Optimal contracts and the manufacturer’s pricing strategies in a supply chain with an inequity-averse retailer

Abstract

Studies in the supply chain literature have typically focused on profit or revenue maximization and assumed that agents within the supply chain are self-interested and only care about their own monetary payoffs. Research in these areas, however, rarely considers an important phenomenon called inequity aversion in which the object pursued by agents within the supply chain is not only their own profit maximization but also the equity of profit allocation. In fact, when agents within a supply chain collaborate with each other to serve a market, the scheme of profit allocation between them usually plays a determinate role in cooperation. Taking into account the impact of agents’ behavior of inequity aversion on the coordination of the supply chain, this paper investigates the optimal contracts and the manufacturer’s pricing strategies in a single-manufacturer and single-retailer supply chain. In this way, we obtain two interesting results: (1) the retailer’s equity aversion largely affects the manufacturer’s decision making, which is not always bad for the manufacturer; and (2) the retailer’s inequity aversion as well as the consumer’s price-sensitive coefficient plays a dominant role in the manufacturer’s decision making.

Appendices

Appendix A: Proof of Lemma 1

In the equity scenario, we have $\pi _R (\hbox {w},\hbox {p})=\gamma \pi _M (\hbox {w},\hbox {p})$. Under the RSC, the retailer’s utility function is $U_{R-RSC} ({w,p})=\pi _{R-RSC} ({w,p})=(\phi p-w)(a-bp)$ and the manufacturer’s profit function is $\pi _{M-RSC} (\hbox {w},\hbox {p})=(w-c+(1-\phi )p)(a-bp)$. Letting $\partial U_{R-RSC} (\hbox {w},\hbox {p})/\partial p=0$, we have $p_{R-RSC}^{*} =(a+bw)/2b$. If the supply chain can be coordinated by the RSC, we have $p_{R-RSC}^{*} =\bar{{p}}=(\hbox {a}+\hbox {bc})/2$, and thus, we have $w=\phi c$. From $\pi _R (\hbox {w},\hbox {p})=\gamma \pi _M (\hbox {w},\hbox {p})$, we can easily obtain $\phi =\gamma /(1+\gamma )$. Therefore, we have the manufacturer’s wholesale price $w=\phi c=\gamma c/(1+\gamma )$. After inserting $w=\gamma c/(1+\gamma )$ and $p_{R-RSC}^{*} =(\hbox {a}+\hbox {bc})/2$ into $\pi _{M-RSC} ({w,p})$, we have $\pi _{M-RSC} =(\hbox {a}+\hbox {bc})^{2}/4b(1+\gamma )$. In the same case under the LQDC, by using the same algebra, we can calculate that when the supply chain is coordinated by the LQDC, we have $w=\theta (\hbox {a}-\hbox {bc})+\hbox {c}$ with $\theta =1/\hbox {b}(1{+}\gamma )$. We can also calculate the manufacturer’s profit under the LQDC that $\pi _{M-LQDC} =(\hbox {a}+\hbox {bc})^{2}/4b(1+\gamma )$. $\square$

Appendix B: Proof of Proposition 1

Because $\pi _{M-RSC} -\pi _{M-LQDC} =(\hbox {b}-1)(\hbox {a}-\hbox {bc})^{2}/4b^{2}(1+\gamma )$, it is easy to find that $\pi _{M-RSC} \ge \pi _{M-LQDC}$ when $b\ge 1$ and $\pi _{M-RSC} <\pi _{M-LQDC}$ when $0<b<1$. Hence, the manufacturer prefers the RSC when $b\ge 1$; therefore, his wholesale price is $w=\gamma c/(1+\gamma )$. On the contrary, the manufacturer will choose the LQDC when $0<b<1$ at a wholesale price $w=c+\theta (\hbox {a}-\hbox {bc})=(a+\gamma bc)/(1+\gamma )\hbox {b}$. $\square$

Appendix C: Proof of Lemma 2

In Case I with $\pi _R (\hbox {w},\hbox {p})<\gamma \pi _M (\hbox {w},\hbox {p})$, under the RSC, we have the retailer’s utility function

$$\begin{aligned} U_R (\left. \hbox {p} \right| \hbox {w})=\pi _R (\hbox {w},\hbox {p})-\alpha \left( \gamma \pi _M (\hbox {w},\hbox {p})-\pi _R (\hbox {w},\hbox {p})\right) , \end{aligned}$$

(16)

where $\pi _{R-RSC} ({w,p})=(\phi p-w)(a-bp)$ and $\pi _{M-RSC} ({w,p})=(w-c+(1-\phi )p)(a-bp)$.

