ANALYSIS OF PRIVATIZATION IN STACKELBERG MIXED OLIGOPOLY

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1 Koj Okuguch Tokyo Metropolta Uversty Departmet of Ecoomcs, Japa ANALYSIS OF PRIVATIZATION IN STACKELBERG MIED OLIGOPOLY JEL classfcato: L3, H4 Abstract Mxed olgopoly wth oe welfare-maxmzg publc ad several proftmaxmzg prvate frms exsts may ecoomes. De Fraja ad Delboo 989) have aalysed mxed olgopoly takg to accout how the publc frm behaves vs-à-vs the prvate frms o the bass of a lear market demad fucto ad symmetrc frms. They have foud that the socal welfare s greater Stackelberg mxed olgopoly where the publc frm acts as a leader tha Courot mxed olgopoly where all frms smultaeously determe ther outputs. A partal publc frm tres to maxmze the weghted average of the socal welfare ad ts profts. Uder some codtos, partal prvatzato of a publc frm leads to greater socal welfare tha Courot mxed olgopoly where the publc frm s fully publc see Matsumura 998) for duopoly ad Okuguch ) for olgopoly). I ths paper we wll prove that ether partal or full prvatzato of a publc frm s optmal a geeral Stackelberg mxed olgopoly where the publc frm acts as a leader ad all prvate frms as followers. Key words: publc frm, Stackelberg mxed olgopoly, prvatzato. INTRODUCTION The exstece of mxed olgopoly where a publc frm ad prvate oes coexst have bee observed ad aalyzed frst by Merrll ad Scheder 966), ad later by Harrs ad Wes 98), Beato ad Mas-Colell 98), Boes 986,99),ad Creamer et al 987) amog others. De Fraja ad Delboo 989) see also De Fraja ad Delboo,99) have compared the welfare of mxed olgopoly cosstg of oe fully publc frm ad several symmetrc prvate

2 MICROECONOMICS 79 frms for four possble cases dstgushed by the publc frm s behavor relatoshp to all prvate frms. They have assumed a lear market demad fucto for a detcal good produced by all frms ad the same quadratc cost fucto for all frms,ad foud, amog other thgs, that the socal welfare s greater Stackelberg mxed olgopoly where the publc frm acts as a leader ad all prvate frms behave as followers tha Courot or Nash) mxed olgopoly where the publc ad prvate frms smultaeously choose ther outputs. Some more recet cotrbutos to Stackelberg mxed olgopoly, especally relatoshp wth the effects of subsdes to frms, clude Poyago- Theotoky), Myles ), Cores ad Sepahvad 3), Fjell ad Heywood 4,7) ad Zkos 7).Myles ) adopts most geeral approach amog them ad assumes away a lear market demad fucto ad quadratc cost fucos.however, he assumes detcal cost fuctos for all frms,cludg the publc oe. A partal publc frm whose maager maxmzes the weghted average of the socal welfare ad ts profts have wdely bee observed may ecoomes. Uder certa codtos, partal prvatzato of a publc frm results greater socal welfare tha Courot mxed olgopoly where the publc frm s uder full cotrol of the govermet. Ths has bee show by Matsumura998) for duopoly ad by Okuguch) for olgopoly. I ths paper we wll systematcally aalyze uder very geeral codtos o the market demad ad frms cost fuctos whether partal or full prvatzato of a publc frm Stackelberg mxed olgopoly wth a publc frm as a leader soptmal or ot. We wll fd that ether partal or full prvatzato of the publc frm s optmal.ths o-optmalty result cocdes wth the oe earler obtaed by De Fraja ad Delboo989) for ther smple case of a lear market demad fucto ad the detcal cost fucto for all frms.we wll be able to derve our result remarkably easly by takg the publc frm s rest of the dustry output as a aalytcal strategc varable, as t s uquely related to ts ow output as show below. Before cocludg ths troductory secto, we would lke to pot out the poeerg paper o Stackelberg olgopoly wth oly prvate frms by Sheral et al.983) from the algorthmc pot of vew of computg the Stackelberg equlbrum.. MODEL AND ANALYSİS Let there be oe publc frmfrm ) ad asymmetrc proft-maxmzg prvate frms. Let π,,,...,,be frm s profts, W be the socal welfare as the sum of frm s profts ad the cosumers surplus, ad U αw + α) π be the partal publc frm s objectve fucto, where the parameter α [,] s the weght the govermet attaches to the socal welfare.

