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Decisions of competing supply chain with altruistic retailer under risk aversion

Jing Liu,
Jing Liu,
Yi Wu,
Yi Wu,
Fuyou Huang et
Fuyou Huang et
Chao Ma
Chao Ma
Publié en ligne: 29 Apr 2022
Volume & Edition: AHEAD OF PRINT
Pages: -
Reçu: 08 Nov 2021
Accepté: 27 Nov 2021
Applied Mathematics and Nonlinear Sciences's Cover Image
Applied Mathematics and Nonlinear Sciences
Détails du magazine
License
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
Langues
Anglais
Introduction

As one of the main themes of global sustainable development, reducing greenhouse gas emissions has been extensively studied. Some scholars consider carbon emission reduction strategies under a low carbon regulation, like cap-and-trade regulation (e.g., [1]), low carbon subsidies (e.g., [2]), and carbon tax (e.g., [3]). Some one demonstrate the coordination in low carbon supply chain (e.g., [4]). We focus on the Stackelberg games in the low carbon decisions. For example, Zhang et al. [5] find out that social welfare is not affected by the power structure of supply chain when manufacturer taking green technology strategy. Wang et al. [6] show that manufacturer Stackelberg (MS) is not conducive to improve manufacturer's profit, while the other Stackelberg games help the improvement of manufacturer's profit. More research references for low-carbon supply chain under power structure see [7, 8].

In reality, competition is everywhere. The existence of multiple manufacturers can prevent monopoly and effectively stimulate product upgrading. Multi-retailer project is conducive to multi-point sales of products. Zhou et al. [9] point out that compared with other competing retailers, the demand for retailers selling more environmentally friendly products will increase as the carbon tax rate increases. Zhang et al. [10] find that different green technologies will ease competition among manufacturers. Liu et al. [11] show that fierce competition for emissions reduction will harm the members of the supply chain and the environment under the cap-and-trade regulation.

In fact, competition will inevitably increase the pressure on companies to resist risks. At the same time, the uncertainty of demand will also increase the company's operational risks. Hence, companies often adopt a risk-averse attitude and adopt conservative decisions to reduce risks. Wang and He [12] constructe a low-carbon supply chain game model considering risk aversion through the mean variance method, and pointe out that the manufacturer's risk aversion has a positive impact on the wholesale price, while the supplier's impact is negative. Bai et al. [13] show that if the underlying manufacturer's degree of risk aversion is not deep, low carbon emissions can be achieved.

In order to promote the long-term development of the transaction and ensure the continuous supply of goods, retailers may bring benefits to their partners. At this point, they often express their altruistic preference attitudes. Fan et al. [14] point out that retailer's altruistic behavior is always beneficial to manufacturer under both Stackelberg game low-carbon model and vertical Nash game low-carbon model. Wang et al. [15] show that the retailer's altruistic preference helps to improve the profit and system efficiency of small and medium-sized manufacturers, but it will reduce the retailer's profit.

In view of the fact of low-carbon supply chain transactions, companies may be competitive and have risk aversion behaviors, moreover retailers are more likely to be altruistic. However, the previous literature has not adequately investigated the low-carbon decisions when these issues are considered altogether. Motivated by the research gap, considering supply chain with altruistic retailer, we investigate four Stackelberg models, including manufacturer-led with competitive/non-competitive, retailer-led with competitive/non-competitive, to analysis the optimal decisions. We contribute to the literature in the following three aspects.

First, we show that the emission reduction level decrease with manufacturers' risk aversion. Second, we indicate that with the transfer of powers from the retailers to the manufacturers, the price will increase. At last, we point out that competition can bring a higher price, a deeper degree of low carbon, and more profits to manufacturers and retailers via a series of numerical experiments.

The remainder of this paper is structured as follows. Section 2 presents the assumptions and notations. Section 3 discusses the models. Section 4 analysis the results. Section 5 investigates the impact of competition. Section 6 concludes the paper.
Assumptions and notations

In the distribution of low-carbon products, manufacturers adopt carbon emission reduction technology to improve the green degree of products. Altruistic retailers sell products purchased from manufacturers to consumers. At this time, the consumer market is affected by retail price, carbon emission reduction level and random demand shock. Following the existing literature, e.g., [16], we denote, the cost of carbon emission reduction is 12ei2 {1 \over 2}e_i^2 , where ei represents the eco-friendly level of the product. Related notions are summarized in Table 1.

The notions and parameters

Symbol Description
p Retail price
w Wholesale price
m Retail margin m = p − w
e Emission reduction level
c Unit production cost
x Demand shock, with expected function E[x] = 0
λ Risk aversion coefficient
π Firms' profit
MS Superscripts, represent manufacturer Stackelberg model
RS Superscripts, represent retailer Stackelberg model
* Superscript, represents the optimality
i Subscript, represents the firm, i ∈ {1,2}
M Subscript, represents the manufacturer
R Subscript, represents the retailer
The models

This section discusses four models, the first model and second model are two stage supply chain composed of a single manufacturer and a single retailer. The third model and fourth model are competitive supply chains composed of two competitive manufacturers and two retailers. The first model and third model are manufacturer Stackelberg game, the manufacturer is a leader and decisions wholesale price and emission reduction level, the retailer acts a follower and sets the retail price. The second model and fourth model are retailer Stackelberg game, the retailer is the leader and sets the retail margin, the manufacturer acts the follower and decides wholesale price and emission reduction level.

