Absorptive Capacity, R&D Spillovers, Emissions Taxes and R&D Subsidies^*

It is widely recognized that (i) the development and diffusion of cleaner technologies play an important role in achieving environmental-quality goals; (ii) firms benefit from each other’s investments in research and development (R&D) through voluntary (e.g., joint ventures) and/or involuntary spillovers; and, (iii) regulators can influence firms’ R&D efforts to reduce emissions through economic incentives (e.g., taxes and subsidies). The aim of this paper is to characterize the socially optimal levels of production (or emissions), investment in inventive R&D and in absorptive R&D, and tax and subsidy rates, in a game played by two polluting duopolists and a regulator.

One of the early studies in the environmental R&D area is Milliman and Prince (1989). The authors considered a competitive industry made up of identical firms, and evaluated the relative merits of different environmental- policy instruments for promoting technological change in pollution control, namely, direct controls, emissions subsidies, emissions taxes, free marketable permits and auctioned marketable permits. They showed that emissions taxes and auctioned permits provide firms with the highest incentives to promote technological change. Jung et al. (1996) extended this comparative approach to a heterogeneous industry. Fischer and Newell (2008) assessed different policies for reducing carbon-dioxide emissions and encouraging innovation and the diffusion of renewable energy. They evaluated the relative performance of policies according to the incentives they provided for emissions reduction, efficiency and other outcomes. They also assessed how the nature of technological progress through learning and R&D, and the degree of knowledge spillovers affected the desirability of different policies. Because of knowledge spillovers, the optimal policy involves a portfolio of different instruments targeted at emissions, learning and R&D. Several other papers have been interested by the investment in cleaner production technologies ( Dosi and Moretto, 1997; Farzin and Kort,2000; Requate and Unold, 2003; Ben Youssef, 2009; Heyes and Kapur, 2011).

In the above literature, the assumption is either that there are no technological spillovers between firms, or that, when they occur, they are free.

As pointed out in many papers in the industrial-organization literature, this assumption may be strong, in the sense that firms need to acquire an absorptive capacity to assimilate and exploit the available information in order to benefit from these technological spillovers.

Cohen and Levinthal (1989) were the first to introduce the idea of absorptive capacity in the (process or cost-reduction) R&D literature. Contrary to the result achieved in the seminal paper by D’Aspremont and Jacquemin (1988) — see also D’Aspremont and Jacquemin (1990), and Kamien et al. (1992) — where R&D spillovers are assumed exogenous and cost-free, Cohen and Levinthal showed that the investment in R&D develops the firm’s ability to identify, assimilate and exploit knowledge from the environment. There is a rich literature interested by endogenous R&D spillovers (Kamien and Zang,2000; Wiethaus, 2005; Jin and Troege, 2006; Leahy and Neary, 2007; Ge and Hu, 2008).

Hammerschmidt (2009) distinguished between two types of R&D: inventive (or original) R&D, which creates new knowledge, and absorptive R&D, which enables a firm to benefit from the inventive research conducted by others. She showed that firms invest more in R&D to strengthen their absorptive capacity when the spillover parameter is higher. Ben Youssef et al. (2013) found that the efficiency of investment in absorptive research has almost no impact on inventive R&D, firms’ profits, consumers’ surplus, and social welfare. Samaniego (2013) used country-industry data to uncover the link between knowledge spillovers and innovative activity. He showed that intellectual property rights enforcement disproportionately increases innovation spending in R&D intensive industries.

We consider a three-stage game consisting of a regulator and two identical firms competing in quantity and producing the same homogeneous good. The production process generates pollution, and firms can invest in R&D to lower their emissions/output ratio. Firms invest in inventive R&D that directly reduces their emissions/output ratios. They also invest in absorptive R&D, which enables a firm to exploit the original research done by others. There are also free R&D spillovers between firms. Since the firms constitute a duopoly and pollute the environment, they are regulated. In the first stage, the regulator announces a tax per-unit of pollution to induce the socially optimal level of pollution and production; a subsidy per-unit of original research, to induce the socially optimal level of inventive R&D; and, a subsidy per-unit of absorptive-capacity research, to induce the socially optimal level of absorptive R&D. In the second stage, firms invest in R&D, and in the third, they compete in quantity on the product market.

