THE EQUILIBRIUM MODELS IN OLIGOPOLY ELECTRICITY MARKET

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1 Iteratoal Coferee The Euroea Eletrty Market EEM-4 etember -, 4, Lodz, Polad Proeedg Volume, THE EQUILIBRIUM MODEL IN OLIGOPOLY ELECTRICITY MARKET Agezka Wyłomańka Wrolaw Uverty of Tehology Wrolaw (Polad) Magdalea Borgoz-Kozwara Ittute of Power ytem Automato Wrolaw (Polad) Abtrat We oder eletrty roduer layg o the eergy market. We model a olgooly market fag uerta demad, where eah frm hooe a t trategy a uly futo relatg quatty to re. uh trategy allow a roduer to adat better to the uerta evromet tha ether ettg a fxed re or a fxed quatty. We aly model of uly futo ometto olgooly uder demad uertaty, develoed by Klemerer ad Meyer [5]. We aume that the uerta demad a futo of the re ad the tohat hok. We derbe the hok by PARMA ytem (erod ARMA model). Our am th aer to rovde a realt a oble model of olgoolt ometto gle erod game. Our belef that the model we aly better rereet olgooly behavour tha tadard Bertrad ad Courot model.. INTRODUCTION.. uly futo Eoomt have log debated whether t make more ee to thk of frm a hoog re or quatte a trateg varable. I the world wth uertaty a frm may ot wat to ommt to ether of thee mle tye of trategy, or a all deo be deferred utl the reoluto of the uertaty. A lot of deo oerg ze ad truture of the orgaato mut be made advae. Thee deo mltly determe a uly futo that relate the quatty the frm wll ell to the re the market wll bear. uh a uly futo allow the frm to adat better to hagg odto tha doe the mly ommtmet to a fxed re (Courot) or quatty (Bertrad). Ay equlbrum oet requre ome tory about how frm adut to uertaty. For examle, a tohat Courot game, frm mut adut ther re to the realato of demad, ad a tohat Bertrad game, frm mut adut ther quatte. Oly uly futo equlbrum do frm adut to the uertaty otmal maer gve ther omettor behavour - wth tohat Courot or Bertrad, frm would wh to alter ther behavour after learg omethg about demad [5]... PARMA model To alulate uly futo the reee of demad uertaty frt we have to derbe exogeou hok. The hok may be oeted wth eaoalty ad ome exogeou eoom odto

2 (e.g. rad eoom growth). Therefore we aled PARMA model (erod ARMA model) to derbe the hok. ere wth erod orrelato hould ot be modelled wth the eaoal autoregreve movg-average (ARMA) la. Beaue ARMA model, otrary to ther ame, are atually tatoary model wth large ( abolute value) autoovarae at lag that are multle of the erod. A flexble la of hortmemory model that have the autoovarae erodte the la of erod autoregreve movg-average model (PARMA). Dete ther alablty, redto ad lkelhood evaluato method for PARMA model rema relatvely uexlored, eeally whe omared wth ther tatoary autoregreve movg-average (ARMA) outerart. If modelled by PARMA( model that mea the equee fulfl the followg odto: q b ( ) a ( ), () where the oeffet are erod wth the ame erod T ad the ovato are deedet wth zero mea. If = the for P b() b()... b( T ) the oluto of the ytem erodally orrelated that mea: E(( E(( E T )( E m T E )( m m T )) E m T Tme ere wth erod orrelato aturally are lmatology, hydrology, gal roeg ad eoom. Perod movg-average roee do ot aear to be ueful eoom. Hee, the eoometr aaly of erodally orrelated tme ere oetrate o PAR model: )). b ( ), () where the oeffet ad the ovato have the ame roerte lke the geeral model [3]..3. Olgooly We oder oooeratve olgooly model, for whh we have to make followg aumto []: oumer are re taker, all frm rodue homogeou rodut, there o etry to the market, frm olletvely have market ower ad take ow re or quatty deo deedetly. Three of the bet-kow olgooly model are the Courot, Bertrad ad takelberg model. I the Courot ad takelberg model frm et outut level (quatty roduto), wherea the Bertrad model frm et re. I the Courot ad Bertrad model all frm lay at the ame tme, wherea the takelberg model oe frm et t quatty roduto before the other. Thee dfferee the ato reult dfferet equlbrum. A lot of reearhe were made to how whh equlbrum the bet for the efed market. But aaretly uh mly tratege very ofte do t ofrm the exetato. That why we deded to aly uly futo equlbrum, whh relate the quatty the frm wll ell to the re the market wll bear. I the world wth exogeou demad uertaty a frm ha a et of roft maxmg ot (oe for eah realato of uertaty). It roved aer [5] that th ettg, a frm a geerally aheve hgher exeted roft by ommttg to a uly futo tha by ommttg to a fxed re or quatty, beaue uly