Letting $\partial U_R (\left. \hbox {p} \right| \hbox {w})/\partial \hbox {p}=0$, we have $p_r^{*} =\frac{\phi a+bw}{2\phi b}+\frac{\alpha \gamma (w-\phi \, \hbox {c})}{2\phi ((1+\alpha )\phi -(1-\phi )\alpha \gamma )}$. From $\pi _R (\hbox {w},\hbox {p})<\gamma \pi _M (\hbox {w},\hbox {p})$, we have $w>\frac{\gamma c-p((1+\gamma )\phi -\gamma )}{1+\gamma }$ or $p>\frac{\gamma c-w(1+\gamma )}{(1+\gamma )\phi -\gamma }$.

Thus, the retailer’s price response in Case I is

$$\begin{aligned} p_r^{*} =\left\{ {\begin{array}{l@{\quad }l} \frac{\phi a+bw}{2\phi b}+\frac{\alpha \gamma (w-\phi c)}{2\phi ((1+\alpha )\phi -(1-\gamma )\alpha \gamma )},&{}\hbox {if }\,w>\frac{\gamma c-p_r^{*} ((1+\gamma )\phi -\gamma )}{1+\gamma } \\ \frac{\gamma c-w(1+\gamma )}{(1+\gamma )\phi -\gamma },&{}\hbox {if }\,\mathrm{otherwise} \\ \end{array}} \right. . \end{aligned}$$

(17)

Similarly, in Case II with$\gamma \pi _M (\hbox {w},\hbox {p})<\pi _R (\hbox {w},\hbox {p})$, we have the retailer’s price response that

$$\begin{aligned} p_r^{*} =\left\{ {\begin{array}{l@{\quad }l} \frac{\phi a+bw}{2\phi b}-\frac{{\upbeta } \gamma (w-\phi \hbox {c})}{2\phi ((1-{\upbeta } )\phi +(1-\gamma ){\upbeta } \gamma )},&{}\mathrm{if}\,w<\frac{\gamma c-p_r^{*} ((1+\gamma )\phi -\gamma )}{1+\gamma } \\ \frac{\gamma c-w(1+\gamma )}{(1+\gamma )\phi -\gamma },&{}\mathrm{if}\,\mathrm{otherwise} \\ \end{array}} \right. . \end{aligned}$$

(18)

From Eqs. (17) and (18), we have the retailer’s price under the RSC that

$$\begin{aligned} p_{r-RSC}^{*} (\hbox {w})=\left\{ {\begin{array}{l@{\quad }l} p_{r-I-RSC}^{*} ,&{}\hbox {if}\,w>(\gamma \hbox {c}+\hbox {p}_{r-I-RSC}^{*} )/(1+\gamma ) \\ p_{r-0-RSC}^{*} ,&{}\hbox {if}\,(\gamma \hbox {c}+\hbox {p}_{r-II-RSC}^{*} )/(1+\gamma )<w\\ &{}\quad <(\gamma \hbox {c}+\hbox {p}_{r-I-RSC}^{*} )/(1+\gamma ) \\ p_{r-II-RSC}^{*} ,&{}\hbox {if}\,w<(\gamma \hbox {c}+\hbox {p}_{r-II-RSC}^{*} )/(1+\gamma ) \\ \end{array}} \right. , \end{aligned}$$

(19)

where $p_{r-0-RSC}^{*} =\frac{(\hbox {w}-\hbox {c})\gamma +\hbox {w}}{(1+\gamma )\phi -\gamma }$. $\square$

