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How to subsidize energy efficiency under duopoly efficiently?
Applied Energy 175 (2016) 31–39
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
How to subsidize energy efficiency under duopoly efficiently? Pu-yan Nie a,⇑, Yong-cong Yang b, You-hua Chen c, Zhao-hui Wang a a
Guangdong University of Finance & Economics (GDUFE), Guangzhou 510320, PR China Guangdong University of Foreign Studies, Guangzhou 510006, PR China c College of Economics & Management, South China Agricultural University, Guangzhou 510642, PR China b
h i g h l i g h t s This article captures the effects of output subsidy. Firms without subsidy are not willing to improve energy efficiency. Subsidy stimulates the subsidized firms’ outputs and deters the others’ outputs. The subsidy intensity depends on firms’ position. Overdue subsidy cannot reach the environmental object.
a r t i c l e
i n f o
Article history: Received 6 January 2016 Received in revised form 17 April 2016 Accepted 24 April 2016
JEL classfication: L1 Q43 Keywords: Energy efficiency Subsidy Duopoly
a b s t r a c t Establishing a game theory model, this paper captures the effects of output subsidy on energy efficiency under Cournot competition and Stackelberg competition. Three types of subsidies are considered in the model, namely without subsidy, unilateral subsidy and bilateral subsidy. The findings indicate that firms without subsidy are not willing to improve energy efficiency. Also, subsidy stimulates the subsidized firms’ outputs while deters the outputs of other firms. Meanwhile, the equilibrium subsidy intensity depends on firms’ position. Furthermore, the minimal subsidy budgets under different situations are presented. Especially, given the fixed subsidy budget, the output of the subsidized firm is the highest if this firm plays the leading position. In addition, certain subsidy can reduce the total emission, while overdue subsidy cannot reach the environmental object. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction International Energy Agency (IEA) defined energy efficiency (EE) as ‘‘something is more energy efficient if it delivers more services for the same energy input” [1]. Furthermore, to promote energy efficiency, IEA recommended the adoption of energy efficiency subsidy policies by governments [2–6]. McKibbin, Morris and Wilcoxen [7] and Wall [8] showed the advantages of energy efficiency subsidy. Therefore, energy efficiency subsidy is attached extensive importance by many researchers, such as the interesting reviewing papers of Gillingham et al. [9] and Allcott and Greenstone [10]. Actually, as one of the most important energy efficiency policies, subsidy plays a key role in the process of transformation from ‘industrial development’ to ‘green development’ [11]. For this ⇑ Corresponding author. E-mail address: [email protected] (P.-y. Nie). http://dx.doi.org/10.1016/j.apenergy.2016.04.105 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved.
reason, many countries and regions subsidize firms or consumers to improve energy efficiency to cope with global climate change [12], like Japan [13], Thailand [14], United Kingdom (UK) [15], and Sweden [16,17]. Fais et al. [18] addressed the regional energy efficiency with subsidy in Europe. There exist different types of energy efficiency subsidies all over the world, such as fixed subsidy and output subsidy. Craig and Allen [19] showed that demographic factors, attitudes, planned purchases, and energy efficiency initiatives of utility provider affect energy efficiency subsidy significantly. Furthermore, market structure also has significant impact on the effects of governmental subsidies on energy efficiency. Under monopoly, considering externalities and price-quality discrimination, Nauleau et al. [20] argued that social optimum could be achieved by different types of subsidies. In practice, since energy efficiency subsidies reduce emission with the support of public finance, it is very important to design
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P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
the suitable measures to subsidize energy efficiency [21]. Recently, Abardi and Cambini [22] developed energy efficiency strategies under incomplete information in competitive environment. Further, Arias and van Beers [23] suggested to subsidizing patents to improve energy efficiency, since they are positively related. Allcott et al. [24] recently gave a new way called as ‘‘tagging energy efficiency subsidies” to subsidize energy efficiency and argued that it would lead to large efficiency gain. Resorted to methods given by Sun and Nie [25], Nie [26] recently compared the effects of fixed subsidies with output subsidies and concluded that output subsidies improve environmental object and consumer surplus more than fixed subsidies. Significantly, Nie [26] proved the output subsidies are more efficient to reach environmental object and suggested output subsidies to improve energy efficiency. Furthermore, it is important to identify how the government should subsidize firms under asymmetric efficiency. Therefore, we follow the idea of Nie [26] and focus on two firms under asymmetric situation. Notice that besides energy efficiency, subsidies are commonly used in many other fields, such as R&D subsidies and investment subsidies. The impact of subsidies under duopoly, however, differs in various sectors and fields. For instance, R&D subsidies stimulate innovation investment effectively [27,28], while investment subsidy may crowd-out private expenditure in investment [29,30]. In related studies, regardless of subsidizing forms, the effects of subsidies are always discussed by game theory and the optimal subsidies could be obtained by backward induction. In this paper, we follow this method to analyze the effects of energy efficiency subsidies, while the results are different from the literature mentioned above because the major concerns are not the same. Under different primary energy efficiency, this article aims to capture the optimal subsidizing strategies. We try to answer the following questions in this paper: What are the effects of subsidy on the subsidized firms and other firms? How are the effects of firms’ position on firms’ strategies? What is the minimal subsidy budget? How are the effects of subsidy on total emission? Taking market structure into account, this paper relates to the studies of Nauleau et al. [20] and Nie [26]. Nauleau et al. [20] discussed the effects of subsidy on the energy efficiency under monopoly and argued that all types of subsidies can reach social optimum. While Nie [26] found that output subsidy is better than the fixed one under oligopoly. The findings of the two researches showed that the effects of energy efficiency depend on market structure. This article continues to capture the effects of market structure on energy efficiency under asymmetry and focus on the effects of firms’ position on energy efficiency subsidy. Specifically, we take Cournot competition and Stackelberg competition into consideration since they are standard and fundamental economic theories in industrial analysis. Cournot competition refers to the situation that firms make production decisions independently at the same time in quantity competition. In the case of Stackelberg competition, the leader firm moves first and then the follower firms move sequentially. In both situations firms compete in quantity to maximize their profits. The equilibrium could be obtained by backward induction. The contribution of this paper is to argue that total subsidy depends on the subsidized firms’ position. The subsidy intensity reaches the lowest if the subsidized firm acts as the follower, while reaches the highest when the subsidized firm plays the leading position. The minimal budget under various types of firms’ position is analyzed. The minimal budget is the lowest if the low energy efficiency firm acts as the follower and is solely subsidized. The minimal budget to subsidize both firms under the condition that the high energy efficiency firm acting as the leader is larger than that the low energy efficiency playing the leading position.
The rest of this article is organized as follows: The model is established in Section 2. In the model, the total subsidies are fixed and government employs output subsidy. The Cournot competition with output subsidies is discussed in Section 3. The Stackelberg competition is addressed in Section 4. In Section 5, Stackelberg case is compared with Cournot competition. Conclusions are remarked in the final section. 2. Model We establish the model of the output subsidies of energy efficiency under duopoly situation. Namely, the energy-efficiency subsidies given to the firms are based on their outputs directly. Assume that there are two firms depending on energies in this industry. Moreover, firms produce the identical products. For convenience, we denote the two firms as i 2 f1; 2g, respectively. To simplify the problem, given the price p and the outputs of firm i to be qi , we assume the inverse demand function is
p ¼ A q1 q2 ;
ð1Þ
where A > 0 means the market size of final products. Similar to [26], the production of these products depends on energies and other inputs. Assume that other inputs are fixed and the initial production function is linear to the energy input. Assume the initial marginal production to be h1 and h2 , which satisfy 1 ¼ h1 < h2 . The two firms invest in energy efficiency that denoted as h1 and h2 , where h1 P h1 and h2 P h2 . Moreover,
hi ¼
hi
No EE investment;
hi þ Dh With EE investment:
ð2Þ
We assume that Dh > 0 is a constant in (2). The corresponding production function is
qi ¼ hi ei :
ð3Þ
Further, with energy inputs ei , we assume that the emission of firm i is EMi ¼ sei , where s > 0 is a constant. Assume that the marginal cost of energies is c, where 1 > c > 0 is a constant standing for the price of energies. The profits of firm i are
pi ¼ phi ei cei hi hi hi ei þ si hi ei ;
ð4Þ
hi ei ¼ c hi ei denotes governmental subsidies to improve where si energy efficiency, and c P 0 is the intensity of the output subsidy. In (4), the first term means the revenues; the second term manifests the costs with energy costs and the third term is the costs to improve energy efficiency. The fourth term indicates the governmental subsidies. The timing of this game is: In the first stage, given 1 ¼ h1 < h2 , government declares the firm(s) to be subsided and the intensity of the output subsidy (c). In the second stage, the two firms simultaneously determine whether to invest energy efficiency or not. In the third stage, the two firms compete in quantity. In the above model, linear demand function is always employed to simplify the model. Furthermore, we use linear production function and it is easy to extend to other complicated production function. Moreover, this article assumes that the information is complete. That is, both firms and government know the costs and the energy efficiency. Moreover, following [26], we also assume that the budget of output subsidies is S0 > 0. Moreover, if a firm is subsidized, we assume that this firm is required to improve energy efficiency. Moreover, by (3) and (4), we assume that the two firms receive the identical increase of marginal production to invest EE, while the costs are different. The firm with higher marginal production
33
P.-y. Nie et al. / Applied Energy 175 (2016) 31–39 Table 1 The definition of variables in the model. Variables
Definition
p qi A hi
Price of the product The outputs of firm i Market size The initial marginal production of firm i Energy efficiency investment of firm i
hi ei
s c
c si S0
Energy inputs of firm i Emission intensity of energy inputs Marginal cost of energy inputs The intensity of the output subsidy Energy efficiency subsidies for firm i The budget of output subsidies
needs higher costs to achieve an increase of energy efficiency than that with lower marginal production. With the assumptions given above, the model is established for further discussion. To sum up, the variables are summarized in Table 1. In addition, we focus on two forms of quantity competition, including Cournot competition and Stackelberg competition. In both situations, we take the following types of subsidies into consideration, namely without subsidy, unilateral subsidy for the firm with low energy efficiency, unilateral subsidy for the firm with high energy efficiency, and bilateral subsidy. Especially, in the case of Stackelberg competition, we distinguish the following two cases: high energy efficiency firm playing the leading position, and low energy efficiency firm playing the leading position. For convenience, all scenarios are listed in Table 2. Then, the effects of subsidies in different cases would be further compared and discussed.
In the third stage, by (1)–(4), we have the following equilibrium under four cases: Case 1 is that two firms neither launch energy efficiency promotion. Case 2 is that both firms promote energy efficiency. Case 3 is that the first firm improves energy efficiency while the second does not. Case 4 is that the second firm promotes energy efficiency while the first one does not.
e1 ¼
8 A2cþhc > 2 > case 1 > 3 > > > > > > Að1þ2cDhÞDhþh2 þc Dh > > case 2 < 3ð1þDhÞ > A2Dhð1þ2cDhÞþhc > > 2 > > 3ð1þDhÞ > > > > > > : AþDh2cþh þc Dh 2
3
Here we consider the case of Cournot competition. Firstly, we give the benchmark model without subsidies and analyze the model. Then, we address the situation to solely subsidize one firm. Finally, we discuss the case to subsidize both firms. Notice that we set Cournot competition as the basis of analysis instead of Bertrand competition. Though both quantity competition and price competition are quite common, it is hard to capture the effects of energy efficiency subsidies that consistent with the reality in a Bertrand model because of ‘Bertrand Paradox’. 3.1. Cournot without subsidies Without subsidies, c ¼ 0 or s1 h1 e1 ¼ s2 h2 e2 ¼ 0 we analyze the model by backward induction. In this case, the first stage is fixed. We first consider the third stage and then the second stage is addressed.