Noting that both the manufacturer and the retailer are risk averse, in line with [17], we consider using the mean variance (MV) method to characterize their respective utility profits. MVπi=E(πi)−λiVar(πi), MV{\pi _i} = E({\pi _i}) - {\lambda _i}\sqrt {Var({\pi _i})}, where E is the expectation operator, and i represent M and R, respectively.

It also note that the retailer holds the attitude of altruistic preference, in line with [18], the retailer's utility function satisfies that UR(πR)=πR+θπM, {U_R}({\pi _R}) = {\pi _R} + \theta {\pi _M}, where θ (0 ≤ θ ≤ 1) is the degree of altruistic preference. The larger the θ is, the stronger the altruistic preference degree of the retailer is.

Non competitive supply chain

In the non competitive supply chain, the supply chain' operation is shown in Figure 1. We consider two Stackelberg games: MS model and RS model. The demand follows that 1 − p + e + x.

Fig. 1

Non competitive supply chain

MS model

In this subsection, the expected profits of the manufacturer and the retailer are as follow: πMMS=(w−c)(1−p+e+x)−12e2, \pi _M^{MS} = (w - c)(1 - p + e + x) - {1 \over 2}{e^2}, πRMS=(p−w)(1−p+e+x). \pi _R^{MS} = (p - w)(1 - p + e + x).

Then the MV's functions of the manufacturer and the retailer are as follow: MVπMMS=E(πMMS)−λMVar(πMMS)=(w−c)(1−p+e)−12e2−λMσ(w−c), MV\pi _M^{MS} = E(\pi _M^{MS}) - {\lambda _M}\sqrt {Var(\pi _M^{MS})} = (w - c)(1 - p + e) - {1 \over 2}{e^2} - {\lambda _M}\sigma (w - c), MVπRMS=E(πRMS)−λRVar(πRMS)=(p−w)(1−p+e)−λRσ(p−w). MV\pi _R^{MS} = E(\pi _R^{MS}) - {\lambda _R}\sqrt {Var(\pi _R^{MS})} = (p - w)(1 - p + e) - {\lambda _R}\sigma (p - w).

Thus, the altruistic retailer's utility function satisfies that URMS=MVπRMS+θMVπMMS=[p−w+θ(w−c)](1−p+e)−λRσ(p−w)−θλMσ(w−c)−θ12e2. U_R^{MS} = MV\pi _R^{MS} + \theta MV\pi _M^{MS} = [p - w + \theta (w - c)](1 - p + e) - {\lambda _R}\sigma (p - w) - \theta {\lambda _M}\sigma (w - c) - \theta {1 \over 2}{e^2}.

First, we consider the retailer's decision. Since ∂2URMS∂p2=−2<0 {{{\partial ^2}U_R^{MS}} \over {\partial {p^2}}} = - 2 < 0 , then by solving ∂URMS∂p=0 {{\partial U_R^{MS}} \over {\partial p}} = 0 , we can obtain that the retail price p(w,e)=12[1+e+θc+(1−θ)w−σλR]. p(w,e) = {1 \over 2}[1 + e + \theta c + (1 - \theta)w - \sigma {\lambda _R}].

Second, we consider the manufacturer's decisions. Substituting Eq. (8) into Eq. (6), we have MVπMMS(w,e)=12(w−c)[1+e−θc−(1−θ)w−2λMσ+σλR−e2]. MV\pi _M^{MS}(w,e) = {1 \over 2}(w - c)[1 + e - \theta c - (1 - \theta)w - 2{\lambda _M}\sigma + \sigma {\lambda _R} - {e^2}].

By assuming that θ<34 \theta < {3 \over 4} , we can obtain that ∂MVπMMS(w,e)∂w2=−(1−θ)<0 {{\partial MV\pi _M^{MS}(w,e)} \over {\partial {w^2}}} = - (1 - \theta) < 0 , |∂2MVπMMS(w,e)∂w2∂2MVπMMS(w,e)∂w∂e∂2MVπMMS(w,e)∂w∂e∂2MVπMMS(w,e)∂e2|=34−θ>0. \left| {\matrix{ {{{{\partial ^2}MV\pi _M^{MS}(w,e)} \over {\partial {w^2}}}{{{\partial ^2}MV\pi _M^{MS}(w,e)} \over {\partial w\partial e}}} \hfill \cr {{{{\partial ^2}MV\pi _M^{MS}(w,e)} \over {\partial w\partial e}}{{{\partial ^2}MV\pi _M^{MS}(w,e)} \over {\partial {e^2}}}} \hfill \cr {} \hfill \cr } } \right| = {3 \over 4} - \theta > 0.

