Question

Cournot Model: Consider a duopoly where 2 firms produce a homogeneous product. Under the assumption that one firm’s decision on output would depend on the other firm’s output, a market demand is given as P = 90 - Q where Q = Q_A + Q_B (Q_A is the quantity of a firm A and Q_B is the quantity of a firm B).

Find the quantity and the price in this duopoly when MC of both firms = 0.

Answer 1

In this case MC_A=MC_B=0

P=90-Q

where Q=Q_A+Q_B

P=90-Q_A-Q_B

Let us consider the case of firm A

Profit of firm A=P*Q_A-MC_A*Q_A

PR_A=(90-Q_A-Q_B)*Q_A-0*Q_A=90Q_A-Q_A²-Q_AQ_B

For profit maximization, take derivative of PR_A with respect to Q_A and equate it to zero

d(PR_A)/dQ_A=90-2Q_A-Q_B=0

2Q_A=90-Q_B

Q_A=45-0.5Q_B -------------------------(1)

Now consider the case of firm B

Profit of firm B=P*Q_B-MC_B*Q_B

PRB=(90-Q_A-Q_B)*Q_B-0*Q_B=90Q_B-Q_B²-Q_AQ_B

For profit maximization, take derivative of PR_B with respect to Q_B and equate it to zero

d(PR_B)/dQ_B=90-2Q_B-Q_A=0

2Q_B=90-Q_A

Put Q_A=45-0.5Q_B

2Q_B=90-45+0.5Q_B

1.5Q_B=45

Q_B=30

So,

Q_A= 45-0.5Q_B=45-0.5*30=30

Total Output =30+30=60 units (Where output of each firm is 30 units)

P=90-Q_A-Q_B=90-30-30=$30