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Simon Fraser University Fall 2012 Econ 302 Quiz — solution Section D200 Thursday Oct. 4, 2012 Due: end of the class, 9:20 am. Write your name, SFU ID number, and tutorial section number on the exam booklet. This is a closed-book exam. You may use a non-graphing calculator. Please leave the room quietly if you finish and want to leave early. I will deduct points off your quiz if you cause too much distractions to your neighbors. 1 1. Multiple-choice / True-false (no penalty for wrong guesses) (a) (2 points) A mixed strategy is a/an (A) strategy profile, (B) action profiles, (C) probability distribution over actions, (D) collection of numbers over action profiles, or (E) none of the above. Answer: (C) (b) (2 points) A Nash equilibrium (A) always involves strictly dominated actions, (B) may not exist in a finite game, (C) is a symmetric strategy profile, (D) makes a coarser prediction than ISD, or (E) none of the above. Answer: (E) (c) (1 points) True or False: Quantity competition usually leads to a lower price than price competition. Answer: False. (d) (1 points) True or False: In a two-player game written out in the “matrix” form, Player 1 controls the columns, and Player 2 controls the rows. Answer: False. For Problem 2 to 4 show all your work. 2. (8 points) The market (inverse) demand function for a good is P (q) = 18 − q. There is a monopolist with a constant marginal cost of 2 for producing each unit of the good. Assume that the government imposes a sales tax of 50%, and the sales tax does not impact the monopolist’s cost. Calculate the monopolist’s price and quantity, and calculate the corresponding consumer’s price. Answer: The monopolist solves 2 max q 1 (18 − q)q − 2q. 1 + 0.5 The first order condition is: 2 (18 − 2qm ) − 2 = 0. 3 Solving this gives the monopolist’s quantity qm = 15/2 = 7.5. The consumer’s price is then 1 P (qm ) = 18 − 7.5 = 10.5. The monopolist’s price is 1+0.5 P (qm ) = 7. 3. (8 points) For the following game, (a) find the strategies that survive iterated deletions of strictly dominated strategies (ISD), and (b) find all Nash equilibria (pure and mixed). W X Y Z A 1, 2 -1, 1 -1, 0 8, 2 B 0, 0 1, 1 -1, 0 7, 1/3 C 2, -1 -1, 0 1, 1 10, 1/3 D 5, 3/2 0, 1 -1/3, 4 8, 2 Answer: Part (a): First round: delete A (strictly dominated by 1/2C + 1/2D). Second round: delete W (strictly dominated by Z) and Z (strictly dominated by 1/2X + 1/2Y ). Third round: delete D (strictly dominated by 0.6B + 0.4C). ISD strategies: {B, C} and {X, Y } Part (b): There are two pure-strategy Nash equilibria: (B, X) and (C, Y ). There is one non-trivial mixed-strategy Nash equilibrium: (1/2B + 1/2C, 1/2X + 1/2Y ). 4. (8 points) The market (inverse) demand function for a good is P (q) = 18 − q. There are 7 firms, each with a constant marginal cost of 2 for producing each unit of the good. The 7 firms compete by setting their quantities of production, and the price of the good is determined by the market demand function given the total quantity. Calculate the symmetric pure-strategy Nash equilibrium in this game and the corresponding market price. Answer: 3 Fix a symmetric strategy profile in which every firm produces q units. The profit of any firm (say firm 1) when he produces q1 and the others each produce q is: Π1 (q1 ) = q1 (P (6q + q1 ) − 2) = q1 (18 − 6q − q1 − 2). Given q, firm 1 wants to maximize Π1 (q1 ). The first order condition with respect to q1 is 18 − 6q − 2q1 − 2 = 0, which gives the best-response function: q1 = 8 − 3q. At a symmetric Nash equilibrium, we must have q1 = q. Therefore, q = 8 − 3q, which gives q = 2. In summary, each firm produces q = 2 is the symmetric pure-strategy Nash equilibrium. The equilibrium price is 18 − 2 × 7 = 4. 4 5