Product greening and pricing strategies of firms under green sensitive consumer demand and environmental regulations

Abstract

Manufacturing firms globally face an increasing consumer demand for environmentally friendly products, along with regulatory changes. These entail significant costs for firms who are unsure about the benefits of greening. In this paper, we aim to answer questions on the economics of greening. We explore various problem settings where we study the impact of product greening costs and Government regulations on a single firm and duopoly, in a green sensitive consumer market. We study firm strategy to derive optimal values of product greening level, price and profits. In addition, we also analyze the impact of Government regulations on firms and society. We find that regulations serve the requisite objective of forcing firms to provide higher greening levels. However, under certain conditions they may have a limited effect. We find that under higher Government penalty or subsidy, a firm with a lower greening cost will offer higher product greening level than its competitor, in turn benefitting in a green consumer market. Under duopoly settings, we find that the relative greening level difference between the competing firms is increasing in the cost of greening difference. Further, the relative greening level difference between the firms is increasing in Government taxation or subsidy as well. We discuss various conditions under which firms would incur Government taxation or subsidy. The key contribution of our work lies in modeling Government regulations and decision making under demand expansion effects while analyzing the resulting decisions of product greening and pricing.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Notes

  1. 1.

    https://www.reuters.com/article/us-india-autos-electric-vehicles/indias-auto-industry-gears-up-for-governments-electric-vehicles-push-idUSKCN1BM02X.

  2. 2.

    http://money.cnn.com/2016/02/17/news/economy/india-new-delhi-air-pollution/index.html?iid=EL.

  3. 3.

    https://economictimes.indiatimes.com/industry/auto/news/passenger-vehicle/cars/ngt-ban-on-diesel-cars-what-you-should-do-if-you-live-in-delhi-ncr/articleshow/60515631.cms.

  4. 4.

    Note however, that recently Toyota has been involved in several product recalls raising questions on the quality standards maintained by the company (www.economist.com/blogs/schumpeter/2014/04/toyota).

  5. 5.

    Note that the case of differential penalty or subsidy can be considered as a simple extension of the parsimonious model here.

  6. 6.

    The FoC’s of optimal price and production quantity w.r.t K are not strictly increasing or decreasing in K, Refer the “Appendix”.

  7. 7.

    The assumption applies to industries like automobiles in which players compete in prices and greening efforts as discussed in the Introduction section of the paper. An interesting extension of the model could be on sequential decision making considering differing market power between competing firms. We thank the reviewer for this suggestion.

  8. 8.

    We thank the reviewers for this suggestion.

References

  1. Atasu, A., Guide, V. D. R., & Wassenhove, L. N. (2008). Product reuse economics in closed-loop supply chain research. Production and Operations Management, 17(5), 483–496.

    Article  Google Scholar 

  2. Banker, R. D., Khosla, I., & Sinha, K. K. (1998). Quality and competition. Management Science, 44(9), 1179–1192.

    Article  Google Scholar 

  3. Barnett, A. J. (1980). The Pigouvian tax rule under monopoly. American Economic Review, 70, 1037–1041.

    Google Scholar 

  4. Bhaskaran, S. R., & Krishnan, V. (2009). Effort, revenue, and cost sharing mechanisms for collaborative new product development. Management Science, 55(7), 1152–1169.

    Article  Google Scholar 

  5. Bonanno, G. (1986). Vertical differentiation with Cournot competition. Economic Notes, 15, 68–91.

    Google Scholar 

  6. Champsaur, P., & Rochet, J. C. (1989). Multi-product duopolists. Econometrica, 57, 533–557.

    Article  Google Scholar 

  7. Chen, C. (2001). Design for the environment: A quality-based model for green product development. Management Science, 47(2), 250–263.

    Article  Google Scholar 

  8. Choi, C. J., & Shin, H. S. (1992). A comment on a model of vertical product differentiation. Journal of Industrial Economics, 40, 229–232.

