A Decision-Making Model of Low-Carbon Supply Chain Based on Government Subsidy

In order to study the role of government subsidy in a low-carbon supply chain, a two-stage game model for the subsidy of low-carbon technology and the subsidy of recycling is built. And the optimized subsidy strategy under the two kinds of subsidy policies is obtained. This paper also studies the impacts of carbon emission abatement, subsidy and the factors of subsidy on the supply chain and the government, so as to compare the two kinds of subsidy strategy. This study shows that the coefficient of environmental benefits has a positive impact on the two kinds of subsidy. The subsidy can stimulate the carbon emission abatement, and the fixed cost has a negative impact on the will of the carbon emission abatement for the enterprises. Besides, the low-carbon preference of the consumers has a positive impact on the carbon emission abatement under the subsidy of low-carbon technology. Furthermore, the government under takes the main costs of the low-carbon technology under the subsidy of low-carbon technology. The environmental benefit also has a positive impact on the subsidy and profits of the government and the profits of the supply chain under the recycling subsidy. Both kinds of subsidy strategies can achieve supply chain coordination.


Introduction
The problems of climate and energy have greatly attracted the attention of the international community.The increasingly stringent international environmental laws, such as the United Nations Framework Convention on Climate Change and the Kyoto Protocol, have made the government, enterprises, and consumers pay more attention to low carbon.However, because of the externality of low carbon, the low-carbon strategies of enterprises driven by the market might not be the best one, and varied strategies of the government are needed to promote the investment of enterprises.Subsidies are more positive compared to restrictive policies, such as carbon tax, carbon emissions trading.
Technical innovation is the core of achieving low-carbon society, so the investment of technology must be sustained and large-scale.Meanwhile, invest in technology can be very risky, so the investment is costly currently.Wang (2011) pointed out that fund shortage has severely restricted the development of low-carbon technology, so the improvement of subsidy system is vital and impending.Zhou (2011) pointed out government policies had significant impacts on the technological innovation of low-carbon technology.And investments from the government subsidy plans on the creation of low-carbon technology innovation platforms and the commercial applications of the achievement can push the developments of the low-carbon technology.Fang (2012) studied a three-stage game model of the government and a duopoly supply chain, and respectively analyzed the impact of government's subsiding enterprises' R&D and product innovation on research investment and social welfare.Meng (2010) also analyzed the impact of subsidy on the carbon emission reduction and social welfare under different emission taxes.
Recycling has made forward supply chains convert to closed-loop supply chains.Sameek et al. (2006) showed that closed-loop supply chains could have the impact of energy saving and emission reduction on environmental protection and resource utilization.Karine et al. (2006) pointed out that closed-loop supply chains could be regarded as an effective carrier to promote the development of low-carbon economy, with the development of low carbon concept and sustainability awareness, and many countries and research institutions had greatly supported closed-loop supply chains.Zhu et al. (2010) studied the efficient of recycling under government subsidy, and found the enterprise's maximum return rate when the government was to maximize, meanwhile the subsidy had positive impact on the profits of enterprises in the supply chains.Zhou et al. (2011) built a two-echelon supply chain model and found the conclusion that recycling subsidy had positive effect on the profits of enterprises in the supply chain, return rates and carbon emissions.
Most of existing literatures had separately studied the impact of low-carbon-technology subsidy on enterprise profits, emission reduction and the impact of recycling subsidy on return rate and enterprise profits.However, there are few literatures which have compared the two kinds of subsidies, and study their factors and influences.Herein, in this paper, we will concentrate on analyzing the decision of the government and the companies in the supply chain, under the two kinds of subsidy strategies.This paper includes three main parts: (1) the factors which would affect low-carbon-technology subsidy, recycling subsidy and carbon abatement; (2) the impact of the low-carbon-technology subsidy and recycling subsidy on low-carbon efficiency and profits of the government and enterprises in the supply chain; (3) the differences of the impacts which low-carbon-technology subsidy and recycling subsidy have on the supply chain and government administration.Based on the study above, this paper may provide some helpful suggestions for supply chain's decision of investment in low-carbon technology, the decision of recycling and government subsidy strategies.

