Accepted Mauscript Determiig the Optimal Carbo Tax Rate based o Data Evelopmet Aalysis Miyue Ji, Xiao Shi, Ali Emrouzejad, Feg Yag PII: DOI: S0959-6526(17)32427-7 10.1016/j.jclepro.2017.10.127 Referece: JCLP 10919 To appear i: Joural of Cleaer Productio Received Date: 01 November 2016 Revised Date: 01 September 2017 Accepted Date: 11 October 2017 Please cite this article as: Miyue Ji, Xiao Shi, Ali Emrouzejad, Feg Yag, Determiig the Optimal Carbo Tax Rate based o Data Evelopmet Aalysis, Joural of Cleaer Productio (2017), doi: 10.1016/j.jclepro.2017.10.127 This is a PDF file of a uedited mauscript that has bee accepted for publicatio. As a service to our customers we are providig this early versio of the mauscript. The mauscript will udergo copyeditig, typesettig, ad review of the resultig proof before it is published i its fial form. Please ote that durig the productio process errors may be discovered which could affect the cotet, ad all legal disclaimers that apply to the joural pertai.
Determiig the Optimal Carbo Tax Rate based o Data Evelopmet Aalysis Miyue Ji 1, Xiao Shi 2*, Ali Emrouzejad 3*, Feg Yag 1 Abstract Carbo tax policy is widely used to cotrol greehouse gases ad how to determie a suitable carbo tax rate is very importat for policy makers cosiderig the trade-off betwee evirometal protectio ad ecoomic developmet. I a idustry regulated by carbo tax policy, we cosider two competig firms who sell ordiary products ad gree products respectively. I order to promote the firm who sells ordiary product to reduce carbo emissios, the govermet of Chia imposes carbo tax o the ordiary products. For the govermet, three objectives are cosidered whe it makes carbo tax policy. They are icreasig the govermet reveue, reducig the govermet expediture ad decreasig the carbo emissios. For the firms, it is importat to explore their pricig strategies take ito accout of the govermet tax policy. To fid a optimal carbo tax rate ad to achieve the three objectives simultaeously, we cosider this as a multiple criteria decisio-makig problem. Hece, we propose to use a cetralized data evelopmet aalysis (DEA) approach to solve it. We fid that whe oe firm produces ordiary products ad the other produces gree products, the govermet may set a high tax rate. While whe both firms sell ordiary products, the optimal tax policy for each firm is differet ad the govermet may impose a higher tax rate for oe firm ad a lower tax rate for the other firm. Keywords: Carbo tax policy; govermet reveue; govermet expediture; carbo emissios; cetralized DEA approach 1
1. Itroductio The recogitio of the climate chage effected by greehouse gases (GHG) has received a large amout of attetio over the last several years (Lee 2011, Al-Mulali ad Sab 2012, Wag et al. 2013). The first treaty aims at reducig carbo emissios, i.e., Uited Natios Framework Covetio o Climate Chage (UNFCCC), etered ito force i 1994. The the Kyoto Protocol exteds the 1994 UNFCCC ad asks differet coutries to fulfill their obligatios of GHG emissios limitatios (Grubb 2004). From the perspective of firms, it is also widely accepted that evirometal issues such as carbo emissios will have bad impacts o shareholders value (McKisey&Compay 2010). Thus, a large umber of firms ad compaies are participatig actively i emissio reductio activities. For istace, Tesla Motors aims at evetually offerig zero emissio electric cars at prices affordable to average cosumers. As aother example cosider Hewlett Packard (HP), HP focuses o materials iovatio, icludes reducig each product carbo emissios durig maufacturig stage ad usage stage (Raz et al. 2013). However, for most firms, maufacturig low-emissios products may eed advaced techology ad require substatial up frot ivestmets, for example, Tesla ivests a large amout of moey i battery techology. This, o the other had, makes firms afraid to move toward evirometally friedly techology. Therefore, it is importat ad ecessary for the govermets ad orgaizatios to implemet regulatios ad laws to motivate firms to adopt gree techology ad produce low-carbo products. The taxatio approach is widely used amog regulators to achieve evirometal beefits (Felder ad Schleiiger 2002, Li ad Li, 2011). A sigificat tax policy ca iduce the firms to adopt pollutio abatemet techologies ad produce evirometally friedly products (Avi-Yoah ad Uhlma 2009). For example, i 1990, the US govermet imposes upo tax o chlorofluoro carbos (CFCs). Because the tax is much higher tha the cost of CFCs, some ew techologies are used to improve the utility ratio of oe uit of CFCs ad the cosumptio of CFCs had dropped by over 70% i the ext 5 years (Krass et al. 2013). Tax policy is perceived 2
