Environmental-regulation pricing strategies for green supply chain management

Environmental-regulation pricing strategies for green supply

chain management

Yenming J. Chen

a,1

, Jiuh-Biing Sheu

b,* a

Department of Logistics Management, National Kaohsiung First University of Science and Technology, 2 Jhuoyue Rd., Nanzih District, Kaohsiung City 811, Taiwan b

Institute of Traffic and Transportation, National Chiao Tung University, 4F, 118, Sec. 1, Chung Hsiao W. Rd., Taipei 100, Taiwan

a r t i c l e

i n f o

Article history:

Received 18 August 2008

Received in revised form 10 February 2009 Accepted 10 April 2009

Keywords:

Extended Production Responsibility Differential game

Optimal control

a b s t r a c t

This paper demonstrates that a proper design of environmental-regulation pricing strate-gies is able to promote Extended Product Responsibility for green supply chain firms in a competitive market. A differential game model comprising Vidale–Wolfe equation has been established in light of sales competition and recycling dynamics as well as regulation related profit function. Analytic solutions of Markovian Nash equilibriums are provided with the necessary condition derived from Hamilton–Jacobi–Bellman equations. We found that governments should opt to gradually raise regulation standards so that rational man-ufacturers will gradually improve its product recyclability, and, in turn, Extended Product Responsibility will get promoted.

Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Competitive strategies for firms and environmental-regulations for governments jointly play an important role in dictat-ing the success of implementdictat-ing Extended Product Responsibility (EPR) policies (Palmer and Walls, 1997; Reijnders, 2003). At the same time, strategic management has long been considered a significant part of business competitiveness. Most of existing reports, however, concentrate only on the impact of policies per se, rather than on the existence of market interac-tion. This paper, therefore, shed new light on recycling policy designs under a more realistic market condition by the help of a differential game model.

Existing analysis of recycling policy – including Design for Environment (DfE) incentives – are mostly based on a single company model (Fullerton and Wu, 1998; Choe and Fraser, 2001; Stavins, 2002). From the literature, however, we under-stand that consequence of incentive behave differently in a multiple companies competition context (Jaffe et al., 1995; Vogelsang, 2002), and thus the interactive effect of incentive policies and regulations needs to be reviewed. Moreover, prod-uct pricing and manufacturing costs mostly determine the profitability of a firm. Manufacturers accrue their profits by set-ting the right pricing strategies with consideration for competitor responses and product characteristics (Reijnders, 2003). Among the environmental policy literature, however, while tax or subsidy pricing is often discussed, little attention is given to product pricing and environmental friendly design policy (Ekins, 1999).

In recent years, EPR has attracted much attention and the notion of EPR has been part of the concept of green supply chain. According toBarde and Stephen (1997), EPR is defined as a strategy designed to promote the integration of environ-mental costs of products throughout their life cycles into the market distribution mechanism so as to reduce product harm to

1366-5545/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2009.04.010

*Corresponding author. Tel.: +886 2 2349 4963; fax: +886 2 2349 4953.

E-mail addresses:yjjchen@ccms.nkfust.edu.tw(Y.J. Chen),jbsheu@mail.nctu.edu.tw(J.-B. Sheu). 1

Tel.: +886 7 601 1000x3214; fax: +886 7 601 1040.

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Transportation Research Part E

the environment. A prosperous green supply chain cannot be substantiated without the help of proper incentives and public policies (Sheu et al., 2005; Sheu, 2008). With the implementation of EPR policies in various supply chains, producer respon-sibilities have been extended from selling products to recycling them, meanwhile pushing waste management issues to up-stream manufacturers and even the entire supply chain (Carter and Jennings, 2002).

In order to promote the concept of EPR, governments around the globe usually provide financial incentives for manufac-turers and encourage them to engage in EPR practices (Palmer and Walls, 1999). Appropriate incentive mechanisms not only internalize externality by changing the cost structure for producers, but they also drive manufacturers to develop more envi-ronmentally friendly products. Moreover, although international prominence has shifted to product sustainability, the sub-ject of product design is still seen as one of the top priorities for governments and manufacturers. When enterprises respond to strict controls regarding their social responsibility, and at the same time begin to take account of competitive pricing and manufacturing costs, it is often considered difficult for them to determine a long-term profit strategy. Existing literature has pointed out that, however, environmentally friendly designs can reduce material use, enhance business competitiveness, and have other benefits, there is no clear suggestions or practical consideration given as to how and to what extent product de-sign can be improved (Avila, 2006).