    Article  Google Scholar 

  9. Drozdenko, R., Jensen, M., & Coelho, D. (2011). Pricing of green products: Premiums paid, consumer characteristics and incentives. International Journal of Business, Marketing, and Decision Sciences, 4(1), 106–116.

    Google Scholar 

  10. Geyer, R., Wassenhove, L. N. V., & Atasu, A. (2007). The economics of remanufacturing under limited component durability and finite product life cycles. Management Science, 53(1), 88–100.

    Article  Google Scholar 

  11. Ghosh, D., & Shah, J. (2012). A comparative analysis of greening policies across supply chain structures. International Journal of Production Economics, 135(2), 568–583.

    Article  Google Scholar 

  12. Gouda, S. K., Jonnalagedda, S., & Saranga, H. (2015). Design for the environment: Impact of regulatory policies on product development. European Journal of Operational Research, 248(2), 558–570.

    Article  Google Scholar 

  13. KPMG. (2016). Global automotive executive survey. Retrieved March 11, 2016 from https://home.kpmg.com/xx/en/home/insights/2015/12/kpmg-global-automotive-executive-survey-2016.html.

  14. Laroche, M., Bergeron, J., & Barbaro-Forleo, G. (2001). Targeting consumers who are willing to pay more for environmentally friendly products. Journal of consumer marketing, 18(6), 503–520.

    Article  Google Scholar 

  15. Letmathe, P., & Balakrishnan, N. (2005). Environmental considerations on the optimal product mix. European Journal of Operational Research, 167(2), 398–412.

    Article  Google Scholar 

  16. Mitra, S., & Webster, S. (2008). Competition in remanufacturing and effects of Government subsidies. International Journal of Production Economics, 111(2), 287–298.

    Article  Google Scholar 

  17. Motta, M. (1993). Endogenous quality choice: Price vs. quantity competition. Journal of Industrial Economics, 41, 113–131.

    Article  Google Scholar 

  18. Nidumolu, R., Prahalad, C. K., & Rangaswami, M. R. (2009). Why sustainability is now the key driver of innovation. Harvard Business Review, 87(9), 56–64.

    Google Scholar 

  19. Parsons, R. (2005). Rentabilite comparee des fermes laitie’res biologiques du Nord-Est. Mimeo, University of Vermont.

  20. PricewaterhouseCoopers, L. L. P. (2010). Green products: Using sustainable attributes to drive growth and value. Retrieved February 13, 2016 from http://www.pwc.com/us/en/corporate-sustainability-climate-change/assets/green-products-paper.pdf.

  21. Ren, J., Bian, Y., Xu, X., & He, P. (2015). Allocation of product-related carbon emission abatement target in a make-to-order supply chain. Computers & Industrial Engineering, 80, 181–194.

    Article  Google Scholar 

  22. Savaskan, C., Bhattacharya, S., & Van Wassenhove, L. N. (2004). Closed-loop supply chain models with product remanufacturing. Management Science, 50(2), 239–252.

    Article  Google Scholar 

  23. Savaskan, C., & Van Wassenhove, L. N. (2006). Reverse channel design: The case of competing retailers. Management Science, 52(1), 1–14.

    Article  Google Scholar 

  24. Schlegelmilch, B. B., Bohlen, G. M., & Diamantopoulos, A. (1996). The link between green purchasing decisions and measures of environmental consciousness. European Journal of Marketing, 30(5), 35–55.

    Article  Google Scholar 

  25. Spence, Michael. (1975). Monopoly, quality, and regulation. Bell Journal of Economics, 6, 417–429.

    Article  Google Scholar 

  26. Swami, S., & Shah, Janat. (2012). Channel coordination in green supply chain management. Journal of the Operational Research Society, 64(3), 336–351.

    Article  Google Scholar 

  27. Vives, X. (1985). On the efficiency of Bertrand and Cournot equilibria with product differentiation. Journal of Economic Theory, 36(1), 166–175.

