An excessive development of green products?

Abstract

This paper examines firms’ incentives to develop a new (green) product, which might compete against the pollutant (brown) good that they traditionally sell. We show that in equilibrium more than one firm might develop a green product, but such an equilibrium outcome is not necessarily efficient. In particular, we predict an excessive amount of green goods under certain conditions, namely, when the green product is extremely clean but both products are not sufficiently differentiated in their attributes, and when the green product is not significantly cleaner than the brown good. We finally provide policies that help regulatory authorities promote equilibrium outcomes yielding the highest social welfare.

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Notes

  1. 1.

    Similarly, Simple Green offers a separate brand, Simple Green Naturals, as “100% naturally derived, with ingredients originating from nature.”

  2. 2.

    More generally, among all newly introduced products in the US, the percentage that claimed to be green increased from 1.1% in 1986 to 9.5% in 1999; see Kirchhoff (2000).

  3. 3.

    Green products could in the long run replace brown products. However, the transition between these goods can still take several years, as the example of hybrid cars suggests. For instance, Toyota has simultaneously produced both the Toyota Prius and fossil-fuel cars since 1997.

  4. 4.

    Solid-recovered fuels provide an example of a good that, despite being relatively green, is still controversial given that its environmental performance is relatively weak; as recognized by the European Recovered Fuel Organization (EN Report 15359). In particular, these fuels are produced by shredding and dehydrating solid waste consisting of largely combustible components of municipal waste. Another highly cited example is that of oil sands, which require an extremely large amount of water for every gallon of oil produced, and that generate more GHGs emissions than standard oil drilling facilities.

  5. 5.

    Spence (1975) considers a monopolist’s decision to invest in quality, and shows that equilibrium outcomes are not necessarily optimal. While we also demonstrate that the monopolist’s decision to develop a green product can be suboptimal, the parameter conditions under which this case arises shrink as brown and green products become more pollutant. Furthermore, our paper considers firms’ incentives to develop a new line of green products in addition to the existing brown good that firms traditionally produce, thus giving rise to BSEs that do not exist in Spence’s model.

  6. 6.

    Similar to our paper, they show that firms competing for socially responsible consumers (e.g., consumers with environmental concerns) can lead to an excessive provision of public goods. Arora and Gangopadhyay (1995) consider two firms, each selling a single good and deciding its degree of cleanness and its price.

  7. 7.

    Their model was extended by Lambertini and Tampieri (2012).

  8. 8.

    See Dosi and Moretto (2001), Cason and Gangadharan (2002), Mason (2006), Hamilton and Zilberman (2006), Greaker (2006), and Ibanez and Grolleau (2008) for the specific practice of ecolabeling, which is often regarded as CSR.

  9. 9.

    Their paper has been extended to settings of environmental externalities by Amachera et al. (2004).

  10. 10.

    For completeness, “Appendix 1” analyzes the case in which both firms simultaneously and independently decide whether to produce a green good.

  11. 11.

    This demand specification is, thus, similar to that of Singh and Vives (1984) for the analysis of firms’ incentives to compete in either quantities or prices when they produce differentiated products.

  12. 12.

    More generally, parameter \(\lambda \) captures the product differentiation between the brown and green goods, thus allowing the parameter to embody both the products distinct instrinsic characteristics (e.g., acceleration and cargo space in hybrid and fossil-fuel cars) and their different environmental properties. However, if parameter \(\lambda \) only captures the intrinsic features of the two products, different cases can arise. For instance, when goods are completely differentiated, i.e., \(\lambda =0\), they exhibit totally different intrinsic characteristics, and thus each of them has its own separate market. In this setting, their environmental features can also be completely different (e.g., if the pollution intensity of the green good is zero), similar (if it is the same as that of the brown product), or take intermediate values (if its pollution intensity is smaller).

  13. 13.

    While the introduction of a green product entails an overall increase in demand, such a development is costly, implying that firm \(i\) does not necessarily find profitable to develop the new product, as we describe in the equilibrium results of Sect. 3.

  14. 14.

    For compactness, we do not include here the expressions of equilibrium profits in each entry setting. Nevertheless, the proof of Lemma 1 provides them.

  15. 15.

