Pollution abatement and reservation prices in a market game

Transcription

1 MPRA Munch Personl RePEc Archve Polluon bemen nd reservon prces n mrke gme George Hlkos nd George Ppgeorgou Unversy of Thessly, Deprmen of Economcs Ocober 2012 Onlne hp://mpr.ub.un-muenchen.de/42150/ MPRA Pper No , posed 23. Ocober :15 UTC

2 1 Polluon bemen nd reservon prces n mrke gme By George E. Hlkos nd George J. Ppgeorgou Lborory of Operons Reserch, Deprmen of Economcs, Unversy of Thessly, Kor 43, 38333, Volos, Greece Absrc In hs pper we se up n olgopolsc mrke model, where frms nves n polluon bemen n order o ncrese he whole mrke sze v n ncrese n he consumers reservon prce. Moreover, we suppose h he demnd funcon s no lner one nd he resulng gme s no usul lner qudrc one. In he consdered model we nvesge he open loop, he memory less closed-loop nd he collusve perns equlbrum. Addonlly, we exmne he socl plnnng perspecve. In he cse of convex demnd we found he surprsng resul h he conrol nd se vrbles hve hgher vlues n he open-loop sedy se equlbrum hn n he closed loop, whle n lner demnd cse he equlbrum s undeermned. In ll cses we fnd h only f he mrke demnd hs concve curvure re he conclusons cler. A number of proposons nd remrks re provded. Keywords: Olgopoly Gme; non-lner demnd; polluon bemen; reservon prce. JEL Clssfcons: Q52; Q58; D43;C61; C62.

3 2 1. Inroducon Brodly spekng, he mn dfference beween he open-loop equlbrum, on he one hnd, nd he feedbck nd closed-loop equlbrum, on he oher hnd, s h he former does no ke no ccoun sregc nercon beween plyers hrough he evoluon of se vrbles over me nd he ssoced djusmen n conrols. Accordng o he open-loop rule, plyers choose her respecve plns he nl de nd comm o hem forever. Therefore, n generl, open-loop equlbrum s no sub-gme perfec, n h s only wekly me conssen, snce plyers mke her con by he clock only. However, here re clsses of gmes where open-loop equlbrum s sub-gme perfec (Clemhou nd Wn, 1974; Dockner, Fechnger nd Jorgensen, 1985; Rengnum, 1982; Mehlmnn, 1988). A furher dsncon cn be mde beween he closed-loop equlbrum nd he feedbck equlbrum, whch re boh srongly me conssen, herefore sub-gme perfec snce, ny de τ, plyers decde by he sock of ll se vrbles. Whle he closed-loop memory less kes no ccoun he nl nd curren levels of ll he ses, he feedbck equlbrum kes no ccoun he ccumuled sock of ech se vrble he curren de. If one plyer decdes ccordng o he feedbck rule, hen s opml for he ohers o do so s well. Hence, he feedbck equlbrum s closedloop equlbrum, whle he oppose s no generlly rue. The m of hs pper s o ssess comprvely he properes of open loop nd closed-loop memory less equlbrum n dynmc olgopolsc polluon model for whch, frsly, he olgopolss fce non-lner mrke demnd nd, secondly, hey emp o ncrese he whole mrke sze v n ncrese n he consumers reservon prce nvesng n polluon bemen.