Effect of EPR incentive on green product design reacts differently from a market with competitors. Member firms in a green supply chain, in every dynamic stage of the decision making process, attempt to estimate the actions of their rivals and then identify what corresponding strategies can be used to drive the firm toward a maximized profit situation. Such strategies, however, are expected to coincide with environmentally friendly design from the views of policy makers. To facil-itate this process, we use a differential game model to derive optimal design trajectories and to illustrate how manufacturers can adopt optimal product green design and pricing strategies for pursuing maximal profit whilst also complying with social responsibility.

Moreover, given that EPR cannot be executed directly, the notion of Design for Environment (DfE) has been suggested instead (Walls, 2003; Spicer and Johnson, 2004). The DfE, however, possesses broad coverage (Calcott and Walls, 2005) and strives to integrate, in a systematic way, various aspects of environment, health, and safety into the design phase of the production process, while at the same time seeking to satisfy simple and easy disassembling design criteria (Calcott

and Walls, 2005; Walls, 2003). Given such broad sentiment, this paper focuses particularly on the recyclability of product

green design in the following three areas: ease of disassembly, usage of toxic materials, and reusability of resources (Calcott and Walls, 2005), i.e., design for recycling (Kriwet et al., 1995). A prevalent definition of recyclability has been known as a rate or percentage of recyclable material in a product composition (Duchin and Lange, 1994; Huisman et al., 2003). This def-inition of recyclability has been adopted in this paper.

There are various regulatory and financial incentive schemes. Globalized organizations – including Apple, Sony, and Mats-ushita – invest a large portion of their budgets in DfE activities in order to green their supply chain. The motivation that drives these firms to implement DfE (Walls, 2003) appears to lie in a combination of regulation and production cost (Palmer

and Walls, 1999; Avila, 2006; Iliyana, 2006; Gottberg et al., 2006). In order to compensate for harm caused by the lack of

flexibility in command and control, incentive mechanisms can be a complement to maintaining industry growth (Jaffe

et al., 1995). Under these mechanisms, manufacturers are charged differently according to their product’s characteristics

in ease of handling. This price discrimination is expected to regulate manufacturers’ environmental responsibility effectively. Among existing incentive designs, product charges or taxes are levied against products that causes environmental pollution prior to production to reflect the externality costs (Barde and Stephen, 1997). We assume that different incentives for firms largely result from differentiated processing fees charged by recycling treatment agencies providing discriminated product recyclability (Duchin and Lange, 1994). In other words, the fee schemes depend on the total amount of scraps as well as the ease of handling in waste treatment and processing.

Comparing to previous literature, we provide a distinctive feature. We extend mixed incentive strategies to a broader view. This paper finds that, for manufacturers in competition, simultaneously offering financial incentives and increasingly stringent regulation is necessary for promoting green product recyclability.

2. Competitive differential game model

In attempting to address the effectiveness of EPR instruments in a competitive environment, our model is built on top of a simplified situation in which an integrated financial incentive and regulation standard is imposed. To manifest the dynamic interaction, and for ease of illustration and analysis, we have constructed a differential game model with sales and recycling dynamics. In our model we assume that, for firms to be environmentally conscious, certain regulation standards need to be imposed to reflect current social responsibility (Foulon et al., 2002). Moreover, a certain amount of capital expenditure also needs to be invested in order to comply with government standards (Cohen, 1999; Foulon et al., 2002).

Fig. 1illustrates the conceptual framework and the game players for constructing our differential equations. xiðtÞ and niðtÞ

represent the market share and recycling rate of producer i at time t, respectively. The incentive is incorporated in recycling treatment fee uiðtÞ, which is charged by the treatment agency and depends on the product’s recyclability involvement diðtÞ,

e.g., the extent of ease of disassembly. To implement a simplified financial incentive in our model, a treatment agency di-rectly charges manufacturers processing fees without involving other third party agencies. In the close-to-real situation, there are other agencies as intermediaries, for example, a Producer Responsibility Organization (PRO) charges EEE

manufac-turers an amount of fees and establishes a fund to operate the system perpetually. These intermediate third part agencies can be incorporated in the future researches.