    While “Appendix 1” analyzes equilibrium development strategies in the simultaneous version of the game, we focus on its sequential version as most real-life examples of firms adding a line of green products to their existing brown goods did it sequentially. For instance, Toyota was the first automaker to offer hybrid cars, the Prius, along with their other (more polluting) cars, in 1997. Other automakers followed by developing their own hybrid cars afterwards: Honda introduced the Insight in 1999, Mitsubishi the Colt in 2005, and Nissan the Leaf in 2010.

  16. 16.

    Figure 1 considers \(\lambda \le \overline{\lambda }\), where cutoff \(\overline{\lambda }\) becomes \(\overline{\lambda }=2/3\) in this parametric example.

  17. 17.

    In addition, note that if products are completely differentiated, \(\lambda =0 \), firm 2 develops the green product if \(K_{2}<\frac{(1-z)^{2}}{9} ( K_{2}<\frac{(1-z)^{2}}{4}\)) upon observing that firm 1 developed (did not develop, respectively) the new good. Our equilibrium analysis at the end of this section provides comparative statics of cutoffs \(K^{A}\) and \(K^{B}\).

  18. 18.

    If firms instead simultaneously choose to develop the green product, the results in Proposition 1 still apply; except for point (3) which holds under different parameter conditions. For more details on the equilibrium of the simultaneous-move version of the game, see “Appendix 1”.

  19. 19.

    A similar representation applies to the second strategy profile, \( (G_{1},NG_{2})\), depicted in areas (2) for firm 1 (in Fig. 2a) and for firm 2 (in Fig. 2b).

  20. 20.

    In particular, if only firm \(i\) had the ability to develop the green good, it would do so when development costs are relatively low, \(K<K^{B}\). However, when firm \(j\) also has the ability to develop the green good, firm \( i\) might decide to develop under more expensive development costs \(K<K^{A}\), where \(K^{A}>K^{B}\).

  21. 21.

    For more details, see http://www.babygearlab.com/Disposable-Diaper-Reviews/Huggies-Pure-Natural.

  22. 22.

    For simplicity, our social welfare function abstracts from the cost of raising public funds. Extended models could consider this cost if the regulator provides subsidies to lower firms’ development costs (such as fixed R&D and capital investments). However, the introduction of these costs would still yield the presence of an excessive/insufficient number of firms under the same parameter conditions as in our model.

  23. 23.

    Following our numerical example in previous sections of the paper, Fig. 4 considers costs \(c=1/4\) and \(z=1/2\). Other parameter values yield similar qualitative results and can be provided by the authors upon request.

  24. 24.

    For this numerical example, outcome \((G_{1},NG_{2})\) arises under condition \( \min \{K_{a},K_{c}\}<K<\max \{K_{b},K_{c}\}\), which in this case implies that \(K\ \)satisfies \(K_{c}<K<K_{b}\); as depicted in the right-hand side of Fig. 4.

  25. 25.

    While the shaded areas describe optimality in terms of the number of firms developing the green good, the externality that both types of products generate is still not addressed by any policy tool (such as taxes or quotas), ultimately implying that these areas only identify second-best optima. (Nevertheless, and for compactness, we refer to shaded regions as optima.)

  26. 26.

    In particular, cutoffs \(K_{a}\) and \(K_{c}\) lie on the negative quadrant. Thus, for all \(K>K_{b}\) the outcome \((NG_{1},NG_{2})\) is socially optimal, which only coincides with the equilibrium outcome in region III (in region I and II one or both firms develop the green good). For all \(K\le K_{b}\), outcome \((G_{1},NG_{2})\) is optimal, thus coinciding with the equilibrium outcome in region II alone.

  27. 27.

    For the parameter values in Table 1, where \(d=1/2\), the welfare benefit from consumer and producer surplus is exactly offset by the environmental damage from the brown product, thus yielding a zero welfare level. Other numerical simulations with \(d<1/2\) yield positive welfare levels, and can be provided by the authors upon request.

  28. 28.

    Note that this is not necessarily the case when goods are relatively differentiated.

  29. 29.

    Graphically, when \(\lambda =0.1\) (in the left-hand side of Fig. 9), moving from point \(A\) to either \(B\) or \(C\) entails a welfare reduction. Similarly, when \(\lambda =0.3\) (in the right-hand side of Fig. 9), moving from \(D\) to either \(E\), \(F\) or \(G\) yields a welfare loss.

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Correspondence to Ana Espínola-Arredondo.

