Green supply chain performance with cost learning and operational inefficiency effects

Journal of Cleaner Production xxx (2015) 1e18

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Journal of Cleaner Production journal homepage: www.elsevier.com/locate/jclepro

Green supply chain performance with cost learning and operational inefficiency effects Qiao Zhang, Wansheng Tang, Jianxiong Zhang* College of Management and Economics, Tianjin University, Tianjin 300072, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 April 2015 Received in revised form 17 October 2015 Accepted 17 October 2015 Available online xxx

Environmental issues are receiving significant attention from the public, accelerating the development of green supply chain in operations management. This study focuses on the green supply chain performance in a single manufacturer-retailer setup, in which the manufacturer determines energy efficiency level and wholesale price, while the retailer decides on sales price. The unit production cost is jointly affected by cost learning and operational inefficiency effects. Open-loop, feedback and myopic equilibria are subsequently derived, and the corresponding supply chain performance and profit distribution are analyzed. The main results show that forward-looking behavior is preferred to myopic one for channel members. Feedback equilibrium is beneficial to the manufacturer, but is harmful to the retailer. The supply chain efficiencies in forward-looking and myopic situations are lower than the static supply chain efficiency, which results from cost learning and operational inefficiency effects, as well as non-price variable. The manufacturer and the retailer extract more profits under certain conditions, i.e., high cost learning effect, large energy efficiency effectiveness, low operational inefficiency effect or discount rate. However, the corresponding supply chain efficiencies decrease, and the manufacturer's profit proportions are less than 2/3. Investigation of supply chain performance with competition from multiple retailers implies that the effects of system parameters on supply chain performance keep the same directions of single manufacturer-retailer setup, while competition improves supply chain efficiency and manufacturer's profit proportion. This work contributes in involving cost learning and operational inefficiency effects simultaneously in a dynamic environment to explore supply chain performance. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Green supply chain Cost learning Operational inefficiency Energy efficiency level Differential games Supply chain performance

1. Introduction During the past few decades, environmental protection has emerged as the hottest topic in global sustainable development. Spurred by the increasing public environmental awareness, many manufacturers have been actively engaging in designing and producing environmentally friendly products. For the manufacturing firms of appliance, the effective way is to produce high energy efficient products through technology investment. In the production process, the firm's unit production cost is mainly affected by two aspects. One is cost learning effect, first reported by Wright (1936), who observes that the direct labor cost of manufacturing an airframe falls by 20% with every doubling of cumulative output. The other is operational inefficiency effect, referring to the

* Corresponding author. E-mail addresses: [email protected] (Q. Zhang), [email protected] (W. Tang), [email protected] (J. Zhang).

phenomenon that improving energy efficiency level embraces large amount of operational challenges such as new control and standards, training, and trials-all of which inevitably increase the unit production cost. As such, on one hand, the manufacturer expects to set a high energy efficiency level to attract more demand aiming at benefiting from cost learning effect; on the other, it tends to avoid a too high energy efficiency level to mitigate operational inefficiency effect. Meanwhile, for the channel members in a green supply chain, there are two behaviors can be adopted: forward-looking and myopic. The former refers to the situation where the channel member takes into account the system dynamics, while the latter denotes that the channel member ignores the dynamic evolution and focuses on short-term profit (Benchekroun et al., 2009). Besides, open-loop and feedback equilibria in forward-looking scenario are alternative, which respectively indicates that the strategies vary with time only, and the strategies are functions of state variable and time.

http://dx.doi.org/10.1016/j.jclepro.2015.10.069 0959-6526/© 2015 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Zhang, Q., et al., Green supply chain performance with cost learning and operational inefficiency effects, Journal of Cleaner Production (2015), http://dx.doi.org/10.1016/j.jclepro.2015.10.069

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With the research background and different games, a few research questions arise to explore the impacts of strategic choice on the supply chain performance: 1) How do the cost learning and operational inefficiency effects influence the supply chain performance? 2) Is feedback equilibrium beneficial to both of the channel members? 3) How the profits of the whole supply chain are distributed between the manufacturer and the retailer? To answer these research questions, a bilateral monopoly setting is developed, of which the manufacturer is responsible to set energy efficiency level and wholesale price, and the retailer is responsible to set sales price. The demand is assumed to be affected by sales price and energy efficiency level, and unit production cost is supposed to be jointly affected by cost learning and operational inefficiency effects. The open-loop, feedback and myopic equilibria are derived, and the impacts of cost learning and operational inefficiency effects on supply chain performance are analyzed, contributing to the following results: 1) The supply chain efficiencies under both forward-looking and myopic behaviors are lower than the counterpart in static supply chain, resulting from cost learning and operational inefficiency effects, and non-price decision variable. 2) Feedback equilibrium is beneficial to the manufacturer, but is harmful to the retailer. 3) The manufacturer and the retailer extract more profits under certain conditions: high cost learning effect, large energy efficiency effectiveness, low operational inefficiency effect or discount rate. However, the corresponding supply chain efficiency decreases. 4) The manufacturer's profit proportion is less than 2/3, i.e., profit of the manufacturer is less than twice that of the retailer. Competition is also introduced to study supply chain performance, and results indicate that the sensitivity analysis of system parameter on supply chain performance is robust, and that competition improves supply chain efficiency and manufacturer's profit proportion. The remainder of this paper is organized as follows. Section 3 describes the formulation of the model, then follows Section 4, which describes the optimal strategies in integrated supply chain. The equilibrium strategies of open-loop, feedback and myopic games in decentralized supply chain are summarized in Section 5, and the pricing and energy efficiency level strategies across different scenarios are compared in Section 6. The numerical analysis of supply chain performance is conducted in Section 7. As an extension, the supply chain performance in competitive environment is discussed in Section 8. This paper is concluded in Section 9, where the main results and future research directions are presented. 2. Literature review The related literature with our work spans across three streams: energy efficiency, cost learning effect and differential games in management science. Energy efficiency refers to using less energy to produce the same amount of services or useful output (Patterson, 1996). For home appliance, high energy efficient products generate less pollution to environment, which enhances the manufacturer's competitive advantage as well (Tseng et al., 2013). To eliminate inefficient appliances, Chinese government established equipment energy efficiency standards in 1989 that mandated maximum allowable energy consumption for 30 types products (Price et al., 2011; Rock, 2012). US “eco-labeling” program (Mason, 2006; Waechter et al., 2015) and Chinese “China Energy Label” (Zhan et al., 2011) were introduced to encourage energy conservation by making energy information easily accessible to consumers. The labeling display and energy efficiency compliance are generally high across regions and most products, pointed out by Khanna et al. (2013). As electric motors account for almost half of the electricity consumption in the

industrial sector in Malaysia, Mahlia and Yanti (2010) conduct the cost analysis and emission reduction by implementing energy efficiency standards for electric motor, and prove the remarkable benefit to consumers, manufacturers, government and environment by implementing energy efficiency standards. Furthermore, Shi (2014) draws lessons from energy efficiency standards and labeling regulations in the AsiaePacific region, and propose necessary components for setting effective regulations: clear liabilities, authoritative administration, open principles for technical systems, and enforceable mechanisms. In response to the government's policy, the manufacturers mainly take green activities in the process of procurement and production. Green procurement (GP), used interchangeably with green purchasing, environmental purchasing, is an instrument able to shape production and consumption trends, thus enlarging the markets for environmentally-friendly products (Li and Geiser, 2005; Rizzi et al., 2014). Appolloni et al. (2014) further carry out a comprehensive literature review on GP in the private sector based on GP's motivation, drivers, barriers and performance impacts. Specifically, machine tool is the basic energy consumed device in manufacturing system. Purchasing high energy efficient machine tool contributes to energy saving and sustainability. As such, Zhou et al. (2015) first discuss the connotation energy efficiency of machine tools, and introduce the design, scheduling, optimization and assessment based on energy efficiency of machine tools. The existing energy consumption models are divided into three categories, the corresponding application characteristics are summarized. In the process of production, advanced technology is the key to improve energy efficiency level of products. Pons et al. (2013) find a significant positive relationship between energy and material saving technologies and environmental performance. Stressing the importance of energy efficiency technologies in heavy industrial sector, Cao et al. (2015) address the ineffectiveness and inefficiency of the one-size-fits-all policy design scheme. Literature above primarily concentrates on energy efficiency from empirical perspective. Meanwhile, considerable amount of theoretical models pertaining to this area are established. Herein, energy efficiency level of product can be regarded as eco-friendly level or greening level. Liu et al. (2012) consider eco-friendly level competition between partially substitutable products from different manufacturers, and price competition between retail stores. The results show that retailers and manufacturers with superior eco-friendly operations benefit from high consumer's environmental awareness. When the eco-friendly level competition is low, the profitability of the inferior eco-friendly firm increases, on the contrary, it tends to decrease. Ghosh and Shah (2014) explore the impact of cost sharing contract on product greening level, price and profits, and the impact of greening costs and consumer sensitivity on green products. They state that cost sharing contract offered by the retailer or obtained through bargaining results in a high greening level for product, and brings about more surplus for the green supply chain. Xie (2015) outlines the impact of the threshold value of energy saving levels set by policy maker on actual energy saving levels and price in integrated and decentralized settings. Wholesale pricing and profit sharing schemes and a lump sum transfer contract are applied to coordinate the supply chain. Zhang et al. (2015) look into the impact of consumer environmental awareness on order quantities and channel coordination in a one-manufacturer and one-retailer setting, where the manufacturer produces two types of products that differ in their price and environmental quality. The closed-form expressions of optimal solutions are derived without production capacity constraint, following the corresponding solutions with constraint. Considering a dual-channel green supply chain, Li et al. (2015a,b) discuss the pricing and greening strategies in both centralized and

