國外博弈論課件lecture(18)-資料下載頁
2024-10-18 12:47本頁面
【正文】 ices are denoted by p1 and p2, respectively. ? The timing of this game as follows. ? Firm 1 chooses a price p1 ?0. ? Firm 2 observes p1 and then chooses a price p2 ?0. ? The quantity that consumers demand from firm 1: q1(p1, p2) = a – p1 + bp2. ? The quantity that consumers demand from firm 2: q2(p1, p2) = a – p2 + bp1. ? The cost to firm i of producing quantity qi is Ci(qi)=cqi. June 10, 2021 73347 Game TheoryLecture 15 23 Sequentialmove Bertrand model of duopoly (differentiated products) ? Solve firm 2’s problem for any p1 ?0 to get firm 2’s best response to p1. ? Max u2(p1, p2)=(a – p2 + bp1 )(p2 – c) subject to 0 ? p2 ? +∞ FOC: a + c – 2p2 + bp1 = 0 p2 = (a + c + bp1)/2 ? Firm 2’s best response, ? R2(p1) = (a + c + bp1)/2 June 10, 2021 73347 Game TheoryLecture 15 24 Sequentialmove Bertrand model of duopoly (differentiated products) ? Solve firm 1’s problem. Note firm 1 can also solve firm 2’s problem. Firm 1 knows firm 2’s best response to p1. Hence, firm 1’s problem is ? Max u1(p1, R2(p1))=(a – p1 + b?R2(p1) )(p1 – c) subject to 0 ? p1 ? +∞ That is, Max u1(p1, R2(p1))=(a – p1 + b?(a + c + bp1)/2 )(p1 – c) subject to 0 ? p1 ? +∞ ? FOC: a – p1 + b?(a + c + bp1)/2+(–1+b2/2) (p1 – c) = 0 p1 = (a+c+(ab+bc–b2c)/2)/(2–b2) June 10, 2021 73347 Game TheoryLecture 15 25 Sequentialmove Bertrand model of duopoly (differentiated products) ? Subgameperfect Nash equilibrium ? ((a+c+(ab+bc–b2c)/2)/(2–b2), R2(p1) ), where R2(p1) = (a + c + bp1)/2 ? Firm 1 chooses a price (a+c+(ab+bc–b2c)/2)/(2–b2), firm 2 chooses a price R2(p1) if firm 1 chooses a price p1. June 10, 2021 73347 Game TheoryLecture 15 26 Summary ? Subgame perfect Nash equilibrium ? Backward induction ? Stackelberg model of duopoly ? Sequentialmove Bertrand model of duopoly (differentiated products) ? Next time ? Dynamic games of plete and imperfect information ? Reading lists ? Sec , of Gibbon
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