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Strategic R&D policy and appropriability
Journal of International Economics 42 (1997) 125-148
ELSEiVIER
Strategic R&D policy and appropriability Kaz Miyagiwa”, Yuka Ohnob “Department of Economics, 2107 CEBA, Louisiana State Universiry, Baton Rouge, LA 70803, USA ‘Department of Economics, Oregon State University, Ballard Extension Hall 303, Corvallis, OR 97331-3612, USA
Received 28 February 1995; accepted I8 April 1996
Abstract The Spencer-Brander argument for strategic R&D policy (Review of Economic Studies, 1983, 50, 702-722) is re-examined within an intertemporal model of stochastic innovation. It is shown that an optimal policy is independent of the choice of strategic variables in product market competition but sensitive to the appropriability of the new technology. At low or high degrees of appropriability R&D taxes are an optimal policy but at intermediate degrees R&D subsidies may be optimal. The paper explains why these results differ from Bagwell and Staiger’s finding (International Economic Review, 1992, 33, 795-816; Journal of International Economics, 1994, 36, 133-150) that R&D subsidies are an optimal policy in the presence of uncertainty. 01997 Elsevier Science B.V. All rights reserved. Key words: R&D subsidy; Innovation; Strategic trade policy; Patent length; Appropriability JEL class$cation:
F13; 031; 032; 033
1. Introduction
This paper reconsiders a strategic role for R&D policy in the presence of international rivalry. The original argument for strategic R&D subsidies, due to Spencer and Brander (1983), is known to be highly sensitive to the model’s assumptions.For example, if firms chooseprices instead of quantities as a strategic variable, an optimal policy switches from a subsidy to a tax.’ Grossman (1988) ‘See Eaton and Grossman(1986). The sensitivity of someresults to the choice of strategic variables however is not unique to the Brander-Spencer model but is endemic in a broader class of oligopoly models. 0022-1996/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII
SOO22-1996(96)01448-l
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finds this fact particularly disturbing when translating theory into trade policy because there is no knowing which strategic variables firms actually choose in real-world situations.* That concern however has been resolved in the recent papersby Bagwell and Staiger (1992), (1994), who introduce uncertainty into the Spencer-Brander model and show that R&D subsidies are an optimal policy regardless of the choice of a strategic variable in product market competition.3 Thus, the introduction of uncertainty not only makes the Spencer-Brander model more realistic but also strengthenstheir case for R&D subsidies. While Bagwell and Staiger focus on uncertainty over the extent of cost reduction under the new technology, innovating firms are also concerned with uncertainty over an actual date of discovery. The latter type of uncertainty is better analyzed in intertemporal models rather than in stage-game models. However, Bagwell and Staiger (1992) show that their stage-game model is equivalent to an intertemporal model of a patent race as long as firms must commit R&D costs at time zero, meaning that R&D subsidiesare still an optimal policy in a model of an uncertain discovery date. This paper extends the above line of research in two directions. First, we consider not only the commitment casebut also the case in which R&D costs are borne over timep Second,implicit in the Bagwell-Staiger model is the assumption that an innovator can forever enjoy exclusive rights to the use of the new technology. In the real world, however, patent protection is neither perfect nor permanent.We therefore investigate the linkage between the degree of appropriability of the new technology and the nature of optimal R&D policy. The key results are the following. Supposefirst that R&D costs are borne over time. Then, an optimal policy at low degreesof appropriability is to tax R&D. As the degreeof appropriability increases,R&D subsidies may emergeas an optimal policy. At sufficiently high degreesof appropriability, however, R&D taxes once again become optimal5 Supposenext that R&D costs must be committed at time zero, as in Bagwell and Staiger. At low degreesof appropriability it is optimal to tax R&D. Even at high degreesof appropriability, R&D taxes can be optimal when firms are not alike. In particular, in the model of battles for monopoly, a special *Taxes may emerge as an optimal policy in the presence of domestic consumption (Dixit and Grossman, 1986). multifirm oligopoly (Dixit, 1984), free entry (Horstmamr and Markusen, 1986). resource constraints (Dixit and Grossman, 1986) and international shareholding (Miyagiwa, 1992). While these papers are concernedwith the optimality of export subsidies,as in Brander and Spencer (1985). their arguments readily apply to the case of R&D subsidies. Brander (1995) provides a comprehensivereview of the literature. ‘A corrective incentive to tax R&D may emerge if there is more than one domestic firm. %-te commitment model and the non-commitmentmodel were first developed by Loury (1979) and Lee and Wilde (1980). respectively. The former is more relevant if R&D projects require substantial setup costs such as building a lab, or are contracted to a researchfirm for a given sum at time zero. ‘Bagwell and Staiger (1992) have conjectured that R&D taxes may be optimal in the noncommitment model.
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caseexamined in Bagwell and Staiger (1992), R&D taxes are an optimal policy if the foreign firm is a more efficient producer than the domestic firm. Since taxing domestic R&D apparently jeopardizes the home country’s chance of winning a patent race, at first blush it looks puzzling that such a strategy can ever be welfare-improving. The key to understanding this puzzle lies in the strategic nature of investments in R&D. The following two extreme casesin the non-commitment model illustrate the general principles involved. In the first case, assumeperfect appropriability so that losing a patent race meansno accessever to the new technology and hence a huge loss in potential profits. In such a case winning a patent race is imperative. When one firm increasesinvestment in R&D, the other firm follows suit. Thus, investments in R&D are strategic complements. Strategic complementarity means that the domestic firm invests more in the presence of R&D rivalry than when it faces no such competition. This excess R&D is wasteful from the domestic country’s point of view. In such a case R&D taxes soften competition and lead to a welfare improvement. In other words, R&D taxes serve as a puppy-dog ployp Going to the other extreme case,assumezero appropriability.7 Then, no matter which firm discovers it, the new technology will equally benefit both firms. Thus, when one firm increases investment in R&D, the other can afford to reduce its investment without sacrificing the expected benefits of the new technology. That makesinvestments strategic substitutes.In such a situation, R&D taxes reduce the level of the domestic firm’s investment, thereby making the foreign firm bear a greater share of R&D costs. While such a strategy improves the foreign firm’s chance of successin R&D, at zero appropriability the domestic country stands to gain as much from the foreign firm’s discovery as from its own. Thus, R&D taxes improve domestic welfare. The remainder of this paper is organized as follows. Section 2 presentsthe basic setup. The non-commitment model is developed in Section 3 and the main results are derived in Section 4. Section 5 applies the non-commitment model to the two special cases common in the industrial organization literature. The commitment model is the focus of analysis in Section 6. The final section concludes the discussion. 2. The setup
Suppose that two firms, one domestic and one foreign, compete for the discovery of a new technology over time. Time is a continuous variable defined %ee Fudenberg and Tirole (1984) for the classification of strategies based on canine and feline characteristics, ‘Innovation can occur even at zero appropriability, since post-discovery profits still exceed prediscovery profits becauseof an induced cost reduction.
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over [O,m).There are two distinct technologies, referred to as the old and the new technology. At time zero, the firms have accessonly to the old technology. The new technology becomesavailable only through investment in R&D, outcomesof which are stochastic. Each firm can improve its probability of discovery by investing more in R&D. When the new technology is discovered, the winner acquires exclusive rights to the use of it for a given time interval, T, after which the new technology becomes available to the other firm. Here, T measuresthe speedof technology diffusion or the degree of appropriability, which may reflect patent length or the difficulty in imitating the new technology. In addition to competing in R&D over time, firms competefor export sharein a third-country market at each point in time r E [O,W).~The demand functions in the third-country market are time-invariant. There is no home market consumption of the export goods. Now we introduce notation. Let ?r(&9*) denote the domestic firm’s (Nash) equilibrium flow profit from export when the domestic firm uses the technology 8 and the foreign firm usesthe technology 0 *P A similar interpretation applies to the foreign firm’s flow profit from export, r*(0,0*). Let an upper bar imply the new technology and a lower bar the old technology. Thus, for example, if the domestic firm discovers the new technology, its flow profit increases from 7r(e,j*) to v(@*) whereas the foreign firm’s flow profit falls from v*(@*) to ‘rr*( 8,$*) for the interval T, after which the domestic and the foreign firm earn 7r(8,8*) and a*(8,8*), respectively. We assumethat these inequalities hold: (14
7r*(e,e*> > n-*(e,B*)> T*@,e,e*)> 7T”(@“).
(lb)
Thus, the firm’s profits are highest when it has exclusive rights to the use of the new technology and lowest when the rival has such rights. Also, profits are greater when firms share the new technology than when neither firm can use it. The above specifications are flexible enough to capture a number of alternative situations. Eqs. (la), (lb) can apply to either Coumot or (differentiated-good) Bertrand competition.” If the new technology is a process innovation, the difference m(t@ *) - ~(0,0*), for example, denotesa gain from a cost reduction. If the new technology isaproduct innovation, the samedifference may denote an increase in profits from new markets. ‘We focus on a Markov-perfect equilibrium, and therefore rule out the possibility of implicit collusion between firms that may arise in a supergame framework. 9An asterisk (*) indicates a variable pertaining to the foreign firm. “In the case of symmetric homogeneous-good Bertrand competition all but the first profits in Eqs. (la), (lb) are equal to zero. Therefore, the last two inequalities in Eqs. (la), (lb) are replaced by weak inequalities.
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3. The non-commitment
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model
3.1. The R&D process
In the non-commitment model, each firm invests in R&D at each point in time until the new technology is discovered, at which time both firms terminate R&D activities. Following Lee and Wilde (1980), we assumethat the domestic firm’s date of discovery, 5, has the exponential cumulative distribution function: Pr(5 5 t) =
1 - exp[-h(k)t],
(2)
where k denotesthe amount the domestic firm invests in R&D per unit of time.” That is, investment k at time t generates the probability h(k) dt of discovery between t and t + dt, given that the new technology has not been discovered before t. The hazard rate function h(k) is twice continuously differentiable and satisfies the following regularity conditions: h(0)
= 0, h’(k) > 0, h”(k) < 0, >ly+h’(k)
= ~0,and lilih’(k) m
= 0.
(3)
The first equality says that there is no chanceof discovery without R&D. The next two conditions statethat investment in R&D improves the probability of discovery with diminishing returns. The limit conditions and Eqs. (la), (lb) guaranteethe existence of an interior solution for k. The same hazard rate function h(.) is assumedto apply to the foreign firm as well.‘* Thus, investment k* in R&D at time t gives the foreign firm the probability h(k*) dt of discovery between t and t+dt. Each firm’s probability of discovery is assumedto be independent. The domestic government subsidizesdomestic R&D activities at the rate s (< 1) so that the domestic firm’s net R&D costs per unit of time are k( l-s). When the new technology is discovered, firms stop investing in R&D, and the subsidy programs also end. 3.2. R&D
competition
We next specify the domestic firm’s intertemporal profit. Supposewe are at time t. Then, with the probability exp[-h(k)t - h(k*)t] neither firm has discovered the new technology before t. Conditional on that, three things can happen between “This assumes the existence of a steady-state solution for k. As long as there has been no discovery firms face the same situation at every moment so this steady-state solution is a correct one even if firms are allowed to vary k at each point in time (see Miyagiwa and Ohno, 1995). ‘*This is only to simplify notation. Even if the foreign firm has a different hazard rate function that satisfies the regularity condition Eq. (3), our conclusions remain intact.