Additional information

We would like to thank Amihai Glazer and two anonymous referees for their insightful comments. In addition, we want to thank the participants of the seminar in the Public Policy Institute at Universidad Diego Portales, Chile, especially Felipe Balmaceda and Gregory Elacqua for their helpful comments and suggestions.

Appendices

Appendices

Appendix 1: Simultaneous-move game

Proposition A

In the production of green goods, Nash equilibrium behavior in the simultaneous-move game is:

  1. 1.

    Both firms develop a green good, \((G_{1},G_{2})\), when \(K_{1},K_{2}<K^{B}\);

  2. 2.

    Only firm 1 develops a green good, \((G_{1},NG_{2})\), when \(K_{1}<K^{A}\) and \(K_{2}\ge K^{B}\);

  3. 3.

    Only firm 2 develops a green good, \((NG_{1},G_{2})\), when \(K_{1}\ge K^{B}\) and \(K_{2}<K^{A}\); and

  4. 4.

    No firm develops a green good, \((NG_{1},NG_{2})\), when \(K_{1},K_{2}\ge K^{A}\).

Hence, strategy profiles (1), (2) and (4) can be supported under the same parameter conditions when firms interact simultaneously and sequentially. However, \((NG_{1},G_{2})\) in which only firm 2 develops green products can be sustained under more general conditions when this firm simultaneously chooses whether to develop the green good than when it acts as the follower upon observing the leader’s decision. In particular, when both firms’ development costs are intermediate, i.e., \(K^{A}>K_{1},K_{2}\ge K^{B}\), the sequential-move game prescribes that a unique equilibrium emerges in which firm 1 develops the green good, \((G_{1},NG_{2})\), while under the simultaneous version of the game, two possible outcomes arise, \((G_{1},NG_{2})\) and \((NG_{1},G_{2})\) in which either firm 1 or 2 develop in equilibrium, thus reflecting that firm 1 benefits from its first-mover advantage.

Proof

As described in Lemma 2, the best response function for any firm \(i=\{1,2\}\) prescribes that, if the rival firm \(j\) develops the green product, firm \(i\) responds developing it if and only if \(K_{i}<K^{B}\). However, if the rival firm does not develop, firm \(i\) responds developing the green good if and only if \(K_{i}<K^{A}\). Let us now examine equilibrium behavior in the nine possible parameter combinations that emerge from this best response functions.

Case 1 (Firm 1’s costs are in region I). Case 1a. When firm 2’s costs are in region I, developing the green product is a strictly dominant strategy for both firms, and thus \((G_{1},G_{2})\) is the unique Nash equilibrium outcome.

Case 1b. When firm 2’s costs are in region II, developing the green product is a strictly dominant strategy only for firm 1, and thus firm 2 responds not developing this good. Hence, \((G_{1},NG_{2})\) arises.

Case 1c. If firm 2’s costs are in region III, developing (not developing) the green product is a strictly dominant strategy for firm 1 (firm 2, respectively). Therefore, \((G_{1},NG_{2})\) also arises in this case.

Case 2 (Firm 1’s costs are in region II). Case 2a. When firm 2’s costs are in region I, developing the green product is a strictly dominant strategy only for firm 2, and thus \((NG_{1},G_{2})\) is the unique Nash equilibrium outcome.

Case 2b. When firm 2’s costs are in region II, two equilibria arise: \((G_{1},NG_{2})\) and \((NG_{1},G_{2})\). In these equilibria, neither firm has incentives to deviate: on one hand, the firm which did not develop the green product cannot increase its profits by developing it, since its rival already developed the good; on the other hand, the firm that developed the product would reduce its profits by deviating towards not developing the product, since it is currently the only producer in the green market.

Case 2c. If firm 2’s costs are in region III, not developing is a strictly dominant strategy for firm 2. Hence, \((G_{1},NG_{2})\) arises in this case.

Case 3 (Firm 1’s costs are in region III). Case 3a. When firm 2’s costs are in region I, developing (not developing) the green product is a strictly dominant strategy for firm 2 (firm 1, respectively). Therefore, \((NG_{1},G_{2})\) is the unique Nash equilibrium outcome.

Case 3b. When firm 2’s costs are in region II, not developing the green product is still a strictly dominant strategy for firm 1, and firm 2 responds developing the green good, i.e., \((NG_{1},G_{2})\).

Case 3c. If firm 2’s costs are in region III, not developing the green product is a strictly dominant strategy for both firms, and \((NG_{1},NG_{2})\) arises.

