ME TUTORIAL 5 SOLUTIONS – OLIGOPOLY AND GAME THEORY
Ans 1. Let 𝑞𝐴 and 𝑞𝐶 be the quantities decided by Andy and Cory, respectively.
a) Revenue for Andy, 𝑅𝐴 = 𝑝 ∗ 𝑞𝐴 = [6 − 0.01𝑞] ∗ 𝑞𝐴 = [6 − 0.01(𝑞𝐴 + 𝑞𝐶 )] ∗ 𝑞𝐴
Marginal Revenue for Andy, 𝑀𝑅𝐴 = 6 − 0.02𝑞𝐴 − 0.01𝑞𝐶
Marginal Cost is given for Andy, 𝑀𝐶𝐴 = 1
Andy will optimally produce at the point where
𝑀𝑅𝐴 = 𝑀𝐶𝐴
=> 6 − 0.02𝑞𝐴 − 0.01𝑞𝐶 = 1
𝑞𝐴 = 250 − 0.5 ∗ 𝑞𝐶
This is the reaction function for Andy.
b) Similarly, Revenue for Cory, 𝑅𝐶 = 𝑝 ∗ 𝑞𝐶 = [6 − 0.01𝑞] ∗ 𝑞𝐶 = [6 − 0.01(𝑞𝐴 + 𝑞𝐶 )] ∗
𝑞𝐶
Marginal Revenue for Cory, 𝑀𝑅𝐶 = 6 − 0.02𝑞𝐶 − 0.01𝑞𝐴
Marginal Cost is given for Cory, 𝑀𝐶𝐶 = 2
Cory will optimally produce at the point where
𝑀𝑅𝐶 = 𝑀𝐶𝐶
=> 6 − 0.02𝑞𝐶 − 0.01𝑞𝐴 = 2
𝑞𝐶 = 200 − 0.5 ∗ 𝑞𝐴
This is the reaction function for Cory.
c) The Cournot Equilibrium is found by equating the two reaction functions
𝑞𝐴 = 250 − 0.5 ∗ 𝑞𝐶
𝑞𝐶 = 200 − 0.5 ∗ 𝑞𝐴
Cournot equilibrium is at qA = 200 and qC = 100
, d) Price set by the market based on the Cournot quantities:
𝑝 = [6 − 0.01𝑞] = [6 − 0.01(𝑞𝐴 + 𝑞𝐶 )] = 3
Revenue for Andy, 𝑇𝑅𝐴 = 𝑝 ∗ 𝑞𝐴 = 3 ∗ 200 = 600
Cost for Andy , 𝑇𝐶𝐴 = 𝑀𝐶𝐴 ∗ 𝑞𝐴 = 1 ∗ 200 = 200
Profit for Andy = 𝑇𝑅𝐴 − 𝑇𝐶𝐴 = 400
Similarly,
Revenue for Cory, 𝑇𝑅𝐶 = 𝑝 ∗ 𝑞𝐶 = 3 ∗ 100 = 300
Cost for Cory , 𝑇𝐶𝐶 = 𝑀𝐶𝐶 ∗ 𝑞𝐶 = 2 ∗ 100 = 200
Profit for Cory = 𝑇𝑅𝐶 − 𝑇𝐶𝐶 = 100
Thus, Andy makes a profit of $2 on each of 200 loaves and Cory makes $1 on each of 100
loaves
Ans 2. a) In a perfectly competitive market,
𝑝 = 𝑀𝐶
100 − 2𝑌 = 4
=> industry output = 48 and industry price = $4
b) If the market is a duopoly and firms are competing on quantities (Cournot Duopoly)
Revenue for Firm 1, 𝑇𝑅1 = 𝑝 ∗ 𝑞1 = [100 − 2𝑌] ∗ 𝑌1 = [100 − 2(𝑌1 + 𝑌2 )] ∗ 𝑌1
Marginal Revenue for Firm 1, 𝑀𝑅1 = 100 − 4𝑌1 − 2𝑌2
Marginal Cost is given for Firm 1, 𝑀𝐶1 = 4
Firm 1 will optimally produce at the point where
𝑀𝑅1 = 𝑀𝐶1
=> 100 − 4𝑌1 − 2𝑌2 = 4
𝑌1 = 24 − 0.5 ∗ 𝑌2
This is the reaction function for Firm 1.
