives (1984) to investigate the choice between price- and quantity-setting behaviour. Aside from algebraic simplicity, a major advantage of this substitutability approach is that it can accommodate both demand substitutes and demand complements. However, there is a limitation: the substitutability parameter is always symmetric between products, so that this type of product differentiation is a public good between firms, namely, a firm’s investment in product differentiation benefits both (all) firms equally. In other words, each firm cannot autonomously choose its product as a point in a product space even when undertaking an independent venture. This modelling technique is suitable in an environment where an RJV leads to standardisation, where products are compatibly interchangeable, as opposed to independent ventures leading to non-standardisation and lack of substitutability, as in Lambertini et al. (1998a).2 To analyse each firm’s product innovation decisions thoroughly, on the other hand, a model with an explicitly defined product space is required. In such an environment, each firm can autonomously choose a product when lodging an independent venture, which makes the firm’s investment in product innovation not entirely a public good across firms. In particular, the product space can be either horizontal or vertical. In this paper we comparatively analyse horizontal and vertical product spaces. We find that independent ventures, when entailing horizontally differentiated products, can facilitate price collusion. When entailing vertical differentiation, on the other hand, firms’ R&D decisions have no effects on the sustainability of price collusion. Also, firms’ decisions between joint and independent ventures can be non-monotone both in the discount factor and in consumers’ willingness to pay, due to the fact that the sustainability of price collusion can be affected by product differentiation. The paper is organised as follows. The general structure of the game is laid out in Section 2. The horizontal differentiation setting is analysed in Section 3. Then, the vertical differentiation model is analysed in Section 4. Section 5 discusses briefly the difference between our findings and existing results in the literature. Finally, Section 6 provides concluding remarks.

2. The model We consider a duopoly with two a priori identical firms playing the following three-stage game. The entire game is embedded in the discrete time structure t 5 0,1,2, . . . . The first two stages take place at t 5 0 . In the first stage, firms 2 In Lambertini et al. (1998a), infinite reversion to one-shot Nash behaviour is used as the punishment in the market supergame. This is known to be inefficient. For this reason, in the present paper we adopt optimal symmetric punishments a` la Abreu (1986).

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choose between independent and joint ventures. An RJV is formed if and only if both firms agree to stay in it; otherwise each firm forms an independent venture. The second stage describes product development. We discuss two separate versions of the model in Sections 3 and 4, considering a horizontal product space and a vertical product space, respectively. In either version: – The two firms jointly develop a single product if they decided on a joint venture in the previous stage. Each firm bears an equal share of the cost of product development. The RJV chooses a product so as to maximise the sum of the two firms’ discounted streams of profits, taking due account of the ensuing price supergame. – Each firm independently chooses a product and develops it if the two firms decided on independent ventures in the first stage. In this case, each firm bears the full development cost of its own product. Note that the noncooperativeness of the firms’ product decisions does not necessarily preclude these decisions being implicitly collusive, by rewarding or penalising particular product profiles through the subsequent pricing actions. The third stage is a Bertrand supergame t 5 1,2, . . . , with the discount factor d [ [0,1) common to both firms. In establishing the critical threshold of the discount factor sustaining price collusion, we follow the optimal punishment strategy as defined by Abreu (1986, 1988). When firms choose a joint venture, there is no product differentiation. In this case, the predictions offered by trigger strategies with one-shot Nash reversion (Friedman, 1971) and by Abreu’s optimal punishment coincide (Lambson, 1987). Namely, for any price collusion to be sustained in the resulting perfect Bertrand market, the discount factor d needs to be ]12 or above. If d , ]12 , there is no prospect of colluding at any prices other than one-shot Bertrand–Nash equilibrium prices. Since we assume constant marginal costs, Bertrand–Nash equilibrium entails zero net profits from the market, which renders costly product development unprofitable, and so only one firm can survive in the market.3 On the other hand, when firms choose independent ventures, their collusive or noncollusive pricing behaviour in the Bertrand supergame can be made contingent upon the product portfolio they have selected. By colluding in prices if and only if a particular product portfolio h1 * , 2 * j has been selected (i.e. firm 1 has chosen product 1 * and firm 2 has chosen 2 * ), firms may be able to sustain the particular product profile as part of a subgame perfect equilibrium. Obviously, there can be countlessly many subgame perfect equilibria of this structure, depending upon

3 The main purpose of this paper is to analyse firms’ strategic venture decisions rather than to discuss entry and exit. All of our qualitative results concerning venture decisions remain intact even if firms were not driven out of the market.

