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Economic problems

5341-Fall20-M1.pdf Question 1 (30 points): Consider the following two person game. Player 1 is row player, Player 2 is column player. L C R T 2,3 1,4 3,2 M 5,1 2,3 1,2 Z 3,7 4,6 5,4 B 4,2 1,3 6,1 a) (10 points) Does player 1 have a strictly dominated strategy? Explain formally by writing player 1’s best response to every possible strategy of player 2. Does player 2 have a strictly dominated strategy? Again explain formally by writing player 2’s best response to every possible strategy of player 1. 2 b) (5 points) After eliminating the strictly dominated strategies for each player, state all pure strategy NE of this game. Note: Just stating the pure strategy NE is sufficient. 3 b) (15 points) Find all mixed strategy NE of this game. 4 Question 2 (20 points) Suppose that two firms (Firm 1 and Firm 2) face an industry demand P = 150 � Q where Q = q1 + q2 is the total industry output. Both firms have the same unit production cost c = 30. Therefore, the profits of each firm is given by p1 (q1, q2) = (150 � q1 � q2)q1 � 30q1 p2 (q1, q2) = (150 � q1 � q2)q2 � 30q2 Each firm is run by managers who care how much more profit their own firm makes relative to the rival firm. • The manager of Firm 1 chooses q1 to maximize p1 (q1, q2)� p2 (q1, q2) • The manager of Firm 2 chooses q2 to maximize p2 (q1, q2)� p1 (q1, q2) . Note that the managers are not maximizing own firm’s profits but the difference between their profit and rival’s profit. The managers are choosing their quantities simul- taneously. 5 a) (15 points) Find the NE pair (q⇤ 1 , q⇤ 2 ) . 6 b) (5 points) What is the Nash equilibrium profit of each firm? 7 Question 4 (30 points): A buyer and a seller simultaneously submit a price, which can be any non-negative number. • Voluntary trade takes place when the price pb chosen by the buyer is at least $5 more than the price ps chosen by the seller, i.e., pb � ps � 5.If voluntary trade occurs, the buyer pays pb and the seller receives pb. • If pb is not at least $5 more than ps, then an arbitrator forces the buyer to pay $50 to the seller and trade takes place at the price $50. • Payoff of the buyer is his value v = 80 minus the price pb he pays. The payoff of the seller is the price pb she receives minus her cost c = 40. The payoffs of players are described below. ub(pb, ps) = ( 80 � pb if pb � ps � 5 30 if pb � ps < 5="" us(pb,="" ps)="(" pb="" �="" 40="" if="" pb="" �="" ps="" �="" 5="" 10="" if="" pb="" �="" ps="">< 5="" 8="" a)="" (10="" points)="" is="" the="" strategy="" profile="" pb="50" and="" ps="50" a="" nash="" equilibrium.="" explain="" formally.="" 9="" b)="" (15="" points)="" are="" there="" any="" nash="" equilibria="" where="" ps="x" and="" pb="x" +="" 5="" where="" x=""> 50 Explain formally. 10 Question 4 (25 points) Two players have to simultaneously decide how much to contribute to the provision of a public good. If player 1 contributes x and player 2 contributes y, then the value of the public good is x + y + 2xy, which both players receive. The players can choose any non-negative number as their contribution. • Player 1 must pay a cost of x2 � xy of contributing x. Note that P1’s cost of contri- bution declines as P2 contributes more (as y increases). Formally, P1 chooses x to maximize u1(x, y) = x + y + 2xy � x2 + xy • Player 2 pays a cost 4y2 if she contributes y. Note that P2’s cost of contribution does not depend on P1’s contribution. Formally, P2 chooses y to maximize u2(x, y) = x + y + 2xy � 4y2 a) (15 points) Find best responses of each player. 11 b) (10 points) Find the Nash equilibrium (x⇤, y⇤) of this game. 12

5341-Fall20-M1.pdf Question 1 (30 points): Consider the following two person game. Player 1 is row player, Player 2 is column player. L C R T 2,3 1,4 3,2 M 5,1 2,3 1,2 Z 3,7 4,6 5,4 B 4,2 1,3 6,1 a) (10 points) Does player 1 have a strictly dominated strategy? Explain formally by writing player 1’s best response to every possible strategy of player 2. Does player 2 have a strictly dominated strategy? Again explain formally by writing player 2’s best response to every possible strategy of player 1. 2 b) (5 points) After eliminating the strictly dominated strategies for each player, state all pure strategy NE of this game. Note: Just stating the pure strategy NE is sufficient. 3 b) (15 points) Find all mixed strategy NE of this game. 4 Question 2 (20 points) Suppose that two firms (Firm 1 and Firm 2) face an industry demand P = 150 � Q where Q = q1 + q2 is the total industry output. Both firms have the same unit production cost c = 30. Therefore, the profits of each firm is given by p1 (q1, q2) = (150 � q1 � q2)q1 � 30q1 p2 (q1, q2) = (150 � q1 � q2)q2 � 30q2 Each firm is run by managers who care how much more profit their own firm makes relative to the rival firm. • The manager of Firm 1 chooses q1 to maximize p1 (q1, q2)� p2 (q1, q2) • The manager of Firm 2 chooses q2 to maximize p2 (q1, q2)� p1 (q1, q2) . Note that the managers are not maximizing own firm’s profits but the difference between their profit and rival’s profit. The managers are choosing their quantities simul- taneously. 5 a) (15 points) Find the NE pair (q⇤ 1 , q⇤ 2 ) . 6 b) (5 points) What is the Nash equilibrium profit of each firm? 7 Question 4 (30 points): A buyer and a seller simultaneously submit a price, which can be any non-negative number. • Voluntary trade takes place when the price pb chosen by the buyer is at least $5 more than the price ps chosen by the seller, i.e., pb � ps � 5.If voluntary trade occurs, the buyer pays pb and the seller receives pb. • If pb is not at least $5 more than ps, then an arbitrator forces the buyer to pay $50 to the seller and trade takes place at the price $50. • Payoff of the buyer is his value v = 80 minus the price pb he pays. The payoff of the seller is the price pb she receives minus her cost c = 40. The payoffs of players are described below. ub(pb, ps) = ( 80 � pb if pb � ps � 5 30 if pb � ps < 5="" us(pb,="" ps)="(" pb="" �="" 40="" if="" pb="" �="" ps="" �="" 5="" 10="" if="" pb="" �="" ps="">< 5="" 8="" a)="" (10="" points)="" is="" the="" strategy="" profile="" pb="50" and="" ps="50" a="" nash="" equilibrium.="" explain="" formally.="" 9="" b)="" (15="" points)="" are="" there="" any="" nash="" equilibria="" where="" ps="x" and="" pb="x" +="" 5="" where="" x=""> 50 Explain formally. 10 Question 4 (25 points) Two players have to simultaneously decide how much to contribute to the provision of a public good. If player 1 contributes x and player 2 contributes y, then the value of the public good is x + y + 2xy, which both players receive. The players can choose any non-negative number as their contribution. • Player 1 must pay a cost of x2 � xy of contributing x. Note that P1’s cost of contri- bution declines as P2 contributes more (as y increases). Formally, P1 chooses x to maximize u1(x, y) = x + y + 2xy � x2 + xy • Player 2 pays a cost 4y2 if she contributes y. Note that P2’s cost of contribution does not depend on P1’s contribution. Formally, P2 chooses y to maximize u2(x, y) = x + y + 2xy � 4y2 a) (15 points) Find best responses of each player. 11 b) (10 points) Find the Nash equilibrium (x⇤, y⇤) of this game. 12

Oct 21, 2021

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