Appl. Math. Inf. Sc. 0 No. 5 657-66 (06) 657 Appled Mathematcs & Informaton Scences An Internatonal Journal http://dx.do.org/0.8576/ams/00503 Envronmental taxaton: Prvatzaton wth Dfferent Publc Frm s Objectve Functons Fernanda A. Ferrera and Flávo Ferrera ESEIG.IPP Appled Management Research Unt (UNIAG) Polytechnc Insttute of Porto Rua D. Sancho I 98 4480-876 Vla do Conde Portugal. Receved: 4 Apr. 06 Revsed: 3 Jun. 06 Accepted: 4 Jun. 06 Publshed onlne: Sep. 06 Abstract: In ths paper we study the effects of envronmental taxes and prvatzaton n a mxed market by consderng that the publc frm ams to maxmze the socal welfare. The model has two stages. In the frst stage the government sets the envronmental tax. Then the frms engage n a Cournot competton choosng output and polluton abatement levels. We also compare the results obtaned wth the ones got when the publc frm ams to maxmze the sum of consumer surplus and the frm s proft. Keywords: Industral Organzaton; Game Theory Cournot model envronment prvatzaton Introducton In recent years study of mxed markets where a welfare-maxmzng publc frm competes wth proft-maxmzng prvate frms are ncreasngly become popular. Mxed markets are common n many countres. Transportaton energy steel telecommuncatons ol and hosptals are good examples of mxed olgopoles. DeFraja and Delbono [4] are poneers n these studes. They showed that n a Cournot competton prvatzaton of the publc frm may mprove socal welfare. Matsumura and Matsushma [7] proved that under certan condtons the partal prvatzaton of the publc frm mproves socal welfare. Some authors (e.g. Ohor [8]) argue that the envronmental qualty can be affected by the polces of prvatzaton of frms. The mposton of restrctve polluton standards also requres the adopton of costly abatement technology by the frms. Yn [0] analysed correctve taxes n an olgopoly model wth nter-frm externaltes and polluton abatement. Bárcena-Ruz and Garzón [] studed envronmental polces n a duopoly model wth ncentves. Ohor (004) examned the nteracton among prvatzaton envronmental and trade polces n nternatonal competton wth envronmental damage. Bárcena-Ruz and Garzón [] nvestgated how the decson on whether to prvatze a publc frm or not nteracts wth envronmental polcy. Wang and Wang [9] explored whether prvatzaton mproves (or deterorates) the envronment n a mxed duopolstc framework wth dfferentated goods and polluton abatement. They showed that f the publc frm s prvatzed less attenton s pad to polluton abatement by all the frms coupled wth less envronment taxes leved by the government and the envronment s more (less) damaged when the goods are less (more) substtutable. Belad and Chao [3] also showed that prvatzaton can have a negatve effect on the envronment. Ferrera and Ferrera [6] examned the same questons as n Wang and Wang s paper by consderng a Stackelberg-type sequental-move game wth homogeneous goods. In our paper we do a smlar analyss as done by Wang and Wang [9] but followng Ohor [8] among others we consder that the publc frm ams to maxmze the socal welfare nstead of the sum of consumer surplus and ts proft. The results that we get are dfferent than the ones from Belad and Chao [3] and Wang and Wang [9]. In fact n our model we conclude that the envronment s more damaged n the mxed than n the prvate market. An extended abstract of a prevous verson of ths paper was publshed n the Proceedngs of the th Internatonal Conference on Numercal Analyss and Appled Mathematcs (see [5]). The remaned of the paper s organzed as follows. In Secton II we present and dscuss the mxed model. Correspondng author e-mal: flavoferrera@eseg.pp.pt
658 F. A. Ferrera F. Ferrera: Envronmental taxaton: prvatzaton wth dfferent publc... Secton III deals wth the prvatzed model. Secton IV yelds the man results ganed by a drect comparson between both the mxed and prvatzed models. Conclusons are presented n Secton V. The mxed duopoly We consder a mxed Cournot duopoly wth one publc frm F and one prvate frm F producng homogeneous goods. We wll use a specfc model whch s a standard settng n ths feld. The nverse demand functon s gven by p=α Q where α > 0 s the demand parameter p s the market prce and Q = q + q s the total output n the market where q and q are the outputs of the publc frm and the prvate frm respectvely. We assume that both frms share dentcal quadratc cost functons: C(q ) = F + q where F = 0 wthout loss of generalty. The total consumer surplus s CS= (q + q ). The producton of the good n both publc and prvate frms leads to polluton e. Envronmental damage functon s gven by ED= ( e ). However each frm can prevent polluton by undertakng abatement measures. Suppose that f frm F chooses polluton abatement level a then the