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which product portfolio is to be sustained. We assume that firms select profit efficient equilibria, i.e. those equilibria yielding the highest possible discounted profits (greater details shall be shown in Sections 3 and 4). Hence, even though their product decisions as well as pricing actions are entirely noncooperative, the firms can effectively collude both in product portfolio and in the ensuing Bertrand supergame, as an outcome of a purely noncooperative subgame perfect equilibrium. A general picture of the decision problem facing the two firms is provided by Fig. 1. The discounted streams of net profits for each firm are listed as (JC) , (M) , (IC) , and (IN), with J, I, C, M and N standing for Joint venture, Independent ventures, price Collusion, Monopoly and one-shot Bertrand–Nash equilibrium, respectively, and k i is the development cost of product i . In the picture, the sub-trees for the supergame in the third stage are suppressed and replaced with binary equilibrium outcomes: either collusion or Bertrand–Nash. Note that both collusive profits and Bertrand–Nash profits depend upon the product portfolio. In the case of undifferentiated products, Bertrand–Nash profits are zero, and in calculating collusive profits ] p Ci we assume that the two firms will set the same price to split demand evenly, thereby equalising profits. In the case of independent ventures, firms collude in prices and earn collusive profits p Ci * if and

Fig. 1. The generic decision tree.

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only if collusive portfolio h1 * , 2 * j has been chosen; otherwise they either repeat one-shot Bertrand–Nash equilibrium, earning p Ni with differentiated products, or either of them gets monopoly profits while the other gets zero profits by exiting.4 Let d * denote the critical discount factor sustaining collusion, which may depend upon the given pair of products. In the following two sections we solve this tree backward to identify pure strategy subgame perfect equilibrium (simply ‘equilibrium’ hereinafter unless otherwise specified).

3. The horizontal differentiation setting We adopt the spatial location model due to d’Aspremont et al. (1979). Consumers are uniformly distributed over the unit line segment [0, 1]. In every period, each consumer buys one unit of the product that maximises his net utility: U 5 s 2 d 2i 2 pi where s is gross surplus, pi is the price charged by firm i and d i is the distance between the consumer and firm i. We assume that, if a consumer is indifferent between the two firms’ products, then he randomises his purchase with probability one half from each firm. This implies that, if the two firms locate at the same site and choose the same price, then they split the demand evenly. We also assume s > 5 / 4 ,5 zero marginal production costs, and that firms also locate on the segment [0,1] . The development cost of a product is a positive constant k, independent of the location. Therefore, at t 5 0 , each firm in an RJV pays k / 2, whereas a firm in an independent venture pays k . The game is solved by backward induction in Sections 3.1–3.3.

3.1. Subgame ensuing independent ventures When firms undertake independent ventures, each of them bears the full development cost k . Although the two firms’ location choices are mutually independent and non-cooperative, as mentioned in Section 2, they may still be able 4

Notice that the noncooperative price subgame following a joint venture produces two pure-strategy equilibria and a mixed-strategy one, where firms randomize over staying in and exiting. In the remainder, we confine ourselves to a pure-strategy equilibrium. It can be shown that this involves no loss of generality. 5 This ensures that full market coverage obtains at the noncooperative one-shot equilibrium in prices, if firms locate at 0 and 1.

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to enforce a particular pair of locations using their ensuing pricing actions as a rewarding / punishing device. Namely, in the supergame, firms collude in prices only if they have chosen a prescribed pair of locations. We assume that, when there are more than one pair of locations enforceable by this means, firms select the symmetric location pair entailing the highest joint profits, characterised as follows: 1. The most profitable symmetric profile of locations and prices, as long as d is sufficiently high in order to sustain such a profile through Abreu’s optimal punishment. 2. If d is too low for (1) above, then the most profitable among those symmetric location pairs starting from which collusion at the joint profit maximal (i.e. monopoly) price level is sustainable by optimal punishment given d . 3. If the set of all those symmetric location pairs in (2) is empty (i.e. if d is so low that collusion at the monopoly price is unsustainable starting from any location pair at all), then the Nash equilibrium location pair anticipating one-shot Bertrand–Nash pricing. ] Let d [s] define the critical threshold between cases 1 and 2, and ]d [s] the threshold between cases 2 and 3, respectively. Then: Lemma 1. ] 1 1. When d >d [s] , firms locate at ]14 , ]34 and price at s 2 ] 16 . ] 2. When d] [s] < d ]54 , where the equality 5 holds only when s 5 ]4 . Proof. See Appendix A. It is algebraically straightforward to verify that, whenever d >d] [s] , each firm has no strict incentive to deviate in location and endure the one-shot Nash equilibrium outcome later, as opposed to complying with the prescribed location and colluding later. h The reason why firms never locate inside the quartiles (i.e. a [ ⁄ s ]14 , ]12 g) is as follows. The critical discount factor for sustaining price collusion increases in a as price deviation (undercutting) becomes more profitable when firms locate closer. On the other hand, collusive profits are the same between the case when firms locate at x and 1 2 x , and the case when firms locate at ]12 2 x and ]12 1 x , for any x [f0, ]14 g. Thus, the former collusion is easier to sustain than the latter, with the 6

The exact form of a cannot be analytically obtained. See Appendix A.