e2 ¼
case 4
> AþDhh2c þ1þcDh > > 2 > > 3h2 > > > > > 2c > > : A2Dhh2 þDhþc 3ðh2 þDhÞ
case 3 case 4: ð5Þ
The corresponding profits are
p1
8 2 > > > A2cþhc2 > > > > >h 9 > i2 > > > 2c c > > ADhð1þDhÞþh2 þDh < 9 ¼ h i2 > > > A2Dhð1þ2cDhÞþhc > 2 > > > 9 > >h i2 > > > c > Aþ D h2cþ > h2 þDh : 9
case 1 case 2
;
case 3 case 4
p2
8 2 > > > Ah2c2 þc > > > > >h 9 > i2 > > > 2c c > > ADhðh2 þDhÞþ1þDh < 9 ¼ h i2 > > > AþDhh2c þ1þcDh > 2 > > > 9 > >h i2 > > > 2c þc > A2 D h > h2 þDh : 9
case 1 case 2
ð6Þ
case 3 case 4:
In the second stage, based on (6), firms determine whether to invest energy efficiency or not. Judged from (6), apparently c < 1 implies the relationship 2 A 2c þ hc2
3. Cournot competition
;
case 3
8 2c Ah þc > 2 > case 1 > 3h2 > > > > > > ADhðh 2c þ c > > 2 þDhÞ 1þDh > case 2 < 3ðh2 þDhÞ
9
h i2 A 2Dh ð1þ2cDhÞ þ hc2 and 9 9 h i2 h i2 A Dh ð1þ2cDhÞ þ h2 þc Dh A þ Dh 2c þ h2 þc Dh < : < 9 9 >
h i2 A 2Dh ð1þ2cDhÞ þ hc2
Therefore, the following results hold. Proposition 1. Under Cournot competition without subsidies, there exists a unique Nash equilibrium (No energy efficiency promotion, No energy efficiency promotion). Remarks. Proposition 1 means that the two firms without energy efficiency promotion strategy is a stable state. Because of high costs to improve energy efficiency, firms are not willing to promote energy efficiency. Without subsidy, the equilibrium demand of energy, outputs, price, profits and the total emission are
Table 2 Summary of the scenarios. Cournot competition
Stackelberg competition
Without subsidy Unilateral subsidy for the firm with low energy efficiency Unilateral subsidy for the firm with high energy efficiency Bilateral subsidy High energy efficiency firm playing the leading position
Low energy efficiency firm playing the leading position
Without subsidy Unilateral subsidy Unilateral subsidy Bilateral subsidy Without subsidy Unilateral subsidy Unilateral subsidy Bilateral subsidy
for the firm with low energy efficiency for the firm with high energy efficiency
for the firm with low energy efficiency for the firm with high energy efficiency
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P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
A 2c þ hc2
e;c1 ¼ 1
3 A 2c þc h2
¼ q;c1 2
¼ e;c1 2
;
;
3 2 ðA 2c þ cÞ h2
p;c1 ¼ 2
9
p;c1 ¼ ;
A 2c þc h2 3h2
A þ c þ hc2 3
A 2c þ hc2
EM;c1 ¼ s
3
3
þ
9 ! A 2c þc h2 3h2
We further discuss the subsidy budget to attract the first firm to take part in the promotion of energy efficiency. The first firm improves the energy efficiency to satisfy the participation constraints
;
ðA 2c þ hc2 Þ2
p;c1 ¼ 1
;
A 2c þ hc2
q;c1 ¼ 1
;
;
:
ð7Þ
Without governmental intervention, firms have no intention to improve energy efficiency. It is therefore important to consider governmental subsidies. We further discuss the unilateral output subsidy and bilateral one, respectively.
2c c c A 2 Dh þ þ 2c;c2 P A 2c þ ð1 þ DhÞ h2 h2 c c;c2 P Dh 1 : 1 þ Dh Therefore, the minimal budget is
S;c2 ¼ 0
A 2c þ hc2 Dh 1 1þcDh 3
3.2. Unilateral output subsidy Here we consider the government subsidizes the unique firm, which includes to solely subsidize the first firm (low energy efficiency) and to solely subsidize the second firm (high energy efficiency). 3.2.1. Output subsidy to the firm with low energy efficiency Here we address the output subsidy to the first firm (or the firm with low energy efficiency). We have s1 h1 e1 ¼ c h1 e1 and h2 e2 ¼ 0. In this situation, the second firm has two choices: pros2 moting energy efficiency or not. For convenience, if the second firm does not promote energy efficiency, we denote case 1. Otherwise, we denote case 2. Similarly, in the third stage, we have
e1 ¼
p1 ¼
8 A2Dhð1þ2cDhÞþhc þ2c > 2 <
case 1
3ð1þDhÞ
> Að1þ2cDhÞDhþh :
c þ2c 2 þDh
case 2
3ð1þDhÞ
8h i2 > A2Dhð1þ2cDhÞþhc þ2c > > 2 < 9
h i2 > > > : Að1þ2cDhÞDhþh2 þc Dhþ2c 9
h
e2 ¼
;
case 1
p2 ¼
;
case 2
8 AþDhh2c þ1þcDhc > 2 <
case 2
3h2
> ADhðh :
2c þ c c 2 þDhÞ 1þDh
case 2:
3ðh2 þDhÞ
8h i2 > AþDhh2c þ1þcDhc > > 2 <
In the second stage, the above profits manifest the relationship i2 h i2
AþDhh2c þ1þcDhc
ADhðh
2c þ c c þDhÞ 1þDh
2 > . As a result, under the subsidy 9 to the first firm, the second firm is not willing to improve energy efficiency. In the first stage, the government determines c according to the 2
e;c3 ¼ 1 ¼ q;c3 1 p;c3 ¼
relationship
2
h
3
c ¼ S0 . That is, the subsidy intensity i rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2
A2Dhð1þ2cDhÞþhc
A2Dhð1þ2cDhÞþhc
þ
þ24S0
. (The other satisfies c;c2 ¼ 4 root is negative and we neglect it.) The equilibrium demand of energy, outputs, price, profits and the total emission are 2
e;c2 ¼ 1 q1;c2 ¼ p;c2 ¼
A 2Dh ð1þ2cDhÞ þ hc2 þ 2c;c2 3ð1 þ DhÞ A 2Dh ð1þ2cDhÞ þ hc2 þ 2c;c2 3 A þ Dh þ ð1þcDhÞ þ hc2 c;c2 3 h
p;c2 ¼ 1
;
e;c2 ¼ 2
;
q;c2 ¼ 2
EM ;c2 ¼ s
3h2 A þ Dh 2c þ 1þcDh c;c2 h2 3
h ;
A 2Dh ð1þ2cDhÞ þ hc2 þ 2c;c2 3ð1 þ DhÞ
A þ Dh 2c þ 1þcDh c;c2 h2
i2
p2;c2 ¼
p
;c3 1
¼
þ
A þ Dh 2c þ 1þcDh c;c2 h2 3h2
3 A þ Dh 2c þ
c h 2 þDh
e2;c3 ¼
;
q2;c3 ¼
A 2Dh h22c þ c þ 2c;c3 þDh
3ðh2 þ DhÞ A 2Dh h22c þ c þ 2c;c3 þDh 3
; ;
;
c
;c3
i2
; 9 c ;c3 A þ Dh 2c þ h2 þDh c 3
h
p
;c3 2
¼
A 2Dh h22c þ c þ 2c;c3 þDh
9 ! A 2Dh þ c þ 2c;c3 : þ 3ðh2 þ DhÞ
i2 ;
2c h2 þDh
In (9), c;c3 meets the relationship
A2Dhh
2c þcþ2c;c3 2 þDh
Similarly, the subsidy intensity satisfies c;c3
c;c3 ¼ S0 . h i P Dh 1 h2 ðh2cþDhÞ 3
and the minimization budget is
S;c3 0
¼
h i ðA 2c þ cÞDh 1 h2 ðh2cþDhÞ h2 3
:
ð11Þ
h i Moreover, if c;c3 P Dh 1 h2 ðh2cþDhÞ , we have q;c3 < q;c1 and 1 1 > q;c1 q;c3 2 2 . By Sections 3.2.1 and 3.2.2, we have the following conclusions.
; ;
Proof. See in Appendix A. h
A þ Dh 2c þ 1þcDh c;c2 h2 9
3 A þ Dh þ c þ h2 þc Dh c;c3
;
Proposition 2. Under unilateral output subsidy, the firm without subsidy is inclined to maintain the initial energy efficiency. Moreover, unilateral output subsidy has stimulating effects on the subsidized firm and preventing effects on the other one. These effects increase with the total budget for subsidization.
;
A 2Dh ð1þ2cDhÞ þ hc2 þ 2c;c2 9
2
3 A þ Dh 2c þ h2 þc Dh c;c3
h
9
A2Dhð1þ2cDhÞþhc þ2c
A þ Dh 2c þ h2 þc Dh c;c3
ð10Þ
case 2:
9
ð9Þ
3.2.2. Output subsidy to the firm with high energy efficiency Here we discuss the situation to uniquely subsidize the second h1 e1 ¼ 0 and s2 h2 e2 ¼ c h2 e2 . Similar to Section 3.2.1, firm, or s1 the equilibrium demand of energy, outputs, price, profits and the total emission are
case 2
h i2 > > > : ADhðh22cþDhÞþ1þcDhc
:
(9) supports the minimal subsidy budget to stimulate the first firm to improve energy efficiency. According to (8), if c;c2 P Dh 1 1þcDh , we have q;c2 > q;c1 and q;c2 < q2;c1 . 1 1 2
EM ;c3 ¼ s
9
or
i2 ;
! : ð8Þ
Remarks. The subsidized firm owns the subsidy advantage and the quantity of products increases with the total subsidy budget. In contrast, the firm who does not receive subsidy would reduce its quantity as a response of the quantity increase of the subsidized firm. Moreover, subsidy improves the profits of the subsidized firm while reduces the profits of that without subsidy. By these two effects, subsidy helps firms to improve energy efficiency.
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P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
Proof. See in Appendix A. h
3.3. Bilateral output subsidy Here we address the situation to subsidize both firms, or s1 h1 e1 ¼ c h1 e1 and s2 h2 e2 ¼ c h2 e2 . Similar to the above analysis with backward induction strategy, we obtain the equilibrium demand of energy, outputs, price, profits and the total emission as follows e;c4 ¼ 1 q;c4 ¼ 1 p;c4 ¼
A ð1þ2cDhÞ Dh þ h2 þc Dh þ c;c4 3ð1 þ DhÞ A ð1þ2cDhÞ Dh þ h2 þc Dh þ c;c4 3
¼ ; e;c4 2 ¼ ; q;c4 2
A Dh ðh22c þ 1þcDh þ c;c4 þDhÞ 3ðh2 þ DhÞ A Dh ðh22c þ 1þcDh þ c;c4 þDhÞ 3
A þ 2Dh þ 1þcDh þ h2 þc Dh 2c;c4
p;c4 ¼ 1
; 3 h i2 A ð1þ2cDhÞ Dh þ h2 þc Dh þ c;c4
EM;c4 ¼ s
h
; 9 " 2c c ;c4 A ð1þDhÞ Dh þ h2 þDh þ c 3ð1 þ DhÞ
p;c4 ¼ 2
; ;
A Dh ðh22c þ 1þcDh þ c;c4 þDhÞ
9 # A Dh ðh22c þ 1þcDh þ c;c4 þDhÞ : þ 3ðh2 þ DhÞ
4. Stackelberg competition
i2 ;
ð12Þ
In (12), c
A
þ
2c ð1þDhÞ
;c4
meets the relationship
Dh þ h2 þc Dh þ c;c4
3
We also discuss it in three situations: without subsidy, unilateral subsidy and bilateral subsidy, respectively. The effects are compared under these situations.
c;c4
¼ S0 : h i Moreover, c;c4 > Dh 1 h2 ðh2cþDhÞ and the minimal budget is
S;c4 ¼ 0
2A 1þcDh hc2 h2 ðhD2hcþDhÞ 3
Here we discuss the output subsidies under Stackelberg cases. We consider this situation in two cases. One is the first firm acting as the leader while the second one is the follower. The other is the second firm playing the leading role while the first one is the follower. Similarly, we consider three types of subsidies for both cases. 4.1. Low energy efficiency firm acting as the leader
c;c4
3 A Dh ðh22c þ 1þcDh þ c;c4 þDhÞ
Remarks. From Proposition 4, it is the lowest expenditure to stimulate the first firm to improve energy efficiency in three types of subsidies. The expenditure is the highest to encourage both firms to promote energy efficiency. Actually, because of the disadvantages of energy efficiency, the first firm has more intention to promote energy efficiency than the second firm. Therefore, it is the easiest measure to stimulate the first firm to improve energy efficiency.
