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ECON 437: Game Theory

Practice Midterm Exam I

Instructions

This exam contains 4 sections. Each section is worth 25 points. For this exam, you may only use pen, pencil, and eraser. Everything else (including calculators, computers, and cell phones) are not allowed. Answer as much as possible in order and show all your work. Make sure your handwriting is clean and easy to read. You have 70 minutes to complete the exam.

Section I: Normal Form Games

Consider the following Normal Form Game in its matrix form:

1.    What is the best response of player 2 to the strategy s1  = (0.5,0,0.5)? (5 points)

2.    What is the payoff of player 1 from the strategy profile (( ,  , ) , (0.5,0.5,0))? (5 points)

3.    Define Nash Equilibrium. (3 points)

4.    Define Normal Form Game. (3 points)

5.    Find the set of Nash equilibria in this game and interpret this result. (5 points)

6.    Find the set of rationalizable actions in this game and interpret this result. (4 points)

Section II: Cournot

Consider a Cournot environment in which the market demand is Q = 100 − P . In this market, there are N identical firms competing. The total cost of producing qi  units for firm i is TCi  = 10qi  + 20.

1.    Define the Cournot equilibrium in this environment. (5 points)

2.    Compute the best response function of firm i . (5 points)

3.    Compute the Cournot equilibrium in this environment. (5 points)

4.    Show that the price in the market approaches 10 as the number of firms increases and provide an explanation for this. (5 points)

5.    Compute the profits of firm i in equilibrium. (5 points)

Section III: Bertrand

Consider a Bertrand environment with N firms. The demand for the commodity produced by firm i is qi  = 10 − pi  + ∑j≠i pj  and the total cost of producing qi  units is TCi  = qi  + 2.

1.    Define the Bertrand equilibrium in this environment. (5 points)

2.    Which is the action set of firm i? (5 points)

3.    Define a Nash equilibrium in this environment. (5 points)

4.    Construct the best response function of firm i . (5 points)

5.    Compute the Bertrand equilibrium in this environment. (5 points)

Section IV: Stackelberg

Consider a Stackelberg environment in which firm 1 chooses q1  first, and firm two chooses q2  after it sees what firm 1 has produced. The market demand is Q = 100 − P, and the total cost of production for both firms is TCi  = 10qi  + 20.

1.    Construct the reaction function for firm 2. (5 points)

2.    Compute the Stackelberg equilibrium in this environment. (5 points)

3.    Compute the profits of firm 2 in equilibrium. (5 points)

4.    Compute the profits of firm 1 in equilibrium. (5 points)

5.    Explain why the profits of the firms are different in this environment. (5 points)