Similarly, the reaction function for Firm 2 will be
𝑌2 = 24 − 0.5 ∗ 𝑌1
Ans 1. Let 𝑞𝐴 and 𝑞𝐶 be the quantities decided by Andy and Cory, respectively.
a) Revenue for Andy, 𝑅𝐴 = 𝑝 ∗ 𝑞𝐴 = [6 − 0.01𝑞] ∗ 𝑞𝐴 = [6 − 0.01(𝑞𝐴 + 𝑞𝐶 )] ∗ 𝑞𝐴
Marginal Revenue for Andy, 𝑀𝑅𝐴 = 6 − 0.02𝑞𝐴 − 0.01𝑞𝐶
Marginal Cost is given for Andy, 𝑀𝐶𝐴 = 1
Andy will optimally produce at the point where
𝑀𝑅𝐴 = 𝑀𝐶𝐴
=> 6 − 0.02𝑞𝐴 − 0.01𝑞𝐶 = 1
𝑞𝐴 = 250 − 0.5 ∗ 𝑞𝐶
This is the reaction function for Andy.
b) Similarly, Revenue for Cory, 𝑅𝐶 = 𝑝 ∗ 𝑞𝐶 = [6 − 0.01𝑞] ∗ 𝑞𝐶 = [6 − 0.01(𝑞𝐴 + 𝑞𝐶 )] ∗
𝑞𝐶
Marginal Revenue for Cory, 𝑀𝑅𝐶 = 6 − 0.02𝑞𝐶 − 0.01𝑞𝐴
Marginal Cost is given for Cory, 𝑀𝐶𝐶 = 2
Cory will optimally produce at the point where
𝑀𝑅𝐶 = 𝑀𝐶𝐶
=> 6 − 0.02𝑞𝐶 − 0.01𝑞𝐴 = 2
𝑞𝐶 = 200 − 0.5 ∗ 𝑞𝐴
This is the reaction function for Cory.
c) The Cournot Equilibrium is found by equating the two reaction functions
𝑞𝐴 = 250 − 0.5 ∗ 𝑞𝐶
𝑞𝐶 = 200 − 0.5 ∗ 𝑞𝐴
Cournot equilibrium is at qA = 200 and qC = 100
, d) Price set by the market based on the Cournot quantities:
𝑝 = [6 − 0.01𝑞] = [6 − 0.01(𝑞𝐴 + 𝑞𝐶 )] = 3
Revenue for Andy, 𝑇𝑅𝐴 = 𝑝 ∗ 𝑞𝐴 = 3 ∗ 200 = 600
Cost for Andy , 𝑇𝐶𝐴 = 𝑀𝐶𝐴 ∗ 𝑞𝐴 = 1 ∗ 200 = 200
Profit for Andy = 𝑇𝑅𝐴 − 𝑇𝐶𝐴 = 400
Similarly,
Revenue for Cory, 𝑇𝑅𝐶 = 𝑝 ∗ 𝑞𝐶 = 3 ∗ 100 = 300
Cost for Cory , 𝑇𝐶𝐶 = 𝑀𝐶𝐶 ∗ 𝑞𝐶 = 2 ∗ 100 = 200
Profit for Cory = 𝑇𝑅𝐶 − 𝑇𝐶𝐶 = 100
Thus, Andy makes a profit of $2 on each of 200 loaves and Cory makes $1 on each of 100
loaves
Ans 2. a) In a perfectly competitive market,
𝑝 = 𝑀𝐶
100 − 2𝑌 = 4
=> industry output = 48 and industry price = $4
b) If the market is a duopoly and firms are competing on quantities (Cournot Duopoly)
Revenue for Firm 1, 𝑇𝑅1 = 𝑝 ∗ 𝑞1 = [100 − 2𝑌] ∗ 𝑌1 = [100 − 2(𝑌1 + 𝑌2 )] ∗ 𝑌1
Marginal Revenue for Firm 1, 𝑀𝑅1 = 100 − 4𝑌1 − 2𝑌2
Marginal Cost is given for Firm 1, 𝑀𝐶1 = 4
Firm 1 will optimally produce at the point where
𝑀𝑅1 = 𝑀𝐶1
=> 100 − 4𝑌1 − 2𝑌2 = 4
𝑌1 = 24 − 0.5 ∗ 𝑌2
This is the reaction function for Firm 1.
Similarly, the reaction function for Firm 2 will be
𝑌2 = 24 − 0.5 ∗ 𝑌1
