Patrice Cassagnard Université Montesquieu Bordeaux IV LAREefi. Abstract

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1 A useful grpicl metod under Cournot competition Ptrice Cssgnrd Université Montesquieu Bordeux IV LAEefi Astrct Tis note proposes grpicl pproc useful in gme teor. Tis metod consists in representing incentives to move strtegicll to grpicl res. Te metod cn e used on severl occsions; we ppl it s n exmple to te model of Bouët (2001). Cittion: Cssgnrd, Ptrice, (2003) "A useful grpicl metod under Cournot competition." Economics Bulletin, Vol. 3, No. 13 pp. 1 5 umitted: June 17, Accepted: Jul 1, UL: ttp:// 03C70013A.pdf

2 1 Te metod We consider Cournot duopol. Two gents A nd B compete in terms of quntit. Te commodities re omogeneous nd te inverse demnd function, in liner model, cn e written s p( + ) = θ k ( + ) θ > 0; k>0 Te mrginl cost of production (c) is supposed constnt; tus te poff of A is π = [θ k ( + )] c (1) impling liner rection function f wose slope is equl to ( 2) = f() Eqution (2) cn esil e otined from te first order condition of (1). π(, ) =k 2 (2) For ec point wic pertins to rection function of A, tereislevelof profit wic depends onl on te quntit produced B ( is fixedonits optiml level, for ec quntit produced te gent B). If we coose two unspecified points (δ nd ε) of te rection function, te vrition of te profits otined A, etween tese two points 1 is written: π δ π ε = k 2 δ 2 ε ten, π δ π ε = k ( δ ε ) ( δ + ε ) (3) Te slope of te rection function is in tis cse ( 2), nd consequentl eqution (3) ecomes µ π δ π ε = k 1 2 δ ε ( δ + ε ) 1 uscript denotes te corresponding point. 1

3 = 1 2 ( ε δ ) ( δ + ε ) is te surfce of te trpezoide ( ε εδ δ ). Tis surfce is sded in figure 1.Ten, π δ π ε = k ε ε δ δ ε δ Figure 1: Grpicl metod: generl cse 2 Appliction to te model of Bouët (2001) Te model of Bouët (2001) corresponds to te frmework clrified in te previous section, ut pplied to Nort-out Cournot duopol. First onl te Nort cn invest in cost reducing &D ctivit nd ten ot compete in quntit. Te issue of te investment is uncertin; on te one nd, pro(c = c )=α(r): te proilit (pro) tt te Nort otins te low mrginl cost (c ) in te lst stge of te gme is function of te volume of &D investment r. On te oter nd, [1 α(r)] is te proilit tt te ig mrginl cost of production (c ) is otined. Tis proilit is endogenousl determined since it depends positivel on te volume of investment 2

4 in &D (r). Depending on te success or te filure of te investment, te rection function is respectivel or. A not ver inding VE (voluntr export restrint) modifies onl one of te equiliri: N is replced N z, s depicted in Figure 2. Te metod, suggested in te first section indictes tt, in cse of filure, te VE induces n increse of te profit of te Nort. Tis increse reduces te incentive to invest in &D ctivit, wic is represented 1 in figure 2 (k =1in tis cse). B incresing onl te profit corresponding to n unfvourle issue of te investment (te profit in cse of success remins te sme), te VE slows down te incentive to invest in &D (proposition 2 of Bouët, 2001). 1 z N N z N * x x x Figure 2: Appliction to proposition 2 (Bouët, 2001: 328) Conversel, in te presence of specific triff, te rection function of te out moves from to τ. Bot free trde Ns equiliri re now modified. Te first one implies decrese of te incentive to innovte ( 2 ), weres te second one induces n increse of te incentive to innovte ( 3 ). 3

5 3 > 2, ten proposition 3 of te pper of Bouët is quickl found. 2 N τ l N 3 τ l x x * τ* x Figure 3: Appliction to proposition 3 (Bouët, 2001: 332 ) eferences [1] Bouët, A., 2001, eserc nd Development, Voluntr Export estriction nd Triffs, Europen Economic eview 45,