Dh 1
c : h2 ðh2 þ DhÞ
ð13Þ
Comparing the four cases above, we obtain the following conclusions Proposition 3. Given S0 , the subsidy intensity satisfies c;c2 > c;c3 > c;c4 . The price of the outputs satisfies p;c2 < p;c3 . If the subsidy intensity satisfies c < Dh, all subsidies reduce the total emission. Proof. See in Appendix A. h
4.1.1. Without subsidy Without subsidy, in the third stage, we have the following equilibrium under four cases: Case 1 is that two firms neither launch energy efficiency promotion. Case 2 is that both firms promote energy efficiency. Case 3 is that the first firm improves energy efficiency while the second one does not. Case 4 is that the second firm promotes energy efficiency while the first one does not.
e1 ¼
8 A2cþ c h2 > > case 1 > 2 > > > 2c c > > > Að1þDhÞDhþh2 þDh case 2 < 2ð1þDhÞ > A2Dhð1þ2cDhÞþhc > 2 > > 2ð1þDhÞ > > > > > : AþDh2cþh2 þc Dh 2
Remarks. Firstly, we argue that the subsidy intensity reaches the highest if government uniquely subsidizes the firm with lower energy efficiency. This comes from the less outputs of the low energy efficiency firm. p;c2 < p;c3 indicates that the consumer surplus under subsidy to the firm with low energy efficiency should be higher than that under subsidy to the firm with high energy efficiency. Moreover, the effects of subsidy on consumer surplus are uncertain, depending on the total budget. Moreover, if the budget is larger than the minimal budget, producer surplus is improved. Under c < Dh, subsidy reduces the total emission. We also note that the assumption c < Dh is rational because the costs to improve energy efficiency are exact equal to hi Dhei . c < Dh means that part of costs to promote energy efficiency is undertaken by firms. Otherwise, under c P Dh, all costs are subsidized by government. Moreover, if subsidy intensity is large enough, subsidy may not reduce the total emission. Or, the overdue subsidy may not achieve the environmental object. We further compare minimal budget to reach the above three states and we have. Proposition 4. The minimal budget under the above three cases < S;c3 < S;c4 satisfies the relationship S;c2 0 0 0 .
e2 ¼
;
case 3 case 4
8 3c Ah þ2c > 2 > case 1 > 4h2 > > > > 3c 2c > ADhðh þDhÞþ1þDh > > 2 < case 2 4ðh2 þDhÞ > Aþ2Dhh3c þ1þ2cDh > 2 > > 4h2 > > > > 3c > > : A3Dhh2 þDhþ2c 4ðh2 þDhÞ
case 3 case 4: ð14Þ
The corresponding profits are
8 2 > > A2cþhc > > 2 > case 1 > > 4 > > > > h i 2 > > > ADhð1þ2cDhÞþh þc Dh > > 2 > < case 2 4
p1 ¼ h > >
i2 > > A2Dhð1þ2cDhÞþhc > 2 > > > 4 > > > > i2 >h > > > AþDh2cþh þc Dh > : 2 4
case 3
case 4
;
8 2 > > Ah3c þ2c > > 2 > case 1 > > 16 > > > > h i 2 > > > ADhðh 3c þ 2c > > 2 þDhÞ 1þDh > < case 2 16
p2 ¼ h > >
i2 > > AþDhh3c þ1þ2cDh > 2 > > > 16 > > > > i2 >h > > > A3Dhh 3c þ2c > þ D h : 2 16
case 3
case 4: ð15Þ
In the second stage, firms determine whether to invest energy efficiency or not. Similar to Section 3.1, there is a unique Nash equilibrium (No energy efficiency promotion, No energy efficiency promotion). Without subsidy under Stackelberg game, if the first firm pays the leading position, the equilibrium demand of energy, outputs, price, profits and the total emission are
36
P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
A 2c þ hc2
e;1s1 ¼ 1
2 A 3c þ 2c h2
¼ q;1s1 2
4
p;1s1 ¼ 2
¼ e;1s1 2
;
A 3c þ 2c h2 4h2 A þ 2c þ hc2
p;1s1 ¼
;
A 3c þ 2c h2
4
2 ;
16
EM;1s1 ¼ s
;
;
q;1s1 ¼ 1
A 2c þ hc2 2
p;1s1 ¼ 1
A 2c þ
c h2
2
þ
ðA 2c þ hc2 Þ2 4 A
3c h2
þ 2c
4h2
;
! :
q2;1s3 ¼
ð16Þ
p;1s3 ¼
¼
¼ e;1s2 2 ¼ q;1s2 1 ¼ q;1s2 2 p;1s2 ¼
2ð1 þ DhÞ
A þ 2Dh 3c þ 1þ2cDh 2c;1s2 h2 4h2 A 2Dh 1þ2cDh þ hc2
A þ 2Dh 3c þ 1þ2cDh 2c;1s2 h2 4
4
p;1s2 ¼ 1
;
;
;
EM
;
2 A þ Dh 2c þ h2 þc Dh c;1s3
; 4 2 A 3Dh h23c þ 2c þ 3c;1s3 þDh 16
¼s
A þ Dh 2c þ h2 þc Dh c;1s3 2
;
þ
A 3Dh h23c þ 2c þ 3c;1s3 þDh 4ðh2 þ DhÞ
A3Dhh 3c þ2cþ3c;1s3 2 þDh 4
3c 3c þ 2c þ 3c;1s3 > A þ 2c h2 þ Dh h2 c Dh > Dh : ðh2 þ DhÞh2
A 3c þ 2c h2 4
2 ;
! :
c;1s3 ¼ S0 ,
or
2ð1 þ DhÞ
;
þ
A þ 2Dh 3c þ 1þ2cDh 2c;1s2 h2
For c;1s2 , it satisfies the relationship The minimal subsidy intensity satisfies
4h2
A2Dh1þ2cDhþhc þ2c;1s2
2c c c A 2 Dh þ þ 2c;1s2 > A 2c þ 1 þ Dh h2 h2 cDh : c;1s2 > Dh 1 þ Dh
! :
2
2
c;1s2 ¼ S0 .
e;1s4 ¼ 1 e;1s4 ¼ 2 q1;1s4 ¼ q2;1s4 ¼
or
p;1s4 ¼
Dh
c Dh : ðh2 þ DhÞh2
A Dh þ c;1s4 1þ2cDh þ h2 þc Dh 2ð1 þ DhÞ A Dh þ c;1s4 h23c þ 1þ2cDh þDh 4ðh2 þ DhÞ
A Dh þ c;1s4 1þ2cDh þ h2 þc Dh A Dh þ c
ð18Þ
If the second firm is solely subsidized, similarly, the first firm is not willing to improve energy efficiency. The equilibrium is
p;1s4 ¼ 1 p;1s4 ¼ 2
2
;1s4
ð20Þ
; ; ;
h23c þ 1þ2cDh þDh
; 4 A þ 3Dh 3c;1s4 þ 1þ2cDh þ h2 þc Dh
cDh If c;1s2 > Dh 1þ , q;1s2 > q;1s1 and q;1s2 < q;1s1 . The corre1 1 2 1 Dh sponding minimal budget is
c Dh Dh : 1 þ Dh
4.1.3. Bilateral output subsidy Here we address the situation to subsidize both firms. Similar to the above analysis with backward induction strategy, the equilibrium demand of energy, outputs, price, profits and the total emission are
2
A 2Dh 1þ2cDh þ hc2 þ 2c;1s2
2
4
S;1s3 ¼ 0
16
A 2c þ
A þ Dh c;1s3 þ 2c þ h2 þc Dh
;
Under c;1s3 > Dh ðh2 þcDDhhÞh2 , q;1s3 < q;1s1 and q;1s3 > q;1s1 . The 1 1 2 1
ð17Þ
S0;1s2 ¼
4
p;1s3 ¼ 2
c;1s3
;
A þ 2Dh 3c þ 1þ2cDh 2c;1s2 h2
c h2
A 3Dh h23c þ 2c þ 3c;1s3 þDh
¼
;1s3
;
A 3 Dh
;
A 2Dh 1þ2cDh þ hc2 þ 2c;1s2
EM ;1s2 ¼ s
2
For c;1s3 satisfying the relationship the minimal subsidy intensity satisfies
4
A þ Dh 2c þ h2 þc Dh c;1s3
;
minimal budget satisfies
A þ 1þ2cDh þ hc2 þ 2Dh 2c;1s2
p;1s2 ¼ 2
þ 2c
4ðh2 þ DhÞ
ð19Þ
;1s2
2
p
;1s3 1
;
A 3Dh h23c þ 2c þ 3c;1s3 þDh
q1;1s3 ¼
4.1.2. Unilateral subsidy We first consider that government solely subsidize the first firm and then discuss that the second firm is uniquely subsidized. If the first firm is uniquely subsidized, similarly, the second firm is not willing to improve energy efficiency. The equilibrium is
A 2Dh 1þ2cDh þ hc2 þ 2c;1s2
2
e;1s3 ¼ 2
Therefore, without subsidy, firms have no intention to improve energy efficiency. This conclusion is the same as that in Section 3.
e;1s2 1
A þ Dh 2c þ h2 þc Dh c;1s3
e;1s3 ¼ 1
;
4 A Dh þ c;1s4 1þ2cDh þ h2 þc Dh 4 A Dh þ c;1s4 h23c þ 1þ2cDh þDh
EM ;1s4 ¼ s
16
A Dh þ c
;1s4
; 2 ; 2 ;
1þ2cDh þ h2 þc Dh
2ð1 þ DhÞ
þ
A Dh þ c;1s4 h23c þ 1þ2cDh þDh 4ðh2 þ DhÞ
! :
ð21Þ
For c;1s4 , it satisfies the relationship
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P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
A Dh þ c;1s4 1þ2cDh þ h2 þc Dh 2
!
þ
A Dh þ c;1s4 h23c þ 1þ2cDh þDh 4
c;1s4
p
;2s2 1
¼
¼ S0 : The minimal subsidy intensity satisfies c;1s4 > Dh ðh2 þcDDhhÞh2 . The
p;2s2 ¼ 2
2 A 3Dh þ 3c;2s2 1þ3cDh þ 2c h2 16 2 A 2c þ 1þcDh þ Dh c;2s2 h2 4
minimal budget meets
S;1s4 ¼ 0
3A þ hc2 h22c 1þ2cDh þDh 4
c Dh : ðh2 þ DhÞh2
Dh
EM ð22Þ
If the low energy efficiency firm plays the leading position, we have the results similar to Propositions 1 and 2 in Section 3 and the following conclusions hold Proposition 5. Given S0 , the subsidy intensity satisfies c;1s4 6 minfc;1s2 ; c;1s3 g. The proof is similar to Proposition 3 and it is neglected. Remarks. Given subsidy budget, if both firms are subsidized, the subsidy intensity reaches the lowest, which is the same as Proposition 3. 4.2. High energy efficiency firm playing the leading position
4.2.1. Without subsidy Without subsidy under Stackelberg game, if the second firm plays the leading position, the equilibrium demand of energy, outputs, price, profits and the total emission are
e;2s1 1
¼
A 3c
e;2s1 2
A 2c þc h2
;
2h2
;
A 3c
¼ ; 4 2 A 3c þ 2c A 2c þ c ;2s1 A þ c þ 2c h2 h2 h2 ;2s1 q;2s1 ¼ ¼ p ¼ ; ; ; p 2 1 4 16 2 2 ! A 2c þc A 3c þ 2c A 2c þc h2 h2 h2 ;2s1 ;2s1 : p2 ¼ ¼s ; EM þ 4 4 2h2 4
¼
q;2s1 1
þ 2c h2
e2;2s2 ¼ q;2s2 ¼ 1 ¼ q;2s2 2 ;2s2
p
¼
A 3Dh þ 3c;2s2 1þ3cDh þ 2c h2 4ð1 þ DhÞ c A 2c þ þ Dh c;2s2 h2 1þDh
; 2h2 A 3Dh þ 3c;2s2 1þ3cDh þ 2c h2 4 A 2c þ 1þcDh þ Dh h2
c
;2s2
2 A þ 1þcDh þ 2c þ Dh c;2s2 h2 4
; ;
;
þ
2h2
! :
3c
þh2c
c Dh the conditions c;2s2 > Dh 1þ , q1;2s2 > q;2s1 and q;2s2 < q;2s1 , the 1 2 2 Dh minimal budget satisfies
S;2s2 ¼ 0
A 3c þ 2c h2 4
c Dh Dh : 1 þ Dh
ð25Þ
If the second firm is solely subsidized, similarly, the first firm is not willing to improve energy efficiency. The equilibrium is
e;2s3 ¼ 2 q;2s3 ¼ 1 q;2s3 ¼ 2
A þ 2Dh 2c;2s3 3c þ h22c þDh 4 A 2Dh þ 2c;2s3 h22c þc þDh 2ðh2 þ DhÞ
p
;2s3 1
p
;2s3 2
EM
;2s3
;
4 A 2Dh þ 2c;2s3 h22c þc þ Dh 2 A þ c þ h22c þ 2Dh 2c;2s3 þ Dh 4
¼
¼
;
A þ 2Dh 2c;2s3 3c þ h22c þDh
;
; ;
2 A þ 2Dh 2c;2s3 3c þ h22c þDh 16
4.2.2. Unilateral subsidy We first consider that the government solely subsidize the first firm and then discuss that the second firm is uniquely subsidized. If the first firm is uniquely subsidized, similarly, the second firm is not willing to improve the energy efficiency. The equilibrium is
4ð1 þ DhÞ
A 2c þ c þ Dh c;2s2 h2
A3Dhþ3c;2s2
Therefore, without subsidy, firms have no intention to improve energy efficiency. This conclusion is the same as that in Section 4.1. Moreover, without subsidy, if the high energy efficiency firm acts as the leader, the consumer surplus is higher than that the low energy efficiency firm playing the leading position.