3 futo allow better adatato to the uertaty..4. Demad ad ot futo P I our model we aly uly futo equlbrum olgooly uder demad uertaty. Therefore we gve ome eoom ba to further oderato. The geeraled demad futo lt varable that fluee demad, for examle: re of rodut, re of related good, exetato of re hage, tate ad referee, advertg exedture, ome er ata. For ue maageral deo-makg, the demad futo mut be made exlt. The relato betwee quatty ad eah demad-determg varable mut be learly ad exltly efed. Q a (3) a a3 3 a4 4 The demad futo efe the relato betwee the quatty demaded ad all varable that determe demad. The demad urve a art of the demad futo that exre the relato betwee the re harged for a rodut ad the quatty demaded, holdg otat the effet of all other varable. To exame the relato betwee re ad the quatty demaded we mut hold otat other varable. Oe of the mot mortat feature of the re elatty oet that t rovde a ueful ummary meaure of the effet of a re hage o reveue. A good etmate of re elatty make t oble to make aurate etmate of the effet of re hage o total reveue. For deo-makg uroe, three ef rage of re elatty have bee detfed: elat demad (ε < -), utary elatty (ε = -) ad elat demad (ε > -) [4]. The horzotal le, whh how Fgure, derbe the rodut that omletely etve to the re (elatty ). Other le how dfferet rage of re elatty. P * ε < - ε > - ε = - Fgure. Pre elatty of demad Cae whe ε = or ε = are rare real world, but the mooole that ell eete uh a harmaeutal eoy relatvely elat demad, wherea frm hghly omettve market uh a groery retalg fae hghly elat demad urve [4]. Aordg to elatty of the demad, whh oeted wth market truture, we aalyed eergy market ad we aumed that the re elatty of demad hould be about,5. All lear demad urve are ubet to varyg elatte at dfferet ot o the urve, ee Fgure. Utary elatty the ot where the effet of a re hage exatly offet by the effet of a hage quatty demaded ad the frm aheve the hghet total reveue. Beaue eah frm ted to aheve the hghet reveue therefore t alway oerate ear the utary elatty. P Elat rage ε < - Utary elatty ε = - Ielat rage ε > - Fgure. Elatte alog demad urve. Q Q

4 The mot revalet olear model for ot etmato the quadrat ot futo [4], whh a be wrtte a follow: TC b b Q b Q b TFC TC (total ot) refer to relevat ot durg a tyal obervato erod; Q the quatty of outut rodued durg that erod; degate all other deedet varable whoe ot effet the aalyt wat to aout for; ad TFC mea total fxed ot. A quadrat total ot futo rodue a lear margal ot futo ad a U- haed average total ot futo. Fgure 3 llutrate the tyal average total ot ad margal ot urve aoated wth a quadrat total ot futo. The TC futo a be wrtte dfferet way a follow: TC a ( Q Q ) b Q * otat value ad oeffet a ad b are grater tha zero. Beaue the total ot futo U-haed o the mmum of the futo Q *. We oder the market where the uly futo deedet o the market re ad the hok, whh a oluto of PARMA ytem. The trategy of eah frm a uly futo. I the aaly we aume: - the market re, { } - the hok, D ) - demad futo, ( ) - uly futo of -th omay. Therefore the exeted roft of -th omay at the momet gve by: w [ ) ) (), ()] (4) where G} mea the exeted value of radom varable G. The maxmum roft for the -th omay aheved by the re * whh fulfl the odto: ( ) dw d w ) ),. The frt odto ha the followg form: C ATC MC [, ) E'{ d, [, ) d Ad the eod odto : )] ]. b Q* Fgure 3. Average total ot urve. UPPLY FUNCTION EQUILIBRIUM IN DOUPOLY Q [ d [ E' {, [ E' '{ d, [, d, ) ) d )] ] )] ].