Appendix D: Proof of Proposition 2

• (1) When $\pi _R (\hbox {w},\hbox {p})<\gamma \pi _M (\hbox {w},\hbox {p})$under the RSC, we have the retailer’s utility function $U_{R{-}RSC} (\left. \hbox {p} \right| \hbox {w})=\pi _R (\hbox {w},\hbox {p})-\alpha (\gamma \pi _M (\hbox {w},\hbox {p})-\pi _R (\hbox {w},\hbox {p}))$ where $\pi _{R-RSC} (\hbox {w},\hbox {p})=(\phi \hbox {p}-\hbox {w})\hbox {D}(\hbox {p})$ and $\pi _{M-RSC} ({w,p})=(w-c+(1-\phi )\hbox {p})(a-bp)$. Letting $\partial U_{R-RSC} (\left. \hbox {p} \right| \hbox {w})/\partial \hbox {p}=0$, we have $p_{r-I-RSC}^{*} =\frac{\phi a+bw}{2\phi b}+\frac{\alpha \gamma (w-\phi \hbox {c})}{2\phi ((1+\alpha )\phi -(1-\phi )\alpha \gamma )}$. If the supply chain can be coordinated under the RSC in this case, we have $p_{r-I-RSC}^{*} =\bar{{p}}=(\hbox {a}+\hbox {bc})/2$; therefore, we get $w=\phi c$. From $\pi _R (\hbox {w},\hbox {p})<\gamma \pi _M (\hbox {w},\hbox {p})$, we obtain $w>(\gamma \hbox {c}+\hbox {p})/(1+\gamma )$. Inserting $w=\phi c$ and $p_{r-I-RSC}^{*} =(\hbox {a}+\hbox {bc})/2$into $w>(\gamma \hbox {c}+\hbox {p})/(1+\gamma )$, we have $\phi <\frac{\gamma }{1+\gamma }$. Based on the above solutions, we have the retailer’s utility $U_{R-RSC} =\frac{(\hbox {a}-\hbox {bc})^{2}}{4b}(\phi +\alpha ((1+\gamma )\phi -\gamma ))$. From $U_{R-RSC} ({w,p})\ge 0$, we have $\phi \ge \frac{\alpha \gamma }{1+\alpha +\alpha \gamma }$. In conclusion, if the supply chain can be coordinated by the RSC in this case, we have $\phi \in [\frac{\alpha \gamma }{1+\alpha +\alpha \gamma },\frac{\gamma }{1+\gamma })$.

• (2) By using the same algebra as used in (1), when $\pi _R (\hbox {w},\hbox {p})>\gamma \pi _M (\hbox {w},\hbox {p})$, we have $p_{r-II-RSC}^{*} =\frac{\phi a+bw}{2\phi b}-\frac{\beta \gamma (\hbox {w}-\phi \hbox {c})}{2\phi ((1-\beta )\phi +(1-\phi )\beta \gamma )}$. After investigating the chain’s coordination condition that $p_{r-II-RSC}^{*} =\bar{{p}}=(\hbox {a}+\hbox {bc})/2$, we have $w=\phi c$. Similarly, from $\pi _R (\hbox {w},\hbox {p})>\gamma \pi _M (\hbox {w},\hbox {p})$, we have $w<(\gamma c+p)/(1+\gamma )$. Inserting $w=\phi c$ and $p_{r-II-RSC}^{*} =(\hbox {a}+\hbox {bc})/2$ into the inequality $w<(\gamma \hbox {c}+\hbox {p})/(1+\gamma )$, we have $\phi >\frac{\gamma }{1+\gamma }$. We can also obtain the retailer’s utility function based on the above solutions that $U_{R-RSC} =\frac{(\hbox {a}-\hbox {bc})^{2}}{4b}(\phi -\beta ((1+\gamma )\phi -\gamma ))$. Letting $U_{R-RSC} ({w,p})\ge 0$, we have $\phi \le \frac{\beta \gamma }{\beta (1+\gamma )-1}$. Thus, the supply chain can be coordinated by the RSC in this case in that $\phi \in (\frac{\gamma }{1+\gamma },\min \{\frac{\beta \gamma }{\beta \gamma -\beta -1},1\})$. Obviously, the inequality $\frac{\beta \gamma }{\beta (1+\gamma )-1}>\frac{\gamma }{1+\gamma }$ should be satisfied, or $\phi \in \varnothing$. Thus, we have $\beta >\frac{1}{1+\gamma }$.

• (3) From the above conclusions, we know that when the supply chain is coordinated under the RSC with $w=\phi c$, we have $p_{r-I-RSC}^{*} =p_{r-II-RSC}^{*} =\bar{{p}}$. Thus, $p_{r-0-RSC}^{*}$ is infeasible, for there is no wholesale price $w$ that satisfies $(\gamma \hbox {c}+\hbox {p}_{r-II-RSC}^{*} )/(1+\gamma )<w<(\gamma \hbox {c}+\hbox {p}_{r-I-RSC}^{*} )/(1+\gamma )$.

