Cournot Equilibrium The dancing mcahine industry is a duopoly. The two firms, Ch
ID: 1167749 • Letter: C
Question
Cournot Equilibrium
The dancing mcahine industry is a duopoly. The two firms, Chuckie B corp and Gene Gene Dancing Machines, compete through Cournot quantity-setting competition. The demand curve for the industry is P = 120-Q, where Q is the total quantity produced by Chuckie B and Gene Gene. Currently, each firm has marginal cost of $60 and not fixed cost.
First question is What is the equilibirum price, quantity, and profit for each firm? I got price:$80, Quantity (each):20, and Profit:$400.00
Part B is where I am struggling
Chuckie B corp. is considering implementing a proprietary technology with a one-time sunk cost of $200. Once this investment is made, marginal cost will be reduced to $40. Gene Gene has no access to this or any other cost saving technology, and its marginal cost will remain at $60. SHould CHuckie B invest in the new technology? (hint: you must compute another cournot equilibrium)
Explanation / Answer
(1)
Let's denote Chuckie B as 1 & Gene Gene as 2.
P = 120 - (Q1 + Q2)
MC = 60
In Cournot model,
MR1 = MC = MR2
TR1 = P x Q1
= 120Q1 - Q12 - Q1Q2
MR1 = dTR1 / dQ = 120 - 2Q1 - Q2
Equating with MC = 60,
120 - 2Q1 - Q2 = 60
2Q1 + Q2 = 60 (1)
TR2 = 120Q2 - Q1Q2 - Q22
MR2 = 120 - Q1 - 2Q2
So,
120 - Q1 - 2Q2 = 60
Q1 + 2Q2 = 60 (2)
Solving (1) & (2):
Q1 = Q2 = 20
P = 120 - 40 = 80
So, TR1 = 20 x 80 = 1600
TC = Q1 x MC = 20 x 60 = 1200
Profit for 1 = 1600 - 1200 = 400
(2)
Now, MC1 = 40 & MC2 = 60
MR1 = MC1 implies
120 - 2Q1 - Q2 = 40
2Q1 + Q2 = 80 (3)
MR2 = MC2 implies
120 - Q1 - 2Q2 = 60
Q1 + 2Q2 = 60 (4)
Solving (3) & (4),
Q1 = 33.33, Q2 = 13.33, Q = 46.66
P = 120 - 46.66 = 73.34
TR1 = P x Q1
= 33.33 x 73.34
= 2,444.42
TC1 = (Q1 x MC1)
= (33.33 x 40)
= 1,333.20
Profit = 1,111.09 > 400 in previous case.
So, it should invest in the technology since profit increases.
So, 1 will invest in new technology, if (a) He wants to get this pro