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Environmental R&D in the Presence of an Eco-Industry

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Abstract

We compare the performance of R&D cooperation and R&D competition within the eco-industry using a model of vertical relationship between a polluting industry and the eco-industry. The polluting industry is assumed perfectly competitive, and the eco-industry is a duopoly in the market for abatement goods and services, with one firm acting as a Stackelberg leader and the other firm as a follower. When there are full information sharing under R&D cooperation and involuntary information leakages under R&D competition, we find that the only case where government intervention is needed is the case where R&D cooperation yields a higher welfare but smaller profits for the follower eco-industrial firm than R&D competition. Furthermore, because of the market power that the eco-industry enjoys, we show that more total R&D efforts under R&D competition do not necessarily translate into more abatement activities and larger social welfare. When there are no involuntary leakages of information under R&D competition, this result occurs because R&D competition can induce more total R&D efforts than R&D cooperation even for significantly high R&D spillovers, if the marginal environmental damage is large.

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Notes

  1. This industrial organization literature focuses on how technological spillovers affect the comparison of R&D cooperation and R&D competition in terms of R&D efforts, profits of the firms, and social welfare [see, e.g., d’Aspremont and Jacquemin [15, 16], Henriques [29], Kamien et al. [32], Suzumura [53], etc.]. This literature argues that R&D cooperation increases R&D efforts and is thus welfare improving, when technological spillovers are sufficiently high. Amir et al. [3] extend this literature to include the comparison of R&D cooperation versus monopoly.

  2. The Organization for Economic Cooperation and Development defines the eco-industry as the set of “(...) activities which produce goods and services to measure, prevent, limit, minimize or correct environmental damage to water, air, and soil, as well as problems related to waste, noise, and eco-systems.” [OECD/Eurostat [43]]

  3. For precise information about the evolution of the eco-industry, as well as a short history of the sector and a discussion of its definition, see Sinclair-Desgagné [52].

  4. The CR&D program provides, on a competitive basis, grants of between 25 and 75 percent of total R&D costs for projects involving two or more collaborators. For more details on the CR&D program, see http://www.innovateuk.org/deliveringinnovation/collaborativeresearchanddevelopment.ashx, (accessed on February 4, 2011).

  5. We thank David Popp for suggesting this example.

  6. As pointed out by Katsoulacos and Ulph [33], spillovers associated with process innovation are in general higher than spillovers from product innovation.

  7. We also analyzed the impact on the organization of environmental R&D of the presence of the eco-industry when the latter engages in Cournot competition in the AGS market. We also find some discrepancy in the comparison of the performance of R&D cooperation and R&D competition in terms of R&D efforts, abatement activities (and thus environmental quality), eco-industry’s profits, and social welfare. However, whenever government intervention is needed, we find that it is always aimed at imposing R&D competition within the eco-industry.

  8. See, e.g., Parry [44], Biglaiser and Horowitz [9], Laffont and Tirole [35, 36], Denicolo [20], Feess and Muehlheusser [22], David and Sinclair-Desgagné [17, 18], Greaker [24], Canton et al. [12], Greaker and Rosendahl [25], Golombek et al. [23], Perino [45], David et al. [19], Nimubona and Sinclair-Desgagné [41, 42], Heyes and Kapur [30], and Greaker and Hoel [26].

  9. See, e.g., Banerjee and Lin [6, 7], Atallah [4], Brocas [11], Ishii [31], Versaevel and Vencatachellum [55], and Chen and Sappington [13].

  10. Atallah [4] develops a model that incorporates two vertically related industries with horizontal spillovers within each industry and vertical spillovers between the two industries.

  11. The previous literature, which analyzes the organization of environmental R&D in the polluting industry, considers instead a Cournot duopoly model (see, e.g., Chiou and Hu [14], Poyago Theotoky [47]). Surely, the Cournot case constitutes an interesting framework for the analysis of a polluting industry. However, as we argue in the introduction section of this paper, typical eco-industrial firms are not expected to behave as Cournot competitors.

