Two firms produce goods that are imperfect substitutes. If firm 1 charges price p1 and firm 2 charges price p2, then their respective demands are q1 = 12 - 2p1 + p2 and q2 = 12 + p1 - 2p2 So this is like Bertrand competition, except that when p1 > p2, firm 1 still gets a positive demand for its product. Regulation does not allow either firm to charge a price higher than 20. Both firms have a constant marginal cost c = 4. (a) Construct the best reply function BR1(p2) for firm 1. That is, p1 = BR1(p2) is the optimal price for firm 1 if it is known that firm 2 charges a price p2. Construct a Nash equilibrium in pure strategies for this game. Are there any Nash equilibria in mixed strategies? If yes, construct one; if no provide a justification.(b) Notice that for any given price p1, firm 1’s demand increases with p2, so firm 1 is better off when firm 2 charges a high price p2. What is the best reply to p2 = 20? What is the best reply to p2 = 0 (c) What prices for firm 1 are not strictly dominated? What prices would survive two rounds of strict dominance? Provide a reason for each strategy that you eliminate.(d) Challenge question: If you continue the iterative elimination of strictly dominated strategies, what strategies will survive? (1) Two firms produce goods that are imperfect substitutes. If firm 1 charges price pi andfirm 2 charges price p2, then their respective demands areq1 = 12 – 2p1 + P2 and q2 = 12+p1 – 2p2.So this is like Bertrand competition, except that when p1 > p2, firm 1 still gets a positivedemand for its product. Regulation does not allow either firm to charge a price higherthan 20. Both firms have a constant marginal cost c = 4.(a) Construct the best reply function BR1(p2) for firm 1. That is, pi =the optimal price for firm 1 if it is known that firm 2 charges a price p2. Construct aNash equilibrium in pure strategies for this game. Are there any Nash equilibria in mixedstrategies? If yes, construct one; if no provide a justification.BR1(p2) is(b) Notice that for any given price P1, firm l's demand increases with p2, so firm 1 is betteroff when firm 2 charges a high price p2. What is the best reply to p2 = 20? What is thebest reply to p2 = 0?(c) What prices for firm 1 are not strictly dominated? What prices would survive tworounds of strict dominance? Provide a reason for each strategy that you eliminate.(d) Challenge question: If you continue the iterative elimination of strictly dominatedstrategies, what strategies will survive?

Demand function for firm 1 : q1 = 12 - 2p1 + p2 Demand function for firm 2 : q2 = 12 + p1 - 2p2…

Two firms produce goods that are imperfect substitutes. If firm 1 charges price p1 and firm 2 charges price p2, then their respective demands are q1 = 12 - 2p1 + p2 and q2 = 12 + p1 - 2p2 So this is like Bertrand competition, except that when p1 > p2, firm 1 still gets a positive demand for its product. Regulation does not allow either firm to charge a price higher than 20. Both firms have a constant marginal cost c = 4. (a) Construct the best reply function BR1(p2) for firm 1. That is, p1 = BR1(p2) is the optimal price for firm 1 if it is known that firm 2 charges a price p2. Construct a Nash equilibrium in pure strategies for this game. Are there any Nash equilibria in mixed strategies? If yes, construct one; if no provide a justification. (b) Notice that for any given price p1, firm 1’s demand increases with p2, so firm 1 is better off when firm 2 charges a high price p2. What is the best reply to p2 = 20? What is the best reply to p2 = 0 (c) What prices for firm 1 are not strictly dominated? What prices would survive two rounds of strict dominance? Provide a reason for each strategy that you eliminate. (d) Challenge question: If you continue the iterative elimination of strictly dominated strategies, what strategies will survive?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter15: Imperfect Competition

Section: Chapter Questions

Problem 15.4P

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Question

Two firms produce goods that are imperfect substitutes. If firm 1 charges price p1 and firm 2 charges price p2, then their respective demands are

q₁ = 12 - 2p₁ + p₂ and q₂= 12 + p₁ - 2p₂

So this is like Bertrand competition, except that when p1 > p2, firm 1 still gets a positive demand for its product. Regulation does not allow either firm to charge a price higher than 20. Both firms have a constant marginal cost c = 4.

(a) Construct the best reply function BR1(p2) for firm 1. That is, p1 = BR1(p2) is the optimal price for firm 1 if it is known that firm 2 charges a price p2. Construct a Nash equilibrium in pure strategies for this game. Are there any Nash equilibria in mixed strategies? If yes, construct one; if no provide a justification.

(b) Notice that for any given price p1, firm 1’s demand increases with p2, so firm 1 is better off when firm 2 charges a high price p2. What is the best reply to p2 = 20? What is the best reply to p2 = 0

(c) What prices for firm 1 are not strictly dominated? What prices would survive two rounds of strict dominance? Provide a reason for each strategy that you eliminate.

(d) Challenge question: If you continue the iterative elimination of strictly dominated strategies, what strategies will survive?

(1) Two firms produce goods that are imperfect substitutes. If firm 1 charges price pi and
firm 2 charges price p2, then their respective demands are
q1 = 12 – 2p1 + P2 and q2 = 12+p1 – 2p2.
So this is like Bertrand competition, except that when p1 > p2, firm 1 still gets a positive
demand for its product. Regulation does not allow either firm to charge a price higher
than 20. Both firms have a constant marginal cost c = 4.
(a) Construct the best reply function BR1(p2) for firm 1. That is, pi =
the optimal price for firm 1 if it is known that firm 2 charges a price p2. Construct a
Nash equilibrium in pure strategies for this game. Are there any Nash equilibria in mixed
strategies? If yes, construct one; if no provide a justification.
BR1(p2) is
(b) Notice that for any given price P1, firm l's demand increases with p2, so firm 1 is better
off when firm 2 charges a high price p2. What is the best reply to p2 = 20? What is the
best reply to p2 = 0?
(c) What prices for firm 1 are not strictly dominated? What prices would survive two
rounds of strict dominance? Provide a reason for each strategy that you eliminate.
(d) Challenge question: If you continue the iterative elimination of strictly dominated
strategies, what strategies will survive?

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