Vol. 21, No. 3, 399–417, September 2007
International R&D Rivalry with
Spillovers and Policy Cooperation in
R&D Subsidies and Taxes
PEI-CHENG LIAO
Department of Accounting, National Taiwan University
ABSTRACT We have investigated non-cooperative and jointly optimal R&D policies in the framework of Spencer & Brander (1983) in the presence of R&D spillovers. When R&D activ-ities are strategic substitutes and the R&D game exhibits a positive externality, the result of Spencer & Brander (1983) reverses: the non-cooperative policy is a tax while the jointly opti-mal policy is a subsidy. Moreover, when R&D activities are strategic complements, the usual result of the prisoners’ dilemma in the strategic subsidy game does not hold, implying that a welfare intervention is preferable over laissez-faire. When spillovers are sufficiently large, the joint welfare increases with subsidies being higher than those under non-cooperation. KEYWORDS: R&D spillovers, R&D subsidy/tax, strategic substitutes/complements, externality JEL CLASSIFICATION: F13; L13
Introduction
This paper investigates the non-cooperative and jointly optimal R&D subsidy policies of two exporting countries in the presence of international tech-nology spillovers. The aim of this study was to improve our understanding of the policy interactions that occur when two governments are setting their R&D subsidy policies.
In a seminal paper, Spencer & Brander (1983) show that an R&D subsidy enables the domestic firm to commit to a higher R&D investment, thereby shifting the profit of the foreign firm to the domestic firm. However, they did not consider the presence of R&D spillovers between firms, and the competing R&D activities
Correspondence Address: Pei-Cheng Liao, Department of Accounting, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., Taipei 106, Taiwan. Email: pcliao@ntu.edu.tw.
1016-8737 Print/1743-517X Online/07/030399–19 © 2007 Korea International Economic Association DOI: 10.1080/10168730701520545
were strategic substitutes. The justification for an R&D subsidy is evident because it enhances the incentive of the domestic firm to invest more in R&D, thus reducing the R&D performed by the foreign firm and increasing the profit of the domestic firm at the expense of the foreign firm. The paper by Spencer & Brander (1983) has received considerable theoretical attention and has inspired a growing body of work addressing other modifications. For example, Bagwell & Staiger (1992, 1994) introduce uncertainty in the R&D process over the extent of cost reduction. Leahy & Neary (1996) focus on the timing of moves and the ability of agents to commit intertemporally. Miyagiwa & Ohno (1997) adopt an intertemporal framework and show that an optimal R&D policy is crucially dependent on the degree of appropriability of the new technology.
The recent rapid growth in intellectual property trade and high-technology products in world markets has led to technological spillovers between firms. Moreover, the increasing number of agreements on international R&D coop-eration and research joint ventures has led to the effect of R&D spillovers being addressed more widely in an international context. Within a country, it is rec-ognized that R&D activities usually involve an externality, which thus calls for government intervention. In the presence of competition in production and tech-nology between firms in different countries and varying degrees of international technology spillovers, it is important to analyze the policies that governments in different countries should use to optimally help their local firms, and whether those policies differ when the governments cooperate in setting them. Although there is a fairly large amount of industrial organization literature on endogenous R&D choices and R&D policies, very few studies have analyzed R&D policy interactions between governments within an international framework, especially cooperation that occurs at the government level. Qiu & Tao (1998) extend Spencer & Brander (1983) to show that both R&D subsidies and taxes can be optimal in the case of R&D collaboration,1and that an R&D subsidy is always optimal in the case of R&D coordination.2Although they also analyze cooperation in R&D subsidies between governments in the case of R&D coordination, the effect of R&D spillovers between firms is not considered. Motta (1996) adopts a four-firm, two-country model to investigate the policy game in which each government can allow or forbid R&D cooperation between its two local firms. DeCourcy (2005) extends Motta (1996) to investigate the policy game in which each government has three choices of cooperative R&D policies: R&D cartel, research joint venture, or research joint venture cartel. She finds that allowing firms to cooperate in R&D is superior to using R&D subsidies. Kang (2006) extends Spencer & Brander (1983) by incorporating an intellectual property rights (IPR) protection policy into his model in which the two exporting countries choose both the R&D subsidies and IPR protection levels simultaneously. At Nash equilibrium, the government of
1This is similar to the ‘noncooperative R&D with spillovers’ model in d’Aspremont & Jacquemin (1988) and Suzumura (1992), and the ‘R&D competition’ model in Kamien et al. (1992). That is, the marginal cost of one firm is decreased by its own R&D, in addition to there being some spillover from the R&D of the other firm.
2This is the same as the model of Spencer & Brander (1983) except that each firm chooses its R&D level to maximize a weighted sum of the profits of both firms.
each country optimally adopts sufficiently weak IPR protection and subsidizes R&D. If both governments cooperate to maximize their joint welfare, they will provide no IPR protection and continue to subsidize R&D.