To study the competitive behavior, i.e., time trajectories, of firms in a market, we denote the opponents’ price decisions and market share as

pi_{ðtÞ ¼ ðp}

1ðtÞ; p2ðtÞ; . . . ; pi1ðtÞ; piþ1ðtÞ; . . . ; pnðtÞÞ;

xi_{ðtÞ ¼ ðx}

1ðtÞ; x2ðtÞ; . . . ; xi1ðtÞ; xiþ1ðtÞ; . . . ; xnðtÞÞ:

We normalize the market share xiðtÞ 2 ½0; 1 such that they sum up to unity at any time instance

Xn

i¼1

xiðtÞ ¼ 1:

The sales dynamics can be suitably described by a set of differential Eq.(1)with the form of Vidale–Wolfe (Prasad and Sethi, 2004). _xiðtÞ ¼ fxiðxiðtÞ; x i_{ðtÞ; pðtÞÞ ¼}X j–i

q

jpjðtÞ ffiffiffiffiffiffiffiffiffiffi xiðtÞ p X j–i

q

ipiðtÞ ffiffiffiffiffiffiffiffiffiffi xjðtÞ q  d xiðtÞ  X j–i xiðtÞ ! ð1Þ

All firms determine their product prices at very time instance in order to conquer maximal market shares. Pricing deci-sions are made by responding competitor reactions of prior price and market share changes. Prices differences between products affect customer purchasing preferences, thereby causing sales and market share deviation. Market share change rate _xiof firm i in(1)constitutes the influence from its own market share xiand the market share xjof other products.

If manufacturers enhance their green product recyclability design, it i.e., the percentage of weight in their products been recycled, their product recycling rate increases proportionately (Huisman et al., 2003). However, when reviewing EPR policy literature, we found that the definition of the recycling rate between countries is not limited to a specific context. Modalities of the recycling vary in countries, but the aim of reducing waste remains consistent. The WEEE Act has had the most far-reaching influence on national laws (Huisman et al., 2003). It clearly regulates that: (1) the re-use and recycling rate be up to 75% and (2) the resource recovery rate be up to 80% of the weight of each recovery (Yamaguchi, 2002). In this case, the recycling rate amounts to the recycled weight percentage with respect to total disposal.

To relate to the EPR, the responsibility elasticity to unfulfilled recycles (Jalal and Rogers, 2002) is defined as

a

¼ @M M @s s ð2Þ

where M ¼ 1 Pn_i¼1nirepresents unfulfilled recyclables, ignored by all manufacturers, and

s

represents producer

responsi-bility in a country. For example,

a

¼ 2 means unfulfilled waste will decrease 2% as responsibility increases 1%. Every coun-try may develop different social responsibility levels. This simply reflects the average environmental consciousness and regulation stringency in a particular society. From the definition in(2), therefore, we have

M ¼

s

a _ð3Þ

Let niðtÞ and diðtÞ represent the recycling rate of product i and the recyclability involvement of product i, respectively.

Moti-vated by diffusion models in marketing and the consequence of new product sales (Dockner and Fruchter, 2004), the recy-cling dynamics can be suitably described through(4)

Treatment agency Consumers

Manufacturer 2 p2(t), d2(t) Manufacturer 1 p1(t), d1(t) Market Share x1 Market Share x2 2 1 Incentive u2(d2) Incentive u1(d1) Recycling Rate Recycling Rate ξ ξ

_niðtÞ ¼ ð

g

þ

ei

diðtÞ=

s

Þ ffiffiffiffiffiffiffiffiffiffi xiðtÞ p 1 X n i¼1 niðtÞ ! ð4Þ

The influence of the dynamics of the recycling rate constitutes recyclability, the producer responsibility acting on market share and any unfulfilled recycling weight. The resulting behavior follows an S-shape dynamics. At lower rates of recycling, the improvement appears to be slow. When the recycling rate, however, increases to some extent, it starts to rise dramatically. Eventually, as most of the materials are recyclable, it becomes more difficult to improve the recycling rate.

The above two dynamics collectively describe the behavior of a recycling system in a competitive environment. The sales dynamic points out that when manufacturers commence a price war in the market, sales volume rises in consequence. More sales, however, leads to more waste, so that manufacturers need to take heavier responsibility for recycling (Barde and Ste-phen, 1997; Sheu et al., 2005). In this case, manufacturers may be more willing to engage in product design recyclability in order to alleviate increasing costs.

