Problem set 6. Solution. The problem of firm 3 is. The FOC is: 2 =0. The reaction function of firm 3 is: = 2

Transcription

1 Pobem set 6 ) Thee oigopoists opeate in a maket with invese demand function given by = whee = + + and is the quantity poduced by fim i. Each fim has constant magina cost of poduction, c, and no fixed cost. The fims choose thei quantities as foows: () fim chooses >; (2) fims 2 and 3 obseve and then simutaneousy choose and. Find the subgame pefect outcome. Soution The pobem of fim 3 is The FOC is: The eaction function of fim 3 is: 2 = = 2 Simiay the eaction function (best esponses) of fim 2 is: = 2 Using symmety ( = we find that the Nash equiibium of the simutaneous game between fims 2 and 3 is: = = 3 We go to find the optima behaviou of fim. But given that Fim anticipates the behaviou of fims 2 and 3 its pobem is: The FOC is Then the optima choice fo fim is: = 2 =

2 epacing in the soution of the subgame between fims 2 and 3 we have = = 6 The backwad induction outcome is: = = =

3 2) Conside the foowing noma fom game whee Paye chooses the ow (eithe T o B), Paye 2 chooses the coumn (eithe o ), chooses the tabe (eithe o ) Paye Paye 2 Paye 2 T,,,,,,,, B,,,,,,,, a) find a Nash equiibia in pue stategies b) assume that paye moves fist, then paye 2 and finay paye 3; evey paye, befoe to pay, obseves the choices of the pedecessos. a. epesent the game using the extensive fom b. Find a subgame pefect Nash equiibia c) Assume that paye 3 is not abe to see the choice of paye 2 a. epesent the game using the extensive fom b. Find a subgame pefect Nash equiibia Soution a) Paye Paye 2 Paye 2 T,,,,,,,, B,,,,,,,, Two Nash equiibia: (T,, ) (B,, )

4 b) Extensive fom epesentation We use backwad induction (in bod the best esponses) Backwad induction outcome: Paye pays B, Paye 2 pays, pays Subgame pefect Nash equiibia i. {(B), (, ), (,,, )} Paye 2 T B Paye Paye 2 T B Paye

5 ii. iii. iv. {(B), (, ), (,,, )} {(B), (, ), (,,, )} {(B), (, ), (,,, )} d) Extensive fom epesentation Paye T B Paye 2 Thee ae 3 subgames: the whoe game, the game between payes 2 and 3 afte T, the game between payes 2 and 3 afte B Afte T the subgame is: Paye 2,,,, Two Nash equiibia: {, } and {, } Afte the subgame is: Paye 2,,,, Two Nash equiibia: {, } and {, }

6 We have to ook fo the best choices of paye fo each possibe combination of Nash equiibia in the two subgames between payes 2 and 3 ) {(T), (, ), (, )} 2) {(), (, ), (, )} 3) {(T), (, ), (, )} ) {(), (, ), (, )} 5) {(), (, ), (, )}

7 3) Thee peiods sequentia bagaining. Two payes, and 2, ae bagaining ove $ using the foowing bagaining pocedue (atenating offes): Peiod : Paye poposes to take a shae s of the doa, eaving s fo paye 2; Paye 2 eithe accepts (game ends) o ejects (Pay goes to peiod 2) Peiod 2: Paye 2 poposes a shae s2 of the doa fo paye, eaving s2 fo paye 2; Paye eithe accepts (game ends) o ejects (Pay goes to peiod 3) Peiod 3: Paye eceives a shae s of the doa, paye 2 eceives s. Payes discount futue payoffs by facto δ pe peiod, < δ <. Find the backwad induction outcome and descibe the subgame pefect Nash equiibium The pobem of paye in peiod 2 is a choice between to have s2 immediatey o s one peiod ate. The best esponse of Paye is to accept s2 if s2 δs, othewise eject (s2 < δs) The pobem of Paye 2 in peiod 2 is a choice between: to offe s2 = δs (paye accepts) and eceive immediatey δs o to offe ess (paye ejects) and eceive s one peiod ate The best esponse of Paye 2 is to popose s2 = δs, because δs > δ ( s) The pobem of paye 2 in peiod is a choice between: To accept s and eceive s immediatey To eject and eceive ( δs) one peiod ate The best esponse of Paye 2 in peiod is to accept s if and ony if s δ( δs ), i.e. s - δ( δs ) The pobem of Paye in peiod is a choice between: To offe s = - δ( δs ) (paye 2 accepts) and eceive - δ( δs ) immediatey To offe ess (paye 2 ejects) and eceive δs one peiod ate The best esponse of Paye in peiod is to popose s = - δ( δs ) because - δ( δs ) > δ 2 s