3 MICROECONOMICS 73 α,the publc frm becomes a prvate proft-maxmzg frm ad f, If t s a fully publc frm.furthermore,let where x be the dustry total output, x s frm s output, p p ) the verse market demad fucto, where p s the prce of a homogeeous good of the dustry ad p < for such that p >, ad C x ) be frm s cost fucto. The, by defto of the socal welfare W, W p x) dx Σ C x ). ) The frm s proft fucto s π ) x p x j ) C x ), j,,,...,. We ow formulate Stackelberg mxed olgopoly wth a partal publc frm as a leader ad all prvate frms as followers. The publc frm s objectve fucto s U defed above ad equals to the weghted average of the socal welfare ad ts profts. We rewrte the frm s proft fucto as π x p x + ) C x ),,,...,, 3) where a aalytcal strategc varable x s the rest of the dustry output for the publc frm. De Fraja ad Delboo 989) have assumed a lear market demad fucto for the good ad a detcal quadratc cost fucto for all frms, whle Beato ad Mas-Colell 98) have used a lear market demad fucto ad geeral cost fuctos for mxed duopolsts. We wll, however, assume geeral demad ad cost fuctos whch are assumed to satsfy: " Assumpto : C >,,,...,, p < C,,,...,. Assumpto : p + x p" <,,,,...,. Now let the leader s output x be gve.the prvate frm maxmzes ts proft wth respect to ts ow output o the bass of the Courot behavorstc assumpto regardg ts rval s outputs. Hece, ts frst order codto for proft maxmzato.

4 MICROECONOMICS 73 π x p x + ) + x p x + ) C x ),,,...,, 4) where we have assumed a teror maxmum.the Assumpto mples that ay two prvate frm s outputs are strategc substtutes each other. Note that the secod order codto holds uder Assumptos ad. We ote passg that the equato 4) shows that the prvate frms are playg a aggregatve game amog themselvessee Okuguch ad Yamazak4)). Solvg 4) wth respect to x as a fucto of x +, we have x ϕ x ), +,,...,, 5) where dϕ ϕ ϕ p + x p" <,,,...,. 6) " d x p C word sce Note that 5) s ot the reacto fucto the tradtoal sese of the ϕ cotas x as oe of ts argumets because of By defto,the rest of the dustry output for the publc frm s + x. ϕ x + ) ϕ x ). 7) Solvg 7) wth respect to the leader output, we have x Ψ ), 8) where vew of 6), Ψ dx ) <. d dϕ d We ca gve a dagrammatc dervato of 8) as follows.if the publc frm s output s,the soluto of 7) correspods to the tersecto E of a dowward-slopg curve for ϕ x + ) ad the 45 degree le orgatg from the org as show the Fgure below. 9)

5 MICROECONOMICS 73 FgureThe soluto of 7) If the publc frm s output creases to,the curve shft " dowwards,ad the ew tersecto becomes E,hece < for <. The publc frm s objectve fucto ow reads x + U α p x) dx C x ) + α) + Ψ ) α p x) dx C Ψ )) C ϕ + Ψ ))) + ) Ψ ) p + Ψ )) C Ψ x )) { } { x p x + ) C x )} α, )

6 MICROECONOMICS 733 where we have take to accout 5) ad 8).Gve α,the maager of the publc frm maxmzes ts objectve fucto U wth respect to ts outpu x,that s,wth respect to ts rest of the dustry output lght of 8).The frst order codto for maxmzato of U wth respect to s rewrtte as V where, α ) αa ) + α) B ) ) A ) p β )) C β )) ϕ β )) + Ψ )) C Ψ )) Ψ ), B ) p β )) Ψ ) + Ψ ) p ) + Ψ C Ψ )) Ψ ), )) β ) + Ψ ). We troduce here the followg secod order codto. U Assumpto 3: <. I order to show the valdty of ths assumpto,cosder the followg case whch the market demad fucto s lear, the publc frm s cost fucto s quadratc ad all prvate frm s cost fuctos are lear ad detcal. c x P a b, C, C x ) cx,,,...,. ) A smple calculato yelds { b + c + ) } + + ) { c a c) bc}, b A ) b < { b + c + ) } + C, b + ) B ) 4) b < { b + c + ) }, ) < 3) A 5)