In order to provide the conceptualization terse and to simplify consequent derivations, we aggregate all niðtÞ to an single

s

ðtÞ (Dockner and Fruchter, 2004). By summing up all _niof(4), the recycling dynamics can be easily transformed to

a

s

_ðtÞ ¼ 

gs

ðtÞ X n i¼1

ei

diðtÞ ffiffiffiffiffiffiffiffiffiffi xiðtÞ p ð5Þ

In order to pursue profit maximization, we assume revenue to be solely generated by selling products, while costs are ac-crued from multiple sources – such as, production cost wiðxiðÞÞ, production process upgrading cost hiðdiðÞÞ, recycling fee

uiðdiðÞÞ paid to the treatment agency, and capital expenditure nð

s

ðÞ; fðÞÞ made to comply with the government regulation

standard fðÞ (Jaffe et al., 1995). Upgrading costs includes R&D investment, costs incurred for altering production processes, and costs associated with consuming recyclable materials (Mukhopadhyay and Setaputra, 2007). In this paper we assume n is linear in f

s

, which represents the environmental-regulation standard determined by producer responsibility in a society. The net profit amounts to the difference between sales revenue and all accrued costs and can be written as(6)with the no-tion of NPV, where riis the discount rate and assumed to be constant.

JiðpiðÞ; diðÞÞ ¼ Z T 0 ert_Fðx iðtÞ;

s

ðtÞ; piðtÞ; diðtÞ; tÞdt ð6Þ where FðxiðtÞ;

s

ðtÞ; piðtÞ; diðtÞ; tÞ ¼

mi

ðxiðtÞ; piðtÞÞ  ciðxiðtÞ;

s

ðtÞ; diðtÞÞ ¼

mi

ðxiðtÞ; piðtÞÞ  wiðxiðtÞÞ  hiðdiðtÞÞ  uiðxiðtÞ; diðtÞÞ  nið

s

ðtÞ; fðtÞÞ

To keep the problem explicit, some assumptions are imposed regarding to the behavior of manufacturers:

1. We are dealing with a differential game with simultaneous decision making (Dockner et al., 2000). Every player is rational and seeks to maximize their objective functional.

2. All products are homogeneous but companies are not. Each firm has its own cost structure and ability to attract custom-ers from its competitors.

3. There is only one representative treatment agency and it makes no profit in our system. It offers incentives by charging manufacturers differently according to the level of recyclability.

With the implementation of incentives and regulations, manufacturers constantly ponder how to re-allocate costs more effectively and select suitable recyclability involvement in order to achieve their own profit maximization. With the optimi-zation problem of competing parties, our differential game model solves the Markovian Nash equilibrium. This occurs when a participant in a game speculates the optimal strategy of other participants to find his own optimal strategy. This strategy gives no motivation for all rational participants to deviate from this equilibrium (Dockner et al., 2000).

Let /i

ðxi;

s

;tÞ denote a Markovian strategy of producer i. A Markovian Nash equilibrium satisfies the Hamilton–Jacobi–

Bellman (HJB) Eq.(7).

riVi¼ max

pi;di

f

mi

ðxi;piÞ  ciðxi;

s

;diÞ þ Vix_xðxi;x1;piÞ þ Vis

s

_ðxi;

s

;diÞg; i ¼ 1; 2; ð7Þ

where the notation Vixpresents the partial derivative of Viwith respect to x, i.e., @Vi=@x. Expand the HJB(7)and(8)

riVi¼ max

mi

ðxi;piÞ  hiðdiÞ  uiðxi;diÞ  nið

s

;fÞ þ Vix

q2

p2 ffiffiffi x p 

q1

p1 ffiffiffiffiffiffiffiffiffiffiffi 1  x p  dð2x  1Þ   n þVis1

a



gs



e1

d1 ffiffiffi x p 

e2

d2 ffiffiffiffiffiffiffiffiffiffiffi 1  x p  o ; i ¼ 1; 2: ð8Þ

@

mi

@pi  Vixqi ffiffiffiffiffiffiffiffiffiffiffiffiffi1  xi p ¼ 0 ð9Þ @hi @di @ui @di  Vis

ei

a

ffiffiffiffi xi p ¼ 0 ð10Þ

The resulting Markovian Nash equilibriums of(9) and (10)represent the optimal pricing and design strategies for each firms. We further assume that the revenue function

m

iðxiðÞ; piðÞÞ is linear in xiðÞ and quadratic in piðÞ and the upgrading cost

of recyclability design hiðdiðÞÞ is quadratic in diðÞ and the processing fee uiðxiðÞ; diðÞÞ is linear in ð1  diðÞÞ

ffiffiffiffiffiffiffiffiffi xiðÞ p , and then we have@hi @di¼ Chidiand @ui @di¼ Cui ffiffiffiffi xi p .