To the best of our knowledge, this paper is the first attempt to integrate costly R&D spillovers and pollution control into a single model. We add costly R&D spillovers to environmental concerns to have the problem better reflect real-world regulatory policies, as shown by the wealth of industrialorganization literature dealing with absorptive capacity.

Our main results are as follows:

1. We do not determine explicitly the socially optimal levels of inventive and absorptive R&D, but we show that they are strictly positive. Using an elegant approach, we determine the three instruments, namely, the perunit emissions tax, the per-unit inventive R&D subsidy, and the per-unit absorptive R&D subsidy, which are used by the regulator to induce competing firms to implement the socially optimal levels of production and research. These three instruments are necessary to our model. Indeed, even if the socially optimal level of pollution can be implemented using only one instrument, such as pollution permits, this does not provide an incentive for the firms to implement the socially optimal levels of production and R&D. Let us notice that both types of R&D serve to reduce the pollution intensity of firms and not serve to reduce the production costs of firms as commonly considered in the industrial-organization literature. Therefore, without the intervention of the regulator, i.e., in the absence of an emission tax and/or R&D subsidies, the market outcome equilibrium implies that firms never invest in both types of R&D.

2. The socially optimal investment cost in innovative R&D is always higher than that in absorptive R&D. This result is true even when the learning parameter is very high encouraging to invest more in absorptive R&D. This result confirms the simulations in Hammerschmidt (2009, Figure 2). It can be attributed to the fact that the investment in absorption takes its economic value from innovation.

3. When the rate of free spillover is sufficiently high, the regulator gives a higher per-unit subsidy for original research; however, and interestingly, when it is low enough and the marginal damage cost of pollution is sufficiently high, he gives a higher per-unit subsidy for absorptive research, to strengthen R&D spillovers. This result differs from the finding of Jin and Troege ( 2006), who showed that, for society and consumers, the marginal value of innovation expenditure is always higher than that of imitation. This difference in the results can be attributed to the use of environmental damages in our model.

4. When the free spillover is low and the marginal damage cost of pollution is high, the regulator then taxes inventive R&D. This is an interesting result from an environmental point of view since it holds only when environmental concern is high.

The paper has the following structure. The next section presents the model, section three studies the firms’ reaction to the regulator’s policy. In section four we derive the socially optimal regulatory instruments and we make some comparisons between innovation and absorption. Then we conclude. Several proofs of propositions are contained in the Appendix.

We consider an industry made up of two firms producing a homogeneous good sold on a market having the following inverse demand function:

The production process generates pollution, and firms can invest in abatement capacity to decrease their emissions per unit of production. We suppose that this abatement capacity requires, and is positively related to, R&D activities. We distinguish between two types of R&D efforts, namely, original or inventive R&D, denoted x°, and absorptive-capacity R&D, denoted x“. To better visualize this, think of inventive R&D as activities related to, e.g., developing better air-filtering systems; whereas absorptive R&D would correspond to efforts to improve the firm’s technological-monitoring capacity through, e.g., hiring engineers and technicians and buying information technology (IT) equipment. As in Ben Youssef et al. (2013), the total knowledge available (also referred to in the literature as the effective R&D level) to firm i is

where $β∈[0,1)$ is a parameter capturing the free and exogenous R&D spillover and $l>0$ is a learning or absorptive parameter. Since a firm cannot gain a research externality that is greater than the original research developed by the competing firm, we impose the constraint $0≤β+lxia≤1.$

Denote by $ei(xio,xia,xjo)$ the emissions per-unit of production. It is assumed that $ei(xio,xia,xjo)$ is decreasing in all its arguments. For simplicity, we adopt the following functional form¹:

Consequently, the total emissions for firm i are given by

The damage cost resulting from these emissions is given by $Di=αEi,$ where $α>0$ is the marginal disutility of pollution.