    Article  Google Scholar 

  28. Walley, N., & Whitehead, B. (1994). It’s not easy being green. Harvard Business Review, 72, 46–52.

    Google Scholar 

  29. Zhang, J. J., Nie, T. F., & Du, S. F. (2011). Optimal emission-dependent production policy with stochastic demand. International Journal of Society Systems Science, 3(1–2), 21–39.

    Article  Google Scholar 

  30. Zhang, B., & Xu, L. (2013). Multi-item production planning with carbon cap and trade mechanism. International Journal of Production Economics, 144(1), 118–127.

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Debabrata Ghosh.

Appendix

Appendix

Greening under demand expansion effects only

The demand faced by the firm is given by

$$\begin{aligned} q = a-b p + \alpha \theta \quad a> bp, \alpha , b >0 \end{aligned}$$

Objective of the firm is

$$\begin{aligned} Max_{p, \theta } \Pi _{G} =(p-c) (a -b p + \alpha \theta )- I \theta ^2 \end{aligned}$$

The first order conditions w.r.t to p and \(\theta \) are given by

$$\begin{aligned} \displaystyle \frac{\partial }{\partial p} \Pi _{G}&= -2bp + a + bc + \alpha \theta \\ \displaystyle \frac{\partial }{\partial \theta } \Pi _{G}&=\alpha (p-c)-2 I \theta \end{aligned}$$

The second order conditions w.r.t to p and \(\theta \) are given by

$$\begin{aligned} \displaystyle \frac{\partial ^2}{\partial p^2} \Pi _{G}&= -2b< 0\\ \displaystyle \frac{\partial ^2}{\partial \theta ^2} \Pi _{G}&= -2 I < 0 \end{aligned}$$

The cross partial derivative is given by

$$\begin{aligned} \displaystyle \frac{\partial ^2}{\partial p \partial \theta } \Pi _{G} = \alpha \end{aligned}$$

So the determinant is \(4Ib- \alpha ^2\). For \(4Ib- \alpha ^2\) > 0, the Hessian H is negative definite. Thus the firm’s profit function is strictly concave in p and \(\theta \). Thus, solving the FoC’s simultaneously, we get,

$$\begin{aligned} \theta _{G}&= \dfrac{\alpha (a-bc)}{4Ib-\alpha ^2} \\ p_{G}&= \dfrac{2Ia + c(2Ib-\alpha ^2)}{4Ib-\alpha ^2}\\&= \left[ \dfrac{2I(a-bc)}{4Ib-\alpha ^2} + c \right] \end{aligned}$$

From the above values we derive the profit of the firm as,

$$\begin{aligned} \Pi _{G}= \dfrac{I(a-bc)^2}{4Ib - \alpha ^2} \end{aligned}$$

Greening under demand expansion effects and government regulation

The demand faced by the firm is given by

$$\begin{aligned} q = a-b p + \alpha \theta \quad a> bp, \alpha , b >0 \end{aligned}$$

Objective of the firm is

$$\begin{aligned} Max_{p, \theta } \Pi _{(P|S)} =(p-c) (a -b p + \alpha \theta )- K (\theta _0-\theta ) (a -b p + \alpha \theta )- I \theta ^2 \end{aligned}$$

The first order conditions w.r.t to p and \(\theta \) are given by

$$\begin{aligned} \displaystyle \frac{\partial }{\partial p} \Pi _{(P|S)}&= -2bp + a + bc + \alpha \theta + Kb(\theta _0-\theta ) \\ \displaystyle \frac{\partial }{\partial \theta } \Pi _{(P|S)}&=K (a+\alpha \theta -b p)+\alpha (p-c)-\alpha K (\theta _0-\theta )-2 I \theta \end{aligned}$$

The second order conditions w.r.t to p and \(\theta \) are given by

$$\begin{aligned} \displaystyle \frac{\partial ^2}{\partial p^2} \Pi _{(P|S)}&= -2b< 0\\ \displaystyle \frac{\partial ^2}{\partial \theta ^2} \Pi _{(P|S)}&= 2 K \alpha -2 I < 0, \quad assuming \quad I > K \alpha \end{aligned}$$