The Markovian Nash equilibriums follow:

p i ¼

q

i Kmi Vix ffiffiffiffiffiffiffiffiffiffiffiffiffi 1  xi p ð11Þ di ¼ eiVis a þ Cui Chi ffiffiffiffi xi p  Fi ffiffiffiffixi p ð12Þ

The HJB condition provides a necessary condition for evaluating the Markovian Nash equilibrium trajectories. This need not be a sufficient condition, as a linear cost function may not be sufficient to ensure unique equilibrium trajectories. It re-mains a goal of future research to consider the sufficient conditions, and the particular restrictions, if any, that need to be imposed to secure both necessary and sufficient conditions (Dockner et al., 2000).

The equilibriums are subgame perfect if they are autonomous (Dockner et al., 2000). From the derivation in the appendix, our solution trajectories are autonomous, that is,

p iðtÞ ¼ / i piðxiðtÞ;

s

ðtÞ; tÞ ¼ / i piðxiðtÞ;

s

ðtÞÞ; ð13Þ diðtÞ ¼ / i diðxiðtÞ;

s

ðtÞ; tÞ ¼ / i diðxiðtÞ;

s

ðtÞÞ: ð14Þ

Applying the Markovian Nash equilibrium(11) and (12)into the HJB equations(7), we are then able to solve the Markov-ian Nash equilibriums with the Hamilton–Jacobi (HJ) equations(15).

riVi¼

mi

ðxi;/ipiðxi;

s

ÞÞ  ciðxi;

s

;/ i diðxi;

s

ÞÞ þ Vix_xiðxi;x 1_;/i piðxi;

s

ÞÞ þ Vis

s

_ðxi;

s

;/ i diðxi;

s

ÞÞ n o ; i ¼ 1; 2: ð15Þ

In a competitive environment, gaining product recyclability is deliberate. A firm often expands its market share by offer-ing prudent price promotion in order not to cause their rivals to fight-back. The small increase in sales gradually costs the manufacture extra fees to process the waste. This excess cost, however, tends to eliminate the benefit of price promotion and give rise to a more conservative promotion strategy. In other words, a producer can choose to sell less in exchange for lower processing fees without engaging in any product design changes, even though an intensive incentive program has been real-ized in a market.

According to the aforementioned assumption, and for the purpose of illustration, we explicitly set the parameter func-tions as

m1

ðx; p1Þ ¼ Cm1x þ 1 2Km1p 2 1 ð16Þ

m2

ðx; p1Þ ¼ Cm2ð1  xÞ þ 1 2Km2p 2 2 ð17Þ h1ðd1Þ ¼ 1 2Ch1d 2 1 ð18Þ h2ðd2Þ ¼ 1 2Ch2d 2 2 ð19Þ u1ðx; d1Þ ¼ Cu1ð1  d1Þ ffiffiffi x p ð20Þ u2ðx; d2Þ ¼ Cu2ð1  d2Þ ffiffiffiffiffiffiffiffiffiffiffi 1  x p ð21Þ nð

s

;fÞ ¼ Enf

s

ð22Þ

where production costs w1and w2have been merged into the expression of Cm1 and Cm2, respectively. Our main problem

therefore can be rewritten explicitly as

max p1;d1 Z 1 0 ert _Cm 1x þ 1 2Km1p 2 1 1 2Ch1d 2 1 Cu1ð1  d1Þ ffiffiffi x p  Enf

s

  dt ð23Þ max p2;d2 Z 1 0 ert _Cm 2ð1  xÞ þ 1 2Km2p 2 2 1 2Ch2d 2 2 Cu2ð1  d2Þ ffiffiffiffiffiffiffiffiffiffiffi 1  x p  Enf

s

  dt Subject to _x ¼

q2

p2 ffiffiffi x p 

q1

p1 ffiffiffiffiffiffiffiffiffiffiffi 1  x p  dð2x  1Þ ð24Þ

a

s

_ ¼ 

gs

ðtÞ 

e1

d1 ffiffiffi x p 

e2

d2 ffiffiffiffiffiffiffiffiffiffiffi 1  x p ð25Þ xð0Þ ¼ x0 ð26Þ

s

ð0Þ ¼

s0

ð27Þ

Proposition 1. For the competition described by(16)–(25), the optimal recyclability in the Markovian Nash equilibrium is a non-decreasing functional of the market share. That is,@diðÞ

@xiðÞP0.

