Asymmetries, Passive Partial Ownership Holdings, and Product Innovation

ESADE WORKING PAPER Nº 265 May 2017 Asymmetries, Passive Partial Ownership Holdings, and Product Innovation Anna Bayona Àngel L. López

ESADE Working Papers Series Available from ESADE Knowledge Web: www.esadeknowledge.com ESADE Avda. Pedralbes, 60-62 E-08034 Barcelona Tel.: +34 93 280 61 62 ISSN 2014-8135 Depósito Legal: B-4761-1992 Supply Function Competition, Private Information, and Market Power: A Laboratory Study

Asymmetries, Passive Partial Ownership Holdings, and Product Innovation Anna Bayona Ángel L. López y May 2017 Abstract We study how asymmetries a ect R&D investments, competition, and welfare in markets with passive partial ownership holdings between rival rms. The asymmetries that we consider can arise either because one rm enjoys an initial competitive advantage, or because the passive partial ownership holdings between rivals are unequal. In contrast to previous ndings, we nd that, due to asymmetries, passive partial ownership holdings can increase total surplus in markets with competition in prices and quality-enhancing R&D with no spillovers. In these markets, our nding suggests that competition authorities should take into account the potential bene cial e ects of asymmetric passive partial ownership holdings. Keywords: partial ownership, minority shareholdings, R&D investments, price competition, competition policy JEL codes: D43, L11, L40, G34 1 Introduction There has been a recent surge in passive partial ownership holdings (hereafter PPO) between rival rms which have attracted the attention of competition authorities around the world. 1 An important characteristic of these are the asymmetries between rms, either because one rm enjoys an initial competitive advantage, or because the PPO holdings between rivals are unequal. Our objective is to study, in the presence of PPO, how asymmetries between rivals a ect R&D investments, competition in the product market, and welfare. In contrast to previous ndings, we nd that, due to asymmetries, PPO can increase total surplus in markets with competition in prices and quality-enhancing R&D with no spillovers. We consider a two-period model with two rms. In the rst period, rms invest in R&D to increase product quality. We examine markets where R&D spillovers are negligible due to high Bayona: ESADE Business School. E-mail address: anna.bayona@esade.edu y López: Departament d Economia Aplicada, Universitat Autònoma de Barcelona, and Public-Private Sector Research Center, IESE Business School. E-mail address: angelluis.lopez@uab.cat 1 These are subject to merger control in the UK, Germany and the US, among others. The European Commission (EC, 2014, 2016) is currently reviewing the issue, speci cally in cases where minority shareholdings acquisitions generate a "competitive signi cant link", which refers to a situation where either of the following conditions is satis ed: a) acquisitions of minority shareholdings in a competitor or vertically related company; b) when either the level of acquired shareholding is around 20% or is greater than 5% but it is accompanied with additional rights. 1

patent protection. In the second period, rms compete in prices à la Hotelling for the demand of a di erentiated product that is purchased by a continuum of consumers. We study how di erent ownership structures a ect equilibrium outcomes, and focus on those in which rms own (small) PPO holdings of rivals with no control rights. Our paper examines how two distinct types of asymmetries a ect incentives to invest in R&D, and as a result, rms competitive position in the second period. One asymmetry concerns unbalanced nancial interests by rivals. We model this asymmetry by supposing that one rm (the acquirer) has a PPO stake in the rival, while the rival (or target rm) does now have a PPO holding in the rm. The other asymmetry is related to situations in which one of the rms enjoys a larger initial competitive advantage since because of brand loyalty or a higher-quality product. We nd that, in markets with symmetric PPO holdings and no competitive advantage by neither rm, an increase in PPO holdings rise equilibrium prices and reduce the total R&D investment. In the rst period, each rm is aware that an increase in its price increases the rival s pro t, which can then be partially internalized by the rm since it owns a fraction of the rm s equity. In addition, because investing in R&D decreases the rival s pro t, and it is costly, rms under-invest in R&D in relation to in markets with no PPO. We then consider markets with symmetric PPO holdings and in which one rm has an initial competitive advantage. We show that the rm with a competitive disadvantage invests less in R&D with symmetric PPO holdings than with no PPO holdings. This is because increasing R&D is less e ective at generating pro ts, and yet because of symmetric PPO, the rm with disadvantage will be able to appropriate a share of the rival s pro ts. The rm with a competitive disadvantage will sell to less than a half of the market. If the competitive advantage is su ciently large, the rm with a competitive advantage may invest more in R&D with symmetric PPO than with no PPO since the bene t of further increasing its market share outweights the cost of investing in R&D. With asymmetric PPO holdings and neither rm enjoying an initial competitive advantage, we nd that the target rm invests more in R&D than the equivalent rm in a market with no PPO holdings, while the acquirer invests less. The target rm has to over-invest in R&D in order to generate a positive R&D di erential which leads to a higher market share, and potentially higher prices, thus augmenting revenues, and pro ts. By contrast, the acquirer has an incentive to reduce the investment in R&D as asymmetric PPO holdings increase since it appropriates a higher proportion of the target s pro ts generated both through the increase in the target s prices and market share. As a result, the acquirer sells to less than half of the market. Similar results hold when the target rm enjoys an initial competitive advantage. However, when the acquirer has a su ciently large competitive advantage, then it can then happen that its R&D investment increases as asymmetric PPO holdings increase: the acquirer invests in R&D in order to generate a product of greater quality that will be enjoyed by a larger proportion of consumers. Our main nding is that asymmetries in markets with PPO may lead to a larger total surplus compared to markets with no PPO. In markets where rms have asymmetric PPO holdings and neither rm enjoys an initial competitive advantage, as PPO holdings increase, the target rm increases its R&D investment but also its market share. This means that aggregate utility 2

increases since more consumers buy the good of higher product quality generated by the R&D investment in relation to markets with no PPO holdings. Total surplus with asymmetric PPO will be larger than with no PPO if the marginal utility due to R&D is su ciently large in relation to cost of investing in R&D and to the aggregate transport cost. Nevertheless, consumer surplus is lower with PPO compared to no PPO, while producer surplus is higher with PPO compared to no PPO. In contrast, total surplus with symmetric PPO and neither rm enjoying a competitive advantage is always lower than in markets with no PPO. This is because rms under-invest in R&D, which implies that a larger proportion of consumers buy a good of lower quality. This decreases the aggregate utility more than the total cost savings due to a lower investment in R&D. However, if one of the rms enjoys a su ciently large initial competitive advantage then total surplus with symmetric PPO is also larger than total surplus with no PPO. The rm with the initial competitive advantage invests more in R&D than the rm with a disadvantage, thus generating a positive R&D di erential, which leads to a higher market share for the rm with the initial competitive advantage. As a result, aggregate utility is larger than in markets with no PPO due to: (1) higher proportion of consumers buying the good with a greater initial gross utility; (2) higher proportion of consumers buying the good of higher quality due to the larger investment in R&D by the rm with the initial competitive advantage. Total surplus with symmetric PPO and an asymmetry in competitive advantage is larger than with no PPO when this increase in aggregate utility outweights transport and R&D investment costs. Total surplus may also be larger with PPO than with no PPO when the two types of asymmetries are present. If the target rm enjoys an initial competitive advantage then both asymmetries reinforce each other and the target rm over-invests in R&D. Hence, total surplus is larger with asymmetric PPO than with no PPO if the marginal utility of investing in R&D is large in relation to transport and R&D investment costs. If the acquirer enjoys an initial competitive advantage then it is also possible that total surplus is larger with PPO than with no PPO since the competitive advantage mechanism will mean that it might invest more in R&D compared to no PPO. However, it is less likely that total surplus is greater with PPO than with no PPO in relation to when the target has a competitive advantage. Our paper is related to three branches of literature. First, it is related to the literature on cooperative R&D and spillovers, which among others includes the seminar articles of d Aspremont and Jacquemin (1988), Suzumura (1992), Ziss (1994), Kamien et al. (1992), and Leahy and Neary (1997). This literature focuses on cost-reducing R&D investments, while we consider quality-enhancing R&D investments. The second branch of literature examines the e ects of passive partial ownership holdings on competition. The seminar articles of Bresnahan and Salop (1986), Reynolds and Snapp (1986), and Allen and Phillips (2000) show that PPO holdings are anti-competitive and result in higher prices. 2 However, in these articles R&D investments are absent. Finally, our paper is related to López and Vives (2016) which consider cost-reducing R&D investment with spillovers, and show that PPO holdings may increase welfare only if markets are not too concentrated, demand is not too convex and moreover spillovers are su ciently large. In contrast, in the presence of quality-enhancing R&D investments, we show that total 2 These results have also been corroborated by the empirical literature, such as in Azar et al (2015). 3