Therefore the Hessian Matrix of MVπMMS(w,e) MV\pi _M^{MS}(w,e) is negative definite regarding w and e, if θ<34 \theta < {3 \over 4} . So we have wMS*=2+(1−4θ)c−4λMσ+2σλR3−4θ, {w^{MS*}} = {{2 + (1 - 4\theta)c - 4{\lambda _M}\sigma + 2\sigma {\lambda _R}} \over {3 - 4\theta }}, eMS*=1−c−2λMσ+σλR3−4θ. {e^{MS*}} = {{1 - c - 2{\lambda _M}\sigma + \sigma {\lambda _R}} \over {3 - 4\theta }}.

Substituting Eqs. (10) and (11) into Eq. (8), we can obtain that pMS*=3−(3+c)θ−(3−2θ)λMσ+θσλR3−4θ. {p^{MS*}} = {{3 - (3 + c)\theta - (3 - 2\theta){\lambda _M}\sigma + \theta \sigma {\lambda _R}} \over {3 - 4\theta }}.

RS model

In this subsection, noting that the retailer's margin m = p − w, then the MV's functions of the manufacturer and the retailer are as follow: MVπMRS=E(πMRS)−λMVar(πMRS)=(w−c)(1−w−m+e)−12e2−λMσ(w−c), MV\pi _M^{RS} = E(\pi _M^{RS}) - {\lambda _M}\sqrt {Var(\pi _M^{RS})} = (w - c)(1 - w - m + e) - {1 \over 2}{e^2} - {\lambda _M}\sigma (w - c), MVπRRS=E(πRRS)−λRVar(πRRS)=m(1−w−m+e)−λRσm. MV\pi _R^{RS} = E(\pi _R^{RS}) - {\lambda _R}\sqrt {Var(\pi _R^{RS})} = m(1 - w - m + e) - {\lambda _R}\sigma m.

Thus, the altruistic retailer's utility function satisfies that URRS=MVπRRS+θMVπMRS=[m+θ(w−c)](1−w−m+e)−λRσm−θλMσ(w−c)−θ12e2. U_R^{RS} = MV\pi _R^{RS} + \theta MV\pi _M^{RS} = [m + \theta (w - c)](1 - w - m + e) - {\lambda _R}\sigma m - \theta {\lambda _M}\sigma (w - c) - \theta {1 \over 2}{e^2}.

First, we consider the manufacturer's decisions. Since ∂MVπMRS∂e2=−1<0 {{\partial MV\pi _M^{RS}} \over {\partial {e^2}}} = - 1 < 0 , |∂2MVπMRS∂w2∂2MVπMRS∂w∂e∂2MVπMRS∂w∂e∂2MVπMRS∂e2|=1>0. \left| {\matrix{ {{{{\partial ^2}MV\pi _M^{RS}} \over {\partial {w^2}}}{{{\partial ^2}MV\pi _M^{RS}} \over {\partial w\partial e}}} \hfill \cr {{{{\partial ^2}MV\pi _M^{RS}} \over {\partial w\partial e}}{{{\partial ^2}MV\pi _M^{RS}} \over {\partial {e^2}}}} \hfill \cr } } \right| = 1 > 0.

Thus the Hessian Matrix of πMRS \pi _M^{RS} is negative definite regarding w and e. So we have w(m)=1−m−σλM, w(m) = 1 - m - \sigma {\lambda _M}, e(m)=1−m−c−σλM. e(m) = 1 - m - c - \sigma {\lambda _M}.

Second, we consider the retailer's decision. Substituting Eq. (16) and Eq. (17) into Eq. (15), we have URRS(m)=12(1−c−m)[m(2−θ)+(1−c)θ]−(1−c−m)θσλM−mσλR+12θσ2λM2. U_R^{RS}(m) = {1 \over 2}(1 - c - m)[m(2 - \theta) + (1 - c)\theta] - (1 - c - m)\theta \sigma {\lambda _M} - m\sigma {\lambda _R} + {1 \over 2}\theta {\sigma ^2}\lambda _M^2.

Considering the first derivative order condition, i.e., ∂URRS(m)∂m=0 {{\partial U_R^{RS}(m)} \over {\partial m}} = 0 , we can obtain that the retail margin mRS*=1−(1−θ)c−θ+θσλM−σλR2−θ. {m^{RS*}} = {{1 - (1 - \theta)c - \theta + \theta \sigma {\lambda _M} - \sigma {\lambda _R}} \over {2 - \theta }}.

Substituting Eqs. (19) into Eqs. (16) and (17), respectively. We can obtain that wRS*=1+(1−θ)c−2λMσ+σλR2−θ, {w^{RS*}} = {{1 + (1 - \theta)c - 2{\lambda _M}\sigma + \sigma {\lambda _R}} \over {2 - \theta }}, eRS*=1−c−2λMσ+σλR2−θ. {e^{RS*}} = {{1 - c - 2{\lambda _M}\sigma + \sigma {\lambda _R}} \over {2 - \theta }}.

Thus, we have pRS*=mRS*+wRS*=1−σλM. {p^{RS*}} = {m^{RS*}} + {w^{RS*}} = 1 - \sigma {\lambda _M}.

Competitive supply chain

In the competitive supply chain, as shown in Figure 2, the demand of supply chain i satisfies that 1 − pi + pj + ei − ej + x, where i ∈ {1, 2} and j = 3 − i.