correspondng emsson level s e = q a. The cost of polluton abatement of frm F s a /. Each frm has to pay an envronmental tax per unt of pollutant emtted and as a result tax revenues collected by the government are T = t e. The envronmental tax s mposed by the government and ts objectve s to maxmze socal welfare whch comprses the consumer surplus CS the producer surplus π + π and the tax revenues collected by the government T less the envronmental damage ED: W = CS+π + π + T ED where the proft of frm F s gven by π = p q q te a =. As mentoned above the publc frm s objectve functon s the socal welfare W and the prvate frm s s to maxmze ts own proft. The model conssts n the followng two-stage game: In the frst stage the government chooses the envronmental tax rate t. In the second stage the frms engage n a Cournot competton choosng smultaneously output and polluton abatement levels. As usual the game s solved by backwards nducton. In the second stage both frms choose smultaneously output and polluton abatement levels. So we dfferentate the functon W wth respect to q and a and the functon π wth respect to q and a : δw δq = α 4q q + a + a = 0 δw δa = q + q a a = 0 δπ δq = α t q 4q = 0 δπ δa = t a = 0. The above frst-order condtons yeld the followng results: q = 5α+ 7t q = 5 5α 8t. () 5 5α 3t a = a = t. () 5 From equatons () we see that the prvate frm abates polluton to the pont where margnal abatement cost equals the tax but ths s not true for the publc frm. Now puttng () and () nto the objectve functon W of the government we get W = 5α + 5αt 9t. 5 Maxmzng ths functon wth respect to the varable t and then substtutng the result back to q and a we get the followng values at equlbrum: Proposton.In equlbrum the envronmental tax the quanttes and the polluton abatement level n the mxed duopoly are gven by : t M = 5α 38 q M = 39α 70 qm = 99α 595 a M = 73α 90 am = 5α 38. From the expressons above we obtan the profts of each frm consumer surplus envronmental damage tax revenues collected by the government and socal welfare as shown below. We use the superscrpt M to refer to the mxed duopoly.
Appl. Math. Inf. Sc. 0 No. 5 657-66 (06) / www.naturalspublshng.com/journals.asp 659 Proposton.In equlbrum the consumer surplus the envronmental damage the proft of each frm the tax revenue collected by the government and the socal welfare are gven by: Proposton 4.In equlbrum the consumer surplus the envronmental damage the proft of each frm the tax revenue collected by the government and the socal welfare n the prvatzed duopoly are gven by: π M = 694α 404600 πm = 744α 8300 CS M = 84α 8300 ED M = 999α 8300 T M = 865α 56644 W M = 50α 380. CS P = 648α 00 ED P = 98α 00 π P = 648α 00 T P = 54α 00 W P = α 0. 4 Effects of prvatzaton 3 Case II: prvate duopoly (post-prvatzaton) We now proceed to the analyss of post-prvatzaton. As usual we regard prvatzaton as a change n the objectve functon of frm F form maxmzng socal welfare to ts own proft : π = p q q te a. Utlzng the same way of calculaton as n the prevous secton we get the followng result. Proposton 3.In equlbrum the envronmental tax the quanttes and the polluton abatement level n the prvatzed duopoly are gven by 3 : t P = α 0 q P = 8α 0 a P = α 0. From the expressons above we obtan the followng result: We suppose that the publc frm s prvatzed wthout cost. 3 We use the superscrpt P to refer to the prvate duopoly. In ths secton we compare the mxed and prvatzed duopoly equlbra. The followng theorem summarzes our results. Theorem.In equlbrum q M < q P < q M Q P < Q M a M < ap < a M t M < t P π M < πp < π M ED P < ED M CS P < CS M T P < T M W P < W M. We observe that n the prvate competton the market s more compettve and thus the prvate frm produces more than the mxed competton (q P > q M ). On the other hand the prvatzed publc frm reduces ts producton snce ts compettor acts more aggressvely (q P < q M ). The overall effect when the publc frm s prvatzed s a decreasng n the aggregate quantty n the market (Q P < Q M ). For the prvate frm the ncrease n the output level ncreases ts proft (π M < πp ). For the prvatzed publc frm the decrease n the output level s not fully compensated by the exchange of ts objectve from socal welfare maxmzaton to proft maxmzaton and so decreases ts proft (π M > π P ). Furthermore hgher (resp. lower) output levels n the mxed competton than n the prvate competton nduces hgher (resp. lower) correspondng abatement levels also n the mxed than n the prvate competton (q M > q P a M > a P ; q M < qp am < ap ). The overall effect s that the envronmental tax s lower n the mxed than n the prvate