We suppose that the cost of R&D activity of type $m=o,a,$ given by $Cm(xim),$ as well as the production cost $gi(qi),$ are given by increasing convex² functions satisfying $Cm(0)=gi(0)=0.$⁠. Hence, we have diminishing returns to scale of R&D. For tractability, we adopt the following quadratic functional forms:

and make the following assumption:

This intuitive assumption simply states that when the investment-cost parameters are relatively very high, it is optimal not to invest in R&D. Let us notice that we do not consider the case $ko,ka→+∞$in this paper.

Nevertheless, we use these limits for comparisons that still remain valid when $kσ$and $ka$ are finite and sufficiently high numbers. Further, the above assumption and/or the choice of a sufficiently low l, guarantee that the constraint $0≤β+lxia≤1$ can always be satisfied.

As firms constitute a polluting duopoly, they are regulated. The regulator maximizes a social-welfare function and uses three regulatory instruments, namely, an emissions tax per-unit of pollution ti, to induce the socially optimal levels of production and pollution; a subsidy per-unit of inventive R&D r°; and, a subsidy per-unit of absorptive R&D r“, to induce the socially optimal levels of effective R&D and emission/output ratio. Note that, as the game is symmetric, we confine our interest to symmetric equilibria.

Remark 1Our assumption that firms distinguish between inventive R&D and absorptive R&D is taken from the literature in industrial organization. See, e.g., Jin and Troege ( 2006) and Hammerschmidt (2009). Further, as the game is of complete information, we also suppose that the regulator can distinguish between the two activities and perfectly observes the level of each one. In practice, this means that the firm states in its tax report all admissible expenses pertaining to either types of R&D, and provide evidence such as receipts, payrolls, etc., in order to collect the subsidy announced by the regulator at the first stage of the game.

The profit of firm i is given by$Πi(qi,qj,xio,xia)=p(qi,qj)qi−qi2−ko(xio)2−ka(xia)2,$⁠, and its net profit by$Vi(qi,qj,xio,xia,xjo)=Πi−tiEi+rioxio+riaxia.$

The consumers’ surplus corresponding to the consumption of $Q=qi+qj$ is

The social welfare is defined as the consumers’ surplus, minus damages and subsidies, plus taxes and net profits of the firms, and is equal, after simplification, to

Note that taxes and subsidies do not appear in the social-welfare function because we suppose that raising public funds is not costly. Indeed, taxes deducted from the firms’ profits are added to the consumers’ welfare, and subsidies added to the firms’ profits are deducted from the consumers’ welfare.

The game has three stages. In the first stage, the regulator announces the socially optimal per-unit emissions tax and per-unit R&D subsidies, i.e., the triplet $(ti,rio,ria),i=1,2.$⁠. In the second stage, the firms choose their levels of R&D; and, finally, in stage 3, the firms select their production levels. To determine a subgame-perfect Nash equilibrium, we solve the game backward. In the third stage, the firms’ first-order conditions are

Solving the system (6) leads to

To interpret the above functions, we compute their partial derivatives, assuming symmetry and considering the case of a positive emissions tax. When a firm increases its level of original or absorptive research, its emissions/output ratio decreases (see (2)), enabling it to expand its production $(∂qi∗∂xio=ti15(4−β−lxia)>$ and $∂qi∗∂xia=415tilxio>0)$⁠. Now consider the derivative $∂qi∗∂xjo=ti15[4(β+lxia)−1]$⁠. When the competing firm increases its original research x^o, then this has two opposite effects on the firm’s production, namely: (i) a positive effect on production, due to the free R&D spillovers and absorptive capacity; and (ii) a negative effect due to competition between firms. (Recall that this is a model a la Cournot, where outputs are strategic substitutes.) When ß and/or l are high enough, the first positive effect dominates. When a competitor increases its absorptive capacity, its emissions ratio decreases, enabling it to expand its production, which in turn, forces the other firm to reduce its production (dx = — filx° 0). The first-order conditions of firm i’s second stage are³