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decentralized settings under a consistent pricing strategy, and further compare the channel members' profits in decentralized single channel and dual-channel respectively. A pioneering study on learning effect is made by Wright (1936), who finds that the time required for the construction of an aircraft decreases with the number of aircrafts already produced. Learning effect facilitates employee's behavioral change, resulting in improved performance (Chen and Chang, 2010). Subsequent studies of learning effect, a fundamental part of management science, extend the learning to cost reduction with regard to cumulative production (Yelle, 1979), which is labeled as cost learning effect. As cost reductions and performance improvements of new technologies are closely linked to policies aiming at increasing production and deployment, Huenteler et al. (2014) use a case study of Thailand's electricity sector to create realistic estimates for the relative contributions of local and global technological learning to reduce costs. From a mathematical modeling point, the formulation of cost learning effect is assumed to be linear for tractability. Gray et al. (2009) assess the impact of linear cost learning effect on outsourcing decisions in the situation where the contract manufacturer has the power to behave strategically, and verify the importance of considering cost learning when making outsourcing decisions. Li et al. (2015a,b) model a two-period supply chain consisting of a manufacturer and a retailer. Specifically, the manufacturer's production cost in the second period is linearly decreasing in the first-period production, but with a random learning rate. They unveil the effects of mean learning rate and learning rate variability on pricing, production, procurement de
cisions. Researchers such as Raman and Chatterjee (1995), Alvarez and Cerd
a (1999), Nemet (2006) adopt linear cost learning effect. However, on one hand, cost learning curves are typically nonlinear in practice, on the other hand, linear learning may lead to negative costs. Therefore, some researchers turn to nonlinear form. Xiao and Gaimon (2013) assume that both the buyer and supplier share the cost learning effect in a power form, and introduce another learning concept-the future value. They highlight the impact of learning effect on buyer's outsourcing level as well as supplier's decisions. Janssens and Zaccour (2014) assume a hyperbolic cost learning effect in the study of price subsidies of new technologies. They obtain the optimal price and government subsidy paths under a budget constraint and focuses on strategic decision making. In addition, Dolan and Jeuland (1981) model cost learning effect as an exponential form, and discuss pricing problem given evolutionary forces for supply and demand. In the differential game, there are two alternative equilibrium strategies: open-loop and feedback. Generally, the feedback equilibrium is preferred for the decision maker. Cellini and Lambertini (2004) dwell on the closed-loop, feedback and open-loop concepts: closed-loop and feedback equilibria are both strongly time consistent and subgame perfect, where players decide ‘by the stock’ of state variables. Open-loop rule means players choose their respective plans at the initial date and commit to them forever, which is only weakly time consistent and not subgame perfect. Xu et al. (2011) study the impacts of cost learning effect on dynamic pricing and channel efficiency in a bilateral monopoly, implying that the feedback retail price is relatively lower than the open-loop price; also the feedback equilibrium improves the channel efficiency relative to the open-loop case. Similar results are also discovered by Chiang (2012), which manifests that the feedback equilibrium is more popular for channel members than the openloop one. However, in a differential duopolistic game where price is sticky and firms can invest in market-enlarging promotional activities, Piga (2000) reveals that the feedback price is higher than the open-loop one under nonlinear Markov perfect feedback strategies, meanwhile, the profits are higher in the open-loop

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equilibrium due to a weak free-ride effect. Dockner and Fruchter (2014) further analyze a decentralized production and marketing decision in a differential game framework, and discuss supply chain coordination through a transfer price in open-loop and feedback structures. Their results state that the dynamic transfer price can fully coordinate decentralized decision making with open-loop strategies, but fails to fully eliminate overall inefficiencies arising from strategic interactions in feedback case. All above works verify that the open-loop equilibrium and feedback equilibrium do not coincide. However, Fershtman (1987) examines the structure of differential games and specifies the conditions where an open-loop equilibrium is also a feedback equilibrium. Lambertini and Palestini (2014) also indicate that when a firm's capital accumulation follows linear growth dynamics, the open-loop equilibrium coincides with the feedback equilibrium. In addition, the study of myopic and forward-looking behaviors also attracts significant attention from researchers in the area of differential game. To identify the conditions under which a myopic pricing behavior is a profit enhancing tool, Benchekroun et al. (2009) formulate a differential game in a bilateral monopoly where channel members determine transfer and retail prices. They conclude that myopia enhances total channel profit when the reference price effect is small enough, and it remains true under competition from manufacturers when products are differentiated enough. Gutierrez and He (2011) examine the manufacturer's preference over the retailer's profitability behaviors: forwardlooking and myopic, yielding that the manufacturer is better off with a forward-looking retailer when the market saturation level is low and is better off with a myopic one when the market saturation level is high. Considering a dynamic two-echelon supply chain where the manufacturer controls wholesale price and quality in
n et al. vestment and the retailer controls retail price, Martín-Herra (2012) investigate the impact of retailer's myopia on pricing strategies and profits of channel members. Although myopic behavior is typically believed to be detrimental, the findings of Chiang (2012) show that firms can benefit from myopic behavior when distributing a durable product in a supply chain; also, decentralization may improve supply chain profitability under myopic pricing. This work is closely related to Xu et al. (2011) and Chiang (2012). Xu et al. (2011) build a Stackelberg differential game in a bilateral monopoly with price-dependent demand dynamics and linear cost learning effect, where the two channel members dynamically determine the wholesale and retail prices to maximize their individual profits. They focus on the impact of cost learning effect on dynamic pricing and channel efficiency, unveiling that cost learning effect is beneficial to both channel members from the view of profitability, but is harmful to the channel efficiency. Chiang (2012) stresses pricing problem in a manufacturer-retailer dyad under open-loop, feedback and myopic games, also further discusses the channel efficiency in different scenarios. This paper differs from theirs in three major respects. First, apart from pricing decision, energy efficiency level decision for the manufacturer is introduced. Second, the unit production cost incurred by the manufacturer not only linearly decreases with the cumulative production quantity, but also increases with energy efficiency level, which are known as cost learning effect and operational inefficiency effect, respectively. Third, pricing and energy efficiency level strategies under various differential games are derived and compared. The impacts of cost learning and operational inefficiency effects on supply chain performance, as well as on manufacturer's profit proportion are explored in detail. 3. Model Consider a two-echelon green supply chain in which a manufacturer produces and distributes products through an exclusive

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retailer. Suppose that the manufacturer controls the energy efficiency level x(t) and the wholesale price w(t), and the retailer determines the sales price p(t). Assume that the manufacturer's production quantity for each time equals to the retailer's order quantity, which accords with the demand rate, denoted by D(t), and the demand dynamics is specified by the following equation:

DðtÞ ¼ Q_ ðtÞ ¼ a  bpðtÞ þ gxðtÞ; Q ð0Þ ¼ 0;

(1)

where Q(t) is the cumulative product sales (quantity) at time t, a > 0 is the market potential, b > 0 symbolizes the effect of sales price on demand, and g > 0 is the energy efficiency effectiveness parameter, which represents the demand expansion effect of the energy efficiency level. Without loss of generality, there is no sale in the initial period. The increasing environmental awareness drives consumer's preference to environmentally friendly products (Sarkis, 2003; Young et al., 2010; Krass et al., 2013), which undoubtedly gives rise to a positive relationship between demand and energy efficiency level. This evolution captures the dependence of demand on both price and non-price variables in a tractable form of deterministic linear expression, which has been widely used in operations management and marketing literature (Gao et al., 2015; Karray, 2015; Zhang et al., 2015). With the cost learning effect, the unit production cost c(t) decreases with the cumulative production quantity Q(t), however, it increases with the improvement of energy efficiency level, meaning that any increase in energy efficiency level leads to a higher production cost. Hence, the unit production cost is characterized as a function of Q(t) and x(t), i.e.,

ZT Jm ¼

 ert ðwðtÞ  c0 þ c1 Q ðtÞ  c2 xðtÞÞða  bpðtÞ þ gxðtÞÞ

0

 k  x2 ðtÞ dt; 2 (4)

ZT Jr ¼

ert ðpðtÞ  wðtÞÞða  bpðtÞ þ gxðtÞÞdt;

(5)

0

with r being discount rate. When r is excessively large to approach infinity, one has to pursue maximum instantaneous profit, which is a myopic behavior, and is equivalent to a static case.

4. Optimal pricing and energy efficiency level strategies In this section, a benchmark is introduced where the manufacturer and the retailer are vertically integrated as a monopolist, and follows a forward-looking strategy. With demand and cost functions in (1)e(2), the objective of the integrated supply chain is to find the optimal sales price and energy efficiency level while maximizing the channel profit, denoted as PF , which is formulated as

ZT F

P ¼ max

pð,Þ;xð,Þ

  k ert ðpðtÞ  c0 þ c1 Q ðtÞ  c2 xðtÞÞQ_ ðtÞ  x2 ðtÞ dt 2

0

s:t: Q_ ðtÞ ¼ a  bpðtÞ þ gxðtÞ; Q ð0Þ ¼ 0: cðtÞ ¼ c0  c1 Q ðtÞ þ c2 xðtÞ;

where c0 is the initial unit cost, c1 is the cost learning coefficient, c2 measures the effect of energy efficiency level on cost, herein is called operational inefficiency coefficient. To ensure positive sopffiffiffiffiffiffiffiffi lutions, c2 should satisfy ðg  2bk=bÞþ < c2 < g=b, and a large value of c2 means a high inefficiency of production process. Similar to Xu et al. (2011), the linear cost learning form is also adopted in order to obtain analytical results. Additionally, the linear assumption with regard to non-price variable, such as quality € ro € s (2006) and De improvement, has been widely taken, such as Vo Giovanni (2011). Specifically, when c1 ¼ c2 ¼ 0, the effects of cost learning and operational inefficiency are disregarded, making the unit production cost convert from a time-varying function to a fixed one. Common in the marketing literature (Eyland and Zaccour, 2014; Jørgensen and Zaccour, 2014; Liu et al., 2014), the cost for investing energy efficiency level is assumed as the following form.

k CðxÞ ¼ x2 ; 2

(6)

(2)

(3)

where k > 0 is the investment cost coefficient. This convex cost implies that, to improve energy efficiency level, the manufacturer obtains “low hanging fruit” initially, but it has to pay more on the further improvement, which reflects the diminishing returns from investment. Assuming a finite planning horizon T and a profit-maximization behavior, the manufacturer's and the retailer's objective functionals are

In order to solve the optimization problem (6), the optimal control theory (Sethi and Thompson, 2000) is applied and the current value Hamiltonian function is formed as

HðQ ; p; x; l; tÞ ¼ ðpðtÞ  c0 þ c1 Q ðtÞ  c2 xðtÞ þ lðtÞÞða  bpðtÞ k þ gxðtÞÞ  x2 ðtÞ; 2 (7) where lðtÞ is the shadow price associated with the state variable Q(t), and satisfies the adjoint equation

vH _ ¼ rlðtÞ  c1 ða  bpðtÞ þ gxðtÞÞ; lðTÞ ¼ 0: lðtÞ ¼ rlðtÞ  vQ

(8)