660 F. A. Ferrera F. Ferrera: Envronmental taxaton: prvatzaton wth dfferent publc... competton (t M < t P ). Another consequence s that the envronment s more damaged n the mxed than n the prvate market (ED M > ED P ). Moreover the consumer surplus and the socal welfare are also hgher n the mxed than n the prvate competton. 5 Comparson wth dfferent publc frm s objectve functons Up to now we have used the socal welfare as the publc frm s objectve functon. An alternatve of that publc frm s objectve functon s the sum of consumer surplus and ts own proft: G= CS+π as t s used by Wang and Wang [9]. They got the followng result: Proposton 5.[9] In equlbrum 4 q MG > q PG > q MG π PG Q MG > Q PG > π MG > π MG a MG = a MG = t MG > t PG = a PG CS MG > CS PG ED PG > ED MG W PG > W MG. From the prevous results we can conclude that the defnton of the publc frm s objectve functon plays an mportant role n the effects of prvatzaton. In fact n our model prvatzaton decreases the envronmental damage and the welfare results that are n contrast wth the ones got through the formalzaton of Wang and Wang [9]: In ther case prvatzaton ncreases envronmental damage and also the welfare. 6 Conclusons In ths paper we analysed the effects of prvatzaton of a publc frm n the envronment by consderng a mxed Cournot duopoly assumng that each frm can prevent polluton by undertakng abatement measures. We concluded that n the mxed market the ndustry output s hgher than n the prvate market. Furthermore the envronmental tax rate n the mxed duopoly s lower than that n the prvatzed duopoly and the envronment s more damaged n the mxed than n the prvate market. The overall effect on the socal welfare s that t wll becomes hgher n the mxed than n the prvate market. We have also compared the results of our model wth the ones got for a dfferent publc frm s objectve functon: the sum of consumer surplus and ts own proft. We concluded that the defnton of the publc frm s objectve functon plays an mportant role n the effects of prvatzaton. In fact n our model prvatzaton decreases the envronmental damage and the socal welfare whch s not the case when the publc frm ams to maxmze the sum of consumer surplus and ts own proft. Acknowledgement We thank ESEIG and Polytechnc Insttute of Porto for ther fnancal support. References [] J.C. Bárcena-Ruz J.C. Garzón Envronmental taxes and strategc delegaton Spansh Economc Revew 4 30-30 (00). [] J.C. Bárcena-Ruz J.C. Garzón Mxed olgopoly and envronmental polcy Spansh Economc Revew 8 39-60 (006). [3] H. Belad C.-C. Chao Does prvatzaton mprove the envronment? Economcs Letters 93 343-347 (006). [4] G. DeFraja F. Delbono Game theoretc models of mxed olgopoles J. Econ. Surveys 4-7 (990). [5] F. A. Ferrera F. Ferrera Prvatzaton n a mxed duopoly wth envronmental taxes AIP Conf. Proc. 558 558-56 (03). [6] F. A. Ferrera F. Ferrera Prvatzaton and government preference n a publc Stackelberg leader duopoly. Proceedngs of the 4th IEEE Internatonal Conference on Nonlnear Scence and Complexty. Budapest Hungary (0). [7] T. Matsumura N Matsushma Endogenous cost dfferentals between publc and prvate entreprses: A mxed duopoly approach Economca London School of Economcs and Poltcal Scence 74 (84) 67-688 (004). [8] S. Ohor Envronmental tax trade and prvatzaton The Kyoto Economc Revew 73 09-0 (004). [9] L.F.S. Wang J. Wang Envronmental taxes n a dfferentated mxed duopoly Economc Systems 33 389-396 (009). [0] X. Yn Correctve taxes under olgopoly wth nter-frm externaltes Envronmental and Resource Economcs 6 69-77 (003). 4 We use the superscrpt MG (resp. PG) to refer to the mxed (resp. prvate) duopoly when the publc frm s objectve functon s G.
Appl. Math. Inf. Sc. 0 No. 5 657-66 (06) / www.naturalspublshng.com/journals.asp 66 Fernanda A. Ferrera s a Professor n the Department of Mathematcs at the School of Management and Industral Studes of the Polytechnc Insttute of Porto Portugal. She earned PhD n Appled Mathematcs from Unversty of Porto. She has publshed journal and conference papers. Fernanda Ferrera s member of the Appled Management Research Unt (UNIAG). Her research nterests nclude ndustral organzaton and game theory. She s the author or co-author of more than 50 journal papers and conference proceedngs. She has gven talks n major nternatonal conferences manly on Mathematcs and ts applcatons n Engneerng and Economcs. Flávo Ferrera s a Full Professor n the Department of Mathematcs and the Dean of the School of Management and Industral Studes of the Polytechnc Insttute of Porto Portugal. He earned PhD n Appled Mathematcs from Unversty of Porto. Flvo Ferrera s member of the Appled Management Research Unt (UNIAG). Hs research nterests nclude ndustral organzaton and game theory. He s co-author of a book publshed by Sprnger-Verlag and author or co-author of more than 40 journal papers and conference proceedngs.