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Econ 101 Problem Set 2 Due in class Monday Feb. (12 pts) It's the Cold War and the US and USSR are facing o over a political dilemma. A Soviet ship...

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Econ 101 Problem Set 2 Due in class Monday Feb. 4 1. (12 pts) It’s the Cold War and the US and USSR are facing off over a political dilemma. A Soviet ship carrying nuclear warheads has been intercepted by a US patrol and both sides are requesting orders from their leaders. The US can order its patrol to back down or to fire on the USSR ship. The USSR can order its ship to try and run the blockade or to retreat. If the US backs down and the USSR retreats, the story is not reported and there are no consequences. If the US opens fire and the USSR retreats, the US gains international prestige and the USSR loses face. If the US backs down and the USSR runs the blockade, then the opposite happens. In the worst case scenario, the US opens fire while the USSR runs the blockade, the USSR ship is sunk, triggering an international crisis of severity K > 1. The payoff matrix is given by: USSR Run Blockade Retreat US Open Fire - K , - K 1, -1 Back Down -1, 1 0, 0 (a) (4 pts) Write down the players and their strategy sets. Write down the payoff function of the US. (b) (2 pts) Find the pure strategy Nash equilibria. (c) (4 pts) Find the mixed strategy Nash equilibrium. What is the probability of an international crisis under this equilibrium? 1
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(d) (1 pts) What are expected payoffs under the mixed strategy Nash equilib- rium? (e) (1 pts) What level of K maximizes expected payoffs? What do you think this model says about the concepts of nuclear deterrence and mutually assured destruction? 2. (10 pts) Nintendo, Sony and Microsoft compete with each other in the market for home video game consoles. The demand for the Nintendo Wii is given by: Q N ( p N ,p S ,p M ) = 100 - 2 p N + p S + p M The demand for the Sony Playstation is given by: Q S ( p S ,p N ,p M ) = 50 - p S - 0 . 5 p N + 0 . 5 p M And the demand for the Microsoft Xbox is given by: Q M ( p M ,p N ,p S ) = 70 - p M - 0 . 5 p N + 0 . 5 p S For simplicity, assume that each producer has zero average and marginal costs. Treat each firm as a price setter. (a) (6 pts) Write down Nintendo, Sony, and Microsoft’s best response func- tions. (b) (4 pts) Find the price of the Nintendo Wii in the Nash equilibrium. 3. (11 pts) Two firms face a market demand curve given by: P = 100 - Q The cost structure for firm 1 is: C 1 ( q ) = 20 q + 1 2 q 2 and the cost structure for firm 2 is: C 2 ( q ) = 20 q + q 2 2
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