PLCY 289 Test TwoSpring 2006

Answer all questions in a blue book. Marks will be based on your explanations—use equations and graphs where possible. This test is being given at multiple sittings; please do not discuss the test questions with anyone until after the examination period has ended. You have two hours.

  1. The mayor of a small town is considering whether to provide $100 worth of a public good (P), but will only do so if the town’s residents derive $100 worth of benefit from P. The mayor is considering the following truth-telling mechanisms to elicit the community’s valuation of P. There are n residents in this town. Which mechanism has truth-telling as a dominant strategy? Which mechanism raises sufficient taxes to build P? Which mechanism would you recommend to the mayor?
  2. Ask each person their valuation of P. Build P if the sum of the valuations exceeds 100 and charge each person (100/n).
  3. Ask each person their valuation of P. Build P if the sum of the valuations exceeds 100 and charge each person their reported valuation (vi).
  4. Ask each person their valuation of P. Build P if the sum of the valuations exceeds 100 and charge each person 100 minus the sum of the valuations reported by the other residents or 0, whichever is greater.
  1. The following questions relate to the stage game depicted in the matrix below.
  2. In the one-shot version of this game, identify the Nash equilibria.
  3. In the infinitely repeated version of the stage game with discounting, can a trigger strategy be sustained? Be specific.
  4. Consider the following strategy profile: {Player 1 alternates between C and D while player 2 always plays C; if anyone deviates from this they both play D forever}. Is this profile a sub-game perfect equilibrium?

1\2 / C / D
C / (5,5) / (0,6)
D / (6,0) / (1,1)
  1. Two students (from an elite private university in the south) are deciding how fast to drive their daddies’ cars. Each chooses a speed xi and derives utility of Ui(xi) from this choice (assume U’(xi)>0). The faster they drive the more likely they are to crash into each other. Let p(x1,x2) be the probability of a crash (increasing in each argument) and ci>0 be the cost of the crash to each individual.
  2. Write down the optimization problem for one of the students. Derive and interpret the FOC. [Hint: This will be an expected utility.]
  3. Write down the optimization problem for ‘society’ (i.e the joint optimization problem). Derive and interpret the FOC.
  4. Explain the difference between these two FOCs. Will each student choose more or less than the ‘optimal’ speed to drive the car? How can we ensure that each student drives at the ‘optimal’ speed?

[The infinite series (a+a2+a3+…) converges to a/(1-a). The infinite series (1+ a+a2+a3+…) converges to 1/(1-a).]