There are many real-world examples of agreements on international R&D cooperation. China and Germany established science and technol-ogy (S&T) cooperation relations in November 1978 by signing the China– Germany S&T cooperation agreement, and this cooperation has been in steady progress since then, with a series of cooperation agreements and protocols having been signed between the corresponding sectors, national S&T organizations, and science foundations in the two countries. To date, the bilateral cooperation covers almost all key areas of S&T, from basic to applied research, from the traditional areas of energy, environment, agriculture, and health to the high-technology areas of aerospace, information technology, biotechnology, new materials, advanced
manufacturing, and automation.3 Israel has also signed bilateral R&D
coop-eration agreements with the United States, Canada, members of the European Union, India, and Singapore. The aim of these agreements is to encourage contacts between Israeli and overseas firms to facilitate joint ventures in R&D, manufac-turing, and marketing. Israel has been operating binational funds for financing cooperative industrial R&D with the US, Canada, Singapore, and Britain. In addi-tion, it has agreements for joint funding of R&D projects with Austria, France,
Germany, The Netherlands, Ireland, Portugal, and Spain.4
In this paper, we focus on non-cooperative and jointly optimal R&D subsidy policies between governments under the framework of Spencer & Brander (1983) in the presence of R&D spillovers. Our work differs from that of Kang (2006) in two main ways. In Kang (2006), the demand is linear and the degree of R&D spillovers (through an IPR protection policy) is endogenous, whereas we use a general demand function and assume that the degree of R&D spillovers is exoge-nous. When the governments in the exporting countries cooperate, Kang (2006) shows that R&D subsidies are positive and no IPR protection is provided (i.e. the spillover effect is completely perfect). Our results are more general because we analyze the R&D subsidy policy for varying degrees of spillovers, and our model is suitable for the cases when the extent of R&D spillovers is not solely deter-mined by IPR protection policy, and when the IPR protection level is not solely controlled by a country itself.5
This paper shows that when spillovers are low (high), the R&D game exhibits a negative (positive) externality, so the jointly optimal policy is to tax (subsidize) R&D. When R&D activities are strategic substitutes and the R&D game exhibits a positive externality, the result in Spencer & Brander (1983) reverses: the non-cooperative policy is a tax while the jointly optimal policy is a subsidy. Moreover, when R&D activities are strategic complements, the usual result of the prisoners’ 3http://www.most.gov.cn/eng/
4http://www.mfa.gov.il/mfa
5For example, in the Uruguay Round of GATT (the General Agreement on Tariffs and Trade), an agreement on the trade-related aspects of IPRs (TRIPs) was passed despite strong resistance from developing countries. Most of the terms of the TRIPs Agreement require that all member countries adopt a minimum IPR protection level. Since developed countries already have tight IPR protection, the main of effect of this agreement is to constrain developing countries.
dilemma in the strategic subsidy game does not hold, implying that a welfare intervention is preferable over laissez-faire. When R&D activities are strategic complements and the R&D game exhibits a positive externality, the jointly opti-mal R&D subsidy is greater than the non-cooperative R&D subsidy, implying that the total welfare will be increased further if the governments provide higher R&D subsidies for the firms in their countries. However, when R&D activities are strategic complements and the R&D game exhibits a negative externality, the total welfare will be increased further if the governments impose higher R&D taxes on the firms in their countries.
The next section presents the basic three-stage model and analyzes the non-cooperative Nash equilibrium in R&D subsidy/tax policy, while the third section analyzes the jointly optimal R&D subsidy/tax policy. The fourth section exam-ines the case of asymmetric spillovers, and the fifth section provides some concluding remarks.
The Model
We consider a one-product, two-firm, three-country model as used by Spencer & Brander (1983). Two of the countries, which are numbered 1 and 2, each have a firm that produces a homogeneous product to be exported to the third country, M.
Demand in country M is given by an inverse demand function, p= p(Q), where
p is the market price. We assume that p(.) is decreasing and is twice continuously
differentiable, with p(Q)Q+ p(Q) < 0. The output of firm i (the one in country i) is denoted by qi, i= 1, 2, and Q = q1+ q2 in equilibrium. Initially, firm i has
a constant marginal cost ci, which is independent of the production level. Each
firm is able to lower its marginal cost through R&D investment xi. Following
d’Aspremont & Jacquemin (1988) and most of the literature, we assume that the R&D production function is linear. The R&D investment is subject to diminish-ing returns, as captured by the quadratic form of the R&D cost, γ x2i/2, γ > 0. The R&D of firm i not only leads to a reduction in its own marginal cost, but also spills over to firm j, i, j= 1, 2, and i = j. For the time being, we assume that the degree of R&D spillovers between firms is identical. We define β ∈ [0, 1] as the degree of R&D spillovers, which represents the proportion of the cost reduction of firm i that enters additively and costlessly into the marginal cost reduction of firm j. Thus, through its own R&D and the spillovers of the R&D of firm j, the marginal cost of firm i becomes MCi= ci− xi− βxj.
We consider the following three-stage, one-shot non-cooperative game. In the first stage, the governments in countries 1 and 2 simultaneously and non-cooperatively announce that they are to subsidize/tax R&D by paying/imposing a given percentage (denoted s1 and s2, respectively) of the R&D expenditures. Thus, the R&D cost of firm i becomes γ x2i(1− si)/2, 0≤ |si| < 1.6In the second
stage, the two firms choose their respective R&D levels, x1and x2, simultaneously
6Alternatively, country i may provide a subsidy/tax s
ifor the per-unit level of R&D investment.
The R&D cost of firm i then becomes γ x2i/2− sixi. The analysis is qualitatively similar to that
and non-cooperatively. In the third stage, the two firms engage in a Cournot com-petition. The game is solved by backward induction. To simplify the analysis, we
concentrate on symmetric solutions.7
In the third stage, the maximization problem for firm i is given by max qi πi = [p(Q) − ci+ xi+ βxj]qi− 1 2γx 2 i(1− si), i, j= 1, 2, and i = j
and the first-order condition is given by
∂πi
∂qi = p
q
i+ p − ci+ xi+ βxj = 0, i, j = 1, 2, and i = j (1)
We define aii ≡ ∂2πi/∂q2i and aij≡ ∂2πi/∂qi∂qj, i, j= 1, 2, and i = j. aii=
pqi+ 2p <0 by the second-order condition, and aij= pqi+ p<0 because
of strategic substitution of outputs. Then the stability condition, 1≡ a11a22−
a12a21= ppQ+ 3(p)2>0, is satisfied. The two first-order conditions in equation (1) can be solved simultaneously to obtain the equilibrium outputs
q∗1(x1, x2)and q∗2(x1, x2). The comparative statics results show that
∂q∗i ∂xi = βaij− ajj 1 = p(βq∗i − q∗j)+ p(β− 2) 1 >0, ∂q∗j ∂xi = aji− βaii 1 = p(q∗j − βq∗i)+ p(1− 2β) 1
Thus, the R&D performed by firm i increases its own output q∗i and the total out-put Q∗. However, the effect of this on q∗j depends on the magnitude of β. We define
˜β = (pq∗
j + p)/(pq∗i + 2p). When β > (<) ˜β, ∂q∗j/∂xi > (<)0, meaning that
an increase in xileads to higher (lower) q∗j.
