Cooperative and Noncooperative R&D in Duopoly with Spillovers and Upstream Input Providers

Cooperative and Noncooperative R&D in Duopoly with Spillovers and Upstream Input Providers LINFENG CHEN ZIJUN LUO RONG ZHU January 11, 2016 Abstract This paper extends the model in d Aspremont and Jacquemin (1988) to include upstream input providers. We show that, for a sufficiently large number of input providers, noncooperative R&D investments are higher vis-à-vis cooperative R&D for any level of spillovers. This contrasts the finding in the literature that cooperative R&D investments are higher with high spillovers. Such result comes as not a surprise in our model because unit cost of production increases as the number of upstream firms goes up. Consequently, both cooperative R&D and noncooperative R&D decline, but the former declines faster than the latter. Keywords: Noncooperative R&D; Input providers; Duopoly JEL Classification Numbers: D2; L5; O3 School of Economics and Finance, University of Hong Kong. Email: mylinfeng@gmail.com. Department of Economics and International Business, Sam Houston State University. Contact information: 237C Smith-Hutson Business Building, Sam Houston State University, Huntsville, TX 77341-2118. Email: luozijun@shsu.edu. National Institute of Labour Studies, Flinders University, Australia. Email: rong.zhu@flinders.edu.au.

Cooperative and Noncooperative R&D in Duopoly with Spillovers and Upstream Input Providers Abstract This paper extends the model in d Aspremont and Jacquemin (1988) to include upstream input providers. We show that, for a sufficiently large number of input providers, noncooperative R&D investments are higher vis-à-vis cooperative R&D for any level of spillovers. This contrasts the finding in the literature that cooperative R&D investments are higher with high spillovers. Such result comes as not a surprise in our model because unit cost of production increases as the number of upstream firms goes up. Consequently, both cooperative R&D and noncooperative R&D decline, but the former declines faster than the latter. Keywords: Noncooperative R&D; Input providers; Duopoly JEL Classification Numbers: D2; L5; O3

1 Introduction Research and Development (R&D) is one of the most vital and intriguing functions of successful modern enterprises. Among topics concerning R&D, one that interests economists the most is arguably the efficiency of cooperative R&D. By cooperative R&D, we refer to the circumstances in which firms, or divisions within the same firm, coordinate in their decisions on R&D investments and efforts, while their interactions in the product market may or may not be competitive. 1 This contrasts the case of a fully noncooperative game where firms make both R&D and output/pricing decisions independently. In their seminal work, d Aspremont and Jacquemin (1988) 2 provide the framework that enables generations of economists to analyze the outcomes, efficiency, and welfare implications of cooperative R&D versus noncooperative R&D. 3 They provide a rational explanation to cooperative R&D by introducing a symmetrical spillovers between the two firms engaged in competition in their model. They find that cooperative R&D is more efficient in terms of reducing costs and enhancing welfare compared to noncooperative R&D when the spillovers are sufficiently high (higher than 1/2 in their paper). Our paper follows closely the analytical framework in d Aspremont and Jacquemin (1988) but with the addition of upstream input providers. We believe the presence of upstream input providers to be important for at least two reasons. First, in cases where cooperative R&D occurs among firms who are competing in the product market, such as the examples mentioned in footnote 1, shared upstream input providers is often a common characteristic. For example, the well-known 1 The automobile industry and the electronic industry provide some of the best examples of cooperative R&D. For instance, in 2005, BMW, DaimlerChrysler, and General Motors signed an agreement to jointly develop a two-mode hybrid drive system (Bourreau and Doğan, 2010). In 2013, DaimlerChrysler, Ford, and Nissan allied to innovate the fuel-cell systems (Bourreau et al., 2014). In 2005, Sony and Samsung collaborated to innovate LCD panels (Bourreau et al., 2014). Other examples of inter-firm cooperative R&D include Texaco and Chevron, Apple and Dell, Texas Instruments and AMD, and Burlington Northern Santa Fe and Union Pacific (Duso et al., 2014). In addition, different models of the same automaker, ipad versus iphone, to name a few, can be considered examples of intra-firm cooperative R&D. 2 Readers are reminded of the 1990 erratum (d Aspremont and Jacquemin, 1990) and comment (Henriques, 1990) on the original publication. 3 Notable earlier efforts were made by Dasgupta and Stiglitz (1980); Brander and Spencer (1983); Spencer and Brander (1983); Spence (1984), and Katz (1986), to name a few. 2

