Econ game theory Exam, 11 hour limit!!!

Exam 2

Rachel Mannahan

Econ 431

Name: (30 points)

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Instructions (read these thoroughly before you begin the exam and again before turning in):

1. This exam is due by 11:59pm on 4/2/20. Turn in your exam by submitting it through the “Assignments” tab in d2l.

2. Your exam must be HANDWRITTEN. I will not accept typed exams (unless the instructor has granted you permission).

3. Make sure you have answered and labeled each question thoroughly (if you turn in your answers on notebook paper). You may not receive points if you write answers without explanation, or if we cannot determine which question you are answering.

4. Do your OWN work. This is not a group exam. You may not speak with other class- mates about the exam until the exam window has passed or share any work regarding the exam. If you violate this rule, you will be reported to the University for academic dishonesty. You may use calculators or notes to help you complete this exam.

5. The instructor and TA will answer emails about clarifications, but will not answer emails regarding course content during the exam window.

6. Check d2l announcements for any updates regarding the exam (i.e. if there is confusing wording that I clear up, etc.).

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1. (15) Two firms compete in a market, simultaneously choosing quantities q1 ≥ 0 and q2 ≥ 0. Firm 1 faces a marginal cost of 3 per unit produced. Firm 2 faces a marginal cost of 1 per unit produced. The market price is given by p = 26 − q1 − 3q2.

(a) Find all pure strategy Nash equilibria. (10)

(b) Calculate each firm’s equilibrium profit. (3)

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(c) Is this a market for homogeneous (exact same) or heterogeneous (very similar) goods? How can you tell? Explain. (2)

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2. (20) Two firms compete in a market. Firm 1 moves first and chooses a quantity q1 ≥ 0 followed by firm 2 (who observes firm 1’s choice) and chooses q2 ≥ 0. Firm 1 faces a marginal cost of 0 per unit produced. Firm 2 faces a marginal cost of 1 per unit produced. The market price is given by p = 24− 3q1− 3q2. Note: This is not the same inverse demand as in problem 1.

(a) Find all subgame perfect equilibria. (10)

(b) Calculate each firm’s equilibrium profits. (4)

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(c) Write down a Nash equilibrium that is not a subgame perfect equilibrium of this game. (2)

(d) Is this game symmetric? Explain. (2)

(e) Is either firm better or worse of than they would be in the corresponding simul- taneous move version of this game? Explain. (2)

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3. (15) Consider a repeated game in which the stage game in the following figure is played in each of two periods with no discounting:

Player 2

D E F

Player 1

A 6, 8 1, 9 0, 0

B 8, 1 0, 0 5, 2

C 2, 3 2, 4 5, 4

(a) Consider the following strategy: In period 1, play (B,D). In period 2, play (A,E) following anything in period 1. Is this a SPE? Explain. (5)

(b) Consider the following strategy: In period 1, play (C,E). In period 2, play (C,E) following anything in period 1. Is this a SPE? Explain. (5)

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(c) Is there a SPE in which (A,D) is played in the first period? Fully describe the strategy that might sustain this SPE. Then, show (by writing out the utilities of each player) whether or not this is a SPE. (5)

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4. (20) Suppose that we have the following game:

Player 2

C D

Player 1 A 1, 2 −1, 4 B 2,−2 0, 0

(a) Suppose this game is repeated twice with no discounting. Describe the subgame perfect equilibrium. (4)

(b) In the infinitely repeated game with discount rate δ (same for each player), con- sider the following strategy. Play (A,C) forever. If either player deviates, play (B,D) forever. Is this a SPE? If so, for which value of δ is this a SPE? If not, show why not. (Hint: This game is asymmetric so you need to show work for 2 players. You need not check deviations from the NE.) (16)

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5. (5 points Extra Credit) Write down the extra credit code word or phrase from our last in-person class. Remember, you will only receive credit if the number of students who write down the correct word is less than the number of students who were in the class that day.

6. (10 points Extra Credit) Consider the p-beauty contest game with a twist. There are two students in the room and both guess an integer between 1 and 5 inclusive. We will then calculate the average of the guesses and multiply that number by p. The student whose guess is closest to p times the average guess wins. The winner’s utility equals the number they guessed (i.e. if there is one winner who guesses 4, the payoff to the winner is 4 utils). If both students submit the same guess, they split the payoff that they guessed (i.e. if both students guess 4, they both receive a payoff of 2 utils). Suppose p = 2

3 . (Hint: Don’t mix up payoffs and strategies in this game.)

(a) Find all pure strategy Nash equilibria in this game. (Show work or explain). (4)

(b) Suppose that this game is repeated infinitely with discount rate δ. Is there a SPE in which we play (5, 5) forever? Try using a grim-trigger strategy to show whether this can be sustained. Since the game is symmetric, you will only need to set up one inequality. (6)

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