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time t and t+dt. First, the domestic firm discovers the new technology with the probability h(k) dt, and earns the discounted sum of profits: cc
f+T e-"7iffQ*) I
dx +
e-"n-(8,8*)
dx,
I t+T
f
(4)
where r denotes an exogenous rate of interest. The first integral in Eq. (4) representsthe domestic firm’s profits while it usesthe new technology exclusively. The second integral represents the domestic firm’s profits when the innovation becomesavailable to the foreign firm after the interval T. Integrating, we write Eq. (4) more compactly: (e-“lr)L
where
(L stands for a ‘leader’).
Second, with the probability h(k*) d? the foreign firm discovers the new technology betweent and t + dt. In this case,the domestic firm’s flow profit falls to n($,t?*) while the foreign firm monopolizes the new technology, and rises to rr( I?$*) when the domestic firm also gains accessto the new technology after T. Therefore, the domestic firm’s profits equal m
f+T e-"n-(&8*) I
dx +
e-"r(t$t?*)
dx,
I z+T
f
which can be written as (e-“lr)F
where F = a,$*)(1
- eLrT) + e-“rr(8,8*)
(F stands for a ‘follower’).
The third possibility is that neither firm makes a discovery between t and t+dr. In that case,the domestic firm enjoys the net flow profit 7T(e,@ *) - k( 1 - s) for the duration of dr, and we move on to time t+ dt, where the same three possibilities just discussedarise once againI Integrating the profits in the three casesover time yields the expression for the domestic firm’s intertemporal profit: “The fourth case, in which both firms innovate between time probability h(k)h(k*) dt2 is of second-order smallness.
t and r+dr, can be ignored since its
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m
v(k,k*,s)
=
e -rhck’+hca*‘l’h(k)(e-“lr)L
I
&
0 m +
co e-[h(k)+h(k*)lth(k*)(e-rt/r)F
&
+
I
e-rte-[h(k)+h(k*)l’[~~,~*) I
0
0
-k(l-s)]dt, which simplifies to y(k,k*,s) =
?r(e,j *> - k( 1 - s) + h(k)Llr + h(k*)Flr r + h(k) + h(k*)
(5)
The domestic firm regards k* and s as given and chooses k so as to maximize v(k,k*,s). The optimal level of-k satisfies the first-order condition: ccl(k,k*.s) = [h’(k)Llr
- 1 + s][r + h(k) + h(k*)]
- [T@,@*) - k( 1 - s) + h(k)Llr + h(k*)Flr]h’(k)
= 0.
(6)
The second-ordercondition is satisfied since tjjj = h”(k)[L - n-(&j *) + k( 1 - s) + h(k*)(L/r
- F/r)] < 0.
(Subscripts denote partial derivatives.) The first-order condition can be given a standardinterpretation if we rewrite Eq. (6) as h’(k){L - [v(@,!,*) - k( 1 - s)] + h(k*)(Llr
- s).
- F/r)} = [r + h(k) + h(k*)]( 1 (7)
A marginal increase in R&D investment will cost the domestic firm the interest rate r, and will also be wasted with the probability h(k) + h(k*) becausewith that probability the new technology will be discovered without the increase in R&D. Thus, the right-hand side of Eq. (7) representsthe (subsidized) marginal costs of R&D. On the left-hand side, L - [ @,e *) - k( 1 - s)] measuresa direct increase in flow profits if R&D is successful at time t. The other term h(k*)(ZJr -F/r) representsthe preemption effect: a successin R&D at time t forever prevents the domestic firm from losing the patent race, which occurs with the probability h(k*) at each point in time, and hence secures for the domestic firm the expected discounted sum of the leader-follower difference in profits, h(k*)(Llr -F/r). Since a unit increasein investment raises the probability of discovery by h’(k), the left-hand side of Eq. (7) amounts to the expected marginal benefits of R&D. Interpreted this way, Eq. (7) is the standard optimality condition: the expected marginal benefits of R&D equal the marginal costs of R&D. We can express the foreign firm’s intertemporal profit similarly:
132
K. Miyagiwa,
v*(k,k*) =
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T*($‘,@*) - k” + h(k)F*lr + h(k*)L*lr r + h(k) + h(k*)
where L* = r*(@*)(l - LrT) + e-%-*(8,8*), F* = 7r*(8,@*)(1 -t?) + e-“r*(8,8*).
and
The first-order condition is +*(k,k*) = [h’(k*)L*lr
- l][r + h(k) + h(k*)]
- [r*@,e,e*) -k*
+ h(k)F*lr
+ h(k*)L*lr]h’(k*)
= 0.
(8)
It is easy to check that the second-ordercondition is satisfied: I@* < 0. The first-order conditions Eqs. (6), (8) define the two firms’ best-response functions: k=b(k*;s) and k* =b*(k). A subgame-perfectNash equilibrium is a pair of R&D expenditures (k,k*) that simultaneously solves these best-response functions. Assumptions Eqs. (la), (lb), (3) guaranteethat b(k*,s) and b*(k) are positive for all k* and k, respectively, so that there is an interior solution.i4 Furthermore, the equilibrium is unique and stable under the assumptions:
which imply
3.3. Properties of the best-response functions
In this subsection we examine the slopes of the best-responsefunctions. Take the domestic firm’s best-responsefunction. Since db(k*;s)lak* = - I,$~/#~, and &CO by the second-order condition, the slope of the domestic firm’s bestresponsefunction takes the sign of erk*.Differentiating Eq. (6) yields I& = h’(k*)[h’(k)(L
- F)lr - (1 - s)].
(9)
r4From Eq. (6). 1/~(0,k*,s) = h’(O)[L - ?r(f,~)] + h’(O)h(k*)(L - F)lr - (1 - s)[r + h(0) + h(k*)I > 0 for all k*, thereby implying that the domestic firm’s best response is always to invest a positive amount in R&D. Similarly, $*(k,O)>O for all k. ‘?f T is sufficiently close to zero, &l&.I. Direct substitution for these expressions shows that h ‘(k)k+ h’(k*)I&.l. Similarly, h ‘(k*)k*+h’(k)l$~l. If T>max{p,T*}, (cl,.>0 and hence &+ &.l$k,l. Direct substitution shows that h‘(k)h(k*)+ h’(k*)h’(k)I$.,l. Similarly, h’(k*)h(k)+ h’(k*)h’(k)
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Notice that the leader-follower L-F
133
difference in post-discovery profits
= [?r(&#*)
- n-@,8*)](1 -em’*) (10) . is strictly monotone-increasing in T, rising from zero when T is zero to ?r( $8 *) ~@,8*) as T approaches infinity.i6 Similarly, the slope of the foreign firm’s best-response function depends on the sign of t,b,*= h’(k)[h’(k*)(L*
- F*)lr
- 11,
where the strictly monotone-increasing function L* -F”
= [r*(j,c?*)
- rr*(t@*)](l
- e-‘r)
takes the value between zero and r*(&e*) The following is a useful lemma. Lemma 1. (9 ~4~ is positive
- rr*(@*).
if T is greater than ?: where
F= (llr){ln[?r(8,8*)
- 77@,8*)] - ln[7r@,tj*) - 7@,8*)]}.
(3 Ax is negative for T su@ciently close to zero. (iii) I,I%~is positive if T is greater than T” , where T* ~(llr){ln[~*(~,f?*)
- 7r*(cQ*)]
- ln[7r*(@*)
- ~*(&j*)]}.
+,* is negative if T is sufficiently close to zero. If we let b denote the domestic firm’s best response, Eq. (6) implies v(b,k*,s)= [h’(b)Llr-(1 -s)]lh’(b), so Eq. (9) can be written as:
Proof.”
(Glk*= h’(k*)h’(b)[v(b,k*,s)
-F/r].
Since b is a best response, v(b,k*,s)s v(O,k*,s). However v(O,k*,s) -F/r
= [rr(J,@*) - F]l[r
+ h(k*)]
where T(@*)
-F = {?T(e,e*) - 7.r@,$*) + eMrT[?r(j,8*) - m-(6,6*)])
is positive if T exceeds i? Therefore, for such a T, Y(b,k*,s) - F/r? v(O,k*,s) Flr>O, and hence &* > 0. On the other hand, since L-F is zero at T =0, we “With zero appropriability there is no advantage in being a leader. As the appropriability increases, the leader can monopolize the new technology for a longer period, implying that the leader-follower difference in profits increases monotonically. “The proof for the case in which T= m is similar to the proof in Reinganum (1983).
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K. Miyagiwa, Y. Ohno I Journal of International Economics 42 (1997) 125148
can choose T as close to zero in Eq. (7) as we like so as to make L-F smaller than 1-s in Eq. (9), so that I,,$* becomesnegative. Similarly, for $f. Cl The next result, which follows from Lemma 1, gives sufficient conditions for determining the slopes of the best-responsefunctions b(k*;s) and b*(k). Proposit& 1. Let T and L@ be as dejined in Lemma 1. Then: 1. if T > T, the best-response function b(k*;s) is positively sloped in k*; if T > F*, b*(k) is positively sloped; 2. if T is sufjiciently close to zero, both b(k*;s) and b*(k) are negatively sloped.
Proposition 1 says that investments in R&D tend to be strategic complements when T is large, and strategic substitutes when T is small.1sHere is an intuitive explanation. At a low degreeof appropriability, the benefits of the new technology spill over to the non-innovating firm quickly so that the preemption effect in Eq. (7) is insignificant. An increasein the foreign firm’s investment therefore has little or no effect on the domestic firm’s marginal benefits from R&D (the left-hand side of Eq. (7)). The marginal cost of R&D (the right-hand side of Eq. (7)) however is now higher, thereby prompting the domestic firm to curtail investment in R&D. That shows that at a low degree of appropriability investments are strategic substitutes. This explains the second part of Proposition 1. At a high degree of appropriability, the preemption effect is so important that an increase in investment by one firm is likely to be challenged by a matching increase by the other. Thus, at high degrees of appropriability investments in R&D are strategic complements.This explains the first part of Proposition 1. Fig. 1 illustrates the case in which T= 0 so that investments are strategic substitutes. In Fig. 2, T>max{F,F*} so that investments are strategic complements. (It is also possible to have mixed cases, in which one best-response function is upward-sloping and the other downward-sloping.) In each figure, b” and b*“, respectively, representthe domestic and the foreign firm’s best-response function when s = 0, and point 1 correspondsto the subgame-perfectequilibrium under laissez-faire policy. (The dotted schedule should be ignored for now.) 4. The effect of R&D subsidies in the non-commitment
model
4.1. The effect on R&D investments
We begin this section by examining how R&D subsidies shift the domestic firm’s best-responsefunction. Applying the implicit function theorem to Eq. (6) yields ‘*The importance of the distinction between strategic substitutes and complements in oligopoly models is discussedthoroughly in Fudenbergand Tirole (1984) and Bulow et al. (1985).
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k’
b0
bS
k Fig. 1. Strategic substitutes [T=O].
ab(k”;s)las = - **I@, where & = r +
h(k) + h(k*) - h’(k)k. k*
Fig. 2. Strategic complements [T>max{T,T*}].