4 3 The exsng lerure n he feld of dynmc gmes devoes consderble moun of enon o denfyng clsses of gmes where he feedbck equlbrum degeneres no he open-loop equlbrum. Degeneron mens h he Nsh equlbrum me ph of he conrol vrbles concdes under he dfferen sregy conceps. In prculr, enls h he resulng open-loop soluon s ndependen of he vecor of he ses. The neres n he concdence beween he equlbrum phs under he dfferen soluon conceps s moved by he followng reson: Whenever open-loop equlbrum s degenere feedbck equlbrum, hen he former s srongly me conssen. Therefore, one cn rely upon he open-loop equlbrum whch s, n generl, much eser o derve hn he closed-loop or feedbck ones. Clsses of gmes where he concdence rses re llusred by number of uhors (Fershmn, 1987; Seersd nd Sydseer, 1977; Rengnum, 1982). As whole, he gmes where open-loop equlbrums re srongly me conssen re known s perfec or se redundn, precsely becuse opml conrols derved from open-loop frs order condons depend only upon me nd no upon ses. A clss of gmes where hs clerly pples s h of lner se gmes, where he Hmlonn funcon s lner n he se vrbles. From he polluon pon of vew, s well known h producon process ccumules polluon s by produc, nd hs exernl economy mus be correced. Nowdys, he menng of exernl dseconomes s essenlly synonymous wh exernly or exernl effecs n he sphere of producon nd consumpon. Exernl effecs n producon re unpd sde effecs of one producer's oupu or npu on oher producers. A n erler sge, exernl dseconomes n he menng now gven were clled echnologcl exernl dseconomes, reflecng he fc h he effecs were

5 4 rnsmed ousde he mrke mechnsm nd lered he echnologcl relonshp beween he recpen frms oupu nd he npus under s conrol. The mos commonly used mehods of correcon found n economc lerure re polluon perms nd Pgouvn xes on pollung frms. Specfclly, he sndrd economc pproch o exernles s ypclly scrbed o Pgou (1920), who devsed sysem of xes nd subsdes o llow for he socl coss whch re no ncluded n prve decson-mkng. In smple erms, x s plced on he polluer o brng hs cos funcon no lne wh he rue socl cos of producon. Ths pproch o exernles hs been chllenged by pro-mrke heorss such s Cose (1960). The free-mrke pproch denfes he problem of exernles s he bsence of mrkes nd he ssoced propery rghs. The ssgnmen of propery rghs o non-mrkeed goods, such s r nd wer, provdes frmework n whch he pres re encourged o resolve he problem of negve exernles rher hn do nohng (Hlkos, 1993, 1994, 1996). In hs pper we consder he bsc mplcons of he polluon n he enre mrke where frms compee under he ssumpon h hey undersnd nd emp o endogenze he exernles of he polluon occurrences. As polluon s very mporn for envronmenl polcy modelng nd nlyss, n emergng reserch kes plce n he ls decdes. In our sudy we dop number of somemes src ssumpons mde n oher reserch res. Frs of ll, we mke he ssumpon h frms re olgopolsc rher hn perfec compeors. The frs jusfcon s h perfec compeon s no pplcble o mjor pollung generng frms whch re olgopolsc n her oupu mrkes. Second, we mke he ssumpon h frms underke he lod of bemen, n conrs o he usul ssumpon where he bemen s underken by he

6 5 governmen. Forser (1980) mkes cler h bemen s governmen polcy or sk. There re lo of endogenzed vrbles h frms could ke no ccoun. For exmple, n he recen lerure n endogenzed vrble could be he cpl ccumulon s n nvesmen decson. Here he proposed model doesn ke he poson of he cpl ccumulon s n nvesmen decson, bu rher kes he perspecve h polluon bemen s n nvesmen decson vrble (Hlkos nd Ppgeorgou, 2012). The hrd ssumpon s h demnd n he enre olgopolsc mrke s no usul lner demnd funcon bu more flexble one. Specfclly, we ccep more generl demnd funcon ffecng he consumers reservon prce, for whch he curvure s deermned by fcor α, o llow vryng forms of concvy, lnery nd convexy. As s shown n he res of he pper, he demnd funcon employed ffecs demnd elscy, consumer surplus nd lso socl welfre. A ls ssumpon for he proposed model s he dynmc envronmen whn whch he gme evolves. Former models of Courno olgopolsc mrkes re sc nd wh or whou lner demnd funcons. Bu dynmc modelng s he more generl modern perspecve h exrcs severl resuls dependng on he nformonl srucure employed by he gme. For hs reson we sudy he wo mn srucures, open nd closed loop, nd drw he fnl conclusons. The srucure of he pper s s follows: Secon 2 revews he exsng relve lerure. In secon 3 he bsc model wh he sysem dynmcs s presened. Secons 4 nd 5 descrbe he Open nd Feedbck Nsh Equlbr. In secon 6 we nvesge he closed-loop memory less Nsh equlbrum, whle n secon 7 he sedy ses of open nd closed-loop memory less ypes of equlbrums re