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same collusive profit level. Hence, without loss of generality, we can confine our attention to the former class of collusion.7

3.2. Subgame ensuing a joint venture In the case of an RJV, the final stage of the game (t 5 1, 2, 3, . . . ) reduces into a perfect Bertrand supergame, where the critical threshold of the discount factor is 1 / 2. As a consequence, the choice between Bertrand–Nash and collusive pricing depends solely on firms’ time preferences. If d [ [1 / 2, 1), the two firms’ joint collusive profits are maximised when they locate at 1 / 2, entailing the profit ] p Ci 5 (s 2 1 / 4) / 2 per firm per period. Otherwise, if d [ [0, 1 / 2), Bertrand–Nash flow profits are nil irrespective of the location. This induces one firm to exit, yielding monopoly profits p M per-period to the other firm, which is of course obliged to bear the full cost k at t 5 0. Define the discounted net flow of profits accruing to the monopolist as:

S

D

d 1 C M ; ]] ? s 2 ] 2 k 12d 4 3.3. Initial venture decisions From Lemma 1 we have observed that, after independent ventures firms can collude whenever d > d * , whereas after a joint venture they can collude when and only when d > 1 / 2 . Therefore, item 4 of Lemma 1 immediately establishes the following. Proposition 1. The range of time discount factors over which price collusion ensuing a joint venture is sustainable is a proper subset of that where collusion ensuing independent ventures is sustainable. In plain words, a joint venture, when it hinders horizontal product differentiation, makes price collusion harder to sustain, by requiring higher levels of the discount factor. The resulting discounted profits per firm are summarised in Fig. 2, where: 1 ] ] if d [ [d [s], 1) a * (d ) 5 4 ] arghd * 5 dj if d [ [d] [s], d [s]) a

5

The dependence of firms’ innovative venture decisions on the time discount factor 7 ¨ As to horizontally differentiated firms’ locations, Hackner (1995, 1996) obtains similar results, although treating locations as parameters in order to conduct comparative statics on the critical discount factor.

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Fig. 2. Discounted profits per firm, horizontal product space.

d, the gross surplus s and the development cost k is identified by the following proposition, using a three-regime taxonomy based upon the level of time preferences. Proposition 2. 1. d [f0, d] [s]d. In this regime, firms repeat the one-shot Bertrand–Nash equilibrium under both independent and joint venture cases. In the latter case, the market is a monopoly, which obtains if exiting is more profitable than ( IN) , i.e.

d k . ]]] 2s1 2 dd ] 2. d [fd] [s] , d [s]d. In this regime, firms collude in prices only under dependent ventures, not if they undertake a joint venture. In the latter case, market is a monopoly, which obtains if exiting is more profitable than ( IC) ,

S

(1) inthe i.e.

D

d 1 k . ]]] s 2 ] 1 a * (d )s1 2 a * (d )d (2) 2s1 2 dd 4 ] 3. d [fd [s], 1d. In this regime, firms always collude. As a result, the joint d venture is undertaken if and only if (JC) . (IC) , i.e. k . ]] a * (d )s1 2 a * (d )d 1 2 d which, since a * (d ) 5 ]14 in this range of d , can be rewritten as 3d k . ]]] 16(1 2 d )

(3)

Fig. 3 plots the venture cost k against the discount factor d, given s. To the left of

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Fig. 3. Comparative statics on firms’ venture decisions in the parametric plane hd, kj.

d the upward sloping curve along which C M 5 0 (i.e. k 5 ]] ss 2 ]41 d), no entry is 1 2 d a dominant strategy equilibrium. Between the curve C M 5 0 and curves (1)–(2), the market is a monopoly. Overall, independent ventures tend to become increasingly attractive as d grows. However, in regime 2, the condition for independent venture is loosened (inequality (2)) as compared to the adjacent areas (inequality conditions (1) and (3)). The driving force is the fact that when d lies in this regime, firms can sustain collusion if and only if they have chosen independent ventures. Consider now the relation between s and d, given k. Fig. 4 plots the gross

Fig. 4. Comparative statics on firms’ venture decisions in the parametric plane hd, sj.

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Fig. 5. Firms’ indifference boundary between joint and independent ventures, drawn on a cost (k)–benefit (s) plane given d.

surplus s against the discount factor d, given the venture cost k. Once again, independent ventures become more profitable in regime 2 relative to the adjacent 3 19 ] areas. When k . ] 16 , the boundary (3) lies to the right of H. Also, when k , 32 , the 5 boundary (2) intersects with the boundary s 5 ]4 at L to the left of H. Thus, as long 3 19 5 ] ] as ] 16 , k , 32 , firms’ venture decisions are non-monotone in d for any s > 4 . Finally, Fig. 5 plots the venture cost k against the gross surplus s. Here, firms’ indifference boundary between joint and independent ventures shifts up as d increases from 0 to 1 / 2. The horizontal portion to the right of the kinks corresponds to regime 1, and the up-sloping portion to the left of the kinks corresponds to regime 2 (the kinked locus is meant to represent a generic d [ (0, 1 / 2)). The boundary jumps down when d reaches 1 / 2, thereafter shifts up again as d increases further from 1 / 2 to 1 (regime 3). This discontinuity reflects the fact that as d exits regime 2 and enters regime 3, the extra benefit of collusive stability offered by product differentiation becomes no longer relevant. These observations imply the following.8

8 A similar non-monotonicity w.r.t. time preferences (Corollary i) also arises in the non-address model of product differentiation (Lambertini et al., 1998a; see, in particular, Fig. 5, p. 309). However, Corollary ii finds no analogue in the non-address model.