135
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Concavity of h(k) and the condition h(0) = 0 imply that h(k) - h’(k)k >O, and hence clr,>O. By the second-order condition, ek is negative. Therefore, &(k*;s)/& is positive, that is, R&D subsidies increase domestic investment in R&D for a given level of k*. In Figs. 1 and 2, that effect is illustrated by the rightward shift in the domestic firm’s best-response function (from the solid line to the dotted line). Note that the result holds whether investments are strategic substitutes or complements. Since s does not directly appear in Pq. (8), R&D subsidies leave the foreign firm’s best-response schedule undisturbed. Therefore, the subgame-perfect equilibrium under R&D subsidies is at point 2 in Figs. 1 and 2. While R&D subsidies always increase the domestic firm’s equilibrium investment, their effect on the foreign firm’s R&D level depends on the slope of the foreign firm’s best-response function. Precise conditions are given in the following. Proposition 2. (i) R&D subsidies increase the domestic jrm ‘s investment in R&D. (ii) If T exceeds T” given in Lemma 1, R&D subsidies increase the foreign jrm ‘s investment. (iii) If T is sufJiciently small, R&D subs’idies decrease the foreign jirm’s investment. Proof. Differentiate Eqs. (6), (8) to get
I,+; dklds + I,$ dk”lds = 0. Solving these simultaneously yields
Since &ID is positive and I,$ is negative by the second-order condition, dklds is positive. This proves (i). On the other hand, dk*lds is positive only if #f is positive. Then, (ii) and (iii) follow from Lemma 1. 0 The intuition underlying Proposition 2 is straightforward in light of Proposition 1. At a low (high) degree of appropriability, investments are strategic substitutes (complements) so an increase in the domestic firm’s investment in R&D induces a decrease (an increase) in the foreign firm’s investment. R&D taxes, of course, have the opposite effect.
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4.2. The welfare efect of R&D subsidies Since there is no home consumption, domestic welfare is the difference between the domestic firm’s intertemporal profit and the present discounted sum of subsidies. The latter is contingent upon non-discovery of the new technology, and equals
e
-rr
e
-[h(k)+h(k*)l+)
&.
I
0
Hence, domestic welfare is written as m u(k,k*,s) =
e-lh(k)+h(k*)“h(k)(err/r)L
I
dt
0
m
+
e-rfe-[h(t)+k(lr*)l’[~~,~*>
-k]
dt,
I
0
where k and k* are equilibrium Integration yields u(k,k*,s) =
R&D investments, i.e. k=k(s) and k* =k*(s).
?r@,@*) - k + h(k)Llr + h(k*)Flr r + h(k) + h(k*) ’
(11)
In order to find the welfare impact of R&D subsidies, we differentiate Eq. (11) totally with respect to s to obtain du[k(s),k*(s),s]lds
= Us dklds + uk* dk”lds
(12)
and evaluate it at s = 0. Noting that u(k,k*,s) = v(k,k*,s) - skl [I + h(k) + h(k*)], and using the envelope theorem yields uk = - s&l[r
+ h(k) + h(k*)]’
so that the first term on the right of Eq. (12) vanishes when evaluated at s = 0. The welfare effect of an R&D subsidy is captured by the second term. By Proposition 2, the sign of dk*lds depends on the degree of appropriability: it is positive when T exceeds p* and negative when T is sufficiently small. As for uk*, one can differentiate Eq. (11) with respect to k* to get
138
K. Miyagiwa,
I’. Ohno I Journal of International
= h’(k*)(Flr)l[r
‘k*
Economics 42 (1997) 125-148
+ h(k) + h(k*)]
- h’(k*)[+?e*) -k + h(k)Llr + h(k*)Flr]l[r + h(k) + h(k*)]’ = h’(k*)[Flr - u(b,k*,s) + sk]l[r + h(k) + h(k*)].
This expression also dependson the degree of appropriability. For T >T, uk*ls=O = h’(k*)[Flr
- u(b,k*,s)]l[r
+ h(k) + h(k*)]
is negative because, as shown in the proof of Lemma 1, F/r- v@,k*,s) is negative, meaning that an increasein the foreign firm’s investment in R&D lowers domestic welfare. Combining the two results, for T>max{7,~*} we have sgn(duld&)
= sgn(u,, dk*ld.s) < 0;
that is, R&D taxes improve domestic welfare at sufficiently high degrees of appropriability. Alternatively, if T is zero, we have dk*lds
+ h(k) + h(k*)]’
- ?T@,$*) + k}l[r + h(k) + h(k*)12 > 0,
i.e. an increase in the foreign firm’s investment in R&D improves domestic welfare. Therefore, at T=O sgn(duldsl,,,) = sgn(u,, dk*ld.r) < 0; that is, R&D taxes increase domestic welfare. Furthermore, since uk* dk*lds is continuous in T, there exists a neighborhood of T=O, in which uk* dk*lds < 0. Thus, if T is sufficiently small, taxing R&D is welfare-improving. We have proved the following (local) result. Proposition 3. (i) Zf T is sujJiciently short (close to zero), or (ii) if T exceeds max{F*,T}, a (locally) optimal policy is to tax R&D.
Fig. 1 depicts Proposition 3(i). With T close to zero, the best-responsefunctions slope downward, and domestic iso-welfare curves are U-shaped, at least in the neighborhood of the initial equilibrium; I9 u” representsdomestic welfare with no government intervention; R&D taxes shift the domestic firm’s best-response ‘?he iso-welfare curves are identical to the iso-profit curves under zero subsidies, with the slope given by dk*ldk= -t+lu,,. For a given k *, let b denote the best responseof the domestic firm. The optimality of b implies that at k= b uk equals zero whereas uk~-0 for k< b, and u,b. Combining this with the results just establishedin the text yields the following. If T is close to zero, uk* > 0, and hence the iso-wefare curve must be U-shaped. If T exceeds max{T,T*}, url.,< 0 (in the neighborhood of k= b), and hence the iso-welfare curve is inverted-U shaped,at least locally.
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function to the left. Given that iso-welfare curves are (locally) U-shaped, there exists a (locally) optimal R&D tax that raises domestic welfare to u’. Proposition 3(ii) is illustrated in Fig. 2. With T sufficiently large, the bestresponse functions slope upward and domestic iso-welfare curves are inverted U-shaped, at least locally. Again, taxing R&D shifts the domestic firm’s bestresponse function to the left, thereby increasing domestic welfare to u’. While R&D taxes are an optimal policy when T is either small or large, the next two propositions characterize an optimal R&D policy for intermediate values of T. Proposition optimal.
4. There exists T E (O,min{~,~*})
such that zero subsidies are locally
Proof Let {k(T),k*(T)} denote the subgame-perfect equilibrium of the R&D game when s =0, and let A(T)= t@[k(T),k*(T),T] and p(T)=t+*[k(T),k*(T),T]. Since dk”lds=@&,lD, where t,kSID>O, sgn(dul&(,=,)=sgn[~(T)~(T)]; h(T) is a continuous function, with h(0) < 0 and A( f *) > 0, as discussed earlier. Therefore, by the intermediate-value theorem there exists at least one T such that A(T)=O. Let T* denote the set of such T. Similarly, since p(T) is continuous, with p(O)>0 and ,x( i;)
Proposition 4 says that there always exists a T at which zero subsidy is an optimal policy. The next proposition gives a sufficient condition under which an optimal policy is actually to subsidize R&D. Proposition 5. If the sets TA and T’” are disjoint, there exists a range of T for which an R&D subsidy is a (locally) optimal policy.2o Proof. If T” and T” are disjoint, there always exists an interval I such that T El implies A(T) so that dulds = uk* dk*lds >O. 0
Propositions 4 and 5 are illustrated in Fig. 3, where we assume A(T) and p(T) are strictly monotone so that T” and T” are singletons: T”={T”} and T’ ={Tp}. For TE [O,T’)U(T”,~), A(T)p(T)O, and hence domestic welfare can be improved by R&D subsidies. Finally, if T equals T” or T’, A(T),u(T)=O so an optimal subsidy is zero. The following example illustrates the case for an optimal subsidy. Consider a symmetric homogeneous-good Bertrand game. Let rr( I@ *) = n-*(&e*) = 5, r= ‘“This is ‘a sufficient condition. Even if T” and T” have common elementsR&D subsidiescan be an optimal policy for some T.
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Fig. 3. A case of optimal R&D subsidies.
1, h(k)=k”*, and T=0.762. Then the equilibrium is given by k = k* =4, uk* < 0, +:O. Actually, the optimal subsidy is s=O.lll. Now it is time to explain Propositions 3 through 5 intuitively. Suppose that T>max{T*,F}. In the absence of R&D competition the domestic firm knows that it will eventually discover the new technology, and invests k M in R&D, as indicated in Fig. 2. This investment generates truly the optimal (monopoly) level of domestic welfare. Since investments are strategic complements for T exceeding max{F*,T}, R&D competition induces more investment than k”. The additional investment is caused purely by the preemption effect, and is socially wasteful. R&D taxes soften competition by allowing the domestic firm to commit itself to low levels of investment in R&D, thereby improving domestic welfare. Thus, R&D taxes serve as a puppy-dog ploy. Suppose next that T is zero. R&D taxes discourage the domestic firm’s R&D effort, and, since R&D investments are strategic substitutes, prompt the foreign firm to invest more in R&D. While such a strategy improves the foreign firm’s chance of discovery, the domestic country benefits as much from it as from its own firm’s discovery because of the large spillover effect. Thus, R&D taxes merely shift R&D costs to the foreign firm without sacrificing the benefits of the new technology, and therefore are welfare-improving. Finally, when T takes an intermediate value, investments may be strategic complements, and yet the domestic country can still benefit from the foreign firm’s discovery because of the spillover effect. In this case, an R&D subsidy is called for to induce the foreign firm to invest more in R&D. Alternatively, investments may be strategic substitutes and yet losing a patent race may prove costly to the domestic country because of slow technology diffusion. In such a case, an R&D
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subsidy can improve the domestic firm’s chance of winning the patent race, and hence is welfare-improving.
5. Applications This section applies the model to the two specific situations examined in the industrial organization literature. In each case R&D taxes emerge as an optimal policy. 5.1. Battles for monopoly
Battles for monopoly are studied by Loury (1979), Lee and Wilde (1980) and Bagwell and Staiger (1992). In the model of battles for monopoly neither firm makes positive flow profits before discovery, and only the winner of the patent race makes a monopoly profit forever after discovery. If we make these adjustments and set T to infinity in our model, it follows from Proposition 3 that R&D taxes are an optimal policy. The result contrasts sharply with the Bagwell and Staiger (1992) finding that R&D subsidies are optimal in the commitment model. The commitment model of battles for monopoly is analyzed in Section 6, where it is shown that R&D taxes can be an optimal policy. 5.2. Monopoly
under the threat of entry
The analysis of the incumbent facing the threat of entry has received attention in the industrial organization literature (e.g. Gilbert and Newbery, 1982; Reinganum, 1983). The basic setup of this literature is as follows. The incumbent initially earns a monopoly profit. If it discovers the new technology first, it retains the monopoly position. If the entrant discovers the new technology, entry occurs and each firm earns a duopoly profit. Implicit in the above setup is the assumption that a winner of the patent race enjoys exclusive rights indefinitely. Thus, set T equal to infinity in our model, and interpret the flow profits to fit the story. For instance, if the domestic firm is the entrant, set its pre-discovery profit r(B,8 *) and the follower’s profit F equal to zero, and interpret the winning profit Las a duopoly profit. Alternatively, if the domestic firm is the incumbent, interpret Q,e*) as a pre-discovery monopoly profit, L as a post-discovery monopoly profit, and F as a duopoly profit. In either case, since T is infinite, investments are strategic complements and so R&D should be taxed by Proposition 3. Again, what is crucial is not the specific form of market competition but the fact that T is infinite.
