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Unformatted text preview: Midterm 2 Solutions Econ 382, Professor Platt, BYU April 8, 2008 1. Suppose Alex and Bob live near a pond where mosquitoes often breed. Ether man can spray the pond with insecticide to reduce mosquito growth. Each has utility U i ( s i ,M ) = (1- s i ) 1 2 (1- M ) 1 2 , where s i represents the time and resources spent by i in spraying, while M represents the fraction of the mosquito population left alive. The way spraying affects the population is determined by: M = 1- s a- 1 2 s b , because Bob uses less effective insecticides. (a) (5 pts) Solve for the voluntary contributions outcome in this model (since mosquito spraying is a public good). In particular, determine the equilibrium size of the mosquito population, and the quantity of spraying by each person. Setup: max s a (1- s a ) 1 2 ( s a + 1 2 s b ) 1 2 max s b (1- s b ) 1 2 ( s a + 1 2 s b ) 1 2 It is necessary to either substitute the production function in for M , or include M as a choice variable and have the production function as a constraint. Solution:- 1 2 s a + 1 2 s b 1- s a ! 1 2 + 1 2 1- s a s a + 1 2 s b ! 1 2 = 0 = s a + 1 2 s b = 1- s a = s a = 1 2- 1 4 s b- 1 2 s a + 1 2 s b 1- s b ! 1 2 + 1 4 1- s b s a + 1 2 s b ! 1 2 = 0 = 2( s a + 1 2 s b ) = 1- s b = s b = 1 2- s a s a = 1 2- 1 4 1 2- s a = 3 4 s a = 3 8 = s a = 1 2 = s b = 0 (b) (5 pts) Set Up the problem to find the efficient outcome in this model. Explain (in one sentence) why mosquito spraying is under or over provided in part (a). Setup: max s a ,s b ,M (1- s a ) 1 2 (1- M ) 1 2 s.t. (1- s b ) 1 2 (1- M ) 1 2 u and M = 1- s a- 1 2 s b 1 Bank / Restaurant Lunch Special Regular Price Close for lunch (0, 20) (0, 10) Open for lunch (-5, 15) (5, 5) Be sure to check that all the choice variables are specified. The explanation should state that mosquito spraying is under provided, and that the reason is that Alex does not consider the benefits of Bob in choosing his spraying (since the reduction in mosquitoes is non-rival), or vice versa. 2. (5 pts) Solve: Suppose that a bank and a restaurant share a parking lot. Thus, the quantity of customers served at one location affects the quantity that can be served at the other. They interact as shown in the game above (payoffs are given as (Bank,Restaurant)). Find the Nash Equilibrium of this game. (To justify your answer, you must define a Nash equilibrium below and show that your solution fits this definition.) Setup: Students need to define a Nash equilibrium (or some practical application of it). For instance, they could say that a Nash equilibrium is a pair of strategies for which each players strategy is a best response to the others strategy. Or they could be more loose in their description, such as strategies are in equilibrium when I wont change my strategy when I know his, and vice versa. Or they can be terse and say that a Nash equilibrium is a pair of mutually compatible best responses....
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