Fig. 2

Competitive supply chain

MS game

In this subsection, the expected profits of the manufacturers and the retailers are as follow: πMiMS=(wi−ci)(1−pi+pj+ei−ej+x)−12ei2, \pi _{{M_i}}^{MS} = ({w_i} - {c_i})(1 - {p_i} + {p_j} + {e_i} - {e_j} + x) - {1 \over 2}e_i^2, πRiMS=(pi−wi)(1−pi+pj+ei−ej+x), \pi _{{R_i}}^{MS} = ({p_i} - {w_i})(1 - {p_i} + {p_j} + {e_i} - {e_j} + x), where i ∈ {1,2} and j = 3 − i.

Then the MV's functions of the manufacturers and the retailers are as follow: MVπMiMS=E(πMiMS)−λMVar(πMiMS)=(wi−ci)(1−pi+pj+ei−ej)−12ei2−λMσ(wi−ci), MV\pi _{{M_i}}^{MS} = E(\pi _{{M_i}}^{MS}) - {\lambda _M}\sqrt {Var(\pi _{{M_i}}^{MS})} = ({w_i} - {c_i})(1 - {p_i} + {p_j} + {e_i} - {e_j}) - {1 \over 2}e_i^2 - {\lambda _M}\sigma ({w_i} - {c_i}), MVπRiMS=E(πRiMS)−λRVar(πRiMS)=(pi−wi)(1−pi+pj+ei−ej)−λRσ(pi−wi). MV\pi _{{R_i}}^{MS} = E(\pi _{{R_i}}^{MS}) - {\lambda _R}\sqrt {Var(\pi _{{R_i}}^{MS})} = ({p_i} - {w_i})(1 - {p_i} + {p_j} + {e_i} - {e_j}) - {\lambda _R}\sigma ({p_i} - {w_i}).

Thus, the altruistic retailers' utility function satisfy that URiMS=MVπRiMS+θMVπMiMS=[pi−wi+θ(wi−ci)](1−pi+pj+ei−ej)−λRσ(pi−wi)−θλMσ(wi−ci)−θ12ei2. \matrix{ {U_{{R_i}}^{MS}} \hfill & { = MV\pi _{{R_i}}^{MS} + \theta MV\pi _{{M_i}}^{MS}} \hfill \cr {} \hfill & { = [{p_i} - {w_i} + \theta ({w_i} - {c_i})](1 - {p_i} + {p_j} + {e_i} - {e_j}) - {\lambda _R}\sigma ({p_i} - {w_i}) - \theta {\lambda _M}\sigma ({w_i} - {c_i}) - \theta {1 \over 2}e_i^2.} \hfill \cr }

Similarly, we also consider the retailers' decisions at first. Since ∂2UR1MS∂p12=−2<0 {{{\partial ^2}U_{{R_1}}^{MS}} \over {\partial p_1^2}} = - 2 < 0 , |∂2UR1MS∂p12∂2UR1MS∂p1∂p2∂2UR2MS∂p2∂p1∂2UR2MS∂p22|=3>0. \left| {\matrix{ {{{{\partial ^2}U_{{R_1}}^{MS}} \over {\partial p_1^2}}{{{\partial ^2}U_{{R_1}}^{MS}} \over {\partial {p_1}\partial {p_2}}}} \hfill \cr {{{{\partial ^2}U_{{R_2}}^{MS}} \over {\partial {p_2}\partial {p_1}}}{{{\partial ^2}U_{{R_2}}^{MS}} \over {\partial p_2^2}}} \hfill \cr } } \right| = 3 > 0.

Then by solving ∂UR1MS∂p1=0 {{\partial U_{{R_1}}^{MS}} \over {\partial {p_1}}} = 0 and ∂UR2MS∂p2=0 {{\partial U_{{R_2}}^{MS}} \over {\partial {p_2}}} = 0 , we can obtain that the retail price pi(wi,wj,ei,ej)=13[3+ei−ej+2(1−θ)wi+(1−θ)wj+θ(2ci+cj)−3σλR], {p_i}({w_i},{w_j},{e_i},{e_j}) = {1 \over 3}\left[{3 + {e_i} - {e_j} + 2(1 - \theta){w_i} + (1 - \theta){w_j} + \theta (2{c_i} + {c_j}) - 3\sigma {\lambda _R}} \right], where i ∈ {1,2} and j = 3 − i.

Second, we consider the manufacturers' decisions. Substituting Eq. (28) into Eq. (26), we have MVπMiMS(w1,w2,e1,e2) MV\pi _{{M_i}}^{MS}({w_1},{w_2},{e_1},{e_2}) .