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6. Optimal R&D policy in the commitment made1 This section examines the strategic role for R&D policy in the commitment model. The issue has already been examined by Bagwell and Staiger (1992). The treatment here generalizestheir model in three respects.First, while Bagwell and Staiger assumeperfect appropriability, we allow T to be finite. Second,they focus on battles for monopoly: that is, they assumethat firms make zero flow profits before discovery and only the winner of the patent race makes positive postdiscovery flow profits. In contrast, we allow each firm to make positive profits in pre-discovery and/or post-discovery competition. Third, while Bagwell and Staiger focus on the symmetry case,we also consider the casesin which firms are asymmetric. 6.1. The setup
The basic setup of the model remains much the sameas in the non-commitment model except for the fact that firms now must commit investments in R&D at time zero. Investment k, sunk at time zero, generatesthe probability h(k) dt of discovery between t and t f dz for all t E [O,w) according to the distribution function Eq. (2), and the hazard rate function h(.) satisfies the assumptionsof Eq. (3). The expression for the domestic firm’s intertemporal profit is similar to the previous definition except that R&D costs k( 1-s) are not a flow: m
V(k,k*,s) = e -th(k)+h(k*)lrh(k)(e-rt/r))L dt
h(k*)(e-“lr)F
dt
+ e-rte-[h(k)+6(k*)lr n@,@*) dt - k( 1 I
s).
0
This expression simplifies to v(k
k.+ s) 9 9
=
7T(e,@*)
+
WWr
+
W*)Flr
- k( 1 - 8).
r + h(k) + h(k*)
The domestic firm takes k* and s as given and choosesk to maximize V(k,k*,s). The first-order condition is given by P(k,k*) = h’(k)[L - nQ,@ *) + h(k*)(Llr
- F/r)]
- (1 - s)[r + h(k) + h(k*)]* = 0,
(13)
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which defines the domestic firm’s best-responsefunction: k = B(k*;s). The slope of the best-responsefunction B(k*,s) dependson the sign of the expression h’(k*)h’(k>{2[7@@*)
- L] + [I + h(k) - h(k*)](L r + h(k) + h(k*)
% =
- F)lr} (14)
Similar relations hold for the foreign firm. In particular, the slope of the foreign firm’s best-responsefunction depends on the sign of the partial derivative Pf, defined analogously to Eq. (14). 6.2. The welfare
analysis
Begin the analysis by investigating the relationship between appropriability and the slope of the best-responsefunctions. Assume zero appropriability first. Then, L-F is zero, and hence the second term in braces on the right hand of Eq. (14) vanishes. The first term is negative since n-(&e*)
- L = @,@*>
- ?r(B,B*) < 0.
Thus, ?& is negative, and so the domestic firm’s best-responsefunction slopes downward. Similarly, the foreign firm’s best-responsefunction is also negatively sloped, meaning that R&D investments are strategic substitutes. It is straightforward to show that under the usual stability assumptiondklds>O and dk*lds< 0. We now turn to the effect on domestic welfare, which can be written as U(k,k”,s) =
@,j*)
+ h(k)Llr
+ h(k*)Flr
_ k
r + h(k) + h(k*)
where k=k(s) and k* = k*(s) constitute a subgame-perfectNash equilibrium in R&D competition. Differentiation of U(k,k*,s) with respect to s yields dU/ds = U, dklds + U,, dk*lds.
(15)
The first term on the right hand side of Eq. (15) vanishes when evaluated at s = 0, so the second term captures the welfare effect of R&D subsidies. In order to evaluate the latter, first differentiate U(k,k*,s) with respect to k* to obtain U,,[r
+ h(k) + h(k*)]‘/h’(k*)
= F - d&J*) + h(k)(F - L)lr = (1 - ,4-rT )?r(e,d*) + e-“77fi@*)
- n-(&j*) - (1 - eCrTMWl[~@*)
(16)
- ?r(B,@Yl.
As T approacheszero, the right-hand side of Eq. (16) converges to ?r(fi,t$*) rr(e,$ *), a positive number, and so U,, > 0. That is, when T is sufficiently close to zero, an increase in the foreign firm’s investment in R&D raises domestic welfare because of spillovers from a discovery by the foreign firm. Rutting the
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above results together, we have dU/dsl,,,=U,, following.
dk*lds
and hence the
Proposition 6. In the commitment model, when T is sufJiciently close to zero, R&D taxes are a (locally) optimal policy.
Alternatively, suppose that T is sufficiently large. Then, the right-hand side of Eq. (16) is negative. Since the spillover benefits from the foreign firm’s discovery are small, an increase in investment by the foreign firm, which raises that firm’s probability of discovery, hurts the domestic country. Therefore, an optimal strategy should discourage the foreign firm from investing in R&D. That goal can be accomplished by R&D subsidies if the foreign firm’s best-response function is negatively sloped, and by R&D taxes if it is positively sloped. Unfortunately, however, the sign of ?& is in general ambiguous.” In order to gain further insight, we focus on the Bagwell and Staiger (1992) model of battles for monopoly. 6.3. Battles for monopoly
It is recalled that in battles for monopoly pre-discovery profits and postdiscovery profits for a loser (F and F *) are equal to zero, and L = n( f@*) and L* = n*(e,e*) represent the monopoly profits, which a winner of R&D competition can enjoy forever. Substituting these values in Eq. (14) and setting T= m yields !4& = h’(k*)h’(k)(Llr)[h(k)
- h(k*) - r] / [r + h(k) + h(k*)]
which is positive if and only if h(k) - h(k*) - r >O. The analogous expression for the foreign firm !Pz = h’(k)h’(k*)(L*lr)[h(k*)
-h(k)
- r]/[r
+ h(k) + h(k*)]
is positive if and only if h(k*) - h(k) - r>O. Fig. 4 depicts the relationship between h(k) and h(k*) on the one hand, and the sign of ?& and ?Pf on the other. There are three areas separated by the lines representing h(k)- h(k*)-r=O and h(k*)- h(k)-r=O. Notice that it is impossible to have both the best-response functions slope upward simultaneously.22 Suppose that the levels of investment between two firms are not too dissimilar (Area 2). Then, ?&‘,*and Pt are both negative, and R&D investments are strategic substitutes. It follows from the earlier discussion that R&D subsidies are an *‘The ambiguity results because unlike in the non-commitment model a firm’s R&D costs are independent of the actual date. of discovery, and hence, of its rival’s R&D effort. **Both Pk, and Pf are positive only if h(k)- h&*)-r>0 and I#*)-h(k)-r>O. Adding the inequalities however yields - 2r>O, a contradiction.
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h(k*)
r=O
Fig. 4. Optimal R&D with asymmetric firms.
optimal policy. Note that this case subsumes the Bagwell and Staiger (1992) model as a special case, in which firms are assumed symmetric. Alternatively, suppose that the domestic firm invests significantly more in R&D than does the foreign firm so that we are inside Area 3. There, the domestic firm’s best-response function is upward-sloping. However, since the foreign firm’s counterpart slopes downward, R&D subsidies are still an optimal policy. As the final case, suppose that the foreign firm invests considerably more than the domestic firm does so that we are inside Area 1. Then, the foreign firm’s best-response function is upward-sloping and the domestic firm’s is downwardsloping. In contrast to the Bagwell and Staiger (1992) result, an optimal policy in this case is to tax R&D. We have established the following generalization of the Bagwell and Staiger (1992) result. Proposition 7. In the commitment model, suppose that the firms battle for a monopoly position and there is perfect appropriability. If the equilibrium levels of investment are such that h(k*) - h(k) - r>O, an optimal policy is to tax R&D. If a diflerence in investments between two firms is such that the inequality holds in reverse, then an optimal policy is to subsidize R&D. If h(k*) -h(k) - r = 0, an optimal subsidy is zero. In order to understand the intuition
behind Proposition 7, notice that in the
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present case the foreign firm’s marginal benefit from doing R&D can be written as: MB = h’(k*)[L*
+ h(k)L*lr]l[r
+ h(k) + h(k*)]‘.
(17)
Since F* =O, the term h(k)L*/r represents the preemption effect alluded to earlier. When there is an increase in k, this effect is magnified, thereby raising the marginal benefit to the foreign firm. However, an increase in k also causes the patent race to end sooner, a prospect which increasesthe effective discount factor (the denominator in Eq. (17)) and thereby reduces the marginal benefit. The relative strength of the preemption and the discount effect determines the net impact of an increase in k on the marginal benefit. When the foreign firm’s investment in R&D exceedsthe domestic firm’s by a significant difference, h(k*) is considerably greater than h(k). Therefore, a marginal increase in k has little effect on the size of the denominator in Eq. (17). With the discount effect dominated by the preemption effect, the foreign firm’s marginal benefit increasesin responseto an increase in k. Since its marginal cost of R&D is unity, and hence is unaffected by an increase in k, an increase in its marginal benefit prompts the foreign firm to increase R&D activities when k is increased; thus, the foreign firm’s best-responsefunction slopes upward. On the other hand, when the foreign firm’s best-responseinvestment in R&D is not much larger (or even smaller) than the domestic firm’s, the discount effect dominatesthe preemption effect. As a consequence,the marginal benefit decreases in response to an increase in k. Thus, the foreign firm’s best-responsefunction slopes downward. The final question then is, when does the foreign firm invest significantly more than the domestic firm? An answer can be found by differentiating Eq. (17). Since d(MB)ldZ, * >O, it follows that the greater the monopoly profit (L*), the greater the incentive to invest in R&D. Thus, one answer to the above question is that the foreign firm invests more in R&D if it is more efficient as a monopoly than the domestic firm.
7. Concluding
remarks
The recent work by Bagwell and Staiger (1992), (1994) incorporatesuncertainty into the deterministic model of international R&D rivalry of Spencerand Brander (1983), and establishesa strong casefor R&D subsidiesregardlessof the choice of the strategic variable in product market competition. While Bagwell and Staiger retain the stage-game setup of Spencer and Brander, and focus on the case of random cost reductions, this paper adopts an intertemporal framework in which a date of discovery is uncertain. Since real-world inventors must take the two kinds
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of uncertainty into consideration, our paper complementsthe work of Bagwell and Staiger. The two types of uncertainty, however, have contrasting implications for strategic R&D policy. In an intertemporal framework, the nature of an optimal policy dependscrucially on the degreeof appropriability of the new technology. If R&D costs are borne over time, it is optimal to tax R&D at low or high degreesof appropriability although the case for R&D subsidies may emerge at intermediate degreesof appropriability. If R&D costs are to be committed at time zero, R&D taxes are still an optimal policy at low degreesof appropriability. At high degrees of appropriability it may (but need not) become optimal to subsidize R&D. A clearer result is obtained for the case of battles for monopoly. R&D subsidies are an optimal policy when the domestic firm is a better producer than the foreign firm or when the two are roughly equal in terms of efficiency. If the foreign firm is a significantly more efficient than the domestic firm, however, it is best to tax R&D.
