1. Let the inverse demand function and the cost function be given by = and = + respectively, where is total industry output and is the price of a product. Suppose the total industry is regulated by a monopolist. Then, find:

a) Profit-maximizing output level of the monopolist, .

b) Price, , the monopolist charges to produce the output level in (a).

c) The maximum profit of the monopolist, .

d) The Consumer Surplus, .

e) The Total Surplus, = + .

f) Now, suppose the industry is regulated by two firms, or duopoly. So, let be the output of firm 1 and the output of firm 2. Then, the total industry output is = + . The cost of each firm is given by ( ) = + for = , . Write down the profit function of each firm. (Cournot Model with identical costs).

g) Derive the reaction function of firm 1, = (). Show what happens to firm 1 output when firm 2 increases . That is, .

h) Find the Nash equilibrium outputs of both firms, and .

i) Find the total industry output, , produced by a duopoly. j) Find the price charged by duopolists, .

k) Find the maximum profit of each firm, and .

l) Find the new consumer surplus, .

m) Find the total surplus, = + + .

2. Cournot Model with different costs: Let again the demand function be = - . But the firms have different costs such as ( ) = + and ( ) = + .

a) Find the Nash equilibrium outputs of both firms, and .

b) Find the total industry output, , produced by a duopoly.

c) Find the price charged by duopolists, .

d) Find the maximum profit of each firm, and .

3. Stackelber Leader-Follower Model with identical costs: Let again the demand function and the cost function be = - and ( ) = + = , . Assume the firm 1 is a leader and the firm 2 is a follower. In other words, first firm 1 enters the market, then firm 2 does.

a) Find the Nash equilibrium outputs of both firms, and .

b) Find the total industry output, , produced by a leader and a follower.

c) Find the maximum profit of each firm, and .

4. Stackelber Leader-Follower Model with different costs: Let again the demand function be = - and But the firms have different costs such as ( ) = + and ( ) = + . Assume the firm 1 is a leader and the firm 2 is a follower. In other words, first firm 1 enters the market, then firm 2 does.

a) Find the Nash equilibrium outputs of both firms, and .

b) Find the total industry output, , produced by a leader and a follower.

c) Find the maximum profit of each firm, and .