At equilibrium, by using (6), (8) and (9) are simplified and the following equations are satisfied for symmetric solution(s)⁴:

where q* is given by (12). The symmetric optimal level of production for each firm is obtained from expression (7):

In the next section, we will show how a suitable choice of policy instruments by the regulator induces the firms to select the socially optimal production and R&D levels, which are at the same time the unique solution to the nonlinear equations system (10) and (11).

In the first stage, the regulator maximizes his social welfare, given by (5), with respect to the decision variables $ti,rio$and $ria,i=1,2.$⁠. Note that solving directly for the optimal per-unit emissions tax and per-unit R&D subsidies is an extremely difficult problem. Therefore, we propose an indirect (and much simpler) method where the regulator maximizes, in the third and second stages respectively, his social welfare with respect to the output and the R&D levels, which then become the new choice variables. Then, by equalizing the socially optimal quantities to those selected by the firms, the regulator determines the socially optimal per-unit emissions tax and per-unit R&D subsidies. In fact, the model is solved as if it were a two-stage game.

The first-order conditions of the regulator’s third stage are

Solving the above system gives

The symmetric socially optimal level of production for each firm is

A sufficient condition for symmetric production quantities to be positive is:

that is, the marginal damage cost of pollution is lower than the maximum willingness to pay for the good. We will assume from now on that this condition is fulfilled.

The first-order conditions of the regulator’s second stage are⁵

At equilibrium, by using conditions (13), Equations (17)-(20) are simplified, and their symmetric solution(s) verify the following equations system:

where q is given by (15), and (21) and (22) are equivalent to

Solving the nonlinear system (23) and (24) gives the symmetric socially optimal R&D levels denoted by x^O and x^a. Unfortunately, we are not able to get an explicit solution. Nevertheless, we will prove the existence of a positive one.

Proposition 1 When $ko$and $ka$are high enough, there is a unique couple of real solutions $xˆo>0$and $xˆa>0$that solves the nonlinear equations system given by (23) and (24); this symmetric solution maximizes the social-welfare function.

Proof: See Appendix.

Condition (16) and the assumption in (3) guarantee that the symmetric socially optimal levels of research, production and pollution are positive, and that $0≤β+lxˆa≤1$⁠,when$ko$and $ka$are high enough.From (21) and (22), we can show that

Proposition 2 The socially optimal investment cost in innovative R&D is always higher than that in absorptive &D.

The above result is true even when the learning parameter is high encouraging to invest more in absorptive$R&D$⁠. One explanation is that a higher learning parameter directly increases the efficiency of the investment in absorptive R&D and indirectly increases the efficiency of the investment in inventive R&D (see expression (1)). This result is similar to that simulated by Hammerschmidt (2009, Figure 2).

The nonlinear system (10) and (11) involves two equations and two unknown variables, which are the optimal symmetric R&D levels for the firms and are denoted by $x*O$and $x*a$⁠. Since the emissions tax and R&D subsidies are set to induce firms to achieve the socially optimal production and R&D levels, then the optimal emissions tax and R&D subsidies should be chosen such that $xˆo$and $xˆa$selected by the regulator are the solution to the equations system (10) and (11). Therefore, from (12), (10), and (11), we have:

From (23) and (24), we can show that

From (15) and (26), we have

Therefore, when the marginal damage of pollution is high enough, the regulator taxes pollution, and when it is low enough, he actually subsidizes production to deal with the duopoly distortion.

The following proposition compares the subsidy rates for efforts in original and absorptive-capacity R&D.

