The inverse market demand in a homogeneous product Cournot duopoly is

P=100−2(Q1+Q2)P=100−2(Q1+Q2)

and the costs are given by

C(Q1)=12Q1C(Q1)=12Q1 and C(Q2)=20Q2C(Q2)=20Q2

The implied marginal costs are $12 for firm 1 and $20 for firm 2.

(a) Determine the reaction function for firm 1.

(b) Determine the reaction function for firm 2.

(c) Calculate the Cournot equilibrium price and quantity.

(d) Suppose firm 1 is a monopoly (firm 2 does not exist), what is firm 1's monopoly output and price?

(e) How does the monopoly price and quantity compare with Cournot equilibrium in part (c)?

a. To find the reaction function of Q1, solve the inverse demand function for Q1.

Solve for Q1

P = 100 - 2(Q1 + Q2)

P = 100 - 2Q1 - 2Q2

2Q1 = 100 - P - 2Q2

Q1 = 50 - 0.5P - Q2

b. To find the reaction function of Q2, solve the inverse demand function for Q2.

Solve for Q2

P= 100 - 2(Q1 + Q2)

P = 100 - 2Q1 - 2Q2

2Q2 = 100 - P - 2Q1

Q2 = 50 - 0.5P - Q1

c. Multiply Q1 by price to get the total revenue of Q1. Take the derivative of the total revenue of Q1 to determine the marginal revenue. In profit maximization, set the marginal revenue of Q1 equal to the marginal cost of Q1 and solve for the profit maximizing volume of Q1 as a function of Q2.

Find MR for Q1

TRQ1 = P * Q1 = 100Q1 - 2Q1^2 - 2Q2*Q1

MRQ1 = 100 - 4Q1 - 2Q2

MCQ1 = 12

Set MR = MC

12 = 100 - 4Q1 - 2Q2

4Q1 = 88 - 2Q2

Q1 = 22 - 0.5Q2

Multiply Q2 by price to get the total revenue of Q2. Take the derivative of the total revenue of Q2 to determine the marginal revenue. In profit maximization, set the marginal revenue of Q2 equal to the marginal cost of Q2 and solve for the profit maximizing volume of Q2 as a function of Q1.

find MR for Q2

TRQ2 = P * Q2 = 100Q2 - 2Q1*Q2 - 2Q2^2

MRQ2 = 100 - 2Q1 - 4Q2

MCQ2 = 20

Set MR = MC

20 = 100 - 2Q1 - 4Q2

4Q2 = 80 - 2Q1

Q2 = 20 - 0.5Q1

Place the profit maximizing equilibrium Q2 into the profit maximizing equilibrium Q1 and solve for Q1. Q1 is equal to 16.

Q1 = 22 - 0.5(20 - 0.5Q1)

Q1 = 22 - 10 + 0.25Q1

0.75Q1 = 12

Q1 = 16

Solve for the profit maximizing volume of Q2 given that Q1 is equal to 16. Q2 is equal to 12.

Q2 = 20 - 0.5(16)

Q2 = 20 - 8 = 12

Q2 = 12

Place the profit maximizing volumes of Q1 and Q2 into the inverse demand function and solve for the equilibrium price. The price is equal to $44.

P = 100 - 2(12 + 16)

P = 100 - 2(28)

P = 100 - 56

P = $44

d. If firm 2 doesn't exist the price becomes equal to P = 100 - 2Q, and the total revenue is equal to the price multiplied by the quantity. Take the derivative of the total revenue to determine the marginal revenue. Set the marginal revenue equal to the marginal cost of Q1 and solve for the equilibrium quantity. The equilibrium quantity is equal to 22. Insert the equilibrium quantity into the new inverse demand function and solve for price. The new price is equal to $56.

TR = P * Q = 100Q - 2Q^2

MR = 100 - 4Q

MC = 12

Set MR = MC

12 = 100 - 4Q

4Q = 88

Q = 22

P = 100 - 2(22)

P = 100 - 44

P = $56

e. The monopoly price is higher than the cournot equilibrium and the monopoly quantity is lower than the cournot equilibrium. The more firms in the market the lower the price and the higher the level of output.