Acknowledgments
We are grateful to two refereesfor helpful comments which led to substantial improvements.We alone are responsible for remaining errors.
References Bagwell, K. and R.W. Staiger, 1992, The sensitivity of strategic and corrective R&D policy in battles for monopoly, International Economic Review 33, 795-816. Bagwell, K. and R.W. Staiger, 1994, The sensitivity of strategic and corrective R&D policy in oligopolistic industries, Journal of International Economics 36, 133-150. Brander, J., 1995, Strategic trade policy, in: GM. Grossman and K. Rogoff, eds., Handbook of international economics,Vol. 3 (North-Holland, Amsterdam) 1395-1455. Brander, J.A. and B.J. Spencer, 1985,Export subsidiesand international market sharerivalry, Journal of International Economics 18, 83-100. Bulow, J.I., J.D. Geanakoplosand PD. Klemperer, 1985, Multimarket oligopoly: Strategic substitutes and complements,Journal of Political Economy 93, 488-5 11. Dixit, A.K., 1984, International trade policies for oligopolistic industries, Economic Journal 94, Supplement, 1-16. Dixit, A.K. and G.M. Grossman,1986,Targeted export promotion with several oligopolistic industries, Journal of International Economics 21, 233-250. Eaton, J. and G.M. Grossman, 1986, Optimal trade and industrial policy under oligopoly, Quarterly Journal of Economics 101, 383-406. Fudenberg,D. and J. Tirole, 1984,The fat cat effect, the puppy dog ploy and the lean and hungry look, American Economic Review, Papers and Proceedings74, 361-368. Gilbert, R.J. and D.M.G. Newbery, 1982, Preemptive patenting and the persistence of monopoly, American Economic Review 72, 514-526. Grossman,G., 1988, Strategic export promotion: A critique, in: Paul R. Krugman, ed., Strategic trade policy and the new international economics (MIT Press,Cambridge) 69-89.
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Horstmann, I. and J.R. Markusen, 1986, Up the average cost curve: Inefficient entry and the new protectionism, journal of International Economics 20, 225-247. Lee, T. and L.L. Wilde, 1980, Market structure and innovation: A reformulation, Quarterly Journal of Economics 94, 429-436. Loury, G., 1979, Market structure and innovation, Quarterly Journal of Economics 93, 395-410. Miyagiwa, K., 1992, International shareholdings and strategic export policy, International Economic Journal 6, 37-42. Miyagiwa, K. and Y. Ohno, 1995, Credible protection and incentives to innovate, Mimeo. Reinganum, J., 1983, Uncertain innovation and the persistence of monopoly, American Economic Review 73, 741-748. Spencer, B.J. and J.A. Brander, 1983, International R&D rivalry and industrial strategy, Review of Economic Studies 50, 702-722.
ELSEiVIER
Strategic R&D policy and appropriability Kaz Miyagiwa”, Yuka Ohnob “Department of Economics, 2107 CEBA, Louisiana State Universiry, Baton Rouge, LA 70803, USA ‘Department of Economics, Oregon State University, Ballard Extension Hall 303, Corvallis, OR 97331-3612, USA
Received 28 February 1995; accepted I8 April 1996
Abstract The Spencer-Brander argument for strategic R&D policy (Review of Economic Studies, 1983, 50, 702-722) is re-examined within an intertemporal model of stochastic innovation. It is shown that an optimal policy is independent of the choice of strategic variables in product market competition but sensitive to the appropriability of the new technology. At low or high degrees of appropriability R&D taxes are an optimal policy but at intermediate degrees R&D subsidies may be optimal. The paper explains why these results differ from Bagwell and Staiger’s finding (International Economic Review, 1992, 33, 795-816; Journal of International Economics, 1994, 36, 133-150) that R&D subsidies are an optimal policy in the presence of uncertainty. 01997 Elsevier Science B.V. All rights reserved. Key words: R&D subsidy; Innovation; Strategic trade policy; Patent length; Appropriability JEL class$cation:
F13; 031; 032; 033
1. Introduction
This paper reconsiders a strategic role for R&D policy in the presence of international rivalry. The original argument for strategic R&D subsidies, due to Spencer and Brander (1983), is known to be highly sensitive to the model’s assumptions.For example, if firms chooseprices instead of quantities as a strategic variable, an optimal policy switches from a subsidy to a tax.’ Grossman (1988) ‘See Eaton and Grossman(1986). The sensitivity of someresults to the choice of strategic variables however is not unique to the Brander-Spencer model but is endemic in a broader class of oligopoly models. 0022-1996/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII
SOO22-1996(96)01448-l
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finds this fact particularly disturbing when translating theory into trade policy because there is no knowing which strategic variables firms actually choose in real-world situations.* That concern however has been resolved in the recent papersby Bagwell and Staiger (1992), (1994), who introduce uncertainty into the Spencer-Brander model and show that R&D subsidies are an optimal policy regardless of the choice of a strategic variable in product market competition.3 Thus, the introduction of uncertainty not only makes the Spencer-Brander model more realistic but also strengthenstheir case for R&D subsidies. While Bagwell and Staiger focus on uncertainty over the extent of cost reduction under the new technology, innovating firms are also concerned with uncertainty over an actual date of discovery. The latter type of uncertainty is better analyzed in intertemporal models rather than in stage-game models. However, Bagwell and Staiger (1992) show that their stage-game model is equivalent to an intertemporal model of a patent race as long as firms must commit R&D costs at time zero, meaning that R&D subsidiesare still an optimal policy in a model of an uncertain discovery date. This paper extends the above line of research in two directions. First, we consider not only the commitment casebut also the case in which R&D costs are borne over timep Second,implicit in the Bagwell-Staiger model is the assumption that an innovator can forever enjoy exclusive rights to the use of the new technology. In the real world, however, patent protection is neither perfect nor permanent.We therefore investigate the linkage between the degree of appropriability of the new technology and the nature of optimal R&D policy. The key results are the following. Supposefirst that R&D costs are borne over time. Then, an optimal policy at low degreesof appropriability is to tax R&D. As the degreeof appropriability increases,R&D subsidies may emergeas an optimal policy. At sufficiently high degreesof appropriability, however, R&D taxes once again become optimal5 Supposenext that R&D costs must be committed at time zero, as in Bagwell and Staiger. At low degreesof appropriability it is optimal to tax R&D. Even at high degreesof appropriability, R&D taxes can be optimal when firms are not alike. In particular, in the model of battles for monopoly, a special *Taxes may emerge as an optimal policy in the presence of domestic consumption (Dixit and Grossman, 1986). multifirm oligopoly (Dixit, 1984), free entry (Horstmamr and Markusen, 1986). resource constraints (Dixit and Grossman, 1986) and international shareholding (Miyagiwa, 1992). While these papers are concernedwith the optimality of export subsidies,as in Brander and Spencer (1985). their arguments readily apply to the case of R&D subsidies. Brander (1995) provides a comprehensivereview of the literature. ‘A corrective incentive to tax R&D may emerge if there is more than one domestic firm. %-te commitment model and the non-commitmentmodel were first developed by Loury (1979) and Lee and Wilde (1980). respectively. The former is more relevant if R&D projects require substantial setup costs such as building a lab, or are contracted to a researchfirm for a given sum at time zero. ‘Bagwell and Staiger (1992) have conjectured that R&D taxes may be optimal in the noncommitment model.
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caseexamined in Bagwell and Staiger (1992), R&D taxes are an optimal policy if the foreign firm is a more efficient producer than the domestic firm. Since taxing domestic R&D apparently jeopardizes the home country’s chance of winning a patent race, at first blush it looks puzzling that such a strategy can ever be welfare-improving. The key to understanding this puzzle lies in the strategic nature of investments in R&D. The following two extreme casesin the non-commitment model illustrate the general principles involved. In the first case, assumeperfect appropriability so that losing a patent race meansno accessever to the new technology and hence a huge loss in potential profits. In such a case winning a patent race is imperative. When one firm increasesinvestment in R&D, the other firm follows suit. Thus, investments in R&D are strategic complements. Strategic complementarity means that the domestic firm invests more in the presence of R&D rivalry than when it faces no such competition. This excess R&D is wasteful from the domestic country’s point of view. In such a case R&D taxes soften competition and lead to a welfare improvement. In other words, R&D taxes serve as a puppy-dog ployp Going to the other extreme case,assumezero appropriability.7 Then, no matter which firm discovers it, the new technology will equally benefit both firms. Thus, when one firm increases investment in R&D, the other can afford to reduce its investment without sacrificing the expected benefits of the new technology. That makesinvestments strategic substitutes.In such a situation, R&D taxes reduce the level of the domestic firm’s investment, thereby making the foreign firm bear a greater share of R&D costs. While such a strategy improves the foreign firm’s chance of successin R&D, at zero appropriability the domestic country stands to gain as much from the foreign firm’s discovery as from its own. Thus, R&D taxes improve domestic welfare. The remainder of this paper is organized as follows. Section 2 presentsthe basic setup. The non-commitment model is developed in Section 3 and the main results are derived in Section 4. Section 5 applies the non-commitment model to the two special cases common in the industrial organization literature. The commitment model is the focus of analysis in Section 6. The final section concludes the discussion. 2. The setup
Suppose that two firms, one domestic and one foreign, compete for the discovery of a new technology over time. Time is a continuous variable defined %ee Fudenberg and Tirole (1984) for the classification of strategies based on canine and feline characteristics, ‘Innovation can occur even at zero appropriability, since post-discovery profits still exceed prediscovery profits becauseof an induced cost reduction.
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over [O,m).There are two distinct technologies, referred to as the old and the new technology. At time zero, the firms have accessonly to the old technology. The new technology becomesavailable only through investment in R&D, outcomesof which are stochastic. Each firm can improve its probability of discovery by investing more in R&D. When the new technology is discovered, the winner acquires exclusive rights to the use of it for a given time interval, T, after which the new technology becomes available to the other firm. Here, T measuresthe speedof technology diffusion or the degree of appropriability, which may reflect patent length or the difficulty in imitating the new technology. In addition to competing in R&D over time, firms competefor export sharein a third-country market at each point in time r E [O,W).~The demand functions in the third-country market are time-invariant. There is no home market consumption of the export goods. Now we introduce notation. Let ?r(&9*) denote the domestic firm’s (Nash) equilibrium flow profit from export when the domestic firm uses the technology 8 and the foreign firm usesthe technology 0 *P A similar interpretation applies to the foreign firm’s flow profit from export, r*(0,0*). Let an upper bar imply the new technology and a lower bar the old technology. Thus, for example, if the domestic firm discovers the new technology, its flow profit increases from 7r(e,j*) to v(@*) whereas the foreign firm’s flow profit falls from v*(@*) to ‘rr*( 8,$*) for the interval T, after which the domestic and the foreign firm earn 7r(8,8*) and a*(8,8*), respectively. We assumethat these inequalities hold: (14
7r*(e,e*> > n-*(e,B*)> T*@,e,e*)> 7T”(@“).