The cross partial derivative is given by

$$\begin{aligned} \displaystyle \frac{\partial ^2}{\partial p \partial \theta } \Pi _{(P|S)} = \alpha - Kb \end{aligned}$$

So the determinant is \(4Ib- (\alpha + Kb)^2\). For \(4Ib- (\alpha +Kb)^2\) > 0, the Hessian H is negative definite. Thus the firm’s profit function is concave in p and \(\theta \). Thus, solving the FoC’s simultaneously, we get,

$$\begin{aligned} \theta _{(P|S)} = \dfrac{(\alpha + Kb)(a - b(c+K \theta _0))}{4Ib -(\alpha +Kb)^2} {\left\{ \begin{array}{ll}< \theta _0 \qquad if \qquad I > \dfrac{(\alpha +Kb)(a-bc+\alpha \theta _0)}{4b \theta _0} \\ \ge \theta _0 \qquad if \qquad \dfrac{(\alpha + Kb)^2}{4b} < I \le \dfrac{(\alpha +Kb)(a-bc+\alpha \theta _0)}{4b \theta _0}\\ \end{array}\right. } \end{aligned}$$

Proof of Proposition 2

$$\begin{aligned} \theta _{(P|S)}&= \dfrac{(\alpha + Kb)(a - b(c+K \theta _0))}{4Ib -(\alpha +Kb)^2} \\ \theta _{G}&= \dfrac{\alpha (a-bc)}{4Ib-\alpha ^2} \end{aligned}$$

We derive,

$$\begin{aligned} \Delta _{\theta }&= \theta _{(P|S)} - \theta _{G} \\&= \dfrac{ b K \left[ (a-b c) (\alpha ^2+ \alpha b K + 4 b I )+ \theta _0 (\alpha ^2- 4 b I ) (\alpha + b K) \right] }{(\alpha ^2- 4 b I )( (\alpha +b K)^2- 4 b I)} \\&= \left[ \dfrac{(\alpha + Kb)\alpha (a-bc+\alpha \theta _0) + 4Ib(a-bc-(\alpha +Kb)\theta _0) }{( 4Ib - \alpha ^2)( 4Ib - (\alpha +b K)^2)} \right] \ge 0 \\&when \quad a \ge \left( bc+(\alpha +Kb)\theta _0\right) \\ \end{aligned}$$

i.e. market demand is sufficiently large and \(I > \dfrac{(\alpha + Kb)^2}{4b}.\) The bound on I maintains the non-negativity of the optimal greening values. Thus, \( \theta _{(P|S)} \ge \theta _{G} \) Therefore, optimal product greening value under penalty or reward scheme is greater than optimal product greening value without penalty or reward.

Additional result of \(\theta _{(P|S)}\)

Deriving first order conditions of \(\theta _{(P|S)}\) with respect to K,

$$\begin{aligned} \dfrac{\partial \theta _{(P|S)}}{\partial K} = \frac{b \left( (\alpha +b K)^2 (a + \alpha \theta _0-b c) + 4 I b (a - \alpha \theta _0 - b (c + 2 K \theta _0))\right) }{\left( (\alpha +b K)^2 - 4 b I\right) ^2} > 0 \end{aligned}$$

Thus, \(\theta _{(P|S)}\) is increasing in K.

Proof of Proposition 3

We derive,

$$\begin{aligned} \Delta _{p}&= price_{(P|S)} - price_{G} \\&= \dfrac{ K \left[ \alpha ^2 (\alpha + Kb) (a-b c + \alpha \theta _0) -2Ib \left\{ 3 \alpha ^2 \theta _0 + Kb(a-bc+2 \alpha \theta _0)\right\} + 8 I^2 b^2 \theta _0 \right] }{\left[ (4I - \alpha ^2)(4 I b - (\alpha +b K)^2)\right] } \end{aligned}$$

Case I: \( \Delta _{p} = 0 \quad \) when \(K =0\) which is the case of no reward or penalization.