as a useful tool to reduce carbo emissios. However, it may hurt firms profit because of its eforcemet. Therefore, whe policy makers make taxatio decisios, they have to cosider the effect of the tax policy from multiple aspects, like eviromet impact, firms profit ad govermet expediture (Makiw et al. 2009). Therefore, our first research questio is how the govermet should make carbo tax policy cosiderig multiple objectives. I practice, a large umber of cosumers are showig iterests i gree products. Accordig to a survey made by the Gallup Orgaizatio i 2009 1, 83% of the Europeas express their willigess to buy a eco-safe product. Bei ad Simpso (1995) idicate that i additio to the utility obtaied directly from a purchased product, gree cosumers ca receive psychological beefits from buyig evirometally friedly products. Basal (2008) study the desig of regulatory istrumets uder the assumptio that the cosumers are willig to pay extra for buyig gree products. Based o these research, our secod research questio explores how the firms make pricig decisios if the cosumers are sesitive to the carbo emissios. I this paper, we discuss the policy decisio of the govermet o how to determie the optimal tax rate cosiderig the firms pricig strategies. The objectives of the govermet iclude three goals, icreasig the govermet reveue, reducig the govermet expediture ad decreasig the carbo emissios. As it ca be show later i this paper, these three objectives are coflictig because icreasig the carbo tax rate may leads to lower carbo emissios but higher govermet expediture. Therefore there is trade-off betwee the three goals. To select the best tax policy, we regard the problem as a multi-criteria decisio-makig problem ad use cetralized DEA approach to solve it. From the umerical example, we fid that whe there is oe firm sellig ordiary products ad the other sellig gree products i the market, the govermet may set a highest tax rate. We fid that whe oe firm produces ordiary products ad the other 1 http://ec.europa.eu/public_opiio/flash/fl_256_e.pdf 3
produces gree products, the govermet may set a high tax rate. While whe both firms sell ordiary products, the optimal tax policy for each firm is differet ad the govermet may impose a higher tax rate for oe firm ad a lower tax rate for the other firm. The rest of this paper is orgaized as follows. We review relevat literature i Sectio 2. I Sectio 3, we describe our model setup ad itroduce the cetralized DEA approach. We aalyze the bechmark model with oe firm sellig ordiary product ad the other sellig gree product i Sectio 4. We study the firms pricig strategies ad govermet carbo tax policies for both firms i Sectio 5. Sectio 6 offers coclusios ad discussios of extesios for future research. 2. Literature review Our work cotributes to the operatios maagemet literature that study evirometal policy (Zhag ad Xu 2013, Qiu et al. 2017, Sabzevar et al. 2017). For example, Drake et al. (2015) study the impact of emissio tax o a compay s logru techology choice ad show that i certai coditios a icrease i tax rate would decrease ivestmet i clea techology. Krass et al. (2013) ivestigate how to use evirometal taxes to motivate firms to adopt greeer techologies ad they fid that over icrease i taxes may have a egative impact o the firms techology choice strategies. Maikas ad Kroes (2015) preset a ew forward buyig method to reduce the impact of emissios allowace acquisitios o firms fiacial performace. From the perspective of supply chai cooperatio, Ji et al. (2017) compare three differet modes of carbo allowace allocatio rules ad show that cap ad trade regulatio based o bechmarkig mechaism ca more effectively push maufacturers to produce low-carbo products. Ad it would be more beeficial to the retailer to cooperate with the maufacturer with lower emissio reductio cost. Zakeri et al. (2015) examie the supply chai performace at the operatioal plaig level uder the carbo taxes policy ad show that carbo tradig mechaisms leads to better supply chai performace i terms of emissios ad cost. Park et al. (2015) study how carbo pealty chages the supply chai structure ad social welfare. 4
