Integrating Corporate Social Responsibility in a closed-loop supply chain under government subsidy and used products collection strategies

Abstract

Due to rapid advancement of the society, recently many manufacturing and retailing companies are showing their interests in Corporate Social Responsibility (CSR) in addition to maximizing their profits. This study introduces CSR activity of the retailer, and develops an integrated model (Model I) and three manufacturer-led decentralized models (Model M, R, and C) depending on different collection options of used products under selling price and CSR effort dependent market demand. The aim of this study is to explore how CSR effort of the retailer can influence the optimal decisions of the supply chain members. In order to stimulate the CSR effort, the government provides CSR dependent subsidy to the retailer. Besides deriving closed-form optimal solutions, this research also determines optimal consumer surplus, environmental damage and social welfare for the proposed models. A comparative study is performed to determine the best sustainable decentralized model. The numerical results show that among the three decentralized models, Model M gives the best performance but fails to challenge with Model I, and government subsidy plays a key role in improving channel performance. A two-part tariff contract is considered to address channel coordination issue. The effects of some key model-parameters on the optimal profitability and the social welfare are investigated through sensitivity analysis, which can help managers to implement CSR activity as well as improve channel performance.

Appendices

Appendix A

Proof of Model I

The Hessian matrix associated with the profit function of the integrated supply chain is given by $H^I = \left( \begin{array}{ccc} - 2 \beta &{} - \beta Z_1 &{} \gamma - \beta \tau _1 Z_1 \\ - \beta Z_1 &{} - 2 \mu &{} \gamma Z_1 - 2 \mu \tau _1 \\ \gamma - \beta \tau _1 Z_1 &{} \gamma Z_1 - 2 \mu \tau _1 &{} - 2 (\lambda + \mu \tau _1^2 - \gamma Z_1 \tau _1) \end{array} \right)$

Now, the principal minors are: $|M_1| = - 2 \beta < 0,~ |M_2| = 4 \beta \mu - Z_1^2 \beta ^2> 0, ~\text{ if }~ \mu > \frac{Z_1^2 \beta }{4}$ and $|H^I| = - 2 [\beta (4 \mu - Z_1^2 \beta ) \lambda - \gamma ^2 \mu ] < 0, ~\text{ if }~ \mu > \frac{Z_1^2 \beta ^2 \lambda }{4 \beta \lambda - \gamma ^2}$. $\mu (4 \beta \lambda - \gamma ^2) - Z_1^2 \beta ^2 \lambda = \beta \lambda (4 \mu - Z_1^2 \beta ) - \gamma ^2 \mu > 0$ implies $\mu > \frac{Z_1^2 \beta }{4}$. Therefore if $\mu > \frac{Z_1^2 \beta ^2 \lambda }{4 \beta \lambda - \gamma ^2}$, the Hessian matrix $H^I$ becomes negative definite. Then, using the first order conditions for optimality of (1), we can get unique optimal solution of the integrated supply chain.

Proof of Model M

The retailer’s reaction

The Hessian matrix of $\Pi _r^M$ is given by $H_R^M = \left( \begin{array}{cc} - 2 \beta &{} \gamma \\ \gamma &{} - 2 \lambda \end{array} \right)$

Now, the principal minors are: $|M_1| = - 2 \beta < 0$ and $|H_R^M| = 4 \beta \lambda - \gamma ^2 > 0$, if $~\lambda > \frac{\gamma ^2}{4 \beta }$. Thus the required conditions for the concavity of $H_R^M$ are given by this condition and it guarantees the positivity of equilibrium solution.

