Spillovers, Appropriability, and R&D

Vol. 75 (2002), No. 1, pp. 1–32

Spillovers, Appropriability, and R&D Stephen Martin Received July 20, 2000; revised version received December 5, 2000

I distinguish the impacts of input spillovers and imperfect appropriability of the revenue generated by cost-saving innovation in a racing model. Comparativestatic relationships with respect to the level of spillovers depend on the levels of post-innovation payoffs; comparative-static relationships with respect to the degree of appropriability depend on changes in the levels of post-innovation payoffs. Simulations illustrate conditions under which private payoffs and net social welfare are maximized for positive spillover levels and incomplete appropriability. The main qualitative results of the basic model hold if it is extended to include endogenous absorptive capacity. Keywords: innovation, spillovers, appropriability. JEL classification: O31.

1 Introduction The inputs to and the outputs from the innovative process have both, in varying degrees, the nature of public goods. This distinguishes private investments in innovation from other kinds of private investments, and the consequences are the subject of extensive empirical and theoretical literatures. On the input side, information about research activities at one firm trickles out and influences the research activities of other firms. In this way, any one firm benefits from research undertaken by other firms, as well as by its own research. Innovation input spillovers affect the cost to a single firm of achieving a given level of effective R&D effort. On the output side, a successful innovator is rarely able to fully control the ability of rivals to profit from his innovation. Rivals imitate new production techniques; they develop differentiated but competing

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varieties of new products. Limited ability to prevent rivals from exploiting an innovation limits the extent to which the innovator can fully appropriate the rents that flow from successful innovation. Although the mechanisms by which input-side and output-side spillovers affect private incentives to innovate are different, they are not always clearly distinguished. Amir (2000) makes the point that the muchgeneralized d’Aspremont and Jacquemin (1988) model of deterministic innovation, which deals with output spillovers, has quite different implications from the model of Kamien et al. (1992), which focuses on input spillovers. Furthermore, particularly in the policy literature, the public-good aspects of innovation inputs and outputs are often simply assumed to be imperfections, undesirable from a social point of view, and it is taken for granted that dynamic market performance would be improved if inputside and output-side leakages could be reduced or eliminated. In Sects. 2 and 3, I present a stochastic innovation model that includes and distinguishes between input-side and output-side spillovers. The use of a stochastic formulation is called for on the ground that uncertainty is an intrinsic element of the innovation process (Tapon and Cadsby, 1996, pp. 389–90, quoting a practitioner in the pharmaceutical sector): ‘‘There are thought experiments that you can do and you can try and predict outcomes but the actuality of the thing is always determined by the data generated. That sometimes is not as predictable as you would like it to be. You cannot think about all the variables that go into even the simplest of systems. . . . But, you’re not going to be able to predict 100%. . . of the outcome. You’re always going to have things that happen that nobody really foresaw and you look back in hindsight and say that there is no way that we could have predicted that outcome. . .’’ Use of a stochastic model also makes it possible to examine the impact of spillovers on the expected time to innovation, a question that cannot be examined in a deterministic framework. In this model, it is not generally true that either firms or society would prefer zero input spillovers or complete appropriability of the rents that flow from successful innovation. In Sect. 4, I examine firms’ incentives to cooperate in R&D, and the consequences of such cooperation for dynamic market performance.

Spillovers, Appropriability, and R&D

3

Results include that R&D joint ventures always leave consumers and society better off, because they ensure stronger product-market rivalry in the post-innovation market, but increase firm value (compared with noncooperative behavior) only if either input spillovers or output appropriability or both are low. In the basic model, I assume that firms are able to take full advantage of such leakages as occur – that firms have complete absorptive capacity (Cohen and Levinthal, 1989; Kamien and Zang, 2000). In Sect. 5, I relax this assumption, and suppose that a firm’s ability either to take advantage of rival R&D effort (before innovation) or to implement the discovery of another firm (after innovation) depends on its own R&D effort, which determines its ability to absorb information flows in a productive way. This yields a model which (for some parameter values) describes markets in which knowledge has a high tacit component. With limited absorptive capacity, pre-innovation R&D effort affects post-innovation payoffs, and firms set higher equilibrium R&D levels than in the basic model, to boost absorptive capacity. The qualitative properties of the basic model continue to hold when absorptive capacity is taken into account. 2 Setup I use a racing model of cost-saving innovation in quantity-setting duopoly.1 The two firms are denoted A and B. Initially, both firms produce with unit cost c1 . There is a target technology that permits production at a lower unit cost c2 , but it can be implemented only upon completion of an R&D project that has a random success date.2 Research efforts by one firm increase not only the probability that it will win the innovation race but also (in general to a lesser extent) the probability that the other firm will win the innovation race. There are thus spillovers on the input side of the R&D process. The first firm to successfully complete an R&D project (‘‘the winner’’) receives a patent on the use of the new technology. If the patent is completely effective, the winner can license use of the new technology to

1 I follow Loury (1979), Lee and Wilde (1980), Reinganum (1981, 1985), and Beath et al. (1988). For general discussions of the racing model of cost-saving innovation, see Reinganum (1989) or the appendix to Martin (1997). 2 Scott (1993, chap. 8) considers the case in which firms run multiple research projects directed toward the same goal.

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the loser at a rate c1 c2 per unit of output produced by the licensing firm.3,4 If the patent is only partially effective, the license fee is something less than c1 c2 per unit of output. In the latter case, there is imperfect appropriability on the output side of the R&D process, with zero appropriability when the loser has full use of the new technology without paying any royalty.5 The model thus allows for all possible combinations of varying degrees of R&D input spillovers and of varying degrees of appropriability of the rents that flow from successful innovation.

2.1 Appropriability and Post-Innovation Product Market Equilibrium Assume that the product is homogeneous and that there is a downwardsloping inverse demand curve p ¼ pðQÞ ;

ð2:1Þ

where Q is total output. In the pre-innovation market, the two firms have identical unit costs (c1 ) and play a noncooperative quantity-setting game. Equilibrium payoffs are pN ðc1 Þ per firm. In the post-innovation market, both firms produce with the new technology. The winner licenses use of the new technology to the loser for a fee aðc1 c2 Þ per unit, where 0 a 1. If the appropriability parameter a equals one, there is complete appropriability of the income generated by use of the new technology. Smaller values of a represent lesser degrees of appropriability. If a equals zero, the loser of the innovation race has full access to the new technology without payment of royalties to the winner.

