Consider now a Bertrand competition environment with one good and two firms, Firm 1 and Firm 2. The (pure) strategy space of Firm 1 is S₁ = [0,5], and the strategy 81 of Firm 1 corresponds to the price they charge for the good. Similarly, the (pure) strategy space of Firm 2 is S₂ = [0,5], and the strategy s2 of Firm 2 corresponds to the price they charge for the good. If Firm 1 were to choose price s₁ and Firm 2 were to choose price s2, then the demand Firm 1 would face is given by if 81 82, D1 (81, 82): the demand Firm 2 would face is given by = 1 if s1 = 82, 0 if 81 82, D2 ($1, $2 82) = if s2 < S1, if 82 = 81, 0 if 82 $1, - the utility of Firm 1 would be their profit, u₁(s1, 82) = (81 − 1)D₁ (81, 82), and the utility of Firm 2 would be their profit, u2(81, 82) = (82 − 1)D2(81, 82). (The marginal cost of production for both firms is 1.) (c) Find the pure-strategy Nash equilibria of this game. Consider now an altered version of the Bertrand competition environment above in which the (pure) strategy space of Firm 1 is S₁ {0, 1, 2, 3, 4, 5}, the (pure) strategy space of Firm 2 is S₂ = {0, 1, 2, 3, 4, 5}, and otherwise the game is the same as the Bertrand competition environment above. = (d) Find the pure-strategy Nash equilibria of this game. 2. Consider a Cournot competition environment with one good and two firms, Firm 1 and Firm 2. The (pure) strategy space of Firm 1 is S₁ = [0,1], and the strategy s₁ of Firm 1 corresponds to the amount of the good they produce. Similarly, the (pure) strategy space of Firm 2 is S₂ = [0, 1], and the strategy 82 of Firm 2 corresponds to the amount of the good they produce. If Firm 1 were to produce quantity 81 and Firm 2 were to produce quantity s2, the prevailing price in the good market would be 1— 81 - 82, the utility of Firm 1 would be their profit, u₁(81, 82) = (1 − 81 − 82 - c) s₁, and the utility of Firm 2 would be their profit, u2 (81, 82) = (1- 81 - 82 - c) s2, where 0 ≤ c < 1 is the marginal cost of production for both firms. (a) Find the pure-strategy Nash equilibria of this game. (b) Are there other Nash equilibria in this game. -
Consider now a Bertrand competition environment with one good and two firms, Firm 1 and Firm 2. The (pure) strategy space of Firm 1 is S₁ = [0,5], and the strategy 81 of Firm 1 corresponds to the price they charge for the good. Similarly, the (pure) strategy space of Firm 2 is S₂ = [0,5], and the strategy s2 of Firm 2 corresponds to the price they charge for the good. If Firm 1 were to choose price s₁ and Firm 2 were to choose price s2, then the demand Firm 1 would face is given by if 81 82, D1 (81, 82): the demand Firm 2 would face is given by = 1 if s1 = 82, 0 if 81 82, D2 ($1, $2 82) = if s2 < S1, if 82 = 81, 0 if 82 $1, - the utility of Firm 1 would be their profit, u₁(s1, 82) = (81 − 1)D₁ (81, 82), and the utility of Firm 2 would be their profit, u2(81, 82) = (82 − 1)D2(81, 82). (The marginal cost of production for both firms is 1.) (c) Find the pure-strategy Nash equilibria of this game. Consider now an altered version of the Bertrand competition environment above in which the (pure) strategy space of Firm 1 is S₁ {0, 1, 2, 3, 4, 5}, the (pure) strategy space of Firm 2 is S₂ = {0, 1, 2, 3, 4, 5}, and otherwise the game is the same as the Bertrand competition environment above. = (d) Find the pure-strategy Nash equilibria of this game. 2. Consider a Cournot competition environment with one good and two firms, Firm 1 and Firm 2. The (pure) strategy space of Firm 1 is S₁ = [0,1], and the strategy s₁ of Firm 1 corresponds to the amount of the good they produce. Similarly, the (pure) strategy space of Firm 2 is S₂ = [0, 1], and the strategy 82 of Firm 2 corresponds to the amount of the good they produce. If Firm 1 were to produce quantity 81 and Firm 2 were to produce quantity s2, the prevailing price in the good market would be 1— 81 - 82, the utility of Firm 1 would be their profit, u₁(81, 82) = (1 − 81 − 82 - c) s₁, and the utility of Firm 2 would be their profit, u2 (81, 82) = (1- 81 - 82 - c) s2, where 0 ≤ c < 1 is the marginal cost of production for both firms. (a) Find the pure-strategy Nash equilibria of this game. (b) Are there other Nash equilibria in this game. -
Chapter15: Imperfect Competition
Section: Chapter Questions
Problem 15.8P
Related questions
Question
please help with problem 2, thank you!
![Consider now a Bertrand competition environment with one good and two firms, Firm 1 and Firm 2.