welfare may increase with PPO holdings even if there are no spillovers in the industry. The paper is organized as follows. Section 2 describes the model. Section 3 nds the equilibrium prices, market shares and R&D investments for di erent ownership structures. Section 4 conducts welfare analysis. Section 5 discusses the strategic incentives to acquire minority shareholdings. Section 6 concludes. 2 Model There are two rms, i; j = A; B with i 6= j, and! i is rm i s share of pro ts in rm j. The pro t of rm i s portfolio of investments is where i = (1! j ) i +! i j ; (1) i = p i s i x 2 i 2 ; (2) where s i is rm i s market share. We are interested in the case where there are asymmetries between the equity interests of the two rms:! A =!;! B = 0. In order to analyze this question, we consider the benchmark of no PPO (! A =! B = 0), and study the case of symmetric PPO, where PPO holdings between rivals are equal (! A =! B =!). 3 We restrict that PPO satis es 0! i ;! j < 1=4. 4 For a given ownership structure (! A ;! B ); we de ne the total equilibrium R&D investment as X (! A ;! B ) = x A (! A;! B ) + x B (! A;! B ). There is a mass of consumers uniformly distributed along the unit line that have a unit demand. Consumers can purchase the good either from rm A or rm B, and we assume full participation. We use Hotelling model for this type of demand. The timing of the model is as follows. At t = 1, each rm chooses an R&D investment, x i, by maximizing the pro ts from its portfolio of investments, i. At t = 2, given an R&D investment, each rm sets a price, p i, for a di erentiated product, and subsequently consumers purchase from either rm. A consumer that buys from rm i obtains the following utility U i (x i 0; x i ) tjq i qj p i ; (3) where t > 0 is the product di erentiation parameter; q i f0; 1g is the location of the rm (without loss of generality assume that q A = 0; q B = 1); and q[0; 1] is the location of a consumer. 3 Passive partial ownership (PPO) stakes are also often called passive minority nancial interests, noncontrolling minority shareholdings, or passive cross-ownership. 4 There is not a commonly agreed thresdhold about what consistute non-controlling minority shareholdings by competitors. However, competition authorities often inspet the non-controlling minority shareholdings by competitors that are between 15% and 25% (Salop and O Brien 2000). 4

Hence, a consumer that is located at q loses tjq i qj units of utility when he purchases the good located q i. The utility function of a consumer that buys from rm i is as follows U i (x i 0; x i ) = x i 0 + x i ; (4) where x i 0 is the initial gross utility of buying good sold by rm i; consumer s utility function increases by > 0 for each unit of R&D invested by rm i. Hence, R&D is of the qualityenhancing type. Let q be the consumer that is indi erent between buying A and B, then: U A (x A 0 ; x i ) tj qj p A = U B (x B 0 ; x i ) tj1 qj p B : Therefore, the market share of rm A is: s A = 1 2t ((xa 0 x B 0 ) + (x A x B ) + (p B p A ) + t); (5) and rm B s market hare is s B = 1 s A. Without loss of generality, denote s A = s and s B = 1 s. De ne the di erence in the initial gross utility from buying from rm A and rm B as: A x A 0 x B 0, and conversely, B = x B 0 x A 0 = A. As explained in the introduction, we consider two distinct types of asymmetries. Asymmetry due to rm i enjoying an initial competitive advantage (disadvantage). In our model this translates to i > 0 ( i < 0), which leads to a positive (negative) di erence in initial gross utility derived from buying from rm i. Asymmetry due to unequal PPO holdings between rms. We model this asymmetry as! i =! and! j = 0, for i 6= j. The equilibrium concept used is the Subgame Perfect Equilibrium. A rm s strategy consists of an investment in R&D and a subsequent pricing strategy which is based on the R&D investment. In terms of welfare, we consider the second-best welfare benchmark such that the planner chooses the R&D that maximizes total surplus, taking as given the equilibrium prices set by the two rms in the second stage. We also compare total surplus at the equilibrium allocation for di erent ownership structures. 3 Equilibrium prices, market shares and R&D Investments 3.1 Benchmark models: No PPO and symmetric PPO We study two benchmark models with respect to the ownership structure: markets with symmetric PPO (! A =! B =!), and markets with no PPO (! = 0). We also allow asymmetries in initial competitive advantage to be present, i.e. i 6= 0. We rst nd the equilibrium of the model for symmetric PPO. Pro ts of rms i; j = A; B with i 6= j are: 5

i (!;!) = (1!) i +! j ; (6) x where i = p i s 2 i i 2, and s i = 1 2t ( i + (x i x j ) + (p j p i ) + t). We solve by backward induction and nd the optimal prices at t = 2. The rst order condition for each rm i satis es @ i @p i = 0. Hence, the best response price equations for rms i; j = A; B are such that for i 6= j: p i (!;!) = (1!) i + (1!)(x i x j ) + p j + (1!)t : 2(1!) For given R&D investments, we observe that the price best-reply function is upward sloping (i.e. competition at t = 2 is of the strategic complements type). Solving for the xed point, we obtain the prices as functions of R&D investments at t = 1: p i (!;!) = (1!) i(1 2!) + t(3 2!) + (x i x j )(1 2!) : (2! 3) (2! 1) Substituting for these optimal prices we nd that the position of the indi erent consumer is: s i = 1 6t (3t + i + (x i x j )). We now solve for the equilibrium R&D investments at t = 1. The rst order condition for rm i satis es: @ i @x i = 0. The second order condition requires that t (2! 3) 2 2 > 0, and the stability of the equilibrium that t (2! 3) 2 2 2 > 0. As a result, we obtain the best response functions for R&D of each rm: i t (1!) (2! 3) x i = (t (2! 3) 2 2 ) (t (2! 3) 2 2 ) x j : We note that R&D investment at t = 1 is of strategic substitutes type. The strategic e ect can be shown to be negative. 5 It follows that, at the interior equilibrium, the direct e ect is positive, in which case rm i under-invests in relation to the direct e ect. Recall that the sign of the strategic e ect depends on whether R&D investment makes rm i tough or soft, and on whether second period actions are strategic substitutes or complements. R&D investment makes rm i tough since @ j(x i ;p i ;p j ) @x i < 0. Furthermore, second period prices are strategic complements. As a result, the strategic e ect is necessarily negative. That is, investment in R&D makes rms tough, which combined with strategic complementarity in the second period yield a negative strategic e ect, which results in under-investment in R&D. Firms under-invest in R&D so as not to trigger an aggressive response from the rival rm, also known as a "puppy dog" strategy (Fudenberg and Tirole, 1984). We now solve for the xed point. The following proposition states the equilibrium outcomes with symmetric PPO. PROPOSITION 1 If the second order and stability conditions are satis ed, i.e. t (2! 3) 2 2 > 0, for each rm i, at equilibrium x i (!;!) = 5 The following are satis ed: @ i @p j dp j dx i i (3 2!) (3 2!) t (2! 3) 2 + (1!) ; (7) 22 < 0; @ i @x i > 0 6