(Please refer to appendix for proof.)

Under the Markovian Nash equilibrium, the market share trajectories are not necessarily increasing, instead, it follows the sales dynamics controlled by optimal pricing, so that recyclability cannot be guaranteed to be improved. In the case of a mar-ket share trajectory not increasing, the government cannot drive producers to a state of higher recyclability without other effective policy. On the other hand, the government can demand all producers take more product responsibility through making the necessary capital investment – for example, production process reconstruction for total waste reduction. This additional expenditure can change the cost structures of manufacturers and force them to reduce costs in other ways, as there is often no room to raise the sales price in a competitive market. In order to meet government standards and take advantage of available incentive programs, a certain degree of product design change needs to be performed – such as easy-disassembly, or increasing the percentage of recyclable components. Observing the behavior of our model, we conjec-ture that if the government forces producers to adopt a higher standard of responsibility in recycling waste, producers ap-pear to be more environmentally conscious.

0 1 2 3 4 5 6 7 8 9 10 0 100 200 300 400 500 periods Profits firm 1 firm 2 firm 1 (stringent) firm 2 (stringent) 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 Market Share 0 1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 Pricing Trajectory 0 1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 Recyclability

Proposition 2. For the competition described by(16)–(25), the optimal recyclability in Markovian Nash equilibrium is a non-decreasing functional of the regulation stringency (negative of f). That is,@diðÞ

@fðÞ 60.

(Please refer to appendix for proof.)

We understand that financial incentives behave differently in a competitive environment (Vogelsang, 2002). This paper explains the elaborate interaction between market share, pricing and product design. We demonstrate our research findings by two experiments – one comparing the effectiveness of fixed versus increasing policy stringency and the other one show-ing the performance with various policy strshow-ingency. Our propositions can be illustrated and reviewed inFig. 2with the re-lated parameter settings inTable 1.

Based on the parameter settings, the optimal state trajectories follows:

_x ¼  2q1R1 ffiffiffiffiT p þ2q2R2 ffiffiffiffiffiffiTX p 1þX þ 2d   x þ 2

q

1R1 ffiffiffiffi T p 1þXþ d;

a

s

_¼ 

gs

 ð

e1

F1

e2

F2Þx 

e2

F2; xð0Þ ¼ x0;

s

ð0Þ ¼

s0

:

There are two designate scenarios expressed inFig. 2. The scenario with fixed policy stringency is of dash lines. The other scenario is of solid lines. Based on the suggestion of the Markovian Nash equilibriums, the market share of firm 1 decreases while that of firm 2 increases. Both of their profit rates, however, are increasing. As the market share of firm 1 decreases, in order to keep suitable profits, its optimal product recyclability strategy will decreases as well. That is, in this case, firm 1 stops making improve to their product Design for Environment.

On the other hand, their behaviors can be altered by a deliberate policy design. We mark the results of the increasing pol-icy stringency scenario as solid lines inFig. 2. Observing this figure, the optimal recyclability for firm 1 increases as the reg-ulations become more stringent, regardless of its losing market share. In this case, producers will to take more responsibility for environmental protection. Therefore, the goal of increasing producer responsibility has been achieved.

In order to manifest the influence of regulation stringency, we conduct another experiment using the parameter set as previous experiment. The Recycling performance changes can be observed by changing the rate of stringency. We let the regulation standard gradually raised by(28).

f¼ f0þ

v

fð1  expðtÞÞ: ð28Þ

The regulation grows with a rate of

v

f. As shown inTable 2, all parameters remain unchanged in the second experiment and

ten levels of rate

v

fhave been employed in this experiment. In spite of profit decreasing as the regulation becomes more

stringent, the recyclability of both firms increases significantly. Under this policy, manufacturers are therefore endowed with motivation to enhance their product design.