7 6 compred. Secon 8 sudes he resulng equlbrum n frm s colluson proposng he socl plnnng (under he colluson ssumpon). The ls secon concludes he pper. 2. Lerure revew Ngurney nd Dhnd (2000) nvesge he modelng nlyss nd compuon of soluons o models of mulproduc, mul-pollun non compln olgopolss who re engged n mrke of polluon perms. In order o compue he prof mxmzed qunes of emssons, hey propose nd pply n lgorhm, long wh he equlbrum llocon of lcenses nd her prces. Txng pollung frms or subsdng frms my be consdered s n ncenve o engge n polluon bemen. The generl concluson produced from xng polcy, found n lerure, s h n opmum x rule mus send o frms he messge h he more hey pollue now, he hgher her fuure lbly wll be. The lerure on xng frms n dynmc conex rses from he sc olgopoly frmework used n Ksoulcos nd Xeppdes (1992) nd Kennedy (1994). The xng lerure consss mnly of wo srems. The frs one focuses on nformonl ssues whou pyng much enon o he problem of sock ccumulon, whle he second one concerns he sock dynmcs nd ssumes perfec nformon. In he second srem of xng, sudyng he wo ypes of equlbrum (open nd closed loop) n dynmc conex Benchekroun nd Long (1998) found h he opml x rules nnounced from he ouse kes smlr forms, bu wh dfferen prmeers. Ths mode of regulon s exogenous o he mrke sysem nd needs boh socl plnnng nd opml x rules mposed by he governmen. One mporn

8 7 reson for he frms o endogenze he polluon exernly s h frms no engged n bemen cves my see her profs nd possbly he mrke sze shrnk due o polluon s well s due o he fll n he consumers reservon prce. In hs pper we consder dynmc olgopolsc model where demnd s no usul lner funcon nd frms underke npolluon cves n order o ncrese he mrke sze. The choce of he non lner demnd funcon s rcble becuse mplcly ffecs consumer surplus nd lso gves he possbly o ncrese he mrke sze. The recen lerure n he cse where frms compee under he demnd consrn nd or bou demnd lnery found s esy o clssfy n cegores. In he frs one, he non lner, Nmzd nd Sbrg (2005) mke he ssumpon of demnd nd qudrc coss n he form of 1 2 p= bq nd C = c + c q + c q, ck 0, k = 0,1, 2 respecvely. Usng he Grden Dynmcs (GD) djusmen process, hey djus he frms producon n he drecon ndced by her (correc) esme of he mrgnl prof. Second, Leonrd nd Nshmur (1999) employ n rbrry non lner demnd whou full nformon. Frms n Courno model do no observe her rvl s cons, mkng msken belefs. The bove ssumpons desroy he sbly of equlbrum nd cree cycles, so he dynmcs of he Courno model re lso ffeced. In dfferen pproch gven by Bylk e l. (2000) he suon where olgopolsc frms compee wh globl demnd consrn s explored. The evoluon of frms mrke demnd s deermned by ll frms prce decsons. Ther neres focuses on he nlyss of some smple clsses of sreges nd on fndng he bes responses o hem. To h end, Huck e l. (2002) repor resuls of he Courno bes reply process. In Courno olgopoly wh four frms, lner demnd nd lner

9 8 cos funcons, he bes reply process explode. They lso nvesge he power of severl lernng dynmcs o expln he unpredced sbly. To h end, he concep of consumers vluon of produc or reservon prce s used by Grdhrn n Berrnd lke model for new produc nroducon. In hs model, ech consumer mkes hs choce from mong he vlble brnds bsed boh on he reservon prce nd cul prces. He shows, n equlbrum, h he prce of he new produc s equl o s men(verge) reservon prce nd hose of he old ones re srcly greer hn he respecve men reservon prces, denong he mpornce of he consumers vluon (reservon prce) for produc mrke. Fnlly, n he proposed model, he nduced dynmc gme for n olgopolsc mrke wh nonlner demnd funcon nd polluon bemen underken by frms s non qudrc dfferenl gme for whch he severl nformonl equlbrum srucures wll be dscussed. 3. Open-loop Nsh Equlbrum (OLNE OLNE) Le us ssume h here re n frms n n olgopoly mrke h produce nd sell homogenous produc. The mrke demnd funcon s non lner one, s n Anderson nd Engers (1992, 1994) [1] whch s used n he sc Sckelberg herrchcl model n order o compre he resuls wh he Courno correspondng sc model nd s expressed s: ( ) ( ) ( ) ( ) q = r p 0 > ( 1 ) [1] To be precse, he mrke demnd used by Anderson nd Engers s n he form Q= 1 p, > 0, where prmeer snds s he curvure of he mrke demnd, nd no reference s mde bou consumers reservon prces.