In the second stage, firm i chooses the R&D level that will maximize its profit, and the first-order condition is given by
∂πi∗ ∂xi = q ∗ i 1+ p ∂q∗j ∂xi − γ xi(1− si)= 0, i, j = 1, 2, and i = j (2)
We define bii≡ ∂2πi∗/∂x2i and bij≡ ∂2πi∗/∂xi∂xj, i, j= 1, 2, and i = j. The second-order conditions, b11<0 and b22 <0, and the stability condition 2≡
b11b22− b12b21>0 are assumed.8 Solving the two first-order conditions in equation (2) simultaneously yields the Nash equilibrium R&D levels x∗1(s1, s2)
7We analyze the case of asymmetric firms with different degrees of spillovers in the fourth section. 8For a linear demand and symmetric case, the second-order condition is 9bγ (1− s
i)− 2(2 − β)2>
0 and the stability condition requires|2(2 − β)(2β − 1)/[9bγ (1 − si)− 2(2 − β)2]| < 1. For a
gen-eral demand function, the second-order and stability conditions are satisfied if bγ is sufficiently large.
and x∗2(s1, s2). Totally differentiating equation (2) with respect to si yields the
following comparative statics results:
∂x∗i ∂si = − γbjjxi∗ 2 >0 and ∂x ∗ j ∂si = γbjix∗i 2
With symmetry, q∗i = q∗j = q∗= Q∗/2 and ∂q∗i/∂xj = ∂q∗j/∂xi, and we have
bji ≡ ∂ 2π∗ j ∂xj∂xi = ∂q∗j ∂xi 1+ p ∂q∗i ∂xj + q∗j p ∂q∗i ∂xi + ∂q∗j ∂xi ∂q∗i ∂xj + p ∂2q∗i ∂xj∂xi = ∂q∗j ∂xi 1+ p ∂q∗i ∂xj + pq∗j ∂q∗i ∂xi + ∂q∗j ∂xi + pq∗j ∂2q∗i ∂xj∂xi = aji− βaii 1 1+ aij− βajj pQ∗+ 3p + pq∗j −1 − β pQ∗+ 3p + pq∗j ∂2q∗i ∂xj∂xi = (1− β)(pq∗+ 2p)− p p(pQ∗+ 3p) (1− β)(pQ∗+ 2p)+ 2p pQ∗+ 3p − pq∗(1+ β)2(pq∗+ 2p) (pQ∗+ 3p)3 (3)
The sign of bji depends on the magnitude of β. It can be shown that if p is
negligible, bji< (>)0 when β = 0(1), and ∂bji/∂β >0.9We define βsas the crit-ical value that makes bji = 0.10 When β < βs, bji<0, meaning that x1 and x2 are strategic substitutes, and an increase in si increases x∗i but decreases x∗j.
When β > βs, bji>0, meaning that x1 and x2 are strategic complements, and an increase in siincreases both x∗i and x∗j.
The strategic nature of x1and x2depends on the two countervailing effects of spillover and rivalry. Suppose that x1is increased by one unit. This will decrease
MC1by one, and thus firm 1 will increase its output by−1/(pq∗1+ 2p)so that its marginal revenue equals its marginal cost. Then the marginal revenue of firm
2 will be decreased by ˜β,11 and the marginal cost of firm 2 will be decreased
9See Appendix A for details.
10If pis negligible, it can be seen that when p> (<)0, ∂2q∗
i/∂xj∂xi> (<)0, so bji|β= ˜β< (>)0,
implying that ˜β < (>)βs.
11The marginal revenue of firm 2 is MR2= pq∗
2+ p, and ∂MR2/∂q1= pq∗2+ p<0. Thus, when the output of firm 1 is increased by−1/(pq∗1+ 2p), the marginal revenue of firm 2 will be decreased by (pq∗2+ p)/(pq∗1+ 2p), which is defined as ˜β.
by β due to the unit increase in x1. If β > ˜β, the marginal revenue of firm 2 is greater than its marginal cost, so firm 2 will increase x2in order to increase its output (the positive spillover effect). On the other hand, due to symmetry, an increase in x2 will increase q∗1(because ∂q∗1/∂x2>0 when β > ˜β), leading to a
lower price. Thus, there is a countervailing incentive for firm 2 to decrease x2
(the negative rivalry effect). If β is sufficiently large (i.e. β > βs> ˜βwhen p>0
or β > ˜β > βs when p<0), the positive spillover effect exceeds the negative
rivalry effect, so x2 increases as x1 increases. Similarly, if β is sufficiently small (i.e. β < ˜β < βswhen p>0 or β < βs< ˜βwhen p<0), the negative spillover
effect exceeds the positive rivalry effect, so x2decreases as x1increases. But there are two cases for intermediate values of β: (1) ˜β < β < βswhen p>0, and (2)
βs< β < ˜β when p<0. In case 1, the positive spillover effect is exceeded by
the negative rivalry effect,12so x2 decreases as x1 increases. Conversely, in case 2, the negative spillover effect is exceeded by the positive rivalry effect, so x2 increases with x1. Therefore, when β < (>)βs, x1and x2are strategic substitutes (complements).
In the first stage, the government of country i chooses the R&D subsidy so as to maximize its welfare, which is defined as the profit of firm i minus the subsidy payments: Wi = πi∗−1 2γx ∗2 i si = p(Q∗)− ci+ x∗i + βxj∗ q∗i − 1 2γx ∗2 i , i, j = 1, 2, and i = j
The first-order condition for maximization is given by13
∂Wi ∂si = ∂πi∗ ∂si − 1 2γx ∗2 i − γ x∗isi ∂x∗i ∂si = ∂πi∗ ∂xj + ∂πi∗ ∂qj ∂q∗j ∂xj ∂x∗j ∂si + 1 2γx ∗2 i − 1 2γx ∗2 i − γ x∗isi ∂x∗i ∂si = q∗i β+ p∂q ∗ j ∂xj ∂x∗j ∂si − γ x ∗ isi ∂x∗i ∂si = 0 (4) 12When p>0 and ˜β < β < βs, ∂2q∗
1/∂x2∂x1>0 and ∂q1∗/∂x2>0, meaning that ∂q∗1/∂x2 increases with x1. This increases the disincentive of firm 2 to invest in R&D, with the increasing negative rivalry effect exceeding the positive spillover effect.
