1)he inverse market demand in a homogeneous product Cournot duopoly is

P=100-2(Q1+Q2), and the costs are given by C(Q1) = 12Q1 and C(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 1s monopoly output and price?

a. Solve for Q1

P= 100-2(Q1+Q2)

P = 100 - 2Q1 - 2Q2

2Q1 = 100 - P - 2Q2

Q1 = 50 - 0.5P - Q2

b. Solve for Q2

P= 100-2(Q1+Q2)

P = 100 - 2Q1 - 2Q2

2Q2 = 100 - P - 2Q1

Q2 = 50 - 0.5P - Q1

c. 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

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

Q1 = 22 - 0.5(20 - 0.5Q1)

Q1 = 22 - 10 + 0.25Q1

0.75Q1 = 12

Q1 = 16

Q2 = 20 - 0.5(16)

Q2 = 20 - 8 = 12

Q2 = 12

P = 100 - 2(12 + 16)

P = 100 - 2(28)

P = 100 - 56

P = 44

The equilibrium price is 44, the equilibrium quantity for good 1 is 16 and the equilibrium quantity for good 2 is 12.

d. If firm 2 doesn't exist P = 100 - 2Q

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

If firm 2 doesn't exist the monopoly equilibrium price is 56 and the equilibrium quantity is 22.