Case II: When \( K \ne 0 \),

$$\begin{aligned} \Delta _{p}&> 0 \quad when \quad K < \left[ \dfrac{2Ib \theta _0 (4Ib-3 \alpha ^2) + \alpha ^3 (a-bc + \alpha \theta _0)}{b \left[ (2Ib-\alpha ^2)(a-bc)+\alpha \theta _0 (4Ib - \alpha ^2) \right] }\right] \\ \Delta _{p}&\le 0 \quad when \quad K \ge \left[ \dfrac{2Ib \theta _0 (4Ib-3 \alpha ^2) + \alpha ^3 (a-bc + \alpha \theta _0)}{b \left[ (2Ib-\alpha ^2)(a-bc)+\alpha \theta _0 (4Ib - \alpha ^2) \right] }\right] \end{aligned}$$

Additional results of\(p_{(P|S)}\)and\(q_{(P|S)}\)

Deriving first order conditions of \(p_{(P|S)}\) with respect to K,

$$\begin{aligned} \dfrac{\partial p_{(P|S)}}{\partial K}= \frac{2 Ib \left( -3 \alpha ^2 \theta _0 + K b (-2 a +2 b c - 4 \alpha \theta _0 + b K \theta _0) \right) +\alpha (\alpha +b K)^2 (a - b c + \alpha \theta _0)+8 b^2 \theta _0 I^2}{\left( (\alpha +b K)^2-4 b I\right) ^2} \end{aligned}$$

It can be observed that the first order condition is quadratic in K, which on equating to zero and solving further gives,

$$\begin{aligned} \dfrac{\partial p_{(P|S)}}{\partial K}&= 0 \quad when,\\ K&= \frac{b \left( 2 I b (a - b c + 2 \alpha \theta _0) - \alpha ^2 (a - b c + \alpha \theta _0) \right) \pm 2 \sqrt{I b^3 \left( I b - \alpha ^2\right) \left( (a+ \alpha \theta _0-b c)^2 - 4 b \theta _0^2 I \right) }}{b^2 (\alpha (a+ \alpha \theta _0-b c)+2 b \theta _0 I)} \end{aligned}$$

Since there is a change in slope at the above values of K, \(\dfrac{\partial p_{(P|S)}}{\partial K}\) is not strictly increasing or decreasing in K.

Deriving first order conditions of \(q_{(P|S)}\) with respect to K,

$$\begin{aligned} \dfrac{\partial q_{(P|S)}}{\partial K}= \frac{2 b^2 I ((\alpha +b K) (2 a+\alpha \theta _0-b (2 c + K \theta _0)) - 4 b \theta _0 I)}{\left( (\alpha + b K)^2 - 4 b I \right) ^2} \end{aligned}$$

Solving the quadratic equation in K, gives,

$$\begin{aligned} \dfrac{\partial q_{(P|S)}}{\partial K}&= 0 \quad when,\\ K&= \frac{b (a-b c) \pm \sqrt{b^2 \left( (a + \alpha \theta _0 - b c)^2 - 4 b \theta _0^2 I \right) }}{b^2 \theta _0} \end{aligned}$$

Since there is a change in slope at the above values of K, \(\dfrac{\partial q_{(P|S)}}{\partial K}\) is not strictly increasing or decreasing in K.

Consumer and social surplus

$$\begin{aligned} CS&= \int _0^{q_{(P|S)}} P(x,\theta _{(P|S)})\, dx\ - p_{(P|S)}q_{(P|S)} \\&= \dfrac{2I^2b[a-b(c+K\theta _0)]^2}{(4Ib-(\alpha +Kb)^2)^2} \end{aligned}$$

where \( P(x,\theta _{(P|S)})\) denotes the inverse demand function and is given by \(\dfrac{(a-x+\alpha \theta _{(P|S)})}{b}\) and x denotes quantity. Substituting the values of \(\theta _{(P|S)}\), \(q_{(P|S)}\) and \(p_{(P|S)}\) from the single firm’s decisions under Government penalty, we obtain consumer surplus as

