The 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 1's monopoly output and price? (e) How does the monopoly price and quantity comparing with Cournot equilibrium in part (c)?

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.

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