By assuming that θ<79 \theta < {7 \over 9} , we can obtain that ∂MVπM1MS(w1,w2,e1,e2)∂w12=−23(1−θ)<0, {{\partial MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial w_1^2}} = - {2 \over 3}(1 - \theta) < 0, |∂2MVπM1MS(w1,w2,e1,e2)∂w12∂2MVπM1MS(w1,w2,e1,e2)∂w1∂e1∂2MVπM1MS(w1,w2,e1,e2)∂e1∂w1∂2MVπM1MS(w1,w2,e1,e2)∂e12|=19(5−6θ)>0, \left| {\matrix{ {{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial w_1^2}}{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {w_1}\partial {e_1}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {e_1}\partial {w_1}}}{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial e_1^2}}} \cr } } \right| = {1 \over 9}(5 - 6\theta) > 0, |∂2MVπM1MS(w1,w2,e1,e2)∂w12∂2MVπM1MS(w1,w2,e1,e2)∂w1∂e1∂2MVπM2MS(w1,w2,e1,e2)∂w2∂w1∂2MVπM1MS(w1,w2,e1,e2)∂e1∂w1∂2MVπM1MS(w1,w2,e1,e2)∂e12∂2MVπM2MS(w1,w2,e1,e2)∂w2∂e1∂2MVπM1MS(w1,w2,e1,e2)∂w1∂w2∂2MVπM1MS(w1,w2,e1,e2)∂e1∂w2∂2MVπM2MS(w1,w2,e1,e2)∂w22|=−127(1−θ)(8−9θ)<0, \left| {\matrix{ {{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial w_1^2}}{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {w_1}\partial {e_1}}}{{{\partial ^2}MV\pi _{{M_2}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {w_2}\partial {w_1}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {e_1}\partial {w_1}}}{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial e_1^2}}{{{\partial ^2}MV\pi _{{M_2}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {w_2}\partial {e_1}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {w_1}\partial {w_2}}}{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {e_1}\partial {w_2}}}{{{\partial ^2}MV\pi _{{M_2}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial w_2^2}}} \cr } } \right| = - {1 \over {27}}(1 - \theta)(8 - 9\theta) < 0, |∂2MVπM1MS(w1,w2,e1,e2)∂w12∂2MVπM1MS(w1,w2,e1,e2)∂w1∂e1∂2MVπM2MS(w1,w2,e1,e2)∂w2∂w1∂2MVπM2MS(w1,w2,e1,e2)∂e2∂w1∂2MVπM1MS(w1,w2,e1,e2)∂e1∂w1∂2MVπM1MS(w1,w2,e1,e2)∂e12∂2MVπM2MS(w1,w2,e1,e2)∂w2∂e1∂2MVπM2MS(w1,w2,e1,e2)∂e2∂e1∂2MVπM1MS(w1,w2,e1,e2)∂w1∂w2∂2MVπM1MS(w1,w2,e1,e2)∂e1∂w2∂2MVπM2MS(w1,w2,e1,e2)∂w22∂2MVπM2MS(w1,w2,e1,e2)∂e2∂w2∂2MVπM1MS(w1,w2,e1,e2)∂w1∂e2∂2MVπM1MS(w1,w2,e1,e2)∂e1∂e2∂2MVπM2MS(w1,w2,e1,e2)∂w2∂e2∂2MVπM2MS(w1,w2,e1,e2)∂e22|=127(7−16θ+9θ2)>0. \left| {\matrix{ {{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial w_1^2}}{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {w_1}\partial {e_1}}}{{{\partial ^2}MV\pi _{{M_2}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {w_2}\partial {w_1}}}{{{\partial ^2}MV\pi _{{M_2}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {e_2}\partial {w_1}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {e_1}\partial {w_1}}}{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial e_1^2}}{{{\partial ^2}MV\pi _{{M_2}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {w_2}\partial {e_1}}}{{{\partial ^2}MV\pi _{{M_2}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {e_2}\partial {e_1}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {w_1}\partial {w_2}}}{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {e_1}\partial {w_2}}}{{{\partial ^2}MV\pi _{{M_2}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial w_2^2}}{{{\partial ^2}MV\pi _{{M_2}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {e_2}\partial {w_2}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {w_1}\partial {e_2}}}{{{\partial ^2}MV\pi _{{M_1}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {e_1}\partial {e_2}}}{{{\partial ^2}MV\pi _{{M_2}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial {w_2}\partial {e_2}}}{{{\partial ^2}MV\pi _{{M_2}}^{MS}({w_1},{w_2},{e_1},{e_2})} \over {\partial e_2^2}}} \cr } } \right| = {1 \over {27}}(7 - 16\theta + 9{\theta ^2}) > 0.

Therefore the Hessian Matrix is negative definite, if θ<79 \theta < {7 \over 9} . So we have wiMS*=21+3(1−θ)ci+(4−13θ+9θ2)cj−27θ−3(7−9θ)σλM7−16θ+9θ2, w_i^{MS*} = {{21 + 3(1 - \theta){c_i} + (4 - 13\theta + 9{\theta ^2}){c_j} - 27\theta - 3(7 - 9\theta)\sigma {\lambda _M}} \over {7 - 16\theta + 9{\theta ^2}}}, eiMS*=7−(1−θ)ci+(1−θ)cj−9θ−(7−9θ)σλM7−16θ+9θ2. e_i^{MS*} = {{7 - (1 - \theta){c_i} + (1 - \theta){c_j} - 9\theta - (7 - 9\theta)\sigma {\lambda _M}} \over {7 - 16\theta + 9{\theta ^2}}}. where i ∈ {1,2} and j = 3 − i.