$$\begin{aligned} CS&= \dfrac{2Ib(a-b(c+K\theta _0))[a(3I-K(\alpha +Kb))+(c+K\theta _0)(b(I-\alpha K)-\alpha ^2)]}{(4Ib-(\alpha +Kb)^2)^2} \\&\quad -[2Ib(a-b(c+K\theta _0))] [\dfrac{2I(a+b(c+K\theta _0))- (aK + \alpha (c+K\theta _0))(\alpha +Kb)]}{{(4Ib-(\alpha +Kb)^2)^2}} \\&= \dfrac{2I^2b[a-b(c+K\theta _0)]^2}{(4Ib-(\alpha +Kb)^2)^2}\\ \max _{q,\theta } SS&= \int _0^q P(x,\theta )\, dx\ - C(q,\theta ) \\&= \int _0^q P(x,\theta )\, dx\ - cq - I\theta ^2 - E(\theta _0-\theta )q \\&= \frac{aq+\alpha \theta q - q^2/2}{b} - cq - I\theta ^2 - E(\theta _0-\theta )q \end{aligned}$$

The first order conditions are

$$\begin{aligned} \dfrac{\partial SS}{\partial q}&= (a+\alpha \theta -q)/b - c - E(\theta _0-\theta ) \\ \dfrac{\partial SS}{\partial \theta }&=\dfrac{\alpha q}{b} - 2I\theta + Eq \end{aligned}$$

The second order conditions are

$$\begin{aligned} \dfrac{\partial ^2 SS}{\partial q^2}&= -\dfrac{1}{b}\\ \dfrac{\partial ^2 SS}{\partial \theta ^2}&= -2I \\ \dfrac{\partial ^2 SS}{\partial q \partial \theta }&= \dfrac{\alpha }{b} + E \end{aligned}$$

The Hessian is positive for \( I > \dfrac{b}{2}(\dfrac{\alpha }{b} + E)^2 \). Thus, equating the first order conditions to zero and solving for the socially optimal \(\theta \), quantity and price gives

$$\begin{aligned} \theta _{SS}&= \dfrac{(\alpha +bE)(a-b(c+E\theta _0))}{2Ib-(\alpha + bE)^2} \\ q_{SS}&= \dfrac{(a-b(c+E\theta _0))2Ib}{2Ib-(\alpha +bE)^2}\\ p_{SS}&= \dfrac{aE(\alpha +bE)+(c+E\theta _0)(2Ib-\alpha (\alpha +bE))}{2Ib-(\alpha +bE)^2} \end{aligned}$$

The case of a duopoly

We employ backward induction method to solve the second problem. We first find out the equilibrium prices given greening levels \(\theta _i, \theta _j\) when penalty is levied. We derive,

$$\begin{aligned} \Pi _i(\theta _i,\theta _j) = (p_i - c-K(\theta _0 - \theta _i))(a-bp_i+ \gamma p_j + \alpha \theta _i - \beta \theta _j) - I_i\theta _{i}^2 \end{aligned}$$

The first order condition is

$$\begin{aligned} \displaystyle \frac{\partial }{\partial p_i} \Pi _i(\theta _i,\theta _j)&= -2b p_i + a + \gamma p_j + \alpha \theta _i - \beta \theta _j + bc + Kb(\theta _0-\theta )\\&= a - 2bp_i + \gamma p_j + \theta _i(\alpha -Kb) - \beta \theta _j + b(c+K\theta _0) \end{aligned}$$

The second order condition is

$$\begin{aligned} \displaystyle \frac{\partial ^2}{\partial p_i^2} \Pi _i(\theta _i,\theta _j) = -2b < 0 \end{aligned}$$

Thus, Firm i’s profit function is strictly concave in ‘\(p_i\)’. Equating the first order condition to zero, we get,

$$\begin{aligned} p_i(\theta _i,\theta _j) = \dfrac{(a+\theta _i(\alpha - Kb)+b(c+K\theta _0)+\gamma p_j - \beta \theta _j)}{2b} \end{aligned}$$