3 For simplicity, I assume an infinitely-lived patent. One might relax this assumption and examine the impact of input spillovers and limited appropriability on optimal patent design. 4 The modeling of spillovers is similar to that of d’Aspremont and Jacquemin (1988), who do not, however, allow for the possibility of licensing. The model of patent royalties used here might be extended to include a fixed fee as well as a fee per unit of output. 5 With completely ineffective patents, the innovator will collect equilibrium duopoly profits in a post-innovation market where both firms have access to the new technology on the same terms. Even with completely effective patents, the innovator cannot prevent the follower from using the old technology, if the follower should find it profitable to do so.


5

a is the degree of appropriability of the rents that flow from innovation; 1 a is the size of spillovers of innovative output. The degree of appropriability is subject to influence of policymakers, but only within limits. The patent system is not as effective in ensuring control of new products or processes as is sometimes assumed in theoretical models. Further, patent rights are not self-enforcing, but require costly legal action by patentholders to assert rights in the event of (alleged) violations. The unit cost of the losing firm is a weighted average of pre- and postinnovation unit costs: cL ¼ c2 þ aðc1 c2 Þ ¼ ac1 þ ð1 aÞc2 :

ð2:2Þ

Post-innovation static payoff functions are6 pW ¼ ðp c2 ÞqW þ aðc1 c2 ÞqL

ð2:3Þ

pL ¼ ðp cL ÞqL :

ð2:4Þ

and

I restrict the discussion to non-drastic innovation. That is, I assume that qL > 0.7 If the losing firm withdraws from the market, incomplete appropriability in the post-innovation market becomes a moot question. It is shown in the Appendix that Lemma 1: opW > 0; oa

opL 0; z00 ðhÞ > 0 :

ð2:8Þ

When firms behave noncooperatively with respect to innovation as well as in the product market, firm i picks its R&D intensity hi to maximize its expected present discounted value, N

Vi ¼

Z

1

ðrþg1 þg2 Þt

e t¼0

gi pW þ gj pL dt pN ðc1 Þ zðhi Þ þ r

ð2:9Þ

(for i, j ¼ A, B and j 6¼ i), taking the R&D level of the other firm as given. The terms under the integral sign can be explained as follows. The probability density that no firm has successfully innovated at time t is exp½ðg1 þ g2 Þt, and in this event firm i has the instantaneous payoff pN ðc1 Þ zðhi Þ. The probability density that firm i is the first to innovate and that this occurs at time t is gi exp½ðg1 þ g2 Þt, and in this event firm i’s value from the moment of innovation is pW =r. The probability density that firm j is the first to innovate and that this occurs at time t is gj exp½ðg1 þ g2 Þt, and in this event firm i’s value from the moment of innovation is pL =r. All expected payoffs are discounted by the factor ert . Carrying out the integration, firm i’s expected present discounted value function is ViN ¼

g pW þgj pL r

pN ðc1 Þ zðhi Þ þ i r þ g1 þ g2

ðp þsp Þh þðspW þpL Þhj

¼

pN ðc1 Þ zðhi Þ þ W L i r r þ ð1 þ sÞðhi þ hj Þ

;

ð2:10Þ

(for i, j ¼ A, B and j 6¼ i), using (2.6) to express the firm’s expected value in terms of research intensities, which are the firms’ choice variables.

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3 Noncooperative R&D 3.1 R&D Best Response Functions The first-order condition to maximize (2.10) can be written9 ðp þsp Þh þðspW þpL Þhj

pN ðc1 Þ zðhi Þ þ W L i r r þ ð1 þ sÞðhA þ hB Þ h i 1 pW þ spL z0 ðhi Þ : ¼ 1þs r

ViN ¼

ð3:1Þ

Differentiating the first-order condition gives the slope of firm i’s R&D-intensity best response function, L ð1 sÞ pW p z0 ðhi Þ ohi r ¼ ; r ohj þ hA þ hB z00 ðhi Þ

ð3:2Þ

1þs

(for i, j ¼ A, B and j 6¼ i). The second-order condition for expected value maximization implies that the denominator is positive. If it is privately profitable for firms to undertake R&D, Eq. (7.19) (in the Appendix) implies that the slope of the R&D best response function is positive for s near zero, negative for s near one. See Fig. 1 for examples of both cases. Stability requires that the slope of the best response function be less than one in absolute value in the neighborhood of equilibrium (Seade, 1980). Setting hi ¼ hj ¼ hN in (3.2), the stability condition is D¼

h i r pW pL þ 2hN z00 ðhN Þ ð1 sÞ z0 ðhN Þ > 0 ; 1þs r ð3:3Þ 10

which is henceforth assumed.

9 The second-order condition for a maximum is that z00 ðhi Þ > 0, and is met by assumption. 10 The stability condition is satisfied for the simulation results presented below.


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Fig. 1. R&D best response curves, alternative spillover levels (ai ¼ 110, c1 ¼ 10, c2 ¼ 5, b ¼ 1, r ¼ 1=10, zðhÞ ¼ 10h þ 1000h2 ; Að23 ; 13Þ indicates A’s best response function for s ¼ 23, a ¼ 13)

3.2 Comparative Statics: R&D Intensity Setting hi ¼ hj ¼ hN in (3.1) and rearranging terms, noncooperative equilibrium R&D intensity satisfies the condensed first-order condition pW þ spL pW pL pN þ ð1 sÞ hN þ zðhN Þ 1þs r r þ 2hN z0 ðhN Þ ¼ 0 : 1þs

ð3:4Þ

It is shown in the Appendix that equilibrium R&D intensity rises as input spillovers fall and as appropriability rises. This is Lemma 2: ohN 1 r pL pW pL hN VN ¼ þ 0: D 1 þ s oa oa oa oa oa r ð3:6Þ Lemma 2 holds for general functional forms. It is illustrated in Figs. 1 and 2 for the case of linear inverse demand,