The economic interpretation of lðtÞ refers to the impact of selling an additional unit on future profit. A positive shadow price implies the monopolist benefits from current sales by lowering the price or increasing energy efficiency level, which sacrifices the current profit for future profit, and vice versa. The following proposition presents the optimal pricing and energy efficiency level strategies. All proofs for subsequent propositions and corollaries are put in Appendix for clarity. Proposition 1. The optimal sales price, energy efficiency level and accumulated sales for a forward-looking monopolist are

pF ðtÞ ¼

r1 k1 ðgðg  bc2 Þ  bkÞer1 t 2

r1 q  c1 b k

þ

r2 k2 ðgðg  bc2 Þ  bkÞer2 t r2 q  c1 b2 k

a þ ; b (9)

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F

x ðtÞ ¼

r1 k1 bðg  bc2 Þer1 t

Q F ðtÞ ¼

r1 q  c1 b2 k k1 b2 ker1 t 2

r1 q  c1 b k

þ

þ

r2 k2 bðg  bc2 Þer2 t r2 q  c1 b2 k

k2 b2 ker2 t 2

r2 q  c1 b k



a  bc0 ; c1 b

;

(10)

(11)

and the shadow price is given by r1 t

lðtÞ ¼ k1 e

r2 t

þ k2 e

;

(12)

where q; r1 ; r2 ; k1 ; k2 are constants respectively given in (57), (60), (63) and (64) in Appendix. As expected, the optimal sales price and energy efficiency level vary over time owing to the cost learning and operational inefficiency effects. Moreover, the optimal solutions share the following properties. Corollary 1. The optimal pricing strategy depends on the system parameters: When bk < gðg  bc2 Þ, the optimal sales price increases over time, i.e., price-penetration strategy; when bk > gðg  bc2 Þ, the optimal sales price decreases over time, i.e., price-skimming strategy; when bk ¼ gðg  bc2 Þ, the optimal sales price is a constant equalling a=b during the entire planning horizon. However, regardless of the parameter values, the energy efficiency level is increasing over time, and the shadow price is positive. This corollary indicates that the pricing strategy depends on the price sensitivity, investment cost, operational inefficiency, and energy efficiency effectiveness (i.e., b; k; c2 ; g). When the parameters satisfy that bk < gðg  bc2 Þ, the sales price monotonically increases over time, i.e., price-penetration strategy. This condition is more likely to hold when g is relatively large, and c2 is relatively small. This suggests that when the demand expansion of energy efficiency level is pronounced, and the operational inefficiency effect is relatively low, the firm prefers to charge a low initial price and then gradually raise price, which enables it to occupy more market share in early periods, and reap more profit in the future due to the increasing sales price and relatively high demand expansion of energy efficiency level. On the contrary, when bk > gðg  bc2 Þ, the sales price decreases over time, i.e., priceskimming strategy. Obviously, a relatively small g and a relatively large c2 make the condition more likely to be satisfied. This shows that a small demand expansion from energy efficiency level and a high operational inefficiency encourage the firm to start with a high initial price to capture the immediate profit rather than sacrificing current profit for future profit. Afterwards, the firm tends to cut sales price to maintain a relatively high demand on account of a relatively small demand expansion of energy efficiency level. However, the optimal energy efficiency level is increasing over the entire planning horizon. More specifically, it can be found from (54) that the initial energy efficiency level xð0Þ ¼ ðbðk1 þ k2 Þ þ a  bc0 Þðg  bc2 Þ=q, which derives that vxð0Þ=vT > 0. This indicates that a forward-looking monopolist would like to set a higher initial energy efficiency level when the planning horizon is relatively long. The intuitive interpretation is that, since the energy efficiency effectiveness is relatively larger than the operational inefficiency, i.e., g > bc2 , a higher initial energy efficiency level leads to higher cumulative sales during a long planning horizon, making the cost learning effect dominate on unit production cost. Meanwhile, the high marginal profit is able to offset the relatively large investment cost from energy efficiency level. Note that the shadow price is positive over the whole planning horizon, implying that the monopolist benefits from selling an additional unit product.

5

In addition, when the integrated supply chain is myopic, the monopolist disregards the evolution of demand and maximizes its instantaneous profit, denoted as PM . Subject to the demand dynamic (1), the optimization problem is given by

k PM ¼ max ðpðtÞ  c0 þ c1 Q  c2 xðtÞÞQ_ ðtÞ  x2 ðtÞ: 2 pð,Þ;xð,Þ

(13)

For the static optimization problem (13), the following proposition characterizes the optimal solutions. Proposition 2. The optimal sales price, energy efficiency level and accumulated sales for a myopic monopolist are

  ða  bc0 Þðgðg  bc2 Þ  bkÞ c1 b2 k t e q 1 qb

pM ¼

þ

xM ¼

ðac2  gc0 Þðg  bc2 Þ þ kða þ bc0 Þ ; q

ða  bc0 Þðg  bc2 Þ c1 b2 k t e q ; q

QM ¼

  a  bc0 c1 b2 k t e q 1 : c1 b

(14)

(15)

(16)

Similarly, the optimal pricing and energy efficiency level strategies with myopic behavior enjoy the same properties with those in forward-looking case. In other words, the monopolist adopts price-penetration strategy if bk < gðg  bc2 Þ, otherwise, it makes price-skimming strategy. In particular, when bk ¼ gðg  bc2 Þ, the firm maintains a constant sales price a=b. Also, the optimal energy efficiency level keeps increasing over time. The following result is generated when comparing pricing, energy efficiency level strategies, as well as cumulative sales across forward-looking and myopic rules. Corollary 2. When bk < gðg  bc2 Þ, the forward-looking and myopic monopolists both take price-penetration strategy, but the forward-looking sales price is always higher than the myopic one; when bk > gðg  bc2 Þ, the monopolists both take price-skimming strategy, but the forward-looking sales price is always lower than the myopic one; when bk ¼ gðg  bc2 Þ, the monopolists both maintain a constant sales price a=b. Besides, the forward-looking energy efficiency level and cumulative sales are always greater than those in the myopic case. Since the shadow price in forward-looking scenario is positive, it can be inferred from (55) that the demand rate with forwardlooking strategy is higher in comparison to that in the myopic case, which leads to higher cumulative sales in forward-looking scenario. Consequently, as seen from (53), the relationship between two distinct pricing rules depends on system parameters. If bk < gðg  bc2 Þ, the forward-looking sales price is relatively high; however, if the opposite of the inequality holds, the myopic sales price is higher. Specifically, when the two terms are equal, there is no difference between sales prices. Likewise, positive shadow price and higher cumulative sales also create higher energy efficiency level for forward-looking case. In a word, although the forwardlooking sales price is higher than the myopic one under certain conditions, the energy efficiency level with forward-looking strategy is much greater than its myopic counterpart, leading to higher demand rate and consequently to higher cumulative sales. This shows that a forward-looking firm is supposed to improve energy efficiency level so as to boost instantaneous demand for more future profit.

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5. Equilibrium pricing and energy efficiency level strategies This section analyzes a decentralized supply chain where the manufacturer controls wholesale price and energy efficiency level while the retailer sets sales price. With profit-maximization behavior, the differential game consisting of the manufacturer and the retailer can be described as

ZT max

wð,Þ;xð,Þ

 ert ðwðtÞ  c0 þ c1 Q ðtÞ  c2 xðtÞÞða  bpðtÞ þ gxðtÞÞ

0

ZT max pð,Þ

 k  x2 ðtÞ dt 2

as the follower, determines its response sales price, its forwardlooking or myopic choice comes to the same thing. With the first-order condition vHr =vp ¼ 0, the response sales price p ¼ a þ gx þ bw=2b. This demonstrates that the sales price is positively affected by the energy efficiency level, meaning that the high energy efficiency level enables the retailer to charge a high sales price. Taking the retailer's best response into account, a forwardlooking manufacturer's optimization problem is given as

ZT Pm ¼ max

wð,Þ;xð,Þ

 ert ðwðtÞ  c0 þ c1 Q  c2 xðtÞÞða  bpðtÞ

0

 k þ gxðtÞÞ  x2 ðtÞ dt; 2

ert ðpðtÞ  wðtÞÞða  bpðtÞ þ gxðtÞÞdt

(21)

0

s:t:Q_ ðtÞ ¼ a  bpðtÞ þ gxðtÞ; Q ð0Þ ¼ 0: (17) This problem is modeled as a Stackelberg differential game where the manufacturer acts as the leader and the retailer as the follower. Three games are discussed to analyze the corresponding equilibrium strategies: open-loop, feedback, and myopic games.

subject to (1). Accordingly, the current Hamiltonian function of the manufacturer is

Hm ðQ ; w; x; lm ; tÞ ¼ ðwðtÞ  c0 þ c1 Q ðtÞ  c2 xðtÞ þ lm ðtÞÞða k  bpðtÞ þ gxðtÞÞ  x2 ðtÞ; 2 (22)

5.1. Open-loop equilibrium With the open-loop equilibrium concept and forward-looking strategy, the manufacturer first announces the trajectories of wholesale price w(t) and energy efficiency level x(t), then the retailer makes sales price decision p(t), which are only weakly time consistent since players make their decisions ‘by the clock’ only (Cellini and Lambertini, 2004). To sustain the equilibrium, an implicit assumption is that both players determine their respective control variables at the initial period and commit to them forever. The retailer's problem is first solved to identify its response function, and the manufacturer's equilibrium strategies are then derived through a backward induction. With the demand dynamics (1), the retailer determines the sales price while maximizing its net discounted profit, i.e.,

ZT Pr ¼ max pð,Þ

ert ðpðtÞ  wðtÞÞða  bpðtÞ þ gxðtÞÞdt:

where lm is the manufacturer's shadow price, and satisfies the following equation.

c l_ m ðtÞ ¼ rlm ðtÞ  1 ða þ gxðtÞ  bwðtÞÞ; lm ðTÞ ¼ 0: 2

The next proposition summaries the open-loop equilibrium strategies for the channel members. Proposition 3. When the manufacturer is forward-looking, the open-loop equilibrium sales price, wholesale price, energy efficiency level and accumulated sales respectively are

s1 k3 ðgðg  bc2 Þ  bkÞes1 t

pOL ðtÞ ¼

2

þ

a ; b

s1 h  c1 b k

þ

s2 k4 ðgðg  bc2 Þ  bkÞes2 t s2 h  c1 b2 k

(24) (18)

0

wOL ðtÞ ¼

s1 k3 ðgðg  bc2 Þ  2bkÞes1 t s1 h  c1 b2 k

The retailer's current value Hamiltonian function is

þ

Hr ðQ ; p; x; lr ; tÞ ¼ ðpðtÞ  wðtÞ þ lr ðtÞÞða  bpðtÞ þ gxðtÞÞ;

s2 k4 ðgðg  bc2 Þ  2bkÞes2 t s2 h  c1 b2 k

(19) where lr is the shadow price associated with the state variable Q(t), satisfying the adjoint equation

vHr ¼ rlr ðtÞ; lr ðTÞ ¼ 0: l_ r ðtÞ ¼ rlr ðtÞ  vQ

(23)

