## 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.

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## Notes

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).

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

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

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.

We thank the reviewers for this suggestion.

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## Appendix

### Appendix

### 1.1 Greening under demand expansion effects only

The demand faced by the firm is given by

Objective of the firm is

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

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

The cross partial derivative is given by

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,

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

### 1.2 Greening under demand expansion effects and government regulation

The demand faced by the firm is given by

Objective of the firm is

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

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

The cross partial derivative is given by

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,

### 1.3 Proof of Proposition 2

We derive,

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*,

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

### 1.4 Proof of Proposition 3

We derive,

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

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

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

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

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

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*,

Solving the quadratic equation in *K*, gives,

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*.

### 1.5 Consumer and social surplus

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

The first order conditions are

The second order conditions are

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

### 1.6 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,

The first order condition is

The second order condition is

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

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

which is further simplified as:

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

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 :

The second order differentiation of the profit function reveals

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

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

Thus,

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

where NC denotes the Nash Equilibrium under competition.

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

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

### 1.7 Proof of Proposition 6

We derive

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.

### 1.8 Proof of Proposition 7

From our previous result we know that

Now,

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

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### 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

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DOI: https://doi.org/10.1007/s10479-018-2903-2

### Keywords

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