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Fig. 2. R&D best response curves, alternative appropriability levels (ai ¼ 110, c1 ¼ 10, c2 ¼ 5, b ¼ 1, r ¼ 1=10, zðhÞ ¼ 10h þ 1000h2 ; Að13 ; 23Þ indicates A’s best response function for s ¼ 13, a ¼ 23)

p ¼ a bQ ;

ð3:7Þ

and a quadratic R&D cost function, zðhÞ ¼ uh þ vh2

ð3:8Þ

(with u 0, v > 0). Details of the algebra of this linear demand-quadratic R&D cost version of the model are given in the Appendix; the parameter values used in Figs. 1 and 2 are given in the captions. The cost saving that is considered for these figures is modest, reducing unit cost from 10 to 5 with a reservation price of 110. The appropriability parameter a equals 1/3 for all the R&D best response curves is shown in Fig. 1. As the input spillover rate s rises from 1/3 to 2/3, firm A’s R&D best response curve shifts left (and, incidently, changes from positive to negative slope), implying a lower R&D level for any R&D level of firm B. Firm B’s best response curve shifts in a similar way. As the spillover level rises, equilibrium R&D intensities fall from the higher levels of equilibrium E1 to the lower levels of equilibrium E2 . This is the first part of Lemma 2. The spillover rate s equals 1/3 for all the best response curves shown in Fig. 2. As a rises from 1/3 to 2/3, firm A’s R&D best response curve shifts right, implying a higher R&D level for any R&D level of firm B.


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Firm B’s R&D best response curve shifts in the same way. As the appropriability level rises, equilibrium R&D intensities rise from the lower R&D levels of equilibrium E1 to the higher levels of equilibrium E3 . This is the second part of Lemma 2. The equilibrium expected time to discovery is E¼

1 : 2ð1 þ sÞh

ð3:9Þ

It follows from Lemma 2 that oE=os is of ambiguous sign, while oE=oa < 0. Increasing post-innovation appropriability reduces the expected time to discovery. With greater R&D input spillovers, equilibrium per-firm R&D effort is less, but such R&D as does occur is more effective, from a social point of view: holding R&D levels constant, greater spillovers mean that it is more likely that some firm will make a discovery, all else equal. Expected time to discovery may be less with greater input spillovers (and with linear demand and quadratic R&D cost there typically being large regions of ðs; aÞ-space where this is the case). 3.3 Comparative Statics: Firm Value From (3.1), equilibrium firm value is VN ¼

i 1 hpW þ spL z0 ðhN Þ : 1þs r

Differentiating (3.10) with respect to a gives oVN opW opL ohN ¼ þs z00 ðhN Þ : ð1 þ sÞ oa oa oa oa

ð3:10Þ

ð3:11Þ

The expression in parentheses on the right is positive for general functional forms (see A.22 in the Appendix). From Lemma 2 and the second-order condition from the firm’s value-maximization problem, the second term is also positive. It follows that oVN =oa is of a priori ambiguous sign. Consideration of simple examples, as in Fig. 3, shows that oVN =oa may in fact take positive and negative values, and may be maximized for values of a that lie strictly between 0 and 1. The qualitative prop-

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Fig. 3. Equilibrium firm value–appropriability relationship, alternative spillover levels (ai ¼ 110, c1 ¼ 10, c2 ¼ 5, b ¼ 1, r ¼ 1=10, u ¼ 10, v ¼ 1000; firm value measured in thousands)

erties of this example are robust to a wide range of variations in parameter values.

With linear demand and quadratic R&D cost, oVoaN a¼0 > 0; increases in appropriability from zero increase expected firm value. But oVN =oa falls as a rises. For low spillover levels, oVN =oa falls so rapidly as a rises that VN reaches a maximum for an interior value of a, then declines. If there are low spillover levels, firm value is maximized when there is incomplete appropriability. When the spillover of R&D effort is low, a firm benefits primarily from its own R&D effort. When there is only partial appropriability of R&D output, a firm will benefit from its rival’s discovery in the post-innovation market, if the rival should discover first, even though it does not benefit from the rival’s R&D effort in the pre-innovation market. If there are low pre-innovation spillovers, firms’ values are maximized with incomplete post-innovation appropriability. Higher spillover levels reduce the magnitude of o2 VN =oa2 . For high spillover levels firm value is maximized when a equals one. If there is substantial leakage of R&D effort among firms, firm value is maximized when there is complete control over the innovation, once it has been realized. If both s and a could be varied, equilibrium firm value would be maximized for a ¼ 1 and s ¼ 0:91. Globally, firm value is maximized when there are high pre-innovation spillovers and high post-innovation appropriability.


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Fig. 4. Equilibrium expected consumers’ surplus–appropriability relationship, alternative spillover levels (ai ¼ 110, c1 ¼ 10, c2 ¼ 5, b ¼ 1, r ¼ 1=10, u ¼ 10, v ¼ 1000; expected consumers’ surplus measured in thousands)

3.4 Comparative Statics: Consumers’ Surplus Figure 4 shows the equilibrium expected consumers’ surplus-appropriability relationship for different spillover levels. Whether spillovers are high or low, consumers’ surplus is maximized at low appropriability levels. If spillovers are low, consumers’ surplus is, maximized at low but positive appropriability levels. If there are high spillover levels, consumers’ surplus is maximized when there is zero appropriability. Consumers are best off if there is only limited control over use of the new technology after discovery, because this ensures that they have access to the product at the lowest possible equilibrium price, taking into account the imperfectly competitive nature of the product market. Overall, consumers’ surplus is maximized when a ¼ 0, s ¼ 1. 3.5 Comparative Statics: Net Social Welfare Firm value is maximized for high R&D input spillovers and intermediate or high R&D output appropriability. Consumers’ surplus is maximized for high R&D input spillovers and low R&D output appropriability. Unless input spillovers are very high, net social welfare (the sum of firm value and the expected present-discounted value of consumers’ surplus) is maximized with intermediate appropriability (Fig. 5). In such environments, high appropriability increases firm value, but leaves consumers so

