Game theory exercises

1. Consider the following Bertrand model with non-homogeneous goods. Two firms face the demand function qi(Pier) = a - m + bpj Fori = 1,2 Marginal cost is equal to average total cost for both firms and it is equal to 6 Hence, the profit function is \"i(Pi:Pj) = Qi(Pi:Pj)[Pi C] a. Derive the Nash equilibrium prices for both firms. b. What would the prices you derived in a. and profits be equal to for a = 10, b = 1, c = 5? c. How do the prices and profits you computed in this exercise compare to the Nash equilibrium prices and profits we get when firms produce a homogeneous good? Explain. 2. Suppose two firms, 1 and 2, produce cell phones. One of the companies uses operating system X and the other operating system Y. The demand for cell phones for firm i = 1,2 is %=m-m+m Assume the production of cell phones is costless, so that the profit function for firm i = 1,2 is \"i = mm a. Derive the reaction function for each firm. b. Derive the price functions. Suppose al- represents advertising expenditures for firms i = 1,2, how do prices change with changes in own advertising expenditures? How do prices change with the "rival\" firm's advertising expenditures? d. Suppose advertising expenditures are equal across the two firms, a1 = (12, would firms compete on prices? Explain. 3. Suppose the inverse demand function in a market composed of two firms is p = 400 6(y1 + 3/2). The cost function for Firm 1 is C1Cy1) = 203/1 and C2012) = 63/2 for Firm 2. a. Discuss a possible outcome for prices and profits for these firms if they were to compete under a Bertrand environment. b. Derive the equilibrium quantities, prices, and profits if the firms compete under a Stackelberg (leading-firm) framework