The Game Model of Closed-Loop Supply Chain Based on Tax and Subsidy

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The Game Model of Closed‐Loop Supply Chain Based on Tax and Subsidy Xin‐ran Li1, Rong Chen*2 Faculty of Management and Economics, Dalian University of Technology, China lixr@dlut.edu.cn; *2dreamcr@163.com

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Abstract This paper discussed a closed‐loop supply chain (CLSC) consisting of a single manufacturer, remanufacturer and retailer under the tax and subsidy policy. We constructed CLSC models with or without tax and subsidy respectively and made a comparative analysis of the two models. We discussed the impact of the tax as well as the subsidy on the decision and profits of the CLSC models based on remanufacturing. Results revealed that, (1) compared with the CLSC models without the tax and subsidy policy, the new product’s market demand will decrease while the remanufactured product’s market demand and the quantity of recycling of waste product will increase. Further the manufacture’s profit will be less and the remanufacturer’s profit will be more; (2) the profit of retailer and CLSC system and the product price vary with the size of the tax and subsidy changes.

Keywords Closed‐Loop Supply Chain; Tax and Subsidy; Decision Strategies

Introduction In recent years, the economic and environmental value of obsolete product remanufacturing has attached more and more attention by researchers and government authorities. Many governments enact environmental laws to reduce the emissions of toxic and harmful substances. On August 7, 2012, Regulations of Government Fund for the Recovery and Disposal of Waste Electrical and Electronic Products was jointly issued in China. The regulations stated that the producer pays taxes according to the quantity of products they sell while the government subsidizes the recycling and remanufacturing enterprises according to the amount they collected and remanufactured.

The effect of the tax and subsidy policy on the promotion of remanufacturing for waste products has been widely confirmed. Bansal et al. (2003) considered two sets of policy instruments—uniform policies for the firms and policies that discriminate between the firms based on their environmental quality. Hong et al. (2011) presented a stackelberg‐type model to determine advanced recycling fees and socially optimal subsidy fees in reverse supply chains. On this basis, Hong et al. (2014) added competition and did further research. Atasu et al. (2013) compared manufacturer‐operated systems and state‐operated systems and found that their impacts on different stakeholders can be significantly different. Shi et al. (2013) investigated two critical environmental factors of the product weight and the collection rate, as well as their environmental consequence of the landfill quantity. Cao et al. (2013) designed incentive contracts between government and manufacturers under the condition that government taxes new products and give subsidies to remanufactured products. Gao et al. (2014) comparatively analyzed pricing strategy of a CLSC with carbon tax, subsidy and a combination of both and found the combination is most effective. Yu et al. (2014) discussed four different recycling models of the government tax and subsidy. Xiao et al.

(2014) studied the government’s incentive mechanism for manufacturers with different policy backgrounds.

Above results have made contributions to the study of CLSC with tax and subsidy, but none has considered price discrimination and competition between remanufacturers and manufacturers. Therefore, this paper attempts to analyze the decision strategies of a CLSC with competition and price discrimination under the tax and subsidy.

Model Description We examine a CLSC system composed of one manufacturer, one remanufacturer and one retailer. The manufacturer manufactures new products with raw materials and the remanufacturer remanufactures with

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obsolete products collected from customers.The retailer sells products wholesaled from the manufacturer to consumers. The government taxes the manufacturer according to the quantity of new products and subsidizes the remanufacturer according to the amount of remanufactured products. The CLSC model is shown in Fig. 1. t

wn

pn

s

wr

pr

A

FIG. 1 THE MODEL OF CLSC WITH TAX AND SUBSIDY

The basic assumptions and variable symbols of this paper are as follows: Assumption 1. The manufacturer, remanufacturer and retailer are risk neutral and completely symmetric with their information. Both the manufacturer and remanufacturer act as the Stackelberg leader and they are mutually independent, and the retailer acts as the Stackelberg follower.

Assumption 2. Waste products in the market are enough for collection and remanufacturing and the only purpose of them is recycling. Obsolete products are homogeneous and can be recycled into remanufactured products. One unit of an obsolete product can be recycled into one unit of a remanufactured product, yielding a remanufacturing rate of 1, so the market demand for remanufactured product is the recycled amount of obsolete products.