Taiwanese manufacturer Foxconn supplies parts for BlackBerry, iphone/ipad, Amazon Kindle, Sony Playstation, Microsoft Xbox, and Wii U, some of which are close competitors. Second, for the special case of intra-firm R&D in multi-product firms, 4 it is inevitable to have shared upstream input providers. Our model of R&D cooperation and competition with upstream input providers is thus suited for the analysis of both inter-firm and intra-firm R&D. 5 Similar to d Aspremont and Jacquemin (1988), we consider two duopolists who engage in cost-reducing R&D investments that have symmetrical spillovers, before competing in the product market in a Cournot model. For the upstream input providers, we make three assumptions. First, we hypothesize that competition among these upstream firms are of the Bertrand fashion. In other words, the upstream input providers simultaneously make their pricing decisions. Second, we assume one unit of output produced by the downstream duopolists requires one unit of input from each of the symmetrical upstream input providers. In addition, no vertical spillovers between upstream and downstream firms are possible. 6 Unlike d Aspremont and Jacquemin (1988), whose conclusion is that R&D investments are higher in the cooperative R&D equilibrium as long as the spillovers are sufficiently high, we find that, given any level of spillovers, the noncooperative game results in higher R&D investments and consequently higher output levels when the number of upstream input providers are sufficiently large. The reason is that input costs that need to be paid to the upstream firms restrain the freedom in the output decisions of the downstream firms. In this case, the high output level resulted from cooperative R&D in the original framework is no longer the optimum. As the number of upstream input providers goes up, 7 the marginal revenue from both noncooperative and cooperative games goes down resulting in lower levels of R&D investments and outputs in both cases. However, the latter declines faster than the former, leading to a reversal of the results in d Aspremont and Jacquemin (1988) when sufficiently large number of upstream input providers are present. 4 See Lin and Zhou (2013) for an analysis of R&D in multiproduct firms. 5 In this paper, only homogeneous product is considered. Products produced by the same firm are naturally differentiated although sometimes may be highly substitutable and have only small difference in their mark-ups. 6 Ishii (2004) considers the possibility of vertical spillovers in addition to horizontal spillovers. 7 For example, when firms need to produce more sophisticated products. 3

While our model fills the gap in the literature by considering upstream input providers, it is instructive to review related papers that contributed important insights to the literature, especially those extend and confirm the basic result of d Aspremont and Jacquemin (1988). 8 Kamien et al. (1992) extend the model in a number of ways, but most notably they consider an alternative form of spillovers. In d Aspremont and Jacquemin (1988), spillovers occur after R&D are realized and hence one firm s higher level of R&D reduces the production costs of the other firm. However, Kamien et al. (1992) assume spillovers to occur during R&D so that one firm s higher R&D expenditure lowers the other firm s expenditure of investing in R&D, which ultimately reduces production costs through a concave function of R&D expenditure. While the interpretation is different, they still find that, for sufficiently high spillovers, R&D investments under cooperation exceeds those in the noncooperative equilibrium. Suzumura (1992), on the other hand, extends d Aspremont and Jacquemin (1988) to the case of oligopoly. The most important contribution of Suzumura (1992) is his consideration of the second-best welfare analysis, although the welfare implications are the same qualitatively regardless of the first-best outcome or the second-best outcome. Suzumura (1992) also finds that when there is no spillovers among the oligopolists, levels of R&D investments could exceed social optimum of both the first- and second-best outcomes. Instead of cost-reducing R&D, Motta (1992) considers quality-improving R&D with spillovers and finds that high spillovers still lead to high R&D investments. In Atallah (2005), spillovers are asymmetric, meaning that the spillover from one firm to the other is not the same as the other way around. Atallah (2005) finds that when the average spillover rate is greater than 1/2, R&D investments are higher in the cooperative equilibrium. Cellini and Lambertini (2009) extend the original model in d Aspremont and Jacquemin (1988) to a dynamic model and find that the result that the cooperative game yields higher R&D investments with high spillovers still holds in the 8 d Aspremont and Jacquemin (1988) have generated a mega size of literature. While we only review the ones that are most relevant to the current paper, a number of papers that have made important contributions to the literature on cooperative R&D investment with spillovers are also worth mentioning. See, for example, De Bondt and Veugelers (1991); Choi (1993); Combs (1993); Ziss (1994); Yi (1996); Amir and Wooders (1999); Petit and Tolwinski (1999); Amir (2000); Cabral (2000); Tesoriere (2008); Kesavayuth and Zikos (2012) and Dawid et al. (2013). For empirical evidence, see recent contributions in Cassiman and Veugelers (2002) and Duso et al. (2014). 4

steady state equilibrium. In a recent paper, Bourreau and Verdier (2014) extend the horizon to two-sided markets and find that the threshold level of spillovers that leads to higher cooperative R&D investments is related to the extent of externalities in a two-sided market. The rest of the paper is organized as follows. Section 2 lays out our model as well as solutions to the noncooperative, cooperative, and collusive equilibria. Section 3 compares the results from the aforementioned equilibria. Section 4 computes the first- and second-best welfare outcomes, before ranking equilibrium levels of R&D investments, outputs, and social welfares in all cases considered. The last section concludes. 2 The Model We consider a model with n( 1) symmetrical upstream input providers and two downstream production firms. The two downstream firms are assumed to be identical. They both invest in cost-reducing R&D, before competing in the output market in the Cournot fashion. The timing of the game is as follows. In the first period, the two firms make R&D investment decisions simultaneously. In the second period, all upstream input providers make pricing decisions simultaneously. In the third and final period, the two downstream firms compete in the Cournot model by choosing profit-maximizing output levels. The game is solved backwardly. Formally, the unit cost of downstream firm i (for i = 1, 2) consists of three components: a constant marginal cost of production that is equal to c; a cost-reduction due to R&D and spillover from the other firm that is equal to x i + sx j (j i); and costs of inputs. For the cost-reducing R&D, x i and x j are investments by own and other firms, with s (0, 1) measuring the size of the spillovers. The cost of R&D is assumed to be a quadratic function of the investment level, x i, and is given by 1 2 rx2 i with r > 0. For costs of inputs, we assume one unit of final output needs one unit of input from all upstream providers, so that the cost of inputs per unit of output is n k=1 w k where w k is the price of input from provider k, to be determined in the second period of the game. 5