with X (!;!) = x A (!;!) + x 2(1!) B (!;!) = (3 2!), p i (!;!) = (1!) t(3 2!) + i (1 2!) 1 + (3 2!) (1 2!) 2 2 (t (2! 3) 2 2 2 ) ; (8) and s i (!;!) = 1 2 1 + i (3 2!) t (2! 3) 2 2 2 : (9) The special case of no PPO can be obtained from Proposition 1 by setting! = 0. COROLLARY 1 (no PPO). If the second order and stability conditions are satis ed, i.e. 9t 2 > 0, for each rm i, at equilibrium x i (0; 0) = 3 i 1 + 3 (9t 2 2 ; ) and X (0; 0) = x A (0; 0) + x 2 B (0; 0) = 3, p i (0; 0) = t 1 + 3 i 9t 2 2 ; (10) and s i (0; 0) = 1 2 1 + 3 i 9t 2 2 : (11) In markets with symmetric PPO and no competitive advantage by neither rm, equilibrium prices are higher than with no PPO since competition is softer in the second stage, and as a result rms invest less in R&D than if rms had no nancial links. The reasoning is as follows. With symmetric PPO, rm i at t = 1 takes into account that at t = 2 an increase in p i increases rm j s pro t, which can be partially internalized by rm i since it owns a fraction! of rm j. In addition, due to symmetry and no initial competitive advantage, each rm will still sell to half of the market (since there is full participation). Because investing in R&D decreases the rival s pro t, and it is costly, rms under-invest in R&D in relation to in markets with no PPO. For symmetric PPO, Table 1 describes the comparisons of equilibrium outcomes with symmetric PPO in relation to markets with no PPO for various signs of i. Table 1. Symmetric PPO: Signs of the comparisons. sign(:::) i = 0 i < 0 i > 0 x i (!;!) x i (0; 0) sign (9t 2 2 )(t(2! 3) 2 2 2 ) i 12t 2 (3 2!)(3!) X (!;!) X (0; 0) p i (!;!) p i (0; 0) + sign (9t 22 )(t(2! 3) 2 2 2 ) (1 2!) i 3t(3 2!) 2 2 (5 2!) s i (!;!) s i (0; 0) 0 + Suppose that rm i has an initial competitive disadvantage, i.e. i < 0. With symmetric PPO, for given levels of R&D and prices, the market share of rm j will be larger than the 7

market share of rm i. More consumers will nd it optimal to purchase from rm j because a larger proportion derive a greater initial gross utility by purchasing from j. As a result, rm j will be able to charge higher prices than rm i, and generally the equilibrium prices of both rms increase with! because of softer competition. Even when! = 0, rm i has a lower level of R&D investment than rm j. In addition, rm i invests less in R&D with symmetric PPO compared to no PPO since it is more costly for this rm to gain market share compared to rm j. With symmetric PPO, rm i can take advantage of the R&D investment by rm j, which is more e ective at generating pro ts. The combined e ect is that rm i has a lower market share than rm j. Suppose now that rm i has a competitive advantage, i.e. i > 0. The argument is the opposite to the one just described in the previous paragraph, except from the sign of how R&D investment with PPO holdings compares to no PPO holdings since it depends on the size of the competitive advantage. If i is large then rm i may invest more in R&D than when there are no PPO holdings. This is because the positive R&D di erential between rms lead to a larger di erential in prices at t = 2, which translate in a larger market share for rm i, and higher corresponding pro ts. In contrast, if i is su ciently small, then the cost of increasing R&D outweights the bene t derived from the potential increase in revenues. Hence, if i is su ciently small then rm i invests less in R&D with symmetric PPO than with no PPO holdings. 3.2 Asymmetric PPO We consider the case when PPO holdings between rivals are unbalanced, and model it by assuming that! A =!;! B = 0. This formulation captures the situation in which the acquirer (A) has a PPO holding of the target rm (B), while the target rm does not have any PPO holdings of the acquirer. Pro ts of rms A and B, are A (!; 0) = A +! B = p A s x 2 A 2 +!p B(1 s) B (!; 0) = (1!) B = (1!) p B (1 s) Next Proposition states the equilibrium with asymmetric PPO holdings.!x 2 B ; (12) 2 x 2 B : (13) 2 PROPOSITION 2 If the second order and stability conditions are satis ed, i.e. t(3!) 2 2 > 0, at equilibrium 0 x A(!; 0) = @ A(3!) 2 (2!) + t (3!)! 2 5! + 3 1 A ; (14) (3!) t (3!) 2 2 2 (3!) (3t x A ) 2 (2!) B(!; 0) = (3!) (t (3!) 2 ; (15) 2 2 ) 8

and X (!; 0) = x A (!; 0) + x (2!) B (!; 0) = (3!), p A(!; 0) = t 0 @ A(3!)(1!) + 2 (! 2 3! 2) + (3!)(3 +!)t t (3!) 2 2 2 1 A ; (16) p B(!; 0) = t 0 @ (3!) A (2!) 2 + 3t(3!) t (3!) 2 2 2 1 A : (17) and s A(!; 0) = 1 2 0 @ A(3!) (2 +!) 2 t (2! 3) (3!) t (3!) 2 2 2 1 A ; (18) while s B (!; 0) = 1 s A (!; 0). Suppose that neither the target (B) nor the acquirer (A) enjoy an initial competitive advantage. In the second stage, for given R&D investments, the acquirer takes into account that it internalizes a share! of the rival s pro t, while the target rm only obtains a share 1! of its own pro ts. For given R&D investments, an increase in p A by one unit triggers an increase in p B by 1 2, while an increase in p B by one unit triggers an increase in p A by 1+! 2. Because of the asymmetry in PPO holdings, prices at t = 2 will be more moderate in relation to the symmetric PPO. At t = 1, the target rm augments R&D as PPO holdings increase as a means of generate a positive R&D di erential which leads to a higher market share, and potentially higher prices, thus increasing revenues and pro ts. By contrast, the acquirer has an incentive to reduce the investment in R&D as! increases since it appropriates a proportion! of the surplus generated by target rm both through the increase in prices and market share. In addition, an increase in the R&D investment by the acquirer decreases the target s pro ts. Table 2 shows the comparison of equilibrium outcomes with asymmetric PPO with respect to the no PPO benchmark. Table 2. Asymmetric PPO: Signs of the comparisons. sign(:::) i = 0 A < 0; B > 0 A > 0; B < 0 x A (!; 0) x i (0; 0) sign ((3!)(9 2!)t 2 )(9t 2 2 ) A 3t 2 (3!)(6!) x B (!; 0) x i (0; 0) + + sign (9t 22 )(t(3!)(6!)+ 2 ) 3t 2 (! 3)(! 6) A X (!; 0) X (0; 0) p A (!; 0) p i (0; 0) sign n 2t 2 sign 2t 2 o 3t(3!) (3!) 2 2 (4!) A (9t 2 2 ) p B (!; 0) p i (0; 0) + + + n o s A (!; 0) s i (0; 0) sign A! 2 +t(3!) (3!) n o s B (!; 0) s i (0; 0) + + sign A! 2 +t(3!) (3!) Table 2 shows that with asymmetric PPO holdings and A 0, the target rms invests more in R&D compared to an equivalent rm in a market with no PPO holdings, while the 9

