Oligopoly Equilibria à la Stackelberg. in Pure Exchange Economies

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1 Oligopoly Equilibria à la Stacklbrg in Pur Excang Economis L.A. JLIEN & F. TRICO EconomiX, nivrsité Paris X-Nantrr Burau K6 00, avnu d la Républiqu 900 Nantrr Cédx Incomplt and prliminary vrsion Abstract. Tis papr introducs two quilibrium concpts wic xtnd t notion of Stacklbrg comptition to covr a gnral quilibrium framwork. In t framwork of a pur xcang conomy, asymptotic idntification and wlfar rsults ar obtaind.. Introduction T Stacklbrg concpt of quilibrium as mainly bn dvlopd in production conomis undr partial quilibrium analysis Tirol 988)). T purpos of tis papr is to insrt t Stacklbrg markt structur in a pur xcang gnral quilibrium framwork, wic nabls to captur t faturs of markt powr and t divrsity of stratgic intractions. W us t framwork of oligopolistic comptition dvlopd by Codognato-Gabszwicz 99), 99)), Gabszwicz-Micl 997) and tn pursud by Gabszwicz 00). According to t comptition à la Walras, all t individuals bav t sam non stratgic way and all t sctors work t sam prfct way. Tis doubl symmtry dos not stand wit t comptition à la Cournot : an asymmtric tratmnt of t sctors is introducd, som bing oligopolistic and otrs staying comptitiv. But t symmtry rmains in t tratmnt of vry individual on a givn sctor. Wn t comptition à la Stacklbrg is introducd, a doubl asymmtry is possibl: btwn t oligopolisitic and t comptitiv sctors, and morovr btwn t ladr and t followrs) in on sam sctor. It is tn possibl to associat a rlativ advantag for on sctor upon anotr unvn distribution of markt powr among t sctors) and a rlativ advantag for an agnt upon anotr unvn distribution of markt powr among t agnts of a givn sctor). Two concpts of Stacklbrg gnral quilibrium ar dvlopd: t Stacklbrg- Walras Equilibrium and t Stacklbrg-Cournot Equilibrium. W compar ts quilibria and obtain svral rsults about asymptotic idntifications and wlfar.

2 . A two-commodity conomy Considr a pur xcang conomy wit two consumption goods and ) and n+ consumrs. It is assumd tat good is takn as t numérair, so p is t pric of good as xprssd in units of good. T prfrncs of vry consumr ar rprsntd by t sam utility function: = xx,. ) T structur of initial ndowmnts in sctor and in sctor is assumd to b, rspctivly: ω = α,0) and ω = α,0), wit α 0,) ) ω = 0,, =,.., n +. ) n In t first sctor, ac agnt is an oligopolist: agnt is t ladr and agnt is t followr. T pur stratgis of agnts and ar dnotd s, wit s [ 0,α ], and, wit [ 0, α ]. In t scond sctor, agnts ar itr pric-takrs or Cournotian oligopolists. T rsulting quilibria ar t Stacklbrg Walras quilibrium SWE) in t formr cas and t Stacklbrg Cournot quilibrium SCE) in t lattr cas. W study ts two concpts of Stacklbrg quilibria for pur xcang conomis and compar tm wit t Cournot quilibrium CE) and t Cournot-Walras quilibrium CWE). In t SWE framwork, it is considrd tat agnts aving ndowmnts in good act comptitivly, wras t otr agnts bav stratgically. T story is solvd by backward induction, considring first t bavior of t Walrasian agnts, tn t dcision of t followr, and finally t coic of t ladr. T comptitiv plans of ownrs of good com from a non-stratgic maximization of t utility function subjct to t budgt constraint, i.. Arg max z z, =,..., n +, wr z rprsnts t comptitiv { z } p n supply of good by agnt, =,..., n +. From ) and ), w dduc t comptitiv individual offr plan z = / ) and t dmand functions x =, x ),, =,..., n +. np n n n+ n+ T aggrgat dmand in good by ownrs of good is = [ ] = / ) np T quilibrium pric is tn givn by = n [ /np ] = s, so [ ] = p / s + ) + ) + x. =. T stratgic plan of t followr is dtrmind by two lmnts: s manipulats t markt pric and s taks t ladr s stratgy as givn. Tus t followr s program is: ) { } Arg max α ), ) s + wic givs t following raction function: s) = s + α ) s s 5) W vrify tat tis function is continuous and incrasing, wit s 0 and / > / s < 0. Morovr, / α < 0. T stratgic plan of t ladr is dtrmind by two lmnts: s manipulats t markt pric and t followr s stratgy. T ladr tus solvs t following program: s ) { } Arg max α s, 6) s s + s) wic givs t following optimal stratgy:

3 s = [ ϕ α) ], 7) wr ϕ α)9 α), wit ϕ 0,). W vrify tat tis function is continuous and incrasing, wit s / α > 0. W can dduc t valu of t followr s stratgy = ) : s = { ψ [ ϕ α) ]}, 8) ψ ϕ α) ϕ + α). Sinc z = /n), it is now possibl to dtrmin t quilibrium pric: p =, ψ 9) T individual allocations ar tus: ϕ α) x, x ) = + α ϕ), ψ 0) ψ ) [ ϕ α) ] x, x ) = ϕ ψ + α), ψ ) x ψ, x ) =, n n, =,.., n + ) T associatd utility lvls ar rspctivly: + α ϕ) ϕ α) ) = 8ψ ) wr ) ) [ ϕ α) ] ψ = ϕ ψ + α) ) ) ψ ψ =, =,.., n + 5) 8n In t SCE framwork, it is considrd tat all agnts bav stratgically, wit agnt as t only ladr. T only diffrnc wit t prvious cas is tat t agnts ndowd in good bav oligopolistically. T story is solvd by backward induction, considring first t dcisions of t n+) Cournotian agnts, and finally t coic of t ladr. W dnot t pur stratgy of agnt, =,..., n +. T n markt pric is givn by p =, wic insurs t markt claring. s + Taking t n ), s, and as givn, ac stratgist of sctor maximizs r utility: s ) + Arg max { } + n ) n, =,..., n +, 6) wic givs t following raction function: n =, =,..., n + nn ) 7) Taking t stratgy s and t n stratgis as givn, t Cournotian followr of sctor maximizs r utility: Arg n + max ) { } s + ), wit [ 0, / n] α, 8)

4 wic givs t following raction function: s) s + ) = α s s. 9) Considring t bst rsponss of all t followrs, t ladr maximizs r utility: Arg n + max s { s } s + s ) s α, 0) ) wic givs t optimal stratgy: s ˆ = [ ϕ α) ] ) T valus of t Cournotian stratgis follow: ˆ = { ψ [ ϕ α) ]} ) n ˆ =, =,..., n + nn ) ) n + ˆ n ) T quilibrium pric pˆ = = can b writtn: sˆ ˆ + n ) ψ n ) pˆ = p n ) T individual allocations ar tus: = n ) x ˆ, ) x, x n 5) = n ) x ˆ, ) x, x n 6) = n n x ˆ ˆ, x ) x, x, =,..., n + n n 7) T utility lvls racd ar rspctivly: ˆ n ) = n 8) ˆ n ) = n 9) n ˆ n )n ) =,..., n + 0) Proposition. Wn t numbr of agnts tnds to infinity, t Stacklbrg-Cournot quilibrium idntifis to t Stacklbrg-Walras quilibrium. Proof. W av to sow tat t quilibrium pric and optimal allocations in sctor convrg toward t Stacklbrg-Walras on wn n bcoms larg. For t quilibrium n ) pric, w av lim p = p. For t individual allocations, as n n = n ), ) x, x for t ladr, = n ) x ˆ, ) x, x for t scond n n agnt and = n n x ˆ ˆ, x ) x, x, =,..., n +, it is obvious tat n n

5 lim, ) = x, x ) for t ladr, lim, ) = x, x ) n n tat lim x ˆ, ) = x, x ) for =,..., n + n. Tis complts t proof. for t scond agnt and Proposition undrlins tat t markt powr of ac oligopolist dcrass wn t numbr of agnts incrass unbounddly. Hnc, wn n gos to infinity, t Cournotian bavior tnds to t Walrasian on. W can also notic tat ts optimal stratgis corrspond to t comptitiv plans. Considr now tat p is takn as givn by ac agnt, =,..., n +. W av to sow tat tis pric is associatd wit t comptitiv plans for t rmaining agnts. T optimal plans com from a non-stratgic maximization of t utility subjct to t budgt constraint, i.. Arg max x x subjct to px + x n for =,..., n +. Tis lads to { z } x x = np,, ), =,..., n +. n / Proposition. Tr is no Parto domination btwn t Stacklbrg-Walras and t Stacklbrg-Cournot quilibria. Proof. From 8), 9) and 0) w av ˆ n ) =, ˆ n = nn ) and n ˆ n ) n =, =,..., n +. As <, < and >, n n nn ) n )n ) w av ˆ >, ˆ > and < ˆ, =,..., n +. Tis complts t proof. Proposition capturs tat stratgic agnts of t first sctor do bttr wn ty fac comptitiv agnts tan wn ty struggl wit stratgic agnts. And tos agnts of t scond sctor compt bttr undr a Cournotian bavior tan undr a Walrasian on. Conclusion Laving t Walrasian quilibrium mans introducing in som way wat is xcludd from prfct comptition: t stratgic intractions. Wras t CWE and t CE introduc only on kind of stratgic bavior, t SWE and t SCE involv two typs of tis gam-tortic bavior: t activ ladr s on and t ractiv followr s on. T SWE is spcially intrsting, as it displays tr kinds of dcision making mod: t comptitiv on, t monopolistic on and t stratgic/paramtric on. Rfrncs Amir R., Grilo I. 999), Stacklbrg vrsus Cournot quilibrium, Gams and Economic Bavior, 6, pp. -. Codognato, J., Gabszwicz, J.J. 99), Equilibrs d Cournot-Walras dans un économi d écang pur, Rvu Economiqu,, pp Cournot A.A. 88), Ls rcrcs sur ls princips matématiqus d la téori ds ricsss, Dunod 00). Fridman, J. 99), Oligopoly tory, in K.J. Arrow and M.D. Intriligator Eds), Handbook of Matmatical Economics, Elsvir, captr, Volum, pp

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