Table 1

Experiment 1 – parameter settings for comparison scenarios.

qi i d a g r Cmi Kmi Chi Cui f x0 s0

0.3 2 0.8 0.08 10 0.8 0.8

Firm1 0.3 1.1 10 0.1 36 18

Firm2 0.3 1.1 10 0.1 36 18

Table 2

Experiment 2 – profit and recyclability increase with stringent rates increased.

Stringent ratevf Profit J1 Profit J2 Final recyclability d1ðTÞ Final recyclability d2ðTÞ

0.0 933 931 5.96 4.21 0.5 892 896 7.03 4.95 1.0 847 859 8.09 5.69 1.5 798 821 9.16 6.43 2.0 744 780 10.2 7.16 2.5 687 738 11.3 7.90 3.0 626 693 12.3 8.64 3.5 561 647 13.4 9.38 4.0 492 599 14.5 10.1 4.5 419 549 15.5 10.8

3. Conclusions

This paper is different from existing works in that it analyzes the interactive effects of financial drivers and environmental policies through a dynamic approach. This paper integrates existing differential game models and establishes a novel dynamics analysis that encourages product recyclability. Taking time and competitors’ reactions into consideration, the con-ditions that drive manufacturers to enhance product recyclability have been identified.

This paper makes a contribution on the EPR effectiveness issue in a competitive market. Based on the results of this paper, governments should opt to gradually raise regulation standards so that rational manufacturers will implement the corre-sponding Markovian strategies, i.e., gradually improve its product recyclability. On the other hand, more incentive benefits nevertheless need to be provided where the regulation standard is fixed, in order to urge businesses to achieve the same level of recyclability as in the case of rising standards. This conclusion cannot be reached without considering the interactive behavior among competitive firms.

Our results further indicate that governments should consider the effectiveness of environmental policy on the premise that it is nature for business to pursue maximal profits. In order to develop EPR among industries, the first priority of the government should be to enact laws or regulations with rising standards to complement available financial incentive pro-grams. Moreover, to make our differential game model closer to reality, future research can be conducted with other types of treatment agencies, such as Producer Responsibility Organization (PRO), private treatment agencies and the issue of illicit disposal of informal sectors.

Acknowledgements

This research was supported by Grant NSC 92-2416-H-009-005, NSC-94-2211-E-327-006, and NSC-97-2410-H-009-042-MY3 from the National Science Council of Taiwan. The authors wish to thank the referees for their helpful comments, which have led the authors to consider more deeply the subject of reverse logistics. The valuable suggestions of Professor Wayne K. Talley to improve this paper are also gratefully acknowledged. Any errors or omissions responsibility of the authors. Appendix.

Proof for Proposition 1 in conditions of recyclability. Given the results of(11), apply the function form(16) to (25), the

Eqs.(11) and (12)expand to

p1¼

q

1 Km1 V1x ffiffiffiffiffiffiffiffiffiffiffi 1  x p ð29Þ p2¼

q2

Km2 V2x ffiffiffix p ð30Þ d1¼ e1 aV1sþ Cu1 Ch1 ffiffiffi x p  F1 ffiffiffix p ð31Þ d2¼ e2 aV2sþ Cu2 Ch2 ffiffiffiffiffiffiffiffiffiffiffi 1  x p  F2 ffiffiffiffiffiffiffiffiffiffiffi 1  x p ð32Þ

Substitute the Markovian strategies(30)–(32)into(8)and then we have the Hamilton–Jacobi equation

rV1¼ Cm1x 

q

2 1 2Km1 V21xð1  xÞ 1 2Ch1F 2 1x  Cu1F1x  Enf

s



q

2 2 Km2 V1xV2xx  V1xdð2x  1Þ 

g

a

V1s

s



e1

a

F1V1sx 

e2

a

F2V1sð1  xÞ; rV2¼ Cm2ð1  xÞ 

q

2 2 2Km2 V2 2xx 1 2Ch2F 2 2x  Cu2F2x  Enf

s



q

2 1 Km1 V1xV2xð1  xÞ  V2xdð2x  1Þ 

g

a

V2s

s



e1

a

F1V2sx 

e2

a

F2V2sð1  xÞ:

We conjecture that the value function Viis linear in the state variables as the sales dynamics(1)are of the Vidale–Wolfe

form (Prasad and Sethi, 2004).