In summary, unit cost of downstream firm i is equal to n w k + c x i sx j k=1 The inverse market demand is assumed to be a linear function: p = a b (q 1 + q 2 ) ; where q 1 and q 2 are output levels of the two downstream firms and the usual assumptions a > 0 and b > 0 are applied. To ensure positive values for solutions of R&D investments and output levels in all cases, we need the following assumptions throughout the paper: Assumption 1. a c nf > 0. Assumption 2. 2br (1 + s) 2 > 0. In Assumption 1, f is the marginal cost of the upstream input providers. The assumption states that the market, measured by a, needs to be large enough. Assumption 2, on the other hand, will hold true when the cost of R&D (r) is sufficiently high. In this paper, we consider three different scenarios of R&D decision-making, the same as those in d Aspremont and Jacquemin (1988). The first scenario is a full noncooperative game, in which the two downstream firms compete noncooperatively in both outputs and R&D investments. In the second scenario, the two downstream firms cooperate in R&D investment but still compete in the output market noncooperatively. However, we will call this the cooperative game since R&D investment is our main concern. The last scenario is the case of collusion, or symmetrical monopoly, in which the two downstream firms cooperate in both the output market and R&D investment. In addition, the socially optimal levels of R&D investments and outputs are computed and compared against results from the aforementioned scenarios. 6

2.1 Noncooperative Equilibrium In the noncooperative game where the two downstream firms make both R&D and output decisions simultaneously and noncooperatively, profit of production firm i is given by ( n ) π i = [a b (q 1 + q 2 )] q i w k + c x i sx j q i 1 2 rx2 i, (1) k=1 for i = 1, 2 and j i. We can solve for the output level of firm i in the Cournot competition as ( ) q i = 1 n a c w k + (2 s) x i + (2s 1) x j 3b k=1 (2) for i = 1, 2 and j i. In the second period, we solve for the input providers optimal pricing strategies. From equation (2), we derive the demand function for an input provider as [ ( q 1 + q 2 = 1 2 a c 3b ) ] n w k + (1 + s) (x 1 + x 2 ). k=1 Assuming that all input providers face the same constant marginal cost f, the profit of input provider m is [ ( π m = 1 2 a c 3b ) ] n w k + (1 + s) (x 1 + x 2 ) (w m f) k=1 for m = 1, 2,..., n. The price that maximizes π m is therefore w m = 2 (a c + f) + (1 + s) (x 1 + x 2 ). (3) 2 + 2n Plugging w m in equation (3) back to the solution in equation (2), we have q i = 1 3b ( 2 (a c nf) n (1 + s) (x1 + x 2 ) 2 + 2n + (2 s) x i + (2s 1) x j ). 7

In the first period of the noncooperative game, the two firms maximize their own profits by choosing R&D investment levels independently and simultaneously. Let superscript N denote variables of the noncooperative game, in equilibrium, we have x N i = 1 (a c nf) (4 2s + 3n 3ns) A (4a) qi N = 3 (a c nf) (1 + n) r A (4b) where A = 9br (1 + n) 2 (1 + s) (4 2s + 3n 3ns). The following comparative statics can be derived from equation (4a): 9 x N i a > 0; xn i b c r f s n < 0. The above comparative statics fully describe the responses of R&D investments to exogenous factors in the noncooperative equilibrium. It is shown that R&D investments are higher in the noncooperative equilibrium with a bigger market ( a ), lower costs (c, r, and f), lower spillovers b (s), and fewer upstream input providers (n). 9 The comparative statics with respect to s and n are less straightforward. For s, we have where x N i s c nf) ( = (a A 2 (2 + 3n) 9br (1 + n) 2 (4 2 + 3n 3ns) 2), (2 + 3n) 9br (1 + n) 2 (4 2 + 3n 3ns) 2 > (2 + 3n) 9br (1 + n) 2 (2 + 3n) (1 + s) (4 2 + 3n 3ns) > 0. As a result, xn i s < 0. For n, we have x N i n = 1 [(4 2s + 3n 3ns) Af + 9br (1 + n) (3n 3ns s + 5) (a c nf)] < 0. A2 8