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Corollary i. For any given k, s such that

S

3 1 1 ],k,] s2] 16 2 16

D

firms’ decisions between joint and independent ventures are non-monotone in the discount factor d. Corollary ii. For any given k, d such that

S

d d 1 ]] , k , ]] arg hd [s] 5 dj 2 ] ] 12d 12d 4 s

D 9

firms’ venture decisions are non-monotone in the gross surplus s.

4. The vertical differentiation setting We consider a model of vertical differentiation in the vein of Gabszewicz and Thisse (1979), Shaked and Sutton (1982), inter alia. A unit mass of consumers are uniformly distributed over the interval [0,Q], representing their marginal willingness to pay for quality. Each consumer buys at most one unit of the product that maximises his net utility: U 5 u qi 2 pi , u [ [0, Q] where qi and pi identify the quality and the price of product i. The market is supplied by two single-product firms producing qualities q1 , q2 [ (0, ]q], where ]q is the highest quality level which is technologically feasible. Without loss of generality we assume q1 > q2 throughout this section. Also, if a consumer is indifferent between the two firms’ products, then he randomises his purchase with probability one half from each firm. This implies that, if the two firms’ qualities and prices are identical, then they split the market evenly. We assume 10

d ]]] Qq] . k 1 2 k 2 8(1 2 d )

(4)

and also that the marginal production cost is zero. The cost of product innovation is a constant k 1 if it is the highest quality being 9

In our current model, when d , ]12 there is no clearcut distinction between venture decisions and entry / exit decisions. Corollary ii is written in a way that reflects the fact that our main interest is in venture decisions not in entry / exit decisions. 10 This assumption, that the total surplus Qq] is sufficiently high, is somewhat parallel to the assumption s > ]54 in the horizontal differentiation model (Section 3), even though full market coverage is no longer guaranteed in the vertical differentiation model. See Appendix A for computational details.

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produced. Otherwise, the development cost is k 2 [ (0, k 1 ). In words, there is a unilateral positive externality from the higher-quality technology to the lowerquality technology.11 The game can be solved backwards, similarly to Section 3.

4.1. Subgame ensuing independent ventures We assume equilibrium selection criteria mostly analogous to those in our horizontal differentiation setting (see Section 3.1) except that, by the nature of vertical differentiation, firms’ product portfolio is not ‘symmetric’ unless they produce an identical quality. Our findings in this vertical differentiation setting is qualitatively quite distinct from those in the horizontal product space in several ways. In particular, the second item in our trichotomy (cf. Section 3.1), the intermediate range of d, turns out to be vacuous. Lemma 2. 1. If d > ]12 , both firms develop ]q in the second stage, and collude in prices in the third stage. 2. If d , ]12 , then the unique pure strategy equilibrium (up to the permutation of ] q 5 ]4 ]q in the second stage followed by Bertrand–Nash the two firms) is q1 5q, 2 7 competition in the third stage. Proof. See Appendix A h.

4.2. Subgame ensuing a joint venture An RJV entails a perfect Bertrand supergame (without product differentiation) where the critical threshold of the discount factor is ]12 in order to sustain price collusion. If d [ [0, 1 / 2), the one-shot Bertrand–Nash equilibrium profits are zero irrespective of the firms’ location in the product space as long as their products are undifferentiated. Otherwise, if d [ [1 / 2, 1), the two firms’ joint collusive profits are maximised as follows. Lemma 3. If firms engage in a joint venture anticipating price collusion in the market supergame, then both firms develop ]q in the second stage. Collusion is sustainable iff d > 1 / 2. Proof. See Step 1 of Appendix A. h 11

Lambertini et al. (1998b) contains further interpretative and observational details of this assumption.

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Lemma 3 implies that, for all d [ [0, 1 / 2), the market is a monopoly where the d firm bears] the full development cost k, net discounted profits being ]] ? p M 2 12d d Qq k 5 ]]] 2 k. The other firm exits, obtaining thus zero profits. 4s1 2 dd

4.3. Initial venture decisions Lemmata 2 and 3 imply: Proposition 3. The critical threshold of the discount factor in sustaining price collusion is always d * 5 1 / 2 irrespective of firms’ initial venture decisions. The relevant per period profits when firms adopt independent ventures and compete a` la Bertrand–Nash are p N1 5 7Qq] / 48 and p N2 5 Qq] / 48. If firms undertook a joint venture and then played Bertrand–Nash, their stage profits in the marketing supergame would be zero. Therefore, when d , 1 / 2, the market is a monopoly and Fig. 6 displays the payoffs associated to one of the pure-strategy equilibria. Define the discounted net profits of a monopolist as: ] d Qq VM ; ]] ? ] 2 k 12d 4 Otherwise, if firms collude in prices, their individual per period profit is p C 5 Qq] / i

8, irrespective of their venture decisions. Hence the discounted profits are summarised in Fig. 6. Clearly, if d [ [1 / 2, 1), firms are going to collude anyway, so that the venture decisions have no relevance as to the quality that is going to be marketed. Otherwise, when d [ [0, 1 / 2), we assume that firms choose independent ventures if and only if they fail to agree on a joint venture at ]q, in which case the firm who disagrees switches to a quality strictly lower than ]q.