The (pure) strategy space of Firm 1 is S₁ = [0,5], and the strategy 81 of Firm 1 corresponds to the
price they charge for the good. Similarly, the (pure) strategy space of Firm 2 is S₂ = [0,5], and the
strategy s2 of Firm 2 corresponds to the price they charge for the good. If Firm 1 were to choose price
s₁ and Firm 2 were to choose price s2, then the demand Firm 1 would face is given by
if 81 82,
D1 (81, 82):
the demand Firm 2 would face is given by
=
1
if s1 = 82,
0
if 81 82,
D2 ($1, $2
82)
=
if s2
< S1,
if 82 = 81,
0 if 82 $1,
-
the utility of Firm 1 would be their profit, u₁(s1, 82) = (81 − 1)D₁ (81, 82), and the utility of Firm 2
would be their profit, u2(81, 82) = (82 − 1)D2(81, 82). (The marginal cost of production for both firms
is 1.)
(c) Find the pure-strategy Nash equilibria of this game.
Consider now an altered version of the Bertrand competition environment above in which the (pure)
strategy space of Firm 1 is S₁ {0, 1, 2, 3, 4, 5}, the (pure) strategy space of Firm 2 is S₂ = {0, 1, 2, 3, 4, 5},
and otherwise the game is the same as the Bertrand competition environment above.
=
(d) Find the pure-strategy Nash equilibria of this game.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F10cd151f-f064-48f6-9113-d3c525d6c2e1%2F45e7d68c-3125-4261-95cf-3272710f6cea%2Frxlwlbo_processed.png&w=3840&q=75)
Transcribed Image Text:Consider now a Bertrand competition environment with one good and two firms, Firm 1 and Firm 2.
The (pure) strategy space of Firm 1 is S₁ = [0,5], and the strategy 81 of Firm 1 corresponds to the
price they charge for the good. Similarly, the (pure) strategy space of Firm 2 is S₂ = [0,5], and the
strategy s2 of Firm 2 corresponds to the price they charge for the good. If Firm 1 were to choose price
s₁ and Firm 2 were to choose price s2, then the demand Firm 1 would face is given by
if 81 82,
D1 (81, 82):
the demand Firm 2 would face is given by
=
1
if s1 = 82,
0
if 81 82,
D2 ($1, $2
82)
=
if s2
< S1,
if 82 = 81,
0 if 82 $1,
-
the utility of Firm 1 would be their profit, u₁(s1, 82) = (81 − 1)D₁ (81, 82), and the utility of Firm 2
would be their profit, u2(81, 82) = (82 − 1)D2(81, 82). (The marginal cost of production for both firms
is 1.)
(c) Find the pure-strategy Nash equilibria of this game.
Consider now an altered version of the Bertrand competition environment above in which the (pure)
strategy space of Firm 1 is S₁ {0, 1, 2, 3, 4, 5}, the (pure) strategy space of Firm 2 is S₂ = {0, 1, 2, 3, 4, 5},
and otherwise the game is the same as the Bertrand competition environment above.
=
(d) Find the pure-strategy Nash equilibria of this game.
![2. Consider a Cournot competition environment with one good and two firms, Firm 1 and Firm 2. The
(pure) strategy space of Firm 1 is S₁ = [0,1], and the strategy s₁ of Firm 1 corresponds to the amount
of the good they produce. Similarly, the (pure) strategy space of Firm 2 is S₂ = [0, 1], and the strategy
82 of Firm 2 corresponds to the amount of the good they produce. If Firm 1 were to produce quantity
81 and Firm 2 were to produce quantity s2, the prevailing price in the good market would be 1— 81 - 82,
the utility of Firm 1 would be their profit, u₁(81, 82) = (1 − 81 − 82 - c) s₁, and the utility of Firm 2
would be their profit, u2 (81, 82) = (1- 81 - 82 - c) s2, where 0 ≤ c < 1 is the marginal cost of production
for both firms.
(a) Find the pure-strategy Nash equilibria of this game.
(b) Are there other Nash equilibria in this game.
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F10cd151f-f064-48f6-9113-d3c525d6c2e1%2F45e7d68c-3125-4261-95cf-3272710f6cea%2F3behr5_processed.png&w=3840&q=75)
Transcribed Image Text:2. Consider a Cournot competition environment with one good and two firms, Firm 1 and Firm 2. The
(pure) strategy space of Firm 1 is S₁ = [0,1], and the strategy s₁ of Firm 1 corresponds to the amount
of the good they produce. Similarly, the (pure) strategy space of Firm 2 is S₂ = [0, 1], and the strategy
82 of Firm 2 corresponds to the amount of the good they produce. If Firm 1 were to produce quantity
81 and Firm 2 were to produce quantity s2, the prevailing price in the good market would be 1— 81 - 82,
the utility of Firm 1 would be their profit, u₁(81, 82) = (1 − 81 − 82 - c) s₁, and the utility of Firm 2
would be their profit, u2 (81, 82) = (1- 81 - 82 - c) s2, where 0 ≤ c < 1 is the marginal cost of production
for both firms.
(a) Find the pure-strategy Nash equilibria of this game.
(b) Are there other Nash equilibria in this game.
-
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