1.2 Oligopoly-exe

1.2 Oligopoly-exe - AMS 335/ECO 355 Game Theory Exercise of...

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Unformatted text preview: AMS 335/ECO 355 Game Theory Exercise of 1.2 Oligopoly Market Fall 2010 Zhen Xu Find the Nash Equilibrium of the following N-firm Oligopoly Market, using both Cournot and Bertrand Models. We assume in Bertrand model, each firm will produce the same quantity. Exercise 1.1.1 N Answer: In Cournot Model, each firm wants to maximize its profit by choosing the optimal production quantity q , i.e. max π P MC q 100 3q 3q 10 q 2, P 100 3Q, MC 10 In the point of local maximum, the first order condition equals 0, i.e. ∂π ∂q 0 6q 90 3q 0 Since it is a symmetric duopoly model, q 6q Thus, in this equilibrium, q q 90 3q 10, Q q . Substitute into the above function, 0 20, P q 10 40, π π 300 To see whether it is a Nash equilibrium or not, we need to see if the firms wants to change its production level given the other firm keep the original one, suppose q 10, q 10 ε, with ε 100 10 10 3Q ε ∞, 0 40 0, ∞ 3ε ε 300 The market price in this situation is P Thus firm 2’s profit π 40 3ε 3 100 Since the game is symmetric, firm1 wouldn’t change its production level given firm 2 will product at 10. Thus the above equilibrium is Nash equilibrium. In Bertrand Model, each firm wants to maximize its profit by choosing the optimal level of price. Since the products of the two firms are homogenous, no consumer will buy from the higher priced firm. The firm charges a lower price will get the whole market. Therefore, both of the firms will charge a price at the marginal cost level in Nash equilibrium, i. e. p p P 10, Q 30, P 40, p p 2, π π 0 It is a Nash Equilibrium because by lowering the price, firms will have negative profit and by increasing the price, firms will receive 0 demand. Page 1 of 2 AMS 335/ECO 355 Game Theory Exercise of 1.2 Oligopoly Market Exercise 1.1.2 N Cournot: q Bertrand: p q p 2, P 20, Q P 100 Q, MC 60, π 60, q q 40 π 400 30, π π 0 Fall 2010 Zhen Xu 40, P 40, Q Exercise 1.1.3 N Cournot: q Bertrand: p q p 3, P q p 333 16, Q P 4Q, MC 48, P 77, Q 77 141, π π q π 1024 π π 0 64, q q 64 ,π 3 Exercise 1.1.4 N Cournot: q Bertrand: p for i 1, 2, 2, Q P ,10 10, P 20, P 42, Q 196 56, π 22, q 7Q, MC 28 2.2, π 42 0 Exercise 1.1.5 N n, n 2, P na bn a b c a bQ, MC c ,P 1 a nb c c ac bn 1 Cournot: q ac ,Q bn 1 a nc ,π n1 Bertrand: p for i 1, 2, P ,n c, Q ,q ,π 0 Page 2 of 2 ...
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