5 If we aume the equlbrum o the ower market ymmetr ad the ot of roduto for eah omay the ame. I that ae we have ) ) ). Moreover the roe { } modelled by PARMA roe wth ovato of zero mea ad ut varae. The frt odto gve by: d [ [ E'{ ) d ]. )] Therefore the futo ( fulfl the dfferee equato: d E' { ) d ad t fulfl the odto: [ E' { d [ [ E' '{ d [ )] d )] ] d ]. (5) Examle. We aume the dutry demad ) m ( m ) ad { } modelled by PAR() whh gve by the equato b ( ) b ( ) (the oeffet are erod wth the erod T), the roduto ot a futo ( q q ) / u, where, q, u (ee [5]). I that ae for eah equato (5) ha the form: d }} m ( q ) (6) The oluto of the above dfferee equato a radom varable uh that t exeted value (whh t take by the dtrbuto of ) equal 4 ( m m m ) q ) alo ot deedet o ad o the oeffet of PARMA model. Examle. We aume the dutry demad ) m ( m ) ad { } modelled by PAR() whh gve by the equato b ( ) b ( ) (the oeffet are erod wth the erod T), the roduto ot a futo ( q q ) / u, where, q, u. I that ae for eah equato (5) ha the form: d ( q m ) }. (7) For fxed we obta the oluto of the dfferetal equato (7): ( m q ). [ m } 4( m ] (8) Therefore th ae the exeted value of uly futo at the momet a futo deedet o the oeffet of the PAR model beuae the eod momet of the hok deedet o PAR oeffet. 3. UPPLY FUNCTION EQUILIBRIUM IN OLIGOPOLY We aume there are k deedet frm, whh ell the ower o the eletrty

6 market. The trategy of -th frm t uly futo. I that ae the roft of -th omay at gve by: w [ ) ) ). )] If we aume that there ext the ymmetr equlbrum o the market: eah of the frm ha the ame uly futo ad the margal ot equal: w [ ) ( k ) ) ( k ) ), )] where ) ),,,.. k. The frt odto ha the form: d [ [ E' { ) d ( k ) )] ] ad t equvalet to the followg d E'{ k (9) d. ))( k ) The eod odto ha the form: [ E' { d [ [ E' '{ d [ )] d ( k ) )] ] d ( k ) ]. Examle 3. We oder the ame aumto a Examle where the demad futo ad the margal ot futo are gve reetvely by: ) m ( q q ) u m, q, u. { } modelled by PAR() whh gve by the equato b ( ) b ( ) (the oeffet are erod wth the erod T). We aume there are k deedet omae o the market ad there ext the ymmetr equlbrum wth uly futo ). I that ae the exeted value of the uly futo fulfl the dfferee equato: d q ( m k () )( k ) The oluto of the dfferetal equato ha the followg form: ( k ) ( m 4. CALCULATION q [ k ( m ( k ) k( m ] We baed our alulato o demad ad re data e Jauary 3 to Augut 3. Frt we oder Examle ad we aume the followg: b ) m ( ) b ( ) ( q q ) / u,