acquirer invests less. In fact, we can show that if A = 0 then the following relationship holds: x B (!; 0) > x i (0; 0) > x i (!;!) > x A (!; 0). However, when the acquirer has a su ciently large competitive advantage, i.e. A > 0, it can then happen that the acquirer increases its R&D investment as asymmetric PPO holdings increase. This is because the acquirer starts with an advantage in market share and quality, and as a result, it can charge higher prices. Because the target rm has a small market share it is more e ective that the acquirer invests in R&D in order to generate a product of greater quality that will be enjoyed by a larger proportion of consumers. Overall, the sum of R&D investments by both rms is smaller with asymmetric PPO holdings than with no PPO holdings. We also observe that the price of the target rm is always larger with asymmetric PPO holdings than with no PPO holdings irrespective of the sign of A. However, the equilibrium price of the acquirer may be larger or smaller than an equivalent rm with no PPO holdings. If A = 0 and 2 is large in relation to t then consumers give a high value to quality and generally prefer to buy the good produced by the target rm since it invests more in R&D, and hence its product is of higher quality. The acquirer then reduces prices in order to be competitive, and not further reduce its market share. In fact we can show that sign (p A (!; 0) p B (!; 0)) = sign t (3!) 2 (4!). Consequently, if A 0 then the market share of the target (acquirer) rm will be larger (smaller) with asymmetric PPO holdings than with no PPO holdings. If the acquirer enjoys a su ciently large competitive advantage (i.e. if A is positive and su ciently large) then market share of the acquirer will be larger for some! > 0 in relation to when! = 0, until the target rm s proposal is more attractive. For an illustration of the comparison between equilibrium outcomes for di erent ownership structures, see Figures 1 and 2. 4 Welfare analysis For a given ownership structure (! A ;! B ), we de ne producer surplus, consumer surplus and total surplus. Producer surplus is P S(! A ;! B ) = A (! A ;! B ) + B (! A ;! B ) = p A s + p B (1 s) 2 (x2 A + x 2 B); (19) where s A = s = 1 2t ((xa 0 x B 0 ) + (x A x B ) + (p B p A ) + t). We notice that producer surplus is equal to the total revenue minus the total cost of investing in R&D. Consumer surplus is equal to CS(! A ;! B ) = Z s 0 Z 1 (x A 0 + x A ty p A )dy + s (x B 0 + x B t(1 y) p B )dy; (20) which is the sum of consumers utility from buying either of the rms minus the total payment transferred to each of the rms and minus the total costs due to product di erentiation. Total surplus is T S(! A ;! B ) = CS(! A ;! B ) + P S(! A ;! B ), and thus equals 10

Figure 1: Equilibrium outcomes with respect to PPO. Figure 2: Equilibrium outcomes with respect to output. 11

T S(! A ;! B ) = (x A 0 s + x B 0 (1 s)) + (x A s + x B (1 s)) t 2 (s2 + (1 s) 2 ) 2 (x2 A + x 2 B): (21) The next Proposition compares total surplus at the equilibrium allocation, T S (! A ;! B ), for di erent ownership structures. PROPOSITION 3 Suppose that i = 0. It always holds that T S (0; 0) > T S (!;!): If is su ciently large and t su ciently small then T S (!; 0) > T S (0; 0) > T S (!;!); while if is su ciently small and t su ciently large then T S (0; 0) > T S (!;!) > T S (!; 0) for some! > 0. The Proposition shows that in markets with no asymmetries due to an initial competitive advantage ( i = 0), total surplus with symmetric PPO is always lower than in markets without PPO. As PPO increases both rms steeply increases prices and under-invest in R&D in relation to markets with no PPO. Yet both rms serve half of the market. Due to PPO, consumer surplus always decreases more than the increase in producer surplus. This is because the underinvestment in R&D causes a larger decrease in aggregate utility (since a larger proportion of consumers buy a good of lower quality) which is not compensated by the cost-savings of both rms due to a lower investment in R&D. The surprising nding is that total surplus with asymmetric PPO may be larger than total surplus with no PPO if the marginal utility due to R&D () is su ciently large in relation to product of the R&D cost parameter,, and the transportation cost associated with product di erentiation, t. This is because, as PPO holdings increase, the target rm increases its R&D investment but also its market share. This leads to a substantial increase in aggregate utility due to more consumers buying a good of higher quality, generated by the R&D investment, in relation to when there are no PPO holdings. When is su ciently large in relation to t, this bene cial allocation e ect outweights the increase in costs due to R&D and transport, and hence total surplus is larger with asymmetric PPO than in markets with no PPO. However, we nd that due to PPO holdings, consumers are always worse o while producers are always better o. This is due to the e ects of prices on both consumer and producer surplus, which is eliminated in total surplus since it is just a transfer from consumers to producers. In contrast, when is su ciently small in relation to t, for some large! total surplus with asymmetric PPO is even smaller than with symmetric PPO since the joint costs of the over-investment in R&D and transport do not compensate the positive increase in aggregate utility. 12

The next proposition studies the comparison of total surplus with symmetric PPO and no PPO when one of the rms enjoys an initial competitive advantage. PROPOSITION 4 If i 6= 0 then sign ft S (!;!) T S (0; 0)g = sign 2 i ; where 4t 2 4! 3 + 12! 2 54! + 27 + 9t 2 2 (6 5!) (2! 3) 2 4 4 4! 2 13! + 12 > 0 for the range of! considered (i.e. 0 <! 1 4 ), and 2 (3!)(9t 2 2 ) 2 (t(2! 3) 2 2 2 ) 2 9t(2! 3) 2 > 0. If either of the rms has a su ciently large initial competitive advantage, i.e. 2 i >, then total surplus with symmetric PPO is larger than with no PPO. Without loss of generality suppose that i > 0. Firm i invests more in R&D than rm j, thus generating a positive R&D di erential, which leads to a higher market share for rm i. As a result, aggregate utility with PPO is larger than with no PPO due to: (1) higher proportion of consumers buying the good with a greater initial gross utility; (2) higher proportion of consumers buying the good of higher quality generated by R&D investment of rm i. Total surplus with symmetric PPO is larger than with no PPO when this increase in aggregate utility outweights the transport and R&D investment costs. We have conducted simulations to analyze the comparison of total surplus with symmetric, asymmetric PPO and with no PPO, which are displayed in the Figure 3. The previous gures show the following: (i) When the target rm enjoys an initial competitive advantage ( B > 0) and is su ciently large in relation to t then total surplus with asymmetric PPO is larger than with no PPO. This is because the mechanism described in Proposition 4 is reinforced since the target rm invests more in R&D, and its market share is greater than if B = 0. In fact, the level of that is required for total surplus to be higher with asymmetric PPO than with no PPO is smaller as B becomes larger. (ii) When the acquirer enjoys an initial competitive advantage ( A > 0), the level of that is required for total surplus to be higher with asymmetric PPO than with no PPO increases as A becomes larger. This is because when! = 0, R&D investment, price and market share are larger for the acquirer than the target rm. As asymmetric PPO holdings increase, the R&D investment, price and market share of the acquirer decrease, while those for the target rm increase. For a given level of, aggregate utility is lower than if A = 0 since on average consumers buy a good of a lower initial gross utility and of a lower quality due to the R&D investment by rms, which does not compensate the aggregate transport and R&D costs. 5 Strategic Incentives to Acquire Minority Shareholdings In the previous sections we have considered exogenous ownership structures. In this section, we examine the strategic incentives to acquire minority shareholdings. 6 Let us consider the case 6 This issue has been previously analyzed by Flath (1991). 13

