Spillovers, rivalry and R&D investment.

I. Introduction

Spillovers of research knowledge are unavoidable. Most technologies have some public good aspects. Moreover, rival firms are able to, through industrial espionage and reverse engineering, access and copy new innovations and are thus able to eat into monopoly profits of innovators. The threat of spillovers affects R&D investments and consequently reduces innovative activity. Imitators spend resources on acquiring or copying existing innovations -- resources that would have otherwise gone into the pursuit and development of new products and processes. Industries with excessive spillovers are likely to have little private R&D investment and might have to rely on public R&D funding.

While patents are designed to provide protection to inventors, they provide only imperfect protection against imitation. Mansfield, Schwartz, and Wagner[13] have shown that a large number of patented innovations are copied in a relatively short time. When patents do not provide adequate protection against spillovers, fewer innovations might be forthcoming and firms might choose to not patent.

Research spillovers, however, are not without their benefits. Whereas perfect appropriability generates static efficiency by conferring a monopoly on an innovator, spillovers might be more efficient in a dynamic sense because of reduced duplication of research effort and perhaps more innovations in the long run. The presence of R&D spillovers thus creates a wedge between private and social returns to R&D.

While most of the literature on the economics of technical change has ignored research spillovers (see Kamien and Schwartz[9] for a review), recently researchers have provided careful accounts of different aspects of R&D spillovers[1; 2; 3; 11]. In an empirical investigation of R&D spillovers, Bernstein and Nadiri[1] find that whereas spillovers are prevalent nearly everywhere, there is substantial variation across industries in the private and social returns to R&D. Some researchers have argued that firms might voluntarily choose to disclose scientific information[3]. Griliches[7] discusses the difficulties with empirically determining the extent of R&D spillovers and provides a nice review of the literature on knowledge spillovers.

This paper's main contribution is to look at the effects of imperfect appropriability of innovation rewards in a dynamic model where innovation is uncertain. Aspects of innovation uncertainty deal with the race to innovate first and/or with R&D investment sufficient to guarantee success. There is some probability that a firm will beat all its competitors and be the first to innovate. If the firm innovates before anyone else, it receives the rewards minus the losses from exogenous spillovers. If, on the other hand, someone else succeeds in innovating first, the firm is still able to benefit from spillovers of successful inventors. Of course, it pays to be an innovator than an imitator. Therefore, the problem facing the firm is to choose that level of R&D investment which maximizes the expected rewards from innovation, plus expected benefits from spillovers in case the firm is unable to innovate first. The effects of research spillovers and R&D rivalry on R&D spending are studied. We are also able to compare our results to similar models of R&D behavior that do not include R&D spillovers and/or innovation uncertainty. Public policy implications of our results are finally discussed.

II. The Model

We study the R&D behavior of a firm pursuing an uncertain innovation. The expected rewards to successful innovation (R) are given. However, there is an industry held estimate that the rate of revenue loss due to knowledge leakage is K per cent. In other words, due to spillovers from imitation the first firm to invent a new product is only able to capture the reward R minus the spillovers RK, where K is the percentage of revenue loss due to imitation. For example, the reduction in innovation benefits might come about because spillovers enable rival firms to copy the innovation relatively early. One could consider the piracy of computer software as an example.

The innovation we consider is a one-shot and are thus able to side-step the inter-temporal strategic issues associated with a sequence of innovations.

For simplicity we consider a noncooperative duopoly where two research labs are competing for an uncertain innovation. Additional rivals can be introduced in the same setup without much difficulty. Let x denote the amount of R&D investment of the ith firm. Rival R&D spending is given by y.

First we model the probability of success in innovation. The general probability distribution regarding the occurrence of an event (i.e., innovation) is given by F(t). The conditional probability density of the first occurrence of an event, given no prior occurrence, is denoted by G(t). This G(t) is referred to as the hazard rate. Formally,

G(t) = [dF(t)/dt]l(1 - F(t)). (1)

Here (1 - F(t)) is the probability of no occurrence. Using F(0) = 0 in solving this differential equation we get

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We can write the hazard rate as the product of a hazard function u(t) [is greater than] 0; and a hazard parameter h(x) [is greater than] 0. Here x is the R&D outlay. Therefore,

G(t;x) = u(t)h(x). (3)(1)

The parameter h(x) directly affects the probability of success. The variable u(t) defines the time path of the hazard rate and h(x) can be interpreted as a shift parameter. One could consider different probability distributions in modelling u(t). Assume that the function h(x) is twice continuously-differentiable and satisfies the following boundary conditions 

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In addition, there are eventually diminishing returns to x, although there may be increasing returns initially.

Using the formulation in (3), the probability F(t;x) from (2) can be written as

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This F(t;x) is the probability of an innovation occurring by time t, given R&D investment x. As expected, the probability of innovation is increasing in h(x).

