HW4-Solutions - Homework Assignment 4 Solutions Econ 382...

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Homework Assignment 4 Solutions Econ 382, Professor Platt, BYU January 23, 2009 1 Warmup Optional questions. Points earned in this section can offset points lost elsewhere (though the maximum score for the assignment is still 100 points). Credit is only given if you show all steps taken to obtain your answer. 1. Suppose there is a single gas station in the rural town of Sanders, AZ, on I-40. Bob, the gas station owner, knows he has two types of clients: local residents, who are less price sensitive since they would have to travel 30 miles to the nearest competitor, and interstate travelers, who are more price sensitive, since they will be traveling onward to more competitive markets anyhow. Bob figures that in a regular business hour, 1 local resident drops in, with individual demand for gasoline given by q r = 19 - 4 P . He also figures that 1 interstate traveler considers stopping in Sanders for gasoline in the same hour, with individual demand given by q t = 11 - P . Bob faces a constant marginal cost of gasoline of two dollar per gallon, so his cost function is C ( Q ) = 2 Q . When entry fees are charged, the membership period only lasts for that one purchase. In each of the following pricing schemes, answer the following list of questions, as well as other specific questions provided: Set up Bob’s maximization problem. What price(s) will he charge? What is the optimal entry fee, if applicable? How many gallons will each type of customer purchase? How much consumer surplus will each type of customer have? Note that CS r = (19 - 4 P ) 2 8 and CS t = (11 - P ) 2 2 How much profit does Bob earn? (a) (4 pts) Suppose Bob operates his business as a single-priced monopoly. Solution: Bob solves the following problem: max P q r P + q t P - 2( q r + q t ) max P ( q r + q t )( P - 2) max P (19 - 4 P + 11 - P )( P - 2) 1
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max P (30 - 5 P )( P - 2) Taking the derivative with respect to P , we obtain 30 - 10 P + 10 = 0, which means P = 4. Placing this in the demand equations, we get q r = 3 and q t = 7. Here, after inserting the price, we get consumer surplus is: CS r = 9 8 = 1 . 12 and CS t = 49 2 = 24 . 5. Total profit is: π = 20. (b) (8 pts) Next, Bob attempts third degree price discrimination, charging a different price to local residents (who he can distinguish, since he knows all of them). Solution: Bob solves the following problem: max p r ,p t q r p r + q t p t - 2( q r + q t ) max p r ,p t (19 - 4 p r )( p r - 2) + (11 - p t )( p t - 2) Taking the derivative with respect to p r , we obtain 19 - 8 p r +8 = 0, which means p r = 27 8 = 3 . 37. Doing the same with p t gives 13 - 2 p t = 0, so p t = 6 . 5. Placing this in the demand equations, we get q r = 5 . 5 and q t = 4 . 5. Consumer surplus is CS r = 121 32 = 3 . 78 and CS t = 81 8 = 10 . 12. Total profit is: π = 445 16 = 27 . 81. (c) (8 pts) Bob then attempts second degree price discrimination. In particular, he charges everyone the same entry fee as well as the same price (where both are set optimally).
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