(20)

Obviously, it can be worked out from (20) that the shadow price lr ðtÞ≡0, meaning that the pricing strategy of the forward-looking retailer is the same with that of the myopic one. In other words, although the retailer has the choice of behaving either myopic or forward-looking, the two strategic choices have no influence on the decisions. Static demand serves as a better explanation for this result. Since the dynamic factor is included in the cost learning effect in the supply side, it only affects the manufacturer's decisions, but not that of the retailer. Undoubtedly, when the retailer,

xOL ðtÞ ¼

s1 k3 bðg  bc2 Þes1 t

Q OL ðtÞ ¼

2

s1 h  c 1 b k k3 b2 kes1 t 2

s1 h  c1 b k

þ

þ

þ

a ; b

s2 k4 bðg  bc2 Þes2 t s2 h  c1 b2 k

k4 b2 kes2 t s2 h  c1 b2 k



a  bc0 ; c1 b

(25)

;

(26)

(27)

and the shadow price is given by

lm ðtÞ ¼ k3 es1 t þ k4 es2 t ;

(28)

where h; s1 ; s2 ; k3 ; k4 are constants respectively given in (71), (75), (78) and (79) in Appendix. Similar to the integrated supply chain, there are two pricing strategies for the channel members: price-penetration and price-

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skimming, and the optimal choice lies on the system parameters. However, the manufacturer and the retailer do not always take the same pricing strategy. Corollary 3. The equilibrium pricing strategies for the manufacturer and the retailer depend on the system parameters: 1) When bk < gðg  bc2 Þ=2, both the manufacturer and the retailer take the price-penetration strategy. 2) When bk > gðg  bc2 Þ, both the manufacturer and the retailer take the price-skimming strategy. 3) When gðg  bc2 Þ=2 < bk < gðg  bc2 Þ, the manufacturer chooses the price-skimming strategy, but the retailer chooses the pricepenetration strategy. Generally, the retailer tends to follow the manufacturer to adopt the same pricing strategy, which is called as strategic complements. Put differently, if one of the channel members increases/decreases its price, correspondingly, the other member will react to increase/ decrease the price. Hence, it is an intuitive result that both the manufacturer and the retailer take price-penetration strategy when 2Þ bk < gðgbc , 2

and

take

price-skimming

strategy

(30)

Anticipating the retailer's sales price, the manufacturer's HJB equation is

rVm 

  vVm vVm ¼ max ðabpþgxÞ wc0 þc1 Q c2 xþ w;x vt vQ  k  x2 ;Vm ðT;Q Þ ¼ 0; 2 (31)

which yields feedback wholesale price and energy efficiency level for the manufacturer.

w¼

when

2Þ bk > gðg  bc2 Þ. However, when bk falls between gðgbc and 2 gðg  bc2 Þ, the manufacturer tends to decrease wholesale price, but the retailer prefers to increase sales price. This is based on the fact that the positive impact of increasing marginal profit outperforms the negative influence of demand shrink brought by sales price enhancement, leading to a final profit increment for the retailer. When bk is greater than gðg  bc2 Þ, the retailer does not raise sales price due to the dominance of negative impact from the decreased demand. Notably, when bk ¼ gðg  bc2 Þ, the open-loop sales price pOL ¼ a=b, equalling to that in the integrated channel, which implies that double marginalization effect is eliminated under this condition. Furthermore, if bk ¼ gðg  bc2 Þ=2, the wholesale price wOL ¼ a=b, showing that the manufacturer prefers to keep a constant wholesale price over the entire planning horizon. Predictably, the energy efficiency level increases over time, and the shadow price of the manufacturer is always positive, which are consistent with the integrated scenario. This means that the manufacturer benefits from high energy efficiency level because the increased profit from demand improvement exceeds the high investment cost. Also, the manufacturer is better off when it produces one more unit product.

  a þ gx 1 vVr þ w : 2b 2 vQ

p¼

7

  1 vVr vVm ðbc2 ðg  bc2 Þ þ 2bkÞ þ ðgðg  bc2 Þ  2bkÞ h vQ vQ   þ c1 Q þ ðac2  gc0 Þðg  bc2 Þ þ 2kða þ bc0 Þ ; (32)

x¼

   1 vVr vVm ðg  bc2 Þ a  bc0 þ b þ þ c1 Q : h vQ vQ

(33)

Therefore, the feedback sales price is given as

p¼

   1 vVr vVm ðgðg  bc2 Þ  bkÞ þ þ c1 Q h vQ vQ  þ ðac2  gc0 Þðg  bc2 Þ þ kð3a þ bc0 Þ :

(34)

Substitute these decisions into HJB equations to solve coefficients of value functions for channel members, which yields the following equilibrium strategies. Proposition 4. When both the manufacturer and the retailer are forward-looking, the feedback equilibrium sales price, wholesale price and energy efficiency level are respectively

1 ððgðg  bc2 Þ  bkÞððAr ðtÞ þ Am ðtÞ þ c1 ÞQ ðtÞ þ Br ðtÞ h

pFB ðtÞ ¼

þ Bm ðtÞÞ þ ðac2  gc0 Þðg  bc2 Þ þ kð3a þ bc0 ÞÞ; (35)

5.2. Feedback equilibrium Compared with the open-loop equilibrium, the feedback equilibrium is known to be time consistent, which allows the channel members dynamically adjust pricing and energy efficiency level strategies based on the status of cumulative demand. In this section, both channel members are assumed to be forward-looking. To investigate the impact of cost learning and operational inefficiency on feedback equilibrium strategies, the corresponding HJB equations for the channel members' optimization problems are established. With Vm and Vr as the value functions for the manufacturer and the retailer, respectively, the retailer's HJB equation is

rVr 

   vVr vVr ða  bp þ gxÞ ; Vr ðT; Q Þ ¼ 0: ¼ max pwþ p vt vQ (29)

Then, the response function of sales price is calculated as

wFB ðtÞ ¼

1 ðððbc2 ðg  bc2 Þ þ 2bkÞAr ðtÞ þ ðgðg  bc2 Þ  2bkÞ h  ðAm ðtÞ þ c1 ÞÞQ ðtÞ þ ðbc2 ðg  bc2 Þ þ 2bkÞBr ðtÞ þ ðgðg  bc2 Þ  2bkÞBm ðtÞ þ ðac2  gc0 Þðg  bc2 Þ þ 2kða þ bc0 ÞÞ; (36)

xFB ðtÞ ¼

1 ðg  bc2 Þða  bc0 þ bðBr ðtÞ þ Bm ðtÞÞ þ bðAr ðtÞ þ Am ðtÞ h þ c1 ÞÞQ ðtÞÞ; (37)

where Ai ; Bi ; i2fr; mg are given in (91) and (92) in Appendix, and the accumulated sales are given as

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Q ðtÞ ¼ eb1 t

ext

cext

þ 1þc

!b2 Z t

bk ða  bc0 h

0

þ bSBi ðuÞÞeb1 u

Proposition 5. When both the manufacturer and the retailer are myopic, the equilibrium sales price, wholesale price and energy efficiency level are respectively

exu þ cexu 1þc

!b2 du;

(38)

þ

where b1 ¼ rh=2ðh þ 2bkÞ; b2 ¼ h=h þ 2bk: Since the analysis of the feedback equilibria poses some degree of complexity, we resort to numerical analysis, and find that the relationship of sales price between feedback and open-loop strategies also rests upon the system parameters: when bk > gðg  bc2 Þ, the feedback sales price decreases over time and is always lower than the open-loop counterpart. When bk < gðg  bc2 Þ, the feedback sales price increases over time and is always higher than the open-loop counterpart. For instance, as seen from Fig. 1(a), bk > gðg  bc2 Þ when k ¼ 2. Under this parameter settings, the feedback sales price is lower than the open-loop price. When k ¼ 0.5 in Fig. 1(b), the opposite result occurs: the feedback sales price is higher than the open-loop counterpart. Cellini and Lambertini (2004) and Xu et al. (2011) state that the feedback equilibrium induces a lower sales price as compared with the openloop solution. Unlike their models, this paper simultaneously considers pricing and energy efficiency level strategies with the assumption of demand extension from energy efficiency level. Generally, one tends to argue that the combined effect of energy efficiency level and demand dynamic may generate a higher sales price in feedback form than the open-loop counterpart, the reason being that in the feedback case the channel members can adjust their strategies flexibly in each period. However, results show that the relationship of sales prices in feedback and open-loop strategies is ambiguous, which reflects a strong link to the parameters. Comparison of the energy efficiency levels under open-loop and feedback concepts unveils that the manufacturer is willing to set a relatively higher energy efficiency level in feedback form. This conforms to the fact that the feedback equilibrium, considering the strategic interactions of the manufacturer and the retailer through the evolution of accumulated sales, is strongly time consistent and therefore, subgame perfect (Chiang, 2012).

5.3. Myopic equilibrium Ignoring the evolution of cumulative sales, myopic equilibrium means that the manufacturer and the retailer only focus on immediate-term profits when they set sales price, wholesale price and energy efficiency level. With the myopic concept, the channel members act as if the planning horizon is reduced to one period, thus, they solve a static optimization problem to maximize the instantaneous profits. Likewise, with backward induction, the retailer's reaction function is

p¼

a þ gx þ bw : 2b

(39)

Taking the retailer's response into consideration, the manufacturer's optimization problem is specified as

max w;x

  ða  bc0 Þðgðg  bc2 Þ  bkÞ c1 bh2 k t e 1 bh

M e p ðtÞ ¼

1 ðw  c0 þ c1 Q  c2 xÞða  bp þ gxÞ  kx2 ; 2

(40)

subject to (1) and (39). The corresponding myopic equilibria are presented in the proposition below.

e M ðtÞ ¼ w

ðac2  gc0 Þðg  bc2 Þ þ kð3a þ bc0 Þ ; h

  ða  bc0 Þðgðg  bc2 Þ  2bkÞ c1 bh2 k t e 1 bh þ

M e x ðtÞ ¼

(41)

ðac2  gc0 Þðg  bc2 Þ þ 2kða þ bc0 Þ ; h

ða  bc0 Þðg  bc2 Þ c1 bh2 k t e ; h

(42)

(43)

and the accumulated sales are

 2  b kc t e M ðtÞ ¼ a  bc0 e h 1  1 : Q bc1

(44)

As compared to the forward-looking equilibria, a few observations can be obtained. First, the myopic sales price, wholesale price and energy efficiency level share the same monotonicity with the open-loop counterparts. Second, the myopic equilibria are insensitive to the discount rate, but the forward-looking equilibria, feedback and open-loop, are closely related to the discount rate.

