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.

Appendix

1.1 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,

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

1.2 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,

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

1.3 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.

1.4 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.

1.5 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

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

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,

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

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 $\theta ^{NC}_{i}$.

1.7 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.

1.8 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, $\frac{\partial \dfrac{\left( \Delta \theta \right) }{\theta _T}}{\partial K} > 0$.