Consider the Cournot duopoly game with demand p = 100 − (q1+ q2) and variable costs ci(qi) = 0 for i ∈ {1, 2}. The twist is that there is now a fixed cost of production k > 0 that is the same for both firms. a. Assume first that both firms choose their quantities simultaneously. Model this as a normal-form game. b. Write down the firm’s best-response function for k = 1000 and solve for a pure-strategy Nash equilibrium. Is it unique? c. Now assume that firm 1 is a “Stackelberg leader” in the sense that it moves first and chooses q1. Then after observing q1 firm 2 chooses q2. Also assume that if firm 2 cannot make strictly positive profits then it will not produce at all. Model this as an extensive-form game tree as best you can and find a subgame-perfect equilibrium of this game for k = 25. Is it unique?
Consider the Cournot duopoly game with demand p = 100 − (q1+ q2) and variable costs ci(qi) = 0 for i ∈ {1, 2}. The twist is that there is now a fixed cost of production k > 0 that is the same for both firms. a. Assume first that both firms choose their quantities simultaneously. Model this as a normal-form game. b. Write down the firm’s best-response function for k = 1000 and solve for a pure-strategy Nash equilibrium. Is it unique? c. Now assume that firm 1 is a “Stackelberg leader” in the sense that it moves first and chooses q1. Then after observing q1 firm 2 chooses q2. Also assume that if firm 2 cannot make strictly positive profits then it will not produce at all. Model this as an extensive-form game tree as best you can and find a subgame-perfect equilibrium of this game for k = 25. Is it unique? d. How does your answer in (c) change for k = 225?
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