A 3Dh þ 3c;2s2 1þ3cDh þ 2c h2
1þDh 2 Likewise, c;2s2 satisfies the relationship c;2s2 ¼ S0 . 4 ;2s2 c Dh > Dh 1þDh. Under The minimal subsidy intensity satisfies c
p;2s3 ¼
ð23Þ
e1;2s2 ¼
¼s
;
ð24Þ
e;2s3 ¼ 1
Here we address the situation that the second firm plays the leading position by backward induction strategies.
þ 2c h2
;2s2
;
A 2Dh þ 2c;2s3 h22c þc þDh 4
¼s
;
2 ;
A þ 2Dh 2c;2s3 3c þ h22c þDh 4
þ
A 2Dh þ 2c;2s3 h22c þc þDh 2ðh2 þ DhÞ
! :
ð26Þ A2Dhþ2c;2s3 h
2c þc þDh
2 For c;2s3 , it satisfies the relationship c;2s3 ¼ S0 . 2 Further, the minimal subsidy intensity satisfies c;2s3 > Dh h2 ðhc2DþhDhÞ. The minimal budget to subsidize meets
S;2s3 ¼ 0
A 2c þc h2 2
Dh
cDh : h2 ðh2 þ DhÞ
ð27Þ
Moreover,
c;2s3 > Dh
c Dh ; h2 ðh2 þ DhÞ
< q;2s1 q;2s3 1 1
and q;2s3 > q;2s1 : 2 2
; 4.2.3. Bilateral output subsidy Here we address the situation that subsidizing both firms. Similar to the above analysis with backward induction strategy, we have the equilibrium demand of energy, outputs, price, profits and the total emission as
38
P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
e;2s4 ¼ 1 ¼ e;2s4 2 ¼ q;2s4 1 ¼ q;2s4 2 ;2s4
p
¼
A Dh þ c;2s4 1þ3cDh þ h22c þDh 4ð1 þ DhÞ A Dh þ c;2s4 h22c þ 1þcDh þDh
2ðh2 þ DhÞ A Dh þ c;2s4 1þ3cDh þ h22c þDh A Dh þ c
p;2s4 ¼ 2 EM
;2s4
Proposition 8. If the subsidy intensity satisfies c < Dh, all subsidies reduce the total emission. Moreover, overdue subsidy may increase the total emission.
; ;
h22c þ 1þcDh þD h
; 2 ;2s4 A þ 1þcDh þ h22c þ 3 D h 3 c þDh
p;2s4 ¼ 1
4
;2s4
Significantly, for the total emission, similar to Proposition 3, we have the following conclusions.
;
; 2
4
A Dh þ c;2s4 1þ3cDh þ h22c þD h 16
;
2
A Dh þ c;2s4 h22c þ 1þcDh þD h 4
¼s
A Dh þ c
;2s4
;
1þ3cDh þ h22c þDh
4ð1 þ DhÞ
þ
A Dh þ c;2s4 h22c þ 1þcDh þDh
!
2ðh2 þ DhÞ
:
ð28Þ For c
;2s4
, it satisfies the following relationship
A Dh þ c;2s4 1þ3cDh þ h22c þDh 4
þ
Proposition 9. S;1s2 > S;c2 > S;2s2 , S0;2s3 > S0;c3 > S;1s3 , S;2s3 > 0 0 0 0 0
!
A Dh þ c;2s4 h22c þ 1þcDh þDh 2
c;2s4
¼ S0 : The minimal subsidy intensity satisfies c;2s4 > Dh h2 ðhc2DþhDhÞ. The minimal budget satisfies
S0;2s4 ¼
3A h2 ðhc2DþhDhÞ 1þcDh 2c h2 4
Dh
c Dh : h2 ðh2 þ DhÞ
ð29Þ
Moreover, the same conclusion as Proposition 5 also holds in this section. 5. Comparison Here the equilibrium in Section 3 is compared with that in Section 4. For the subsidy intensity, we have. Proposition 6. Given the total budget S0 and c 6 Dh, we have c;1s2 < c;c2 < c;2s2 , c;2s3 < c;c3 < c;1s3 and c;2s4 < c;1s4 . Proof. See in Appendix A. h Remarks. Under
fixed
subsidy
budget
to
the
sole
firm,
c;1s2 < c;c2 < c;2s2 and c;2s3 < c;c3 < c;1s3 manifest that the subsidy intensity reaches to the lowest if the uniquely subsidized firm plays the leading position. In this case, the solely subsidized firm owns the first-move advantage and the outputs are the maximum. Similarly, the subsidy intensity is the highest if this firm acts as the follower. About the outputs, similar to the method in Proposition 6, we have the following conclusions. Proposition 7. Given the total budget S0 ; q;1s2 > q;c2 > q;2s2 , 1 1 1 q;2s3 > q;c3 > q;1s3 and q;2s4 þ q;2s4 > q;1s4 þ q;1s4 . 2 2 2 1 2 1 2 Table 3 Comparison of the equilibrium under different scenarios. Variables
Results
c
c;1s2 < c;c2 < c;2s2 , c;2s3 < c;c3 < c;1s3 , c;2s4 < c;1s4
qi
q;1s2 > q1;c2 > q;2s2 , q;2s3 > q;c3 > q;1s3 , 1 1 2 2 2
S0
> >
S0;c2 > S;2s2 , S;2s3 > S;c3 0 0 0 ;2s2 ;2s4 ;2s3 S0 , S0 > S0 , S;2s4 0
> >
S;1s3 , 0 S;1s4 0
S;1s2 , S;1s3 > S;2s2 , S;2s4 > S;2s3 and S0;2s4 > S;1s4 . 0 0 0 0 0 0 Remarks. This proposition captures the minimal budget. The minimal budget is the lowest if the low energy efficiency firm acts as the follower and is solely subsidized. The minimal budget to subsidize both firms under the condition that the high energy efficiency firm acting as the leader is larger than that the low energy efficiency playing the leading position. In summary, the optimal subsidies and policy implications under Cournot competition and Stackelberg competition are quite different. Also, the equilibrium subsidy intensity depends on firms’ position, implying that the optimal choice of subsidizing strategy relies on market structure. As a result, other variables such as production and emission vary in accordance to the change of subsidy intensity under various scenarios. The major results of the comparison are summarized in Table 3. 6. Concluding remarks This article addresses the output subsidy to improve energy efficiency. In general, subsidy reduces the total emission, while overdue subsidy may increase the total emission. For the uniquely subsidized firm, the subsidy intensity reaches the lowest if it acts as the follower, while reaches the highest when it plays the leading position. The conclusions in this article are useful for decision-makers and some policy implications are represented. Firstly, the subsidy should be lower than the costs incurred by energy efficiency promotion. Secondly, given the fixed subsidy budget, the outputs of the subsidized firm is the highest if this firm plays the leading position. Finally, to guarantee the subsidized firms to improve energy efficiency, the minimal subsidy budget should be set at appropriate level. Under Stackelberg case, the subsidy expenditure of the follower seems less than that in other cases. Some researching topics arise. On one hand, the detail of emission reduction is not fully considered because it seems too difficult to handle. On the other hand, we assume that the promotion of energy efficiency is fixed. It is interesting to extend to general situation. Acknowledgments
q;2s4 þ q;2s4 > q;1s4 þ q;1s4 1 2 1 2 S;1s2 0 S;1s3 0
Remarks. c < Dh means that the costs to improve energy efficiency are jointly undertaken by the firms and the government. Therefore, the total energy consumption is reduced and the total emission is correspondingly reduced. Under c > Dh, the costs to promote energy efficiency are all undertaken by government and the firms receive the extra subsidy to improve the outputs and the energy consumption. Therefore, overdue subsidy cannot reduce the total emission. The result is consistent with the study of Brockway et al. [31], and Nie [32]. Similar to Proposition 6, the minimal budget in the above cases satisfies the following conclusions.
S;2s3 0
>
S;1s2 , 0
This work is partially supported by Foundation for High-level Talents in Higher Education of Guangdong, GDUPS (2012) and
P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
National Natural Science Foundation of PRC (71271100 and 71401057), The Guangdong Social Science Foundation (GD13YLJ02), The Soft Science Project of Guangdong Province (2014A070704008), Collaborative Innovation Center of Scientific Finance & Industry, and Innovative Group Foundation (Humanities and Social Sciences) for Higher Education of Guangdong Province (2015WCXTD009). Appendix A A.1. Proof of Proposition 2 Based on the above analysis, we have: Under unilateral output subsidy, the firm without subsidy is inclined to maintain the initial energy efficiency. Moreover, from qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c;c2 ¼
2
A2Dhð1þ2cDhÞþhc þ
½A2Dhð1þ2cDhÞþhc þ24S0
2
2
4
, we have
@S0 @ c;c2
> 0. Accord-
ing to (8), by comparative static analysis technique, we have @e;c2 1
@ c;c2 @e;c2 2 @ c;c2
@e;c2 1 @S0
> 0;
@e;c2 2
< 0;
@S0
> 0; < 0;
@q;c2 1 @ c;c2
@q;c2 2 @ c;c2
> 0; < 0;
@q;c2 1 @S0 @q;c2 2 @S0
@ p;c2 1
> 0;
@ c;c2
@ ;c2 2 @ c;c2
p
< 0;
> 0; < 0;
@ p;c2 1 @S0 ;c2 2
@p @S0
> 0; < 0:
Similar conclusions hold for (10). Therefore, unilateral output subsidy has stimulating effects on the subsidized firm and preventing effects on the opposite. These effects increase with the total budget to subsidize. h A.2. Proof of Proposition 3 A2Dhð1þ2cDhÞþhc þ2c;c2 2
3
2c þcþ2c;c3 2 þDh
A2Dhh
c;c2 ¼ S0 ,
3
c;c3 ¼ S0
and
2c þ c þc;c4 2 þDhÞ 1þDh
c þc;c4 2 þDh
ADhðh
Að1þ2cDhÞDhþh
c;c4 þ c;c4 ¼ S0 jointly manifest 3 3 that c;c2 > c;c3 > c;c4 . From c;c2 > c;c3 > c;c4 , (8) and (10), we have
p;c3 p;c2 ¼ ¼
A þ Dh þ c þ h2 þc Dh c;c3
A þ Dh þ ð1þcDhÞ þ hc2 c;c2
3 h i 1 cDh ð1þDhÞ ðh2 þ1DhÞh2 þ ðc;c2 c;c3 Þ 3
3 > 0:
Thus, the price of the outputs satisfies the relationship p;c2 < p;c3 . Obviously, under the condition c < Dh, from (8), (10) and (12), all subsidies reduce the total emission. h A.3. Proof of Proposition 4 By (9), (11) and (13), conclusions are obvious. Results are achieved and the proof is complete. h A.4. Proof of Proposition 6 We
first
A2Dhð1þ2cDhÞþhc þ2c;c2 3
A3Dhþ3c;2s2 1þ3cDhþh2c 2
4
c
A2Dh1þ2cDhþhc þ2c;1s2
c ¼ S0 and c;1s2 < c;c2 based on 2
achieve
c;1s2 < c;c2 < c;2s2 .
show
;c2
;c2
2
2
c 6 Dh.