(lb)
Thus, the firm’s profits are highest when it has exclusive rights to the use of the new technology and lowest when the rival has such rights. Also, profits are greater when firms share the new technology than when neither firm can use it. The above specifications are flexible enough to capture a number of alternative situations. Eqs. (la), (lb) can apply to either Coumot or (differentiated-good) Bertrand competition.” If the new technology is a process innovation, the difference m(t@ *) - ~(0,0*), for example, denotesa gain from a cost reduction. If the new technology isaproduct innovation, the samedifference may denote an increase in profits from new markets. ‘We focus on a Markov-perfect equilibrium, and therefore rule out the possibility of implicit collusion between firms that may arise in a supergame framework. 9An asterisk (*) indicates a variable pertaining to the foreign firm. “In the case of symmetric homogeneous-good Bertrand competition all but the first profits in Eqs. (la), (lb) are equal to zero. Therefore, the last two inequalities in Eqs. (la), (lb) are replaced by weak inequalities.
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3. The non-commitment
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model
3.1. The R&D process
In the non-commitment model, each firm invests in R&D at each point in time until the new technology is discovered, at which time both firms terminate R&D activities. Following Lee and Wilde (1980), we assumethat the domestic firm’s date of discovery, 5, has the exponential cumulative distribution function: Pr(5 5 t) =
1 - exp[-h(k)t],
(2)
where k denotesthe amount the domestic firm invests in R&D per unit of time.” That is, investment k at time t generates the probability h(k) dt of discovery between t and t + dt, given that the new technology has not been discovered before t. The hazard rate function h(k) is twice continuously differentiable and satisfies the following regularity conditions: h(0)
= 0, h’(k) > 0, h”(k) < 0, >ly+h’(k)
= ~0,and lilih’(k) m
= 0.
(3)
The first equality says that there is no chanceof discovery without R&D. The next two conditions statethat investment in R&D improves the probability of discovery with diminishing returns. The limit conditions and Eqs. (la), (lb) guaranteethe existence of an interior solution for k. The same hazard rate function h(.) is assumedto apply to the foreign firm as well.‘* Thus, investment k* in R&D at time t gives the foreign firm the probability h(k*) dt of discovery between t and t+dt. Each firm’s probability of discovery is assumedto be independent. The domestic government subsidizesdomestic R&D activities at the rate s (< 1) so that the domestic firm’s net R&D costs per unit of time are k( l-s). When the new technology is discovered, firms stop investing in R&D, and the subsidy programs also end. 3.2. R&D
competition
We next specify the domestic firm’s intertemporal profit. Supposewe are at time t. Then, with the probability exp[-h(k)t - h(k*)t] neither firm has discovered the new technology before t. Conditional on that, three things can happen between “This assumes the existence of a steady-state solution for k. As long as there has been no discovery firms face the same situation at every moment so this steady-state solution is a correct one even if firms are allowed to vary k at each point in time (see Miyagiwa and Ohno, 1995). ‘*This is only to simplify notation. Even if the foreign firm has a different hazard rate function that satisfies the regularity condition Eq. (3), our conclusions remain intact.
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time t and t+dt. First, the domestic firm discovers the new technology with the probability h(k) dt, and earns the discounted sum of profits: cc
f+T e-"7iffQ*) I
dx +
e-"n-(8,8*)
dx,
I t+T
f
(4)
where r denotes an exogenous rate of interest. The first integral in Eq. (4) representsthe domestic firm’s profits while it usesthe new technology exclusively. The second integral represents the domestic firm’s profits when the innovation becomesavailable to the foreign firm after the interval T. Integrating, we write Eq. (4) more compactly: (e-“lr)L
where
(L stands for a ‘leader’).
Second, with the probability h(k*) d? the foreign firm discovers the new technology betweent and t + dt. In this case,the domestic firm’s flow profit falls to n($,t?*) while the foreign firm monopolizes the new technology, and rises to rr( I?$*) when the domestic firm also gains accessto the new technology after T. Therefore, the domestic firm’s profits equal m
f+T e-"n-(&8*) I
dx +
e-"r(t$t?*)
dx,
I z+T
f
which can be written as (e-“lr)F
where F = a,$*)(1
- eLrT) + e-“rr(8,8*)
(F stands for a ‘follower’).
The third possibility is that neither firm makes a discovery between t and t+dr. In that case,the domestic firm enjoys the net flow profit 7T(e,@ *) - k( 1 - s) for the duration of dr, and we move on to time t+ dt, where the same three possibilities just discussedarise once againI Integrating the profits in the three casesover time yields the expression for the domestic firm’s intertemporal profit: “The fourth case, in which both firms innovate between time probability h(k)h(k*) dt2 is of second-order smallness.
t and r+dr, can be ignored since its
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m
v(k,k*,s)
=
e -rhck’+hca*‘l’h(k)(e-“lr)L
I
&
0 m +
co e-[h(k)+h(k*)lth(k*)(e-rt/r)F
&
+
I
e-rte-[h(k)+h(k*)l’[~~,~*) I
0
0
-k(l-s)]dt, which simplifies to y(k,k*,s) =
?r(e,j *> - k( 1 - s) + h(k)Llr + h(k*)Flr r + h(k) + h(k*)
(5)
The domestic firm regards k* and s as given and chooses k so as to maximize v(k,k*,s). The optimal level of-k satisfies the first-order condition: ccl(k,k*.s) = [h’(k)Llr
- 1 + s][r + h(k) + h(k*)]
- [T@,@*) - k( 1 - s) + h(k)Llr + h(k*)Flr]h’(k)
= 0.
(6)
The second-ordercondition is satisfied since tjjj = h”(k)[L - n-(&j *) + k( 1 - s) + h(k*)(L/r
- F/r)] < 0.
(Subscripts denote partial derivatives.) The first-order condition can be given a standardinterpretation if we rewrite Eq. (6) as h’(k){L - [v(@,!,*) - k( 1 - s)] + h(k*)(Llr
- s).
- F/r)} = [r + h(k) + h(k*)]( 1 (7)
A marginal increase in R&D investment will cost the domestic firm the interest rate r, and will also be wasted with the probability h(k) + h(k*) becausewith that probability the new technology will be discovered without the increase in R&D. Thus, the right-hand side of Eq. (7) representsthe (subsidized) marginal costs of R&D. On the left-hand side, L - [ @,e *) - k( 1 - s)] measuresa direct increase in flow profits if R&D is successful at time t. The other term h(k*)(ZJr -F/r) representsthe preemption effect: a successin R&D at time t forever prevents the domestic firm from losing the patent race, which occurs with the probability h(k*) at each point in time, and hence secures for the domestic firm the expected discounted sum of the leader-follower difference in profits, h(k*)(Llr -F/r). Since a unit increasein investment raises the probability of discovery by h’(k), the left-hand side of Eq. (7) amounts to the expected marginal benefits of R&D. Interpreted this way, Eq. (7) is the standard optimality condition: the expected marginal benefits of R&D equal the marginal costs of R&D. We can express the foreign firm’s intertemporal profit similarly:
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K. Miyagiwa,
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T*($‘,@*) - k” + h(k)F*lr + h(k*)L*lr r + h(k) + h(k*)
where L* = r*(@*)(l - LrT) + e-%-*(8,8*), F* = 7r*(8,@*)(1 -t?) + e-“r*(8,8*).
and
The first-order condition is +*(k,k*) = [h’(k*)L*lr
- l][r + h(k) + h(k*)]
- [r*@,e,e*) -k*
+ h(k)F*lr
+ h(k*)L*lr]h’(k*)
= 0.
(8)
It is easy to check that the second-ordercondition is satisfied: I@* < 0. The first-order conditions Eqs. (6), (8) define the two firms’ best-response functions: k=b(k*;s) and k* =b*(k). A subgame-perfectNash equilibrium is a pair of R&D expenditures (k,k*) that simultaneously solves these best-response functions. Assumptions Eqs. (la), (lb), (3) guaranteethat b(k*,s) and b*(k) are positive for all k* and k, respectively, so that there is an interior solution.i4 Furthermore, the equilibrium is unique and stable under the assumptions:
which imply
3.3. Properties of the best-response functions
In this subsection we examine the slopes of the best-responsefunctions. Take the domestic firm’s best-responsefunction. Since db(k*;s)lak* = - I,$~/#~, and &CO by the second-order condition, the slope of the domestic firm’s bestresponsefunction takes the sign of erk*.Differentiating Eq. (6) yields I& = h’(k*)[h’(k)(L
- F)lr - (1 - s)].
(9)
r4From Eq. (6). 1/~(0,k*,s) = h’(O)[L - ?r(f,~)] + h’(O)h(k*)(L - F)lr - (1 - s)[r + h(0) + h(k*)I > 0 for all k*, thereby implying that the domestic firm’s best response is always to invest a positive amount in R&D. Similarly, $*(k,O)>O for all k. ‘?f T is sufficiently close to zero, &
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Notice that the leader-follower L-F
133
difference in post-discovery profits
= [?r(&#*)
- n-@,8*)](1 -em’*) (10) . is strictly monotone-increasing in T, rising from zero when T is zero to ?r( $8 *) ~@,8*) as T approaches infinity.i6 Similarly, the slope of the foreign firm’s best-response function depends on the sign of t,b,*= h’(k)[h’(k*)(L*
- F*)lr
- 11,
where the strictly monotone-increasing function L* -F”
= [r*(j,c?*)
- rr*(t@*)](l
- e-‘r)
takes the value between zero and r*(&e*) The following is a useful lemma. Lemma 1. (9 ~4~ is positive
- rr*(@*).
if T is greater than ?: where
F= (llr){ln[?r(8,8*)
- 77@,8*)] - ln[7r@,tj*) - 7@,8*)]}.
(3 Ax is negative for T su@ciently close to zero. (iii) I,I%~is positive if T is greater than T” , where T* ~(llr){ln[~*(~,f?*)
- 7r*(cQ*)]
- ln[7r*(@*)
- ~*(&j*)]}.
+,* is negative if T is sufficiently close to zero. If we let b denote the domestic firm’s best response, Eq. (6) implies v(b,k*,s)= [h’(b)Llr-(1 -s)]lh’(b), so Eq. (9) can be written as:
Proof.”
(Glk*= h’(k*)h’(b)[v(b,k*,s)
-F/r].
Since b is a best response, v(b,k*,s)s v(O,k*,s). However v(O,k*,s) -F/r
= [rr(J,@*) - F]l[r
+ h(k*)]
where T(@*)
-F = {?T(e,e*) - 7.r@,$*) + eMrT[?r(j,8*) - m-(6,6*)])
is positive if T exceeds i? Therefore, for such a T, Y(b,k*,s) - F/r? v(O,k*,s) Flr>O, and hence &* > 0. On the other hand, since L-F is zero at T =0, we “With zero appropriability there is no advantage in being a leader. As the appropriability increases, the leader can monopolize the new technology for a longer period, implying that the leader-follower difference in profits increases monotonically. “The proof for the case in which T= m is similar to the proof in Reinganum (1983).
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can choose T as close to zero in Eq. (7) as we like so as to make L-F smaller than 1-s in Eq. (9), so that I,,$* becomesnegative. Similarly, for $f. Cl The next result, which follows from Lemma 1, gives sufficient conditions for determining the slopes of the best-responsefunctions b(k*;s) and b*(k). Proposit& 1. Let T and L@ be as dejined in Lemma 1. Then: 1. if T > T, the best-response function b(k*;s) is positively sloped in k*; if T > F*, b*(k) is positively sloped; 2. if T is sufjiciently close to zero, both b(k*;s) and b*(k) are negatively sloped.
