Microeconomics I Undergraduate Programs
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1 Game-theoretcal olgooly models Quantty multaneous equental tackelberg Fernando Branco econd emester essons 8 and 9 Prce fferent. trategc Comlements omnant Frm tackelberg olgooly Equlbrum n the tackelberg olgooly There are a few sulers; q Oututs are dfferentated or homogeneous; One suler (the leader decdes ts outut before the others. q R ( q The other sulers (followers decde after observng the leader s decson. q q Follower s decson eader s decson The followes sets quantty to maxmze ts roft: P ( q, q q C ( q, q CMg ( q RMg Reacton functon: q R q ( ( q Otmal outut level of the follower, gven the leader s decson. In general, t sloes downward. The leader maxmzes ts roft, takng n consderaton the follower s decson: P ( q, R ( q q C ( q RMg ( q CMg( q
2 tackelbergversus Game-theoretcal olgoy models multaneous equental q R ( q Quantty tackelberg C q q C q q R ( q q Prce fferent. trategc Comlements omnant Frm olgooly uler decsons There are a few sulers; Oututs are homogeneous (and rovded wth constant margnal cost; ulers take rce decsons smultaneously; Buyers have comlete nformaton and no transacton costs; There are entry barrers. Buyers urchase the outut from the lowest rce suler. If one suler knew other s rce, what rce would lke to choose? A rce just slghtly below the other s, as long as t s not below the margnal cost. Equlbrum Cost dfferences The equlbrum s reached f all frms rce at margnal cost. Two sulers are enough to rovde the same outcome as erfect cometton. What would haen f suler have dfferent margnal costs? Examle If the dfference s small, the equlbrum rce s equal to the margnal cost of the less effcent suler; otherwse, the more effcent suler rce at ts monooly level.
3 emand: Costs: Examle wth dfferent costs Q Equlbrum: If c < 6: ( 0 c, c c c If c 6: 6, c olgooly n ratce It s not easy to fnd examle of olgooles n ractce. The sulers rofts would be zero. Hence, slers try to fnd eays to ncrease ther rofts (for examle dfferentaton of colluson. Game-theoretcal olgoy models The domnant frm olgooly Quantty Prce fferent. multaneous trategc Comlements equental tackelberg omnant Frm The are a few sulers; Oututs are homogeneous; One sulyer sets the rce (the domnant frm; Other sulers are rce takers, and adust ther suly to the rce set. Followers decsons ulers from the comettve frnge follow the domnant frm s rce. Each maxmzes ts roft. Hence, these sulers decde as n erfect cometton: They adjust the suly to equalze margnal cost to rce. omnant frm s decson The domnant frm sets the rce to maxmze ts roft: Max Q ( Q ( C Q ( Q ( The domnant frm acts as a monoolst on the resdual demand (the demand left after subtractng the suly of the followers. 3
4 A numerc examle Examle: grahcal analyss Market demand: omnant s cost: Q ( 0 C ( q q ( q q Costs of the followers: C Followers suly: Q ( Resdual demand: 0 Prce chosen: ules: q 4; q q. 5 P P R MC Q Q Q Q MC Q Game-theoretcal olgoy models Prce choces wth dfferentaton Quantty Prce fferent. multaneous trategc Comlements equental tackelberg omnant Frm The strategc nteractons are smlar to those n olgooles (smultaneous choces or tackelberg olgooles (sequental choces Man dfference: reacton curves sloe urwards. Algebrac examle. multaneous choce: algebrac examle multaneous choce: grahcal analyss emands: q α β + γ j Costs: C ( q cq ( α β + γ ( c α + βc γj Reacton functons: + Equlbrum: α + βc γ j R R 4
5 equental choces: algebrac examle equental choces: grahcal analyss emands: q α β + γj Costs: C ( q cq Follower s reacton functon: α + βc γ + eader s objectves: α + βc γ α β + γ + ( c Equlbrum: ( + γ ( α + ( β γ c ( ( γ ( a + βc + γ ( R R Colluson and cartels Incentves to cartels ulers may collude to decrease comettve ressure. Ina cartle, sulers decde roducton quotas and/or set rces, to ncrease rofts. Examle: OPEC. ules may ncrease rofts. Problem smlar to that of a mult-unt suler that needs to decde how to slt the actvtes. Addtonally, comensatory ayments may be needed to enforce the cartel. The cartel objectve mts to cartels q q egal lmts: Work mlct colluson. Instablty of cartels: If one suler knows the quantty of the other, t wll want to make ts choce to be on ts reacton curve. Ths creates an nstablty n the cartels. 5
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