# ps2 - Economics 310-2 Spring 2016 Professor Jeff Ely...

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Economics 310-2 Spring 2016 Professor Jeff Ely Problem Set 2 Due April 22 1. Two roomates Xavier and Yolanda share an apartment with lovely shag carpeting. Neither of them owns a vacuum cleaner. In order to keep the carpet clean, they must enlist the services of their neighbor Zeke. If Zeke spends a total of h hours a day vacuuming their carpet, Xavier and Yolanda would obtain value v x ( h ) and v y ( h ) respectively, where v x ( h ) = 3 h - h 2 v y ( h ) = 2 h - 3 h 2 Zeke’s value, on the other hand is negative v z ( h ) = - h 2 (vacuuming is a chore for Zeke and since it is not his apartment he gets no benefit from the clean carpet.) So consider the problem of deciding how many hours h Zeke should spend cleaning the carpet. Any h 0 is a possible alternative. (a) Suppose to begin with there is no money, so that the agents utility functions are just their value functions v x , v y , v z . With these preferences, what levels of h are Pareto efficient? (b) Write down the formula that shows for a given h the total value for all individ- uals. (c) The utilitarian social welfare function would choose the h which maximizes the expression you found above. How many hours a day should Zeke vacuum ac- cording to this utilitarian rule? Call this the utilitarian solution h * . (d) Now suppose that money is available and the agents have utility functions of the form discussed in class: U i ( h, t i ) = v i ( h ) - t i for i = x, y, z .

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