Additioal, Liu et al. (2015) focus o the impact of carbo emissio regulatio o firms remaufacturig strategies ad Wag et al. (2015) study how carbo taxes affect the selectio of trasportatio modes ad social welfare. While i these papers, they assume that the govermet has sigle goal, i.e. maximize social welfare or miimize the total pollutio. I our paper, we cosider multidimesioal missios of the govermet, which iclude icreasig the govermet reveue, decreasig the govermet expediture ad reducig carbo emissios, ad cetralized DEA approach to fid out the best carbo tax policy. A umber of studies examie evirometal efficiecy evaluatio based o DEA method (Bia ad Yag 2010, Wag et al. 2013, Ha et al. 2015). Xie et al. (2014) ivestigate the evirometal efficiecies of electric power idustries i 26 OECD from 1996 to 2010 ad fid that fuel structure chage ad ecoomic situatio ad eergy prices are mai drivig forces that affect evirometal total factor productivity. Sog et al. (2014) cosider the iteral structure of productio system ad propose a two-stage DEA approach to evaluate the evirometal efficiecy of Chia. Furthermore, Wu et al. (2016) combies idividual iterest preferece with efficiecy evaluatio to reflect the curret evirometal situatio i Chia. Che ad Jia (2017) itroduce big data theory to select iput ad output data of differet regio ad apply Slack-based measure model to evaluate the evirometal efficiecies of Chia's idustry. Recetly, Lozao et al. (2004), Lozao et al. (2011), ad Yu et al. (2013) utilize cetralized DEA models to deal with cetralized resource allocatio problems. I these models, the objective of the cetralized decisio-makers is to maximize the overall output productio by all DMUs or optimize the combied resource cosumptio of all uits, rather tha cosiderig them separately (Feg 2013, Yu et al. 2013). To evaluate the evirometal efficiecy from the overall view, cetralized DEA approach is also applied to model udesirable variables, e.g., carbo emissios (Zhou et al. 2014, Su et al. 2015). Much differet from the literature above, i this paper, we focus o how to choose a best carbo tax policy for the govermet cosiderig the firms carbo 5
reductio activities ad the govermet multiple goals. I order to solve the problem, we use backward iductio combied with cetralized DEA approach to select the optimal carbo tax rate. To the best of our kowledge, we are the first to use cetralized DEA method to cosider multidimesioal problems of the govermet alog with pricig decisios of the firms. 3. Model set-up ad assumptios We cosider two competig firms i the market ( i 1,2 ), each of which sells a kid of product. The products ca be differetiated by the sellig prices, productio costs ad carbo emissios per uit. Except these, the two products are of o differece. There is a local govermet i the market. I order to promote the firms to reduce carbo emissios, the govermet decides to impose a carbo tax o the products which exceeds the gree stadard. Next, we preset the decisios of each players ad the sequece of the evet. Notatios are summarized i Table 1. Table 1 Summary of Notatios Symbol e i c i p i ti Defiitio Carbo emissios of PRO i per uit Productio cost of PRO i per uit Sellig price of PRO i per uit The turover tax imposed by the govermet for each uit of product The carbo tax rate for PRO i i The govermet expediture for each uit of PRO i 3. 1 Firms There are two competig firms F 1 ad F 2 i the market. The subscript 1 represets F 1 ad the subscript 2 represets F 2. The products are deoted by PRO 1 ad 6
PRO 2. We assume that each cosumer buys at most oe uit of either PRO 1 or PRO 2. The et utility of a cosumer ca be stated as follows (Bi et al. 2016): U = u i e i p i, i=1, 2 (1) Where u i (i=1, 2) is the cosumer s potetial utility i possessig a product if e i =0 ad p i =0 ad is large eough to guaratee o-empty market coverage. I other words, oe cosumer ca choose to buy a PRO 1 or a PRO 2, ad he caot choose to buy othig. Without loss of geerality we ormalize that u1 u2 u, ei (i=1,2) is the amout of carbo emissios per uit of product ad we assume e e 1 2. is the sesitivity coefficiet of a particular cosumer to the carbo emissios ad is assumed to be uiformly distributed o [, ]. Obviously a higher implies that the cosumer prefers products with lower carbo emissios better. p i (i=1,2) is the sellig prices of product i. By calculatig the utility fuctio, we obtai that cosumers with * * [0, ] would buy PRO 1, while cosumers with [,1] would buy PRO 2, * where ( p p ) ( e e ). 