10 9 wh q( ) denong he ggrege demnd n q nd = 1 q he ndvdul frm s demnd. The posve prmeer shows he curvure of he demnd funcon, so s convex f ( 0,1), s concve f > 1 nd s lner f = 1. The r prmeer lso mesures he reservon prce (2), r j, of ech j consumer, ggregng cross ll consumers. The r j vrble s mplcly n ndex o he consumer surplus nd, consequenly, o socl welfre; h s, he hgher he reservon prce, he hgher he consumer surplus. Wh he ls observon, he reservon prce r clerly ffecs ggrege demnd n he sense h ggrege demnd s he number of he consumers whose reservon prce s les r for one un of he homogenous produc. The nverse demnd funcon of he ndusry, hen, s esly obned s ( ) 1 ( ) = ( ) ( ) ( 2 ) p r q nd he prce p s lwys posve,.e. p( ) > 0 The prce elscy of demnd, n bsolue erms, equls o ε q, p q = = = ( r ) p p q r p q p q nd he consumer surplus (CS) s evlued he reservon prce q mesures he socl welfre, h s, r ( + 1) 1 r CS = ( r q) dr= = r, whch lso wh he bove relons verfyng h elscy, consumer surplus nd reservon prce depend on he curvure of demnd. (2) For more dels on consumers reservon prce, see Vrn (1992).

11 10 Frms n he ndusry seek o ncrese he ggrege reservon prce r( ) nd consequenly he ffeced mrke fcors, hrough nvesmen n polluon bemen (e.g. recyclng). Every bemen underken me by frm s formlly denoed by A ( ). Ths mples h frm s ble o ncrese he mrke sze s resul of he frm s overll bemen. The reservon prce s lso subjec o consn deprecon reδ. sysem Then he followng dfferenl equon could descrbe he dynmcs of he dr d where he dfferenl dr d r( ) n rɺ A δr ( 3 ) = = = 1 =ɺ represens he evoluon of he reservon prce (ggregng cross consumers) n he ndusry nd wh n A we denoe he sum of = 1 he polluon bemen underken by ll nvolved frms. Every frm engged n bemen fces qudrc cos of he form ( ) = ( ) 2, > 0 ( 4 ) c A b A b In he dynmc problem, hen, frms seek o mxmze he presen vlue of he fuure profs,.e. ol revenues mnus ol coss ncurred from he nvesmen n bemen. So, frm s profs re: π ( r q q ) q b( A ) = 1 2 nd frm s problem cn be expressed s he mxmzon of he presen vlue funcon: mxj = e r q q q b A d ρ 1 ( ) 2 ( 5 ) 0

