Collaboration and sharing mechanisms in improving corporate social responsibility

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.

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:

$$\begin{aligned} H^V&= \left( \begin{array}{c@{\quad }c@{\quad }c} \frac{\partial ^2\,\varPi _B^V}{\partial \,m^2}&{} \frac{\partial ^2\,\varPi _B^V}{\partial \,m\,\partial \,q_S}&{} \frac{\partial ^2\,\varPi _B^V}{\partial \,m\,\partial \,w}\\ \\ \frac{\partial ^2\,\varPi _S^V}{\partial \,q_S\,\partial \,m}&{} \frac{\partial ^2\,\varPi _S^V}{\partial \,q_S^2}&{} \frac{\partial ^2\,\varPi _S^V}{\partial \,q_S\,\partial \,w}\\ \\ \frac{\partial ^2\,\varPi _S^V}{\partial \,w\,\partial \,m}&{} \frac{\partial ^2\,\varPi _S^V}{\partial \,w\,\partial \,q_S}&{} \frac{\partial ^2\,\varPi _S^V}{\partial \,w^2}\\ \end{array} \right) = \phi \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) . \end{aligned}$$

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

$$\begin{aligned} \xi ^{R}&\equiv \frac{2 (c-\alpha ) }{\left( \gamma ^2 \phi -2 (2+\phi ) \eta _B \psi _B\right) {}^2}<0,\quad \xi ^{V}\equiv \frac{2 (c-\alpha ) }{\left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) {}^2}<0,\quad \text{ and }\\ \xi ^{J}&\equiv \frac{ (c-\alpha ) }{\left( \gamma ^2 \phi \eta _S \psi _S+\eta _B \psi _B \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) \right) {}^2}<0. \end{aligned}$$

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:

$$\begin{aligned} \frac{\partial \,{w^R}^*}{\partial \,\eta _B}&= \gamma ^2 \phi ^2 \psi _B \xi ^{R}<0,\quad \frac{\partial \,{m^R}^*}{\partial \,\eta _B}=\gamma ^2 \phi \psi _B \xi ^{R}<0,\\ \frac{\partial \,{q_B^R}^*}{\partial \,\eta _B}&= \gamma \phi \psi _B (2+\phi ) \xi ^{R}<0.\\ \frac{\partial \,{w^V}^*}{\partial \,\eta _S}&= \gamma ^2 \phi \psi _S \xi ^{V}<0,\quad \frac{\partial \,{m^V}^*}{\partial \,\eta _S}=\gamma ^2 \psi _S \xi ^{V}<0,\quad \text{ and }\\ \frac{\partial \,{q_S^V}^*}{\partial \,\eta _S}&= \gamma \psi _S (2+\phi ) \xi ^{V}<0.\\ \end{aligned}$$

For model $J$, computing the first derivatives of the equilibrium decisions with respect to $\eta _i$ ($i=S,B$) result in the following:

$$\begin{aligned} \frac{\partial \,{w^J}^*}{\partial \,\eta _B}&= 2 \gamma ^2 \phi ^2 \eta _S^2 \psi _B \psi _S^2 \xi ^{J}<0,\quad \frac{\partial \,{m^J}^*}{\partial \,\eta _B}=2 \gamma ^2 \eta _S^2 \psi _B \psi _S^2 \xi ^{J}<0,\\ \frac{\partial \,{q_B^J}^*}{\partial \,\eta _B}&= -\gamma \phi \eta _S \psi _B \psi _S \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) \xi ^{J}<0,\\ \frac{\partial \,{q_S^J}^*}{\partial \,\eta _B}&= \gamma ^3 \psi \eta _S \psi _B \psi _S \xi ^{J}<0,\\ \frac{\partial \,{w^J}^*}{\partial \,\eta _S}&= 2 \gamma ^2 \phi \eta _B^2 \psi _B^2 \psi _S \xi ^{J}<0,\quad \frac{\partial \,{m^J}^*}{\partial \,\eta _S}=2 \gamma ^2 \eta _B^2 \psi _B^2 \psi _S \xi ^{J}<0,\\ \frac{\partial \,{q_B^J}^*}{\partial \,\eta _S}&= \gamma ^3 \phi \eta _B \psi _B \psi _S \xi ^{J}<0,\\ \frac{\partial \,{q_S^J}^*}{\partial \,\eta _S}&= -\gamma \phi \eta _B \psi _B \psi _S \left( \gamma ^2 \phi -2 (2+\phi ) \eta _B \psi _B\right) \xi ^{J}<0. \end{aligned}$$

