QuestionsSimulMoveCont - ECON 201 - GAMES AND STRATEGY...

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ECON 201 - GAMES AND STRATEGY EXERCISES ON GAMES WITH CONTINUOUS STRATEGIES 1. On a street of length 100 meters, the consumers are uniformly distributed : 0 x 100 Seller A Seller B “Uniformly distributed” means that there is the same number of consumers per unit distance: in other words, the number of consumers at the left of point x is simply x and the number of consumers located to the right of point x is 100 – x. Each consumer will buy only one unit of ice cream. There are two sellers: Seller A is located at point “0” and Seller B is located at point “100.” For each seller, the cost of producing one unit of ice cream is “ c .” When a consumer decides to buy one ice cream from seller A or B, she pays the price announced by that seller and also “pays” a transportation cost that is proportional to the distance of that seller from her location. The transportation cost is “t” per unit distance . So, if the consumer at “x” buys from seller B, she pays the price p B plus the transpotation cost (100 – x)t. If she buys from the Seller A, she pays p A plus the transpotation cost x.t . Thus, the consumer located at “x” will buy from seller A if: p A + x.t . < p B + (100 – x)t. (cost of buying at A) < (cost of buying at B) Obviously, consumers prefer to buy from the seller that is closer to their location, but they also care about the price. Consider a situation where the two sellers are sharing the street market: there must be a consumer who is indifferent between buying from seller A and seller B. Let that consumer be located at some point like “x.” Since she is indifferent between the two sellers, it must be that p A + x.t . = p B + (100 – x)t. (i) Obtain the expression of the location “x” in terms of the prices p A , p B and the unit transportation cost t , using the equality above. The closer “x” is to the point 100, the larger is seller A’s market share. How does “x” change if Seller A changes his price? If seller B changes his price?
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(ii) How does “x” change if the transportation cost t increases slightly? On what does your answer depend? Comment (based on the expression of x and your understanding of the model). (iii) The profit of seller A depends on his own price p A , as well as the price announced by Seller B, p B , as follows: Profit A = revenues – costs = price times output – unit cost times output = = ( p A ) x – c.x. Profit B = ( p B )(100 – x) – c. (100 – x). The sellers will simultaneously announce their prices (note that x depends on the prices as well). Show that the best response functions of the sellers are, respectively: p A = [ p B + 100t + c]/2 and p B = [ p A + 100t + c]/2.
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This note was uploaded on 03/16/2012 for the course FENS 101 taught by Professor Selçukerdem during the Fall '12 term at Sabancı University.

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QuestionsSimulMoveCont - ECON 201 - GAMES AND STRATEGY...

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