2 1 1 2 Evirometal tax is a effective way to help reducig products carbo emissios (Krass et al. 2013). It is ituitively believed that the reductio of carbo emissios has a positive correlatio with govermet carbo tax. Therefore, we use t ei e ei k i i to represet the impact of carbo tax o the firm s emissios reductio of PRO i per uit. Here, is the elasticity parameter of carbo tax ad 1. I other words, decreasig the emissios becomes icreasigly ecessary with the augmet of the carbo tax. k i deotes the correlatio coefficiet betwee the carbo emissios ad tax rate. Higher k i meas that the govermet carbo tax has a greater impact o the firm s emissios reductio activities. If k, the e e. That meas the carbo emissios of PRO i is very close to the gree stadard. We abado i i the situatio of k 0 i because the carbo emissios of a regular product per uit 7
would always be a defiite value ad could ot be. I order to make it easier to ti calculate, we let k 1/ ki ad the e ca be writte as ei kt i. k i 3. 2 Govermet I practice, may idicatios impact the govermet decisio of tax rate. I this paper, we cosider three objectives, i.e., govermet reveue, govermet expediture ad carbo emissios. The defiitios of the three objectives are give as follows. Govermet reveue Govermet reveue refers to the reveue of the govermet fiace from its impositio of a tax, a fee or ay other charge. Taxatio is the mai source of govermet reveue ad ca help to stabilize the ecoomy ad promote desirable behavior. Therefore it is ecessary for policy makers to cosider the govermet reveue whe they implemet a carbo tax policy. The govermet reveue is calculated by the followig equatio R ( p c )( ) ( p c )( ) t ( e e)( ) t ( e e)( ) * * * * 1 1 2 2 1 1 2 2 (2) I Equatio (2), the first two terms are turover tax from PRO i, the secod two terms are carbo tax of PRO i. Govermet expediture The govermet expediture icludes the cost that the govermet pays for supervisig the productio process of each uit of product, like persoel, iformatio techology ad telecommuicatios (Che 2001). For differet types of the products, the spedig may be differet. We assume that govermet spedig of PRO 1 per uit is 1 ad that of PRO 2 per uit is 2. Based upo this, we idetify govermet expediture as follows E=( * ) 1 ( * ) 2 (3) The total govermet expediture cosists of two parts, i.e., the spedig geerated from PRO 1 ad the spedig geerated from PRO 2. 8
Carbo emissios Carbo emissio, particular CO 2, is the major factor that causes potetially irreversible chages i global climate. Therefore, we use carbo emissios as the third idicator for the govermet to implemet the tax policy. The total carbo emissio is expressed as Q ( ) e +( ) e * * 1 2 (4) Where the first term is the total carbo emissios of PRO 1 ad the secod term is the total carbo emissios of PRO 2. The sequece of the evets ca be described as follows: First, the local govermet determies the tax policy for each firm. Upo seeig the tax policy, the two firms, simultaeously, choose their product sellig prices p i. We use backward iductio to characterize the problem. I the first step we preset the firms pricig decisios give the govermet tax rates, while i the secod step, we offer a budle of feasible carbo tax rates, ad use cetralized DEA approach to select the govermet optimal tax rates proposed to each firm. 3. 3 Method Suppose there are a budle of feasible choices, deoted by t 1, t 2,, t. The correspodig govermet reveue, govermet expediture ad carbo emissio for t i are R i, E i ad Q i respectively. I the decisio makig process, the govermet aims to icrease the reveue ad decrease the govermet expediture ad carbo emissios. The to determie the optimal tax rate becomes a multiple criteria decisio-makig problem, which is to rak or select from a set of alterative courses of actio i the presece of coflictig criteria. Correspodigly i our problem, the govermet reveue R is positive criteria while the govermet expediture E ad carbo emissios Q are egative oes. Next, we give the method for the govermet o selectio of best carbo tax policy based o cetralized DEA approach. 9