6. Strategy comparison across scenarios Based on the results above, the pricing and energy efficiency level strategies across five scenarios are compared through numerical simulations in this section. Figs. 1 and 2 present pricing and energy efficiency level strategies across five scenarios under different parameter values, respectively. As seen from Fig. 1, there exist two different pricing strategies: price-skimming, price-penetration, and the choice depends on the relationship of bk and gðg  bc2 Þ. When bk > gðg  bc2 Þ, the channel members choose the price-skimming strategy; otherwise, they prefer price-penetration strategy. Furthermore, there are a few observations on the basis of Fig. 1(a). First, the myopic sales price is always higher than the forward-looking one, regardless of integration or decentralization. Second, the sales price of the decentralized channel is always higher than that in the integrated case, either with forward-looking or myopic behavior, which is labeled as double marginalization effect. Third, in the decentralized supply chain, the open-loop sales price is higher than the feedback counterpart, which implies that the open-loop rule aggravates double marginalization. However, Fig. 1(b) presents the opposite results. That is, the myopic sales prices are always lower than the forward-looking prices in integrated and decentralized channels. Decentralization does not necessarily lead to high sales price, on the contrary, the sales price in decentralized channel is lower relative to the integrated one. In addition, the open-loop sales price is relatively lower in comparison to the feedback price. The opposite results reflected from the figures show that pricing strategies are heavily dependent on system parameters b; k; c2 ; g. When energy efficiency effectiveness is relatively low, i.e., g ¼ 1 (see Fig. 1(a)), the channel members tend to start a high initial sales price to extract more profits. Meanwhile, the forward-looking sales prices are lower than the myopic prices, deriving from the fact that the forward-looking member benefits from selling one more

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Fig. 1. Pricing strategies across scenarios (FI: forward-looking integration; MI: myopic integration; MD: myopic decentralization).

product. However, with a high energy efficiency effectiveness, i.e., g ¼ 2 (see Fig. 1(b)), the channel members prefer to set a low initial sales price to boost demand. More specifically, the forward-looking prices are always higher than the myopic counterparts. For the forward-looking member, setting a high sales price not only capturers a high marginal profit, but also reaches a high demand due to high energy efficiency effectiveness. The following observations are obtained from Fig. 2. First, the energy efficiency level under forward-looking behavior is larger than the counterpart under myopic behavior, regardless of integration and decentralization. Since the shadow price is positive, the forward-looking supply chain benefits from selling one more product. As such, high energy efficiency level is provided to stimulate demand. Second, the integrated supply chain shares high

energy efficiency level in both forward-looking and myopic scenarios. It is intuitive that the integrated supply chain chooses a high energy efficiency level to pursue its maximum profits as a whole. Third, the energy efficiency level with myopic behavior in integrated case is much larger than that with forward-looking behavior under decentralized channel. Last, high energy efficiency level is much favored in feedback equilibrium than open-loop one. 7. Supply chain performance Given our focus on the impacts of cost learning and operational inefficiency effects, non-price decision, as well as strategic behavior (myopic or forward-looking) on supply chain profitability and channel performance, it is clear that the five most important parameters are c1 ; c2 ; g; T and r. More specifically, this section explores the effects of the five parameters on decentralized supply chain performance across different equilibrium solutions: feedback and open-loop equilibria with forward-looking rule, myopic equilibria. There follows discussions of how the five parameters affect the profit distribution between the manufacturer and the retailer. However, the intricacies of this problem throw up great challenges in acquiring intuitive results. Hence, numerical simulation is conducted to gain some valuable managerial insights. Denote e PFi ; POL i by the feedback and open-loop profits of player i, Pi by the myopic profit of player i, where i2fm; r; sg representing manue I denote the profits of facturer, retailer and supply chain. Also, PI ; P integrated supply chain in forward-looking and myopic strategies, respectively. For the subsequent numerical analysis, the following basic parameters are set as:

a ¼ 50; b ¼ 1; g ¼ 1; r ¼ 0:1; c0 ¼ 10; c1 ¼ 0:02; c2 ¼ 0:2; k ¼ 2; T ¼ 10:

7.1. Profitability of decentralization Fig. 2. Energy efficiency level strategies across (a ¼ 50; b ¼ 1; g ¼ 1; r ¼ 0:1; c0 ¼ 10; c1 ¼ 0:02; c2 ¼ 0:2; k ¼ 2; T ¼ 10).

scenarios

In this section, the performance of decentralization is assessed from the views of long term (forward-looking) and short term

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(myopic). Table 1 summaries the impacts of parameters c1 ; c2 ; g; T; r on profits of integrated and decentralized supply chains with both forward-looking and myopic behaviors. Figs. 3e7 depict the impacts of c1 ; c2 ; g; T; r on supply chain efficiency. Herein, supply chain efficiency is defined as the ratio of the decentralized channel's profit to the integrated channel's profit. The observations drawn out of Table 1 are the following:  From the perspective of integrated supply chain, the profit of forward-looking monopolist is higher in comparison to the myopic one, and the profit discrepancy increases with c1 ; g; T, decreases with r, but is insensitive to parameter c2. Specifically, with the value of c1 increasing from 0.01 to 0.04, the profit discrepancy increases greatly from 2.4 to 55.1, which indicates that forward-looking behavior benefits more from cost learning effect than the myopic one does. The increasing profit discrepancy with large g also reflects that, relative to myopic behavior, forward-looking strategy takes more advantages from high energy efficiency effectiveness. When the planning horizon T increases from 10 to 40, the profit discrepancy increases sharply from 10.9 to 111.3, implying that the longer the planning horizon is, the more profits the forward-looking monopolist gains. With the increase of discount rate r, the profit discrepancy decreases from about 11 to 0.1. When the discount rate is large enough, i.e., r ¼ 0.9, there is little difference between forward-looking profit and myopic one. Put another way, facing with a relatively high discount rate, channel members' choice upon forward-looking or myopic strategy reaches nearly the same performance. Indeed, the decision maker is inclined to choose myopic strategy with a large discount rate because it is difficult to accommodate intertemporal effects (forward-looking) in decision making, according to behavioral research (Chakravarti et al., 1979). However, the profit discrepancy is insensitive to operational inefficiency effect, keeping at about 11 with the increase of c2. In the decentralized supply chain, the individual profits generated by the forward-looking manufacturer and retailer are higher than those of the myopic channel members, regardless of which equilibrium concept, feedback or openloop, is applied. In brief, forward-looking behavior is preferred to myopic behavior for channel members, which

Fig. 3. Impact of c1 on supply chain efficiency.

comes down to the fact that the shadow price is positive, leading to a lower production quantity under myopic behavior, and eventually to a low profit.  The whole supply chain's profit with feedback equilibrium is higher than the open-loop counterpart, i.e., PFs > POL s . Specifically, the manufacturer benefits from the feedback equilibrium, i.e., PFm > POL m . However, the retailer is better off with the open-loop as compared with the feedback equilibrium, i.e., F POL r > Pr . Furthermore, the profit discrepancy of manufacturer between feedback and open-loop equilibria increases with c1 ; g; T, and decreases with r, but is insensitive to c2 . The profit discrepancy of retailer between open-loop and feedback equilibria follow the same rule. This shows that the manufacturer with feedback equilibrium gains more when the cost learning effect is large; the energy efficiency effectiveness is high; the planning horizon is long or the discount rate is low.

Table 1 Impacts of parameters c1 ; c2 ; g; T; r on channel members' profits. PV c1

c2

g

T

r

0.01 0.02 0.03 0.04 0.19 0.20 0.21 0.22 0.4 0.7 1.0 1.3 10 20 30 40 0.1 0.3 0.5 0.7 0.9

PFm

PFr

PFs

POL m

POL r

POL s

PI

em P

er P

es P

eI P

1421.9 1473.8 1530.1 1591.9 1477.4 1473.8 1470.3 1466.9 1352.2 1392.3 1473.8 1611.9 1473.8 2121.7 2406.2 2531.8 1473.8 722.3 446.7 318.9 247.1

772.8 801.0 830.9 863.8 804.7 801.0 797.4 793.8 679.0 718.1 801.0 949.8 801.0 1150.5 1306.6 1375.3 801.0 392.5 243.1 173.3 134.3

2194.7 2274.8 2361.0 2455.7 2282.1 2274.8 2267.7 2260.7 2031.2 2110.4 2274.8 2561.7 2274.8 3272.2 3712.8 3907.1 2274.8 1114.8 689.8 492.2 381.4

1406.1 1439.6 1474.8 1511.8 1443.0 1439.6 1436.4 1433.2 1326.4 1363.9 1439.6 1566.8 1439.6 2032.2 2276.5 2377.3 1439.6 710.3 441.2 315.1 244.5

782.0 819.7 860.3 904.0 823.5 819.7 816.0 812.4 695.8 735.8 819.7 970.9 819.7 1194.4 1364.9 1441.9 819.7 398.3 244.9 174.0 134.5

2188.1 2259.3 2335.1 2415.8 2266.5 2259.3 2252.4 2245.6 2022.2 2099.7 2259.3 2537.7 2259.3 3226.6 3641.4 3819.1 2259.4 1108.6 686.1 489.1 379.0

3167.9 3343.5 3540.1 3761.8 3361.4 3343.5 3326.0 3308.9 2790.0 2961.6 3343.5 4120.0 3343.5 4932.8 5696.2 6064.6 3343.5 1617.3 992.0 703.6 543.5

1405.9 1438.7 1472.6 1507.7 1442.0 1438.7 1435.5 1432.3 1325.6 1363.1 1438.7 1565.6 1438.7 2028.8 2270.8 2370.0 1438.7 710.1 441.1 315.1 244.4

764.1 781.9 800.4 819.4 785.4 781.9 778.4 775.1 666.2 703.6 781.9 922.3 781.9 1102.6 1234.1 1288.1 781.9 385.9 239.7 171.3 132.9

2170.0 2220.6 2273.0 2327.1 2227.4 2220.6 2213.9 2207.4 1991.8 2066.7 2220.6 2487.9 2220.6 3131.4 3504.9 3658.1 2220.6 1096.0 680.8 486.4 377.3

3165.5 3332.6 3512.7 3706.7 3350.4 3332.6 3315.3 3298.4 2783.6 2954.0 3332.6 4100.2 3332.6 4888.2 5614.4 5953.3 3332.6 1615.2 991.4 703.4 543.4

Note: PV denotes parameter value.