The unit production cost of the new product and the remanufactured product are cn and cr and cr  cn , where cn  cr   signifies a remanufacturing cost advantage. The unit wholesale price of the new product and the remanufactured product are wn and wr .The unit sale price of the new and remanufactured products are pn and pr .

It must be satisfied that wn  wr and pn  pr . The unit recycling price of the obsolete product that is given to the consumers is A and A   .

The new products and remanufactured products are alternative. We supposed that the demand functions for new products and remanufactured products are qn  n  pn   pr and qr  r  pr   pn respectively and

n  r , 0    1 .(Ferrer et al. 2006) In the following, the variables that were given with the superscript “NG” and “YG” represent the CLSC with or without the tax and subsidy, and the variables with the superscript ”*” represent the optimal decision results.

CLSC Decision-making Model CLSC Model without Tax and Subsiy In this case, the manufacturer decides the wholesale price of the new product and the remanufacturer decides it of the remanufactured product based on their own profit maximization. The retailer decides the sale price based on its own profit maximization. The profit functions of the manufacturer, remanufacturer and retailer are as follows:

 M ( wn )  ( wn  cn )(n  pn   pr ) ,  r ( wr )  ( wr  cr  A)(r  pr   pn ) ,  R ( pn , pr )  ( pn  wn )(n  pn   pr )  ( pr  wr )(r  pr   pn ) Given that the manufacturer and remanufacturer are leaders and the retailer is the follower, the retailerʹs decisions depend on the two leaders’s decisions. Using backward induction, we have the following proposition: Proposition 1. Without tax and subsidy,  R ( pn , pr ) is a strictly concave function on pn and pr . So formula exists in only the optimal solution. Therefore, the optimal decisions of the closed‐loop supply chain are as follows: wnNG* 

2(n  cn )   (r  cr  A) 4  2 prNG* 

, wrNG* 

 (n  cn )  2(r  cr  A) NG* 3(2   2 )n   (5  2 2 )r 2cn   (cr  A) , pn 

 2(4   2 )(1   2 ) 2(4   2 ) 4  2

3(2   2 )r   (5  2  2 )n  2(4   2 )(1   2 )



 cn  2(cr  A) NG* 2n   (r  cr  A)  (2   2 )cn , qn  , 2(4   2 ) 2(4   2 )

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qrNG* 

2r   (n  cn )  (2   2 )(cr  A) 2(4   2 )

The optimal profits of the manufacturer, remanufacturer and retailer are:

 MNG* 

 RNG* 

[2n   (r  cr  A)  (2   2 )cn ]2 2(4   )

2 2

,  rNG* 

[2r   (n  cn )  (2   2 )(cr  A)]2 2(4   2 ) 2

(  2  2)n  3r  2(1   2 )cn   (1   2 )(cr  A) 2n   (r  cr  A)  (2   2 )cn  2(4   2 )(1   2 ) 2(4   2 )

(  2  2)r  3n   (1   2 )cn  2(1   2 )(cr  A) 2r   (n  cn )  (2   2 )(cr  A)   2(4   2 )(1   2 ) 2(4   2 )

CLSC Model with Tax and Subsidy Based on the model without tax and subsidy, this section considers CLSC model with tax and subsidy. The government taxes per unit of new product by t and subsidizes per unit of remanufactured product by s . The profit functions of the manufacturer, remanufacturer and retailer are as follows:

 M ( wn )  ( wn  cn  t )(n  pn   pr ) ,  r ( wr )  ( wr  cr  A  s)(r  pr   pn ) ,  R ( pn , pr )  ( pn  wn )(n  pn   pr )  ( pr  wr )(r  pr   pn ) The same with the model of Section 3.1, we have the following proposition: Proposition 2. With tax and subsidy,  R ( pn , pr ) is a strictly concave function on pn and pr . So formula exists in only the optimal solution. Therefore, the optimal decisions of the CLSC are as follows: wnYG*  pnYG* 

2(n  cn  t )   (r  cr  A  s ) 4  2

3(2   2 )n   (5  2 2 )r 2(4   )(1   ) 2

qnYG* 

2



2(cn  t )   (cr  A  s ) 2(4   ) 2

2n   (r  cr  A  s )  (2   2 )(cn  t ) 2(4   2 )