Fig. 6. Discounted profits per firm, vertical product space.

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Proposition 4. 1. d [ [0,1 / 2). In this regime, firms are unable to collude in prices. Hence, the market is a duopoly only under independent ventures. The relevant condition evaluates the relative profitability of independent venture vs. exiting for the low-quality firm. Exit is preferred, and thus the market is a monopoly, iff ] d Qq ]] ? ] 2 k 2 , 0 (5) 1 2 d 48 2. d [ [1 / 2, 1). In this regime, firms collude in prices regardless of their venture decisions, so that the market is always duopolistic. Therefore, as long as the venture cost is strictly positive, a joint venture always dominates independent ventures. Fig. 7 is a vertical-differentiation analogue of Fig. 3, plotting the minimum independent product development cost k 2 against the discount factor d, given the ] Along VM 5 0, the profit flow of a monopolist is just sufficient gross surplus Qq. to cover the high-quality development cost k 1 , so that above this curve no firm enters the market. Within the range 0 < d , 1 / 2, the two firms are more likely to undertake independent ventures as d grows from 0 towards 1 / 2, according to the inequality condition (5). This reflects the fact that, as firms become increasingly forward looking, the reduction in initial venture costs made possible by an RJV decreases its importance. Once d > 1 / 2, on the other hand, a joint venture is unambiguously more profitable than independent ventures. In brief, as we see from Propositions 3 and 4, there is no interaction between venture decisions and tacit collusion in the case of the vertical product space.

Fig. 7. Comparative statics on firms’ venture decisions in the parametric plane hd, k 2 j.

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5. Discussion relating to literature

5.1. Horizontal product space Connoisseurs may have noticed that the subgame ensuing independent ventures in the horizontal product space is reminiscent of Friedman and Thisse (1993). Our observation in Lemma 1, which also affects Propositions 1 and 2, is nevertheless different from Friedman and Thisse. The reason is as follows. The key difference between their analysis and ours is the timing when collusive behaviour commences. Friedman and Thisse stands on the assumption (or, in other words, equilibrium selection criterion) that, in the marketing supergame, firms collude in prices so as to maximise their joint profits given any location pair they have chosen. Based upon this premise, back in the second stage, each firm locates according to individual incentives. Thereby any location decision is not subject to punishment through pricing actions. In this sense, collusive behaviour does not commence until the third (marketing) stage. In our paper, on the contrary, we focus on such equilibria that, if a firm deviates from the prescribed location in the second stage, then both firms compete by repeating the one-shot Bertrand–Nash equilibrium. This serves as a punishment against the location deviation. In this sense, collusion commences in the second (location) stage onwards. The reason why we consider this class of equilibria is because this can entail a more profit-efficient subgame perfect equilibrium outcome. If we applied a similar analysis to Friedman and Thisse, then our results would be altered accordingly. Firstly, Lemma 1 would be replaced with the following characterisation of the subgame after independent ventures. Lemma 1 * . – When d > ]12 , the two firms’ equilibrium locations coincide at ]12 to raise collusive profits p iC 5 ]21 ss 2 ]41 d per firm per period. – Otherwise, when d , ]21 , they locate at the endpoints of the unit segment and play the one-shot Bertrand–Nash equilibrium at each t 5 1, 2, . . . . It is algebraically straightforward to verify that this result, including the critical discount factor d * 5 ]12 , stands unaffected by the difference in penal codes to sustain price collusion — one-shot Nash reversion in Friedman and Thisse, and Abreu’s optimal punishment in our analysis.12 Also see d’Aspremont et al. (1979) as for the second half of Lemma 1 * . 12

These two penal schemes lead to the same critical discount factor in a Bertrand supergame when products are perfect substitutes (i.e. located at the same point). See Lambertini and Sasaki (1999).

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Consequently, Propositions 1, 2 and Figs. 2 and 3 would be replaced with the following. Proposition 1 * . The range of time discount factors over which the price collusion in the binary equilibrium is sustainable is d [ [1 / 2, 1] irrespective of firms’ venture decisions in the first stage (Fig. 2 * ). Proposition 2 * . 1. d [ [0, 1 / 2). In this regime, firms are unable to collude. Hence, independent d ventures are preferred if and only if (IN) < (M), i.e. if k < ]]]. 2(1 2 d ) 2. d [ [1 / 2, 1). In this regime, firms always collude. As a result, since a joint venture has a cost-saving effect vis a` vis independent ventures, while both choices ensure the same stream of operative (collusive) profits, a joint venture is always preferred. Note that the dependence of firms’ innovative venture decisions on the gross surplus s would disappear if we assumed against collusion in location as Friedman and Thisse does. The border between no entry and monopoly is C M 5 0, as in Fig. 3. It is particularly remarkable that Fig. 3 * is qualitatively equivalent to Fig. 7 in Section 4 which represented the vertical, not the horizontal, product space.