Figure 3: Total surplus with symmetric and asymmetric PPO. of asymmetric shareholdings: A = A +! B and B = (1!) B, and include a stage 0 in which the acquirer, A, decides the PPO holdings to invest in the target,!. If the market is e cient then rm A will pay for the equity! exactly the amount! B. Therefore, the net pro t is A = A! B = A, and rm A will acquire! from B only if as a result of the acquisition its operating pro t ( A ) increases. Second-period equilibrium yields p i (x i; x j ), p j (x i; x j ), while rst-period equilibrium gives x i (!) and x j (!), which replaced into the former expressions gives p i (!), p j (!). Therefore, at equilibrium we can write i as a function of! as follows: A (p A(!); p B(!); x A(!); x B(!)). Note that A is the operating pro t, and therefore @ A = = 0. We have 0 1 0 1 d A d! = @ @ A dp A @p A d! + @ A dx A A + B @ A dp B @x A d! @ @p {z } B d! + @ A dx B C @x B d! A 0, {z } STRATEGIC EFFECT if d A =d! < 0, then! = 0. DIRECT EFFECT If the direct e ect is negative, then the incentives to acquire minority shareholdings depend on the sign of the strategic e ect. Consider the rst-order conditions of the second-period: @ A @p A +! @ B @p A = 0. 14

Since @ B =@p A > 0, it is clear that @ A =@p A < 0. Clearly, @ A =@p B > 0. Note also that, in the rst period for i (x i ; x j ; p i ; p j ), the rst-order condition is @ A @x A +! @ B @x A = 0, since the game is tough with respect to R&D investments: @ B =@x A < 0, necessarily at equilibrium @ A =@x A > 0. 7 For t su ciently large, evaluated at! = 0, we get sign dx A d! sign dp A d! < 0, sign!=0 > 0, sign!=0 dx B d! dp B d! > 0,!=0 @A > 0 and sign!=0 @x B < 0.!=0 Therefore, the direct e ect is negative (as in the standard game), whereas the strategic e ect has one positive component and one negative component. For t su ciently large: d A d! 0 1 0 ( ) (+) B@ A dp (+) ( ) A = @!=0 @p A d! + @ A dx (+) (+) A C A + B @ A dp B @x A d! @ @p {z } B d! + @x B DIRECT EFFECT ( ) @ A (+) dx B d! {z } STRATEGIC EFFECT 1 C A 0. While the R&D investment choices are strategic substitutes and therefore introduce a negative strategic e ect, price choices are strategic complements and therefore introduce a positive strategic e ect. This positive strategic e ect, (@ A =@p B ) (dp B =d!), can make PPO acquisition pro table. The intuition is as follows. At period 2, for some given x i ; x j, a higher! softens the competitive pressure of rm A, which results in a higher p A. Since price choices are strategic complements, rm B also increases its price: p B. Speci cally, @p A = 23t A (x A x B ) (3!) 2 = 2 @p B. That is, the increase in p A is twice the increase in p B. Since R&D investments make rm A tough (@ B =@x A < 0) and prices are strategic complements, at period 1 rm A adopts a puppy dog strategy: A under-invests so as not to trigger an aggressive response from its rival. A higher! makes the second-period game less tough since prices increase with!, but since the increase in p B is half of the increase in p A, rm A under-invests even more to reduce the competitive pressure in period 2. Thus, x A decreases and because of R&D investments are strategic substitutes, x B increases. The impact of the changes of the choice variables on A s pro t is intuitively clear: The increase in p A decreases A because A is overpricing (recall that A maximizes A +! B ) 7 In particular, for i =! = 0, @ A=@x A = =6 > 0. 15

The decrease in x A decreases A because A is under-investing (because of the tough nature of the subgame) The increase in p B increases A because for the same price p A, rm A obtains a higher market share The increase in x B decreases A because it has a negative direct e ect on A market share, and moreover it makes competition more tough in the second period Therefore, only the positive strategic e ect of! on rival price can induce A to acquire some nancial interests in rm B. 6 Concluding Remarks This paper has studied how asymmetries in passive partial ownership (PPO) holdings or asymmetries in initial competitive advantage a ect R&D investments, competition, and welfare. We build a two-period model with two rms, such that in the rst period rms invest in R&D to increase product quality, and in the second period rms compete in prices à la Hotelling for the demand of a di erentiated product that is purchased by continuum of consumers that buy from either of the rms. We characterize the Subgame Perfect Equilibrium of the game, and we consider the second-best welfare benchmark. Our model applies to markets in which rival rms have PPO holdings for purely nancial reasons, and in which PPO holdings are small so that rms do not obtain control rights. In addition, our model is relevant in markets where R&D spillovers are negligible due to high patent protection. In contrast to previous ndings, we nd that PPO can increase total surplus in markets with competition in prices and quality-enhancing R&D with no spillovers. This is due to asymmetries in either PPO holdings between rivals, or due to one of the rms enjoying a su ciently large initial competitive advantage. The target rm or the rm with the initial competitive advantage over-invest in R&D in relation to when there are no PPO holdings, leading to a higher market share, which means that a larger proportion of consumers buy a good of higher quality. This leads to greater aggregate utility than if there were no PPO holdings. If consumers give su cient high value to quality in relation to cost of R&D and transport then total surplus will be larger. Our nding suggests that competition authorities should take into account the potential bene cial e ects of asymmetric PPO holdings in markets with competition in prices, and qualityenhancing R&D even if there are no spillovers. Our paper suggests a few open questions for future research. Future work could study the welfare e ects of asymmetries in active partial ownership stakes. Furthermore, the paper calls for the development of further empirical work which investigates the role of asymmetries (either in PPO stakes or in initial competitive advantage) on prices and R&D investments. 16

