Question

2. (Cournot Model) Consider a Cournot duopoly. The market demand is p=160 - q2. Firm 1's marginal cost is 10, and firm 2's marginal cost is also 10. There are no fixed costs. A. Derive each firm's best response function B. What is the Nash equilibrium of this model? Find the equilibrium market price. C. Find the equilibrium profit for each firm D. Find the equilibrium consumer surplus in this market. 3. (Bertrand Model) Consider a Bertrand duopoly. The market demand is q=160p. Consumers only buy from the firm whose price is lower. If two firms charge the same price, they share the market equally. The marginal cost for firm 1 is 20, and the marginal cost for firm 2 is also 20. There are no fixed costs. A. Find the Nash equilibrium of this Bertrand game. B. Find the equilibrium output and profit for each firm. (Extra Credit) Now suppose that the marginal cost for firm 2 is 30 rather than 20. What are your new answers to question part A and part B?

Answer 1

Question 2

a) Find the profit functions

π1 = TR1 - TC1   π2 = TR2 - TC2

= (160 - q1 - q2)q1 - 10q1 = (160 - q1 - q2)q1 - 10q1

Find the derivative of profit (marginal profit) to maximize it

π'1 = 0 π'2 = 0

150 - 2q1 - q2 = 0 150 - 2q2 - q1 = 0

Best response functions are

q1 = 75 - 0.5q2 and q2 = 75 - 0.5q1

b) NE occurs when

q1 = 75 - 0.5*(75 - 0.5q1)

0.75q1 = 37.5

q1 = q2 = 50

This is the required NE. Market price = 160 - 50 - 50 = $60

c) Profits are

π1 = (160 - q1 - q2)q1 - 10q1 = (160 - 50 - 50)*50 - 10*50 = 2500

π2 = (160 - q1 - q2)q1 - 10q1 = (160 - 50 - 50)*50 - 10*50 = 2500

d) CS = 0.5*(Max price - current price)*current qty = 0.5*(160 - 60)*100 = 5000.