7 MICROECONOMICS ) ) { b + c + ) } < B, 6) { ab + c a c) + ) bc} C ). 7) + I vew of Iequalt s 5) ad 6),we kow tha the secod order codto s satsfed for the model gve by ). Uder the Assumpto 3,we solve ) wth respect to as a fucto of the parameter α. α), 8) where we have vew of the Assumpto 3 d B dα U α < > > accordg as B. 9) Furthermore, we have lght of 9) ad 8), d d > + Ψ) dα dα < α) < > accordg as B. ) Sce the govermet s objectve fucto s α ) W, ) < p x) dx C Ψ )) C ϕ α))) dfferetato of t wth respect to α yelds dw d p Cϕ ) + Ψ) CΨ dα dα α ) B d > α dα for B. ) Ths proves that f that s, the publc frm should be ether partally or fully prvatzed. B, the socal welfare s maxmzed for α,

8 MICROECONOMICS CONCLUSION I ths paper we have aalyzed whether partal or full prvatzato of a publc frm coexstg wth several proft-maxmzg prvate frms s optmal the sese of socal welfare maxmzato.we have gve the role of leadershp to the publc frm whch s assumed to be maxmzg ts objectve fucto as the weghted sum of the socal welfaread ts profts, ad the followershp role to all prvate frms. We have foud wthout assumg a lear market demad fucto ad quadratc cost fuctos for all frms that ether paral or full prvatzato of the publc frm s optmal. Ths fdg s sharp cotrast wth that of the optmalty of partal prvatzato of the publc frm Courot mxed olgopoly where all frms are assumed to act as Courot olgopolsts. REFERENCES Beato, P. Ad Mas-Colell,A.984).The Margal Cost Prcg as a Regulato Mechasm Mxed Markets.I Marchad,M.et al.eds.).the Performace of Publc Eterprses.Amsterdam:North-Hollad. Boes, D.986).Publc Eterprse Ecoomcs.Amsterdam:North-Hollad. Boes, D.99). Prvatzato;A Theoretcal Aalyss.Oxford;Claredo Press. Coress, R.C.,Sepahvad, M.3)..Courot vs Stackelberg Equlbra wth a Publc Eterprse ad Iteratoal Competto.Uversty of Notttgham Ecoomcs Dscusso Paper.No 3/. Cremer,H.,Marchad,M.,Thsse,J.F. 989).The Publc Frm as a Istrumet for Regulatg a Olgopolstc Market. Oxford Ecoomc Papers,4,pp De Fraja,G.,Delboo,F.989).Alteratve Strateges of a Publc Eterprse Olgopoly.Oxford Ecoomc Papers,4,pp.3-3. De Fraja,G.,Delboo,F.99). Joural of Ecoomc Surveys,4,pp.-7. Fjell,K.,Heywood,J.S.4).Mxed Olgopoy,Subsdzato ad the Order of Frm s Motves:The Relevace of Prvatzato. Ecoomcs Letters,83,pp Fjell,K.,Heywood,J.S.7).Publc Stackelberg Leadershp a Mxed Duopoly wth Foreg Frms.Australa Ecoomc Papers,4,pp Harrs,R.G.,Wes,E.G.98).Govermet Eterprse:A Istrumet for the Iteral Regulator of Idustry.Caada Joural of Ecoomcs, 3,pp.5-3. Matsumura,T.998).Partal Prvatzato Mxed Duopoly,Joural of Publc Ecoomcs, 7,pp Merrll,W.C.,Scheder,N.966).Govermet Frms Olgopoly Idustry:A Short Ru Aalyss.Quarterly Joural of Ecoomcs, 8,pp.4-4. Myles,G.).Mxed Olgopoly,Subsdzato ad the Order of Frm s Moves:A Irrelevace Result for the Geeral Case.Ecoomcs Bullet,,pp.-6.

9 MICROECONOMICS 736 Okuguch,K.).Geeral Aalyss of Courot Mxed Olgopoly wth Partal Prvatzato.Eurasa Ecoomc Revew,,pp Okuguch,K.,Yamazak,T.4).Global Stablty of Nash Equlbrum Aggregatve Games.Iteratoal Game Theory Revew, pages). Poyago-Theotoky,J.).Mxed Olgopoly,Subsdzato ad the Order of Frms Moves:A Irrelevace Result.Ecoomcs Bullet,pp.-5. Sheral,H.D.,Soyster,A.L.,Murphy,H.983).Stackelberg-Nash-Courot Equlbra:Characterzato ad Computatos.Operatos Research,3,pp Zkos,V.7).Stackelberg Mxed Olgopoly wth Asymmetrc Subsdes.Ecoomcs Bullet,,pp.-5.