Question

Question 6

In a homogeneous product duopoly, each firm has constant marginal cost equal to 10.  The market inverse demand curve is p = 250 – 2Q  where Q = q₁ + q₂ is the sum of the outputs of firms 1 and 2, and p  is the price of the good. Marginal and average cost for each firm is 10.  

(a) What are the Cournot and Bertrand equilibrium quantities and prices in this market?  In a two period version of the model in which each firm can observe the other’s behaviour in the first period, will the firms collude?

(b) Find the outcome for the Stackelberg model, assuming firm 1 is the “leader” (i.e., moves first) and firm 2 is the “follower” (i.e., moves second).

Answer 1

p = 250 – 2Q

P = 250 - 2q1 - 2q2

(a)

In Cournot,

Profit of Firm 1:

1 = (P - MC)*q1

1 = (250 - 2q1 - 2q2 - 10)*q1

1 = (240 - 2q2)*q1 - 2q1^2

FOC: d1/dq1 = 0

240 - 2q2 = 4q1

q1* = 60 - 0.5q2 -----------equation 1

Similarly, q2* = 60 - 0.5q1 ----------equation 2

Solving equation 1 and 2 gives,

q1 = 60 - 0.5(60 - 0.5q1)

q1 = 60 - 30 + 0.25q1

0.75q1 = 30

q1* = q2* = 40

Aggregate Output, Q = q1+q2 = 40 + 40 = 80

Price, P = 250 - 2*80

P = 90

Profit of firm 1: 1 = (90-10)*40 = 3200 = 2

Total Profit = 3200 + 3200 = 6400

In Bertrand Model, equilibrium occurs when P1 = P2 = MC

So, Equilibrium Prices: P1 = P2 = 10

When Firms collude, then at equilibrium, MR = MC

TR = p*Q = 250Q - 2Q2

MR = dTR/dQ = 250 - 4Q

So, 250 - 4Q = 10

Q* = 60

P* = 250 - (2*60)

P* = 130

Profit, = (130 - 10)*60 = 7200

Since, proift in collusion > Proft in Cournot. So, the two firms will collude to fetch higher profits.

(b)

Stackelberg:

Max 1 s.t. q2* = 60 - 0.5q1

1 = (250 - 2q1 - 2q2* - 10)*q1

1 = (240 - 2q1 -2*(60 - 0.5q1)) * q1

1 = (240 - 2q1 - 120 + q1)*q1

1 = 120q1 - q1^2

FOC: d1/dq1 = 0

120 = 2q1

q1* = 60

and q2 = 60 - (0.5*60)

q2 = 30

Output of Leader = 60

Output of Follower = 30

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