Problem Description and Parameters
This paper considers a two-echelon supply chain which consists of one manufacturer and one retailer.And the manufacturer and retailer have an equal partnership.The manufacturer generates carbon emissions during the production, and the retailer generates carbon emissions during the circulation of products.The enterprises can abate carbon emissions by investing in low-carbon technology and recycling.Both low-carbon technology and recycling will have a strong effect on enterprises' costs and profits.The government would give proper subsidies to encourage the enterprises in carbon abatement.
By building the model of low-carbon technology investment and the model of recycling, under government subsidy, this paper studied the optimized subsidy strategy and carbon-abating strategy.So, a two-echelon game model under two ways of carbon abatement is built, and the model includes: (1) the government makes the subsidy policy at the first stage; (2) the enterprises in the supply chain make the decision of carbon abatement.The major notations in our following model are summarized in Table 1.Assumption 4. The manufacturer and retailer have an equal partnership.There is exists information symmetry in the three parties of the manufacturer, retailer and government, i.e., the government knows the supply chain's investment in low carbonization.The government gives proper subsidies to the manufacturer and retailer for the expenditure of low carbon, and the subsidy coefficients are λ m , λ r respectively.
Assumption 5.The manufacturer supply products according to the maximum demand D+ΔD.

Modeling and Analysis
Base on the above description and assumptions, the supply chain optimization model under low-carbon-technology subsidy is built.In the model, the government gives proper subsidies to the manufacturer and retailer for the expenditure of low carbon, and the subsidy coefficients are λ m , λ r respectively.θ m , θ r are decision variable, and objective functions are the enterprises in the supply chain and the government's profits.

1) Profit function of the manufacturer
The profit of the manufacturer includes the income from selling products to the retailer,w ∆ , and government's subsidy λ m C m .The cost of the manufacturer includes production cost, c ∆ , and the investment in low-carbon technology, C m.So the profit function can be described as the formula (1).

( )( )
2) Profit function of the retailer The profit of the retailer includes the income from selling products to the consumers, p ∆ , and government subsidy λ r C r .The costs of the retailer includes cost of purchasing products from the manufacturer, w ∆ , and the investment in low-carbon technology, C r .So the profit function of the retailer is as the formula (2). (2) 3) Profit function of the government The profit of the government includes environmental benefits from the enterprises' investment in low carbon, (C m + C r ) RF, and total profit of the supply chain, П m +П r.The cost of government is the subsidy for low-carbon technology, λ m C m +λ r C r .So the profit function of government is as the formula (3).
In the above formula, RF represents the environmental profits coefficient of enterprise's investment in carbon abatement, and 0<RF<1.
Proposition 1.On the condition of 0<RF<1，0<α,β<0.5,there exists the unique equilibrium solution of subsidy rates which the government invest in enterprises and emission reduction rates of enterprises, λ m * , λ r * , θ m * , θ r * , as follows.
Proof of Proposition 1.Using the method of inverse solution, in the second stage of the game, the government has set the subsidy rate, and both the manufacturer and retailer try to maximize their own benefits.Then the problem could be changed as follows: For formula (4), ( 5), the first-order conditions (FOC) are as follows: By solving the system of the two first-order conditions, formula (6), (7), the following solution is gained.1)

(
In the first stage of the game, the government set the optimal subsidy rate to maximize its profit, so the objective function is as follow. Take formula (8), (9) into formula (10), and take the first derivative of formula ( 6) with respect to λ m ，λ r respectively.The subsidy rates, λ m ， λ r , can be obtained.Then take formula (11), (12) into formula(8), ( 9), so θ m *, θ r * can be obtained.Thus, proposition 1 is true.
Property 1. From formula ( 8) and ( 9), we can find that 0 , 0 , 0 , 0 , 0 , 0 the government subsidies and consumers' preference for low carbon have positive impact on the enterprises' carbon abatement, however, the cost coefficient of low-carbon technology has negative cost coefficient of low-carbon technology.
RF has a positive impact on λr, i.e. the environment benefits more from the low-carbon technology, the government would be more willing to encourage enterprises in carbon abatement.
Property 3. From formula (11), ( 12), we can know , so w is the determining factor in profit distribution of the manufacturer and retailer, when p and c are settled.When w is bigger, the manufacturer would get more benefit, then the government would give less subsidy to the manufacturer and give the retailer more subsidy, and vice versa.So the government's subsidy for low-carbon technology can balance the manufacturer and retailer's investment in carbon abatement, and the two enterprises can obtain the best benefit.