Under exponential probability distribution and rival R&D given by [y,.sub.3] the ith firm's probability of obtaining the patent by innovating before anyone else is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e., the probability that the ith firm is first to innovate and that no one else has already innovated.(4) Although the exponential distribution is analytically more tractable, this distribution leads to a time-independent innovation hazard rate (i.e., G(t;x) = h(x)).

Research costs include costs of R&D equipment and personnel. Following Lee and Wilde [10] we model research costs as being variable and non-contractual. R&D costs are an increasing function of investment such that c(X) > 0; c' > 0; c" > 0. Kamien and Schwartz [8] and Loury [12], on the other hand, use contractual R&D costs.

With r as the discount rate, a firm chooses R&D expenditure so as to maximize the net present discounted value of expected profits from innovation plus the expected benefits from imitation in case someone else beats it in the race to innovate first. Formally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here T is the time of uncertain innovation and K is the expected percentage revenue loss due to knowledge spillovers. In the absence of spillovers (i.e., with K = 0), our objective function reduces to the winner-take-all formulation of Lee and Wilde [10]. In a model with differentiated products but no innovation uncertainty, De Bondt, Slaets, and Cassiman [2] consider a situation where a firm loses through spillovers of its own knowledge but gains from the leakage of its rivals' knowledge.

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Employing Cournot conjectures with respect to rival R&D,(5) the first-order condition for a maximum is

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The first-order condition equates the expected marginal R&D benefits to marginal research costs. Since R&D affects only the probability of success, h'(x) can be seen as the marginal product of R&D.

An interesting aspect to be studied involves the effect of a change in spillovers on the firm's R&D. Increased spillovers affect a firm's incentives to conduct R&D. The degree of spillovers might change when industrial spying or reverse engineering become easier. Whereas spillovers in our model are exogenous, in a different model, where the degree of spillovers is an endogenous variable, De Fraja [3] shows that competing firms might choose on their own to disclose scientific information.

Assuming that the second-order condition for a maximum is satisfied (i.e., ([[delta].sup.2]II/[[delta].sup.2]) < 0 ), and employing the implicit-function rule to (6), we have

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When the second-order condition for a maximum is satisfied, the sign of ([Delta]x/[Delta]K) will be that of ([[Delta].sup.2]II/[Delta]x[Delta]K). From (6),

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The sign of ([[Delta].sup.2]II/[Delta]x[Delta]K) is negative when h(y) [is equal to or greater than] h(x), and so is the sign of ([Delta]x/[Delta]K). When rival R&D is no riskier than own R&D (i.e., the hazard rate from rival R&D is at least as large as the hazard rate from own R&D), an increase in spillovers has a negative impact on own R&D. On the other hand, when h'(x) is sufficiently low and rival R&D is more risky h(y) [is less than] h(x)) then increased spillovers might increase R&D spending. Recall that the hazard rate, h(x), can be looked upon as the benefit of R&D in this model.

Evidence regarding the effect of spillovers on R&D investment is mixed in the literature. In a static model of process innovations without innovation uncertainty, Goel [4] has shown that when a Stackelberg innovating leader faces an imitating fringe, an increase in spillovers reduces the leader's R&D and output. Further, contrary to earlier findings of Spence [14], Levin and Reiss [11] use a static model, where own and rival R&D are imperfect substitutes, and find that diminished appropriability of innovation rewards does not necessarily diminish R&D spending. De Bondt, Slaets, and Cassiman [2] find support for Levin and Reiss's results in a model of differentiated oligopoly. In our dynamic model, where innovation is uncertain, diminished appropriability reduces R&D spending when rival hazard rate is at least as large as own hazard rate. This result suggests that reduced appropriability of innovation rewards might not reduce R&D spending when own probability of innovation is greater than that of rivals.

Let us now consider the effect of a change in rival R&D on the firm's own R&D. Rival R&D affects the probability of success. Proceeding in the same manner as before and employing the implicit-function rule to determine the effect of rivalry on R&D from (6), we get

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The sign of ([Delta]x/[Delta]y) will be opposite that of the numerator in (9).

From the first-order condition given in (6),

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([[Delta].sup.2]II/[Delta]x[Delta]y), and therefore, ([Delta]x/[Delta]y) will be positive when the following two conditions are satisfied:

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These conditions appear to place a lower bound on the hazard rate h(x). Clearly, condition (i) is a sufficient condition (with strict inequality) for rival R&D to increase own R&D.

Let us briefly discuss the economic interpretation of condition (i). The firm increases its R&D investment in response to greater rival R&D when own probability of innovation is high, the innovation is sufficiently delayed or when spillovers are low.(6) In these situations increased rival R&D does not have a dampening effect on own R&D investment. Therefore, arguments for policy measures that promote R&D competition are strengthened.