By
c
;1s2
From ¼ S0 ,
c 6 Dh
c;2s2 ¼ S0 ,
;2s2
we and
we have c;c2 < c;2s2 . Thus, ;2s3 . Similarly, we have c < c;c3 < c;1s3 and
Conclusions are achieved and the proof is complete. h
39
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Contents lists available at ScienceDirect
Applied Energy journal homepage: www.elsevier.com/locate/apenergy
How to subsidize energy efficiency under duopoly efficiently? Pu-yan Nie a,⇑, Yong-cong Yang b, You-hua Chen c, Zhao-hui Wang a a
Guangdong University of Finance & Economics (GDUFE), Guangzhou 510320, PR China Guangdong University of Foreign Studies, Guangzhou 510006, PR China c College of Economics & Management, South China Agricultural University, Guangzhou 510642, PR China b
h i g h l i g h t s This article captures the effects of output subsidy. Firms without subsidy are not willing to improve energy efficiency. Subsidy stimulates the subsidized firms’ outputs and deters the others’ outputs. The subsidy intensity depends on firms’ position. Overdue subsidy cannot reach the environmental object.
a r t i c l e
i n f o
Article history: Received 6 January 2016 Received in revised form 17 April 2016 Accepted 24 April 2016
JEL classfication: L1 Q43 Keywords: Energy efficiency Subsidy Duopoly
a b s t r a c t Establishing a game theory model, this paper captures the effects of output subsidy on energy efficiency under Cournot competition and Stackelberg competition. Three types of subsidies are considered in the model, namely without subsidy, unilateral subsidy and bilateral subsidy. The findings indicate that firms without subsidy are not willing to improve energy efficiency. Also, subsidy stimulates the subsidized firms’ outputs while deters the outputs of other firms. Meanwhile, the equilibrium subsidy intensity depends on firms’ position. Furthermore, the minimal subsidy budgets under different situations are presented. Especially, given the fixed subsidy budget, the output of the subsidized firm is the highest if this firm plays the leading position. In addition, certain subsidy can reduce the total emission, while overdue subsidy cannot reach the environmental object. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction International Energy Agency (IEA) defined energy efficiency (EE) as ‘‘something is more energy efficient if it delivers more services for the same energy input” [1]. Furthermore, to promote energy efficiency, IEA recommended the adoption of energy efficiency subsidy policies by governments [2–6]. McKibbin, Morris and Wilcoxen [7] and Wall [8] showed the advantages of energy efficiency subsidy. Therefore, energy efficiency subsidy is attached extensive importance by many researchers, such as the interesting reviewing papers of Gillingham et al. [9] and Allcott and Greenstone [10]. Actually, as one of the most important energy efficiency policies, subsidy plays a key role in the process of transformation from ‘industrial development’ to ‘green development’ [11]. For this ⇑ Corresponding author. E-mail address: [email protected] (P.-y. Nie). http://dx.doi.org/10.1016/j.apenergy.2016.04.105 0306-2619/Ó 2016 Elsevier Ltd. All rights reserved.
reason, many countries and regions subsidize firms or consumers to improve energy efficiency to cope with global climate change [12], like Japan [13], Thailand [14], United Kingdom (UK) [15], and Sweden [16,17]. Fais et al. [18] addressed the regional energy efficiency with subsidy in Europe. There exist different types of energy efficiency subsidies all over the world, such as fixed subsidy and output subsidy. Craig and Allen [19] showed that demographic factors, attitudes, planned purchases, and energy efficiency initiatives of utility provider affect energy efficiency subsidy significantly. Furthermore, market structure also has significant impact on the effects of governmental subsidies on energy efficiency. Under monopoly, considering externalities and price-quality discrimination, Nauleau et al. [20] argued that social optimum could be achieved by different types of subsidies. In practice, since energy efficiency subsidies reduce emission with the support of public finance, it is very important to design
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P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
the suitable measures to subsidize energy efficiency [21]. Recently, Abardi and Cambini [22] developed energy efficiency strategies under incomplete information in competitive environment. Further, Arias and van Beers [23] suggested to subsidizing patents to improve energy efficiency, since they are positively related. Allcott et al. [24] recently gave a new way called as ‘‘tagging energy efficiency subsidies” to subsidize energy efficiency and argued that it would lead to large efficiency gain. Resorted to methods given by Sun and Nie [25], Nie [26] recently compared the effects of fixed subsidies with output subsidies and concluded that output subsidies improve environmental object and consumer surplus more than fixed subsidies. Significantly, Nie [26] proved the output subsidies are more efficient to reach environmental object and suggested output subsidies to improve energy efficiency. Furthermore, it is important to identify how the government should subsidize firms under asymmetric efficiency. Therefore, we follow the idea of Nie [26] and focus on two firms under asymmetric situation. Notice that besides energy efficiency, subsidies are commonly used in many other fields, such as R&D subsidies and investment subsidies. The impact of subsidies under duopoly, however, differs in various sectors and fields. For instance, R&D subsidies stimulate innovation investment effectively [27,28], while investment subsidy may crowd-out private expenditure in investment [29,30]. In related studies, regardless of subsidizing forms, the effects of subsidies are always discussed by game theory and the optimal subsidies could be obtained by backward induction. In this paper, we follow this method to analyze the effects of energy efficiency subsidies, while the results are different from the literature mentioned above because the major concerns are not the same. Under different primary energy efficiency, this article aims to capture the optimal subsidizing strategies. We try to answer the following questions in this paper: What are the effects of subsidy on the subsidized firms and other firms? How are the effects of firms’ position on firms’ strategies? What is the minimal subsidy budget? How are the effects of subsidy on total emission? Taking market structure into account, this paper relates to the studies of Nauleau et al. [20] and Nie [26]. Nauleau et al. [20] discussed the effects of subsidy on the energy efficiency under monopoly and argued that all types of subsidies can reach social optimum. While Nie [26] found that output subsidy is better than the fixed one under oligopoly. The findings of the two researches showed that the effects of energy efficiency depend on market structure. This article continues to capture the effects of market structure on energy efficiency under asymmetry and focus on the effects of firms’ position on energy efficiency subsidy. Specifically, we take Cournot competition and Stackelberg competition into consideration since they are standard and fundamental economic theories in industrial analysis. Cournot competition refers to the situation that firms make production decisions independently at the same time in quantity competition. In the case of Stackelberg competition, the leader firm moves first and then the follower firms move sequentially. In both situations firms compete in quantity to maximize their profits. The equilibrium could be obtained by backward induction. The contribution of this paper is to argue that total subsidy depends on the subsidized firms’ position. The subsidy intensity reaches the lowest if the subsidized firm acts as the follower, while reaches the highest when the subsidized firm plays the leading position. The minimal budget under various types of firms’ position is analyzed. The minimal budget is the lowest if the low energy efficiency firm acts as the follower and is solely subsidized. The minimal budget to subsidize both firms under the condition that the high energy efficiency firm acting as the leader is larger than that the low energy efficiency playing the leading position.
The rest of this article is organized as follows: The model is established in Section 2. In the model, the total subsidies are fixed and government employs output subsidy. The Cournot competition with output subsidies is discussed in Section 3. The Stackelberg competition is addressed in Section 4. In Section 5, Stackelberg case is compared with Cournot competition. Conclusions are remarked in the final section. 2. Model We establish the model of the output subsidies of energy efficiency under duopoly situation. Namely, the energy-efficiency subsidies given to the firms are based on their outputs directly. Assume that there are two firms depending on energies in this industry. Moreover, firms produce the identical products. For convenience, we denote the two firms as i 2 f1; 2g, respectively. To simplify the problem, given the price p and the outputs of firm i to be qi , we assume the inverse demand function is
p ¼ A q1 q2 ;
ð1Þ
where A > 0 means the market size of final products. Similar to [26], the production of these products depends on energies and other inputs. Assume that other inputs are fixed and the initial production function is linear to the energy input. Assume the initial marginal production to be h1 and h2 , which satisfy 1 ¼ h1 < h2 . The two firms invest in energy efficiency that denoted as h1 and h2 , where h1 P h1 and h2 P h2 . Moreover,
hi ¼
hi
No EE investment;
hi þ Dh With EE investment:
ð2Þ
We assume that Dh > 0 is a constant in (2). The corresponding production function is
qi ¼ hi ei :
ð3Þ
Further, with energy inputs ei , we assume that the emission of firm i is EMi ¼ sei , where s > 0 is a constant. Assume that the marginal cost of energies is c, where 1 > c > 0 is a constant standing for the price of energies. The profits of firm i are
pi ¼ phi ei cei hi hi hi ei þ si hi ei ;
ð4Þ
hi ei ¼ c hi ei denotes governmental subsidies to improve where si energy efficiency, and c P 0 is the intensity of the output subsidy. In (4), the first term means the revenues; the second term manifests the costs with energy costs and the third term is the costs to improve energy efficiency. The fourth term indicates the governmental subsidies. The timing of this game is: In the first stage, given 1 ¼ h1 < h2 , government declares the firm(s) to be subsided and the intensity of the output subsidy (c). In the second stage, the two firms simultaneously determine whether to invest energy efficiency or not. In the third stage, the two firms compete in quantity. In the above model, linear demand function is always employed to simplify the model. Furthermore, we use linear production function and it is easy to extend to other complicated production function. Moreover, this article assumes that the information is complete. That is, both firms and government know the costs and the energy efficiency. Moreover, following [26], we also assume that the budget of output subsidies is S0 > 0. Moreover, if a firm is subsidized, we assume that this firm is required to improve energy efficiency. Moreover, by (3) and (4), we assume that the two firms receive the identical increase of marginal production to invest EE, while the costs are different. The firm with higher marginal production
33
P.-y. Nie et al. / Applied Energy 175 (2016) 31–39 Table 1 The definition of variables in the model. Variables
Definition
p qi A hi
Price of the product The outputs of firm i Market size The initial marginal production of firm i Energy efficiency investment of firm i
hi ei
s c
c si S0
Energy inputs of firm i Emission intensity of energy inputs Marginal cost of energy inputs The intensity of the output subsidy Energy efficiency subsidies for firm i The budget of output subsidies
needs higher costs to achieve an increase of energy efficiency than that with lower marginal production. With the assumptions given above, the model is established for further discussion. To sum up, the variables are summarized in Table 1. In addition, we focus on two forms of quantity competition, including Cournot competition and Stackelberg competition. In both situations, we take the following types of subsidies into consideration, namely without subsidy, unilateral subsidy for the firm with low energy efficiency, unilateral subsidy for the firm with high energy efficiency, and bilateral subsidy. Especially, in the case of Stackelberg competition, we distinguish the following two cases: high energy efficiency firm playing the leading position, and low energy efficiency firm playing the leading position. For convenience, all scenarios are listed in Table 2. Then, the effects of subsidies in different cases would be further compared and discussed.
In the third stage, by (1)–(4), we have the following equilibrium under four cases: Case 1 is that two firms neither launch energy efficiency promotion. Case 2 is that both firms promote energy efficiency. Case 3 is that the first firm improves energy efficiency while the second does not. Case 4 is that the second firm promotes energy efficiency while the first one does not.
e1 ¼
8 A2cþhc > 2 > case 1 > 3 > > > > > > Að1þ2cDhÞDhþh2 þc Dh > > case 2 < 3ð1þDhÞ > A2Dhð1þ2cDhÞþhc > > 2 > > 3ð1þDhÞ > > > > > > : AþDh2cþh þc Dh 2
3
Here we consider the case of Cournot competition. Firstly, we give the benchmark model without subsidies and analyze the model. Then, we address the situation to solely subsidize one firm. Finally, we discuss the case to subsidize both firms. Notice that we set Cournot competition as the basis of analysis instead of Bertrand competition. Though both quantity competition and price competition are quite common, it is hard to capture the effects of energy efficiency subsidies that consistent with the reality in a Bertrand model because of ‘Bertrand Paradox’. 3.1. Cournot without subsidies Without subsidies, c ¼ 0 or s1 h1 e1 ¼ s2 h2 e2 ¼ 0 we analyze the model by backward induction. In this case, the first stage is fixed. We first consider the third stage and then the second stage is addressed.