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Fig. 4. Impact of c2 on supply chain efficiency.

On the contrary, facing with the same situations, the retailer with open-loop equilibrium obtains more profit than the counterpart with feedback equilibrium. Unlike the conclusion of Chiang (2012) which states that both chain members are better off with the feedback equilibrium, this work indicates that the manufacturer benefits from feedback, while the retailer benefits from open-loop. To sum up, in comparison to open-loop equilibrium, feedback equilibrium improves the profits of the manufacturer and the whole supply chain, but damages the retailer's profit. In order to enhance supply chain performance, Nash bargaining (Nash, 1950) can be adopted to split the extra-profits, which provides incentives for channel members to participate in feedback equilibrium.

Fig. 5. Impact of g on supply chain efficiency.

11

Fig. 6. Impact of T on supply chain efficiency.

 The profits of the manufacturer, the retailer and the whole supply chain with forward-looking or myopic strategy, regardless of integrated or decentralized channel, increase with cost learning effect, energy efficiency effectiveness and planning horizon, whereas decrease with operational inefficiency and discount rate. Intuitively, the channel members take advantage of cost learning effect due to large cost savings. Additionally, the demand is expanded by high energy efficiency effectiveness, creating more profits for the manufacturer and the retailer. When the planning horizon is relatively long, the large production quantity creates more cost savings and hence, making more profits for channel members. One thing to note is that with the increase of planning horizon, the growth in profitability shrinks heavily. This indicates that although planning horizon plays a positive role on channel member's profits, the role is

Fig. 7. Impact of r on supply chain efficiency.

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weakened when the planning horizon is relatively long, which is attributed to the fact that the discounted profit is smaller corresponding to a distant future. Moreover, operational inefficiency effect damages their profits, but the decreasing speed of the profits is almost stable. Obviously, the discount rate poses a highly negative effect on channel members' profits, suggesting that patient members (small r) make much more profits, which far outstrips the impatient counterparts.  Although forward-looking strategy dominates myopic one in both integrated and decentralized channels, the chain-wide profit with forward-looking behavior in decentralized scenario is much smaller than the integrated profit with myopic behavior, which indicates that the myopic integration is superior to the forward-looking decentralization. Note from Figs. 3e7 that supply chain efficiencies under forward-looking (feedback and open-loop) and myopic behaviors decrease with c1 ; g and T, but increase with c2 and r. Combined with the results in Table 1, it is found that although the manufacturer and the retailer extract more profits from high cost learning effect, large energy efficiency effectiveness, long planning horizon, low operational inefficiency effect or discount rate, the supply chain efficiency declines instead. Therefore, compared with the decentralized supply chain, the integrated supply chain gains more when the cost learning effect is high; the energy efficiency effectiveness is large; the planning horizon is long; the operational inefficiency effect or discount rate is low. In an intuitive sense, the production quantity of the integrated channel is relatively large with high learning effect, begetting more cost savings and consequently more profits. Besides, the falling supply chain efficiency with respect to energy efficiency effectiveness stems from the fact that the energy efficiency level in integrated setting is larger than that in decentralized case on one hand, and on the other hand, high energy efficiency effectiveness further enlarges the gap in demand, which collectively contribute to the higher profit for integrated supply chain. In the long run, the longer the planning horizon is, the larger the production quantity is, especially in integrated channel, coming about significant cost savings, which serves to a better explanation of the decreasing supply chain efficiency with a longer planning horizon. When the operational inefficiency effect is considerably high, although the profit of decentralized channel shrinks, the decrease of the integrated channel's profit is more notable, leading to a high supply chain efficiency. In addition, the high supply chain efficiency arising out of a large discount rate reflects that an impatient decentralized channel (high r) performs much more efficiently than a patient one. As seen from the figures above, the supply chain efficiency of forward-looking behavior is always larger relative to myopic one. Specifically, with forward-looking rule, the supply chain efficiency of feedback equilibrium is greater than that of open-loop one. Such results show that, in terms of supply chain efficiency, forwardlooking behavior outperforms myopic one, and feedback equilibrium outperforms open-loop one. Moreover, it is found from Fig. 3 that the supply chain efficiency of myopic situation declines faster than those of feedback and open-loop scenarios. In other words, the advantages of forwardlooking behavior, especially feedback equilibrium, is highlighted if the cost learning effect is relatively high. From Fig. 4, it is seen that the discrepancy of supply chain efficiency upon forward-looking and myopic behaviors is insensitive to the operational inefficiency effect. In terms of the impact of planning horizon on supply chain efficiency, Fig. 6 shows that, the supply chain efficiency in feedback equilibrium decreases faster than the open-loop counterpart, and the latter decreases faster than the myopic one. This emphasizes the efficient performance of forward-looking

strategy from the point of long-term. Fig. 7 illustrates that the discrepancy of efficiency between feedback and open-loop equilibria enlarges when the discount rate is large. This suggests that the feedback equilibrium has more efficient supply chain performance than the open-loop one when facing a high discount rate. Meanwhile, the gap of channel efficiency between forward-looking (feedback or open-loop) and myopic rules decreases with discount rate, indicating that the dominance of forward-looking behavior as compared to myopic one is weakened with the increase of discount rate. Under this situation, there is minor difference of being myopic or forward-looking. Generally, supply chain efficiencies presented in the above figures lie in around 60%e70%. More precisely, when the energy efficiency effectiveness is relatively low, i.e., g ¼ 0:4, the supply chain efficiency in feedback equilibrium reaches up to about 73%. However, these supply chain efficiencies are lower than the one in a static decentralized channel, 75%, as proposed by Spengler (1950). By contrast, this work simultaneously considers cost learning and operational inefficiency effects in a dynamic environment, and assumes that the manufacturer is responsible for energy efficiency level strategy, besides wholesale price decision. As a result, cost learning and operational inefficiency effects, as well as non-price control variable jointly lead to supply chain inefficiencies. 7.2. Profit distribution across channel members Given the analysis above, it is important to investigate how the profits of decentralized channel are distributed between the manufacturer and the retailer, and how cost learning effect, operational inefficiency effect, energy efficiency effectiveness, planning horizon and discount rate affect the profit distribution. Table 2 generalizes the manufacturer's proportion of channel profits under forward-looking (feedback and open-loop) and myopic strategies, which well answers these research questions. The following findings are obtained on the basis of Table 2:  The manufacturer's profit proportion is always less than 2/3, regardless of forward-looking (feedback or open-loop) or myopic behavior. In other words, profit of the manufacturer is less than twice that of the retailer. However, Ingene et al. (2012) point out that the leader reaps twice the net revenue of its follower in a Stackelberg game in a static setting. Extending it to
n and Taboubi a dynamic setup, Chiang (2012) and Martín-Herra (2015) also find the same result. The essence is that their works only consider pricing problem with a fixed production cost. Nevertheless, this paper combines pricing and energy efficiency level decisions in the context of cost learning and operational inefficiency effects, and postulates that the manufacturer bears the costs of energy efficiency level, which jointly leads to a low profit proportion in the supply chain for the manufacturer.  The manufacturer's profit proportion shows tiny difference between feedback and myopic equilibria, which indicates that, from the perspective of profit distribution, myopic behavior and forward-looking behavior (feedback equilibrium) achieve almost identical performance. Besides, with forward-looking strategy, the manufacturer's profit proportion under feedback equilibrium is always larger than open-loop counterpart, implying that the manufacturer benefits from feedback equilibrium. In a word, although myopic behavior produces the lowest profits for the manufacturer and the retailer, it creates relatively high dominance for the manufacturer in the decentralized supply chain.  In myopic scenario, the manufacturer's profit proportions remain almost unaltered with the increase of c1 ; T; r, equivalently, profit proportion of the manufacturer is insensitive to

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13

Table 2 Impacts of parameters c1 ; c2 ; g; T; r on manufacturer's proportion (%). PV 0.01 0.02 0.03 0.04 0.19 0.20 0.21 0.22 0.4 0.7 1.0 1.3 10 20 30 40 0.1 0.3 0.5 0.7 0.9

c1

c2

g

T

r

PFm PFs

POL m POL s

em P

64.7879 64.7881 64.8073 64.8246 64.7386 64.7881 64.8366 64.8869 66.5714 65.9733 64.7881 62.9231 64.7881 64.8402 64.8082 64.7999 64.7881 64.7928 64.7613 64.7901 64.7874

64.2612 63.7163 63.1579 62.5797 63.6692 63.7160 63.7719 63.8226 65.5919 64.9569 63.7163 61.7409 63.7160 62.9827 62.5172 62.2476 63.7160 64.0736 64.2982 64.4314 64.5090

64.7889 64.7886 64.7885 64.7886 64.7387 64.7893 64.8479 64.8864 66.5528 65.9553 64.7886 62.9286 64.7893 64.7889 64.7893 64.7877 64.7887 64.7885 64.7891 64.7889 64.7885

es P

Note: PV denotes parameter value.











parameters c1 ; T; r. Apparently, T and r have no impact on decision making with myopic behavior. Also, when the evolution of cumulative sales is ignored, the associated parameter c1 exerts little effect on strategies. As a consequence, the impacts of these three parameters on profit distribution are trivial, as verified in Table 2. With the increase of c1, the manufacturer's profit proportion in open-loop equilibrium decreases, while the counterpart in feedback equilibrium is insensitive. Such outcome indicates that although the profits of channel members increase with a high cost learning effect, the manufacturer obtains less in open-loop case. The manufacturer's profit proportions in feedback, open-loop and myopic equilibria all increase with operational inefficiency parameter c2. In the presence of a large operational inefficiency effect, the manufacturer tends to decrease energy efficiency level to cut cost, leading to a relatively small profit loss relative to the retailer, and consequently to a high profit proportion for the manufacturer. It is shown that the increase of energy efficiency effectiveness g lowers the manufacturer's profit proportions in feedback, openloop and myopic scenarios. Attracted by the high energy efficiency effectiveness, the manufacturer would ideally like to set a high energy efficiency level to boost demand, which comes about more profits for the manufacturer and the retailer. However, the manufacturer has to incur all the costs of energy efficiency level, resulting in a low profit proportion in the decentralized supply chain. When the planning horizon T increases, the manufacturer's profit proportion of open-loop equilibrium decreases, whereas the counterpart in feedback equilibrium fluctuates around 64.8%. This indicates that under open-loop strategy, the manufacturer's dominance in the supply chain is impaired by the relatively long planning horizon. On the contrary, the retailer, as the follower, benefits from long planning horizon, accounting for an increased profit proportion. The manufacturer's profit proportion in open-loop equilibrium increases with discount rate r, but the counterpart in feedback is insensitive to that parameter. Although high discount rate damages the profits of the manufacturer and the retailer in open-loop equilibrium, it improves the manufacturer's dominance in the supply chain.