Substituting Eqs. (29) and (30) into Eq. (28), we can obtain that piMS*=28+(3−5θ)ci+4(1−θ)cj−(7−9θ)(3λMσ+σλR)7−9θ, p_i^{MS*} = {{28 + (3 - 5\theta){c_i} + 4(1 - \theta){c_j} - (7 - 9\theta)(3{\lambda _M}\sigma + \sigma {\lambda _R})} \over {7 - 9\theta }}, where i ∈ {1,2} and j = 3 − i.
RS game

In this subsection, the MV's functions of the manufacturers and the retailers are as follow: MVπMiRS=(wi−ci)(1−wi+wj+−mi+mj+ei−ej)−σλM(wi−ci), MV\pi _{{M_i}}^{RS} = ({w_i} - {c_i})(1 - {w_i} + {w_j} + - {m_i} + {m_j} + {e_i} - {e_j}) - \sigma {\lambda _M}({w_i} - {c_i}), MVπRiRS=mi(1−wi+wj+−mi+mj+ei−ej)−σλMmi, MV\pi _{{R_i}}^{RS} = {m_i}(1 - {w_i} + {w_j} + - {m_i} + {m_j} + {e_i} - {e_j}) - \sigma {\lambda _M}{m_i}, where i ∈ {1,2} and j = 3 − i.

Therefore, the altruistic retailers' utility function are URiRS=MVπRiRS+θMVπMiRS=[mi+θ(wi−ci)](1−wi−mi+wj+mj+ei−ej)−λRσmi−θλMσ(wi−ci)−θ12ei2, \matrix{ {U_{{R_i}}^{RS}} \hfill & { = MV\pi _{{R_i}}^{RS} + \theta MV\pi _{{M_i}}^{RS}} \hfill \cr {} \hfill & { = [{m_i} + \theta ({w_i} - {c_i})](1 - {w_i} - {m_i} + {w_j} + {m_j} + {e_i} - {e_j}) - {\lambda _R}\sigma {m_i} - \theta {\lambda _M}\sigma ({w_i} - {c_i}) - \theta {1 \over 2}e_i^2,} \hfill \cr } where i ∈ {1,2} and j = 3 − i.

Similar to subsection 3.1.2, we consider the manufacturers' decisions at first, then consider the retailers' decisions. Since second derivative order of MVπMiRS MV\pi _{{M_i}}^{RS} regarding to wi and ei are as follow ∂MVπM1RS∂w12=−2<0, {{\partial MV\pi _{{M_1}}^{RS}} \over {\partial w_1^2}} = - 2 < 0, |∂2MVπM1RS∂w12∂2MVπM1RS∂w1∂e1∂2MVπM1RS∂w1∂e1∂2MVπM1RS∂e12|=1>0, \left| {\matrix{ {{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial w_1^2}}{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {w_1}\partial {e_1}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {w_1}\partial {e_1}}}{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial e_1^2}}} \cr } } \right| = 1 > 0, |∂2MVπM1RS∂w12∂2MVπM1RS∂w1∂e1∂2MVπM2RS∂e2∂w1∂2MVπM1RS∂w1∂e1∂2MVπM1RS∂e12∂2MVπM2RS∂e2∂e1∂2MVπM1RS∂w1∂e2∂2MVπM1RS∂e1∂e2∂2MVπM2RS∂e22|=−1<0, \left| {\matrix{ {{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial w_1^2}}{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {w_1}\partial {e_1}}}{{{\partial ^2}MV\pi _{{M_2}}^{RS}} \over {\partial {e_2}\partial {w_1}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {w_1}\partial {e_1}}}{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial e_1^2}}{{{\partial ^2}MV\pi _{{M_2}}^{RS}} \over {\partial {e_2}\partial {e_1}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {w_1}\partial {e_2}}}{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {e_1}\partial {e_2}}}{{{\partial ^2}MV\pi _{{M_2}}^{RS}} \over {\partial e_2^2}}} \cr } } \right| = - 1 < 0, |∂2MVπM1RS∂w12∂2MVπM1RS∂w1∂e1∂2MVπM2RS∂e2∂w1∂2MVπM2RS∂w2∂w1∂2MVπM1RS∂w1∂e1∂2MVπM1RS∂e12∂2MVπM2RS∂e2∂e1∂2MVπM2RS∂w2∂e1∂2MVπM1RS∂w1∂e2∂2MVπM1RS∂e1∂e2∂2MVπM2RS∂e22∂2MVπM2RS∂w2∂e2∂2MVπM1RS∂w1∂w2∂2MVπM1RS∂e1∂e2∂2MVπM2RS∂e2∂w2∂2MVπM2RS∂w22|=1>0. \left| {\matrix{ {{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial w_1^2}}{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {w_1}\partial {e_1}}}{{{\partial ^2}MV\pi _{{M_2}}^{RS}} \over {\partial {e_2}\partial {w_1}}}{{{\partial ^2}MV\pi _{{M_2}}^{RS}} \over {\partial {w_2}\partial {w_1}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {w_1}\partial {e_1}}}{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial e_1^2}}{{{\partial ^2}MV\pi _{{M_2}}^{RS}} \over {\partial {e_2}\partial {e_1}}}{{{\partial ^2}MV\pi _{{M_2}}^{RS}} \over {\partial {w_2}\partial {e_1}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {w_1}\partial {e_2}}}{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {e_1}\partial {e_2}}}{{{\partial ^2}MV\pi _{{M_2}}^{RS}} \over {\partial e_2^2}}{{{\partial ^2}MV\pi _{{M_2}}^{RS}} \over {\partial {w_2}\partial {e_2}}}} \cr {{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {w_1}\partial {w_2}}}{{{\partial ^2}MV\pi _{{M_1}}^{RS}} \over {\partial {e_1}\partial {e_2}}}{{{\partial ^2}MV\pi _{{M_2}}^{RS}} \over {\partial {e_2}\partial {w_2}}}{{{\partial ^2}MV\pi _{{M_2}}^{RS}} \over {\partial w_2^2}}} \cr } } \right| = 1 > 0.