  12. Amir et al. [2] compare the outcomes of simultaneous versus sequential moves in the R&D stage of the R&D competition game. They characterize the R&D leader and follower behaviors while assuming simultaneous-move at the final good production stage. In contrast, this paper features the first-move advantages at the AGS production stage while keeping simultaneous move in the R&D stage. Lambertini et al [37] model the production process as a Stackelberg game. However, they do not consider R&D outsourcing as they examine the case where R&D is undertaken by the producers of the final good.

  13. Considering R&D spillovers as endogenous, Poyago-Theotoky [46] argues that it is always optimal for duopolistic firms to set ω=0 and ω=1 when they compete and cooperate in the R&D stage, respectively.

  14. It can be shown that this inverse demand function for abatement is decreasing in A, i.e., p A <0 . Moreover, a tax rise will generate a clockwise rotation of the inverse demand curve with respect to its horizontal intercept, i.e., p t >0 and p A t ≤0 [see David et al. [19]].

  15. It is worth noting that the case in which R&D contributions are asymmetric can also be envisaged. As suggested by Lambertini et al. [37] for the distribution of profits, R&D expenditures could be alternatively shared according to a Nash bargaining solution or in proportion to the asymmetric firms’ market shares. However, this would entail significant difficulties as one will need to take into account the firms’ commitment and agreement to share R&D costs.

  16. If condition (ii) is verified, then the second-order condition for optimal R&D investment, given by 16t−3(ω+1)2>0, is verified for any ω 𝜖[0,1]. We thank an anonymous referee for bringing this to our attention.

  17. If condition (ii) is verified, then the leader’s and the follower’s second-order conditions for optimal R&D investment, respectively given by 4t−(ω−2)2>0 and 8t−(2ω−3)2>0, are verified for any ω 𝜖[0,1].

  18. In region A, R&D cooperation occurs in equilibrium, an outcome which is desirable from both the leader eco-industrial firm and social standpoints. In region D, it is R&D competition that takes place instead, as the preferred R&D regime by both sides.

  19. See, e.g., d’Aspremont and Jacquemin [15], Kamien et al. [32]. In these papers, however, firms that conduct R&D compete a la Cournot in the final good market. In a different setting from ours, Amir and Wooders [1] also find that total profits can be higher with R&D competition than with R&D cooperation due to cost asymmetry in the R&D competition case.

  20. Recall that α=10 for the graphical analysis.

  21. Recall that α=10 for the graphical analysis.

  22. For example, Canada and the USA have been offering for the last few years expedited processing for green technology patent applications. For more information on these programs, see http://www.cipo.ic.gc.ca/eic/site/cipointernet-internetopic.nsf/eng/wr02462.htmland http://www.uspto.gov/patents/init_events/green_tech.jsp (accessed on January 31, 2013).

  23. Recall that α=10 for the graphical analysis.

  24. We are grateful to an anonymous referee for this suggestion.

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Authors and Affiliations

  1. Department of Economics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Alain-Désiré Nimubona

  2. CIRANO, Montreal, H3A 2M8, Canada

    Alain-Désiré Nimubona

  3. Department of Economics, McGill University, Montreal, Quebec, H3A 2T7, Canada

    Hassan Benchekroun

  4. CIREQ, Montreal, Quebec, H3T 1N8, Canada

    Hassan Benchekroun

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  1. Alain-Désiré Nimubona
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Correspondence to Alain-Désiré Nimubona.

Appendices

Appendix A: Equilibria of the R&D Cooperation and Competition Cases

The following equilibrium obtains for our game in the R&D cooperation case:

$$\begin{array}{@{}rcl@{}} y^{CS^{\ast }} &=&\frac{3\left( 1+\omega \right) \left( t\xi -g\right) }{ 16t-3\left( \omega +1\right)^{2}} \\ Y^{CS^{\ast }} &=&\frac{6\left( 1+\omega \right) \left( t\xi -g\right) }{ 16t-3\left( \omega +1\right)^{2}} \\ a_{1}^{CS^{\ast }} &=&\frac{8\left( t\xi -g\right) }{16t-3\left( \omega +1\right)^{2}} \end{array} $$
$$\begin{array}{@{}rcl@{}} a_{2}^{CS^{\ast }} &=&\frac{4\left( t\xi -g\right) }{16t-3\left( \omega +1\right)^{2}} \\ A^{CS^{\ast }} &=&\frac{12\left( t\xi -g\right) }{16t-3\left( \omega +1\right)^{2}}\\[-2pt] \pi_{1}^{CS^{\ast }} &=&\frac{\left( t\xi -g\right)^{2}\left[ 64t-9\left( \omega +1\right)^{2}\right] }{2\left[ 16t-3\left( \omega +1\right)^{2} \right]^{2}} \\ \pi_{2}^{CS^{\ast }} &=&\frac{\left( t\xi -g\right)^{2}\left[ 32t-9\left( \omega +1\right)^{2}\right] }{2\left[ 16t-3\left( \omega +1\right)^{2} \right]^{2}} \\[-2pt] x^{CS^{\ast }} &=&\frac{\alpha -t}{2} \\ e^{CS^{\ast }} &=&Max\left\{ 0,\frac{\alpha -t}{2}-12\right.\\[-2pt] &&\left.\times\left[ \frac{\xi \left( t\xi -g\right) }{16t-3\left( \omega +1\right)^{2}}-\frac{6\left( t\xi -g\right)^{2}}{\left( 16t-3\left( \omega +1\right)^{2}\right)^{2}} \right] \right\}. \end{array} $$
(A.1)

In turn, the equilibrium of our game in the R&D competition case is given by:

$$\begin{array}{@{}rcl@{}} y_{1}^{NC^{\ast }} &=&\frac{4\left( 2-\omega \right) \left[ \left( 2t-3\right) +\omega \left( 5-2\omega \right) \right] \left( t\xi -g\right) }{ \left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) } \\ y_{2}^{NC^{\ast }} &=&\frac{4\left( 3-2\omega \right) \left[ \left( t-2\right) +\omega \left( 3-\omega \right) \right] \left( t\xi -g\right) }{ \left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) } \\ Y^{NC^{\ast }} &=&\frac{4\left[ \left( 7-4\omega \right) \left( t-2\right) +2\omega \left( 2\omega -3\right) \left( \omega -3\right) +2\right] \left( t\xi -g\right) }{\left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) } \\ [10pt] a_{1}^{NC^{\ast }} &=&\frac{8\left( t\xi -g\right) \left[ 2t-\left( \omega -1\right) \left( 2\omega -3\right) \right] }{\left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) } \\ [10pt] a_{2}^{NC^{\ast }} &=&\frac{8\left( t\xi -g\right) \left[ t-\left( \omega -1\right) \left( \omega -2\right) \right] }{\left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) } \\ A^{NC^{\ast }} &=&\frac{8\left( t\xi -g\right) \left[ 3t-\left( \omega -1\right) \left( 3\omega -5\right) \right] }{\left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) } \\ \pi_{1}^{NC^{\ast }} &=&\frac{8\left( t\xi -g\right)^{2}\left[ 2t-\left( \omega -1\right) \left( 2\omega -3\right) \right]^{2}\left[ 4t-\left( \omega -2\right)^{2}\right] }{\left \{ \left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) \right \}^{2}} \\ \pi_{2}^{NC^{\ast }} &=&\frac{8\left( t\xi -g\right)^{2}\left[ t-\left( \omega -1\right) \left( \omega -2\right) \right]^{2}\left[ 8t-\left( 2\omega -3\right)^{2}\right] }{\left \{ \left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) \right \}^{2}} \\ x^{NS^{\ast }} &=&\frac{\alpha -t}{2} \\ e^{NS^{\ast }} &=&Max\left \{ \begin{array}{l} 0,\frac{\alpha -t}{2}-\left[ \frac{8\xi \left( t\xi -g\right) \left[ 3t-\left( \omega -1\right) \left( 3\omega -5\right) \right] }{\left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) }\right. \\ \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left. -\frac{32\left( t\xi -g\right)^{2}\left[ 3t-\left( \omega -1\right) \left( 3\omega -5\right) \right]^{2}}{\left \{ \left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) \right \}^{2}}\right] \end{array} \right \} . \end{array} $$
(A.2)