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Fig. 5. Equilibrium expected net social welfare–appropriability relationship, alternative spillover levels (ai ¼ 110, c1 ¼ 10, c2 ¼ 5, b ¼ 1, r ¼ 1=10, u ¼ 10, v ¼ 1000; net social welfare measured in thousands)

much worse off that overall welfare is maximized by intermediate appropriability. When input spillovers are very high, net social welfare is maximized with zero appropriability. If high spillovers mean that firms have only a limited incentive to invest in cost-saving innovation, society is best off if the cost-saving technology is widely disseminated after discovery. Two factors combine to make it profitable for firms to invest in innovation even if patents are completely ineffective in giving property rights to an innovation ða ¼ 0Þ. First, if a ¼ 0 each firm will profit if some firm discovers – duopoly profit with lower unit cost is greater than duopoly profit with higher unit cost. Second, for any R&D level of the other firm (and in particular if the other firm does no R&D), with uncertain innovation there is a positive marginal value to the firm making some R&D effort of its own, which reduces the expected time to discovery.11

11 Reinganum (1982) works with a finite-horizon model, and in this case even if both firms undertake R&D, there is some probability that innovation does not occur. If one firm does R&D, R&D by the other reduces the probability that innovation does not occur (Reinganum, 1982, p. 681). With an infinite horizon and the assumed distribution for discovery time, the probability that discovery occurs at some time is one. One firm could in principle wait for the other to discover and free ride on the innovation, once it occurs. For a given level of R&D by one firm, however, the most profitable option for the other is to carry out some R&D of its own, reducing the expected time to discovery.


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If a and s could both be varied, net social welfare would be maximized for a ¼ 0, s ¼ 1.

4 Spillovers, Appropriability, and R&D Joint Ventures Ouchi (1989), and Vonortas (1994) distinguish two types of R&D joint ventures. In duopoly, the two types are:12 • an operating entity RJV: the two firms establish a common R&D laboratory, each pays half the cost, and both firms have access to the results; • a secretariat RJV: the two firms carry out independent R&D activities and share the results. In each case, both firms have access to the new technology after discovery, neither firm pays royalties to the other, and firms compete as Cournot duopolists in the post-innovation market. Here I present results for secretariat R&D joint ventures. Secretariat joint ventures always yield a greater expected present discounted firm value than operating entity joint ventures.13 The region of ðs; aÞ-space in which operating entity joint ventures yield a greater expected present discounted value than noncooperative R&D is a strict subset of the corresponding region for secretariat joint ventures. Expected net social welfare with a secretariat joint venture is always greater than expected net social welfare with an operating entity joint venture.14 I assume also that formation of an R&D joint venture means that the input spillover level changes from its noncooperative level to s ¼ 1: if firms form a secretariat

12 The literature contains several typologies of R&D joint ventures, defined among other ways according to the number of R&D operations maintained by cooperating firms, on whether or not formation of an R&D joint venture means an increase in the spillover parameter, and on whether or not firms cooperate in production as well as R&D. See d’Aspremont and Jacquemin (1988), Hagedoorn (1990), Kamien et al. (1992), and Amir (2000). 13 Given the increasing-cost nature of the R&D cost function, this is to be expected; even leaving the spillover unchanged from its ‘‘natural’’ value, the cost to the two firms of achieving a given effective level of R&D effort is less if there are two R&D operations rather than one. 14 An appendix describing ðs; aÞ-performance relationships for operating entity joint ventures is available on request from the author.

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joint venture, they fully share information generated during the R&D process.15

4.1 Firm Value With a secretariat joint venture, each firm maintains its own R&D project. There are complete R&D input spillovers. Since both firms have equal access to the innovation, once it is realized, the degree of appropriability has no impact on post-innovation payoffs. Firm A’s expected present discounted value is pN ðc2 Þ dt ¼ e pN ðc1 Þ zðhA Þ þ ðgSA þ gSB Þ r t¼0 1 pN ðc2 Þ ¼ ; ð4:1Þ pN ðc1 Þ zðhA Þ þ 2ðhA þ hA Þ r þ 2ðhA þ hB Þ r Z

VSA

1

ðrþgSA þgSB Þt

where firm A’s effective R&D intensity is gSA ¼ hA þ hB :

ð4:2Þ

Firm 1 maximizes VSA by choice of hA , taking hB as given. This determines firm 1’s R&D best response function. Firm 2 behaves in an analogous way. Equilibrium R&D levels are found at the intersection of the R&D best response curves. Equilibrium R&D levels, firm value, consumer surplus, and net social welfare with a secretariat R&D joint venture depend on neither s nor a. Noncooperative equilibrium R&D and welfare levels do depend on s and a. Figure 6 compares noncooperative equilibrium firm value and equilibrium firm value with a secretariat joint venture in ðs; aÞ-space. Cooperative R&D yields firms a greater equilibrium firm value in two approximately triangular, overlapping regions, low appropriability for all

15 See Katsoulacos and Ulph (1998), Cassiman and Veugelers (1998). An alternative specification is that s is unchanged by formation of an R&D joint venture. There is some case study evidence consistent with this; see Sigurdson’s (1986, pp. 45–6) discussion of Japan’s VLSI project. The alternative specification is analyzed in an appendix available on request from the author.