Solving for \(p_i\) and \(p_j\) simultaneously, we obtain the equilibrium price for each firm:

$$\begin{aligned} p^{*}_i(\theta _i,\theta _j) = \dfrac{(2b+\gamma )(a+b(c+k\theta _0)) + \theta _i (2b(\alpha -Kb)-\gamma \beta ) - \theta _j(2b\beta - \gamma (\alpha -Kb))}{4b^2-\gamma ^2} \end{aligned}$$

which is further simplified as:

$$\begin{aligned} p^{*}_i(\theta _i,\theta _j)&= \dfrac{(A_1 + S_1 \theta _i - T \theta _j)}{W} \qquad \text{ where } \\&W = (4b^2-\gamma ^2)\\&A_1 = (2b+\gamma )(a+b(c+K\theta _0))\\&S_1 = (2b(\alpha -Kb)-\gamma \beta )\\&T = 2b\beta - \gamma (\alpha - Kb) \\&i \ne j,\; i,j=1,2 \end{aligned}$$

The corresponding values of quantities and profits at the equilibrium prices are:

$$\begin{aligned} q^{*}_i(\theta _i,\theta _j)&= \dfrac{b(A_2 + S_2 \theta _i - T \theta _j)}{W} \qquad \text{ where } \\&W = (4b^2-\gamma ^2)\\&A_2 = (2b+\gamma )(a-(b-\gamma )(c+K\theta _0))\\&S_2 = 2b(\alpha +Kb)-\gamma ( \beta + K \gamma )\\&T = 2b\beta - \gamma (\alpha - Kb) \\&i \ne j, i,j=1,2 \\ \Pi ^{*}_i(\theta _i,\theta _j)&= \dfrac{b(A_1 -Wc+\theta _i S_1 - \theta _j T)(A_2+\theta _i S_2 - \theta _j T)}{W^2} - I_i\theta _i^2 \\&\quad \ - \dfrac{bK(\theta _0-\theta _i)(A_2+\theta _i S_2 - \theta _j T)}{W} \end{aligned}$$

We need the following assumptions:

Assumption

When \(\theta _i = \theta _j = 0\), we should have positive quantity and prices. Hence, \(A_1 >0\) and \(A_2 >0\).

Assumption

We observe that if \(T<0\), the Firm i’s prices and quantities increase in the greening level of its competitor Firm j, which is not the market scenario. Hence, \(T>0\).

Assumption

The impact of Firm i’s own greening level on its prices and quantities should be higher than that of its competitor. Hence, \(S_1> T\) and \(S_2>T\).

To solve for the optimum ‘level of greening’ , we differentiate the profit function of the firm with respect to \(\theta _i\) and equating it to zero, obtain the best action for Firm i given that Firm j chooses \(\theta _j\). The equilibrium ‘level of greening’ for Firm i is :

$$\begin{aligned} \theta _i = \dfrac{b[S_2(A_1-W(c+K\theta _0))+ A_2(S_1+KW)-\theta _jT(S_1+S_2+KW)]}{2(I_iW^2-bS_1S_2-KbS_2W)}; \qquad i \ne j, i,j = 1,2 \end{aligned}$$

The second order differentiation of the profit function reveals

$$\begin{aligned} \displaystyle \frac{\partial ^2}{\partial ^2 \theta _i} \Pi _i = 2(\dfrac{bS_1S_2}{W^2}-I_i+\dfrac{KbS_2}{W}) \end{aligned}$$

The profit of the Firm is strictly concave in the level of greening \(\theta _i\) when

$$\begin{aligned} \mathbf Condition \!: I_i > \dfrac{bS_2(S_1+KW)}{W^2}, \qquad i \ne j, i,j =1,2 \end{aligned}$$

To simplify the expression for the equilibrium value of \(\theta _i\) further, let

$$\begin{aligned}&X = S_2(A_1-W(c+K\theta _0))+A_2(S_1+KW) \\&Y = T(S_1+S_2+KW) \\&B = b/2 \\&Z = bS_2(S_1+KW) \end{aligned}$$

Thus,

$$\begin{aligned} \theta _i = \dfrac{B[X-\theta _j Y]}{I_iW^2-Z} \qquad i \ne j, i,j =1,2 \end{aligned}$$

Now, solving the two simultaneous equations in \(\theta _i\) and \(\theta _j\), we get the equilibrium ‘levels of greening’ as:

$$\begin{aligned} \theta ^{NC}_{i} = \dfrac{BX[(I_jW^2-Z)-BY]}{(I_iW^2-Z)(I_jW^2-Z)-B^2Y^2} \end{aligned}$$

where NC denotes the Nash Equilibrium under competition.