2.2 Cooperative Equilibrium In the cooperative game, the two firms maximize joint profit π 1 + π 2 : [ ] n Π = a b(q 1 + q 2 ) w k c (q 1 + q 2 ) + (q 1 + sq 2 )x 1 + (sq 1 + q 2 )x 2 1 2 rx2 1 1 2 rx2 2. (5) k=1 The solutions to the second and third periods of the cooperative game is the same as those of the noncooperative game. Let superscript C denote variables of the cooperative game, in equilibrium, we have x C i = 2 (a c nf)(1 + s) (6a) B qi C = 3 (a c nf)(1 + n)r (6b) B where B = 9br(1 + n) 2 2(1 + s) 2. The following comparative statics can be derived from equation (6a): 10 x N i a > 0; xn i b c r f s > 0; xn i n < 0. The above comparative statics fully describe the responses of R&D investments to exogenous factors in the cooperative equilibrium. It is shown that R&D investments are higher in the noncooperative equilibrium with a bigger market ( a ), lower costs (c, r, and f), higher spillovers b (s), and fewer upstream input providers (n). This set of comparative statics is the same compared to the noncooperative game except for the response to spillovers. 10 The comparative static with respect to n is less straightforward. From equation (6a), we have x C i n = 1 [Bf + 18br(1 + n)(a c nf)] < 0. B2 9

2.3 Collusive Equilibrium Let x M (= x M 1 = x M 2 ), q M (= q M 1 +q M 2 ) and π M (= π M 1 +π M 2 ) denote R&D investment, output, and profit under collusion, 11 the profit maximization problem is given by max π M = (a bq M )q M q M ( n ) w k + c q M + (1 + s)x M q M r(x M ) 2, (7) k=1 with optimal output level in the first period as q M = 1 2b ( a c ) n w k + (1 + s)x M. (8) k=1 In the second period, profit of input provider m is π m = 1 2b ( a c with the solution of input prices given by ) n w k + (1 + s)x M (w m f), k=1 w m = a c + f + (1 + s)xm 1 + n. (9) Plugging equations (8) and (9) back to equation (7), we can solve for the optimal R&D investment level in collusion as x M = 1 (a c nf)(1 + s), (10) D where D = 4br(1 + n) 2 (1 + s) 2 > 0. In addition, we have qi M = qm 2 = 1 (a c nf)(1 + n)r. (11) D 11 We consider the same collusive game, or symmetrical monopoly, as in d Aspremont and Jacquemin (1988). In other words, in equilibrium, x M 1 = x M 2, q M 1 = q M 2 = 1 2 qm, and π M 1 = π M 2 = 1 2 πm. 10

The following comparative statics can be derived from equation (10): 12 x N i a > 0; xn i b c r f s n < 0. The above comparative statics fully describe the responses of R&D investments to exogenous factors in the collusive equilibrium. It is shown that R&D investments are higher in the noncooperative equilibrium with a bigger market ( a ), lower costs (c, r, and f), lower spillovers b (s), and fewer input providers (n). This set of comparative statics is the same as that from the noncooperative game. 3 Comparison of Equilibrium Outcomes In this section, we will compare the equilibrium outcomes of the three scenarios considered above. The focus is on the comparison of R&D investments, which we put into propositions. Comparisons of output levels are concluded as corollaries. 3.1 Noncooperative versus Cooperative The equilibrium R&D investment levels in the noncooperative and cooperative games, from equations (4a) and (6a), allow us to establish the following lemma and proposition: Lemma 1. R&D investments are higher in the cooperative equilibrium than in the noncooperative equilibrium (x C i > x N i ) if, and only if, spillovers are sufficiently high (s > 3n+2 3n+4 ). Proposition 1. R&D investments are higher in the noncooperative equilibrium than in the cooperative equilibrium (x N i > x C i ) if, and only if, the number of upstream input providers is sufficiently large (n > 4s 2 3 3s ). 12 The comparative static with respect to n is less straightforward. From equation (10), we have x M n = (1 + s) D 2 [Df + 8br(1 + n)(a c nf)] < 0. 11

Proof. Subtracting (6a) from (4a) gives x N i x C i = 1 AB 9br(a c nf)(1 + n)2 (4s 2 3n + 3ns), which implies x N i > x C i 4s 2 3n + 3ns < 0. The solution of s to the reversal of the above inequality (x C i > x N i ) proves Lemma 1, while the solution of n to the above inequality proves Proposition 1. Note that 4s 2 3 3s 5 7 s < 1. In other words, n > 4s 2 3 3s is always satisfied when s < 5 7 holds. 1 if and only if Lemma 1 is similar to the result of d Aspremont and Jacquemin (1988), which states that when spillovers are sufficiently large (s > 1 2 in their model), cooperative R&D investments are higher than noncooperative R&D investment. Note that 3n + 2 3n + 4 > 1 2, for any n > 1, which suggests that our model requires higher spillovers for cooperative R&D investment to exceed noncooperative R&D investment. The intuition is that the presence of the input providers makes it less profitable when firms invest more in R&D and also produce more because they need to pay for the costs of upstream inputs. Added to this explanation, Proposition 1 further states that given any size of spillovers, as long as the number of upstream input providers is sufficiently large (n > 4s 2 ), the result in d Aspremont and Jacquemin (1988) is reversed. 3 3s To better understand this result, which runs in contrary to d Aspremont and Jacquemin (1988) and many of the extensions in the literature, we hypothesize that initially 4s 2 3n+3ns = 0. This implies x N i = x C i and q N i = q C i. Suppose now the number of input providers, n, has increased. 13 13 This may happen when firms need to produce more sophisticated products. 12