13To ensure the existence of the equilibrium solution, we assume that the second-order and stability conditions are satisfied in the region close to the equilibrium point. Equation (4) indicates that when si is zero, a small increase (decrease) in this parameter will increase the welfare if [β + p(∂q∗j/∂xj)](∂x∗j/∂si) > (<)0. However, a subsidy (tax) that is too high will decrease the welfare.
Thus, we conclude that there is a subsidy (tax) that will maximize the welfare, and the equilibrium solution exists.
where β+ p∂q ∗ j ∂xj = β + p(βq∗j − q∗i)+ p(β− 2) pQ∗+ 3p = p[2βq∗j − (1 − β)q∗i] + 2p(2β− 1) pQ∗+ 3p (5)
From equation (4) we have
s∗i = q
∗
i[β + p(∂q∗j/∂xj)](∂x∗j/∂si)
γx∗i(∂x∗i/∂si)
(6)
Thus, s∗i > (<)0 if β+ p(∂q∗j/∂xj)and ∂x∗j/∂sihave the same (opposite) signs.
The sign of β+ p(∂q∗j/∂xj)is equivalent to that of dπi∗/dxj, which is the total
effect of the R&D performed by one firm on the profit of the other firm, and equation (5) indicates that it depends on the magnitude of β. We define βeas the
critical value that makes β+ p(∂q∗j/∂xj)= 0, and
βe =
pq∗i + 2p
2pq∗j + pq∗i + 4p (7)
If β < βe, then β+ p(∂q∗j/∂xj) <0; that is, dπi∗/dxj <0. Thus, the R&D game exhibits a negative externality. If β > βe, dπi∗/dxj >0, and the R&D game exhibits a positive externality.
The sign of ∂x∗j/∂si is equivalent to that of bji. When β < (>)βs, bji< (>)0, so ∂x∗j/∂si < (>)0. The ranking of the two critical values, βe and βs, is indeter-minate, since it depends on the curvature of the demand function. We have the following lemma.
Lemma 1 If the demand function is concave and p≥ 0 then βe < βs; that is,
strategic complementarity of R&D activities guarantees a positive externality of the R&D game. If the demand function is linear, βe = βs= 0.5.
Proof. For the proof see Appendix B.
We have the following proposition.
Proposition 1 When the R&D spillover effect is sufficiently small or suf-ficiently large (i.e. β ∈ [0, min(βe, βs)) or β ∈ (max(βe, βs), 1]), the
govern-ments should subsidize R&D. For intermediate R&D spillover effects (i.e. β ∈ (min(βe, βs), max(βe, βs))), the governments should tax R&D. When β = βeor
Although Qiu & Tao (1998) have already showed that it is possible for either R&D subsidies or taxes to be optimal,14Proposition 1 clearly indicates that the governments should tax R&D when the degree of spillovers is moderate (between the two critical values) and that they should not intervene when the degree of spillovers equals either critical value. When β is sufficiently small (i.e. less than min(βe, βs)), an R&D subsidy is optimal for the reason given by Spencer & Bran-der (1983). However, when β is sufficiently large (i.e. greater than max(βe, βs)),
an R&D subsidy is optimal for a different reason from that given by Spencer & Brander (1983): due to strategic complementarity and a positive externality, a domestic R&D subsidy increases the foreign R&D investment, which in turn increases the profit of the domestic firm. There are two cases when β is between the two critical values: (1) if βs< β < βe, a domestic R&D subsidy increases
the foreign R&D investment (due to strategic complementarity), which in turn decreases the profit of the domestic firm (due to a negative externality); and (2) if βe < β < βs, a domestic R&D subsidy decreases the foreign R&D investment
(due to strategic substitution), which in turn decreases the profit of the domestic firm (due to a positive externality). In either case, the profit-shifting effect of a domestic R&D subsidy is negative, so an R&D tax is optimal. When β = βsor
β = βe, either a domestic R&D subsidy has no effect on the foreign R&D
invest-ment, or the foreign R&D investment does not affect the profit of the domestic firm. Therefore, the profit-shifting effect is zero, and laissez-faire is optimal.
To illustrate the general result of Proposition 1, we consider the following three demand functions in closed forms.
(1) Linear demand with p= a − bQ. We know that βe= βs= 0.5 from
Lemma 1. Thus, the governments should subsidize R&D for any value of
βexcept β = 0.5, and adopt a laissez-faire policy when β = 0.5.
(2) Concave demand with p= a − bQ2. In Appendix C we show that βe= 5/11
and βs= 0.5312. This confirms Lemma 1 that βe< βswhen p<0 and p ≥
0. Thus, when 0≤ β < 5/11 or 0.5312 < β ≤ 1, the governments should
subsidize R&D; when 5/11 < β < 0.5312, they should tax R&D; and when
β = 5/11 or β = 0.5312, the optimal policy is laissez-faire.
(3) Convex demand with p= a − b ln Q. In Appendix D we show that βe= 0.6
and βs= 0.4776. Thus, when 0 ≤ β < 0.4776 or 0.6 < β ≤ 1, the
govern-ments should subsidize R&D; when 0.4776 < β < 0.6, they should tax R&D;
and when β = 0.4776 or β = 0.6, the optimal policy is laissez-faire.
Jointly Optimal Policy: Subsidy or Tax?
Spencer & Brander (1983) show that the jointly optimal policy for the two export-ing countries is for their governments to tax R&D. In this section, we examine the robustness of their result in the presence of R&D spillovers.