Regarding the trends of the demand, we obtain the following:

$$\begin{aligned} \frac{\partial d^R}{\partial \eta _B}&= \gamma ^2 \phi \psi _B \xi ^R<0,\quad \frac{\partial d^V}{\partial \eta _S}=\gamma ^2 \psi _S \xi ^V<0,\\ \frac{\partial d^J}{\partial \eta _B}&= \gamma ^2 \phi \eta _S^2 \psi _B \psi _S^2 \xi ^J<0,\quad \frac{\partial d^J}{\partial \eta _S}= \gamma ^2 \eta _B^2 \psi _B^2 \psi _S \xi ^J<0. \end{aligned}$$

The proof is completed. $\square$

Proof of Proposition 5

For part (i), we have

$$\begin{aligned} {m^V}^*-{m^O}^*&= \frac{(c-\alpha ) \left( \gamma ^2-2 (-1+\phi ) \eta _S \psi _S\right) }{3 \gamma ^2-6 (2+\phi ) \eta _S \psi _S},\\ {w^V}^*-{w^O}^*&= \frac{(c-\alpha ) \left( \gamma ^2+4 (-1+\phi ) \eta _S \psi _S\right) }{3 \gamma ^2-6 (2+\phi ) \eta _S \psi _S},\\ {p^V}^*-{p^O}^*&= \frac{2 (c-\alpha ) \left( \gamma ^2+(-1+\phi ) \eta _S \psi _S\right) }{3 \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) }. \end{aligned}$$

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

$$\begin{aligned} \zeta _i^k\equiv \frac{\alpha -c}{\left( \gamma ^2 \phi -2 (2+\phi ) \eta _i \psi _i\right) \left( \gamma ^2 \phi \eta _S \psi _S+\eta _B \psi _B \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) \right) }, \end{aligned}$$

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:

$$\begin{aligned} \Lambda _{q_B}^{R}&= \gamma ^3 \phi \eta _B \psi _B \zeta _B^R>0,\quad \Lambda _{q_S}^{V}=\gamma ^3 \phi \eta _S \psi _S \zeta _S^V>0\\ \Lambda _{m}^{R}&= \gamma ^2 \eta _B^2 \psi _B^2 \zeta _B^R>0,\quad \Lambda _{m}^{V}=\phi \gamma ^2 \eta _S^2 \psi _S^2 \zeta _S^V>0,\\ \Lambda _{w}^{R}&= \phi \Lambda _{m}^{R}>0,\quad \text{ and }~~\Lambda _{w}^{R}=\phi \Lambda _{m}^{R}>0. \end{aligned}$$

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:

$$\begin{aligned} {q_S^V}^*-{q_B^R}^*&= 2\gamma (2+\phi ) \left( \eta _B \psi _B-\phi \eta _S \psi _S\right) \hat{\zeta },\\ {m^V}^*-{m^R}^*&= {d^V}-{d^R}=2\gamma ^2 \left( \eta _B \psi _B-\phi \eta _S \psi _S\right) \hat{\zeta },~~\text{ and }\\ {w^V}^*-{w^R}^*&= 2\gamma ^2 \phi \left( \eta _B \psi _B-\phi \eta _S \psi _S\right) \hat{\zeta }. \end{aligned}$$