Substituting (A.1) into (16) and after some manipulations, we find the following expression of the equilibrium level of social welfare under R&D cooperation:

$$ W^{CS^{\ast }}=\frac{\left( \alpha -d\right)^{2}}{2}+\frac{\left[ 120d-9\left( 1+\omega \right)^{2}\right] \left( d\xi -g\right)^{2}}{\left[ 16d-3\left( \omega +1\right)^{2}\right]^{2}}. $$
(A.3)

Similarly, we obtain the expression of the equilibrium level of social welfare under R&D competition by substituting the equilibrium levels of polluting output, abatement, and R&D efforts, as given in Eq. A.2, into Eq. 16.

Appendix B: Proof of Lemma 1

First, recall that t=d by assumption. When ω max=ω∈[0,1] and ω min=0, we get from Eq. A.1 and A.2 the following expressions of the variation of the profits of the leader and follower eco-industrial firms, respectively.

$$\pi_{1}^{CS^{\ast }}-\pi_{1}^{NS^{\ast }}= \frac{\left( d\xi -g\right)^{2}\left \{ \left[ 64d-9\left( \omega +1\right)^{2}\right] \left[ \left( 8d-9\right) \left( d-1\right) -3\right]^{2}-4 \left[ 16d-3\left( \omega +1\right)^{2}\right]^{2}\left( d-1\right) \left( 2d-3\right)^{2}\right \} }{2\left[ 16d-3\left( \omega +1\right)^{2}\right]^{2}\left[ \left( 8d-9\right) \left( d-1\right) -3\right]^{2}} $$
$$\pi_{2}^{CS^{\ast }}-\pi_{2}^{NS^{\ast }}= \\\frac{\left( d\xi -g\right)^{2}\left \{ \left[ 32d-9\left( 1+\omega \right)^{2}\right] \left[ \left( 8d-9\right) \left( d-1\right) -3\right]^{2}-\left[ 16d-3\left( \omega +1\right)^{2}\right]^{2}\left( d-2\right)^{2}\left( 8d-9\right) \right \} }{2\left[ 16d-3\left( \omega +1\right)^{2}\right]^{2} \left[ \left( 8d-9\right) \left( d-1\right) -3\right]^{2}}. $$

Based on our assumptions, the sign of \(\pi _{1}^{CS^{\ast }}-\pi _{1}^{NS^{\ast }}\) is given by the sign of

$$\begin{array}{@{}rcl@{}}&&\left[ 64d-9\left( \omega +1\right)^{2}\right] \left[ \left( 8d-9\right) \left( d-1\right) -3\right]^{2}\\ &&{\kern1pc}-4\left[ 16d-3\left( \omega +1\right)^{2} \right]^{2}\left( d-1\right) \left( 2d-3\right)^{2}. \end{array} $$

For ω∈[0,1] and d∈]3,α[, the graphical analysis in Fig. 1 and simple calculus based on the latter expression reveal that \(\pi _{1}^{CS^{\ast }}-\pi _{1}^{NS^{\ast }}<0\), when: (i) \(\omega <\omega _{\pi _{1}}\), where \(\omega _{\pi _{1}}\) depends on the value of α or (ii) \(\omega _{\pi _{1}}<\omega <0.42\) and d is sufficiently low. The opposite holds, i.e., \(\pi _{1}^{CS^{\ast }}-\pi _{1}^{NS^{\ast }} >0\), when: (i) ω>0.42 or (ii) \(\omega _{\pi _{1}}<\omega <0.42\) and d is sufficiently large.

The sign of \(\pi _{2}^{CS^{\ast }}-\pi _{2}^{NS^{\ast }}\) is in turn given by the sign of

$$\begin{array}{@{}rcl@{}} &&\left[ 32d-9\left( 1+\omega \right)^{2}\right] \left[ \left( 8d-9\right) \left( d-1\right) -3\right]^{2}\\ &&{\kern1pc}-\left[ 16d-3\left( \omega +1\right)^{2} \right]^{2}\left( d-2\right)^{2}\left( 8d-9\right) , \end{array} $$

which can be shown to be always positive for ω∈[0,1] and d∈[3,α], using simple calculus and graphical analysis.

