1. A monopolist faces a market demand curve given by Q = 53- P. Its cost function is given by C = 5Q + 50, i.e. its  MC =$5.

(a) Calculate the profit-maximizing price and quantity for this monopolist. Also calculate its optimal profit.

(b) Suppose a second firm enters the market. Let q₁ be the output of the first firm and q₂ be the output of the second. There is no change in market demand, which is given by Q = 53- P, but note that Q = q₁ + q₂. The second firm has the same MC as the first (incumbent) firm, but its fixed cost is $80. Assuming that both firms behave as Cournot duopolists who want to maximize individual profits, determine the best response function for each firm. Calculate the optimal outputs of each firm at the Nash Cournot equilibrium. Calculate the market price and individual and industry profits.

(c) Suppose firm 1 can invest in a new technology that can lower its MC to $3.50. Assuming that firm 2's cost structure remains unchanged, calculate optimal output of firm 1 if it can successfully lower its MC to $3.50. Also calculate firm 2's output and the market price at the new Nash Cournot equilibrium. What is the maximum amount that firm 1 will be willing to invest in this cost saving technology?

(d) Suppose that both firms have the same cost structure described in part (a) or (b), i.e. the MC of each firm is $5. The two firms now form a cartel to maximize joint profits. Determine the total output produced by the Cartel, if the Cartel wants to maximize joint profit. Determine the individual outputs, the market price and the individual as well as the Cartel's profit.