Summarizing, the symmetric equilibrium outcomes \((G_{1},G_{2})\) can only be supported when \(K_{1},K_{2}<K^{B}\), and \((NG_{1},NG_{2})\) can be sustained when \(K_{1},K_{2}\ge K^{A}\). However, equilibrium outcome \((G_{1},NG_{2})\) can be supported under four different settings: (1) \(K_{1}<K^{B}\) and \( K^{A}>K_{2}\ge K^{B}\); (2) \(K_{1}<K^{B}\) and \(K_{2}\ge K^{A}\); (3) \( K^{A}>K_{1},K_{2}\ge K^{B}\); and (4) \(K^{A}>K_{1}\ge K^{B}\) and \(K_{2}\ge K^{A}\). Hence, cases (1) and (2) can be collapsed into \(K_{1}<K^{B}\) and \( K_{2}\ge K^{B}\), while cases (3) and (4) can be expressed as \( K^{A}>K_{1}\ge K^{B}\) and \(K_{2}\ge K^{B}\). Finally, these two conditions can be summarized as \(K_{1}<K^{A}\) and \(K_{2}\ge K^{B}\). Similarly, equilibrium outcome \((NG_{1},G_{2})\) can be sustained under four different parameter conditions: (a) \(K^{A}>K_{1}\ge K^{B}\) and \(K_{2}<K^{B}\); (b) \( K^{A}>K_{1},K_{2}\ge K^{B}\); (c) \(K_{1}\ge K^{A}\) and \(K_{2}<K^{B}\); and d) \(K_{1}\ge K^{A}\) and \(K^{A}>K_{2}\ge K^{B}\). Therefore, cases (a) and (b) can be collapsed into \(K^{A}>K_{1}\ge K^{B}\) and \(K_{2}<K^{A}\), whereas cases (c) and (d) can be expressed as \(K_{1}\ge K^{A}\) and \(K_{2}<K^{A}\). Finally, these two conditions can be summarized as \(K_{1}\ge K^{B}\) and \( K_{2}<K^{A}\). \(\square \)

Appendix 2: Convex environmental damage

When the environmental damage is convex in output, i.e., \(ED=d(Q+\alpha X)^{2}\), the welfare comparisons of Sect. 5.4 still hold. The following two figures evaluate the welfare arising in each of the three equilibrium outcomes for similar parameter values as in Figs 8, 9, 10 and 11, i.e., \(d=1/2\). Similarly as in Sect. 5.4, when the green product is completely clean, \( \alpha =0\), social welfare is the highest when both firms develop the green product when goods are relatively differentiated, \(\lambda =0.1\), as depicted in point \(C\); but becomes the highest when only one firm develops the green product if goods are more homogeneous, \(\lambda =0.4\), as illustrated in point \(E\). However, when the green product exhibits a poor environmental performance, \(\alpha =0.8\), social welfare is the highest when no firm develops the green good, both when products are differentiated (as depicted at point \(A\) in the case that \(\lambda =0.1\)) and when they are undifferentiated (as illustrated by point \(D\) in the case that \(\lambda =0.3\), Table 3).

Fig. 10
figure10

Low pollution intensity, \(\alpha =0\)

Fig. 11
figure11

High pollution intensity, \(\alpha =0.8\)

Table 3 Welfare comparisons: low pollution intensity (a) and high pollution intensity (b)

Appendix 3: Welfare gains

The difference in welfare gains \(\left( SW_{G_{1}NG_{2}}-SW_{NG_{1}NG_{2}}\right) -\left( SW_{G_{1}G_{2}}-SW_{G_{1}NG_{2}}\right) \) is

$$\begin{aligned}&\frac{1}{36(1-\lambda ^{2})^{2}}\left( 11-18\alpha d-4\lambda +32d\lambda -15\lambda ^{2}-50d\lambda ^{2}+4\lambda ^{3}+4d\lambda ^{3}-5\lambda ^{4}\right. \nonumber \\&\qquad \left. +\,14d\lambda ^{4}+ c^{2}\left( 4(3+4d)\lambda ^{3}-36-18\alpha d(-2+\lambda )^{2}-12\lambda +(11-16d)\lambda ^{2}\right. \right. \nonumber \\&\qquad \left. \left. +\,(14d-11)\lambda ^{4}\right) -4z -20\lambda z-48d\lambda z+4\lambda ^{2}z+36d\lambda ^{2}z+2\lambda ^{3}z+12d\lambda ^{3}z \right. \nonumber \\&\qquad \left. -\,16z^{2}+16d\lambda z^{2}+7\lambda ^{2}z^{2}-2d\lambda ^{2}z^{2}- 2c \left( 18+18\alpha d(\lambda -2)+2(7d-4)\lambda ^{4} \right. \right. \nonumber \\&\qquad \left. \left. -\,36z+\lambda \left( 8z+8d(1-z)-35\right) +2\lambda ^{2}\left( 9z-5+8d(z-2)\right) \right. \right. \nonumber \\&\qquad \left. \left. +\,\lambda ^{3}\left( 17-8z+2d(5+3z)\right) \right) \right) \end{aligned}$$