References [1] Allen, J. W., & Phillips, G. M. (2000). Corporate equity ownership, strategic alliances, and product market relationships. The Journal of Finance, 55(6), 2791-2815. [2] Azar, J., Schmalz, M. C., & Tecu, I. (2015). Anti-competitive e ects of common ownership. Ross School of Business Paper, No. 1235. [3] Bresnahan, T. F., & Salop, S. C. (1986). Quantifying the competitive e ects of production joint ventures. International Journal of Industrial Organization, 4(2), 155-175. [4] d Aspremont, C., & Jacquemin, A. (1988). Cooperative and noncooperative R & D in duopoly with spillovers. The American Economic Review, 78(5), 1133-1137. [5] Flath, D. (1991). When is it rational for rms to acquire silent interests in rivals?. International Journal of Industrial Organization, 9(4), 573-583. [6] Fudenberg, D., & Tirole, J. (1984). The fat-cat e ect, the puppy-dog ploy, and the lean and hungry look. The American Economic Review, 74(2), 361-366. [7] Kamien, M. I., Muller, E., & Zang, I. (1992). Research joint ventures and R&D cartels. The American Economic Review, 1293-1306. [8] Leahy, D., & Neary, J. P. (1997). Public policy towards R&D in oligopolistic industries. The American Economic Review, 642-662. [9] López, Á. L., & Vives, X. (2016). Cross-ownership, R&D Spillovers, and Antitrust Policy. CESifo Group Munich, No. 5935. [10] Reynolds, R. J., & Snapp, B. R. (1986). The competitive e ects of partial equity interests and joint ventures. International Journal of Industrial Organization, 4(2), 141-153. [11] Salop, S. C., & O Brien, D. P. (2000). Competitive e ects of partial ownership: Financial interest and corporate control. Antitrust Law Journal, 67(3), 559-614. [12] Suzumura, K. (1992). Cooperative and Noncooperative R&D in an Oligopoly with Spillovers. The American Economic Review, 1307-1320. [13] Ziss, S. (1994). Strategic R & D with Spillovers, Collusion and Welfare. The Journal of Industrial Economics, 375-393. 7 Appendix The following appendix provides a proof of the propositions in the text. 17

t = 2: are Proof of Proposition 1 @ i @p i We start solving backwards and use the rst order condition at = 0 for i = A; B. Hence, the reaction functions for for rms i; j = A; B and i 6= j p i (!;!) = (1!) i + (1!)(x i x j ) + p j + (1!)t : 2(1!) Finding the xed point we note that p i (!;!) = (1!) (2! 3) (2! 1) ( i(1 2!) + t(3 2!) + (x i x j )(1 2!)); with s i = 1 2t ( i + (x i x j ) + (p j p i ) + t) and p j p i = 2 ( i + (x i x j )) 1! 3 2!. Substituting these expressions into pro ts for each rm, we can then solve at t = 1 for the optimal amount of R&D investment: @ i @x i = 0 for i; j = A; B and i 6= j such that the second order condition, t (2! 3) 2 2 > 0 is satis ed and that the stability condition requires that obtain @ 2 A @x 2 A @ 2 B @x 2 B > @ 2 A @ 2 B @x A @x B @x B @x A ; which is equivalent that t (2! 3) 2 2 2 > 0 applies. From the rst order condition, we x i (!;!) = t (1!) (2! 3) + i t (2! 3) 2 2 2 (1!) (1!)(t (2! 3) 2 2 ) x i: Hence, nding the xed point, nd the results stated in Proposition 2. Proof of Corollary A Let us draw comparative statics of the equilibrium allocation with symmetric PPO with respect to!. and @s (!) = i t (2! 3) 2 + 2 2 (t (2! 3) 2 2 2 ) 2, @p i (!) = t (t (2! 3)2 2 2 ) 2 i (2! 1) 2 (t (2! 3) 2 2 2 (5 4!)) (1 2!) 2 (t (2! 3) 2 2 2 ) 2 ; and @x i (!) = Let us consider several cases: If i = 0 then @s (!) = 0, @p i (!) > 0, @x i (!) < 0. If i < 0 then @s (!) < 0, @x i (!) < 0, and 9t 2 2 12t! + 4t! 2 2 4t 2 i (3 2!) 3 (2! 3) 2 ( 4t! 2 + 12t! + 2 2 9t) 2 : sign( @p i (!) ) = sign((t (2! 3)2 2 2 ) 2 i (2! 1) 2 (t (2! 3) 2 2 2 (5 4!))): 18

If i > 0 then @s (!) > 0, and sign( @x i (!) ) = sign(4t2 i (3 2!) 3 9t 2 2 12t! + 4t! 2 2 ); sign( @p i (!) ) = sign((t (2! 3)2 2 2 ) 2 i (2! 1) 2 (t (2! 3) 2 2 2 (5 4!))): @ B @p B Proof of Proposition 2 = 0 imply The rst order conditions for rm A, @ A @p A = 0, and for rm B p A (!; 0) = 1 2 ( A + (x A x B ) + p B (1 +!) + t): Substituting the expression of p B into A s price reaction function we obtain: p A (!; 0) = 1 3! (t(3 +!) + (1!)( A + (x A x B ))); and from the earlier proposition we know that p B (!; 0) = 1 3! (3t A (x A x B )); 1 with s(!; 0) = 2t(3!) (t(3 2!) + A + (x A x B )). Now, we can solve t = 1 optimal amount of R&D. The rst order condition for A satis es: We obtain that @ A @x A = 0 and the second order condition is satis ed if t(3!) 2 2 > 0, and the stability condition requires that t(3!) 2 2 2 > 0. The rst order condition, implies the following reaction function: x A (!; 0) = A + t(! 2 5! + 3) x B (t (3!) 2 ; 2 ) and the rst order condition for B and reaction function are analogous to the benchmark model without PPO. Hence, 3t A x B (!; 0) = (t (! 3) 2 2 ) 2 x A (t (! 3) 2 2 ) ; and hence by nding the xed point the results of Proposition 3follow. Proof of Corollary B asymmetric PPO with respect to!. Let us draw comparative statics of the equilibrium allocation with @x A (!) = (9!) (3!) 3 t 2 2 2 (3!) 3 t 2 A (7 2!) (3!) 2 2 t + 2 4 2 ; (! 3) 2 t (3!) 2 + 2 2 19

@x B (!) = 2 (3!) 3 A t 2 6t 2 2 (3!) 3 + (3 2!) (3!) 2 t 2 + 2 4 2 ; (3!) 2 t (3!) 2 + 2 2 @s A (!) = 3 (! 3)2 t 2 2 + t 2 (! 2 4! 3) + A (2 2 + t (! 3) 2 ) + 2 4 2 ; 2 t (3!) 2 + 2 2 @p A (!) = t 24 (3 2!) + 6 (! 3) 2 t 2 2 + 2 t( 3! 2 + 26! 39) + 2 A (2 2 (2!) (! 3) 2 t) 2 ; t (3!) 2 + 2 2 @p B (!) = t ( 3 (3!)2 t 2 2 + t 2 (! 2 4! 3) + 2 4 + A ((3!) 2 t + 2 2 )) 2. t (3!) 2 + 2 2 Let us consider several cases: When A = 0 then @p A (!) @p B (!) = t 2t 2 (3t (! 3) 2 2 2 (3 2!)) (9t 2 2 6t! + t! 2 ) 2 ; = 3t2 2 (! 3) 2 t 2! 2 4! 3 2 4 (9t 2 2 6t! + t! 2 ) 2, which means that @p A (!) > 0 and @p B (!) < 0. For R&D investments, we observe that @x A (!) = (9!) (3!) 3 t 2 2 + (7 2!) (3!) 2 t 2 2 4 (! 3) 2 t (3!) 2 + 2 2 2 ; @x B (!) = 6t 2 2 (3!) 3 (3 2!) (3!) 2 t 2 2 4 2 ; (3!) 2 t (3!) 2 + 2 2 which means that under most circumstances @x A (!) < 0 and @x B (!) > 0. @s A (!) = 3 (! 3)2 t 2 2 + t 2 (! 2 4! 3) + 2 4 2 ; 2 t (3!) 2 + 2 2 which implies that when is high in relation to t then @s A (!) > 0. Otherwise, @s A (!) < 0. When A 6= 0 then 20