5.2. Vertical product space Analogously to the horizontal case, we can either allow or prohibit collusion in location when the product space is vertical. However, the vertical differentiation game does not entail observationally distinct outcomes between these two forms of collusion.

Fig. 2 * . Discounted profits per firm, horizontal product space.

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Fig. 3 * . Comparative statics on firms’ venture decisions in the parametric plane hd, kj.

The intuitive reason why these two forms of collusion yield observationally distinct outcomes in the horizontal differentiation game is because it is profitable to cover the whole market, thereby it is joint profit efficient for independent ventures to locate far apart from each other so as to cover separate parts of the horizontal segment. In the vertical differentiation game, it is no longer profitable to cover the low end of the consumers’ distribution, so that independent ventures cannot enhance their joint collusive profits analogously by differentiating away from each other.

6. Concluding remarks We have analysed the interplay between R&D and the ensuing product market under horizontal and under vertical differentiation. We have mapped the effects of time discounting, the technology of product development, and consumers’ willingness to pay, on firms’ venture decisions as well as on price behaviour over the entire parameter space. In particular, we have established that the interaction between firms’ R&D decisions and the prospect of collusion in the product market hinges crucially upon the form of collusion — more concretely, whether they are to collude in locations and prices, or in prices only. Insofar as firms are to collude whenever possible, given any product portfolio they have chosen, the set of those discount factors under which collusion is strategically sustainable (d > ]12 in our model; see Section 5) stands entirely unaffected by the firms’ initial choice between joint and independent ventures. This result holds in both horizontal and vertical differentiation settings. On the other hand, if the firms are to collude more efficiently, then their decisions in product innovation may influence their collusive prospects, through the effect that horizontal differentiation can enhance the sustainability of collusion.

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Namely, by choosing that subgame perfect equilibrium which prescribes price collusion only in the particular subgame commencing from that location pair which maximises the discounted sum of the firms’ total profits, the firms can effectively enforce such a profit-efficient location pair as part of the collusive equilibrium path. This enforcement of horizontal differentiation enhances not only the firms’ collusive profits, but also the sustainability of collusion by lowering the critical discount factor. This effect is present only with horizontal differentiation; in the vertical differentiation game, there is no hope in the direction of lowering the critical discount factor by this means. Finally, we have also established that the relationship between product differentiation and the discount factor can indeed be non-monotone. This seemingly counterintuitive result stems from the balance between cost consideration in product development and firms’ ability to sustain future collusion, be there any interactive forces between these two effects or not.

Acknowledgements The first version of this paper was written when we were at the Institute of Economics and Centre for Industrial Economics, University of Copenhagen. We would like to thank Simon Anderson, two anonymous referees, and the audience at the conference Topics in Microeconomics and Game Theory, Copenhagen, June 1997, for useful comments and suggestions. Generous financial assistance from Faculty of Economics and Commerce, University of Melbourne, is also gratefully acknowledged. The usual disclaimer applies.

Appendix A Proof of Lemma 1 Firms 1 and 2 locate at a and 1 2 b. Without loss of generality we assume a < 1 2 b. It is straightforward to verify that, insofar as s > 5 / 4, it is always profitable for firms, whether pricing collusively or competitively, to cover the entire market, i.e. all consumers in [0, 1] should prefer buying to not buying. The generic location x of the consumer who is indifferent between the two products is defined by s 2 (x 2 a)2 2 p1 5 s 2 (1 2 b 2 x)2 2 p2 Whenever there is a unique x satisfying this condition, which occurs only if a , 1 2 b, the following demand system obtains:

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p2 2 p1 12b1a y 1 5 ]]] 1 ]]]]; 2 2(1 2 b 2 a)

y2 5 1 2 y1

Otherwise, if there is no such x [ [0, 1], one of the firms will take over the whole market. Hence the complete demand system is:

H H H H

JJ JJ

p2 2 p1 12b1a y 1 5 max 0, min ]]] 1 ]]]], 1 2 2(1 2 b 2 a)

(A.1)

p1 2 p2 12a1b y 1 5 max 0, min ]]] 1 ]]]], 1 2 2(1 2 a 2 b)

(A.2)

The two firms are to choose that subgame perfect equilibrium which yields the highest joint discounted profits. The most profitable outcome consists of a location pair a C , 1 2 b C and the price pair p 1C , p 2C . The subgame perfect equilibrium sustaining this outcome, when d is sufficiently high, is as follows. – The two firms collude at p C1 , p C2 in the marketing supergame if and only if a 5 a C , b 5 b C has been selected in the second stage. Otherwise, if either a ± a C or b ± b C has been detected, then they simply repeat the one-shot Bertrand–Nash equilibrium resulting from the location pair a, 1 2 b. – Once the equilibrium location decisions a 5 a C , b 5 b C have been observed, then the firms play p C1 , p C2 until either firm deviates. Once a deviation is detected, then Abreu’s optimal punishment is invoked. The next step is to examine the sustainability of such collusion. From the symmetric structure of the game, it is apparent that the most profitable subgame perfect equilibrium must be a symmetric profile. When Abreu’s optimal punishment is considered, finding the optimal punishment price p p as well as the critical threshold of the discount factor d * involves solving the following system of simultaneous equations:

p di ( p C ) 2 p iC 5 d * (p iC 2 pi ( p p))