Proof of Lemma 1

No firm develops a green good. When no firm produces green goods, every firm \(i\)’s production level of brown goods under duopoly is \(q_{i}^{BB}=\frac{1-c}{3}\), for all \(i=\{1,2\}\), entailing equilibrium profits of \(\pi _{i}^{B}(NG_{1},NG_{2})=\frac{(1-c)^{2}}{9}\).

Only firm i develops a green good. In this case, firm \(i\) and \(j\)’s profit maximizing output of brown goods are

$$\begin{aligned} q_{i}^{GB}=\frac{\left[ 2+(\lambda -3)\lambda +3\lambda z\right] -\left( 2+\lambda ^{2}\right) c}{6(1-\lambda ^{2})}\,\quad \text {and }\,\quad q_{j}^{GB}=\frac{ 1-c}{3} \end{aligned}$$

while firm \(i\)’s production of green good is

$$\begin{aligned} x_{i}^{GB}=\frac{1-c-(1-\lambda )c}{2(1-\lambda ^{2})}, \end{aligned}$$

since best response functions are given by \(q_{i}\left( q_{j},x_{i}\right) = \frac{1-c-2\lambda x_{i}}{2}-\frac{q_{j}}{2}\), \(q_{j}\left( q_{i},x_{i}\right) =\frac{1-z-2x_{i}}{\lambda }-2q_{i}\), and \(x_{i}\left( q_{i},q_{j}\right) =\frac{1-z-\lambda \left( 2q_{i}+q_{j}\right) }{2}\), respectively. Hence, firm \(i\)’s equilibrium profits from the brown good are

$$\begin{aligned} \pi _{i}^{B}(G_{i},NG_{j})=\frac{\left[ 1-c\right] \left[ (\lambda -2)(\lambda -1)+3\lambda z-(2+\lambda ^{2})c\right] }{18(1-\lambda ^{2})}, \end{aligned}$$

firm \(i\)’s profits from the green product are

$$\begin{aligned} \pi _{1}^{G}(G_{i},NG_{j})=\frac{\left[ 1-z-(1-c)\lambda \right] \left[ 3-\lambda -3z+\lambda c\right] }{12(1-\lambda ^{2})}, \end{aligned}$$

which is positive for all \(\lambda \le \frac{1-z}{1-c}\equiv \overline{ \lambda }\), and firm \(j\)’s profits are \(\pi _{j}^{B}(G_{i},NG_{j})=\frac{ (1-c)^{2}}{9}\). (The equilibrium profits in which only firm \(j\) invests are analogous.) The output difference \(q_{i}^{GB}-x_{i}^{GB}\) is positive and increasing in \(\lambda \) since

$$\begin{aligned} \frac{\partial \left( q_{i}^{GB}-x_{i}^{GB}\right) }{\partial \lambda }= \frac{z-c}{2(1-\lambda )^{2}} \end{aligned}$$

is positive given that \(z>c\).

Both firms develop a green good. In this case, firm \(i\)’s profit maximizing outputs from producing brown and green goods are, respectively,

$$\begin{aligned} q_{i}^{GG}=\frac{1-c-\lambda (1-z)}{3(1-\lambda ^{2})}\quad \text { and }\quad x_{i}^{GG}=\frac{1-z-\lambda (1-c)}{3(1-\lambda ^{2})} \end{aligned}$$

entailing equilibrium profits of

$$\begin{aligned} \pi _{i}^{B}(G_{i},G_{j})=\frac{1-c}{3}q_{i}^{GG}=\frac{\left[ 1-c\right] \left[ 1-c-(1-\lambda )c\right] }{9(1-\lambda ^{2})}, \end{aligned}$$