• (4) Inserting $w=\phi c$ and $p=(\hbox {a}+\hbox {bc})/2$ into $\pi _{M-RSC} ({w,p})=(w-c+(1-\phi )p)(a-bp)$, we have $\pi _{M-RSC} =(1-\phi _i )\frac{(\hbox {a}-\hbox {bc})^{2}}{4b}$ where $i=a$ or $d$. $\square$

Appendix E: Proof of Lemma 3

By using the same approach as the one used in “Appendix C”, Lemma 3 can be proven. $\square$

Appendix F: Proof of Proposition 3

By using the same algebra as used in Proposition 2, this proposition can be proven. It should be especially mentioned here that in Case II with $\gamma \pi _{M} ({w,p})<\pi _{R} ({w,p})$, we can calculate the retailer’s utility function under the LQDC in the following by using the above algebra

$$\begin{aligned} U_{R-LQDC} (\hbox {w},\hbox {p})=\frac{(\hbox {a}-\hbox {bc})^{2}}{4b}((1-\beta )(1-\theta \, \hbox {b})+\beta \gamma \theta \, \hbox {b}) \end{aligned}$$

Letting $U_{R-LQDC} ({w,p})\ge 0$, we have $\theta <\frac{1-\beta }{(1-\beta (1+\gamma ))\hbox {b}}$. After inserting optimal solutions into $\gamma \pi _M ({w,p})<\pi _R ({w,p})$, we have $\theta <\frac{1}{(1+\gamma )\hbox {b}}$. Thus, if the supply chain can be coordinated in this case, we have $\theta \in (0,\min \{\frac{1}{(1+\gamma )\hbox {b}},\frac{1-\beta }{(1-\beta (1+\gamma ))\hbox {b}}\}]$. Here, the inequality $\frac{1-\beta }{(1-\beta (1+\gamma ))\hbox {b}}>0$ should be satisfied, or $\theta \in \varnothing$. Thus, we have $\beta <\frac{1}{1+\gamma }$. Because $\frac{1}{(1+\gamma )\hbox {b}}-\frac{1-\beta }{(1-\beta (1+\gamma ))\hbox {b}}=\frac{\gamma }{b(1+\gamma )(\beta (1+\gamma )-1)}<0$ with $\beta <\frac{1}{1+\gamma }$, we have $\theta \in (0,\frac{1}{(1+\gamma )\hbox {b}})$. $\square$

Appendix G: Proof of Proposition 4

From Propositions 2 and 3, we know that when $\pi _R ({w,p})<\gamma \pi _M ({w,p})$, we have

$$\begin{aligned} \left\{ {\begin{array}{l} \max \pi _{M-RSC} =(1-\min \phi _d )\frac{(\hbox {a}-\hbox {bc})^{2}}{4b}=\left( 1-\frac{\alpha \gamma }{1+\alpha +\alpha \gamma }\right) \frac{(\hbox {a}-\hbox {bc})^{2}}{4b}\\ \quad =\left( \frac{1+\alpha }{1+\alpha +\alpha \gamma }\right) \frac{(\hbox {a}-\hbox {bc})^{2}}{4b} \\ \max \pi _{M-LQDC} =\max \theta _d \frac{(\hbox {a}-\hbox {bc})^{2}}{4b}=\frac{1+\alpha }{(1+\alpha +\alpha \gamma )b}\frac{(\hbox {a}-\hbox {bc})^{2}}{4b} \\ \end{array}} \right. . \end{aligned}$$

(20)

From Eq. (20), we have $\max \pi _{M-RSC} -\max \pi _{M-LQDC} =(\hbox {b}-1)\frac{(1+\alpha )(\hbox {a}-\hbox {bc})^{2}}{4b^{2}(1+\alpha +\alpha \gamma )}$. Thus, when $\pi _R ({w,p})<\gamma \pi _M ({w,p})$, we have

$$\begin{aligned} \left\{ {\begin{array}{ll} \max \pi _{M-RSC} \ge \max \pi _{M-LQDC} ,&{}\mathrm{if}\,b\ge 1 \\ \max \pi _{M-RSC} <\max \pi _{M-LQDC} ,&{}\mathrm{if}\, b<1 \\ \end{array}} \right. . \end{aligned}$$

(21)

In a similar way, we have the same conclusion when $\pi _R ({w,p})>\gamma \pi _M ({w,p})$. $\square$