The manufacturer’s reaction

With the optimal decisions of the retailer, the Hessian matrix $\Pi _m^M$ is given by $H_M^M =\left( \begin{array}{ccccc} -2 \mu &{} \frac{2 \mu (\gamma \mu \tau _1 - Z_1 \beta \lambda )}{4 \beta \lambda - \gamma _1^2} \\ \frac{2 \mu (\gamma \mu \tau _1 - Z_1 \beta \lambda )}{4 \beta \lambda - \gamma _1^2} &{} \frac{2 \beta ^2 [- 8 \beta \lambda ^2 + 2 Z_1 \beta \gamma \lambda \tau _1 + \gamma ^2 (2 \lambda - \mu \tau _1^2)]}{(4 \beta \lambda - \gamma ^2)^2} \end{array} \right)$

Now, the principal minors are: $|M_1| = - 2 \mu < 0$ and $|H_M^M| = \frac{4 \beta ^2 \lambda X_1}{(4 \beta \lambda - \gamma ^2)^2} > 0$, if $\mu > \frac{\beta _1^2 Z_1^2 \lambda }{2 (4 \beta \lambda - \gamma ^2)}$. Under this conditions, the Hessian matrix $H_M^M$ becomes negative definite. Thus the required conditions for the concavity of $H_M^M$ are given by this condition and using first order optimality conditions optimal decisions can be obtained.

Proof of Model R

The retailer’s reaction

The Hessian matrix of $\Pi _r^R$ is given by $H_R^R = \left( \begin{array}{ccc} - 2 \beta &{} - (B - A) \beta &{} \gamma - \beta \tau _1 (B - A) \\ - (B - A) \beta &{} - 2 \mu &{} (B - A) \gamma - 2 \mu \tau _1 \\ \gamma - \beta \tau _1 (B - A) &{} (B - A) \gamma - 2 \mu \tau _1 &{} - 2 [\lambda + \mu \tau _1^2 - (B - A) \gamma \tau _1] \end{array} \right)$

Now, the principal minors are: $|M_1| = - 2 \beta < 0, |M_2| = 4 \beta \mu - (B - A)^2 \beta ^2> 0, ~\text{ if }~ \mu > \frac{(B - A)^2 \beta }{4}$, and $|H_R^R| = - 2 [\beta \lambda [4 \mu - \beta (B - A)^2] - \gamma ^2 \mu ] < 0$, if $~\mu > \frac{\beta ^2 \lambda (B - A)^2}{4 \beta \lambda - \gamma ^2}$. Thus the required conditions for the concavity of $H_R^R$ are given by these condition and it guarantees the positivity of equilibrium solution.

The manufacturer’s reaction

$\frac{\partial ^2\Pi _m^{R}}{\partial w^2} = - \frac{4 \beta ^2 \mu \lambda X_2}{[\beta \lambda [4 \mu - \beta (B - A)^2] - \gamma ^2 \mu ]^2} < 0$, if $\mu > \frac{(B - A) Z_1 \beta ^2 \lambda }{4 \beta \lambda - \gamma ^2}$. Under this condition, $\Pi _m^R$ will be negative definite and unique optimal solution can be obtained by solving first order optimality condition of $\Pi _m^{R}$.

Proof of Model C

The retailer’s reaction

The Hessian matrix of $\Pi _r^C$ is given by $H_R^C = \left( \begin{array}{cc} - 2 \beta &{} \gamma \\ \gamma &{} - 2 \lambda \end{array} \right)$

Now, the principal minors are: $|M_1| = - 2 \beta < 0$, and $|H_R^C| = 4 \beta \lambda - \gamma ^2 < 0$, if $~\lambda > \frac{\gamma ^2}{4 \beta }$. Thus the required conditions for the concavity of $H_R^C$ are given by this condition and it guarantees the positivity of equilibrium solution.

The collector’s reaction

As $\frac{\partial ^2\Pi _c^{C}}{\partial \tau _0^2} = - 2 \mu < 0$, $\Pi _c^C$ is negative definite and unique optimal solution can be obtained by solving first order optimality condition of $\Pi _c^{C}$.

The manufacturer’s reaction

As $\frac{\partial ^2\Pi _m^{C}}{\partial w^2} = - \frac{4 \beta ^2 \lambda X_3}{\mu (4 \beta \lambda - \gamma ^2)^2} < 0$, $\Pi _m^C$ is negative definite and unique optimal solution can be obtained by solving first order optimality condition of $\Pi _m^{C}$.