when producing brown goods, and

$$\begin{aligned} \pi _{i}^{G}(G_{i},G_{j})=\frac{1-z}{3}q_{i}^{GG}=\frac{(1-z)\left[ 1-z-(1-\lambda )c\right] }{9(1-\lambda ^{2})} \end{aligned}$$

when producing green goods. The output difference \(q_{i}^{GG}-x_{i}^{GG}\) is positive and increasing in \(\lambda \) since

$$\begin{aligned} \frac{\partial \left( q_{i}^{GG}-x_{i}^{GG}\right) }{\partial \lambda }= \frac{z-c}{3(1-\lambda )} \end{aligned}$$

is positive given that \(z>c\).

Proof of Lemma 2

Let us analyze the production decision of the second mover (firm 2). If firm 1 does not develop green goods, then firm 2 responds producing them if its profits from brown goods and its profits from green goods (net of investment costs) exceed those from staying out,

figurea

Note that the difference \(EGB_{2}(NG_{1})\equiv \pi _{2}^{B}(NG_{1},G_{2})-\pi _{2}^{B}(NG_{1},NG_{2})\) captures the effect that the development of green goods produces on sales of the brown good \((EGB)\). Hence, condition \((C_{2}^{A})\) can be compactly expressed as

$$\begin{aligned} \pi _{2}^{G}(NG_{1},G_{2})+EGB_{2}(NG_{1})\equiv K^{A}>K_{2} \end{aligned}$$

where, in particular, cutoff \(K^{A}=\frac{(1-z-(1-c)\lambda )^{2}}{ 4(1-\lambda ^{2})}\). Note that when products are completely differentiated, \( \lambda =0\), this cutoff coincides with the profits that firm 2 obtains from the green product, i.e., \(K^{A}=\frac{(1-z)^{2}}{4}\).

If, instead, firm 1 enters, firm 2 responds producing green goods as well if

figureb

which can similarly be expressed as \(\pi _{2}^{G}(G_{1},G_{2})+EGB_{2}(G_{1})\equiv K^{B}>K_{2}\), where \(EGB_{2}(G_{1})\equiv \pi _{2}^{B}(G_{1},G_{2})-\pi _{2}^{B}(G_{1},NG_{2})\). Cutoff \(K^{B}=\frac{(1-z-(1-c)\lambda )^{2}}{9(1-\lambda ^{2})}\) and, when \(\lambda =0\), it coincides with the profits that firm 2 obtains from the green product, i.e., \(K^{B}=\frac{(1-z)^{2}}{9}\).

Proof of Proposition 1

In the case that firm 2 responds producing green goods regardless of firm 1’s action, i.e., region I of Fig. 1, i.e., \(K_{2}<K^{B}\), firm 1 develops green goods if

figurec

or \(K^{B}>K_{1}\). Therefore, if \(K^{B}>K_{1}\) both firms produce green goods, \((G_{1},G_{2})\). However, if \(K^{B}\le K_{1}\) only firm 2 produces green goods, \((NG_{1},G_{2})\), since its investments costs are low while those of firm 1 are relatively high.

If firm 2 responds developing green goods only after observing that firm 1 does not produce them, i.e., region II, i.e., \(K^{A}>K_{2}\ge K^{B}\), firm 1 chooses to develop green goods if

figured

or \(K^{A}>K_{1}\). Hence, when \(K^{A}>K_{1}\) the subgame perfect equilibrium (SPNE) predicts that firm 1’s production decision deters firm 2 from producing green goods, \((G_{1},NG_{2})\), since firm 1’s investment costs are relatively low, while its opponent’s are high. In contrast, when \(K^{A}\le K_{1}\) the opposite strategy profile can be sustained, in which firm 1 does not produce them and, hence, firm 2 responds developing green goods, i.e., \( (NG_{1},G_{2})\).

