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Unformatted text preview: Second Midterm Solutions Econ 382, Professor Platt, BYU Winter 2009 1. Suppose Alex and Bob have homes that share a long private driveway. During the winter months, when ice forms on the driveway, each is able to sprinkle salt on the driveway to reduce the ice. Each has utility u i = α ln( s a + s b ) cs i , where i = a or b . (a) (5 pts) — Solve for the voluntary contributions outcome in this model with public goods. In particular, determine the total amount of salt they will put on the driveway. Solution: max s a α ln( s a + s b ) cs a and max s b α ln( s a + s b ) cs b Taking the derivative w.r.t. s a , we get 1 s a + s b = c , and the same from Bob’s maximization. In other words, they will jointly ensure that 1 c units of salt end up on the road. This is not necessary for full credit, but in particular, Alex’s best reponse is to add enough salt to whatever Bob has sprinkled to ensure that we get 1 c units. (b) (5 pts) — Solve for the efficient outcome in this model. Explain (in one sentence) why salt is under or overprovided in part (a). Solution: max s a ,s b α ln( s a + s b ) cs a s.t. α ln( s a + s b ) cs b ≥ ¯ u Taking firstorder conditions, we find: α s a + s b c + λ s a + s b = 0 and α s a + s b + λ s a + s b λc = 0 We then solve and substitute for λ : c ( s a + s b ) 1 α = λ = α c ( s a + s b ) 1 So c ( s a + s b ) 1 α 2 = 1 = ⇒ s a + s b = 2 α c . Salt is underprovided, because each of the neighbors are free riding on the salt contributions of the other. Or, the benefits of salting are nonexcludable; hence, each neighbor neglects the benefit to the other in deciding how much to sprinkle. 1 Alex \ Bob Newpaper TV Radio Newspaper (1, 1) (1, 2) (3,3) Radio (0, 2) (4, 0) (2,1) 2. (5 pts) — Solve: Suppose Alex and Bob run competing pizza. Each is deciding how to advertise. The outcome for each possible choice is shown in the normal form game above. Payoffs are (Alex, Bob). Find the Nash Equilibrium of this game if it is only played once. Briefly explain why it is a Nash Equilibrium. Solution: The Nash equilibrium is that both Alex will choose “Newspaper” and Bob will choose “Radio.” This may be found in several ways: iterated elimination, picking a strategy pair and verifying, or solving for the mixed strategy (and finding that one strategy is dominant). Some explanation of why this qualifies as a Nash equilibrium, such as: “If Alex plays News, Bob will get 1 or 2 units more profit by playing Radio; and vice versa as well.” The iterated elimination would have to occur in the following order: For Bob, Ra dio dominates TV. For Alex, Newspaper would dominate Radio. For Bob, Radio dominates Newspaper....
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 Spring '08
 Mcdonald,J
 Econometrics, Game Theory, pts, Alex

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