Question

Suppose that 2 firms are competing against each other in Cournot (output) competition and that the market demand curve is given by P = 60 – Q or Q = 60 – P. In addition, assume the marginal cost for each firm is equal to 0 as we did in class.

a. Solve for firm 1’s total revenue. Note that this should not require any calculus.

b. If you take the derivative of firm 1’s total revenue, you should find that firm 1’s marginal revenue is MR1 = 60 – 2 * q1 – q2. Use this along to solve for firm 1’s reaction function.

c. Given your answer to part b, what would firm 2’s reaction function look like?

d. Given both firm’s reaction functions, solve for the Cournot equilibrium price and quantity for each firm. How much does each firm make in profits? e. Now assume the two firms collude and act as a single monopoly where they figure out the monopoly output level and then each make half. Solve for the new equilibrium price and quantity for each firm. How much does each firm make in profits?

f. Now assume the two firms act as if they are in a purely competitive market with many sellers. Solve for the new equilibrium price and quantity for each firm. How much does each firm make in profits?

Answer 1

a)

P = 60 – Q1 - Q2

TR = P*Q

= (60 - Q1 - Q2)*Q1

= 60Q1 - Q1^2 - Q2Q1

b)

MR1 = 60 - 2Q1 - Q2

Equilibrium Condition:

60 -2Q1 - Q2 = 0

Q1 = 30 - 0.5Q2 ......(1)

c)

Since cost function is symmetric

Reaction function of Q2

Q2 = 30 - 0.5Q1 ..........(2)

d)

Substitute value of (2) in equation (1)

Q1 = 30 - 0.5 (30 - 0.5Q1)

Q1 = 30 - 15 +0.25Q1

0.75Q1 = 15

Q1 = 20

Q2 = 20

Total output =40

P = 60 -Q

= 60 - 40

= 20

Profit = 40*20 - 0

= 800

Each firm profit = 400 ( 800/2)

e)

Monopoly:

P = 60 - Q

TR = (60-Q)Q

= 60Q - Q^2

MR = 60 - 2Q

MC= 0

Equilibrium:

60 -2Q = 0

Q = 60/2

= 30

P = 60 - 30

= 30

Profit = TR - TC

= 30*30 * 0*30

= 900

each firm profit = 450 ( 900/2)

f)

Competitive output

P = MC

60-Q = 0

Q = 60

P =0

Profit - 0