For \(\theta ^{NC}_{i} < \theta _0\), we derive the condition

$$\begin{aligned} \mathbf Condition \!: I_i> \dfrac{B(X - \dfrac{B Y (X - Y \theta _0)}{(I_j W^2 - Z)} + \theta _0 Z)}{W^2 \theta _0} \end{aligned}$$

All other equilibrium values are derived using the optimal value of \(\theta ^{NC}_{i}\).

Proof of Proposition 6

We derive

$$\begin{aligned} \dfrac{\left| \theta _i^{NC} - \theta _j^{NC}\right| }{\theta _i^{NC} + \theta _j^{NC}}&= \dfrac{\left| \Delta \theta \right| }{\theta _T}\\&= \dfrac{W^2\left| (I_j - I_i)\right| }{W^2 (I_i + I_j)- 2(Z+BY)}\\&=\dfrac{W^2\left| (I_j - I_i)\right| }{2[W^2\dfrac{(I_i + I_j)}{2}- (Z+BY)]} \end{aligned}$$

Keeping the cost averages constant \(\dfrac{(I_i + I_j)}{2}\), it is observed that the relative greening difference is increasing in the cost of greening difference.

Proof of Proposition 7

From our previous result we know that

$$\begin{aligned} \dfrac{|\theta _i^{NC} - \theta _j^{NC}|}{\theta _i^{NC} + \theta _j^{NC}} = \dfrac{\left| \Delta \theta \right| }{\theta _T} =\dfrac{W^2\left| I_j - I_i\right| }{2[W^2\dfrac{(I_i + I_j)}{2}- (Z+BY)]} \end{aligned}$$

Now,

$$\begin{aligned} \frac{\partial \dfrac{\left( \Delta \theta \right) }{\theta _T}}{\partial K}= & {} \Bigg [b(2 b- \gamma ) (2 b+ \gamma )^2 |I_j - I_i|\Bigg ]\\&\Bigg [\Bigg (\dfrac{\alpha (4 b^2+2 b \gamma - \gamma ^2)+4 b^3 K }{2 (b (\alpha +K (b+\gamma )+\beta ) (2 \alpha b+2 b^2 K-\gamma (\beta +\gamma K))-\dfrac{(I_i + I_j)}{2} (2 b-\gamma ) (2 b+\gamma )^2)^2}\Bigg )\\&+ \Bigg (\dfrac{2 b^2 (\beta + 2 \gamma K)-b \gamma (\beta +2 \gamma K)-2 \gamma ^2 (\beta +\gamma K))}{2 (b (\alpha +K (b+\gamma )+\beta ) (2 \alpha b+2 b^2 K-\gamma (\beta +\gamma K)-\dfrac{(I_i + I_j)}{2} (2 b-\gamma ) (2 b+\gamma )^2)^2} \Bigg )\Bigg ] \end{aligned}$$

Since \( b >\gamma \) and the denominator is a squared term, the above expression is positive. Hence, \(\frac{\partial \dfrac{\left( \Delta \theta \right) }{\theta _T}}{\partial K} > 0\).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ghosh, D., Shah, J. & Swami, S. Product greening and pricing strategies of firms under green sensitive consumer demand and environmental regulations. Ann Oper Res 290, 491–520 (2020). https://doi.org/10.1007/s10479-018-2903-2

Download citation

Keywords

  • Environmental operations
  • Game theory
  • Green product
  • Government regulations
  • Pricing