Proposition 4 When$ko$and $ka$are high enough, then

• $ro>ra$⁠, for $β≥1/19$⁠, or $β<1/19$and $α$is low enough;

• $ro<ra$⁠, for $β<1/19$and $α$is high enough.

Proof: See Appendix. The results in the above proposition are to some extent unexpected. Indeed, consider the case where the free spillover $β$is zero, $l$is close to zero, and the marginal damage cost of pollution is high. In this case, where free spillover is absent and spillover benefits are nearly absent, we expect the regulator to subsidize original research at a higher rate to prevent environmental damage. The result in item (ii) however actually shows the reverse. This result is very interesting from an environmental point of view. It differs from the finding of Jin and Troege ( 2006), who showed that, for society and consumers, the marginal value of innovation expenditure is always higher than that of imitation. This difference in the results is due to the inclusion of environmental damages in the present paper. To summarize, the regulator's subsidy consists of trying to induce a minimum level of R&D externalities. Indeed, when the free spillover is high enough, he supports original research, and when it is low enough and the marginal damage of pollution is sufficiently high, he supports absorptive research.

Since $limk∘⁡$⁠,$k2→+∞ra=0$from Proposition 4, we can know when the subsidy for original research is positive or negative (in such a case the regulator actually taxes inventive R&D). Indeed, when the free spillover is high enough, original research is subsidized. When the free spillover and the marginal damage cost of pollution are low enough, we know that the regulator subsidizes production; this may induce firms to underinvest in inventive R&D with respect to the socially optimal level, and that is why it is subsidized. However, when the marginal damage cost of pollution is high enough, pollution is taxed, which may induce firms to overinvest in inventive $R&D$⁠, and that is why it is actually taxed.

We consider a duopoly competing in quantity, where firms can invest in both original (inventive) and absorptive R&D to control their pollution emissions. The regulator induces firms to implement the first best levels of production and R&D by means of three instruments: a tax per-unit of pollution, a subsidy per-unit of inventive R&D, and a subsidy per-unit of absorptive R&D. Our objective is to compare the socially optimal investment cost levels for original and absorptive R&D, and to study the behavior of the regulator with regard to these two types of research.

The socially optimal investment cost in innovative R&D is always higher than that in absorptive R&D. This is true even when the learning parameter is high encouraging greater investment in absorptive R&D. One explanation for this is that a higher absorptive parameter directly increases the efficiency of the investment in absorptive R&D and indirectly increases the efficiency of the investment in inventive R&D.

Interestingly, when the free spillover is equal to zero, the marginal damage cost of pollution is high enough and the learning parameter is close to zero, we would expect the regulator to subsidize original research at a higher rate to prevent environmental damage, but we obtain the opposite result. In fact, through this subsidy policy, and because there is a possibility of investing in absorptive research, the regulator tries to induce a minimum level of R&D externalities. Indeed, when the free spillover is high enough, he gives a higher per-unit subsidy for inventive research, and when it is low enough and the marginal damage cost of pollution is sufficiently high, he gives a higher per-unit subsidy for absorptive research. Moreover, the investment in inventive R&D is actually taxed when the free spillover is sufficiently low and the marginal damage cost of pollution is high enough. Clearly, these two last results holding for high marginal damage cost are interesting from the environmental point of view.

Finally, we mention three extensions to our work for future research, namely: (i) to allow the possibility for firms to licence their inventions when these latter serve to reduce their pollution intensity; (ii) to suppose that the regulator is imperfectly informed about the R&D activities of firms and cannot distinguish between inventive R&D and absorptive R&D carried out by firms; and (iii) to study in an empirical setting the more realistic second- or third-best outcomes. As there are many possibilities, one would necessarily need to focus on only some of these outcomes.

Firms' Second-order Conditions of the Second Stage

Consider the Hessian matrix:

By using the first-order conditions given by (8) and (9), we can determine the second derivatives constituting matrix $HVi$⁠, which can be written as

where $gi,i=1,2,3$are polynomial functions in $ti,tj,xio,xjo,xia$⁠, and $xja$⁠.