Finally, if firm 2 responds not producing green goods regardless of firm 1’s production decision, region III, i.e., \(K_{2}\ge K^{A}\), firm 1 chooses to produce green goods if

figuree

or \(K^{A}>K_{1}\). Hence, when \(K^{A}>K_{1}\) the SPNE predicts that only firm 1 produces green goods, \((G_{1},NG_{2})\); whereas when \(K^{A}\le K_{1}\) no firm develops green products, i.e., \((NG_{1},NG_{2})\). Finally, note that the case in which condition \(C_{2}^{B}\) holds but \(C_{2}^{A}\) does not, cannot be sustained since \(\lambda <\overline{\lambda }\), which implies that cutoff \(K^{A}\) lies above \(K^{B}\). Therefore, equilibrium \((G_{1},NG_{2})\) can be sustained when \(K_{1}<K^{A}\) and \(K_{2}\ge K^{A}\), and when \( K_{1}<K^{A}\) and \(K^{A}>K_{2}\ge K^{B}\). We can, hence, collapse both cases as \(K_{1}<K^{A}\) and \(K_{2}\ge K^{B}\).

Proof of Proposition 2

Both firms develop a green good. The social welfare when both firms produce green goods, \(SW_{G_{1}G_{2}}\), is defined as

$$\begin{aligned} SW_{G_{1}G_{2}}= & {} CS(Q)+CS(X)+\pi _{1}^{B}(G_{1},G_{2})+\pi _{1}^{G}(G_{1},G_{2})-K_{1} \\&\quad +\,\pi _{2}^{B}(G_{1},G_{2})+\pi _{2}^{G}(G_{1},G_{2})-K_{2}-d\left( Q^{2}+\alpha X^{2}\right) . \end{aligned}$$

where \(CS(Q)=\frac{2(1-c)(c+\lambda (1-z)-1)}{9(\lambda ^{2}-1)}\), \(CS(X)= \frac{2(1-z)\left[ (1-c)\lambda +z-1\right] }{9(\lambda ^{2}-1)}\), \(\pi _{i}^{B}(G_{1},G_{2})=\frac{(1-c)(c+\lambda (1-z)-1)}{9(\lambda ^{2}-1)}\) for all firm \(i=\{1,2\}\), \(\pi _{i}^{G}(G_{1},G_{2})=\frac{(1-z)\left[ (1-c)\lambda +z-1\right] }{9(\lambda ^{2}-1)}\), \(Q=\frac{2(1-c-\lambda (1-z)) }{3(1-\lambda ^{2})}\), and \(X=\frac{2(1-z-\lambda (1-c))}{3(1-\lambda ^{2})}\).

Only firm 1 develops a green good. The equilibrium in which only firm 1 produces green goods yields a social welfare,

$$\begin{aligned} SW_{G_{1}NG_{2}}= & {} CS(Q)+CS(x_{1})+\pi _{1}^{B}(G_{1},NG_{2})+\pi \quad _{1}^{G}(G_{1},NG_{2})\nonumber \\&\quad -\,K_{1}+\pi _{2}^{B}(G_{1},NG_{2})-d\left( Q^{2}+\alpha x_{1}^{2}\right) \end{aligned}$$

where \(CS(Q)=\frac{[4+c(\lambda ^{2}-4)-\lambda A][2cB+4+\lambda (3z-\lambda )]}{72(1-\lambda ^{2})^{2}}\) where \(A\equiv 3(1-z)+\lambda \) and \(B\equiv (1+2\lambda )(\lambda -2)\). In addition, \(CS(X)=\frac{[1+c(\lambda -2)][3+cC-\lambda (\lambda A-4)]}{24(1-\lambda ^{2})^{2}}\) where \(C\equiv \lambda ^{3}-\lambda -6\). Profits are \(\pi _{1}^{B}(G_{1},NG_{2})=\frac{ D[c(2+\lambda ^{2})-2-\lambda (\lambda -3(1-z))]}{36(1-\lambda ^{2})^{2}}\), where \(D\equiv 2-5\lambda ^{2}+2c[\lambda (3+\lambda )-1]-3\lambda z\), and \( \pi _{2}^{B}(G_{1},NG_{2})=\frac{(c-1)D}{18(\lambda ^{2}-1)}\) from the brown product, and

$$\begin{aligned} \pi _{1}^{G}(G_{1},NG_{2})=\frac{[1+c(\lambda -2)][3+cC-6z+\lambda (\lambda (\lambda -3(1-z))-4)]}{12(1-\lambda ^{2})^{2}} \end{aligned}$$

from the green product for the only firm that develops such a good (firm 1). Finally, aggregate output levels are \(Q=\frac{2+(\lambda -3)\lambda +3\lambda z-(2+\lambda ^{2})c}{6(1-\lambda ^{2})}+\frac{1-c}{3}\), and \(X= \frac{1-c-(1-\lambda )c}{2(1-\lambda ^{2})}\).