(A.3)

p di ( p p) 2 pi ( p p) 5 d * (p Ci 2 pi ( p p))

(A.4)

where p C is the collusive profit per firm, per period, p C is the collusive price, and p di ( p2i ) is the profit of firm i from playing the one-shot best response when the ¨ other firm is pricing at p2i . As in Chang (1991, 1992), Ross (1992) and Hackner (1995, 1996), we define the collusive profile in terms of a generic pair of symmetric locations a C and 1 2 a C and solve the system (A.3)–(A.4) by plugging a 5 b 5 a C into (A.1)–(A.2), to obtain p p and d * . First consider item 1 of Lemma 1. The most profitable outcome (whether it is an equilibrium outcome or not) in marketing is:

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1 a 5 b 5 ], 4

1 p1 5 p2 5 s 2 ] 16

849

(A.5)

as Bonanno (1987) proves. In Section 3.1 we define the sustainability condition for ] ] this price collusion as d >d [s]. It can be verified from (A.3)–(A.4) that d [s] ] increases in the total surplus s. In particular, d [s]↑ ]12 as s↑`. On the other hand, 25 when a C [f0, ]14 g and ]54 < s < ] 16 , the quantity sold by the deviator from the collusive price is not bound by the upperlimit ( 5 unity), hence the solution to (A.3)–(A.4) is: (4s 1 8a C 2 5)2 p p 5 2 ]]]]] ; 64(1 2 2a C )

(4s 1 8a C 2 5)2 d * 5 ]]]]] (4s 2 8a C 1 3)2

(A.6)

with the deviation output: 3 2 4((a C )2 1 a C 2 s) y di ( p p) 5 ]]]]]] (A.7) 16(1 2 2a C ) ] ] 5 4s 2 3 2 1 ] ] Letting a C 5 ]14 in (A.6) we obtain d [s] 5s ]] 4s 1 1 d . Especially, df 4 g 5 9 . Now proceed to item 2 of in Lemma 1. From (A.1)–(A.2) and (A.3)–(A.4), d * strictly increases in a C [f0, ]41 g given s. Hence d] [s] is the critical discount factor ] when a C 5 0. This directly implies that, when d] [s] < d ,d [s], firms locate at a C and 1 2 a C so that d * 5 d in (A.6). This a C increases in d. In particular, firms choose a C 5 0 when d 5d] [s]. Plugging a C 5 0 into (A.6), it can be verified that 13 d] [s] 5s4s 2 5 / 4s 1 3d 2 . The deviation output y di ( p p) , 1 for all s , ] 4 , with d p 5 13 d]f ]4 g 5 0. For all s > ] 4 , the deviation output is the entire market, y i ( p ) 5 1. When d ,d] [s], price collusion at maximum differentiation is unsustainable by means of Abreu’s optimal punishment. Hence item 3 of Lemma 1 comes in effect. When firms are unable to collude, they play the well known two-stage subgame perfect equilibrium yielding maximum differentiation (d’Aspremont et al., 1979). Hence, firms choose to collude whenever d >d] [s]. In summary, the foregoing discussion establishes that price collusion ensuing independent ventures becomes easier to sustain when (i) product differentiation is large, and (ii) gross individual surplus s is low. Finally, if we considered the possibility for the firms to collude at a price level lower than the joint profit maximal (i.e. monopoly) level (sometimes referred to as ‘partial price collusion’), the following would emerge. It is obvious that item 1 of ] Lemma 1, when d [s] < d , 1, would stand unaffected by this modification. Also,slightly lengthier algebra could show that locating sufficiently far apart for monopoly collusion turns out always more profitable than locating closer with partial price collusion. Hence, insofar as we accept that firms choose the most profitable collusion, item 2 of Lemma 1 would also stay unaffected when ] d] [s] < d ,d [s]. Partial price collusion would arise only over the low discount ] factor range 0 < d ,d [s], whereby affecting the firms’ price decisions in item 3 of

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Fig. 8. Endogenous horizontal differentiation (independent ventures).

Lemma 1 (location decisions illustrated in Fig. 8 would not be affected). From the comparative statics illustrated in Figs. 3–5, this alteration would affect the first regime in Proposition 2 as well as Corollary ii (which depends upon the distinction between regimes 1 and 2). However, the more substantive part of our findings, i.e. the latter two regimes in Proposition 2 and Corollary i (which reflects the distinction between regimes 2 and 3, not between regimes 1 and 2) would not be qualitatively altered. Proof of Lemma 2 In each period of the market supergame, the demand functions obtain as follows. When q1 . q2 (which can occur only when the two firms develop their products independently), we identify the marginal willingness to pay of the consumer who is indifferent between buying the high-quality good and buying the low-quality good, denoted by h, and that between buying the low-quality good and not buying at all, denoted by l, as: p1 2 p2 h 5 ]]; q1 2 q2

p2 l5] q2

(A.8)