We have already established in Sections 2.1 and 2.2 that x N i n < 0; xc i n < 0; which indicates that as the number of input providers increases, R&D investment levels in both the noncooperative and cooperative equilibria decrease. However, the rates of decline in the two cases are not the same. From equations (4a) and (6a), we further find that x N i n xc i n = 3 (1 s)(a c nf) (1 + A ) 2(1 + s)2 > 0, (12) A with the inequality sign follows from Assumptions 1 and 2. In addition, the derivation of equation (12) utilizes the condition 4s 2 3n + 3ns = 0, i.e., A = B. According to equation (12), the decline in the cooperative game is faster than that of the noncooperative game with more input providers. In other words, other things being equal, we have x C i n < xn i n < 0. This explains the result in Proposition 1 that x N i > x C i for n > 4s 2 3 3s. In fact, if the production of input is costless, at the limit, R&D investments in the noncooperative equilibrium will be at least as high as R&D investments in the cooperative equilibrium. Formally, we can establish: Corollary 1. When the marginal cost of upstream input providers is zero (f = 0) and the number of input providers approaches infinity (n ), R&D investments in the cooperative equilibrium will never exceed R&D investments in the noncooperative equilibrium (x C i x N i ). Last, comparison of output levels is established in the following corollary: Corollary 2. q N i q C i if and only if x N i x C i. 13

Proof. Subtracting (6b) from (4b) gives qi N qi C = 3 (1 + n)(a c nf)(4s 2 3n + 3ns) AB which implies q N i > q C i 4s 2 3n + 3ns < 0, the same condition as the comparison between x N i and x C i. 3.2 Noncooperative versus Collusion Comparing equations (4a) and (10) gives x M x N i = 1 AD br(a c nf)(1 + n)2 (17s 7 12n + 12ns), (13) which allows us to establish the following lemma and proposition: Lemma 2. R&D investments are higher in collusion than in the noncooperative equilibrium (x M > x N i ) if, and only if, spillovers are sufficiently high (s > 12n+7 12n+17 ). Proposition 2. R&D investments are higher in the noncooperative than in collusion (x N i > x M ) if, and only if, the number of upstream input providers is sufficiently large (n > 17s 7 12 12s ). Proof. From equation (13), we have x M < x N i 17s 7 12n + 12ns < 0. The solution of s to the reversal of the above inequality proves Lemma 2, while the solution of n to the above inequality proves Proposition 2. Note that 17s 7 12 12s is less than 5 4s 2, the critical value for 7 3 3s 1 in Section 3.1. 1 if and only if s 19 29, which Lemma 2 is similar to d Aspremont and Jacquemin (1988). They show that, for sufficiently high spillovers (s > 0.41 in their model), R&D investments are higher in the collusive equilibrium 14

than in the noncooperative equilibrium. However, similar to the result and intuition in Section 3.1, 14 Proposition 2 suggests that when the number of upstream providers is sufficiently large, with any size of spillovers, R&D investments in the noncooperative equilibrium can exceed those in collusion. Comparison of output levels is given in the following corollary: Corollary 3. Output level is always higher in the noncooperative equilibrium than in collusion (q N i > q M i ). Proof. From equations (4b) and (11), we have q N i q M i = (1 + n)(a c nf)r AD [ 3br(1 + n) 2 2(1 + s) 2 + 3(1 + n)(1 n 2 ) ] > 0, which is positive according to Assumptions 1 and 2. This proves Corollary 3. 3.3 Cooperative versus Collusion From equations (6a) and (10), we find x M x C i = 1 BD br(a c nf)(1 + n)2 (1 + s) > 0, which allows us to establish the following lemma that is the same as d Aspremont and Jacquemin (1988): Lemma 3. R&D investments are always higher in collusion than in the cooperative equilibrium (x M x C i ). Additionally, the comparison of output levels is given in the following corollary: 14 For the comparison between xn i n xm and n, we have x N i n xm n br(1 + n)(4 2s + 3n 3ns)(a c nf) = (1 + s)a 2 (24n 24ns 29s + 19) > 0. 15

Corollary 4. Output levels are always higher in the cooperative equilibrium than in collusion (q C i > q M i ). Proof. From equations (6b) and (11), we have q C i q M i = (1 + n)(a c nf) BD ( 3br(1 + n) 2 (1 + s) 2) > 0, which is positive according to Assumptions 1 and 2. This proves Corollary 4. 4 Welfare Analysis In this Section we follow the literature to compare our equilibrium levels of R&D investments and outputs in all three scenarios to the ones that are socially optimum. In other words, we find levels of R&D investments and outputs that maximize social welfare (SW ), which is defined as the sum of consumer surplus (CS) and producer surplus (P S). Such approach constitutes the first-best outcome. In addition, we consider the second-best outcome as proposed by Suzumura (1992). 4.1 Solutions to First- and Second-Best Let x denote the first-best R&D investment level for each firm and q the first-best total output level (q 1 + q 2). In our model, consumer surplus is CS = 1 2 (a p)q = 1 2 [a (a bq )]q = b 2 (q ) 2. (14) Recognizing the fact that once the socially optimal level of R&D is settled in the first period, marginal cost of production is constant in the third period, we have producer surplus to be P S = (p MC)q r(x ) 2 (15) 16