7 where m =.4655, =.5, q = 3.43, u = 73, q a mea of the demad data, u a mmum of the re data ad m alulated o the re ad demad data. I th ae the exeted value of the uly futo a lear futo of (ee Fgure 4) ad t equal: where the oeffet m,, q, u are the ame a the revou examle. I the aaly we take the erod T=7 (7 day). The Yule-Walker method of the PAR oeffet etmato gve the reult: 4m ( m m ) q ) uly futo Fgure 4. The uly futo. Fgure 5. Total ot futo. ot futo Table. The PAR oeffet b () b () I th ae the exeted value of the uly futo for a fxed ha the followg form: ( ) (.4655 } 4( ) I the Table we reet the exeted value of the uly futo for three eleted value of. To obta the eod momet of the hok we ue realato of for eah. 4 Table. The exeted value of the uly futo I Examle we have the followg aumto: ( ) b ( ) ) m, b q ( q q ) / u, ( day). The exeted value of uly futo Aordg to Examle 3 we oder the market wth k roduer. We aume that

8 the demad ad ot futo are the ame a the revou examle. O Fgure 6 we reet how the exeted value of the uly futo deed o the umber of layer. I the aaly we take =7. For omaro, o Fgure 7 we how the exeted value of uly futo ae wthout the hok. uly futo k= k=3 k= k=3 the margal ot Fgure 6. The uly futo for k roduer ad the margal ot k= k=3 k= k=3 the margal ot uly futo Fgure 7. The uly futo for k roduer ad the margal ot ae wthout the hok. 5. CONCLUION We rovded three uly futo equlbrum for duooly ad olgooly market. I ae of duooly we obtaed two uly futo equlbrum gve by oluto of dfferetal equato (6) ad (7). Whe there are a lot of roduer o the market the uly futo equlbrum gve by a oluto of equato ().

9 We ued the mathematal ad eoomal theory reeted eto ad 3 to aalye the real re ad demad data. The uly futo equlbrum deedet o wa obtaed ae wthout the hok. Next we aalyed the demad urve, whh a futo of the hok ad etmated the PAR oeffet for the real re ad demad data, ee Table. I th ae the uly futo reae tme, ee Table. We aalyed the market wth k roduer to omare the reult ad reeted them o Fgure 6 ad 7. Moreover, the ae of k roduer the uly futo reae the umber of layer. The margal ot, whh a futo of exeed the uly futo ether the ae wth hok or wthout hok. The loe of the uly futo wth k ted to the loe of the margal ot urve. 6. REFERENCE. Brozkewz-uwa E., Makago A., Wero R., Wyłomańka A.: O detetg ad modelg erod orrelato faal data. Phya A, o. 336, 4, Carlto D.W., Perloff J.M.: Moder Idutral Orgazato. eod Edto. HarerColl CollegePublher, New York Ghyel E., Obor D.R.: The eoometr Aaly of eaoal Tme ere, Cambrdge Uverty Pre, Cambrdge. 4. Hrhey M., Paa J.L., Whgham D.: Maageral Eoom. Euroea Edto. The Dyre Pre, Lodo Klemerer P.D., Meyer M.A.: uly futo equlbra olgooly uder uertaty. Eoometra, vol. 57, o. 6, November 989, Lud R., Baawa I.V.: Reurve redto ad lkelhood evaluato for erod ARMA model, J. Tme er. Aal., o., 998, Wero A., Wyłomańka A. O ARMA (, model wth bouded ad erodally orrelated oluto. Probablty Mathematal tatt to aear. Agezka Wyłomańka wa bor 978 Nowa Ruda, Polad. he reeved M.. degree mathemat ee from the Wrolaw Uverty of Tehology. At reet he takg PhD dertato. Her area of teret lude ARMA model wth varyg oeffet. Malg addre: Agezka Wyłomańka Wrolaw Uverty of Tehology Ittute of Mathemat 4 Jazewkego t., 5-37 Wrolaw POLAND Phoe: (+48)(7) e-mal: wyloma@m.wr.wro.l Magdalea Borgoz-Kozwara wa bor 97 wda, Polad. he reeved M.. degree from the Wrolaw Uverty of Tehology. Preetly he takg PhD dertato. Her area of teret oer alato of game theory to eergy market. Malg addre: Magdalea Borgoz-Kozwara Ittute of Power ytem Automato Wytawowa t., 5-68 Wrolaw POLAND Phoe: (+48)(7) e-mal: magda@ae.wro.l