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Fig. 6. Comparison of noncooperative and secretariat joint venture equilibrium firm value (a ¼ 110, c1 ¼ 10, c2 ¼ 5, b ¼ 1, r ¼ 1=10, zðhÞ ¼ 10h þ 1000h2 )

spillover levels and low spillover levels for all appropriability levels. In the bulk of ðs; aÞ-space, firms have a greater value with independent than with cooperative R&D, and one would not expect firms to voluntarily form R&D joint ventures. One benefit of a secretariat joint venture is that spillovers rise from its noncooperative level to one. This benefit becomes smaller as s rises. One cost of a secretariat joint venture is the giving up of expected royalties, in the event that a firm should discover first. This cost is greater, the greater is a. Noncooperative R&D is therefore privately preferred to cooperative R&D for sufficiently high values of s and/or a. 4.2 Consumer Surplus and Net Social Welfare Consumer surplus is greater with a secretariat R&D joint venture than with noncooperative R&D for all s and a. In large measure, this reflects the fact that with a secretariat joint venture, both firms have equal access to the new technology in the post-innovation market. Post-innovation product market performance is better with a secretariat joint venture, and this leaves consumers better off than they would be with noncooperative R&D. Net social welfare is also greater with a secretariat R&D joint venture than with noncooperative R&D for all s and a. This implies that in much

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of ðs; aÞ-space, firms will not voluntarily form R&D joint ventures that would be socially beneficial. This result is one justification for policy measures to encourage the formation of R&D joint ventures.16

5 Absorptive Capacity To this point, I have assumed that firms are able to fully take advantage of the input- and output-leakages that come their way. But as argued by Cohen and Levinthal (1989), and Kamien and Zang (2000), we can think of a firm’s own research efforts as generating a firmspecific stock of knowledge that determines its capacity to absorb preinnovation information flows and apply them to its own R&D projects or to apply post-innovation technology produced by other firms. To extend the basic model to include absorptive capacity, suppose that a firm’s ability to take advantage of information flows is measured by an absorption function mðhÞ, with m0 ðhÞ 0, m00 ðhÞ 0, 0 mðhÞ 1, and limh!1 mðhÞ ¼ 1. A firm can absorb only a fraction of the information that comes its way from other firms, and that fraction depends on the intensity of its own R&D efforts.17 In the pre-innovation market, effective R&D intensities are gA ¼ hA þ mðhA ÞshB and gB ¼ hB þ mðhB ÞshA :

ð5:1Þ

Thus a fraction shB of firm B’s research effort spills over to the public domain. Firm A is able to absorb a fraction mðhA Þ of this spillover and effectively augment its own research effort accordingly. In the post-innovation market, the winner’s unit cost is c2 < c1 . The winner’s cost reduction is c1 c2 . The losing firm is able to absorb a fraction mðhL Þ of this cost reduction, reducing its production cost by mðhL Þðc1 c2 Þ. The winner is able to appropriate a fraction a of this cost reduction in license fees. The loser’s unit cost in the post-innovation market is

16 The result that formation of a secretariat joint venture improves net social welfare depends on the assumption that firms behave noncooperatively in the post-innovation market. 17 In a more general specification, one might allow a firm’s ability to absorb R&D input spillovers to differ from its ability to absorb R&D output spillovers.


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c1 ð1 aÞmðhL Þðc1 c2 Þ ¼ c2 þ ½1 ð1 aÞmðhL Þðc1 c2 Þ ; ð5:2Þ with amðhL Þðc1 c2 Þ of this unit cost representing a payment from the loser to the winner. Unit costs determine payoffs in the post-innovation market. In the basic model, R&D efforts determine the expected time of transition from the pre-innovation to the post-innovation market, but do not affect the magnitude of payoffs in the post-innovation market. In the model with absorptive capacity, there is a link between pre-innovation R&D efforts and post-innovation payoffs: the greater a firm’s pre-innovation R&D intensity, the greater its post-innovation payoff if it should happen to lose the innovation race. Write • pAA ¼ firm A’s post-innovation payoff if firm A wins the innovation race; • pAB ¼ firm A’s post-innovation payoff if firm B wins the innovation race. Firm A’s expected present discounted value is then

VA ¼

¼

B pAB pN ðc1 Þ þ gA pAA þg r r þ gA þ gB ogB opAB 1 ogA 0 p þ p þ g AA AB B r ohA ohA ohA z ðhA Þ

o ohA

ðgA þ gB Þ

;

ð5:3Þ

where the equation given by the first equality is the definition of the firm’s expected present discounted value and the equation given the second equality is the first-order condition to maximize expected value with respect to hA . 5.1 Noncooperative R&D 5.1.1 Equilibrium R&D Intensity A one-parameter functional form with the properties assumed for the absorption function is

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Table 1. Nash equilibrium characteristics, a ¼ 0:25, / ¼ 1, alternative spillover levels (a ¼ 110, c1 ¼ 10, c2 ¼ 5, b ¼ 1, r ¼ 1=10, zðhÞ ¼ 10h þ 1000h2 ) s

h

m

V

CS

NSW

0 1/3 2/3 1

0.36 0.42 0.49 0.53

0.27 0.30 0.33 0.37

11624 11645 11665 11686

23409 23466 23522 23576

46658 46755 46853 46949

mðhÞ ¼ 1

1 /h ¼ : 1 þ /h 1 þ /h

ð5:4Þ

Greater R&D intensity h increases mðhÞ and the ability of the firm to take advantage of input spillovers or realized cost savings. The parameter / determines the efficiency of R&D effort in creating absorptive capacity. Large values of / mean that relatively low levels of R&D effort create near-complete ability to absorb information flows: the absorption model approaches the basic model as / goes to infinity. With low values of /, increases in the spillover rate s result in modest increases in equilibrium firm value, consumers’ surplus, and net social welfare. This is apparent from Table 1, which reports equilibrium characteristics for a ¼ 0:25, / ¼ 1, and four different values of the innovation effort spillover level s. The qualitative properties of the various equilibria as a varies are similar for all s, and I discuss here changes in equilibrium characteristics as a varies for s ¼ 2=3.18 Figure 7 shows the equilibrium R&D intensity, absorptive capacity, and expected time to discovery for s ¼ 2=3 as post-innovation appropriability a varies from zero to one. Equilibrium R&D intensity rises as a rises, which is the same relationship as for the basic model (Lemma 2), and for the same underlying reason: as the ability of the innovator to collect royalties for use of the innovation rises, the profit from successful innovation rises as well. Absorptive capacity also rises as a rises, which follows from m0 ðhÞ > 0. Expected time to discovery19 falls as appropriability rises.