12 11 subjec o dr d δ = rɺ = A + A r where we hve se frm (plyer) nd A ( ) q s he quny produced by ll oher frms (plyers) excep he bemen of ll oher plyers excep. In hs seng, he decson vrbles re he qunes q nd he nvesmen decsons A, whle he se vrble s he reservon prce, r( ). If we decde o work wh he curren vlue Hmlonn funcon denong by µ he co se vrble, hen plyer s Hmlonn s he followng funcon: 1 2 H= ( r q q ) q b( A ) λ( A A δr ) e ρ + + ( 6 ) where λ = µ e ρ s he curren vlue of he co se vrble. Le us nex consder he Open-loopNsh Equlbrum, h s, he Nsh equlbrum n whch every frm drws he me phs of he conrol vrbles ndependenly of he ses of he gme. The only knowledge of he se s s vlue he nl me= 0, h s, he nformonl srucure of hs ype of equlbrum s he sngleon of he nl vlue of he se. The Closed-loop Nsh Equlbrum s by defnon he concep of equlbrum n whch he choce of plyer s curren con s condoned on curren me nd on he se vecor, oo. Imposng hs ssumpon on he nformonl srucure of he gme, he hsory of he gme clerly becomes mporn nd s refleced n he curren vlue of he se vecor. Consequenly, plyer ' s opml me phs ke no ccoun he conrol vrbles of he oher plyers ny pon of me. Ths ype of equlbrum ffecs he se vrbles, requrng revson of he plyer s conrols ny me nsn. Here

13 12 we pply he so clled memory less (3) closed-loop equlbrum, ccordng o whch every plyer needs o know only he curren vlue nsed of he whole hsory of he se vrble. The bsc dfference beween he wo ype of equlbrums s h he frs one, he open loop, s wekly me conssen, whle he closed-loop s srongly me conssen one. Here he me conssen propery s n he sense of sub gme perfecness (4). 4. Open-loop Nsh Equlbrum (OLNE) The OLNE s obned by he followng sysem of FOC equons H r q H A = 0 = 0 H dλ = ρλ d ( 7 ) nd he rnsversly condon r λ ( 7 ) cn be expressed s H q lm e ρ = 0. The frs equon of sysem ( ) ( 1 ) 1 r q q q ( r q q ) = + = 0 Mkng use of he symmery of frms, by seng q ( ) = Q( ) nd ( 1) bove frs order condon (FOC) s smplfed o H q ( ( ) ) ( 1 ) 1 r Q n 1 Q nq 1 q n Q =, he ( r Q ( n ) Q ) = + 1 = 0 1 (3) The memoryless perfec se nformon pern s defned by Olsder nd Bsr (1999) s he nformonl se whch hs wo elemens; one s he nl vlue of he se nd he oher elemen s he curren vlue of he se vrble. (4) See Dockner e l. (2000) for more dels bou he sub gme perfecness nd he me conssency equlbrum sreges.

14 13 nd he soluon wh respec o Q( ) s Q r = ( 8 ) + The second equon of ( 7 ) fer he ppropre lgebrc mnpulons gves H A or λ = 2bA 1 n λ = 2bA + = 0 Dfferenon of he ls wh respec o me yelds da The hrd equon of sysem ( 7 ) gves 1 dλ = ( 9 ) d 2b d ( ) ( ) δλ q 1 1 dλ r q q + = ρλ d or dλ d q ( ) ( 1 ) 1 = r q q + ( δ+ ρλ ) ( 10 ) Combnng equons (9)-(10) we hve da d q ( ) ( 1 ) 1 = r q q + ( δ+ ρλ ) ( 11 ) Hvng he sysem of he wo dfferenl equons descrbed by dr d da d δ = rɺ = A + A r q ( ) ( 1 ) 1 = r q q + ( δ+ ρλ ) nd mkng use of he symmery of frms h s: q ( ) = Q( ), A( ) A( ) relon ( 8 ), he sysem fnlly urns ou o be = nd he

15 14 dr d δr = na 1 da 1 r = ( δ+ ρ) A d 2b 1 n + ( 12) Sysem ( ) 12 n he sedy ses s defned s d r 0 d = nd da = 0. The sedy se d soluon of sysem ( 12 ) s hen gven by: * δr A = n r * ( 1) n 1 = 2bδδ ( + ρ) ( ) ( 1 n ) ( 13 ) nd he producon level he sedy se s gven by expresson ( 8 ). Now we re ble o nlyze he sbly properes of equlbrum. 4.1 Sbly properes of equlbrum In order o deermne he sbly of he dynmc sysem ( 12 ) we lnerze hs round he sedy ses,.e. where we hve se nd dr d r =Φ da A d f f δ n r A ( 1 ) Φ= = r g g δ+ ρ 1 2b( n+ 1) r A