Then we can obtain the wholesale prices and emission reduction levels satisfy that wi(mi,mj)=1+cj−mi+mj−σλM, {w_i}({m_i},{m_j}) = 1 + {c_j} - {m_i} + {m_j} - \sigma {\lambda _M}, ei(mi,mj)=1−ci+cj−mi+mj−σλM, {e_i}({m_i},{m_j}) = 1 - {c_i} + {c_j} - {m_i} + {m_j} - \sigma {\lambda _M}, where i ∈ {1,2} and j = 3 − i.

Substituting Eq. (35) and Eq. (36) into Eq. (34), we have UR1RS(m1,m2) U_{{R_1}}^{RS}({m_1},{m_2}) and UR2RS(m1,m2) U_{{R_2}}^{RS}({m_1},{m_2}) , respectively. Considering the first derivative order condition, i.e., ∂UR1RS(m1,m2)∂m1=0 {{\partial U_{{R_1}}^{RS}({m_1},{m_2})} \over {\partial {m_1}}} = 0 and ∂UR2RS(m1,m2)∂m2=0 {{\partial U_{{R_2}}^{RS}({m_1},{m_2})} \over {\partial {m_2}}} = 0 , we can obtain that the retail margins miRS*=(1−θ)(3−ci+cj−2θ)3−2θ+θσλM−σλR, m_i^{RS*} = {{(1 - \theta)(3 - {c_i} + {c_j} - 2\theta)} \over {3 - 2\theta }} + \theta \sigma {\lambda _M} - \sigma {\lambda _R}, where i ∈ {1,2} and j = 3 − i.

Substituting Eqs. (37) into Eqs. (35) and (36), respectively. We can obtain that wiRS*=1+ci−σλM−ci−cj3−2θ, w_i^{RS*} = 1 + {c_i} - \sigma {\lambda _M} - {{{c_i} - {c_j}} \over {3 - 2\theta }}, eRS*=1−σλM−ci−cj3−2θ, {e^{RS*}} = 1 - \sigma {\lambda _M} - {{{c_i} - {c_j}} \over {3 - 2\theta }}, where i ∈ {1,2} and j = 3 − i.

Thus, we have piRS*=miRS*+wiRS*=(2−θ)(3−2θ+cj)−(1−θ)ci3−2θ−σ[(1−θ)λM−λR], p_i^{RS*} = m_i^{RS*} + w_i^{RS*} = {{(2 - \theta)(3 - 2\theta + {c_j}) - (1 - \theta){c_i}} \over {3 - 2\theta }} - \sigma [(1 - \theta){\lambda _M} - {\lambda _R}], where i ∈ {1,2} and j = 3 − i.
Results analysis

In this section, we discuss the impact of risk aversion and power structure on the decisions.

By assuming that c1 = c2 = c, we can obtain the equilibrium strategies of the competitive supply chain. For MS game, the equilibrium retail price, the equilibrium wholesale price and the equilibrium emission reduction level are pMSN*=4+c−3σλM−σλR, {p^{MSN*}} = 4 + c - 3\sigma {\lambda _M} - \sigma {\lambda _R}, wMSN*=3+(1−θ)c−3σλM1−θ, {w^{MSN*}} = {{3 + (1 - \theta)c - 3\sigma {\lambda _M}} \over {1 - \theta }}, eMSN*=1−σλM1−θ. {e^{MSN*}} = {{1 - \sigma {\lambda _M}} \over {1 - \theta }}.

For RS game, the equilibrium retail price, the equilibrium wholesale price and the equilibrium emission reduction level are pRSN*=2+c−θ−(1−θ)σλM−σλR, {p^{RSN*}} = 2 + c - \theta - (1 - \theta)\sigma {\lambda _M} - \sigma {\lambda _R}, wRSN*=1+c−σλM, {w^{RSN*}} = 1 + c - \sigma {\lambda _M}, eRSN*=1−σλM. {e^{RSN*}} = 1 - \sigma {\lambda _M}.

Theorem 1

Considering the influence of retailers' risk aversion degree, it holds that wMS*, eMS*, wRS* and eRS* are all increasing with λR, while pMSN* and pRSN* decrease with λR.