14Proposition 1 of Qiu & Tao (1998) states that in the case of international R&D collaboration, (i) when λR is small enough, governments subsidize R&D investments; and (ii) when λRis large
Now suppose that in the first stage, countries 1 and 2 choose s1and s2 simulta-neously and cooperatively so as to maximize their total welfare. The first-order condition is given by:
∂(Wi+ Wj) ∂si = q ∗ i β+ p ∂q∗j ∂xj ∂x∗j ∂si − γ x ∗ isi ∂x∗i ∂si + q∗j β+ p∂q ∗ i ∂xi ∂x∗i ∂si − γ x ∗ jsj ∂x∗j ∂si = q∗j β + p∂q ∗ i ∂xi − γ x∗isi ∂x∗i ∂si + q∗i β+ p∂q ∗ j ∂xj − γ x∗jsj ∂x∗j ∂si = q∗j β + p∂q ∗ i ∂xi − γ x∗isi ∂xi∗ ∂si + ∂x∗j ∂si (with symmetry) = 0 (8) We denote the jointly optimal R&D subsidy rates as˜si, i= 1, 2. Since (∂x∗i/∂si)+
(∂x∗j/∂si)= γ x∗i(bji− bjj)/2>0,˜si and β+ p(∂q∗i/∂xi)have the same sign.
Thus, when β < (>)βe, ˜si < (>)0, and when β= βe, ˜si = 0. This leads to the following proposition.
Proposition 2 When the R&D game exhibits a negative externality (i.e. 0≤ β < βe), the jointly optimal policy is to tax R&D. When the R&D game exhibits
a positive externality (i.e. βe < β≤ 1), the jointly optimal policy is to subsidize
R&D. When β = βe, the jointly optimal policy is laissez-faire.
To compare˜siwith s∗i, evaluating ∂(Wi+ Wj)/∂siat si = s∗i and sj = s∗j yields
∂(Wi+ Wj) ∂si si=s∗i,sj=s∗j = q∗j[β + p(∂q∗i/∂xi)] (∂x∗j/∂sj) × ∂x∗i ∂si ∂x∗ j ∂sj − ∂x∗i ∂sj ∂x∗ j ∂si (9) Since ∂x∗j/∂sj >0 and [(∂xi∗/∂si)(∂x∗j/∂sj)− (∂x∗i/∂sj)(∂x∗j/∂si)] = γ2x∗1x∗2/ 2>0, equation (9) and β+ p(∂q∗i/∂xi)have the same sign. Thus, when β < βe,
∂(Wi+ Wj)/∂si<0 at si = s∗i and sj = s∗j, implying ˜si <s∗i; when β > βe, ˜si >s∗i; and when β = βe,˜si = s∗i. Combining this with the results of Propositions
Proposition 3 When the R&D spillover effect is sufficiently small (i.e. β ∈
[0, min(βe, βs)),˜si<0 < s∗i. When the R&D spillover effect is sufficiently large
(i.e. β ∈ (max(βe, βs), 1]), 0 < s∗i <˜si. For intermediate R&D spillover effects,
s∗i ≤ 0 < ˜si if βe < β≤ βsand ˜si <s∗i ≤ 0 if βs≤ β < βe, where the equalities
hold when β = βs. When β = βe,˜si = s∗i = 0.
Proposition 3 includes some interesting results. When R&D activities are strate-gic substitutes (i.e. when 0≤ β < βs), s∗i and ˜si have opposite signs. Therefore,
both countries are better off under a laissez-faire equilibrium than under a Nash equilibrium, which is the well-known result of the prisoners’ dilemma in the strategic subsidy game. Moreover, when β ∈ [0, min(βe, βs)), ˜si <0 < s∗i, and
the result of Spencer & Brander (1983) is still applicable in the presence of a low spillover effect. But if βe < β < βs, s∗i <0 < ˜si, implying a reversal of the result
of Spencer & Brander (1983).
On the other hand, when R&D activities are strategic complements (i.e. when
βs< β ≤ 1), s∗i and ˜si have the same sign, and both countries are better off
when their governments intervene than under a laissez-faire equilibrium. Thus, when the competing variables are strategic complements, the usual character of the prisoners’ dilemma does not exist in the strategic subsidy game. Moreover, when β ∈ (max(βe, βs), 1], 0 < s∗i <˜si, implying that the total welfare will be
increased further if the governments provide higher R&D subsidies for the firms in their countries. But if βs< β < βe,˜si <s∗i <0, implying that the total welfare
will be increased further if the governments impose higher R&D taxes on these firms.
Asymmetric Spillovers
The model has presumed so far that the two firms are identical, with symmetric R&D spillovers. Now we investigate the robustness of the general results if this
symmetry assumption is relaxed. The marginal cost of firm i is MCi = ci− xi−
βjxj, where βj ∈ [0, 1] denotes the leakage from the R&D of firm j, i, j = 1, 2,
and i= j. To simplify the analysis and provide more insights into the asymmetric
case, we assume that the demand is linear and c1= c2. It is easy to show that
βi = 0.5 is the critical value for determining the strategic nature of R&D activities
and the sign of externality. That is, if βi > (<)0.5, then ∂x∗j/∂si > (<)0 and βi+
p(∂q∗i/∂xi) > (<)0. With asymmetric spillovers, the non-cooperative subsidy in equation (6) becomes
s∗i = q
∗
i[βj+ p(∂q∗j/∂xj)](∂x∗j/∂si)
γx∗i(∂x∗i/∂si)
The sign of s∗i depends on the signs of βj+ p(∂q∗j/∂xj)and ∂x∗j/∂si, which in turn
depend on the values of βj and βi, respectively. Table 1 lists the non-cooperative
subsidy policies for various values of β1and β2. The governments should subsidize R&D if both β1and β2are less than or greater than 0.5, they should tax R&D if one of these variables is less than 0.5 and the other is greater than 0.5, and they should adopt a laissez-faire policy if either one is equal to 0.5.