3. 3. 1 Evirometal DEA Techology The joit productio of desirable ad udesirable outputs ca be described as evirometal productio techology. It depicts the process of covertig iput vectors (E) to desirable output vectors (R) ad udesirable output vectors (Q) (Fare et al. 1989). The productio techology set (T) is assumed to satisfy the followig assumptios: (1) If ( E, R, Q) T ad 0 1, the ( x, E, Q) T (weak disposability for desirable ad udesirable outputs). (2) If ( E, R, Q) T ad Q 0, the R 0 (ull-joitess of desirable ad udesirable outputs). The coceptual defiitio of evirometal productio techology ca be approximated by piecewise liear combiatios of the observed data, which is similar theoretically to DEA form; as such, it ca also be called evirometal DEA techology. Uder CRS assumptio, evirometal productio techology T ca be approximately formulated as follows: T W E, R, Q : jeij Ei, i 1,..., m j1 jrrj Rr, r 1,..., s j1 jqdj Qd, d 1,..., D j1 j 0, j 1,..., (5) The formulatio of Shephard eviromet DEA techology is as follows: 10
E, R, Q : jeij Ei, i 1,..., m j1 jrrj Rr, r 1,..., s j1 TS jqdj Qd, d 1,..., D j1 0 1, j 1 j1 j 0, j 1,..., (6) Where is the abatemet factor, which esures that T satisfies the above two assumptios. The secod ad third costraits i Model (6) are o-liear, which they s ca be trasformed to liear form by defiig a ew variable ˆj j to replace the old variables. The coverted model ca be expressed as follow: T S E,R,Q : j1 j1 j1 0 mi Ei,i 1,...,m Rr,r 1,...,s Q d,d 1,...,D ˆ 1, j j1 ˆ 0, j 1,..., E j i ˆ E j j j ˆ R ij rj ˆ Q It is worth otig that the reaso to add costrai dj E ij Ei mieij (7) is to prevet the coditio 0 (which cause ˆ 0 ), tha ay activity E,0,0 will be feasible (This j is because accordig to the first costrait i Model (3), iput vectors E caot be smaller tha the observed practical value). 3. 3. 2 Cetralized Model for Evirometal DEA Techology Based o evirometal DEA techology, the cetralized models (6) ad (7) are 11
employed i order to make sure that all firms act i a way which optimize the performace for the govermet as a whole. After cetralized optimizatio, all DMU activities are as efficiet as possible uder the potetial desirable outputs ad allocated udesirable outputs set by the model. The, the cetralized DEA model (Lozao ad Villa, 2005; Yu et al. 2013) ca be formulated as follows: Mi s. t. E E, i 1,..., m jr ij ir r1 j1 r1 R R \, r 1,..., s jr tj tr r1 j1 r1 Q Q, d 1,..., D jr dj dr r1 j1 r1 0 1, 1 jr i j1 0, j 1,...,, r 1,... E mi E ij jr (8) This is a liear optimizatio model with 2 1 variables. By solvig this model above, the efficiecy of each DMU ca be obtaied. I sectio 4, we first explai a bechmark model cosiderig the situatio where there is oly oe firm that produce ordiary products with carbo emissios exceed the gree stadard. We the exteded this, i sectio 5, to the case where the govermet imposes carbo taxes o both firms. 4. Bechmark model: tax policy o oe firm I this sectio, we assume that there is oly oe firm i the market whose carbo emissios of its product exceed the gree stadard. I other words, the other firm that produce gree product do ot eed to pay for carbo tax. Next, we derive the pricig strategies of the two competig firms ad give govermet optimal carbo tax policy cosiderig the govermet three goals. 12
4.1 The pricig decisios of the two firms As ca be see above, the demad of the ordiary product PRO 1 is demad of the gree product PRO 2 is. The, F 1 s optimizatio problem is give by * ad max =( p c )( )(1 ) t( e e )( ) p1, p2 1 1 1 1 2 (10) Where c 1 is F 1 s variable cost per uit of product ad p 1 is the sellig price. is the turover tax rate for each product that sells out. The first term represets the et sale profit ad the secod term represets the govermet carbo tax. The profit of F 2 ca be writte as follows: max ( p c )( )(1 ) p1, p2 2 2 2 (11) Where c 2 is F 2 s variable cost per uit of product ad p 2 is the sellig price. The products that F 2 maufactures ca meet the gree stadard. Therefore it does ot eed to pay carbo taxes. To fid the optimal p 1 ad p 2 to maximize ( p, p ) ad ( p, p ), we 1 1 2 2 1 2 differetiate ( p ) with respect to ad ( p ) with respect to p, which yield the first-order coditios: 1 1 p1 2 2 2 1 1 1 =( k t ( p2 p1 ) )(1 ) k t ( p1 c1 )(1 ) t p 1 2 p 2 =( + k t ( p p ))(1 ) k t ( p c )(1 ) 1 1 1 2 2 2 (12) (13) Sice it ca be verified that the two fuctios are covex i p 1 ad p 2 respectively, the optimal p 1 ad p 2 ca be obtaied by forcig 1 p1 0 ad 2 p2 0. The we have +1 1 2kt p1 ( t)= ( kt 2kt 2 c1 c2 ) 3 1 1 1 kt p2( t)= (2kt kt c1 2 c2 ) 3 1 13 (14) (15)
