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Consider a variant version of the Bertrand model. Two firms sell differentiated products that are substitutes. Specifically, if Firm 1 charges p1 and Firm 2 charges p2, Firm 1’s demand will be D1 (p1, p2) = 20 p1 + p2 and Firm 2’s demand will be D2(p1, p2) = 20 p2 + p1. Both firms have a marginal cost of 4 and no fixed cost. Each firm chooses its own price to maximize its profit. Formulate this into a strategic game and find the Nash equilibrium of the game.

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Joshua Stredder
Joshua StredderLv10
21 Jul 2021
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Related questions

Consider the following Bertrand competition model. Two firms choose their prices simultaneously and the market demand is Q(p). A firm with the lowest price serves the entire market. They split the demand equally if prices are equal. Each firm i = 1, 2 has constant unit of production (i.e., marginal cost) ci > 0 where c1 > c2,and so the firms marginal costs are different. a) Show that p1 = p2 = c1 is not a Nash equilibrium. b) Show that there is no Nash equilibrium where the firms pick the same price (i.e., p1 = p2). c) Is there a Nash equilibrium of this game? d) Suppose now that the prices are discrete and so firm is price must be a multiple of a cent: That is, pi - {0.01, 0.02, 0.03, ..., 0.01n, ...} where n is some positive integer. What is the Nash equilibrium now?

Assume that two firms, say firm 1 and firm 2, sell differentiated goods and face the following demand functions:

q1 = 4/3 - 2/3a - 4/3 p1 + 2/3 p2 and q2 = 4/3a - 2/3 + 2/3 p1 - 4/3 p2

where a > 1. In addition assume that the firms target own profit maximization, compete (simultaneously) in quantities and have marginal costs equal to c1 = c2 = c. Assume that 1 > c > or equal to 0. Describing the necessary mathematical steps, and providing the related mathematical intuition,

1. Find the inverse demands faced by the two firms (Hint: Solve the two demands simultaneously with respect to prices.

2. Derive each firm's best response function. Are quantities strategic substitutes or complements.

3. Derive the equilibrium in quantities.

4. Explain the economic meaning of parameter a. What happens to the equilibrium quantities and profits if a increases?

The instruction for this question is that I have to make sure to use proper mathematical notation used.

Firms 1 and 2 produce horizontally differentiated products. The demand for firm 1’s product is given by the equation, Q1 = 100 − P1 + P2 2 . The demand for firm 2’s product is given by the equation, Q2 = 200 − 4P2 + 2P1. Firm 1’s marginal cost is MC1 = $10, while firm 2’s marginal cost is MC2 = $20. The two firms compete in Bertrand competition, by simultaneously selecting prices. Question 1: Is firm 2’s product a substitute or a compliment for firm 1’s product? Briefly explain. Your answer must reference firm 1’s demand function. (2 Marks) Question 2: Does the demand for firm 2’s product satisfy the law of demand? Briefly explain. Your answer must reference firm 2’s demand function. (2 Marks) Question 3: What is the equation of firm 1’s best-response function? (4 Marks) Question 4: What is the equation of firm 2’s best-response function? (4 Marks) Question 5: Find the equilibrium prices. (2 Marks) Question 6: Find the equilibrium profits. (3 Marks) Question 7: Which firm enjoys the greater market power? (3 Marks)Firms 1 and 2 produce horizontally differentiated products. The demand for firm 1’s product is given by the equation, Q1 = 100 − P1 + P2 2 . The demand for firm 2’s product is given by the equation, Q2 = 200 − 4P2 + 2P1. Firm 1’s marginal cost is MC1 = $10, while firm 2’s marginal cost is MC2 = $20. The two firms compete in Bertrand competition, by simultaneously selecting prices. Question 1: Is firm 2’s product a substitute or a compliment for firm 1’s product? Briefly explain. Your answer must reference firm 1’s demand function. (2 Marks) Question 2: Does the demand for firm 2’s product satisfy the law of demand? Briefly explain. Your answer must reference firm 2’s demand function. (2 Marks) Question 3: What is the equation of firm 1’s best-response function? (4 Marks) Question 4: What is the equation of firm 2’s best-response function? (4 Marks) Question 5: Find the equilibrium prices. (2 Marks) Question 6: Find the equilibrium profits. (3 Marks) Question 7: Which firm enjoys the greater market power? (3 Marks)

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