18 Other cases are described in an appendix that is available on request from the author. 19 Here, 1=½2ð1 þ smðhÞÞh.


21

Fig. 7. Equilibrium R&D intensity, absorption capacity, expected time to discovery; (a ¼ 110, c1 ¼ 10, c2 ¼ 5, b ¼ 1, r ¼ 1=10, zðhÞ ¼ 10h þ 1000h2 , s ¼ 23, / ¼ 1)

Equilibrium firm value rises as a rises: with endogenous absorptive capacity, firms prefer complete appropriability. Expected consumers’ surplus falls as a rises: with endogenous absorptive capacity, consumers prefer zero appropriability. For these parameter values, expected net social welfare is maximized for internal values of a, approximately a ¼ 0:25 for all values of s. With low values of /, the impact of changes in a on these equilibrium values is consistent in its nature, but slight in magnitude. An increase in a from 0 to 1 raises equilibrium firm value almost 1 per cent, with a fall in expected consumers’ surplus of slightly more than 1 per cent. This is illustrated in Table 2, which reports equilibrium firm value, consumer surplus, and net social welfare for s ¼ 2=3, / ¼ 1, and different values of the appropriability parameter a. 5.2 Secretariat Joint Venture Suppose once again that formation of a secretariat joint venture implies complete spillover of pre-innovation R&D effort and that the firm that discovers first does not receive royalty payments from the other firm for access to information about the innovation. The logic of the absorption model nonetheless implies that the firm which does not discover first will be able to utilize the information it

22

S. Martin

Table 2. Equilibrium values, s ¼ 2=3, / ¼ 1, alternative appropriability levels (a ¼ 110, c1 ¼ 10, c2 ¼ 5, b ¼ 1, r ¼ 1=10, zðhÞ ¼ 10h þ 1000h2 ) a

V

CS

NSW

0.0 0.1 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

11641 11651 11660 11665 11670 11681 11691 11702 11713 11725 11736 11748

23566 23550 23532 23522 23511 23487 23460 23431 23398 23363 23325 23285

46849 46852 46853 46853 46852 46848 46843 46835 46824 46812 46798 46781

receives about the innovation only to the extent that it has the absorptive capacity to do so. With a secretariat joint venture, the non-discovering firm’s unit cost in the post-innovation market is then c1 mðhL Þðc1 c2 Þ ¼ c2 þ ½1 mðhL Þðc1 c2 Þ :

ð5:5Þ

Firm A’s expected outputs in the post-innovation market are qSAA ¼

a c2 þ ½1 mðhB Þðc1 c2 Þ 3b

ð5:6Þ

a c2 2½1 mðhA Þðc1 c2 Þ 3b

ð5:7Þ

if it discovers first, and qSAB ¼

if firm B discovers first, where the superscript S denotes the secretariat joint venture case. The equilibrium outputs determine equilibrium instantaneous payoffs on the post-innovation market. Firm A’s expected present discounted value is S

V ðhA ; hB Þ ¼

b

ac 2 1

3b

g pSAA þgSB pSAB r

uhA vh2A þ SA r þ gSA þ gSB

;

ð5:8Þ


23

Table 3. /-equilibrium relationship, secretariat joint venture (a ¼ 110, c1 ¼ 10, c2 ¼ 5, b ¼ 1, r ¼ 1=10, zðhÞ ¼ 10h þ 1000h2 ) /

h

mðhÞ

VS

CS S

NSW S

0.5 1 2.5 5 10 100

0.3248 0.5052 0.6259 0.5793 0.5050 0.3476

0.14 0.34 0.61 0.74 0.83 0.97

11588 11660 11801 11897 11969 12073

23354 23627 23961 24108 24201 24315

46530 46946 47564 47902 48138 48461

where pSAj denotes firm A’s instantaneous payoff in the post-innovation market if firm j discovers first, for j ¼ A, B. Each firm sets its R&D level noncooperatively to maximize its expected present discounted value, taking the R&D level of the other firm as given. The equilibrium R&D levels determine equilibrium firm values, expected consumer surplus, and expected net social welfare. These values depend neither on s nor on a. Table 3 reports equilibrium characteristics for selected values of /. For very low values of /, knowledge is highly tacit and a firm benefits very little from any R&D effort except its own. Equilibrium h levels are small, as are equilibrium absorption levels. As / rises from low levels, equilibrium R&D intensity rises as well. Increases in / and increases in h both act to increase equilibrium absorptive capacity. This allows the firm that does not discover first to take greater advantage of the new technology. As / rises from about 2.5, equilibrium R&D intensity begins to fall. Equilibrium absorptive capacity continues to rise as / rises. Equilibrium firm value, consumer surplus, and net social welfare all rise as / rises. This is to be expected, since larger values of / mean the R&D effort is more effective in creating absorptive capacity, all else equal. In a market system, firms would voluntarily form a secretariat joint venture (competition policy permitting) if it yields a greater expected value than noncooperative R&D. Firms’ incentives to form R&D joint ventures in the large-/ case will be similar to those of the basic model (complete absorptive capacity). Figure 8, which corresponds to Fig. 6 for the basic model, compares noncooperative and cooperative equilibrium firm payoffs for / ¼ 1. Equilibrium firm value with a secretariat R&D joint venture is 11,660.

24

S. Martin

Fig. 8. Comparison of noncooperative and secretariat joint venture equilibrium firm value, absorptive capacity model (a ¼ 110, c1 ¼ 10, c2 ¼ 5, b ¼ 1, r ¼ 1=10, u ¼ 10, v ¼ 1000, / ¼ 1)

With s ¼ 0, non-cooperative R&D equilibrium firm value is 11,660 for a ¼ 0:69; the VN ðs; aÞ ¼ 11660 locus slopes down as s rises. For s ¼ 1, noncooperative R&D yields the same expected firm value as cooperative R&D for a ¼ 0, and greater otherwise. For s ¼ 1, firms will prefer noncooperative R&D if the degree of post-innovation appropriability is positive. When appropriability is low, a first innovator has little to gain in terms of post-innovation royalty fees. It is the part of expected postinnovation profit that consists of such royalty fees that is given up by formation of a secretariat joint venture. The benefit that comes from setting up a secretariat joint venture is the increase in spillovers from the independent level s to 1, and this benefit is greatest for small values of s. For any s < 1, the benefit of forming a secretariat joint venture outweighs the cost for sufficiently low a. For s ¼ 1, firms get full innovation input spillovers without an R&D joint venture, and prefer noncooperative R&D. As in the basic model, expected consumers’ surplus and expected net social welfare are greater with a secretariat joint venture than with noncooperative R&D. The underlying reason is, once again, that a secretariat joint venture implies greater rivalry and better static market performance in the post-innovation market.