Proposition 1 says that investments in R&D tend to be strategic complements when T is large, and strategic substitutes when T is small.1sHere is an intuitive explanation. At a low degreeof appropriability, the benefits of the new technology spill over to the non-innovating firm quickly so that the preemption effect in Eq. (7) is insignificant. An increasein the foreign firm’s investment therefore has little or no effect on the domestic firm’s marginal benefits from R&D (the left-hand side of Eq. (7)). The marginal cost of R&D (the right-hand side of Eq. (7)) however is now higher, thereby prompting the domestic firm to curtail investment in R&D. That shows that at a low degree of appropriability investments are strategic substitutes. This explains the second part of Proposition 1. At a high degree of appropriability, the preemption effect is so important that an increase in investment by one firm is likely to be challenged by a matching increase by the other. Thus, at high degrees of appropriability investments in R&D are strategic complements.This explains the first part of Proposition 1. Fig. 1 illustrates the case in which T= 0 so that investments are strategic substitutes. In Fig. 2, T>max{F,F*} so that investments are strategic complements. (It is also possible to have mixed cases, in which one best-response function is upward-sloping and the other downward-sloping.) In each figure, b” and b*“, respectively, representthe domestic and the foreign firm’s best-response function when s = 0, and point 1 correspondsto the subgame-perfectequilibrium under laissez-faire policy. (The dotted schedule should be ignored for now.) 4. The effect of R&D subsidies in the non-commitment
model
4.1. The effect on R&D investments
We begin this section by examining how R&D subsidies shift the domestic firm’s best-responsefunction. Applying the implicit function theorem to Eq. (6) yields ‘*The importance of the distinction between strategic substitutes and complements in oligopoly models is discussedthoroughly in Fudenbergand Tirole (1984) and Bulow et al. (1985).
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k’
b0
bS
k Fig. 1. Strategic substitutes [T=O].
ab(k”;s)las = - **I@, where & = r +
h(k) + h(k*) - h’(k)k. k*
Fig. 2. Strategic complements [T>max{T,T*}].
135
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Concavity of h(k) and the condition h(0) = 0 imply that h(k) - h’(k)k >O, and hence clr,>O. By the second-order condition, ek is negative. Therefore, &(k*;s)/& is positive, that is, R&D subsidies increase domestic investment in R&D for a given level of k*. In Figs. 1 and 2, that effect is illustrated by the rightward shift in the domestic firm’s best-response function (from the solid line to the dotted line). Note that the result holds whether investments are strategic substitutes or complements. Since s does not directly appear in Pq. (8), R&D subsidies leave the foreign firm’s best-response schedule undisturbed. Therefore, the subgame-perfect equilibrium under R&D subsidies is at point 2 in Figs. 1 and 2. While R&D subsidies always increase the domestic firm’s equilibrium investment, their effect on the foreign firm’s R&D level depends on the slope of the foreign firm’s best-response function. Precise conditions are given in the following. Proposition 2. (i) R&D subsidies increase the domestic jrm ‘s investment in R&D. (ii) If T exceeds T” given in Lemma 1, R&D subsidies increase the foreign jrm ‘s investment. (iii) If T is sufJiciently small, R&D subs’idies decrease the foreign jirm’s investment. Proof. Differentiate Eqs. (6), (8) to get
I,+; dklds + I,$ dk”lds = 0. Solving these simultaneously yields
Since &ID is positive and I,$ is negative by the second-order condition, dklds is positive. This proves (i). On the other hand, dk*lds is positive only if #f is positive. Then, (ii) and (iii) follow from Lemma 1. 0 The intuition underlying Proposition 2 is straightforward in light of Proposition 1. At a low (high) degree of appropriability, investments are strategic substitutes (complements) so an increase in the domestic firm’s investment in R&D induces a decrease (an increase) in the foreign firm’s investment. R&D taxes, of course, have the opposite effect.
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4.2. The welfare efect of R&D subsidies Since there is no home consumption, domestic welfare is the difference between the domestic firm’s intertemporal profit and the present discounted sum of subsidies. The latter is contingent upon non-discovery of the new technology, and equals
e
-rr
e
-[h(k)+h(k*)l+)
&.
I
0
Hence, domestic welfare is written as m u(k,k*,s) =
e-lh(k)+h(k*)“h(k)(err/r)L
I
dt
0
m
+
e-rfe-[h(t)+k(lr*)l’[~~,~*>
-k]
dt,
I
0
where k and k* are equilibrium Integration yields u(k,k*,s) =
R&D investments, i.e. k=k(s) and k* =k*(s).
?r@,@*) - k + h(k)Llr + h(k*)Flr r + h(k) + h(k*) ’
(11)
In order to find the welfare impact of R&D subsidies, we differentiate Eq. (11) totally with respect to s to obtain du[k(s),k*(s),s]lds
= Us dklds + uk* dk”lds
(12)
and evaluate it at s = 0. Noting that u(k,k*,s) = v(k,k*,s) - skl [I + h(k) + h(k*)], and using the envelope theorem yields uk = - s&l[r
+ h(k) + h(k*)]’
so that the first term on the right of Eq. (12) vanishes when evaluated at s = 0. The welfare effect of an R&D subsidy is captured by the second term. By Proposition 2, the sign of dk*lds depends on the degree of appropriability: it is positive when T exceeds p* and negative when T is sufficiently small. As for uk*, one can differentiate Eq. (11) with respect to k* to get
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= h’(k*)(Flr)l[r
‘k*
Economics 42 (1997) 125-148
+ h(k) + h(k*)]
- h’(k*)[+?e*) -k + h(k)Llr + h(k*)Flr]l[r + h(k) + h(k*)]’ = h’(k*)[Flr - u(b,k*,s) + sk]l[r + h(k) + h(k*)].
This expression also dependson the degree of appropriability. For T >T, uk*ls=O = h’(k*)[Flr
- u(b,k*,s)]l[r
+ h(k) + h(k*)]
is negative because, as shown in the proof of Lemma 1, F/r- v@,k*,s) is negative, meaning that an increasein the foreign firm’s investment in R&D lowers domestic welfare. Combining the two results, for T>max{7,~*} we have sgn(duld&)
= sgn(u,, dk*ld.s) < 0;
that is, R&D taxes improve domestic welfare at sufficiently high degrees of appropriability. Alternatively, if T is zero, we have dk*lds
+ h(k) + h(k*)]’
- ?T@,$*) + k}l[r + h(k) + h(k*)12 > 0,
i.e. an increase in the foreign firm’s investment in R&D improves domestic welfare. Therefore, at T=O sgn(duldsl,,,) = sgn(u,, dk*ld.r) < 0; that is, R&D taxes increase domestic welfare. Furthermore, since uk* dk*lds is continuous in T, there exists a neighborhood of T=O, in which uk* dk*lds < 0. Thus, if T is sufficiently small, taxing R&D is welfare-improving. We have proved the following (local) result. Proposition 3. (i) Zf T is sujJiciently short (close to zero), or (ii) if T exceeds max{F*,T}, a (locally) optimal policy is to tax R&D.
Fig. 1 depicts Proposition 3(i). With T close to zero, the best-responsefunctions slope downward, and domestic iso-welfare curves are U-shaped, at least in the neighborhood of the initial equilibrium; I9 u” representsdomestic welfare with no government intervention; R&D taxes shift the domestic firm’s best-response ‘?he iso-welfare curves are identical to the iso-profit curves under zero subsidies, with the slope given by dk*ldk= -t+lu,,. For a given k *, let b denote the best responseof the domestic firm. The optimality of b implies that at k= b uk equals zero whereas uk~-0 for k< b, and u,
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function to the left. Given that iso-welfare curves are (locally) U-shaped, there exists a (locally) optimal R&D tax that raises domestic welfare to u’. Proposition 3(ii) is illustrated in Fig. 2. With T sufficiently large, the bestresponse functions slope upward and domestic iso-welfare curves are inverted U-shaped, at least locally. Again, taxing R&D shifts the domestic firm’s bestresponse function to the left, thereby increasing domestic welfare to u’. While R&D taxes are an optimal policy when T is either small or large, the next two propositions characterize an optimal R&D policy for intermediate values of T. Proposition optimal.
4. There exists T E (O,min{~,~*})
such that zero subsidies are locally
Proof Let {k(T),k*(T)} denote the subgame-perfect equilibrium of the R&D game when s =0, and let A(T)= t@[k(T),k*(T),T] and p(T)=t+*[k(T),k*(T),T]. Since dk”lds=@&,lD, where t,kSID>O, sgn(dul&(,=,)=sgn[~(T)~(T)]; h(T) is a continuous function, with h(0) < 0 and A( f *) > 0, as discussed earlier. Therefore, by the intermediate-value theorem there exists at least one T such that A(T)=O. Let T* denote the set of such T. Similarly, since p(T) is continuous, with p(O)>0 and ,x( i;)
Proposition 4 says that there always exists a T at which zero subsidy is an optimal policy. The next proposition gives a sufficient condition under which an optimal policy is actually to subsidize R&D. Proposition 5. If the sets TA and T’” are disjoint, there exists a range of T for which an R&D subsidy is a (locally) optimal policy.2o Proof. If T” and T” are disjoint, there always exists an interval I such that T El implies A(T) so that dulds = uk* dk*lds >O. 0
Propositions 4 and 5 are illustrated in Fig. 3, where we assume A(T) and p(T) are strictly monotone so that T” and T” are singletons: T”={T”} and T’ ={Tp}. For TE [O,T’)U(T”,~), A(T)p(T)
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Fig. 3. A case of optimal R&D subsidies.
1, h(k)=k”*, and T=0.762. Then the equilibrium is given by k = k* =4, uk* < 0, +:
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141
subsidy can improve the domestic firm’s chance of winning the patent race, and hence is welfare-improving.
5. Applications This section applies the model to the two specific situations examined in the industrial organization literature. In each case R&D taxes emerge as an optimal policy. 5.1. Battles for monopoly
Battles for monopoly are studied by Loury (1979), Lee and Wilde (1980) and Bagwell and Staiger (1992). In the model of battles for monopoly neither firm makes positive flow profits before discovery, and only the winner of the patent race makes a monopoly profit forever after discovery. If we make these adjustments and set T to infinity in our model, it follows from Proposition 3 that R&D taxes are an optimal policy. The result contrasts sharply with the Bagwell and Staiger (1992) finding that R&D subsidies are optimal in the commitment model. The commitment model of battles for monopoly is analyzed in Section 6, where it is shown that R&D taxes can be an optimal policy. 5.2. Monopoly
under the threat of entry
The analysis of the incumbent facing the threat of entry has received attention in the industrial organization literature (e.g. Gilbert and Newbery, 1982; Reinganum, 1983). The basic setup of this literature is as follows. The incumbent initially earns a monopoly profit. If it discovers the new technology first, it retains the monopoly position. If the entrant discovers the new technology, entry occurs and each firm earns a duopoly profit. Implicit in the above setup is the assumption that a winner of the patent race enjoys exclusive rights indefinitely. Thus, set T equal to infinity in our model, and interpret the flow profits to fit the story. For instance, if the domestic firm is the entrant, set its pre-discovery profit r(B,8 *) and the follower’s profit F equal to zero, and interpret the winning profit Las a duopoly profit. Alternatively, if the domestic firm is the incumbent, interpret Q,e*) as a pre-discovery monopoly profit, L as a post-discovery monopoly profit, and F as a duopoly profit. In either case, since T is infinite, investments are strategic complements and so R&D should be taxed by Proposition 3. Again, what is crucial is not the specific form of market competition but the fact that T is infinite.