where MC = nf + c (1 + s)x. (16) Combining equations (15) and (16) gives P S = [a bq (nf + c (1 + s)x )] q r(x ) 2 =aq b(q ) 2 (nf + c (1 + s)x )q r(x ) 2. (17) Note that equation (16) postulates the fact that equilibrium of the upstream input providers game is such that they all price at marginal cost. This result could be rationalized by the fact that price equals to marginal cost eliminates dead weight loss in the input market that consists of n upstream input providers. Summing equations (14) and (17) gives SW = aq b 2 (q ) 2 (nf + c (1 + s)x )q r(x ) 2. (18) To solve for the socially optimal output level in the last period of the game, we take the derivative of equation (23) with respect to q, set it to zero, and solve for q taken x as given. The socially optimal q can be computed as q = a c nf + (1 + s)x. (19) To solve for the socially optimal level of R&D (x ), we plug equation (19) back to equation (23), then take the derivative of SW with respect to x and set it to zero. The resulting x is x = 1 (a c nf)(1 + s), (20) E where 15 E = 2br (1 + s) 2. 15 This is the equation that Assumption 2 is based upon. 17

Plugging x back to equation (19) gives each firm s socially optimal output level as qi = 1 2 q = 1 (a c nf)r. (21) E Suzumura (1992) argues that the first-best welfare analysis may not be appropriate in the cases of R&D because the second-stage quantity game lies beyond the regulatory power of the nonomnipotent government (pp.1314). Following his suggestion, we also calculate the second-best R&D investment level, which we denote as x, and the second-best firm output level, which we denote as q i. The second-best R&D and output levels are x = 1 (a c nf)(4 + 6n)(1 + s); F (22a) q i = 3 (a c nf)(1 + n)r; F (22b) where F = 9br(1 + n) 2 (4 + 6n)(1 + s) 2. 4.2 Comparing R&D Investments A comparison among R&D investment levels, from equations (4a), (6a), (10), (20), and (22a), allows us to establish the following proposition: Proposition 3. The complete ranking of all R&D investment levels in our model is: (i) x > x > x M x C i x N i if and only if s 3n+2 for any given n; 3n+4 n 4s 2 for any given s; 3 3s (ii) x > x > x N i x M x C i if and only if s 12n+7 for any given n; 12n+17 n 17s 7 for any given s; 12 12s (iii) x > x > x M > x N i > x C i if and only if 18

12n+7 < s < 3n+2 for any given n; 12n+17 3n+4 4s 2 < n < 17s 7 for any given s; 3 3s 12 12s Proof. The conditions for comparisons among x N i, x C i, and x M are followed from Lemmas 1-3, and Propositions 1 and 2. In addition, from equations (4a), (6a), (10), (20), and (22a), we get x x = br EF (a c nf)(1 + 3n)2 (1 + s) > 0; x x N i x x C i = 27br AF (a c nf)(1 + n)2 (n + 2s + 3ns) > 0; = 18br BF (a c nf)(1 + n)2 (1 + 3n)(1 + s) > 0; x x M = br DF (a c nf)(1 + n)2 (7 + 24n)(1 + s) > 0. These together prove Proposition 3. We draw two important conclusions from Proposition 3. First, when either spillovers are sufficiently high (s 3n+2 4s 2 ) or the number of input providers is sufficiently small (n ), 3n+4 3 3s that is, in cases (i), more cooperations and coordination between the two downstream production firms leads to higher levels of R&D investments, as x > x > x M x C i x N i is the resulted ranking. Second, although noncooperative R&D investment level does not exceed that of social optimum, it overshoots the cases of both cooperative and collusive equilibria when either spillovers are sufficiently low (s 12n+7 17s 7 ) or the number of input providers is sufficiently large (n ). 12n+17 12 12s The result arisen from sufficiently low spillovers is similar to Suzumura (1992). 4.3 Comparing Output Levels Comparison among output levels, from equations (4b), (6b), (11), (21), and (22b), leads to the following proposition: Proposition 4. The complete ranking of output levels in our model is: (i) q i > q i > q C i q N i > q M i if and only if s 3n+2 for any given n; 3n+4 19