Winner-take-all models of innovation, on the other hand, generally find the effect of rival R&D on own R&D to be negative. Using exponential probability of innovation and variable R&D costs, Lee and Wilde [10] show that greater rival R&D leads to lower own R&D when fixed R&D costs are more significant than variable costs. Goel [5] uses a general probability distribution of innovation with contractual research costs and finds a negative impact of rival R&D on own R&D.

III. Conclusions

This paper uses a duopoly setup incorporating an uncertain innovation with imperfect appropriability. Our results show that increased spillovers diminish R&D investment when rival innovation hazard rate is at least as large as own hazard rate. However, research spillovers are not always R&D inhibiting. They start hurting own R&D when rival R&D is at least as productive as own R&D. Another important result is that when innovation is sufficiently delayed or when spillovers are low relative to R&D benefits, increased R&D rivalry will increase R&D investment. In other words, with low spillovers or delayed innovation, aggressive rival R&D behavior does not intimidate firms into lowering R&D spending. The effect of R&D rivalry is at odds with the results found in some winner-take-all models of innovation. In fact, when own innovation hazard rate is sufficiently high, both increased research spillovers and higher rival R&D lead to higher own R&D. A firm's comparative advantage in R&D can thus overcome the R&D dampening effects of higher spillovers and greater R&D rivalry.

Research consortia permit firms to spread risk by pooling resources and allow the internalizing of innovation externalities [6]. Our results suggest that research consortia are more likely to be forthcoming when R&D is risky and knowledge spillovers are high. R&D policy should, therefore, encourage cooperative research when R&D is risky. On the other hand, firms with productive R&D ventures are less likely to join such consortia. In this instance, costly policy measures to protect against competition or to protect against increases in spillovers might not be needed.

Finally, although this paper has overcome the time-invariance of static models of the kind used by Levin and Reiss [11], the possibility of endogenous research spillovers is not considered here.

(1.) See Kamien and Schwartz [8] and Loury [12] for earlier game-theoretic models of R&D.

(2.) Here v(t) [equivalent] [??.sup.t.sub.0]u(b)db.

(3.) Under exponential probability distribution of innovation, u(t) = 1 and v(t) = t. Like Lee and Wilde [101 and Loury [12] we make this assumption to limit the number of variables. See Kamien and Schwartz [8] for an alternate formulation and Goel [5] for an application.

(4.) The probability that no other firm has innovated by time t is given by (1 - F(t;y)) = [e.sup.-h(y)t]. The probability density function with respect to (4) is h(x)[e.sup.-h(x)t].

(5.) More sophisticated conjectures are difficult to sign in this model.

(6.) Note that under exponential probability distribution, since u(t) = 1, the hazard parameter is the conditional probability density of occurrence of an event, given no prior occurrence (i.e., G(t;x) = h(x)).

References

[1.] Bernstein, Jeffrey I. and M. Ishaq Nadiri, "Research and Development and Intra-Industry Spillovers: An Empirical Application of Dynamic Duality." Review of Economic Studies, April 1989, 249-69.

[2.] De Bondt, Raymond, Patrick Slaets, and Bruno Cassiman, "The Degree of Spillovers and the Number of Rivals for Maximum Effective R&D." International Journal of Industrial Organization, March 1992, 35-54.

[3.] De Fraja, Giovanni, "Strategic Spillovers in Patent Races." International Journal of Industrial Organization, March 1993, 139-46.

[4.] Goel, Rajeev K., "Innovation, Market Structure, and Welfare: A Stackelberg Model." Quarterly Review of Economics and Business, Spring 1990, 40-53.

[5.] _____, "On Vertical Integration into R&D." Quarterly Review of economics and Finance, Autumn 1992, 54-59.

[6.] _____, "Research Joint Ventures and Uncertain Innovation." Economic Notes, forthcoming.

[7.] Griliches, Zvi, "The Search for R&D Spillovers." Scandinavian Journal of Economics, Supplement, 1992, 29-47.

[8.] Kamien, Morton I. and Nancy L. Schwartz, "A Generalized Hazard Rate." Economics Letters, 5, 1980, 245-49.

[9.] _____. Market Structure and Innovation. Cambridge: Cambridge University Press, 1982.

[10.] Lee, Tom and Louis L. Wilde, "Market Structure and Innovation: A Reformulation." Quarterly Journal of Economics, March 1980, 429-36.

[11.] Levin, Richard C. and Peter C. Reiss, "Cost-Reducing and Demand-Creating R&D with Spillovers." Rand Journal of Economics, Winter 1988, 538-56.

[12.] Loury, Glenn C., "Market Structure and Innovation." Quarterly Journal of Economics, August 1979, 395-410.

[13.] Mansfield, Edwin, Mark Schwartz, and Samuel Wagner, "Imitation Costs and Patents: An Empirical Study." Economic Journal, December 1981, 907-18.

[14.] Spence, Michael, "Cost Reduction, Competition, and Industry Performance." Econometrica, January 1984, 101-21.