V1¼ A1þ B1x þ C1s;

V2¼ A2þ B2ð1  xÞ þ C2s:

Therefore V1x¼ B1;V1s¼ C1;V2x¼ B2and V2s¼ C2. The HJ equations expand to

rA1þ rB1x þ rC1s¼ 

q

2 1 2Km1 B21þ dB1

e2

a

F2C1 þ

q

2 1 2Km1 B21 2dB1

q

2 2 Km2 B1B2 1 2Ch1F 2 1   Cu1þ

e1

a

C1   F1þ

e2

a

F2C1þ Cm1  x þ 

g

a

C1 Enf  

s

; rA2þ rB2x þ rC2s¼ 

q

2 2 2Km2 B22 dB2

e1

a

F1C2 þ

q

2 2 2Km2 B22þ 2dB2

q

2 1 Km1 B1B2 1 2Ch2F 2 2   Cu2þ

e2

a

C2   F2þ

e1

a

F1C2þ Cm2  ð1  xÞ þ 

g

a

C2 Enf  

s

:

Equating powers of x and

s

, some of the unknowns can be easily solved as

A1¼  1 r

q

2 1 2Km1 B21 dB1þ

e2

a

F2C1   ; A2¼  1 r

q

2 2 2Km2 B22þ dB2þ

e1

a

F1C2   ; C1¼ C2¼  Enaf

a

r þ

g

; Let R1¼

q

2 1 2Km1 ; R2¼

q

2 2 2Km2 ; W ¼ r þ 2d; H1¼

e1

f

a

r þ

g

; H2¼

e2

f

a

r þ

g

; Z1¼  3 2Ch1 ðCu1 H1Þ 2  1 Ch2 ðCu2 H2ÞH2þ Cm1; Z2¼  3 2Ch2 ðCu2 H2Þ 2  1 Ch1 ðCu1 H1ÞH1þ Cm2: To solve B1and B2, R1B21 WB1 2R2B1B2þ Z1¼ 0;  R2B22 WB2þ 2R1B1B2þ Z2¼ 0; or WðB1þ B2Þ 2  ðZ1þ Z2Þ 2 ¼ 0 R1B21þ R2B22 2ðR1þ R2ÞB1B2þ ðZ1 Z2Þ ¼ 0 Let B1¼ r cos h; B2¼ r sin h;

Applying the parameterization approach, the system of nonlinear equations transforms to r2_{ð1 þ sin 2hÞ ¼ ððZ} 1þ Z2Þ=WÞ2; ð33Þ r2 _{1 þ}1 2 R2 R1 2R1þ R2ð1  cos 2hÞ   ¼ ðR1þ R2Þ  ððZ1þ Z2Þ=WÞ2 ðZ1 Z2Þ: ð34Þ Set S ¼ ððZ1þ Z2Þ=WÞ 2 ; T ¼ ðR1þ R2ÞððZ1þ Z2Þ=WÞ2 ðZ1 Z2Þ: Divide(33)by(34)as T2R2þ R1 2R1þ R2 S   tan2_{h 2S tan h þ T  S ¼ 0:} Therefore tan h ¼ S  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 T2R2þR1 2R1þR2 S   ðT  SÞ r T2R2þR1 2R1þR2 S  X and r ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 1 þ sin 2 tan1_X s ¼  ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tð1 þ X2Þ ð1 þ XÞ2 s

Transform back to B1and B2,

B1¼  ffiffiffiffiffi T p 1 þ X; B2¼  ffiffiffiffiffiffiffiffi TX p 1 þ X;

The Markov Nash equilibriums follow

p 1¼ 2R1 ffiffiffiffiffi T p 1 þ X ffiffiffiffiffiffiffiffiffiffiffi 1  x p ; p 2¼ 2R2 ffiffiffiffiffiffiffiffi TX p 1 þ X ffiffiffi x p ; d1¼ Ene1f arþgþ Cu1 Ch1 ffiffiffi x p  F1 ffiffiffi x p d2¼ Ene2f arþgþ Cu2 Ch2 ffiffiffiffiffiffiffiffiffiffiffi 1  x p  F2 ffiffiffiffiffiffiffiffiffiffiffi 1  x p :

Therefore, the derivative of optimal recyclability diwith respect to the market share x becomes

@di

@x ¼ FiP0 

Proof for Proposition 2 with respect to stringency. Follow the results in Proposition 1, the derivative of optimal recycla-bility diwith respect to f becomes

@di @f ¼ Enei

a

r þ

g

60; since

a

;

g

6_{0, and r; En;}



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