16 15 dr d (, ) δ = f A r = na r 1 da 1 r = g( A, r) = ( δ+ ρ) A d 2b 1 n + The rce nd he deermnn of he Jcobn mrx Φ, evlued he sedy se level, hve he followng vlues f g r( Φ ) = + = δ+ δ+ ρ= ρ r A In order o compue he deermnn of he Jcobn mrx Φ we mke use of he vlue of * r prevously found from sysem( 12 ). So he deermnn s: de ( ) δδ ( ρ) ( 1 ) f g g f r Φ = = + + n r A r A 2b n+ 1 ( ) Subsung bck he vlue of r no he ler nd mkng he res of he lgebrc mnpulons, he fnl resul of he deermnn s: de δ ( ) ( ) ( 1 Φ = ρ+ δ ) For he sddle pon ph exsence, suffces he deermnn of Jcobn mrx o be less hn zero, h s, he followng nequly o hold: δ ( ) ( ρ δ ) + 1 < 0 Ths s clerly verfed when he mrke demnd s concve, h s, when > 1 s lredy menoned. From he prevous resonng he followng proposon s obvous. 1

17 16 Proposon 1. The sedy se equlbrum of he gme descrbed by he soluons of he conrol vrbles A * δr = nd n Q * * r = nd he se vrble + ( 1 n) r * ( 1) n 1 = 2bδδ ( + ρ) ( ) ( 1 n ) s sddle pon ph f nd only f he mrke demnd s concve. Remrk 1. In he oher wo cses where he mrke demnd s convex or lner he equlbrum s unsble nd degenered respecvely. In he frs cse of convex demnd, boh he rce nd he deermnn of he Jcobn mrx re posve, so equlbrum s unsble. Smlrly, n he second cse where he vlue of he se vrble s n undeermned mgnude or he deermnn of he Jcobn mrx s zero, we hve degenered equlbrum. In oher words, he soluon of he dynmcl sysem ( 12 ) s collecon of prllel lnes. The phse dgrms for he hree curvures of he demnd funcon re shown n he followng pnels of he fgure 1 sedy se. A A da d= 0 dr d= 0 da d= 0 dr d= 0 r r () (b)

18 17 A dr d= 0 da d= 0 r (c) Fgure 1: Phse dgrms for he hree curvures of he demnd funcon. In pnel (c) he doed lne denfes he sddle ph. 5. The Collusve problem In hs secon we sudy equlbrum under he hypohess h frms collude n order o mxmze he presen vlue of he jon profs. Moreover, we ssume symmery of frms. Then he dynmc problem s defned s follows. ρ 1 2 mx J= e n( r nq ) q nb( A ) d 0 dr = na d δr ( 24) Ths s sndrd opml conrol problem nd cn be solved usng conrol heorec mehods. Workng wh he curren vlue Hmlonn, hs funcon s expressed s 1 2 ( ) λ δ c H = r nq nq nb A + na r e ρ ( 25 )

19 18 Where c H represens he collusve Hmlonn funcon. The frs order condons of ( 25 ) ccordng o Ponrygn mxmum prncple gve he followng condons c c c H n q 1 = 0 r nq + n r nq = 0 q H A ba ( ) = 0 λ = 2 26 H dλ = ρλ r d nd he rnsversly condon r λ lm e ρ = 0 Solvng he frs condon of sysem ( 26 ) wh respec o q( ), he soluon s q = n r ( + 1) ( 27 ) whch, compred wh he open-loop soluon ( 8 ), s smller. From he wo remnng condons of sysem ( 26 ) we obn he evoluon equon of he reservon prce s da 1 1 = + d 2b 1+ 1 ( ρ δ) A ( r ) 1 ( 28 ) The dynmc sysem of he collusve frms s chrcerzed, n he sedy ses, by he followng equons dr d da d where he soluon of he frs s = 0 = 0 δr A= n nd he soluon of he second s

