Abstract
This paper investigates the effects of the collaboration between an upstream and a downstream firm regarding their decisions of prices and levels of corporate social responsibility (CSR) efforts. The firms collaborate with each other by sharing their costs or benefits to improve their profitabilities and CSR performances. Three collaborative models are developed for considering that collaboration may be undertaken by either or both firms, and each model has both profit- and cost-sharing mechanisms. We derive and characterize the consumer valuation and the firms’ decisions at equilibrium with respect to the changes in the sharing scheme, and further identify the impacts of each sharing mechanism. Moreover, a Nash bargaining game is developed for examining the choices of sharing scheme under the negotiation between the firms. Finally, we provide economic and managerial insights for socially concerned companies.
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Acknowledgments
The author thanks the anonymous reviewer for the constructive comments, which have helped to improve the quality of the paper significantly. This research was supported by the National Science Council, Taiwan, R.O.C. under Grant #NSC-101-2410-H-224-003-MY2.
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Appendices
Appendix 1: Summary of Notation
Subscripts | |
\(S\), \(B\) | Upstream firm and downstream firm, respectively |
Superscripts | |
\(O\) | Basic model in which CSR activities are absent |
\(R\), \(V\) | Collaborative models in which CSR activities are handled by |
the downstream firm and the upstream firm, respectively | |
\(J\) | Collaborative model in which CSR activities are jointly handled |
by the downstream firm and the upstream firm | |
\(*\) | Equilibrium value |
Symbols | |
\(i\) | Indicator of firm \(i\in \{S,B\}\) |
\(k\) | Indicator of collaborative model \(k\in \{V,R,J\}\) |
\(c\) | Upstream firm production cost per unit |
\(d\) | Market demand |
\(m\) | Downstream firm margin per unit |
\(p\) | Downstream firm price per unit to the market \(p\equiv m+w\) |
\(q\), \(q_0\) | The additional CSR effort and the base effort, respectively |
\(w\) | Wholesale price per unit chosen by the upstream firm |
\(\alpha \) | Market base |
\(\eta _i\) | Firm \(i\) investment cost related to CSR activities |
\(\gamma \) | Effort elasticity |
\(\phi \) | Proportion of the profit retained by the downstream firm |
\(\psi _i\) | Proportion of firm \(i\) investment cost born by itself |
\(\varPi \) | Profit function |
\(\varOmega ^k\) | Sharing scheme in model \(k\) |
Appendix 2: Proofs of Propositions
Proof of Proposition 3
The uniqueness of the chain members’ equilibrium decisions in model \(V\) can be obtained by examining the Hessian matrix, as follows:
The non-negativity of the equilibrium decisions asserts that the numerators are negative, namely, \(\gamma ^2-2 (2+\phi ) \eta _S \psi _S<0\); hence, we obtain that the Hessian is negative definite \(H^V<0\), which shows the uniqueness of the chain members’ equilibrium decisions in model \(V\).
By the same token, if the equilibrium decisions in model \(R\) and model \(J\) are non-negative, \((\gamma ^2 \phi -2 (2+\phi ) \eta _B \psi _B)<0\) and \(\gamma ^2 \phi \eta _S \psi _S+\eta _B \psi _B \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) <0\) hold, respectively. Consequently, we obtain that the Hessian matrix \(H^R=\phi \left( \gamma ^2 \phi -2 (2+\phi ) \eta _B \psi _B\right) <0\) and \(H^J=-2 \phi \left( \gamma ^2 \phi \eta _S \psi _S+\eta _B \psi _B \left( \gamma ^2-2 (2+\phi )\right. \right. \) \(\left. \left. \eta _S \psi _S\right) \right) >0\), such that the equilibrium decisions in model \(R\) and \(J \) are unique. \(\square \)
Proof of Proposition 4
We let
Taking the first derivative of the equilibrium decisions in model \(k\) (\(k=R,V\)) with respect to \(\eta _i\) (\(i=S,B\)) gives the following:
For model \(J\), computing the first derivatives of the equilibrium decisions with respect to \(\eta _i\) (\(i=S,B\)) result in the following:
Regarding the trends of the demand, we obtain the following:
The proof is completed. \(\square \)
Proof of Proposition 5
For part (i), we have
From Proposition 3, \(\gamma ^2-2 (2+\phi ) \eta _S \psi _S<0\), and thus \({m^V}^*>{m^O}^*\). The sign of \({w^V}^*-{w^O}^*\) depends on the term \((\gamma ^2+4 (-1+\phi ) \eta _S \psi _S)\); specifically, we obtain that \({w^V}^*<{w^O}^*\) if \(\phi <1-\frac{\gamma ^2}{4 \eta _S \psi _S}\). Because \(\gamma ^2/2\eta _i\approx 0\) yields \(\phi <1-\frac{\gamma ^2}{4 \eta _S \psi _S}\approx 1\), \({w^V}^*<{w^O}^*\) always holds. The sign of \({p^V}^*-{p^O}^*\) depends on the term \((\gamma ^2+(-1+\phi ) \eta _S \psi _S)\); specifically, \({p^V}^*<{p^O}^*\) if and only if \(\phi <1-\gamma ^2/\eta _S\psi _S\). Because the proofs of parts (ii) and (iii) follow from arguments analogous to that of part (i), we omit the details. \(\square \)
Proof of Proposition 6
Let
where \(k\in \{R,V\}\), and \(i=B\) for \(k=R\) and \(i=S\) for \(k=V\). From Proposition 3, we know that \(\zeta _i^k>0\). Moreover, we define \(\Lambda ^{k}_{x}\equiv {x^J}^*-{x^k}^*\) be the difference in the equilibrium decisions \(x\) (\(x=q_i,w,m\) and \(i=B,S\)) between model \(J\) and model \(k\) (\(k=R,V\)), which are given by the following:
The proof of part (i) is completed.
Likewise, we let \(\hat{\zeta }\equiv (\alpha -c)/\!\left( \gamma ^2 \phi -2 (2+\phi ) \eta _B \psi _B\right) \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) >0\) and obtain the difference in the equilibrium decisions between model \(R\) and model \(V\) as follows:
The signs of the above equations depend on the term \((\eta _B \psi _B-\phi \eta _S \psi _S)\). Solving \((\eta _B \psi _B-\phi \eta _S \psi _S)=0\) for \(\eta _s\) gives a threshold \(\eta _B\,\psi _B/\phi \,\psi _s\). If and only if \(\eta _s<\eta _B\,\psi _B/\phi \,\psi _s\), \({d^V}>{d^R}\), \({q_S^V}^*>{q_B^R}^*\), \({m^V}^*>{m^R}^*\), and \({w^V}^*>{w^R}^*\). Therefore, the proof of part (ii) is completed. \(\square \)
Proof of Proposition 7
Taking the equilibrium decisions into consumer utility function gives the following results:
The discrepancies of the utilities are given by
With the aid of Proposition 3, we acknowledge that the above discrepancies are positive, and thus the proof of part (i) is obtained.
It is straightforward to show that the derivatives of the above discrepancies with respect to \(\phi \) and \(\psi _i\) are negative, given by:
Hence, the proof of Proposition 7 is completed. \(\square \)
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Wu, CH. Collaboration and sharing mechanisms in improving corporate social responsibility. Cent Eur J Oper Res 24, 681–707 (2016). https://doi.org/10.1007/s10100-014-0377-0
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DOI: https://doi.org/10.1007/s10100-014-0377-0