Hence, the demand functions are unravelled as follows: Q 2 minhh, Qj y 1 5 ]]]]; Q

minhh, Qj 2 minhl, Qj y 2 5 ]]]]]] Q

(A.9)

On the other hand, if the two firms produce an identical quality q1 5 q2 , if p1 ± p2 then whichever firm charging the higher price sells nil; otherwise if p1 5 p2 they split the market evenly, attracting

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S

H JD

p1 1 y 1 5 y 2 5 ] Q 2 min ], Q 2Q q1

851

(A.10)

customers each. Under the assumption that unit variable cost of production is nil, the per period profit of firm i is pi 5 pi y i . The first half of Lemma 2 can be proven through the following three steps. Step 1. When q1 5 q2 , the joint profit of the two firms is

S

minh p1 , p2 j minh p1 , p2 j p1 1 p2 5 ]]]] Q 2 ]]]] Q q1

D

(A.11)

minh p1 , p2 j per period, as long as ]]]] < Q (see (A.10)). The first-order condition: q1 ≠(p1 1 p2 ) ]]]] 5 0 ≠ minh p1 , p2 j Qq1 is satisfied at minh p1 , p2 j 5 ]], with ]which the joint profit (A.11) increases in 2 Qq ] minh p , p j 5 ] q1 . Hence q1 5 q2 5q, is joint profit maximal.] We assume that 1 2 2 Qq colluding firms split the demand evenly by setting p1 5 p2 5 ]. 2 Step 2. We now need to prove that there is no equilibrium with q1 . q2 . By (A.8), when q1 . q2 each firm attracts a strictly positive demand iff p p1 2 p2 ]2 , ]] ,Q q2 q1 2 q2

(A.12)

Insofar as these conditions are satisfied, by demand functions (A.9), per period profit functions are

S

D

p1 p1 2 p2 p1 5 ] Q 2 ]] , Q q1 2 q2

S

p2 p1 2 p2 p2 p2 5 ] ]] 2 ] Q q1 2 q2 q2

D

(A.13)

Within the range (A.12), the system of first-order conditions: ≠ ≠ ] (p1 1 p2 ) 5 ] (p1 1 p2 ) 5 0 ≠p1 ≠p2 has no interior solution. The only solution is the limiting solution on the boundary of the range (A.12): Qq1 p1 → ]], 2

Qq2 p2 → ]] 2

which implies that the lower quality attracts zero demand. It is also straightforward

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852

to verify that the symmetric joint profit (A.11) is no less than the supremum of the sum of (A.13) over the range (A.12). Step 3. Finally, we need to ascertain that a firm does not have a strict incentive to deviate from ]q to a lower quality without expecting any positive demand. When both firms produce ]q, each of them earns the discounted net profit:

d 2 k 1 1 ]]] Qq] 8(1 2 d )

(A.14)

If a firm deviates to a lower quality ]q 2 ´, the firm’s net discounted profit becomes simply 2 k 2 , which is strictly lower than (A.14) under the assumption (4). Hence, the deviation is unprofitable. Note also that, whenever q1 5 q2 (whether they are at ]q or not) collusion is sustainable if and only if d > ]12 . This, in conjunction with above step 2, implies that there is no range of d where firms resort to partial collusion in qualities in the case of vertical differentiation, as they did in item 2 in Lemma 1 (see Section 3.1) in the horizontal differentiation case. This completes the proof of the first half of Lemma 2. On the other hand, the second half of the lemma can be proven using in part the following Lemma 2-i. Lemma 2-i. If firms undertake independent ventures and anticipate Bertrand– Nash competition in marketing, then any pure-strategy equilibrium must have q2 5 ]47 q1 in the second stage. Proof. The profit functions at the first stage are (cf. Choi and Shin, 1992): 4Qq 21 (q1 2 q2 ) p1 5 ]]]] ; (4q1 2 q2 )2

Qq1 q2 (q1 2 q2 ) p2 5 ]]]]] (4q1 2 q2 )2

It can be immediately verified that, as ≠p2 / ≠q2 5 0 if q2 5 4q1 / 7, the solution to the leader’s problem defined as:

SU

D

4 max p1 q1 q2 5 ] q1 q1 7 ] is q1 5q, i.e. it coincides with the Nash best reply identified by Choi and Shin. h The remainder of the proof of the second half of Lemma 2 is to ascertain that, given q2 5 ]47 ]q, firm 1 does not have a strict incentive to deviate from q1 5q] to 16 ] q1 5 ] 49 q , in the latter case firms indeed switch labels since we always refer to the higher quality firm as ‘firm 1’. ] If q1 5q,

4 7 q2 5 ] ]q , then p1 5 ] Qq] 7 48

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4 If q1 5 ] ]q, 7

853

16 1 ] q2 5 ] ]q, then p2 5 ] Qq 49 84

Hence, the condition for no deviation is:

d 7 d 1 ]] ? ] Qq] 2 k 1 > ]] ? ] Qq] 2 k 2 1 2 d 48 1 2 d 84 which simplifies into: 15d ]]]] Qq] > k 1 2 k 2 112(1 2 d ) Obviously, this is always satisfied under assumption (4). This completes the proof of the second half of Lemma 2.

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