As $limk∘⁡$⁠,$ka→+∞xio$⁠,$limko,ka→+∞⁡xia$and $limko,ka→+∞⁡ti,i=1,2,$⁠,are finite numbers, then $gi$ takes finite values when $ko$ and $ka$ tend to $+∞$⁠.

Therefore, when $ko$and $ka$are sufficiently high,

• $d2Vidxio2<0$and $d2Vidxia2<0$⁠,

• $detHVi=(g1−2ko)(g3−2ka)−g22>0$⁠.

Therefore, each function $Vi$is strictly concave with respect to its arguments when $ko$and $ka$are high enough, implying that the critical point verifying system(8) and (9) maximizes function$Vi$⁠.

Regulator's Second-order Conditions of theSecond Stage

Consider the Hessian matrix:

By using the first-order conditions given by (17)-(20), we can compute the second derivatives constituting matrix$HS$⁠, which can be written as

where $fi,i=1,…,10$⁠, are polynomial functions in $xio,xjo,xia$⁠, and $xja$⁠. Since lim?(k^o,k^a?+?)?x_i^o=lim?(k^o,k^a?+?)?x_i^a=0,i=1,2, then $fi$ take finite values when $ko$and $ka$tend to $+∞$⁠.

Therefore, when $ko$and $ka$are high enough,

• $Δ1=f1−2ko<0$

  ,
• $Δ2=|f1−2kof2f2f5−2ka|>0$
• $Δ3=|f1−2kof2f3f2f5−2kaf6f3f6f8−2ko|$

• We have $limko,ka→+∞⁡Δ3ko2ka=limko,ka→+∞⁡−8ko2kako2ka=−8$ Thus, $Δ3<0$

• $Δ4=|f1−2kof2f3f4f2f5−2kaf6f7f3f6f8−2kof9f4f7f9f10−2ka|$

• We have $limko,ka→+∞⁡Δ4ko2ka2=limko,ka→+∞⁡16ko2ka2ko2ka2=16.$⁠. Thus, $Δ4>0$⁠.Therefore, function$S$is strictly concave in its arguments when $ko$and $ka$are sufficiently high, implying that the critical point verifying system (17)-(20) maximizes function S.

Proof of Proposition 1

Expression (23) can be developed as

From (24), we have

By using the expression of $xa$given by (33) in (32) and then multiplying by $(8ka−α2l2xo2)2$we get a polynomial function of degree 5 in $xo:Q(xo)=0$⁠.

The coefficient of $xo5$is $−8α4l4ko$⁠, and the constant term is $64α(1+β)(a−α)ka2$⁠.

Since $Q(0)>0$and $limx∘→+∞⁡Q(xo)=−∞$⁠, then $Q(xo)$admits at least one real and positive root $xˆo>0$⁠, and admits at most five roots. Since function $S$is strictly concave, then $Q(xo)$admits a unique real and positive root $xˆo>0$⁠. Because of (3), (16), and (33), $xˆa>0$when $ko$and $ka$are high enough.

Proof of Proposition 4

From (30) and (31): $limk∘,ka→+∞⁡ro>limk∘,k∘→+∞⁡ra⇔5(4β−1)α+(4−β)a>0.$

• The above inequality is always satisfied when $β⩾1/4$⁠.

• Suppose that $β<1/4$⁠:

• $limka,ka→+∞⁡ro>limko,ka→+∞⁡ra⇔α(4−β)a5(1−4β);$ because of condition (16), this last inequality is always verified when $β≥1/19$⁠; and when $β<1/19$⁠, we need $α$to be sufficiently low.

• $limka,ka→+∞⁡rolimka,ka→+∞⁡ra⇔α>(4−β)a5(1−4β);$⁠; this last inequality is not in contradiction with (16) when $β<1/19$⁠.