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Unformatted text preview: Second Midterm Solutions Econ 382, Professor Platt, BYU Fall 2008 1. Suppose Alex and Bob are neighbors, and form a neighborhood watch organization to reduce crime. Alex has utility u a = (1 t a ) 1 2 S 1 2 , where t a is the fraction of his day that Alex spends on patrol and S is the amount of safety produced by the organization. Bob has a stronger preference for for safety: u b = (1 t b ) 1 3 S 2 3 . The amount of safety is directly proportional to the amount of total time spent by either neighbor: t a + t b = S . (a) (5 pts) Solve for the voluntary contributions outcome in this model with public goods. In particular, report the amount of safety that will be provided, and the fraction of time contributed by each person. Solution: The maximization problem for Alex is: max t a (1 t a ) 1 2 ( t a + t b ) 1 2 Taking the derivative with respect to t a , we get: 1 2 t a + t b 1 t a 1 2 + 1 2 1 t a t a + t b 1 2 = 0 = t a = 1 2 (1 t b ). The maximization problem for Bob is: max t b (1 t b ) 1 3 ( t a + t b ) 2 3 Taking the derivative with respect to t b , we get: 1 3 t a + t b 1 t b 2 3 + 2 3 1 t b t a + t b 1 3 = 0 = t b = 1 3 (2 t a ). The Nash Equilibrium is then where the best response functions coincide: t a = 1 2 1 6 (2 t a ) = t * a = 1 5 and t * b = 3 5 . Thus S * = 4 5 . (b) (5 pts) Setup the maximization problem to find the efficient outcome in this model. Solution: The needed maximization problem is: max t a ,t b ,S (1 t a ) 1 2 S 1 2 s.t. (1 t b ) 1 3 S 2 3 u and S t a + t b They could also substitute in for S and have only the first constraint. 1 Arbys / Wendys Sale No Sale Sale (1, 1) (6, 0) No Sale (0, 6) (5, 5) (c) (3 pts) Discuss one specific method to improve efficiency in provision of safety, as well as the difficulties of perfectly implementing it. Solution: Several options could be discussed: Public funding or public provision. Government determines the amount of safety, and requires a certain contribution from each neighbor (or taxes them and hires someone else to do it). More specifically, they could use a Lindahl tax system to discover the efficient level of public goods. Perhaps one could find a technical solution to make the contributions ex cludable. One could educate the neighbors or try to apply guilt. They should provide some detail. It would be helpful if they indicate that what ever program they indicate would need to increase the amount of safety. They should talk about the difficulty of knowing the right level of provision, or the limited efficacy of education, or the practical problem of people gaming the Lindahl system rather than revealing true preferences. For technical solutions, there would still be the inefficiency caused by charging more than marginal cost....
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 Spring '08
 Mcdonald,J
 Econometrics, Utility

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