25

6 Conclusion The stochastic innovation model presented here distinguishes and considers simultaneously input spillovers and output spillovers. High R&D input spillovers and low appropriability of the rents that flow from successful innovation both reduce the profit that flows from successful innovation, but for different reasons. High input spillovers mean that a firm’s R&D effort contributes to its rivals’ R&D programs. Low output appropriability means that a firm profits less from successful innovation if it is the first to innovate, all else equal. In the post-innovation market, a successful innovator’s payoff increases if there is greater appropriability, all else equal. But in industries where input spillovers are low, firm value is maximized at low appropriability levels: a greater chance to benefit from another firm’s discovery after innovation compensates for a limited possibility of benefiting from another firm’s R&D effort before innovation. Consumer welfare is generally greatest when appropriability is low – zero, in fact, unless spillovers are also low. Consumers are better off if firms have access to the new technology on nearly equal terms in the postinnovation market. With linear demand and quadratic R&D cost, net social welfare is maximized when input spillovers are high, ensuring that R&D effort by one firm is socially effective in the sense that it implies a great increase in the probability that some firm will innovate, and when appropriability is low, ensuring rivalry on nearly equal terms in the post-innovation market. Joint R&D always improves expected social welfare, because it increases R&D input spillovers and ensures greater product-market rivalry in the post-innovation market. There are large parts of parameter space – high spillovers, high appropriability – in which firms will not noncooperatively form secretariat joint ventures that would be socially beneficial. The qualitative properties of the basic model continue to hold if the model is generalized so that a firm’s absorptive capacity depends on its own R&D efforts. In this case, firm value with noncooperative R&D is maximized with complete appropriability, expected consumer surplus with zero appropriability, and expected net social welfare for low values of appropriability. Policy measures that improve firms’ ability to create absorptive capacity will improve net social welfare.

26

S. Martin

Appendix 1. Proof of Lemma 1 The first- and second-order conditions for profit maximization are opW ¼ pðqW þ qL Þ c2 þ qW p0 ¼ 0 ; oqW

ðA:1Þ

o2 pW ¼ 2p0 þ qW p00 < 0 oq2W

ðA:2Þ

for the winner and opL ¼ pðqW þ qL Þ ac1 ð1 aÞc2 þ qL p0 ¼ 0 ; oqL o2 pL ¼ 2p0 þ qL p00 < 0 oq2L

ðA:3Þ

ðA:4Þ

for the loser. Assume that firms’ outputs are strategic substitutes: o2 pW < 0; oqW oqL

o2 pL 0: D1 ¼ oqW oqL oqW oqL oq2W oq2L

ðA:7Þ


27

Then from (A.6) oqW c 1 c 2 o 2 pW >0 ¼ oa D1 oqW oqL

oqL c1 c2 o2 pW < 0 : ðA:8Þ ¼ oa D1 oq2W

Now write pW ¼ pW ½qW ðaÞ; qL ðaÞ; a

ðA:9Þ

pL ¼ pL ½qW ðaÞ; qL ðaÞ; a :

ðA:10Þ

Differentiate (A.9) with respect to a to obtain opW opW oqW opW oqL opW oqL ¼ þ þ pW ;3 ¼ þ pW ;3 oa oqW oa oqL oa oqL oa

ðA:11Þ

(using the envelope theorem). Since opW ¼ p0 qW þ aðc1 c2 Þ ¼ ðp c2 Þ þ aðc1 c2 Þ oqL (using the first-order condition for qW , (A.1)) ¼ ðp cL Þ < 0 ;

ðA:12Þ

pW ;3 ¼ ðc1 c2 ÞqL > 0 ;

ðA:13Þ

opW opW oqL ¼ þ pW ;3 > 0 : oa oqL oa

ðA:14Þ

and

we have

In like manner opL ¼ ðp cL Þ < 0; oqW

pL;3 ¼ ðc1 c2 ÞqL < 0 :

ðA:15Þ

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S. Martin

opL opL oqW opL oqL opL oqW ¼ þ þ pL;3 ¼ þ pL;3 < 0 : oa oqW oa oqL oa oqW oa

( ðA:16Þ

2 Proof of Lemma 2 (i) Differentiate (3.4) with respect to s and rearrange terms to obtain D

hp p i p p oh r W L W L 0 ¼ z h: ðhÞ þ os ð1 þ sÞ2 r r

ðA:17Þ

Rewrite the second equality on the right in (3.1), the equation that defines equilibrium research intensity, as ViN ¼

i pL 1 hpW pL þ z0 ðhi Þ : 1þs r r

ðA:18Þ

If it is privately optimal for the firm to undertake R&D, ViN > pL =r. Assuming this to be the case, we have pW pL pL z0 ðhi Þ ¼ ð1 þ sÞ ViN >0 r r

ðA:19Þ

along firm i’s first-order condition and in particular in equilibrium. Substituting (A.19) into ( A.17) gives (3.5). (ii) Differentiate (3.4) with respect to a and rearrange terms to obtain (3.6). From (A.14) and (A.16), opW opL opW oqL opL oqW þs ¼ þ pW ;3 þ s þ spL;3 oa oa oqL oa oqW oa oqL oqW þs þ ð1 sÞðc1 c2 ÞqL : ¼ ðp cL Þ oa oa

ðA:20Þ

The final term on the right is nonnegative. From (A.8), oqL oqW c1 c2 o2 pW o2 pW þs ¼ s 0 :

ðA:22Þ

This is used in discussion of (3.11). It follows from (A.14), (A.16), and (A.22) that oh 1 r opW opL opW opL h ¼ >0: þs þ ð1 sÞ oa D 1 þ s oa oa oa oa r

