Midterm Advanced Economic Theory, ECO326F1H Marcin Pęski February, 2015

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1 Midterm Advanced Economic Theory, ECO326F1H Marcin Pęski February, 2015 There are three questions with total worth of 100 points. Read the questions carefully. You must give a supporting argument and an answer in words to get full credit. If you don t know the answer to any of the parts, try to solve the next one. You do not need to compute the exact values of algebraic formulas (for example, it is OK to say that x = instead of x = 4.) 5 You have 120 minutes. Good luck! 1

2 2 (1) (25 points) Consider the following game. Player 1 \ Player 2 L C R U 5, 0 3, 3 3, 5 M 0, 5 3, 3 5, 2 D 2, 3 2, 4 2, 5 (a) What can you predict about the game if you know that player 1 is rational? (b) What can you predict about the game if you know that player 2 is rational? (c) What can you predict about the game if you know that both players are rational and both players know that they are rational? (d) Find (all) Nash equilibria of the game..

3 (2) (40 points) A recent outbreak of measles is associated with a report of declining vaccination rates among certain groups of population. In this question, we will illustrate vaccine avoidance with a simple model. 3 Imagine that there is a population of N individuals. Each one of them simultaneously chooses between taking a vaccine (V ) or not (N). All individuals have the same preferences. The payoff from not taking the vaccine depends on the number of unvaccinated people n and it is equal to u i (N, n) = α 1 N n. The above equation is a simple illustration of herd immunity : the more unvaccinated people, the larger the probability that individual i gets infected with a disease. We can refer to 1 1 n as the vaccination rate. The utility of N taking the vaccine is equal to u i (V, n) = c β 1 N n, where c > 0 is a constant real or imagined cost of taking vaccination (for example, price of the vaccine at the pharmacy, perceived probability that it leads to complications, etc.), and β < α. The last part of the utility reflects the fact that no vaccines are perfect and even vaccinated people are at risk from a disease. Additionally, we assume that α β > c > 1 N α. (a) Find the best response function of an individual. (b) Imagine an individual who lives in a society which consists of only unvaccinated people would like to take a vaccine. individual would always want to take the vaccine? Explain that such an (c) Does this game have any (strictly or weakly) dominated or dominant actions? (d) Find a Nash equilibrium of the vaccination game. unique? Is the equilibrium (e) For what values of the parameters are there individuals who do not take vaccine in the Nash equilibrium? Explain this fact in the light of your answer to question (b).

4 4 (f) Compute the welfare of the society (i.e., the sum of individuals utilities) in the equilibrium. (g) Suppose that c < β. What outcome of the game (i.e., what profile of individual actions) maximize the social welfare? Is the optimal vaccination rate smaller or larger than the equilibrium rate? (h) Suppose that the government wants to induce 100% vaccination rate by imposing an extra penalty cost c 0 > 0 that is paid by the unvaccinated. (For example, monetary cost, compulsory medical exams, school suspensions, etc.) How big the cost must be to ensure that all individuals take a vaccine?

5 (3) (35 points) We analyze a variant of the voting process. There are N 100 voters in the society choosing from m 2 alternatives. Each voter has preferences over all alternatives. In the voting process, each voter ranks all the alternatives from top to bottom. Then, he or she submits the entire ranking (instead of a single alternative, as in the classic voting model). Each alternative receives points that depend on its place in the ranking: if alternative x is ranked at the top of the list, it receives m points, if it is ranked at the 2nd place, it receives m 1 points,..., if it is ranked at the last place, it receives 1 point. The alternative that receives the largest number of points wins. If two or more alternatives tie, the outcome is chosen randomly. (a) Suppose first that m = 2 and there are two alternatives, a and b. Consider a voter with preferences a b. Show, for this voter, it is a weakly dominant strategy to report a at the top and b at the bottom of the list, i.e., to tell the truth. Remember to explain your answer carefully. (b) Is the truth-telling strategy strictly dominant? (c) From now on, suppose that m = 3 and there are three alternatives, a, b and c. Consider a voter with preferences a b c. Is reporting 1a, 2b, 3c (i.e., truth-telling) a weakly or strictly dominant strategy? Show that it is so if yes, find a counterexample if not. (d) Consider the same voter as in the previous question. Does she have a weakly or strictly dominated strategy? 5

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