Appendix B

Proof of Proposition 1

(i) For collection rate of used products

$\tau ^I - \tau ^M = \frac{Z_1 \beta \mu (4 \beta \lambda - \gamma ^2) [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{2 X X_1} > 0$;

$\tau ^M - \tau ^R = \frac{\beta \mu (2 Z_2 - Z_1)(4 \beta \lambda - \gamma ^2) [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{4 X_1 X_2 } > 0$, if $B < \frac{\Delta + A}{2}$;

$\tau ^R - \tau ^C = \frac{(B - A)^3 \beta ^3 \lambda [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{4 X_2 X_3} > 0$.

(ii) For wholesale price

$w^C - w^R = \frac{(B - A)^3 Z_2 \beta ^3 \lambda [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{4 X_2 X_3 } > 0$,

$w^R - w^M = \frac{\beta [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )][\mu (4 \beta \lambda - \gamma ^2) \big ((Z_1 - Z_2)^2 + Z_2^2\big ) - Z_1^2 \beta ^2 \lambda (B - A)^2]}{4 X_1 X_2 } > 0$, $\mu$ being sufficiently large.

(iii) For selling price

$p^C - p^R = \frac{(B - A)^2 \beta \mu (2 \beta \lambda - \gamma ^2) [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{4 X_2 X_3 } > 0$;

$p^R - p^M = \frac{\beta \mu Z_1 (2 Z_2 - Z_1)(2 \beta \lambda - \gamma ^2) [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{4 X_1 X_2 } > 0$, if $B < \frac{\Delta + A}{2}$;

$p^M - p^I = \frac{\mu ^2 (4 \beta \lambda - \gamma ^2)(2 \beta \lambda - \gamma ^2) [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{2 \beta \lambda X X_1} > 0$.

(iv) For CSR

$y^I - y^M = \frac{\gamma \mu ^2 (4 \beta \lambda - \gamma ^2) [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{2 \beta \lambda X X_1} > 0$;

$y^M - y^R = \frac{\beta ^2 \gamma \mu Z_1 (2 Z_2 - Z_1)(2 \beta \lambda - \gamma ^2) [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{4 X_1 X_2 } > 0$, if $B < \frac{\Delta + A}{2}$;

$y^R - y^C = \frac{(B - A)^2 \beta ^2 \gamma \mu [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{4 X_2 X_3} > 0$. $\square$

Proof of Proposition 2

$\Pi _m^M - \Pi _m^R = \frac{\beta ^2 \mu \lambda Z_1 (2 Z_2 - Z_1) [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]^2}{8 \lambda X_1 X_2 } > 0$, if $B < \frac{\Delta + A}{2}$;

$\Pi _m^R - \Pi _m^C = \frac{(B - A)^2 \beta ^2 \mu [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]^2}{8 X_2 X_3} > 0$. $\square$

Proof of Proposition 3

$CS^I - CS^M = \frac{\beta \mu ^3 (4 \beta \lambda - \gamma ^2) [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]^2 (X + X_1)}{2 X^2 X_1^2} > 0$,

$CS^M - CS^R = \frac{\beta ^3 \mu ^2 \lambda Z_1 (2 Z_2 - Z_1) (2 X_2 + X_1)[s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]^2}{8 X_1^2 X_2^2 } > 0$, if $B < \frac{\Delta + A}{2}$;

$CS^R - CS^C = \frac{(B - A)^2 \beta ^3 \mu ^2 \lambda [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]^2}{8 X_2^2 X_3^2} > 0$. $\square$

Proof of Proposition 4

(i) For wholesale price

$w^M - w^M_{S0} = \frac{s y_0 \gamma [\mu (4 \beta \lambda - \gamma ^2) - Z_1^2 \beta ^2 \lambda ]}{2 \beta \lambda X_1 } > 0$; $w^R - w^R_{S0} = \frac{s y_0 \gamma [\mu (4 \beta \lambda - \gamma ^2) - \beta ^2 \lambda (B - A)(Z_1 + Z_2)]}{4 \beta \lambda X_2 } > 0$,

$w^C - w^C_{S0} = \frac{s y_0 \gamma [\mu (4 \beta \lambda - \gamma ^2) - 2 \beta ^2 \lambda (B - A) Z_2]}{4 \beta \lambda X_3} > 0$.