e2 ¼
case 4
> AþDhh2c þ1þcDh > > 2 > > 3h2 > > > > > 2c > > : A2Dhh2 þDhþc 3ðh2 þDhÞ
case 3 case 4: ð5Þ
The corresponding profits are
p1
8 2 > > > A2cþhc2 > > > > >h 9 > i2 > > > 2c c > > ADhð1þDhÞþh2 þDh < 9 ¼ h i2 > > > A2Dhð1þ2cDhÞþhc > 2 > > > 9 > >h i2 > > > c > Aþ D h2cþ > h2 þDh : 9
case 1 case 2
;
case 3 case 4
p2
8 2 > > > Ah2c2 þc > > > > >h 9 > i2 > > > 2c c > > ADhðh2 þDhÞþ1þDh < 9 ¼ h i2 > > > AþDhh2c þ1þcDh > 2 > > > 9 > >h i2 > > > 2c þc > A2 D h > h2 þDh : 9
case 1 case 2
ð6Þ
case 3 case 4:
In the second stage, based on (6), firms determine whether to invest energy efficiency or not. Judged from (6), apparently c < 1 implies the relationship 2 A 2c þ hc2
3. Cournot competition
;
case 3
8 2c Ah þc > 2 > case 1 > 3h2 > > > > > > ADhðh 2c þ c > > 2 þDhÞ 1þDh > case 2 < 3ðh2 þDhÞ
9
h i2 A 2Dh ð1þ2cDhÞ þ hc2 and 9 9 h i2 h i2 A Dh ð1þ2cDhÞ þ h2 þc Dh A þ Dh 2c þ h2 þc Dh < : < 9 9 >
h i2 A 2Dh ð1þ2cDhÞ þ hc2
Therefore, the following results hold. Proposition 1. Under Cournot competition without subsidies, there exists a unique Nash equilibrium (No energy efficiency promotion, No energy efficiency promotion). Remarks. Proposition 1 means that the two firms without energy efficiency promotion strategy is a stable state. Because of high costs to improve energy efficiency, firms are not willing to promote energy efficiency. Without subsidy, the equilibrium demand of energy, outputs, price, profits and the total emission are
Table 2 Summary of the scenarios. Cournot competition
Stackelberg competition
Without subsidy Unilateral subsidy for the firm with low energy efficiency Unilateral subsidy for the firm with high energy efficiency Bilateral subsidy High energy efficiency firm playing the leading position
Low energy efficiency firm playing the leading position
Without subsidy Unilateral subsidy Unilateral subsidy Bilateral subsidy Without subsidy Unilateral subsidy Unilateral subsidy Bilateral subsidy
for the firm with low energy efficiency for the firm with high energy efficiency
for the firm with low energy efficiency for the firm with high energy efficiency
34
P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
A 2c þ hc2
e;c1 ¼ 1
3 A 2c þc h2
¼ q;c1 2
¼ e;c1 2
;
;
3 2 ðA 2c þ cÞ h2
p;c1 ¼ 2
9
p;c1 ¼ ;
A 2c þc h2 3h2
A þ c þ hc2 3
A 2c þ hc2
EM;c1 ¼ s
3
3
þ
9 ! A 2c þc h2 3h2
We further discuss the subsidy budget to attract the first firm to take part in the promotion of energy efficiency. The first firm improves the energy efficiency to satisfy the participation constraints
;
ðA 2c þ hc2 Þ2
p;c1 ¼ 1
;
A 2c þ hc2
q;c1 ¼ 1
;
;
:
ð7Þ
Without governmental intervention, firms have no intention to improve energy efficiency. It is therefore important to consider governmental subsidies. We further discuss the unilateral output subsidy and bilateral one, respectively.
2c c c A 2 Dh þ þ 2c;c2 P A 2c þ ð1 þ DhÞ h2 h2 c c;c2 P Dh 1 : 1 þ Dh Therefore, the minimal budget is
S;c2 ¼ 0
A 2c þ hc2 Dh 1 1þcDh 3
3.2. Unilateral output subsidy Here we consider the government subsidizes the unique firm, which includes to solely subsidize the first firm (low energy efficiency) and to solely subsidize the second firm (high energy efficiency). 3.2.1. Output subsidy to the firm with low energy efficiency Here we address the output subsidy to the first firm (or the firm with low energy efficiency). We have s1 h1 e1 ¼ c h1 e1 and h2 e2 ¼ 0. In this situation, the second firm has two choices: pros2 moting energy efficiency or not. For convenience, if the second firm does not promote energy efficiency, we denote case 1. Otherwise, we denote case 2. Similarly, in the third stage, we have
e1 ¼
p1 ¼
8 A2Dhð1þ2cDhÞþhc þ2c > 2 <
case 1
3ð1þDhÞ
> Að1þ2cDhÞDhþh :
c þ2c 2 þDh
case 2
3ð1þDhÞ
8h i2 > A2Dhð1þ2cDhÞþhc þ2c > > 2 < 9
h i2 > > > : Að1þ2cDhÞDhþh2 þc Dhþ2c 9
h
e2 ¼
;
case 1
p2 ¼
;
case 2
8 AþDhh2c þ1þcDhc > 2 <
case 2
3h2
> ADhðh :
2c þ c c 2 þDhÞ 1þDh
case 2:
3ðh2 þDhÞ
8h i2 > AþDhh2c þ1þcDhc > > 2 <
In the second stage, the above profits manifest the relationship i2 h i2
AþDhh2c þ1þcDhc
ADhðh
2c þ c c þDhÞ 1þDh
2 > . As a result, under the subsidy 9 to the first firm, the second firm is not willing to improve energy efficiency. In the first stage, the government determines c according to the 2
e;c3 ¼ 1 ¼ q;c3 1 p;c3 ¼
relationship
2
h
3
c ¼ S0 . That is, the subsidy intensity i rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2
A2Dhð1þ2cDhÞþhc
A2Dhð1þ2cDhÞþhc
þ
þ24S0
. (The other satisfies c;c2 ¼ 4 root is negative and we neglect it.) The equilibrium demand of energy, outputs, price, profits and the total emission are 2
e;c2 ¼ 1 q1;c2 ¼ p;c2 ¼
A 2Dh ð1þ2cDhÞ þ hc2 þ 2c;c2 3ð1 þ DhÞ A 2Dh ð1þ2cDhÞ þ hc2 þ 2c;c2 3 A þ Dh þ ð1þcDhÞ þ hc2 c;c2 3 h
p;c2 ¼ 1
;
e;c2 ¼ 2
;
q;c2 ¼ 2
EM ;c2 ¼ s
3h2 A þ Dh 2c þ 1þcDh c;c2 h2 3
h ;
A 2Dh ð1þ2cDhÞ þ hc2 þ 2c;c2 3ð1 þ DhÞ
A þ Dh 2c þ 1þcDh c;c2 h2
i2
p2;c2 ¼
p
;c3 1
¼
þ
A þ Dh 2c þ 1þcDh c;c2 h2 3h2
3 A þ Dh 2c þ
c h 2 þDh
e2;c3 ¼
;
q2;c3 ¼
A 2Dh h22c þ c þ 2c;c3 þDh
3ðh2 þ DhÞ A 2Dh h22c þ c þ 2c;c3 þDh 3
; ;
;
c
;c3
i2
; 9 c ;c3 A þ Dh 2c þ h2 þDh c 3
h
p
;c3 2
¼
A 2Dh h22c þ c þ 2c;c3 þDh
9 ! A 2Dh þ c þ 2c;c3 : þ 3ðh2 þ DhÞ
i2 ;
2c h2 þDh
In (9), c;c3 meets the relationship
A2Dhh
2c þcþ2c;c3 2 þDh
Similarly, the subsidy intensity satisfies c;c3
c;c3 ¼ S0 . h i P Dh 1 h2 ðh2cþDhÞ 3
and the minimization budget is
S;c3 0
¼
h i ðA 2c þ cÞDh 1 h2 ðh2cþDhÞ h2 3
:
ð11Þ
h i Moreover, if c;c3 P Dh 1 h2 ðh2cþDhÞ , we have q;c3 < q;c1 and 1 1 > q;c1 q;c3 2 2 . By Sections 3.2.1 and 3.2.2, we have the following conclusions.
; ;
Proof. See in Appendix A. h
A þ Dh 2c þ 1þcDh c;c2 h2 9
3 A þ Dh þ c þ h2 þc Dh c;c3
;
Proposition 2. Under unilateral output subsidy, the firm without subsidy is inclined to maintain the initial energy efficiency. Moreover, unilateral output subsidy has stimulating effects on the subsidized firm and preventing effects on the other one. These effects increase with the total budget for subsidization.
;
A 2Dh ð1þ2cDhÞ þ hc2 þ 2c;c2 9
2
3 A þ Dh 2c þ h2 þc Dh c;c3
h
9
A2Dhð1þ2cDhÞþhc þ2c
A þ Dh 2c þ h2 þc Dh c;c3
ð10Þ
case 2:
9
ð9Þ
3.2.2. Output subsidy to the firm with high energy efficiency Here we discuss the situation to uniquely subsidize the second h1 e1 ¼ 0 and s2 h2 e2 ¼ c h2 e2 . Similar to Section 3.2.1, firm, or s1 the equilibrium demand of energy, outputs, price, profits and the total emission are
case 2
h i2 > > > : ADhðh22cþDhÞþ1þcDhc
:
(9) supports the minimal subsidy budget to stimulate the first firm to improve energy efficiency. According to (8), if c;c2 P Dh 1 1þcDh , we have q;c2 > q;c1 and q;c2 < q2;c1 . 1 1 2
EM ;c3 ¼ s
9
or
i2 ;
! : ð8Þ
Remarks. The subsidized firm owns the subsidy advantage and the quantity of products increases with the total subsidy budget. In contrast, the firm who does not receive subsidy would reduce its quantity as a response of the quantity increase of the subsidized firm. Moreover, subsidy improves the profits of the subsidized firm while reduces the profits of that without subsidy. By these two effects, subsidy helps firms to improve energy efficiency.
35
P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
Proof. See in Appendix A. h
3.3. Bilateral output subsidy Here we address the situation to subsidize both firms, or s1 h1 e1 ¼ c h1 e1 and s2 h2 e2 ¼ c h2 e2 . Similar to the above analysis with backward induction strategy, we obtain the equilibrium demand of energy, outputs, price, profits and the total emission as follows e;c4 ¼ 1 q;c4 ¼ 1 p;c4 ¼
A ð1þ2cDhÞ Dh þ h2 þc Dh þ c;c4 3ð1 þ DhÞ A ð1þ2cDhÞ Dh þ h2 þc Dh þ c;c4 3
¼ ; e;c4 2 ¼ ; q;c4 2
A Dh ðh22c þ 1þcDh þ c;c4 þDhÞ 3ðh2 þ DhÞ A Dh ðh22c þ 1þcDh þ c;c4 þDhÞ 3
A þ 2Dh þ 1þcDh þ h2 þc Dh 2c;c4
p;c4 ¼ 1
; 3 h i2 A ð1þ2cDhÞ Dh þ h2 þc Dh þ c;c4
EM;c4 ¼ s
h
; 9 " 2c c ;c4 A ð1þDhÞ Dh þ h2 þDh þ c 3ð1 þ DhÞ
p;c4 ¼ 2
; ;
A Dh ðh22c þ 1þcDh þ c;c4 þDhÞ
9 # A Dh ðh22c þ 1þcDh þ c;c4 þDhÞ : þ 3ðh2 þ DhÞ
4. Stackelberg competition
i2 ;
ð12Þ
In (12), c
A
þ
2c ð1þDhÞ
;c4
meets the relationship
Dh þ h2 þc Dh þ c;c4
3
We also discuss it in three situations: without subsidy, unilateral subsidy and bilateral subsidy, respectively. The effects are compared under these situations.
c;c4
¼ S0 : h i Moreover, c;c4 > Dh 1 h2 ðh2cþDhÞ and the minimal budget is
S;c4 ¼ 0
2A 1þcDh hc2 h2 ðhD2hcþDhÞ 3
Here we discuss the output subsidies under Stackelberg cases. We consider this situation in two cases. One is the first firm acting as the leader while the second one is the follower. The other is the second firm playing the leading role while the first one is the follower. Similarly, we consider three types of subsidies for both cases. 4.1. Low energy efficiency firm acting as the leader
c;c4
3 A Dh ðh22c þ 1þcDh þ c;c4 þDhÞ
Remarks. From Proposition 4, it is the lowest expenditure to stimulate the first firm to improve energy efficiency in three types of subsidies. The expenditure is the highest to encourage both firms to promote energy efficiency. Actually, because of the disadvantages of energy efficiency, the first firm has more intention to promote energy efficiency than the second firm. Therefore, it is the easiest measure to stimulate the first firm to improve energy efficiency.