Theorem 1 indicates that the higher the risk aversion of the retailer, the higher the wholesale price and emission reduction level of the non-competitive supply chain, and the lower the retail price of the competitive supply chain.
Theorem 2

Considering the influence of manufacturers' risk aversion degree, it holds that as the increases of λM, all the pMS*, wMS*, eMS*, pRS*, wRS*, eRS*, pMSN*, wMSN*, eMSN*, pRSN*, wRSN* and eRSN* also increase.

Theorem 2 reflects that the more risk aversion manufacturers are, the more they will restrain price increases and reduce emissions reduction inputs at the same time.
Theorem 3

By comparing the prices of MS and RS, it holds that pMS* > pRS*, wMS* > wRS*, pMSN* > pRSN* and wMSN* > wRSN*.

Theorem 3 shows that with the transfer of powers from the retailers to the manufacturers, the price will increase.
Theorem 4

By comparing the emission reduction levels of MS and RS, it holds that i) eMSN* eRSN*; ii) eMS* > eRS* if 34>θ>13 {3 \over 4} > \theta > {1 \over 3} ; iii) eMS* ≤ eRS* if 0≤θ≤13 0 \le \theta \le {1 \over 3} .

Theorem 4 reflects that compared with the retailer-led model, the manufacturer-led model is conducive to improving the level of carbon emission reduction. However, in a non-competitive supply chain, when the degree of altruistic behavior of the retailer is low, the emission reduction effect led by the manufacturer is not as good as the one led by the retailer.
Impact of competition

In this section, a series of numerical studies will be conducted. We assume that the variables follow c = 0.3, λM = 0.4, λR = 0.2 and σ = 1. We alter θ to discuss the difference from competitive model and non-competitive model.

Figure 3 shows that no matter in RS modes or in MS modes, the retail price in competitive supply chain is higher than it in the non-competitive model, i.e. pRSN* > pRS* and pMSN* > pMS*. Figure 4 reflects that in the RS and MS models, the wholesale price of the competitive supply chain is higher than that of the non-competitive situation, i.e., wRSN* > wRS* and wMSN* > wMS*. Figure 5 revels that whether in the RS model or the MS model, the carbon emission reduction level under the competitive supply chain is higher than that under the non-competitive situation, i.e., eRSN* > eRS* and eMSN* > eMS*. All the figures from Figure 3 to Figure 5 indicates that competition can make products have a higher price and a deeper degree of low carbon.

Fig. 3

Impact of altruistic preference on retail price

Fig. 4

Impact of altruistic preference on wholesale price

Fig. 5

Impact of altruistic preference on emission reduction level

Figure 6 and Figure 7 respectively points out that the MV function of the manufacturer and the altruistic utility of the retailer in a competitive supply chain are both higher than that in the case of non-competition supply chain. Figure 6 and Figure 7 also demonstrate that competition can bring more profits to manufacturers and retailers.

Fig. 6

Impact of altruistic preference on manufacturers' MV function

Fig. 7

Impact of altruistic preference on retailers' utility function
Concluding remarks

Considering a low carbon supply chain with risk aversion, this paper establishes four models under retailer's altruistic preference. The first is that manufacturer Stackelberg of non-competitive supply chain, the second is retailer Stackelberg of non-competitive supply chain, the third is that manufacturer-led competitive supply chain, and the last is that retailer-led competitive supply chain. The results are as follows:

First, we analyze the role of risk aversion. Results show that the more risk aversion manufacturers are, the lower the price and emission reduction level are; The more risk averse retailers are, the higher the wholesale price and emission reduction level in the non competitive model, and the lower the retail price in the competitive model.

Second, we compare the difference in power structure, and find that with the transfer of powers from the retailers to the manufacturers, the price will increase.

At last, we make a series of numerical experiments. It points out that competition can bring a higher price, a deeper degree of low carbon, and more profits to manufacturers and retailers.

Fig. 1

Non competitive supply chain
Non competitive supply chain

Fig. 2

Competitive supply chain
Competitive supply chain

Fig. 3

Impact of altruistic preference on retail price
Impact of altruistic preference on retail price

Fig. 4

Impact of altruistic preference on wholesale price
Impact of altruistic preference on wholesale price

Fig. 5

Impact of altruistic preference on emission reduction level
Impact of altruistic preference on emission reduction level

Fig. 6

Impact of altruistic preference on manufacturers' MV function
Impact of altruistic preference on manufacturers' MV function

Fig. 7

Impact of altruistic preference on retailers' utility function
Impact of altruistic preference on retailers' utility function

The notions and parameters

Symbol Description
p Retail price
w Wholesale price
m Retail margin m = p − w
e Emission reduction level
c Unit production cost
x Demand shock, with expected function E[x] = 0
λ Risk aversion coefficient
π Firms' profit
MS Superscripts, represent manufacturer Stackelberg model
RS Superscripts, represent retailer Stackelberg model
* Superscript, represents the optimality
i Subscript, represents the firm, i ∈ {1,2}
M Subscript, represents the manufacturer
R Subscript, represents the retailer

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