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• Test Prep
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page 2 1. St. George is a street with a length of one mile. Two hotdog shops are potential entrants and they are considering locating their outlets. Each of them can open one outlet. Suppose both these two shops have constant marginal cost c . A mass 1 UT students are uniform distributed on St. George st. and they are looking for hotdogs. Their reservation value for hotdog is very high with V >> c . However, there is a transportation cost of walking to the hotdog shop which is a quaratic function of the distance between the consumer and the shop. For example, consumer who is located at x needs to suffer a cost of t ( x a ) 2 in order to walk to a . And we assume V >> t . (a) [5] Suppose these two shops are located in ( a, 1 b ), with a bracketleftbig 0 , 1 2 bracketrightbig . They are involved in a price competition with shop 0 charges p 0 and shop 1 charges p 1 . Find the marginal consumer. Given any location choice ( a, 1 b ), suppose the marginal consumer is x m V p 0 t ( x a ) 2 = V p 1 t (1 b x ) 2 Solve and get x m = a + 1 a b 2 + p 1 p 0 2 t (1 a b ) (b) [5] Suppose we can solve for the best response function of the two firms as a function of locations and competitor’s price. BR 0 ( p 1 ) = at (1 a b ) + t (1 a b ) 2 2 + p 1 + c 2 BR 1 ( p 0 ) = bt (1 a b ) + t (1 a b ) 2 2 + p 0 + c 2 If firm 0 is going to charge p 0 = 0, what is the optimal price for shop 1? If p 0 = 0, according to the best response function, BR 1 ( p 0 = 0) = bt (1 a b ) + t (1 a b ) 2 2 + 0 + c 2 = bt (1 a b ) + t (1 a b ) 2 2 + c 2 (c) [5] If firm 0 is going to charge p 0 + , what is the optimal price for shop 1? Firm 1 now is not going to decide price according to the best response function. Actually, Firm 1 will behave as the monopolist in the market. Since the V is high ( V >> t ), Firm 0 will maximize its profit by serve the whole market (Proof see Question 2 in PS2, no need to show here). So Firm 1 will set price such that the consumer at x = 1 is indifferent from purchase or no purchase, i.e., p 1 = V t (1 a ) 2 (d) [5] Find the Nash Equilibrium price for the two firms.
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• Fall '12
• CarlosSerrano
• Game Theory, Tax Returns, Stackelberg competition

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