Table 1. The non-cooperative subsidy policy
0≤ β1<0.5 β1= 0.5 0.5 < β1≤ 1 0.5 < β2≤ 1 s∗1<0, s∗2<0 s1∗= 0, s∗2= 0 s∗1>0, s∗2>0 β2= 0.5 s∗1= 0, s∗2= 0 s1∗= 0, s∗2= 0 s∗1= 0, s∗2= 0 0≤ β2<0.5 s∗1>0, s∗2>0 s∗1= 0, s∗2= 0 s∗1<0, s∗2<0
Now we investigate the jointly optimal policy in the presence of asymmetric spillovers. Condition (8) becomes
∂(Wi + Wj) ∂si = q ∗ i βj + p ∂q∗j ∂xj ∂x∗j ∂si − γ x ∗ isi ∂x∗i ∂si + q∗j βi+ p ∂q∗i ∂xi ∂x∗i ∂si − γ x ∗ jsj ∂x∗j ∂si = q∗j βi+ p ∂q∗i ∂xi − γ x∗isi ∂x∗i ∂si + q∗i βj+ p ∂q∗j ∂xj − γ x∗jsj ∂x∗j ∂si = 0 Evaluating ∂(Wi+ Wj)/∂si at si = sj = 0 yields ∂(Wi+ Wj) ∂si si=sj=0 = q∗j βi+ p ∂q∗i ∂xi ∂x∗i ∂si + q∗i βj+ p ∂q∗j ∂xj ∂x∗j ∂si (10) When βi >0.5 and βj ≥ 0.5, ∂(Wi+ Wj)/∂si|si=sj=0 >0 because the first term
in equation (10) is positive and the second term is non-negative. When
βi >0.5 and βj <0.5, ∂(Wi+ Wj)/∂si|si=sj=0>0.15 When βi = 0.5, ∂(Wi+
Wj)/∂si|si=sj=0= 0 because both the first and second terms in equation (10) are
zero. When βi<0.5, and βj ≥ 0.5, the first term is negative and the second term
is non-positive, so ∂(Wi+ Wj)/∂si|si=sj=0<0. When βj ≤ βi <0.5, ∂(Wi+
Wj)/∂si|si=sj=0<0, but when βi < βj <0.5, the sign of ∂(Wi+ Wj)/∂si|si=sj=0
is ambiguous.16Since the analysis is symmetric relative to country j, we know that when βi <0.5 and βj <0.5 there is at least one country whose jointly optimal policy is an R&D tax. Table 2 lists the jointly optimal subsidy policies for various values of β1and β2.
15See Appendix E. 16See Appendix E.
Table 2. The jointly optimal subsidy policy 0≤ β1<0.5 β1= 0.5 0.5 < β1≤ 1 0.5 < β2≤ 1 ˜s1<0,˜s2>0 ˜s1= 0, ˜s2>0 ˜s1>0,˜s2>0 β2= 0.5 ˜s1<0,˜s2= 0 ˜s1= 0, ˜s2= 0 ˜s1>0,˜s2= 0 at least one 0≤ β2<0.5 ˜si<0 ˜s1= 0, ˜s2<0 ˜s1>0,˜s2<0
Table 3. A comparison of s∗i and˜si
0≤ β1<0.5 β1= 0.5 0.5 < β1≤ 1
0.5 < β2≤ 1 ˜s1<s∗1<0, s∗2<0 <˜s2 ˜s1= s∗1= 0, 0 = s∗2<˜s2 0 < s∗1<˜s1, 0 < s∗2<˜s2 β2= 0.5 ˜s1<s∗1= 0, ˜s2= s∗2= 0 ˜s1= s1∗= 0, ˜s2= s∗2= 0 0= s∗1<˜s1,˜s2= s∗2= 0 0≤ β2<0.5 ˜s1<s∗1,˜s2<s∗2 ˜s1= s∗1= 0, ˜s2<s∗2= 0 s∗1<0 <˜s1,˜s2<s∗2<0
To compare˜siwith s∗i, evaluating ∂(Wi+ Wj)/∂siat si = s∗i and sj = s∗j yields
∂(Wi+ Wj) ∂si si=s∗i,sj=s∗j = q ∗ j[βi+ p(∂q∗i/∂xi)] (∂x∗j/∂sj) × ∂x∗i ∂si ∂x∗ j ∂sj − ∂x∗i ∂sj ∂x∗ j ∂si
which has the same sign as βi+ p(∂q∗i/∂xi). Thus, ˜si > (<)s∗i if βi > (<)0.5,
and˜si = s∗i if βi = 0.5. Combining this with Tables 1 and 2 yields Table 3, which
compares s∗i and˜si for various values of β1and β2. Thus, we have the following proposition.
Proposition 4 Suppose that the demand function is linear and that R&D spillovers differ between firms:
(i) If 0.5 < β1≤ 1 and 0.5 < β2≤ 1, 0 < s∗i <˜si, i= 1, 2.
(ii) If β1= β2= 0.5, ˜si = s∗i = 0, i = 1, 2. (iii) If 0≤ β1= β2<0.5,˜si <0 < s∗i, i= 1, 2.
(iv) If 0≤ βj < βi <0.5,˜si <0 < si∗, 0 < s∗j, and˜sj <s∗j, i, j= 1, 2 and i = j.
(v) If 0≤ βi <0.5 and 0.5 < βj ≤ 1, ˜si <si∗<0, s∗j <0 <˜sj, i, j= 1, 2 and
i= j.
(vi) If βi = 0.5 and 0 ≤ βj <0.5, ˜si = s∗i = 0, ˜sj <s∗j = 0, i, j = 1, 2 and i= j.
When both spillovers are greater than 0.5, 0 < s∗i <˜si, so the usual character
of the prisoners’ dilemma does not exist, and the joint welfare increases further with the R&D subsidies being higher than under non-cooperation. When one spillover rate is less than 0.5 but the other is greater than 0.5, the governments of the two countries should adopt opposite jointly optimal policies; that is, one
should apply a tax and the other a subsidy.17When both spillovers are less than
0.5, the jointly optimal policy of at least one of the governments is an R&D tax.