n 4s 2 for any given s; 3 3s (ii) q i > q i > q N i > q C i > q M i if and only if s < 3n+2 for any given n; 3n+4 n > 4s 2 for any given s. 3 3s Proof. The conditions for comparisons among q N i, q C i, and q M are followed from Corollaries 2-4, and Propositions 1 and 2. In addition, from equations (4b), (6b), (11), (21), and (22b), we get q i q i = q i q N i q i q C i r EF (a c nf)(1 + 3n) [ 3br(1 + n) (1 + s) 2] > 0; = 9r (a c nf)(1 + n)(1 + s)(n + 2s + 3ns) > 0; BF = 6r AF (a c nf)(1 + n)(1 + 3n)(1 + s)2 > 0; q i q M i = r DF (a c nf)(1 + n) [ (3 + 12s)br(1 + n) 2 + (1 + 6n 3s)(1 + s) 2] > 0. These together prove Proposition 4. The ranking of output levels is similar to the ranking of R&D investments with one exception: output levels in the collusive equilibrium are always the lowest among all cases. This is to be expected because firms work as a joint/symmetrical monopoly in collusion. With higher market power, they produce lower combined levels of outputs. 4.4 Comparing Social Welfares Finally, we compute values of social welfares as 16 SW (4 + 6n) = (a c nf) 2 r, F (23a) SW N = [18br(1 + n)2 (2 + 3n) (4 2s + 3n 3ns) 2 ] (a c nf) 2 r, A 2 (23b) SW C = [18br(1 + n)2 (2 + 3n) 4(1 + s) 2 ] B 2 (a c nf) 2 r, (23c) SW M = [2br(1 + n)2 (3 + 4n) (1 + s) 2 ] D 2 (a c nf) 2 r. (23d) 16 Note that we have ignored SW because by definition SW is the maximum possible welfare. 20

A comparison among these values allow us to establish the following proposition: Proposition 5. The complete ranking of social welfares in our model is: (i) SW > SW > SW C SW N > SW M if and only if s 3n+2 for any given n; 3n+4 n 4s 2 for any given s; 3 3s (ii) SW > SW > SW N > SW C > SW M if and only if s < 3n+2 for any given n; 3n+4 n > 4s 2 for any given s. 3 3s Proof. First, we note that SW N = SW C if and only if 4 2s + 3n 3ns = 2(1 + s). In other words, SW N SW C if and only if x N i x C i or q N i q C i. Second, taking the difference between SW C and SW M gives SW C SW M = (1 + n)2 (a c nf) 2 br 2 B 2 D ( 2 2(12n + 5) [ (1 + s) 2 3br(1 + n) 2] 2 [ + (1 + n)(1 + s) 2 5br(1 + n) 2(1 + s) 2]) > 0, where the inequality sign is derived according to Assumption 2. Third, proof of SW N > SW M is presented in the Appendix. Lastly, we find SW SW N = 1 A 2 F SW SW C = 1 B 2 F [ 81(1 + n) 2 (2s + n + 3ns) 2 (a c nf) 2 br 2] > 0; [ 36(1 + n) 2 (1 + 3n) 2 (1 + s) 2 (a c nf) 2 br 2] > 0. These together prove Proposition 5. Taken together, Propositions 3, 4, and 5 show that, for the three cases of R&D considered in this paper, both R&D investments and output levels are under-supplied compared to social optimum of either first- or second-best. Consequently, the social welfares from these three cases are below the social optimal levels. The comparison among results of second-best, noncooperative 21

game, and cooperative game is interesting. Our calculations indicate that full competition, as in the case of the noncooperative game, does not automatically lead to social optimum. On the other hand, even the cooperative game in which firms coordinate in their R&D decisions is sub-optimal compared to the second-best outcome. Keep in mind that the second-best outcome has already recognized the fact that the last period of the game will be Cournot competition, as proposed by Suzumura (1992). Together, the welfare conclusions from these propositions indicate that stronger government intervention and coordination efforts in R&D investments may be beneficial. 5 Concluding Remarks Since the seminal work of d Aspremont and Jacquemin (1988), many extensions have shown that with sufficiently high spillovers, R&D investments in the cooperative equilibrium are higher than those in the noncooperative equilibrium. With the addition of upstream input providers, we demonstrate in this paper that the aforementioned result can reverse, so long as the number of upstream input providers is sufficiently large. We show that the presence of the input providers has changed the decision making of the downstream firms. Given any level of spillovers, as the number of input providers goes up, the marginal revenue from more R&D investment and more output decreases at a faster rate in the cooperative game than in the noncooperative game. As a result, when the number of input providers is sufficiently high, firms invest more in R&D in the noncooperative equilibrium compared to the cooperative equilibrium. Consequently, the downstream firms also produce more output in the noncooperative equilibrium. We believe the contribution in this paper is important because as manufactured goods become more sophisticated, more components and/or steps are needed to finish a product. More sophisticated products will call for more upstream input providers. The connection between R&D spillovers and upstream input providers is also natural. When a new technology is being introduced as the outcome of cost-reducing R&D, a firm may look for new input providers who have the ability to handle the new technology, or it may teach its current input providers the new procedure. Both 22