4.2 Determiig the optimal tax policy based o cetralized DEA I this sectio we use a umerical example to illustrate our method. Some parameters are give i Table 2. Table 2. Parameters settig i the umerical example c 1 =1 c 2 =3 e 2 =10 =0.1 = 0 1 1 =4 2 =3 k=1.5 α=1.5 We let tax rate t to be 0.4, 0.55,,6.25 with commo differece of 0.15, ad thus 40 feasible choices o the tax rate are obtaied, as show i the first colum of Table 3. Accordig to Equatios (2)-(4), all values of the govermet reveue, govermet expediture ad carbo emissios are obtaied, as ca be see i Table 3. Table 3. Govermet reveue, govermet expediture ad carbo emissios No. t R i E i Q i No. t R i E i Q i 1 0.4 1.13 3.30 11.765 21 3.4 1.78 4.86 10.445 2 0.55 0.91 3.31 11.143 22 3.55 1.87 4.99 10.447 3 0.7 0.81 3.33 10.856 23 3.7 1.96 5.13 10.448 4 0.85 0.77 3.37 10.702 24 3.85 2.06 5.26 10.450 5 1 0.76 3.41 10.611 25 4 2.15 5.41 10.451 6 1.15 0.77 3.46 10.554 26 4.15 2.25 5.55 10.453 7 1.3 0.80 3.51 10.517 27 4.3 2.34 5.70 10.455 8 1.45 0.84 3.57 10.492 28 4.45 2.44 5.86 10.457 9 1.6 0.88 3.64 10.475 29 4.6 2.54 6.01 10.458 10 1.75 0.94 3.71 10.463 30 4.75 2.64 6.18 10.460 11 1.9 1.00 3.79 10.455 31 4.9 2.74 6.34 10.462 12 2.05 1.06 3.88 10.449 32 5.05 2.85 6.51 10.464 13 2.2 1.13 3.97 10.445 33 5.2 2.95 6.68 10.465 14 2.35 1.20 4.06 10.443 34 5.35 3.05 6.85 10.467 15 2.5 1.28 4.16 10.442 35 5.5 3.16 7.03 10.469 16 2.65 1.36 4.27 10.441 36 5.65 3.26 7.21 10.470 17 2.8 1.44 4.38 10.441 37 5.8 3.37 7.39 10.472 18 2.95 1.52 4.49 10.442 38 5.95 3.48 7.58 10.473 19 3.1 1.61 4.61 10.443 39 6.1 3.59 7.77 10.475 20 3.25 1.69 4.73 10.444 40 6.25 3.70 7.96 10.476 14
I order to get a more ituitive result, we use Figs. 1-3 to show the relatioship betwee the tax rate ad the three objectives. As show i Fig. 1, whe the tax rate icreases, the govermet reveue decreases rapidly at first, the icreases. While i Fig. 2, the govermet expediture icreases quickly. I Fig. 3, the total carbo emissios decrease quickly ad the icreases slowly ad smoothly with icreasig t. Next we use DEA aalysis to evaluate each tax rate s efficiecy to select the optimal carbo tax rate. Fig. 1. Govermet reveue uder the carbo tax policy 15
Fig. 2. Govermet expediture uder the carbo tax policy Fig. 3. Total carbo emissios uder the carbo tax policy Usig model (7), we obtai the DEA efficiecy scores of all feasible cadidates. Results show that, policy 40 is the best choice, which leads to highest govermet reveue ad govermet expediture, while it maitai lower carbo emissios. Table 4. DEA efficiecy scores of all feasible policies No. i No. i No. i No. i 1 0.737 11 0.568 21 0.788 31 0.930 16
2 0.591 12 0.588 22 0.806 32 0.942 3 0.523 13 0.612 23 0.822 33 0.950 4 0.492 14 0.636 24 0.843 34 0.958 5 0.479 15 0.662 25 0.855 35 0.967 6 0.479 16 0.685 26 0.872 36 0.973 7 0.490 17 0.707 27 0.883 37 0.981 8 0.506 18 0.728 28 0.896 38 0.988 9 0.520 19 0.751 29 0.909 39 0.994 10 0.545 20 0.769 30 0.919 40 1.000 I the ext sectio, we aalyze the situatio where both of the firms produce ordiary products ad should pay for carbo tax. Similar to Sectio 4, we first obtai the firms pricig strategies give the govermet carbo tax rate. The we aalyze how the govermet makes differet carbo tax policies for the two firms. 5. Tax policy o two firms I this sectio, we assume that there are two firms i the market ad the carbo emissios of both of their products exceed the gree stadard. I other words, both of the firms produce ordiary products ad eed to pay for carbo tax. Next, we derive the pricig strategies of the two competig firms ad give govermet optimal carbo tax policy cosiderig the govermet three goals. 5.1 The pricig decisios of the two firms Similar to Sectio 4.1, we obtai the demad of product PRO 1 as demad of product PRO 2 as. The, F 1 s optimizatio problem is give by * ad max =( p c )( )(1 ) t ( e e)( ) p1, p2 1 1 1 1 1 (16) Where e is the gree stadard for both of the products, c 1 is F 1 s variable cost per uit of product ad p 1 is the sellig price. is the turover tax rate for each uit of product that sells out. The first term represets the et sale profit ad the secod term represets the govermet carbo tax. The profit of F 2 ca be writte as follows: 17