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:

$$\begin{aligned} U^O&= \frac{1}{3} (-c-2 \alpha +3 \theta ),\\ U^R&= \frac{\gamma ^2 (-\alpha +\theta ) \phi +2 (c+\alpha (1+\phi )-\theta (2+\phi )) \eta _B \psi _B}{\gamma ^2 \phi -2 (2+\phi ) \eta _B \psi _B},\\ U^V&= \frac{\gamma ^2 (-\alpha +\theta )+2 (c+\alpha (1+\phi )-\theta (2+\phi )) \eta _S \psi _S}{\gamma ^2-2 (2+\phi ) \eta _S \psi _S},\\ U^J\!&= \!\frac{\gamma ^2 (-\alpha \!+\!\theta ) \phi \eta _S \psi _S\!+\!\eta _B \psi _B \left( \gamma ^2 (-\alpha \!+\!\theta )\!+\!2 (c+\alpha (1\!+\!\phi )-\theta (2\!+\!\phi )) \eta _S \psi _S\right) }{\gamma ^2 \phi \eta _S \psi _S+\eta _B \psi _B \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) }. \end{aligned}$$

The discrepancies of the utilities are given by

$$\begin{aligned} U^R-U^O&= \frac{(c-\alpha ) \left( \gamma ^2 \phi +2 (1-\phi ) \eta _B \psi _B\right) }{3 \gamma ^2 \phi -6 (2+\phi ) \eta _B \psi _B}>0,\\ U^V-U^O&= \frac{(c-\alpha ) \left( \gamma ^2+2 (1-\phi ) \eta _S \psi _S\right) }{3 \gamma ^2-6 (2+\phi ) \eta _S \psi _S}>0,\\ U^J-U^O&= \frac{(c-\alpha ) \left( \gamma ^2 \phi \eta _S \psi _S+\eta _B \psi _B \left( \gamma ^2+2 (1-\phi ) \eta _S \psi _S\right) \right) }{3 \gamma ^2 \phi \eta _S \psi _S+3 \eta _B \psi _B \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) }>0.\\ \end{aligned}$$

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:

$$\begin{aligned} \frac{\partial \,(U^R-U^O)}{\partial \,\phi }&= -\frac{2 (c-\alpha ) \eta _B \psi _B \left( \gamma ^2-2 \eta _B \psi _B\right) }{\left( \gamma ^2 \phi -2 (2+\phi ) \eta _B \psi _B\right) {}^2}<0\\ \frac{\partial \,(U^R-U^O)}{\partial \,\psi _B}&= \frac{2 (c-\alpha ) \gamma ^2 \phi \eta _B}{\left( \gamma ^2 \phi -2 (2+\phi ) \eta _B \psi _B\right) {}^2}<0,\\ \frac{\partial \,(U^V-U^O)}{\partial \,\phi }&= \frac{4 (c-\alpha ) \eta _S^2 \psi _S^2}{\left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) {}^2}<0,\\ \frac{\partial \,(U^V-U^O)}{\partial \,\psi _S}&= \frac{2 (c-\alpha ) \gamma ^2 \eta _S}{\left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) {}^2}<0,\\ \frac{\partial \,(U^J-U^O)}{\partial \,\phi }&= -\frac{2 (c-\alpha ) \eta _B \eta _S^2 \psi _B \left( \gamma ^2-2 \eta _B \psi _B\right) \psi _S^2}{\left( \gamma ^2 \phi \eta _S \psi _S+\eta _B \psi _B \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) \right) {}^2}<0\\ \frac{\partial \,(U^J-U^O)}{\partial \,\psi _S}&= \frac{2 (c-\alpha ) \gamma ^2 \phi \eta _B \eta _S^2 \psi _S^2}{\left( \gamma ^2 \phi \eta _S \psi _S+\eta _B \psi _B \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) \right) {}^2}<0,\\ \frac{\partial \,(U^J-U^O)}{\partial \,\psi _B}&= \frac{2 (c-\alpha ) \gamma ^2 \eta _B^2 \eta _S \psi _B^2}{\left( \gamma ^2 \phi \eta _S \psi _S+\eta _B \psi _B \left( \gamma ^2-2 (2+\phi ) \eta _S \psi _S\right) \right) {}^2}<0. \end{aligned}$$

Hence, the proof of Proposition 7 is completed. $\square$