8 ) Taiffs and impefect intenationa competition. Thee ae two identica counties denoted by i =, 2. One homogeneous good is poduced in each county by a fim, fim i in county i. A shae h i of this poduct is sod in the home maket and a shae e i is expoted in the othe county. Govenments choose taiffs, i.e. a tax on the impot. Govenment of county i chooses taiff t i In county i the invese demand function is P i (Q i ) = a Q i whee Q i = h i + e j. The fim s payoff (pofits) is p i = [a h i e j ]h i + [a h j e i ]e i c[h i + e i ] t j e i whee c> is the magina cost. The govenment s payoff is W i =.5 Q i 2 + p i + t i e j Timing: Govenments simutaneousy choose taiffs (t,t 2 ); Fims obseve (t,t 2 ) and simutaneousy choose quantities (h, e ) (h 2, e 2 ). Find the backwad induction outcome and descibe the subgame pefect Nash equiibium (Hint: suppose that govenments have chosen taiffs (t, t 2 ) and find the optima behaviou of fims as function of (t, t 2 ). Assume that govenments coecty pedict the optima behaviou of fims fo each possibe combination of (t, t 2 ) and find the optima taiff ates) We suppose that govenments have chosen taiffs (t, t 2 ) and we find the optima behaviou of fims as function of (t, t 2 ). Max (h, e) p whee p =[a h e 2 ]h + [a h 2 e ]e c[h + e ] t 2 e Fim s FOCs: [a 2h e 2 ] c = [a h 2 2e ] c t 2 = h = (a e 2 c) / 2 e = (a h 2 c t 2 ) / 2 Fo Fim 2: Max (h2, e2) p 2 whee p 2 =[a h 2 e ]h 2 + [a h e 2 ]e 2 c[h 2 + e 2 ] t e 2 Fim 2 s FOCs: [a 2h 2 e ] c = [a h 2e 2 ] c t = h 2 = (a e c) / 2 e 2 = (a h c t ) / 2 We have to sove a system of equations in unknowns:. h = (a e 2 c) / 2 2. e = (a h 2 c t 2 ) / 2

9 Soutions: 3. h 2 = (a e c) / 2. e 2 = (a h c t ) / 2. h *= (a c + t ) / 3 2. e *= (a c 2t 2 ) / 3 3. h 2 *= (a c + t 2 ) / 3. e 2 *= (a c 2t ) / 3 We assume that govenments coecty pedict the optima behaviou of fims fo each possibe combination of (t, t 2 ) and we find the optima taiff ates. The pobem of county s govenment is: Max (t) W =.5 (Q *) 2 + p *+ t e * whee Q * = h *+ e 2 * = (a c + t ) / 3 + (a c 2t ) / 3 = (2a 2c t ) / 3 p * =[a h * e 2 *] h * + [a h 2 * e *] e * c[h * + e *] t 2 e * Using ageba: W = (2 (a c) t ) 2 /8+ (a c + t ) 2 /9 + (a c 2t 2 ) 2 /9 + t (a c 2t )/3 Simiay we can wite the pobem of county 2 s govenment We compute the govenments FOCs and we find: t * = (a c)/ 3 t 2 * = (a c)/ 3 Then Fim wi poduce: h * = (a c)/ 9 e * = (a c)/ 9 Fim 2 wi poduce: h 2 * = (a c)/ 9 e 2 * = (a c)/ 9 Backwad Induction outcome t * = (a c)/ 3 t 2 * = (a c)/ 3 h * = (a c)/ 9 e * = (a c)/ 9 h 2 * = (a c)/ 9 e 2 * = (a c)/ 9 Subgame Pefect Nash Equiibium (SPNE): Note: One info set fo govenments infinite numbe of info set fo fims, i.e. each possibe combination of t t 2 t * = (a c)/ 3 t 2 * = (a c)/ 3 h *= (a c + t ) / 3 e *= (a c 2t 2 ) / 3 h 2 *= (a c + t 2 ) / 3 e 2 *= (a c 2t ) / 3

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