 Independent of the values of c1 ; c2 ; T; r, the manufacturer's profit proportion is about 64%, but when g is relatively low, i.e., g ¼ 0:4, the manufacturer's profit proportion reaches up to 66.57%, which is close to 2/3 in static setting, while when g is relatively large, i.e., g ¼ 1:3, the manufacturer's profit proportion decreases to 62.92%.

8. Extension In previous sections, the green supply chain performance with cost learning and operational inefficiency effects under the single manufacturer-retailer setup has been discussed. It is also of interest to investigate supply chain performance under competition, that is, taking into account a green supply chain with multiple players. Here, the marketing competition with two retailers is first considered for tractability, and afterwards, the competition is expanded to the situation with multiple retailers. Considering a supply chain composed of one manufacturer and two price-competing retailers, the manufacturer is responsible to distribute the products with energy efficiency level x(t) to retailers, Retailer 1 and Retailer 2, at wholesale price w1(t) and w2(t) respectively, and Retailers 1 and 2 sell the products to consumers at sales price p1(t) and p2(t) respectively. The retailers' demand rates at time t are respectively

Q_ 1 ðtÞ ¼ fða þ gxðtÞÞ  bp1 ðtÞ þ mp2 ðtÞ; Q1 ð0Þ ¼ 0;

(45)

Q_ 2 ðtÞ ¼ ð1  fÞða þ gxðtÞÞ  bp2 ðtÞ þ mp1 ðtÞ; Q2 ð0Þ ¼ 0;

(46)

where Q1(t),Q2(t) are the cumulative product sales at time t for Retailers 1 and 2, f2ð0; 1Þ is the market share of Retailer 1, m > 0 represents price competition, i.e., the sensitivity of demand to the competing retailer's price. It is assumed that b > m to ensure a higher price effect on own demand relative to competitor's. Specifically, the market potential a þ gxðtÞ is partitioned by Retailers 1 and 2 at proportion f and 1  f respectively. When f ¼ 0:5, the two retailers are identical, equally sharing the market potential. The manufacturer's cumulative product quantity Q(t) equals to the retailers' cumulative demand, i.e., Q ðtÞ ¼ Q1 ðtÞ þ Q2 ðtÞ. The

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corresponding unit production cost and investment cost functions are Eqs. (2) and (3) respectively. With price competition, the objective functionals of the manufacturer, Retailers 1 and 2 in the finite planning horizon [0,T] are

ZT Jm ¼

 ert ðw1 ðtÞ  c0 þ c1 Q ðtÞ  c2 xðtÞÞðfða þ gxðtÞÞ

0

 bp1 ðtÞ þ mp2 ðtÞÞ þ ðw2 ðtÞ  c0 þ c1 Q ðtÞ  c2 xðtÞÞ  k  ðð1  fÞða þ gxðtÞÞ  bp2 ðtÞ þ mp1 ðtÞÞ  x2 ðtÞ dt; 2 (47) ZT Jr1 ¼

ert ðp1 ðtÞ  w1 ðtÞÞðfða þ gxðtÞÞ  bp1 ðtÞ þ mp2 ðtÞÞdt;

0

(48) ZT Jr2 ¼

Apart from myopic equilibrium, forward-looking equilibrium (open-loop and feedback) is also of value to derive. However, it is indeed a great challenge. To get open-loop equilibrium, for example, a system consisting of four differential equations in terms of Q_ 1 ; Q_ 2 ; l_ 1 ; l_ 2 is needed to solve, where l1 ; l2 are the adjoint variables associated with Q1,Q2. Although the four equations with four boundary conditions imply a solution, it is very difficult to derive analytical solutions for all the variables as functions of time and system parameters (Gutierrez and He, 2011). As a result, the analysis of forward-looking equilibrium remains an important future work in our agenda. Moreover, with myopic strategy, when there are more than two retailers, it is expected that the sensitivity analysis of system parameters on supply chain performance also holds and when the number of retailers increases, the corresponding supply chain efficiency rises. A similar observation that the channel efficiency can be improved with the emergence of horizontal competition has been verified by Xu et al. (2011), which investigates the impacts of the supply-side cost learning effect on pricing strategies and the channel efficiency in a decentralized supply chain. 9. Concluding remarks and future research directions

ert ðp2 ðtÞ  w2 ðtÞÞðð1  fÞða þ gxðtÞÞ  bp2 ðtÞ

0

þ mp1 ðtÞÞdt:

(49)

Likewise, the problem is modeled as a Stackelberg game where the manufacturer is the leader and the retailers are the followers. The sequence is the following: the manufacturer first announces wholesale prices and energy efficiency level, and then the retailers play a Nash game and simultaneously set sale prices. The myopic behavior is first considered, and the corresponding equilibria are obtained with the same method. To examine whether the impacts of system parameters on supply chain performance are robust or not, numerical simulations in Table 3 are carried out. As shown from Table 3, the profits of manufacturer, retailers, and the supply chain as a whole in integrated and decentralized channels increase with cost learning effect c1, energy efficiency effectiveness g and planning horizon T, but decrease with operational inefficiency c2 and discount rate r. The corresponding supply chain efficiency experiences the opposite trend. This is in line with the results without competition, which implies that the effects of system parameters on supply chain performance are robust. Furthermore, Figs. 8e9 describe the impacts of competition and market share on supply chain efficiency, of which the parameter values are given in Table 3. Fig. 8 shows that the supply chain efficiency peaks at about 82.8% when f ¼ 0:5 but then decreases with the increase of f. This indicates that when the competing retailers share the same market potential, the supply chain efficiency reaches its maximum. It is shown from Fig. 3 that the supply chain efficiency without competition is about 60%e70%. By comparison, the supply chain efficiencies with price competition are higher with f varying from 0.1 to 0.9, larger than the static supply chain efficiency 75%, which means that competition improves supply chain efficiency. When f ¼ 0:5, Fig. 9 shows the effect of competition intensity on supply chain efficiency, unveiling that the higher the competition intensity is, the higher supply chain efficiency is. Additionally, it is easy to acquire that the manufacturer's profit proportion is about 72.82% when f ¼ 0:5, and it increases with competition intensity, which is larger than the counterpart without competition (about 65% shown in Table 2), implying that competition improves manufacturer's profit proportion in the supply chain.

Consider a two-echelon green supply chain composed of a manufacturer and a retailer, of which the manufacturer controls energy efficiency level and wholesale price, and the retailer controls sales price. Energy efficiency level, on one hand, directly increases unit production cost, namely, operational inefficiency effect. On the other hand, it raises demand and consequently indirectly decreases unit production cost through cost learning effect. Based on the model, the corresponding pricing and energy efficiency level strategies under different scenarios are figured out, and strategy comparison across scenarios are conducted. This paper focuses on the impacts of cost learning and operational inefficiency effects on supply chain performance, and emphasizes the manufacturer's profit proportion in the decentralized supply chain, which yields a number of significant insights that have not been identified. Overall, the main findings are collected as follows: 1) Forwardlooking behavior is always preferred for channel members rather than myopic one. From the view of supply chain efficiency, forwardlooking strategy is better than myopic counterpart, and feedback equilibrium is better than open-loop one. 2) The supply chain efficiencies under both forward-looking and myopic behaviors are lower than the static supply chain efficiency, owing to cost learning and operational inefficiency effects, as well as non-price decision variable in our model. 3) The entire supply chain's profit with feedback equilibrium is higher than the open-loop one, but only the manufacturer benefits from the feedback equilibrium. For the retailer, open-loop equilibrium is more favorable. 4) The manufacturer and the retailer reap more profits under certain conditions,

Table 3 Impacts of parameters on supply chain performance with competition under myopic strategy.

em P e r1 P e r2 P es P eI P r

c1

c2

g

T

r

[ [ [ [ [ Y

Y Y Y Y Y [

[ [ [ [ [ Y

[ [ [ [ [ Y

Y Y Y Y Y [

e I representing supply chain efficiency. Basic parameters: e s =P Note: r ¼ P a ¼ 50; b ¼ 1, g ¼ 1; m ¼ 0:5; f ¼ 0:8; r ¼ 0:1; c0 ¼ 10; c1 ¼ 0:02; c2 ¼ 0:2; k ¼ 2; T ¼ 10.

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15

who set individual energy efficiency level for their products to capture market. Indeed, for the different products from manufactures, price competition is accompanied by energy efficiency level competition, adding another dimension to complexity. It is worthwhile to study supply chain performance under the combined effects of market competition and operational competition in the future. In addition, the network structure with multi-echelon and multi-nodes in each echelon is a new research paradigm to discuss the impact of supply chain structure on performance. It is of great interest to uncover the impact of competition from different echelons and players on supply chain performance, which may be one direction for our future research. Acknowledgments This work was supported by National Natural Foundation of China No. 61473204, Humanity and Social Science Youth Foundation of Ministry of Education of China No. 14YJCZH204, and the Program for New Century Excellent Talents in Universities of China No. NCET-11-0377. Fig. 8. Impact of f on supply chain efficiency.

Appendix i.e., high cost learning effect, large energy efficiency effectiveness, low operational inefficiency effect or discount rate. However, the corresponding supply chain efficiencies decline. 5) The manufacturer's profit proportion is less than 2/3, that is, profit of the manufacturer is less than twice that of the retailer. 6) The manufacturer's profit proportion shows little difference between feedback and myopic cases. Moreover, the profit proportion of the manufacturer is insensitive to cost learning effect, planning horizon and discount rate. 7) In presence of competition in the supply chain, it is found that the effects of system parameters on supply chain performance are robust and that, competition improves supply chain efficiency and manufacturer's profit proportion. Potential extensions of the paper can be made according to the following directions. The demand can be extended to a dynamic one involving saturation effect and diffusion effect, which may present different effects on supply chain performance. It is interesting to explore how competition from market and operational aspects affect supply chain performance. For example, one may consider operational competition from multiple manufacturers

For simplification, the time argument is omitted in Appendix. A. Proof of Proposition 1. The current value Hamiltonian for the integrated channel is given in (7). According to the optimal control theory, the following optimality conditions are given as

vH vH ¼ 0; ¼ 0; vp vx

(50)

Q_ ¼ a  bp þ gx;

(51)

l_ ¼ rl  c1 ða  bp þ gxÞ;

(52)

which yields

p¼

ðc1 Q þ lÞðgðg  bc2 Þ  bkÞ þ ðac2  gc0 Þðg  bc2 Þ þ kða þ bc0 Þ ; q (53)

x¼

ðbðc1 Q þ lÞ þ a  bc0 Þðg  bc2 Þ ; q

(54)

b2 c1 k b2 k bkða  bc0 Þ Q_ ¼ Qþ lþ ; q q q b2 c21 k Qþ l_ ¼  q

(55)

! b2 c1 k bc kða  bc0 Þ l 1 ; r q q

(56)

where

q ¼ 2bk  ðg  bc2 Þ2 > 0: Note from (55) and (56) that

Fig. 9. Impact of m on supply chain efficiency.