, wrYG*  , prYG*  , qrYG* 

 (n  cn  t )  2(r  cr  A  s ) 4  2 3(2   2 )r   (5  2  2 )n 2(4   )(1   ) 2

2



 (cn  t )  2(cr  A  s) 2(4   2 )

2r   (n  cn  t )  (2   2 )(cr  A  s ) 2(4   2 )

The optimal profits of the manufacturer, remanufacturer and retailer are:

 MYG* 

 RYG* 

[2n   (r  cr  A  s )  (2   2 )(cn  t )]2 2(4   )

2 2

,  rYG* 

[2r   (n  cn  t )  (2   2 )(cr  A  s )]2 2(4   2 ) 2

(  2  2)n  3r  (1   2 )[2(cn  t )   (cr  A  s )] 2n   (r  cr  A  s )  (2   2 )(cn  t )  2(4   2 )(1   2 ) 2(4   2 )

3n  (  2  2)r  (1   2 )[  (cn  t )  2(cr  A  s)] 2r   (n  cn  t )  (2   2 )(cr  A  s )   2(4   2 )(1   2 ) 2(4   2 )

Model Comparison and Analysis We will compare the two models and analyse the effectiveness of the tax and subsidy policy and the effect on the optimal decision and profit of the CLSC. Theorem 1. When 0  t s   2 , wnYG*  wnNG* , pnYG*  pnNG* ; When 0  t s  2  , wrYG*  wrNG* , prYG*  prNG* .

It shows that both the size and the ratio of tax and subsidy are important for effectiveness of the government policy. The appropriate ratio is concerned with the alternative coefficient between the new product and remanufactured product. The government can control the price of the product by adjusting the size and the ratio of tax and subsidy to achieve the goal of promoting the waste product collection and remanufacturing.

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Theorem 2. qnYG*  qnNG* , qrYG*  qrNG* , and when t s  1 , the total market demand increases.

With the tax and subsidy, the market demand for the new product always decreases while the market demand for the remanufactured product and the quantity of waste product recycled increase. It shows that the tax and subsidy policy can promote the collection and remanufacturing of the discarded product. Besides, the CLSC scale also develops as the total market demand increases when the ratio of tax and subsidy satisfies a certain condition.

YG * Theorem 3.  M   MNG* ,  rYG*   rNG* , and there exists 1 and  2 that when t s  min{1 ,  2 } ,

 RYG*   RNG* ,  TYG*   TNG* .

It shows that the remanufacturer benefits from the tax and subsidy as its profit improved and the manufacturer suffers damages from the policy as its profit decreased. So the policy can improve the enthusiasm of the remanufacturer to remanufacture. The government can ensure the interests of the retailer and CLSC through adjusting the ratio between tax and subsidy, which is important to maintain the stability of CLSC and the enthusiasm of its members.

Theorem 4. When tqnYG*  sqrYG* ,  TYG*   TNG* .

It shows that the total profit of CLSC will decrease if the government pursues balance or surplus of the tax and subsidy. In other words, the government should set size of the tax and subsidy starting with the interests of the whole CLSC and its members instead of the surplus or balance of the tax and subsidy, which should take the market size, production costs, substitution coefficient of the new product and remanufactured product into account.

Conclusions Based on the tax and subsidy policy, we studied the decision strategies of a CLSC consisting of a single manufacturer, remanufacturer and retailer with competition and price discrimination. Through a comparative analysis of CLSC models with or without the policy, we found that the remanufactured product’s market demand and the quantity of recycling will increase with tax and subsidy, which means that the government can promote the waste product collection and remanufacturing as hoped. But most of all, the government should take all factors of CLSC into consideration and set right size of the tax and subsidy to ensure the interests and stability of the CLSC on the premise of promoting the waste product collection and remanufacturing.

ACKNOWLEDGMENT

This research is supported by the National Social Science Foundation of China under numbers 14BGL063.

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Li Xin‐ran, born in Dalian, China, on September 27, 1963,got masterʹs degree in Dalian University of Technology,china in June 1989. Main areas of research include operations management, logistics and supply chain management.

He is an associate professor in Faculty of Management and Economics, Dalian University of Technology. He published ʺPricing and Contract Coordination of Closed Loop Supply Chainʺ( monograph) in 2014.

2

Chen Rong, born in Yichang, China, on January 3 , 1989, currently studying for Masterʹs degree in Dalian University of Technology. The research direction is CLSC management.

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