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 s₁ 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.
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 s₁ 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.
Chapter1: Making Economics Decisions
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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
S₁ 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 (S1, S2) = (1 − 81 S2 - 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%2F16422919-81b5-4cbd-b847-25e864aab8a7%2F5dw9tc_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
S₁ 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 (S1, S2) = (1 − 81 S2 - 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 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
1
D1 ($1, $2)
=
{
if 81 82,
if s₁ = 82,
0
if 81 82,
the demand Firm 2 would face is given by
if 82
< S1,
1
D2 (81, 82)
=
if 82 = 81,
0
if 82 > $1,
=
the utility of Firm 1 would be their profit, u₁ (S1, S2) (811)D1 (S1, S2), 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%2F16422919-81b5-4cbd-b847-25e864aab8a7%2F66l4fcd_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
1
D1 ($1, $2)
=
{
if 81 82,
if s₁ = 82,
0
if 81 82,
the demand Firm 2 would face is given by
if 82
< S1,
1
D2 (81, 82)
=
if 82 = 81,
0
if 82 > $1,
=
the utility of Firm 1 would be their profit, u₁ (S1, S2) (811)D1 (S1, S2), 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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