Model Assumptions
Assumption 1.The manufacturer can not only use new materials to make new products, but also can use recycling products to make remanufacturerd products.And the market doesn't differentiate the two kinds of products.
Assumption 2. The retailer sells the products and recycles used products from consumers through its selling channels, and unit recycling price is p 1 .The manufacturer recycles used products from the retailer to make remanufacturerd products, and the unit recycling price is w 1 , the unit cost of remanufacturing is c mr .the remanufacturerd products are also sold to retailer at the price of w, and p 1 < w 1 < c-c mr .The government makes subsidy coefficient of recycling subsidies, s, according to the rate of recycling τ.
Assumption 3. The fixed cost of recycling is A(τ), and it is related to τ, and A(τ)would increase when τ increases.The functional relationship between A(τ)and τ is as follows.

( )
A e

  
The e in the function above is cost coefficient of recycling, and e > 0.
Assumption 4. The carbon emission during the manufacturer's remanufacturing process is ξ times of carbon emission during production with new materials, and 0<ξ<1, so the change in carbon emissions is E  . (

Modeling and Analysis
Base on the description and assumptions above, the low-carbon supply chain decision model under recycling subsidy is built.In the model, the government gives proper subsidies to the manufacturer and retailer according to rate of recycling, and rate of recycling τ and subsidy coefficient of recycling s are decision variables and objective functions are the enterprises in the supply chain and the government's profits.This subsidy strategy is aimed to encourage the enterprises in recycling.

1) Profit function of the manufacturer
The profit of the manufacturer includes the income from selling products to the retailer, wD.The cost of the manufacturer includes production cost, but because of recycling the unit cost turns into (1-τ)c+τ(c mr +w 1 ), after simplification it is c-τc+τc mr +τw 1.So the total production cost is(c-τc+τc mr +τw 1 )D.So the profit function is as the formula (13).

( )
2) Profit function of the retailer The profit of the retailer includes the income from selling products to the consumers,pD, the income from selling recovered products to the manufacturer, w 1 τD, and government's subsidy sτ.The cost of the manufacturer includes production cost, cD, the fixed cost of recycling A, and the cost of recycling used products from the consumer p 1 τD.So the profit function of the retailer is as the formula ( 14).

3) Profit function of the government
The profit of the government includes environmental benefits from the enterprises' investment in low carbon ARF, and total profit of the supply chain, П m +П r.The cost of government is the subsidy for recycling.So the profit function of government is as the formula (15).
Proposition 2. On the condition of 0<RF<1，there exists the unique equilibrium solution of governmental subsidy rates and rate of recycling.
Proof of Proposition 2. From formula (13) we can know τ has positive effect on profit of the manufacturer, as c >w 1 +c mr.Using the method of inverse solution, in the second stage of the game, the government has set the subsidy rate s, and the retailer make decisions to maximize its own benefit.Then the problem could be changed as follow: Take the first derivative of formula ( 16) with respect to , make the derivative equals to zero, and solve the equation.Then we can get formula (17).
In the first stage of the game, consider τ is known, the objective function is as follow.
Take formula (17) into formula (18), and take the first derivative of formula ( 18) with respect to s.
Take formula (19) into formula ( 17), then *  can be obtained as formula (20).Thus, proposition 2 is true.(3) As RF increases, the profits of the members just change a little under low-carbon-technology subsidy strategy, and the government undertakes most of the investments in low-carbon technology.However, when 0.3 ≤ RF < 0.65, the government is willing to give subsidies, and all the members' profits increase as RF increase.
(4) The government subsidies are given according to the profit distribution of the enterprises, which can balance the manufacturer and retailer's investment in carbon abatement under low-carbon technology subsidy strategy.The recycling subsidy makes the retailer make decisions considering the whole supply chain instead of its own profits.

Figure
Figur

Table 1 .
Model parameters and the definitions

m ,П r ,П g Expected
profits of the retailer, manufacturer and the supply chain

CO m ,CO r
impacts on both emission reduction rate and recycling rate.Consumer's preference to low-carbon products also has positive impact on emission reduction rate.So the government should help consumers to enhance their awareness of low-carbon life concepts to improve the emission reduction rate.