Only firm 2 develops a green good. In this case, social welfare is given by

$$\begin{aligned} SW_{NG_{1}G_{2}}= & {} CS(Q)+CS(x_{2})+\pi _{1}^{B}(NG_{1},G_{2})+\pi _{2}^{B}(NG_{1},G_{2})+\pi _{2}^{G}(NG_{1},G_{2})\nonumber \\&\quad -\,K_{2}-d\left( Q^{2}+\alpha x_{2}^{2}\right) \end{aligned}$$

and hence \(SW_{NG_{1}G_{2}}=SW_{G_{1}NG_{2}}\).

No firm develops a green good. Finally, when no firm produces green goods social welfare is just given by

$$\begin{aligned} SW_{NG_{1}NG_{2}}=CS(Q)+\pi _{1}^{B}(NG_{1},NG_{2})+\pi _{2}^{B}(NG_{1},NG_{2})-dQ^{2} \end{aligned}$$

where \(CS(Q)=\frac{2(1-c)^{2}}{9}\), \(\pi _{i}^{B}(NG_{1},NG_{2})=\frac{ (1-c)^{2}}{9}\) for all firm \(i=\{1,2\}\), and aggregate output is \(Q=\frac{ 2(1-c)}{3}\).

Welfare comparison. Comparing \(SW_{G_{1}G_{2}}\) and \(SW_{G_{1}NG_{2}}\), we obtain that \(SW_{G_{1}G_{2}}>SW_{G_{1}NG_{2}}\) for all \(K<K_{a}\), where

$$\begin{aligned} K_{a}\!\equiv \! \frac{1}{72A^{2}}\left[ \begin{array}{c} \chi +c^{2}(36-\lambda (-A(5\lambda +12))+2d(36+\lambda (\lambda -11)(\lambda +4))))+\phi \\ +z^{2}(32-23\lambda ^{2})-16\alpha dA(4\lambda +4z-1)+\eta \end{array} \right] \end{aligned}$$

where \(A=(\lambda -1)(\lambda +1)\), \(\chi =(5+\lambda (-28-\lambda (\lambda +3)(11\lambda +5)+12d(\lambda -1)^{2}(1+\lambda ))\), \(\phi =-28z+2\lambda z(26+6dA+\lambda (14-17\lambda ))\) and \(\eta =2c(18-36z+-16\alpha dA(6+\lambda )+\lambda (19-8z+\lambda (8\lambda ^{2}+\lambda +8\lambda z+18z-26-6d\lambda ^{2}+6d)))\).

Similarly, comparing \(SW_{G_{1}NG_{2}}\) and \(SW_{NG_{1}NG_{2}}\), we obtain that \(SW_{G_{1}NG_{2}}>SW_{NG_{1}NG_{2}}\) for all \(K<K_{b}\), where

$$\begin{aligned} K_{b}\equiv & {} \frac{1}{24A^{2}}\left[ 9+\Phi -12z+\lambda (-5\lambda +4(z-3)+12dA(\lambda +z-1)\right. \nonumber \\&\quad \left. -\,\lambda \left( 7\lambda +3(-4+z)(\lambda +z)\right) +\kappa \right] \end{aligned}$$

where \(\Phi =c^{2}(-12+\lambda (4-9\lambda )A)+12\alpha dA\) and \(\kappa =2c(-6+6\alpha d(\lambda -2)A+12z+\lambda \) \((17-8z+\lambda (-2-6dA-6z+\lambda (8z+8\lambda -11))))\).

Finally, comparing \(SW_{G_{1}G_{2}}\) and \(SW_{NG_{1}NG_{2}}\), we obtain that \(SW_{G_{1}G_{2}}{>}SW_{NG_{1}NG_{2}}\) for all \(K<K_{c}\), where

$$\begin{aligned} K_{c}\equiv \frac{2(1+B\lambda -z)(3d(\alpha -\lambda )+2(z-\lambda B-1))}{9A} \end{aligned}$$

where \(B=(c-1)\).

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Espínola-Arredondo, A., Muñoz-García, F. An excessive development of green products?. Econ Gov 17, 101–129 (2016). https://doi.org/10.1007/s10101-015-0161-1

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Keywords

  • Excessive entry
  • Product differentiation
  • Pollution intensity
  • Environmental damage

JEL Classification

  • L12
  • D82
  • Q20
  • D62