Concluding Remarks
This paper has investigated the non-cooperative and jointly optimal R&D subsidy policies under the framework of Spencer & Brander (1983) in the presence of R&D spillovers. When the two governments choose their policies non-cooperatively, they may tax R&D when the degree of spillovers is moderate. When they choose their policies cooperatively, the jointly optimal policy depends on the R&D exter-nality. When spillovers are low, the R&D performed by one firm reduces the profit of the other firm, so the jointly optimal policy is to tax R&D in order to internal-ize the negative externality. On the other hand, when spillovers are high, the R&D game exhibits a positive externality, so the jointly optimal policy is to subsidize R&D.
We find that the result in Spencer & Brander (1983) is still applicable if the spillover effect is sufficiently small so that R&D activities are strategic substitutes and the R&D game exhibits a negative externality. However, if R&D activi-ties are strategic substitutes and the R&D game exhibits a positive externality, the result of Spencer & Brander (1983) is reversed: the non-cooperative policy is a tax while the jointly optimal policy is a subsidy. Our results also reveal that the usual result of the prisoners’ dilemma in the strategic subsidy game does not hold when the competing variables are strategic complements. Thus, a welfare intervention is preferable over laissez-faire. When R&D activities are strategic complements and the R&D game exhibits a positive externality, the jointly optimal R&D subsidy is greater than the non-cooperative R&D subsidy, implying that the total welfare will be increased further if the governments pro-vide higher R&D subsidies for the firms in their countries. However, when R&D activities are strategic complements and the R&D game exhibits a negative exter-nality, the total welfare will be increased further if the governments impose R&D taxes on the firms in their countries that are higher than the non-cooperative taxes.
The case of asymmetric spillovers between firms has also been analyzed. With a linear demand, the jointly optimal policy for the government in each country will differ when one spillover rate is less than 0.5 and the other is greater than
17Park (2001) and Zhou et al. (2002) also demonstrate the case where countries may choose R&D policies that are in opposite directions where firms engage in R&D activities to improve quality and they produce differentiated products due to asymmetric R&D costs.
0.5. The government of a country with a spillover rate that is less (greater) than 0.5 should tax (subsidize) R&D.
Under the Agreement of Subsidies and Countervailing Measures (SCM Agree-ment), members of the World Trade Organization (WTO) may, under certain conditions specified in the Agreement, apply ‘non-actionable subsidies’ to assist the R&D activities of firms. However, the provisions concerning non-actionable subsidies of the SCM Agreement expired at the end of 1999, and there has been ongoing discussion about reintroducing them into the Agreement. The results of this paper indicate that the desirability of allowing R&D subsidies varies with the degree of R&D spillovers, and hence WTO members should take R&D spillovers into consideration when negotiating the issue of non-actionable subsidies.
Acknowledgements
I am grateful to the journal editor, Jaymin Lee, and two anonymous referees for their valuable comments and suggestions. I gratefully acknowledge the financial support of the National Science Council, Taiwan, under grant no. NSC 94-2415-H-002-021.
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Appendix A Since ∂q∗i/∂xj = p(q∗i − βq∗j)/1+ (1 − 2β)/(pQ∗+ 3p), we have ∂2q∗i ∂xj∂xi = p ∂Q∗ ∂xi (q∗i − βq∗j)+ p ∂q∗i ∂xi − β ∂q∗j ∂xi 1 −p (q∗ i − βq∗j) −(1+β)(p2Q∗+7pp+ppQ∗) pQ∗+3p (1)2 − 1− 2β (pQ∗+ 3p)2 −(1+ β)(4p+ pQ∗) pQ∗+ 3p
With symmetry and some mathematical manipulation, we obtain
∂2q∗i ∂xj∂xi = −
(1+ β)2(pq∗+ 2p) (pQ∗+ 3p)3
Substituting ∂2q∗i/∂xj∂xi into bji yields equation (3). When β = 0, bji in equation (3) becomes
bji|β=0=
(pq∗+ p)(pQ∗+ 4p)(pQ∗+ 3p)− (p)2q∗(pq∗+ 2p) p(pQ∗+ 3p)3
If pq∗+ 2p≥ 0, bji|β=0<0. If pq∗+ 2p<0, since p is negligible,
bji|β=0<0. When β = 1, bjiin equation (3) becomes
bji|β=1= −2(p
)2(pQ∗+ 3p)− 4(p)2q∗(pq∗+ 2p)
p(pQ∗+ 3p)3
= −2p[3(pQ∗+ p)+ pq∗Q∗]
(pQ∗+ 3p)3 >0
for pbeing negligible.
Differentiating equation (3) with respect to β yields
∂bji ∂β = −(p q∗+ 2p) p(pQ∗+ 3p) (1− β)(pQ∗+ 2p)+ 2p pQ∗+ 3p
+ (1− β)(pq∗+ 2p)− p p(pQ∗+ 3p) −pQ∗+ 2p pQ∗+ 3p − 2pq∗(1+ β)(pq∗+ 2p) (pQ∗+ 3p)3 = −(pQ∗+ 3p)[2(1 − β)(pq∗+ 2p)(pQ∗+ 2p)+ 2(p)2] −(1 + β)(p)2Q∗(pq∗+ 2p) p(pQ∗+ 3p)3 > −2(p )2(pQ∗+ 3p)− (1 + β)(p)2Q∗(pq∗+ 2p) p(pQ∗+ 3p)3 If pq∗+ 2p≤ 0, ∂bji/∂β >0. If pq∗+ 2p>0, − 2(p)2(pQ∗+ 3p)− (1 + β)(p)2Q∗(pq∗+ 2p) >−2(p)2(pQ∗+ 3p)− 2(p)2Q∗(pq∗+ 2p) = −2(p)2[3(pQ∗+ p)+ pq∗Q∗] >0
for p being negligible, and thus ∂bji/∂β >0. Therefore, we have bji|β=0<0,
bji|β=1>0, and ∂bji/∂β >0.
