Question

A) Two firms operate as Cournot competitors. Inverse demand is P = 120 - 2Q. Firm 1 has total cost function c(q1) = 10q1 and Firm 2 has total cost function c(q2) = 20q2. Solve for the Nash equilibrium in quantities and determine the equilibrium price. B) Now assume the rms are Bertrand competitors, simultaneously choosing prices. Solve for the Nash equilibrium in prices and determine the equilibrium quantities.

Answer 1

A. In cournot game firms chooses the quantity.

Let's solve for the corunot competition.

The best response function for firm 1 will be equal to,

Profit of firm 1= [120 - 2(q1 + q2)] × q1 - 10q1

Now maximizing the profit of firm 1.

= 120 - 2(q1 + q2) + q1(-2) - 10

= 120 - 4q1 - 2q2 - 10

4q1 + 2q2 = 110

2q1 + q2 = 55

q1 = (55 - q2)/2

Similarly the best response function for firm 2 will be equal to,

Profit of firm 2,

= [120 - 2(q1 + q2)] × q2 - 20q2

Maximizing the profit

= 120 - 2(q1 + q1) + q2(-2) - 20

= 120 - 2q1 - 4q2 - 20

4q2 + 2q1 = 100

2q2 + q1 = 50

q2 = (50 - q1)/2

Now we can solve two best response functions simultaneously to get cournot equilibrium.

q2 = (50 - (55 - q2)/2]/2

q2 = (100 - 55 + q2)/4

4q2 = 45 + q2

3q2 = 45

q2 = 45/3

q2 = 15

q1 = (55 - 15)/2

q1 = 40/2 = 20

So the cournot nash equilibrium is that firm one produces 20 units and firm 2 produces 15 units.

And cournot price will be equal to,

P = 120 - (15 + 20)

P = 120 - 35

P = 85.

B. In bertrand competition firms chooses prices simultaneously and with different marginal costs the firm with lower marginal cost will capture the whole market by charging a price lower the marginal cost of the other firm.

Here firm 1 has a marginal cost of 10 and marginal cost of firm 2 is 20.

So firm 1 will charge a price slightly lower than the marginal cost of firm 2 which will make firm 2 leave the market since the price will be lower than its marginal cost and firm 2 will be making a loss if it instead chooses to remain in market, so it will definitely leave the market and leaving the whole market to firm 1.

So firm 1 will charge a price of 19 which is slightly less than the marginal cost of firm 2.

At this price total demand will be equal to,

19 = 120 - 2q1

2q1 = 120 - 19

2q1 = 101

q1 = 101/2

q1 = 50.5.

So the bertrand equilibrium price and quantity will be equal to 19 and 50.5 respectively.