max =( p c )( )(1 ) t ( e e)( ) p1, p2 2 2 2 2 2 (17) Where c 2 is F 2 s variable cost per uit of product ad p 2 is the sellig price. We obtai the firms optimal sellig prices as followers: 1 1 2 1 2 1 1 2 31 t t kt t t c c t k t p ( t, t ) 2 1 2 1 2 1 2 2 1 2 1 1 2 2 2 1 2 1 31 t t t k t c c t kt t p ( t, t ) 1 2 2 1 1 2 1 1 2 (18) (19) Next, we use a umerical example to aalyze the govermet optimal tax policy for each of the firm. 5.2 Determiig the optimal tax policies based o cetralized DEA I this sectio we use a umerical example to illustrate our results. We give some of the parameters i Table 5. Table 5. Parameters settig i the umerical example c 1 =0.5 c 2 =0.8 e=50 =0.3 = 0 1 1 =4 2 =2 k=3.5 α=1.5 The govermet may use differet tax policies for the two firms respectively. Here we let tax rate t 1 be 0.1, 0.2 1.0 with commo differece of 0.1, ad let t 2 be 0.2, 0.4 2.0 with commo differece of 0.2. There are 100 tax portfolios of t 1 ad t 2. Elimiatig the results that caot meet the coditio, we get 95 feasible choices o each firm s tax rates, as ca be show i the first colum of Table 6. Table 6 also shows the correspodig govermet reveue, govermet expediture ad carbo emissios to each t 1 ad t 2. 18
Table 6. Govermet reveue, govermet expediture ad carbo emissios No. t 1 t 2 R i E i Q i No. t 1 t 2 R i E i Q i 1 0.1 0.2 24.86 2.63 111.54 49 0.6 0.2 4.20 2.56 80.27 2 0.1 0.4 27.02 2.61 93.58 50 0.6 0.4 6.36 2.48 62.32 3 0.1 0.6 27.15 2.61 88.90 51 0.6 0.8 6.30 2.52 55.58 4 0.1 0.8 27.03 2.60 86.85 52 0.6 1 6.12 2.48 54.46 5 0.1 1 26.88 2.60 85.72 53 0.6 1.2 5.94 2.45 53.76 6 0.1 1.2 26.74 2.60 85.02 54 0.6 1.4 5.78 2.43 53.28 7 0.1 1.4 26.61 2.60 84.54 55 0.6 1.6 5.64 2.41 52.93 8 0.1 1.6 26.50 2.60 84.19 56 0.6 1.8 5.51 2.40 52.67 9 0.1 1.8 26.40 2.59 83.93 57 0.6 2 5.40 2.39 52.46 10 0.1 2 26.31 2.59 83.73 58 0.7 0.2 3.78 2.56 79.91 11 0.2 0.4 13.31 2.59 71.28 59 0.7 0.4 5.94 2.48 61.96 12 0.2 0.6 13.41 2.57 66.59 60 0.7 0.6 6.04 2.33 57.27 13 0.2 0.8 13.28 2.56 64.54 61 0.7 0.8 5.90 2.61 55.22 14 0.2 1 13.12 2.56 63.42 62 0.7 1 5.71 2.48 54.10 15 0.2 1.2 12.96 2.55 62.71 63 0.7 1.2 5.53 2.44 53.40 16 0.2 1.4 12.82 2.55 62.23 64 0.7 1.4 5.37 2.42 52.92 17 0.2 1.6 12.70 2.54 61.89 65 0.7 1.6 5.22 2.40 52.57 18 0.2 1.8 12.59 2.54 61.63 66 0.7 1.8 5.09 2.38 52.31 19 0.2 2 12.49 2.54 61.42 67 0.7 2 4.98 2.37 52.10 20 0.3 0.2 7.44 2.58 83.97 68 0.8 0.2 3.47 2.55 79.68 21 0.3 0.4 9.58 2.58 66.02 69 0.8 0.4 5.64 2.47 61.73 22 0.3 0.6 9.67 2.55 61.33 70 0.8 0.6 5.74 2.37 57.04 23 0.3 0.8 9.53 2.54 59.28 71 0.8 1 5.41 2.53 53.87 24 0.3 1 9.36 2.52 58.16 72 0.8 1.2 5.23 2.45 53.16 25 0.3 1.2 9.19 2.51 57.45 73 0.8 1.4 5.07 2.41 52.68 26 0.3 1.4 9.05 2.51 56.97 74 0.8 1.6 4.92 2.38 52.34 27 0.3 1.6 8.92 2.50 56.63 75 0.8 1.8 4.79 2.36 52.08 28 0.3 1.8 8.80 2.50 56.37 76 0.8 2 4.67 2.35 51.87 29 0.3 2 8.70 2.49 56.16 77 0.9 0.2 3.23 2.55 79.52 30 0.4 0.2 5.76 2.57 81.89 78 0.9 0.4 5.41 2.47 61.57 31 0.4 0.6 7.99 2.54 59.25 79 0.9 0.6 5.52 2.38 56.88 32 0.4 0.8 7.84 2.52 57.20 80 0.9 0.8 5.36 2.15 54.83 33 0.4 1 7.66 2.50 56.08 81 0.9 1 5.19 2.70 53.71 34 0.4 1.2 7.49 2.49 55.37 82 0.9 1.2 5.01 2.48 53.01 35 0.4 1.4 7.34 2.47 54.89 83 0.9 1.4 4.84 2.42 52.53 36 0.4 1.6 7.21 2.47 54.55 84 0.9 1.6 4.70 2.38 52.18 37 0.4 1.8 7.09 2.46 54.28 85 0.9 1.8 4.56 2.35 51.92 38 0.4 2 6.98 2.45 54.08 86 0.9 2 4.44 2.33 51.71 39 0.5 0.2 4.81 2.57 80.86 87 1 0.2 3.05 2.55 79.41 40 0.5 0.4 6.95 2.47 62.90 88 1 0.4 5.24 2.46 61.46 41 0.5 0.6 7.05 2.58 58.22 89 1 0.6 5.34 2.38 56.77 42 0.5 0.8 6.90 2.51 56.17 90 1 0.8 5.19 2.24 54.72 19
43 0.5 1 6.72 2.48 55.04 91 1 1.2 4.84 2.56 52.90 44 0.5 1.2 6.54 2.46 54.34 92 1 1.4 4.67 2.44 52.42 45 0.5 1.4 6.39 2.45 53.86 93 1 1.6 4.52 2.39 52.07 46 0.5 1.6 6.25 2.44 53.51 94 1 1.8 4.39 2.35 51.81 47 0.5 1.8 6.12 2.43 53.25 95 1 2 4.27 2.32 51.60 48 0.5 2 6.01 2.42 53.05 Figs 4-6 preset the impact of govermet differet carbo tax policies o each of the three goals. Differet form the figures i Sectio 4, we use three-dimesioal graphs to show how the tax portfolios of t 1 ad t 2 impact the govermet multiple objectives. I Fig. 4, we fid that i most cases, the govermet reveue may icrease with icreasig t 2 while decrease with icreasig t 1. However, i case 11-13, or 63-66, the govermet reveue would decrease i t 2. It is difficult to fid a best carbo tax policy from the three figures. Therefore, i what followers, we use DEA to evaluate the efficiecy of each tax portfolio of t 1 ad t 2 ad select the optimal carbo tax portfolio for the govermet. Fig. 4. Govermet reveue uder differet carbo tax policies 20
