Question

There is a Cournot game consisting of firm Red and firm Green, which produce the same good.

Quantity produced by firm Red = qR

Quantity produced by firm Green = qG

Marginal cost for firm Red is equal to average cost, which is 3.

Marginal cost for firm Green is equal to average cost, which is 4.

Inverse demand curve for market is P = 70 - (qR + qG).

Answer the following questions, and show work:

1. What is the profit function of firms Red and Green?

2. What is the ideal reply function of firms Red and Green?

3. What is the Nash equilibrium of the firms?

4. What are the equilibrium profits of the firms, and which is making a better (higher) profit?

Answer 1

1. Profit function of Red = (p - MCR) *qR

= (70 - qR-qG - 3)*qR

= (67 - qG) qR - qR2

Profit function of Green = (70 - qR-qG - 4)*qG

= (66 - qR)qG - qG2

2.

For ideal reply function of Red, we set dprofit/dqR = 0

67-qG = 2qR

qR = 33.5 - 0.5qG

Similarly best reply function of a green firm:

66 - qR= 2qG

qG = 33 - 0.5qR

3. Solving two ideal functions

qR= 33.5 - 0.5(33 - 0.5qR)

0.75qR = 17

qR* = 22.66

qG* = 33 - 0.5*22.66 = 21.67

Nash equilibrium = (22.66, 21.67)

4.

Total output = 22.66 + 21.67 = 44.33

P = 70 - 44.33 = 25.67

Profit of Red = (25.67 - 3)22.66 = 513.7

Profit of Green = (25.67 - 4)21.67 = 469.6

Firm red earns higher profit