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 s₁ 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 D1 (81, 82) = if 81 < 82, 1 if 8₁ = 82, if 81 82, the demand Firm 2 would face is given by D2 (81, 82) = 1 12 0 the utility of Firm 1 would be their profit, u₁ (S1, S2) if s2 < $1, if 82 81, = if s2 > S1, = - (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.
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 s₁ 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 D1 (81, 82) = if 81 < 82, 1 if 8₁ = 82, if 81 82, the demand Firm 2 would face is given by D2 (81, 82) = 1 12 0 the utility of Firm 1 would be their profit, u₁ (S1, S2) if s2 < $1, if 82 81, = if s2 > S1, = - (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.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![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 s₁ 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
D1 (81, 82)
=
if 81 < 82,
1 if 8₁ = 82,
if 81 82,
the demand Firm 2 would face is given by
D2 (81, 82)
=
1
12
0
the utility of Firm 1 would be their profit, u₁ (S1, S2)
if s2 < $1,
if 82 81,
=
if s2 > S1,
=
-
(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%2F33efa0ee-e3c4-4640-bf2a-d63d72536f00%2F15c83274-a7fc-41fc-98c5-4d47de8e2a1f%2Fi1641q_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 s₁ 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
D1 (81, 82)
=
if 81 < 82,
1 if 8₁ = 82,
if 81 82,
the demand Firm 2 would face is given by
D2 (81, 82)
=
1
12
0
the utility of Firm 1 would be their profit, u₁ (S1, S2)
if s2 < $1,
if 82 81,
=
if s2 > S1,
=
-
(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.
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