@p sign A (!) = sign(2 4 (3 2!)+6 (! 3) 2 t 2 2 + 2 t( 3! 2 +26! 39) 2 A ((! 3) 2 t 2 2 (2!))); sign( @p B (!) ) = sign((3 (3!)2 t 2 2 t 2 (! 2 4! 3) 2 4 A ((3!) 2 t + 2 2 ))), sign( @s A (!) ) = sign( 3 (! 3)2 t 2 2 + t 2 (! 2 4! 3) + 2 4 + A (2 2 + t (! 3) 2 )): Notice that sign( @p B (!) ) = sign( @s A (!) ), and sign( @x A (!) ) = sign( t2 2 (9!) (3!) 3 + (7 2!) (3!) 2 t 2 2 4 ); @x B (!) = sign(6t 2 2 (3!) 3 (3 2!) (3!) 2 t 2 2 4 ): Proof of Corollary 2 The comparison of prices when the second order conditions are satis ed and i = 0 is as follows: p i (0; 0) p (3!)t + 2 B(!; 0) = t! 9t 2 2 6t! + t! 2 < 0; p i (!;!) p! i (0; 0) = t 1 2! > 0; p i (!;!) p B(!; 0) = t! t (! + 2) (3!) 2 (3 2!) (1 2!) (9t 2 2 6t! + t! 2 ) ; p i (!;!) p A(!; 0) = t! t (3! + 1) (3!) + 2 2! 2 7! + 1 (1 2!) (9t 2 2 6t! + t! 2 ; ) p A(!; 0) p i (0; 0) = t! (3!) 2t 2 9t 2 2 6t! + t! 2 ; p A(!; 0) p B(!; 0) = t! t (3!) 2 (4!) 9t 2 2 6t! + t! 2 : and the results from the Corollary follow since we also apply the second order conditions. Let us now analyze the investment in R&D 21

sign(x i (!;!) 0 9t 2 2 1 t (2! 3) 2 2 2 x i (0; 0)) = sign @ i A 12t 2 ; (3 2!) (3!) sign(x A(!; 0) x t! (5!) B(!; 0)) = sign A ; 2 sign(x A(!; 0) x (3!) (9 2!) t 2 9t 2 2! A(0; 0)) = sign A 3t 2 ; (3!) (6!) and ((3!)(9 2!)t 2 )(9t 2 2 ) 3t 2 (3!)(6!) > 0 is positive. Furthermore, sign(x B(!; 0) x B(0; 0)) = sign 9t 2 2 t (3!) (6!) + 2 3t 2 (! 3) (! 6) A! ; sign(x A(!; 0) x A(!;!)) = sign 0 @ (3!)! 2 6! + 6 t + 2 + 3t2 A (2!) (3!) (3 2!) t (2! 3) 2 2 2 1 A : Then, we note that: sign(x B(!; 0) 0 x B(!;!)) = sign @ A + + (24 +t 2 2 (3!)(! 2 7!+9)(2! 3) 2 +t 2 (2! 3 24! 2 +72! 63)) 3t 2 (2!)(3!)(3 2!) 1 A : 2! 3 Notice that when A = 0 then x B (!; 0) x B (!;!) =! (t(3!)(! 2 7!+9) 2 ) (3 2!)(3!)(t(2! 3) 2 2 2 ) > 0. For the combined R&D of both rms, we notice that for all i : X (!;!) X (0; 0) = 3 2! < 0, X (!; 0) X (!;!) =! (3 2!)(3!) > 0, and X (!; 0) X (0; 0) =! 3(3!) < 0. Proof of proposition 3 Let us rst assume that i = 0 then x A 0 = xb 0 and 2 T S (0; 0) = x B 2 0 + 9 t ; 4 T S (!;!) = x B 0 + 2 (1!) 2! (3 2!) 3 2! t 4 ; 22

T S (!; 0) = x B 0 + + t3 3 2! 2 6! + 9 (3!) 4 + t 4 2! 3 + 5! 2 + 24! 36 (! 3) 2 4 (3!) 2 (t (3!) 2 2 2 ) 2 + +4 6 (2!) (4!) + 2t 2 2 2! 4 8! 3 + 28! 2 63! + 54 (! 3) 2 4 (3!) 2 (t (3!) 2 2 2 ) 2 : Now, let us compute the di erences when i = 0. We rst note that in fact note that @T S (!;!) = T S (!;!) T S (0; 0) =!2 (3!) 9 (2! 3) 2 < 0; 2 (3 2!) 3 < 0. Hence, T S (!; 0) T S (0; 0) =! 46 (6!) + t 4 18! 2 77! 24 (! 3) 2 36 (! 3) 2 (9t 2 2 6t! + t! 2 ) 2 + +! 2t2 2 2 18! 24! 2 + 5! 3 27 (! 3) 2 9t 3 3! (! 3) 4 36 (! 3) 2 (9t 2 2 6t! + t! 2 ) 2 : Hence, sign(t S (!; 0) T S (0; 0)) = sign( 4 6 (6!) t 4 18! 2 77! 24 (! 3) 2 +2t 2 2 2 5! 3 24! 2 + 18! 27 (! 3) 2 9t 3 3! (! 3) 4 ): The rst, third and fourth terms are negative for the range of! considered. The second term is positive for the range of! considered. Hence when is large in relation to t then total surplus with asymmetric PPO may be larger than with no PPO, and hence also than with symmetric PPO. Another comparison is as follows: T S (!; 0) T S (!;!) =! 126 (2!) t 4 93! + 126! 2 60! 3 + 8! 4 + 24 (! 3) 2 4 (2! 3) 2 (! 3) 2 (9t 2 2 6t! + t! 2 ) 2 + +! 2t2 2 2 36! + 25! 3 14! 4 + 2! 5 + 27 (! 3) 2 t 3 3! (2! 3) 2 (! 3) 4 4 (2! 3) 2 (! 3) 2 (9t 2 2 6t! + t! 2 ) 2 : Note that the rst, second and third terms are positive while the fourth term is negative. Then, if is small in relation to t are large then the last term could potentially dominate the previous two terms. Hence, if some conditions are satis ed then total surplus with asymmetric PPO may be smaller than with symmetric PPO. 23