Finally, the expressions of the equilibrium levels of social welfare can be used to compute the following expression for ω max=ω∈[0,1] and ω min=0,

$$W^{CS^{\ast }}-W^{NS^{\ast }}= \frac{\left( d\xi -g\right)^{2}\left \{ \left[ 120d-9\left( \omega +1\right)^{2}\right] \left[ \left( 8d-9\right) \left( d-1\right) -3\right]^{2}-\left( 30d^{3}-\frac{225}{2}d^{2}+126d-36\right) \left[ 16d-3\left( \omega +1\right)^{2}\right]^{2}\right \} }{\left[ 16d-3\left( \omega +1\right)^{2}\right]^{2}\left[ \left( 8d-9\right) \left( d-1\right) -3 \right]^{2}}. $$

The sign of \(W^{CS^{\ast }}-W^{NS^{\ast }}\) is given by the sign of

$$\begin{array}{@{}rcl@{}}&&\left[ 120d-9\left( \omega +1\right)^{2}\right] \left[ \left( 8d-9\right) \left( d-1\right) -3\right]^{2}\\&&-\left( 30d^{3}-\frac{225}{2} d^{2}+126d-36\right) \left[ 16d-3\left( \omega +1\right)^{2}\right]^{2}. \end{array} $$

For ω∈[0,1] and d∈[3,α], we can show that on the one hand, \(W^{CS^{\ast }}-W^{NS^{\ast }}<0\) when (i) ω<ω W , where ω W depends on the value of α, or (ii) ω W <ω<0.37 and d is sufficiently low. On the another hand, \(W^{CS^{\ast }}-W^{NS^{\ast }}>0\) when (i) ω>0.37 or (ii) ω W <ω<0.37 and d is sufficiently large.

Appendix C: Proof of Lemma 2

First, recall that t=d by assumption. When ω max=1 and ω min=ω∈[0,1],and based on our assumptions, the sign of \(\pi _{1}^{CS^{\ast }}-\pi _{1}^{NS^{\ast }}\) is given by the sign of

$$\begin{array}{@{}rcl@{}} &&\left( 16t-9\right) \left \{ \left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \right.\\ &&\quad\times\left.\left( 2\omega -1\right) \left( 2\omega -3\right) \right \}^{2}-64\left( 4t-3\right)^{2}\left[ 2t-\left( \omega -1\right) \left( 2\omega -3\right) \right]^{2}\\ &&\quad\times\left[ 4t-\left( \omega -2\right)^{2}\right] , \end{array} $$

which can be shown to be always positive for ω∈[0,1] and d∈]3,α[, using simple calculus and graphical analysis. The sign of \(\pi _{2}^{CS^{\ast }}-\pi _{2}^{NS^{\ast }}\) is in turn given by the sign of

$$\begin{array}{@{}rcl@{}} &&\left( 8t-9\right) \left \{ \left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \right.\\&&\quad\times\left. \left( 2\omega -1\right)\left( 2\omega -3\right) \right \}^{2} -64\left( 4t-3\right)^{2}\left[ t-\left( \omega -1\right) \left( \omega -2\right) \right]^{2}\\&&\quad\times\left[ 8t-\left( 2\omega -3\right)^{2}\right] .\end{array} $$

For ω∈[0,1] and d∈]3,α[, the graphical analysis in Fig. 2 and simple calculus based on the latter expression reveal that \(\pi _{2}^{CS^{\ast }}-\pi _{2}^{NS^{\ast }}<0\), when (i) \(\omega >\omega _{\pi _{2}}\), where \(\omega _{\pi _{2}}\) depends on the value of α or (ii) \(0.63<\omega <\omega _{\pi _{2}}\) and d is sufficiently low. The opposite holds, i.e., \(\pi _{2}^{CS^{\ast }}-\pi _{2}^{NS^{\ast }}>0\) when (i) ω<0.63 or (ii) \(0.63<\omega <\omega _{\pi _{2}}\) and d is sufficiently large.

