#1. The inverse market demand in a homogeneous product Cournot duopoly is P=100-

Question

#1. The inverse market demand in a homogeneous product Cournot duopoly is

P=100-2(Q1+Q2), and the costs are given by C(Q1) = 12Q1 and C(Q2) = 20Q2. The implied marginal costs are $12 for firm 1 and $20 for firm 2.

Determine the reaction function for firm 1.

P*Q1 = 100Q1 – 2Q1^2 – 2Q1Q2

HR= 100-4Q1-2Q2 = 12

88-2Q2/4 = Q1

Determine the reaction function for firm 2.

P*Q2 = 100Q2-2Q1Q2 – 2Q2

HR= 100 – 2Q1 - 4Q2 = 20

80 – 2Q1 / 4 = Q2

Calculate the Cournot equilibrium price and quantity.

Suppose firm 1 is a monopoly (firm 2 does not exist), what is firm 1’s monopoly output and price?

How does the monopoly price and quantity comparing with Cournot equilibrium in part (c)?

Explanation / Answer

Answer.

The best response functions have been defined above. Using those for both the firms, we have:

Q1= 22-Q2/2

Q2= 20-Q1/2

Substituting the value of Q1 in above equation:

Q2= 20- ((22-Q2/2)/2)

Q2= 20-11+Q2/4

3Q2/4=9

Q2= (9*4)/3

Q2= 12.

Substitute value of Q2:

Q1= 22- Q2/2

Q1= 22-6

Q1=16.

Now P= 100-2(Q1+Q2)

P= 100-2(16+12)

P=100- 56

P= 44

In case firm 1 is a monopoly: (Q2 is 0 now)

Profit maximisation is at MR=MC

as per the equation P= 100-2Q1

P.Q1= 100Q1-2Q1^2

MR= 100-4Q1

MC= 12

Thus , 100-4Q1=12

4Q1=88

Q1= 22

and P= 100-2(22)= 56.

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