Appendix B
To compare βewith βs, substitute βe into equation (3). If bji|β=βe < (>)0, then
βe < (>)βs. If bji|β=βe = 0, then βe = βs. We have bji β=βe = (1− βe)(pq∗+ 2p)− p p(pQ∗+ 3p) (1− βe)(pQ∗+ 2p)+ 2p pQ∗+ 3p − pq∗(1+ βe)2(pq∗+ 2p) (pQ∗+ 3p)3 = pq∗ p(3pq∗+ 4p) pQ∗+ 4p 3pq∗+ 4p − 8ppq∗+ 4ppq∗2 (pQ∗+ 3p)(3pq∗+ 4p)2
= q∗[p(pQ∗+ 4p)(pQ∗+ 3p)− 8(p)2p− 4(p)2pq∗]
p(pQ∗+ 3p)(3pq∗+ 4p)2 = q∗{p[(pQ∗)2+ 7ppQ∗+ 4(p)2] − 4(p)2pq∗}
p(pQ∗+ 3p)(3pq∗+ 4p)2
The sign of bji|β=βeis indeterminate. When p<0 (concave demand) and p ≥ 0,
bji|β=βe <0, so βe < βs. When p= p = 0 (linear demand), bji|β=βe = 0, so
βe = βs. Moreover, it is obvious from equation (7) that βe = 0.5 when p= 0.
Appendix C
Suppose that p= a − bQ2. Thus, p = −2bQ, p= −2b, and p = 0. The
second-order condition aii= pqi+ 2p= −2bqi− 4bQ < 0 and the stability
condition 1= ppQ+ 3(p)2 = 16b2Q2>0 are satisfied. With symmetry,
q∗i = q∗j = q∗= Q∗/2, and from equation (7) we have
βe = pq∗+ 2p 3pq∗+ 4p = −2bq∗− 4bQ∗ −6bq∗− 8bQ∗ = −5bQ∗ −11bQ∗ = 5 11
To derive βs, substitute p = −2bQ, p= −2b, and p= 0 into equation (3) and
let it equal zero. We can derive bji= (−29β2+ 70β − 29)/128bQ∗= 0. Thus,
βs= 0.5312.
This example confirms Lemma 1: when p= −2b < 0 and p= 0, βe < βs.
Appendix D
Suppose that p= a − b ln Q. Thus, p = −b/Q, p= b/Q2, and p= −2b/Q3. The second-order condition aii= pqi+ 2p = (b/Q2)qi − (2b/Q) < 0 and the
stability condition 1 = ppQ+ 3(p)2 = 2b2/Q2>0 are satisfied. With sym-metry, q∗i = q∗j = q∗= Q∗/2, and from equation (7) we have
βe= pq∗+ 2p 3pq∗+ 4p = b 2Q∗ − 2b Q∗ 3b 2Q∗ − 4b Q∗ = − 3b 2Q∗ − 5b 2Q∗ = 3 5 = 0.6
To derive βs, substitute p = −b/Q, p= b/Q2, and p = −2b/Q3into equation
(3) and let it equal zero. We can derive bji = [(−7β2+ 18β − 7)Q∗]/(16b) = 0.
Thus, βs= 0.4776.
Appendix E
Under the assumption that the demand is linear with p= a − bQ and c1 = c2=
R&D levels in the second stage as follows: q∗i = 1 3b[a − c + (2 − βi)xi+ (2βj− 1)xj], x∗i = 2(a− c)(2 − βi)[3bγ (1 − sj)− 2(1 − βj)(2− βj)] where = 27b2γ2(1− s1)(1− s2)− 6bγ (1 − s1)(2− β2)2− 6bγ (1 − s2)(2−
β1)2+ 4(2 − β1)(2− β2)(1− β1β2) >0 by the stability condition. The second-order conditions for the profit maximization problem in the second stage are 9bγ (1− si)− 2(2 − βi)2>0, i= 1, 2. Moreover, we assume that 3bγ (1 − si)−
2(1− βi)(2− βi) >0 for a non-negative x∗i, i= 1, 2.
It can be shown that ∂x∗i/∂si = 3bγ xi∗[9bγ (1 − sj)− 2(2 − βj)2]/ > 0 and
∂x∗j/∂si = 6bγ x∗i(2βi− 1)(2 − βj)/, whose sign depends on βi. Thus, we have
∂(Wi+ Wj) ∂si si=sj=0 = q∗j βi+ p ∂q∗i ∂xi ∂x∗i ∂si + q∗i βj+ p ∂q∗j ∂xj ∂x∗j ∂si = 2bγ q∗jx∗i(2βi− 1)[9bγ − 2(2 − βj)2] +4bγ q∗ixi∗(2βi− 1)(2βj− 1)(2 − βj) = 2bγ x∗i(2βi− 1) q∗j[9bγ − 2(2 − βj)2] + 2q∗i(2βj − 1)(2 − βj)
When βj ≤ βi, it can be shown that x∗i ≤ x∗j and q∗i ≤ q∗j, and we have
q∗j[9bγ − 2(2 − βj)2] + 2q∗i(2βj− 1)(2 − βj)
≥ q∗i[9bγ − 2(2 − βj)2+ 2(2βj− 1)(2 − βj)]
= 3q∗i[3bγ − 2(1 − βj)(2− βj)]
>0
Thus, when βj <0.5 < βi, ∂(Wi+ Wj)/∂si|si=sj=0>0, and when βj ≤ βi <0.5,
∂(Wi+ Wj)/∂si|si=sj=0<0.
However, when βi < βj <0.5, q∗j <q∗i and the sign of q∗j[9bγ − 2(2 − βj)2] +
2q∗i(2βj− 1)(2 − βj)is ambiguous because it is possible that the (negative) second
term is greater than the (positive) first term. If bγ is sufficiently large so that the first term is greater than the second term, then q∗j[9bγ − 2(2 − βj)2] + 2q∗i(2βj−