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6. Optimal R&D policy in the commitment made1 This section examines the strategic role for R&D policy in the commitment model. The issue has already been examined by Bagwell and Staiger (1992). The treatment here generalizestheir model in three respects.First, while Bagwell and Staiger assumeperfect appropriability, we allow T to be finite. Second,they focus on battles for monopoly: that is, they assumethat firms make zero flow profits before discovery and only the winner of the patent race makes positive postdiscovery flow profits. In contrast, we allow each firm to make positive profits in pre-discovery and/or post-discovery competition. Third, while Bagwell and Staiger focus on the symmetry case,we also consider the casesin which firms are asymmetric. 6.1. The setup
The basic setup of the model remains much the sameas in the non-commitment model except for the fact that firms now must commit investments in R&D at time zero. Investment k, sunk at time zero, generatesthe probability h(k) dt of discovery between t and t f dz for all t E [O,w) according to the distribution function Eq. (2), and the hazard rate function h(.) satisfies the assumptionsof Eq. (3). The expression for the domestic firm’s intertemporal profit is similar to the previous definition except that R&D costs k( 1-s) are not a flow: m
V(k,k*,s) = e -th(k)+h(k*)lrh(k)(e-rt/r))L dt
h(k*)(e-“lr)F
dt
+ e-rte-[h(k)+6(k*)lr n@,@*) dt - k( 1 I
s).
0
This expression simplifies to v(k
k.+ s) 9 9
=
7T(e,@*)
+
WWr
+
W*)Flr
- k( 1 - 8).
r + h(k) + h(k*)
The domestic firm takes k* and s as given and choosesk to maximize V(k,k*,s). The first-order condition is given by P(k,k*) = h’(k)[L - nQ,@ *) + h(k*)(Llr
- F/r)]
- (1 - s)[r + h(k) + h(k*)]* = 0,
(13)
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which defines the domestic firm’s best-responsefunction: k = B(k*;s). The slope of the best-responsefunction B(k*,s) dependson the sign of the expression h’(k*)h’(k>{2[7@@*)
- L] + [I + h(k) - h(k*)](L r + h(k) + h(k*)
% =
- F)lr} (14)
Similar relations hold for the foreign firm. In particular, the slope of the foreign firm’s best-responsefunction depends on the sign of the partial derivative Pf, defined analogously to Eq. (14). 6.2. The welfare
analysis
Begin the analysis by investigating the relationship between appropriability and the slope of the best-responsefunctions. Assume zero appropriability first. Then, L-F is zero, and hence the second term in braces on the right hand of Eq. (14) vanishes. The first term is negative since n-(&e*)
- L = @,@*>
- ?r(B,B*) < 0.
Thus, ?& is negative, and so the domestic firm’s best-responsefunction slopes downward. Similarly, the foreign firm’s best-responsefunction is also negatively sloped, meaning that R&D investments are strategic substitutes. It is straightforward to show that under the usual stability assumptiondklds>O and dk*lds< 0. We now turn to the effect on domestic welfare, which can be written as U(k,k”,s) =
@,j*)
+ h(k)Llr
+ h(k*)Flr
_ k
r + h(k) + h(k*)
where k=k(s) and k* = k*(s) constitute a subgame-perfectNash equilibrium in R&D competition. Differentiation of U(k,k*,s) with respect to s yields dU/ds = U, dklds + U,, dk*lds.
(15)
The first term on the right hand side of Eq. (15) vanishes when evaluated at s = 0, so the second term captures the welfare effect of R&D subsidies. In order to evaluate the latter, first differentiate U(k,k*,s) with respect to k* to obtain U,,[r
+ h(k) + h(k*)]‘/h’(k*)
= F - d&J*) + h(k)(F - L)lr = (1 - ,4-rT )?r(e,d*) + e-“77fi@*)
- n-(&j*) - (1 - eCrTMWl[~@*)
(16)
- ?r(B,@Yl.
As T approacheszero, the right-hand side of Eq. (16) converges to ?r(fi,t$*) rr(e,$ *), a positive number, and so U,, > 0. That is, when T is sufficiently close to zero, an increase in the foreign firm’s investment in R&D raises domestic welfare because of spillovers from a discovery by the foreign firm. Rutting the
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above results together, we have dU/dsl,,,=U,, following.
dk*lds
and hence the
Proposition 6. In the commitment model, when T is sufJiciently close to zero, R&D taxes are a (locally) optimal policy.
Alternatively, suppose that T is sufficiently large. Then, the right-hand side of Eq. (16) is negative. Since the spillover benefits from the foreign firm’s discovery are small, an increase in investment by the foreign firm, which raises that firm’s probability of discovery, hurts the domestic country. Therefore, an optimal strategy should discourage the foreign firm from investing in R&D. That goal can be accomplished by R&D subsidies if the foreign firm’s best-response function is negatively sloped, and by R&D taxes if it is positively sloped. Unfortunately, however, the sign of ?& is in general ambiguous.” In order to gain further insight, we focus on the Bagwell and Staiger (1992) model of battles for monopoly. 6.3. Battles for monopoly
It is recalled that in battles for monopoly pre-discovery profits and postdiscovery profits for a loser (F and F *) are equal to zero, and L = n( f@*) and L* = n*(e,e*) represent the monopoly profits, which a winner of R&D competition can enjoy forever. Substituting these values in Eq. (14) and setting T= m yields !4& = h’(k*)h’(k)(Llr)[h(k)
- h(k*) - r] / [r + h(k) + h(k*)]
which is positive if and only if h(k) - h(k*) - r >O. The analogous expression for the foreign firm !Pz = h’(k)h’(k*)(L*lr)[h(k*)
-h(k)
- r]/[r
+ h(k) + h(k*)]
is positive if and only if h(k*) - h(k) - r>O. Fig. 4 depicts the relationship between h(k) and h(k*) on the one hand, and the sign of ?& and ?Pf on the other. There are three areas separated by the lines representing h(k)- h(k*)-r=O and h(k*)- h(k)-r=O. Notice that it is impossible to have both the best-response functions slope upward simultaneously.22 Suppose that the levels of investment between two firms are not too dissimilar (Area 2). Then, ?&‘,*and Pt are both negative, and R&D investments are strategic substitutes. It follows from the earlier discussion that R&D subsidies are an *‘The ambiguity results because unlike in the non-commitment model a firm’s R&D costs are independent of the actual date. of discovery, and hence, of its rival’s R&D effort. **Both Pk, and Pf are positive only if h(k)- h&*)-r>0 and I#*)-h(k)-r>O. Adding the inequalities however yields - 2r>O, a contradiction.
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h(k*)
r=O
Fig. 4. Optimal R&D with asymmetric firms.
optimal policy. Note that this case subsumes the Bagwell and Staiger (1992) model as a special case, in which firms are assumed symmetric. Alternatively, suppose that the domestic firm invests significantly more in R&D than does the foreign firm so that we are inside Area 3. There, the domestic firm’s best-response function is upward-sloping. However, since the foreign firm’s counterpart slopes downward, R&D subsidies are still an optimal policy. As the final case, suppose that the foreign firm invests considerably more than the domestic firm does so that we are inside Area 1. Then, the foreign firm’s best-response function is upward-sloping and the domestic firm’s is downwardsloping. In contrast to the Bagwell and Staiger (1992) result, an optimal policy in this case is to tax R&D. We have established the following generalization of the Bagwell and Staiger (1992) result. Proposition 7. In the commitment model, suppose that the firms battle for a monopoly position and there is perfect appropriability. If the equilibrium levels of investment are such that h(k*) - h(k) - r>O, an optimal policy is to tax R&D. If a diflerence in investments between two firms is such that the inequality holds in reverse, then an optimal policy is to subsidize R&D. If h(k*) -h(k) - r = 0, an optimal subsidy is zero. In order to understand the intuition
behind Proposition 7, notice that in the
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present case the foreign firm’s marginal benefit from doing R&D can be written as: MB = h’(k*)[L*
+ h(k)L*lr]l[r
+ h(k) + h(k*)]‘.
(17)
Since F* =O, the term h(k)L*/r represents the preemption effect alluded to earlier. When there is an increase in k, this effect is magnified, thereby raising the marginal benefit to the foreign firm. However, an increase in k also causes the patent race to end sooner, a prospect which increasesthe effective discount factor (the denominator in Eq. (17)) and thereby reduces the marginal benefit. The relative strength of the preemption and the discount effect determines the net impact of an increase in k on the marginal benefit. When the foreign firm’s investment in R&D exceedsthe domestic firm’s by a significant difference, h(k*) is considerably greater than h(k). Therefore, a marginal increase in k has little effect on the size of the denominator in Eq. (17). With the discount effect dominated by the preemption effect, the foreign firm’s marginal benefit increasesin responseto an increase in k. Since its marginal cost of R&D is unity, and hence is unaffected by an increase in k, an increase in its marginal benefit prompts the foreign firm to increase R&D activities when k is increased; thus, the foreign firm’s best-responsefunction slopes upward. On the other hand, when the foreign firm’s best-responseinvestment in R&D is not much larger (or even smaller) than the domestic firm’s, the discount effect dominatesthe preemption effect. As a consequence,the marginal benefit decreases in response to an increase in k. Thus, the foreign firm’s best-responsefunction slopes downward. The final question then is, when does the foreign firm invest significantly more than the domestic firm? An answer can be found by differentiating Eq. (17). Since d(MB)ldZ, * >O, it follows that the greater the monopoly profit (L*), the greater the incentive to invest in R&D. Thus, one answer to the above question is that the foreign firm invests more in R&D if it is more efficient as a monopoly than the domestic firm.
7. Concluding
remarks
The recent work by Bagwell and Staiger (1992), (1994) incorporatesuncertainty into the deterministic model of international R&D rivalry of Spencerand Brander (1983), and establishesa strong casefor R&D subsidiesregardlessof the choice of the strategic variable in product market competition. While Bagwell and Staiger retain the stage-game setup of Spencer and Brander, and focus on the case of random cost reductions, this paper adopts an intertemporal framework in which a date of discovery is uncertain. Since real-world inventors must take the two kinds
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of uncertainty into consideration, our paper complementsthe work of Bagwell and Staiger. The two types of uncertainty, however, have contrasting implications for strategic R&D policy. In an intertemporal framework, the nature of an optimal policy dependscrucially on the degreeof appropriability of the new technology. If R&D costs are borne over time, it is optimal to tax R&D at low or high degreesof appropriability although the case for R&D subsidies may emerge at intermediate degreesof appropriability. If R&D costs are to be committed at time zero, R&D taxes are still an optimal policy at low degreesof appropriability. At high degrees of appropriability it may (but need not) become optimal to subsidize R&D. A clearer result is obtained for the case of battles for monopoly. R&D subsidies are an optimal policy when the domestic firm is a better producer than the foreign firm or when the two are roughly equal in terms of efficiency. If the foreign firm is a significantly more efficient than the domestic firm, however, it is best to tax R&D.
Acknowledgments
We are grateful to two refereesfor helpful comments which led to substantial improvements.We alone are responsible for remaining errors.
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