cases cause spillovers horizontally among downstream production firms, as an input provider can serve multiple downstream production firms. Since our model builds on the framework of d Aspremont and Jacquemin (1988), it can potentially be applied to extensions of d Aspremont and Jacquemin (1988) considered in the literature. We believe the results derived in the current paper will continue to hold because the logic in the decision-makings of the downstream firms will continue to be valid. References Amir, R. (2000). Modelling Imperfectly Appropriable R&D via Spillovers. International Journal of Industrial Organization 18, 1013 1032. Amir, R. and J. Wooders (1999). Effects of One-Way Spillovers on Market Shares, Industry Price, Welfare, and R&D Cooperation. Journal of Economics & Management Strategy 8, 223 249. Atallah, G. (2005). R&D Cooperation with Asymmetric Spillovers. Canadian Journal of Economics 38, 919 936. Bourreau, M. and P. Doğan (2010). Cooperation in Product Development and Process R&D between Competitors. International Journal of Industrial Organization 28, 176 190. Bourreau, M., P. Doğan, and M. Manant (2014, July). Size of RJVs with partial cooperation in product development. Institute for Advanced Study, School of Social Science, Economics Working Papers. Bourreau, M. and M. Verdier (2014). Cooperative and Noncooperative R&D in Two-Sided Markets. Review of Network Economics 13, 175 190. Brander, J. A. and B. J. Spencer (1983). Strategic Commitment with R&D: The Symmetric Case. Bell Journal of Economics 14, 225 235. Cabral, L. M. B. (2000). R&D Cooperation and Product Market Competition. International Journal of Industrial Organization 18, 1300 1047. Cassiman, B. and R. Veugelers (2002). R&D Cooperation and Spillovers: Some Empirical Evidence from Belgium. American Economic Review 92, 1169 1184. Cellini, R. and L. Lambertini (2009). Dynamic R&D with Spillovers: Competition vs Cooperation. Journal of Economic Dynamics & Control 33, 568 582. Choi, J. P. (1993). Cooperative R&D with Product Market Competition. International Journal of Industrial Organization 11, 553 571. 23

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Spencer, B. J. and J. A. Brander (1983). International R&D Rivalry and Industrial Strategy. Review of Economic Studies 50, 707 722. Suzumura, K. (1992). Cooperative and Noncooperative R&D in an Oligopoly with Spillovers. American Economic Review 82, 1307 1320. Tesoriere, A. (2008). Endogenous R&D Symmetry in Linear Duopoly with One-way Spillovers. Journal of Economic Behavior & Organization 66, 213 225. Yi, S.-S. (1996). The Welfare Effects of Cooperative R&D in Oligopoly with Spillovers. Review of Industrial Organization 11, 681 698. Ziss, S. (1994). Strategic R&D with Spillovers, Collusion and Welfare. Journal of Industrial Economics 42, 375 393. Appendix This appendix provides detailed proof of the result that SW N SW M > 0 in Proposition 5. From equations (23b) and (23d), we find where SW N SW M = (1 + n)2 (a c nf) 2 br 2 A 2 D 2 (Z 1 + Z 2 + Z 3 + Z 4 + Z 5 ), Z 1 =954bnr ( bn 3 r s 2) + 288bn 4 r(2s + 1)(1 s), Z 2 =90b 2 r 2 + 216b 2 n 5 r 2 br ( 31 58s + 487s 2) + 4 ( 20s 2 + s 1 ) (1 + s) 2, Z 3 =n ( 2 1404b 2 r 2 6 (1 s) (29 13s) (s + 1) 2 br ( 3535s 2 346s 425 )), Z 4 =12n ( 3 138b 2 r 2 6 (1 + s) 2 (s + 1) 2 br ( 199s 2 48s 55 )), Z 5 =2n ( 288b 2 r 2 br ( 628s 2 58s 11 ) + 2 ( 25s 2 + 41s 20 ) (1 + s) 2). With Assumption 1, it can be verified that Z 1 > 0, Z 2 (br) > 0, Z 3 (br) > 0, Z 4 (br) > 0, Z 5 (br) > 0. For Z 2 to Z 5, replacing br with (1+s)2 2 results in Z 2 >90b 2 r 2 + 216b 2 r 2 br ( 31 58s + 487s 2) + 4 ( 20s 2 + s 1 ) (1 + s) 2, ( ) (1 + s) 2 2 ( ) (1 + s) 2 (31 >306 ) 58s + 487s 2 + 4 ( 20s 2 + s 1 ) (1 + s) 2 2 2 =3 (1 + s) 2 ( 19 + 62s 29s 2) > 0. 25

( ) Z 3 >n 2 (1 + s) 1404 2 2 ( ) 6 (1 s) (29 13s) (s + 1) 2 (1 + s) 2 (3535s 2 346s 425 ), 2 2 = 1 2 (1 + ( s)2 779 + 2254s 2989s 2) > 0. ( ) Z 4 >12n 3 (1 + s) 138 2 2 ( ) 6 (1 + s) 2 (s + 1) 2 (1 + s) 2 (199s 2 48s 55 ), 2 2 = (1 + s) 2 ( 56 + 105s 71s 2) > 0. ( ) (1 + s) Z 5 > 288 2 2 ( ) (1 + s) 2 (628s 2 58s 11 ) + 2 ( 25s 2 + 41s 20 ) (1 + s) 2, 2 2 = 3 2 (1 + s)2 ( 25 + 170s 128s 2) > 0. All the inequality signs above utilize Assumption 1 and the fact that 0 < s < 1. Together these prove that SW N SW M > 0. 26