Finally, the sign of \(W^{CS^{\ast }}-W^{NS^{\ast }}\) for ω max=1 and ω min=ω∈[0,1] is given by the sign of

$$\begin{array}{@{}rcl@{}} \left( 30d-9\right) \left \{ \left[ 4d-\left( \omega -2\right)^{2}\right] \left[ 8d-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) \right \}^{2} \\ -32\left( 4d-3\right)^{2}\left( 2-\omega \right) \left[ d\left( 4+6\omega \right) +\left( 1-\omega \right) \left( 6\omega^{2}-7\omega -6\right) \right] \left[ \left( 2d-3\right) +\omega \left( 5-2\omega \right) \right] \\ -32\left( 4d-3\right)^{2}\left( 3-2\omega \right) \left[ d\left( 1+10\omega \right) +\left( 1-\omega \right) \left( 10\omega^{2}-15\omega -2\right) \right] \left[ \left( d-2\right) +\omega \left( 3-\omega \right) \right] \\ -128\left( 4d-3\right)^{2}\left \{ \left[ 5d-3\left( \omega -2\right)^{2} \right] d-\left( \omega -1\right) \left( \omega -2\right) \left( 2\omega -3\right) \left( \omega +1\right) \right \} \left[ 3d-\left( \omega -1\right) \left( 3\omega -5\right) \right] , \end{array} $$

which can be shown to be always positive for ω∈[0,1] and d∈]3,α[, using simple calculus and graphical analysis.

Appendix D: Proof of Lemma 3

First, recall that t=d by assumption. When ω max=ω∈[0,1] and ω min=0, subtracting the expressions of the equilibrium levels of total R&D efforts in Eqs. A.1 and A.2 yields:

$$Y^{CS^{\ast }}-Y^{NS^{\ast }}=\frac{\left( d\xi -g\right) \left \{ 6\left( 1+\omega \right) \left[ \left( 8d-9\right) \left( d-1\right) -3\right] -\left( 7d-12\right) \left[ 16d-3\left( \omega +1\right)^{2}\right] \right \} }{\left[ 16d-3\left( \omega +1\right)^{2}\right] \left[ \left( 8d-9\right) \left( d-1\right) -3\right] }\text{ .} $$

Given our assumptions, the sign of \(Y^{CS^{\ast }}-Y^{NS^{\ast }}\) corresponds to the sign of

$$\begin{array}{@{}rcl@{}}&&6\left( 1+\omega \right) \left[ \left( 8d-9\right) \left( d-1\right) -3 \right] -\left( 7d-12\right) \\&&\quad\left[ 16d-3\left( \omega +1\right)^{2}\right] . \end{array} $$

For ω∈[0,1] and d∈]3,α[, we can use the graphical analysis in Fig. 3 and simple calculus to show that \(Y^{CS^{\ast }}<Y^{NS^{\ast }}\).

From the equilibrium levels of total abatement activities in Eqs. A.1 and A.2, when ω max=ω∈[0,1] and ω min=0, we can get

$$A^{CS^{\ast }}-A^{NS^{\ast }}=\frac{2\left( d\xi -g\right) \left \{ 6\left[ \left( 8d-9\right) \left( d-1\right) -3\right] -\left( 3d-5\right) \left[ 16d-3\left( \omega +1\right)^{2}\right] \right \} }{\left[ 16d-3\left( \omega +1\right)^{2}\right] \left[ \left( 8d-9\right) \left( d-1\right) -3 \right] }\text{ .} $$

The sign of \(A^{CS^{\ast }}-A^{NS^{\ast }}\) corresponds to the sign of

$$6\left[ \left( 8d-9\right) \left( d-1\right) -3\right] -\left( 3d-5\right)\left[ 16d-3\left( \omega +1\right)^{2}\right] . $$

For ω∈[0,1] and d∈]3,α[, we can check that, on one hand, \(A^{CS^{\ast }}<A^{NS^{\ast }}\) when (i) ω<ω A , where ω A depends on the value of α, or (ii) ω A <ω<0.58 and d is sufficiently low. On another hand, \(A^{CS^{\ast }}>A^{NS^{\ast }}\) when (i) ω>0.58 or (ii) ω A <ω<0.58 and d is sufficiently large.