(ii) For selling price

$p^I - p^I_{S0} = \frac{s y_0 \gamma (2 \mu - Z_1^2 \beta )}{2 [\mu (4 \beta \lambda - \gamma ^2) - Z_1^2 \beta ^2 \lambda ]} > 0$; $p^M - p^M_{S0} = \frac{s y_0 \gamma [\mu (6 \beta \lambda - \gamma ^2) - Z_1^2 \beta ^2 \lambda ]}{2 \beta \lambda X_1 } > 0$,

$p^R - p^R_{S0} = \frac{s y_0 \gamma [\mu (6 \beta \lambda - \gamma ^2) - 2 \beta ^2 \lambda (B - A)Z_1]}{4 \beta \lambda X_2 } > 0$,

$p^C - p^C_{S0} = \frac{s y_0 \gamma [\mu (6 \beta \lambda - \gamma ^2) - 2 \beta ^2 \lambda (B - A) Z_2]}{4 \beta \lambda X_3 } > 0$.

(iii) For CSR

$y^I - y^I_{S0} = \frac{s y_0 \beta (4 \mu - Z_1^2 \beta )}{2 [\mu (4 \beta \lambda - \gamma ^2) - Z_1^2 \beta ^2 \lambda ]} > 0$; $y^M - y^M_{S0} = \frac{s y_0 [\mu (8 \beta \lambda - \gamma ^2) - Z_1^2 \beta ^2 \lambda ]}{2 \lambda X_1 } > 0$,

$y^R - y^R_{S0} = \frac{s y_0 [\mu (8 \beta \lambda - \gamma ^2) - 2 \beta ^2 \lambda (B - A) Z_1]}{4 \lambda X_2 } > 0$; $y^C - y^C_{S0} = \frac{s y_0 [\mu (8 \beta \lambda - \gamma ^2) - 2 \beta ^2 \lambda (B - A) Z_2]}{4 \lambda X_3 } > 0$.

(iv) For collection rate of used products

$\tau ^I - \tau ^I_{S0} = \frac{s y_0 \gamma \beta Z_1}{2 [\mu (4 \beta \lambda - \gamma ^2) - Z_1^2 \beta ^2 \lambda ]} > 0$; $\tau ^M - \tau ^M_{S0} = \frac{s y_0 \gamma \beta Z_1}{2 \lambda X_1 } > 0$,

$\tau ^R - \tau ^R_{S0} = \frac{s y_0 \gamma \beta (B - A)}{4 \lambda X_2 } > 0$; $\tau ^C - \tau ^C_{S0} = \frac{s y_0 \gamma \beta (B - A)}{4 \lambda X_3 } > 0$. $\square$

Proof of Proposition 5

(i) For profit of the manufacturer

$\Pi _m^M - \Pi ^M_{mS0} = \frac{s y_0 \gamma \mu [s y_0 \gamma + 4 \lambda (\alpha - c_m \beta )]}{4 \lambda X_1 } > 0$; $\Pi _m^R - \Pi ^R_{mS0} = \frac{s y_0 \gamma \mu [s y_0 \gamma + 4 \lambda (\alpha - c_m \beta )]}{8 \lambda X_2 } > 0$,

$\Pi _m^C - \Pi ^C_{mS0} = \frac{s y_0 \gamma \mu [s y_0 \gamma + 4 \lambda (\alpha - c_m \beta )]}{8 \lambda X_3 } > 0$.

The proof for the retailer’s profit being similar, we have omitted those proofs. $\square$

Proof of Proposition 6

$CS^I - CS^I_{S0} = \frac{\beta \mu ^2 s y_0 \gamma [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{2 X^2} > 0$; $CS^M - CS^M_{S0} = \frac{\beta \mu ^2 s y_0 \gamma [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{2 X_1^2} > 0$,

$CS^R - CS^R_{S0} = \frac{\beta \mu ^2 s y_0 \gamma [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{8 X_2^2} > 0$; $CS^C - CS^C_{S0} = \frac{\beta \mu ^2 s y_0 \gamma [s y_0 \gamma + 2 \lambda (\alpha - c_m \beta )]}{8 X_3^2} > 0$. $\square$