20 19 ( 1 ) r A= 2b 1 ( )( ) ( 1 ρ+ δ + ) The opml nvesmen n bemen effor corresponds o curve, whch my be ( ) ( ) convex ( 0,1), concve ( 1, + ) or lner ( 1) = dependng gn on he curvure prmeer. As consequence, he collusve soluon exss he sedy ses nd s dynmc properes re he sme s n he Courno cses,.e. hs sedy se s sddle pon n he cse of ( 1, + ),.e. he demnd funcon s concve. From he prevous resonng he followng resul s esly obned. Proposon 2. In he collusve cse of frms, he sedy ses equlbrum of he orgnl model s gn sddle pon ph f ( 1, + ) nd,n hs cse, he sedy ses levels of boh r nd A re lrger hn he sedy ses levels of open nd closed-loop equlbrum. A hs pon we cn compre he collusve soluons wh he decenrlzed ones. In he sedy se equlbrum, he reservon prce n boh open nd closedloop s lower hn n he collusve sedy se equlbrum. Ths resul s subjec o he free rdng problem. The economc nerpreon of hs resul s srghforwrd: Snce frms enjoy uly from he hgher mrke reservon prce r( ), nd -s lredy menoned- ffecs he bsc mrke fcors.e. mrke sze, hey hve no ncenve o nves n bemen bu o benef from he nvesmen of he oher frms The socl opmum We dop smlr procedure o fnd he presen vlue of he socl welfre flows. The socl welfre me s defned s he sum of he consumer surplus plus

21 20 he frm s ne profs. Tkng no ccoun he symmery hypohess, he consumer surplus nd ol profs me re respecvely: Q 1 ( ) ( + 1) ( ) ( + 1) CS = r s ds = r r Q ( 29 ) 0 1 = 1 ( ) 2 ( 30 ) nπ nq r Q nb A Supposng h he socl plnner uses he sme dscoun fcor ρ for he socl welfre (SW) flows, he dynmc problem s hen expressed s: ( ) ( 1) ( ) ( 1) 1 mx e ρ + + SW = r r Q nq r Q + nb( A ) 2 d 1 0 subjec o dr = na δr ( 31 ) d The soluon of problem ( 26 ) he sedy ses, ( ) da d = 0, s 1 r A= 2bρ δ ( + ) ( 32 ) Ths s he curve h lys bove ll s counerprs n he prevous regmes for concve curvure of he demnd funcon. Then he followng resul s obvous: Proposon 3 The soclly opml nvesmen n polluon bemen, n he sedy ses, s hgher hn ny oher lernve nformonl srucure, bu only for concve demnd funcon,.e. for ll > 1. Remrk 2. In he cse of convex demnd funcon ( 0,1), he sedy ses vlues of he reservon prce, r, nd bemen, A, re boh smller hn he respecve vrbles of he Courno gme nd he sedy se of lner demnd, = 1, does no exs.

22 21 6. Concludng remrks In hs pper we se up n olgopolsc model where frms re engged n bemen n order o ncrese he whole mrke sze v n ncrese n he consumers reservon prce. We ddonlly ssumed h he demnd funcon s no lner one nd he resulng gme s no usul lner qudrc one. In he consdered model we nvesged he open-loop, he memory less closed-loop nd he collusve perns equlbrum, whle n he feedbck cse only he lner demnd s nvesged mkng use of he dynmc progrmmng Hmlon Jcob Bellmn equon. Moreover, we exmned he socl plnnng perspecve. In he model formulon nd n he cse of convex demnd we found s resul h he conrol nd se vrbles hve hgher vlues n he open-loop sedy se equlbrum compred o he memory less closed-loop sedy ses, whle n lner demnd cse he equlbrum s undeermned. In ll cses, we found h only f he mrke demnd hs concve curvure, he conclusons re cler. Specfclly, we found h he socl plnnng cse hs ll he vlues of he conrol vrbles hgher hn n ll he oher cses. In he model presened we consdered he reservon prce s knd of publc good. Ths mens h every frm nsde he mrke hs ccess o hs prce nd benefs from s hgher vlue. As s shown, comprng he collusve sedy se equlbrum vlues of he reservon prce r( ) o he open nd closed-loop vlues, he oucome s h n ndvdul frm hs no ncenve o nves n npolluon effor, bu benefs from he nvesmen of he oher frms. Th s, n mplcon mong he plyers of he gme s he well known free-rdng problem.

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