Finally, subtracting the expressions of the equilibrium levels of net emissions in Eqs. A.1 and A.2 when ω max=ω∈[0,1] and ω min=0 gives rise to

$$\begin{array}{@{}rcl@{}} e^{CS^{\ast }}-e^{NS^{\ast }}&=& \frac{1}{2}\left[ \frac{12\left( d\xi -g\right) }{16d-3\left( \omega +1\right)^{2}}-\frac{2\left( 3d-5\right) \left( d\xi -g\right) }{\left( 8d-9\right) \left( d-1\right) -3}\right]\\ && \left[ \frac{12\left( d\xi -g\right) }{16d-3\left( \omega +1\right)^{2}}+\frac{2\left( 3d-5\right) \left( d\xi -g\right) }{\left( 8d-9\right) \left( d-1\right) -3}-2\xi \right], \end{array} $$

which can also be written as

$$e^{CS^{\ast }}-e^{NS^{\ast }}=\frac{1}{2}\left( A^{CS^{\ast }}-A^{NS^{\ast }}\right) \left( A^{CS^{\ast }}+A^{NS^{\ast }}-2\xi \right) . $$

Since \(\xi >A^{CS^{\ast }}\) and \(\xi >A^{NS^{\ast }}\) from our assumptions, it is straightforward that \(A^{CS^{\ast }}+A^{NS^{\ast }}-2\xi <0\). Therefore, \(e^{CS^{\ast }}-e^{NS^{\ast }}\) and \(A^{CS^{\ast }}-A^{NS^{\ast }} \) always have opposite signs.

Appendix E: Proof of Lemma 4

First, recall that t=d by assumption. When ω max=1 and ω min=ω∈[0,1], and given our assumptions, the sign of \(Y^{CS^{\ast }}-Y^{NS^{\ast }}\) corresponds to the sign of

$$\begin{array}{@{}rcl@{}} &&3\left\{ \left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right]\right. \\ &&-\left.\left( \omega -2\right) \left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) \right \} \\ &&-4\left( 4t-3\right) \left[ \left( 7-4\omega \right) \left( t-2\right) +2\omega \left( 2\omega -3\right) \left( \omega -3\right) +2\right] . \end{array} $$

For ω∈[0,1] and d∈]3,α[, we can use the graphical analysis in Fig. 4 and simple calculus to show that, on one hand, \(Y^{CS^{\ast }}>Y^{NS^{\ast }}\) when (i) ω>ω Y , where ω Y depends on the value of α, or 0<ω<ω Y and d is sufficiently low. On another hand, \( Y^{CS^{\ast }}<Y^{NS^{\ast }}\) when 0<ω<ω Y and d is sufficiently large.

From the equilibrium levels of total abatement activities in Eqs. A.1 and A.2, when ω max=1, ω min=ω∈[0,1], the sign of \(A^{CS^{\ast }}-A^{NS^{\ast }}\) corresponds to the sign of

$$\begin{array}{@{}rcl@{}} &&3\left \{ \left[ 4t-\left( \omega -2\right)^{2}\right] \left[ 8t-\left( 2\omega -3\right)^{2}\right] -\left( \omega -2\right)\right. \\ &&\quad\left.\times\left( 3\omega -2\right) \left( 2\omega -1\right) \left( 2\omega -3\right) \right \} \\ &&\quad-8\left( 4t-3\right) \left[ 3t-\left( \omega -1\right) \left( 3\omega -5\right) \right] . \end{array} $$

For ω∈[0,1] and d∈]3,α[, we can check that \(A^{CS^{\ast }}>A^{NS^{\ast }}\) by using simple calculus and graphical analysis. Finally, it is direct to show that \(e^{CS^{\ast }}-e^{NS^{\ast }}\) and \(A^{CS^{\ast }}-A^{NS^{\ast }}\) always have opposite signs (see Proof of Lemma 3 above).

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Nimubona, AD., Benchekroun, H. Environmental R&D in the Presence of an Eco-Industry. Environ Model Assess 20, 491–507 (2015). https://doi.org/10.1007/s10666-014-9439-x

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