Question

3.) Imagine a Cournot model in the market for soda sold by Coke and Pepsi (two firms with identical goods, that face the same demand curve, same constant marginal and average costs, and each choose quantity simultaneously) that face the following price function and marginal cost: P(Q) = 100 – 2q1 – 2q2 MC = 20

a. What is the best response function for Coke?

b. What is the best response function for Pepsi?

c. How much soda should Coke produce to maximize profit? How much soda should Pepsi produce to maximize profit?

d. What price should the firms charge for soda? How much profit does each firm make?

e. Explain why your above answers are the Nash Equilibrium for this game. How is this different from a Bertrand model?

Answer 1

P(Q) = 100 – 2q1 – 2q2

MC = 20

a. For best response function for Coke:

Total revenue function of coke= TR1= P*q1= 100q1-2q12-2q2q1

Marginal revenue of coke= MR1= differentiate TR1 wrt q1= 100-4q1-2q2

MC=20

Profit maxizing condition:

MR1=MC

100-4q1-2q2=20

80=4q1+2q2 equation 1

(80-2q2)/4=q1 Best response function for coke

b. For best response function for Pepsi:

Total revenue function of Pepsi= TR2= P*q2= 100q2-2q22-2q2q1

Marginal revenue of Pepsi= MR2= differentiate TR2 wrt q2= 100-4q2-2q1

MC=20

Profit maxizing condition:

MR2=MC

100-4q2-2q1=20

80=4q2+2q1 equation 2

(80-2q1)/4=q2 Best response function for Pepsi

c. To find out the quantity of both firm that how much should they produce:

Solve equation 1 and 2

80=4q1+2q2 equation 1

80=4q2+2q1 equation 2

Multiply 2 in equation 1

160=8q1+4q2 equation 3

Now subtract equation 2 from equation 3

160-80= 8q1-2q1+4q2-4q2

80=6q1

q1= 80/6= 40/3

Put q1=40/3 in BRS of Pepsi, then

q2= 40/3

d. For price:

P=100 – 2q1 – 2q2= 100-80/3-80/3= (300-80-80)/3= 140/3

Profit of firm 1= (P-MC)q1= (140/3 - 20) x40/3= 80/3 x 40/3= 3200/9

Profit of firm 2= (P-MC)q2= (140/3 - 20) x40/3= 80/3 x 40/3= 3200/9