Question

A monopolist can produce a constant average (and marginal) cost of AC=MC=$5. It faces a market demand curve given by Q=53-P.

A)

Consider Problem 2 in the Homework 12 Supplement.  Suppose a second firm enters the market. Let Q1 be the output of the first firm and Q2 be the output of the second. Market demand is now given by Q1+Q2 = 53-P. Assuming that this second firm has the same costs as the first, how much will each firm produce (Cornot equilibrium)?

B)

Consider Problem 2 in the Homework 12 Supplement.  Suppose a second firm enters the market. Let Q1 be the output of the first firm and Q2 be the output of the second. Market demand is now given by Q1+Q2 = 53-P. Assuming that this second firm has the same costs as the first, what price will the equilibrium price be (Cornot equilibrium)?

Answer 1

A) Now a second firm enters the market. So now the profit function of first firm ( firm 1) can be written as:

profit₍₁₎ = TR₍₁₎ - TC₍₁₎

   = P*Q1 - AC₍₁₎*Q1

= (53- Q1 - Q2)*Q1 - 5Q1 [ as Q1+Q2 = 53-P implies

_Profit(1) = 48*Q1 - (Q1)²- Q1*Q2 P = 53-Q1-Q2]

Under the Cournot assumption, each firm treats the output of the other firm as a constant in its maximization calculations. Therefore, Firm 1 chooses Q1 to maximize its profit with Q2 being treated as a constant.

d(profit₍₁₎)/dQ1 = 48 - 2Q1 - Q2

Put it equal to 0, we get

Q1 = 24 - Q2/2

Similarly we can find ?₍₂₎

_Profit(2) = P*Q2 - 5Q2

= (53-Q1-Q2)*Q2 - 5*Q2

Profit₍₂₎ = 48*Q2 - (Q2)² - Q1*Q2

d(profit₍₂₎)/dQ2 = 48 - 2Q2 - Q1,

Put it equal to 0, we get

Q2 = 24 - Q1/2

Now solve Q1 and Q2 , by substitute Q2 in Q1 we get

Q1 = 24 - (1/2)*(24 - Q1/2)

Q1 = 16, similarly we get

Q2 = 16 .

B) Now for equilibrium price put Q1 and Q2 in demand curve we get,

P = 53 - 16 -16

P = $21 (cournot equilibrium price)