(

ðA:23Þ 3 Linear Demand, Quadratic R&D Cost Function Let the inverse demand curve be linear, (3.7), and the R&D cost function quadratic, (3.8). Equilibrium pre-innovation outputs and payoffs are qN ðc1 Þ ¼

a c1 ; 3b

pN ðc1 Þ ¼ bq2N ðc1 Þ :

ðA:24Þ

Equilibrium post-innovation outputs and payoffs are qW ¼

a c2 þ aðc1 c2 Þ ; 3b

qL ¼

a c2 2aðc1 c2 Þ 3

ðA:25Þ

and pW ¼ bq2W þ aðc1 c2 ÞqL ;

pL ¼ bq2L :

ðA:26Þ

and opW opL 5 4s ðc1 c2 ÞqL > 0 ; þs ¼ 3 oa oa

ðA:27Þ

opW opL ¼ 3ðc1 c2 ÞqL > 0 : oa oa

ðA:28Þ

30

S. Martin

With linear demand and quadratic R&D cost, the condensed best response function (3.4), which determines equilibrium R&D intensity, is quadratic:

pW pL 2rv h u 3vh ð1 sÞ 1þs r pW þ spL ru pN ¼ 0 : 1þs 2

ðA:29Þ

From (3.1), equilibrium firm value is VN ¼

1 pW þ spL u 2vh : 1þs r

ðA:30Þ

In obvious notation, the expected present discounted value of consumers’ surplus is SN ¼

CSN ðc1 Þ þ 2ð1 þ sÞh CSN ðcr1 ;c2 ;aÞ : r þ 2ð1 þ sÞh

ðA:31Þ

Net social welfare is the sum of firm value and consumers’ surplus: NSW ¼ 2VN þ SN :

ðA:32Þ

Acknowledgments I am grateful for comments received from Bruno Cassiman, Jeroen Hinloopen, at the Fundacion Empresa Pubblica, Madrid, the Swedish School of Economics, Helsinki, the Tinbergen Institute, from two anonymous referees, and to Morton Kamien both for comments on an earlier draft and for nudging me to pursue the question of absorptive capacity. Responsibility for errors is my own.

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Beath, J., Katsoulacos, Y., and Ulph, D. (1988): ‘‘R&D Rivalry vs. R&D Cooperation Under Uncertainty.’’ Recherches Economiques de Louvain 54: 373–384. Cassiman, B., and Veugelers, R. (1998): ‘‘R&D Cooperation and Spillovers: Some Empirical Evidence.’’ Working Paper 9829, Katholieke Universiteit Leuven. Cohen, W. M., and Levinthal, D. A. (1989): ‘‘Innovation and Learning: the Two Faces of R&D.’’ Economic Journal 99: 569–596. Fo¨lster, S. (1995): ‘‘Should Cooperative R&D be Subsidized? An Empirical Analysis.’’ In Market Evolution: Competition and Cooperation, edited by A. van Witteloostuijn. Dordrecht: Kluwer Academic Publishers: 53–68. Hagedoorn, J. (1990): ‘‘Organizational Modes of Inter-firm Co-operation and Technology Transfer.’’ Innovation 10: 17–30. Henderson, R., and Cockburn, I. (1996): ‘‘Scale, Scope and Spillovers: The Determinants of Research Productivity in Drug Discovery.’’ Rand Journal of Economics 27: 32–39. Irwin, D. A., and Klenow, P. J. (1996): ‘‘High-tech R&D Subsidies: Estimating the Effects of Sematech.’’ Journal of International Economics 40(3–4): 323–344. Kamien, M. I., Muller, E., and Zang, I. (1992): ‘‘Research Joint Ventures and R&D Cartels.’’ American Economic Review 82: 1293–1306. Kamien, M. I., and Zang, I. (2000): ‘‘Meet Me Halfway: Research Joint Ventures and Absorptive Capacity.’’ International Journal of Industrial Economics 18: 995–1012. Katsoulacos, Y., and Ulph, D. (1998): ‘‘Endogenous Spillovers and the Performance of Research Joint Ventures.’’ Journal of Industrial Economics 46: 333–358. Lee, T., and Wilde, L. L. (1980): ‘‘Market Structure and Innovation: A Reformulation.’’ Quarterly Journal of Economics 94: 429–436. Loury, G. C. (1979): ‘‘Market Structure and Innovation.’’ Quarterly Journal of Economics 93: 395–10. Martin, S. (1997): ‘‘Public Policies Towards Cooperation in Research and Development.’’ In Competition Policy in the Global Economy. London and New York: Routledge. Ouchi, W. G. (1989): ‘‘The New Joint R&D.’’ Proceedings of the IEEE, Vol. 77, no. 9: 1318–1326. Reinganum, J. F. (1982): ‘‘A Dynamic Game of R and D: Patent Protection and Competitive Behavior.’’ Econometrica 50: 671–688. — (1985): ‘‘Corrigendum.’’ Journal of Economic Theory 35: 196–197. — (1989): ‘‘The Timing of Innovation: Research, Development, and Diffusion.’’ In Handbook of Industrial Organization. Amsterdam: North-Holland, Vol. 1, 849–908. Scott, J. T. (1993): Purposive Diversification and Economic Performance. Cambridge: Cambridge University Press. Seade, J. (1980): ‘‘The Stability of Cournot Revisited.’’ Journal of Economic Theory 23: 15–27. Sigurdson, J. (1986): Industry and State Partnership in Japan: the Very Large Scale Integrated (VLSI) Circuit Project. Lund: Swedish Research Policy Institute, University of Lund.

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Tapon, F., and Cadsby, C. B. (1996): ‘‘The Optimal Organization of Research: Evidence from Eight Case Studies of Pharmaceutical Firms.’’ Journal of Economic & Behavioral Organization 31: 381–399. Vonortas, N. S. (1994): ‘‘Inter-firm Cooperation with Imperfectly Appropriable Research.’’ International Journal of Industrial Organization 12: 413–435. Address of author: Stephen Martin, FEE/F&O, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands (e-mail: smart@fee. uva.nl.)