Fig. 5. Govermet expediture uder differet carbo tax policies Fig. 6. Total carbo emissios uder differet carbo tax policies Usig model (5)-(7), we obtai the DEA efficiecy scores of all feasible policies respectively i Table 7. 21
Table 7. DEA efficiecy scores of all feasible policies No. i No. i No. i No. i 1 0.909 25 0.508 49 0.166 73 0.306 2 0.995 26 0.505 50 0.324 74 0.299 3 1.000 27 0.500 51 0.360 75 0.292 4 1.000 28 0.496 52 0.357 76 0.286 5 1.000 29 0.492 53 0.351 77 0.129 6 1.000 30 0.223 54 0.345 78 0.279 7 1.000 31 0.428 55 0.339 79 0.308 8 1.000 32 0.435 56 0.332 80 0.311 9 0.999 33 0.434 57 0.327 81 0.307 10 0.998 34 0.430 58 0.150 82 0.300 11 0.593 35 0.425 59 0.305 83 0.293 12 0.640 36 0.420 60 0.335 84 0.286 13 0.654 37 0.415 61 0.339 85 0.279 14 0.657 38 0.410 62 0.335 86 0.273 15 0.657 39 0.189 63 0.329 87 0.122 16 0.654 40 0.351 64 0.322 88 0.271 17 0.652 41 0.385 65 0.315 89 0.299 18 0.649 42 0.390 66 0.309 90 0.301 19 0.646 43 0.388 67 0.304 91 0.291 20 0.282 44 0.382 68 0.138 92 0.283 21 0.461 45 0.377 69 0.290 93 0.276 22 0.501 46 0.371 70 0.320 94 0.269 23 0.511 47 0.365 71 0.319 95 0.263 24 0.511 48 0.360 72 0.313 Table 7 shows the DEA efficiecy scores of all 95 feasible policies. It ca be foud that policy 3, 4, 5, 6, 7 ad 8 may all be good choices for the govermet. Furthermore, much differet from the tax policy for oe firm that highest tax rate would be chose, the govermet would use the strategy to set relatively lower tax rate for oe firm ad higher tax rate for the other to achieve a comprehesive effect. 6. Coclusios Policy makers always operate with multiple goals, ad a give tax policy may beefit some goals while hider the others. I this paper, we examie the role the carbo tax policy plays i reducig carbo emissios, icreasig govermet reveue ad decreasig govermet expediture. I order to select the optimal carbo tax rate 22
that ca achieve a comprehesive effect, we regard the problem as multiple criteria decisio makig problem ad use a evaluatig method based o cetralized DEA approach to solve the problem. The cotributio of this paper ca be summarized as follows: we cosider multiple goals of a govermet whe makig evirometal policy decisio. I most literature that discuss the govermet evirometal policy, like pollutio tax (Krass et al. 2013), gree subsidy (Huag et al. 2013, Cohe et al. 2015) ad cap-ad-trade regulatio (Xu et al. 2016), they assume that the govermet makes policy strategies oly cosiderig oe goal, i.e., maximize the social warfare or miimize the carbo emissios. While i this paper, we cosider govermet reveue, govermet expediture ad carbo emissios, ad use cetralized DEA method to fid a most suitable tax rate to achieve comprehesive beefit of the three goals. The, we preset a three-step decisio framework to determie the optimal carbo tax rate for the govermet. I the first step, we aalyze the firms pricig strategies assumig that the tax rate is give. I the secod step, we idetify the govermet three goals ad deduce the qualitative estimatios to help us uderstad the coflicts amog the three goals. I the last step, we regard the problem as multiple criteria decisio makig problem ad use cetralized DEA approach to obtai the optimal carbo tax rate. We use a game-based method to probe ito the competitio betwee the ordiary products ad the gree products, ad a equilibrium poit is obtaied to determie the optimal decisios of the two firms o how to price the products for the maximal profit. Furthermore, we cosider two differet cases i this paper. I the first case, we focus o the govermet carbo tax policy for oe firm. We fid that the govermet may set a high tax rate. While i the secod case, we exped the curret model to focus o the govermet differet policies for two firms. We fid that the govermet would use the strategy to set relatively lower tax rate for oe firm ad higher tax rate for the other to achieve a comprehesive effect. We suggest that three possible extesios to our curret work are worth ivestigatio. First, some other assumptios e.g. stochastic market demad, multiple 23
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