Proof of Corollary 2 T S (0; 0) = 1 2 Let us rst assume that i 6= 0 then (x A 0 + x B 0 ) + 32 A 9t 2 2 + 1 + 92 2 A 2 2 (9t 2 2 ) 2 9 t ; 4 T S (!;!) = (xa 0 + xb 0 ) + 2 (1!) 2 (3 2!) 2 2 (1!) t (2! 3) 2 + 4 2 2 A (2! 3)2 (t (2! 3) 2 2 2 ) 2 2! 3 2! t 4 + 2 A (3 2!) 2(t (2! 3) 2 2 2 ) : Total surplus with asymmetric PPO is beyond the scope of this paper. The comparison of the previous two suggests that T S (!;!) T S (0; 0) =!2 (3!) 9 (2! 3) 2 + +!2 i t2 4t 2 4! 3 + 12! 2 54! + 27 (9t 2 2 ) 2 (t (2! 3) 2 2 2 ) 2 +! 2 i t2 9t 2 2 (6 5!) (2! 3) 2 4 4 4! 2 13! + 12 + (9t 2 2 ) 2 (t (2! 3) 2 ; 2 2 ) 2 it sign can be re-written as sign(t S (!;!) T S (0; 0)) = sign 2 i where = 4t 2 4! 3 + 12! 2 54! + 27 + 9t 2 2 (6 5!) (2! 3) 2 4 4 4! 2 13! + 12 and = 2 (3!)(9t 2 2 ) 2 (t(2! 3) 2 2 2 ) 2 9t(2! 3) 2. Proof of Proposition 4. i) Symmetric PPO We rst need to calculate the second-best welfare optimal R&D investment taking as given the equilibrium prices at t=2. As such, we have that s = t(2! 3)+ A+(x A x B ) 2t(3 2!), and hence @s(!) @x A = 2t(2! 3) and @s(!) @x B = 2t(2! 3) :The rst order condition is then @T S @x A = @s @x A A + (x A x B ) @s @x A + s t @s (2s 2(1 s)) x A ; 2 @x A and the second order condition is @ 2 T S @x 2 A = 2 (5 4!) 2t (2! 3) 2 2t (2! 3) 2 < 0; which is equivalent to 2 2t(2! 3)2 @ < (5 4!) and note that 2 T S @x A @x B = 2 (5 4!) < 0. If the stability 2t(2! 3) 2 condition t (2! 3) 2 2 (5 4!) > 0. Hence, we obtain the reaction functions in the rst period 24

x i = i (5 4!) + t (2! 3) 2 2t (2! 3) 2 2 (5 4!) 2 (5 4!) 2t (2! 3) 2 2 (5 4!) x j: Then, nding the xed point we obtain the second-best e cient R&D investments: x i (!;!) = 1 + 2 As a special case, we nd that with no PPO i (5 4!) (t (2! 3) 2 2 (5 4!)) x i (0; 0) = 1 + A 2 (9t 2 ) ii) Comparison of the equilibrium and e cient R&D investments. Comparing the expression of the second best optimal R&D and expression with the equilibrium one, we obtain that x (!;!) which implies x i (!;!) = t2 2 (2! 3) 4 2 4 (5 4!)+t 2 (7 4!)(2! 3) 2 t 2 i (3 4!)(3 2!) 3, 2(3 2!)(t(2! 3) 2 2 2 )(t(2! 3) 2 2 (5 4!)) sign(x i (!;!) where s i = (t(2! 3)2 2 2 )(t(2! 3) 2 2 (5 4!)) t 2 (3 4!)(3 2!) 3 > 0. With no-cross ownership we obtain that x i (0; 0) x which implies x i (!;!)) = sign( s i + i ) i (0; 0) = 104 +63t 2 81t 2 ( i +t) 2(3 2!)(9t 5 2 )(9t 2 2 ), sign(x i (0; 0) x i (0; 0)) = sign( s0 i + i ); to! where s0 i = (9t 52 )(9t 2 2 ) > 0. 81t 2 We also notice that do comparative statics of the second-best optimal quantity with respect sign( @x i (!;!) ) = sign( i ): d! This implies that when i > 0 then increasing PPO increases the amount of e cient R&D investment. iii) Asymmetric PPO. We rst nd the e cient R&D investment under asymmetric PPO. Note that and hence @s(!) @x A = s = 2t(3!) 1 2t (3!) (t(3 2!) + A + (x A x B )) and @s(!) @x B = 2t(3!). The rst order condition is then @T S @x A = @s @x A A + (x A x B ) @s @x A + s t @s (2s 2(1 s)) 2 @x A x A 25

and hence @T S = A(5 @x A 2!) + 2 (x A x B )(5 2!) + t 2! 2 8! + 9 2t (3!) 2 2tx A (3!) 2 the second order condition is satis ed if @ 2 T S @x 2 A = 2 (5 2!) 2t (3!) 2 2t (3!) 2 < 0 and the stability condition requires that t(3!) 2 2 (5 2!) > 0. Note that @ 2 T S @x A @x B = 2 (5 2!) < 0. Hence, the best reaction function is 2t(3!) 2 x A = A(5 2!) + t 2! 2 8! + 9 (2t (3!) 2 2 (5 2!)) For the acquired rm we obtain the following 2 (5 2!) (2t (3!) 2 2 (5 2!)) x B @T S = A(5 2!) 2 (x A x B )(5 2!) t(9 4!) 2t (3!) 2 x B @x B 2t (3!) 2 2t (3!) 2 + 2t (3!) 2 2t (3!) 2 @ 2 T S @x 2 B = 2 (5 2!) 2t (3!) 2 2t (3!) 2 < 0 Then, from the rst order condition, we obtain the reaction function x B = A(5 2!) + t(9 4!) 2t (3!) 2 2 (5 2!) 2 (5 2!) 2t (3!) 2 2 (5 2!) x A Then, substituting one into the other, we obtain the e cient R&D investments with asymmetric PPO. x A (!; 0) = (5 2!)( A 2 ) + t 2! 2 8! + 9 2(t (3!) 2 2 (5 2!)) B (!; 0) = ( A + 2 )(5 2!) + t(9 4!) 2 (t(3!) 2 2 : (5 2!)) x The comparison leads to sign(x A(!; 0) x A (!; 0)) = sign( a A A ) where a A = t2 2 (2!+3)(3!) 3 +t 2 (!+1)(7 2!)(3!) 2 2 4 (5 2!) t 2 (3 2!)(3!) 3 and sign(x B(!; 0) x B (!; 0)) = sign ( B + a B) 26

where a B = t2 2 (3 4!)(3!) 3 +2 4 (5 2!) t 2 (1!)(7 2!)(3!) 2 t 2 (3 2!)(3!) 3 : Let us study the derivative of these two expressions with respect to!: and @x A (!; 0) = t 2 (! 2 5! + 5) + A (2!) (3!) t (3 2!) (3!) d! (t (3!) 2 2 (5 2!)) 2 @x B (!; 0) = t 2! 2 5! + 5 + A (2!) (3!) t (3 2!) (3!) d! (t (3!) 2 2 (5 2!)) 2 Hence, @x A (!; 0) @x B (!; 0) sign = sign = d! d! = sign 2 (! 2 5! + 5) t (3 2!) (3!) + A (2!) (3!) 27

Campus Barcelona Pedralbes Av. Pedralbes, 60-62 08034 Barcelona (España) T. +34 932 806 162 F. +34 932 048 105 Campus Barcelona Sant Cugat Av. de la Torreblanca, 59 08172 Sant Cugat del Vallés Barcelona (España) T. +34 932 806 162 F. +34 932 048 105 Campus Madrid Mateo Inurria, 25-27 28036 Madrid (España) T. +34 913 597 714 F. +34 917 030 062 www.esade.edu