A¼

 1 b2 c1 k q b2 c21 k

b¼

  bkða  bc0 Þ 1 : c1 q

 b2 k ; 2 rq  b c1 k



Q_ l_





Q ¼A l

(57)

 þ b, where

(58)

(59)

The eigenvalues of A are

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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 0 2 2 1@ 4b c kr 1 1 A; r2 ¼ @r  r2  4b c1 krA; r þ r2  r1 ¼ 2 q 2 q (60) with rq > 4b2 c1 k to have real root, and the corresponding matrix of eigenvectors of A is

2

b2 k 6 H ¼ 4 r1 q  b2 c1 k

7 r2 q  b c1 k 5:



Q l



Q l

 rt e1 ¼H 0 2

0 er2 t

b2 kk1 er1 t



6 ¼ 4 r1 q  b2 c1 k

k1 k2

þ



 A1 b

b2 kk2 er2 t r2 q  b2 c1 k

(61) p¼ 

3 a  bc0 bc1 7 5:

! b2 c1 k bc kða  bc0 Þ lm  1 ; r h h

(70)

   ða  bc0 Þ r1 q  c1 b2 k r2 q  c1 b2 k eðr2 r1 ÞT  ;   k1 ¼  b3 kc1 r1 q  c1 b2 k  r2 q  c1 b2 k eðr2 r1 ÞT   ða  bc0 Þ r1 q  c1 b k r2 q  c1 b2 k  :   k2 ¼ b3 kc1 r1 q  c1 b2 k  r2 q  c1 b2 k eðr2 r1 ÞT

(71)

ðc1 Q þlm Þðgðgbc2 ÞbkÞþðac2 gc0 Þðgbc2 Þþkð3aþbc0 Þ : h (72) 

Note from (69) and (70) that

(62)

Since Q(0) ¼ 0 and lðTÞ ¼ 0, k1,k2 are generated as

(63)

B¼

 1 c1 b2 k h c21 b2 k

d¼

  bkða  bc0 Þ 1 : c1 h

 b2 k ; rh  c1 b2 k

   Q_ ¼B Q þd, where lm l_ m

(73)

(74)

The eigenvalues of B are

2

(64)

As such, accumulated sales and shadow price are acquired in (11) and (12). Substituting them into (53) and (54), yields the optimal sales price and energy efficiency level in (9) and (10). B. Proof of Corollary 1. It is easy to verify that r1 q  c1 b2 k > r2 q  c1 b2 k > 0, yielding k1 < 0; k2 > 0. According to Eq. (52), it is found that l_ l¼0 < 0, with lð0Þ ¼ k1 þ k2 > 0 and lðTÞ ¼ 0, which ensures that lðtÞ > 0 for all t2½0; TÞ. With (51)e(53), it is found that _ _ rlðgðgbc2 ÞbkÞ 2 ÞbkÞ _ p_ ¼ ðc1 Q þlÞðgðgbc ¼ . Since l > 0, p > 0 when q q gðg  bc2 Þ > bk, and p_ < 0 when gðg  bc2 Þ < bk. Likewise, it is verified that x_ > 0 during the entire planning horizon. C. Proof of Proposition 3. To derive the equilibrium solutions, the retailer's optimization problem is first solved, and the response sales price is obtained as p ¼ aþgxþbw . With the retailer's response function, the Hamiltonian 2b of the manufacturer is given in (22), and the optimality conditions are as follows.

vHm vHm ¼ 0; ¼ 0; vw vx

(65)

c l_ m ¼ rlm  1 ða þ gx  bwÞ; 2

(66)

which yields

w¼

(69)

h ¼ 4bk  ðg  bc2 Þ2 > 0:

k1 er1 t þ k2 er2 t



b2 c1 k b2 k bkða  bc0 Þ Qþ lm þ ; Q_ ¼ h h h

where

1

Therefore,



(68)

2

1 

ðbðc1 Q þ lm Þ þ a  bc0 Þðg  bc2 Þ ; h

b2 c21 k l_ m ¼  Qþ h

3

b2 k

x¼

ðc1 Q þlm Þðgðgbc2 Þ2bkÞþðac2 gc0 Þðgbc2 Þþ2kðaþbc0 Þ ; h (67)

0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 1@ 4b c kr 1 4b2 c1 krA 1 A r þ r2  ; s2 ¼ @r  r2  ; s1 ¼ 2 h 2 h (75) and the corresponding matrix of eigenvectors of B is

2

b2 k

6 H ¼ 4 s1 h  b2 c1 k

b2 k

3

7 s2 h  b2 c1 k 5:

1

1

Hence,





Q lm

Q lm





 st e1 ¼H 0 2

0 es2 t

b2 kk3 es1 t



6 ¼ 4 s1 h  b2 c1 k

k3 k4

þ



 B1 d

b2 kk4 es2 t s2 h  b2 c1 k s1 t

k3 e

(76)



3 a  bc0 bc1 7 5:

(77)

s2 t

þ k4 e

Since Q(0) ¼ 0 and lm ðTÞ ¼ 0, k3,k4 are given as

   ða  bc0 Þ s1 h  c1 b2 k s2 h  c1 b2 k eðs2 s1 ÞT  ;   k3 ¼  b3 kc1 s1 h  c1 b2 k  s2 h  c1 b2 k eðs2 s1 ÞT    ða  bc0 Þ s1 h  c1 b2 k s2 h  c1 b2 k  :   k4 ¼ b3 kc1 s1 h  c1 b2 k  s2 h  c1 b2 k eðs2 s1 ÞT

(78)

(79)

Then, the accumulated sales and shadow price are obtained in (27) and (28). Substituting them into (67), (68) and (72), the equilibrium results are given in (24)e(26). D. Proof of Proposition 4.

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When the manufacturer and the retailer are both forwardlooking, the value functions of the manufacturer and the retailer could be conjectured as

Vi ¼

Am ðtÞ ¼ ¼

1 A ðtÞQ 2 þ Bi ðtÞQ þ Ci ðtÞ; 2 i

(80)

where i2fr; mg and Ai(t),Bi(t),Ci(t) follow from (80) that

vVi ¼ Ai ðtÞQ þ Bi ðtÞ: vQ

(81)

(82)

 2  vVm k vVr vVm a  bðc0  c1 Q Þ þ b ¼ þ : 2h vt vQ vQ

(83)

With (81), (33) and (34), it is obtained that

 1 2 b kðSAi þ c1 ÞQ þ bkða  bc0 þ bSBi Þ : Q_ ¼ a  bp þ gx ¼ h (84)

(91)

whereqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2kb2 c1 ðhþ2bkÞ 2xT 1 e .Simir2 h2  4rkb2 c1 ðh þ 2bkÞ; c ¼ ð2xþrÞh x ¼ 2h ð2xrÞh2 þ2kb2 c ðhþ2bkÞ

h Br ðtÞ 2bk 0 r ZT h @ e2 t ð1 þ cÞ a1 ¼ 2bk ext þ cext t     1

xt ! x2r t x2r t xt x e  ce e þ ce dtA; þ a2 xt 1þc e þ cext

Bm ðtÞ ¼

(92)

0Þ 0Þ where a1 ¼ rkðabc ; a2 ¼ 2kðabc : hþ2bk hþ2bk

Accordingly, Cr and Cm are

0 T 1 Z h h rt @ bk2 ða  bc0 þ bSBi ðvÞÞerv A Cr ðtÞ ¼ e Cm ðtÞ ¼ dv : 2bk 2bk h2

Substituting (84) into (82) and (83), yields

  r 1 _ Ar ðtÞQ 2 þ rBr ðtÞQ þ rCr ðtÞ  Ar ðtÞQ 2 þ B_ r ðtÞQ þ C_ r ðtÞ 2 2 ¼

 2xh2 ext  cext h

 2bk bðh þ 2bkÞ2 ext þ cext ! 2kb2 c1 ðh þ 2bkÞ  rh2  ; bðh þ 2bkÞ2

larly, comparing the coefficient of Q, it can be generated that

2   vVr bk2 vVr vVm ¼ 2 a  bðc0  c1 Q Þ þ b þ ; vt vQ vQ h

rVm 

h Ar ðtÞ 2bk

1

Substituting (32), (33) and (34) into (29) and (31), yields

rVr 

17

t

(93)

bk2 ða  bðc0  c1 Q Þ þ bðSAi Q þ SBi ÞÞ2 ; h2 (85) References

 r 1 _ Am ðtÞQ 2 þ rBm ðtÞQ þ rCm ðtÞ  Am ðtÞQ 2 þ B_ m ðtÞQ 2 2  þ C_ m ðtÞ ¼

k ða  bðc0  c1 Q Þ þ bðSAi Q þ SBi ÞÞ2 : 2h

(86)

Equating the coefficients of Q2 on both sides of (85) and (86) respectively gives

r 1 k2 b3 Ar ðtÞ  A_ r ðtÞ ¼ 2 ðSAi þ c1 Þ2 ; 2 2 h

(87)

r 1 kb2 ðSAi þ c1 Þ2 ; Am ðtÞ  A_ m ðtÞ ¼ 2 2 2h

(88)

which lead to

  r 1 2bk r 1 Ar ðtÞ  A_ r ðtÞ ¼ Am ðtÞ  A_ m ðtÞ : 2 2 h 2 2 Since Vr(T,Q) ¼ 0,Vm(T,Q) ¼ 0, it is Ar(T) ¼ Am(T) ¼ 0. Therefore, it is found that

Am ðtÞ ¼

h Ar ðtÞ; 2bk

and the solution is

(89) computed

that

(90)


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