Dh 1
c : h2 ðh2 þ DhÞ
ð13Þ
Comparing the four cases above, we obtain the following conclusions Proposition 3. Given S0 , the subsidy intensity satisfies c;c2 > c;c3 > c;c4 . The price of the outputs satisfies p;c2 < p;c3 . If the subsidy intensity satisfies c < Dh, all subsidies reduce the total emission. Proof. See in Appendix A. h
4.1.1. Without subsidy Without subsidy, in the third stage, we have the following equilibrium under four cases: Case 1 is that two firms neither launch energy efficiency promotion. Case 2 is that both firms promote energy efficiency. Case 3 is that the first firm improves energy efficiency while the second one does not. Case 4 is that the second firm promotes energy efficiency while the first one does not.
e1 ¼
8 A2cþ c h2 > > case 1 > 2 > > > 2c c > > > Að1þDhÞDhþh2 þDh case 2 < 2ð1þDhÞ > A2Dhð1þ2cDhÞþhc > 2 > > 2ð1þDhÞ > > > > > : AþDh2cþh2 þc Dh 2
Remarks. Firstly, we argue that the subsidy intensity reaches the highest if government uniquely subsidizes the firm with lower energy efficiency. This comes from the less outputs of the low energy efficiency firm. p;c2 < p;c3 indicates that the consumer surplus under subsidy to the firm with low energy efficiency should be higher than that under subsidy to the firm with high energy efficiency. Moreover, the effects of subsidy on consumer surplus are uncertain, depending on the total budget. Moreover, if the budget is larger than the minimal budget, producer surplus is improved. Under c < Dh, subsidy reduces the total emission. We also note that the assumption c < Dh is rational because the costs to improve energy efficiency are exact equal to hi Dhei . c < Dh means that part of costs to promote energy efficiency is undertaken by firms. Otherwise, under c P Dh, all costs are subsidized by government. Moreover, if subsidy intensity is large enough, subsidy may not reduce the total emission. Or, the overdue subsidy may not achieve the environmental object. We further compare minimal budget to reach the above three states and we have. Proposition 4. The minimal budget under the above three cases < S;c3 < S;c4 satisfies the relationship S;c2 0 0 0 .
e2 ¼
;
case 3 case 4
8 3c Ah þ2c > 2 > case 1 > 4h2 > > > > 3c 2c > ADhðh þDhÞþ1þDh > > 2 < case 2 4ðh2 þDhÞ > Aþ2Dhh3c þ1þ2cDh > 2 > > 4h2 > > > > 3c > > : A3Dhh2 þDhþ2c 4ðh2 þDhÞ
case 3 case 4: ð14Þ
The corresponding profits are
8 2 > > A2cþhc > > 2 > case 1 > > 4 > > > > h i 2 > > > ADhð1þ2cDhÞþh þc Dh > > 2 > < case 2 4
p1 ¼ h > >
i2 > > A2Dhð1þ2cDhÞþhc > 2 > > > 4 > > > > i2 >h > > > AþDh2cþh þc Dh > : 2 4
case 3
case 4
;
8 2 > > Ah3c þ2c > > 2 > case 1 > > 16 > > > > h i 2 > > > ADhðh 3c þ 2c > > 2 þDhÞ 1þDh > < case 2 16
p2 ¼ h > >
i2 > > AþDhh3c þ1þ2cDh > 2 > > > 16 > > > > i2 >h > > > A3Dhh 3c þ2c > þ D h : 2 16
case 3
case 4: ð15Þ
In the second stage, firms determine whether to invest energy efficiency or not. Similar to Section 3.1, there is a unique Nash equilibrium (No energy efficiency promotion, No energy efficiency promotion). Without subsidy under Stackelberg game, if the first firm pays the leading position, the equilibrium demand of energy, outputs, price, profits and the total emission are
36
P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
A 2c þ hc2
e;1s1 ¼ 1
2 A 3c þ 2c h2
¼ q;1s1 2
4
p;1s1 ¼ 2
¼ e;1s1 2
;
A 3c þ 2c h2 4h2 A þ 2c þ hc2
p;1s1 ¼
;
A 3c þ 2c h2
4
2 ;
16
EM;1s1 ¼ s
;
;
q;1s1 ¼ 1
A 2c þ hc2 2
p;1s1 ¼ 1
A 2c þ
c h2
2
þ
ðA 2c þ hc2 Þ2 4 A
3c h2
þ 2c
4h2
;
! :
q2;1s3 ¼
ð16Þ
p;1s3 ¼
¼
¼ e;1s2 2 ¼ q;1s2 1 ¼ q;1s2 2 p;1s2 ¼
2ð1 þ DhÞ
A þ 2Dh 3c þ 1þ2cDh 2c;1s2 h2 4h2 A 2Dh 1þ2cDh þ hc2
A þ 2Dh 3c þ 1þ2cDh 2c;1s2 h2 4
4
p;1s2 ¼ 1
;
;
;
EM
;
2 A þ Dh 2c þ h2 þc Dh c;1s3
; 4 2 A 3Dh h23c þ 2c þ 3c;1s3 þDh 16
¼s
A þ Dh 2c þ h2 þc Dh c;1s3 2
;
þ
A 3Dh h23c þ 2c þ 3c;1s3 þDh 4ðh2 þ DhÞ
A3Dhh 3c þ2cþ3c;1s3 2 þDh 4
3c 3c þ 2c þ 3c;1s3 > A þ 2c h2 þ Dh h2 c Dh > Dh : ðh2 þ DhÞh2
A 3c þ 2c h2 4
2 ;
! :
c;1s3 ¼ S0 ,
or
2ð1 þ DhÞ
;
þ
A þ 2Dh 3c þ 1þ2cDh 2c;1s2 h2
For c;1s2 , it satisfies the relationship The minimal subsidy intensity satisfies
4h2
A2Dh1þ2cDhþhc þ2c;1s2
2c c c A 2 Dh þ þ 2c;1s2 > A 2c þ 1 þ Dh h2 h2 cDh : c;1s2 > Dh 1 þ Dh
! :
2
2
c;1s2 ¼ S0 .
e;1s4 ¼ 1 e;1s4 ¼ 2 q1;1s4 ¼ q2;1s4 ¼
or
p;1s4 ¼
Dh
c Dh : ðh2 þ DhÞh2
A Dh þ c;1s4 1þ2cDh þ h2 þc Dh 2ð1 þ DhÞ A Dh þ c;1s4 h23c þ 1þ2cDh þDh 4ðh2 þ DhÞ
A Dh þ c;1s4 1þ2cDh þ h2 þc Dh A Dh þ c
ð18Þ
If the second firm is solely subsidized, similarly, the first firm is not willing to improve energy efficiency. The equilibrium is
p;1s4 ¼ 1 p;1s4 ¼ 2
2
;1s4
ð20Þ
; ; ;
h23c þ 1þ2cDh þDh
; 4 A þ 3Dh 3c;1s4 þ 1þ2cDh þ h2 þc Dh
cDh If c;1s2 > Dh 1þ , q;1s2 > q;1s1 and q;1s2 < q;1s1 . The corre1 1 2 1 Dh sponding minimal budget is
c Dh Dh : 1 þ Dh
4.1.3. Bilateral output subsidy Here we address the situation to subsidize both firms. Similar to the above analysis with backward induction strategy, the equilibrium demand of energy, outputs, price, profits and the total emission are
2
A 2Dh 1þ2cDh þ hc2 þ 2c;1s2
2
4
S;1s3 ¼ 0
16
A 2c þ
A þ Dh c;1s3 þ 2c þ h2 þc Dh
;
Under c;1s3 > Dh ðh2 þcDDhhÞh2 , q;1s3 < q;1s1 and q;1s3 > q;1s1 . The 1 1 2 1
ð17Þ
S0;1s2 ¼
4
p;1s3 ¼ 2
c;1s3
;
A þ 2Dh 3c þ 1þ2cDh 2c;1s2 h2
c h2
A 3Dh h23c þ 2c þ 3c;1s3 þDh
¼
;1s3
;
A 3 Dh
;
A 2Dh 1þ2cDh þ hc2 þ 2c;1s2
EM ;1s2 ¼ s
2
For c;1s3 satisfying the relationship the minimal subsidy intensity satisfies
4
A þ Dh 2c þ h2 þc Dh c;1s3
;
minimal budget satisfies
A þ 1þ2cDh þ hc2 þ 2Dh 2c;1s2
p;1s2 ¼ 2
þ 2c
4ðh2 þ DhÞ
ð19Þ
;1s2
2
p
;1s3 1
;
A 3Dh h23c þ 2c þ 3c;1s3 þDh
q1;1s3 ¼
4.1.2. Unilateral subsidy We first consider that government solely subsidize the first firm and then discuss that the second firm is uniquely subsidized. If the first firm is uniquely subsidized, similarly, the second firm is not willing to improve energy efficiency. The equilibrium is
A 2Dh 1þ2cDh þ hc2 þ 2c;1s2
2
e;1s3 ¼ 2
Therefore, without subsidy, firms have no intention to improve energy efficiency. This conclusion is the same as that in Section 3.
e;1s2 1
A þ Dh 2c þ h2 þc Dh c;1s3
e;1s3 ¼ 1
;
4 A Dh þ c;1s4 1þ2cDh þ h2 þc Dh 4 A Dh þ c;1s4 h23c þ 1þ2cDh þDh
EM ;1s4 ¼ s
16
A Dh þ c
;1s4
; 2 ; 2 ;
1þ2cDh þ h2 þc Dh
2ð1 þ DhÞ
þ
A Dh þ c;1s4 h23c þ 1þ2cDh þDh 4ðh2 þ DhÞ
! :
ð21Þ
For c;1s4 , it satisfies the relationship
37
P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
A Dh þ c;1s4 1þ2cDh þ h2 þc Dh 2
!
þ
A Dh þ c;1s4 h23c þ 1þ2cDh þDh 4
c;1s4
p
;2s2 1
¼
¼ S0 : The minimal subsidy intensity satisfies c;1s4 > Dh ðh2 þcDDhhÞh2 . The
p;2s2 ¼ 2
2 A 3Dh þ 3c;2s2 1þ3cDh þ 2c h2 16 2 A 2c þ 1þcDh þ Dh c;2s2 h2 4
minimal budget meets
S;1s4 ¼ 0
3A þ hc2 h22c 1þ2cDh þDh 4
c Dh : ðh2 þ DhÞh2
Dh
EM ð22Þ
If the low energy efficiency firm plays the leading position, we have the results similar to Propositions 1 and 2 in Section 3 and the following conclusions hold Proposition 5. Given S0 , the subsidy intensity satisfies c;1s4 6 minfc;1s2 ; c;1s3 g. The proof is similar to Proposition 3 and it is neglected. Remarks. Given subsidy budget, if both firms are subsidized, the subsidy intensity reaches the lowest, which is the same as Proposition 3. 4.2. High energy efficiency firm playing the leading position
4.2.1. Without subsidy Without subsidy under Stackelberg game, if the second firm plays the leading position, the equilibrium demand of energy, outputs, price, profits and the total emission are
e;2s1 1
¼
A 3c
e;2s1 2
A 2c þc h2
;
2h2
;
A 3c
¼ ; 4 2 A 3c þ 2c A 2c þ c ;2s1 A þ c þ 2c h2 h2 h2 ;2s1 q;2s1 ¼ ¼ p ¼ ; ; ; p 2 1 4 16 2 2 ! A 2c þc A 3c þ 2c A 2c þc h2 h2 h2 ;2s1 ;2s1 : p2 ¼ ¼s ; EM þ 4 4 2h2 4
¼
q;2s1 1
þ 2c h2
e2;2s2 ¼ q;2s2 ¼ 1 ¼ q;2s2 2 ;2s2
p
¼
A 3Dh þ 3c;2s2 1þ3cDh þ 2c h2 4ð1 þ DhÞ c A 2c þ þ Dh c;2s2 h2 1þDh
; 2h2 A 3Dh þ 3c;2s2 1þ3cDh þ 2c h2 4 A 2c þ 1þcDh þ Dh h2
c
;2s2
2 A þ 1þcDh þ 2c þ Dh c;2s2 h2 4
; ;
;
þ
2h2
! :
3c
þh2c
c Dh the conditions c;2s2 > Dh 1þ , q1;2s2 > q;2s1 and q;2s2 < q;2s1 , the 1 2 2 Dh minimal budget satisfies
S;2s2 ¼ 0
A 3c þ 2c h2 4
c Dh Dh : 1 þ Dh
ð25Þ
If the second firm is solely subsidized, similarly, the first firm is not willing to improve energy efficiency. The equilibrium is
e;2s3 ¼ 2 q;2s3 ¼ 1 q;2s3 ¼ 2
A þ 2Dh 2c;2s3 3c þ h22c þDh 4 A 2Dh þ 2c;2s3 h22c þc þDh 2ðh2 þ DhÞ
p
;2s3 1
p
;2s3 2
EM
;2s3
;
4 A 2Dh þ 2c;2s3 h22c þc þ Dh 2 A þ c þ h22c þ 2Dh 2c;2s3 þ Dh 4
¼
¼
;
A þ 2Dh 2c;2s3 3c þ h22c þDh
;
; ;
2 A þ 2Dh 2c;2s3 3c þ h22c þDh 16
4.2.2. Unilateral subsidy We first consider that the government solely subsidize the first firm and then discuss that the second firm is uniquely subsidized. If the first firm is uniquely subsidized, similarly, the second firm is not willing to improve the energy efficiency. The equilibrium is
4ð1 þ DhÞ
A 2c þ c þ Dh c;2s2 h2
A3Dhþ3c;2s2
Therefore, without subsidy, firms have no intention to improve energy efficiency. This conclusion is the same as that in Section 4.1. Moreover, without subsidy, if the high energy efficiency firm acts as the leader, the consumer surplus is higher than that the low energy efficiency firm playing the leading position.
