Question

Consider a Cournot duopoly with the following inverse demand function:p(Q) =a−Q where p is the price of the product and Q is the total amount of goods exchanged in the market. The total costs areC(q1) = 300q1, C(q2) = 300q2 for firm 1 and firm 2, respectively. But the demand is uncertain (i.e., a new product may be introduced soon which will decrease the demand drastically). Firm 1 learns whether demand will be high (a =1800) or small (a=900) before it makes its quantity decision. However, firm 2 knows just the probability of high demand (1/4) and the probability of low demand (3/4). All of this is common knowledge. In particular, firm 2 knows that firm 1 knows the demand for certain. The two firms simultaneously choose quantity. What is the Bayesian equilibrium of the game (price in both states of the world and quantity produced by each firm)?

Answer 1

This question needs a bit to work out. It is been given that when firm 1 has a demand of 1800 the other has a demand of 900.So the demand function of say firm 1 can be written as:

P(Q)= 1800-Q

where Q= q1+q2

Total cost (TC)= 300q (for both under cournot duopoly model)

P= 1800-(q1+q2)

= 1800-q1-q2

Total revenue (TR1)= price *quantity(P(Q)*q1)

=(1800-q1-q2) q1

=1800q1-q1²-q2q1

Now the marginal revenue(MR1)=

= 1800-2q1-q2

Marginal cost (MC)=

= 300

Since Cournot duopoly is operating at MR=MC, we have:

1800-2q1-q2=300

2q1=1800-300-q2

2q1=1500-q2

q1=750-1/2q2

Now firm 2 follows the same pattern:

P(Q)= 900-Q

where Q= q1+q2

Total cost (TC)= 300q (for both under cournot duopoly model)

P= 900-(q1+q2)

= 900-q1-q2

Total revenue (TR2)= price *quantity(P(Q)*q2)

=(900-q1-q2) q2

=900q1-q1q2-q2

Now the marginal revenue(MR1)=

= 900-q1-2q2

Marginal cost (MC)=

= 300

Since Cournot duopoly is operating at MR=MC, we have:

900-2q2-q1=300

2q2=900-300-q1

2q2=600-q1

q2=300-1/2q1

Substituting this to q1

q1=750-1/2(300-1/2q1)

q1=750-150+1/4q1

  600+1/4q1

q1=4/3*600

q1=800

substituting this to q2:

q2=300-1/2*800

q2=-100

q1 and q2 are the equilibrium quantities of firm 1 and firm 2 respectively at prices 1800 and 900.