A 3Dh þ 3c;2s2 1þ3cDh þ 2c h2
1þDh 2 Likewise, c;2s2 satisfies the relationship c;2s2 ¼ S0 . 4 ;2s2 c Dh > Dh 1þDh. Under The minimal subsidy intensity satisfies c
p;2s3 ¼
ð23Þ
e1;2s2 ¼
¼s
;
ð24Þ
e;2s3 ¼ 1
Here we address the situation that the second firm plays the leading position by backward induction strategies.
þ 2c h2
;2s2
;
A 2Dh þ 2c;2s3 h22c þc þDh 4
¼s
;
2 ;
A þ 2Dh 2c;2s3 3c þ h22c þDh 4
þ
A 2Dh þ 2c;2s3 h22c þc þDh 2ðh2 þ DhÞ
! :
ð26Þ A2Dhþ2c;2s3 h
2c þc þDh
2 For c;2s3 , it satisfies the relationship c;2s3 ¼ S0 . 2 Further, the minimal subsidy intensity satisfies c;2s3 > Dh h2 ðhc2DþhDhÞ. The minimal budget to subsidize meets
S;2s3 ¼ 0
A 2c þc h2 2
Dh
cDh : h2 ðh2 þ DhÞ
ð27Þ
Moreover,
c;2s3 > Dh
c Dh ; h2 ðh2 þ DhÞ
< q;2s1 q;2s3 1 1
and q;2s3 > q;2s1 : 2 2
; 4.2.3. Bilateral output subsidy Here we address the situation that subsidizing both firms. Similar to the above analysis with backward induction strategy, we have the equilibrium demand of energy, outputs, price, profits and the total emission as
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P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
e;2s4 ¼ 1 ¼ e;2s4 2 ¼ q;2s4 1 ¼ q;2s4 2 ;2s4
p
¼
A Dh þ c;2s4 1þ3cDh þ h22c þDh 4ð1 þ DhÞ A Dh þ c;2s4 h22c þ 1þcDh þDh
2ðh2 þ DhÞ A Dh þ c;2s4 1þ3cDh þ h22c þDh A Dh þ c
p;2s4 ¼ 2 EM
;2s4
Proposition 8. If the subsidy intensity satisfies c < Dh, all subsidies reduce the total emission. Moreover, overdue subsidy may increase the total emission.
; ;
h22c þ 1þcDh þD h
; 2 ;2s4 A þ 1þcDh þ h22c þ 3 D h 3 c þDh
p;2s4 ¼ 1
4
;2s4
Significantly, for the total emission, similar to Proposition 3, we have the following conclusions.
;
; 2
4
A Dh þ c;2s4 1þ3cDh þ h22c þD h 16
;
2
A Dh þ c;2s4 h22c þ 1þcDh þD h 4
¼s
A Dh þ c
;2s4
;
1þ3cDh þ h22c þDh
4ð1 þ DhÞ
þ
A Dh þ c;2s4 h22c þ 1þcDh þDh
!
2ðh2 þ DhÞ
:
ð28Þ For c
;2s4
, it satisfies the following relationship
A Dh þ c;2s4 1þ3cDh þ h22c þDh 4
þ
Proposition 9. S;1s2 > S;c2 > S;2s2 , S0;2s3 > S0;c3 > S;1s3 , S;2s3 > 0 0 0 0 0
!
A Dh þ c;2s4 h22c þ 1þcDh þDh 2
c;2s4
¼ S0 : The minimal subsidy intensity satisfies c;2s4 > Dh h2 ðhc2DþhDhÞ. The minimal budget satisfies
S0;2s4 ¼
3A h2 ðhc2DþhDhÞ 1þcDh 2c h2 4
Dh
c Dh : h2 ðh2 þ DhÞ
ð29Þ
Moreover, the same conclusion as Proposition 5 also holds in this section. 5. Comparison Here the equilibrium in Section 3 is compared with that in Section 4. For the subsidy intensity, we have. Proposition 6. Given the total budget S0 and c 6 Dh, we have c;1s2 < c;c2 < c;2s2 , c;2s3 < c;c3 < c;1s3 and c;2s4 < c;1s4 . Proof. See in Appendix A. h Remarks. Under
fixed
subsidy
budget
to
the
sole
firm,
c;1s2 < c;c2 < c;2s2 and c;2s3 < c;c3 < c;1s3 manifest that the subsidy intensity reaches to the lowest if the uniquely subsidized firm plays the leading position. In this case, the solely subsidized firm owns the first-move advantage and the outputs are the maximum. Similarly, the subsidy intensity is the highest if this firm acts as the follower. About the outputs, similar to the method in Proposition 6, we have the following conclusions. Proposition 7. Given the total budget S0 ; q;1s2 > q;c2 > q;2s2 , 1 1 1 q;2s3 > q;c3 > q;1s3 and q;2s4 þ q;2s4 > q;1s4 þ q;1s4 . 2 2 2 1 2 1 2 Table 3 Comparison of the equilibrium under different scenarios. Variables
Results
c
c;1s2 < c;c2 < c;2s2 , c;2s3 < c;c3 < c;1s3 , c;2s4 < c;1s4
qi
q;1s2 > q1;c2 > q;2s2 , q;2s3 > q;c3 > q;1s3 , 1 1 2 2 2
S0
> >
S0;c2 > S;2s2 , S;2s3 > S;c3 0 0 0 ;2s2 ;2s4 ;2s3 S0 , S0 > S0 , S;2s4 0
> >
S;1s3 , 0 S;1s4 0
S;1s2 , S;1s3 > S;2s2 , S;2s4 > S;2s3 and S0;2s4 > S;1s4 . 0 0 0 0 0 0 Remarks. This proposition captures the minimal budget. The minimal budget is the lowest if the low energy efficiency firm acts as the follower and is solely subsidized. The minimal budget to subsidize both firms under the condition that the high energy efficiency firm acting as the leader is larger than that the low energy efficiency playing the leading position. In summary, the optimal subsidies and policy implications under Cournot competition and Stackelberg competition are quite different. Also, the equilibrium subsidy intensity depends on firms’ position, implying that the optimal choice of subsidizing strategy relies on market structure. As a result, other variables such as production and emission vary in accordance to the change of subsidy intensity under various scenarios. The major results of the comparison are summarized in Table 3. 6. Concluding remarks This article addresses the output subsidy to improve energy efficiency. In general, subsidy reduces the total emission, while overdue subsidy may increase the total emission. For the uniquely subsidized firm, the subsidy intensity reaches the lowest if it acts as the follower, while reaches the highest when it plays the leading position. The conclusions in this article are useful for decision-makers and some policy implications are represented. Firstly, the subsidy should be lower than the costs incurred by energy efficiency promotion. Secondly, given the fixed subsidy budget, the outputs of the subsidized firm is the highest if this firm plays the leading position. Finally, to guarantee the subsidized firms to improve energy efficiency, the minimal subsidy budget should be set at appropriate level. Under Stackelberg case, the subsidy expenditure of the follower seems less than that in other cases. Some researching topics arise. On one hand, the detail of emission reduction is not fully considered because it seems too difficult to handle. On the other hand, we assume that the promotion of energy efficiency is fixed. It is interesting to extend to general situation. Acknowledgments
q;2s4 þ q;2s4 > q;1s4 þ q;1s4 1 2 1 2 S;1s2 0 S;1s3 0
Remarks. c < Dh means that the costs to improve energy efficiency are jointly undertaken by the firms and the government. Therefore, the total energy consumption is reduced and the total emission is correspondingly reduced. Under c > Dh, the costs to promote energy efficiency are all undertaken by government and the firms receive the extra subsidy to improve the outputs and the energy consumption. Therefore, overdue subsidy cannot reduce the total emission. The result is consistent with the study of Brockway et al. [31], and Nie [32]. Similar to Proposition 6, the minimal budget in the above cases satisfies the following conclusions.
S;2s3 0
>
S;1s2 , 0
This work is partially supported by Foundation for High-level Talents in Higher Education of Guangdong, GDUPS (2012) and
P.-y. Nie et al. / Applied Energy 175 (2016) 31–39
National Natural Science Foundation of PRC (71271100 and 71401057), The Guangdong Social Science Foundation (GD13YLJ02), The Soft Science Project of Guangdong Province (2014A070704008), Collaborative Innovation Center of Scientific Finance & Industry, and Innovative Group Foundation (Humanities and Social Sciences) for Higher Education of Guangdong Province (2015WCXTD009). Appendix A A.1. Proof of Proposition 2 Based on the above analysis, we have: Under unilateral output subsidy, the firm without subsidy is inclined to maintain the initial energy efficiency. Moreover, from qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c;c2 ¼
2
A2Dhð1þ2cDhÞþhc þ
½A2Dhð1þ2cDhÞþhc þ24S0
2
2
4
, we have
@S0 @ c;c2
> 0. Accord-
ing to (8), by comparative static analysis technique, we have @e;c2 1
@ c;c2 @e;c2 2 @ c;c2
@e;c2 1 @S0
> 0;
@e;c2 2
< 0;
@S0
> 0; < 0;
@q;c2 1 @ c;c2
@q;c2 2 @ c;c2
> 0; < 0;
@q;c2 1 @S0 @q;c2 2 @S0
@ p;c2 1
> 0;
@ c;c2
@ ;c2 2 @ c;c2
p
< 0;
> 0; < 0;
@ p;c2 1 @S0 ;c2 2
@p @S0
> 0; < 0:
Similar conclusions hold for (10). Therefore, unilateral output subsidy has stimulating effects on the subsidized firm and preventing effects on the opposite. These effects increase with the total budget to subsidize. h A.2. Proof of Proposition 3 A2Dhð1þ2cDhÞþhc þ2c;c2 2
3
2c þcþ2c;c3 2 þDh
A2Dhh
c;c2 ¼ S0 ,
3
c;c3 ¼ S0
and
2c þ c þc;c4 2 þDhÞ 1þDh
c þc;c4 2 þDh
ADhðh
Að1þ2cDhÞDhþh
c;c4 þ c;c4 ¼ S0 jointly manifest 3 3 that c;c2 > c;c3 > c;c4 . From c;c2 > c;c3 > c;c4 , (8) and (10), we have
p;c3 p;c2 ¼ ¼
A þ Dh þ c þ h2 þc Dh c;c3
A þ Dh þ ð1þcDhÞ þ hc2 c;c2
3 h i 1 cDh ð1þDhÞ ðh2 þ1DhÞh2 þ ðc;c2 c;c3 Þ 3
3 > 0:
Thus, the price of the outputs satisfies the relationship p;c2 < p;c3 . Obviously, under the condition c < Dh, from (8), (10) and (12), all subsidies reduce the total emission. h A.3. Proof of Proposition 4 By (9), (11) and (13), conclusions are obvious. Results are achieved and the proof is complete. h A.4. Proof of Proposition 6 We
first
A2Dhð1þ2cDhÞþhc þ2c;c2 3
A3Dhþ3c;2s2 1þ3cDhþh2c 2
4
c
A2Dh1þ2cDhþhc þ2c;1s2
c ¼ S0 and c;1s2 < c;c2 based on 2
achieve
c;1s2 < c;c2 < c;2s2 .
show
;c2
;c2
2
2
c 6 Dh.
By
c
;1s2
From ¼ S0 ,
c 6 Dh
c;2s2 ¼ S0 ,
;2s2
we and
we have c;c2 < c;2s2 . Thus, ;2s3 . Similarly, we have c < c;c3 < c;1s3 and
Conclusions are achieved and the proof is complete. h
39
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